Essays on Insurance-Linked Capital Market Instruments ...FILE/dis3870.pdf · Essays on...

Essays on Insurance-Linked Capital Market Instruments,Solvency Measurement, and Insurance Pricing

DISSERTATIONof the University of St.Gallen,

School of Management,Economics, Law, Social Sciences

and International Affairsto obtain the title of

Doctor of Philosophy in Management

submitted by

Alexander Braun

from

Germany

Approved on the application of

Prof. Dr. Hato Schmeiser

and

Prof. Dr. Manuel Ammann

Dissertation no. 3870

Braun Druck & Medien GmbH, Tuttlingen, 2011

The University of St. Gallen, School of Management, Economics, Law,

Social Sciences and International Affairs hereby consents to the printing

of the present dissertation, without hereby expressing any opinion on the

views herein expressed.

St. Gallen, May 13, 2011

The President:

Prof. Dr. Thomas Bieger

To my dear parents/Meinen lieben Eltern

Marliese & Herbert Braun

Acknowledgements

The dissertation at hand would not have been feasible without the con-

tinuous encouragement and support of a number of people, both before

and during the time of its writing.

To begin with, I wish to express my sincere gratitude to my super-

visor, Prof. Dr. Hato Schmeiser, who always offered valuable advice and

suggestions and created an excellent research environment with excep-

tional working conditions. In addition, I thank my co-supervisor, Prof.

Dr. Manuel Ammann, for his constructive feedback and his thoughtful

comments, which helped me to further improve this dissertation.

Moreover, I am grateful to my co-authors as well as my colleagues of

the Institute of Insurance Economics of the University of St. Gallen for

inspiring discussions, exciting joint research projects, and an extraordi-

nary pleasant working atmosphere. They all contributed to the unique

experience that I enjoyed during my doctoral studies.

Finally, I would like to wholeheartedly thank my long-time girlfriend,

my brother, my grandmother, and my parents. They are the most impor-

tant pillars in my life and accompanied me through highs and lows over

all those years. In particular, this achievement would be inconceivable

without the generosity, devotion, and unremitting support from my dear

parents, to whom I owe so much.

St. Gallen, July 2011

Alexander Braun

Vorwort

Die vorliegende Dissertation ware ohne die andauernde Forderung und

Unterstutzung einer Reihe von Personen sowohl vor als auch wahrend

ihrer Entstehungsphase nicht realisierbar gewesen.

Zunachst gilt mein aufrichtiger Dank meinem Referenten und Be-

treuer Prof. Dr. Hato Schmeiser, der mir stets wertvolle Ratschlage und

Anregungen gab und eine herausragende Forschungsumgebung mit ex-

zellenten Arbeitsbedingungen schuf. Ausserdem danke ich meinem Ko-

referenten Prof. Dr. Manuel Ammann fur sein konstruktives Feedback

und seine durchdachten Kommentare, die mir dabei halfen, diese Disser-

tation noch weiter zu verbessern.

Ferner bin ich meinen Koautoren und meinen Kollegen am Institut

fur Versicherungswirtschaft der Universiat St. Gallen fur inspirierende

Diskussionen, spannende gemeinsame Forschungsprojekte und eine aus-

serordentlich angenehme Arbeitsatmosphare dankbar. Sie alle trugen da-

zu bei, dass mein Doktoratsstudium zur einmaligen Erfahrung wurde.

Schliesslich mochte ich von ganzem Herzen meiner langjahrigen Freun-

din, meinem Bruder, meiner Grossmutter und meinen Eltern danken.

Sie sind die wichtigsten Saulen in meinem Leben und haben mich in all

den Jahren durch Hohen und Tiefen begleitet. Insbesondere ware das

Erreichte undenkbar ohne die Grosszugigkeit, Hingabe und unablassige

Unterstutzung meiner lieben Eltern, denen ich so viel zu verdanken

habe.

St. Gallen im Juli 2011

Alexander Braun

Outline iii

Outline

I Performance and Risksof Open-End Life Settlement Funds 1

II Pricing Catastrophe Swaps:A Contingent Claims Approach 59

III Solvency Measurementof Swiss Occupational Pension Funds 119

IV Stock vs. Mutual Insurers:Who Does and Who Should Charge More? 163

Curriculum Vitae 221

iv Contents

Contents

Contents iv

List of Figures vii

List of Tables ix

Summary xi

Zusammenfassung xiii

I Performance and Risksof Open-End Life Settlement Funds 1

1 Introduction 2

2 Market overview and fund business model 52.1 The U.S. life settlement market: an overview . . . . . . . 52.2 Closed-end vs. open-end life settlement funds . . . . . . . 72.3 The anatomy of open-end life settlement funds . . . . . . 11

3 Empirical analysis 163.1 Data and sample selection . . . . . . . . . . . . . . . . . . 163.2 The return distribution of open-end life settlement funds 223.3 Performance measurement and correlation analysis . . . . 263.4 Analysis of individual funds . . . . . . . . . . . . . . . . . 28

4 Risks of open-end life settlement funds 364.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Valuation risk . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Longevity risk . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Liquidity risk . . . . . . . . . . . . . . . . . . . . . . . . . 404.5 Policy availability risk . . . . . . . . . . . . . . . . . . . . 424.6 Operational risks . . . . . . . . . . . . . . . . . . . . . . . 434.7 Credit risk . . . . . . . . . . . . . . . . . . . . . . . . . . 444.8 Changes in regulation and tax legislation . . . . . . . . . 45

5 Summary and conclusion 47

6 Appendix 486.1 Index descriptions (from the providers) . . . . . . . . . . 486.2 Performance measures . . . . . . . . . . . . . . . . . . . . 51

References 53

Contents v

II Pricing Catastrophe Swaps:A Contingent Claims Approach 59

1 Introduction 60

2 Literature review 62

3 Catastrophe swaps 64

3.1 Contract design . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Market development . . . . . . . . . . . . . . . . . . . . . 67

3.3 Areas of application . . . . . . . . . . . . . . . . . . . . . 68

3.4 Accounting and regulation . . . . . . . . . . . . . . . . . 69

3.5 Comparison to other risk transfer instruments . . . . . . 70

4 Pricing model 70

4.1 Risk-neutral valuation of catastrophe derivatives . . . . . 72

4.2 Pricing catastrophe swaps ex-ante . . . . . . . . . . . . . 74

4.3 Pricing catastrophe swaps in the loss reestimation phase . 78

5 Empirical analysis 87

5.1 Severity distributions for natural disasters in the U.S. . . 87

5.2 Derivation of implied Poisson intensities . . . . . . . . . . 94

5.3 The stochastic process of implied Poisson intensities . . . 101


7 Appendix: The market price of cat risk 110

References 112

III Solvency Measurementof Swiss Occupational Pension Funds 119

1 Introduction 120

2 The particularities of the Swiss pension system 123

3 The model framework 125

4 The traffic light approach 131

5 Implementation and calibration 132

5.1 Input data . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

vi Contents

6 Sensitivity analysis 1426.1 Equity allocation . . . . . . . . . . . . . . . . . . . . . . . 1436.2 Asset concentration . . . . . . . . . . . . . . . . . . . . . 1456.3 Misestimation of liabilities . . . . . . . . . . . . . . . . . 1476.4 Coverage ratio . . . . . . . . . . . . . . . . . . . . . . . . 1486.5 Lowest acceptable coverage ratio . . . . . . . . . . . . . . 1506.6 Exchange rate risk . . . . . . . . . . . . . . . . . . . . . . 151

7 Supervisory review and actions 154

8 Notes on a potential introduction in Switzerland 156

9 Conclusion 158

References 159

IV Stock vs. Mutual Insurers:Who Does and Who Should Charge More? 163

1 Introduction 164



4 Model framework 1754.1 Stock insurer claims structure . . . . . . . . . . . . . . . 1754.2 Mutual insurer: full equity participation . . . . . . . . . . 1774.3 Mutual insurer: partial equity participation . . . . . . . . 1874.4 Claims structure relationships . . . . . . . . . . . . . . . 192

5 Numerical analysis 1965.1 Option pricing formulae . . . . . . . . . . . . . . . . . . . 1965.2 The impact of recovery option and equity participation . 1995.3 Premium, safety level, and equity capital . . . . . . . . . 203

6 Economic implications 209

7 Conclusion 213

References 215

Curriculum Vitae 221

List of Figures vii

List of Figures

Performance and Risksof Open-End Life Settlement Funds

1 Stylized mechanics of an open-end life settlement fund . . 122 Life settlements in comparison to other asset classes

(12/2003–06/2010) . . . . . . . . . . . . . . . . . . . . . . 253 Individual life settlement funds in comparison

(01/2007–06/2010) . . . . . . . . . . . . . . . . . . . . . . 30

Pricing Catastrophe Swaps:A Contingent Claims Approach

4 Illustration of the barrier option pricing approach . . . . . 865 Natural disaster losses in the U.S. (1900–2005) . . . . . . 916 Histograms of normalized natural disaster losses . . . . . 927 Monthly implied intensities (08/2005–09/2010) . . . . . . 998 Intensity factor scores and out-of-sample forecast example 1049 ACF and PACF for the hurricane factor . . . . . . . . . . 10610 ACF and PACF for the earthquake factor . . . . . . . . . 107

Solvency Measurementof Swiss Occupational Pension Funds

11 Sensitivity analysis: equity allocation . . . . . . . . . . . . 14412 Sensitivity analysis: asset concentration . . . . . . . . . . 14613 Sensitivity analysis: misestimation of liabilities . . . . . . 14914 Sensitivity analysis: coverage ratio . . . . . . . . . . . . . 14915 Sensitivity analysis: minimum coverage ratio . . . . . . . 15116 Sensitivity analysis: actual and minimum coverage ratio . 15217 Sensitivity analysis: FX hedging . . . . . . . . . . . . . . 153

Stock vs. Mutual Insurers:Who Does and Who Should Charge More?

18 Stock insurer equity and policyholder payoff . . . . . . . . 17819 Mutual insurer default put option payoff . . . . . . . . . . 18120 Mutual insurer recovery option payoff . . . . . . . . . . . 18221 Mutual insurer equity payoff: full participation . . . . . . 18522 Mutual insurer policyholder stake payoff . . . . . . . . . . 18623 Mutual insurer (expected) equity payoff . . . . . . . . . . 190

viii List of Figures

24 Theoretical premium comparison . . . . . . . . . . . . . . 19325 Equity-premium combinations:

full equity participation and no recovery option . . . . . . 20426 Equity-premium combinations:

full equity participation and recovery option . . . . . . . . 20727 Equity-premium combinations:

partial equity participation and recovery option . . . . . . 208

List of Tables ix

List of Tables

Performance and Risksof Open-End Life Settlement Funds

1 Closed-end vs. open-end life settlement funds . . . . . . . 92 Life settlement funds in the original dataset . . . . . . . . 19

3 Sample details . . . . . . . . . . . . . . . . . . . . . . . . 224 Descriptive statistics: index return distributions

(12/2003–06/2010) . . . . . . . . . . . . . . . . . . . . . . 275 Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . 296 Descriptive statistics: fund return distributions

(01/2007–06/2010) . . . . . . . . . . . . . . . . . . . . . . 327 Summary statistics: 14 fund return distributions . . . . . 34

Pricing Catastrophe Swaps:A Contingent Claims Approach

8 Catastrophe risk transfer instruments in comparison . . . 719 Descriptive statistics for (non-zero) disaster losses . . . . . 9010 Estimates: lognormal, Burr, and Pareto distribution . . . 95

11 Estimates: Weibull, gamma, and exponential distribution 9512 Descriptive statistics: time series of cat swap prices . . . . 97

13 Descriptive statistics: implied intensity time series . . . . 10014 Correlation matrices and factor loadings . . . . . . . . . . 10015 Hurricane intensity factor: unit root tests and estimates . 108

16 Earthquake intensity factor: unit root tests and estimates 108

Solvency Measurementof Swiss Occupational Pension Funds

17 Input parameters for the sample funds in 2007 . . . . . . 135

18 Input parameters for the sample funds in 2008 . . . . . . 13519 Asset allocations of the sample funds in 2007 . . . . . . . 13620 Asset allocations of the sample funds in 2008 . . . . . . . 136

21 Annualized means and standard deviations . . . . . . . . 13822 Correlation matrix . . . . . . . . . . . . . . . . . . . . . . 138

23 One-year default probabilities for different rating classes . 13924 Probabilities and test outcomes for 2007 . . . . . . . . . . 14125 Probabilities and test outcomes for 2008 . . . . . . . . . . 141

26 Parameters for a representative pension fund . . . . . . . 143

x List of Tables

Stock vs. Mutual Insurers:Who Does and Who Should Charge More?

27 Descriptive statistics of the data . . . . . . . . . . . . . . 17328 Estimation results . . . . . . . . . . . . . . . . . . . . . . 17629 Input parameters and values for the stock insurer . . . . . 20030 Impact of recovery option and equity participation . . . . 20131 Impact of the maximum amount of additional premiums . 203

Summary xi

Summary

This dissertation consists of four parts, each of which comprises anindividual research paper. The first two parts pertain to the field ofinsurance-linked capital market instruments. The paper ”Performanceand Risks of Open-End Life Settlements” is a comprehensive analysisof open-end funds, which exclusively invest in so-called senior life set-tlements, that is, U.S. life insurance policies traded in the secondarymarket. It comprises an explanation of the business model of open-endlife settlement funds, an empirical analysis of their return distributions,a performance measurement, and an in-depth risk analysis. Althoughthe empirical results suggest that life settlement funds offer attractivereturns paired with low volatilities and are virtually uncorrelated withestablished asset classes, investors need to be aware of latent risk factorssuch as liquidity, longevity and valuation risks, which are largely notreflected by the examined historical data. Yet, these aspects should betaken into account in order not to overestimate the performance of thisasset class.

The second research paper, ”Pricing Catastrophe Swaps: A Contin-gent Claims Approach”, centers around the catastrophe (cat) swap, afinancial instrument through which natural disaster risks can be trans-ferred between counterparties. It begins with a discussion of the typicalcontract design, the current state of the market, and major areas of ap-plication. In addition, a two-stage option-theoretic pricing approach isproposed, which distinguishes between the main risk drivers ex-ante andduring the loss reestimation phase. Catastrophe occurrence is modeled asa doubly stochastic Poisson process (Cox process) with mean-revertingOrnstein-Uhlenbeck intensity. Moreover, by fitting various parametricdistributions to historical loss data from the U.S., the heavy-tailed Burrdistribution is found to be the most adequate representation for lossseverities. The pricing model is then applied to market quotes for hur-ricane and earthquake contracts to derive implied Poisson intensities.Since a first order autoregressive process provides a good fit to the re-sulting time series, its continuous-time limit, the Ornstein-Uhlenbeckprocess should be well suited to represent the dynamics of the Poissonintensity in a cat swap pricing model.

With the research paper ”Solvency Measurement of Swiss Occupa-tional Pension Funds”, the focus in the third part of the dissertationis on solvency measurement in the occupational pension system. Based

xii Summary

on the combination of a stochastic pension fund model and a trafficlight signal approach, a solvency test for occupational pension funds inSwitzerland is proposed. Being designed as a regulatory standard model,the set-up is intentionally kept parsimonious and, assuming normally dis-tributed asset returns, a closed-form solution can be derived. Despite itssimplicity, the framework comprises the essential risk sources needed insupervisory practice. Due to its ease of calibration, it is additionally wellsuited for the fragmented Swiss market, keeping costs of solvency testingat a minimum. To illustrate its application, the model is calibrated andimplemented for a small sample of ten Swiss pension funds. Moreover,a sensitivity analysis is conducted to identify important drivers of theshortfall probabilities for the traffic light conditions.

Finally, the fourth and last part of the dissertation contains the re-search paper ”Stock vs. Mutual Insurers: Who Does and Who ShouldCharge More?”, which is an empirical and theoretical analysis of therelationship between the premiums of insurers in the legal form of stockand mutual companies. An evaluation of panel data for the German mo-tor liability insurance sector does not provide indications that mutualscharge significantly higher premiums than stock insurers. Subsequently,a comprehensive model framework for the arbitrage-free pricing of in-surance contracts is employed to compare stock and mutual insurancecompanies with regard to the three central magnitudes premium size,safety level, and equity capital. Although, from a normative perspective,there are certain circumstances in which the premiums of stock and mu-tual insurers should be equal, these situations would generally requirethe mutual to hold comparatively less capital. This being inconsistentwith the empirical results, it appears that the observed insurance pricesare not arbitrage-free.

Zusammenfassung xiii

Zusammenfassung

Diese Dissertation besteht aus vier Teilen, die jeweils eine in sich ge-schlossene Forschungsarbeit enthalten. Kapitel eins und zwei sind in dasThemenfeld der versicherungsgebundenen Kapitalmarktinstrumente ein-zuordnen. Die Arbeit

”Performance and Risks of Open-End Life Settle-

ments“ stellt eine umfassende Analyse offener Fonds dar, die in am Zweit-markt gehandelte US-amerikanische Lebensversicherungen, sogenannteSenior Life Settlements, investieren. Sie umfasst eine Erlauterung desGeschaftsmodells und der Funktionsweise offener Lebensversicherungs-fonds, eine empirische Analyse ihrer Renditeverteilungen, eine Perfor-mancemessung sowie eine detaillierte Risikoanalyse. Obwohl die empi-rischen Ergebnisse darauf hindeuten, dass offene Lebensversicherungs-fonds attraktive Renditen bei geringer Volatilitat bieten und praktischunkorreliert mit etablierten Anlageklassen sind, sollten sich Investorenlatenter Risikofaktoren wie beispielsweise Liquiditats-, Langlebigkeits-und Bewertungsrisiken bewusst sein, welche sich grosstenteils nicht inden untersuchten historischen Daten widerspiegeln. Eine Berucksichtig-ung dieser Risiken ist jedoch erforderlich, um die Performance dieserAnlageklasse nicht zu uberschatzen.

Im Mittelpunkt der zweiten Forschungsarbeit”Pricing Catastrophe

Swaps: A Contingent Claims Approach“ steht der Katastrophen-Swap,ein Finanzinstrument, welches eine Ubertragung der Risiken von Natur-katastrophen zwischen Marktteilnehmern ermoglicht. Die Ausfuhrungenbeginnen mit einer Erlauterung der ublichen Vertragsgestaltung, der ge-genwartigen Marktlage sowie der wichtigsten Anwendungsgebiete. Da-ruber hinaus wird ein zweistufiger Bewertungsansatz vorgeschlagen, derzwischen den Hauptrisikotreibern ex-ante und wahrend der Aktualisie-rung der zugrunde liegenden Verlustschatzungen unterscheidet. Das Auf-treten von Katastrophen wird mit einem doppelt stochastischen Poisson-Prozess (Cox-Prozess) modelliert, dessen Intensitat einem Ornstein-Uhlenbeck-Prozess folgt. Durch das Anpassen verschiedener parametri-scher Verteilungsfunktionen an historische US-amerikanische Verlustda-ten kann des Weiteren die endlastige Burr-Verteilung als angemessensteReprasentation der Verlustschwere identifiziert werden. Anschliessendwird das Bewertungsmodell auf Marktpreise von Orkan- und Erdbeben-kontrakten angewendet, um implizite Poisson-Intensitaten abzuleiten.Da ein autoregressiver Prozess erster Ordnung die resultierenden Zeitrei-hen gut abbildet, sollte sein zeitstetiges Aquivalent, der Ornstein-Uhlen-

xiv Zusammenfassung

beck-Prozess, zur Darstellung der Dynamik der Poisson-Intensitaten ineinem Bewertungsmodell fur Katastrophen-Swaps gut geeignet sein.

Mit der Forschungsarbeit”A Traffic Light Approach to Solvency Mea-

surement of Swiss Occupational Pension Funds“ liegt der Fokus im drit-ten Teil der Dissertation auf der Solvenzmessung in der beruflichen Vor-sorge. Es wird ein Solvenztest fur Schweizer Vorsorgeeinrichtungen vor-geschlagen, welcher auf der Kombination eines stochastischen Pensions-kassenmodells mit einem Ampelsignalansatz basiert. Im Sinne eines regu-latorischen Standardmodells wird der Aufbau bewusst einfach gehalten.Zudem kann unter der Annahme normalverteilter Anlagerenditen eine ge-schlossene Losung abgeleitet werden. Trotz der relativ geringen Komple-xitat deckt das System die wesentlichen in der aufsichtlichen Praxis erfor-derlichen Risikoquellen ab. Aufgrund der einfach durchzufuhrenden Ka-librierung ist es zusatzlich gut zur Anwendung im fragmentierten Marktfur Schweizer Vorsorgeeinreichtungen geeignet und halt die Kosten derSolvenzregulierung so gering wie moglich. Zur Veranschaulichung derAnwendung des Modells wird es mittels einer kleinen Stichprobe vonzehn Vorsorgeeinrichtungen kalibriert und umgesetzt. Daruber hinauswird eine Sensitivitatsanalyse durchgefuhrt, um wichtige Einflussfakto-ren der Unterschreitungswahrscheinlichkeiten fur die Ampelbedingungenzu identifizieren.

Der vierte und letzte Teil der Dissertation schliesslich beinhaltetdie Forschungsarbeit

”Stock vs. Mutual Insurers: Who Does and Who

Should Charge More?“, welche eine empirische und theoretische Analy-se des Zusammenhangs zwischen den Pramien von Versicherern in derRechtsform der Aktiengesellschaft und des Versicherungsvereins auf Ge-genseitigkeit darstellt. Eine Auswertung von Paneldaten aus dem Bereichder deutschen Kfz-Haftpflichtversicherung liefert keinerlei Anhaltspunk-te dafur, dass Versicherungsvereine signifikant hohere Pramien berechnenals Aktiengesellschaften. Im Anschluss wird ein umfassender Modellrah-men fur die arbitragefreie Bewertung von Versicherungsvertragen ein-gesetzt, um Versicherer in Form von Aktiengesellschaften und Vereinenhinsichtlich der drei zentralen Grossen Pramienhohe, Sicherheitsniveauund Eigenkapital zu vergleichen. Obwohl es aus normativer Sicht be-stimmte Umstande gibt, in denen die Pramien einer Aktiengesellschaftund eines Versicherungsvereins auf Gegenseitigkeit gleich sein sollten,ware es hierfur erforderlich, dass der Verein vergleichsweise weniger Ka-pital vorhalt. Da dies den empirischen Ergebnissen widerspricht, scheinendie beobachteten Versicherungspreise nicht arbitragefrei zu sein.

1

Part I

Performance and Risks of

Open-End Life Settlement

Funds

Abstract

In this paper, we comprehensively analyze open-end funds dedicated to

investing in U.S. senior life settlements. We begin by explaining their

business model and the roles of institutions involved in the transactions

of such funds. Next, we conduct the first empirical analysis of life settle-

ment fund return distributions as well as a performance measurement,

including a comparison to other asset classes. Since the funds contained

in our dataset cover a large fraction of this relatively young segment

of the capital markets, representative conclusions can be derived. Even

though the empirical results suggest that life settlement funds offer at-

tractive returns paired with low volatility and are virtually uncorrelated

with other asset classes, we find latent risk factors such as liquidity,

longevity and valuation risks. Since these risks did generally not mate-

rialize in the past and are hence largely not reflected by the historical

data, they cannot be captured by classical performance measures. Thus,

caution is advised in order not to overestimate the performance of this

asset class.1

1Alexander Braun, Nadine Gatzert, and Hato Schmeiser (2009), Performance andRisks of Open-End Life Settlement Funds, Working Papers on Risk Management and

Insurance, No. 73. This paper has been accepted for publication in the Journal of

Risk and Insurance.

2 I Life Settlement Funds

1 Introduction

In the secondary market for life insurance, policyholders sell their con-

tracts to life settlement providers, which usually pass them on to in-

vestors or, in some cases, hold them on their own balance sheet. Such

transactions are termed ”life settlements”. The payment to the selling

policyholder is above the surrender value of the life insurance policy of-

fered by the primary insurer. The investor continues to pay premiums

until the contract is either resold or until it matures due to death or

reaching a fixed term and, in turn, receives the associated payoff. The

life settlement asset class, which emerged towards the end of the last

century, is not entirely new. Larger volumes of life insurance policies,

primarily those of terminally ill AIDS patients, had already been traded

in the so-called viatical settlements market of the 1980s. Most recently,

however, the asset class has begun to attract increasing attention from

the capital markets, since its return characteristics of low volatility and

virtually no correlation with other asset classes are appealing to a wide

range of investors. In addition, several Wall Street banks explore ways

to enter this business.

Since life settlements are a rather young asset class, literature on the

topic is still scarce and mainly practitioner-oriented. One of the early

analyses of the life settlement industry was provided by Giacalone (2001),

followed by Doherty and Singer (2002), who discuss benefits and welfare

gains arising from the secondary market for life insurance policies. Fur-

thermore, Kamath and Sledge (2005) review the characteristics of the

market for U.S. life settlements and the main drivers of its growth. While

Ingraham and Salani (2004), Freeman (2007) and Leimberg et al. (2008)

describe the decision making and due diligence process, McNealy and

Frith (2006) focus on the sourcing process for life settlements and point

out major product-flow constraints. In addition, Ziser (2006) and Smith

and Washington (2006) focus on transactional aspects, such as the di-

versification of life settlement portfolios in order to reduce risks. Seitel

(2006) and Seitel (2007) examine the industry from an institutional in-

vestor’s and a life settlement provider’s viewpoint, respectively. Other

1 Introduction 3

studies of market development, size, participants, regulatory environ-

ment, and future prospects include Moodys (2006), Conning & Company

(2007), and Ziser (2007). Moreover, a special report by Fitch Ratings

(2007) identifies selected risks associated with the market. Casey and

Sherman (2007) discuss whether life settlements should be regarded as

a security, Gatzert et al. (2009) analyze the effects of a secondary mar-

ket on the surrender profits of life insurance providers and Katt (2008)

discusses direct sales without intermediaries. Finally, Gatzert (2010) pro-

vides a comprehensive overview and discussion of benefits and risks of

the secondary markets for life insurance in the U.K., Germany, and

the U.S.

Apart from these publications, which focus on the market conditions

and their implications, the literature has presented other topics related

to life settlements. Regulation and tax aspects are reviewed by Doherty

and Singer (2003), Kohli (2006), and by Gardner et al. (2009). A study

by Deloitte (2005) features an actuarial analysis of the value generated

for the seller in a life settlement transaction. Russ (2005) examines the

quality of life expectancy estimates and Milliman Inc. (2008) offers in-

sights on mortality experience for two U.S. providers. Further publica-

tions include Zollars et al. (2003) and Mason and Singer (2008) who

address the valuation of life settlements. Perera and Reeves (2006) and

Stone and Zissu (2007) explore the sensitivity of life settlement returns to

life expectancy estimates and possibilities of risk mitigation, respectively.

Finally, Stone and Zissu (2006) as well as Ortiz et al. (2008) consider

the securitization of life settlements, a likely and natural future direc-

tion for the asset class when considering that the agencies A.M. Best

and DBRS have already provided their views on rating methodologies

for such transactions (see A.M. Best, 2009; DBRS, 2008).

To the best of our knowledge, no empirical analysis of investment

return characteristics and performance for the life settlement asset class

has been conducted in the literature yet. In addition, a comprehensive

analysis of its risks from an investor’s perspective is still missing. A ma-

jor reason for the lack of empirical work in this context is the scarcity


of publicly available data on life settlement transactions. In the last few

years, however, a growing number of open-end funds exclusively dedi-

cated to investing in U.S. life settlements has emerged. These funds

determine their portfolio values on a monthly basis, thus providing the

possibility for a performance analysis based on time series data. Con-

sequently, in this paper, we contribute to the literature by conducting

the first empirical analysis of life settlement fund return distributions, a

general performance measurement, and a comparison to established as-

set classes. In addition, we put the empirical results into perspective by

extensively elaborating on the risks associated with open-end life settle-

ment funds. Our dataset has been provided by AA-Partners, a private

consulting firm specialized in U.S. life settlements, and is, in its entirety,

not publicly available. Since the dataset largely covers this segment

of the capital markets, we believe it to be a unique opportunity to gain

early insights into the return characteristics of this rather new asset class.

The remainder of the paper is structured as follows. In Section 2 we

give a brief overview of the secondary market for life insurance in the

U.S. and discuss key aspects of the structure and business model of life

settlement funds, which are essential to an understanding of their risk

profile. Section 3 is the empirical section, comprising the examination

of the funds’ return characteristics, the performance measurement and

the correlation analysis both on an aggregate level and for the individ-

ual funds in the dataset. A discussion of the risks associated with life

settlements in general and open-end life settlement funds in particular

is presented in Section 4. Finally, in Section 5 we conclude.

2 Market overview and fund business model 5

2 Life settlements: market overview

and fund business model

2.1 The U.S. life settlement market: an overview

Not many countries have a secondary market for life insurance policies,

because of the dependence on a sufficiently large primary market and

available target policies. The primary market in the U.S. represents the

largest life insurance market worldwide, making up 24.17 percent of the

global premium volume in 2007 (see SwissRe, 2007). In particular, ap-

proximately 160 million individual life insurance policies are currently

in force, with an aggregate face amount of more than 10 trillion USD

(see ACLI, 2007).2 According to the U.S. Individual Life Insurance Per-

sistency Study 2009 by the Life Insurance Marketing and Research As-

sociation (LIMRA) and the Society of Actuaries (SOA), this figure can

be broken down into 51.8 percent whole life, 23.9 percent term life, 14.5

percent universal life, and 9.8 percent variable universal life policies (see

LIMRA, 2009).

In the U.S. senior life settlement market, life insurance policies of in-

sureds above the age of 65 with below-average life expectancy—typically

2-12 years—and impaired health are purchased.3 Traded target policies

mainly include lifelong contracts with death benefit payment such as uni-

versal or whole life insurance, with universal life being by far the largest

segment.4 These contracts differ in their premium payment method,

which may be an important criterion with regard to the attractiveness

2For comparison, the U.S. equity market capitalization as of June 2010 is 12.4trillion USD (source: S&P), U.S. government bond notional outstanding as of Au-gust 2010 amounts to 8.4 trillion USD (source: U.S. Treasury), U.S. corporate bondnotional outstanding as of Q1/2010 is 7.2 trillion USD (source: Securities Industryand Financial Markets Association), global hedge fund assets under management asof Q2/2010 amount to 1.5 trillion USD (source: Credit Suisse Asset Management),and global commodity derivative notional outstanding as of December 2009 is 2.4trillion USD (source: Bank for International Settlements).

3This is in contrast to the life settlement markets in the U.K. or Germany, whereendowment contracts with a fixed maturity are traded. For an overview of the sec-ondary market for life insurance, see Gatzert (2010).

4In the partial study of Life Policy Dynamics LLC (LPD) (2007a,b), the share ofuniversal life among purchased policies is approximately 80-85 percent.


for investors. While whole life contracts have constant level premiums,

universal life policies offer the possibility of flexible premium payments

as long as the cash value (policyholder’s reserve) remains positive. When

selling the policy to a life settlement provider, the policyholder receives a

payment that exceeds the surrender value but is less than the death ben-

efit. The provider determines the offer price by subtracting the present

value of expected future costs from the present value of expected future

benefits associated with the contract. The actual amount depends in

large part on the insureds estimated life expectancy. Thus, an impor-

tant yield driver from the investors perspective is the quality of the life

expectancy estimates provided by medical underwriters. Life settlement

providers commonly sell the policies on to investors who continue to pay

the premiums necessary to keep the policy in force and, in turn, receive

the death benefit (face value) when the insured person dies.5 Hence,

while the payment amount—the face value of the policy—is known when

a policy is purchased, the payment date is stochastic. The shorter the

insured lives after having sold the policy, the higher the return for the

investor, since only few premiums have to be paid and the death benefit

is received earlier.

In line with the large primary market, the U.S. life settlement market

has ample potential.6 In its Data Collection Report 2006, the Life In-

surance Settlement Association (LISA) published data it collected from

11 life settlement providers, which were estimated to represent about 50

percent of the industry. Those figures show that the annual death bene-

fits settled increased by around 65 percent from 3.9 billion USD in 2005

to more than 6.4 billion USD in 2007, and the number of settled policies

rose by 54 percent from 2,025 to 3,138 (see LISA, 2008). Other market

estimates include Conning & Company (2007) (5.5 billion USD in 2005;

6.1 billion USD in 2006) and Kamath and Sledge (2005) (total market

size: 13 billion USD in 2005).

5According to a special report by Moodys (2006), life settlements are primarilypurchased by institutional investors.

6The market volume is commonly reported in terms of the aggregated face valueof purchased policies.

2.2 Closed-end vs. open-end life settlement funds 7

2.2 Closed-end vs. open-end life settlement funds

The first life settlement funds appeared between 2002 and 2004, offer-

ing investors access to this relatively young asset class, which emerged

during the late 1990s in the aftermath of the fading market for viatical

settlements.7 For most investors, an investment through funds is sig-

nificantly more attractive and convenient than a direct purchase of the

underlying life insurance policies due to diversification benefits and the

reliance on professional expertise to determine the portfolio composition.

In addition, the complex acquisition process of a life insurance policy, in-

cluding legal requirements and transaction costs, are a major constraint

to direct investments.

Thus, the popularity of funds investing in U.S. life settlements has

grown continuously in recent years. During this period, two types of such

funds have evolved.8 Closed-end life settlement funds in the legal form of

limited partnerships with a fixed maturity strongly resemble structures

that are well-known from other illiquid asset classes such as investments

in real estate, aircraft or ships. In these cases, the fund management

company or a special-purpose subsidiary typically acts as the fund’s gen-

eral partner, while investors participate in the fund as limited partners.

The fund shares are therefore virtually an entrepreneurial equity holding

for which a premature redemption is not intended. These closed-end life

settlement funds are domiciled in the country of their primary investor

base, which are currently mainly Germany, the U.K., Ireland, and Lux-

embourg (see, e.g., Seitel, 2006; Moodys, 2006). They follow a classical

buy-and-hold investment style, generally do not use leverage and have

a rather moderate fee schedule, comparable to common mutual funds,

where the manager receives a fixed percentage of the assets under man-

7Viatical settlements are life insurance contracts of terminally ill policyholders,which are sold in the secondary market. The viaticals business surged during theAIDS epidemic in the late 1980s (see, e.g., Fitch Ratings, 2007.

8The information in this section is largely based on offering memorandums aswell as marketing material of a number of funds, which in some cases was publiclyavailable on their websites, whereas in other cases was received upon request. Webelieve that the typology we offer adequately captures the key characteristics andmain differences of these investment vehicles.


agement.9 Another distinctive feature is the liquidity reserve most of

them build up from subscription payments in order to handle liquidity

risks arising from a lack of cash inflows after the final close of the fund.

Money returning from maturing policies is usually distributed to the

limited partners instead of reinvested. Closed-end life settlement funds

provide an annual report on their operations but refrain from delivering

portfolio valuations on a regular basis.

In contrast to their closed-end counterparts, open-end life-settlement

funds are perpetual and generally offer ongoing subscriptions and re-

demptions in either monthly or quarterly intervals. Liquidity from an

investor’s point of view is usually restricted by notice periods between

30 and 90 days, lock-ups of up to 3 years, and so-called gates: limits

on the amount which can be withdrawn in a given period. This type of

life settlement fund is almost exclusively domiciled in offshore banking

places and thus features a variety of legal forms consistent with local par-

ticularities. Active trading of the portfolio and leverage is possible and

the fee structure is hedge fund-like with management fees of one to two

percent and performance fees of up to twenty percent for which in some

cases hurdle rates and high water marks apply. The death benefit pro-

ceeds from matured policies are almost exclusively reinvested in order to

acquire new life settlement assets, whereas distributions to investors are

rather exceptional. Taking these characteristics into account, together

with targeted absolute returns of between eight and fifteen percent p.a.,

these funds have structural similarities to hedge funds.10 Open-end life

settlement funds provide valuations on a regular basis. Since the sec-

ondary market for life insurance policies is not as large and developed

as other capital markets, the underlying of life settlement funds is es-

sentially illiquid. Accordingly, a marking-to-market of their portfolios is

usually not possible and the need for mark-to-model valuation mecha-

nisms arises. On each valuation date, the funds employ their valuation

9A few exceptions exist with regard to these characteristics. Those resemble pri-vate equity funds, combining a limited partnership structure domiciled in an offshorebanking location with the possibility to actively trade policies as well as performancefees and leverage.

10Due to their specific underlying, however, life settlement funds are long-only.

2.2 Closed-end vs. open-end life settlement funds 9

Type Closed-end Open-end

Domicilecountry of primaryinvestor base

offshorebanking locations

Legal form limited partnerships depends on domicile

Regulationsubject tonational regulation

virtually unregulated

Maturity fixed perpetual

Subscriptions not after final closeongoing(usually monthly)

Redemptions at maturityongoing(monthly or quarterly)

Lock-Up period n/a up to 3 years

Notice period n/a 30 - 90 days

Redemption limits n/a common

Investment style buy-and-hold active trading possible

Leverage none possible

Fee schedulefixed percentageof capital

management/performancefees; hurdle rates; highwater marks

Liquidity reserve common not common

Death benefits distribution reinvestment

Valuations annual report on a monthly basis

Table 1: Closed-end vs. open-end life settlement funds


methodology in order to determine the net asset value (NAV) of their

portfolio, i.e., the value of their assets less the value of their liabilities,

which then forms the basis for subscriptions and redemptions of fund

shares.11 As a result, time series of monthly NAVs for open-end life

settlement funds exist and can be used to conduct an empirical per-

formance analysis. Table 1 summarizes the main structural differences

between closed-end and open-end life settlement funds.

Although closed-end and open-end life settlement funds are currently

quite common, it is uncertain whether both of these formats will prevail

throughout the next decade. From their emergence until they become

established, asset classes usually traverse an evolutionary process with

regard to their wrapping, beginning with rather illiquid structures such

as closed-end funds and successively migrating to more liquid ones as

the market grows larger, more transparent, and increasingly standard-

ized. The advent of derivatives as well as securitization are commonly

seen as indications of a maturing asset class. Against this background,

industry experts expect open-end funds to dominate the life settlement

market in the future. Early signs of this development are already be-

coming apparent: there are a number of initiatives to promote standard-

ization, transparency and the diffusion of information pertaining to life

settlement transactions. One example is the Institutional Life Markets

Association (ILMA), which was founded by institutional investors such

as Credit Suisse, Goldman Sachs, and Mizuho International PLC.12 Fur-

thermore, according to AA-Partners, half a dozen new open-end funds

are currently being prepared for launch. In contrast to that, there seems

to be humble activity in the closed-end segment.

11Funds can either apply the ”investment method” or the ”fair value method” forthe ongoing valuation of their life insurance policies. The choice needs to be made onan instrument-by-instrument basis and is binding for the entire term of the contract.These methods will be described in further detail in Section 4.2.

12For more information refer to www.lifemarketsassociation.org.

2.3 The anatomy of open-end life settlement funds 11

2.3 The anatomy of open-end life settlement funds

To interpret empirical results for open-end life settlement funds and ana-

lyze their risk profile, it is of critical relevance that one first understands

their mechanics. To the best of our knowledge, neither the structure nor

the business model of open-end life settlement funds has been compre-

hensively described before. Consequently, the remainder of this section

explains how these funds operate. For the sake of clarity it is organized

based on the roles of the various involved parties. A stylized representa-

tion of an open-end life settlement fund is depicted in Figure 1.

As any other collective investment scheme, life settlement funds de-

pend on so-called trustees, i.e., certain institutions which hold their prop-

erty and facilitate their transactions. Hence, before entering business,

the fund management company needs to appoint a custodian (deposi-

tary) in its country of domicile. The primary function of the custodian

is to hold the fund’s assets. In general, the custodian administers any liq-

uid assets, such as government bonds or cash and assigns the safekeeping

of life settlements to a sub-custodian in the United States. Furthermore,

the custodian is responsible for the administration of the fund shares

(units), for receiving and holding application money, and for redistribut-

ing funds to investors in the course of redemptions.

Whenever life insurance policies are acquired, the custodian transfers

the necessary amount of money to the sub-custodian, which, in turn,

uses it to settle transactions. With regard to policy purchases, the sub-

custodian also serves as an escrow agent, facilitating the acquisition by

retaining the payment for the respective life settlement in an escrow

account while the transfer documents are sent to the insurance company

in order to change ownership rights and beneficiaries. Once the amended

documents have been returned by the insurance company, the money is

released to the seller. The original life insurance contracts as well as

the transfer and assignment documents are subsequently held by the

sub-custodian on behalf of the fund. Whenever due, regular premiums

are paid by the sub-custodian. In some cases, the sub-custodian is also

12

ILifeSettlementFunds

Secondary market regulatory environment and U.S. tax legislation

Insureds

Life settlement fund(open-end)

Liquid assets*

Premium reserve

Other third party services

Life settlementportfolio

Medicalunderwriters

Original

beneficiaries

Servicer

Lifesettlementproviders**

Lifeinsurancecompanies Auditor Actuarial Legal

advisors advisors

Investors’funds

Sub-custodian

(Escrow agent)

Custodian

(Depositary)

Investors

Banks

Leverage

Liquidity

* Note that in case the fund retains any liquid assets such as government bonds or cash, those are usually held by the custodian.** Some policy sellers are represented by life settlement brokers, who negotiate with several life settlement providers to obtain the best offer.

Figure 1: Stylized mechanics of an open-end life settlement fund


responsible for building up a premium reserve account for the fund in

order to be able to mitigate potential liquidity shortages.

