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Variance Advancing the Science of Risk
4350 North Fairfax Drive Suite 250 Arlington, Virginia 22203 www.variancejournal.org
2010 V
OLU
ME
04
IS
SU
E 0
2
2010121 Bootstrap Estimation of the Predictive
Distributions of Reserves Using Paid and Incurred Claims by Huijuan Liu and Richard Verrall
136 Robustifying Reserving by Gary G. Venter and Dumaria R. Tampubolon
155 Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach by Jackie Li
170 The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium by Stephen P. D’Arcy, Richard A. Derrig, and Herbert I. Weisberg
191 Optimal Layers for Catastrophe Reinsurance by Luyang Fu and C. K. “Stan” Khury
VOLUME 04 ISSUE 02
VARIANCE MissionVariance is a peer-reviewed journal published by the Casualty Actuarial Society to disseminate work of interest to casualty actuaries worldwide. The focus of Variance is original practical and theoretical research in casualty actuarial science. Significant survey or similar articles are also considered for publication. Membership in the Casualty Actuarial Society is not a prerequisite for submitting papers to the journal and submission by non-CAS members is encouraged.
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Table of Contents115 A Note from the Editor by Roger W. Bovard
116 Contributors to this Issue
121 Bootstrap Estimation of the Predictive Distributions of Reserves Using
Paid and Incurred Claims by Huijuan Liu and Richard Verrall
This paper presents a bootstrap approach to estimate the prediction distributions of reserves produced by the Munich chain ladder (MCL) model. The MCL model was introduced by Quarg and Mack (2004) and takes into account both paid and incurred claims information. In order to produce bootstrap distributions, this paper addresses the application of bootstrapping methods to dependent data, with the consequence that correlations are considered. Numerical examples are provided to illustrate the algorithm and the prediction errors are compared for the new bootstrapping method applied to MCL and a more standard bootstrapping method applied to the chain-ladder technique.
136 Robustifying Reserving by Gary G. Venter and Dumaria R. Tampubolon
Robust statistical procedures have a growing body of literature and have been applied to loss severity fitting in actuarial applications. An introduction of robust methods for loss reserving is presented in this paper. In particular, following Tampubolon (2008), reserving models for a development triangle are compared based on the sensitivity of the reserve estimates to changes in individual data points. This measure of sensitivity is then related to the generalized degrees of freedom used by the model at each point.
155 Prediction Error of the Future Claims Component of Premium Liabilities
under the Loss Ratio Approach by Jackie Li
In this paper we construct a stochastic model and derive approximation formulae to estimate the standard error of prediction under the loss ratio approach of assessing premium liabilities. We focus on the future claims component of premium liabilities and examine the weighted and simple average loss ratio estimators. The resulting mean square error of prediction contains the process error component and the estimation error component, in which the former refers to future claims variability while the latter refers to the uncertainty in parameter estimation. We illustrate the application of our model to public liability data and simulated data.
114 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Variance Advancing the Science of Risk
170 The Economics of Insurance Fraud Investigation: Evidence of a Nash
Equilibrium by Stephen P. D’Arcy, Richard A. Derrig, and Herbert I. Weisberg
The behavior of competing insurance companies investigating insurance fraud follows one of several Nash Equilibria under which companies consider the claim savings, net of investigation cost, on a portion, or all, of the total claim. This behavior can reduce the effectiveness of investigations when two or more competing insurers are involved. Cost savings are reduced if the suboptimal equilibrium prevails, and may instead induce fraudulent claim behavior and lead to higher insurance premiums. Alternative cooperative and noncooperative arrangements are examined that could reduce or eliminate this potential inefficiency.
191 Optimal Layers for Catastrophe Reinsurance by Luyang Fu and C. K. “Stan” Khury
Insurers purchase catastrophe reinsurance primarily to reduce underwriting risk in any one experience period and thus enhance the stability of their income stream over time. Reinsurance comes at a cost and therefore it is important to maintain a balance between the perceived benefit of buying catastrophe reinsurance and its cost. This study presents a methodology for determining the optimal catastrophe reinsurance layer by maximizing the risk-adjusted underwriting profit within a classical mean-variance framework.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 115
Variance Advancing the Science of Risk
When I started off as an actuarial student, my mentors in life insurance introduced me to the technique known as “general reasoning.” One of the first examples I encountered was a derivation for the present value (P) of an ordinary annuity certain. The general reasoning derivation goes as follows: The sum of $1 invested at rate i will yield annual income of i per year for n years (present value i*P) with the $1 principal still intact at the end of n years (present value vn). Thus 1=iP+vn. From this expression, the well-known formula for P follows by simple algebra. I recall being impressed with the way this technique gets to the heart of the matter while bypassing the somewhat messier algebra associated with the mathematical derivation.
There are a number of such general reasoning demonstrations in actuarial mathematics on the life side, but I have yet to encounter any in property and casualty insurance. However, I do occasionally encounter another technique for deriving useful results in an elegant manner while avoiding tedious algebraic manipulation. The technique I have in mind is transition to a higher level of abstraction. An example of this technique can be found in the very readable paper “Credibility Formulas of the Updating Type” by Jones and Gerber. The derivation in this paper is somewhat abstract, but accessible to most actuaries. The benefit of abstraction is a smoother flow resulting in increased understanding of the subject matter. A reduction in the level of abstraction would probably force a substantial increase in algebraic manipulation, which is actually more distracting than illuminating. In an appendix to the paper, the authors present an alternate derivation that is even shorter, but more abstract. Readers familiar with linear algebra concepts are able to appreciate the shorter derivation.
These observations are intended for readers as well as authors. It is the na-ture of those who author technical papers to build on other technical papers previously published. Thus, knowledge is advancing as well as increasing. As a practical matter, I encourage readers to invest time mastering some of the more advanced tools required to stay abreast.
Jones, D.A., and H.U. Gerber, “Credibility Formulas of the Updating Type,” Transactions of the
Society of Actuaries 27, 1975, pp. 31-46. Available for download from the Society of Actuaries
Web Site: http://www.soa.org/library/research/transactions-of-society-of-actuaries/1975/
january/tsa75v274.pdf
Roger Bovard
A Note from the Editor
116 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Variance Advancing the Science of Risk
Contributors to this Issue
Stephen P. D’Arcy
Stephen P. D’Arcy, FCAS, MAAA, Ph.D., is professor emeritus of fi-nance at the University of Illinois, a visiting lecturer at California State University Fullerton’s Mihaylo Col-lege of Business and Economics, and President of D’Arcy Risk Consult-ing, Inc. He is a past president of the Casualty Actuarial Society and of the American Risk and Insurance Asso-ciation.
Luyang Fu
Luyang Fu, FCAS, MAAA, is the di-rector of predictive modeling for State Auto Insurance Companies, where he leads the developments of personal lines pricing models, com-mercial lines underwriting models, and corporate DFA and ERM mod-els. Prior to joining State Auto, he served in various actuarial roles with both Grange Insurance and Bristol West Insurance. He holds a B.S. and an M.S. in economics from Renmin University of China, and a M.S. in finance and a doctorate in agricul-tural and consumer economics from University of Illinois at Urbana-Champaign.
Richard A. Derrig
Richard Derrig, Ph.D., is president of OPAL Consulting LLC, which provides research and regulatory support to the P&C insurance indus-try. For over 27 years, Dr. Derrig held various posts at the Massachu-setts Bureau of Automobile Insurers and the Massachusetts Bureau of In-surance Fraud. He has won several prizes, including the CAS Ratemak-ing Prize (1993 co-winner), the ARIA Prize (2003), the RIMS Edith F. Li-chota Award (1998), and ARIA’s Mehr Award (2005). Dr. Derrig has coedited three books on solvency and coauthored papers applying fuzzy set theory to insurance.
Roger W. Bovard Editor in Chief
Editorial BoardEDITORS: Shawna S. Ackerman Avraham Adler Todd Bault Morgan Haire Bugbee Daniel A. Crifo Susan L. Cross Stephen P. D’Arcy Enrique de Alba Ryan M. Diehl Robert J. Finger
Steven A. Gapp Emily Gilde Annette J. Goodreau Richard W. Gorvett David Handschke Philip E. Heckman Daniel D. Heyer John Huddleston Ali Ishaq Eric R. Keen Ravi Kumar
ASSISTANT EDITORS: Joel E. Atkins Gary Blumsohn Frank H. Chang Clive L. Keatinge Dmitry E. Papush Christopher M. Steinbach
Richard Fein Associate Editor— Peer Review
Dale R. Edlefson Associate Editor—Copyediting
Gary G. Venter Associate Editor—Development
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 117
Variance Advancing the Science of Risk
Contributors to this Issue
Jackie Li
Dr. Jackie Li is currently an assistant professor in the Division of Banking and Finance at Nanyang Business School (NBS), Nanyang Technologi-cal University (NTU), Singapore. Dr. Li obtained his Ph.D. degree in actu-arial studies from the University of Melbourne, Australia, and is a Fel-low of the Institute of Actuaries of Australia (FIAA). He has been lec-turing and tutoring various actuarial courses and his main research areas are stochastic loss reserving for prop-erty/casualty insurance and mortality projections. Before joining NBS, Dr. Li worked as an actuary in the areas of property/casualty insurance and superannuation.
Huijuan Liu
Dr. Huijuan Liu earned a Ph.D. at Cass Business School, London, in 2008. Her research was sponsored by Lloyd’s of London and she was su-pervised by Professor Richard Ver-rall. Together, they have recently published two papers in the ASTIN Bulletin. Dr. Liu now works for the Financial Services Authority in Lon-don.
Yin Lawn Pierre Lepage Martin A. Lewis Xin Li Cunbo Liu Kevin Mahoney Donald F. Mango Leslie Marlo Stephen J. Mildenhall Christopher J. Monsour Roosevelt C. Mosley Jr.
Mark W. Mulvaney Prakash Narayan Adam Niebrugge Darci Z. Noonan Jonathan Norton A. Scott Romito David Ruhm Theodore R. Shalack John Sopkowicz John Su James Tanser
Neeza Thandi George W. Turner Jr. Trent R. Vaughn Cheng-Sheng Peter Wu Satoru Yonetani Navid Zarinejad Yingjie Zhang Alexandros Zimbidis
COPY EDITORS: Nathan J. Babcock Laura Carstensen Hsiu-Mei Chang Andrew Samuel Golfin Jr. Mark Komiskey William E. Vogan
C. K. “Stan” Khury
C. K. ‘Stan’ Khury, FCAS, MAAA, CLU is a principal with Bass & Khury, an independent actuarial con-sulting firm located in Las Vegas, Nevada. Stan is a past president of the CAS and has written numerous papers and article in CAS publica-tions over a period spanning nearly 40 years. He provides a wide range of actuarial consulting services to in-surers, reinsurers, intermediaries, regulators, and law firms.
CAS STAFF: Elizabeth A. Smith Manager of Publications
Donna Royston Publication Production Coordinator
Sonja Uyenco Desktop Publisher
118 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Variance Advancing the Science of Risk
Dumaria R. Tampubolon
Dumaria Rulina Tampubolon com-pleted her S1 degree (similar to an honors degree) in mathematics, ma-joring in statistics, at Institut Teknologi Bandung in Bandung, In-donesia. She holds a master of sci-ence degree in statistics from Monash University, Melbourne, Australia; and a Ph.D. degree in actuarial stud-ies from Macquarie University, Syd-ney, Australia. Her research interest is in general insurance and applied statistics. Currently she teaches at the Faculty of Mathematics and Nat-ural Sciences, Institut Teknologi Bandung. She can be reached at [email protected] and [email protected].
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© 2010 Casualty Actuarial Society.
Gary G. Venter
Gary G. Venter, FCAS, CERA, ASA, MAAA, is head of economic capital modeling at Chartis and teaches graduate courses in actuarial science at Columbia University. His 35+ years in the insurance and reinsur-ance industry included stints at the Instrat group, which migrated from EW Payne through Sedgwick to Guy Carpenter; the Workers Compensa-tion Reinsurance Bureau; the Na-tional Council on Compensation In-surance; Prudential Reinsurance; and Fireman’s Fund. Gary has an under-graduate degree in mathematics and philosophy from UC-Berkeley and a master’s degree in mathematics from Stanford University. He has served on a number of CAS committees and is on the editorial team of several ac-tuarial journals.
Contributors to this Issue
Richard Verrall
Richard Verrall has been at City Uni-versity since 1987. He is an Honor-ary Fellow of the Institute of Actuar-ies (1999), an Associate Editor of the British Actuarial Journal, the North American Actuarial Journal, and In-surance: Mathematics and Econom-ics, and a principle examiner for The Actuarial Profession (U.K.). Courses for industry include “Statistics for Insurance,” an introductory course aimed at non-specialists, such as un-derwriters, in the uses of statistics in risk assessment; “Stochastic Claims Reserving,” a specialist course for actuaries and statisticians on how to apply statistical methods to reserving for non-life companies; and “Bayes-ian Actuarial Models,” an introduc-tory course in Bayesian methods for premium rating and reserving.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 119
Variance Advancing the Science of Risk
Contributors to this Issue
Herbert I. Weisberg
Herbert I. Weisberg received his Ph.D. in statistics from Harvard in 1970. He is the president of Correla-tion Research, Inc., a consulting firm specializing in litigation support and analysis of insurance fraud. He has published numerous articles and re-ports related to application and de-velopment of statistical methodolo-gy, and is a co-author of Statistical Methods for Comparative Studies: Techniques for Bias Reduction. Re-cently, his research has related to causal inference in statistics, draw-ing on and extending the burgeoning literature in this area. He has recently written a new book titled Bias and Causation: Models and Judgment for Valid Comparisons.
120 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Variance Advancing the Science of Risk
Bootstrap Estimation of thePredictive Distributions of ReservesUsing Paid and Incurred Claims
by Huijuan Liu and Richard Verrall
ABSTRACT
This paper presents a bootstrap approach to estimate the
prediction distributions of reserves produced by the Mu-
nich chain ladder (MCL) model. The MCL model was in-
troduced by Quarg and Mack (2004) and takes into account
both paid and incurred claims information. In order to pro-
duce bootstrap distributions, this paper addresses the appli-
cation of bootstrapping methods to dependent data, with
the consequence that correlations are considered. Numeri-
cal examples are provided to illustrate the algorithm and the
prediction errors are compared for the new bootstrapping
method applied to MCL and a more standard bootstrapping
method applied to the chain ladder technique.
KEYWORDS
Bootstrap, Munich chain ladder, correlation, simulation
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 121
Variance Advancing the Science of Risk
1. IntroductionBootstrapping has become very popular in sto-
chastic claims reserving because of the simplic-
ity and flexibility of the approach. One of the
main reasons for this is the ease with which it
can be implemented in a spreadsheet in order to
obtain an approximation to the estimation error
of a fitted model in a statistical context. Further-
more, it is also straightforward to extend it to
obtain the approximation to the prediction error
and the predictive distribution of a statistical pro-
cess by including simulations from the underly-
ing distributions. Therefore, bootstrapping is a
powerful tool for the most popular subject for
reserving purposes in general insurance, the pre-
diction error of the reserve estimates. It should
be emphasized that to obtain the predictive dis-
tribution, rather than just the estimation error, it
is necessary to extend the bootstrap procedure by
simulating the process error. It is also important
to realize that bootstrapping is not a “model,”
and therefore it is important to ensure that the
underlying reserving models are correctly cali-
brated to the observed data. In this paper, we do
not address the issue of model checking, but sim-
ply show how a bootstrapping procedure can be
applied to the Munich chain ladder model.
In the area of non-life insurance reserving,
there are primarily two types of data used. In
addition to the paid claims triangle, there is fre-
quently a triangle of incurred data also available.
The traditional approach is to fit a model to either
paid or incurred claims data separately, and one
of the most popular methods in this context is the
chain ladder technique. While we do not believe
that this is the most appropriate approach for all
data sets, it has retained its popularity for a num-
ber of reasons. For example, the parameters are
understood in a practical context; it is flexible;
and it is easy to apply. This paper concentrates on
methods which have a chain ladder structure, and
in this context, two types of approaches exist:
deterministic methods such as chain ladder, and
the recently developed stochastic chain ladder re-
serving models. When the chain ladder technique
is used (either as a deterministic approach or us-
ing a stochastic model), one set of data will be
omitted–either the paid or the incurred data can
be used, but not both at the same time. Obviously,
this does not make full use of all the data avail-
able and results in the loss of some information
contained in those data.
This leads us to consider whether it is possible
to construct a model for both data sets, and to a
consideration of the dependency between the two
run-off triangles. A related issue also arises when
traditional methods are applied separately to each
triangle, which produces inconsistent predicted
ultimate losses. In response, Quarg and Mack
(2004) proposed a different approach within a
regression framework, considering the likely cor-
relations between paid and incurred data. Quarg
and Mack (2004) called this new method the
Munich chain ladder (MCL) model. It is this
model that is the subject of this paper, and we
show how the predictive distribution may be es-
timated using bootstrapping. Thus, in this paper
an adapted bootstrap approach is described, com-
bined with simulation for two dependent data
sets. The spreadsheets used in this paper can be
used in practice for any data sets, and are avail-
able on request from the authors.
The paper is set out as follows. Section 2
briefly describes the MCL model using a nota-
tion appropriate for this paper. In Section 3, the
basic algorithm and methodology of bootstrap-
ping is explained. Section 4 shows how to obtain
the estimates of the prediction errors and the em-
pirical predictive distribution using the adapted
bootstrapping and simulation methods. In Sec-
tion 5, two numerical examples are provided, in-
cluding the data from Mack (1993) as well as
some real London market data. Finally, Section 6
contains a discussion and conclusion.
122 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims
2. The Munich chain laddermethodThe MCL model aims to produce a more con-
sistent prediction of ultimate claims when model-
ing both paid and incurred claim data. It is spe-
cially designed to deal with the correlation be-
tween paid and incurred claims. The traditional
models, such as the chain ladder model, some-
times produce unsatisfactory results by ignoring
this dependence. It should be emphasized that
the paid and incurred claims from the same cal-
endar years are not correlated. It is that the paid
claims (incurred claims) are correlated with the
incurred claims (paid claims) from the next (pre-
vious) calendar year.
The fundamental structure of the MCL model
is the same as Mack’s distribution-free chain lad-
der model (Mack 1993). In other words, the chain
ladder development factors in the MCL model
are obtained by Mack’s distribution-free ap-
proach. However, the MCL model adjusts the
chain ladder development factors using the corre-
lations between the observed paid and incurred
claims. The adjusted chain ladder development
factors therefore become individual not only for
different development years but also for differ-
ent accident years. The adjustment is explained
in more detail in Sections 2.1 and 2.2.
2.1. Notation and assumptions
For ease of notation, we assume that we have
a triangle of data. Although the data could be
classified in different ways, we refer to the rows
as “accident years” and the columns as “devel-
opment years.”
Denote CPij as cumulative paid claims and CIij
as cumulative incurred claims occurred in acci-
dent year i, development year j, where 1· i · nand 1· j · n¡ i+1 for the observed data. Theaim of the chain ladder technique and of MCL
is to estimate the data up to development year n.
This produces estimates for CPij and CIij , where
1· i · n and n¡ i+2· j · n, and we therefore
refer to the complete rectangle of data in the as-
sumptions: 1· i, j · n.Mack’s distribution-free chain ladder method
models the pattern of the development factors,
which are defined as FPij = CPi,j+1=C
Pij , for paid
claims and FIij = CIi,j+1=C
Iij , for incurred claims.
Also the ratios of paid divided by incurred claims
and the inverse are introduced as Qij = CPij =C
Iij
and Q¡1ij = CIij=C
Pij , respectively.
Furthermore, define the observed data for
accident year i, up to development year k as Pi(k)
= fCPij : j · kg, Ii(k) = fCIij : j · kg and Bi(k) =fCPij ,CIij : j · kg, for paid claims, incurred claimsand both, respectively.
The following assumptions are taken from
Quarg and Mack (2004), Section 2.1.2.
Assumption A (Expectations)
(A1) For 1· j · n there exists a constant fPjsuch that (for i= 1, : : : ,n)
E[FPij j Pi(j)] = fPj :This assumption is for paid claims. It is neces-
sary to make another analogous assumption for
incurred claims since both data sets are taken into
account.
(A2) For 1· j · n, there exists a constant fIjsuch that (for i= 1, : : : ,n)
E[FIij j Ii(j)] = fIj :In order to analyze the two run-off triangles
dependently, the following assumptions are also
required.
(A3) For 1· j · n, there exists a constant q¡1jsuch that (for i= 1, : : : ,n)
E[Q¡1ij j Pi(j)] = q¡1j :(A4) For 1· j · n, there exists a constant qj
such that (for i= 1, : : : ,n)
E[Qij j Ii(j)] = qj:Note that assumptions (A3) and (A4) will ap-
ply that imply that Qij is constant, which is con-
tradictory–see Section 3.1.2 of Mack and Quarg
(2004) for a discussion of this point.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 123
Variance Advancing the Science of Risk
Assumption B (Variances)
(B1) For 1· j · n, there exists a constant ¾Pjsuch that (for i= 1, : : : ,n)
Var[FPij j Pi(j)] =(¾Pj )
2
CPij:
Again, the analogous assumption for the in-
curred claims is made as follows.
(B2) For 1· j · n, there exists a constant ¾Ijsuch that (for i= 1, : : : ,n)
Var[FIij j Ii(j)] =(¾Ij )
2
CIij:
Also, for the ratios of incurred to paid and
vice versa, the following variance assumptions
are made.
(B3) For 1· j · n, there exists a constant ¿Pjsuch that (for i= 1, : : : ,n)
Var[Q¡1ij j Pi(j)] =(¿Pj )
2
CPij:
(B4) For 1· j · n, there exists a constant ¿ Ijsuch that for (i= 1, : : : ,n)
Var[Qij j Ii(j)] =(¿ Ij )
2
CIij:
Assumption C (Independence) The usual as-
sumptions for individual triangles are as follows:
(C1) The random variables pertaining to dif-
ferent accident years for paid claims, i.e., fCP1j jj = 1,2, : : : ,ng, : : : ,fCPnj j j = 1,2, : : : ,ng, are sto-chastically independent.
(C2) The random variables pertaining to dif-
ferent accident years for incurred claims, i.e.,
fCI1j j j = 1,2, : : : ,ng, : : : ,fCInj j j = 1,2, : : : ,ng,are stochastically independent.
In fact, a stronger assumption is used (see Sec-
tion 3.2 of Quarg and Mack 2004), which is in-
dependence of accident years across both paid
and incurred claims:
(C3) The random sets fCP1j ,CI1j j j =1,2, : : : ,ng, : : : ,fCPnj ,CInj j j = 1,2, : : : ,ng, are stochasticallyindependent.
Using assumptions A to C, the Pearson resid-
uals used in the MCL model can be defined as
shown in Equations (2.1) to (2.4). These residu-
als are a crucial part of the bootstrapping proce-
dures described in Section 4.
rPij =FPij ¡E[FPij j Pi(j)]qVar[FPij j Pi(j)]
, (2.1)
rQ¡1
ij =Q¡1ij ¡E[Q¡1ij j Pi(j)]q
Var[Q¡1ij j Pi(j)], (2.2)
rIij =FIij ¡E[FIij j Ii(j)]qVar[FIij j Ii(j)]
, (2.3)
rQij =Qij ¡E[Qij j Ii(j)]qVar[Qij j Ii(j)]
: (2.4)
Assumption D (Correlations)
(D1) There exists a constant ½P such that (for
1· i, j · n)E[rPij j Bi(j)] = ½PrQ
¡1ij : (2.5)
The following equation states that the constant
½P is in fact the correlation coefficient between
the residuals. Note that since the residuals have
variance 1, the correlation is equal to the covari-
ance. The proof can be found in Quarg and Mack
(2004).
Cov[rPij ,rQ¡1ij j Pi(j)] = Corr[rPij ,rQ
¡1ij j Pi(j)]
= Corr[FPij ,Q¡1ij j Pi(j)] = ½P
(2.6)
Quarg and Mack (2004) derives expected MCL
paid development factors adjusted by the corre-
lation as shown in Equation (2.7).
E[FPij j Bi(j)]
= E[FPij j Pi(j)] +vuut Var[FPij j Pi(j)]Var[Q¡1ij j Pi(j)]
£Corr[FPij ,Q¡1ij j Bi(j)](Q¡1ij ¡E[Q¡1ij j Pi(j)]):(2.7)
124 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims
(D2) Analogous to assumption (D1), for the
incurred claims it is assumed that there exists a
constant ½I such that (for 1· i, j · n)E[rIij j Bi(j)] = ½IrQij : (2.8)
Similarly, the constant ½I measures the corre-
lation coefficient or the covariance between the
residuals. i.e.,
Cov[rIij ,rQij j Bi(j)] = Corr[rIij ,rQij j Bi(j)]
= Corr[FIij ,Qij j Bi(j)] = ½I
(2.9)
Hence, the expected MCL incurred develop-
ment factors adjusted by the correlation can be
derived as follows:
E[FIij j Bi(j)]
= E[FIij j Ii(j)]+vuut Var[FIij j Ii(j)]Var[Qij j Ii(j)]
£Cov[FIij ,Qij j Bi(j)](Qij ¡E[Qij j Ii(j)]):(2.10)
2.2. Unbiased estimates of theparameters
In this section, we summarize the unbiased es-
timates of the parameters derived by Quarg and
Mack (2004). For the paid data, estimates are
required for the parameters of the development
factors, the variances and also the correlation co-
efficient.
The estimates of the paid development factor
parameters can be interpreted as weighted aver-
ages of the observed development factors FPij or
Q¡1ij , using CPij as the weights
fPj =
Pn¡ji=1 C
Pi,j+1Pn¡j
i=1 CPij
=
n¡jXi=1
CPijPn¡ji=1 C
Pij
FPij
(2.11)and
q¡1j =
Pn¡j+1i=1 CIijPn¡j+1i=1 CPij
=
n¡j+1Xi=1
CPijPn¡j+1i=1 CPij
Q¡1ij :
(2.12)
The unbiased estimates of the variances are as
follows:
(¾Pj )2 =
1
n¡ j¡ 1n¡jXi=1
CPij (FPij ¡ fPj )2
(2.13)and
(¿ Pj )2 =
1
n¡ jn¡j+1Xi=1
CPij (Q¡1ij ¡ q¡1j )2
(2.14)Hence the Pearson residuals are
rPij =FPij ¡ fPj¾Pj
qCPij (2.15)
and
rQ¡1
ij =Q¡1ij ¡ q¡1j
¿Pj
qCPij : (2.16)
Finally, the estimate of the correlation coeffi-
cient is given in Equation (2.17).
½P =
Pi,j r
Q¡1ij rPijP
i,j(rQ¡1ij )2
: (2.17)
Similarly, for incurred data, the estimates of
the development factor parameters can be inter-
preted as weighted averages of the development
factors FIij or Qij , using CIij as the weights:
fIj =
Pn¡ji=1 C
Ii,j+1Pn¡j
i=1 CIij
=
n¡jXi=1
CIijPn¡ji=1 C
Iij
FIij
(2.18)and
qj =
Pn¡j+1i=1 CPijPn¡j+1i=1 CIij
=
n¡j+1Xi=1
CIijPn¡j+1i=1 CIij
Qij:
(2.19)
Also, the unbiased estimates for the variance
parameters are as follows:
(¾Ij )2 =
1
n¡ j¡ 1n¡jXi=1
CIij(FIij ¡ fIj )2:
(2.20)and
(¿ Ij )2 =
1
n¡ jn¡j+1Xi=1
CIij(Qij ¡ qj)2:
(2.21)
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Variance Advancing the Science of Risk
Hence the Pearson residuals are
rIij =FIij ¡ fIj¾Ij
qCIij (2.22)
and
rQij =Qij ¡ qj¿ Ij
qCIij: (2.23)
And finally, the estimate of the correlation co-
efficient is given in Equation (2.24).
½I =
Pi,j r
Qij r
IijP
i,j(rQij )2: (2.24)
Assumptions A in Section 2.1 have defined the
expectations of the development factors, given
just the data in the respective triangles. In or-
der to produce a single estimate based on the
data from both paid and incurred data, Quarg
and Mack (2004) also considers the expectations
of the development factors given both triangles,
and defines E[FPij j Bi(j)] = ¸Pij and E[FIij j Bi(j)]= ¸Iij . Using plug-in estimates from Equations
(2.11) to (2.17), the estimates of the paid MCL
development factors are calculated using Equa-
tion (2.7):
ˆPij = f
Pj + ½
P¾Pj¿Pj(Q¡1ij ¡ q¡1j ): (2.25)
Similarly, plug-in estimates from Equations
(2.18) to (2.24) are used in Equation (2.10) so
that the estimates of the incurred development
factors are
ˆ Iij = f
Ij + ½
I¾Ij¿ Ij(Qij ¡ qj): (2.26)
3. Bootstrapping and claimsreserving
Bootstrapping is a simulation-based approach
to statistical inference. It is a method for produc-
ing sampling distributions for statistical quan-
tities of interest by generating pseudo samples,
which are obtained by randomly drawing, with
replacement, from observed data. It should be
emphasized that bootstrapping is a method rather
than a model. Bootstrapping is useful only when
the underlying model is correctly fitted to the
data, and bootstrapping is applied to data which
are required to be independent and identically
distributed. The bootstrapping method was first
introduced by Efron (1979) and a good introduc-
tion to the algorithm can be found in Efron and
Tibshirani (1993).
For the purpose of clarity we begin by giving
a general bootstrapping algorithm and briefly re-
viewing previous applications of bootstrapping
to claims reserving. In Section 4, we show how
an algorithm of this type can be applied to the
MCL. Suppose we have a sample ~X and we re-
quire the distribution of a statistic μ. The follow-
ing three steps comprise the simplest bootstrap-
ping process:
1. Draw a bootstrap sample ~XB1 = fXB1 ,XB2 , : : : ,XBn g1 from the observed data ~X = fX1,X2, : : : ,Xng.
2. Calculate the statistic of interest μB1 for the
first bootstrap sample ~XB1 = fXB1 ,XB2 , : : : ,XBn g1.3. Repeat steps 1 and 2 N times.
By repeating steps 1 and 2 N times, we obtain
a sample of the unknown statistic μ, calculated
fromN pseudo samples, i.e.,~μB = fμB1 , μB2 , : : : , μBNg.When N ¸ 1000, the empirical distribution con-structed from ~μB = fμB1 , μB2 , : : : , μBNg can be takenas the approximation to the distribution for the
statistic of interest μ. Hence all the quantities of
the statistic of interest μ can be obtained, since
such a distribution contains all the information
related to μ.
The above bootstrapping algorithm can be ap-
plied to the prediction distributions for the best
estimates in stochastic claims reserving. England
and Verrall (2007) contains an excellent review
of the application. In addition, Lowe (1994),
England and Verrall (1999) and Pinheiro (2003)
are also good resources for more details. Eng-
land and Verrall (2007) showed how bootstrap-
ping can be used for recursive models, follow-
126 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims
ing from the earlier papers (England and Verrall
1999; England 2002) which applied bootstrap-
ping to the over-dispersed Poisson model.
It should be noted here that the Pearson resid-
uals are commonly used rather than the origi-
nal data in the generalized linear model (GLM)
framework. The Pearson residuals are required
in order to scale the response variables in the
GLM so that they are identically distributed. This
is necessary because the bootstrap algorithm re-
quires that the response variables are indepen-
dent and identically distributed.
Other papers in the actuarial literature that con-
sider triangles of dependent data include Taylor
and McGuire (2007) and Kirschner et al. (2008).
It should be noted here that a model taking ac-
count of all information available could be very
valuable, even when the data is dependent in
practice. The dependence makes it difficult to
calculate the prediction error theoretically. For
these reasons, we believe that adopting bootstrap
method for these models is worthy of investiga-
tion, particularly in order to obtain the predictive
distribution of the estimates of outstanding lia-
bilities.