Medical underwriters review the medical records of the insureds and,

based on the information contained therein, prepare mortality profiles

that comprise a summary of the medical conditions, a mortality sched-

ule, and an estimation of the life expectancy for each insured.13 For this

purpose, they assess how certain characteristics and medical conditions

affect the insureds mortality relative to a ”standard” or reference mor-

tality (see A.M. Best, 2009). The outcome is a specific multiplier (also

called mortality rating) which modifies the reference mortality. Method-

ologies for the derivation of the multiplier as well as standard mortality

tables depend on the medical underwriters. However, within the last

few years, many medical underwriters have opted for the Valuation Ba-

sic Tables (VBT), which are prepared by a task force of the Society of

Actuaries (SOA).14 These tables include mortality rates for ages up to

ninety years over time horizons from one to twenty-five years, which

have been derived from historical data and are differentiated according

to simple characteristics such as smoking status and gender.15 Although

life expectancy estimates have systematically increased over the last few

years, the figures provided by different medical underwriters for the same

lives can vary substantially, implying a potential for misestimation (see

A.M. Best, 2009; Gatzert, 2010). This has implications both for the

pricing of life settlements and for the fund returns. Consequently, some

funds seek to mitigate the impact of misestimation by demanding at

least two life expectancy estimates and then applying the longer one or

a (weighted) average of the two.

The servicer (tracking agent) performs a wide variety of supporting

services in the context of premium and claims administration for the pool

13The four largest medical underwriters in the U.S. life settlement market are 21stServices, AVS, EMSI, and Fasano (see Russ, 2005; Gatzert, 2010).

14See www.soa.org.15Mortality rates are commonly denoted by qx (where x stands for the current

age of the group under consideration) and measure the number of deaths per 1000individuals of a population in a certain time period (typically one year).


of lives in the portfolio.16 Its goal is to ensure a smooth and on-time

transfer of legal paperwork, notifications, and cash flows. The servicer

notifies the trustees and provides them with disbursement instructions

for the regular premium payments and maintains close contact with the

insurance company to obtain the latest information on developments of

each policy (e.g., cash surrender values). Moreover, it is responsible for

ordering the policyholder’s medical records and life expectancy estima-

tions from the medical examiners and then archiving them. Another key

duty is the tracking of the insured, i.e., the maintenance of registers with

their contact details as well as the verification of their life/death status.

For this purpose, the servicer relies on routines which resemble those em-

ployed in consumer loan servicing such as subscribed database services,

mailings, and telephone calls. In addition, it matches social security

numbers to death indices on a regular basis. Whenever the servicer be-

comes aware of the death of a policyholder, it immediately informs the

fund manager and the trustees and obtains the death certificate. After

the signed insurance claim package has been provided by the trustee, the

servicer forwards it to the insurance company and follows up until the

claim is paid so as to facilitate the prompt collection of death benefits.

Life settlement providers (life settlement companies) source life in-

surance contracts from policyholders or licensed brokers in order to pass

them on to the funds. For this purpose, the funds usually set certain

investment criteria, which reflect the cornerstones of their portfolio di-

versification approach. Life settlement providers can also act as invest-

ment advisors, pitching life settlements to the manager and participat-

ing in the policy-picking and portfolio structuring process. Whereas

some funds rely on a so-called single-source approach, thus exclusively

collaborating with just one life settlement company, others deliberately

maintain business relationships with several. Such a multi-source ap-

proach is meant to improve the funds’ access to life settlement assets,

especially in times of greater product-flow constraints or less active mar-

kets (see, e.g., McNealy and Frith, 2006). An important aspect to be

16Note that the average fund portfolio in our dataset comprises 193 lives, while themaximum number of lives in a portfolio is 567 (see Table 2 in Section 3.1).


considered with regard to life settlement providers are their incentives

to act in the interest of investors. Since their fees are paid upfront and

generally depend on number and volume of the policies rather than their

long-term investment performance, the degree of diligence that can be

expected from life settlement providers during the acquisition process

is questionable.17 More specifically, to increase their chance of prevail-

ing in the competitive bidding process for a policy they could, e.g., be

tempted to avoid medical underwriters which issue rather conservative

life expectancy estimates, since those would be associated with a lower

offer price. Once acquired, the policy is then resold by the life settlement

provider to the fund whose investors ultimately have to bear the risk of

a misestimated life expectancy.

In addition, the general mechanics of open-end life settlement funds

are usually complemented by third-party service providers. Auditors ad-

vise on accounting and tax implications, inspect the funds’ balance sheets

and income statements, and issue annual reports with their opinion of

the funds’ financial situation. Moreover, actuarial advisors assist with

the pricing of transactions as well as the valuation of life settlements in

the portfolio and review actuarial models used by the funds. Similarly,

legal advisors offer counseling with regard to the legal form, draft all

the contracts, and ensure the completeness of documentation packages

in addition to compliance with the applicable legislation and regulation.

Banks are involved either by providing medium- to longer-term debt fi-

nancing, which some funds use to leverage their investments, or through

a liquidity facility, which is commonly employed to bridge life settlement

purchases or premium payments in the absence of other cash inflows.

Finally, life insurance companies originally issued the policies and must

be notified about the transfer of ownership. They continue to receive

the premiums after the sale has been completed and pay out the death

benefits to the fund’s sub-custodian after the insured has passed away.

17This is quite similar to the incentive problem that ultimately led to the demiseof the U.S. subprime market where the common practice of instantly selling-on ini-tiated mortgages to third parties, such as investment banks (originate-to-distribute),created a lack of long-term financial incentives and instigated originators to an ex-treme relaxation of lending standards.


3 Empirical analysis

3.1 Data and sample selection

We obtained our data on open-end life settlement funds from AA-Partners

AG, a Zurich-based investment boutique specialized in this asset class.18

Analogously to providers of hedge fund data, AA-Partners maintains an

extensive network in the life settlement industry, through which it is

in a position to collect performance data directly from fund managers.

Using a variety of sources, it carries out regular cross-checks and veri-

fications of its fund database to ensure correct classification, reliability

and representativeness. The original dataset comprises monthly NAVs of

17 open-end funds, which, according to AA-Partners, largely cover this

market.19 Each fund is USD denominated, subject to an independent

audit conforming to international standards and almost all are purely

dedicated to investing in U.S. senior life settlements, i.e., mixed strategy

funds are excluded.20 In our view, this dataset is a valid opportunity for

an empirical analysis, as we are not aware of any other sources of such

comprehensive time series data for life settlement funds.

Table 2 provides additional information with regard to inception, size,

fee structure, and liquidity profile of the fund shares.21 While the oldest

fund in the dataset began operations in late 2003, other funds emerged

just as recently as 2007/2008. This suggests that the asset class has

gradually attracted the attention of the investment industry throughout

the last decade. Interestingly, the funds are quite different in size as

reflected by investment volumes, number of policies in the portfolio, and

the sum of face values. This can be an important factor with regard to

potential policy availability issues which will be discussed in Section 4.5.

18AA-Partners acts as an independent third party advisor with regard to investmentsolutions for U.S. life settlements. Its main services include investment advice relatedto open-end funds, valuation of life settlement portfolios, market research, and datacollection. See www.aa-partners.ch for more information.

19The dataset consists of single funds. To our knowledge, life settlement fund offunds do currently not exist.

20However, one fund has a minor position in U.K. endowment policies and anotherone holds a small fraction of viatical settlements.

21For confidentiality reasons the fund names have been substituted with numbers.

3.1 Data and sample selection 17

While there is some variation in the fee structures, most funds seem to

charge a management fee of around two percent and a performance fee

of around twenty percent. The majority of life settlement funds in the

dataset offers subscriptions and redemptions on a monthly basis with a

notice period of 30 days. Furthermore, several funds partially protect

themselves against excessive cash outflows by imposing redemption gates

and lock-up periods on their investors.

Since the market is still in an early stage of its development, not all of

the funds feature time series of sufficient length for statistical inference.

To capture the risks and returns of open-end life settlement funds as

comprehensively as possible, we have created a custom index, beginning

with the oldest fund, which appeared in December 2003. Whenever the

inception date of another fund is reached, it is added to the index and

whenever the return time series of a fund ceases prematurely (e.g., due to

suspended reporting or liquidation), it drops out of the index. The index

time series ends in June 2010 and comprises 79 monthly returns. At any

point in time, the returns of all index constituents are equally weighted.22

A further analysis of the individual funds will be conducted in Section

3.4. In addition to the custom life settlement index, we have selected

broad indices as representatives for various other asset classes in order to

conduct performance comparisons and correlations analyses.23 In this

context, the U.S. stock market is represented by the S&P 500 while

the FTSE U.S. Government Bond Index as well as the DJ Corporate

Bond Index have been selected as proxies for the respective bond mar-

kets. Furthermore, the HFRI Fund Weighted Composite Index serves as

a broad measure for the hedge fund universe, while real estate returns

are provided through the S&P/Case-Shiller Home Price Index (Compos-

22Note that this approach of calculating the index assumes an investor with a naıvediversification approach, assigning the same target portfolio weight to all availablelife settlement funds at any point in time. We believe this procedure to be an ade-quate way of reflecting the development of an open-end life settlement fund portfoliobetween 2003 and 2010.

23Indices are sufficiently diversified portfolios. Thus, an analysis based on indicesis well suited to examine the risk-return profile of aggregate asset classes.


ite of 20).24 Finally, the S&P GSCI, a recognized measure of general

commodity price movements, is used as indicator for the global com-

modity markets. The selection is completed by the S&P Listed Private

Equity Index. Since congruent time series are required for our analysis,

the scarcity of available life settlement fund data constrains the choice

of time period and return interval for the other asset classes. Hence,

monthly index returns from December 2003 to June 2010 have been col-

lected for those as well.25 Wherever available, total return indices have

been used to account for coupons and dividends, which would otherwise

not be reflected in prices. Table 3 summarizes the sample characteristics.

As with hedge fund data, our sample suffers from certain biases,

which have to be considered when interpreting the empirical results in

the following section.26 Self-selection bias arises from the rather opaque

nature of the funds which, in contrast to mutual funds, are not obliged to

disclose return data to the public. This bias is likely to be particularly

large if non-reporting funds significantly underperform their reporting

counterparts. However, we are aware of 17 funds that essentially make

up the market. Of these 17 funds, only two suspended their reporting

during the time period under consideration. Hence, we consider this bias

not to be material.27 In addition, survivorship bias arises when funds,

which ceased to exist, are not included in a database. If these funds ter-

minated operations as a result of poor performance, the available data

is likely to overstate historical returns and understate risk. AA-Partners

knows of three funds, which were shut down, but have never been part

of their database.28 Apart from that, two of the 17 funds in our dataset

24We deliberately chose the S&P/Case-Shiller Index instead of publicly listed RealEstate Investment Trust (REIT) indices, since the latter are strongly influenced bygeneral stock market dynamics and due to this noisiness only partly reflect the per-formance of the true underlying real estate assets. This phenomenon with regard toREITs has been described by Giliberto (1993) and Ling et al. (2000).

25The data has been downloaded from the Bloomberg database.26For a more detailed discussion of these biases, see L’Habitant (2007).27Self-selection bias cannot be quantified as the returns for non-reporting funds

remain unobservable.28Due to the over-the-counter character of the market for open-end life settlement

funds, data collection is a very challenging and time consuming task. Even insti-tutions with extensive connections into the life settlement industry, such as AA-Partners, are unable to obtain return data in certain cases.

3.1

Data

and

sam

ple

selection19

Fund 100 Fund 101 Fund 102 Fund 103 Fund 104 Fund 105

Currency USD USD USD USD USD USD

Inception Apr 03 June 06 Aug 03 June 07 Nov 05 Dec 03

Volume (mn) 385 950 428 102 466 62

Sum of facevalues (mn)

770 2367 720 362 619 108

No. of policies 261 567 447 183 406 65

Management fee 0.75% 1.95% 2.00% 2.00% 1.50% 1.50%

Performance fee n/a 20.00% n/a 20% 75% n/a

Hurdle Rate n/a 10.00% n/a 9.00% 8.00% n/a

Style passive passive passive passive passive passive

Subscriptions monthly monthly monthly monthly monthly monthly

Redemptions monthly monthly monthly monthly monthly monthly

Notice Period 30 days 30 days 30 days 90 days 45 days 30 days

Redemption fees n/a

year 2: 4%year 3: 4%year 4: 3%year 5: 3%

10%,decreasingby 0.33%per month

year 2: 8%year 3: 7%year 4: 4%nil after

5%,decreasingby 1%per year

year 1: 7%year 2: 5.5%year 3: 4%year 4: 2.5%year 5: 1%

Lock-up Period n/a 1 year n/a 1 year n/a n/a

Redemptionlimits (gates)

10% ofoutstandingshares p.a.

n/a20% ofoutstandingshares p.a.

30(60)% ofinvestmentin year 2(3)

10% of sharesper redemptiondate

10% of sharesper redemptiondate

Table 2: Life settlement funds in the original dataset

20


Fund 201 Fund 202 Fund 203 Fund 204 Fund 205 Fund 208

Currency USD USD USD USD USD USD

Inception Jan 05 July 06 Feb 05 March 04 Dec 04 Nov 06

Volume (mn) 31 494 unknown unknown 10 57


98 unknown unknown unknown 45 283

No. of policies 126 unknown unknown unknown 40 113

Management fee 1.50% 2% 1.75% 1.75% 1.50% 1.25%

Performance fee n/a n/a 25% 20% 20% 15%

Hurdle Rate n/a n/a 8% 6% n/a 7%

Style passive passive passive passive passive passive

Subscriptions weekly monthly monthly monthly monthly monthly

Redemptions weekly monthly Monthly monthly quarterly monthly

Notice Period 30 days 30 days 30 days 30 days 60 days 90 days

Redemption fees

5%,decreasingby 1%per year

7% untilyear 8,4% after

deferredsales charge(5 years)

year 1: 8.6%year 2: 8.6%year 3: 7.5%year 4: 6%year 5: 5%

3%

17.5%,decreasingby 2.5%per year

Lock-up Period n/a 3 years 1 year 6 months n/a n/a


n/a 20% p.a. 20% p.a. 10% p.a. n/a 20% p.a.

Table 2: Life settlement funds in the original dataset - continued

3.1

Data

and

sam

ple

selection21

Fund 210 Fund 212 Fund 216 Fund 217 Fund 514

Currency USD USD USD USD USD

Inception July 04 Dec 07 Jan 07 Jan 08 June 06

Volume (mn) 100 8 43 5 178


178 20 130 20 344

No. of policies 242 17 58 8 179

Management fee 0.30% 1.25% 2.00% 2.00% 0%

Performance fee n/a 10% 20% 20% 30%

Hurdle Rate n/a 10% 8% n/a 6.5%

Style passive passive active passive passive

Subscriptions monthly monthly monthly monthly monthly

Redemptions monthly monthly quarterly quarterly monthly

Notice Period 30 days 30 days 90 days 90 days 90 days

Redemption fees

year 1: 3%year 2: 2%year 3: 1%nil after

n/a2% afterfirst year

n/a8%, decreasingby 1.6%per year

Lock-up Period n/a n/a 1 year 1 year n/a


n/a 5% p.a.10% oftotal assets p.a.

10% oftotal assets p.a.

20% p.a.

Table 2: Life settlement funds in the original dataset - continued


Observed variables8 indices(see Appendix for additional information)

Selection criterionBroad market indices, i.e., diversified portfolios(as representative as possible for each asset class)

Return interval monthly returns

Sample period12/2003–06/2010(79 data points)

Source of dataAA-Partners AG for life settlement fundsBloomberg for indices of other asset classes

Table 3: Sample details

are currently being liquidated and the final proceeds to investors are

unknown at this time. According to AA-Partners, those liquidation pro-

ceeds can be expected to be considerably smaller than the last NAV

published by the funds. These considerations imply that survivorship

bias could, to some extent, be an issue in the context of our empirical

analysis. Since the return time series of those funds which were not in-

cluded in the database as well as the liquidation proceeds for the two

terminated funds are not available to us, it is not possible to measure and

consequently explicitly control for survivorship bias. Finally, illiquidity

bias is an issue with regard to life settlement funds. Life settlements are

highly illiquid assets. Thus, a marking-to-market is difficult due to the

absence of regularly quoted market prices. Accordingly, the fund man-

agers have considerable flexibility when determining NAVs, which they

could use to smooth monthly returns. This bias is of major importance

as we will see in the detailed risk analysis of the funds in Section 4.

3.2 The return distribution

of open-end life settlement funds

Based on the fact that the main underlying risks are biometric in nature

rather than originating from the broader capital markets, academics and

practitioners have repeatedly emphasized that life settlements should

offer attractive returns paired with a conservative risk profile and are

3.2 The return distribution of open-end life settlement funds 23

uncorrelated with other asset classes (see, e.g., Stone and Zissu, 2007).

In order to verify this, we conduct the first empirical analysis of this

asset class.29 We begin with a characterization of the empirical return

distributions, which forms the basis for subsequent comparisons. Figure

2 plots the performance of all previously mentioned asset classes, ex-

cept commodities and private equity, between December 2003 and June

2010.30 All time series have been indexed to 100 at the beginning of the

period under consideration, thus reflecting the development of the value

of a hypothetical investment of 100 USD over time.

At a first glance, the graph of open-end life settlement funds looks

excellent. It dominates both bond indices at almost every point in time

and has only been exceeded by stocks and real estate until the subprime

crisis in the U.S. struck in summer 2007 and spread into the global capital

markets in 2008. Over the whole period, only a hedge fund investment

would have yielded a higher value. These observations are also reflected

in the figures characterizing the return distribution, which can be found

Table 4. The portfolio of life settlement funds represented by our cus-

tom index exhibits generally respectable positive returns and very low

volatility. Furthermore, it has only suffered a comparatively moderate

drawdown31 during the financial crisis of 2007 to 2009. With the substan-

tial quantity of 37.30 percent, life settlement funds generated the third

highest total return of all analyzed asset classes from December 2003

to June 2010. Only hedge funds (45.90 percent) and government bonds

(37.38 percent) provided higher total returns over this period. Apart

from corporate bonds, which yielded a mere 2.00 percent, the remaining

asset classes even exhibited negative total returns. An investment in

stocks, for example, would have lost 2.60 percent of its original value.

29To the best of our knowledge, the scarcity of NAV data did not allow for anyearlier empirical analysis.

30The S&P GSCI as well as the S&P Listed Private Equity Index with their com-paratively high volatility have been excluded from this figure in order to enhance thereadability. Please refer to Table 4 for the respective data.

31That is, a loss incurred over a certain time period.


Studying the means of the monthly return distributions reveals a

similar pattern. With 0.40 percent (4.85 percent p.a.), open-end life

settlement funds had a higher mean return than all other asset classes

except for hedge funds (0.50 percent; 5.98 percent p.a.) and government

bonds (0.41 percent; 4.91 percent p.a.). While private equity (0.37 per-

cent; 4.44 percent p.a.) and commodities (0.25 percent, 2.95 percent p.a.)

also exhibited positive mean returns over the period under consideration,

those of the remaining asset classes were close to zero. Moreover, life set-

tlement funds were by far the least volatile investment, as represented by

their return standard deviation of 0.66 percent (2.28 percent p.a.). Even

government bond returns with a standard deviation of 1.10 percent (3.80

percent p.a.) were almost twice as volatile, let alone stocks, commodi-

ties and private equity, where the multiplier is more than six, eleven,

and thirteen, respectively. Maximum and minimum returns are furthest

apart for the asset classes with the highest volatilities, i.e., private equity,

commodities, and stocks, while the empirical return distribution for life

settlements merely spans 5.94 percent from a maximum of 2.79 percent

to a minimum of -3.15 percent.

The remarkable impression provided by the portfolio of life settlement

funds is further bolstered by taking into account the small number of

negative returns: only 9 during the whole examination period of 79

months (see row 11 of Table 4). All remaining asset classes experienced

many more negative months, ranging from 26 to 33. However, the life

settlement fund return distribution exhibits the comparatively largest

negative skewness (-1.97) and positive excess kurtosis (12.66), implying

a long and heavy left tail. These values for the third and fourth moments

lead to an exceptionally high Jarque-Bera test statistic (578.55), meaning

the null hypothesis of normality has to be rejected on all reasonable

significance levels.32

32Although for almost all other asset classes, the null hypothesis under the Jarque-Bera test is rejected on the one percent significance level as well, their test statisticsare considerably smaller.

3.2

The

return

distrib

ution

ofop

en-en

dlife

settlemen

tfu

nds

25

6080

100

120

140

160

time

index

lev

el

2004 2005 2006 2007 2008 2009 2010

Life Settlement Fund IndexS&P 500FTSE U.S. Government Bond IndexDJ U.S. Corporate Bond IndexHFRI Fund Weighted Composite IndexS&P/Case−Shiller Home Price Index

Figure 2: Life settlements in comparison to other asset classes (12/2003–06/2010)


3.3 Performance measurement

and correlation analysis

To elaborate on the risk return profile of open-end life settlement funds,

we apply four common performance measures.33 Apart from the proba-

bly most classic performance measure in finance literature, the Sharpe

Ratio, we calculate the Sortino Ratio, the Calmar Ratio and the Excess

Return on Value at Risk (VaR) for the asset classes under consider-

ation.34 Based on these indicators, we establish a rank order for all

asset classes with positive excess returns.35 The results are displayed

in the lower part of Table 4. With a Sharpe Ratio of 0.3327, life settle-

ment funds clearly rank first with a considerable distance to the second-

ranked government bonds (0.2039). Hedge funds (0.1589), private equity

(0.0211), and commodities (0.0079) on ranks 3, 4, and 5 also feature a

positive Sharpe Ratio, which, however, in the latter case is close to zero.

Negative Sharpe Ratios for the remaining investment alternatives reflect

their poor performance over the analyzed time period, falling short of a

possible investment at the risk-free rate. Looking at their Sortino Ra-

tio of 0.4580 and Excess Return on VaR of 0.2889, we gather the same

picture: life settlement funds outperformed the runners-up government

bonds and hedge funds by far.36 The performance ranking based on the

Calmar Ratio is a slight exception. With a value of 0.0695, the life settle-

ment fund index ends up on the second position, just behind government

bonds (Calmar Ratio of 0.0813).

Certainly, the choice of the time period for the analysis—including

the financial crisis 2008/2009—negatively influences the image of almost

all established asset classes. Nevertheless, two important factors should

33The definitions for these performance measures can be found in the Appendix.34The average 1-month U.S. Treasury Bill rate between December 2003 and June

2010 has been used as a proxy for the risk-free interest rate rf . The figures can beaccessed on www.ustreas.gov. With regard to the Sortino Ratio, we choose rf as thethreshold return τ . The Excess Return on VaR is based on the 95 percent VaR.

35The applied performance measures are not meaningful for negative excess returnssince, in that case, a higher value of the risk measure in the denominator leads to abetter result (less negative ratio).

36Our results are in line with the findings of Eling and Schuhmacher (2007) in thatthe different performance measures lead to almost the same rank order.

3.3

Perfo

rmance

measu

remen

tan

dcorrelation

analy

sis27

LSFI S&P 500 FTSEUSGBI

DJ CBI HFRIFWI

S&P/CSHPI

S&PGSCI

S&PLPEI

Total return 37.30% -2.60% 37.38% 2.00% 45.90% -0.84% -4.97% -1.90%

Mean return 0.40% 0.07% 0.41% 0.04% 0.50% 0.00% 0.25% 0.37%

annualized 4.85% 0.78% 4.91% 0.53% 5.98% -0.03% 2.95% 4.44%

Standard deviation 0.66% 4.39% 1.10% 1.95% 1.97% 1.27% 7.77% 8.74%

annualized 2.28% 15.20% 3.80% 6.75% 6.84% 4.40% 26.91% 30.26%

Maximum 2.79% 9.39% 3.24% 7.63% 5.15% 1.99% 19.67% 30.54%

Minimum -3.15% -16.94% -2.75% -6.43% -6.84% -2.79% -28.20% -30.33%

Skewness -1.97 -1.08 -0.21 0.11 -1.11 -0.50 -0.64 -0.43

Excess kurtosis 12.66 2.48 0.75 4.05 2.65 -0.61 1.41 3.91

Jarque-Bera test 578.55 35.74 2.45 54.18 39.41 4.58 11.92 52.83

*** *** - *** *** - *** ***

Negative months 9 30 26 33 26 39 33 27

SHR (rank) 0.33 (1) -0.03 0.20 (2) -0.07 0.16 (3) -0.15 0.01 (5) 0.02 (4)

SOR (rank) 0.46 (1) -0.03 0.33 (2) -0.10 0.22 (3) -0.18 0.01 (5) 0.03 (4)

CAR (rank) 0.07 (2) -0.01 0.08 (1) -0.02 0.05 (3) -0.07 0.00 (5) 0.01 (4)

ERVaR (rank) 0.29 (1) -0.01 0.20 (2) -0.06 0.12 (3) -0.08 0.01 (5) 0.01 (4)

Indices: Life settlement funds index (LSFI); Standard & Poor’s 500 (S&P 500); FTSE U.S. Government Bond Index (FTSE USGBI);Dow Jones Corporate Bond Index (DJ CBI); HFRI Fund Weighted Composite Index (HFRI FWI); Standard & Poor’s/Case-ShillerHome Price Index (S&P/CS HPI); Standard & Poor’s Goldman Sachs Commodities Index (S&P GSCI); Standard & Poor’s ListedPrivate Equity Index (S&P LPEI). Significance levels: *** = 1%, ** = 5%, * =10%. Performance measures: Sharpe Ratio (SHR);Sortino Ratio (SOR); Calmar Ratio (CAR); Excess Return on VaR (ERVaR).

Table 4: Descriptive statistics: index return distributions (12/2003–06/2010)


be considered. First, as mentioned in Section 3.1, the choice of time

period was not arbitrary but determined by the availability of data for

the life settlement fund market. Second, the rather weak performance of

some of the indices representing the other asset classes under considera-

tion underscores even more strongly how extraordinary these empirical

observations for life settlements are. This finding should trigger addi-

tional questions as to why this asset class has seemingly been able to

withstand the major dislocations in the world’s capital markets.

Finally, to complete the empirical analysis on the portfolio basis, we

examine the correlation structure between life settlement funds and the

other indices in our sample. Table 5 displays the correlation matrix as

well as the significance levels for the correlation t-test. Only one of the

tested Bravais-Pearson correlation coefficients between the returns on the

custom life settlement fund index and the other indices turned out to be

statistically significant. In particular, life settlement fund and corporate

bond returns seemed to be negatively correlated during our examination

period. Overall, it appears as if life settlements rightly have the repu-

tation of being virtually uncorrelated with other asset classes.37 To put

further emphasis on this result, we provided the correlation coefficients

among the remaining asset classes. Apart from two exceptions involving

corporate bonds, those are all significantly different from zero. Particu-

larly, all correlations of the HFRI Fund Weighted Composite Index with

the traditional asset classes are highly significant, raising doubts about

the suitability of hedge funds as a means for portfolio diversification. Life

settlement funds, on the contrary, seem to offer excellent diversification

qualities.

3.4 Analysis of individual funds

Due to the extraordinary performance of the life settlement fund index

revealed in the previous section, we deem it necessary to conduct further

analyses on a disaggregate level. Thus, we examine return distributions

and performance for the individual life settlement funds in the sample.

37To be more precise, we cannot reject the null hypothesis that life settlementreturns are uncorrelated with the returns of the other asset classes.

3.4

Analy

sisofin

div

idual

funds

29

LSFI S&P500

FTSEUSGBI

DJ CBI HFRIFWI

S&P/CSHPI

S&PGSCI

S&PLPEI

(I) (II) (III) (IV) (V) (VI) (VII) (VIII)

(I) 1.0000 -0.1231 -0.0414 -0.2683 ** -0.0606 -0.1679 -0.0292 -0.0834

(II) 1.0000 -0.2575 ** 0.3487 *** 0.8015 *** 0.2982 ** 0.3877 *** 0.8779 ***

(III) 1.0000 0.3639 *** -0.3795 *** -0.2681 ** -0.2361 * -0.2228 *

(IV) 1.0000 0.3603 *** 0.0427 0.1057 0.2794 **

(V) 1.0000 0.2712 ** 0.5866 *** 0.7665 ***

(VI) 1.0000 0.2442 * 0.3149 ***

(VII) 1.0000 0.4124 ***

(VIII) 1.0000

Indices: Life settlement funds index (LSFI); Standard & Poor’s 500 (S&P 500); FTSE U.S. Government Bond Index (FTSE USGBI);Dow Jones Corporate Bond Index (DJ CBI); HFRI Fund Weighted Composite Index (HFRI FWI); Standard & Poor’s/Case-ShillerHome Price Index (S&P/CS HPI); Standard & Poor’s Goldman Sachs Commodities Index (S&P GSCI); Standard & Poor’s ListedPrivate Equity Index (S&P LPEI). Significance levels: *** = 1%, ** = 5%, * =10%.

Table 5: Correlation Matrix

30


7080

9010

011

012

013

0

time

fund lev

el

2007 2008 2009 2010

Fund 101Fund 204Fund 205

Figure 3: Individual life settlement funds in comparison (01/2007–06/2010)

3.4 Analysis of individual funds 31

To ensure congruent time series, we selected the period from January

2007 until June 2010.38 This enables us to include as many funds from

the original dataset as possible, while still retaining a total of 42 monthly

returns in the time series. As a consequence, we removed three funds,

which did not yet exist in January 2007. Also note that due to vari-

ous reasons (see fund status in Table 6) the time series for some of the

remaining 14 funds stop before June 2010. The results for each fund

are reported in Table 6, Table 7 provides some summary statistics, and

Figure 3 displays the development of an investment of 100 USD in each

of the life settlement funds over the considered time period. While we

observe a solid growth in value for most funds, there are some exceptions

that differ from the pack.

In particular, we notice that Fund 202 and Fund 216 experienced a

comparatively larger number of negative months and Fund 101, Fund

204, as well as Fund 205 exhibited a large drawdown. The magnitude of

this remarkable negative monthly return is -18.97 percent, -16.68 percent,

and an enormous -51.12 percent for Fund 101, Fund 204, and Fund 205,

respectively. As a result, the return volatilities (standard deviations)

of 3.05 percent, 3.48 percent, and 9.36 percent for these three funds

are much higher than the average of 1.41 percent and the variation in

maximum and minimum returns as well as skewness and excess kurtosis

across all individual funds appears substantial (see Table 7). The highest

maximum return of 9.41 percent (Fund 204) in one month compares to a

mere 0.76 percent for Fund 514. More alarming for investors, however, is

the discrepancy in minimum returns. While those are equal to or greater

than zero for 7 of the 14 funds and Fund 102 still generated 0.54 percent

in its worst month, the previously mentioned devastating drawdown of

Fund 205 (-51.12 percent) marks the lower bound of the range. Industry

experts point out a variety of explanations for the sudden collapse in

38For the fund performance figures to be comparable, they need to be calculatedbased on congruent time periods. Although the chosen period is relatively short, ithelps to understand two important questions: Does the performance of certain indi-vidual funds considerably differ from the results we observed for the index (portfolioof funds) in the previous section? Did the financial crisis have an impact on individualfunds (this did not really seem to be the case on the aggregate level)?

32


Fund100

Fund101

Fund102

Fund104

Fund105

Fund201

Fund202

Fund status active active active active active merged active

Sample size (months) 42 42 42 42 42 28 42

Total return 31.85% 0.96% 37.27% 35.97% 32.49% n/a 7.11%

Mean return 0.66% 0.07% 0.76% 0.73% 0.67% 0.46% 0.17%

annualized 7.94% 0.89% 9.09% 8.81% 8.08% 5.47% 1.99%

Standard deviation 0.52% 3.05% 0.10% 0.14% 0.55% 0.58% 0.71%

annualized 1.81% 10.56% 0.35% 0.47% 1.90% 2.02% 2.47%

Maximum 2.62% 2.05% 0.92% 1.05% 3.95% 3.03% 2.26%

Minimum -0.94% -18.97% 0.54% 0.46% 0.25% 0.00% -1.57%

Skewness 1.02 -6.22 -0.34 0.37 5.46 3.42 0.27

Excess kurtosis 6.13 39.76 -0.97 0.32 32.45 14.46 0.98

Negative months 1 3 0 0 0 0 15

SHR (rank) 1.01 (6) -0.02 6.14 (2) 4.38 (5) 0.98 (7) 0.55 (9) 0.04 (10)

SOR (rank) n/a -0.02 n/a n/a n/a n/a 0.06 (1)

CAR (rank) n/a -0.00 n/a n/a n/a n/a 0.02 (1)

ERVaR (rank) n/a -0.22 n/a n/a n/a n/a 0.04 (1)

Performance measures: Sharpe Ratio (SHR); Sortino Ratio (SOR); Calmar Ratio (CAR); Excess Return on VaR (ERVaR).

Table 6: Descriptive statistics: individual fund return distributions (01/2007–06/2010)

3.4

Analy

sisofin

div

idual

funds

33

Fund203

Fund204

Fund205

Fund208

Fund210

Fund216

Fund514

Fund status suspendedreporting

liquidated suspendedreporting

active active active active

Sample size (months) 20 32 38 42 42 42 42

Total return n/a n/a n/a 31.14% 33.74% 2.05% 29.43%

Mean return 0.45% -0.10% -1.89% 0.65% 0.69% 0.05% 0.62%

annualized 5.41% -1.16% -22.71% 7.77% 8.34% 0.59% 7.39%

Standard deviation 0.46% 3.48% 9.36% 0.11% 0.11% 0.43% 0.07%

annualized 1.61% 12.05% 32.42% 0.37% 0.37% 1.49% 0.24%

Maximum 1.70% 9.41% 2.68% 0.88% 0.88% 1.10% 0.76%

Minimum -0.12% -16.68% -51.12% 0.47% 0.40% -1.57% 0.44%

Skewness 1.67 -3.04 -4.70 0.65 -0.70 -0.66 -0.78

Excess Kurtosis 2.37 19.12 22.78 -0.17 0.62 4.50 0.51

Negative months 1 4 8 0 0 20 0

SHR (rank) 0.6761 (8) -0.0669 -0.2167 4.8178 (4) 5.1965 (3) -0.2027 6.7865 (1)

SOR (rank) n/a -0.0886 -0.2259 n/a n/a -0.2442 n/a

CAR (rank) n/a -0.0139 -0.0397 n/a n/a -0.0555 n/a

ERVaR (rank) n/a -0.1773 -0.3397 n/a n/a -0.2429 n/a

Performance measures: Sharpe Ratio (SHR); Sortino Ratio (SOR); Calmar Ratio (CAR); Excess Return on VaR (ERVaR).

Table 6: Descriptive statistics: individual fund return distributions (01/2007–06/2010) - continued


Mean Standarddeviation

Maximum Minimum

Mean return 0.29% 0.69% 0.76% -1.89%

Standard deviation 1.41% 2.53% 9.36% 0.07%

Maximum return 2.38% 2.25% 9.41% 0.76%

Minimum return -6.32% 14.41% 0.54% -51.12%

Skewness -0.26 3.00 5.46 -6.22

Excess Kurtosis 10.21 13.40 39.76 -0.97

Table 7: Summary statistics: 14 fund return distributions

the NAVs of Fund 101, 204, and 205. The introduction of the 2008 VBT

tables by the Society of Actuaries (SOA) is certainly an important deter-

minant in this regard. In comparison to the 2001 release, which had been

widely applied in the life settlement industry, life expectancies associated

with the new tables are generally longer. In some cases, these differences

necessitated substantial policy devaluations. Another important factor

is the turmoil in the wake of the financial crisis, which significantly inten-

sified in September 2008 after the bankruptcy of Lehman Brothers and

the AIG bail-out. Due to the great extent of uncertainty in the capital

markets, many open-end life settlement fund investors began to redeem

their fund shares. In combination with a lack of subscriptions, these

excessive redemptions resulted in a liquidity shortage for some funds.

Particularly those with a substandard cash management were suddenly

forced to sell policies at fire sale prices in order to avoid complete dis-

tress. Finally, those funds, which opted for fair value accounting of life

settlements, had to substantially write down their assets to reflect the

changed market environment in late 2008 and early 2009.39

39The different valuation methods and their consequences for the evolution of fundNAVs over time are explained in more detail in the following section.

3.4 Analysis of individual funds 35

Since notable discrepancies between individual funds seem to exist,

careful selection of the fund manager can be crucial. This finding is

supported by the four performance measures we discussed earlier.40 For

instance, we observe negative Sharpe Ratios for the Funds 101, 204, 205,

and 216, implying an average monthly return below the risk-free rate. In

addition, the positive Sharpe Ratios of the remaining funds range from

6.7865 down to 0.0419, a figure that is worse than those for government

bonds, corporate bonds, and hedge funds over the same time period.41

Furthermore, it should be noted that the current status of four of the

analyzed funds is an alarming sign. Fund 203 and Fund 205 suspended

their reporting during the period under consideration, Fund 201 was

merged with Fund 103 (which had been excluded from the analysis in

this section due to its short time series and is currently being liquidated),

and Fund 204 has been terminated. Consequently, the performance fig-

ures derived from the available data for these funds can be expected to

be still upward biased.42

Overall, according to the empirical analysis of the life settlement

index return profile, the asset class indeed appears to be an interest-

ing investment opportunity, offering solid returns comparable to those

provided by government bonds, complemented by an extraordinary low

volatility as well as virtually no correlation with other asset classes. Nev-

ertheless, an examination on the individual fund instead of the index level

revealed anomalies. Although half of the funds under consideration did

not experience a single negative month and, even for the weaker perform-

40The average 1-month U.S. Treasury Bill rate between January 2007 and June2010 has been used as a proxy for the risk-free interest rate. Note that for mostfunds, Sortino Ratios are unavailable since returns did extremely rarely or not at alldrop below the threshold, i.e., the risk-free rate. Thus, the Lower Partial Momentin the denominator is either very close to or exactly zero and the ratio consequentlymeaningless or not defined. Additionally, Calmar Ratios have been omitted wheneverthe lowest return in the series was positive (or negative but very close to zero),rendering a drawdown-based measure pointless. Finally, for those life settlementfunds with no more than a single negative return, an informative 95 percent VaRcannot be derived and therefore Excess Returns on VaR are not available.

41Sharpe Ratios (01/2007 - 06/2010) of the other asset classes for comparisonpurposes: stocks: -0.1057; government bonds: 0.4760; corporate bonds: 0.0851; hedgefunds: 0.0741; real estate: -0.6212; commodities: -0.0466; private equity: -0.0875.

42This was already pointed out in the discussion of potential biases in Section 3.1.


ers such an occasion appears to be rare relative to the established asset

classes, a negative month—if it actually occurs—can in fact cause a seri-

ous (Fund 101 and Fund 205) or even fatal drawdown (Fund 204). While

the observed performance of life settlement funds could be a result of the

market being inefficient and providing arbitrage opportunities because

many investors have not yet discovered the asset class’ attractiveness,

a more likely explanation is that considerable risks embedded in those

funds are largely not reflected in historical performance data. Therefore,

we will conduct an in-depth risk analysis in the following section, taking

into account the structural insights that we elaborated on in Section 2.3.

4 Risks of open-end life settlement funds

4.1 Overview

During the recent financial crisis, investments with attractive returns

and presumably low risk, such as higher rated tranches of so-called sub-

prime residential mortgage-backed securities turned out to be very risky,

whereas those risks, which finally materialized, had not been reflected by

ex ante risk analyses.43 In combination with our empirical results, this

raises a degree of suspicion. Hence, in the following section, we focus on

latent risks associated with the asset class and, in particular, open-end

life settlement funds.44 Since most of the risks can hardly be quanti-

fied, one needs to rely on a comprehensive qualitative risk analysis. The

discussion offers an explanation for the observed unusual performance of

open-end life settlement funds. We identify the following key risk drivers

in descending order of their severity, as determined by their expected

43See, e.g., studies by the Senior Supervisors Group (2008), the Financial StabilityForum (2008), and the International Institute of Finance (2008).

44Note that most of these risks arise from the characteristics of the underlyinglife settlement assets. Consequently, closed-end life settlement funds as describedin Section 2.2 are generally exposed to them as well. Due to structural differences,however, closed-end funds are better equipped to cope with some of the risk factorsexplained in this section. Liquidity reserves, the absence of leverage, and the factthat investors are locked in until maturity, e.g., mitigate the impact of liquidity risks.Similarly, closed-end funds are less likely to run into policy availability issues andthe associated pricing pressure, since, after the so-called ramp-up period, they do notneed to permanently acquire new policies.

4.2 Valuation risk 37

detrimental impact on an investment in life settlement funds: valuation

risk, longevity risk, liquidity risk, policy availability risk, operational

risk, credit risk, and changes in regulation and tax legislation.

4.2 Valuation risk

The most severe risk factor associated with life settlement funds is ar-

guably valuation risk. As described in Section 2.2, the valuation of a life

settlement portfolio is commonly conducted on a mark-to-model basis.

This means that due to the absence of objective market values, fund

shares are dealt based on model values determined by the fund manage-

ment, even though it is not clear whether the assets can in fact be sold

at those values. In addition, not all models are reviewed by an actuarial

advisor, implying the necessity of a profound actuarial know-how of the

fund management.

The Financial Accounting Standards Board (FASB) guidelines for

life settlements distinguish two valuation approaches: the investment

method and the fair value method (see FASB, 2006). While, in both

cases, initial measurement is based on the purchase (transaction) price,

the NAV development of a life settlement fund materially depends on

the methodology used for subsequent measurement. The purchase price

is agreed upon by the counterparties of a life settlement transaction.

Through the life settlement provider, the fund typically submits an offer

to the policyholder, which he or she can accept or reject. The offer price

is commonly calculated as the present value of expected future payoffs

less the present value of expected premium payments and other costs.