4. Bootstrapping the Munich chainladder modelThis section considers bootstrapping the MCL
model. In Section 4.1 we describe the method-
ology and in Section 4.2 we give the algorithm
that is used.
4.1. MethodologyThe method of bootstrapping stochastic chain
ladder models can be seen in a number of dif-
ferent contexts. England and Verrall (2007) cat-
egorize the models as recursive and nonrecur-
sive and show how bootstrapping methods can
be applied in either case. Since we are dealing
with recursive models here, we follow England
and Verrall and consider the observed develop-
ment link ratios rather than the claims data them-
selves. In other words, for Mack’s distribution-
free chain ladder model the link ratios Fij are
randomly drawn against Xij , noting that
E[Fij j Xij] = E"Xi,j+1Xij
¯¯ Xij
#= fj:
Here, Xij is used to represent observed claims
data in general. Note that the bootstrap estimates
of the development factors fBj which are obtained
by taking weighted averages of the bootstrapped
observed link ratios FBij , use Xij rather than XBij
as the weights.
However, this method cannot be simply ex-
tended to the MCL method, since this model is
designed for dealing with two sets of correlated
data, the paid and incurred claims. This means
that it is not possible to use the normal bootstrap
approach because the independence assumption
cannot be met.
In order to address the problem of how to
adapt the existing bootstrap approach in order to
cope with the MCL method for dependent data
sets, the consideration of the correlation is cru-
cial. It should be noted that the correlation which
is observed in the data represents real depen-
dence between the paid and incurred claims, and
the model is specifically designed for this depen-
dence. Therefore, it should remain unchanged
within any resampling procedure. The straight-
forward solution is to draw samples pairwise so
that the correlation between the two dependent
original data sets will not be broken when gen-
erating a sampling distribution for a statistic of
interest.
Obviously, when bootstrapping the recursive
MCL method, the residuals of the paid and in-
curred link ratios are required instead of the
raw data. The question arises of how to deal
with these residuals in order to meet the require-
ment of not breaking the observed dependence
between paid and incurred claims. The answer
is to group all four sets of residuals calculated
in the MCL method, i.e., the paid and incurred
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Variance Advancing the Science of Risk
development link ratios, the ratios of incurred
over paid claims from the previous years, and its
inverse, individually. There are two reasons for
this. First, the paid claims (incurred claims) are
correlated to the incurred to the paid claims ratio
(paid to incurred claims ratio) from the previous
year, and doing this will preserve the required
dependence. Second, the correlation coefficient
of paid and incurred claims is equal to the corre-
lation coefficient of those residuals, as stated in
Equations (2.6) and (2.10).
Thus, in the case of the paid claims data, the
triangles (which have the same dimensions) con-
taining the residuals of the observed paid link
ratios and the residuals of the ratios of incurred
over paid (except the first column), are paired
together. The same procedure is used for the in-
curred claims data. We do this for convenience,
even though the ratios of the paid over incurred
claims and the inverse give the same information.
Note that these ratios should remain unchanged
when pairing them with paid and incurred claims
with the same dimensions. The consequence of
this is that all four sets of residuals for paid, in-
curred link ratios and the ratios of incurred over
paid claims and the inverse are all grouped to-
gether. (Note here that an alternative approach
would be to group three sets of residuals: the
residuals of the paid and incurred link ratios and
either the residuals of the paid over incurred ra-
tios or the inverse. This would produce the same
results as grouping four sets of residuals, as the
residuals of paid over incurred ratios and the in-
verse can always be calculated from each other.
However, it is simpler to group the four sets, as
the calculation of the fourth set of residuals is
naturally skipped in this case.)
This combines the four residuals triangles into
one new triangle that consists of these grouped
residuals and we name it as the grouped resid-
ual triangle. In each unit from this triangle of
quadruples, the residuals are from the same ac-
cident and development year and correspond to
paid and incurred claims. Therefore, the new tri-
angle of quadruples contains all the information
available and meanwhile maintains the observed
dependence.
When applying bootstrapping, this triangle is
considered as the observed sample. The new gen-
erated pseudo samples are obtained by random
drawing, with replacement, from the triangle of
quadruples.
The resampled incurred and paid triangles can
be obtained by separating the pairs in the pseudo
sample generated as above and backing out the
residual definition. The MCL approach can then
be applied to calculate all the statistics of inter-
est for the resampled paid and incurred triangles,
i.e., the correlation coefficient for paid and in-
curred, the paid and incurred development fac-
tors, the ratios of paid over incurred or the in-
verse, and the variances. Finally, adjusting the
paid and incurred development factors by the
correlation coefficient using the MCL approach,
the bootstrapped MCL reserve estimates are ob-
tained. This completes a single bootstrap itera-
tion.
Again, the bootstrap method provides only
the estimation error of the MCL method. In
order to include the prediction error and esti-
mate the predictive distribution for the MCL esti-
mates of outstanding liabilities, an additional step
is added at the end of each bootstrap iteration,
which is to add the process variance to the esti-
mation error.
Note that we apply the final simulation for the
process variance to paid and incurred claims, in-
dependently. This is because, for a particular ac-
cident and development year, paid and incurred
claims are actually independent. Under the as-
sumptions of the MCL model, paid (incurred)
claims are only correlated with previous incurred
(paid) claims, and the forecasts produced by the
bootstrapping will pick up this dependency.
In order to obtain a reasonable approximation
to the predictive distribution, at least 1000 pseudo
samples are required. For each of the pseudo
128 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims
samples, the row totals and overall total of out-
standing liabilities are stored so that the sample
means, sample variances and the empirical dis-
tributions can be calculated and plotted. They
are taken as the approximations to the best es-
timates of outstanding liabilities, the prediction
errors and the predictive distributions of the out-
standing liabilities. Also, an estimate of any re-
quired percentile and confidence interval can be
calculated from the predictive distribution.
In order to satisfy the assumption that the sam-
ple is identically distributed in the bootstrapping
procedure, the Pearson residuals are calculated
and used. As in England and Verrall (2007), we
use the Pearson residuals of the observed de-
velopment factors rather than those for the ac-
tual claims, since we are using recursive mod-
els. Note that a bootstrap bias correction is also
needed, and the simplest way to do this is to mul-
tiply the residuals byp(n¡ j)=(n¡ j¡ 1).
In addition to drawing the grouped sample for
bootstrapping correlated data sets, there are also
two other practical points that should be men-
tioned. The first is to note that the fitted values
are obtained by starting from the final diagonal
in each triangle and working backwards, by di-
viding by the development factors. The second
is that the zero residuals which appear in both
triangles are also left out.
4.2. Algorithm
This section provides the algorithm, step by
step, which is needed in order to implement the
bootstrap process introduced in Section 4.1.
– Apply the MCL method to both the cumula-
tive paid and incurred claims data to obtain
the residuals for all four sets of ratios: the
paid, incurred link ratios, the paid over in-
curred ratios and the reverse. They can be
obtained from following equations:
rPij =FPij ¡ fPj¾Pj
qCPij ,
rQ¡1
ij =Q¡1ij ¡ q¡1j
¿ Pj
qCPij ,
rIij =FIij ¡ fIj¾Ij
qCIij and
rQij =Qij ¡ qj¿ Ij
qCIij :
– Adjust the Pearson residual estimates by mul-
tiplying byp(n¡ j)=(n¡ j¡ 1) to correct the
bootstrap bias.
– Group all four residuals together, i.e., rPij , rQ¡1ij ,
rIij and rQij . We write this as Uij = f(rPij ), (rQ
¡1ij ),
(rIij), (rQij )g.
– Start the iterative loop to be repeated N times
(N ¸ 1000). This consists of the followingsteps:
1. Randomly sample from the grouped resid-
uals with replacement, denoted as UBij =
f(rPij )B, (rQ¡1
ij )B , (rIij)B , (rQij )
Bg, from the group-
ed triangle so that a pseudo sample of the
grouped residuals is created.
2. Calculate the pseudo samples of the four
triangles for the paid, incurred link ratios, the
ratios of paid over incurred and the inverse by
inverting the Pearson residuals definition as
follows:
(FPij )B =
(rPij )B¾PjqCPi,j
+ fPj ,
(Q¡1ij )B =
(rQ¡1
ij )B¿PjqCPi,j
+ q¡1j ,
and
(FIij)B =
(rIij)B¾IjqCIi,j
+ fIj ,
(Qij)B =
(rQij )B¿ IjqCIi,j
+ qj :
3. Calculate the CPi,j-weighted and CIi,j-
weighted average of the bootstrap paid and
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Variance Advancing the Science of Risk
incurred development factors as follows:
(fPj )B =
n¡jXi=1
CPi,jPn¡ji=1 C
Pi,j
(FPij )B ,
(q¡1j )B =
n¡j+1Xi=1
CPijPn¡j+1i=1 CPij
(Q¡1ij )B
and
(fIj )B =
n¡jXi=1
CIi,jPn¡ji=1 C
Ii,j
(FIij)B ,
(qj)B =
n¡j+1Xi=1
CIijPn¡j+1i=1 CIij
(Qij)B:
Note that the weights used here are from the
original data sets and not from the pseudo
samples.
4. Calculate the corresponding correlation
coefficient for the resampled data using the
pseudo residuals (rPij )B , (rQ
¡1ij )B, (rIij)
B and
(rQij )B as follows,
(½P)B =
Pi,j(r
Q¡1ij )B(rPij )
BPi,j((r
Q¡1ij )B)2
and
(½I)B =
Pi,j(r
Qij )B(rIij)
BPi,j((r
Qij )B)2
:
5. Calculate the variances for the bootstrap
data as follows:
((¾Pj )2)B =
1
n¡ j¡ 1n¡jXi=1
CPij ((FPij )
B ¡ (fPj )B)2
((¿ Pj )2)B =
1
n¡ j¡ 1n¡jXi=1
CPij ((Q¡1ij )
B ¡ (q¡1j )B)2
((¾Ij )2)B =
1
n¡ j¡ 1n¡jXi=1
CIij((FIij)B ¡ (fIj )B)2
((¿ Ij )2)B =
1
n¡ j¡ 1n¡jXi=1
CIij((Qij)B ¡ (qj)B)2:
Note that all the sums here are from 1 to n¡ jbecause the last diagonals of paid to incurred
(and incurred to paid) are not included in the
resampling procedure.
6. Calculate the bootstrap development fac-
tors adjusted by the correlation coefficient
between the pseudo samples as follows:
( ˆPij)B = (fPj )
B +(½P)B(¾Pj )
B
(¿ Pj )B((Q¡1ij )
B ¡ (q¡1j )B)
and
( ˆ Iij)B = (fIj )
B +(½I)B(¾Ij )
B
(¿ Ij )B((Qij)
B ¡ (qj)B),
for the resampled bootstrap paid and incurred
run-off triangles, respectively.
7. Simulate a future payment for each cell in
the lower triangle for both paid and incurred
claims, from the process distribution with the
mean and variance calculated from the previ-
ous step. To do this, the following steps are
required:
² For the one-step-ahead predictions fromthe leading diagonal, a normal distribution
is assumed, i.e., for 2· i · n,XPi,n¡i+2 »Normal(( ˆPi,n¡i+1)BXPi,n¡i+1,
((¾Pn¡i+1)2)BXPi,n¡i+1)
for paid claims and
XIi,n¡i+2 »Normal(( ˆ Ii,n¡i+1)BXIi,n¡i+1,((¾In¡i+1)
2)BXIi,n¡i+1)
for incurred claims.
² For the two-step-ahead predictions up tothe n-step-ahead predictions, normal dis-
tributions are still assumed, but with the
mean and variance calculated from previ-
ous predictions instead of from the observ-
ed data, i.e., for 3· k · n and n¡ k+3·j · n,XPkl »Normal(( ˆPi,l¡1)BXPk,l¡1, ((¾Pl¡1)2)BXPk,l¡1)
for paid claims, and
XIkl »Normal(( ˆ Ii,l¡1)BCIk,l¡1, ((¾Il¡1)2)BXIk,l¡1)
for incurred claims.
130 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims
Table 1. Paid claims from Quarg and Mack (2004)
j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7
i = 1 576 1804 1970 2024 2074 2102 2131i = 2 866 1948 2162 2232 2284 2348i = 3 1412 3758 4252 4416 4494i = 4 2286 5292 5724 5850i = 5 1868 3778 4648i = 6 1442 4010i = 7 2044
Table 2. Incurred claims from Quarg and Mack (2004)
j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7
i = 1 978 2104 2134 2144 2174 2182 2174i = 2 1844 2552 2466 2480 2508 2454i = 3 2904 4354 4698 4600 4644i = 4 3502 5958 6070 6142i = 5 2812 4882 4852i = 6 2642 4406i = 7 5022
8. Sum the simulated payments in the future
triangle by origin year and overall, to give
the origin year and total reserve estimates re-
spectively.
9. Store the results, and return to the start of
the iterative loop.
5. Examples
This section illustrates the bootstrapping ap-
proach to the MCL and uses two numerical ex-
amples to assess the results. The first example
uses the data from Quarg and Mack (2004). Ex-
ample 2 uses market data from Lloyd’s which
have been scaled for confidentiality reasons.
These data relate to aggregated paid and incurred
claims for two Lloyd’s syndicates, categorized at
risk level.
Example 1 is included in order to illustrate the
results for the original set of data used by Quarg
and Mack (2004). The purpose of example 2 is
to illustrate that the MCL model does not neces-
sarily provide better results in all situations. Our
results indicate that it performs better when the
data have less inherent variability and are less
“jumpy.”
Table 3. A Comparison of methods for reserves projected onpaid and incurred claims
Bootstrap MCL Mack
Paid Incurred Paid Incurred Paid Incurred
i = 1 0 43 0 43 0 43i = 2 37 94 35 96 32 97i = 3 109 131 103 135 158 88i = 4 277 321 269 326 331 277i = 5 299 296 289 302 407 191i = 6 657 651 646 655 919 465i = 7 5492 5646 5505 5606 4063 6380
Overall Total 6871 7182 6846 7163 5911 7540
Table 4. Comparison of bootstrap prediction errors for MCLand CL Mack chain ladder methods
MCL Mack
Paid Incurred Paid Incurred
i = 1 0 0 0 0i = 2 5 5 15 9i = 3 48 70 53 82i = 4 61 86 68 105i = 5 72 104 72 117i = 6 215 208 289 216i = 7 735 716 897 869
Overall Total 776 782 991 980
EXAMPLE 1. In this section, we apply the boot-
strapping methodology with 10,000 bootstrap
simulations to the data from Quarg and Mack
(2004).
Tables 1 and 2 show the data. In order to il-
lustrate the nature of the run-off of the data, Fig-
ures 1 and 2 are the plots of the data from Ta-
bles 1 and 2, respectively. From Figures 1 and
2, it can be seen that the data are stable and not
excessively spread out.
The results of applying the bootstrap method-
ology described in this paper are shown be-
low, and are compared with the results from
the straightforward chain ladder technique and
Mack’s method for the prediction errors. Table 3
shows that the theoretical MCL reserves (from
Quarg and Mack 2004) and the mean of the boot-
strap distributions, together with the chain ladder
reserves when the triangles are considered sep-
arately. It can be seen that the bootstrap means
are close to the theoretical values, for both the
paid and incurred claims.
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Variance Advancing the Science of Risk
Figure 1. Paid claims
Figure 2. Incurred claims
Figure 3. Comparison of predictive distributions of overall reservesfor CL and MCL reserves for paid and incurred claims
Table 4 displays the bootstrap prediction error
of the MCL reserves projected by both paid and
incurred claims. Also shown are the prediction
errors for the Mack method. It can be seen that
the MCL prediction errors are lower than those
of the Mack method.
Since the purpose of the MCL method is to
use more data to improve the estimation of the
reserves, it is expected that the prediction errors
should be lower than the Mack model. This is
confirmed for these data by Table 5, which shows
that the prediction error, as a percentage of the re-
serve, is lower for the MCL reserves than the pre-
diction error of CL the reserves from the Mack
model.
In Figure 3, the distributions of the MCL and
CL reserve projections for paid and incurred
claims are plotted. Figure 3 shows that the paid
and incurred best reserve estimates are very close
when using the MCL approach. In contrast, the
132 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims
Table 5. Comparison of bootstrap prediction errors % forMCL and CL methods
MCL Mack
Paid Incurred Paid Incurred
i = 1 — 0% — 0%i = 2 14% 5% 45% 9%i = 3 44% 53% 33% 93%i = 4 22% 27% 21% 38%i = 5 24% 35% 18% 61%i = 6 33% 32% 31% 46%i = 7 13% 13% 22% 14%
Total 11% 11% 17% 13%
paid and incurred best reserve estimates projected
by the chain ladder method are much further
apart. Furthermore, the CL method provides a
much more spread-out distribution than the MCL
approach, in the case of both paid and incurred
claims.
EXAMPLE 2. In this section, a set of aggregate
data from Lloyd’s syndicates is considered. In
this case, the data are not as stable or well-be-
haved and the results are quite different. Tables 6
and 7 show the data, which are plotted in Fig-
Table 6. Scaled aggregate paid claims at risk level from Lloyd’s Market
j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7 j = 8 j = 9 j = 10
i = 1 1139 5680 6906 7069 7205 7350 7421 7487 7506 7518i = 2 1101 6223 8038 8652 9064 9249 9343 9421 9455i = 3 1215 8058 10593 11638 12346 12784 12978 13161i = 4 949 5324 7608 8257 8719 8972 9103i = 5 638 4107 6367 7099 7489 7586i = 6 647 4166 6231 7029 7335i = 7 1198 4660 7303 7791i = 8 1194 6540 9251i = 9 1248 6062i = 10 1083
Table 7. Scaled aggregate incurred claims at risk level from Lloyd’s Market
j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7 j = 8 j = 9 j = 10
i = 1 2170 6941 7709 7403 7452 7508 7514 7547 7555 7563i = 2 2184 7822 9182 9368 9445 9520 9508 9547 9585i = 3 2759 10947 12649 12947 13090 13283 13328 13360i = 4 1958 8398 9814 9800 9306 9370 9272i = 5 1376 6177 7699 7799 7984 7904i = 6 1464 5861 7546 7679 7687i = 7 2405 6385 8151 8234i = 8 3128 8772 10265i = 9 2980 8045i = 10 2722
ures 4 and 5. It can be seen from these figures
that the data are much more unstable and more
spread out compared with the previous two ex-
amples.
The MCL method still produces consistent ul-
timate loss predictions for this data set, as shown
in Table 8. However, the prediction error con-
tained in Table 9, estimated by the bootstrap
MCL approach, appears not to offer such an im-
provement as was seen in Example 1.
Table 10 shows a comparison of the predic-
tion errors as a percentage of the reserve, and
again it can be seen that the results do not in-
dicate that the MCL method is a significant im-
provement over the CL model. The conclusion
from this is that although the MCL method uses
more data, and should be expected to produce
lower prediction errors, this is not always the
case in practice. We believe that the reason for
this is that the assumptions made by the MCL
method–the specific dependencies assumed–
are not as strong as expected in this case. A
conclusion from this is that the data have to be
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 133
Variance Advancing the Science of Risk
Figure 4. Scaled paid claims from Lloyd’s Market
Figure 5. Scaled incurred claims from Lloyd’s Market
Table 8. Comparison of methods for reserves projected onpaid and incurred claims
Bootstrap MCL CL
Paid Incurred Paid Incurred Paid Incurred
i = 1 0 45 0 45 0 0i = 2 24 138 19 139 15 15i = 3 78 245 71 245 63 62i = 4 146 236 139 237 143 142i = 5 252 357 246 354 220 215i = 6 383 455 373 454 400 395i = 7 590 614 579 624 829 825i = 8 1345 1355 1318 1366 1850 1820i = 9 3758 3811 3707 3787 4081 4042i = 10 9874 9962 9740 9840 8765 8698
Overall Total 16451 17218 16192 17092 16367 16214
examined carefully before the MCL method is
applied.
This conclusion is reinforced by Figure 6,
which shows the predictive distributions.
5. ConclusionThis paper has shown how a bootstrapping ap-
proach can be used to estimate the predictive
distribution of outstanding claims for the MCL
model. The model deals with two dependent data
sets, the paid and incurred claims triangles, for
Table 9. Comparison of bootstrap prediction errors for MCLand CL methods
MCL CL
Paid Incurred Paid Incurred
i = 1 0 0 0 0i = 2 10 10 2 2i = 3 16 32 11 11i = 4 32 27 39 38i = 5 47 62 43 44i = 6 78 97 92 96i = 7 204 249 166 168i = 8 324 372 391 382i = 9 573 592 987 973i = 10 1762 1818 1940 1963
Overall Total 1911 1994 2277 2305
general insurance reserving purposes. We believe
that bootstrapping is well-suited for these pur-
poses from a practical point of view, since it
avoids complicated theoretical calculations and
is easily implemented in a simple spreadsheet.
This paper adapts the method by taking account
of the dependence observed in the data and re-
sampling pairwise.
A number of examples have been given, which
show that the MCL model does not always pro-
duce superior results to the straightforward chain
134 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims
Figure 6. Comparison of predictive distributions of CL andMCL reserves predicted on paid and incurred claims
Table 10. Comparison of bootstrap prediction errors for MCLand CL methods using scaled data
MCL CL
Paid Incurred Paid Incurred
i = 1 — — — —i = 2 43% 7% 69% 14%i = 3 21% 13% 27% 18%i = 4 22% 11% 28% 27%i = 5 19% 17% 20% 20%i = 6 20% 21% 23% 24%i = 7 35% 41% 20% 20%i = 8 24% 27% 21% 21%i = 9 15% 16% 23% 24%i = 10 18% 18% 22% 23%
Overall Total 12% 12% 14% 14%
ladder model. As a consequence, we believe that
it is important for the data to be carefully checked
to test whether the dependency assumptions of
the MCL model are valid for each data set be-
fore it is applied.
AcknowledgmentFunding for this project from Lloyd’s of Lon-
don is gratefully acknowledged.
ReferencesEfron, B., “Bootstrap Methods: Another Look at the Jack-
knife,” Annals of Statistics 7, 1979, pp. 1—26.
Efron, B., and R. J. Tibshirani, An Introduction to the Boot-
strap, New York: Chapman and Hall, 1993.
England, P. D., and R. J. Verrall, “Analytic and Bootstrap
Estimates of Prediction Errors in Claims Reserving,” In-
surance: Mathematics and Economics 25, 1999, pp. 281—
293.
England, P. D., Addendum to “Analytic and Bootstrap Es-
timates of Prediction Errors in Claims Reserving,” Insur-
ance: Mathematics and Economics 31, 2002, pp. 461—466.
England P. D., and R. J. Verrall, “Predictive Distributions
of Outstanding Liabilities in General Insurance,” Annals
of Actuarial Science 1, 2007, pp. 221—270.
Kirschner, G. S., C. Kerley, and B. Isaacs, “Two Approaches
to Calculating Correlated Reserve Indications Across
Multiple Lines of Business,” Variance 2, 2008, pp. 15—
38.
Lowe, J., “A Practical Guide to Measuring Reserve Variabil-
ity Using Bootstrapping, Operational Time and a Distri-
bution Free Approach,” Proceedings of the 1994 General
Insurance Convention, Institute of Actuaries and Faculty
of Actuaries, 1994.
Mack, T., “Distribution-free Calculation of the Standard Er-
ror of Chain Ladder Reserve Estimates,” ASTIN Bulletin
23, 1993, pp. 214—225.
Pinheiro, P. J. R., J. M. Andrade e Silva, and M. L. C.
Centeno, “Bootstrap Methodology in Claim Reserving,”
Journal of Risk and Insurance 70, 2003, pp. 701—714.
Quarg, G., and T. Mack, “Munich Chain Ladder,” Blatter
DGVFM 26, 2004, pp. 597—630.
Taylor, G., and G. McGuire, “A Synchronous Bootstrap to
Account for Dependencies between Lines of Business in
the Estimation of Loss Reserve Prediction Error,” North
American Actuarial Journal 11 (3), 2007, pp. 70—88.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 135
Robustifying Reserving
by Gary G. Venter and Dumaria R. Tampubolon
ABSTRACT
Robust statistical procedures have a growing body of lit-
erature and have been applied to loss severity fitting in
actuarial applications. An introduction of robust methods
for loss reserving is presented in this paper. In particu-
lar, following Tampubolon (2008), reserving models for a
development triangle are compared based on the sensitiv-
ity of the reserve estimates to changes in individual data
points. This measure of sensitivity is then related to the
generalized degrees of freedom used by the model at each
point.
KEYWORDS
Loss reserving; regression modeling; robust, generalized degrees of freedom
136 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Robustifying Reserving
All models are wrong but some are useful.
–Christian Dior (or maybe George E. P. Box)
1. Introduction
The idea of this paper is simple. For models
using a loss development triangle, the robustness
of the model can be evaluated by comparing the
derivative of the loss reserve with respect to each
data point. All else being equal, models that are
highly sensitive to a few particular observations
are less preferred than ones that are not. This is
supported by the fact that individual cells can be
highly volatile. This general approach, based on
Tampubolon (2008), is along the lines of robust
statistics, so some background into robust statis-
tics will be the starting point. Published models
on three data sets will be tested by this method-
ology. For two of them, unsuspected problems
with the previously best-fitting models are found,
leading to improved models.
The sensitivity of the reserve estimate to in-
dividual points is related to the power of those
points to draw the fitted model towards them.
This can be measured by what Ye (1998) calls
generalized degrees of freedom (GDF). For a
model and fitting procedure, the GDF at each
point is defined as the derivative of the fitted
point with respect to the observed point. If any
change in a sample point is matched by the same
change in the fitted, the model and fitting proce-
dure are giving that point full control over its fit,
so a full degree of freedom is used. GDF does
not fully explain the sensitivity of the reserve to
a point, as the position of the point in the triangle
also gives it more or less power to change the re-
serve estimate, but it adds some insight into that
sensitivity.
Section 2 provides some background of robust
analysis and Section 3 shows some previous ap-
plications to actuarial problems. These help to
place the current proposal into perspective in that
literature. Sections 4, 5, and 6 apply this ap-
proach to some published loss development mod-
els. Section 7 concludes.
2. Robust methods in general
Classical statistics takes a model structure and
tries to optimize the fit of data to the model un-
der the assumption that the data is in fact gener-
ated by the process postulated in the model. But
in many applied situations, the model is a con-
venient simplification of a more complex pro-
cess. In this case, the optimality of estimation
methods such as maximum likelihood estimation
(MLE) may no longer hold. In fact, a few obser-
vations that do not arise from the model assump-
tions can sometimes significantly distort the esti-
mated parameters when standard techniques are
used. For instance, Tukey (1960) gives examples
where even small deviations from the assumed
model can greatly reduce the optimality proper-
ties. Robust statistics looks for estimation meth-
ods that in one way or another can insulate the
estimates from such distortions.
Perhaps the simplest such procedure is to iden-
tify and exclude outliers. Sometimes outliers
clearly arise from some other process than the
model being estimated, and it may even be clear
when current conditions are likely to generate
such outliers, so that the model can then be ad-
justed. If the parameter estimates are strongly in-
fluenced by such outliers, and the majority of the
observations are not consistent with those esti-
mates, it is reasonable to exclude the outliers and
just be cautious about when to use the model.
An example is provided by models of the U.S.
one-month treasury bill rates at monthly inter-
vals. Typical models postulate that the volatility
of the rate is higher when the rate itself is higher.
Often the volatility is proposed to be proportional
to the pth power of the rate. The question is–
what is p? One model, the CIR or Cox, Inger-
soll, Ross model, assumes a p value of 0.5. Other
models postulate p as 1 or even 1.5, and others
try to estimate p as a parameter. An analysis by
Dell’Aquila, Ronchetti, and Troiani (2003) found
that when using traditional methods, the estimate
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 137
Variance Advancing the Science of Risk
of p is very sensitive to a few observations in the
1979—82 period, when the U.S. Federal Reserve
bank was experimenting with monetary policy.
Including that period in the data, models with
p = 1:5 cannot be rejected, but excluding that
period finds that p = 0:5 works just fine. That
period also experienced very high values of the
interest rate itself, so their analysis suggests that
using p = 0:5 would make sense unless the inter-
est rate is unusually high.
A key tool in robust statistics is the identifica-
tion of influential observations, using the influ-
ence function defined by Hampel (1968). This
procedure looks at statistics calculated from a
sample, such as estimated parameters, as func-
tionals of the random variables that are sampled.
The influence function for the statistic at any ob-
servation is a functional derivative of the statis-
tic with respect to the observed point. In practice,
analysts often use what is called the empirical in-
fluence. For instance, Bilodeau (2001) suggests
calculating the empirical influence at each sam-
ple point as the sample size times the decrease
(which may be negative) in the statistic from ex-
cluding the point from the sample. That is, the
influence is n times [statistic with full sample mi-
nus statistic excluding the point]. If the statistic is
particularly sensitive to a single or a few obser-
vations, its accuracy is called into question. The
gross error sensitivity (GES) is defined as the
maximum absolute value of the influence func-
tion across the sample.
The effect on the statistic of small changes in
the influential observations is also a part of ro-
bust analysis, as these effects should not be too
large either. If each observation has substantial
randomness, the random component of influen-
tial observations has a disproportionate impact
on the statistic. The approach used below in the
loss reserving case is to identify observations for
which small changes have large impacts on the
reserve estimate.
Exclusion is not the only option for dealing
with outliers. Estimation procedures that use but
limit the influence of the outliers are also an im-
portant element of robust statistics. Also, find-
ing alternative models which are not dominated
by a few influential points and estimating them
by traditional means can be an outcome of a ro-
bust analysis. In the interest rate case, a model
with one p parameter for October 1979 through
September 1982 and another elsewhere does this.
Finding alternative models with less influence
from a few points is what we will be attempt-
ing in the reserve analysis.
3. Robust methods in insuranceSeveral papers on applying robust analysis to
fitting loss severity distributions have appeared
in recent years. For instance, Brazauskas and Ser-
fling (2000a) focus on estimation of the simple
Pareto tail parameter ® assuming that the scale
parameter b is known. In this notation the sur-
vival function is S(x) = (b=x)®. They compare
several estimators of ®, such as MLE, matching
moments or percentiles, etc. One of their tests is
the asymptotic relative efficiency (ARE) of the
estimate compared to MLE, which is the factor
which when applied to the sample size would
give the sample size needed for MLE to give
the same asymptotic estimation error. Due to the
asymptotic efficiency of MLE, these factors are
never greater than unity, assuming the sample is
really from that Pareto distribution.
The problem is that the sample might not be
from a simple Pareto distribution. Even then,
however, you would not want to identify and
eliminate outliers. Whatever process is generat-
ing the losses would be expected to continue, so
no losses can be ignored.1 The usual approach to
this problem is to find alternative estimators that
have low values of the GES and high values of
ARE. Brazauskas and Serfling (2000a) suggest
1A related problem is contamination of large losses by a non-
recurring process. The papers on robust severity also address this,
but it is a somewhat different topic than fitting a simple model to
a complex process.
138 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Robustifying Reserving
estimators they call generalized medians (GM).
The kth generalized median is the median of all
MLE estimators of subsets of size k of the origi-
nal data. That can be fairly calculation-intensive,
however, even with small k of 3, 4, or 5.
Finkelstein, Tucker, and Veeh (2006) define an
estimator they call the probability integral trans-
form statistic (PITS) which is quite a bit easier
to calculate but not quite as robust as the GM.
It has a tuning parameter t in (0,1) to control
the trade-off between efficiency and robustness.