However, the discount rate in this context is not derived from a term

structure but determined by the internal rate of return the fund aims

to achieve on the investment, which is generally a function of its cost

of capital (see, e.g., Zollars et al., 2003). The key factor in determining

the expected cash flows from a life insurance contract is an insureds life

expectancy. After the initial examination, further life expectancy esti-

mates are carried out at each fund’s own discretion.


When using the investment method, the initial recognition of the pol-

icy in the books is given by the purchase price plus initial direct costs

(legal costs, commissions paid, etc.). Further valuation has to be con-

ducted by capitalizing any continuing costs such as premiums to keep the

policy in force. Gains may only be recognized in case of a policy resale

or in case of the insured’s death, and are then given by the difference be-

tween the sales proceeds or the death benefit payment and the carrying

amount of the life settlement contract. In contrast, a loss must be rec-

ognized for impairments, i.e., if there is updated information available,

indicating that the expected policy payoff does not suffice to cover the

carrying amount of the contract plus all projected undiscounted future

premiums. This can occur if an increase in the life expectancy becomes

evident or if the creditworthiness of the primary insurer deteriorates sub-

stantially. As an alternative, the FASB proposes the fair value method,

where the initial value of a life settlement investment is also determined

by the purchase price and, after that, ongoing valuation is based on the

fair value, i.e., the sales price that the asset is likely to achieve in the mar-

ket less transactions costs, with value changes being directly recognized

in periodic earnings. However, due to the illiquid nature of life settle-

ments, a mark-to-market approach is typically not practicable. Thus,

it is prevalent to estimate fair values by marking-to-model. Since these

valuation models are based on extensive assumptions and there is little

oversight as to their validity, the fair value method implies a largely sub-

jective assessment.

Overall, the solid performance that could be observed in Section 3 is

likely to be all but a mere by-product of the accounting oriented valua-

tion methodology for life settlements implied by the widespread invest-

ment method. This approach leaves room for large price movements only

if death benefits are received or life expectancy estimates are renewed

and differ significantly from the original ones. In all other cases, one

should observe an almost linear growth path. Hence, it is quite likely

that most funds, which displayed more stable returns over the considered

time period, tend to avoid the fair value method. However, although life

settlements are acquired at a large discount of their face value and the

4.3 Longevity risk 39

purchase price tends to understate the fair value on the transaction date,

the investment method still involves the risk of an incorrect purchase

price due to model errors or misestimated life expectancy (see Perera

and Reeves, 2006). If the whole industry would be obliged to dispose

the investment method and switch to fair values, life settlement fund re-

turns would probably become considerably more volatile than suggested

by our empirical observations over the last years. Moreover, the fair

value method is associated with a further pitfall. Since fund managers

are in a position to change their mark-to-model estimation methodol-

ogy over time, they could on the one hand smooth returns and on the

other hand evaluate fund shares at fire sale prices in the case of exten-

sive redemptions by investors. Based on these considerations, erroneous

valuation is, as already mentioned in Section 3.4, a likely cause for the

major drawdowns in the time series of Fund 101, 204, and 205.

4.3 Longevity risk

Another key risk factor is longevity risk, which describes the possibil-

ity that the insured lives longer than originally expected. To measure

the sensitivity of senior life settlement portfolios to changes in mortality

rates and longevity risk, also called life extension risk, Stone and Zissu

(2006) propose to use a life expectancy duration. The more the actual

lifetime exceeds the expected lifetime, the less valuable the policy be-

comes for the fund and its investors. The reason is that initial pricing

assumptions turned out to be incorrect in that premium payments have

to be made longer and the death benefit is received later than expected.

Longevity risk is particularly important in its systematic form, i.e., if the

life expectancy of the whole portfolio is simultaneously prolonged. The

discovery of a cure or a mitigating treatment for a common illness, e.g.,

implies a substantial increase in the correlation between those lives in

the portfolio, which had been suffering from that particular disease (see

Perera and Reeves, 2006). To cope with longevity risk, life settlement

funds diversify their portfolios across different types of diseases, purchase

insurance coverage (if available) or employ innovative risk management

tools such as longevity swaps.


Assessing the quality of life expectancy estimates is challenging and

results are rarely disclosed to the public. According to Milliman Inc.

(2008), which examined the mortality experience data of two providers

gained from filings with the Texas Department of Insurance, the actual

number of deaths recorded from 2004 to 2006 was only 60 percent of those

that had been expected. This provides an indication of the fundamen-

tal longevity risk that is inherent in life settlement portfolios. Realized

investor returns in this case are likely to be considerably smaller than

originally projected. In line with these findings, A.M. Best (2009) de-

scribed how five year old portfolios showed signs that the life expectancy

estimates had historically been too short and that since 2005, medical

underwriters have issued more conservative ones. Furthermore, as in-

dicated by industry experts, some of the largest medical underwriters,

which have been able to steadily expand their influence in the market,

seem to systematically underestimate life expectancies.

Thus, the importance of longevity risk should not be misjudged, par-

ticularly against the background of the potential incentive problems of

life settlement providers, which were discussed in Section 2.3. It should

be of central interest to investors with which medical underwriters fund

managers cooperate and whether they require more than one life ex-

pectancy estimate to be at least partially protected against major errors

in medical underwriting. Taking these considerations into account, it

may well be that a large number of insureds in the portfolios of Fund

101, 204, and 205 turned out to live much longer than initially expected,

forcing the funds to realize substantial losses on the respective life set-

tlement assets.

4.4 Liquidity risk

After the initial sale of fund shares, there are in principle two sources

of cash inflows on the fund level—new subscriptions and death benefit

payments—neither of which occur on a regular basis or are easy to fore-

cast. In addition, open-end funds typically reinvest death benefits in

order to purchase new policies or use them to pay due premiums. Some

4.4 Liquidity risk 41

fund managers maintain a position in liquid assets, a reserve account,

or can draw on short-term debt financing through a liquidity facility.45

Cash outflows, in contrast, occur on a regular basis due to premium

payments, redemptions, and potentially interest plus repayment in case

the fund is leveraged. In combination with the illiquid nature of the

underlying, this implies that life settlement funds are fairly vulnerable

to becoming liquidity strained. The consequences for investors could be

devastating. If a fund falls short of sufficient cash to cover due redemp-

tions, it has no choice but to sell off assets to make up for the missing

amount unless a reserve account has been set up or short-term debt fi-

nancing is attainable. Then again, the fund will probably not be able

to sell life settlements from its portfolio at an acceptable value at short

notice due to the mediocre permanent trading activity in the market

as well as the complexity and length of the transactions. Moreover, a

distressed life settlement fund is highly likely to default on the ongo-

ing premium payments of at least some of its policies, causing them to

lapse. Evidently, these risks increase disproportionately with the degree

of leverage applied by a life settlement fund since it also has to bear

the debt service. The same is true if policies are premium financed, i.e.,

if the fund takes out loans to fund premium payments. As with hedge

funds, some life settlement funds partially protect themselves against the

problem of illiquid assets and excessive redemptions by imposing lock-up

periods, gates, and redemption fees. As a last resort, most fund man-

agers reserve the right to suspend redemptions. While these measures

reduce liquidity risk at the fund level, they clearly hamper liquidity of

the fund shares at the investors’ level and should thus be carefully fac-

tored into an investment decision if one does not want to find his money

locked into a life settlement fund in major distress. Inevitably, the ma-

jor dislocations in the capital markets during the peak of the financial

crisis in 2008 have led to an imbalance between subscriptions and re-

demptions, which revealed severe liquidity issues of a number of funds.

This is another likely cause for the observed losses of Fund 101, 204, and

205.

45The reader is referred to the structural overview in Section 2 to identify thesesources of liquidity.


4.5 Policy availability risk

Along with valuation, longevity, and liquidity risk, there is also availabil-

ity risk and competitive pricing pressure, because the secondary market

is limited by the size of the primary market as well as the number of avail-

able target policies. Evidently, the identification of suitable policies is a

critical success factor for an investment in life settlements (see Moodys,

2006). In addition, funds will have to consider the number of contracts

and the policy mix in their portfolios, including different types of diseases

and different primary insurers to diversify risks. Target policies typically

satisfy specific criteria such as a reduced policyholder life expectancy of

on average 113 months, a high face value of on average 1.8 million USD,

and a policyholder age of approximately 76 years. Moreover, ideally the

insured would have otherwise surrendered the policy (see Milliman Inc.,

2008). Such contracts are not plentiful. According to Moodys (2006),

around one percent of the permanent policies in force in the U.S. market

match the characteristics commonly targeted by life settlement funds.

This is one reason for the fact that only about fifteen to twenty-five per-

cent of the policies presented to life settlement providers are actually

purchased (see, e.g., McNealy and Frith, 2006). Other reasons include

the inability of the policyholder to qualify for renewed coverage and the

failure of the transaction partners to agree on the purchase price.

It is imperative to take these potential availability constraints into

account, since the supply-demand-situation on the life settlement mar-

ket substantially influences acquisition prices. Problems for the funds

can occur if a large inflow of capital into the asset class is not met by

a sufficient supply of adequate policies or if the market activity in gen-

eral freezes. The resulting competitive pressure implies a reduction in

achievable returns due to higher purchase prices. Furthermore, even for

fund managers which have performed well to date, there may be adverse

changes in the portfolio composition if the number of valuable life settle-

ment investment opportunities noticeably decreases. In such a scenario,

it is of importance whether a fund runs a single or multi-source approach

with regard to life settlement providers since those fund managers with

access to a larger number of life insurance policies are likely to be in a

4.6 Operational risks 43

better position when supply in general is short. Apart from that, fund

size can play an important role because smaller funds may find it easier

to source enough policies that fit their investment criteria and offer an

attractive risk-return perspective. Larger funds, on the contrary, could

face situations where they need to relax their policy picking standards to

be able to invest all of their investor money. Since market participants

have not reported any supply shortages during the last two years, it is

rather unlikely that Fund 101, 204, and 205 experienced drawdowns due

to constrained policy availability. Nevertheless, investors should bear

this potential risk in mind.

4.6 Operational risks

Among the less severe but still noteworthy risk factors are operational

risks: insured fraud risk, litigation or legal risks, and operational risks

originating from third-party service providers. Insured fraud risk could

mean a misrepresentation of one’s health status in order to achieve a

higher price for the policy. In rare cases the policyholder also might not

disclose all original beneficiaries or fraudulently sell the same policy to

multiple buyers. Furthermore, insureds may use sales proceeds to im-

prove their living standard and medical care, which can increase their

life expectancy and, in turn, reduce investor returns.

Litigation and legal risks can arise due to the high complexity of con-

tractual agreements, despite the fact that sales processes are becoming

increasingly standardized. Life insurance companies may possibly con-

test the policy and refuse to pay the death benefit, e.g., due to lack of

insurable interest. In addition, payments are typically held back if the

insureds body is missing. This can be done by insurers for up to seven

years (see Perera and Reeves, 2006). Furthermore, former beneficiaries

could initiate lawsuits, accusing life settlement firms of unethical sales

practice or invalid transfer with the intent to claim the death benefit

for themselves. As a consequence, the payment may be substantially de-

layed or not transferred at all. In such a case, legal expenses may even

exceed the return from the policy.


Further operational risks arise from the reliance on third-party ser-

vice providers. The tracking agent, for instance, might fail to service the

policy properly such that the insured’s death is reported late or he cannot

be located posthumously, thus delaying the collection of death benefits.

However, most servicers are insured up to some amount against such

operational risks. Another important risk factor with respect to the in-

volved third parties is fraud. In particular, life settlement providers may

collude with brokers in order to discourage competitive bids. In 2006,

one of the largest life settlement companies, Coventry First, was sued by

New York Attorney General Eliot Spitzer and accused of bid-rigging to

keep policy purchase prices low. The provider was believed to have made

secret payments to life settlement brokers in exchange for which they al-

legedly suppressed competitive bids from other life settlement companies.

The lawsuit was settled in October 2009 with Coventry First paying an

additional 1.4 million USD to policyholders to adequately compensate

them for the appropriate market value of their life insurance policies. Fur-

thermore, the company agreed to pay 10.5 million USD to the state of

New York to end the litigation. As a corollary of this settlement, no fine

or penalty was issued against Coventry. Another prominent case is Mu-

tual Benefits Corporation, which was alleged to have made substantial

misrepresentations to investors in its marketing material, prospectuses,

as well as through its network of sales people and failed to disclose focal

information over several years. In particular, life expectancy estimates

for a large number of its policies were fraudulently assigned at the discre-

tion of its directors. As a consequence, around 90 percent of the policies

needed to be maintained significantly beyond their life expectancy esti-

mates, inflicting high losses on investors. In the particular cases of Fund

101, 204, and 205, however, we deem it unlikely that losses occurred due

to a manifestation of operational risk factors.

4.7 Credit risk

Life settlement funds also face credit risk due to a potential default of pri-

mary insurers. Although such a credit event was thought to be virtually

impossible before the financial crisis, the AIG bail-out in 2008 provides

4.8 Changes in regulation and tax legislation 45

evidence that the default of an insurer, no matter what ultimately causes

it, can be an issue. Yet, since the average rating of the insurance com-

panies in the portfolios of our sample funds is ”AA” and policyholders’

claims rank most senior in the case of insolvency, we believe that credit

risk has been irrelevant at least in the past with regard to the problems

of Fund 101, 204, and 205. In addition, in the unlikely case of an insurer

default, there are still state-dependent insurance guarantee funds in the

U.S., which provide protection to policy owners.46

4.8 Changes in regulation and tax legislation

Finally, there is a risk of adverse amendments to regulatory frameworks

and tax legislation. Until recently, regulation of the U.S. life settle-

ment market was partially lax and inconsistent (see Fitch Ratings, 2007).

While this has changed, regulation still varies by state. Few states do

not regulate transactions at all, other states regulate viatical transac-

tions but not senior life settlements, and still others require that brokers

and providers be licensed (see Gatzert, 2010). One often discussed prob-

lem in the United States is stranger-originated (or investor-initiated) life

insurance (STOLI), as it contradicts the principle of insurable interest

which had already been established in the early 19th century before it

was confirmed by the U.S. Supreme Court in 1911 in Grigsby v. Russell

(see Katt, 2008). The principle of insurable interest distinguishes insur-

ance from speculation. It was designed to protect the insured, since, if

allowed to purchase insurance on the lives of strangers, the holder of

the policy has a financial interest in the death of the insured. The main

feature of a STOLI process is that the policy is not initiated by the policy-

holder, but by an investor or third-party lender who provides the insured

with cash to cover the premium payments and ultimately receives the

death benefit (see, e.g, Fitch Ratings, 2007; Ziser, 2007; Gatzert, 2010).

STOLI must be distinguished from the common practice of non-recourse

46However, in most cases an insurance guarantee fund would probably not coverthe full death benefit of the policies due to the high face values in the case of seniorlife settlements. In addition, to the best of our knowledge, there has not been aprecedent yet. Hence, it is not clear from a legal point of view, whether an insuranceguarantee fund would need to pay for life settlement fund investors.


premium financing, which allows policyholders who do have an insurable

interest to fund their premium payments with a loan that is collateral-

ized by the insurance policy (see Freedman, 2007).

To introduce transparency and clear rules in the market, the National

Association of Insurance Commissioners (NAIC) proposed the Viatical

Settlements Model Act, which would ban life settlements of non-recourse

premium financed policies during the first five years of the contract (see,

e.g, Fitch Ratings, 2007; Ziser, 2007; Gatzert, 2010). In November 2007,

the National Conference of Insurance Legislators (NCOIL) introduced

the Life Settlement Model Act, which does not include the five-year

ban proposed by the NAIC, but explicitly defines STOLI as a fraudu-

lent life settlements act (see NCOIL, 2007). In addition, the NCOIL

proposal prohibits premium financing companies from owning or being

financially involved in policies they finance (see Gatzert, 2010). The frag-

ile legal status of STOLI appears to have an impact on the demand by

institutional investors in that they generally avoid purchasing premium

financed policies (see Beyerle, 2007). Overall, both proposals by NAIC

and NCOIL are still criticized and may be refined, thus implying ongo-

ing uncertainty in respect to the regulatory treatment of life settlements

(see, e.g., Freedman, 2007). Another risk factor relates to tax legislation.

As Fitch Ratings (2007) points out, the absence of insurable interest be-

tween policyholder and beneficiary may affect tax advantages associated

with life insurance. Moreover, in 2009 the U.S. Internal Revenue Service

(IRS) determined that death benefit payments to foreign life settlement

investors will be subject to withholding tax. Although these aspects

distinctly affect the market’s legal environment, we believe that adverse

changes in regulation and tax legislation did not cause the distress of

Fund 101, 204, and 205.


5 Summary and conclusion

We comprehensively analyze open-end funds dedicated to U.S. senior life

settlements, explaining their business model and the roles of institutions

involved in the transactions of such funds. In addition, we contribute to

the literature by conducting the first empirical analysis of life settlement

fund return distributions as well as a performance measurement, includ-

ing a comparison to established asset classes. Since the funds contained

in our dataset largely cover this young segment of the capital markets,

representative conclusions can be derived. Based on these findings, we

elaborate on the risk profile of life settlement assets in general and open-

end life settlement funds in particular.

Although our empirical results suggest that life settlements generally

offer attractive returns paired with low volatility and are uncorrelated

with other asset classes, we find substantial latent risks associated with

the funds, such as liquidity, longevity and valuation risks. Since these did

generally not materialize in the past and are hence largely not reflected

by the historical data, they cannot be captured by classical performance

measures. Therefore, investors should not be misled by a superficial first

impression of the asset class. Caution is advised and the expected re-

turn on life settlement funds should be regarded as a compensation for

investors who decide to bear those risks.

It is advisable to perform extensive due diligence on life settlement

funds, focusing on valuation methodology, cash management, asset pipe-

lines as well as business partners. Wherever possible, independent third

parties such as auditors and rating agencies can be involved for cross-

checking and to deliver additional information such that the investor is

able to balance the expected returns against a comprehensive qualitative

assessment of latent risks before deciding on the portfolio weight he

would like to allocate to life settlement funds. Nonetheless, our results

also illustrated that—within reasonable limits—life settlements certainly

provide a suitable means for diversification as they seem to be genuinely

uncorrelated with the broader capital markets.


6 Appendix

6.1 Index descriptions (from the providers)

- Life Settlement Funds Index:

A custom index of open-end life settlement funds. This index is an

equally weighted portfolio consisting of all available funds at any

point in time. The aim of the index is to track the development of

a portfolio of life settlement funds between 12/2003 and 06/2010

as adequately as possible.

Bloomberg Ticker: -

Further information: -

- S&P 500:

The S&P 500 is widely regarded as the best single measure of

the U.S. equities market and includes 500 leading companies in

the major industries of the U.S. economy. Although the S&P 500

focuses on the large cap segment of the market, with approximately

75 percent coverage of U.S. equities, it is also well suited to assess

the total market.

Bloomberg Ticker: SPX <Index> <Go>

Further information: www.standardandpoors.com

- FTSE U.S. Government Bond Index:

FTSE Global Government Bond Indices comprise central govern-

ment debt from 22 countries, denominated in the domestic currency

or Euros for Eurozone countries. These are total return indices,

taking into account the price changes as well as interest accrual

and payments of each bond.

Bloomberg Ticker: FGGVUSP5 <Index> <Go>

Further information: www.ftse.com/Indices

www.standardandpoors.com

6.1 Index descriptions (from the providers) 49

- DJ Corporate Bond Index:

The Dow Jones Corporate Bond Index is an equally weighted bas-

ket of 96 recently issued investment-grade corporate bonds with

laddered maturities. The objective of this index is to capture the

return of readily tradable, high-grade U. S. corporate debt.

Bloomberg Ticker: DJCBP <Index> <Go>

Further information: www.djindexes.com/mdsidx

- HFRI Fund Weighted Composite Index:

The HFRI Monthly Indices are designed to reflect hedge fund in-

dustry performance by means of equally weighted composites of

constituent funds. They range from the industry-level view of the

HFRI Fund Weighted Composite Index, which encompasses over

2000 funds, to the increasingly specific-level of sub-strategy classi-

fications. Fund of Funds are not included.

Bloomberg Ticker: HFRIFWI <Index> <Go>

Further information: www.hedgefundresearch.com

- S&P/Case-Shiller Home Price Index (Composite of 20):

The S&P/Case-Shiller Home Price Indices are designed to gauge

the value growth of residential real estate in various regions across

the United States. The underlying methodology to measure house

price movements has been developed in the 1980s and is still con-

sidered to be the most accurate way to measure this asset class.

Bloomberg Ticker: SPCS20 <Index> <Go>


www.hedgefundresearch.com



- S&P GSCI (USD, Total Return):

The S&P GSCI provides investors with a reliable and publicly avail-

able benchmark for investment performance in the commodity mar-

kets. The index is designed to be tradable, readily accessible to

market participants, and cost efficient to implement. The S&P

GSCI is widely recognized as the leading measure of general com-

modity price movements and inflation in the world economy.

Bloomberg Ticker: SPGSCITR <Index> <Go>


- S&P Listed Private Equity Index (USD, Total Return):

In the last few years increasing numbers of private equity businesses

have listed on stock exchanges to meet investor requirements for

liquidity and transparency. The S&P Listed Private Equity In-

dex is comprised of 30 leading listed private equity companies that

meet size, liquidity, exposure, and activity requirements. It is de-

signed to provide tradable exposure to the leading publicly listed

companies in the private equity space.

Bloomberg Ticker: SPLPEQTR <Index> <Go>




6.2 Performance measures 51

6.2 Performance measures

The Sharpe Ratio (see Sharpe, 1966) is given by

Sharpe Ratioi =µi − rfσi

, (1)

where µi is the average monthly return on asset i, rf is the risk

free monthly interest rate and σi represents the standard deviation of

monthly returns. The Sharpe Ratio has often been criticized because

of its apparent inability to capture all characteristics of non-normal re-

turn distributions. Thus it is viewed as a misleading indicator for the

risk return profile of certain investments (see, e.g., Amin and Kat, 2003).

Consequently, complementary performance indicators utilize alterna-

tive risk measures in order to avoid the alleged problems associated with

the Sharpe Ratio. One of these measures is the Sortino Ratio (see Sortino

and Van Der Meer, 1991), which employs the Lower Partial Moment of

order 2 (LPM2) instead of the standard deviation, i.e.,

Sortino Ratioi =µi − τ

√

LPM2i(τ). (2)

The nth order LPM for asset i is defined as:

LPMni(τ) =1

T

T∑

t=1

max [τ − rit, 0]n.

In general, Lower Partial Moments quantify risk through negative

deviations from a certain threshold return τ (e.g., the mean return, the

risk free interest rate or 0). The order n governs the weighting for this

downside risk and should therefore be higher, the more risk averse the

investor is (see Fishburn, 1977).

Other modern performance measures are based on drawdown, i.e.,

the loss incurred over a certain time period. The Calmar Ratio, which

has become common among practitioners, particularly in the context of

hedge fund performance measurement, is given by:


Calmar Ratioi =µi − rf−MDi

. (3)

It relates the excess return over the risk free interest rate to the

maximum drawdown MDi, which represents the lowest return over the

period under consideration. Since the lowest return is usually negative,

the formula for the Calmar ratio contains a minus sign in the denomina-

tor, turning it into a positive risk figure.

Finally, performance measures can also be based on Value at Risk

figures. The Value at Risk for an asset i (V aRi) is the loss over a certain

period, which is not exceeded with a prespecified probability (1−α), i.e.,

the α-quantile of the return distribution under consideration. One such

indicator is Excess Return on Value at Risk:

Excess Return on VaRi =µi − rfV aRi

. (4)

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59

Part II

Pricing Catastrophe Swaps:

A Contingent Claims

Approach

Abstract

In this paper, we comprehensively analyze the catastrophe (cat) swap,

a financial instrument which has attracted little scholarly attention to

date. We begin with a discussion of the typical contract design, the

current state of the market, as well as major areas of application. Sub-

sequently, a two stage contingent claims pricing approach is proposed,

which distinguishes between the main risk drivers ex-ante and during the

loss reestimation phase. Catastrophe occurrence is modeled as a doubly

stochastic Poisson process (Cox process) with mean-reverting Ornstein-

Uhlenbeck intensity. In addition, we fit various parametric distributions

to normalized historical loss data for hurricanes and earthquakes in the

U.S. and find the heavy-tailed Burr distribution to be the most adequate

representation for loss severities. Applying our pricing model to market

quotes for hurricane and earthquake contracts, we derive implied Poisson

intensities which are subsequently condensed into a common factor for

each peril by means of exploratory factor analysis. Further examining

the resulting factor scores, we show that a first order autoregressive pro-

cess provides a good fit. Hence, its continuous-time limit, the Ornstein-

Uhlenbeck process should be well suited to represent the dynamics of

the Poisson intensity in a cat swap pricing model.47

47Alexander Braun (2010), Pricing Catastrophe Swaps: A Contingent Claims Ap-proach, Working Papers on Risk Management and Insurance, No. 78.

60 II Catastrophe Swaps

1 Introduction

Since the early 1990s, insurance and reinsurance companies have been

using innovative financial instruments to lay off natural disaster risk in

the capital markets. To date the most popular of these alternative risk

transfer tools is the catastrophe (cat) bond, a security which pays regu-

lar coupons to the investor unless a catastrophic event occurs, leading to

full or partial loss of principal. The principal is held by a special purpose

vehicle (SPV) in the form of highly-rated securities and paid out to the

hedging (re)insurer to cover its losses if the trigger condition, which has

been defined in the bond indenture, is fulfilled. In addition to cat bonds,

catastrophe derivatives can be employed to access the capital markets.

In 1992 the Chicago Board of Trade (CBOT) initiated exchange-traded

catastrophe futures and options based on its own loss index (see Swiss

Re, 2009). Due to humble trading activity, these contracts were soon

replaced by options based on Property Claim Services (PCS) indices, a

measure for catastrophe losses in nine geographical regions of the United

States. However, PCS-options, which paid off for the buyer in case the

underlying index exceeded the strike price, were eventually also discon-

tinued in 2000 (see Cummins and Weiss, 2009). Despite their earlier

demise, catastrophe derivatives with an exclusive focus on U.S. hurricane

risk have recently been re-launched by several exchanges. Catastrophe

event-linked futures (ELFs), which are co-offered by Deutsche Bank and

the Insurance Futures Exchange (IFEX), feature a binary payoff contin-

gent on regional PCS losses.48 Similar contracts are listed on the Eu-

ropean Exchange (EUREX). Apart from insurance futures, the Chicago

Mercantile Exchange (CME) and the New York Mercantile Exchange

(NYMEX) provide catastrophe options as well. While the former refer-

ence the so-called CME Hurricane Index,49 NYMEX options settle based

on the Re-Ex index by Gallagher Re (see Cummins, 2008).

48IFEX contacts are traded on the Chicago Climate Futures Exchange. Refer tothe IFEX website for more information.

49This index, which had been developed by the reinsurance intermediary Carvill,was formerly known as the Carvill Hurricane Index (CHI). In 2009, CME grouppurchased the rights and renamed it to CME Hurricane Index. See www.artemis.bm.

www.theifex.com

www.artemis.bm

1 Introduction 61

Although there is a growing body of literature on catastrophe bonds,

futures and options, it seems that another innovative risk transfer in-

strument of increasing importance for risk managers and investors has

been neglected so far: the catastrophe swap. Cat swaps are over-the-

counter (OTC) contracts, allowing (re)insurers to tap additional risk

capacity by synthetically passing a portion of their insurance risk on to

a counterparty. The latter could be an investor, who, in turn, gains

unfunded exposure to natural disaster risk. As indicated by industry ex-

perts, catastrophe swaps have been steadily gaining ground over the past

few years and due to recent progress with regard to contract documenta-

tion and standardization, the market outlook is fairly promising. Early

references to catastrophe swaps can be found in Borden and Sarkar (1996)

and Canter et al. (1997), who mentioned the instrument in their articles

on insurance-linked securities. Furthermore, Cummins (2008) and Cum-

mins and Weiss (2009) briefly describe the general mechanism behind a

catastrophe swap contract. Apart from these publications, however, the

instrument has, to the best of our knowledge, not attracted scholarly

attention. This paper is intended to fill this gap.

The remainder of the paper is structured as follows. In Section 2, we

briefly review the extant literature on the pricing of catastrophe-linked

instruments. Furthermore, Section 3 contains a discussion of the main

characteristics of cat swap contracts and an overview of the current state

of the market as well as major areas of application. A two-stage approach

for the pricing of catastrophe swaps ex-ante and in the loss reestimation

phase is presented in Section 4. In Section 5, we fit different parametric

distributions to normalized historical hurricane and earthquake loss data

for the U.S. to select an adequate loss severity distribution. Subsequently,

the pricing model is applied to back out implied Poisson intensities from

cat swap market data which we condense into a common factor for each

peril by means of exploratory factor analysis. The resulting factor score

times series are then used to estimate a first order autoregressive process

and evaluate its fit, thereby shedding some light on the adequacy of a

mean-reverting process for the Poisson intensity in our cat swap model.

Finally, in Section 6, we conclude.


2 Literature review

While no article has been devoted to potential pricing approaches for

catastrophe swaps yet, a few authors have discussed the pricing of cat

bonds, futures, and options. In this regard, models based on three dif-

ferent theoretical foundations have been brought forward.50 First of all,

within their empirical examination of cat bond prices, Lane (2000) as well

as Lane and Mahul (2008) apply actuarial pricing methodology, thereby

acknowledging the resemblance of catastrophe-linked capital market in-

struments to traditional reinsurance.

In contrast to that, utility-based approaches are centered around the

notion that insurance markets are generally incomplete, implying that it

is not possible to find a unique equivalent martingale measure by merely

ruling out arbitrage opportunities. Embrechts and Meister (1997) pro-

vide a generic discussion of catastrophe futures pricing in a utility max-

imization context. Furthermore, Aase (1999) treats catastrophe risk as

systematic and resorts to a partial equilibrium framework with constant

absolute risk aversion to derive pricing formulae for cat futures, caps,

call options, and spreads. Similarly, Cox and Pedersen (2000) derive a

pricing approach for cat bonds in an incomplete markets setting based

on equilibrium pricing theory and time separable utility. Christensen

and Schmidli (2000) introduce an exponential utility model for cat fu-

tures which includes loss reporting lags. Amending his earlier work on

cat derivatives pricing by employing a Markov model for the dynamics

of underlying, Aase (2001) proposes a competitive equilibrium approach

which assumes constant relative risk aversion of the representative agent.

In addition, Young (2004) computes the indifference price of cat bonds

based on exponential utility investor preferences. Utility-based pricing

of cat bonds is also considered by Egami and Young (2008). Moreover,

Dieckmann (2009) proposes a dynamic equilibrium model for cat bonds

with an external habit process as in Campbell and Cochrane (1999).

50Galeotti et al. (2009) provide an empirical comparison of some of these ap-proaches.


However, the majority of papers on the pricing of cat bonds and

derivatives proposes preference free no-arbitrage frameworks. Cummins

and Geman (1994, 1995) value cat futures and call spreads with an Asian

option approach, assuming a jump-diffusion process with constant jump

amplitude for the claim dynamics. Besides, Chang et al. (1996) develop

a cat option model based on a stochastic time change linked to insurance

futures transactions, allowing them to convert a compound Poisson into

a pure diffusion process for which risk-neutral valuation is readily appli-

cable and a parsimonious closed formula can be derived. Similarly, by

means of stochastic time change and Laplace transform, Geman and Yor

(1997) present a semi-analytical solution for the price of cat options on a

loss index which follows a jump-diffusion process. Having priced simple

cat bonds under the Black and Scholes (1973) assumptions in the first

section of their paper, Louberge et al. (1999) subsequently consider a

compound Poisson process in combination with a simple binomial model

for the interest rate. Another no-arbitrage pricing model for cat bonds

built upon a compound Poisson process is presented by Baryshnikov et al.

(2001). Lee and Yu (2002) additionally contemplate default risk of the

cat bond issuer as well as issues of moral hazard and basis risk, adopt-

ing a structural credit model with stochastic interest rates as in Cox

et al. (1985). Furthermore, Bakshi and Madan (2002) provide a closed-

form solution for (PCS) cat option prices based on the assumption that

losses follow a mean-reverting Markov process with one-sided jumps. A

compound doubly stochastic Poisson process (Cox process) is used by

Burnecki and Kukla (2003) to value zero-coupon and coupon cat bonds

and by Dassios and Jang (2003) to model stop-loss reinsurance contracts

and cat derivatives. Muermann (2003) assumes a compound Poisson loss

process and values cat derivatives relative to observed premiums of in-

surance contracts on the same underlying risks. Moreover, in his model

for options on a PCS index, Schmidli (2003) distinguishes between catas-

trophe occurrence and loss development period, which are governed by

a compound Poisson process and a Geometric Brownian Motion, respec-

tively. A barrier option framework for the price of a cat bond is proposed

by Vaugirard (2003a,b, 2004), who assumes a jump-diffusion process for

the underlying physical index and stochastic interest rates based on a Va-


sicek (1977) model. In addition, Cox et al. (2004) consider the valuation

of double trigger catastrophe put options when losses are generated by a

compound Poisson process and Jaimungal and Wang (2006) generalize

their work by incorporating stochastic interest rates. While Lee and Yu

(2007) apply insights from their earlier work on cat bonds to reinsurance

contracts, Biagini et al. (2008) use Fourier transform to derive an ana-

lytical solution for the price of an option with catastrophe occurrence

and loss development period. Muermann (2008) applies a cat call option

model based on a compound Poisson process for the underlying loss index

to extract the market price of insurance risk from market quotes of traded

cat derivatives. Chang et al. (2008, 2010) generalize their concept from

the mid-1990s from a complete market continuous-time to an incomplete

market discrete-time framework to price Asian-style cat options with a

doubly-binomial model. Additionally, they consider stochastic Poisson

intensities described by a mean-reverting Ornstein-Uhlenbeck process

and reduce the computational effort through a stochastic time change

from calendar to claim time. Hardle and Cabrera (2010) price a hybrid

cat bond for earthquakes, assuming a doubly stochastic Poisson process

for the flow of catastrophic events. Finally, Wu and Chung (2010) employ

a doubly stochastic Poisson process with Ornstein-Uhlenbeck intensity

in combination with a Cox et al. (1985) model for the interest rate and

the framework of Jarrow and Yu (2001) for counterparty default risk to

price catastrophe bonds, futures, and options.

3 Catastrophe swaps

3.1 Contract design

Catastrophe swaps are financial instruments through which natural dis-

aster risk can be transferred between counterparties. In a typical con-

tract, the protection buyer (fixed payer) agrees to make periodic pre-

mium payments to the protection seller (floating payer) in exchange for

a predetermined binary compensation payment51 contingent on the oc-

51This is usually the full notional value. Alternatively, the payoff profile can belinearly increasing in the underlying losses within the layer between an attachmentlevel and a cap.

3.1 Contract design 65

currence of a trigger event (covered event) in the covered territory (see,

e.g., Swiss Re, 2006). While the covered territory defines the country

or geographic region in which a catastrophe has to strike in order to

be relevant under the swap transaction, a trigger event is determined

by the so-called reference peril (reference catastrophe), the associated

reference losses (reference amount), and the contract’s thresholds. The

term reference peril means the type of disaster which is covered under

the swap, e.g., wind storms including named hurricanes. Whenever such

a catastrophe occurs, the appointed loss report provider assigns a serial

number to it and publishes an initial estimate of the resulting insurance

industry losses. This loss estimate is subsequently refreshed on a regular

basis, with the final loss report usually being released no later than six

months after the event. An important characteristic in this regard is

that the reference losses from different natural disasters are not aggre-

gated but tracked separately. Hence, the trigger mechanism relates to

the losses of individual catastrophes, not a sum of losses. Cat swaps typ-

ically exhibit two thresholds: an event and a slightly higher acceleration

threshold.52 If, during the term of the contract, a final loss estimate

for a reference peril reaches the event threshold (attachment level), it

results in an immediate payoff to the protection buyer and the subse-

quent termination of the contract. Similarly, the payoff under the swap

is triggered instantaneously by an interim loss estimate in excess of the

acceleration threshold. Finally, the protection buyer receives a payoff at

maturity if an interim loss estimate is equal to or higher than the event

threshold.

A concrete example for cat swap contracts offered in current market

practice are ”Deutsche Bank Event Loss Swaps” (ELS).53 ELS for U.S.

wind storms, i.e., hurricanes and tornadoes, have been launched in late

2006 and are available with thresholds of USD 20 billion, USD 30 billion

or USD 50 billion, while the attachment levels for earthquake-based con-

tracts can be set at USD 10 billion and USD 15 billion. The standard

52The acceleration threshold is commonly set ten percent above the event threshold.See ISDA documentation template.

53This information is based on a press release by Deutsche Bank.

http://www.isda.org/publications/isdacommderivdefsup.aspx

www.db.com/presse/en/content/press_releases_2006_3263.htm


maturity is one calendar year and notional amounts are staggered in lots

of USD 5 million. Similar standardized contracts for U.S. wind and earth-

quake events called ”Swiss Re Natural Catastrophe Swaps” (SNaCSTM)

have been launched by Swiss Re (see Swiss Re, 2009).

For U.S. transactions, the Property Claim Services (PCS) division of

Insurance Services Office, Inc. (ISO) acts as loss report provider. PCS

has access to a nationwide network of industry representatives, claim

departments and adjusters, insurance agents, meteorologists, and pub-

lic authorities through which it gathers loss information. In January

2009, several leading firms in the insurance industry have founded the

European index provider PERILS AG, which collects insurance data and

provides a benchmark measure for losses caused by natural catastrophes

in Europe.54

An alternative to the above mentioned contract structure is a trans-

action format termed pure risk swap (or portfolio swap). In a pure risk

swap, two (re)insurance companies exchange uncorrelated catastrophe

risk exposures from their existing books in order to improve portfolio di-

versification and potentially reduce regulatory capital requirements (see

Bruggeman, 2007). Thereby, insurers whose business is locally concen-

trated in an area which is particularly susceptible to natural disasters

can replace a portion of their core risk with another type of peril that

they may not be able to access directly. Pure risk swaps can be executed

through intermediaries, via the web-based Catastrophic Risk Exchange

(CATEX) or directly in the OTC market (see Mutenga and Staikouras,

2007; Cummins, 2008). Like standard catastrophe swaps, these con-

tracts are usually set up such that the present values of the two swap

sides exactly balance and there are no up-front payments between the

counterparties. Instead, money under the swap is only exchanged in

case of a qualifying event. This requires an alignment of the triggers as

well as precise risk modeling in order to match expected losses through

54Founding shareholders of PERILS AG include Allianz SE, AXA, AssicurazioniGenerali, Groupama, Guy Carpenter, Munich Re, Partner Re, Swiss Re, and ZurichFinancial Services. Fore more information refer to the company website.

http://www.perils.org/web.html

3.2 Market development 67

the configuration of the terms and conditions of the contract. A promi-

nent risk swap example is a 2003 transaction in which Mitsui Sumitomo

Insurance swapped USD 100 million of Japanese typhoon risk against

USD 50 million of North Atlantic hurricane risk and USD 50 million of

European windstorm risk with Swiss Re (see Cummins, 2008). Since

pure risk swaps reference the counterparties’ insurance portfolios, they

are indemnity-based contracts and not financial instruments. Pure risk

swaps are not the focus of this paper.

3.2 Market development

Although cat bonds and insurance derivatives have been around for al-

most two decades, the market for catastrophe swaps is very young and,

due to its OTC character, information on transactions is currently largely

anecdotal (see Cummins and Weiss, 2009). Nevertheless, industry ex-

perts claim that the market size is increasing rapidly (see, e.g., Cum-

mins, 2008) and the World Economic Forum (2008) estimates that cat

swaps, together with industry loss warranties (ILWs), currently account

for about USD 10 billion in outstanding notional. In May 2009, the

International Swaps and Derivatives Association (ISDA) released a doc-

umentation template for catastrophe swap transactions referencing U.S.

windstorm events.55 The goal of these standardized definitions for key

terms is to reduce uncertainty, improve liquidity and transparency, and

encourage growth in the market. ISDA documentation for other refer-

ence catastrophes, such as California earthquakes, is already planned.

The introduction of ISDA standards is an important step with regard

to the development of the catastrophe swap market as well as the ac-

ceptance of the instrument among investors and is expected to result in

increasing trading volumes (see Swiss Re, 2009). To date, swap counter-

parties are mainly insurance and reinsurance companies. Yet, as for in-

surance futures, new participants, such as investment banks, hedge funds,

and other institutional investors, could soon be encouraged to establish

themselves as market makers.56 Illiquidity of swaps has been cited as

55This template can be accessed on the ISDA website.56See IFEX website for additional information on the insurance futures market

structure.

http://www.isda.org/publications/isdacommderivdefsup.aspx


a significant shortcoming relative to tradable catastrophe securities and

insurance options (see, e.g., Cummins and Weiss, 2009). However, in-

creasing standardization and the introduction of ISDA standards now

also enables swap counterparties to assign a contract to other investors

in order to close out their position, thus enhancing liquidity.

3.3 Areas of application

By no-arbitrage reasoning, catastrophe swaps should behave similar to

cat bonds, i.e., they should share their properties of comparatively high

yields and immaterial return correlation with other asset classes.57 Con-

sequently, from an investor’s perspective catastrophe swaps are an at-

tractive means to gain synthetic exposure to natural disaster risk, i.e.,

without requiring to fund the purchase of a cat bond. In addition, it is

not necessary for the protection buyer to actually hold a book of insur-

ance contracts, nor does the protection seller need to have the status of

a regulated insurance entity to be eligible as swap counterparty. There-

fore, apart from hedging insurance risks, catastrophe swaps could also be

applied for investment purposes. An example are negative basis trades

between cat swaps and bonds. This risk-arbitrage strategy, which is com-

mon in the credit markets, aims at exploiting price discrepancies between

the cash and derivative instrument. If a cat bond spread is sufficiently

larger than the spread on an adequately matching catastrophe swap, i.e.,

if the basis is negative, a positive carry can be locked-in by buying the

bond and simultaneously buying protection under the swap agreement.58

The idea is that the occurrence of an event should trigger both instru-

ments such that the loss on the bond is (at least partially) compensated

by the payoff from the swap. Another potential field of application for

cat swaps are synthetic Collateralized Debt Obligations (CDOs) of catas-

trophic risks. In general, a CDO is a securitization of a pool of assets.