Since (b=x)® is a probability and so a number be-
tween zero and one, it should be distributed uni-
formly [0,1]. Thus (b=x)t® should be distributed
like a uniformly raised to the t power. The av-
erage of these over a sample is known to have
expected value 1=(t+1), so the PITS estimator
is the value of ¯ for which the average of (b=x)t¯
over the sample is 1=(t+1). This is a single-
variable root finding exercise. Finklestein,
Tucker, and Veeh give values of the ARE and
GES for the GM and PITS estimators, shown in
Table 1. A simulation suggests that the GES for
MLE for ®= 1 is about 3.9, and since its ARE
is 1.0 by definition, PITS at 0.94 ARE is not
worthwhile in this context. In general the gener-
alized median estimators are more robust by this
measure.
Other robust severity studies include Brazau-
skas and Serfling (2000b), who use GM estima-
tion for both parameters of the simple Pareto;
Gather and Schultze (1999), who show that the
best GES for the exponential is the median scaled
to be unbiased (but this has low ARE); and Ser-
fling (2002), who applies GM to the lognormal
distribution.
4. Robust approach to lossdevelopmentOmitting points from loss development trian-
gles can sometimes lead to strange results, and
not every development model can be automat-
ically extended to deal with this, so instead of
calculating the influence function for develop-
Table 1. Comparative efficiency and robustness of tworobust estimators of Pareto ®
ARE GM-k PITS-t GM-GES PITS-GES
0.88 3 0.531 2:27® 2:88®0.92 4 0.394 2:60® 3:54®0.94 5 0.324 2:88® 4:08®
ment models, we look at the sensitivity of the
reserve estimate to changes in the cells of the
development triangle, as in Tampubolon (2008).
In particular, we define the impact of a cell on
the reserve estimate under a particular develop-
ment methodology as the derivative of the esti-
mate with respect to the value in the cell. We
do this for the incremental triangle, so a small
change in a cell affects all subsequent cumula-
tive values for the accident year. This seems to
make more sense than looking at the derivative
with respect to cumulative cells, whose changes
would not continue into the rest of the triangle.
If you think of a number in the triangle as its
mean plus a random innovation, the derivative
with respect to the random innovation would be
the same as that with respect to the total, so a
high impact of a cell would imply a high impact
of its random component as well. Thus models
with some cells having high impacts would be
less desirable. One measure of this is the maxi-
mum impact of any cell, which would be anal-
ogous to the GES, but we will also look at the
number of cells with impacts above various
thresholds in absolute value.
This is just a toe in the water of robust analysis
of loss development. We are not proposing any
robust estimators, and will stick with MLE or
possibly quasi-likelihood estimation. Rather we
are looking at the impact function as a model
selection and refinement tool. It can be used to
compare competing models of the same develop-
ment triangle, and it can identify problems with
models that can guide a search for more robust
alternatives. This is similar to finding models that
work for the entire history of interest rate changes
and are not too sensitive to any particular points.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 139
Variance Advancing the Science of Risk
To help interpret the impact function, we will
also look at the generalized degrees of freedom
(GDF) at each point. This is defined as the deriva-
tive of the fitted value with respect to the ob-
served value. If this is near 1, the point’s initial
degree of freedom has essentially been used up
by the model. The GDF is a measure of how
much a point is able to pull the fitted value to-
wards itself. Part of the impact of a point is this
power to influence the model, but its position in
the triangle also can influence the estimated re-
serve. Just like with the impact function, high
values of the GDF would be a detriment.
For the chain-ladder (CL) model, some obser-
vations can be made in general. All three corners
of the triangle have high impact. The lower left
corner is the initial value of the latest accident
year, and the full cumulative development applies
to it. Since this point does not affect any other
calculations, its impact is the development factor,
which can sometimes be substantial. The upper
right corner usually produces a development fac-
tor which, though small, applies to all subsequent
accident years, so its impact can also be substan-
tial. When there is only one year at ultimate, this
impact is the ratio of the sum of all accident years
not yet at ultimate, developed to the penultimate
lag, to the penultimate cumulative value for the
oldest accident year. The upper left corner is a
bit strange in that its impact is usually negative.
Increasing it will increase the cumulative loss at
every lag, without affecting future incrementals,
so every incremental-to-previous-cumulative ra-
tio will be reduced. The points near the upper
right corner also tend to have high impact, and
those near the upper left tend to have negative
impact, but the lower left point often stands alone
in its high impact.
The GDFs for CL are readily calculated when
factors are sums of incrementals over sums of
previous cumulatives. The fitted value at a cell
is the factor applied to the previous cumulative,
so its derivative is the product of its previous
cumulative and the derivative of the factor with
respect to the cell value. But that derivative is just
the reciprocal of the sum of the previous cumu-
latives, so the GDF for the cell is the quotient
of its previous cumulative and the sum. Thus
these GDFs sum down a column to unity, so
each development factor uses up a total GDF of
1.0. Essentially each factor uses 1 degree of free-
dom, agreeing with standard analysis. The aver-
age GDF in a column is thus the reciprocal of
the number of observations in that column. Thus
the upper right cell uses 1 GDF, the previous
column’s cells use 12each on average, etc. Thus
the upper right cells have high GDFs and high
impact.
We will use ODP, for over-dispersed Poisson,
to refer to the cross-classified development mod-
el in which each cell mean is modeled as a prod-
uct of a row parameter and a column parame-
ter, the variance of the cell is proportional to its
mean, and the parameters are estimated by the
quasi-likelihood method. It is well known that
this model gives the same reserve estimate as CL.
Thus if you change a cell slightly, the changed
triangle will give the same reserve under ODP
and CL. Thus the impacts of each cell under
ODP will be the same as those of CL. The GDFs
will not be the same, however, as the fitted val-
ues are not the same for the two models. The
CL fitted value is the product of the factor and
the previous cumulative, whereas the ODP cu-
mulative fitted values are backed down from the
latest diagonal by the development factors, and
then differenced to get the incremental fitted. It
is possible to write down the resulting GDFs ex-
plicitly, but it is probably easier to calculate them
numerically.
It may be fairly easy to find models that re-
duce the impact of the upper right cells. Usually
the development factors at those points are not
statistically significant. Often the development is
small and random, and is not correlated with the
previous cumulative values. In such cases, it may
be reasonable to model a number of such cells as
a simple additive constant. Since several cells go
140 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Robustifying Reserving
Table 2. Incremental loss development triangle
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
11,305 18,904 17,474 10,221 3,331 2,671 693 1,145 744 112 40 138,828 13,953 11,505 7,668 2,943 1,084 690 179 1,014 226 16 6168,271 15,324 9,373 11,716 5,634 2,623 850 381 16 28 5587,888 11,942 11,799 6,815 4,843 2,745 1,379 266 809 128,529 15,306 11,943 9,460 6,097 2,238 493 136 11
10,459 16,873 12,668 9,199 3,524 1,027 924 1,1908,178 12,027 12,150 6,238 4,631 919 435
10,364 17,515 13,065 12,451 6,165 1,38111,855 20,650 23,253 9,175 10,31217,133 28,759 20,184 12,87419,373 31,091 25,12018,433 29,13120,640
Table 3. Impact of CL
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
AY0 ¡1:21 ¡0:34 0.04 0.39 0.73 1.10 1.48 1.85 2.46 3.35 4.61 7.31AY1 ¡1:21 ¡0:34 0.04 0.39 0.73 1.10 1.48 1.85 2.46 3.35 4.61 7.31AY2 ¡1:17 ¡0:29 0.08 0.44 0.78 1.14 1.53 1.89 2.51 3.39 4.66AY3 ¡1:15 ¡0:27 0.10 0.46 0.80 1.16 1.55 1.91 2.53 3.41AY4 ¡1:14 ¡0:27 0.11 0.46 0.80 1.17 1.56 1.92 2.54AY5 ¡1:10 ¡0:23 0.15 0.50 0.84 1.21 1.59 1.96AY6 ¡1:07 ¡0:20 0.18 0.53 0.87 1.24 1.62AY7 ¡1:03 ¡0:16 0.22 0.57 0.91 1.28AY8 ¡0:95 ¡0:08 0.30 0.65 0.99AY9 ¡0:73 0.14 0.52 0.87AY10 ¡0:31 0.57 0.95AY11 0.70 1.58AY12 4.95
into the estimation of this constant, the impact of
some of them is reduced. Alternatively, the fac-
tors in that region may follow some trends, linear
or not, that can be used to express them with a
small number of parameters. Again, this would
limit the impact of some of the cells.
The lower left point is more difficult to deal
with in a CL-like model. One alternative is a
Cape-Cod type model, where every accident year
has the same mean level. This can arise, for in-
stance, if there is no growth in the business, but
also can be seen when the development triangle
consists of on-level loss ratios, which have been
adjusted to eliminate known differences among
the accident years. In this type of model, all the
cells go into estimating the level of the last ac-
cident year, so the lower left cell has much less
impact. This reduction in the impact of the ran-
dom component of this cell is a reason for using
on-level triangles.
The next three sections illustrate these con-
cepts using development triangles from the ac-
tuarial literature. The impacts and GDFs are cal-
culated for various models fit to these triangles.
The impacts are calculated by numerical deriva-
tives, as are the GDFs except for those for the
CL, which have been derived above.
5. A development-factor example5.1. Chain ladder
Table 2 is a development triangle used in Ven-
ter (2007a). Note that the first two accident years
are developed all the way to the end of the tri-
angle, at lag 11. Table 3 shows the impact of
each cell on the reserve estimate using the usual
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 141
Variance Advancing the Science of Risk
Figure 1. Impact of chain ladder by diagonal
Table 4. GDFs of CL
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
AY0 1.0 0.080 0.093 0.114 0.133 0.151 0.177 0.201 0.245 0.306 0.394 0.581AY1 1.0 0.063 0.070 0.082 0.097 0.110 0.128 0.145 0.174 0.221 0.285 0.419AY2 1.0 0.059 0.073 0.079 0.103 0.124 0.147 0.167 0.202 0.250 0.321AY3 1.0 0.056 0.061 0.076 0.089 0.106 0.128 0.148 0.177 0.223AY4 1.0 0.061 0.073 0.086 0.104 0.126 0.149 0.168 0.202AY5 1.0 0.074 0.084 0.096 0.113 0.130 0.149 0.170AY6 1.0 0.058 0.062 0.077 0.089 0.106 0.123AY7 1.0 0.074 0.086 0.098 0.123 0.146AY8 1.0 0.084 0.100 0.134 0.149AY9 1.0 0.122 0.141 0.158AY10 1.0 0.138 0.156AY11 1.0 0.131AY12 1.0
sum/sum development factors. In the CL model
an explicit formula can be derived for these im-
pacts, but it is easier to do the derivatives numeri-
cally, simply by adding a small value to each cell
separately and recalculating the estimated reserve
to get the change in reserve for the derivative.
As discussed, the impacts are highest in the up-
per right and lower left corners, and the upper left
has negative impact. The impacts increase mov-
ing to the right and down. The last four columns
and the lower left point have impacts greater
than 2, and six points have impacts greater than
4. Table 4 shows the GDFs for the chain ladder
using the formula previous cumulative/sum pre-
vious cumulatives derived in Section 4. L0’s
GDFs are shown as identically 1.0. Like the im-
pact function, except for lag 0, these increase
from one column to the next. Within each col-
umn the sizes depend on the volume of the year.
Figure 1 graphs the impacts by lag along the
diagonals of the triangle. After the first four lags,
the impacts are almost constant across diagonals.
142 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Robustifying Reserving
Table 5. Impact of regression model
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
AY0 ¡1:36 0.02 0.42 0.67 0.10 0.87 1.35 1.35 0.97 1.35 0.97 1.73AY1 ¡1:56 0.22 0.66 ¡0:04 0.67 1.28 1.35 0.97 1.35 0.97 1.73 1.35AY2 ¡1:53 0.52 ¡0:39 0.38 1.02 1.27 0.97 1.35 0.97 1.73 1.35AY3 ¡0:51 ¡0:64 0.15 0.78 1.07 0.90 1.35 0.97 1.73 1.35AY4 ¡1:24 ¡0:31 0.45 0.76 0.64 1.27 0.97 1.73 1.35AY5 ¡1:38 0.11 0.47 0.32 1.00 0.89 1.73 1.35AY6 ¡1:61 0.22 0.18 0.80 0.68 1.66 1.35AY7 ¡0:89 ¡0:36 0.35 0.24 1.34 1.25AY8 ¡1:34 0.00 ¡0:12 0.87 0.94AY9 0.29 ¡0:44 0.61 0.57AY10 ¡0:18 0.66 0.43AY11 1.11 1.04AY12 4.31
Table 6. GDFs of regression model
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
AY0 1.0 0.071 0.089 0.127 0.352 0.195 0.034 0.034 0.055 0.034 0.055 0.076AY1 1.0 0.047 0.056 0.305 0.107 0.099 0.034 0.055 0.034 0.055 0.076 0.034AY2 1.0 0.046 0.299 0.084 0.098 0.123 0.055 0.034 0.055 0.076 0.034AY3 1.0 0.297 0.067 0.058 0.074 0.107 0.034 0.055 0.076 0.034AY4 1.0 0.064 0.056 0.072 0.120 0.128 0.055 0.076 0.034AY5 1.0 0.062 0.073 0.110 0.118 0.149 0.076 0.034AY6 1.0 0.040 0.067 0.061 0.095 0.140 0.034AY7 1.0 0.082 0.075 0.118 0.182 0.172AY8 1.0 0.077 0.134 0.212 0.207AY9 1.0 0.198 0.239 0.246AY10 1.0 0.245 0.253AY11 1.0 0.192AY12 1.0
5.2. Regression model
Venter (2007a) fit a regression model to this
triangle, keeping the first 5 development factors
but including an additive constant. The constant
also represents development beyond lag 5. By
stretching out the incremental cells to be fitted
into a single column Y, this was put into the formof a linear model Y=X¯+ ", which assumes anormal distribution of residuals with equal vari-
ance (homoscedasticity) across cells. X has the
previous cumulative for the corresponding incre-
mentals, with zeros to pad out the columns, a
column of 1s for the constant. There were also
diagonal (calendar year) effects in the triangle.
Two diagonal dummy variables were included in
X, one with 1s for observations on the 4th di-agonal and 0 elsewhere, and one equal to 1 on
the 5th, 8th, and 10th diagonals, ¡1 on the 11th
diagonal, and 0 elsewhere. The diagonals are
numbered starting at 0, so the 4th is the one
beginning with 8,529 and the 10th starts with
19,373. The variance calculation used a hetero-
scedasticity correction. This model with 8 param-
eters fit the data better than the development
factor model with 11 parameters. Here we are
only addressing the robustness properties, how-
ever.
Table 5 gives the impact function for this mod-
el. It is clear that the large impacts on the right
side have been eliminated by using the constant
instead of factors to represent late development.
The effects of the diagonal dummies can also be
seen, especially in the right of the triangle. Now
only one point has impact greater than 2, and one
greater than 4.
Table 6 shows the GDFs for the regression
model. For regression models the GDFs for the
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Figure 2. Impact of regression model by diagonal
observations in the Y vector are known to be
calculable as the diagonal of the “hat” matrix,
where hat =X(X0X)¡1X0, e.g., see Ye [2]. How-ever in development triangles, changing an in-
cremental value also changes subsequent cumu-
latives, so the X matrix is a function of lags of
Y. This requires the derivatives to be done nu-merically. The total of these, excluding lag 0, is
8.02, which is a bit above the usual number of
parameters, due to the exceptions to normal lin-
ear models. Compared to the CL, the GDFs are
lower for lag 6 onward, but are somewhat higher
along the modeled diagonals. They are especially
high for diagonal 4, which is short and gets its
own parameter.
Figure 2 graphs the impacts. Note that due to
the diagonal effects, diagonal 11 has higher im-
pact than diagonal 12 after the first two lags.
5.3. Square root regression modelAs a correction for heteroscedasticity, regres-
sion courses sometimes advise dividing both Yand X by the square root of Y, row by row. This
makes the model Y1=2 = (X=Y1=2)¯+ ", wherethe " are IID mean zero normals. Then Y=X¯+Y1=2", so now the variance of the residuals is pro-portional to Y. This sounds like a fine idea, but itis a catastrophe from a robust viewpoint. Table 7
shows the impact function. There are 12 points
with impact over 2, seven with impact over 4,
five with impact over 10, and three with impact
over 25.
Part of the problem is that the equation Y=X¯+Y1=2" is not what you would really want.The residual variance should be proportional to
the mean, not the observations. This setup gives
the small observations small variance, and so the
ability to pull the model towards them. But the
observations might be small because of a neg-
ative residual, with a higher expected value. So
this formulation gives the small values too much
influence.
Table 8 shows the related GDFs. It is unusual
here that some points have GDFs greater than 1.
A small change in the original value can make a
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Table 7. Impact of square root regression model
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
AY0 ¡0:94 ¡0:08 0.16 0.68 0.15 0.56 0.01 0.00 0.01 0.38 4.57 15.61AY1 ¡1:06 ¡0:10 0.28 ¡0:30 2.19 1.86 0.01 0.15 0.00 0.15 10.21 0.01AY2 ¡0:58 0.12 ¡0:09 0.20 0.68 0.39 0.01 0.03 28.26 3.09 0.02AY3 ¡0:20 ¡0:50 0.13 0.66 0.69 0.27 0.00 0.11 0.00 32.67AY4 ¡0:90 ¡0:15 0.33 0.41 0.59 0.56 0.03 0.14 37.14AY5 ¡1:28 ¡0:36 0.17 0.37 2.05 2.87 0.00 0.00AY6 ¡1:20 ¡0:09 0.01 0.77 0.71 2.34 0.02AY7 ¡1:02 ¡0:18 0.36 0.23 0.76 1.97AY8 ¡0:86 ¡0:07 ¡0:01 1.23 0.46AY9 ¡0:91 ¡0:06 0.59 1.02AY10 ¡0:45 0.48 0.89AY11 0.50 1.46AY12 4.56
Table 8. GDFs of square root regression model
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
AY0 1.0 0.082 0.078 0.129 0.697 0.074 0.000 0.000 0.000 0.010 0.356 1.102AY1 1.0 0.071 0.076 0.230 0.227 0.175 0.000 0.004 0.000 0.011 0.720 0.000AY2 1.0 0.053 0.287 0.031 0.074 0.042 0.000 0.001 2.199 0.218 0.000AY3 1.0 0.201 0.045 0.081 0.064 0.025 0.000 0.008 0.000 0.906AY4 1.0 0.051 0.077 0.061 0.066 0.061 0.002 0.010 1.030AY5 1.0 0.076 0.102 0.089 0.252 0.322 0.000 0.000AY6 1.0 0.073 0.045 0.104 0.072 0.221 0.001AY7 1.0 0.069 0.103 0.053 0.106 0.249AY8 1.0 0.075 0.051 0.246 0.068AY9 1.0 0.117 0.193 0.208AY10 1.0 0.145 0.166AY11 1.0 0.144AY12 1.0
greater change in the fitted value, but due to the
non-linearity the fitted value is still not exactly
equal to the data point. The sum of the GDFs
is 13.0, which is sometimes interpreted as the
implicit number of parameters.
5.4. Gamma-p residuals
Venter (2007b) fits the same regression model
using the maximum likelihood with gamma-p
residuals. The gamma-p is a gamma distribution,
but each cell is modeled to have the variance pro-
portional to the same power p of the mean. This
models the cells with smaller means as having
smaller variances. However, the effect is not as
extreme as in the square root regression, where
the variance is proportional to the observation,
not its expected value.
In this case, p was found to be 0.71. The im-
pacts are shown in Table 9 and graphed in Fig-
ure 3. It is clear that these are not nearly as dra-
matic as the square root regression, but worse
than the regular regression, and perhaps compa-
rable to the chain ladder. Diagonals 10 and 11
can be seen to have a few significant impacts.
These are at points with small observations that
are also on modeled diagonals. Even with the
variance proportional to a power of the expected
value, these points still have a strong pull. The
GDFs are in Table 10.
Again this is less dramatic than for the square
root regression, but the small points on the mod-
eled diagonals still have high GDFs. The total of
these is 11.3, which is still fairly high. This is
somewhat troublesome, as the gamma-p model
fit the residuals quite a bit better than did the
standard regression. The fact that the problems
center on small observations on the modeled di-
agonals suggests that additive diagonal effects
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Figure 3. Impact of gamma-p residual model
Table 9. Impact of gamma-p residual model
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
AY0 ¡0:59 ¡0:07 0.24 0.59 ¡0:03 1.47 1.37 1.37 1.23 1.25 ¡1:45 7.97AY1 ¡0:90 ¡0:05 0.28 0.10 0.90 1.11 1.30 0.77 1.37 0.91 6.73 1.36AY2 ¡0:46 0.08 ¡0:07 0.56 0.94 1.43 1.22 1.45 ¡5:62 4.33 1.35AY3 ¡0:29 ¡0:58 0.21 0.47 1.31 1.37 1.21 0.98 1.47 0.10AY4 ¡0:68 ¡0:15 0.19 0.51 0.94 1.48 1.24 1.96 0.02AY5 ¡1:04 ¡0:18 0.20 0.49 0.96 1.07 1.43 1.38AY6 ¡1:00 0.09 0.22 0.45 1.28 1.13 1.41AY7 ¡1:02 ¡0:18 0.50 0.50 0.95 1.17AY8 ¡0:71 ¡0:12 0.12 0.66 0.96AY9 ¡0:85 ¡0:02 0.80 0.86AY10 ¡0:44 0.48 0.88AY11 0.46 1.45AY12 4.43
Table 10. GDFs of gamma-p residual model
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
AY0 1.0 0.082 0.078 0.129 0.697 0.074 0.000 0.000 0.000 0.010 0.356 1.102AY1 1.0 0.071 0.076 0.230 0.227 0.175 0.000 0.004 0.000 0.011 0.720 0.000AY2 1.0 0.053 0.287 0.031 0.074 0.042 0.000 0.001 2.199 0.218 0.000AY3 1.0 0.201 0.045 0.081 0.064 0.025 0.000 0.008 0.000 0.906AY4 1.0 0.051 0.077 0.061 0.066 0.061 0.002 0.010 1.030AY5 1.0 0.076 0.102 0.089 0.252 0.322 0.000 0.000AY6 1.0 0.073 0.045 0.104 0.072 0.221 0.001AY7 1.0 0.069 0.103 0.053 0.106 0.249AY8 1.0 0.075 0.051 0.246 0.068AY9 1.0 0.117 0.193 0.208AY10 1.0 0.145 0.166AY11 1.0 0.144AY12 1.0
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Table 11. Impact of gamma-p multiplicative model
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
AY0 ¡0:94 ¡0:03 0.22 0.58 0.09 1.16 1.43 1.43 1.42 1.36 0.55 2.31AY1 ¡1:02 0.00 0.32 0.17 0.56 1.02 1.43 1.26 1.43 1.30 2.14 1.43AY2 ¡0:74 0.15 ¡0:46 0.39 0.98 1.30 1.42 1.42 ¡0:78 1.82 1.42AY3 ¡0:25 ¡0:50 ¡0:02 0.46 0.97 1.26 1.43 1.33 1.43 0.69AY4 ¡0:68 ¡0:39 0.23 0.51 0.83 1.26 1.39 1.50 0.64AY5 ¡1:09 ¡0:10 0.33 0.26 0.93 0.69 1.43 1.43AY6 ¡1:02 0.05 0.00 0.45 0.79 1.12 1.42AY7 ¡0:72 ¡0:37 0.31 0.29 1.11 1.07AY8 ¡0:81 ¡0:01 ¡0:21 0.92 0.99AY9 ¡0:76 ¡0:25 0.85 0.88AY10 ¡0:58 0.56 0.94AY11 0.35 1.50AY12 4.34
Table 12. GDFs of gamma-p multiplicative model
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
AY0 1.0 0.079 0.087 0.125 0.323 0.136 0.034 0.033 0.038 0.040 0.093 0.074AY1 1.0 0.063 0.069 0.191 0.210 0.132 0.034 0.048 0.033 0.046 0.066 0.034AY2 1.0 0.053 0.410 0.079 0.085 0.068 0.038 0.035 0.175 0.050 0.034AY3 1.0 0.361 0.105 0.070 0.071 0.063 0.033 0.044 0.031 0.101AY4 1.0 0.107 0.070 0.067 0.111 0.084 0.040 0.034 0.106AY5 1.0 0.079 0.094 0.158 0.185 0.276 0.030 0.033AY6 1.0 0.066 0.106 0.081 0.104 0.117 0.035AY7 1.0 0.143 0.093 0.127 0.108 0.200AY8 1.0 0.080 0.200 0.220 0.102AY9 1.0 0.355 0.281 0.208AY10 1.0 0.316 0.196AY11 1.0 0.163AY12 1.0
may not be appropriate for this data. They do
fit into the mold of a generalized linear model,
but that is not too important when fitting by MLE
anyway. As an alternative, the same model but
with the diagonal effects as multiplicative factors
was fit. The multiplicative diagonal model can be
written:
EY=X[,1 : 6]¯[1 : 6] ¤¯[7]X[,7] ¤¯[8]X[,8],which means that the first six columns of X aremultiplied by the first six parameters, which in-
cludes the constant term, and then the last two di-
agonal parameters are factors raised to the power
of the last two columns of X. These are nowthe diagonal dummies, which are 0, 1, or ¡1.Thus the same diagonals are higher and the same
lower, but now proportionally instead of by an
additive constant. It turns out that this model
actually fits better, with a negative loglikelihood
of 625, compared to 630 for the generalized lin-
ear model. This solves the robustness problems
as well. The impacts are in Table 11, the GDFs
in Table 12, and the impacts are graphed in
Figure 4.
Diagonal 11 still has more impact than the oth-
ers, but this barely exceeds 2.0 at the maximum.
The sum of the GDFs is 8.67. There are eight
parameters for the cell means but two more for
the gamma-p. It has been a question whether or
not to count those two in determining the number
of parameter used in the fitting. The answer to
that from the GDF analysis is basically to count
each of those as 1/3 in this case. Here the robust
analysis has uncovered a previously unobserved
problem with the generalized linear model, and
lead to an improvement.
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Figure 4. Impact of gamma-p multiplicative model
Table 13. Incremental triangle (Taylor and Ashe 1983)
Lag 0 L1 L2 L3 L4 L5 L6 L7 L8 L9
357,848 766,940 610,542 482,940 527,326 574,398 146,342 139,950 227,229 67,948352,118 884,021 933,894 1,183,289 445,745 320,996 527,804 266,172 425,046290,507 1,001,799 926,219 1,016,654 750,816 146,923 495,992 280,405310,608 1,108,250 776,189 1,562,400 272,482 352,053 206,286443,160 693,190 991,983 769,488 504,851 470,639396,132 937,085 847,498 805,037 705,960440,832 847,631 1,131,398 1,063,269359,480 1,061,648 1,443,370376,686 986,608344,014
6. A multiplicative fixed-effectsexample
A multiplicative fixed-effects model is one
where the cell means are products of fixed factors
from rows, columns, and perhaps diagonals. The
most well-known is the ODP model discussed in
Section 4, where there is a factor for each row, in-
terpreted as estimated ultimate, a factor for each
column, interpreted as fraction of ultimate for
that column, and the variance of each cell is a
fixed factor times its mean. This model if esti-
mated by MLE gives the same reserve estimates
as the chain ladder and so the same impacts for
each cell, but the GDFs are different, due to the
different fitted values.
The triangle for this example comes from Tay-
lor and Ashe (hereafter TA; 1983) and is shown
in Table 13. The CL = ODP impacts are in Ta-
ble 14 and are graphed in Figure 5.
Because the development factors are higher,
the impacts are higher than in the previous ex-
ample. Even though it is a smaller triangle, 14
points have impacts with absolute values greater
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Figure 5. Impact of CL = ODP on TA
Table 14. Impact of CL = ODP on TA
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9
AY0 ¡3:11 ¡1:62 ¡1:01 ¡0:45 0.01 0.51 1.16 2.27 4.54 12.59AY1 ¡2:87 ¡1:38 ¡0:77 ¡0:20 0.25 0.76 1.40 2.51 4.78AY2 ¡2:43 ¡0:93 ¡0:33 0.24 0.69 1.20 1.85 2.95AY3 ¡2:21 ¡0:72 ¡0:11 0.45 0.91 1.41 2.06AY4 ¡1:95 ¡0:46 0.15 0.71 1.17 1.67AY5 ¡1:67 ¡0:18 0.43 0.99 1.45AY6 ¡1:25 0.25 0.85 1.42AY7 ¡0:14 1.35 1.96AY8 2.07 3.57AY9 13.45
Table 15. GDFs of CL on TA
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9
AY0 1.0 0.108 0.110 0.115 0.120 0.153 0.208 0.272 0.423 1.0AY1 1.0 0.106 0.121 0.144 0.182 0.211 0.258 0.365 0.577AY2 1.0 0.087 0.126 0.147 0.175 0.222 0.259 0.363AY3 1.0 0.093 0.138 0.146 0.204 0.224 0.275AY4 1.0 0.133 0.111 0.141 0.157 0.189AY5 1.0 0.119 0.130 0.145 0.162AY6 1.0 0.132 0.126 0.161AY7 1.0 0.108 0.139AY8 1.0 0.113AY9 1.0
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Table 16. GDFs of ODP on TA
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9
AY0 0.154 0.261 0.273 0.295 0.229 0.224 0.253 0.301 0.459 1.0AY1 0.186 0.295 0.308 0.333 0.276 0.281 0.325 0.400 0.612AY2 0.187 0.300 0.312 0.338 0.278 0.282 0.324 0.398AY3 0.188 0.304 0.317 0.344 0.280 0.282 0.323AY4 0.184 0.309 0.322 0.348 0.275 0.271AY5 0.197 0.331 0.346 0.374 0.293AY6 0.221 0.375 0.391 0.423AY7 0.284 0.498 0.519AY8 0.370 0.747AY9 1.0
Table 17. Impact of 6-parameter gamma- 12 on TA
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9
AY0 0.65 ¡0:82 ¡1:08 ¡2:07 ¡0:87 0.97 ¡0:32 0.33 0.53 12.06AY1 1.45 ¡0:02 0.68 0.60 ¡0:25 1.90 1.40 1.61 1.57AY2 1.64 0.75 ¡0:19 0.84 0.90 1.93 1.66 1.36AY3 1.26 0.43 ¡0:21 0.97 ¡0:36 1.70 1.71AY4 1.62 0.08 0.67 0.37 0.63 1.35AY5 1.19 ¡0:11 0.57 0.51 1.17AY6 2.56 1.19 0.91 1.13AY7 2.18 1.27 1.49AY8 1.72 0.92AY9 1.59
Table 18. GDFs of 6-parameter gamma- 12 on TA
L0 L1 L2 L3 L4 L5 L6 L7 L8 L9
AY0 0.046 0.152 0.211 0.288 0.150 0.017 0.248 0.095 0.082 0.938AY1 0.044 0.031 0.057 0.155 0.014 0.115 0.018 0.051 0.043AY2 0.055 0.041 0.134 0.027 0.102 0.114 0.046 0.049AY3 0.045 0.078 0.062 0.028 0.181 0.052 0.064AY4 0.078 0.057 0.026 0.119 0.011 0.037AY5 0.037 0.147 0.083 0.032 0.026AY6 0.254 0.200 0.095 0.100AY7 0.111 0.527 0.250AY8 0.047 0.031AY9 0.047
than 2, four are greater than 4, and two are great-
er than 12. The CL GDFs are in Table 15. These
sum to 9, excluding the first column, and are
fairly high on the right where there are few ob-
servations per column. The ODP GDFs are in
Table 16. These sum to 19, and are fairly high
near the upper right and lower left corners.
The GDFs can be used to allocate the total
degrees of freedom of the residuals of n¡ p. Then is allocated equally to each observation, and the
p can be set to the GDF of each observation. This
would give a residual degree of freedom to each
observation which could be used in calculating a
standardized residual that takes into account how
the degrees of freedom vary among observations.