These assets are purchased and held by an SPV, which funds the transac-

57These characteristics of cat bond returns have been documented by several au-thors, see, e.g., Litzenberger et al. (1996), Bantwal and Kunreuther (2000), GuyCarpenter (2008) or Cummins and Weiss (2009).

58In analogy to the credit markets, the basis can be defined as catastrophe swapminus catastrophe bond spread.

3.4 Accounting and regulation 69

tion through the issuance of rated security tranches of differing seniority.

In contrast to a so-called true sale or cash CDO structure, a synthetic

CDO does not involve the physical transfer of assets. Instead, the SPV

gains risk exposure by selling protection under swap contracts. While of-

fering substantial risk transfer capacity to (re)insurers, cat CDOs enable

investors to take a position in a diversified portfolio of natural disaster

risks by selecting a tranche which matches their specific risk appetite. A

concrete example is the USD 200 million transaction ”Fremantle 2007-I”

arranged by ABN AMRO, which featured three classes of notes rated

AAA, BBB+ and BB− by Fitch Ratings and utilized cat swaps to trans-

fer the risk to the SPV. For a further discussion of insurance risk CDOs

refer to Forrester (2008).

3.4 Accounting and regulation

In contrast to ILWs, catastrophe swaps are financial instruments, not

insurance contracts. Their regulatory and accounting treatment is un-

ambiguous: Under International Financial Reporting Standards (IFRS)

as well as U.S. GAAP, pure index contracts, such as catastrophe swaps,

are not eligible for reinsurance accounting and consequently do not influ-

ence the underwriting result. Instead, they have to be accounted for at

fair value, which can lead to elevated volatility in the income statement

of the (re)insurer, since technical liabilities are currently not marked to

market (see, e.g., World Economic Forum, 2008). In addition, the cur-

rent Solvency framework in Europe as well as the National Association

of Insurance Commissioners (NAIC) regulation in the U.S. only accept

instruments with an indemnity trigger (i.e., instruments without basis

risk) as (re)insurance contracts. However, it appears that under the

upcoming Solvency II, which is currently still being refined, all instru-

ments which accomplish an effective economic risk transfer could result

in regulatory capital relief (see, e.g., Swiss Re, 2009). In the U.S., on

the contrary, new reserving rules, employing a more economic stance

with regard to risk mitigation instruments, are currently not envisioned.

Klein and Wang (2009) believe this to be a major impediment for U.S.

insurers to use swaps on a larger scale.


3.5 Comparison to other risk transfer instruments

Although they do not include an indemnity trigger, catastrophe swaps

are economically largely equivalent to ILWs. In solely referencing indices,

they are not subject to issues of asymmetric information and moral haz-

ard and, as a result, do not require an intensive underwriting process.

Thus, they can serve as cost-effective substitutes for traditional reinsur-

ance contracts. Very much like ILWs and most cat bonds, catastrophe

swaps do expose the protection buyer to basis risk if the transaction

is aimed at hedging a specific portfolio of underwritten insurance con-

tracts. Basis risk arises as the industry index is typically not perfectly

correlated with the losses which the (re)insurer suffers on his book of

business. Cat swaps are highly standardized and avoid the structural

complexities and costs associated with the issuance of full-fledged insur-

ance securitizations, such as setting up an SPV and entering a total

return swap. Consequently, from the perspective of a (re)insurance com-

pany, they are simple to initiate and can be executed much more rapidly

than a cat bond transaction. Compared to ILWs, cat swaps bear the

additional potential of becoming more liquidly tradable once the market

fully takes off. Swaps in general are unfunded transactions and thus by

design not fully collateralized. However, since fluctuations in a contract’s

mark-to-market value will effect regular margin calls between the swap

partners, counterparty default risk for the protection buyer is limited.

Table 1 summarizes the main points of this comparison.

4 Pricing model

Below we introduce a two-stage pricing framework for catastrophe swaps.

Ex-ante the main risk drivers of the instrument are the random num-

ber of natural disasters, their timing, and the associated final loss esti-

mates.59 Apart from determining the fair cat swap spread before the

occurrence of a catastrophe, however, market participants will also need

to value their contracts in the special situation when a catastrophe has

already struck and an initial loss estimate has been published. In this

59Refer to Section 3.1.

4P

ricing

model

71

CatastropheSwaps

Cat Bonds ILWsReinsuranceContracts

triggers industry index

indemnity-basedindustry indexpure parametricparametric indexmodeled loss

double trigger:industry indexindemnity trigger

indemnity-based

moral hazard nonenone if pureindex trigger

low high

basis risk highnone if pureindemnity trigger

high none

standardization very high low high low

transaction cost low high low high

counterpartydefault risk

partiallycollateralized

negligible(collateralized)

collateralizationpossible

collateralizationpossible

accountingtreatment

financialinstrument

dependson trigger

reinsurance reinsurance

Table 8: Catastrophe swaps, cat bonds, ILWs, and reinsurance contracts in comparison


case, which will be termed loss reestimation phase in the following, the

timing of the catastrophe and an approximate range for the correspond-

ing losses are known. Yet, there is still uncertainty as to the further

development of the loss estimates until the final loss report or maturity.

Consequently, we distinguish between the pricing of cat swaps before a

catastrophe (ex-ante) and during the reestimation phase.60 Both model

components will be developed in a continuous-time contingent claims

framework without bid-ask spreads, transaction costs, short-selling con-

straints, taxes, or other market frictions.

4.1 Risk-neutral valuation of catastrophe derivatives

Since catastrophe swaps are not insurance contracts, it seems adequate

to price them with financial rather than actuarial approaches.61 Based

on the no-arbitrage principle modern option pricing theory as constituted

by Black and Scholes (1973) as well as Merton (1973) has established the

preference free pricing of derivative instruments under the risk-neutral

(equivalent martingale) measure. The absence of arbitrage opportunities

in the capital markets implies the existence of such an equivalent mar-

tingale measure. For a single unique equivalent martingale measure to

arise, however, markets also need to be complete, meaning that all contin-

gent claims are replicable through available securities (see Harrison and

Kreps, 1979). Hence, the general incompleteness of insurance markets

prevents the uniqueness of the equivalent martingale measure. Neverthe-

less, the literature on catastrophe derivatives and bonds is dominated by

contingent claims valuation frameworks, irrespective of the non-traded

underlying and the fact that natural disaster risk cannot be hedged with

traditional securities.62 Different solutions have been proposed to tackle

the ambiguity with regard to the change of measure.

60Practitioners call exchange traded catastrophe instruments ”live cat” beforea catastrophic event and ”dead cat” during the loss reestimation phase. Seewww.theifex.com

61Although a discussion with industry experts indicated that actuarial pricing cur-rently seems to prevail in practice.

62Refer to the literature review in Section 1.

4.1 Risk-neutral valuation of catastrophe derivatives 73

Some authors assume that market completeness is preserved due to

the existence of other tradable and sufficiently liquid instruments which

are driven by the same source of randomness.63 Consequently, repli-

cating portfolios can be formed and a unique risk-neutral measure is

obtainable by recovering a market price of catastrophe risk from ob-

served quotes.64 While early attempts by CBOT to establish a liquid

market for exchange-traded catastrophe derivatives have failed, several

exchanges have recently re-introduced various types of contracts. Among

these are options and futures on the CME hurricane index and futures on

the PCS indices (see Swiss Re, 2009). Although the market is still juve-

nile, its long-term outlook is promising.65 Alternatively, investors could

turn to OTC catastrophe derivatives or cat bonds in order to mimic

movements of cat swap positions. In addition, as argued by Muermann

(2003), insurance and reinsurance contracts permit indirect trading in

the underlying catastrophe risk. Lastly, Cummins and Geman (1995)

and Vaugirard (2003a,b, 2004) mention that certain energy, commodity,

and weather derivatives could be suitable to track continuous changes in

catastrophe losses, since geological and meteorological determinants of

insurance claims impact the value of these instruments, too.

Another common approach to specify a unique pricing measure de-

spite an incomplete markets set-up dates back to Merton (1976). Anal-

ogous to his reasoning, natural disasters can be treated as unsystematic

shocks to the overall economy which are fully diversifiable, implying

risk-neutrality of the market participants.66 Thus, there is no cat risk

premium and model parameters under the physical and equivalent mar-

tingale measure are identical. This stance is supported by the empir-

63Examples are Cummins and Geman (1994, 1995), Chang et al. (1996), Gemanand Yor (1997), Baryshnikov et al. (2001), Muermann (2003), Vaugirard (2003a,b,2004), Muermann (2008), Chang et al. (2008, 2010).

64A brief illustration of the reasoning behind this proceeding is given in AppendixA. Concrete examples for arbitrage portfolios in the context of PCS options can befound in the empirical study of Balbas et al. (1999), who demonstrate the generalapplicability of financial theory to the sphere of catastrophe derivatives.

65However, due to currently still restricted trading volumes and liquidity, thesemay not be ideal instruments to approximate an instantaneous riskless portfolio yet.

66This stance is adopted in Bakshi and Madan (2002), Lee and Yu (2002), Coxet al. (2004), Jaimungal and Wang (2006), as well as Lee and Yu (2007).


ical studies of Hoyt and McCullough (1999) as well as Cummins and

Weiss (2009), who provide evidence for the zero-beta characteristics of

catastrophe-linked instruments.

Finally, the issue of incomplete markets can be overcome by selecting

a particular change of measure such as the well-known Esscher trans-

form (see Gerber and Shiu, 1996).67 Another such distortion operator

for the risk-neutral valuation of insurance contracts has been introduced

by Wang (2000).

We follow Biagini et al. (2008) as well as Wu and Chung (2010) in

not further discussing the choice of martingales and the change from

the physical measure P to the risk-neutral measure Q. Instead, we will

assume that Q has been predetermined according to one of the above-

mentioned alternatives and directly proceed to a risk-neutral formula-

tion of our model framework. The equivalent martingale measure Q is

restricted to the class which only corrects parameters, while stochastic

processes and distributions retain the same characteristics as under P.

4.2 Pricing catastrophe swaps ex-ante

The first, more general component of our pricing approach is meant to

be applied before a natural disaster has occurred in the covered territory.

As mentioned above, the major uncertainty in this phase of a cat swap

transaction centers around the stochastic number and timing of events

during the term of the contract as well as the ultimate losses they cause.

Hence, we will focus on these underlying sources of randomness, while

abstracting from any uncertainty surrounding the reestimation process

of claims. Let (Ω,F ,Q) denote a probability space with the set of all

possible outcomes Ω, a filtration F for the relevant subsets of Ω, and the

equivalent probability measure Q. In line with the prevailing practice

in the literature on catastrophe derivatives, we assume that the number

67Examples are Embrechts and Meister (1997), Schmidli (2003), and Dassios andJang (2003).

4.2 Pricing catastrophe swaps ex-ante 75

of natural disasters nt,t+∆t in any time interval (t, t + ∆t] is Poisson

distributed with intensity λt,t+∆t:

nt,t+∆t ∼ P (λt,t+∆t) , ∀t ∈ (0, T − ∆t], (5)

where λt,t+∆t =∫ t+∆t

tλ(u)du. Furthermore, we follow Chang et al.

(2008, 2010) as well as Wu and Chung (2010) and allow for cyclicality

in the occurrence of catastrophes by incorporating a stochastic Poisson

intensity, which is assumed to adhere to the mean-reverting Ornstein-

Uhlenbeck process

dλ(t) = κ (µλ − λ(t)) dt+ σλdWQλ (t), (6)

with mean reversion rate κ, long-term mean µλ, volatility of the inten-

sity σλ, and a standard Brownian motion dWQλ (t). Thus, the arrival

of natural disasters is governed by a doubly stochastic Poisson process

(Cox process), i.e., two-stage randomization procedure: the Ornstein-

Uhlenbeck process in Equation (6) generates the intensity for the Pois-

son distribution of nt,t+∆t. Intuitively this makes particularly sense for

periodic climate patterns such as the El Nino phenomenon which recurs

on average in five year intervals or the annual Atlantic hurricane season

in the U.S. from June to November. However, it also seems suitable to

capture such as the typical clustering of earthquakes. We aim to provide

some empirical support for this assumption in Section 5.3.

Each catastrophe i is associated with a stochastic final loss estimate,

represented by positive independent and identically distributed (i.i.d.)

random variables Yi with distribution function FY (x). We further as-

sume that nt,t+∆t and Yi are stochastically independent and that there is

no time delay between the occurrence of the catastrophe and the issuance

of the final loss report.68 Consequently, the aggregate final loss estimates

68It is straightforward to extend the model with a deterministic or stochastic wait-ing time between occurrence of the catastrophe and the issuance of the final lossreport. We abstain from this additional layer of complexity as it is neither empha-sized in the literature nor central to the cat swap pricing problem.


due to natural disasters in any time interval (t, t+ ∆t] can be expressed

as compound Poisson process with expected value λt,t+∆tEQ(Yi):

Lt,t+∆t =

nt,t+∆t∑

i=1

Yi, ∀t ∈ (0, T − ∆t]. (7)

Recall from Section 3.1 that the payoff of a cat swap transaction is

triggered immediately when the final loss estimate for a single natural

disaster equals or exceeds the event threshold. Hence, in contrast to

the usual procedure for other cat instruments, we must not aggregate

losses from different events over the whole term of the contract. Instead,

each final loss estimate is separately compared to the event threshold at

the time of its occurrence. Since the instrument terminates prematurely

if a trigger event has been identified, its timing is crucial for valuation

purposes and a pricing model needs to capture path dependency. We

achieve this by sequentially reevaluating the loss process in Equation (7)

for infinitesimally small time steps dt from t = 0, i.e., the outset of the

contract until its maturity t = T :

lim∆t→0

Lt,t+∆t = Lt,t+dt ≡ dLt, ∀t ∈ (0, T − ∆t]. (8)

Consequently, instead of one process for the whole term, we generate

a series of compound Poisson processes. Under this set-up, arrivals in

non-overlapping intervals are independent, the probability of exactly one

catastrophe per time step is approximately λt,t+dt, and the probability

of more than one catastrophe in a marginal interval of length dt is vir-

tually zero.69

As described in Section 2, a cat swap consists of a fixed leg, which

comprises the stream of regular premiums by the protection buyer, and a

floating leg, which is the compensation payment by the protection seller

contingent on a trigger event. Swap pricing generally entails the separate

69Alternatively the the occurrence of a catastrophe at each time step could bemodeled as a Bernoulli trial, implying a binomially distributed sum. With regard tototal number of events until maturity this differentiation is less relevant, since, for alarge number of trials, the Binomial converges to the Poisson distribution.

4.2 Pricing catastrophe swaps ex-ante 77

valuation of each leg in a transaction, with the goal of balancing their

present values the through the fair spread scat:

PVfloating = PVfixed(scat). (9)

We define the first passage time (or stopping time) τ for the series of

compound Poisson processes as the earliest instant in which a final loss

estimate is equal to or higher than the event threshold ET :

τ ≡ inft | Lt,t+dt ≥ ET. (10)

Consider a catastrophe swap contract with maturity T , notional N ,

and a payoff which is determined as a preset percentage α of the no-

tional. Assume that a trigger event can occur at any given point in time

and causes an immediate payoff under the swap contract.70 Then the

following is an expression for the present value of the floating leg:

PVfloating = EQ0 [e−rταN1τ≤T ]. (11)

Here, 1τ≤T is the indicator function which equals 1 if τ ≤ T and 0

otherwise. In addition, r is the instantaneous risk-free rate, i.e., the term

structure is assumed to be flat and deterministic. Although relatively

recent publications in the context of cat bond valuation have introduced

stochastic interest rates based on the well-known term structure models

of Vasicek (1977) or Cox et al. (1985) (see, e.g, Lee and Yu, 2002; Vaugi-

rard, 2003a,b; Wu and Chung, 2010), we decide to dismiss this possibility

in favor of computational efficiency. Since catastrophe swap contracts

are exclusively available with one year maturities, the effect of random

changes in the risk-free term structure should have a notably lesser im-

pact than on medium or longer-term instruments. Consequently, this

assumption does not severely influence our results.71

70Since we model the final loss estimate for each catastrophe, we do not need toallow for a development period of the losses after τ . However, through this assumptionwe do abstract from delays in payment collection.

71Young (2004) argues that term structure models are appropriate for instrumentswith a maturity of more than one year.


Furthermore, the fixed leg of a cat swap consists of regular premiums

as well as an accrual payment in case the first passage time does not

coincide with a premium payment date:

PVfixed(scat) = PVpremiums(scat) + PVaccrual(scat). (12)

Given a final loss estimate has not exceeded the event threshold, the

buyer of cat swap protection makes payments of scatN∆ti on premium

dates ti, where i= 1,...,n, ∆ti is the length of a premium period (ti−ti−1),

and scat is the fair spread we are looking for. Consequently, we can value

the premium part of the fixed leg as follows:

PVpremiums(scat) =

n∑

i=1

e−rtiEQ0 [scatN∆ti1τ>ti ]. (13)

Finally, the present value of the expected accrual payment for the

time period since the last premium date (τ − ti−1) can be expressed as:

PVaccrual(scat) = EQ0 [e−rτscatN(τ − ti−1)1ti−1≤τ≤ti ]. (14)

Evidently, pricing the cat swap involves solving a first passage time

problem. Due to the compound Poisson process in Equation (7), however,

a closed-form solution for the first passage time cannot be derived (see

Kou and Wang, 2003). Therefore, one needs to resort to Monte Carlo

techniques for an estimation of the fair spread scat.

4.3 Pricing catastrophe swaps

in the loss reestimation phase

After a cat swap has been entered, its mark-to-market value will fluc-

tuate in accordance with the occurrence of natural disasters as well as

the development of their associated loss estimates. More specifically, a

particularly sharp increase in value should be observed when interim

loss estimates approach the contract’s thresholds. However, the ex-ante

pricing approach introduced above does not reflect this sensitivity of

the instrument with regard to interim loss reports. Therefore, it is less

4.3 Pricing catastrophe swaps in the loss reestimation phase 79

suitable to price cat swaps during the reestimation phase, i.e., after a

catastrophe has occurred and an initial loss estimate has been issued.

At that time it is still unclear what the final loss estimate will be and

whether the preset acceleration threshold will be exceeded by an interim

loss estimate before maturity. In the remainder of this section, we aim

to capture this uncertainty with regard to the reestimation of losses by

means of a parsimonious barrier option framework under which closed-

form expressions for the cat swap spread can be derived. Assume that,

under the risk-neutral measure Q, the dynamics of the interim loss esti-

mates Li(t) referenced by the catastrophe swap are adequately described

by a Geometric Brownian Motion:

dLi(t)

Li(t)= rdt+ σdWQ

Li(t), (15)

with volatility σ and a standard Q-Wiener process dWQ

Li(t). The choice

of a diffusion process for the development of catastrophic loss estimates

is common in the literature on cat bond and derivative pricing (see, e.g.,

Bakshi and Madan, 2002; Schmidli, 2003; Biagini et al., 2008). The Ge-

ometric Brownian motion in particular has been applied by quite a few

authors to model the accrual of losses and unpredictability in reporting

over time.72 Apart from that, it ensures analytical tractability and the

resulting lognormally distributed estimates are in line with the empirical

findings of Levi and Partrat (1991) and Burnecki et al. (2000) for PCS

loss data.

We define the distance to the acceleration threshold AT (≥ ET ) at

time t as follows:

DAT (t) =AT

Li(t). (16)

Similarly, we will refer to the ratio of ET to the loss estimate Li(t)

as the distance to the event threshold:

DET (t) =ET

Li(t). (17)

72Examples are Cummins and Geman (1994, 1995), Geman and Yor (1997), Lou-berge et al. (1999), Gatzert et al. (2007), and Wu and Chung (2010).


Again, we need to match both legs of the swap transaction through scat:

PVfloating = PVfixed(scat). (18)

We begin with the floating leg. Assume that at time s (0 ≤ s ≤T ) an initial or interim loss estimate Li(s) for a specific catastrophe

is available. Although Li(s) is unknown at the outset of a contract,

i.e., in t = 0, it is deterministic during the reestimation phase. The

interim loss process determined by Equation (15) starts at Li(s) and

the protection buyer receives a fixed payment whenever an interim loss

estimate Li(t) during the remaining term of the contract breaches the

acceleration threshold for the first time, i.e., when DAT (t) hits unity.

This payoff profile equals a binary up-and-in one-touch barrier option

on the interim loss estimates. Define the first passage time τd for the

diffusion process as:73

τd ≡ inft | Li(t) ≥ AT. (19)

From Rubinstein and Reiner (1991a,b) we know that in the present

setting, the following analytic expression for the first passage time den-

sity applies:

h(τd) =ln(DAT (s))√

2πστ3/2d

exp

−1

2

(

− ln(DAT (s)) + (r − σ2

2 )τd

σ√τd

)2

. (20)

Now, recall from Section 3.1, that a compensation payment αN by

the protection seller can also be due at maturity T in case an interim loss

estimate is equal to or higher than the event threshold, i.e., Li(T ) ≥ ET

which implies DET (T ) ≤ 1. Thus, ET can be interpreted as the strike

price of a binary European-type call option. However, in order not to

price certain states twice, this option also needs to include a knock-out

73Due to the design of typical catastrophe swap contracts with a binary payoff assoon as the acceleration threshold is reached, we do not need to allow for a develop-ment period of the losses after τd has occurred. Again, we do abstract from delays inpayment collection. Also note that the model ignores so-called extension thresholdswhich, if exceeded by the interim loss estimates, cause a prespecified extension of thecontract’s maturity to allow for a further accumulation of losses.


feature so that it lapses whenever Li(t) ≥ AT , i.e., when the payoff

from the previously discussed one-touch option is triggered. To see this

consider the case where Li(T ) = AT ≥ ET . Here both the up-and-

in one-touch and a simple binary European call option would pay off.

Hence, in order to value the floating leg, we need to combine the up-

and-in one-touch (UAIone touch) with a binary up-and-out call (UAOcall)

option which pays off if and only if Li(T ) ≥ ET and the acceleration

threshold has not been hit during the term of the contract:

PVfloating = UAIone touch + UAOcall. (21)

Applying results from Rubinstein and Reiner (1991a,b), the price of

the up-and-in one touch (binary) in t = s can be expressed as:

UAIone touch =

∫ T

s

αNe−rτdh(τd)dτd

= αN

∫ T

s

e−rτdh(τd)dτd

= αNQ(T ),

(22)

with

Q(u) =

∫ u

s

e−rτdh(τd)dτd = DAT (s)a+bΦ(d1) +DAT (s)a−bΦ(d2),

where s < u, Φ(x) is the standard normal cumulative distribution func-

tion (cdf), and

a =r

σ2− 1

2, b =

√

(r − σ2

2 )2 + 2rσ2

σ2,

d1 =− ln(DAT (s)) − bσ2u

σ√u

,

d2 =− ln(DAT (s)) + bσ2u

σ√u

.


Furthermore, the up-and-out cash-or-nothing call (binary) in t = s

is worth (see Rubinstein and Reiner, 1991a,b):

UAOcall = αN (B1(T ) −B2(T ) +B3(T ) −B4(T )) (23)

with

B1(T ) = e−rTΦ(x1 − σ√T ), B2(T ) = e−rTΦ(x2 − σ

√T ),

B3(T ) = e−rTDAT (s)2aΦ(−y1 + σ√T ),

B4(T ) = e−rTDAT (s)2aΦ(−y2 + σ√T ),

and

x1 =− ln(DET (s)) + (a+ 1)σ2T

σ√T

, x2 =− ln(DAT (s)) + (a+ 1)σ2T

σ√T

,

y1 =ln(AT 2/(Li(s)ET )) + (a+ 1)σ2T

σ√T

,

y2 =ln(DAT (s)) + (a+ 1)σ2T

σ√T

.

The fixed leg again comprises the stream of spread payments and

an accrual which accounts for the fact that the contract can be triggered

in between two scheduled premium dates:

PVfixed(scat) = PVpremiums(scat) + PVaccrual(scat). (24)

The protection buyer pays scatN∆ti on the remaining premium dates ti(s < ti ≤ T ) with ∆ti being the length of a premium period (ti − ti−1).

Defining the survival probability of the contract from time s to time

u (i.e., the probability that a trigger event does not occur before u) as


(1 −H(u)) with H(u) =∫ u

sh(τd)dτd, the stream of premium payments

has the following value in t = s:

PVpremiums(scat) =

n∑

i=1

scatN∆tie−rti (1 −H(ti))

= scatN

n∑

i=1

∆tie−rti (1 −H(ti))

= scatNΣpremiums,

(25)

where

Σpremiums =

n∑

i=1

∆tie−rti (1 −H(ti)),

and (see, e.g., Vaugirard, 2003b)

H(u) =

∫ u

s

h(τd)dτd = DAT (s)2aΦ(z1) + Φ(z2),

with

z1 =− ln(DAT (s)) − (r − σ2

2 )u

σ√u

, z2 =− ln(DAT (s)) + (r − σ2

2 )u

σ√u

.

Moreover, the present value in t = s of the expected accrual payment

can be expressed as:

PVacc(scat) =

n∑

i=1

∫ ti

ti−1

scatN(τd − ti−1)e−rτdh(τd)dτd

= scatN

[n∑

i=1

∫ ti

ti−1

τde−rτdh(τd)dτd −

n∑

i=1

ti−1

∫ ti

ti−1

e−rτdh(τd)dτd

]

= scatN

[n∑

i=1

(J(ti) − J(ti−1)) −n∑

i=1

ti−1(Q(ti) −Q(ti−1))

]

= scatN

[

J(T ) −n∑

i=1

ti−1(Q(ti) −Q(ti−1))

]

= scatNΣaccrual, (26)


where

Σaccrual =

[n∑

i=1

(J(ti) − J(ti−1)) −n∑

i=1

ti−1 (Q(ti) −Q(ti−1))

]

,

and (see, e.g, Gil-Bazo, 2006)

J(u) =

∫ u

s

τde−rτdh(τd)dτd

=− ln(DAT (s))

bσ2

(DAT (s)a+bΦ(d1) −DAT (s)a−bΦ(d2)

).

Inserting (21) to (26) into Equation (18), the fair cat swap spread

can be calculated as follows:

PVfloating = PVfixed(scat)

(UAIone touch + UAOcall) = scatN (Σpremiums + Σaccrual)

scat =(UAIone touch + UAOcall)

N (Σpremiums + Σaccrual). (27)

The decision for a switch from the ex-ante to this barrier option pric-

ing approach must depend on the size of the loss estimate Li(s). If it

is too low, the subsequent reestimation process does not constitute a

major risk driver and the instrument will be worthless under the barrier

option approach. This is due to the fact that small Li(s) are associated

with large distances to the thresholds, which implies low probabilities of

the up-and-in one touch being triggered and the up-and-out call ending

up in the money. Consequently, Li(s) needs to be sufficiently close to

ET so that a substitution of the model is sensible. As a rule of thumb

we suggest to switch whenever the spread under the barrier option ap-

proach associated with a current loss estimate Li(s) is at least equal to

the spread under the ex-ante approach. This criterion is illustrated in

Figure 4(a) by means of a simple numerical example. It is based on a


cat swap contract with ET = 20 bn, AT = 22 bn, and up-front premium

instead of regular spread payments, i.e., we simply price the floating leg

as shown in Equation (21). The following parameter values have been

assumed: σ = 0.2, r = 0.02, α = 1, N = 1, and λt,t+∆t = λ = 2. The

dotted line in the center shows the prices from the ex-ante approach for

different T , while the solid line at the top and the dashed line at the bot-

tom represent prices from the barrier option approach for a current loss

estimate of Li(s) = 18 bn and Li(s) = 17 bn, respectively. We observe

that for Li(s) = 17 bn the prices from the ex-ante and the barrier option

approach are quite similar across all T . For Li(s) = 18 bn, in contrast,

the barrier option approach already results in higher cat swap spreads

than the ex-ante approach, which is insensitive to the level of initial and

interim loss estimates. Therefore, in this situation, we would suggest to

consider a substitution of the pricing model if an initial or interim loss

estimate is about 15 percent below the event threshold ET . Finally, to

complete our discussion of cat swap valuation during the reestimation

phase, in Figure 4(b) we have plotted the sensitivity of the up-front pre-

mium and its option components with regard to Li(t) for a remaining

time to maturity of T = 0.5 and the above-mentioned parameter values.

Evidently, an increase in Li(t), i.e., a shrinking distance to ET and AT

inflates the cat swap spread exponentially since it results in a higher

probability of a payoff to the protection buyer.


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

time to maturity T (years)

up−

fron

t pre

miu

m

Barrier option approach for Li(s) = 18 bnEx−ante approach

Barrier option approach for Li(s) = 17 bn

(a) Barrier option vs. ex-ante pricing approach

15 16 17 18 19 20 21 22

0.0

0.1

0.2

0.3

0.4

interim loss estimate Li(t)

up−

fron

t pre

miu

m

Cat swapUp−and−in one touchUp−and−out Call

(b) Sensitivity with regard to interim loss estimates

Figure 4: Illustration of the barrier option pricing approach



5.1 Severity distributions

for natural disasters in the U.S.

In this section we want to select a distribution for the severity of final

losses Yi, which is a crucial component of our ex-ante pricing model

(see Section 4.2). The class of heavy-tailed distributions is particularly

relevant in the context of catastrophe-related claims, since it allows to

properly account for the low frequency high severity character of natu-

ral disasters by assigning comparatively large probabilities to extreme

losses.74

Although some authors have applied distributions with lighter tails,

such as the gamma and exponential distribution,75 extant empirical ev-

idence indicates that those are outperformed by heavy-tailed distribu-

tions. Levi and Partrat (1991), e.g., estimate the lognormal, Pareto,

and exponential distribution based on U.S. hurricane losses between 1954

and 1986 and conclude that the former provides the best fit. Their re-

sult is confirmed by the work of Burnecki et al. (2000), who examine the

time series of the quarterly national PCS loss index from 1950 to 2000,

fitting lognormal, Pareto, Burr, and gamma distributions. Consistent

with this empirical evidence, the lognormal distribution remains by far

the most common choice for the modeling of catastrophe loss amounts in

the literature.76 In contrast to that, Milidonis and Grace (2008) analyze

catastrophe loss data for the state of Florida between 1949 and 2004

and find that the lognormal is outperformed by the Pareto as well as the

74Heavy-tailed distributions have a density function which converges to zero moreslowly than an exponential function, i.e., compared to the exponential distributionthey exhibit more probability mass in the tails. Typical examples are the lognormal,Weibull (with a shape parameter < 1), Pareto, and Burr distribution. For a moredetailed discussion of this class of distributions see Bryson (1974).

75Bakshi and Madan (2002) as well as Dassios and Jang (2003), e.g., use the expo-nential distribution and Aase (1999) as well as Jaimungal and Wang (2006) employthe gamma distribution.

76Examples are Chang et al. (1996), Louberge et al. (1999), Schmidli (2003), Leeand Yu (2002), Burnecki and Kukla (2003), Vaugirard (2003a,b, 2004), Lee and Yu(2007), as well as Wu and Chung (2010).


Burr distribution. Based on a likelihood ratio test, they subsequently

decide to employ the latter. Since their samples differ in terms of time

period, geographic focus, and loss normalization method, there is some

ambiguity as to which of these previous empirical studies best suits our

purpose. The article of Milidonis and Grace (2008) is the most recent

and at least partially covers the last decade which has been particularly

prominent with regard to Atlantic hurricane activity. However, it exclu-

sively focuses on the state of Florida while Levi and Partrat (1991) and

Burnecki et al. (2000) use national data. In addition, despite their larger

dataset, Milidonis and Grace (2008) only report estimation results for

the shorter time series from 1990 to 2004, thus ignoring once in a century

events such as the 1906 San Francisco Earthquake or the Great Miami

Hurricane of 1926. Yet, on a normalized scale, damages caused by these

extreme incidents have been shown to considerably exceed those of more

recent natural disasters.77

Hence, we aim to conduct an empirical investigation ourselves in or-

der to select a distribution which is capable of adequately capturing the

tail characteristics of catastrophe losses. We use the datasets of Pielke

Jr. et al. (2008) and Vranes and Pielke Jr. (2009), who, in their recent

empirical work, normalize estimates of economic losses from all major

U.S. hurricanes and earthquakes between 1900 and 2005 to 2005 Dollars.

They have published their results in extensive appendices to the articles,

thereby providing a reliable basis for the estimation of catastrophe loss

distributions.78 Their normalization methodology is based on inflation,

wealth, and population growth in the affected areas and has proven its

capability to effectively adjust historical loss data for societal factors.

The inflation and wealth adjustment are based on the implicit gross do-

mestic product price deflator and the fixed assets and consumer durable

goods statistic, respectively. Both magnitudes are available from the U.S.

77In 2005 Dollars, the 1906 San Francisco Earthquake and the 1926 Great MiamiHurricane would have caused losses of USD 284 billion and USD 161 billion, respec-tively (see Pielke Jr. et al., 2008; Vranes and Pielke Jr., 2009). This compares to USD116 billion for Katrina, which was the second most severe hurricane in U.S. history.

78Note, however, that economic losses regularly exceed insured losses. As a con-sequence, our fitted loss distributions must be viewed as quite conservative for thepurpose of pricing of cat instruments.

5.1 Severity distributions for natural disasters in the U.S. 89

Bureau of Economic Analysis (BEA). Moreover, the population adjust-

ment is conducted with county-level statistics from the U.S. Department

of Census. Figure 5 shows the time series of normalized annual U.S.

hurricane and earthquake losses. The corresponding histograms can be

found in Figure 6. While the histogram for earthquake losses is a little

more pointy than for hurricanes, the shape of both indicates an asym-

metrical distribution with a particularly heavy right tail. Hence, we will

fit the lognormal, Burr, Pareto, and Weibull distribution to the data

by obtaining estimates for the parameters of their respective cumulative

distribution function (cdf):

- lognormal cdf for µ ∈ R and σ > 0

F (x) =

∫ x

0

1√2πσu

e−(lnx−µ)2

2σ2 = Φ

(lnx− µ

σ

)

, (28)

- Burr cdf with a (shape1), b (shape2), and c (scale) > 0 (also called

Burr XII distribution)

F (x) = 1 −(

1 +(x

c

)b)−a

, (29)

- Pareto cdf with a (shape) and c (scale) > 0 (also called Pareto

type II or Lomax distribution)79

F (x) = 1 −(

1 +x

c

)−a

, (30)

- Weibull cdf with a (shape) and c (scale) > 0

F (x) = 1 − e−( xc)a . (31)

In addition, the gamma and exponential distribution will be consid-

ered for comparison purposes:

79Note that this is a special case of the Burr XII distribution with shape parameterb = 1. The Pareto distribution is often used in a generalized form with a third, so-called location parameter which allows for a lower limit above zero. We waive thatgenerality here.


- gamma cdf with a (shape), c (scale), and β = 1/c (rate) > 0

F (x) =

∫ x

0

ua−1e−u/c

Γ(a)cadu, (32)

- exponential cdf for β (rate) > 0

F (x) = 1 − e−βx. (33)

While some of these distributions are only defined for strictly positive

values, our dataset also includes years without damages. To tackle this

issue we follow Burnecki et al. (2000), remove the characteristic spike

at zero (see histograms), and estimate the cdfs on the subset of positive

observations. Table 9 provides some descriptive statistics with regard to

the empirical distribution of non-zero losses.

Hurricane losses (USD bn)

max. 161.3311 quantiles

min. 0.0190 0% 0.0190

mean 12.1362 25% 0.4003

median 3.0555 50% 3.0555

s.d. 24.7117 75% 13.2370

skewness 3.7786 100% 161.3311

excess kurtosis 16.7313

Earthquake losses (USD bn)

max. 283.7353 quantiles

min. 0.0019 0% 0.0019

mean 9.0238 25% 0.0526

median 0.1707 50% 0.1707

s.d. 41.2054 75% 1.3028

skewness 6.1166 100% 283.7350

excess kurtosis 37.5699

Table 9: Descriptive statistics for (non-zero) disaster losses (1900–2005)


1900 1920 1940 1960 1980 2000

050

100

150

Year

Nor

mal

ized

hurr

ican

e lo

sses

(U

SD

billion

)

(a) Normalized annual hurricane losses

1900 1920 1940 1960 1980 2000

050

100

150

Year

Nor

malize

d e

arth

quake

loss

es (

USD

billion

)

(b) Normalized annual earthquake losses

Figure 5: Natural disaster losses in the U.S. (1900–2005)


Normalized hurricane losses (USD billion)

Fre

quen

cy

0 2 4 6 8 10

01

23

4

(a) Histogram of normalized hurricane losses

Normalized earthquake losses (USD billion)

Fre

quen

cy

0 2 4 6 8 10

01

23

4

(b) Histogram of normalized earthquake losses

Figure 6: Histograms of normalized natural disaster losses


In order to assess the fit of the above parametric distributions, we

apply the Kolmogorov-Smirnov and the Anderson-Darling test, which

are based on a comparison of the empirical distribution function F (x) =1n

∑ni=1 1xi≤x and the theoretical distribution function F (x, θ), where θ

represents a vector of parameter estimates. The null hypothesis for both

tests is that the sample at hand comes from the specified distribution

(H0: F (x) = F (x, θ)). Although the chi-square goodness of fit test is

also very common, it will not be considered due to its sensitivity to the

binning of the data and its low power for small sample sizes. Since empir-

ical samples such as our dataset of cat losses generally contain a rather

small amount of extreme observations, it is questionable whether the

chi-square test would reveal a severe misfit of the tail. The Kolmogorov-

Smirnov test, in contrast, is more suitable for small samples. In addition,

the Anderson-Darling test, a modification of the Kolmogorov-Smirnov

test, is one of the globally most powerful goodness of fit tests (see Levi

and Partrat, 1991). It puts a higher weight on the tail of the distribution

and is therefore the most adequate statistic for our purpose.

Tables 10 and 11 contain maximum likelihood estimation (MLE) re-

sults for the cdf parameters as well as the Kolmogorov-Smirnov (KSn)

and Anderson-Darling (ADn) goodness of fit test statistics and their cor-

responding p-values. The lower the respective test statistic (the higher

the p-value), the better the fit of the distribution. As expected, the

exponential distribution is rejected on all reasonable significance levels.

While the gamma distribution does very poorly with respect to the earth-

quake dataset (p-values < 0.01), it fits the hurricane losses surprisingly

well. In fact, its ADn value for the earthquake dataset is even smaller

than that of the more heavy-tailed Pareto distribution. Furthermore,

the lognormal distribution seems to be a reasonable choice for both sam-

ples, although it is outperformed in the tail by the Burr and the Weibull

distribution for hurricane losses and by the Burr and the Pareto distri-

bution for earthquake losses (see respective ADn values). Overall, the

Burr distribution exhibits the lowest Anderson-Darling statistics for both

datasets, leading us to conclude that it is the most suitable candidate

for the severity of natural disaster damages. Note that this confirms the


empirical results of Milidonis and Grace (2008). Thus, in the context of

the following analysis, we adopt a Burr loss severity distribution with

the parametrizations as shown in Table 10.

5.2 Derivation of implied Poisson intensities

The vast majority of authors opts for a simple homogeneous Poisson

process to represent the arrival of claims due to catastrophes.80 The

adequacy of this choice has been underlined by the empirical studies of

Levi and Partrat (1991) and Milidonis and Grace (2008), who test the

goodness of fit of the Poisson distribution with ISO/PCS data and find it

to be superior to the alternative binomial distribution. Hence, the choice

of a general Poisson distribution for the frequency of natural disasters

seems hardly questionable. However, more advanced modeling frame-

works have been based on the time-inhomogeneous Poisson process81 or

the doubly stochastic Poisson process (Cox process).82 Just recently,

e.g., Chang et al. (2010) as well as Wu and Chung (2010) suggested to

employ a mean-reverting intensity process, which we also adopted within

our ex-ante model framework in Section 4.2. Since, to the best of our

knowledge, there is little empirical evidence to support these dynamics

for the Poisson intensity as of yet, in this section we employ our model to

derive intensities implied by cat swap market quotes as a basis for a time

series analysis. In analogy to implied volatilities in option markets we

define the implied intensity as the fixed value λt,t+∆t = λ ∀t ∈ (0, T−∆t]

which, if used in our cat swap pricing framework, generates a theoretical

spread equal to the observed market spread.

80See Cummins and Geman (1994, 1995), Chang et al. (1996), Embrechts andMeister (1997), Geman and Yor (1997), Aase (1999), Louberge et al. (1999), Chris-tensen and Schmidli (2000), Bakshi and Madan (2002), Lee and Yu (2002), Vaugirard(2003a,b, 2004), Cox et al. (2004), Jaimungal and Wang (2006), Lee and Yu (2007),as well as Muermann (2008).

81See Embrechts and Meister (1997), Schmidli (2003), and Biagini et al. (2008)82Christensen and Schmidli (2000), Basu and Dassios (2002), and Dassios and

Jang (2003), e.g., use a Cox process with gamma, lognormal, and shot noise intensity,respectively

5.2

Deriva

tion

ofim

plied

Poisson

inten

sities95

lognormal Burr Pareto

Earthquake Hurricane Earthquake Hurricane Earthquake Hurricane

µ -1.3778 0.8064 a 0.4027 4.8358 a 0.4602 0.6620

σ 2.5835 2.1346 b 1.1018 0.5886 c 0.0503 1.1300

c 0.0426 70.1560

KSn 0.1218 0.0683 KSn 0.0712 0.0784 KSn 0.0703 0.0926

p-value 0.4038 0.7699 p-value 0.9420 0.6101 p-value 0.9472 0.3982

ADn 0.8275 0.6403 ADn 0.3264 0.4958 ADn 0.3451 1.5995


Table 10: Parameter estimates and test statistics for the lognormal, Burr, and Pareto distribution

Weibull gamma exponential

Earthquake Hurricane Earthquake Hurricane Earthquake Hurricane

a 0.3557 0.5252 a 0.2050 0.3909 β 0.1108 0.0824

c 0.9774 6.3610 β 0.0227 0.0322

KSn 0.1673 0.0787 KSn 0.2440 0.0993 KSn 0.6546 0.3084


ADn 1.9962 0.5264 ADn 4.3858 1.4651 ADn 81.275 27.4791


Table 11: Parameter estimates and test statistics for the Weibull, gamma, and exponential distribution


Due to their OTC character, market quotes for catastrophe-linked

instruments are generally scarce and hardly publicly available. Yet, we

obtained ILW quotes from the BMS pricing grid which is published by

the Thomson Reuters Insurance Linked Securities community on a reg-

ular basis. Where possible, the figures have been cross-checked with

expert judgment based on various sources as well as direct market in-

telligence. Interviews with industry practitioners revealed that, from

a pricing perspective, ILWs and cat swaps are currently not differenti-

ated, which confirms the suitability of ILW premiums for our research

purpose. The dataset comprises time series of monthly up-front prices

from August 2005 to September 2010 for U.S. hurricane and earthquake

contracts with one year maturities and event thresholds of USD 20 bn,

USD 25 bn, USD 30 bn, USD 40 bn, as well as USD 50 bn. In case of a

trigger event each contract pays off its full notional. Table 12 contains

some descriptive statistics for the time series. We notice that, for each

attachment level, the prices of hurricane contracts exhibit a higher mean

and standard deviation than those of earthquake contracts. To acquire

protection against hurricane losses in excess of USD 20 bn, e.g., a protec-

tion buyer needed to pay an average up-front premium of 26.80 percent

of the notional between 08/2005 and 09/2010. The USD 20 bn earth-

quake contract, in contrast, was available for an average price of 14.33

percent. Furthermore, while the minimum premiums for corresponding

hurricane and earthquake contracts differ only slightly, the maximums

for the former are considerably larger. Finally, both hurricane and earth-

quake contracts are on average more expensive for lower thresholds.

In the following, we employ the the ex-ante pricing framework from

Section 4.2 to back out implied intensity time series from the market

quotes.83 Recall that the model has been formulated under the risk-

neutral measure Q. Thus, for it to be applicable to real data, we need

to determine a change of measure as discussed in Section 4.1. Due to

extant empirical support for the zero beta characteristics of cat instru-

83Note that a similar proceeding is applied by Hardle and Cabrera (2010), who backout the implied intensity for a Mexican cat bond and compare it to figures derivedfrom the reinsurance market as well as historical data.

5.2 Derivation of implied Poisson intensities 97

Hurricane contracts

attachment 20 bn 25 bn 30 bn 40 bn 50 bn

mean 0.2680 0.2203 0.1870 0.1437 0.1170

s.d. 0.0509 0.0467 0.0452 0.0386 0.0324

max. 0.3750 0.3167 0.2667 0.2083 0.1750

min. 0.1200 0.0800 0.0575 0.0450 0.0350

Earthquake contracts

attachment 20 bn 25 bn 30 bn 40 bn 50 bn

mean 0.1433 0.1159 0.0924 0.0786 0.0676

s.d. 0.0239 0.0231 0.0197 0.0162 0.0146

max. 0.2000 0.1750 0.1500 0.1200 0.1000

min. 0.1000 0.0800 0.0600 0.0500 0.0425

Table 12: Descriptive statistics: time series of cat swap prices

ments, we assume that catastrophe risk is unsystematic. As a result, the

model parameters under Q remain the same as under the physical mea-

sure P. Since a closed-form solution for the cat swap price is unavailable,

our calculations are based on Monte Carlo simulations. For this pur-

pose, we discretize the model and evaluate the underlying loss process of

Equation (7) for a total of 10,000 sample paths, each one consisting of

252 trading days per year (i.e., ∆t = 1/252). The intuition is that, while

a catastrophe can occur on any day, the official declaration of a trigger

event, i.e., a final loss estimate in excess the of event threshold, and

the resulting transfer of cash flows will only take place on trading days.

Furthermore, the risk free interest rate is the monthly yield on 1-year

U.S. T-Bills84 and, as suggested by our results in the previous section,

we employ a Burr distribution with the parametrizations from Table 10

for the respective loss severities Yi of hurricanes and earthquakes. The

proceeding to capture the implied intensities works as follows: instead

of the stochastic process of Equation (6), we assume a deterministic an-

nual λ which corresponds to an intensity of λ∆t per trading day. Then

84The rates can be accessed on www.ustreas.gov.


we embed the valuation framework into a one-dimensional optimization

algorithm which searches for λ by recalculating the model price until it

matches the monthly market quote for each contract. In doing so, we

extract ten time series of annualized implied intensities, i.e., one for each

hurricane and earthquake contract. The results are displayed in Figure

7. Table 13 contains some descriptive statistics.

From a theoretical perspective, the market implied intensities for each

type of peril at any point in time should not vary across event thresholds

(attachment levels), since all contracts are driven by the same catastro-

phes. Yet, there seem to be slight differences in the cross sections: we

observe a tendency for the means and standard deviations of hurricane

implied intensities to increase and those of earthquake implied intensi-

ties to decrease with the attachment level (see Table 13). Unreported

results of Welch’s t-test indicate that the differences in all pairs of means

of the implied intensities for hurricane contracts are insignificant.85 For

earthquake contracts, in contrast, the means of the implied intensity

time series seem to differ significantly, suggesting that the model might

be somewhat less suitable to capture the characteristics of cat swaps on

earthquake risk. From the sample of Pielke Jr. et al. (2008) we derive

a historical number of 1.95 U.S. hurricanes per year between 1900 and

2005. Similarly, records of the Insurance Information Institute show that

an average of 1.80 hurricanes per year made landfall between 1990 and

2009.86 These figures lie well within the overall range of implied inten-

sities for each hurricane cat swap contract in Table 13. In contrast to

that, historical experience from the Vranes and Pielke Jr. (2009) dataset

indicates that, between 1900 and 2005, 0.76 major earthquakes occurred

in the U.S. per year. This figure is at the lower bound of the ranges of

implied intensities we derived from the market prices of the earthquake

contracts, thus again alluding to some refinement potential.

85Welch’s test allows to compare the means of two samples without assuming thattheir variances are equal. The null hypothesis is equality of the means.

86Refer to www.iii.org.

5.3 The stochastic process of implied Poisson intensities 99

time

2006

20072008

20092010

event threshold

20

25

30

3540

4550

implied

inten

sity 01234

5

(a) Hurricane contracts

time

2006

20072008

20092010

event threshold

20

25

30

3540

4550

implied

inten

sity 01234

5

(b) Earthquake contracts

Figure 7: Monthly implied intensities (08/2005–09/2010)

100

IICatastropheSwaps

Hurricane contracts Earthquake contracts

attachment 20 bn 25 bn 30 bn 40 bn 50 bn 20 bn 25 bn 30 bn 40 bn 50 bn

mean 2.1221 2.0840 2.0951 2.1742 2.3011 1.5346 1.4059 1.3077 1.1472 1.0675

s.d. 0.4718 0.5009 0.5658 0.6355 0.7113 0.3627 0.3126 0.3084 0.2472 0.2370

max. 3.2618 3.2778 3.2561 3.4430 3.8274 2.7309 2.3719 2.1101 1.6771 1.5221

min. 0.8609 0.7055 0.5829 0.5870 0.6346 0.9860 0.8704 0.7806 0.7361 0.6423

JB p-value 0.7438 0.4999 0.3289 0.8600 0.9867 0.0070 0.0528 0.1715 0.0709 0.0859

KSn p-value 0.3339 0.5438 0.4606 0.2829 0.6632 0.2900 0.2316 0.3923 0.0432 0.0635

ADn p-value 0.4789 0.6039 0.4283 0.3647 0.7672 0.3956 0.4169 0.5682 0.1104 0.0790

Table 13: Descriptive statistics and p-values of normality tests for the implied intensity time series

Hurricane contracts Earthquake contracts

attachment 20 bn 25 bn 30 bn 40 bn 50 bn 20 bn 25 bn 30 bn 40 bn 50 bn

20 bn 1.0000 1.0000

25 bn 0.9734 1.0000 0.9525 1.0000

30 bn 0.9546 0.9606 1.0000 0.9126 0.9446 1.0000

40 bn 0.9672 0.9679 0.9667 1.0000 0.8975 0.8996 0.9122 1.0000

50 bn 0.9125 0.9237 0.9370 0.9645 1.0000 0.7639 0.7712 0.8188 0.8789 1.0000

loading 0.9775 0.9802 0.9767 0.9914 0.9578 0.9609 0.9765 0.9645 0.9377 0.8260

Table 14: Correlation matrices and factor loadings for the implied intensity time series


5.3 The stochastic process

of implied Poisson intensities

Having considered the cross-sectional characteristics of the implied in-

tensities, we now want to focus on our actual research goal: the time

series analysis. More specifically, we aim to determine whether a mean-

reverting Ornstein-Uhlenbeck type process is an adequate assumption

for the dynamics of cat swap implied intensities. A first indication is

provided through Figure 7. We observe a clear sign of cyclicality: im-

plied intensities seem to adhere to some sort of wave pattern over time.

Since an Ornstein-Uhlenbeck process is the continuous-time limit of a

discrete-time first order autoregressive process, i.e., an AR(1), we could

now simply apply the Box-Jenkins methodology to every single implied

intensity time series and assess the fit of different models. Instead, how-

ever, we choose a slightly more efficient way to tackle the issue. Table

14 shows the correlation matrices for the implied intensity time series

of hurricane and earthquake cat swaps from 08/2005 through 09/2010.

Evidently, the implied intensities for contracts on the same type of peril

are highly correlated,87 suggesting that there is at least one common

underlying driver which can be revealed by means of exploratory factor

analysis (EFA). EFA is a statistical technique that describes the covari-

ance (correlation) structure of observed random variables in terms of a

smaller number of latent variables called factors. Once we have identi-

fied these factors for the two perils and derived their respective factor

scores, we can focus our analysis on their time series instead of those for

the individual contracts. The following is an analytical representation

of the general EFA model:

X = Λξ + δ (34)

where X is the vector of observed variables (indicators), Λ represents

the matrix of factor loadings, ξ is the vector of latent variables (factors),

and δ stands for the vector of unique factors (residuals). Applying matrix

algebra, one can derive the covariance matrix Σ implied by the model:

Σ = ΛΦΛ′ + Ψδ. (35)

87All correlations are significant on the one percent level.


with Φ being the covariance matrix of the factors and Ψδ being the covari-

ance matrix of the residuals. The parameters (factor loadings and resid-

ual variances) for the EFA model are determined by means of maximum

likelihood estimation (MLE) such that the model implied covariance (cor-

relation) matrix fits its empirically observed counterpart as closely as

possible. Standard EFA assumes multivariate normality of the indicator

variables. In order to check this prerequisite the well-known Jarque-

Bera test as well as the previously introduced Kolmogorov-Smirnov and

Anderson-Darling tests have been applied to the implied intensity time

series (see Table 13 for the respective p-values). Since all test results but

two are insignificant on the five per cent level, i.e., we cannot reject the

null hypothesis that the sample has been drawn from a normal distribu-

tion, we reason that multivariate normality is given.88 The adequacy of

our sample for an EFA is underlined by a Kaiser-Mayer-Olkin (KMO)

Measure of 0.88 for the hurricane and 0.86 for the earthquake implied

intensities. While 0 < KMO < 1, KMO > 0.5 indicates that an EFA can

be performed and if KMO > 0.8, the sample is particularly well suited

for the analysis (see Kaiser, 1974). Furthermore, conducting Bartlett’s

test of sphericity, we reject the null hypothesis of all pairwise correlations

being zero on all reasonable significance levels with a χ2 test statistic of

679.05 for hurricanes and 474.57 for earthquakes.89 Consequently, we

can proceed and apply EFA to the sample.

Initial factor extraction is conducted by means of principal compo-

nents analysis, which provides as many factors as there are indicator vari-

ables, i.e., five for each peril. We find that, for the hurricane contracts,

the first factor explains 96.23 percent of the variance of the implied in-

tensity series. Similarly, for the earthquake contracts, it explains 90.10

percent of the variance. Thus, as previously suspected, a one-factor solu-

tion is an adequate choice with regard to the dimensionality of the model.

The last row of Table 14 contains the factor loadings we obtained for the

88A vector of random variables is multivariate normally distributed if all of itselements follow a univariate normal distribution.

89Note that Bartlett’s test also requires multivariate normality.


one-factor EFA based on the previously generated correlation matrices.90

Apart from one exception all factor loadings are higher than 0.90, un-

derlining a strong influence of the common factor for each peril on the

implied intensity time series for the individual contracts.91 In addition,

factor score estimates ξ have been computed by means of the so-called

regression method, which employs the sample covariance matrix Σ and

the estimated factor loadings matrix Λ as follows:

ξ = Λ′Σ−1X. (36)

The time series of the standardized factor scores for hurricane and

earthquake contracts are plotted in Figure 8(a). As for the individual

implied intensities we observe a clear cyclical pattern. Having the time

series of the factor scores at hand, it is now straightforward to test which

stochastic process provides a satisfactory fit to the data. Before estimat-

ing the parameters, one commonly preselects reasonable model specifi-

cations through the patterns observed in the autocorrelation function

(ACF) and partial autocorrelation function (PACF) of the time series,

which have been plotted in Figures 9 and 10. Both the factor scores for

hurricane and for earthquake contracts exhibit the theoretical character-

istics of an AR(1): an ACF which successively decays towards zero and

a PACF with a significant spike at the first lag while all other lags are

insignificant. Since the spike in the PACF is positive, we can expect a

positive coefficient in the autoregressive process.92 The high value of the

partial autocorrelation at lag one suggests that the process is near inte-

grated or might have a unit root. Hence, we conduct the Dickey-Fuller

unit root test with asymptotic and small sample (MacKinnon) critical

values as well as the Philipps Perron test.93 In contrast to the Dickey-

Fuller test, the Philipps Perron test is robust to serial correlation and

heteroskedasticity in the error terms of the test regression. In addition,

90EFA is commonly performed with standardized variables. Hence, all indicatorvariables are demeaned and divided by their standard deviation before the analysis.

91Note that the factor loadings can be interpreted as correlation coefficients betweenindicator variables and factors.

92In the absence of a negative autoregressive coefficient, the oscillating decline inthe ACF could be a sign for seasonality.

93Since the factor score series are standardized, we do not include an intercept inthe equation of the test regression.


−4

−2

02

4

time

stan

dar

diz

ed fac

tor

scor

e

2006 2007 2008 2009 2010

hurricane contractsearthquake contracts

(a) Time series of factor scores (standardized)

−4

−2

02

4

time

fact

or s

core

fore

cast

2009 2010

actual valueAR(1) forecast

(b) One month forecasts for the hurricane factor

Figure 8: Intensity factor scores and out-of-sample forecast example


the KPSS test for stationarity is considered. The p-values for these tests

with a lag length of one can be found in the left part of Tables 15 and

16. Since all of the unit root tests are statistically significant at least on

the five per cent level (i.e., we reject the null hypothesis of a unit-root)

and the KPSS stationarity tests are insignificant, we conclude that there

does not seem to be a unit root problem.

We now estimate an AR(1) on the hurricane and earthquake factor

score series. Tables 15 and 16 contain the results. For comparison pur-

poses we have also included an AR(2). Generally, if a model is suitable

to capture the pattern inherent in a time series, its residuals should be

white noise. To test the null hypothesis of independent residuals, we

calculate the Ljung-Box (LB) Q-statistic with a lag of 3 and 10.94 Since

these tests turn out to be insignificant (p-values > 0.1000), the residuals

of both the AR(1) and the AR(2) should not be autocorrelated. Due

to their different parameter specifications, the in-sample performance of

these models is compared by means of two common goodness of fit cri-

teria: the Akaike information criterion (AIC) and the Schwarz Bayesian

criterion (SBC) which incorporates a relatively larger penalty term for

the number of parameters. For both criteria the model with the smaller

value is considered superior. Although the AR(1) appears slightly worse

under the AIC, it is associated with a better SBC for the hurricane and

earthquake factor. In combination with the fact that the second coeffi-

cient of the AR(2) is insignificant on the five per cent level, this leads us

to decide in favor of the AR(1).95 Consequently, the Ornstein-Uhlenbeck

process, which is the continuous-time equivalent of the AR(1) seems to

be an adequate choice for the intensity dynamics in a cat swap pricing

model. In addition, it could be considered for out-of-sample forecasts

of implied intensities and, in turn, cat swap spreads (see Figure 8(b)

for an example). Before such an application, however, one should con-

duct further analyses of the short and long term forecasting performance.

94It is common to conduct the Ljung-Box test for a short and a long lag length.95Unreported results for pure moving average (MA), higher order AR, and com-

bined ARMA models indicate their inferiority due to insignificant coefficients, non-white noise residuals, or worse AIC/SBC values.


10 20 30 40

−0.

50.

00.

51.

0

Lag

AC

F

(a) Autocorrelation function (ACF)

10 20 30 40

−0.5

0.0

0.5

1.0

Lag

Part

ial A

CF

(b) Partial autocorrelation function (PACF)

Figure 9: ACF and PACF for the hurricane intensity factor


10 20 30 40

−0.

50.

00.

51.

0

Lag

AC

F

(a) Autocorrelation function (ACF)

10 20 30 40

−0.5

0.0

0.5

1.0

Lag

Part

ial A

CF

(b) Partial autocorrelation function (PACF)

Figure 10: ACF and PACF for the earthquake intensity factor

108

IICatastropheSwaps

p-values for unit root tests AR(1) AR(2)

Dickey-Fuller 0.0075 coefficients 0.9463 1.1249 -0.2008

MacKinnon 0.0031 p-value 0.0000 0.0000 0.0606

Phillips-Perron 0.0000 LB Q(3)/Q(10) p-value 0.1124 0.2835 0.2293 0.4341

KPSS (stationarity test) > 0.1000 AIC/SBC 67.6732 71.9275 67.3087 73.6901

Abbreviations: AR(1): first order autogregressive process; AR(2): second order autoregressive process; LB Q(t): Ljung-Box Q-statisticfor lag t; AIC: Akaike information criterion; SBC: Schwarz Bayesian criterion.

Table 15: Hurricane intensity factor: unit root tests and model estimation results

p-values for unit root tests AR(1) AR(2)

Dickey-Fuller 0.0241 coefficients 0.8751 1.0470 -0.2027

MacKinnon 0.0089 p-value 0.0000 0.0000 0.0548

Phillips-Perron 0.0000 LB Q(3)/Q(10) p-value 0.5643 0.5577 0.5715 0.3637

KPSS (stationarity test) > 0.1000 AIC/SBC 94.4151 98.6694 93.9122 100.2936

Abbreviations: AR(1): first order autogregressive process; AR(2): second order autoregressive process; LB Q(t): Ljung-Box Q-statisticfor lag t; AIC: Akaike information criterion; SBC: Schwarz Bayesian criterion.

Table 16: Earthquake intensity factor: unit root tests and model estimation results


6 Summary and conclusion

In this paper, we contribute to the literature through a comprehensive

analysis of the catastrophe swap, a relatively new financial instrument

which has attracted little scholarly attention to date. We begin with a

brief discussion of the typical contract design, the current state of the

market, as well as major areas of application. Subsequently, a two-stage

contingent claims pricing approach for catastrophe swaps is proposed,

which distinguishes between the main risk drivers ex-ante and during

the loss reestimation phase. The occurrence of catastrophes is modeled

as a doubly stochastic Poisson process with mean-reverting Ornstein-

Uhlenbeck intensity. In addition, we fit various parametric distributions

to normalized historical loss data for hurricanes and earthquakes in the

U.S. and find the heavy-tailed Burr distribution to be the most adequate

representation for loss severities. Applying our ex-ante pricing model to

market quotes for hurricane and earthquake contracts, we then derive

implied intensities which are subsequently condensed into a common

factor for each peril by means of exploratory factor analysis. Further

examining the resulting factor scores, we show that an AR(1) provides a

good fit. Hence, its continuous-time limit, i.e., the Ornstein-Uhlenbeck

process should be well suited to represent the dynamics of the Poisson

intensity in a cat swap pricing model.

Future research could be centered around refinements of the pricing

model such as the inclusion of stochastic interest rates which is reason-

able in case longer term cat swap contracts begin to be traded. More-

over, it would be interesting to use implied intensities from cat swap

transactions in valuation models for other, less standardized and liquid

catastrophe-linked instruments such as cat bonds or even reinsurance.

In doing so, one could ensure consistent pricing across different markets

for catastrophe risk and eliminate potential arbitrage opportunities. Fi-

nally, the AR(1) could be applied to produce implied intensity and cat

swap spread forecasts, the accuracy of which would need to be assessed

relative to natural competitors such as the random walk or an AR(1)

with seasonality.


7 Appendix: The market price of cat risk

In the case of derivatives on non-traded underlyings market completeness

can be preserved if other liquidly tradable instruments exist, which are

driven by the same source of risk. Hence, replicating portfolios can be

formed and a unique risk-neutral measure is obtainable by recovering a

market price of risk from observed quotes. Below, we briefly recapitulate

the reasoning of Hull (2008) for the simple case where the underlying

process is a Geometric Brownian Motion. Consider two derivatives G1(ξ)

and G2(ξ) on the variable ξ. Suppose the dynamics of ξ, which by itself

is not a traded asset, are adequately described through the following

diffusion process, with drift a and volatility b:

dξt = aξtdt+ bξtdWt. (37)

where dWt is a standard Wiener process. Further, assume that G1(ξ)

and G2(ξ) adhere to the processes:

dG1(ξt) = µ1G1(ξt)dt+ σ1G1(ξt)dWt, (38)

dG2(ξt) = µ2G2(ξt)dt+ σ2G2(ξt)dWt, (39)

such that the Wiener process dWt, which originates from the dynamics of

the underlying, is the only source of uncertainty affecting the derivative

prices. By buying σ2G2(ξt) of the first derivative and selling σ1G1(ξt) of

the second derivative, an investor could now form a portfolio Ω, which

would be instantaneously risk-free (i.e., dWt is eliminated):

Ω(ξt) = σ2G2(ξt)G1(ξt) − σ1G1(ξt)G2(ξt). (40)

Using (2) and (3), the marginal change in value of this portfolio can

be expressed as follows:

dΩ = σ2G2(ξt)dG1 − σ1G1(ξt)dG2

= [σ2G2(ξt)µ1G1(ξt) − σ1G1(ξt)µ2G2(ξt)] dt.(41)

7 Appendix: The market price of cat risk 111

Consequently, this portfolio must yield the risk-free interest rate over

the next marginal time period:

dΩ = rΩdt. (42)

Inserting (4) and (5) and eliminating G(ξ)1G(ξ)2dt, this is equivalent

to σ2µ1 − σ1µ2 = rσ2 − rσ1, or

µ1 − r

σ1=µ2 − r

σ2= λξ. (43)

λξ is the market price of risk for the underlying ξ. If the no-arbitrage

principle holds, λξ at any given point in time has to be the same for any

derivative dependent on ξ, regardless of its specification. The market

price of risk gauges the risk-return tradeoff for financial instruments

based on ξ. Multiplied by the volatility (i.e., the quantity of risk) of the

asset under consideration, it represents the risk premium over and above

the risk-free rate, which investors require to hold this asset: µ− r = λξσ.


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119

Part III

Solvency Measurement of

Swiss Occupational Pension

Funds

Abstract

In this paper, we combine a stochastic pension fund model with a traffic

light approach to solvency measurement of occupational pension funds in

Switzerland. Assuming normally distributed asset returns, a closed-form

solution can be derived. Despite its simplicity, we believe the model

comprises the essential risk sources needed in supervisory practice. Due

to its ease of calibration, it is well suited for a regulatory application

in the fragmented Swiss market, keeping costs of solvency testing at a

minimum. We calibrate and implement the model for a small sample

of ten Swiss pension funds in order to illustrate its application and the

derivation of traffic light signals. In addition, a sensitivity analysis is

conducted to identify important drivers of the shortfall probabilities for

the traffic light conditions. Although our analysis concentrates solely

on Switzerland, the approach could also be applied to similar pension

systems.96

96Alexander Braun, Przemys law Rymaszewski, and Hato Schmeiser (2010), A Traf-fic Light Approach to Solvency Measurement of Swiss Occupational Pension Funds,Working Papers on Risk Management and Insurance, No. 74. This paper has beenpublished in: Geneva Papers on Risk and Insurance - Issues and Practice, 2011,36(2):254-282.

120 III Pension Fund Solvency

1 Introduction

The recent crisis in the global financial markets hit not only banks and

insurers but also the pension fund industry. The resulting underfunding

of a large number of pension schemes triggered a discussion about the

rearrangement of prudential regulation and supervision for occupational

pension funds in Switzerland. The obligatory character of occupational

pension plans for the majority of Swiss employees, the large volume in-

vested through them (according to the Swiss Federal Statistical Office,

in 2008 the aggregated book value of assets was approximately equal

to the Swiss GDP), as well as significant social costs linked to poten-

tial insolvencies demonstrate that this debate is not exclusively political.

Instead, a solvency test for pension funds is of considerable relevance

to employees, employers, and pensioners. Supervision and regulation of

pensions in Switzerland is currently conducted at the cantonal level (see,

e.g., Gugler, 2005). The main task of these regulators is to ensure that

the pension funds comply with the legal requirements. Besides, they re-

ceive the annual reports and the report of an independent occupational

pension expert, whose duty is the valuation of a fund’s technical liabili-

ties. The expert also examines whether or not a fund is able to cover its

liabilities. Comprehensive solvency regulation, however, is not present

for occupational pension funds in Switzerland, although banks and in-

surance companies have to adhere to Basel II and the Swiss Solvency

Test (SST), respectively (see, e.g., Eling et al., 2008).

This paper is an attempt to address this issue. We suggest an effi-

cient solvency test for occupational pension funds, providing condensed

information for the stakeholder groups instead of prescribing regulatory

capital. For this purpose, we adopt a model for pension funds under

stochastic rates of return and combine it with a traffic light approach,

allowing an efficient comparison of the risks inherent in different funds

as well as a comprehensible communication of results of the solvency

test. This signal based approach can be used not only to support the

supervisory process, but also to facilitate an increased level of market

discipline. However, more transparency within the pension fund market

1 Introduction 121

can intensify the latter only if insureds are both capable of interpreting

the received signal and of taking actions as a consequence of the infor-

mation they receive.

The literature with regard to stochastic pension fund modeling has

been strongly influenced by the work of O’Brien (1986, 1987) and

Dufresne (1988, 1989, 1990). While the former proposes a continuous-

time approach, the early model of Dufresne operates in a discrete-time

environment. This original discrete-time model has subsequently been

applied and extended in several papers. Haberman (1992) introduces

time delays with regard to additional contributions for unfunded liabili-

ties and, in a consecutive paper, Haberman (1993a) examines the effects

of changes in the valuation frequency for the pension fund’s assets and

liabilities. Furthermore, Zimbidis and Haberman (1993) use the model

with time delays to derive expectations and variances for fund and con-

tribution level distributions. In two additional publications, Haberman

(1993b, 1994) drops the assumption of independent and identically dis-

tributed (iid) asset returns in favor of a first-order autoregressive process

and utilizes the model to compare different pension funding methods. In

contrast to the discrete-time focus of the majority of papers, Haberman

and Sung (1994) present and employ a continuous-time model to simulta-

neously minimize an objective function for contribution rate and solvency

risk. Haberman (1997) reverts to a discrete-time version with iid asset

returns and analyses funding approaches to control contribution rate risk

of defined benefit pension funds. Cairns (1995) extends previous work

by turning to the fund’s asset allocation strategy as a means of control-

ling funding level variability. In a later paper, Cairns (1996) presents

a pension fund model in continuous-time with continuous adjustments

to the asset allocation and contribution rate. A similar model but with

stochastic benefit outgo is discussed in Cairns (2000), while Cairns and

Parker (1997) apply a discrete-time approach and compare the effect of

a change from iid to autoregressive returns on the variability of funding

level and contribution rates. Finally, Bedard and Dufresne (2001) show

that the dependence of successive rates of return can have a considerable

effect on the model results in a multi-period setting.


The model we present is based on the discrete-time framework which

has been frequently employed in the literature in order to analyze issues,

such as contribution rate risk or behavior of the funding level over time.

However, it has not been previously considered in the context of solvency

measurement. We adapt the model as to capture the particularities as-

sociated with the occupational pension fund system in Switzerland and

demonstrate that its simplicity and ease of calibration are advantages

for an application as a regulatory standard model in this fragmented

market. The model enables us to estimate shortfall probabilities which

are then funneled into a traffic light approach in order to send a signal to

stakeholder groups, which carries condensed information about a fund’s

financial strength and is straightforward to interpret, even for less sophis-

ticated claim holders. Although the scope of our analysis is limited to

Switzerland, both the model itself and the insights from its application

can be transferred to similar pension systems.

The remainder of this paper is organized as follows. Section 2 sets

the stage with a brief introduction to the particularities of Switzerland’s

occupational pension fund system. The stochastic pension fund model

which forms the basis for the proposed solvency test is presented in

Section 3, while Section 4 explains the traffic light approach to solvency

measurement. Section 5 comprises an exemplary calibration of the model

and illustrates its application by computing shortfall probabilities and

deriving the traffic lights for a small sample of ten Swiss pension funds. A

sensitivity analysis is then conducted in Section 6 in order to identify im-

portant drivers of the shortfall probabilities for the traffic light conditions.

Section 7 focuses on the supervisory review process. Some additional

considerations concerning a potential implementation in Switzerland are

provided in Section 8. Finally, in Section 9, we conclude.

2 The particularities of the Swiss pension system 123

2 The particularities

of the Swiss pension system

The Swiss pension system comprises three pillars. The first pillar is

earnings-related and embedded in the public social security scheme; the

second pillar relates to the mandatory occupational pension fund sys-

tem;97 the third pillar consists of additional benefits that need to be

accumulated individually (see, e.g., see, e.g., Brombacher Steiner, 1999;

OECD, 2009). Our paper focuses on the second pillar which is governed

by the Swiss occupational pension law (abbreviated in German: BVG

and BVV2). At the heart of the second pillar, which, apart from retire-

ment pensions, also provides widow(er) and invalidity pensions, are the

occupational pension funds (in German: Vorsorgeeinrichtungen).

The vast majority of occupational pension funds in Switzerland takes

the legal form of private trusts, where the employees have a right of par-

ity participation in the administrative council (Art. 55 BVG).98 Apart

from single-employer pension funds, which are run exclusively for the em-

ployees of one company, the specific structure of the Swiss economy with

many small and medium-sized businesses necessitates so-called multi-

employer pension funds (in German: Sammeleinrichtungen; see Swiss

Federal Statistical Office, 2009). This relieves small businesses from the

burden of setting up their own pension fund, because they can join a

multi-employer fund which bundles the occupational pension schemes

of several independent firms.99 A change of pension fund can only be

completed by the employer with the agreement of the majority of em-

ployees. The second pillar is covered by a guarantee fund (in German:

Sicherheitsfonds BVG), with the main purpose of subsidizing schemes

with an adverse age structure and guaranteeing the obligatory payments

of defaulted funds.

97Participation in the occupational pension system is mandatory for all employeesof age 18 or older who earn a minimum annual salary of 20’520 CHF (Art. 7 BVG).

98Pension funds of the federation, cantons, and municipalities are institutions underpublic law.

99Employers are obliged to either establish a firm-specific pension fund or to joinmulti-employer fund with the consent of their employees (Art. 11 BVG).


Compulsory pension contributions are based on the so-called coordi-

nated salary100 (in German: koordinierter Lohn) of the employee and the

employer has to bear at least half of each installment (Art. 8 and Art. 66

BVG).101 These regular payments are credited to a pension account (in

German: Altersguthaben) and at least compounded with an obligatory

minimum rate of return (currently 2 percent). Once the insured reaches

the retirement age of 65 for men or 64 for women (Art. 13 BVG), the

obligatory pension annuity is calculated by multiplying the annuity con-

version rate, which is currently 6.8 percent, with the final balance of

the pension account (Art. 14 BVG).102 The Swiss Federal Council (in

German: Der Schweizerische Bundesrat) determines both the minimum

interest rate and the conversion rate at two- and ten-year intervals, re-

spectively.103 In general, Swiss occupational pension funds can be set

up either as defined contribution or as defined benefit plans.

One important aspect of the occupational pension fund system in

Switzerland is that funds are legally allowed to temporarily operate with

a deficit of assets relative to liabilities (Art. 65c BVG). Such an under-

funding of liabilities is indicated by the coverage ratio, i.e., the proportion

of the market value of assets over technical liabilities, falling below 100

percent (Art. 44 BVV2). However, the tolerance of a temporary under-

funding is strictly linked to the condition that a pension fund continues

its ongoing obligatory pension payments and takes action to restore full

coverage within an adequate time horizon. In addition, the pension fund

has to promptly inform the regulator, the employer, the employees, and

the pensioners about the magnitude and causes of the asset shortage as

well as countermeasures that have been initiated. The pension fund has

to eliminate the deficit itself as the guarantee fund can merely intervene

100Currently the coordinated salary is the part of an employee’s annual incomebetween 23’940 and 82’080 CHF.101Voluntary payments in excess of the compulsory contributions are possible.102The pension funds can provide annuities over and above the obligatory level.103When determining the minimum interest rate, the Swiss Federal Council takes

into account the recent development of the returns of marketable investments, witha particular focus on government bonds, corporate bonds, equities, and real estate(Art. 15 BVG). Mortality improvements are accounted for through an adjustment ofthe conversion rate.


in case of insolvency (Art. 65d BVG). For this purpose, the fund can

raise additional contributions from the employer and the employees to

rectify the deficit. If and only if all other actions prove insufficient, the

fund is allowed to go below the obligatory minimum interest rate by up

to 0.5 percent for no longer than 5 years.

3 The model framework

We suggest building a solvency framework for occupational pension funds

around underfunding probabilities, at the center of which we need a

stochastic pension fund model. While advanced internal models could

be allowed for the supervision of pension funds with sophisticated risk

management know-how and processes, the requirements of a regulatory

standard model suggest an approach that concentrates on the most es-

sential risk drivers. The complexity of such a standard model should be

kept within adequate limits so that the introduction of the solvency reg-

ulation does not cause an unjustifiably large increase in personnel and

infrastructure cost, especially for smaller occupational pension funds.

Apart from that, a properly developed simple model is capable of cap-

turing the main determinants of pension fund activity (see Cairns and

Parker, 1997). Moreover, the feasibility of the whole concept depends on

sufficient data being available for calibration. This is more likely to be

the case for an approach which entirely relies on observable variables such

as accounting figures. With these considerations in mind, we decide in

favor of a discrete-time model that ensures universal applicability, cost-

efficient implementation, and straightforward calibration.104 The model

we present is based on the work of Cairns and Parker (1997). In the fol-

lowing, we adapt it to the specific characteristics of occupational pension

funds in Switzerland and combine it with a traffic light approach for the

assessment of shortfall probabilities in order to construct a pragmatic

solvency test.

104Equivalent formulations in continuous time can be found in the literature (see,e.g., Cairns, 1996).


Consider a one-period evaluation horizon and continuous compound-

ing. If the occupational pension fund is assumed to have a stationary

membership and all cash flows are exchanged at the beginning of the

period, the asset process of the pension fund can be described as follows:

A1 = exp(r1) (A0 + C0 −B0) , (44)

where

- A1: stochastic market value of the assets in t = 1,

- r1: stochastic return on the assets between t = 0 and t = 1,

- A0: assets in t = 0,

- C0: contributions for the period between t = 0 and t = 1,

- B0: benefit payments for the period between t = 0 and t = 1.

The aggregated asset return consists of normally distributed returns

for each asset class in the fund’s portfolio:

r1 =

n∑

i=1

wiri, (45)

with ri ∼ N (µi, σi) , ∀i ∈ 1, . . . , n, where

- wi: portfolio weight for asset class i,

- ri: return of asset class i between t = 0 and t = 1,

- n: number of asset classes in the portfolio.

Note that for some asset classes, the assumption of normally dis-

tributed returns is merely an approximation (see, e.g., Officer, 1972).

However, it will enable us to derive a closed-form solution, which we

consider a very valuable aspect of a standard solvency model.

Since occupational pension funds commonly have a large pool of em-

ployees and pensioners, their liabilities are fairly well diversified and

consequently relatively stable. Hence, the crucial source of risk is consti-

tuted by a pension fund’s asset allocation and a deterministic approach

for the liabilities is justifiable. In general, the value of the life insurance


liabilities in t = 0 is calculated as the present value of expected future

payments to those insured less the present value of expected premium

inflows. These cash outflows are estimated actuarially, taking into ac-

count the age structure and mortality profile of the fund as well as the

targeted rate of return, which needs to be equal to or greater than the

obligatory minimum. Although an actuarial technical interest rate is

commonly used in this context, it is more adequate to apply the current

interest rate term structure. Therefore, we incorporate the market value

of the liabilities into our model and define the corresponding yield as

the valuation rate of interest iv. Issues resulting from a potential mises-

timation of the pension liabilities will be addressed in Section 6. If the

liabilities are assumed to be continuously compounded at iv, we have

the following relationship:

L1 = exp(iv) (L0 +RC0 −B0) , (46)

where

- L1: market value of the liabilities in t = 1,

- iv: interest rate for the valuation of the liabilities,

- L0: market value of the liabilities in t = 0,

- RC0: regular contributions for the period between t = 0 and t = 1.

The assumptions of normally distributed asset returns and determin-

istic liabilities could be relaxed by resorting to numerical solutions, e.g.,

via a Monte-Carlo simulation framework. In that case, many different

distributional assumptions and dependency structures could be incorpo-

rated. Similarly, a numerical solution would allow the introduction of

a longer time horizon and intermediate time steps or a continuous-time

framework.105

The contributions between t = 0 and t = 1, C0, consist of two distinct

elements:

C0 = RC0 +AC0, (47)

105See Buhlmann (1996) for the calculation of ruin probabilities in a similar context,applying a multi-dimensional geometric Brownian motion for the asset dynamics.


with

AC0 = αmax [L0 −A0, 0] , 106 (48)

where

- AC0: additional contributions between t = 0 and t = 1 for the

recovery of a deficit in t = 0,

- α: fraction of the deficit in t = 0, which will be covered between

t = 0 and t = 1.

At the beginning of each period due additional contributions are

determined based on the current deficit of assets relative to liabilities.

Hence, AC0 also accounts for additional contributions remaining from

prior deficits. Consider, e.g., a deficit in t = −1. The resulting addi-

tional contribution AC−1 will increase the value of the assets in t = 0,

A0, which then forms the basis for the calculation of AC0. Therefore,

if AC−1 together with the development of the assets and liabilities be-

tween t = −1 and t = 0 was sufficient to eliminate the deficit, there will

be no need for further additional contributions and AC0 will be zero.

Additional contributions are subject to two restrictions. First of all,

α ≥ αmin =1

θ, (49)

which implies

ACmin0 = αmin max [L0 −A0, 0] , (50)

where

- θ: maximum number of years for the elimination of the deficit (set

by the regulator),

- αmin: minimum fraction of the deficit in t = 0, which needs to be

covered between t = 0 and t = 1.

The restriction in Inequality (49) implies that deficits have to be

eliminated within an adequate time horizon (see Section 2), which will

106Note that a negative value of L0 −A0 implies a positive fluctuation reserve or apositive amount of uncommitted funds.


be set by the regulator through the choice of θ.107 As a consequence,

additional contributions in the period under consideration must not fall

below a certain minimum, ACmin0 , as defined in (50), since otherwise

the elimination of the deficit would take too long. Intuitively, the fewer

years available for the fund to restore its coverage ratio at least to unity,

the higher the scheduled additional contributions for each year have to

be. Furthermore,

A0 ≥ Amin0 = βL0, (51)

which implies

ACmax0 = max [L0 − βL0, 0] = max [(1 − β)L0, 0] , (52)

with

- β: lowest acceptable coverage ratio Amin0 /L0 (set by the regulator).

Excessive additional contributions are disputable, since they transfer

the investment risk from the pensioners to the employees and employers.

Accordingly, Inequality (51) accounts for the fact that deficits can only be

healed by means of additional contributions up to a certain amount. For

instance, consider a case in which the value of assets falls to zero. Clearly,

a restructuring of the pension fund is not feasible in this case. Hence, in

order to protect those insured from having to pay an unacceptably large

amount of additional contributions into a pension fund in major distress,

we define a lower limit for the assets Amin0 (a fixed percentage of the pen-

sion fund’s liabilities), which puts a cap on additional contributions per

period. This amount, termed ACmax0 , is defined in Equation (52) and

based on β, i.e., the lowest coverage ratio acceptable by the regulator.