Venter (2007a) looked at reducing the number
of parameters in this model by setting parameters
equal if they are not significantly different, and
using trends, such as linear trends, between pa-
rameters. Also, diagonal effects were introduced.
The result was a model where each cell mean is a
product of its row, column, and diagonal factors.
There are six parameters overall. For the rows
there are three parameters, for high, medium, and
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Figure 6. Impact of gamma- 12 on TA
low accident years. Accident year 0 is low, year
7 is high, year 6 is the average of the medium
and high levels, and all other years are medium.
There are two column factors: high and low. Lags
1, 2, and 3 are high, lag 4 is an average of high
and low, lag 0 and lags 5 to 8 are low, and lag
9 is 1 minus the sum of the other lags. Finally
there is one diagonal parameter c. Diagonals 4
and 6 have factors 1+ c, lag 7 has factor 1¡ c,and all the other diagonals have factor 1.
With just six parameters this model actually
provides a better fit to the data than the 19 param-
eter model. The combining of parameters does
not degrade the fit much, and adding diagonal ef-
fects improves the fit. An improved fit over that
in Venter (2007a) was found by using a gamma-
p distribution with p = 12so the variance of each
cell is proportional to the square root of its mean.
The impacts and GDFs of this model are shown
in Tables 17 and 18, and the impacts are graphed
in Figure 6, this time along accident years.
The impacts are now all quite well contained
except for one point–the last point in AY0. Pos-
sibly because AY0 gets its own parameter, lag 9
influences the level of the other lags’ parameters,
and this is a small point with a small variance,
this model only slightly reduces the high level
of impact that point has in ODP. The same thing
can be seen in the GDFs as well, where this point
has slightly less than a whole GDF. The points
on AY7 and the modeled diagonals also have rel-
atively high GDFs, as do some small cells. The
total of the GDFs is 6.14. There are six param-
eters affecting the means, plus one for the vari-
ance of the gamma. That one can affect the fit
slightly, so counting it as 1/7th of a parameter
seems reasonable.
In an attempt to solve the problem of the
upper-right point, an altered model was fit: lag 9
gets half of the paid in the low years. This can
be considered a trend to 0 for lag 10. Making the
lags sum to 1.0 now eliminates a parameter, so
there are five. The negative loglikelihood (NLL)
is slightly worse, at 722.40 vs. 722.36, but that is
worth saving a parameter. The robustness is now
much better, with only two impacts greater than
2.0, the largest being 2.35.
7. Paid and incurred exampleVenter (2008), following Quarg and Mack
(2004), builds a model for simultaneously es-
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Table 19. Quarg-Mack paid increments
L0 L1 L2 L3 L4 L5 L6
AY0 576 1228 166 54 50 28 29AY1 866 1082 214 70 52 64AY2 1412 2346 494 164 78AY3 2286 3006 432 126AY4 1868 1910 870AY5 1442 2568AY6 2044
Table 20. Quarg-Mack unpaid
L0 L1 L2 L3 L4 L5 L6
AY0 402 300 164 120 100 80 43AY1 978 604 304 248 224 106AY2 1492 596 446 184 150AY3 1216 666 346 292AY4 944 1104 204AY5 1200 396AY6 2978
Table 21. Average reserve impact of paid
L0 L1 L2 L3 L4 L5 L6
A0 ¡0:68 ¡:02 0.32 0.86 2.32 5.95 13.99A1 ¡0:45 0.20 0.54 1.08 2.54 6.17A2 ¡0:41 0.24 0.58 1.12 2.59A3 ¡0:36 0.30 0.64 1.18A4 ¡0:32 0.33 0.67A5 ¡0:20 0.46A6 1.37
timating paid and incurred development, where
each influences the other. The paid losses are part
of the incurred losses, so the separate effects are
from the paid and unpaid triangles, shown in Ta-
bles 19 and 20.
First, the impacts on the reserve (7059.47) from
the average of the paid and incurred chain ladder
reserves are calculated, where the paids at the last
lag are increased by the incurred-to-paid ratio at
that lag. Tables 21 and 22 show the impacts of
the paid and unpaid triangles, and Tables 23 and
24 show the GDFs.
The impacts of the lower left are not great,
mostly because the development factors are fairly
low in this example. The impacts on the upper
right of both paid and unpaid losses are quite
high, however. The unpaid losses other than the
last diagonal have a negative impact, because
Table 22. Average reserve impact unpaid
L0 L1 L2 L3 L4 L5 L6
A0 ¡0:29 ¡0:15 ¡0:26 ¡0:72 ¡1:76 ¡4:01 14.99A1 ¡0:29 ¡0:15 ¡0:26 ¡0:72 ¡1:76 3.57A2 ¡0:29 ¡0:15 ¡0:26 ¡0:72 1.77A3 ¡0:29 ¡0:15 ¡0:26 1.08A4 ¡0:29 ¡0:15 0.82A5 ¡0:29 0.68A6 0.84
Table 23. Average reserve GDF of paid
0 L1 L2 L3 L4 L5 L6
A0 1 0.068 0.109 0.140 0.233 0.476 1A1 1 0.102 0.117 0.153 0.257 0.524A2 1 0.167 0.227 0.301 0.509A3 1 0.271 0.319 0.406A4 1 0.221 0.228A5 1 0.171A6 1
Table 24. Average reserve GDF unpaid
0 L1 L2 L3 L4 L5 L6
A0 1 0.067 0.106 0.139 0.232 0.464 1A1 1 0.126 0.129 0.160 0.269 0.536A2 1 0.198 0.219 0.306 0.499A3 1 0.239 0.300 0.395A4 1 0.192 0.246A5 1 0.180A6 1
they lower subsequent incurred development fac-
tors, but do not have factors applied to them. The
GDFs are similar to CL in general.
The model in Venter (2008) used generalized
regression for both the paid and unpaid triangles,
where regressors could be from either the paid
and unpaid triangles or from the cumulative paid
and incurred triangles. Except for the first couple
of columns, the previous unpaid losses provided
reasonable explanations of both the current paid
increment and the current remaining unpaid. The
paid and unpaid at lags 3 and on were just mul-
tiples of the previous unpaid, with a single fac-
tor for each. That is, expected paids were 33.1%,
and unpaids 72.3%, of the previous unpaid. Since
these sum to more than 1, there is a slight upward
drift in the incurred. The lag 2 expected paid was
68.5% of the lag 1 unpaid. The best fit to the lag
152 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Robustifying Reserving
Table 25. Weibull model impact of paid
L0 L1 L2 L3 L4 L5 L6
A0 0.09 ¡0:18 ¡1:58 4.38 0.38 7.67 5.45A1 0.04 0.26 0.59 1.90 2.75 2.32A2 ¡:37 0.33 0.42 0.57 ¡0:28A3 ¡:13 0.17 0.67 1.26A4 ¡:02 0.20 0.31A5 ¡:94 0.70A6 1.25
Table 26. Weibull model impact unpaid
L0 L1 L2 L3 L4 L5 L6
A0 0.06 0.67 ¡1:02 ¡1:45 ¡1:82 0.51 4.14A1 ¡0:17 ¡0:44 ¡1:80 ¡0:73 0.52 2.56A2 ¡0:20 ¡0:16 0.47 ¡1:17 3.63A3 ¡0:09 ¡0:32 ¡1:17 2.51A4 ¡0:10 ¡0:34 1.89A5 ¡0:32 1.47A6 0.65
Table 27. Weibull model GDF of paid
0 L1 L2 L3 L4 L5 L6
A0 1 0.938 0.725 0.235 0.268 0.125 .143A1 1 0.451 0.057 0.052 0.066 0.065A2 1 0.192 0.347 0.377 0.290A3 1 0.137 0.250 0.145A4 1 0.094 0.277A5 1 0.269A6 1
Table 28. Weibull model GDF unpaid
0 L1 L2 L3 L4 L5 L6
A0 1 0.824 0.172 ¡0:058 0.015 .072 .054A1 1 0.357 0.700 ¡0:044 0.115 .052A2 1 0.152 0.465 ¡0:173 0.113A3 1 0.050 0.136 0.123A4 1 0.507 0.089A5 1 0.687A6 1
2 expected unpaid was 9.1% of the lag 1 cumu-
lative paid. For lag 1 paid, 78.1% of the lag 0
incurred was a reasonable fit. Lag 1 unpaid was
more complicated, with the best fit being a re-
gression, with constant, on lag 0 and lag 1 paids.
There were also diagonal effects in both mod-
els. The residuals were best fit with a Weibull
distribution. Tables 25—28 show the fits.
The two highest impacts for the average of
paid and incurred are 14 and 15. For the Weibull
they are 7.7 and 5.5. The average has two other
points with impacts greater than 5, whereas the
Weibull has none. Below 5 the impacts are rough-
ly comparable. Since the Weibull has variance
proportional to the mean squared, small obser-
vations have lower variance, and so a stronger
pull on the model and higher impacts. In total,
excluding the first column, the GDFs sum to 9.9,
but including the diagonals (see Venter 2008 for
details) there are 12 parameters plus two Weibull
shape parameters. The form of the model appar-
ently does not allow the parameters to be fully
expressed. The Weibull model still has more high
impacts than would be desirable, but it is a clear
improvement over the average of the paid and
incurred. The reserve is quite a bit lower for the
better fitting Weibull model as well: 6255 vs.
7059.
8. ConclusionRobust analysis has been introduced as an
additional testing method for loss development
models. It is able to identify points that have a
large influence on the reserve, and so whose ran-
dom components would also have a large influ-
ence. Through three examples, customized mod-
els were found to be more robust than standard
models like CL and ODP, and in two of the ex-
amples, even better models were found as a re-
sponse to the robust analysis.
ReferencesBilodeau, M., Discussions of Papers Already Published.
North American Actuarial Journal 5(3), 2001, pp. 123—
28. (Online version included in Brazauskas, Vytaras, and
Robert Serfling, 2000).
Brazauskas, V., and R. Serfling, “Robust and Efficient Es-
timation of the Tail Index of a Single-Parameter Pareto
Distribution,” North American Actuarial Journal 4(4),
2000a, pp. 12—27.
Brazauskas, V., and R. Serfling, “Robust Estimation of Tail
Parameters for Two-Parameter Pareto and Exponential
Models via Generalized Quantile Statistics,” Extremes 3,
2000b, pp. 231—249.
Dell’Aquila, R., E. Ronchetti, and F. Troiani, “Robust GMM
Analysis of Models for the Short Rate Process,” Journal
of Empirical Finance 10, 2003, pp. 373—397.
Finkelstein, M., H. G. Tucker, and J. A. Veeh, “Pareto Tail
Index Estimation Revisited” North American Actuarial
Journal 10(1), 2006, pp. 12—27.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 153
Variance Advancing the Science of Risk
Gather, U., and V. Schultze, “Robust Estimation of Scale of
an Exponential Distribution,” Statistica Neerlandica 53,
1999, pp. 327—341.
Hampel, F. R., “Contributions to the Theory of Robust Esti-
mation,” Ph.D. thesis, University of California, Berkeley,
1968.
Quarg, G., and T. Mack, “Munich Chain-Ladder–A Re-
serving Method that Reduces the Gap between IBNR
Projections Based on Paid Losses and IBNR Projections
Based on Incurred Losses,” Blatter DGVFM 26, 2004,
pp. 597—630.
Serfling, R., “Efficient And Robust Fitting of Lognormal
Distributions,” North American Actuarial Journal 6(4),
2002, pp. 95—116.
Tampubolon, D. R., “Uncertainties in the Estimation of Out-
standing Claims Liability in General Insurance,” Ph.D.
thesis, Macquarie University, 2008.
Taylor, G. C., and F. R. Ashe, “Second Moments of Esti-
mates of Outstanding Claims,” Journal of Econometrics
23, 1983, pp. 37—61.
Tukey, J. W., “A Survey of Sampling from Contaminated
Distributions,” in I. Olkin (Ed.), Contributions to Prob-
ability and Statistics, pp. 448—485. Palo Alto: Stanford
University Press, 1960.
Venter, G. G., “Refining Reserve Runoff Ranges,” CAS
E-Forum, August 2007a.
Venter, G. G., “Generalized Linear Models beyond the Ex-
ponential Family with Loss Reserve Applications,” CAS
E-Forum, August 2007b.
Venter, G. G., “Distribution and Value of Reserves Using
Paid and Incurred Triangles,” CAS Reserve Call Papers
2008.
Ye, J., “On Measuring and Correcting the Effects of Data
Mining and Model Selection,” Journal of the American
Statistical Association 93, 1998, pp. 120—131.
154 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Prediction Error of the Future ClaimsComponent of Premium Liabilitiesunder the Loss Ratio Approach
by Jackie Li
ABSTRACT
In this paper we construct a stochastic model and derive
approximation formulae to estimate the standard error of
prediction under the loss ratio approach of assessing pre-
mium liabilities. We focus on the future claims compo-
nent of premium liabilities and examine the weighted and
simple average loss ratio estimators. The resulting mean
square error of prediction contains the process error com-
ponent and the estimation error component, in which the
former refers to future claims variability while the latter
refers to the uncertainty in parameter estimation. We illus-
trate the application of our model to public liability data
and simulated data.
KEYWORDS
Premium liabilities, loss ratio, standard error of prediction, mean square
error of prediction
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 155
Variance Advancing the Science of Risk
1. IntroductionThere has been extensive literature on loss re-
serving models over the past 25 years, includ-
ing the Mack (1993) model. While the focus has
been largely on how to tackle outstanding claims
liabilities, relatively few materials have been pre-
sented for premium liabilities. Some references
include Cantin and Trahan (1999), Buchanan
(2002), Collins and Hu (2003), and Yan (2005),
which focus on the central estimate (i.e., the
mean) of premium liabilities but not on the un-
derlying variability.
As noted in Clark et al. (2003), the Interna-
tional Accounting Standards Board (IASB) has
proposed a new reporting regime for insurance
contracts, in which both outstanding claims li-
abilities and premium liabilities should be as-
sessed at their fair values. It is generally under-
stood that this fair value includes a “margin” al-
lowing for different types of variability for insur-
ance liabilities. Accordingly, the Australian Pru-
dential Regulation Authority (APRA) has pre-
scribed an approach similar to the fair value ap-
proach. Under Prudential Standard GPS 310, a
“risk margin” has to be explicitly calculated such
that outstanding claims liabilities and premium
liabilities are assessed at a sufficiency level of
75%, subject to a minimum of the mean plus one-
half the standard deviation. Australian Account-
ing Standard AASB 1023 also requires inclusion
of a risk margin, though there is no prescription
on the adequacy level. No matter what approach
one takes, it is obvious that urgency for develop-
ing proper tools to measure liability variability
exists not only for outstanding claims liabilities
but also for premium liabilities. In addition, ac-
cording to Yan (2005), premium liabilities ac-
count for around 30% of insurance liabilities for
direct insurers and 15% to 20% for reinsurers in
Australia from 2002 to 2004. Premium liabilities
represent a significant portion of an insurer’s li-
abilities and proper assessment of the underlying
variability should not be overlooked.
The definition of premium liabilities varies for
different countries. Broadly speaking, premium
liabilities refer to all future claim payments and
associated expenses arising from future events
after the valuation date which are insured under
the existing unexpired policies. Buchanan (2002)
notes that there are two main methods of de-
termining the central estimate of premium lia-
bilities. The first method is prospective in na-
ture and involves a full actuarial assessment from
first principles. Yan (2005) calls this method the
claims approach and differentiates it into the loss
ratio approach and historical claims approach.
The loss ratio approach is the most common one
for premium liability assessment in practice and
is essentially an extension of the outstanding
claims liability valuation. It applies a projected
loss ratio to the unearned premiums or number
of policies unexpired. The historical claims ap-
proach uses the number of claims and average
claim size and is more suitable for short-tailed
lines of business where data is sufficient. While
the historical claims approach has been studied
extensively under the classical risk theory, the
loss ratio approach has received relatively little
attention in the literature. In this paper we follow
the loss ratio approach and attempt to supplement
this knowledge gap.
On the other hand, the second method noted in
Buchanan (2002) is retrospective in nature and
involves an adjustment of the unearned premi-
ums to take out the profit margin. As discussed
in Cantin and Trahan (1999) and Yan (2005),
both Canadian and Australian accounting stan-
dards require a reporting of this unearned premi-
ums item, in which a premium deficiency reserve
is added if this item is less than the premium li-
ability estimate determined by the first method.
Obviously the first method above plays a key role
in premium liability assessment, and we focus
on the loss ratio approach under this prospective
method.
156 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach
In this paper we construct a stochastic model
to estimate the standard error of prediction under
the loss ratio approach of assessing premium lia-
bilities. We focus on modeling the future claims
which form the largest component in premium li-
abilities (about 85% according to Collins and Hu
2003). We look at the weighted average ultimate
loss ratio and simple average ultimate loss ratio,
and derive approximation formulae to estimate
the corresponding mean square error of predic-
tion with respect to the accident year following
the valuation date. As similarly reasoned in Tay-
lor (2000), the resulting mean square error of
prediction is composed of the process error com-
ponent and the estimation error component, and
no covariance term exists as one part is related
only to the future while the other only to the past.
We also illustrate the application of our model to
Australian private-sector direct insurers’ public
liability data and some hypothetical data simu-
lated from the compound Poisson model.
The outline of the paper is as follows. In Sec-
tion 2 we introduce the basic notation and as-
sumptions of our model. In Section 3 we present
the formulae for estimating the standard error of
prediction for premium liabilities. In Section 4
we apply the model to public liability data and
simulated data and analyze the results. In Sec-
tion 5 we set forth our concluding remarks. Ap-
pendices A to D furnish the proofs for the for-
mulae stated in this paper.
2. Notation and assumptions
Let Ci,j (for 1· i · n+1 and 1· j · n) bea random variable representing the cumulative
claim amount (either paid or incurred) of acci-
dent year i and development year j. Assuming
all claims are settled in n years, Ci,n represents
the ultimate claim amount of accident year i.
We consider the case where a run-off triangle
of Ci,j’s is available for i+ j · n+1. In effect,the valuation date is at the end of accident year
n, Ci,j’s for i+ j > n+1 and 1· i· n refer to
the future claims of outstanding claims liabili-
ties, and Cn+1,j’s refer to the future claims of
premium liabilities. Let Ei (for 1· i · n+1) bethe premiums of accident year i. The premiums
are assumed to be known. The term Ci,n=Ei then
becomes the ultimate loss ratio of accident year i.
It is also assumed that exposure is evenly dis-
tributed over each year, and the exposure dis-
tribution of accident year n+1 is the same as
that of the past accident years. In reality, the
future exposure relating to premium liabilities
would arise more from the earlier part of accident
year n+1, while the past exposure would spread
more uniformly across the whole year. Although
the timing of claims development is actually dif-
ferent between the two cases, the way that the
claims develop to ultimate remains basically the
same. As our focus is on the ultimate loss ratio,
this approximation is reasonable and represents
a convenient simplification for the model setting.
As mentioned in the Introduction, the loss ratio
approach for the premium liability valuation is
basically an extension of the outstanding claims
liability valuation. Hence we start with the struc-
ture of the chain ladder method, which is the
most common method for assessing outstanding
claims liabilities in practice and is linked to a
distribution-free model in Mack (1993). Incor-
porating Ei into the three basic assumptions of
the Mack (1993) model, we deduce the follow-
ing for 1· i · n+1:
E
ÃCi,j+1Ei
¯¯Ci,1,Ci,2, : : : ,Ci,j
!=Ci,jEifj ;
(for 1· j · n¡ 1) (2.1)
Var
ÃCi,j+1Ei
¯¯Ci,1,Ci,2, : : : ,Ci,j
!=Ci,j
E2i¾2j ;
(for 1· j · n¡ 1) (2.2)
Ci,j and Cg,h are independent.
(for i 6= g) (2.3)
The parameter fj is the development ratio and
the parameter ¾2j is related to the conditional vari-
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 157
Variance Advancing the Science of Risk
ance of Ci,j+1. These two parameters are un-
known and need to be estimated from the claims
data.
As of the valuation date, there is no claims
data for accident year n+1. In order to model
the future claims in the first development year
of accident year n+1, we add the following two
assumptions for 1· i · n+1, which are anal-ogous to those for new claims in Schnieper
(1991):
E
μCi,1Ei
¶= u; (2.4)
Var
μCi,1Ei
¶=v2
Ei: (2.5)
Rearranging assumptions (2.4) and (2.5) into
E(Ci,1) = Eiu and Var(Ci,1) = Eiv2, we can see
that the mean and variance of the claim amount
of the first development year is effectively as-
sumed to be proportional to the premiums. This
is analogous to assumptions (2.1) and (2.2), in
which the conditional mean and variance of the
claim amount Ci,j for 2· j · n depends on theprevious development year’s claim amount
Ci,j¡1. The parameters u and v2 are unknown andcan be estimated from the claims and premiums
data.
Mack (1993) suggests the following unbiased
estimators for fj and ¾2j and proves that fj and
fh are uncorrelated for j 6= h:
fj =
Pn¡jr=1Cr,j+1Pn¡jr=1Cr,j
=
Pn¡jr=1Cr,j
Cr,j+1Cr,jPn¡j
r=1Cr,j;
(for 1· j · n¡ 1) (2.6)
¾2j =1
n¡ j¡ 1n¡jXr=1
Cr,j
ÃCr,j+1Cr,j
¡ fj!2;
(for 1· j · n¡ 2)
¾2n¡1 = minþ4n¡2¾2n¡3
, ¾2n¡3
!: (2.7)
We now introduce two unbiased estimators for
u and v2 as follows, which are again based on
Schnieper (1991):
u=
Pnr=1Cr,1Pnr=1Er
=
Pnr=1Er
Cr,1ErPn
r=1Er; (2.8)
v2 =1
n¡ 1nXr=1
Er
μCr,1Er
¡ u¶2: (2.9)
It can be seen that both formulae (2.6) and
(2.8) are weighted averages and that both formu-
lae (2.7) and (2.9) use weighted sums of squares.
The proofs for unbiasedness of u and v2 are givenin Appendix A.
In effect, we integrate the model assumptions
in Mack (1993) with those of development year
one for new claims in Schnieper (1991). The
overall structure is based on the chain ladder
method. It then becomes possible to assess the
next accident year’s ultimate loss ratio using the
observed run-off triangle. As shown in the next
section, the results of the outstanding claims lia-
bility valuation (i.e., projected ultimate loss ratios
of the past accident years) are carried through
to the premium liability valuation (regarding the
expected ultimate loss ratio of the accident year
following the valuation date).
3. Standard error of predictionIn practice, actuaries often examine the pro-
jected ultimate loss ratios of past accident years
and compare these figures with target or bud-
get ratios or industry ratios to obtain an esti-
mate of the next accident year’s ultimate loss
ratio. Here we assume no such prior knowledge
or objective information is available and inves-
tigate the following two estimators for the next
accident year’s expected ultimate loss ratio q=E(Cn+1,n=En+1):
q=
Pni=1Ci,n+1¡ifn+1¡ifn+2¡i : : : fn¡1Pn
i=1Ei
=
Pni=1Ci,n+1¡iSn+1¡i,n¡1Pn
i=1Ei=
Pni=1 Ci,nPni=1Ei
=
Pni=1Ei
Ci,nEiPn
i=1Ei; (3.1)
158 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach
q¤ =1
n
nXi=1
Ci,n+1¡ifn+1¡ifn+2¡i : : : fn¡1Ei
=1
n
nXi=1
Ci,n+1¡iSn+1¡i,n¡1Ei
=1
n
nXi=1
Ci,nEi:
(3.2)
Let Sj,h = fj fj+1 : : : fh (for j · h; equal to oneotherwise) and Ci,j = Ci,n+1¡iSn+1¡i,j¡1 (for i+j > n+1 and 1· i · n). C1,n is read as equalto C1,n. Formula (3.1) gives a weighted average
while formula (3.2) provides a simple average.
Both estimators are unbiased and the proofs are
set forth in Appendix B. For the expected future
claims component of premium liabilities of the
next accident year E(Cn+1,n), we define its esti-
mator as
Cn+1,n = En+1q: (3.3)
For now we deal with (3.1) and, as shown later,
the results of (3.1) can readily be extended to the
use of (3.2). We will also look at the effects of
excluding some accident years when calculating
q, as a practitioner may exclude or adjust a few
years’ projected loss ratios that are regarded as
inconsistent with the rest, out of date, or irrele-
vant. Such circumstances arise when there have
been past changes in, for example, the regula-
tions, underwriting procedures, claims manage-
ment, business mix, or reinsurance arrangements.
Using the idea in Chapter 6 of Taylor (2000),
we define the mean square error of prediction of
the estimator q as follows:
MSEP(q) = E
ÃμCn+1,nEn+1
¡ q¶2!
= E
ÃμCn+1,nEn+1
¡ q+ q¡ q¶2!
= E
ÃμCn+1,nEn+1
¡ q¶2!
+E((q¡ q)2)
(Cn+1,n and q are independent due to (2.3))
= Var
μCn+1,nEn+1
¶+Var(q): (q is unbiased)
(3.4)
The mean square error of prediction above
consists of two components: the first allows for
process error and the second for estimation er-
ror. The process error component refers to future
claims variability and the estimation error com-
ponent refers to the uncertainty in parameter esti-
mation due to sampling error. As similarly noted
in Taylor (2000), there is no covariance term in
(3.4) because at the valuation date, Cn+1,n is en-
tirely related to the future while q is completely
based on the past observations.
The corresponding standard error of prediction
can then be calculated as
SEP(q) =qMSEP(q): (3.5)
For the next accident year’s expected ultimate
claim amount, we compute the standard error of
prediction of its estimator as
SEP(Cn+1,n) = En+1 SEP(q) = En+1
qMSEP(q):
(3.6)
We derive the process error component as fol-
lows and the proof is given in Appendix C:
Var
μCn+1,nEn+1
¶=
1
En+1E
μCn+1,nEn+1
¶
£n¡1Xj=1
¾2jfjfj+1fj+2 : : :fn¡1
+v2
En+1f21 f
22 : : :f
2n¡1, (3.7)
which can be estimated by
dVarμCn+1,nEn+1
¶=
q
En+1
n¡1Xj=1
¾2j
fjSj+1,n¡1
+v2
En+1S21,n¡1: (3.8)
The estimation error component requires more
computation. We derive the following approxi-
mation for this component and the proof is pro-
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 159
Variance Advancing the Science of Risk
vided in Appendix D:
Var(q)¼ 1¡Pn
i=1Ei¢2 n¡1X
j=1
ÃnX
i=n+1¡j
E(Ci,n)
fj
!2
Var(fj)
+1¡Pn
i=1Ei¢2 nX
i=1
f2n+1¡if2n+2¡i : : :f
2n¡1Var(Ci,n+1¡i)
+2¡Pn
i=1Ei¢2 n¡1X
j=1
n¡jXi=1
ÃnX
r=n+1¡j
E(Cr,n)
fj
!
£ (fn+1¡ifn+2¡i : : :fn¡1)Cov(fj ,Ci,n+1¡i), (3.9)
which can be estimated by
dVar(q) = 1¡Pni=1Ei
¢2 n¡1Xj=1
0@ nXi=n+1¡j
Ci,n
fj
1A2dVar(fj)+
1¡Pni=1Ei
¢2 nXi=1
S2n+1¡i,n¡1dVar(Ci,n+1¡i)+
2¡Pni=1Ei
¢2 n¡1Xj=1
n¡jXi=1
nXr=n+1¡j
Cr,n
fj
£ Sn+1¡i,n¡1dCov(fj ,Ci,n+1¡i),(3.10)
where the estimators of the variance and covari-
ance terms are derived as
dVar(fj) = ¾2jPn¡jr=1Cr,j
; (3.11)
dVar(Ci,n+1¡i) = Ci,n+1¡i n¡iXj=1
¾2j
fjSj+1,n¡i+Eiv
2S21,n¡i;
(3.12)
dCov(fj ,Ci,n+1¡i) = Ci,n+1¡iPn¡jr=1Cr,j
¾2j
fj: (3.13)
By now we have shown all the formulae that
are needed to calculate the standard error of pre-
diction of (3.1). Note that the term fjfj+1 : : :fn¡1for j > n¡ 1 is read as equal to one in the sum-mations. In the next section we will apply these
formulae to some real claims data and simulated
data.
4. Illustrative examples
We first apply the formulae shown previously
to some public liability data. We use the aggre-
gated claim payments and premiums (both gross
and net of reinsurance) of the private-sector di-
rect insurers from the “Selected Statistics on the
General Insurance Industry” (APRA) for acci-
dent years 1981 to 1991 (n= 10). Adopting the
approach as described in Hart et al. (1996), all
the figures have been converted to constant dol-
lar values in accordance with the average weekly
ordinary time earnings (AWOTE) before the cal-
culations. This procedure is common in practice
and is based on the assumption that wage infla-
tion is the ‘normal’ inflation for the claims.
The inflation-adjusted claims (incremental)
and premiums data are presented in Table 1 be-
low for gross of reinsurance and in Table 2 for
net of reinsurance. All the figures are in thou-
sands.
The two run-off triangles show that it takes
several years for public liability claims to de-
velop and this line of business is generally re-
garded as a long-tailed line of business. We use
formulae (2.6) to (2.9), (3.1), and (3.3) to es-
timate the parameters, accident year 1991’s ex-
pected ultimate loss ratio, and so the expected fu-
ture claims of premium liabilities. We then adopt
formulae (3.4) to (3.13) to compute the corre-
sponding standard error of prediction. Table 3
below presents our results both gross and net of
reinsurance.
As shown in Table 3, the estimated gross and
net expected ultimate loss ratios for accident year
1991 are 49.2% and 53.6%. The standard error of
prediction for the future claims of premium lia-
bilities, expressed as a percentage of the mean, is
greater for gross than for net. The gross and net
percentages are 47.1% and 33.1% respectively.
This feature is in line with the general percep-
tion that gross liability variability is greater than
its net counterpart. In both cases the process er-
ror component is larger than the estimation error
component.
160 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach
Table 1. Public liability (gross of reinsurance)
Claims 1 2 3 4 5 6 7 8 9 10 Premiums
1981 15,898 20,406 17,189 19,627 35,034 12,418 8,922 12,555 8,965 6,693 289,7321982 16,207 21,518 17,753 18,780 19,113 18,634 15,857 13,050 9,362 319,2161983 14,141 20,315 16,458 25,473 16,427 92,888 18,698 15,295 314,6071984 14,649 21,162 19,084 23,857 20,171 15,098 17,637 344,4461985 21,949 26,455 23,285 25,251 22,286 23,424 418,3581986 18,989 28,741 32,754 30,240 28,443 535,6581987 19,367 36,420 31,204 27,487 639,1301988 26,860 39,550 33,852 751,8971989 23,738 52,683 780,6691990 34,567 719,1811991 334,566
Table 2. Public liability (net of reinsurance)
Claims 1 2 3 4 5 6 7 8 9 10 Premiums
1981 13,451 16,801 12,947 13,752 13,802 8,583 6,847 9,237 5,641 3,784 168,9751982 13,533 17,489 13,111 13,541 13,603 11,937 10,524 8,609 5,987 186,9901983 11,808 17,525 12,644 15,609 11,821 17,305 10,524 11,061 200,4751984 13,309 17,806 14,777 17,295 15,340 12,060 11,752 222,8431985 19,546 22,786 19,686 21,860 19,268 18,692 262,7481986 17,865 25,888 28,194 25,578 22,985 333,7161987 17,797 33,517 24,182 24,337 410,4291988 24,591 33,398 28,512 502,8691989 21,567 46,146 532,2981990 30,343 545,2181991 234,659
All accident years’ estimated ultimate loss ra-
tios are fairly consistent with one another ex-
cept the gross loss ratio of accident year 1983.