Usually, 0 ≤ β ≤ 1, and the lower β, the higher the maximum amount of

additional contributions that can be charged by the pension fund in any

given period.108 If the assets fall below the threshold Amin0 , the fund will

ceteris paribus be unable to rectify the deficit within a single period. In

107In Switzerland this time period is not legally defined. In current practice, however,a five-year span seems to have emerged as convention.108Note that theoretically β could also exceed one. In such a case, additional con-

tributions would be ruled out by our model framework. To see this, refer to Equa-tion (52).


addition, if ACmin0 exceeds ACmax

0 , which is theoretically possible, par-

ticularly for high values of α (low values of θ) and β, the pension fund

faces an existential funding problem, since it would be required to collect

a larger amount of additional contributions than it is actually allowed to.

Under the above assumptions, the assets at the end of the evaluation

period are log-normally distributed with:

E[

A1

]

= E [exp (r1) (A0 + C0 −B0)]

= exp

(

E [r1] +var [r1]

2

)

(A0 + C0 −B0) ,(53)

and

var[

A1

]

= var [exp (r1) (A0 + C0 −B0)]

= (A0 + C0 −B0)2exp

(2E [r1] + var [r1]

)

·(exp (var [r1]) − 1

). (54)

Hence, in order to calculate the first two central moments, which

entirely determine the asset distribution in t = 1 under the assumption of

normally distributed returns, estimates for E [r1] and var [r1] are required.

Using Equation (45), mean and variance for the returns of the aggregated

asset portfolio can be calculated in the following manner:

E [r1] = E

[n∑

i=1

wiri

]

=

n∑

i=1

wiE [ri] (55)

and

var [r1] = σ2r1 = var

[n∑

i=1

wiri

]

=

n∑

i=1

n∑

j=1

wiwjρri,rjσriσrj , (56)

where ρri,rj denotes the correlation coefficient between the returns of

asset class i and j.

4 The traffic light approach 131

4 The traffic light approach

There are several ways to implement a solvency framework. The regu-

lator could, for example, prescribe that each pension fund needs to set

aside regulatory capital based on the outcome of a solvency test. Such

an approach is common in the banking and insurance industries. In the

case of occupational pension funds, however, which do not posses equity

capital, this is rather problematic as the funds would need to build up

reserves from contributions. On the other hand, pension funds have the

possibility to demand additional contributions from employers and em-

ployees, which is similar to authorized equity capital of corporations that

can be drawn in predefined cases. The risk of not being able to raise

this capital when needed is negligible, since it resembles a tax levied by

the government. Thus, we believe that the prescription of regulatory

capital is not the most suitable approach for pension funds. Instead, our

proposal is oriented towards early alert. For solvency measurement pur-

poses, we combine the previously introduced pension fund model with a

concept akin to a value-at-risk framework and funnel the results into a

so-called traffic light approach.

As discussed in the previous section, the model delivers a determinis-

tic value for the liabilities at the end of the analyzed period. Using this

value as a threshold in conjunction with the asset distribution, we can

derive shortfall probabilities for the pension fund under consideration.

These probabilities could be compared to reference probabilities ψ, e.g.,

default rates from rating agency data, in order to generate a signal for

the regulator and the insured. Various categorizations for such a signal

are conceivable. As a straightforward solution, we suggest the following:

- green:

Pr(

A1 ≤ L1

)

≤ ψ, (57)

- yellow:

Pr(

A1 +ACmax1 ≤ L1

)

≤ ψ, (58)


- red:

Pr(

A1 +ACmax1 ≤ L1

)

> ψ, (59)

where ACmax1 denotes the maximum amount of additional contributions

which can be charged by the pension fund in t = 1. ACmax1 is deter-

ministic, since it is based on the value of the liabilities in t = 1.109 If

the probability of underfunded liabilities in t = 1 is smaller than the

preset reference probability ψ, the pension fund is assigned a green light.

In addition, if the assets and the maximum additional contributions in

t = 1 are only insufficient to cover the liabilities with a probability lower

than ψ, the light is yellow. In this case the fund is able to suppress the

probability of underfunded liabilities in t = 1 below the reference prob-

ability through its option of charging additional contributions. Finally,

the red light comes up if the probability that the assets plus ACmax1 fall

short of the liabilities exceeds ψ.

5 Implementation and calibration

5.1 Input data

A major advantage of the model is its low implementation cost due to

the use of readily available data. In this section, we illustrate that even

for smaller occupational pension funds with less sophisticated risk man-

agement techniques in place, it should be straightforward to calibrate

and implement the model. For the purpose of calibration, we rely on

accounting figures from the funds’ annual reports. In practice, pension

funds and regulators would be able to use superior data from their man-

agement accounting and financial planning units or databases. As such

internal data is not available to us, we deem annual reports to be the

most adequate and reliable source. Note that this approach is subject to

certain limitations. As defined in Section 3, a solvency test for pension

funds should theoretically be based on market values of assets and lia-

bilities. This is in line with the latest developments in risk management

109Alternatively, different reference probabilities could be chosen for all three condi-tions.

5.1 Input data 133

practice as well as supervisory frameworks for the insurance sector (Sol-

vency II and the SST). Yet, figures derived from annual reports are, in

general, not consistent with market values. In particular, reported pen-

sion liabilities are commonly valued using a technical interest rate, i.e.,

an actuarial rate instead of the prevailing term structure. Consequently,

we substitute iv in Equation (46) with the technical interest rate itec ap-

plied by each fund. The appropriate level of the technical interest rate

is currently controversially discussed in Switzerland. More specifically,

some pension funds seem to be reluctant to reduce it as to reflect the

low interest rate environment which resulted from the financial crisis

2007/2008, implying an even greater discrepancy between market and

book values of the liabilities. Nonetheless, we believe that the following

illustration of the proposed solvency framework offers useful insights.

Tables 17 to 20 show the parameter values we collected for ten occu-

pational pension funds in Switzerland.110 It is important to note that

coverage ratios, assets, liabilities, technical interest rates, and portfolio

weights for 2007 and 2008 have been extracted from annual reports of

the same year. In contrast to that, 2008 and 2009 figures have been used

for contributions and benefits of 2007 and 2008, respectively, assuming

that the funds can perfectly forecast these magnitudes at the beginning

of the period.111 Since the market values of the funds’ assets could not

be directly obtained from their annual reports, they have been estimated

by multiplying the reported coverage ratios (A0/L0) with the book val-

ues of the liabilities. Furthermore, we decided to conduct the analysis

based on seven broad asset classes. Tables 19 and 20 contain the port-

folio weights each fund assigns to the these asset classes.112 Note that

the asset allocation of some pension funds is fairly concentrated. The

implications of this issue together with the effect of insufficient diversi-

110The funds were made anonymous.111This proceeding has been chosen since our model treats contributions and benefits

as deterministic (see Section 3). If the funds are unable to produce reliable forecasts,however, the model could be revised by incorporating benefits and contributions asstochastic variables.112If deemed necessary, the solvency test could be based on a more detailed catego-

rization of the asset side.


fication within the subportfolio for each asset class will be addressed in

Section 6.2. While the market environment in 2007 was still relatively

stable, the 2008 figures reflect the major turbulences caused by the global

financial crisis. Thus, this dataset enables us to apply the solvency test

in two different economic settings. In addition, we included single as

well as multi-employer funds to further increase the informative value of

our calculations.

It could be discussed whether the parameters for the asset class return

distributions should be preset by the regulator, thereby reducing discre-

tionary competencies to a minimum. However, taking into account the

ease of estimation and regulatory verification of these parameter values,

we suggest they should be determined by the pension funds themselves.

Therefore, means, volatilities, and pair-wise correlations for the return

distributions of the seven asset classes have been estimated from capital

market time series data. To this end, we have chosen broad indices as

representatives for each asset class.113 The S&P U.S. Treasury Bond

Index and the SBI Swiss Government Bond Index have been selected as

proxies for the international and Swiss government bond markets, respec-

tively. International equities are represented by the MSCI World, while

the Swiss Market Index (SMI) is employed for the Swiss equity market.

Real estate returns are provided through the Rued Blass Swiss REIT

Index and the HFRI Fund Weighted Composite Index serves as a broad

measure for the alternative investments universe. Finally, the Swiss 3M

Money Market Index is used as an indicator for the development of cash

holdings. Distribution moments as well as a correlation matrix based

on monthly returns for these indices from January 1997 to December

2007 are exhibited in Tables 21 and 22. Based on the the simplifying

assumption that the pension funds can perfectly hedge exchange rate

fluctuations at a negligible cost, we have not converted the time series

of the three U.S. Dollar denominated indices into Swiss Francs. Since

hedging foreign currency investments against exchange rate risk is very

113Wherever available, total return indices have been used to account for couponsand dividends.

5.1

Input

data

135

CHF mn Fund 1 Fund 2 Fund 3 Fund 4 Fund 5 Fund 6 Fund 7 Fund 8 Fund 9 Fund 10

A0/L0 111% 103% 104% 130% 115% 104% 110% 116% 107% 102%

A0 11’591.90 3’136.98 247.65 14’585.39 16’996.47 1’173.10 1’498.20 6’582.35 34’703.20 13’589.43

L0 10’415.01 3’048.57 237.67 11’176.54 14’792.40 1’130.16 1’360.16 5’688.67 32’524.09 13’309.92

C0 1’175.49 260.93 50.42 631.57 750.00 179.18 61.21 1’200.24 1’360.62 891.41

RC0 919.13 249.60 50.42 631.57 750.00 79.45 61.21 1’200.24 1’183.44 891.41

AC0 256.36 12.33 0.00 0.00 0.00 99.72 0.00 0.00 177.18 0.00

B0 787.37 201.88 15.58 837.47 797.20 87.57 90.87 711.13 2’359.56 952.76

itec 3.75% 4.00% 3.00% 4.00% 4.00% 3.00% 3.50% 3.50% 4.00% 4.00%

Table 17: Input parameters for the sample funds in 2007

CHF mn Fund 1 Fund 2 Fund 3 Fund 4 Fund 5 Fund 6 Fund 7 Fund 8 Fund 9 Fund 10

A0/L0 100% 88% 85% 105% 97% 88% 86% 98% 96% 88%

A0 10’934.19 2’866.88 230.28 11’712.17 14’822.70 1’109.74 1’174.94 6’207.89 30’290.15 11’704.35

L0 10’923.27 3’257.81 270.60 11’186.41 15’265.40 1’261.07 1’373.40 6’317.17 31’611.52 13’285.30

C0 951.45 275.08 47.18 615.18 720.80 168.87 53.28 1’000.23 1’357.44 983.90

RC0 937.62 261.17 47.18 615.18 720.80 106.50 48.56 1’000.23 1’357.44 983.90

AC0 13.83 13.91 0.00 0.00 0.00 62.37 4.72 0.00 0.00 0.00

B0 816.38 213.94 13.39 759.69 787.90 95.05 108.06 782.97 2’113.04 882.45

itec 3.75% 4.00% 3.50% 4.00% 4.00% 3.00% 3.50% 3.50% 3.50% 3.50%

Table 18: Input parameters for the sample funds in 2008

136

IIIPensionFund

Solvency

% Fund 1 Fund 2 Fund 3 Fund 4 Fund 5 Fund 6 Fund 7 Fund 8 Fund 9 Fund 10

Bonds (intl.) 9.7 14.4 33.5 13.2 18.3 23.0 8.9 12.3 23.5 15.3

Bonds (CH) 24.1 23.9 8.1 24.1 15.6 32.0 8.9 34.6 46.3 30.5

Stocks (intl.) 15.8 12.7 21.9 30.6 17.4 16.6 34.3 20.1 13.2 9.2

Stocks (CH) 4.4 19.0 6.2 7.8 11.2 13.6 5.5 15.9 7.3 18.3

Real Estate 11.4 15.4 6.5 9.0 23.8 2.5 25.0 12.7 5.4 10.3

Alternatives 9.7 4.0 10.3 12.5 4.3 6.9 7.7 2.2 0.0 8.1

Cash 25.0 10.7 13.4 2.9 9.4 5.4 9.6 2.2 4.3 8.4

Table 19: Asset allocations of the sample funds in 2007


Bonds (intl.) 9.2 15.8 15.0 22.4 5.2 22.7 6.4 13.5 19.4 18.5

Bonds (CH) 33.4 28.0 29.8 31.8 31.7 29.5 10.0 37.3 53.1 37.0

Stocks (intl.) 8.5 11.9 18.4 15.9 11.0 14.4 25.1 13.2 12.6 6.7

Stocks (CH) 2.4 13.4 5.1 3.7 9.2 12.5 5.1 11.7 6.9 13.5

Real Estate 10.8 16.6 7.8 9.1 28.1 7.4 30.5 14.8 6.1 10.5

Alternatives 17.1 2.4 7.7 10.2 4.5 9.2 7.5 4.7 0.0 8.3

Cash 18.6 11.9 16.4 6.9 10.2 4.4 15.4 4.7 2.0 5.6

Table 20: Asset allocations of the sample funds in 2008

5.2 Results 137

common114 in the asset management industry and the trading costs for

foreign exchange (FX) futures and options are relatively small,115 we

believe this to be an acceptable approach for our purpose. The effect of

imperfect FX hedging will be considered in Section 6.6.

5.2 Results

In order to be able to interpret the shortfall probabilities with the traffic

light approach presented in Section 4, we need to determine reference

probabilities. A straightforward approach is to refer to historic default

rate data as commonly collected and maintained by the large rating

agencies. Consequently, we suggest constructing probability intervals

for rating categories based on default rate experience. The regulator

could then set a minimum target rating for occupational pension funds,

which is linked to the threshold probability.

In the following we use global corporate cumulative default rates from

1981 to 2008 provided by Standard & Poor’s (2009) and establish inter-

vals for the one-year default probabilities as shown in Table 23. While

the use of specific default rates for the investment industry in general

or the pension fund market segment in particular would be preferable,

we need to rely on the rather high-level data available to us. Neverthe-

less, in case of an introduction in practice, it would be advisable for the

regulator to cooperate with rating agencies in order to access a more pre-

cise database. As a reasonable minimum target rating for pension funds,

we propose the lowest investment grade rating category: BBB. Partic-

ipation in the occupational pension fund system in Switzerland is not

voluntary.116 In addition, the volume invested through contributions of

employers and employees is significant. Therefore, occupational pension

114As an example, consider an investment in a foreign currency denominated gov-ernment bond. If left completely unhedged, this would be an outright speculation onthe exchange rate, as the returns in the investor’s home currency will be dominatedby exchange rate movements, implying that the asset does not exhibit the typicalcharacteristics of a government bond.115Flat fees for FX futures trades at the Chicago Mercantile Exchange (CME), the

largest regulated FX marketplace worldwide, can be as low as 0.11 USD, dependingon membership and volume. For more information see http://www.cmegroup.com.116Refer to Section 2.

http://www.cmegroup.com

138

IIIPensionFund

Solvency

S&P USTI SIX SBI MSCI WO SIX SMI RBREI HFRI SMMI

Currency USD CHF USD CHF CHF USD CHF

Mean return (%) 5.83 3.63 7.02 8.50 4.98 10.32 1.60

Volatility (%) 3.71 3.51 14.04 17.22 7.67 7.19 0.29

Indices: S&P US Treasury Bond Index (S&P USTI); Swiss Government Bond Index (SIX SBI); MSCI World (MSCI WO); Swiss MarketIndex (SIX SMI); Rued Blass Real Estate Index (RBREI); HFRI Fund Weighted Composite Index (HFRI); Swiss 3M Money MarketIndex (SMMI).

Table 21: Annualized means and standard deviations for the seven asset classes

S&P USTI SIX SBI MSCI WO SIX SMI RBREI HFRI SMMI

S&P USTI 1.00

SIX SBI 0.57 1.00

MSCI WO -0.27 -0.22 1.00

SIX SMI -0.30 -0.17 0.74 1.00

RBREI 0.00 0.21 0.27 0.26 1.00

HFRI -0.19 -0.16 0.77 0.50 0.22 1.00

SMMI 0.21 0.12 -0.21 -0.14 -0.07 -0.14 1.00

Indices: S&P US Treasury Bond Index (S&P USTI); Swiss Government Bond Index (SIX SBI); MSCI World (MSCI WO); Swiss MarketIndex (SIX SMI); Rued Blass Real Estate Index (RBREI); HFRI Fund Weighted Composite Index (HFRI); Swiss 3M Money MarketIndex (SMMI).

Table 22: Correlation matrix for the seven asset classes

5.2 Results 139

% AAA AA A BBB BB B

lower bound 0.00 0.03 0.08 0.24 0.99 4.51

upper bound 0.03 0.08 0.24 0.99 4.51 25.67

Table 23: One-year S&P default probabilities for different rating classes

funds bear much responsibility for an individual’s retirement provisions.

Against this background, their financial strength should be demanded to

be investment grade. Otherwise the uncertainty for those insured would

be considerable, while they are not free to entrust their money with other

financial institutions of their choice. Moreover, from the perspective of

regulators and financial market participants, it would be very difficult to

argue why pension funds should be allowed a notably lower credit quality

than other financial institutions such as banks or insurance companies.

Having determined the reference probability ψ to be 0.99 percent

(lower bound of BBB), we can now run the model calculations and inter-

pret the results. For each fund, the probabilities for the traffic light condi-

tions in 2007 and 2008 as well as the associated test outcomes (pass/fail)

are presented in Tables 24 and 25, respectively. The calculations for the

yellow condition have been conducted based on a β of 0.95 and 0.90, i.e.,

we limited the maximum additional contributions per period to 5 and 10

percent of the liabilities.117 First, we observe that four out of ten funds

fail the green condition in 2007, although none has underfunded liabili-

ties at the outset (the lowest coverage ratio among the sample funds in

2007 was 102 percent, see Table 17). When inspecting the coverage ratio

these four funds actually reported in 2008 (see Table 18), we find that

all of them in fact suffer from underfunded liabilities, ranging from 2 to

an alarming 15 percent. A fund with a 15 percent deficit of assets rela-

tive to liabilities is in a serious state, since, even for the lower β of 0.90,

it cannot be restructured through additional contributions in a single

period. Taking into account that the current convention in Switzerland

117Recall the definition of β from Equation (52). As discussed, it is ultimately upto the regulator to set a value for β.


is a maximum of 5 years to eliminate the deficit, the fund needs addi-

tional contributions of at least 3 percent per year. These are already

close to the 5 percent upper limit which we applied in the analyses for

the yellow condition, underscoring the severity of this situation. Hence,

a failure of the green condition is only acceptable in exceptional cases

and should instantly trigger heightened attention from the supervisor as

well as those insured. In this context it should also be emphasized, that

a need for refinements to the traffic light approach is not automatically

constituted by the fact that Fund 3 ends up with a deficit in excess of

ACmax1 in 2008, although it was assigned a yellow light in the previous

period. The proposed solvency test is exclusively based on probabilities.

Therefore, by all means, a fund can pass one or both traffic light con-

ditions and still end up with substantial unfunded liability at the end

of the period. Being assigned a green or yellow light only means that

the probability for the respective event is sufficiently low. However, if

similar discrepancies are detected for in the context of a comprehensive

quantitative impact study prior to the introduction of the solvency test,

its overall configuration and calibration could be reconsidered.

The second point we learn from Table 24 is that a β of 0.95 is more

than enough to compress the probabilities for the yellow condition to very

low levels for almost all funds. Evidently, this effect is even stronger for

β = 0.90. While the probabilities are virtually zero for the financially

sounder pension funds, even those which did not conform to the green

condition seem to be able to comply with the yellow condition without

difficulties. This illustrates an important point, which had already been

mentioned in Section 4: the option to demand additional contributions

implies that a large part of the pension funds’ investment risk is ulti-

mately borne by employees and employers. Consequently, in case of

a practical implementation of this approach, the supervisory authority

should carefully determine the upper limit on additional contributions.

A further insight we gain from Table 17 is that merely comparing the

coverage ratios, as currently done in supervisory practice in Switzerland,

is generally insufficient to capture the risk profile of pension funds. To

5.2

Resu

lts141


green condition 0.01 18.36 7.29 0.00 0.10 0.07 3.28 0.24 0.70 21.42

pass fail fail pass pass pass fail pass pass fail

yellow condition 0.00 3.02 0.47 0.00 0.00 0.00 0.45 0.01 0.00 3.06

β = 0.95 pass fail pass pass pass pass pass pass pass fail


β = 0.90 pass pass pass pass pass pass pass pass pass pass

Table 24: Probabilities and test outcomes for the traffic light conditions in 2007


green condition 29.46 99.44 99.96 4.36 66.93 83.99 99.32 48.31 81.83 99.71

fail fail fail fail fail fail fail fail fail fail


β = 0.95 fail fail fail pass fail fail fail fail fail fail


β = 0.90 pass fail fail pass fail fail fail pass fail fail

Table 25: Probabilities and test outcomes for the traffic light conditions in 2008


see this, compare Fund 3 and Fund 6. Both are characterized by an

equal coverage ratio of 104 percent in 2007 (see Table 19). However,

only Fund 6 passes the green condition of our proposed solvency test

(see Table 24). The reason is simple: a comparison of the coverage ratio

does not take into account differences in asset allocation and the option

to charge additional contributions.

Finally, when examining the results of the solvency test for 2008 in

Table 25, we notice that the financial crisis has strongly influenced the

condition of the pension funds in our sample. Since all funds except

Fund 1 and Fund 4 already exhibit underfundings at the beginning of

the period, their probabilities for both the green and yellow condition

have increased considerably. As a result, not a single pension fund is

able to pass the green condition and only one (Fund 4) passes the yellow

condition based on a β of 0.95. Although the analyzed sample is rather

small, this illustrates that the financial crisis has left the Swiss pension

fund sector in a dramatic situation.

6 Sensitivity analysis

In this section, we explore the main drivers of the shortfall probabilities

for the traffic light approach. These are relevant for the regulator in

various ways, including the political discussion about the state of the

Swiss occupational pension fund sector, the supervisory determination

of model variables, and the decision about measures in case a pension

fund fails the green or yellow condition of the solvency test. We base the

analysis on a standard (representative) pension fund, the input data for

which can be found in Table 26.118 This data is mainly based on 2007

average figures from the Swisscanto (2008) pension fund survey, compris-

ing 265 occupational pension funds in Switzerland, and has been comple-

mented and cross-checked with annual report data from our sample.119

The standard pension fund under consideration is financially sound at

118Unless noted otherwise, β has been set to 0.95.119The Swisscanto series of surveys is published on an annual basis and contains

representative data with regard to the structure, performance, capitalization andportfolio allocation of the participating funds.

6.1 Equity allocation 143

Input parameters Asset Allocation

A0/L0 110% Bonds (intl.) 13%

A0 11’000 Bonds (CH) 27%

L0 10’000 Stocks (intl.) 18%

C0 1’000 Stocks (CH) 10%

RC0 1’000 Real Estate 15%

AC0 - Alternatives 7%

B0 750 Cash 10%

itec 4%

Table 26: Parameters for a representative pension fund

the beginning of the period with a coverage ratio of 110 percent and a

fairly balanced asset allocation.

6.1 Equity allocation

The first sensitivity we examine is related to the proportion of equities

in the pension fund’s portfolio. In this context we proceed as follows:

from the original asset allocation in Table 26, we calculate the weight

of each asset class with regard to the remaining part of the portfolio if

stocks (international and Swiss) are excluded. As an example, consider

the category alternative investments: aside from stocks, the remaining

asset classes together make up 72 percent of the portfolio, 7 percent of

which are alternative investments. Consequently, alternative investments

are assigned a ”residual” weight of 7/72 = 9.7 percent for the analysis.

In the same fashion, we get 18.1 percent for international bonds, 37.5

percent for Swiss bonds, 20.8 percent for real estate, 9.7 percent for al-

ternatives, and 13.9 percent for cash. We then successively calculate

the shortfall probabilities associated with the traffic light conditions for

an increasing portfolio weight of stocks, beginning with zero and ending

with the legal limit of 50 percent. In every case, the percentage is equally

shared between Swiss and international equities. For each allocation, the

remainder of the portfolio is distributed among the other asset classes

according to the previously calculated residual weights.


0.1 0.2 0.3 0.4 0.5

0.00

0.01

0.02

0.03

0.04

portfolio weight allocated to equities

shor

tfal

l pro

bab

ility

green conditionyellow conditionreference probabilitygreen (original parameters)yellow (original parameters)

Figure 11: Sensitivity analysis: equity allocation

Figure 11 shows the results. As one would expect, the shortfall prob-

abilities generally increase in the portfolio weight assigned to equities.

The probabilities associated with the pension fund’s original portfolio

composition as shown in Table 26 are represented through a point and a

triangle on the curves at the 0.28 position,120 while the threshold prob-

ability of 0.99 percent has been indicated by the dotted horizontal line.

In its current state this average pension fund evidently passes the green

condition with ease. On the one hand, we observe that the increase of

the probability curve for the green condition is quite strong, revealing

a critical portfolio weight for equities of 0.34, i.e., well below the legal

limit of 0.5. On the other hand, Figure 11 reveals that the fund would in

no case fail the yellow condition, even for the highest possible allocation

to stocks. Hence, as already suspected in the previous section, allowing

additional contributions up to 5 percent of the liabilities within a single

120Note that these points are in fact slightly off the curve since at the position 0.28the curve has been calculated with 0.14 allocated to international and 0.14 allocatedto Swiss stocks, while the original asset allocation shows 0.18 and 0.1, respectively.

6.2 Asset concentration 145

year bears a considerable potential to suppress the shortfall probabilities.

These results have an important implication for the regulator. One of

the prevalent regulatory actions in case of a failure of the green traffic

light condition should be an in-depth analysis of the portfolio compo-

sition of the respective pension fund with a focus on the more volatile

asset classes such as equities. This could be followed by a dialog between

the fund and the regulator to agree on an optimization of the portfolio

to lower the probability of failing the green condition while still retaining

reasonable return potential.

6.2 Asset concentration

In Section 5.1 we calibrated the model based on indices (well-diversified

portfolios), representing various asset classes. This approach implicitly

assumes that pension funds adequately diversify their investments within

the subportfolio of each asset class. In practice, a basic degree of diversi-

fication should be ensured, since pension funds have to obey mandatory

limits on their asset positions. The equity portfolio, e.g., cannot account

for more than 50 percent of a fund’s total assets. In addition, within this

equity portfolio, the maximum investment per individual stock (domes-

tic or international) is currently limited to 5 percent of the total assets.

Yet, with its calibration relying on indices, the solvency test might not be

well suited for an application to pension funds which hold insufficiently

diversified subportfolios. Thus, we briefly illustrate the impact of con-

centration issues within subportfolios, using domestic equity holdings as

an example.

For this purpose, we form an equally weighted portfolio (naıve di-

versification), consisting of an increasing number of stocks which are

drawn from the constituents of the SMI Index. First, the portfolio only

contains a single stock. Additional stocks are then successively added

in random order until the portfolio contains a total of eight stocks.121

For each step, we recalculate mean and volatility of the domestic equity

121The final portfolio consists of the following equities, mentioned in the sequencein which they have been added: Credit Suisse Group, Adecco SA, Roche Holding AG,Holcim Ltd., SGS SA, Nestle SA, Swatch Group AG, and Swiss Re.


1 2 3 4 5 6 7 8

number of stocks in domestic stock portfolio

shor

tfal

l pro

bab

ility

0.00

00.

005

0.01

00.

015

0.02

0

yellow conditiongreen conditionreference probability

Figure 12: Sensitivity analysis: asset concentration

portfolio (Table 21, column 4) as well as the correlations with the other

asset classes (Table 22, line 4 and column 4) and recalibrated the model

accordingly. The resulting shortfall probabilities for the solvency test

are illustrated in Figure 12, together with the original case based on the

complete SMI (20 stocks). As expected, the shortfall probabilities tend

to decrease with a rising number of equally weighted stock holdings in

the portfolio, i.e., with a decreasing asset concentration.122 More specif-

ically, for the relatively small number of six equities in the portfolio,

the shortfall probabilities are already fairly close to those of the original

case with the SMI as domestic equity portfolio. Therefore, only very

high degrees of asset concentration in the subportfolios should result in

a notable distortion of the proposed solvency test. However, if a pension

fund naıvely diversifies its holdings over at least half a dozen stocks, the

122Note that while we observe a general decrease in the shortfall probability for agrowing number of stocks in the portfolio, it can sometimes slightly increase when anew stock is added, depending on the order of inclusion. This effect occurs due tochanges on the overall correlation structure in the portfolio.

6.3 Misestimation of liabilities 147

use of indices for calibration purposes seems to be a valid approach. To

further underscore this, note that the standard pension fund underlying

the sensitivity analyses in this section has a total of 18 (international)

+ 10 (domestic) = 28 percent invested in equities (see Table 26). Taking

into account the legal limit of 5 percent per individual stock, this implies

that the fund needs to have at least 28/5 = 5.6 ≈ 6 different stocks in

its portfolio. Similarly, consider a hypothetical pension fund which in-

vested the legal maximum of 50 percent of its portfolio in equities. As

a result, its equity holdings would need to consist of a minimum of 50/5

= 10 different stocks. Nevertheless, if for some reason a subportfolio is

extremely concentrated, the model should be recalibrated accordingly.

6.3 Misestimation of liabilities

Another interesting question centers around the valuation of liabilities.

As explained in Sections 4 and 5, the model at the heart of our approach

to measuring pension fund solvency treats the liabilities as deterministic

and relies on input figures which are reported by the pension funds them-

selves. A current discussion in the Swiss pension fund system revolves

around the technical interest rate, which serves as a discount rate for the

stochastic future cash outflows in the context of an actuarial valuation of

the liabilities. It has repeatedly been stated that many funds hesitated

to lower their technical interest rate in lockstep with the development

of the term structure, thereby understating the present value of their

technical liabilities (see, e.g., Swisscanto, 2008). In addition, despite var-

ious hedging techniques broadly applied in practice (see, e.g., Mao et al.,

2008; Yang and Huang, 2009), there is a remaining uncertainty about fu-

ture mortality improvements and their modeling. Obviously, a potential

misestimation of the liabilities will have consequences for the results of

the proposed solvency test. Thus, Figure 13 displays the sensitivity of

the shortfall probabilities with regard to the estimation error of the tech-

nical liabilities. We observe a pattern similar to the effect of a change

in the portfolio weight of stocks examined in Section 6.1. Again, the

graph comprises a point and a triangle, representing the shortfall prob-

abilities for the original value of the liabilities. In the area to the left of


these points, where the liabilities are found to have been overestimated

its liabilities, the actual shortfall probabilities rapidly decline towards

zero. The opposite is true for an underestimation, however. If the liabil-

ities were a mere 1.6 percent higher than originally estimated, the fund

would already breach the threshold for condition green. Beyond that

estimation error, the increase of the probabilities becomes even steeper.

Again, the whole curve for the yellow condition lies below the reference

probability. A practical insight associated with these results is that the

supervisory review should include an in-depth analysis of the method-

ology, assumptions, and database which the pension funds employ to

estimate their liabilities. In case the supervisor has reasons to doubt the

precision of the estimates, the outcome of the solvency test would have

to be adjusted.

6.4 Coverage ratio

Next, we consider the probabilities’ sensitivity to the coverage ratio of the

pension fund at the beginning of the period. Figure 14 shows the results

for coverage ratios varying from 1.1 down to 0.85. Again, the values of

0.3803 percent for the green condition and 0.0085 percent for the yellow

condition associated with the original coverage ratio of 1.1 (see Table 26)

are represented by a point and a triangle on the curves.123 The results

for coverage ratios over and above 1.1 are not particularly interesting

as the shortfall probabilities quickly become very small. Similarly, we

observe that for coverage ratios of below 0.9, the probabilities are very

close to 1 and thus far beyond any reasonable threshold. Some more

attention should be devoted to the region between 0.9 and 1.1. Just

below 1.1, both curves initially exhibit a slightly negative slope, which

then sharply increases in magnitude below 1.05 for the curve representing

the green and below 1.0 for the curve representing the yellow condition.

This is an important result: pension funds with a coverage ratio of below

1.05 are relatively likely to end up with underfunded liabilities at the end

of the period. In addition, if their liabilities are just about covered at

the beginning of the period, even the probability for failing the yellow

123Due to their small difference relative to the scale chosen for the overall graph,these points appear as one.

6.4 Coverage ratio 149

−0.04 −0.02 0.00 0.02 0.04

0.00

0.01

0.02

0.0

30.0

4

estimation error for technical liabilities in t = 1

shor

tfal

l pro

bab

ility

green conditionyellow conditionreference probabilitygreen (original parameters)yellow (original parameters)

Figure 13: Sensitivity analysis: misestimation of liabilities

0.85 0.90 0.95 1.00 1.05 1.10

0.0

0.2

0.4

0.6

0.8

1.0

coverage ratio in t = 0

shor

tfall p

robab

ility

green conditionyellow conditiongreen (original parameters)yellow (original parameters)

Figure 14: Sensitivity analysis: coverage ratio


condition grows to levels where it begins to be perceivable. This has

important implications for the Swiss occupational pension fund system.

In particular, the common practice of letting pension funds continue their

business with dramatically underfunded liabilities without a specifically

tight supervisory review and careful amendments to their overall strategy

has to be considered inadequate from the point of view of modern risk

management and solvency regulation principles.

6.5 Lowest acceptable coverage ratio

Furthermore, we examine the sensitivity of the shortfall probability for

the yellow condition with regard to β, i.e., the lowest coverage ratio

acceptable by the supervisor. The results are depicted in Figure 15.

Consistent with our previous analyses, the curve begins at a β of 0.95

which is associated with a near zero probability (0.0085 percent) of an

underfunding after additional contributions (marked by a triangle). For

an increasing β, however, we observe a non-linear rise in the probabil-

ity. A β of 0.97, for example, is already associated with a probability of

0.046 percent, which is more than five times the above value. When β

approaches 1, i.e., additional contributions are ruled out, the probability

reaches the value of 0.3803 percent associated with the green condition

(marked by a point). This suggests that the impact of each percentage

of additional contributions allowed to fix deficits is relatively strong.

Since both the current and lowest acceptable (minimum) coverage

ratio have a strong influence on the shortfall probability for the yellow

condition, we finally want to consider their joint impact in order to assess

which combinations have counterbalancing or strengthening effects (see

Figure 16). A very important observation is, that for β below approx-

imately 0.96, the yellow condition becomes rapidly less binding, even

if we assume that the fund already begins the period with a relatively

weak coverage ratio of around 1. For a β of 0.90, the shortfall probabil-

ity for condition yellow is virtually negligible until the coverage reaches

0.95 where it begins to rise sharply. In contrast, if β is set to 1 (no

additional contributions allowed), even a relatively small underfunding

6.6 Exchange rate risk 151

leads to large shortfall probabilities, which reach 100 percent around the

coverage ratio of 0.90. Thus, as previously suspected, the regulator’s

choice of β has a crucial impact on the bindingness of the yellow traffic

light condition. Simply allowing pension funds with a low coverage ratio

to draw on large amount of additional contributions per period provides

them with a convenient means to continue business without significant

revisions to their asset or risk management practices. This somewhat

contradicts the purpose of a solvency test and essentially means that

premium payers subsidize pensioners, an effect which is generally not

intended within the second pillar of the Swiss pension system.

0.95 0.96 0.97 0.98 0.99 1.00

0.00

00.

001

0.00

20.

003

0.00

4

lowest acceptable coverage ratio

shor

tfal

l pro

bab

ility

green (original parameters)yellow (original parameters)

Figure 15: Sensitivity analysis: minimum coverage ratio

6.6 Exchange rate risk

In Section 5.1 we mentioned that the U.S. Dollar denominated indices

have not been converted to Swiss Francs for the model calibration. For

this to be an adequate approach, pension funds would need to hedge

out major exchange rate fluctuations in their asset portfolios at an im-

152

IIIPensionFund

Solvency

coverage ratio in t = 0

0.90

0.95

1.00

1.05

lowes

t accep

table co

vera

ge ra

tio

0.90

0.92

0.94

0.96

0.98

1.00

shortfall p

robab

ility

0.2

0.4

0.6

0.8

Figure 16: Sensitivity analysis: actual and minimum coverage ratio

6.6 Exchange rate risk 153

1 0.75 0.5 0.25 0

FX hedged fraction of U.S. Dollar denominated portfolio

shor

tfal

l pro

bab

ility

0.00

00.

010

0.02

00.0

30yellow conditiongreen condition

reference probability

Figure 17: Sensitivity analysis: FX hedging

material cost. In this section, we relax the assumption of a perfectly

FX hedged portfolio and analyze the impact of exchange rate risk on the

shortfall probabilities. To this end, we convert the time series of the three

U.S. Dollar denominated indices124 to Swiss Francs and compute the as-

sociated returns. For each index, we then calculate weighted averages

of the returns of the original time series (U.S. Dollars) and the returns

of the converted time series (Swiss Francs), applying weights of 100, 75,

50, 25, and 0 percent. These weights are meant to reflect the percentage

of the foreign currency denominated portfolio which has been hedged

against exchange rate risk.125 Accordingly, a 100 percent weight on the

returns of the U.S. Dollar index time series reflects a situation where the

whole portfolio is immune to exchange rate fluctuations, whereas a 100

percent weight on the returns of the Swiss Franc converted time series

implies no FX hedging activity at all. For all combinations in between,

the exchange rate risk is assumed to be partially hedged. Subsequently,

means, volatilities, correlation matrix, and the resulting shortfall proba-

bilities are recalculated for each case (see Figure 7 for the results). For


both the green and yellow condition the shortfall probabilities expect-

edly rise with the exchange rate exposure. If foreign investments in the

pension fund’s portfolio remain entirely unhedged, the probability for

the green condition turns out to be more than seven times higher than

in the case of a perfect FX hedge. However, we also see that the pension

fund would still be assigned a green light if it protects only half of its

foreign asset holdings against exchange rate risk. In addition, the yellow

condition is passed in every case, even without any FX hedging activi-

ties. From these insights we conclude that a model calibration based on

foreign currency denominated indices should be valid, as long as pension

funds hedge a large part of their foreign asset portfolio against exchange

rate fluctuations. If this is not the case, a model recalibration should be

requested and monitored by the supervisory authority.

7 Supervisory review and actions

In analogy to Basel II and the SST, the approach we introduced and

illustrated throughout the previous sections should be embedded into

a comprehensive supervisory review process. As part of this process,

occupational pension funds should be obliged to report and comment

on certain key figures resulting from the application of the supervisory

model in regular intervals. This quantitative solvency report could be

accompanied by a qualitative judgment of risks which are not explicitly

covered by the model framework, such as credit and operational risk.

In order to react properly to the risk and solvency situation of pension

funds, the regulator should possess a variety of competencies. According

to the degree of compliance with the traffic light conditions, a certain

catalog of measures could be decided. For pension funds which are as-

124These are the S&P U.S. Treasury Bond Index, the MSCI World, and the HFRIFund Weighted Index. See Table 21.125While this is a rather general analysis, abstracting from a detailed characteri-

zation of the associated transactions with regard to strategy, timing, instruments,volumes, strike prices, etc., we believe it to be satisfactory in this context. A moreelaborate treatment of FX hedging issues in the asset management industry is beyondthe scope of this paper.

7 Supervisory review and actions 155

signed a green light, the regulator could stick to periodic reviews focused

on the adequacy of the regulatory standard model. As illustrated in Sec-

tion 6.2, a recalibration could become necessary in certain cases.

If a pension fund hands in a regulatory report with a yellow light, it

should be subjected to closer scrutiny. This could, for instance, comprise

a comprehensive check-up of the fund’s assets, liabilities, liquidity, and

cash flow profile with a particular focus on valuation methodologies and

assumptions. In addition, such funds could be put on a regulatory watch

list, resulting in a shortened reporting interval. The requirement to de-

sign a concept for financial restructuring is also a potential measure to

be imposed on funds in the yellow category. Such a concept would need

to cover the asset and liability side, demonstrating how a solid solvency

situation can be restored through a combination of portfolio adjustments

as well as capital replenishment by means of additional contributions. In

any case, the regulator would have to ensure that the lower and upper

limit for additional contributions is obeyed.

If a fund is assigned a red light, more drastic consequences would be

necessary. These could comprise constraints to the management’s ability

to choose its asset allocation with the aim of preventing the fund from

incurring additional investment risks. Otherwise the problem of ”gam-

bling for resurrection” could arise, meaning that the fund management

tries to rescue the institution through large bets. Furthermore, the regu-

lator should be authorized to issue directives to the management of red

light funds. Moreover, the inclusion of additional contributors should

be suspended until the fund has been restored to an acceptable level of

solvency. This protects prospective fund members from the excessive

subsidization of current pensions through their contributions. Finally,

the regulator should have the ability to replace the board and fund man-

agement of highly distressed pension funds with a special administrator.

Beyond that, rules with regard to the publication and dispersion of

these easily interpretable solvency signals could increase transparency,

and, given the receivers can appropriately react to the information, pro-


mote market discipline. Hence, apart from the supervisory authority,

receivers of the signal should be employers, employees, transaction part-

ners, and the general public. Due to a reduction of information asymme-

tries, pension funds with an abnormally high shortfall probability would

thus have to face public scrutiny and reactions of their business partners.

8 Some notes on a potential introduction

in Switzerland

An important organizational requirement for pension funds which would

arise from a concrete introduction of the solvency test is the recruit-

ment of personnel with an adequate background for the application and

maintenance of stochastic pension fund models. Further requirements

relate to the necessary infrastructure for running the model, including

databases and software. In order to align the fund manager’s interest

with that of the insured, the former should benefit from the prevention

of yellow signals. This could, for example, be achieved by linking his

variable compensation to a combination of fund performance and traffic

light signals.

As explained in Section 2, Swiss occupational pension funds take

the legal form of private trusts, which have very limited possibilities

of self-supervision. A corporation, in contrast, has bodies such as the

board and annual meeting, which serve supervisory purposes. Conse-

quently, the introduction of a regulatory framework for occupational

pension funds could be complemented with a fundamental reformation

of the legal forms they can adopt. The recommended traffic light ap-

proach would then receive additional disciplinary weight through board

and shareholders of the corporation as receivers of the signal.