A closer look at the claims data reveals that the
gross claim payments made at accident year 1983
and development year 6 are $92,888 thousand,
the amount of which is much larger than the
other figures in the same development year. We
find that if the amount is changed to say $18,000
thousand, then the standard error of predic-
tion reduces significantly from 47.1% to 35.5%.
Whether to allow for this extra variability or ad-
just the data is a matter of judgment and in prac-
tice requires further investigation into the under-
lying features of those claims.
As mentioned in the previous section, one can
exclude some accident years’ loss ratios when
calculating q if those loss ratios are considered
inconsistent, out of date, or irrelevant. This com-
putation can readily be done by setting an indica-
tor variable for each accident year, in which the
indicator is one if the loss ratio of that accident
year is included and zero otherwise. Table 4 be-
low demonstrates some results of using different
numbers of accident years in computing q and
SEP(q) with (3.1), (3.8), and (3.10).
For each case of a particular number of acci-
dent years being included, Table 4 sets out the
average figures across all the possible combina-
tions of accident years in that case. It can be seen
that the estimation error component and so the
standard error of prediction decreases when more
accident years (i.e., more data) are used. The pro-
cess error component is stable because in our
analysis, the indicator adjustments are only
applied to (3.1) and (3.10) but not (2.6) to
(2.9).
Hitherto we have been focusing on the use of
(3.1). In many situations one may prefer using
a simple average of loss ratios as in (3.2). We
only need to replace (3.9) and (3.10) with the
following, the proof of which is analogous to
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 161
Variance Advancing the Science of Risk
Table 3. Estimated results using weighted average loss ratio
Gross of Reinsurance Net of Reinsurance
Accident Ultimate Loss Accident Ultimate LossYear Premiums Claims Ratio Year Premiums Claims Ratio
1981 289,732 157,705 54.4% 1981 168,975 104,844 62.0%1982 319,216 156,934 49.2% 1982 186,990 112,391 60.1%1983 314,607 244,292 77.6% 1983 200,475 118,959 59.3%1984 344,446 159,365 46.3% 1984 222,843 124,138 55.7%1985 418,358 192,494 46.0% 1985 262,748 165,031 62.8%1986 535,658 247,328 46.2% 1986 333,716 191,706 57.4%1987 639,130 259,865 40.7% 1987 410,429 195,706 47.7%1988 751,897 313,187 41.7% 1988 502,869 227,747 45.3%1989 780,669 364,832 46.7% 1989 532,298 264,892 49.8%1990 719,181 421,727 58.6% 1990 545,218 297,641 54.6%1991 334,566 164,750 49.2% 1991 234,659 125,678 53.6%
Var
μCn+1,n
En+1
¶Var(q) SEP(q) SEP(Cn+1,n)
E(Cn+1,n)Var
μCn+1,n
En+1
¶Var(q) SEP(q) SEP(Cn+1,n)
E(Cn+1,n)
0.0481 0.0058 0.2322 47.1% 0.0292 0.0022 0.1773 33.1%
Gross 1 2 3 4 5 6 7 8 9
fj 2.5556 1.5283 1.3761 1.2773 1.3170 1.1148 1.0886 1.0648 1.0443
¾2j 2,227.06 242.72 235.27 720.66 13,377.69 166.44 35.49 0.78 0.02
u 0.0404v2 42.1016
Net 1 2 3 4 5 6 7 8 9
fj 2.5075 1.4858 1.3431 1.2323 1.1744 1.1167 1.1043 1.0588 1.0374
¾2j 1,992.25 206.88 36.77 11.43 157.84 32.84 11.97 0.02 0.00
u 0.0546v2 50.2089
Table 4. Average results with different number of accident years included
Gross of Reinsurance Net of ReinsuranceNo. ofAccidentYearsIncluded
q Var
μCn+1,n
En+1
¶Var(q) SEP(q) SEP(Cn+1,n)
E(Cn+1,n)q Var
μCn+1,n
En+1
¶Var(q) SEP(q) SEP(Cn+1,n)
E(Cn+1,n)
1 50.7% 0.0490 0.0340 0.2852 57.0% 55.5% 0.0295 0.0245 0.2311 41.7%2 49.9% 0.0485 0.0159 0.2532 51.2% 54.4% 0.0293 0.0110 0.2006 36.9%3 49.6% 0.0483 0.0112 0.2438 49.4% 54.1% 0.0293 0.0071 0.1906 35.3%4 49.4% 0.0482 0.0091 0.2394 48.6% 53.9% 0.0292 0.0053 0.1858 34.5%5 49.4% 0.0482 0.0080 0.2369 48.1% 53.8% 0.0292 0.0042 0.1829 34.1%6 49.3% 0.0482 0.0072 0.2353 47.8% 53.7% 0.0292 0.0036 0.1810 33.7%7 49.3% 0.0482 0.0067 0.2342 47.5% 53.6% 0.0292 0.0031 0.1797 33.5%8 49.3% 0.0481 0.0063 0.2333 47.4% 53.6% 0.0292 0.0027 0.1787 33.3%9 49.3% 0.0481 0.0060 0.2327 47.3% 53.6% 0.0292 0.0025 0.1779 33.2%
10 49.2% 0.0481 0.0058 0.2322 47.1% 53.6% 0.0292 0.0022 0.1773 33.1%
162 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach
Table 5. Estimated results using simple average loss ratio
Gross of Reinsurance Net of Reinsurance
Accident Ultimate Loss Accident Ultimate LossYear Premiums Claims Ratio Year Premiums Claims Ratio
1991 334,566 169,752 50.7% 1991 234,659 130,184 55.5%
Var
μCn+1,n
En+1
¶Var(q¤) SEP(q¤) SEP(Cn+1,n)
E(Cn+1,n)Var
μCn+1,n
En+1
¶Var(q¤) SEP(q¤) SEP(Cn+1,n)
E(Cn+1,n)
0.0490 0.0063 0.2353 46.4% 0.0295 0.0027 0.1794 32.3%
Appendix D:
Var(q¤)¼ 1
n2
n¡1Xj=1
0@ nXi=n+1¡j
E(Ci,n)
Eifj
1A2
Var(fj)
+1
n2
nXi=1
f2n+1¡if2n+2¡i : : :f
2n¡1
E2iVar(Ci,n+1¡i)
+2
n2
n¡1Xj=1
n¡jXi=1
0@ nXr=n+1¡j
E(Cr,n)
Erfj
1A£μfn+1¡ifn+2¡i : : :fn¡1
Ei
¶Cov(fj ,Ci,n+1¡i),
(4.1)
which can be estimated by
dVar(q¤) = 1
n2
n¡1Xj=1
0@ nXi=n+1¡j
Ci,n
Eifj
1A2dVar(fj)+1
n2
nXi=1
S2n+1¡i,n¡1E2i
dVar(Ci,n+1¡i)+2
n2
n¡1Xj=1
n¡jXi=1
nXr=n+1¡j
Cr,n
Erfj
£ Sn+1¡i,n¡1Ei
dCov(fj ,Ci,n+1¡i):(4.2)
The estimated results using (3.2), (3.8), and
(4.2) are shown in Table 5. The resulting ulti-
mate loss ratios are slightly larger than previ-
ously while the standard error of prediction esti-
mates are slightly larger in magnitude but smaller
in percentage.
Finally we apply our formulae to some hypo-
thetical data simulated from the compound Pois-
son model Xi,j =PNi,jk=1Yi,j,k. Let Xi,j be a ran-
dom variable representing the incremental claim
amount of accident year i and development year
j and so Ci,j = Ci,j¡1 +Xi,j for 2· j · 10 andCi,1 = Xi,1. Let Ni,j and Yi,j,k be independent ran-
dom variables representing the number of claims
and the size of the kth claim of accident year i
and development year j. Let Ni,j » Pn(Ei¸j) andYi,j,k » LN(¹,¾) where ¸j’s are equal to 9£ 10¡6,8£ 10¡6, 7£ 10¡6, 6£ 10¡6, 5£ 10¡6, 5£ 10¡6,4£ 10¡6, 3£ 10¡6, 2£ 10¡6, 1£ 10¡6 respec-tively for 1· j · 10, ¹= 8:8638, and ¾ =
0:8326 (E(Yi,j,k) = 10,000 and SD(Yi,j,k) =
10,000). We assume Ei grows from 1,000,000 at
10% each year and the unearned premiums are
half of E11. Effectively, accident year 11’s ulti-
mate loss ratio has a mean of 50% and a variance
of 0.0077. We then simulate a run-off triangle
based on this compound Poisson model and ap-
ply our formulae (3.2), (3.8), and (4.2) to this
triangle.
Under the compound Poisson model above,
Xi,j’s are independent while under our model,
Ci,j+1 depends on Ci,j . Hence we expect our for-
mulae to produce a process error estimate larger
than the true variance underlying the simulated
data. The simulated run-off triangle and estimat-
ed results are presented in Tables 6 and 7.
As expected, the process error estimate of
0.0259 is larger than the underlying variance of
0.0077. In dealing with real claims data, one
should check the underlying assumptions thor-
oughly regarding the conditional relationships or
independence between different development
years.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 163
Variance Advancing the Science of Risk
Table 6. Simulated data (10% growth)
Claims 1 2 3 4 5 6 7 8 9 10 Premiums
1 80,946 97,396 43,469 40,208 52,068 19,518 14,644 1,692 12,429 1,964 1,000,0002 63,077 76,181 46,565 68,880 26,412 44,620 53,513 14,540 3,577 1,100,0003 93,688 112,399 87,149 133,804 17,549 14,814 91,392 35,367 1,210,0004 116,704 224,930 87,005 61,843 101,357 42,731 27,057 1,331,0005 192,542 147,366 85,361 61,776 90,964 103,829 1,464,1006 118,717 97,519 83,964 114,058 89,192 1,610,5107 156,966 172,695 139,843 156,225 1,771,5618 175,068 116,656 100,157 1,948,7179 164,691 112,805 2,143,589
10 239,127 2,357,94811 1,296,871
Table 7. Estimated results using simple average loss ratio
Accident Ultimate LossYear Premiums Claims Ratio
11 1,296,871 581,948 44.9%
Var
μCn+1,n
En+1
¶Var(q¤) SEP(q¤) SEP(Cn+1,n)
E(Cn+1,n)
0.0259 0.0030 0.1699 37.9%
5. Concluding remarksIn this paper we examine the weighted and
simple average loss ratio estimators and construct
a stochastic model to derive some simple ap-
proximation formulae to estimate the standard
error of prediction for the future claims compo-
nent of premium liabilities. Based on the idea in
Taylor (2000), we deduce the mean square er-
ror of prediction as comprising the process error
component and the estimation error component,
and no covariance term exists as the first part
is associated only with the future while the sec-
ond part only with the past observations. We ap-
ply these formulae to some public liability data
and simulated data and the results are reason-
able in general. Since the starting part of our
model follows the structure of the chain ladder
method, one may apply the various tests stated
in Mack (1994) to check whether the model as-
sumptions are valid for the claims data under in-
vestigation.
The formulae derived in this paper appear to
serve as a good starting point for assessment of
premium liability variability in practice. Never-
theless, there are other practical considerations
in dealing with premium liabilities such as the
insurance cycle, claims development in the tail,
catastrophes, superimposed inflation, multi-year
policies, policy administration and claims han-
dling expenses, future recoveries, future reinsur-
ance costs, retrospectively rated policies, un-
closed business, refund claims, and future
changes in reinsurance, claims management, and
underwriting. To deal with these issues, a prac-
titioner needs to judgmentally adjust the data or
make an explicit allowance, based on managerial,
internal, and industry information.
Acknowledgments
The author thanks the editors and the anony-
mous referees for their helpful and constructive
comments. This research was partially support-
ed by Nanyang Technological University AcRF
Grant, Singapore.
164 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach
Appendix A
In this appendix we prove that the estimators u and v2 of (2.8) and (2.9) are unbiased:
E(u) = E
μPnr=1Cr,1Pnr=1Er
¶=
Pnr=1E(Cr,1)Pnr=1Er
=
Pnr=1EruPnr=1Er
= u; (from (2.4))
Var(u) = Var
μPnr=1Cr,1Pnr=1Er
¶=
Pnr=1Var(Cr,1)¡Pn
r=1Er¢2 =
Pnr=1Erv
2¡Pnr=1Er
¢2 = v2Pnr=1Er
; (from (2.3) and (2.5))
E(v2) = E
"1
n¡ 1nXr=1
Er
μCr,1Er
¡ u¶2#
=1
n¡ 1nXr=1
ErE
"μCr,1Er
¡ u¶2#
=1
n¡1nXr=1
Er
(E
ÃC2r,1E2r
!¡ 2E
ÃCr,1Er
Png=1Cg,1Png=1Eg
!+E(u2)
)
=1
n¡1nXr=1
Er
(Var
μCr,1Er
¶+E
μCr,1Er
¶2¡ 2
Er
"Png=1E(Cr,1)E(Cg,1)Pn
g=1Eg+Var(Cr,1)Pn
g=1Eg
#+Var(u)+E(u)2
)
(from (2.3))
=1
n¡1nXr=1
Er
(v2
Er+ u2¡ 2
Er
"Png=1ErEgu
2Png=1Eg
+Erv
2Png=1Eg
#+
v2Png=1Eg
+ u2
)
(from (2.4), (2.5), and above)
=1
n¡1nXr=1
Ãv2¡ Erv
2Png=1Eg
!= v2:
Appendix B
We prove in this appendix that both (3.1) and (3.2) give unbiased estimators. To start with, we have
to show the following with repeated use of the law of total expectation:
E
μCn+1,nEn+1
¶= E
μE
μCn+1,nEn+1
¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡1
¶¶= E
μCn+1,n¡1En+1
fn¡1¶
(from (2.1))
= E
μE
μCn+1,n¡1En+1
fn¡1
¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡2
¶¶= E
μCn+1,n¡2En+1
fn¡2fn¡1¶
(from (2.1) again)
= ¢ ¢ ¢= EμCn+1,1En+1
f1f2 : : :fn¡1¶
(repeat above)
= uf1f2 : : :fn¡1: (from (2.4))
The above results can readily be extended to accident years and development years other than n+1
and n shown here.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 165
Variance Advancing the Science of Risk
Mack (1993) proves that fj is unbiased and that fj and fh are uncorrelated for j 6= h. First we lookat the expected value of the weighted average estimator of (3.1):
E(q) = E
0@Pni=1Ci,n+1¡ifn+1¡ifn+2¡i : : : fn¡1Pn
i=1Ei
1A=
Pni=1E(Ci,n+1¡i)fn+1¡ifn+2¡i : : :fn¡1Pn
i=1Ei(from Mack (1993))
=
Pni=1Eiuf1f2 : : :fn¡1Pn
i=1Ei= uf1f2 : : :fn¡1 = q: (from above)
Similarly, the expected value of the simple average estimator of (3.2) is as follows:
E(q¤) = E
0@1n
nXi=1
Ci,n+1¡ifn+1¡ifn+2¡i : : : fn¡1Ei
1A=1
n
nXi=1
E
μCi,n+1¡iEi
¶fn+1¡ifn+2¡i : : :fn¡1 (from Mack (1993))
=1
n
nXi=1
uf1f2 : : :fn¡1 = uf1f2 : : :fn¡1 = q: (from above)
Appendix CWe derive the process error component in (3.7) of the mean square error of prediction as follows,
with repeated use of the law of total variance:
Var
μCn+1,nEn+1
¶= E
μVar
μCn+1,nEn+1
¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡1
¶¶+Var
μE
μCn+1,nEn+1
¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡1
¶¶
= E
μCn+1,n¡1E2n+1
¾2n¡1
¶+Var
μCn+1,n¡1En+1
fn¡1
¶(from (2.2) and (2.1))
= E
μCn+1,n¡1E2n+1
¾2n¡1
¶+E
μVar
μCn+1,n¡1En+1
fn¡1
¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡2
¶¶
+Var
μE
μCn+1,n¡1En+1
fn¡1
¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡2
¶¶= E
μCn+1,n¡1E2n+1
¾2n¡1
¶+E
μCn+1,n¡2E2n+1
¾2n¡2f2n¡1
¶+Var
μCn+1,n¡2En+1
fn¡2fn¡1
¶(from (2.2) and (2.1) again)
= ¢ ¢ ¢= EμCn+1,n¡1E2n+1
¾2n¡1
¶+E
μCn+1,n¡2E2n+1
¾2n¡2f2n¡1
¶+ ¢ ¢ ¢+E
μCn+1,1E2n+1
¾21f22 f
23 : : :f
2n¡1
¶
+Var
μCn+1,1En+1
f1f2 : : :fn¡1
¶(repeat above)
=1
En+1
·E
μCn+1,n¡1En+1
¶¾2n¡1 +E
μCn+1,n¡2En+1
¶¾2n¡2f
2n¡1 + ¢ ¢ ¢+E
μCn+1,1En+1
¶¾21f
22 f
23 : : :f
2n¡1
¸
+v2
En+1f21 f
22 : : :f
2n¡1 (from (2.5))
166 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach
=1
En+1
"E
μCn+1,nEn+1
¶¾2n¡1fn¡1
+E
μCn+1,nEn+1
¶¾2n¡2fn¡2
fn¡1 + ¢ ¢ ¢+EμCn+1,nEn+1
¶¾21f1f2f3 : : :fn¡1
#
+v2
En+1f21 f
22 : : :f
2n¡1 (from Appendix B)
=1
En+1E
μCn+1,nEn+1
¶ n¡1Xj=1
¾2jfjfj+1fj+2 : : :fn¡1 +
v2
En+1f21 f
22 : : :f
2n¡1:
Appendix DIn the following we derive the estimation error component in (3.9) of the mean square error of
prediction. We first apply the Taylor series expansion to the estimator q of (3.1):
q=
Pni=1Ci,n+1¡ifn+1¡ifn+2¡i : : : fn¡1Pn
i=1Ei¼Pni=1E(Ci,n+1¡i)fn+1¡ifn+2¡i : : :fn¡1Pn
i=1Ei
+1Pni=1Ei
n¡1Xj=1
(fj ¡fj)nXi=1
@
@fjCi,n+1¡ifn+1¡ifn+2¡i : : : fn¡1
¯¯Ci,n+1¡i=E(Ci,n+1¡i); fj=fj
+1Pni=1Ei
nXi=1
(Ci,n+1¡i¡E(Ci,n+1¡i))nXr=1
@
@Ci,n+1¡iCr,n+1¡rfn+1¡rfn+2¡r : : : fn¡1
¯Ci,n+1¡i=E(Ci,n+1¡i); fj=fj
= E(q)+1Pni=1Ei
n¡1Xj=1
(fj ¡fj)nX
i=n+1¡j
E(Ci,n)
fj(from Appendix B)
+1Pni=1Ei
nXi=1
(Ci,n+1¡i¡E(Ci,n+1¡i))fn+1¡ifn+2¡i : : :fn¡1:
Moving the term E(q) to the left-hand side of the equation and then taking the expectation on the
square of the resulting equation, we deduce that:
Var(q)¼ 1¡Pni=1Ei
¢2 E2640@n¡1Xj=1
(fj ¡fj)nX
i=n+1¡j
E(Ci,n)
fj
1A2375
+1¡Pni=1Ei
¢2 E24Ã nX
i=1
(Ci,n+1¡i¡E(Ci,n+1¡i))fn+1¡ifn+2¡i : : :fn¡1!235
+2¡Pni=1Ei
¢2 E240@n¡1X
j=1
(fj ¡fj)nX
i=n+1¡j
E(Ci,n)
fj
1AÃ nXi=1
(Ci,n+1¡i¡E(Ci,n+1¡i))fn+1¡ifn+2¡i : : :fn¡1!35
=1¡Pni=1Ei
¢2 n¡1Xj=1
0@ nXi=n+1¡j
E(Ci,n)
fj
1A2
Var(fj) (fj’s are unbiased and uncorrelated)
+1¡Pni=1Ei
¢2 nXi=1
f2n+1¡if2n+2¡i : : :f
2n¡1Var(Ci,n+1¡i) (from (2.3))
+2¡Pni=1Ei
¢2 n¡1Xj=1
n¡jXi=1
0@ nXr=n+1¡j
E(Cr,n)
fj
1A (fn+1¡ifn+2¡i : : :fn¡1)Cov(fj ,Ci,n+1¡i):(fj is unbiased; fj and Ci,n+1¡i are independent for j > n¡ i due to (2.3)).
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 167
Variance Advancing the Science of Risk
As shown in Mack (1993), E(fj j Bj) = fj and Var(fj j Bj) = ¾2j =Pn¡jr=1Cr,j where Bj represents all the
past claims data to development year j. We then deduce the following:
Var(fj) = E(Var(fj j Bj))+Var(E(fj j Bj))
= E
0@ ¾2jPn¡jr=1Cr,j
1A+Var(fj)= E
0@ ¾2jPn¡jr=1Cr,j
1A ,which can be approximated by dVar(fj) = ¾2j =Pn¡j
r=1Cr,j .
Repeatedly using the law of total variance as in Appendix C, we derive that Var(Ci,n+1¡i) = E(Ci,n+1¡i)¢Pn¡i
j=1(¾2j =fj)fj+1fj+2 : : :fn¡i+Eiv
2f21 ¢ f22 : : :f2n¡i, which can then be estimated by dVar(Ci,n+1¡i) =Ci,n+1¡i
Pn¡ij=1(¾
2j =fj)Sj+1,n¡i+Eiv
2 ¢ S21,n¡i.Finally, we derive the covariance between fj and Ci,n+1¡i for j · n¡ i as follows:
Cov(fj ,Ci,n+1¡i) = E(fjCi,n+1¡i)¡E(Ci,n+1¡i)fj (fj is unbiased)
= E(E(fjCi,n+1¡i j Bn¡i))¡E(Ci,n+1¡i)fj
= E(fjCi,n¡ifn¡i)¡E(Ci,n+1¡i)fj (from (2.1))
= ¢ ¢ ¢= E(fjCi,j+1fj+1fj+2 : : :fn¡i)¡E(Ci,n+1¡i)fj (repeat above)
= E
ÃPn¡jr=1Cr,j+1Ci,j+1Pn¡j
r=1Cr,jfj+1fj+2 : : :fn¡i
!¡E(Ci,n+1¡i)fj (from (2.6))
= E
ÃPn¡jr=1 E(Cr,j+1Ci,j+1 j Bj)Pn¡j
r=1Cr,jfj+1fj+2 : : :fn¡i
!¡E(Ci,n+1¡i)fj
= E
ÃPn¡jr=1 E(Cr,j+1 j Bj)E(Ci,j+1 j Bj) +Var(Ci,j+1 j Bj)Pn¡j
r=1Cr,jfj+1fj+2 : : :fn¡i
!¡E(Ci,n+1¡i)fj
(from (2.3))
= E
ÃPn¡jr=1Cr,jCi,jf
2j +Ci,j¾
2jPn¡j
r=1Cr,jfj+1fj+2 : : :fn¡i
!¡E(Ci,n+1¡i)fj (from (2.1) and (2.2))
= E
ÃCi,jf
2j fj+1fj+2 : : :fn¡i+
Ci,j¾2jPn¡j
r=1Cr,jfj+1fj+2 : : :fn¡i
!¡E(Ci,n+1¡i)fj
= E
ÃCi,n+1¡iPn¡jr=1Cr,j
!¾2jfj, (from Appendix B)
which can be approximated by dCov(fj ,Ci,n+1¡i) = (Ci,n+1¡i=Pn¡jr=1Cr,j)(¾
2j =fj).
168 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach
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forum/99fforum/99ff021.pdf.
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tralia 15th General Insurance Seminar, Sydney, Australia,
2005.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 169
The Economics of Insurance FraudInvestigation: Evidence of a Nash
Equilibriumby Stephen P. D’Arcy, Richard A. Derrig, and Herbert I. Weisberg
ABSTRACT
The behavior of competing insurance companies investi-
gating insurance fraud follows one of several Nash Equilib-
ria under which companies consider the claim savings, net
of investigation cost, on a portion, or all, of the total claim.
This behavior can reduce the effectiveness of investigations
when two or more competing insurers are involved. Cost
savings are reduced if the suboptimal equilibrium prevails,
and may instead induce fraudulent claim behavior and lead
to higher insurance premiums. Alternative cooperative and
noncooperative arrangements are examined that could re-
duce or eliminate this potential inefficiency. Empirically, an
examination of Massachusetts no-fault auto bodily injury
liability claim data for independent medical examinations
shows that (1) investigation produces a net total savings
as high as eight percent; (2) the investigation frequency is
likely in excess of the theoretical optimal; and (3) predic-
tive modeling of claim suspicion scores can significantly
enhance the net savings arising from independent medical
examinations.
KEYWORDS
Insurance fraud, Nash equilibrium, automobile medical payments, liability
insurance, independent medical examination
170 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium
1. IntroductionFraud is a major problem for the insurance in-
dustry. Although the true cost of fraud for the
industry, and subsequently for insurance policy-
holders who bear this cost through higher pre-
miums, cannot be known, the FBI estimates the
annual cost of fraud to be $40 billion (FBI 2009).
Insurance fraud comes in two varieties: hard and
soft fraud are the operational terms. Hard fraud
applies to claims for fictitious accidents and in-
juries, while soft fraud denotes the increase of
claimed loss through unnecessary and/or inflated
values of claimants’ loss costs. The former is
criminal and is the purview of the criminal jus-
tice system; the latter is generally a civil matter
that is the larger of the two in dollar terms and
is the purview of the insurers (Derrig 2002).
Insurers can take steps to reduce the amount
of fraud, especially soft fraud, but these steps
are costly and these costs have to be weighed
against the expected savings. Insurance fraud has
been the subject of considerable research from a
variety of angles. This paper examines how the
number of insurers in the market and how the
different laws regarding subrogation in liability
claims affect the incentives to investigate, and
therefore reduce, fraud.
A number of recent studies have examined
claim settlement behavior by insurers as it re-
lates to insurance fraud (Artis, Ayuso and Guillen
2002; Crocker and Tennyson 2002; Derrig 2002;
Derrig and Weisberg 2004; Loughran 2005;
Dionne, Giuliano and Picard 2009). Several stud-
ies have utilized a Nash Equilibrium framework
(Nash 1951) between insurers and policyhold-
ers to examine auditing strategies for fraudulent
claiming behavior by policyholders (Boyer 2000;
Boyer 2004; Shiller 2006). In this paper, a model
of insurance company behavior combining the
cost of claims, the cost of investigating claims
and the potential for reducing claim costs is de-
veloped and analyzed in a game theoretic ap-
proach in which the other players are insurers,
rather than policyholders. The presence of a Nash
Equilibrium, in which no player in a simulta-
neous noncooperative game can unilaterally im-
prove its position by shifting its strategy for in-
vestigating claims, is observed under a variety of
different market conditions.
For a simple example of a Nash Equilibrium
consider Jack and Jill, two very young entrepre-
neurs operating lemonade stands in front of ad-
jacent houses. They have an unlimited supply
of their product from their parents’ kitchen at
no cost (to Jack and Jill) and they consider the
time they spend staffing their stands to be fun,
so there are no labor costs involved. They know
the thirst level and financial position of each of
their potential customers, so they can determine
how the demand for lemonade is affected by the
price. Both sellers and buyers can see what each
competitor is charging, and buyers will get their
lemonade from the lower cost seller, so the mar-
ket is fully competitive. The battle of the sexes,
junior edition, prevents the sellers from pricing
their product cooperatively, so this is a classic
noncooperative game to which Nash’s work ap-
plies. The sellers’ decision is whether to charge
1¢ or 2¢ per cup. Demand is such that 10 cups
will be sold if the price is 1¢, but only 8 cups
will be sold if the price is 2¢. All sales will be
made by the lemonade stand charging the lower
price. If both stands charge the same price, each
will get 1/2 of the sales.
Game theory often utilizes payoff matricies to
illustrate the results from different strategies. In
the payoff matrix below, and the subsequent ones
in this paper, the choices of one competitor are
shown along the top of the matrix and the choices
of the other along the left side. To determine
the payoff from a particular strategy for the top
competitor, look at the column representing their
choice and move down to the row representing
its competitor’s choice. The top-right half of the
appropriate box is the payoff to the competitor
listed along the top, and the bottom-left half of
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 171
Variance Advancing the Science of Risk
Table 1. Payoff matrix
that box shows the payoff to the competitor listed
along the side.
The payoff matrix that illustrates the lemonade
pricing decision facing Jack and Jill is shown in
Table 1.
If both children charge 2¢ per cup, each will
sell 4 cups of lemonade (half of the total de-
mand) and earn a profit of 8¢. However, this is
not a Nash Equilibrium since if either Jack or
Jill (but not both) lowered the price to 1¢, that
child would sell 10 cups and earn a total profit
of 10¢. In this example, the situation where each
child charges 1¢ a cup is a Nash Equilbrium, be-
cause neither child can unilaterally change the
price and earn a higher profit. By raising the
price alone to 2¢ a cup, they would not sell any
lemonade and their total profit would be 0. For a
more complete description of game theory eco-
nomics1 and Nash Equilibrium, see chapter 6 of
Miller (2003). In most insurance cases, the Nash
Equilibrium is not at the globally optimal claim
investigation strategy.
Claims presented to an insurance company for
payment may include a variety of different com-
ponents. One component is a valid expense that
should be paid in full by the insurer, since both
the amount is appropriate and the coverage is
applicable. Another component could be an ex-
cessive charge on a claim that would otherwise
1Miller characterizes a Nash Equilibrium as “a no regrets outcome
in which all the players are satisfied with their strategy given what
every other player has done.”
be covered. A charge is considered excessive if
it is judged by the insurer to be “unreasonable”;
most insurance policies cover only “reasonable”
charges with reasonability defined by context and
ultimately determined by negotiation, arbitration
or, if necessary, a court. A third component could
be a claim for a service that is not covered al-
though other services would be covered. A final
component could be for an incident that is not
covered in its entirety by the insurance policy.
Sorting out the different components of a claim
efficiently is a constant process within a claims
department.
For automobile insurance coverage in the
United States, bodily injury claims can consist
of two different insurance coverages. Medical
expenses incurred by the policyholder or any-
one else insured under the policy (family mem-
bers, anyone occupying the covered vehicle) as
the result of an automobile accident are covered,
subject to policy limits, by the insurance com-
pany providing medical payments (MedPay) or
personal injury protection (PIP) coverage with-
out regard to fault.2 If someone is injured as
the result of the fault of another person, then
that injured party could pursue a liability claim
against the responsible party, depending on the
tort threshold applicable under the policy (IRC
2003, chapter 2, 2004a, 2004b). The insurance
company of the responsible party would be liable
2PIP coverage in no-fault states includes compensation for wage
loss and other benefits.
172 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium
for the damages incurred by the injured person,
subject to policy limits and degree of fault, under
the liability insurance policy. Bodily injury lia-
bility damages consist of such tangible expenses
as medical expenses and loss of income, which
are termed special damages, and intangible com-
ponents such as pain-and-suffering, loss of con-
sortium or hedonic damages, which are termed
general damages.3 The insurer that paid the med-
ical expenses under the medical payments or PIP
policy may also be able to recoup its payments
from the liability insurer under subrogation. Si-
mon (1989) examined rules for allocating loss
adjustment expenses between primary insurers
and reinsurers when subrogation was involved.
Similar complexities are generated for insurers
when determining the cost of claim investigation
when two policies are involved.
In some cases the same insurer is responsible
for both the medical expenses and the bodily in-
jury liability payment. This would occur when
the driver is responsible for an injury to a pas-
senger, or if the same insurer covered the injured
person under medical payments coverage and the
responsible party under a different liability insur-
ance policy. When a single insurer is responsi-
ble for all payments, determining the appropriate
level of fraud investigation considers the entire
cost of all claims.
2. Claim investigation for injuryclaimsSeveral types of claim investigation are com-
monly used by automobile insurers in addition
to the routine gathering and evaluation of the
circumstances of the accident and the cost of
the treatment for the injury. The most common
method is an Independent Medical Examination
(IME), in which a doctor selected by the insurer
examines the injured claimant and develops an
independent assessment of the injury and the ap-
3See Loughran 2005 for an extensive analysis of auto BI liability
general damage settlements.
propriate treatment. If the IME indicates a more
moderate level of injury or treatment than the
claimant has reported through his or her medi-
cal care provider, then the claims department has
a stronger case for denying some or all of the
medical expenses that have been, or are likely to
be, submitted. Another type of investigation is a
Medical Audit (MA), in which the medical ex-
penses are reviewed by a specialist or an expert
system. Unusual factors that appear in the medi-
cal audit may provide the claims department with
justification to reduce the claim payment. A third
alternative is to refer the claim to a Special In-
vestigation Unit (SIU), where specifically trained
personnel are assigned to investigate claims with
unusual questions in order to determine whether,
and how much of, the claim should be paid.4
Derrig and Francis (2008) examined a collection
of objective factors in a predictive model for re-
ferring Massachusetts auto injury claims for an
IME or a special investigation for fraud, along
with the likelihood of success at reducing the
claim amount. Such predictive models should al-
low for more efficient, i.e., less costly by reduc-
ing false positives, selection of claims to investi-
gate.
IMEs and MAs can be used to reduce the
amount of claim payments for medical expenses.
SIUs can also reduce these expenses, but can
also impact other expenses or even determine if
the claim is valid at all. One level of investiga-
tion would be to investigate each claim for which
the expected savings from the investigation ex-
ceed the cost of the investigation. We call that
approach “tactically optimal.” Another level of
investigation would vary according to the char-
acteristics of the claim so that the savings net of
costs for the entire portfolio of claims is optimal
in some way. We call this approach “strategically
optimal.” In order to measure the expected sav-
ings, the insurer needs to ascertain the chance of
4The Insurance Research Council provides countrywide claims han-
dling outcome data for these three techniques for a sample of 2002
bodily injury claims (IRC 2003, pp. 92—104).
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 173
Variance Advancing the Science of Risk
finding unreasonable or fraudulent activity and
the potential savings if that activity is discov-
ered. We now turn to a formalization of the cost/
savings process when total claim payments con-
sist of first party PIP and, when applicable, third
party liability.5
3. Savings versus costThe following notation will be used:
Cost of claim without any investigation:
PIP claim = P
Liability claim (excess of PIP) = L
Total Compensation = P+L
Subscripts on P and L:
First subscript indicates company responsible for
PIP
Second subscript indicates company responsible
for Liability (0 if no liability)
P1,0 represents a PIP claim where company 1 has
the PIP coverage and there is no liability claim
P1,1 represents a PIP claim where company 1 has
the PIP coverage and the liability coverage
P1,2 represents a PIP claim where company 1 has
the PIP coverage and company 2 has the liability
coverage
P1,¢ represents the sum of all PIP claims where
company 1 has the PIP coverage
L1,1 represents a liability claim where company
1 has the PIP coverage and the liability coverage
L2,1 represents a liability claim where company
2 has the PIP coverage and company 1 has the
liability coverage
L¢,1 represents the sum of all liability claims
where company 1 has the liability coverage
Savings from investigations:
Savings on PIP claims = SP
Savings on Liability claims = SL
Savings on Total claim = ST = SP+SL
5The Insurance Research Council provides an analysis of their 2002
claim sample for four no-fault states: Colorado (now a full tort
state), Florida, New York, and Michigan (IRC 2004a).
Level of investigation:
No investigation = 0
Optimal investigation based upon information on
first party claims = A
Optimal investigation based upon information on
both first party and liability claims = B
Subscripts on SPA, SPB, SLA and SLB:
First subscript indicates company responsible for
PIP
Second subscript indicates company responsible
for Liability (0 if no liability claims)
SPA1,0 represents the savings on PIP claims from
an A level investigation where company 1 has the
PIP coverage and there is no liability claim
SPA1,1 represents the savings on PIP claims from
an A level investigation where company 1 has the
PIP coverage and the liability coverage
SLA2,1 represents the savings on liability claims
from an A level investigation where company 2
has the PIP coverage and company 1 has the li-
ability coverage
Investigation cost:
Cost of an A level investigation = IA
Cost of an B level investigation = IB
Subscripts on IA and IB:
First subscript indicates company responsible for
PIP
Second subscript indicates company responsible
for Liability (0 if no liability claims)
IA1,0 represents the cost of an A level investiga-
tion where company 1 has the PIP coverage and
there is no liability claim
IA1,1 represents the cost of an A level investiga-
tion where company 1 has the PIP coverage and
the liability coverage
IA1,2 represents the cost of an A level investiga-
tion where company 1 has the PIP coverage and
company 2 has the liability coverage
The relationships between the cost of investi-
gation and expected savings, as well as the de-
174 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium
Figure 1. Optimal level of claim investigation
termination of the optimal levels of investigation
under different circumstances, are illustrated in
Figure 1. The x axis represents the number of
claims. The y axis indicates dollar values. The
claims are ordered in decreasing size of expected
savings from claim investigations. The use of ex-
ante expectations of savings from investigation
is important and differs strongly from the ex-
post ordering of claims with savings from inves-
tigation. In practice, actual investigations will be
taken from a random or targeted draw of claims
that yield the largest expected savings. The cost
of investigations (I) function is a straight line un-
der the assumption that each investigation has
the same expected cost.6 Two concave functions
represent the expected savings from an investi-
gation. The lower curve, labeled SP, represents
the savings on first party claims and the higher
curve, labeled ST, represents the savings on the
total claim including both PIP and Liability pay-
ments. Both SP and ST have positive slopes and
negative curvature. A point will be reached where
all the remaining claims are expected to be com-
pletely valid, so no additional savings are achiev-
ed by additional investigation.
6Insurers generally pay, for example, a fixed amount for an IME. If
the claimant does not appear for the examination, the fee is reduced,
but the insurer would not know, when requesting the IME, if the
claimant will appear for it or not. SIU investigations cost more than
IMEs and Medical audits cost less. The use of multiple techniques
is relatively small. Thus, the assumption is made that the expected
cost of an investigation is the same for each claim, and the function
is linear.
The optimal level of investigation is determin-
ed when the slopes of the cost of investigation
line and the savings are equal. The tactically op-
timal number of claims to investigate, based on
information in first party claims, is A. At this
point, SP¡ I is maximized. The cost of this in-vestigation is IA, the savings on first party claims
is SPA, and the savings on total claims is STA =
SPA+SLA.7 The strategically optimal number
of first party claims to investigate, based on total
claim savings is B, an amount in excess of A.
Some of the relationships that develop from
this approach are:
SPB> SPA
IB> IA
SPA> IA
STB> IB
SPB¡SPA< IB¡ IA:
4. Single insurer caseWhen a single insurer writes the entire auto-
mobile insurance market, this company will be
responsible for paying both the PIP expenses and
the liability award resulting from every automo-
bile accident. In this case, the company can weigh
the potential cost savings on the total claim
against the cost of this investigation. The tacti-
cally optimal level of investigation would be to
investigate all claims where the expected savings
from the investigation exceed the cost of the in-
vestigation. This is the situation we will consider
first.
The three choices a single insurer faces regard-
ing the level of claim investigation are displayed
in Table 2. The insurer can perform no investiga-
tions and simply pay the amount claimed. This
situation is displayed in the first box. Alterna-
tively, the insurer can investigate A claims. The
7Dionne, Guiliano, and Picard (2009) derive varying optimal levels
of investigation depending on a (fraud) risk class partition of the
set of claims.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 175
Variance Advancing the Science of Risk
Table 2. Single insurer case, net cost of claim and investigations
Level of Claim Investigation
None (0) PIP Based (A) Total Claim Based (B)
P1,0 +P1,1 +L1,1 P1,0 +P1,1 +L1,1¡SPA1,0¡SPA1,1¡SLA1,1 + IA1,0 + IA1,1 P1,0 +P1,1 +L1,1¡SPB1,0¡SPB1,1¡SLB1,1 + IB1,0 + IB1,1
additional cost is IA1,0 + IA1,1 and the associ-
ated savings are SPA1,0 +SPA1,1 +SLA1,1. Since
the savings on the PIP claims alone, SPA1,0 +
SPA1,1 exceed the cost of the investigations, the
insurer would prefer this option over the case
of no investigations. The third choice, though,
where the insurer investigates B claims, is the
optimal choice. The cost of this additional inves-
tigation is IB¡ IA. The additional savings areSPB+SLB¡SPA¡SLA. Since the slope of theTotal Savings curve exceeds the slope of the cost
of investigations curve over the range from A to
B, then the savings exceed the costs, and the in-
surer would minimize net claim costs by investi-
gating B claims.
This strategy will have the benefit of reduc-
ing the cost of unreasonable medical treatment
to the lowest feasible level considering the cost
to investigate claims. This strategy also reduces
liability awards and the cost of automobile insur-
ance to the lowest level feasible given the cost
of investigating these claims and the ability to
lower awards through negotiation (Derrig and
Weisberg 2004). Additional reductions in claims
costs could be obtained, but the additional inves-
tigation expenses would exceed the claim cost
savings, so insurance premiums would actually
increase. The other expenses of the insurer, in-
cluding underwriting expenses and normal loss
adjustment expenses (other than investigating for
fraud), are not included in this analysis, since
they will be the same regardless of the level of
investigation for claims fraud.
5. Two insurer case: NosubrogationAssume the market consists of two compet-
ing insurers of equal size, with similar claim dis-
tributions (the SP and ST curves are the same
for each insurer). Assume the claim settlement
system does not permit the recovery of the PIP
claim payment and adjustment expense from any
at-fault party through subrogation.8 Then they
would each face a decision about the appropri-
ate level of investigation of claims fraud, but
their net claim costs would depend both on their
own investigation level decision and the decision
of their competitor. The outcomes, in the case
where there is no subrogation, are shown on Ta-
ble 3. The upper segment of each cell denotes
the position of insurer 1; the lower segment that
of insurer 2.
If both insurers were to investigate optimally
based on aggregate claim costs, then each insurer
would bear the cost of investigating B claims,
and benefit from the savings in claim costs on
both PIP and Liability claims. This situation is
represented in cell (B,B) and resembles the op-
timal position for the single insurer case. Unrea-
sonable medical expenses are reduced to the low-
est economically efficient level, liability costs are
minimized, and the total cost of auto insurance
is kept at the lowest level.9
However, this is not a stable situation. Insur-
er 1 might be better off if it only investigated
8Medical payments excess of PIP in a no-fault state, for example.9This insurer might prefer to investigate the claims it knows it has
the liability insurance coverage on up to the aggregate level, and
only investigate the remaining claims on which there is either no
liability coverage or coverage provided by the other insurer, if it
could identify those claims. However, there are several problems
with this strategy. First, an insurer may not know if another com-
pany will be liable for a claim or not early enough in the claim
process to make this distinction. Second, adopting a claim process
that requires claims adjusters to have different strategies for investi-
gation can complicate the process and increase overall costs. Based
on discussions with claims personnel, such differential strategies
are not common.
176 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium
Table 3. Two insurer case, net cost of claim and investigations, no subrogation
Table 4. Two insurer case, no subrogation
Nash Equilibrium
Insurer 1 Insurer 2
(0,0) IA1,¢ > SPA1,¢ + SLA1,1 IA2,¢ > SPA2,¢ + SLA2,2
(A,A) IA1,¢ < SPA1,¢ + SLA1,1 IA2,¢ < SPA2,¢ + SLA2,2
IB1,¢ ¡ IA1,¢ > SPB1,¢ ¡SPA1,¢ + SLB1,1¡SLA1,1 IB2,¢ ¡ IA2,¢ > SPB2,¢ ¡SPA2,¢ + SLB2,2¡SLA2,2
(B,B) IB1,¢ ¡ IA1,¢ < SPB1,¢ ¡SPA1,¢ + SLB1,1¡SLA1,1 IB2,¢ ¡ IA2,¢ < SPB2,¢ ¡SPA2,¢ + SLB2,2¡SLA2,2
claims at the A level, which would lower its
cost of investigations by (IB¡ IA), and only in-crease claim costs by (SPB1,¢ ¡SPA1,¢+SLB1,1¡SLA1,1). If insurer 2 were to continue to investi-
gate claims at the B level, then insurer 1 would
benefit on its liability claims for which insurer
2 had the PIP coverage (SLB2,1). For the two
insurer example, the lower investigation costs
may or may not exceed the savings. Although
(IB¡ IA)> (SPB1,¢ ¡SPA1,¢), whether it also
exceeds (SPB1,¢ ¡SPA1,¢+SLB1,1¡SLA1,1) de-pends on the relationship between the SP and ST
curves and the cost of the claims where insurer 1
has both PIP and Liability. The cost savings on
liability claims must be included in the decision
of which level of investigation to pursue. How-
ever, if it is advantageous for insurer 1 to move
to a lower level of investigation, then it would
also benefit insurer 2 to move to that level, so
the resulting position would be that displayed in
cell (A,A).
If the insurers move to cell (A,A), that will
prove to be a Nash Equilibrium. Neither insurer
can move unilaterally to another position that
benefits itself. Insurer 1 will not stop investigat-
ing claims at the A level and move to the no
investigation level. If it were to do so, the sav-
ings would be IA and the cost would be SPA1,¢+SLA1,1. Since IA< SPA alone, this change
would increase the net cost of claims. Although
the overall optimal position would be cell (B,B),
that is not a stable equilibrium since one com-
pany might benefit by reducing the level of in-
vestigations.
Table 4 describes the conditions that lead to
each claim investigation strategy for the insur-
ers. Cell (B,B) is a Nash Equilibrium if IB1,¢ ¡IA1,¢ < SPB1,¢ ¡SPA1,¢+SLB1,1¡SLA1,1. Since
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 177
Variance Advancing the Science of Risk
both insurers are assumed to be the same size and
have the same distribution of claims and costs,
then if this relationship holds for insurer 1, it
should also apply to insurer 2. This equilibrium
would apply if the cost savings for each insurer
on claims where it had both the PIP and the
liability coverage exceeded the additional cost
of investigating claims at the B level. Each in-
surer would not be assured of receiving the sav-
ings of a B level investigation on its liability
claims where the other insurer has the PIP claim,
since that insurer might elect a lower level of in-
vestigation. Alternatively, cell (A,A) would be
the Nash Equilibrium if IB1,¢ ¡ IA1,¢ > SPB1,¢ ¡SPA1,¢+SLB1,1¡SLA1,1 and IA1,¢ < SPA1,¢+SLA1,1. Since SPA1,¢ > IA1,¢ by itself, then cell(0,0) will never be the Nash Equilibrium if there
is no subrogation. Note also that the off-diagonal
investigation levels in Table 4 exhibit elements
of the free rider problem; namely, one insurer
reaps the liability benefit of the higher PIP in-
vestigation level of the other insurer without the
additional cost.
In this paper, we assume that insurers follow
the same approach for determining the level of
investigation for all claims, regardless of whether
they are providing the liability coverage or a
competitor is providing this coverage. There are
several reasons for this assumption. The most im-
portant reason is that asking claims personnel to
follow different approaches for PIP claims de-
pending on which insurer will bear the liability
costs would significantly complicate and poten-
tially delay the claims process. PIP claims de-
velop quickly and must be covered regardless of
fault, so PIP claim files may not contain enough
information to determine whether liability cover-
age will apply and, if it does apply, which insurer
will provide this coverage. Decisions about how
to investigate potential fraud cannot be delayed
until all the information is available, or it could
be too late to reduce total economic loss. Having
a single process in place allows for a more ef-
fective decision-making process. There are also
some cases in which it is unknown which insurer
will ultimately be held liable, such as when a pas-
senger is injured in a two car accident and it is
not known which driver will be held liable un-
til the claim is finally settled. Another reason is
that insurers know the levels of investigation that
other insurers adopt, through subrogation cases
and through hiring each other’s former employ-
ees. If an insurer followed a suboptimal claim
investigation process that put other insurers at a
disadvantage, it could trigger retaliation. Finally,
regulators may object, and even fine, a company
if the claims department had a policy of know-
ingly underinvestigating fraud in cases where the
BI liability lies with another carrier. Some regu-
lators have already disallowed a portion of rate
requests based on assumed inadequate fraud in-
vestigation; if a company had a specific policy in
place not to investigate claims where savings are
expected to exceed the costs, this approach could
provide a strong case for this type of regulatory
reaction.
6. Two insurer case: SubrogationThis situation differs from the no subrogation
case in several ways.10 First, each Liability in-
surer is responsible for paying the PIP or Med-
Pay claims of the other insurer when liability at-
taches and the PIP or MedPay insurer and the Li-
ability insurer are different (Pi,j where i 6= j). Onepossible situation is to allow subrogation for the
claim, but not for loss adjustment expense. The
rationale for this approach is that claim payments
are more easily verifiable than loss adjustment
expenses. In the case that only claim payments
are subrogated, if insurer 1 investigates claims at
the A level but insurer 2 does not investigate, in-
surer 1 does not benefit from the savings on the
PIP claims where insurer 2 has the PIP claim but
10Table 7 shows that 35 of the 51 jurisdictions allow subrogation
of PIP and/or MedPay to the liability carrier.
178 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium
Table 5. Two insurer case, net cost of claim and investigations, subrogation
Table 6. Two insurer case, subrogation
Nash Equilibrium
Insurer 1 Insurer 2
(0,0) IA1,0 + IA1,1 > SPA1,0 + SPA1,1 + SLA1,1 IA2,0 + IA2,2 > SPA2,0 + SPA2,2 + SLA2,2
(A,A) IA1,0 + IA1,1 < SPA1,0 + SPA1,1 + SLA1,1 IA2,0 + IA2,2 < SPA2,0 + SPA2,2 + SLA2,2
IB1,0 + IB1,1¡ IA1,0¡ IA1,1 >
SPB1,0¡SPA1,0 + SPB1,1¡SPA1,1 + SLB1,1¡SLA1,1
IB2,0 + IB2,2¡ IA2,0¡ IA2,2 >
SPB2,0¡SPA2,0 + SPB2,2¡SPA2,2 + SLB2,2¡SLA2,2
(B,B) IB1,0 + IB1,1¡ IA1,0¡ IA1,1 <
SPB1,0¡SPA1,0 + SPB1,1¡SPA1,1 + SLB1,1¡SLA1,1
IB2,0 + IB2,2¡ IA2,0¡ IA2,2 <
SPB2,0¡SPA2,0 + SPB2,2¡SPA2,2 + SLB2,2¡SLA2,2
insurer 1 has the liability (SPA2,1). Insurer 2 ben-
efits from the savings on PIP claims, however,
where insurer 1 has the PIP claim and insurer 2
has the liability (SPA1,2).11 Thus, the free rider
problem may be more severe when subrogation
is considered. In this situation, the Nash Equilib-
rium could be no claims investigation, since the
insurer bears the cost of investigating its own PIP
claims, but benefits only on those claims where
there is no liability or if the same company has
the liability coverage, unless the other insurer in-
vestigates all its own PIP claims. The outcomes,
11In cases where the liability insurer negotiates an overall fair set-
tlement independent of the PIP claim investigation result, and pays
that settlement less the PIP payment to the claimant, there would
be no effect from PIP levels of investigation on the liability insurer.
Generally, however, a favorable PIP investigation may curtail treat-
ment, limit both PIP and overall economic damages and, thus, lower
the (total) liability settlement. The latter is the situation we assume.
given this approach to subrogation, are shown in
Table 5.
Table 6 describes the conditions that lead to
each claim investigation strategy for the two in-
surers when subrogation is introduced. In this
case, cell (0,0) may be a Nash Equilibrium, since
each insurer only saves money on claims where
there either is no liability or it has the liability
claim as well. Insurers no longer save money
on PIP claims if another insurer has the liabil-
ity, since those payments would be reimbursed
under subrogation. Thus, subrogation introduces
a disincentive to investigating claims for fraud
unless the cost of investigation is also subject to
the subrogation recovery.
An alternative approach to subrogation would
be to allow subrogation for both the claim pay-
ment and any allocated loss adjustment expense,
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 179
Variance Advancing the Science of Risk
which would include IMEs and MAs, but not
SIU costs which are internal. In this approach,
the insurer responsible for paying subrogation
would have to trust, or be able to audit, the cod-
ing of allocated loss adjustment expenses by the
other insurer to assure that the investigation costs
do apply to the appropriate claim. Subrogation of
allocated claim expense increases the incentive to
investigate PIP claims since the (allocated) costs
to investigate PIP claims would be reimbursed
if another insurer has the liability coverage. In
this case, if one insurer conducted an IME that
results in cost savings for a second insurer, the
second insurer would have reimbursed the first
insurer for the cost of the IME.
A third approach would be to allow subroga-
tion for unallocated loss adjustment expenses as
well as allocated loss adjustment expenses.12 Un-
allocated LAE (ULAE) are the claim expenses
that cannot be assigned to a particular claim,
which would consist of the cost of running a
company’s claim department, including salaries,
supplies and office expenses. In Massachusetts,
where this approach to subrogation is applied,
ULAE is calculated as 10 percent of the claim
cost. If a claim adjuster is considering investi-
gating a claim in which the expected savings will
exceed the cost of the investigation, but another
company is likely to be liable for the loss, the
insurer is saving the other insurer money and re-
ducing its ULAE reimbursement. For example,
assume that a claim on which one insurer had the
PIP coverage and the other insurer had the lia-
bility coverage generated $2200 in claimed med-
ical expenses. The PIP insurer could request an
IME that is expected to cost $300 and that would
reduce the medical expenses by $800, to $1400.
12Recent changes in annual statement reporting have two new cat-
egories: Defense and Cost Containment Expenses (DCCE), which
parallels the allocated expense category, and Other Adjusting Ex-
pense (OAE) that parallels the unallocated expense category. Our
paper continues to use the prior terminology of allocated and un-
allocated expense for expenses assignable to particular claims and
those expenses that are not, respectively.
The PIP insurer may not do this investigation un-
der a tactically optimal strategy. If the claim were
to qualify for subrogation, then the reimburse-
ment for ULAE declines from $220 (10 percent
of $2200) to $140 (10 percent of $1400) even
though the claim department puts in additional
effort to request and review the IME and then
negotiate with the claimant to reduce the claim
payment. On the other hand, under a strategi-
cally optimal strategy, the PIP insurer may well
investigate reimbursable PIP claims to reinforce
a “hard-line” attitude on unreasonable medical
charges in order to maximize savings on its own
claims.
Subrogation rules can have a significant effect
on the incentives for investigating claims. Table 7
summarizes the different subrogation regulations
by state, and also indicates the type of compensa-
tion system in effect in each state. In some states,
including California and Florida, no subrogation
is allowed for either Medical Payments or PIP.
In other states, both Medical Payments and PIP
are eligible for subrogation. Most states allow
subrogation for either Medical Payments or PIP,
but not both. Massachusetts allows subrogation
of PIP claims but not Medical Payments excess
of PIP.
7.1. Multiple insurer caseA more realistic situation arises where there
are many insurers in the market. Some insurers
may write a major share of the market within an
individual state, in a few cases in excess of 30
percent. However, in most states a large number
of insurers compete and the market share of most
companies represents only a small share of the
market. Thus, it is relatively rare that the same
insurer provides PIP coverage under one policy
involved in a claim and liability coverage under
another policy by covering both cars involved in
a two car accident.13 In this situation, the Nash
13In Massachusetts with only 18 active personal auto insurers, ap-
proximately 80 percent of liability claims have different PIP and
liability insurers.
180 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium
Table 7. Tort type and subrogation laws by state
2006 Subrogation
State Tort Type Med Pay PIP
Alabama Tort Yes NoAlaska Tort Yes NoArizona Tort No NoArkansas Add-on Yes YesCalifornia Tort No NoColorado Tort No YesConnecticut Tort No NoDelaware Add-on No YesDC Add-on No YesFlorida No-Fault No NoGeorgia Tort No NoHawaii No-Fault No YesIdaho Tort Yes NoIllinois Tort Yes NoIndiana Tort Yes NoIowa Tort Yes NoKansas No-Fault No YesKentucky Choice No-Fault No YesLouisiana Tort Yes NoMaine Tort Yes NoMaryland Add-on No NoMassachusetts No-Fault No YesMichigan No-Fault No YesMinnesota No-Fault No NoMississippi Tort Yes NoMissouri Tort No NoMontana Tort No NoNebraska Tort Yes NoNevada Tort No NoNew Hampshire Tort No NoNew Jersey Choice No-Fault No NoNew Mexico Tort Yes NoNew York No-Fault No YesNorth Carolina Tort No NoNorth Dakota No-Fault Yes YesOhio Tort Yes NoOklahoma Tort Yes YesOregon Add-on No YesPennsylvania Choice No-Fault No NoRhode Island Tort Yes NoSouth Carolina Add-on No NoSouth Dakota Add-on Yes YesTennessee Tort Yes NoTexas Add-on Yes NoUtah No-Fault No YesVermont Tort Yes NoVirginia Add-on No NoWashington Add-on No YesWest Virginia Tort Yes NoWisconsin Add-on Yes NoWyoming Tort Yes No
Sources: IRC (2003),Insurance Information Institute (2006),Mattheisen, Wickert, and Lehrer (2006)
Table 8. Multiple insurer case, subrogation
Nash Equilibrium
Insurer k
(0,0) IAk,¢ > SPAk,0 + SPAk,k + SLAk,k
(A,A) IAk,¢ < SPAk,0 + SPAk,k + SLAk,k
IBk,¢ ¡ IAk,¢ >SPBk,0¡SPAk,0 + SPBk,k ¡SPAk,k + SLBk,k ¡SLAk,k
(B,B) IBk,¢ ¡ IAk,¢ <SPBk,0¡SPAk,0 + SPBk,k ¡SPAk,k + SLBk,k ¡SLAk,k
When the number of insurers, n, increases:The total investigation cost is IAk,¢ =
Pn
j=1 IAk,j + IAk,k + IAk,0 (k 6= j).The share of efficient part (IAk,k + IAk,0) in the investigation, whichis spent on SPAk,0 + SPAk,k + SLAk,k , is (IAk,k + IAk,0)=IAk,¢. Whenn!1,
Pn
j=1 IAk,j=IAk,¢ ! 1, (IAk,k + IAk,0)=IAk,¢ ! 0. That meansthat little of the investigation cost is spent to improve savings fromSPAk,0 + SPAk,k + SLAk,k . Thus, no insurer would be likely to investi-gate claims for fraud. The Nash Equilibrium would tend to be (0,0).
Equilibrium position is even more likely to be the
No Investigation level, since most of the benefits
of the investigations will accrue to other insur-
ers. The simple relationships for a market with
multiple insurers and subrogation are described
in Table 8.
7.2. Example
The decision process facing each insurer can
be illustrated by an example. A PIP claimant is
visiting a physical therapist for treatment. The
current cost of the claim is $2000 for medical
expenses. Another driver is expected to be held
liable for the accident. Based on past experience
for that type of injury with that physical thera-
pist, the PIP insurer expects the total claim for
medical treatment will be $2200. If the PIP in-
surer orders an IME, which costs $300, the in-
surer expects to be able to determine that no addi-
tional physical therapy is needed, limiting med-
ical expenses to $2000. The liability award for
noneconomic losses (pain and suffering) is ex-
pected to be $4000 if no additional treatment is
received, but $4360 if additional treatment is pro-
vided. Assume that the liability insurer is not in
a position to undertake this investigation and re-
duce its costs because at the time a determination
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 181
Variance Advancing the Science of Risk
of liability is made the full treatment of physical
therapy has been be completed.14
The cost of the IME, $300, exceeds the PIP
savings of $200 on this claim, but is less than the
total of the PIP and liability savings ($560). In
the single insurer case, the insurer will request an
IME on this claim and curtail the additional costs.
In the two-insurer case, if there is no subroga-
tion, the PIP insurer spends $300, saves $200 on
the PIP, and has a 50% chance of saving $360
more on the liability claim (with only two insur-
ers, the PIP insurer has a 1 in 2 chance of writing
the responsible party’s liability insurance in sim-
ple two-car collisions). Therefore, the PIP insurer
would also request the IME on this claim. In the
two-insurer case where there is subrogation, the
PIP insurer faces a 50% chance of saving on the
PIP claim and on the noneconomic losses (if it
also has the liability), so the expected savings
would be $280 (half of the $560 total savings).
Thus, the PIP insurer would not investigate this
claim unless the allocated LAE is reimbursable.
If LAE is not reimbursable, the cost of investi-
gating the claim is $300. If LAE is reimbursable,
then the expected cost of the IME is reduced to
$150, which would encourage the PIP insurer to
undertake this investigation in order to save an
expected value of $280. If unallocated LAE is
covered by subrogation as a percentage of the
PIP claim, the insurer would be slightly less in-
clined to perform this investigation, as it would
reduce the PIP payment by $200, and the reim-
bursement from the other carrier by one-half of
the ULAE subrogation rate times $200 (one-half
since there is an equal chance that each insurer
will be the one responsible for the liability). In
Massachusetts, where the unallocated LAE reim-
bursement rate is 10% of a PIP claim, this would
reduce the value of investigating this claim by
$10 (:5£ 10%£ 200).14The liability insurer may not be able to reduce the claimed medi-
cal expenses but the negotiated award may be lower if the additional
treatment is known or suspected of being unnecessary (Derrig and
Weisberg 2004).
In the multiple insurer case, the PIP insurer
will have a lower chance of providing the lia-
bility coverage on this claim. In this example,
with no subrogation, if the chance of covering
both the PIP and liability is less than 28%, then
the expected savings on the noneconomic losses
would not be enough to compensate the PIP in-
surer to undertake this investigation. (The cost
of the investigation is $300, the savings on the
PIP are $200, and the expected savings on the
liability would be 28% of $360.) If there is sub-
rogation of losses, but not of LAE, then the PIP
insurer would never investigate this claim unless
the chance of covering both PIP and liability is
greater than 54% (300/560), as the expected sav-
ings would be the market share times $560 and
the cost of investigating would be $300. If al-
located LAE is also covered under subrogation,
then the PIP insurer would have the incentive
to investigate this claim, as the expected savings
would be the chance of having both PIP and li-
ability times $560 and the expected cost of in-
vestigating would be that chance times $300. As
long as the total expected savings exceeds the
expected cost of the investigation, the PIP in-
surer should perform the investigation. However,
reimbursement of unallocated LAE can change
the decision again. For example, if the chance
of covering both PIP and liability is only 5%,
then the expected savings from this investiga-
tion is $13 (5%£ (560¡ 300)), while the reduc-tion in expected ULAE reimbursement is $19
(95%£ 10%£ $200).Thus, the incentive for insurers to be strategi-
cally optimal is much lower when a large num-
ber of insurers compete. There is more room
for some insurers to exploit a free rider prob-
lem when more than two insurers are involved
in splitting the costs and benefits of adjusting
claims. This would be one disadvantage of hav-
ing a hybrid no-fault-limited tort system rather
182 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium
than a simple tort or no-fault system of compen-
sation.15
8. Alternative arrangementsIncentives to underinvestigate claims can be
addressed in several ways. If the claim inves-
tigation strategy is viewed as a repeated game,
with each insurer monitoring the performance of
the other insurers for free riding and adapting
their own behavior based on what other insurers
are doing, then rules can be established to pro-
vide incentives to investigate claims more fully
to the mutual benefit of all, leading to the optimal
(B,B) equilibrium. The prior strategy described
in this paper assumes that insurers make only one
choice of investigation after considering the ex-
pected costs and savings. Alternatively, insurers
can switch levels of investigation depending on
the behavior of the other insurer, making this sit-
uation a repeated, noncooperative game. In this
situation, negotiation and monitoring might be
able to move the equilibrium position back to cell
(B,B). Liability insurers will know, when paying
the claim to the injured person and the subroga-
tion costs to the other insurer, whether the claim
has been investigated fully, especially if ALAE is
covered under subrogation. If a company is not
investigating an appropriate proportion of claims
(each insurer would know this, since the opti-
mal level of investigation is assumed to be the
same among all insurers), other insurers could
retaliate against the offending insurer by treating
that company’s PIP claims differently For exam-
ple, they might be less cooperative when deter-
mining subrogation payments or provoke regu-
latory oversight. Therefore, competing insurers
could investigate claims at the strategically op-
timal level in order to reduce claim costs, and
premiums, to the lowest feasible level and then
monitor competitors to make sure they are liv-
15IRC 2004a provides contrasting medical expense and total claim
cost in four no-fault states, one of which (Colorado) has subse-
quently changed to a system of tort liability only.
ing up to this standard. However, in the case of
an insurer that expects to become insolvent in
the near future, there is no expectation of the re-
peated game. Such an insurer may revert to no
investigation for at least those claims with sav-
ings accruing to other insurers without fear of
future retaliation.16 Thus, observing an insurer’s
claim investigation pattern could also prove to be
an early warning sign of financial problems.