The degree of market discipline emanating from the traffic light ap-

proach strongly depends on its familiarity to stakeholder groups and

the expected consequences of bad signals such as the potential threat of

many insured wanting to change their pension fund. However, employees

8 Notes on a potential introduction in Switzerland 157

are currently not free in their choice, which greatly reduces this sort of

pressure. Therefore it needs to be discussed whether the introduction of

the solvency regulation should be accompanied by a liberalization of the

market itself, enabling a free choice of the pension fund. The downside

would be, that the situation of an already distressed fund could further

deteriorate in case a large number of insureds wants to redeem their

holdings. Nevertheless, we believe that more flexibility in this regard is

warranted and would be an important step towards an efficient regula-

tion of Swiss occupational pension funds.

Finally, the regulator could conceive of establishing higher barriers to

entry for pension schemes. These could, for example, be fit and proper

conditions for the individuals managing the pension fund, as common

for employees in other branches of the financial services industry such

as banking. Participation in the Swiss pension fund market is currently

not tied to specific criteria. Setting prerequisites would likely lead to a

consolidation, reducing the current number of funds from approximately

2’500 in 2008126 to a number which can be supervised more efficiently.

126See Swiss Federal Statistical Office under http://www.bfs.admin.ch.

http://www.bfs.admin.ch


9 Conclusion

We adopt a stochastic pension fund model and combine it with a traf-

fic light approach for solvency measurement purposes. The calibration

and implementation of the model with a small sample of ten pension

funds illustrates its application for the computation of probabilities and

derivation of traffic light signals. The model adequately captures the

particularities associated with the occupational pension fund system in

Switzerland. Due to its efficiency and ease of calibration it is well suited

as a regulatory standard model in this very fragmented market, keeping

costs of the solvency test at a minimum, even for small pension funds

with less sophisticated risk management know-how and infrastructure.

In addition, the sensitivity analysis identifies important drivers of the

shortfall probabilities and can thus assist the regulator with regard to

specific decisions associated with the configuration of the framework.

However, some questions remain in respect to model design and cali-

bration. First, we did not explicitly account for credit risk in the fund’s

asset portfolio. Therefore, the supervisory authority should exercise ad-

ditional care with regard to solvency test results for pension funds with a

relatively high proportion of default-able instruments, such as corporate

bonds, in their portfolio. Second, an incorporation of stochastic liabil-

ities and different statistical distributions for the modeled asset classes

could be discussed, although a departure from the associated assump-

tions would necessitate a switch from the closed-form to a numerical

solution. Third, portfolio diversification and foreign currency exposure

have to be borne in mind as critical factors with regard to the proposed

calibration procedure. Finally, a practical implementation would need

to be preceded by a comprehensive quantitative impact study for the

majority of Swiss pension funds. Overall, we consider this straightfor-

ward framework to be an adequate first step towards a state-of-the-art

solvency regulation of occupational pension funds in Switzerland.

References 159

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163

Part IV

Stock vs. Mutual Insurers:Who Does and Who ShouldCharge More?

Abstract

In this paper, we empirically and theoretically analyze the relationship

between the insurance premium of stock and mutual companies. Eval-

uating panel data for the German motor liability insurance sector, we

do not find evidence that mutuals charge significantly higher premiums

than stock insurers. If at all, it seems that stock insurer policies are

more expensive. Subsequently, we employ a comprehensive model frame-

work for the arbitrage-free pricing of stock and mutual insurance con-

tracts. Under the chosen set-up, the formulae for the premium and the

present value of the equity of a stock insurer are nested in our more

general model. Based on a numerical implementation of the framework,

we then compare stock and mutual insurance companies with regard

to the three central magnitudes premium size, safety level, and equity

capital. Although we identify certain circumstances under which the mu-

tual’s premium should be equal to or smaller than the stock insurer’s,

these situations would generally require the mutual to hold less capital

than the stock insurer. This being inconsistent with our empirical results,

it appears that policies offered by stock insurers are overpriced relative

to policies of mutuals. While our analysis focuses on the insurance con-

text, the insights can be transferred to other industries where mutual

companies are an established legal form.127

127Alexander Braun, Przemys law Rymaszewski, and Hato Schmeiser (2011), Stockvs. Mutual Insurers: Who Does and Who Should Charge More?, Working Papers on

Risk Management and Insurance, No. 87.

164 IV Stock vs. Mutual Insurers

1 Introduction

Private insurance firms in many insurance markets can be organized ei-

ther as mutual or stock insurance companies. Similar to policyholders of

a stock insurance company, those of a mutual insurer are obliged to pay

the insurance premium which, in turn, entitles them to an indemnity

payment contingent on the occurrence of a loss. Apart from that, how-

ever, several important differences between these two legal forms exist

(see, e.g., Smith and Stutzer, 1990). First of all, in contrast to stock in-

surers, mutuals are in fact owned by their policyholders. By paying the

respective premium, the buyer of a mutual policy becomes a so-called

member, which is economically equivalent to acquiring a policyholder

and an equityholder stake in the firm.128 As a result, those insured by a

mutual are usually granted direct or indirect participation in the admin-

istrative bodies and should thus be able to exert influence on business

decisions. To establish a similar position, policyholders of stock insur-

ance companies would need to acquire ownership rights by purchasing

the company’s common stock. Unlike the shareholders of a stock insurer,

however, members of a mutual cannot simply sell their equity stake. This

is due to the fact that, in practice, it is not explicitly differentiated from

the policyholder stake and a secondary market does not exist. Hence,

the only way to fully realize the value of the equity are liquidation or

demutualization of the company, which would need to be enacted col-

lectively by a majority of the members.129 A further difference to stock

insurers is, that mutual members can expect occasional premium refunds

if the company is profitable. These payouts are economically akin to the

dividends a stock insurer distributes to its shareholders. Finally, stock

insurance companies cannot draw on their policyholders to recover fi-

nancial deficits, whereas the membership in a mutual insurer might be

associated with the obligation to make additional premium payments

contingent on the firm being in financial distress. These additional pre-

miums are virtually authorized capital, i.e., equity which has not been

128Rasmusen (1988) describes rights and obligations resulting from a membershipin savings and loan associations, credit unions, and mutual savings banks.129In the course of a demutualization, the insurer changes its legal form and is

transformed into a stock company.

1 Introduction 165

paid in yet (see, e.g., Mayers and Smith, 1988). Since the legal form de-

termines these rights and obligations associated with the purchase of an

insurance contract, it should ceteris paribus result in different arbitrage-

free prices for policies, covering identical claims.

While there is a large body of literature, dealing with various aspects

of mutual and stock companies, to the best of our knowledge, there has

not yet been a rigorous empirical and theoretical analysis of the rela-

tionship between the premium of stock and mutual insurers. Therefore,

in this paper, we want to shed some light on this research question by

evaluating panel data for the German motor liability insurance sector.

In addition, we contribute to the literature by employing a contingent

claims model framework to consistently price stock and mutual insurance

contracts. For this purpose, we split the arbitrage-free mutual insurance

premium into an ownership and policyholder stake, both of which are

further decomposed into distinct option-theoretic building blocks. The

model explicitly takes into account the restricted ability of members to

realize the value of their equity stake as well as the mutuals’ right to

charge additional premiums in times of financial distress, which will be

termed recovery option in the course of this paper. Under the chosen

set-up, the formulae for the premium and the present value of the equity

of a stock insurer are nested in our more general model. Moreover, we

derive conditions, under which the premiums of a stock and a mutual

insurance company should theoretically be equal. Finally, combining our

empirical and theoretical results, we are able to derive relevant economic

implications. While we apply our model within the insurance context, its

insights can be transferred to other industries where mutual companies

are an established legal form such as credit unions and pension funds.

The remainder of this paper is organized as follows. Section 2 con-

tains a comprehensive overview of previous literature on issues surround-

ing stock and mutual insurance companies. In Section 3, we apply panel

data methodology to provide some empirical evidence with regard to the

relationship between the premiums of stock and mutual insurers. Aim-

ing to explain these empirical results by means of normative theory, in


Section 4 we develop our contingent claims model framework, beginning

with the simple and well-established case of the stock insurance com-

pany. Subsequently, we consider a mutual insurer with recovery option

and fully realizable equity, before formally describing the general case

with partial participation in future equity payoffs. Section 5 comprises

a comprehensive numerical analysis which forms the basis for our nor-

mative findings. In Section 6, we integrate our empirical and theoretical

results and discuss relevant economic implications. Finally, in Section 7,

we conclude.

2 Literature review

The literature comparing stock and mutual insurance companies has

predominantly dealt with agency issues associated with the legal form.

Coase (1960) argues that the ownership structure of a company, which

is determined by property rights constituting the discretionary power of

control, is relevant only in the presence of transaction costs. This is due

to the fact that conflicts of interest between different stakeholders may

arise and entail costs, which depend on the extent of discretion as well

as established control mechanisms. Ownership structure is identified as

one possible means of control. In this spirit, Mayers and Smith (1981)

develop a positive theory on insurance contracting, extending the fun-

damental work of Jensen and Meckling (1976) on agency theory. They

analyze incentives resulting from the different ownership arrangements

of stock and mutual insurers and discuss two kinds of potential conflicts

between parties brought together in an insurance firm. On the one hand,

asymmetric information and the call option-like payoff profile associated

with the shareholder position in a stock insurer imply that the equity

value increases with the risk inherent in the company.130 At the same

time, however, riskier assets are detrimental to the position of the poli-

cyholders, giving rise to the so-called owner-policyholder conflict.

130The notion that the equity stake in a company can be interpreted as a call optionon its assets, struck at the face value of the liabilities, was introduced by Merton(1974).


Against this background, the company’s owners will seek to estab-

lish efficient sanction mechanisms, ensuring that the management acts

in their interest. Consequently, agency costs occur and impair economic

efficiency compared to a setting without transaction costs. Since owners

and policyholders within a mutual insurance company coincide, agency

costs can be reduced.131 On the other hand, stock insurers provide more

efficient sanction mechanisms to tackle the so-called owner-management

conflict, which results from diverging incentives between shareholders

and company executives. In addition to being held responsible by the

organizational bodies of the insurance company, which are controlled by

its owners, poorly performing executives of a stock insurer must fear

market discipline such as, for instance, hostile takeovers.132 The reason

is that, in contrast to a mutual insurer, the equity of a stock insurer is

freely tradable and not linked to a particular insurance policy. Hence,

agency costs resulting from the so-called owner-manager conflict can be

expected to be higher for mutuals.133 Assuming that a large number

of decision makers (owners) cannot coordinate as efficiently as single en-

tity or individual, the costs of control can be expected to rise with the

granularity of the equity stake. While the majority of shares of publicly

listed corporations are frequently owned by large blockholders, only a

marginal fraction of the ownership rights is allocated to each member of

a mutual firm. Thus, internal sanction mechanisms are likely to be more

effective for stock than for mutual insurers. Accordingly, from the poli-

cyholder perspective, the optimal choice of legal form should depend on

the trade-off between agency costs arising from the owner-policyholder

and the owner-manager conflict. Therefore, Mayers and Smith (1981,

1988, 1994) argue that stock firms should be more prevalent in activi-

ties that involve significant managerial discretion, since, in this context,

potential owner-manager conflicts are most severe (also see Pottier and

131Also see Garven (1987). Similar incentives can be achieved by including partici-pation rights in the stock insurance contracts (see, e.g., Garven and Pottier, 1995).132The new owner normally exchanges the board of the company (see, e.g., Mayers

and Smith, 1988).133Fama and Jensen (1983a,b) argue that a further mechanism to control manage-

ment is the fact that assets of all mutual financial institutions need to be redeemed ondemand of their members. However, we assent to the arguments raised by Smith andStutzer (1990), who suggest that this is not the case within the insurance context.


Sommer, 1997). In contrast to that, mutuals should theoretically prevail

in the long-term lines of business that are usually encumbered with a

more significant owner-policyholder conflict potential, such as the life

insurance sector (see Hansmann, 1985; Mayers and Smith, 1988).

A number of empirical articles support the previously explained

agency-theoretic considerations. Lamm-Tennant and Starks (1993) pro-

vide evidence for the owner-policyholder conflict by showing that stock

insurers are generally riskier than mutual insurance companies. This

is coherent with the results of Lee et al. (1997), who analyze both le-

gal forms in the context of insurance guaranty funds. Furthermore, the

greater potential for the owner-manager conflict in mutuals is illustrated

by Greene and Johnson (1980), who conduct a survey in which they

analyze policyholder awareness of the rights resulting from the owner-

ship stake in a mutual insurance company. Compared to the holders of

publicly traded stock, members of the analyzed mutual companies were

less aware of their voting rights and appeared to exercise less control.

Similarly, Wells et al. (1995) find that, in contrast to managers of stock

insurers, those of mutuals have a higher free cash flow at their disposal,

implying a greater opportunity to waste cash on unprofitable invest-

ments. Further evidence for the owner-manager conflict in the context

of mutual and stock insurers is provided by Mayers and Smith (2005),

who document that mutual company charters are more likely to contain

provisions which limit the range of operating policies of the firm. Zou

et al. (2009) observe that, probably owing to their inferior management-

control mechanisms, mutuals tend to pay significantly lower dividends

than stock insurers. Finally, analyzing data from the property-liability

insurance sector, He and Sommer (2010) find that, compared to stock

insurers, the board of mutuals generally comprises a larger fraction of

outside directors. They argue that additional monitoring through out-

side directors is necessary since ownership and control in mutuals are

separated to a greater extent, thus increasing agency costs arising from

the owner-manager conflict.134

134The owner-manager conflict in the context of mutual and stock banks has, e.g.,been considered by Gropper and Hudson (2003) who provide evidence for considerableexpense-preference behavior in mutual savings and loans associations based on a U.S.wide sample.


Another major strand of literature deals with changes in the le-

gal form of an insurer. Fletcher (1966) as well as Mayers and Smith

(1986) focus on mutualization issues. However, much more research has

been conducted on the demutualization process. A survey by Fitzger-

ald (1973) identifies economic pressure as the main reason for the con-

version of small property-liability insurers into stock companies, while

Viswanathan and Cummins (2003) view access to capital as a major

driver for demutualization. Furthermore, Carson et al. (1998) find the

level of free cash flow to be significantly related to the probability of mu-

tual firms transforming into stock companies. Zanjani (2007) analyzes

macroeconomic and regulatory conditions under which mutual insurance

companies have been formed in order to explain the observed evolution of

the whole U.S. life insurance industry from the mutual towards the stock

insurer form. He concludes that tight state regulation did not coincide

with a demise of the mutual form. Instead, a general rise in founding

capital requirements seems to have harmed mutuals due to their very

limited access to external funding. Moreover, Erhemjamts and Leverty

(2010) argue that the incentive to demutualize differs by the type of con-

version: full demutualization versus mutual holding company. Finally,

in their empirical study of U.S. life insurers, McNamara and Rhee (1992)

find that increased efficiency seems to be an important reason for demu-

tualization.

The question of efficiency differences between stock and mutual firms

has been further examined by several other authors. Spiller (1972)

finds evidence that ownership structure is a determinant of performance.

While Jeng et al. (2007) present mixed results with regard to efficiency

improvements implied by changes of the legal form, Cummins et al.

(1999) find mutuals to be less cost-efficient.135 Furthermore, in their

study based on Spanish insurance market data, Cummins et al. (2004)

identify differences in efficiency between stocks and mutuals only for

small mutual insurance companies. Harrington and Niehaus (2002) fo-

135Iannotta et al. (2007) conducted a similar study for the banking industry. Con-trolling for company characteristics as well as geography and time, they find thatmutual banks are less profitable than stock banks. Moreover, they provide evidencefor a higher loan quality among mutuals compared to stock and public sector banks.


cus on dissimilarities concerning capital structure, which may result from

the costs of raising new capital and Viswanathan (2006) finds initial pub-

lic offerings of mutuals to be significantly underpriced. The latter result

is confirmed by Lai et al. (2008).

Besides agency-theoretic considerations, (de)mutualization, and effi-

ciency implied by the legal form, various other topics related to stock

and mutual insurers have been explored in the literature. Differences in

the contractual structure of policies offered by mutual and stock insurers

are examined by Smith and Stutzer (1990, 1995). They argue that infor-

mation asymmetries rather than agency problems are the major determi-

nant for the types of contracts offered by mutuals. The parallel existence

of different legal forms of insurance companies is justified, amongst oth-

ers, by self-selection of those insured. In addition, Cass et al. (1996)

consider how a Pareto optimal risk allocation can be achieved through

mutual insurance in the presence of individual risk. Ligon and Thistle

(2005) point out that issues arising from asymmetric information can

restrict the size of mutual institutions. Using an equilibrium model in

which mutuals can exclusively offer fully participating policies, Friesen

(2007) show that stock companies can only provide partially participat-

ing insurance when their shareholders require premiums that ensure a

fair return on equity. Finally, Laux and Muermann (2010) demonstrate

that, by linking policies to the provision of capital, mutuals can resolve

free-rider and commitment issues faced by stock insurers.



In this section, we want to empirically investigate whether the legal form

of an insurance company is a determinant of the premium it charges. To

ensure comparability, the insurance product under consideration needs

to be as homogeneous as possible. Therefore, our sample is based on

annual accounting figures for the German motor vehicle liability insur-

ance sector.136 The data has been obtained from Hoppenstedt, a major

provider of company information for a wide variety of industries of the

German economy. To ensure consistency, we have carried out cross-

checks with the annual reports of the respective insurers. The sample

consists of 99 stock and 14 mutual insurers for which repeated obser-

vations over a differing number of time periods between 2000 and 2006

are available. Hence, we are working with unbalanced panel data, cov-

ering 532 and 87 firm years for stock and mutual insurance companies,

respectively. Table 27 contains some descriptive statistics on the panel

dataset. We measure the price of insurance by means of the average an-

nual gross premium (AvPrem), which is obtained by dividing the total

annual premium volume in the motor liability business line of each firm

by the respective number of contracts.137 Within the analysis, we control

for various additional factors which are likely to influence the insurance

price. The average annual loss (AvLoss), defined as the amount of losses

in the motor insurance line divided by the number of contracts, is used

as a proxy for underwriting risk. In a similar manner, the average an-

nual costs of the motor liability business line (AvCosts) are employed to

account for differences in the efficiency of the companies. Furthermore,

we include the equity ratio (EqR), i.e., the book value of equity divided

136Specialty insurers have been excluded.137An alternative measure for the insurance price is the economic premium ratio

(EPR) which has been suggested by Winter (1994) and is frequently used in theliterature (see, e.g., Gron, 1994; Cummins and Danzon, 1997; Phillips et al., 2006).For a given business line of an insurer, the EPR is the ratio of premium revenuesnet of expenses and policyholder dividends relative to the estimated present value oflosses (see Phillips et al., 2006). Since, in the subsequent chapters, we are interestedin the mutual premium which includes an equityholder and a policyholder stake,policyholder dividends cannot be excluded. In addition, our data does not cover line-specific estimates for the present value of losses. Hence, we control for underwritingrisk by incorporating average annual losses into our regression equations.


by the book value of the assets, as well as the log total premium volume

in a given year (LTP ) to control for capital structure and size effects.

The Lagrange multiplier (pooling) test, conducted in line with

Gourieroux et al. (1982), suggests significant cross-sectional and time

effects in our data.138 In this case, the pooled ordinary least squares

(OLS) estimator is known to be inefficient: it does not fully exploit the

information inherent in panel datasets (see, e.g., Petersen, 2008). In-

stead, more sophisticated models are needed to make the most effective

use of our data. Based on the Hausman test (see Hausman, 1978) with a

χ2 test statistic of 483.70 and four degrees of freedom, we reject the ran-

dom effects (RE) model. A likely reason for this outcome are significant

correlations between unit-specific components and regressors, implying

an inconsistent RE (and pooled OLS) estimator. While a fixed effects

(FE) model with unit-specific intercept terms could handle this sort of

correlation, the so-called FE within estimator is based on a transforma-

tion of the regression equation into deviations from individual means

and is thus incapable of capturing the impact of time-invariant variables

(see, e.g., Wooldridge, 2010). This a serious issue since our analysis is

focused on the legal form, which, if at all, changes very rarely.

Therefore, we decide to apply the Hausman-Taylor estimator, an in-

strumental variables approach combining characteristics of FE and RE

models (see Greene, 2007; Verbeek, 2008). It is capable of handling

correlations between independent variables and unobserved unit-specific

effects and enables us to estimate coefficients for time-invariant regres-

sors. Consider the following linear regression equation:

AvPremit = µ+ β1AvLossit + β2AvCostsit + β3EqRit

+ β4LTPit + β5Stocki + ui + ǫit. (60)

where µ is the intercept and Stocki is a time-invariant dummy variable

representing the legal form of insurer i, which is set to one for stock

and zero for mutual companies. The ui are N − 1 (here: 112) unit-

specific fixed effects and ǫit denotes the independent and identically dis-

138We compute a χ2 test statistic of 2, 134.13, with two degrees of freedom.

3E

mpirica

lanaly

sis173

Panel A: Pooled stocks and mutuals

abbreviation mean std. dev. minimum maximum skewness kurtosis

Average annual premium AvPrem 256.6744 57.8718 110.8053 577.1687 0.5238 5.5277

Average annual loss AvLoss 225.6424 63.3354 54.6619 514.9426 0.7426 4.5793

Average annual costs AvCosts 40.4934 23.1798 5.8454 287.5176 3.1525 26.7360

Equity ratio EqR 0.2244 0.1127 0.0387 0.7432 1.3999 5.5725

Log-Total premiums LTP 19.0946 1.4775 13.4902 22.9681 -0.2079 2.8635

Panel B: Mutuals




Average annual costs AvCosts 27.1455 13.3401 5.8454 53.9230 -0.0729 1.9150

Equity ratio EqR 0.2798 0.1253 0.1138 0.5804 0.8573 2.7524


Panel C: Stocks




Average annual costs AvCosts 42.6762 23.7181 9.3966 287.5176 3.2660 26.9900

Equity ratio EqR 0.2154 0.1079 0.0387 0.7432 1.5321 6.5753


Descriptive statistics for the variables which enter the empirical analysis. In Panel A, all available data has beenpooled, whereas Panel B and C refer to the separate subsamples of mutual and stock insurers. The underlyingcurrency is Euro.

Table 27: Descriptive statistics of the data


tributed error term. In order to estimate this model, Hausman and Tay-

lor (1981) propose the following instruments: exogenous regressors, i.e.,

those explanatory variables that are uncorrelated with the unit-specific

effects, are their own instruments. In addition, endogenous time-varying

and time-invariant regressors are instrumented by their own individual

means (over time) and those of the exogenous time-varying regressors,

respectively.139 Hence, the analysis requires at least as many exogenous

time-varying as there are endogenous time-invariant regressors, i.e., one

in our case. Based on a correlation test between the above explanatory

variables and their unit-specific components from a fixed effects model,

we identify EqR as exogenous.

An alternative three-stage procedure for estimation of time-invariant

variables in panel data models named fixed effects vector decomposition

(FEVD) has been proposed by Plumper and Troeger (2007). Originated

in the empirical political science literature, FEVD quickly gained popu-

larity among researchers in various fields. Although the authors provided

Monte Carlo simulation results to underline the apparent favorable char-

acteristics of their estimator, it has recently been severely criticized. In

particular, FEVD standard errors have been shown to be systematically

too small and the estimator is inconsistent if time-invariant variables are

correlated with unit-specific effects (see Breusch et al., 2010 and Greene,

2010). Despite these major shortcomings, we decide to additionally ap-

ply this method for comparison purposes.

Table 28 contains the estimation results for the Hausman-Taylor ap-

proach, the FEVD procedure, as well as a simple FE model.140 Apart

from EqR, all time-varying regressors seem to be key determinants

of the insurance premium, since they are associated with statistically

significant coefficients for each of the three estimators. For the time-

invariant variable Stock, in contrast, we get diverging results. While

the Hausman-Taylor estimator does not indicate a significant difference

139A more detailed treatment of the Hausman-Taylor estimator is beyond the scopeof this paper. The reader is referred to advanced panel data texts such as Hsiao(2002), Baltagi (2005), and Wooldridge (2010).140The heteroskedasticity and autocorrelation consistent (HAC) covariance matrices

of Andrews (1991) as well as Driscoll and Kraay (1998) have been applied.

4 Model framework 175

in the average premium of stock and mutual insurers, the FEVD coef-

ficient suggests that mutuals tend to charge less. Taking into account

the above-mentioned limitations of FEVD, we are evidently more confi-

dent in the Hausman-Taylor estimate. For our purpose, however, it is

sufficient to conclude that observed premiums are either approximately

equal, or stock insurer policies tend to be more expensive. To put it

differently, we do not find evidence that mutuals charge higher premi-

ums than stock insurers. Throughout the remainder of this paper we

want to adopt a normative stance and explore whether this empirical

phenomenon is consistent with fair insurance prices as suggested by con-

tingent claims theory.

4 Model framework

In this section we present a general contingent claims model framework

for insurance companies based on the seminal work of Merton (1974)

as well as Doherty and Garven (1986). Assume that the firm runs for

a single period and all stakes are paid in full at the outset. The econ-

omy is characterized by perfect capital markets, i.e., there are no bid-ask

spreads, transaction costs, short-selling constraints, taxes or other mar-

ket frictions. We begin with the relatively simple case of the stock in-

surance company (Section 4.1), which is then incrementally generalized

to include the specifics of mutual insurers. In Section 4.2, we introduce

the recovery option, i.e., the right to demand additional payments in

times of financial distress. Similarly, in Section 4.3, we further extend

our model by allowing for incomplete participation of members in the

mutual’s equity payoffs.

4.1 Stock insurer claims structure

Equity stake

An insurance firm in the legal form of a corporation (stock insurer) is

bankrupt, if the market value of the assets A1 available at the end of the

period is insufficient to cover its claims costs (losses) L1, i.e., A1 < L1.


Hausman-Taylor FEVD Procedure Fixed Effects Model

(Intercept) -213.4151*** -237.3012*** —

(-2.6692) (-12.1466)

AvLoss 0.3420*** 0.3469*** 0.3420***

(15.4295) (9.9042) (10.9533)

AvCosts 0.6053*** 0.5994*** 0.6053***

(7.3825) (6.1891) (3.9955)

EqR 20.0231 15.7489* 20.0231

(1.0095) (1.9075) (0.5184)

LTP 19.2463*** 18.7959*** 19.2463***

(7.0319) (17.3699) (7.3742)

Stock -3.9429 33.7803*** —

(-0.0470) (14.7292)

Coefficients and t-statistics (in parentheses) for Hausman-Taylor estimator, theFEVD procedure, and the standard FE model. The average annual premium(AvPrem) is regressed on the following set of explanatory variables: averageannual losses (AvLoss), average annual costs (AvCosts), equity ratio (EqR),and logged total premium (LTP ). Hausman-Taylor and FEVD additionallyinclude the time-invariant variable legal form (Stock). ***, **, and * denotestatistical significance on the 1, 5, and 10 percent confidence level.

Table 28: Estimation results

Due to the limited liability of the owners, the equity in t = 1 is worth

zero in this case. Therefore, the payoff profile of the equity stake equals

that of a European call option on the company’s assets, struck at the

value of the claims. Hence, the present value of the equity of a publicly

traded stock insurer EC0, which is a function a parameter set P, can be

expressed as follows

ECS0 = e−rEQ

0 [max (A1 − L1; 0)]

= e−rEQ0 (A1 − L1) +DPOS

0 , (61)

where EQ0 denotes the conditional expectation in t = 0 under the risk-

neutral measure Q, r is the riskless interest rate, and P contains the

relevant parameters for any specific option pricing framework.141 The

call option payoff is equivalent to a long position in the assets and a

141Under the Black and Scholes (1973) model, e.g., the parameter set P wouldcontain the initial value of the assets, the level of claims costs (i.e., the option’s strikeprice), the asset volatility, the risk-free interest rate, as well as the time to maturity.

4.2 Mutual insurer: full equity participation 177

short position in the claims costs (A1 − L1) plus the value of the so-

called default put option of the stock insurer (DPOS). To see this refer to

Figure 18. The default put option is a proxy for the expected bankruptcy

cost and therefore a measure for the safety level of the firm from the

policyholder perspective (see Doherty and Garven, 1986). Its present

value DPOS0 = DPOS

0 (P) is equal to

DPOS0 = e−rEQ

0 [max (L1 −A1; 0)] . (62)

Policyholder stake

If the stock insurer is solvent at time t = 1, the insurance company fully

indemnifies policyholders for their incurred losses. In case of bankruptcy,

however, policyholders only receive the part of their claims which is

covered by the remaining market value of the assets in t = 1. Based on

this payoff profile, the present value of the policyholder stake and thus

the fair premium πS0 of a stock insurer, P S

0 = P S0 (P), is:

P S0 = πS

0 = e−rEQ0 (L1) −DPOS

0 . (63)

The first term represents the present value of expected future claims

costs and corresponds to a default-free insurance premium. The second

term is the value of default put option. This relation implies that stock

insurers with a higher (lower) default risk should charge lower (higher)

premiums πS0 . In the absence of arbitrage, the contribution of the equi-

tyholders and policyholders in t = 0 will be equal to ECS0 and P S

0 = πS0 ,

respectively, implying that the purchase of each stake is associated with

a net present value of zero. The insurance company then invests the sum

A0 = ECS0 + πS

0 in the capital markets.

4.2 Mutual insurer claims structure:

full participation in equity payoff

Equity stake

As discussed in Section 1, one important aspect in which mutuals may

differ from stock insurance companies is their potential right to demand

178

IV

Stock

vs.MutualInsurers

ECS1

L1

PS1

A1 − L1

L1 A1

450

DPOS1

Figure 18: Payoff to the equityholders ECS1 (solid line) and policyholders P S

1 (dotdashed line) of a stock insurancecompany in t = 1. The dashed lines illustrate the elements of the replicating portfolio (A1 − L1 and DPOS

1 ).


additional premiums in times of financial distress. Provided a mutual

insurer exhibits such a recovery option and its members fully participate

in the payoff profile of the equity, the present value of the mutual’s equity

stake, ECMf0 , can be expressed as

ECMf0 = e−rEQ

0 (A1 − L1) +RO0 +DPOM0 , (64)

where RO0 equals the present value of the recovery option and DPOM0

denotes the present value of the default put option of the mutual insurer.

Comparing Equations (61) and (64), we notice that these two option

components replace DPOS0 . Due to the recovery option, the default put

option of the mutual insurer ceteris paribus differs from its stock insurer

counterpart (see Figure 19 for a graphical illustration). In particular,

the mutual insurer remains solvent as long as the recovery option has

not been fully exhausted. Accordingly, the assets in t = 1 have to fall

under a lower default threshold X = L1−Cmax than for the stock insurer

before bankruptcy is declared and the remaining assets are distributed

among those members with valid claims. Cmax denotes the upper limit

on additional payments which can be charged through the recovery op-

tion.142 Formally, the present value of the mutual insurer’s default put

option, DPOM0 = DPOM

0 (P , Cmax), is defined as

DPOM0 = POX

0 +BPO0 (65)

where

POX0 = e−rEQ

0

(POX

1

)= e−rEQ

0 [max (X −A1; 0)] , (66)

and

BPO0 = e−rEQ0 (Cmax1A1<X) . (67)

1 is the indicator function, which equals one if A1 < X and zero

otherwise. POX0 is a simple European put option with strike price X

and BPO0 is a cash-or-nothing binary put option which reflects the fact

that, in the instance in which the mutual insurer becomes insolvent, the

assets will have already dropped below the claims by an amount of Cmax.

142Cmax is usually defined in a company’s charter. In our model, it can be easilyadjusted to account for members’ potential default risk or reluctance to pay additionalpremiums.


In other words, in case of a mutual insurer bankruptcy, losses on the poli-

cyholder stake will be at least Cmax. By comparing the respective payoff

profiles in Figure 19, we notice that generally POX0 ≤ DPOM

0 ≤ DPOS0 .

In addition, the smaller Cmax, the more valuable DPOM0 and, in the spe-

cial case of Cmax = 0 (i.e., X = L1), we get POX0 = DPOM

0 = DPOS0

and BPO0 = 0.

Figure 20, depicts the payoff profile for two different specifications of

the recovery option. We define the standard (basic) recovery option as

one which allows to raise no more than the exact amount of the missing

capital. Its present value, RO0 = RO0(P , Cmax), can be expressed as

RO0 = DPOS0 −DPOM

0

= DPOS0 − POX

0 −BPO0. (68)

and thus equals a long position in DPOS0 and a short position in DPOM

0 .

To put it differently, instead of the stock insurer’s default put option, the

owners of a mutual insurer hold a combination of the recovery option

and the default put option of the mutual, implying that the value DPOS0

is perfectly decomposed into RO0 and DPOM0 , i.e., DPOS

0 = RO0 +

DPOM0 . Consequently, the equity of the stock and the mutual insurer

do not differ in value. However, ceteris paribus mutual members enjoy

a higher safety level of their policies since the probability that their

insurance claims in t = 1 are paid in full is greater than for the stock firm.

Intuitively, the recovery option works as follows: whenX ≤ A1 ≤ L1, i.e.,

if the assets in t = 1 fall below the claims by an amount less than Cmax

such that the recovery option is sufficient to rectify the deficit, L1−A1 is

demanded from policyholders. This is exactly enough additional capital

to eliminate the shortage. Note that the lower Cmax, the less valuable

RO0 and for Cmax = 0, RO0 is worthless. In contrast to that, Cmax =

L1 is associated with the maximum value of the recovery option, while

the default put option of the mutual insurer has no value in this case.

Therefore, Cmax determines how the value of the stock insurer default

put option is split into DPOM0 and RO0.

4.2

Mutu

alin

surer:

full

equity

particip

ation181

L1 A1

45

L1

X

DPOM1

0

Cmax

X

BPO1

DPOS1

POX1

Cmax

Figure 19: Mutual insurer default put option payoff in t = 1 (DPOM1 , solid line). The dashed lines indicate the

elements of the replicating portfolio (BPO1 and POX1 ). For comparison purposes, the dotdashed line illustrates

the default put option of a stock insurer with an identical claim structure (DPOS1 ).


L1 A1

45

arctan(λ)

RO1(P, Cmax, λ > 1)

RO1(P, Cmax, λ = 1)

L1

Cmax

−Cmax

X X⋆

λDPOS1

−λPOX⋆

1

−BPO1

DPOS1

−POX1

0

Cmax

1

λCmax

Figure 20: Mutual insurer recovery option payoff in t = 1: RO1(P ,Cmax, λ = 1) (bold dotdashed line) and RO1(P , Cmax, λ > 1) (boldsolid line). The thin dotdashed and dashed lines illustrate the respectivereplicating portfolios.


Theoretically, a distressed mutual insurer might be allowed to collect

more than just the missing capital from its members, implying that the

firm can build up a reserve. By adjusting Equation (68) we can extend

our model framework to account for this special case, which will be called

excess of loss recovery option. The following is a more general expression

for the present value of the recovery option, RO0 = RO0(P , Cmax, λ),

RO0 = λDPOS0 − λPOX⋆

0 −BPO0, (69)

where

POX⋆

0 = e−rEQ0

(

POX⋆

1

)

= e−rEQ0 [max (X⋆ −A1; 0)] , (70)

with X⋆ = L1 − 1λC

max and λ ∈ [1;∞). Consequently, in the gen-

eral case, the recovery option is a position of λ units of DPOS long,

λ units of POX⋆

short, and BPO short. The parameter λ constitutes

a straightforward charging rule and denotes the multiple of additional

payments over the deficit. For λ = 2, e.g., the mutual is able to charge

policyholders twice the deficit and build up a reserve from the surplus.

The impact of λ on the recovery option payoff profile is illustrated

in Figure 20. Intuitively, the higher this multiple, the lower the dis-

tance between L1 and X⋆, the steeper the slope of RO1 in this interval,

and the smaller the amount by which the assets have to fall below the

claims so that the mutual will simply collect Cmax.143 Analogously to

POX, POX⋆

is a European put option with strike X⋆, which depends

on λ. Clearly, if λ = 1 we have POX⋆

0 = POX0 . In Figure 20, we

see that RO1(P , Cmax, λ > 1) > RO1(P , Cmax, λ = 1), which implies

RO0(P , Cmax, λ > 1) > RO0(P , Cmax, λ = 1).

To sum up, if the recovery option is designed to simply eliminate a

given deficit (λ = 1), the payoff profile of the overall equity stake of a

143For λ → ∞, the slightest deficit will induce the mutual insurer to charge addi-tional payments of Cmax.


mutual insurer will ceteris paribus be the same as for the stock insurer.

However, if the recovery option is specified so that the mutual can charge

multiples of a given deficit (λ > 1), its equity stake will be relatively more

valuable. This can be easily seen by comparing the payoff profile for the

equityholders of a mutual insurer with excess of loss recovery option

(Figure 21) to that plain call option shape we saw for the stock insurer

in Figure 18. In any case, the value of the equity stake ECMf0 depends

not only on the parameter set P but also on the specific characteristics

of the recovery option as represented by Cmax and λ.

Policyholder stake

Consistent with its equity, we define the present value of the policyholder

stake of a mutual insurer as

PM0 = e−rEQ

0 (L1) −RO0 −DPOM0 . (71)

Again, e−rEQ0 (L1) is the fair insurance premium without default risk

and instead of DPOS0 we have a short position in the combination of the

recovery option and the default put option of the mutual. If, at the end

of the period, the assets have fallen below the claims costs L1 but not

the mutual’s default threshold X, i.e., X < A1 ≤ L1, the policyholder

stake of the mutual insurance company is associated with an equal or

a higher financial loss than that of a stock insurer. This is due to the

fact that the mutual charges λ(L1 − A1) through the recovery option,

while the insolvency of the stock insurer results in a policyholder deficit

of L1−A1. Therefore, generally DPOS0 ≤ RO0+DPOM

0 and PM0 ≤ PS

0 .

More specifically, the policyholder stake of a mutual insurance company

is less valuable than that of an otherwise identical stock insurer when

it contains a recovery option with λ > 1 (see Figure 22). Similarly,

we know from Equation (64) that the present value of the equity stake

increases for more expensive recovery options. Hence, an excess of loss

recovery option essentially redistributes value from the policyholder to

the equity stake. If λ = 1, in contrast, we have DPOS0 = RO0 +DPOM

0

and consequently PS0 = PM

0 (refer to Equation (68)).

4.2

Mutu

alin

surer:

full

equity

particip

ation185

ECMf1

A1 − L1

L1 A1

4545

arctan(λ)

arctan(λ− 1)

RO1DPOM

1

Cmax

X X⋆0

Zone I Zone II Zone III

Figure 21: Mutual insurer equity payoff (full participation) in t = 1 (ECMf1 ) for λ > 1 (solid line). The dashed

lines indicate the elements of the replicating portfolio.

186

IV

Stock

vs.MutualInsurers

PM1

ECMf1

L1

L1

A1

45

arctan(λ) − 45

DPOM1

+ RO1

Cmax

X X⋆0


Figure 22: Mutual insurer policyholder stake payoff in t = 1 (PM1 ) for λ > 1 (solid line). The payoff profile of the

default put option (DPOM1 ), the recovery option (RO1), and the equity stake given full participation (ECMf

1 )are drawn as dashed lines for comparison purposes.

4.3 Mutual insurer: partial equity participation 187

Arbitrage-free premium

Since, through the purchase of a policy in a mutual company, one ac-

quires the equity and the policyholder stake at the same time, the fair

premium of a mutual insurer must comprise the arbitrage-free price of

both components. Accordingly, we define ΠM0 = ΠM

0 (P , Cmax, λ) as

ΠM0 = PM

0 + ECMf0

= e−rEQ0 (L1) −RO0 −DPOM

0︸︷︷︸

policyholder stake

+ e−rEQ0 (A1 − L1) +RO0 +DPOM

0︸︷︷︸

equityholder stake

= e−rEQ0 (A1) . (72)

Thus, if members fully participate in the equity payoffs, purchasing a

policy from a mutual insurer is equivalent to acquiring a position the

company’s assets. Policyholders of an otherwise identical stock insurer

additionally would have to buy the common stock of the company in

order to establish the same payoff profile.

4.3 Mutual insurer claims structure:

partial participation in equity payoff

Equity stake

There is generally no secondary market for ownership stakes in mutual

insurance companies. As a consequence, payoffs from the equity stake of

a mutual insurer and thus its present value crucially depend on the pre-

mium refund policy of the management and the ability of the members

to prompt an initial public offering (IPO) or break-up of the company.

Let α be the payout (premium refund) ratio and pL the probability of

demutualization or liquidation of the company.144 The impact of these

144Under agency-theoretic considerations the firm’s management generally has apreference to retain as much capital in the company as possible. This aspect of theso-called owner-manager conflict lowers the premium refund ratio α. Furthermore, incontrast to a corporation, there are no blockholders in a mutual insurer. Therefore,pL will depend on the members’ ability to coordinate an agreement on the demutu-alization or liquidation of the firm.


parameters on the payoff profile of the equity stake depends on the zones

indicated in Figures 21, 22, and 23 which are determined by the realiza-

tions of assets and claims in t = 1. If A1 < X, i.e., the assets have

fallen below the default threshold (Zone I), the mutual is insolvent, bro-

ken up and the remaining assets are distributed to its members. Thus,

the equity stake is worthless and neither α nor pL are relevant in Zone

I. Furthermore, if X < A1 < L1 (Zone II) the mutual insurer exercises

the recovery option to charge additional payments (via the policyholder

stake). It is safe to assume that a mutual in financial distress will re-

frain from premium refunds, implying that members can only fully re-

alize the equity payoff via an IPO or the liquidation of the company.