A second approach to addressing the underin-
vestigation problem would be to develop a sys-
tem under which the claim investigation costs
are shared among all insurers. This approach is
similar to that recommended by Picard (1996)
for dealing with claim audit costs. One possi-
ble approach would be to handle claim investiga-
tions in a manner similar to a reinsurance pool,
where bills are submitted to the pool and any
market share adjustments necessary are made at
the pool level. Each company is required to pay a
proportionate cost of claim investigations, based
on market share, regardless of its own investi-
gation strategy. This strategy may introduce in-
creased overall system costs above those of the
market monitoring strategy and, thereby, be less
efficient. Another (partial) method of doing this
would be to establish a separate fraud investiga-
tion unit, with the costs shared by all insurers,
to decides which claims to investigate based on
the total cost savings impact, regardless of which
insurer would benefit from these savings.17
9.1. Empirical evidenceThere is evidence in Massachusetts auto injury
claims that insurers follow the strategy of using
independent medical examinations (IMEs) to in-
vestigate claims at least to the extent that they de-
16Additionally, a failing insurer will attempt to minimize the sub-
rogation payments to other insurers giving yet another sign of fi-
nancial weakness.17Separate insurance fraud bureaus in the United States are, how-
ever, chiefly concerned with reducing criminal fraud with the sav-
ings accruing to the policyholders of all insurers just as the costs
are shared among all policyholders.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 183
Variance Advancing the Science of Risk
rive (1) positive savings net of investigation costs
overall and (2) no net loss on PIP investigations
(Derrig and Weisberg 2003). Massachusetts is a
no-fault state, with all auto insurance companies
required to offer first-party PIP coverage to pol-
icyholders. This coverage provides up to $8000
of coverage for economic losses such as medical
expenses, loss of income, compensation for loss
of services, and other expenses related to an in-
jury caused by an automobile accident. These can
be the bulk of the expenses that typically serve as
the special damages in a tort claim, the remain-
der being general damages or pain and suffer-
ing. There is also a $2000 medical expense tort
threshold for liability claims in Massachusetts.18
This threshold can be met by eligible medical ex-
penses including ambulance, hospital, physician,
chiropractor, or physical therapy bills. An injured
person can only recover noneconomic losses if
the accident is the fault of another party and med-
ical expenses exceed $2000. Since medical ex-
penses are covered by the PIP insurance, there is
an incentive for a claimant to incur at least this
amount in medical bills (Weisberg, Derrig, and
Chen 1994).
If the PIP insurer can contain the medical ex-
penses below $2000, not only will the PIP claim
be lower but lower (or no) payments will be
made for any noneconomic losses. Even if med-
ical expenses exceed the threshold, limiting the
total claimed medical expenses can have an ad-
ditional impact on the liability claim, since the
noneconomic losses included in liability settle-
ments tend to be directly related to claimed med-
ical expenses. Although demonstrating that the
total liability settlement is not simply a multi-
ple of the medical expenses, Derrig and Weis-
18There is also a verbal threshold (Mass C351 s6D) listing particu-
lar injuries that can be compensated by general damages but those
injuries generally incur medical expenses in excess of $2,000, as
well. Those compensatory injuries are (1) cause of death, (2) con-
sists in whole or in part of loss of a body member, (3) consists in
whole or in part of permanent and serious disfigurement, (4) result
of loss of sight or hearing and (5) consists of a fracture.
berg (2004) and Loughran (2005) found that the
settlements for noneconomic losses do increase
with the cost of the medical expenses incurred
but are reduced in other circumstances (such as
high suspicion of fraud or positive findings from
a BI IME) by negotiation. Thus, any impact the
PIP insurer can have to restrain medical expenses
will have an additional cost savings on the non-
economic losses and level B investigation may
raise the return to investigation for all insurers.
9.2. Massachusetts independentmedical examinationsThe Automobile Insurers Bureau of Massachu-
setts conducted a study of three data sets of
claims involving IMEs, the primary tool used by
insurers to control auto injury costs (Derrig and
Weisberg 2003), that we summarize here. The
methodology is a “tabular” analysis that simply
compares the mean payments for four subgroups
of claims:
² IME not requested² IME requested but not completed (no-show)² IME completed and positive outcome (positiveIME)
² IME completed and negative outcome (nega-tive IME)
The estimated gross savings for each of the
first three subgroups above is the difference be-
tween the average payment for that category and
the average for the last subgroup (completed with
a negative outcome). An average IME cost is
then subtracted to obtain an estimate of net sav-
ings. The following results are taken from the
AIB Derrig and Weisberg (2003) report.
Table 9 displays the results for the three sets
of tabular analyses:
² 1993 AIB sample (claims from a prior AIB
study)
² 1996 DCD sample (claims from AY 1996 in
the AIB detailed claim database of all auto in-
jury claims19)
19A random sample of all reported claims.
184 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium
Table 9. Summary results of Massachusetts IME Study
PIP
Sample: 1993 AIB 1996 DCD 1996 CSE
Total Net Savings (PIP) 0.2% ¡0:2% ¡0:8%Savings from IME Requested but not Completed 0.7% 0.3% 0.7%Savings from Positive IMEs 0.7% 0.4% ¡0:4%Cost of Negative IMEs ¡1:3% ¡0:9% ¡1:1%
PIP + BI
Sample: 1993 AIB 1996 DCD 1996 CSE
Total Net Savings (PIP + BI) 3.8% 5.7% 8.7%Savings from IME Requested but not Completed* 4.4% 2.8% 4.3%Savings from Positive IMEs 0.1% 3.2% 4.9%Cost of Negative IMEs ¡0:7% ¡0:3% ¡0:5%
¤Inclusion of All PIP claims with IME requested but not completed. 4.2% of savings for 1993 AIB comes from PIPs withno matching BIs where IME requested but not completed. 2.1% savings for 1996 DCD. 2.7% savings for 1996 CSE.
² 1996 CSE sample (claims from AY 1996 in
the claim screen experiment20)
Results are shown for both the PIP payment and
for the total payment (PIP+BI). The results sug-
gest that IMEs as currently employed represented
roughly a break-even proposition on PIP, al-
though for the CSE sample the cost of IMEs
slightly outweighed their benefit.
The bottom half, however, tells a different
story. Here the overall net savings for BI and PIP
payments are combined. These savings are based
on the outcome for the “best” IME, whether car-
ried out on the PIP or BI claim. The average
gross savings for the CSE sample was 9.2%, with
a net savings of 8.7%. Nearly half of the gross
savings (4.3%) is attributable to IMEs requested
but not actually completed. That is, the claimant
fails to show for the exam. In that case, savings
can result either if a potential BI claim is never
made, or if the BI settlement is reduced through
negotiation.
Somewhat more than half (4.9%) results from
a positive IME outcome (reduction of medical
20The Claim Screen Experiment (CSE) tracked about 3,000 PIP
claims arising at four large carriers during May—September, 1996.
Each carrier tracked the arrival of a preset collection of “red flags”
which in turn generated a running suspicion score for the adjusters
and their supervisors. Outcomes were recorded for all PIP claims
and for any associated BI tort liability claim.
Figure 2. Level of claim investigations inMassachusetts
expenses or curtailment of medical treatment) on
the PIP or BI IME. In the case of a BI IME or a
PIP IME used in the BI settlement, it may be too
late to have a meaningful impact on any ongoing
treatment. However, evidence of excess treatment
uncovered during the IME may provide leverage
to the adjuster in negotiating a lower settlement
by eliminating those medicals from any proposed
settlement.
Figure 2 illustrates the placement of the Mas-
sachusetts IME investigations relative to the the-
oretically optimal levels of Figure 1. PIP savings
equal to the costs, as shown in Table 9, would
place the Massachusetts investigation level, with
negative net PIP savings, to the right of both the
A and B optimal levels. In general, this would
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 185
Variance Advancing the Science of Risk
Table 10. Net savings by suspicion level
imply that Massachusetts carriers were investi-
gating more claims than the B optimal level. As
we will see next, the judicious use of the suspi-
cion score could have resulted in fewer IMEs by
limiting investigations to only claims with mod-
erate scores.21
9.3. Net savings by suspicion level
The CSE PIP adjusters collected data that was
used to calculate and return a suspicion score
on a 10-point scale. The suspicion score was
based on a linear regression analysis (Weisberg
and Derrig 1998). The net savings effects of the
IMEs are analyzed separately by the level of sus-
picion in Table 10. A positive net saving on PIP
occurred only for claims with a moderate level
of suspicion (4—6). The results of the PIP IMEs
are shown in the top line of Table 10. These re-
21This, of course, is easier said from hindsight than done in real
time. The complication in making the decision to investigate is
the tension between the timing of the arrival of the red flags that
determine the risk class and the ongoing treatment.
sults, based on the all PIP claims with IMEs, in-
dicate a modest 2.6% net savings for moderately
suspicious claims, and a net negative for other
categories.
A subset of the PIP claims was found to re-
sult in BI claims. When attention is restricted to
these BI-bound PIP claims, we obtain the results
in line 3 (highlighted). For moderately suspicious
claims, the estimated savings remains relatively
unchanged at 3.4%. The other categories appear
to change, but it should be noted that the num-
bers of claims with zero suspicion or high suspi-
cion are fairly small in the BI sample. So, these
changes might in part be attributable to random
variation.
The bottom part of the table is similar, except
that the suspicion breakdown is based on a dif-
ferent measure of suspicion. Since there was no
suspicion score model for BI claims within the
CSE, an alternative external scoring model de-
veloped by National Healthcare Resources, Inc.,
186 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium
Table 11. IME performance data
% of Claims with IME Requested
Strain/Sprain Other Injury
1993 AIB 1996 DCD 1996 CSE 1996 CSE 1996 CSE
PIP IME (PIP Claims) 18% 23% 26% 32% 14%PIP IME (BI Claims) 34% 35% 52% 54% 47%PIP or BI IME (BI Claims) 41% 40% 57% 58% 53%
% of Completed IMEs with Positive Outcomes
Strain/Sprain Other Injury
1993 AIB 1996 DCD 1996 CSE 1996 CSE 1996 CSE
PIP IME (PIP Claims) 34% 59% 58% 59% 55%PIP IME (BI Claims) 32% 60% 59% 59% 57%PIP or BI IME (BI Claims) 36% 60% 70% 71% 62%
(NHR) is used.22 This NHR model was used to
obtain a suspicion score for the BI claims, based
on the data extracted by coders from the BI claim
files.
The critical line is the last in Table 10, which
is highlighted. This line presents results for the
best outcome (PIP or BI) for the matched sample.
In effect, this analysis attempts to estimate the
overall impact of IMEs, taking into account both
the PIP payment and BI settlement. For moder-
ately suspicious claims, there is a 14.4% net sav-
ings, which accounts for most of the total sav-
ings. For claims with slight suspicion, IMEs rep-
resent effectively a break-even proposition, and
for the very low or very high suspicion a nega-
tive impact. It might appear counterintuitive that
IMEs do not have a positive value for claims with
high suspicion. Our explanation is that
such claims are not very amenable to reduction
through negotiation based on IME results. IMEs
are used primarily to constrain the total amount
of medical treatment, not to question the validity
of the injury itself or the circumstances of the
accident. To deal with “hard fraud” requires the
techniques of special investigation (e.g., exam-
ination under oath (EUO), accident reconstruc-
tion, surveillance).
22This scoring product was originally developed by Correlation Re-
search, Inc., while it was owned by National Healthcare Resources,
Inc. (NHR). NHR subsequently became part of Concentra, Inc.
9.4. Comparison across samples
Table 9 showed an overall 8.7% net savings for
the 1996 CSE claims, taking into account both
BI and PIP payments and IMEs. This outcome
is higher than the 5.7% for all claims (DCD) in
1996, which is in turn higher than the 3.8% reg-
istered in 1993. We now turn to the factors that
produce the IME savings.
In general, there are two factors that determine
the savings:
² Percent of claims on which an IME was re-quested
² Percent of completed IMEs with a positive out-come
Table 11 displays these percentages for each of
the three cohorts reported in Table 9. Compar-
ing first the 1993 sample versus the 1996 DCD,
we find the total number of IME requests has
remained essentially constant. However, the ap-
parent effectiveness of the IMEs performed has
increased dramatically. While this may be an ar-
tifact of the different coding patterns in the two
samples, it could reflect increased sophistication
on the part of adjusters regarding the selection
and/or utilization of the performed IMEs.
The results for the 1996 CSE claims pinpoint
where additional savings above the DCD esti-
mates were being derived. During the CSE, IMEs
were requested on a much higher percentage of
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 187
Variance Advancing the Science of Risk
those PIP claims destined to result in BI claims.
For example, 40% of the DCD BI claims had ei-
ther a PIP or BI IME requested, compared with
57% of the CSE sample. Since this increase oc-
curred regardless of experimental or control sta-
tus, the feedback of suspicion scores to the ex-
perimental group cannot explain this increase.
Rather, we suspect that a “Hawthorne effect” may
have resulted from the awareness by adjusters
that a study was happening.23
Interestingly, the increased IMEs did not re-
sult in a diminution of IME effectiveness. The
IMEs still produced positive outcomes at effec-
tively the same rate, or perhaps somewhat higher.
So the adjusters may have been quite discrimi-
nating in their selection of claims. In any event,
it is encouraging that the CSE intervention may
have in some manner generated an improvement
in performance.
Table 11 also shows an IME requested (au-
dit) ratio for strains and sprains of 32%, more
than twice the 14% ratio for the remaining in-
juries. This large difference is indicative of a gen-
eral strategy of auditing riskier (for fraud and
buildup) classes of claims more often than less
risky claims (Dionne et al. 2009). The similar-
ity of positive outcome percentages for the two
classes (59%, 55%) indicates that this differen-
tial auditing strategy is playing a role in deterring
fraudulent and build-up claims as well as detect-
ing them (Tennyson and Salsas-Forn 2002, pp.
304—306).
A comparison with the national data on IME
use in auto injury coverages is instructive. The
IRC (2004b, pp. 93—98) study of 2002 claims
countrywide shows that IMEs are used on about
40 percent of the less than 20 percent of PIP
claims with an appearance of fraud or build-up
or about 10 percent overall. By way of contrast,
the Massachusetts claims in the IRC sample had
23“Paying attention to people, which occurs in placing them in
an experiment, changes their behavior. This rather unpredictable
change is called the Hawthorne effect.” (Kruskal, W. H. and
J. M. Tanur,, International Encyclopedia of Statistics, Free Press,
New York, vol. 4, p. 210, 1978).
about 50 percent more PIP claims (29%) with the
appearance of fraud or buildup than the overall
sample, suggesting an IME rate between 15 and
20 percent overall (IRC 2004b). The CSE anal-
ysis of Massachusetts PIP claims in Table 11 in-
dicates a somewhat higher 1996 IME rate of 26
percent. For BI claims, the countrywide rate of
IME use is less than 10 percent overall (IRC,
2003), consistent with the approximate 4 to 8
percent use of BI IMEs in Table 11 (compare
upper table rows 2 and 3). As noted previously,
the IMEs on PIP claims in Massachusetts pro-
duce a positive outcome (favorable to the insurer)
slightly less than 60 percent of the time. Coun-
trywide, over 80 percent of PIP claims are miti-
gated by the use of an IME, 90 percent on claims
with the appearance of fraud or buildup. These
comparative data reinforce the observation that
Massachusetts insurers may be conducting IMEs
at a rate in excess of the desired near-optimal
level, perhaps because of the higher or broader
levels of suspicious auto injury claims, and that
a more judicious choice of claims, based upon
suspicion scoring methods, would produce more
cost-efficient results.
10. ConclusionThe optimal level of claim investigation de-
pends on how many insurers are in the market
and what the subrogation rules are for loss ad-
justment expenses. Viewing claim investigation
strategy in a game theoretic framework demon-
strates the incentives and disincentives that cur-
rently exist to investigate automobile insurance
claims for excessive claim behavior. A frame-
work for establishing Nash equilibria was devel-
oped for the monopoly and two-insurer cases.
When insurers can choose two levels of inves-
tigation or none at all, the equilibrium is estab-
lished by the relationship of the savings to the
cost of investigation. Circumstances are identi-
fied for the case where Nash equilibrium may be
inefficient. In no-fault systems, when subrogation
188 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium
of PIP claims exists, subrogation of allocated ex-
pense provides an incentive for investigation, but
a percentage reimbursement for unallocated ex-
pense provides a disincentive.
Empirical results from Massachusetts personal
auto injury claims were examined. Analyses of
three data sets show that carriers are generat-
ing substantial net savings from IME investiga-
tions, but those savings accrue mostly to the tort
carrier, indicating the workings of a noncoop-
erative game near equilibrium. A closer look at
the suspicion levels of the Massachusetts claims
shows that (1) insurers may have been conduct-
ing too many IMEs and (2) that a better selec-
tion (more toward the optimal equilibrium) could
be obtained with the use of a suspicion scoring
model.
Based on this analysis, additional cooperative
behavior should be encouraged in order to more
effectively reduce excessive medical treatment
and overall insurance costs. Subrogation rules
should cover allocated loss adjustment expenses.
If unallocated loss adjustment expenses are also
subject to subrogation, these payments should be
a set amount for each claim, and not a function
of claim size, as claim size adjustment provides a
disincentive to spend time and money to reduce
fraudulent claim costs. Other methods to encour-
age insurers to engage in strategically optimal ap-
proaches to investigating claims should be devel-
oped that consider the long-term, industry-wide
impact of reducing fraud.
Finally, the empirical data used above in the
study of Massachusetts claims was examined as
presented without optimizing beyond what indi-
vidual companies procedures produced for claim
investigation at that time. It is clear from Table 10
results that suspicion scores can be used to select
better candidates for investigation with higher net
savings and the application of so-called predic-
tive models can increase efficiency through bet-
ter claim selection methods (Derrig and Francis
2008). Many such procedures have been covered
in the annual CAS Ratemaking Seminars and
in the published literature, for example, Dionne,
Guilliani, and Picard (2009) and Artis, Ayuso,
and Guillen (2002).
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Automobile Insurance Fraud with Discrete Choice Mod-
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ance 69:3, pp. 325—340.
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tigation,” Geneva Papers on Risk and Insurance Theory
25:2, pp. 159—178.
Boyer, M. M., 2004, “Overcompensation as a Partial So-
lution to Commitment and Renegotiation Problems: The
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Derrig, R., A., and H. I. Weisberg, 2003, Auto Bodily In-
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cion of Fraud,” Insurance and Risk Management 71, pp.
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diting with Scoring: Theory and Application to Insurance
Fraud,” Management Science 5:1, pp. 58—70.
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Subrogation-In-All-50-States.asp.
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Mathematics 54(2), pp. 286—295.
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190 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Optimal Layers for CatastropheReinsurance
by Luyang Fu and C. K. “Stan” Khury
ABSTRACT
Insurers purchase catastrophe reinsurance primarily to re-
duce underwriting risk in any one experience period and
thus enhance the stability of their income stream over time.
Reinsurance comes at a cost and therefore it is important to
maintain a balance between the perceived benefit of buying
catastrophe reinsurance and its cost. This study presents a
methodology for determining the optimal catastrophe rein-
surance layer by maximizing the risk-adjusted underwrit-
ing profit within a classical mean-variance framework.
From the perspective of enterprise risk management, this
paper improves the existing literature in two ways. First, it
considers catastrophe and noncatastrophe losses simultane-
ously. Previous studies focused on catastrophe losses only.
Second, risk is measured by lower partial moment which
we believe is a more reasonable and flexible measure of
risk compared to the traditional variance and Value at Risk
(VaR) approaches.
KEYWORDS
Catastrophe reinsurance layer, downside risk, lower partial moment,
semivariance, utility function, enterprise risk management
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 191
Variance Advancing the Science of Risk
1. Introduction1
Catastrophe reinsurance serves to shield an in-
surer’s surplus against severe fluctuations aris-
ing from large catastrophe losses. By purchas-
ing catastrophe reinsurance, the reinsured trades
part of its profit to gain stability in its under-
writing and financial results.2 If catastrophe risks
are independent of other sources of risk and di-
versifiable in equilibrium, Froot (2001) argued
that, under the assumption of a perfect financial
market, the reinsurance premium for catastrophe
protection should equal the expected reinsurance
cost and that a risk-averse insurer would seek
the protection against large events over hedging
low-retention layers. Gajek and Zagrodny (2004)
concluded that if a reinsured has enough money
to buy full protection against bankruptcy (ruin),
the optimal reinsurance arrangement is the aggre-
gate stop-loss contract with the largest possible
deductible.
In practice, financial markets are not perfect,
reinsurance prices are significantly higher than
those indicated by the theory noted above, and
it does not make economic sense for insurers
to buy full protection against ruin. Froot (2001)
showed that the ratios of the catastrophe reinsur-
ance premium to the expected catastrophe loss
can be as high as 20 for certain high-retention
layers that have low penetration probabilities.3
Facing the reality of relatively high reinsurance
prices and the practical economic constraints on
the catastrophe reinsurance buyer, primary insur-
ers are reluctant to surrender a large portion of
their profit for limited catastrophe reinsurance
protections. Thus the reinsured often purchases
1This study is jointly sponsored by the Actuarial Foundation and
Casualty Actuarial Society.2While it is a minor point, if these concepts are extended to a line of
business that is not expected to be profitable (for whatever reason)
during the prospective reinsurance exposure period, the reduction
in profit referenced here becomes an increase in the projected loss.
The ideas are still the same.3Froot (2001), Figure 4, p. 540.
low reinsurance layers that are subject to a high
probability of being penetrated.
Economists have offered many explanations
for the inefficiency of the reinsurance market.
Borch (1962) investigated the equilibrium of the
reinsurance market and found that the reinsur-
ance market will, in general, not reach a Pareto
Optimum if each participant seeks to maximize
his utility. Froot (2001) identified eight theoreti-
cal reasons to explain why insurers buy relatively
little reinsurance against large catastrophe events.
He found the supply restrictions associated with
capital market imperfections (insufficient capital
in reinsurance, market power of reinsurers, and
inefficient corporate forms for reinsurance) pro-
vide the most powerful explanation. As practic-
ing actuaries, the authors believe there are signif-
icant limitations on reinsurers’ ability to diversify
risk. In the context of catastrophe reinsurance, no
reinsurer is big enough to fully diversify away
catastrophe risk. To support the risk, a reinsurer
needs to hold a prohibitively large amount of ad-
ditional capital and will, in turn, need to realize
an adequate return on such capital.
Froot (2007) found that product-market sen-
sitivity to risk and exposure asymmetry tend to
make insurers more conservative in accepting un-
systematic risks, more eager to diversify under-
writing exposures, and more aggressive in hedg-
ing. Even though reinsurance prices are high rel-
ative to expected loss, insurers are still willing to
seek reinsurance protections.
In this study, the authors do not explore the
level of reinsurance price and its reasonableness.
We also do not investigate why insurance firms
are risk averse to catastrophe losses which are
unsystematic and uncorrelated with aggregate
market return. Instead, we treat reinsurance price
as a predetermined variable in the overall strate-
gic decision-making process and develop an opti-
mal reinsurance strategy for insurers conditioned
by their risk appetite, prevailing reinsurance
192 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Optimal Layers for Catastrophe Reinsurance
prices at the time a decision is made, and the
overall profitability of the enterprise.
This introductory discussion would not be
complete if we did not explicitly point out that
a reinsurance arrangement, in fact, is a combina-
tion of art and science. It often depends on gen-
eral economic conditions in the state or coun-
try of the reinsured and worldwide, the recent
history of catastrophe reinsurance in the state or
country of the reinsured and worldwide, and the
risk characteristics of the reinsured. Both rein-
surer and reinsured usually are well informed
and are free to negotiate in the spirit of open
competition. In the negotiation, it is quite com-
mon that certain terms of the treaties would be
modified, such as changing the retention level
and the size of the layer. In pursuing the opti-
mization process outlined in this paper, we are
not attempting to deny or ignore this general
art/science characteristic of the reinsurance ar-
rangement. Instead our hope is that knowledge
of the optimal balance between profit and risk,
as measured using the process outlined in this pa-
per, in the particular circumstances of a reinsured
vis-a-vis the then-prevailing market prices, will
serve to enhance the quality of the reinsurance
decision, ceteris paribus.
Reinsurance arrangements have been studied
extensively because of their strategic importance
to the financial condition of insurance compa-
nies. However, previous studies on optimal catas-
trophe reinsurance only utilized partial informa-
tion in the reinsurance decision-making process.
Gajek and Zagrodny (2000) and Kaluszka (2001)
investigated the optimal reinsurance arrangement
by way of minimizing the consequential variance
of an insurer’s portfolio. Gajek and Zagrodny
(2004) discussed the optimal aggregate stop-loss
contract from the perspective of minimizing the
probability of ruin. Those studies focus on the
risk component, but ignore the profit side of the
equation. Bu (2005) developed the optimal rein-
surance layer within a mean-variance framework.
Insurers are assumed to minimize the sum of the
price of reinsurance, the catastrophe loss net of
reinsurance recoveries, and the risk penalty. Bu
used both the profit and risk components in the
optimization. However, his method focused on
the catastrophe loss only and ignored the concur-
rent effect of noncatastrophe underwriting per-
formance on the financial results of the reinsured.
In practice, the overall profitability is an impor-
tant factor impacting the reinsurance strategy be-
cause, among other things, it can enhance an in-
surer’s capability to assume risk.
Lampaert and Walhin (2005) studied the opti-
mal proportional reinsurance that maximizes
RORAC (return on risk-adjusted capital). The ap-
proach requires the estimation of economic cap-
ital based on VaR or TVaR (tail value at risk)
at a small predetermined probability. VaR and
TVaR are popular in insurance generally and in
actuarial circles specifically. VaR is the point at
which a “bad” outcome can occur at a prede-
termined probability, say 1%. TVaR is the mean
of all outcomes that are “worse” than the pre-
determined “bad” outcome. VaR, and especially
TVaR, has some convenient features as a risk
measure.4 TVaR only contemplates severe losses
having a probability at or lower than a given
probability as the central risk drivers, and it treats
those losses linearly. For example, if an insurer
has a 5% probability of a loss of $3 million, a 4%
probability of a loss of $5 million, a 0.9% proba-
bility of a loss of $10 million, and a 0.1% proba-
bility of a loss of $100 million, VaR at 1% is $10
million and TVaR is $19 million (10 ¤ 0:9%+100 ¤ 0:1%)=1%. VaR and TVaR are not consis-tent with common risk perception from two per-
spectives: (1) fear is not just of severe losses, it is
also of smaller losses (Bodoff 2009). In the case
above, $3- and $5-million losses will not con-
tribute to VaR because VaR only considers 1%
probability at which risk is generated; (2) risk-
4For example, Meyers (2001) discussed that TVaR satisfies the four
criteria of coherent risk measures.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 193
Variance Advancing the Science of Risk
bearing entities do not weigh the risk of loss in
a linear manner and are more concerned about
the incidence of large losses than smaller ones.
In other words, risk perception is exponentially,
not linearly, increased with the size of loss.
In practice, the RORAC method has been pop-
ular in calculating the optimal catastrophe rein-
surance layer. In this study, we improve the pop-
ular mean-variance approach advocated in aca-
demic studies by using lower partial moment
(LPM) as the measure of risk, and provide an
alternative method for determining optimal rein-
surance layers. Compared with the RORAC ap-
proach, our method has three advantages. First, it
does not involve the calculation of the necessary
economic capital, which has no universally ac-
cepted definition. Second, by VaR or TVaR, true
risk exists only at the tail of the distribution. By
LPM, on the other hand, all the losses are con-
sidered as generating risk to the risk-bearer, but
severe losses contribute to LPM disproportion-
ately. Third, the estimation of variance and semi-
variance is relatively robust compared to VaR
and TVaR in the context of catastrophe losses.
The tail estimation of remote catastrophe losses
generally is not robust, and is very sensitive to
the assumptions about the underlying distribu-
tion, especially at high significance levels. The
limitations of the proposed method are the limita-
tions inherent to the mean-variance framework. It
can be difficult to estimate the risk-penalty coef-
ficient, as the parameter is often time-dependent
and subject to management’s specific risk
appetite.
This paper improves the previous mean-var-
iance optimal reinsurance studies from two per-
spectives. First, it considers noncatastrophe and
catastrophe losses simultaneously. Second, the
risk is measured by LPM (semivariance), which
is a more reasonable and appropriate risk mea-
sure than the traditional risk measures, such as
total variance, used in previous studies (i.e.,
Borch 1982; Lane 2000; Kaluszka 2001; Bu
2005). Even though the authors investigate the
optimal layers in the context of catastrophe rein-
surance, the proposed method can be easily ap-
plied to aggregate excess-of-loss (XOL) treaties
and occurrence XOL treaties that cover shock
losses at individual occurrence/claim levels.
2. Risk-adjusted profit modelInsurance companies buy catastrophe reinsur-
ance to reduce potential volatility in earnings and
to provide balance sheet protection from catas-
trophic events. However, reinsurance comes at a
cost, and therefore attaining an optimal balance
between profit levels after the effect of catastro-
phe reinsurance and the reduction in their risk
exposure is important. Buying unnecessary or
“excessive” reinsurance coverage would give up
more of the reinsured’s underwriting profit than
is necessary or desirable. Buying inadequate rein-
surance coverage would still expose the reinsured
to the volatility engendered by the risk of large
catastrophe events, the reinsurance cover not-
withstanding. The value of reinsurance is the sta-
bility gained or the risk reduced and the cost
is the premium paid less the loss recovered. As
Venter (2001) pointed out, the analysis of a rein-
surance arrangement is the process of quantify-
ing this cost/benefit relationship. It is self-evident
that in an insurance company’s decision-making
process, a relatively certain, but maximal, profit
is preferable over other, perhaps higher profit po-
tentials that are also exposed to the risk of large
catastrophic losses. Following the classic mean-
variance framework in financial economics, a
reinsured will buy reinsurance to maximize its
risk-adjusted profit, defined as
RAP = E(r)¡ μ ¤Var(r) (1)5
where r is the net underwriting profit rate, E(r) is
the mean of r and Var(r) is its variance. μ ¤Var(r)
5In financial economics, it is often referred to as the expected util-
ity function with the formula E(r)¡ 0:5A ¤Var(r), where A is theso-called risk aversion coefficient. In this paper, the risk-penalty
coefficient μ = 0:5A.
194 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Optimal Layers for Catastrophe Reinsurance
is the penalty on risk. μ is the risk-penalty coef-
ficient: the higher the risk-penalty μ, the greater
is the reinsured’s risk aversion. If μ = 0, the rein-
sured is risk-neutral. It will try to maximize profit
and not care about risk. In this scenario, it will
not give up any profit to purchase reinsurance.