Hence, in Zone II only pL has an influence on the present value of the

equity. Finally, in Zone III, where the company is solvent and does

not need to exercise the recovery option, members receive the whole eq-

uity value with probability pL, or a premium refund of α(A1 −L1) with

probability (1 − pL). We summarize these two cases in the parameter

γ = pL + (1 − pL)α, which can be interpreted as the expected value of

the equity stake in Zone III, normalized to unity. Since α ∈ [0; 1] and

pL ∈ [0; 1], we get γ ∈ [0; 1]. Under this set-up, the present value of a

mutual’s equity stake in the general case (recovery option and partially

realizable equity), ECM0 = ECM

0 (P , Cmax, λ, pL, α), can be described as

follows:

ECM0 = 0

︸︷︷︸

Zone I

+ pL

[

(λ− 1)DPOS0 − λPOX⋆

0 + POX0

]

︸︷︷︸

Zone II

+ γe−rEQ0 [max (A1 − L1; 0)]

︸︷︷︸

Zone III

= γ[

e−rEQ0 (A1 − L1) +DPOS

0

]

+ pL

(

λDPOS0 − λPOX⋆

0 + POX0 −DPOS

0

)

= γe−rEQ0 (A1 − L1) + γDPOS

0

+ pL(RO0 +DPOM

0 −DPOS0

)

= γe−rEQ0 (A1 − L1) − (pL − γ)DPOS

0

+ pL(RO0 +DPOM

0

). (73)


For a graphical verification of this expression refer to Figure 23. As

mentioned previously, the equity payoff in the interval [0, X] (Zone I) is

zero. Furthermore, the payoff profile between X and L1, i.e., in Zone

II, is characterized by an asymmetric butterfly spread, consisting of

(λ − 1) units of DPOS0 long, λ units of POX⋆

0 short and one unit of

POX0 long.145 Finally, the equity payoff in Zone III is equal to a long

stake in γ units of a simple call option on the assets with strike price

L1. Recalling Equation (61), we realize that this call option is exactly

the one describing the equity value of a stock insurance company. As

also illustrated in Figure 23, the consideration of the parameters pLand γ, which were introduced above, results in a flattening of the pay-

off of the mutual insurer’s equity stake in Zones II and III. In the ab-

sence of arbitrage, members of a mutual insurance company anticipate

that they can only partially access future cash flows arising from the

equity stake, implying a reduction of its present value. The difference

between ECMf0 and ECM

0 – represented by the shaded area in Figure 23

– is the discount in the present value of the equity stake resulting from

the incomplete participation of the current members in its future pay-

off. In our contingent claims framework, this ”non-realizable” equity,

ECMn0 = ECMn

0 (P , Cmax, λ, pL, α), has a price in t = 0 equal to

ECMn0 = ECMf

0 − ECM0

= e−rEQ0 (A1 − L1) +RO0 +DPOM

0︸︷︷︸

equity given full participation

− pL(RO0 +DPOM

0

)

− γe−rEQ0 (A1 − L1)

+ (pL − γ)DPOS0

realizable equity

= (1 − γ) e−rEQ0 (A1 − L1) + (pL − γ)DPOS

0

+ (1 − pL)(RO0 +DPOM

0

). (74)

In order to comprehensively understand the effect of pL, α, and, in

turn, γ, on the value of the equity stake of the mutual insurance company,

we consider two special cases. First of all, the expression for the present

145Note that for λ = 1, we have a standard put option butterfly spread.

190

IV

Stock

vs.MutualInsurers

ECM1

ECMf1

ECMn1

L1 A1

45

arctan(λ− 1)

arctan[pL(λ− 1)]

arctan(γ)

X X⋆

0


Figure 23: Mutual insurer (expected) equity payoff in t = 1 in case of partial equity participation (ECM1 ) for

λ > 1, pL, and α < 1 (solid line). The dashed line illustrates the equity stake given full participation (ECMf1 )

for comparison purposes.


value of the equity stake under full participation, i.e., Equation (64), is

nested in the more general Equation (73). The exact payoff profile we

saw in Figure 21 (ECMf1 ) in the previous section can only be realized

in the special case of pL = 1, i.e., full participation in the equity payoff

stream in all three zones. To see this note that pL = 1 directly results

in γ = 1 such that Equation (73) becomes Equation (64) and Equa-

tion (74) collapses to zero: there is no non-realizable equity component.

Apart from that, pL < 1 and α = 1 also results in γ = 1: the mutual

distributes the whole equity to its members when it is solvent, but when-

ever the recovery option is exercised, there are no premium refunds and

participation in the equity payoff is contingent on the probability of liq-

uidation pL. Consequently, the first term in Equation (74) disappears

and the remainder reduces to (1 − pL)(RO0 +DPOM

0 −DPOS). This

means that only the excess value of the recovery and default put option

of a mutual over the default put option of a stock insurer constitutes

non-realizable equity.146 While, in this case, the payoff profile in Zone

III is the same as for full equity participation, we get a flatter curve in

Zone II. Overall, in the arbitrage-free setting, realizable equity ECM0 and

non-realizable equity ECMn0 will always sum up to ECMf

0 , while pL and

α govern the size of these components relative to each other.

Policyholder stake

While participation in the future cash flows of the equity stake might be

limited, there are no such restrictions associated with the policyholder

stake. In other words, the present value of the policyholder stake re-

mains the same as in Section 4.2, comprising the default-free premium

e−rEQ0 (L1) as well as a short position in the recovery and the default

put option of the mutual. Thus, we have the same expression as in

Equation (71):

PM0 = e−rEQ

0 (L1) −RO0 −DPOM0 . (75)

146In case there is no such excess value, i.e., for λ = 1, the non-realizable equity iszero and the equity stake of the mutual equals that of the stock insurer.


Arbitrage-free premium

As explained at the end of Section 4.2, the arbitrage-free premium of

a mutual insurer must comprise the present values of both the equity

and the policyholder stake. In the general case of partial participation

in the equity payoff, however, the price of the equity stake splits into a

realizable and a non-realizable component. Yet, the overall level of the

mutual insurance company premium remains unchanged and equals the

expected discounted value of the firm’s assets in t = 1. Consequently, in

the general case, we replace Equation (72) with an alternative expression

for ΠM0 = ΠM

0 (P , Cmax, λ):

ΠM0 = PM

0 + ECM0 + ECMn

0

= e−rEQ0 (L1) −RO0 −DPOM

0︸︷︷︸

policyholder stake

+ γe−rEQ0 (A1 − L1)

− (pL − γ)DPOS0

+ pL(RO0 +DPOM

0

)

realizable equity stake

+ (1 − γ) e−rEQ0 (A1 − L1)

+ (pL − γ)DPOS0

+ (1 − pL)(RO0 +DPOM

0

)

non-realizable equity

= e−rEQ0 (A1) . (76)

4.4 Claims structure relationships

Below we briefly illustrate the theoretical impact of recovery option and

limited participation in equity payoffs on the premium of a mutual in-

surer relative to a comparable stock insurer. Imagine two insurance firms

with the exact same underlying assets and claims: one is founded as a

corporation and the other one adopts the legal form of a mutual. Fig-

ure 24 depicts the relationship between the claim structures of these two

companies in four distinct cases, characterized by different configurations

of recovery option and equity participation. As defined in Equation (63),

the marginal premium charged by a stock insurer equals the value of its

4.4

Cla

ims

structu

rerelation

ship

s193

stock insurer mutual insurer

PS0

πS0

ECS0

ΠM0

πM0

πM0

RO0 + DPOM0

−DPOS0

case

equityparticipation

excess of lossrecoveryoption

PM0

ECMf0

I

fullγ = 1

noλ = 1

PM0

ECMn0

ECM0

II

partialγ < 1

noλ = 1

PM0

ECMf0

III

fullγ = 1

yesλ > 1

PM0

ECMn0

ECM0

IV

partialγ < 1

yesλ > 1

Figure 24: Theoretical premium comparison


policyholder stake. The premium of the mutual insurer corresponding

to this reference case, however, depends on the appointed setting.

In Case I, the mutual insurance company is either not allowed to

charge additional premiums at all (i.e., Cmax = 0, which results in

DPOS0 = DPOM

0 )147 or the amount of additional premiums is restricted

to the actual deficit L1 − A1 (i.e., λ = 1, which results in DPOS0 =

DPOM0 + RO0)

148. In addition, the equity stake of the mutual is fully

realizable (pL = 1 and, hence, γ = 1). Comparing Equation (63) and

(71), we see that under these circumstances P S0 = PM

0 : there is no dif-

ference between the value of the policyholder stakes of a stock and a

mutual insurer. Moreover, comparing Equation (61) and (64), we no-

tice that both equity stakes have the same value, i.e., ECS0 = ECMf

0 .

Due to the fact that the equity of the mutual can be entirely realized

and there are no additional contributions in excess of a loss, the rights

of mutual members are economically identical to those of the combined

policyholder and ownership stake of the stock insurer. In other words,

since policyholders and owners coincide, the position in a mutual could

be replicated by simply purchasing both an insurance contract and an

appropriate amount of shares of the stock insurer. Hence, the aggregate

premium ΠM0 charged by a mutual should equal the premium of a stock

insurer, πS0 , plus the value of its equity ECS

0 .

In Case II, the mutual insurer’s company charter excludes additional

premiums in excess of a loss (λ = 1). However, its equity stake can-

not be fully realized (γ < 1). Since, in an arbitrage-free market, ra-

tional individuals anticipate this, the ownership stake of the mutual

insurer is separated into a realizable and a non-realizable component

and prospective mutual members are generally not willing to provide

the latter. Consequently, the mutual premium is now πM0 , i.e., ΠM

0 net

of the non-realizable equity ECMn0 . The full premium ΠM

0 can only be

demanded if members are being compensated for ECMn0 , e.g., through a

binding right to payments from future policyholders upon the beginning

147Refer to Equations (65) to (68).148Refer to Equation (68).

4.4 Claims structure relationships 195

of their membership in the mutual.149 It is important to note that, in

any case, the non-realizable equity needs to be paid in for the company

to be founded at all. This is due to the fact that less initial equity than

ECMf0 is associated with a lower expected payoff in t = 1. In anticipation

of this consequence, individuals will further reduce their willingness to

pay, eventually reaching an equilibrium where the value of both stakes

is zero. In this situation, the mutual insurance company cannot find

customers if it charges a positive arbitrage-free premium, since every

insurance policy would be associated with a negative net present value.

Thus, if the members do not provide ECMn0 , an external third party such

as a founding capital provider, whose capital repayment is contractually

guaranteed, would need to step in instead.

Case III represents the claims structure of the mutual if its equity is

fully realizable (pL = 1 and, hence, γ = 1) and its recovery option allows

to charge additional premiums over and above the actual loss (λ > 1).

Due to the excess of loss recovery option, the value of the equity position

increases and the value of the policyholder position decreases compared

to the stock insurer (and Case I) by an amount equal to the difference

between RO0 + DPOM0 and DPOS

0 . The higher λ, the bigger the shift

between both stakes. Since the recovery option solely redistributes value

between the stakes, the overall amount of assets within the company is

unchanged. Therefore, the overall mutual premium remains equal to ΠM0 .

Finally, the combined effect of partially realizable equity (γ < 1) and

excess of loss recovery option (λ > 1) is illustrated in Case IV. Again,

the equity stake splits into ECMn0 plus ECM

0 and πM0 denotes the full

mutual premium less the present value of the non-realizable equity. In

contrast to Case II, however, both equity components are slightly more

expensive, since value is shifted from the policyholder to the equity stake

via the excess of loss recovery option. As before, a non-zero arbitrage-

149In a multiperiod framework such compensation payments could be conducted atthe end of each period. For instance, the current members (from t = 0 to t = 1) wouldneed to receive the right to be paid an amount of ECMn

1in t = 1 by the members of

the following period (t = 1 to t = 2). This right is worth ECMn0

= e−rEQ0

(

ECMn1

)

today.


free solution can only be achieved if the non-realizable equity is paid in

as well. Hence, more expensive recovery options (for a higher λ), are

ceteris paribus associated with a more valuable (non-realizable) equity

and a lower πM0 .

5 Numerical analysis

In this section, we concretely describe assets and claims costs as con-

tinuous-time stochastic processes and present closed-form solutions for

the various option prices on which our model framework is based. Sub-

sequently, we provide a brief numerical example to further illustrate the

model mechanics as well as the effect of recovery options and equity par-

ticipation on the premium of a mutual insurer. In addition, based on the

numerical implementation of our model framework, we derive normative

insights with regard to feasible combinations of premium, safety level,

and capital structure of stock and mutual insurance companies.

5.1 Option pricing formulae

Suppose that assets are traded continuously in time and that the term

structure of interest rates is flat and deterministic. The insurance com-

panies’ assets are assumed to be stochastic and their dynamics is mod-

eled by the following Geometric Brownian Motion under the risk-neutral

measure Q:

dAt

At= rdt+ σAdWQ

At, (77)

where the drift is given by the risk-free interest rate r and σA denotes the

volatility of assets and dWAt is a standard Wiener process under Q.150

150In this set-up, asset returns are normally distributed. While, in most cases, this ismerely an approximation of the empirically observed distributions (see, e.g., Officer,1972; Akgiray and Booth, 1988; Lau et al., 1990), it simplifies matters by allowing usto apply closed-form solutions. Since insurance companies tend to hold a considerablefraction of bonds in their investment portfolios, an alternative set-up could includeterm structure models (see, e.g., Vasicek, 1977; Cox et al., 1981).

5.1 Option pricing formulae 197

The insurer’s claims are assumed to be deterministic.151 Under these

assumptions, closed-form solutions for the present values of the various

European options described in Section 4 are available; see Black and

Scholes (1973). In line with the one-period model from Section 4, the

present value of the stock insurer default put option, DPOS0 , can be

computed as follows:

DPOS0 = e−rEQ

0

(DPOS

1

)= e−rEQ

0 [max (L1 −A1; 0)]

= e−rL1Φ(−d1) −A0Φ(−d2), (78)

where Φ(x) is the cumulative distribution function of the standard nor-

mal distribution and

d1 =ln(A0/L1) + r − σ2

A/2

σA,

d2 =ln(A0/L1) + r + σ2

A/2

σA.

In addition, the present value of the put option POX0 in Equation

(66), which is one of the two building blocks of the default put option of

a mutual insurer (DPOM), can be calculated using the following formula:

POX0 = e−rEQ

0

(POX

1

)= e−rEQ

0 [max (X −A1; 0)]

= e−rXΦ(x1) −A0Φ(x2)

= e−r(L1 − Cmax)Φ(−x1) −A0Φ(−x2), (79)

where

x1 =ln [A0/(L1 − Cmax)] + r − σ2

A/2

σA,

x2 =ln [A0/(L1 − Cmax)] + r + σ2

A/2

σA.

151This decision is made for reasons of computational simplicity. Since the modelframework in Section 4 has been deliberately kept on a general level, different as-sumptions for the asset and claims dynamics as well as associated option-pricingframeworks can be applied without loss of generality.


The second building block of the DPOM is a cash-or-nothing binary

put option which pays Cmax if A1 < X and zero otherwise. Rubinstein

and Reiner (1991) show that the price of this option is equal to:

BPO0 = e−rCmaxΦ(−x1). (80)

Using Equations (79) and (80), the formula for the present value of

the default put option of the mutual insurer, DPOM0 , can be derived:

DPOM0 = POX

0 +BPO0

= e−r(L1 − Cmax)Φ(−x1) −A0Φ(−x2) + e−rCmaxΦ(−x1)= e−rL1Φ(−x1) −A0Φ(−x2). (81)

This formula somehow resembles Equation (78), which describes the

price of the default put option of a stock insurer. Yet, the probabili-

ties with which the parameters e−rL1 and A0 are weighted differ. To

grasp the intuition behind this, recall from Section 4.2 that the assets

A1 have to fall below the threshold X before the default put option of

the mutual insurer is in the money. Contingent on A1 < X, however,

the payoff profiles of DPOM and DPOS are congruent (refer back to

Figure 19): in the area A1 < X, both options pay L1 −A1. As a result,

the formula for DPOM0 includes e−rL1 and A0, but weighted with the

probabilities Φ(−x1) and Φ(−x2) instead of Φ(−d1) and Φ(−d2).

Finally, to calculate the value of the recovery option in the general

case (i.e., for λ > 1), we additionally need the closed-form solution for

the put option POX⋆

0 . Following the same rationale as above, we get

POX⋆

0 = e−rEQ0

(

POX⋆

1

)

= e−rEQ0 [max (X⋆ −A1; 0)]

= e−rX⋆Φ(z1) −A0Φ(z2)

= e−r(L1 −1

λCmax)Φ(−z1) −A0Φ(−z2), (82)

with

z1 =ln[A0/(L1 − 1

λCmax)

]+ r − σ2

A/2

σA,

5.2 The impact of recovery option and equity participation 199

z2 =ln[A0/(L1 − 1

λCmax)

]+ r + σ2

A/2

σA.

Combining Equations (78), (80), and (82), the value of the recovery

option can be expressed as:152

RO0 = λDPOS0 − λPOX⋆

0 −BPO0,

= λ[e−rL1Φ(−d1) −A0Φ(−d2)

]

− λ[e−r(L1 −1

λCmax)Φ(−z1) −A0Φ(−z2)]

− e−rCmaxΦ(−x1)= λe−rL1Φ(−d1) − λA0Φ(−d2) − λe−rL1Φ(−z1)

+ λA0Φ(−z2) + e−rCmaxΦ(−z1) − e−rCmaxΦ(−x1)= λ

e−rL1 [Φ(−d1) − Φ(−z1)] −A0 [Φ(−d2) − Φ(−z2)]

+ e−rCmax [Φ(−z1) − Φ(−x1)] . (83)

5.2 The impact of recovery option

and participation in equity payoff

Having determined asset and claims dynamics as well as the associated

option pricing formulae, the equity and policyholder stake of mutual and

stock insurance companies can now be valued. Table 29 contains the ba-

sic input parameters used in our numerical examples and the resulting

present values (PVs) for the stock insurer.

The first three columns of Table 30 illustrate the impact of the recov-

ery option in a mutual insurance company with full participation in the

equity payoff stream (pL = 1 and α = 1).153 For λ = 1, i.e., no excess

of loss recovery option, the value of the default put option of the stock

insurer DPOS0 (0.2481) perfectly splits into RO0 (0.2463) and DPOM

0

(0.0018). In addition, equity ECMf0 (30.2481) and policyholder stake PM

0

152Note that [Φ(−d1) − Φ(−z1)] is Pr(X⋆ < A1 < L1) and [Φ(−z1) − Φ(−x1)] isPr(X < A1 < X⋆).153These numerical results correspond to Case I and III in Section 4.4.


A0 100 initial value of the assets

L0 70 initial value of the liabilities

σA 0.20 volatility of the asset returns

r 0.03 risk free rate

DPOS0

0.2481 PV of the stock insurer’s default put option

ECS0

30.2481 PV of the stock insurer’s equity

PS0

= πS0

69.7519 PV of the stock insurer’s policyholder claims

Table 29: Input parameters and values for DPOS0 , ECS

0 , and P S0

(69.7519) of the mutual are worth the same as those of the stock insurer

shown in Table 29: although the two companies differ in terms of le-

gal form, they are economically identical in this case. Since, through

a membership in the mutual, one acquires both stakes, the mutual pre-

mium (ΠM0 = 30.2481 + 69.7519 = 100) equals the present value of the

assets. For an increasing λ, however, we observe a non-linear growth in

RO0, resulting in a value of 0.2708 in case 110 percent of a deficit can

be demanded from mutual members, i.e., λ = 1.1. In this case, the sum

RO0+DPOM0 (0.2726) is almost ten percent higher than DPOS

0 (0.2481).

Furthermore, in Table 30 we see that the excess of loss recovery option

redistributes value from the policyholder to the equityholder stake since

PM0 falls and ECMf

0 , ECMn0 , as well as ECM

0 rise in λ. Analogously to

the default put option of the stock insurer, DPOM0 measures the safety

level of a mutual insurance company’s policyholder stake. As DPOM0

remains the same (0.0018) for all values of λ and is always lower than

DPOS0 (0.2481), the mutual insurer with recovery option has a higher

safety level than the otherwise identical stock insurer.

The three columns in the center of Table 30 show the case where

mutual members partially participate in the equity payoff (pL = 0.1 and

α = 0.1).154 Again, for λ = 1, we have RO0+DPOM0 = DPOS

0 = 0.2481.

This time, however, the total equity value ECMf0 (30.2481) splits into

a realizable component ECM0 (5.7471) and a non-realizable component

ECMn0 (24.5010). The former is considerably lower than the latter, since

154See Case II and IV in Section 4.4.

5.2

The

impact

ofrecovery

option

and

equity

particip

ation201

pL = 1, α = 1 pL = 0.1, α = 0.1 pL = 0, α = 0

λ 1.00 1.05 1.10 1.00 1.05 1.10 1.00 1.05 1.10

DPOM0

0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018

RO0 0.2463 0.2586 0.2708 0.2463 0.2586 0.2708 0.2463 0.2586 0.2708

DPOM0

+ RO0 0.2481 0.2604 0.2726 0.2481 0.2604 0.2726 0.2481 0.2604 0.2726

ECM0

30.2481 30.2604 30.2726 5.7471 5.7484 5.7496 0.0000 0.0000 0.0000

ECMn0

0.0000 0.0000 0.0000 24.5010 24.5120 24.5230 30.2481 30.2604 30.2726

ECMf0

30.2481 30.2604 30.2726 30.2481 30.2604 30.2726 30.2481 30.2604 30.2726

PM0

69.7519 69.7396 69.7274 69.7519 69.7396 69.7274 69.7519 69.7396 69.7274

ΠM0

100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000

Table 30: Impact of recovery option and equity participation given Cmax = 25


the figures are based on a fairly low premium refund rate and probabil-

ity of liquidation. As in the previous case, a rise in λ implies a more

expensive recovery option. The associated value redistribution reduces

PM0 and increases both components of the equity stake. Consistent with

our choice of pL and α, however, ECMn0 absorbs a relatively larger share.

Although the ten percent likelihood of liquidation assumed for this nu-

merical example probably has to be considered relatively high from a

real world perspective, the realizable equity stake has already become

quite small. Consequently, even lower values for pL, which are perfectly

conceivable, would result in a situation where virtually the whole eq-

uity is attributed to the non-realizable component. As explained in

Section 4.4, the capital in such a case would need to be provided by

a third party, since, under the arbitrage-free framework applied, mutual

members would not be prepared to incur a negative net present value

investment. Again, DPOM0 = 0.0018 < DPOS

0 = 0.2481 for all λ. There-

fore, as in the previous example, the mutual insurer’s policyholder stake

exhibits a higher safety level than that of the stock insurer. The last

three columns of Table 30 contain the numerical results when the equity

stake is not realizable at all (pL = 0 and α = 0). Obviously, in this case,

the whole equity value is attributed to the non-realizable component.

Apart from λ, the maximum amount of additional premiums Cmax is

a key determinant of the recovery option value and has a direct impact on

the safety level of the firm. Table 31 illustrates that a recovery option

does not exist if Cmax = 0.155 Instead, the default put option of the

mutual insurance company is exactly the same as for a stock insurer

(DPOM0 = DPOS

0 = 0.2481). The higher Cmax, i.e., the less binding

the upper limit on additional payments, the more valuable becomes the

recovery option. In addition, an increase in Cmax simultaneously results

in a decline of DPOM0 , implying an improving safety level. For Cmax =

40, we get RO0 = 0.2726 and the mutual’s default put option is (almost)

worthless because the value of the assets in t = 1 would have to drop by

more than 40 below the value of the claims for it to be in the money. As

in Table 30, the decrease in PM0 due to the incremental growth in RO0

155See Figure 20 in Section 4.2.

5.3 Premium, safety level, and equity capital 203

Cmax 0 10 20 30 40

DPOM0 0.2481 0.1016 0.0103 0.0002 0.0000

RO0 0.0000 0.1604 0.2612 0.2727 0.2729

DPOM0 +RO0 0.2481 0.2620 0.2715 0.2729 0.2729

ECM0 5.7471 5.7485 5.7495 5.7496 5.7496

ECMn0 24.5010 24.5135 24.5221 24.5233 24.5233

ECMf0 30.2481 30.2620 30.2716 30.2729 30.2729

PM0 69.7519 69.7380 69.7284 69.7271 69.7271

ΠM0 100.0000 100.0000 100.0000 100.0000 100.0000

Table 31: Impact of the maximum amount of additional contributions(Cmax) given partial participation in the equity payoffs of the mutualfirm and an excess of loss recovery option: pL = 0.1, α = 0.1, λ = 1.1

is counterbalanced by an increased value of the equity stake (realizable

and non-realizable component).

5.3 Stock vs. mutual insurers:

premium, safety level, and equity capital

In the following, we compare a stock and a mutual insurer with identical

underlying assets and claims with regard to the three central magnitudes

premium size, safety level, and equity capital, considering cases with and

without recovery option as well as full and partial participation in equity

payoffs. Again, the calculations have been based on the parameter values

in Table 29. While other configurations would change the magnitude of

the observed effects, their direction remains the same.

We begin with the case where the mutual insurer does not have a

recovery option (Cmax = 0) and its equity stake can be fully realized

by the members. In Figure 25, the arbitrage-free mutual and stock in-

surer premiums have been plotted against the value of the respective

equity stakes. Under the arbitrage-free framework used, both curves

must start at zero. Let us first look at the solid curve, which rep-

204

IV

Stock

vs.MutualInsurers

0 5 10 15 20 25

6065

7075

80

85

EC0S, EC0

Mf

P0S=

π0S, P

0M, Π

0M

Curves:Π0

M (Mutual premiums)L0 (PV of claims costs)L0 −DPO0

M (Safety levels of mutuals with RO)P0

M= P0

S = π0S (PV of policyholder stakes)

Points:Π0

M= L0

Figure 25: Equity-premium combinations for full equity participation and no recovery option

5.3 Premium, safety level, and equity capital 205

resents equity-premium-combinations for the stock insurer and equity-

policyholder-stake-combinations for the mutual insurer. As the amount

of initial equity capital is raised, the stock insurer premium converges

towards the present value of the claims costs L0 (represented by a dot-

ted horizontal line).156 For any amount of equity capital, the distance

between L0 and the solid curve equals the present value of the stock

insurer’s default put option (DPOS0 ), which, in this case, is identical

to that of the mutual insurer (DPOM0 ) since it is assumed that the lat-

ter does not have a recovery option. The vertical dotted line is meant

to serve as a concrete example. As a consequence, if they are identi-

cally capitalized, mutual and stock insurer offer contracts with the same

safety level. In addition, more equity capital is associated with a de-

cline in DPOS0 (= DPOM

0 ) due to the fact that a larger equity buffer

reduces the likelihood of the assets dropping below the claims costs at

the end of the period. The dashed curve represents premiums of the

mutual insurer. Since members have to purchase both stakes, it lies

strictly above the solid curve. Thus, in the absence of a recovery option,

if both companies hold the same amount of equity capital and members

of the mutual insurer can fully participate in its equity stake, then they

should be charged higher premiums than the policyholders of the stock

insurer. Another relevant observation is related to the point where the

ΠM0 -curve intersects the L0-line (marked by a small circle). If the mutual

insurer holds more initial equity capital than associated with this point,

its premium must be strictly higher than that of the stock insurer, no

matter how well capitalized the latter is. This is due to the fact that the

πS0 -curve converges to but never exceeds L0.

Next, we introduce a basic recovery option (Cmax > 0, λ = 1), while

still allowing for full participation in the equity payoffs of the mutual

insurer. As discussed in Section 4.2, the recovery option enables mutual

insurers to stay solvent and satisfy all claims, even if their equity cap-

ital is fully exhausted. More specifically, a mutual insurer is bankrupt

156In Section 4.1 we explained that the fair stock insurance premium equals thepresent value of the policyholder stake, i.e., πS = PS

0. Besides, L0 is the default-free

premium.


only if the deficit of assets relative to liabilities exceeds the limit on ad-

ditional premiums (Cmax), which implies DPOS0 = RO0 + DPOM

0 or

DPOM0 < DPOS

0 . This is illustrated in Figure 26, where we now have

an additional dotdashed curve, reflecting the safety levels of the mutual

insurer. While DPOS0 is still represented by the distance from L0 to

the solid curve, the distance between L0 and the dotdashed curve equals

DPOM0 . Since the dotdashed lies strictly above the solid curve, the mu-

tual insurer with recovery option exhibits a strictly better safety level

than the identically capitalized stock insurer. In other words, the mu-

tual insurer with recovery option needs less equity capital to achieve the

same safety level as the stock insurer.157 Furthermore, analogously to

Figure 25, the mutual must charge a higher premium than the stock in-

surer if it holds more equity capital than associated with the intersection

of the ΠM0 -curve and the L0-line. Consequently, safety level and premium

of a well-capitalized insurance company should be higher if it adopts the

legal form of a mutual. In contrast to the results in Figure 25, however,

we now find capitalizations for which the premium of the mutual can be

equal to or lower than that of the stock insurer with an identical safety

level. To see this, we focus on the intersection between the dashed (ΠM0 )

and dotdashed curve (L0 − DPOM0 ), which has been highlighted by a

black dot. If the mutual insurance company holds precisely this much

equity capital, it exhibits the same safety level and charges the same

premium as the stock insurer with the amount of equity capital which

corresponds to the black triangle.158 Right of the black dot, the mutual

charges more and left of the black dot it charges less than the stock in-

surer with the same safety level.

In Figure 27, we account for limited participation in the equity payoff

stream of the mutual insurer by splitting its capital into the realizable

and the non-realizable component.159 However, as explained in Sec-

157Intuitively, the recovery option can be interpreted as an equity substitute. Hence,in most jurisdictions mutual insurers can—to some extent—account for their recoveryoption when calculating solvency capital charges.158To find the latter, follow an imaginary horizontal line from the black dot to the

right until it reaches the πS0

-curve.159The non-realizable equity is calculated based on pL = 0.1 and α = 0.1.

5.3

Prem

ium

,safety

level,an

deq

uity

capital

207

0 5 10 15 20 25

6065

7075

8085

EC0S, EC0

Mf

P0S=

π0S, P

0M, Π

0MCurves:Π0

M (mutual premiums in PV terms)L0 (PV of claims costs)L0 −DPO0

M (safety levels of mutuals with RO)P0

M= P0


Points:Π0

M= L0 −DPO0

M Π0

M= L0

Figure 26: Equity-premium combinations for full equity participation and recovery option

208

IV

Stock

vs.MutualInsurers

0 5 10 15 20 25

6065

7075

80

85

EC0S, EC0

Mf

P0S=

π0S, P

0M, π

0M

Curves:π0

M (Mutual premiums)L0 (PV of claims costs)L0 −DPO0

M (safety levels of mutuals with RO)P0

M= P0


Points:π0

M= L0 −DPO0

M π0

M= L0

Figure 27: Equity-premium combinations for partial equity participation and recovery option


tion 4.4, both components need to be paid in for the company to be able

to begin business. Therefore, the x-axis is still based on the full value of

the mutual’s equity ECMf0 and the dotdashed curve, representing safety

levels of the mutual insurer, is unaffected by this change. In contrast

to that, however, the non-realizable equity is excluded from the mutual

premium, meaning that πM0 instead of ΠM

0 is shown on the y-axis. The

intuition behind this proceeding is that members are either compensated

by an amount equal to the present value of the non-realizable equity, or

the latter is provided by a third party, e.g., a founding capital provider.

Since, for each amount of initial equity capital, the mutual premium is

now lower than in the case of full equity participation (Figures 25 and

26), the πM0 -curve has a smaller slope than the ΠM

0 -curve (plotted in light

grey). Hence, for a decreasing probability of liquidation pL, the premi-

ums of the mutual insurance company converge to those of the identically

capitalized stock insurer as the non-realizable equity is not borne by the

members. Besides, the dashed curve (ΠM0 ) now intersects the dotted line

(L0) further to the right such that we have a broader range of capitaliza-

tions, which allow the mutual to match the premium of the stock insurer.

Similarly, the intersection between the dashed (πM0 ) and the dotdashed

curve (L0 −DPOM0 ) has been shifted to the right, implying a larger set

of stock insurer capital structures for which the mutual is able to provide

less expensive policies with the same safety level.160

6 Economic implications

Due to competition in insurance markets one might expect the premi-

ums of stock and mutual companies not to differ significantly (see, e.g.,

Mayers and Smith, 1988). This view is partially supported by the empir-

ical evidence we presented in Section 3. Despite the different results for

two common estimators we were able to conclude that, in any case, mu-

tuals do not charge higher premiums than stock firms. If at all, it seems

that stock insurer policies are more expensive. We can now combine

160The black triangle, representing that particular capitalization of the stock insurerfor which the mutual can chose to match both its safety level and its premium, isnow outside the scale of Figure 27.


these empirical findings with the normative results from the previous

section to derive economic implications with regard to the relationship

of stock and mutual insurer premiums. To begin with, we sum up un-

der which specific circumstances the contingent claims model framework

supports the equality of premiums.

First of all, in the absence of a recovery option and if members can

fully participate in a mutual insurer’s equity payoffs, its premium can

only be similar to that of a stock insurer when its capitalization and

safety level are very low (Figure 25). However, such a scenario is un-

likely to occur in practice since, in most jurisdictions, solvency regulation

frameworks ensure a minimum safety level for insurance companies.161

If the equity of a mutual insurer without recovery option is only partially

realizable, i.e., its premium curve in Figure 25 becomes flatter, there are

more likely to be capitalizations which allow the mutual to charge the

same or a lower premium than the stock insurer, while still conforming

to the applicable solvency standards. By holding less equity capital than

the stock insurer, however, the mutual would also maintain a compara-

tively lower safety level.

Secondly, even in the presence of a recovery option, the mutual com-

pany with fully realizable equity is only able to match or undercut the

prices of the stock insurer when featuring less initial equity capital. Yet,

despite the generally smaller equity buffer, the mutual’s safety level could

be lower than, similar to, or even higher than that of the stock insurer

with the same premium, depending on its capitalization.162 Again, the

practical relevance of this scenario depends on the lower limit for the

safety level as established by the applicable solvency regulation. How-

ever, due to the recovery option, it is less likely that all capital structures

which enable mutuals to charge less than stock insurers are ruled out.

161Within our model framework, a minimum safety level is equivalent to an upperlimit on the present value of the default put option. Hence, it could be reflected inFigures 25 to 27 by means of a vertical line, the area to the left of which would notbe admissible under the prevailing solvency standards.162Consider the area around the black dot in Figure 26.


Finally, reconsider the situation where the equity of mutuals with

recovery option is only partially realizable (Figure 27). As before, of-

fering policies for the same or a lower premium than the stock insurer

requires that the mutual commands less equity. However, in this case

it will be more likely that the mutual also complies with the respective

solvency standards since for any given capital structure, a larger fraction

of non-realizable equity is associated with a lower mutual premium.163

As mentioned in Section 1, there are generally no liquid secondary mar-

kets for ownership stakes of mutuals. Consequently, the non-realizable

equity should be rather large, leading us to believe that this might the

most relevant case from a practical perspective.

To sum up, while our arbitrage-free model does not generally exclude

the possibility of the mutual premium being lower than the stock insurer

premium, in any case, such an outcome would require the mutual to

hold less equity capital than the stock insurer. Within the empirical

analysis, however, we explicitly controlled for capital structure effects

as well as other premium determinants such as underwriting risk and

administration costs. Thus, it appears that the empirically observed

prices are not arbitrage-free in the sense of the applied contingent claims

approach. In other words, from a normative perspective, policies offered

by stock insurers seem to be overpriced relative to policies of mutuals.

Since this situation is not a theoretical equilibrium, it can only prevail

due to further factors which are exogenous to our model. One such

aspect might be that we consider stakes in present value terms while

observed mutual premiums are quoted as up-front cash flows, i.e., net of

the recovery option value which can be viewed as an ex-post premium

component. However, due to its rather low value compared to the overall

mutual premium (see numerical analysis in Section 5), it is safe to assume

that the recovery option has a minor impact on the results. Moreover,

the deviation from the theoretical premium relationship could be caused

by superior marketing and sales efforts of stock companies. Although

163Recall that, relative to Figure 26, the black dot and triangle in Figure 27 areshifted to the right.


this might be a reason for the persistence of economic rents, its impact

is difficult to assess in the absence of empirical work on the subject.

Another point to be taken into account is that asymmetric informa-

tion can be an important issue in insurance markets. As explained in

Section 4.4, the notion of perfectly informed individuals which underlies

our contingent claims context implies that a mutual insurance company

would not be able to attract customers if its premium includes the present

value of non-realizable equity. Yet, in a situation where prospective mu-

tual members are unaware of economic differences associated with the

legal form of insurance companies, they are unable to correctly assess

the value of a policy. Therefore, the deviation from our arbitrage-free

results might occur because mutual members do not have enough infor-

mation or are not financially literate enough to determine the fair price

of both stakes included in the mutual premium. Asymmetric information

could lead to a scenario in which individuals actually pay for all or part

of the non-realizable equity without being compensated in some form.

Evidently, this would imply a transfer of wealth to an unknown group

of future profiteers such as, e.g., a generation of policyholders which

participates in the liquidation or demutualization of the firm. However,

such a violation of the no-arbitrage condition does not need to be re-

curring. Since most of the mutual insurance companies in our sample

are rather old and well-capitalized firms (see Table 27), wealth transfers

could have taken place in the past. Some of the affected individuals

might have already left the company without adequate compensation.

Current members benefit from this development as an accumulation of

equity reserves through violations of the no-arbitrage condition in the

past would imply that mutuals are now able to offer policies for a lower

premium than stock insurers. Alternatively, wealth transfers could also

persist between the policyholders and owners of the stock insurance com-

panies, implying that the former overpay for their contracts. Finally, a

combination of these sorts of wealth transfers within stock and mutual

organizations is conceivable.

7 Conclusion 213

7 Conclusion

In this paper, we empirically and theoretically analyze the relationship

between the insurance premium of stock and mutual companies. Eval-

uating panel data for the German motor liability insurance sector, we

do not find evidence that mutuals charge significantly higher premiums

than stock insurers. If at all, it seems that stock insurer policies are more

expensive. Subsequently, we employ a comprehensive model framework

for the arbitrage-free pricing of stock and mutual insurance contracts.

Based on a numerical implementation of our model, we then compare

stock and mutual insurance companies with regard to the three central

magnitudes premium size, safety level, and equity capital. Although

we identify certain circumstances under which the mutual’s premium

should be equal to or smaller than the stock insurer’s, these situations

would generally require the mutual to hold less capital than the stock

insurer. This being inconsistent with our empirical results, it appears

that policies offered by stock insurers are overpriced relative to policies

of mutuals.

Although various reasons for the observed deviation of our empiri-

cal and theoretical results are conceivable, we believe a violation of the

no-arbitrage principle due to asymmetric information to be the most

plausible explanation. Therefore, we argue that the documented dis-

crepancies are an indicator for likely wealth transfers between different

stakeholder groups of mutual and stock companies. A more detailed

identification of the size and direction of these wealth transfers could be

an interesting avenue for future research. Since such an analysis would

need to be based on a separate consideration of the different stakes, our

contingent claims model framework is well suited for an application in

this context. On the empirical side, however, more detailed insurance

company information would be required. Another interesting research

question centers around the coexistence of stock and mutual insurance

companies. Our normative results could be a starting point for a further

consideration of this topic. As previously discussed, an arbitrage-free

market implies that rational individuals would not be willing to pay for


the non-realizable component of the equity stake. Hence, we suggested

that mutual companies can only come into existence if, e.g., their initial

members are granted the right to compensation payments for the non-

realizable equity by future member generations or if a third party acts

as founding capital provider. Since both alternatives are rarely observed

in practice, it would be interesting to explore other possibilities which

enable mutuals to coexist with stock companies.

References 215

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Curriculum Vitae

Personal InformationName: Alexander Braun

Date of Birth: 13th of April 1981

Place of Birth: Tuttlingen, Germany

Nationality: German

Education02/2009–present University of St. Gallen (HSG), Switzerland

Doctoral Studies in Finance

07/2010–08/2010 University of Michigan, Ann Arbor, United States

Summer Program in Quantitative Research Methods

10/2003–05/2007 University of Mannheim, Germany

Diplom-Kaufmann (Majors: Banking and Finance, Marketing)

02/2004–12/2004 Monash University, Melbourne, Australia

Study Abroad Program (Major: International Management)

10/2001–08/2003 University of Mannheim, Germany

Pre-Diploma in Management and Intercultural Studies

09/1991–06/2000 Immanuel-Kant-Gymnasium, Tuttlingen, Germany

Abitur (A-Levels)

Work Experience

02/2009–present Institute of Insurance Economics

University of St. Gallen, Switzerland

Project Manager and Research Associate

07/2007–09/2008 Lehman Brothers Ltd., London, United Kingdom

Senior Analyst (full-time), Capital Markets Division

02/2006–04/2006 Deutsche Bank AG, Frankfurt am Main, Germany

Intern, Global Markets

09/2005–01/2006 SAP Deutschland AG & Co. KG, Walldorf, Germany

Working Student (part-time), Marketing Analysis

01/2005–04/2005 SAP Deutschland AG & Co. KG, Walldorf, Germany

Intern, Strategic Marketing and Project Management

09/2003–10/2003 BASF AG, Ludwigshafen, Germany

Intern, Management Accounting

05/2001–07/2001 Karl Storz GmbH & Co. KG, Tuttlingen, Germany

Intern, Accounting and Product Management

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