The most common measurement of risk is the
variance associated with a particular outcome.
Variance reflects the magnitude of uncertainty
(variability) in underwriting results, and how
widely spread the values of the profit rate are
likely to be around the mean. Therefore, within
a variance framework, all the variations, both de-
sirable and undesirable, are viewed as manifes-
tations of risk. Large favorable swings will lead
to a large variance, but insurers certainly have
no problem with such favorable underwriting re-
sults. Markowitz (1959) pointed out the draw-
backs of using total variance as a measure of risk,
as there is implicitly and directly a cost to both
upside and downside movements.
Fishburn (1977) argued that risk should be
measured in terms of only below-target returns.
Hogan and Warren (1974) and Bawa and Lin-
denberg (1977) suggested using LPM to replace
total variance as the risk measure:
LPM(T,k) =
Z T
¡1(T¡ r)kdF(r) (2)
where T is the minimum acceptable profit rate, k
is the moment parameter which measures one’s
risk perception sensitivity to large loss, and F(r)
is the probability function of r. Unlike total vari-
ance, LPM only measures the unfavorable vari-
ation (e.g., when r < T) as risk. Because LPM
does not change with favorable deviations, it
would seem to be a superior measure of risk.
When T is triggered at the 1% probability level
and k = 1, LPM is equal to 0:01 ¤TVaR. Whenthe distribution is symmetric, T is the mean, and
k = 2, it is equal to 0:5 ¤ variance. LPM com-
bines the advantages of variance and TVaR. It
is superior to variance by not treating the favor-
able outcomes as risks. It is superior to TVaR be-
cause (1) it considers small and medium losses
as risk components, and (2) it provides nonlin-
ear increasing penalties on larger losses when
k > 1. In the example above, suppose a $100 mil-
lion loss will cause a financial rating downgrade
while a $10 million loss merely causes a bad
quarter. Management inevitably will perceive a
$100 million loss to be more than 10 times as
bad as a $10 million loss. By VaR, a $100 mil-
lion loss is 10 times as bad as a 10 million loss.
By LPM with k = 1:5, the risk of a $100 million
loss is 31.6 times that of a $10 million loss; and
by LPMwith k = 2, it is 100 times. The k value is
a direct measure of risk aversion to large losses.
When k = 2, LPM is often called “semivari-
ance” (it excludes the effects of variance associ-
ated with desirable outcomes in the measurement
of risk) and has been gaining greater acceptance.
By formula, semivariance is defined as
SV(T) =
Z T
¡1(T¡ r)2dF(r): (3)
A growing number of researchers and practi-
tioners are applying semivariance in various fi-
nancial applications. For example, Price, Price,
and Nantell (1982) showed that semivariance
helps to explain the puzzle of Black, Jensen, and
Scholes (1972) and Fama and MacBeth (1973)
that low-beta stocks appear systematically un-
derpriced and high-beta stocks appear systemat-
ically overpriced. However, to date, the casualty
actuarial literature has seldom used the semivari-
ance as a risk management tool and neither does
it appear much in practice.6
Generally, a decision-maker can be expected
to be more concerned with the semivariance than
with the total variance. Using downside risk in-
stead of total variance, the downside-risk-adjust-
ed profit (DRAP) becomes
DRAP =Mean(r)¡ μ ¤LPM(T,k), (4)
6Berliner (1977) studied a special case of semivariance with the
mean as the minimum acceptable value, against variance as risk
measures. He concluded that although the semi-variance is more
theoretically sound, variance provides a better risk measure.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 195
Variance Advancing the Science of Risk
where μ is the penalty coefficient on downside
risk.
Three parameters T, k, μ in the DRAP for-
mula interactively reflect risk perception and risk
aversion. With these three parameters, the DRAP
method provides a comprehensive and flexible
capacity to capture risk tolerance and appetite.
T is the benchmark point of profit below which
the performance would be considered as “down-
side” (lower than is minimally unacceptable). T
can be a target profit rate, the risk-free rate, zero,
or even negative, depending on one’s risk per-
ception. When T is at the very right tail of r,
only large losses contribute to the downside risk.
T can vary by the mix of lines of business. For
example, for long tail lines, negative underwrit-
ing profits may be tolerable because of antici-
pated investment income on loss reserves.
The moment parameter k reflects one’s risk
perception as the size of loss grows: k > 1 im-
plies exponentially increasing loss perception to
large losses; 0< k < 1 represents concavely in-
creasing loss perception to large losses; k = 1 im-
plies linearly increasing loss perception. In gen-
eral, k is larger than 1 since fear of extreme
events that can lead to a financial downgrade is
greater than the fear of multiple smaller losses.
Because semivariance is the most popular LPM,
we choose k = 2 to illustrate our approach in the
case study presented below.
The risk aversion level is represented by μ,
which is a function of T and varies according to
its values. For example, when T = 0, all the un-
derwriting losses contribute to LPM(0,k); when
T =¡10%, only losses exceeding the 10% loss
rate contribute to LPM(¡0:1,k). LPM(¡0:1,k)represents a much more severe risk than
LPM(0,k). μ is also a function of k. For example,
LPM(T,1) and LPM(T,2) are at different scales
because the former is at the first moment and the
latter is at the second moment. μ should vary with
k because k changes the scale of risk measure.
Note that μ may not be constant across loss
layers. For example, when k = 1, LPM is a lin-
ear function of loss. For a run of smaller losses
that cause a “bad” quarter, μ may be very small.
For losses that cause a “bad” year or eliminate
management’s annual bonus, μ may be larger.
For losses that lead to a financial downgrade or
the replacement of management, μ will be even
larger. Interested readers can expand the models
in this paper by adding a series of risk-penalties
upon predetermined loss layers with various
risk aversion coefficients. When k ¸ 2, becauseLPM increases exponentially with extreme
losses, it may not be necessary to impose higher
risk-penalty coefficients on higher layers.
In addition, k may not be constant across loss
layers. The scale of loss impacts the value of k.
If k is constant, say k = 2, it implies that $100
million loss is 100 times worse than $10 million
loss. It also indicates that $100 loss is 100 times
worse of $10 loss. The latter, in general, is not
true because of linear risk perception when view-
ing a smaller nonmaterial loss. In the context of
reinsurance, k might be closer to one at a working
layer (low retention with high probability of pen-
etration) and would increase for higher excess
layers. Interested readers can expand the models
in this paper by adding a series of risk-penalties
upon predetermined loss layers with various mo-
ment parameters.
The academic tradition in financial economics
has been to set μ as a constant and k = 2.7 Assum-
ing an individual has a negative exponential util-
ity function u(r) =¡exp(¡A ¤ r), where A > 0.If r is normally distributed, the expected utility
is E[u(r)] =¡exp[¡A ¤E(r) +0:5A2 ¤Var(r)].Maximizing E[u(r)] is equivalent to maximizing
E(r)¡ 0:5A ¤Var(r). Also, ¡u00(r)=u0(r), which isequal to A in this specific case, is often referred
as the “Arrow-Pratt measure of absolute risk
7In recent years, academics have found increasing evidences of
higher moments of risk aversion. For example, Harvey (2000)
showed that skewness (3rd moment) and kurtosis (4th moment)
are priced in emerging stock markets but not in developed markets.
196 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Optimal Layers for Catastrophe Reinsurance
aversion.” Constant μ and k = 2 are built-in fea-
tures under the assumptions of negative expo-
nential utility function and normality. Alterna-
tively, interested readers can use negative expo-
nential utility, logarithmic utility, or define their
own concavely increasing utility curves, and se-
lect the optimal reinsurance layer to maximize
the expected utility function.8 To simplify the il-
lustration and to be consistent with academic tra-
dition, we use a constant μ and k = 2 in the case
study.
An inherent difficulty in mean-variance type of
analysis is the need to estimate the risk-penalty
coefficient empirically. The key is to measure
how much risk premium one is willing to pay
for hedging risk. The flip counter-party ques-
tion is how much investors would require for
assuming that risk. For overall market risk pre-
mium, one can obtain the market risk premium
by subtracting the risk-free treasury rate from
market index return. For example, if the market
return is 10%9 and the risk-free rate is 5.5%, the
risk premium is 4.5%, or 45% of total “profit.”
For the risk premium in the insurance/reinsur-
ance market, one can use the market index for in-
surance/reinsurance companies. For the risk pre-
mium in catastrophe reinsurance, one can com-
pare catastrophe bond rates to the risk-free rate.
For example, if the catastrophe bond yield is
12%,10 the treasury rate yield is 5.5%, and the
expected loss from default is 0.5%, then the risk
premium11 is 6%, consisting of 50% of total
yield.
The methods above provide objective estima-
tion of μ assuming that management’s risk ap-
8Insurance profit is generally not normally distributed. It can be
positive or negative, but has a fatter tail on left side. Reinsurance
layers complicate the distribution of r. It may be very difficult
to derive an analytical solution for E[u(r)]. However, a numerical
solution maximizing E[u(r)] can be obtained easily.9According to Vanguard, its S&P 500 index returns 10.34% annu-
ally from its inception (08/31/1976) to 01/31/2010.10In 1997, USAA issued its one-year catastrophe bond at LIBOR
plus 576 basis points, which was close to 12%.11Risk premium is equal to cat bond yield¡risk-free rate¡expectedloss. For details, please refer to Bodoff and Gan (2009).
petite is consistent with the market. In reality, μ
varies by risk-bearing entity. Each risk-bearing
entity has its own risk perception and tolerance.
To measure management’s risk aversion, one can
obtain μ by asking senior executives, “In order to
reduce downside risks, how much of the other-
wise available underwriting profit per unit of risk
are you willing to pay?” The answer to a single
question may not be sufficient to pin down the
value of theta. Most likely management would
require information about expected results under
optimal reinsurance programs at various values
of theta to fully understand the implications of
the final theta value selected. To replicate the sen-
sitivity tests that management may perform when
determining the utility function, the case study
provides optimal insurance solutions at various
values of theta.
For the same management within the same in-
stitution, μ often is time-variant12 as the risk ap-
petite often changes to reflect macro economic
conditions or micro financial conditions. For ex-
ample, after a financial crisis, insurance compa-
nies may become more risk averse. μ also varies
by the mix of business. For lines with little catas-
trophe potential, such as personal auto, the tol-
erance on downside risk might be higher and μ
would be smaller. For lines with higher catas-
trophe potential, such as homeowners, μ can be
larger. μ is difficult to estimate because of its sub-
jective nature. Actuarial judgment plays an im-
portant role when determining the risk-penalty
coefficient.
In the context of catastrophe reinsurance, the
layers are bands of protection associated with
catastrophe-triggered loss amounts. Outside of
price, the main parameters of a catastrophe layer
are the retention, the coverage limit, and the ces-
sion percentage within the layer of protection.
Retention is the amount that the total loss from
12The authors tried to estimate a constant risk penalty coefficient
based on management past reinsurance decisions. The result clearly
indicates that the risk coefficient is time-variant.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 197
Variance Advancing the Science of Risk
a catastrophe event must exceed before reinsur-
ance coverage attaches. The limit is the size of
the band above the retention that is protected by
reinsurance. The cession percentage is the per-
centage of the catastrophe loss within the rein-
surance layer that will be covered by the rein-
surer. The limit multiplied by the cession per-
centage is the maximum reinsurance recovery
from a catastrophe event within that particular
band of loss. The coverage period of catastro-
phe reinsurance contracts is typically one year.
Let xi denote the gross incurred loss from the
ith catastrophe event within a year, and Y be the
total gross non-catastrophe loss of the year. Let
R be the retention level of the reinsurance, L be
the coverage layer of the reinsurance immedi-
ately above R,13 and Á be the coverage percent-
age within the layer.
The loss recovery from reinsurance for the ith
catastrophe event is
G(xi,R,L) =
8>><>>:0 if xi · R
(xi¡R) ¤Á if R < xi · R+LL ¤Á if xi > R+L
:
(5)
Let EP be the gross earned premium, EXP be
the expense of the reinsured, N be the total num-
ber of catastrophe events in the reinsurance con-
tract year, RP(R,L) be the reinsurance premium,
which is a decreasing function of R and an in-
creasing function of L, and RI be the reinstate-
ment premium.
The underwriting profit gross of reinsurance is
¼ = EP¡EXP¡Y¡NXi=1
xi: (6)
Reinstatement premium is a pro rata reinsur-
ance premium charged for the reinstatement of
the amount of reinsurance coverage that was
“consumed” as the result of a reinsurance loss
13In general, reinsurance is structured in multiple layers. The for-
mulation works for continuous layers. For disruptive layers, one
needs to introduce additional retentions and limits.
payment under a catastrophe cover. The stan-
dard practice is one or two reinstatements. The
number of reinstatements imposes an aggregate
limit on catastrophe reinsurance. The reinstate-
ment premium after the ith catastrophe event is
RI(xi,R,L) = RP(R,L) ¤G(xi,R,L)=L: (7)
The underwriting profit net of reinsurance is
¼ = EP¡EXP¡Y¡NXi=1
xi¡RP(R,L)
+NXi=1
G(xi,R,L)¡NXi=1
RI(xi,R,L): (8)
The underwriting profit rate net of reinsurance
is
r = 1¡ EXP+Y+RP(R,L)EP
¡PNi=1 xi¡G(xi,R,L) +RI(xi,R,L)
EP:
(9)
Thus the optimal layer is that combination of
R and L which maximizes DRAP:
MaxR,L
Mean(r)¡ μ ¤ r), subject to C:14
(10)
Capital asset pricing model suggests that firms in
a perfect market are risk neutral to unsystematic
risks. If reinsurers are adequately diversified and
price as though they are risk neutral, the reinsur-
ance premium would be equal to the expected
reinsurance cost.15 That is, the total premium
paid by reinsured, RP(R,L)+PNi=1RI(xi,R,L),
is equal to the expected reinsurer’s loss cost,
14C is the constraint for the optimization, which can be (a) a budget
constraint on reinsurance premium; (b) a risk tolerance constraint
such as the probability of downgrade is less than 1%, or the PML
will reduce surplus by 15% no more than 1 in 100 years; or (c) any
number of other possible constraints that are relevant to the partic-
ular reinsured. In the case study, to simplify the analysis, we do not
impose constraints on the optimization. The same framework can
be applied to optimal aggregate reinsurance strategies. Actuaries
would select four parameters: event deductible, event cap, aggre-
gate retention, and limit to maximize the risk-adjusted profit.15According to financial economics theory, risk premium only ap-
plies to nondiversifiable systematic risk. Catastrophe risk is diver-
sifiable in theory so that the risk premium of assuming catastrophe
risk is zero. The theory is extensively discussed in Froot (2001).
198 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
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Optimal Layers for Catastrophe Reinsurance
PNi=1G(xi,R,L) plus reinsurer’s expense. In this
case, reinsurancewill significantly reduce the vol-
atility of the underlying results of the insured
over time, but slightly reduce the expected profit
by the reinsurance expense over the same period
of time. Figure 1 shows reinsurance optimiza-
tion under the assumption of perfect market di-
versification. A is the combination of profit and
downside risk without any reinsurance. B is the
profit and risk with full reinsurance. B is not
downside-risk-free because noncatastrophe loss
could cause the profit to fall below the minimum
acceptable level. Line AB is the efficient frontier
with all possible reinsurance layers. Closer to B,
it represents buying a great deal of reinsurance
coverage. Closer to A, it represents buying mini-
mal reinsurance coverage. Under the assumption,
the reinsurance premium only covers reinsurer’s
costs, and the line is relatively flat.16 U1, U2, and
U3 are the utility curves. The slope of those lines
is the risk-penalty coefficient μ. The steeper the
curve, the more risk-averse. All the points on a
given curve provide the exact same utility. The
higher utility curve represents the higher utility.
The utilities on line U1 are higher than those on
lines U2 and U3. An insurance company gains
the highest utility at point B. Thus maximizing
the risk-adjusted profit is equivalent to minimiz-
ing the downside risk. The optimal solution oc-
curs when R = 0 and L=+1. A retention equalto zero coupled with an unlimited reinsurance
layer will completely remove the volatility from
catastrophe events with a low cost (reinsurance
premium-recovery). Under the perfect market di-
versification assumption, the proposed method
yields a solution consistent with Froot (2001).
In practice, however, because of the need to
reward reinsurers for taking risk, the reinsurance
price RP(R,L) is always larger than the expected
reinsurer’s loss cost. The expected loss/premium
ratio is generally a decreasing function of reten-
tion R. The higher the retention R, the lower the
16Line AB would be flat assuming zero reinsurer’s expense.
Figure 1. Reinsurance optimization under theassumption of perfect diversification
Figure 2. Reinsurance optimization in reality
expected reinsurance loss ratio. From the rela-
tionship between risk transfer and reinsurance
premium, a higher layer implies a higher level
of risk being transferred to the reinsurer. To sup-
port the risk associated with higher layers, the
reinsurer needs more capital and thus requires a
higher underwriting margin. Therefore, a rein-
sured has to pay a larger risk premium on higher
layers to hedge its catastrophe risk. In practice,
E(PNi=1G(xi,R,L))=RP(R,L) is often less than
40%, and even below 10% for high retention
treaties. The relatively high prices associated with
high retentions often deter the reinsured from
purchasing coverage at those levels. Subject to
the constraints imposed by reinsurance prices and
the willingness of the reinsured to pay, as Froot
(2001) discussed, the optimal solutions are of-
ten low reinsurance retentions at a relatively low
price and a high probability of being penetrated.
Figure 2 shows reinsurance optimization in real-
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 199
Variance Advancing the Science of Risk
ity. As in Figure 1, curve AB represents the ef-
ficient frontier. Because reinsurance companies
cannot fully diversify catastrophe risk and re-
quire higher returns to assume the risk on higher
layers, AB is a concave curve: from A to B, the
slope becomes steeper to reflect higher risk pre-
miums associated with higher layers. Close to
point B, the slope is very steep to reflect the ex-
tra capital surcharge at the top layers. Of all the
possible reinsurance layers, point C provides the
highest utility (or downside-risk-adjusted profit)
to the reinsured.
3. A case study
3.1. Key parameters
Suppose an insurance company with $10 bil-
lion17 gross earned premium plans to purchase
catastrophe reinsurance. Within one year, the
number of covered catastrophe events is normally
distributed18 with a mean of 39.731 and a stan-
dard deviation of 4.450; and the gross loss from a
single catastrophe event is assumed to be lognor-
mally distributed. The logarithm of the catastro-
phe loss has a mean of 14.478 and a standard de-
viation of 1.812,19 which imply a mean of $10.02
million and a standard deviation of $50.77 mil-
lion for the catastrophe loss from one event. The
mean of the aggregate gross loss from all the
catastrophe events within a year is $397.94 mil-
17The premium is for all lines of business. The catastrophe losses
are from property lines. All the catastrophe loss parameters in this
case study are estimates drawn from Applied Insurance Research
(AIR) data simulated based on the property exposures of an insur-
ance company and are scaled accordingly to be consistent with $10
billion earned premium.18The number of catastrophe events (severe storm and hurricane)
at state level generally fits Poisson distributions better than normal.
At aggregate company level, the number of catastrophe events is
asymptotically normal by AIR data.19The frequency and severity are estimated using the data from
AIR. In this study we randomly generate the loss data to avoid
revealing proprietary information. The AIR data has a longer tail
than the fitted lognormal distribution. In practice, actuaries can use
catastrophe data directly from AIR, RMS (Risk Management So-
lutions), or EQECAT, or generate loss data based on proprietary
catastrophe models.
lion and the standard deviation is $322.92 mil-
lion. The expense ratio of the insurance company
is assumed to be 33.0%. The aggregate gross
noncatastrophe loss is also assumed to be log-
normally distributed with a logarithm of non-
catastrophe loss mean of 22.497 and standard de-
viation of 0.068. This implies that the mean of
gross noncatastrophe loss is $5.91 billion and the
standard deviation is $402.10 million. Assum-
ing catastrophe and noncatastrophe losses are in-
dependent,20 the mean of the aggregate gross
loss is $6.30 billion and the standard deviation
is $515.72 million. The mean underwriting profit
rate is 3.93% and the standard deviation is 5.16%.
The numerical study is based on this hypothetical
company using the simulated noncatastrophe and
catastrophe loss data. The simulation is repeated
10,000 times. In each simulation, the noncatas-
trophe loss and catastrophe losses within a year
are generated, the losses covered by the reinsur-
ance treaty are calculated by Equation (5), and
the total profit rate is calculated by Equation (9)
assuming two reinstatements. Let rm be the profit
rate from the mth round of simulation. The semi-
variance is
R,L) =1
10000
10000Xm=1
(min(rm¡T, 0))2:
(11)
Equation (11) is a discrete formula of semi-
variance, which is an approximation of Equa-
tion (3).
Let us assume that reinsurance will cover 95%
of the layer (R,L) and UL is the upper limit of
the covered reinsurance layer (R,L), UL = R+L.
For reinsurance prices, we fit a nonlinear curve
using actual reinsurance price quotes.21 The fit-
20This is a simplifying assumption for the case study. In practice,
the correlation is weakly positive from the perspective of the pri-
mary insurer because catastrophe events cause claims that are oc-
casionally categorized as noncatastrophe losses.21The reinsurance prices are proportionally scaled by the premium
adjustment.
200 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
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Optimal Layers for Catastrophe Reinsurance
Table 1. Distributional summaries of loss covered from reinsurance in a year for quoted reinsurance layers
Retention Upper Limit Recovery/reinsurance Penetration(million) (million) Mean Standard Deviation Premium Probability
305 420 8,859,074 29,491,239 42.59% 10.18%420 610 8,045,968 35,917,439 37.08% 6.04%610 915 6,496,494 41,009,356 32.81% 3.15%610 1,030 7,923,052 51,899,244 31.44% 3.15%
1,030 1,800 4,858,545 55,432,115 16.93% 1.11%1,800 3,050 2,573,573 48,827,021 6.58% 0.40%
ted reinsurance price of layer (R,L) is22
RP(R,L) = 1:2300 ¤ (UL¡R)+ 1:2978 ¤ 10¡4
¤ (UL2¡R2)¡ 1:3077 ¤ 10¡8
¤ (UL3¡R3)¡ 0:1835¤ (UL ¤ log(UL)¡R ¤ log(R))+45:4067 ¤ (log(UL)¡ log(R)):
Appendix contains both the quoted reinsur-
ance prices and the fitted reinsurance prices. The
actual prices below layer ($1,800 million, $3,050
million) are derived by combining the six lay-
ers with known quotes. Simon (1972) and Khury
(1973) discussed the importance of maintaining
the logical consistency among various alter-
natives, especially on pricing. The fitted price
curve is logically consistent in two ways: (1) the
rate-on-line is strictly decreasing with reten-
tion and consistent with actual observations; (2)
for two adjacent layers, the sum of prices is
equal to the price of the combined layer, that is,
RP(R,L1 +L2) = RP(R,L1)+RP(R+L1,L2).
The minimum acceptable profit rate T and risk-
penalty coefficient μ vary by business mix and
by risk-bearing entity. In this case study, 0% is
selected as the minimum acceptable profit rate
for illustrative purposes. So, only underwriting
losses contribute to the risk calculation. For μ, we
use three values, 16.71, 22.28, and 27.85. Those
coefficients represent management’s willingness
22The curve fitting of reinsurance price quotes is discussed in Ap-
pendix. In practice, actuaries can select rate-on-line by judgment,
or fit their own curves by regression or use interpolation. In the
case study included in this paper, the three options do not produce
significantly different results.
to pay 30%,23 40%, and 50% of underwriting
profits to hedge downside risk, respectively. The
risk-penalty coefficients in the case study are se-
lected solely for illustrative purposes.
3.2. Numerical results
In the simulation, we generate catastrophe loss
and noncatastrophe loss for 10,000 years. In the
instant case, 397,257 catastrophe events are gen-
erated. Table 1 summarizes the losses covered by
reinsurance for quoted layers; Table 2 reports the
distribution summary of underwriting profit rates
net of reinsurance and the risk-adjusted profit
rates for quoted layers and their continuous com-
binations.
As illustrated in Table 1, a catastrophe loss has
a 10.18% chance to penetrate the retention level
of $305 million within one year. So, roughly in 1
of 10 years, the reinsured will obtain recoveries
by purchasing the reinsurance for this layer. The
higher the retention level, the lower the prob-
ability that the catastrophe loss penetrates the
layer. For example, the catastrophe loss has only
a 0.40% chance of penetrating a retention level
of $1,800 million.24 This is expected because the
frequency of a very large catastrophe loss is rel-
atively small. For the layer ($305 million, $420
million), the reinsurance price is $20.8 million,
while the mean of loss recovered from the rein-
23The mean profit and semivariance without any reinsurance are
3.93% and 0.07%, respectively. If a primary insurer would like to
use 30% of its gross profit to hedge downside risk, the risk penalty
coefficient is 16:71 = 3:93% ¤ 0:3=0:07%.24By using the catastrophe losses from AIR, the probability is
higher because the AIR models produce catastrophe losses with
larger tails than the fitted lognormal distribution.
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 201
Variance Advancing the Science of Risk
Table 2. Distributional summary of underwriting profit rates for selected reinsurance layers when μ = 22:28
Retention Upper Limit Risk-adjusted(million) (million) Probr < 0% Probr <¡15% Mean Variance Semivariance25 Profit
No Reinsurance 18.41% 0.48% 3.916% 0.263% 0.070% 2.350%305 420 19.02% 0.42% 3.781% 0.253% 0.067% 2.291%420 610 19.17% 0.35% 3.771% 0.249% 0.064% 2.341%610 915 19.31% 0.30% 3.779% 0.247% 0.061% 2.412%610 1030 19.53% 0.27% 3.739% 0.243% 0.059% 2.428%
1030 1800 19.95% 0.26% 3.676% 0.243% 0.057% 2.397%1800 3050 20.44% 0.41% 3.551% 0.247% 0.061% 2.186%
305 610 19.63% 0.33% 3.637% 0.241% 0.061% 2.268%305 915 20.50% 0.25% 3.503% 0.228% 0.055% 2.287%305 1,030 20.76% 0.22% 3.465% 0.224% 0.053% 2.293%305 1,800 22.31% 0.13% 3.231% 0.210% 0.045% 2.231%305 3,050 24.77% 0.04% 2.869% 0.200% 0.042% 1.934%420 915 19.85% 0.25% 3.634% 0.235% 0.057% 2.373%420 1,030 20.06% 0.22% 3.595% 0.232% 0.054% 2.382%420 1,800 21.79% 0.14% 3.358% 0.216% 0.046% 2.330%420 3,050 24.25% 0.05% 2.995% 0.206% 0.043% 2.038%610 1,800 21.05% 0.16% 3.500% 0.226% 0.049% 2.402%610 3,050 23.35% 0.11% 3.135% 0.215% 0.045% 2.124%915 1,030 18.63% 0.40% 3.877% 0.258% 0.067% 2.380%915 1,800 20.14% 0.21% 3.637% 0.239% 0.055% 2.407%915 3,050 22.44% 0.17% 3.272% 0.226% 0.050% 2.155%
1030 3,050 22.15% 0.20% 3.311% 0.230% 0.052% 2.156%
680 1,390 20.00% 0.21% 3.667% 0.237% 0.055% 2.451%
Layers are rounded to 5 million.Layers below (1800, 3050) and above (680, 1390) are all the continuous combinations of quoted layers.
Figure 3. Reinsurance efficient frontier
25Semivariance using the original AIR catastrophe losses are larger
because the AIR models produce catastrophe losses with larger tails
than the lognormal distribution.
surance is $8.9 million. The ratio of reinsurance
recovery to reinsurance premium is 42.59%. The
reinsurance is costly, especially for the higher
202 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Optimal Layers for Catastrophe Reinsurance
Table 3. Optimal reinsurance layers when μ = 16:71, 22.28, 27.85
Theta Retention Upper Limit Risk-Adjusted Profit Risk-Adjusted Profit Risk-Adjusted ProfitTheta (million) (million) Mean Semivariance theta = 16:71 theta = 22:28 theta = 27:85
16.71 795 1220 3.771% 0.060% 2:768% 2.434% 2.100%22.28 680 1390 3.667% 0.055% 2.755% 2:451% 2.147%27.85 615 1460 3.610% 0.052% 2.736% 2.445% 2:154%
The optimal layers are rounded to 5 million.
layers. For the top layer ($1,800 million, $3,050
million), the reinsurance price is $39.1 million
while the mean of loss recovered by reinsurance
is $2.6 million. The ratio of recovery to premium
is 6.58%. So, the capital charge on the top layer
of reinsurance tower is very high.
Table 2 reports the probability of net under-
writing loss, the probability of severe loss (de-
fined as more than 15% of net underwriting loss),
mean of profit, variance of profit, semivariance
of profit, and risk-adjusted profits at μ =
22:28. The scattered dots (except for A, C, D, and
E) in Figure 3 represent the quoted reinsurance
layers and all possible continuous combinations
of those layers. A represents the no reinsurance
scenario and B represents the maximal reinsur-
ance scenario of stacking all quoted layers. The
slope from A to B becomes steeper and reflects
the reality of reinsurance pricing. The concave
curve in Figure 3 represents the efficient frontier.
Not unexpectedly, some of the quoted reinsur-
ance layers are not at the frontier: one can find
another layer to produce a higher return at the
same downside risk or a lower risk at the same
return. For example, layer ($305 million, $1,030
million) is not efficient with a mean profit and a
semivariance 3.465% and 0.053%, respectively.
Layer ($610 million, $1,800 million) is clearly
superior because it increases average return
(3.500%) while reducing risk (0.049%).
As shown in Table 2, the reinsured will maxi-
mize its downside-risk-adjusted profit by select-
ing the layer ($680 million, $1,390 million) as-
suming a 22.28 risk-penalty coefficient, which
implies that the management would be willing
to pay 1.567% of gross premium (40% of gross
underwriting profit) to hedge downside risk. For
a lower layer, even though the layer has a greater
chance to be penetrated, the potential risk of
catastrophe loss is tolerable by the reinsured. For
a higher layer, the reinsurance price is too high
compared to the risk mitigation it provides. Point
C in Figure 3 represents this optimization op-
tion. The straight tangent line represents the util-
ity curve at μ = 22:28. All other possible layers
are below the line and therefore have lower util-
ity values.
It is also clear from Table 2 that catastrophe
reinsurance does not increase the probability of
being profitable in the instant case. Without rein-
surance, the probability of underwriting loss is
18.41%. With reinsurance of various layers, the
probabilities of underwriting loss are over 19%.
The purpose of reinsurance is to buy protections
against large events. Without reinsurance, the
chance of severe loss is 0.48%, or roughly one
in 200 years. With a minimal reinsurance layer
($305 million, 420 million), it reduces to 0.42%,
or roughly one in 250 years. With the optimal
reinsurance layer ($680 million, $1,390 million),
the chance of severe loss reduces to 0.21%, or
roughly one in 500 years.
If the reinsured is less risk-averse, the opti-
mal layer will be narrower and the retention level
will be higher. As shown in Table 3, when μ =
16:71, or when the management would like to
pay 1.175% of gross premium (30% of total un-
derwriting profit) to hedge its downside risk, the
optimal layer is ($795 million, $1,220 million).
On the contrary, if the reinsured is more risk-
averse, the optimal layer will be wider and the
retention level will be lower. For example, when
μ = 27:85, or the management would like to pay
1.958% of gross premium (50% of total under-
writing profit) to hedge its downside risk, the
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 203
Variance Advancing the Science of Risk
optimal layer is ($615 million, $1,460 million).
Point D in Figure 3 represents the optimization
at reduced risk aversion while Point E represents
the optimization at higher risk aversion.
In practice, actuaries may not be able to choose
reinsurance layers from an unlimited pool of op-
tions. They often need to select a layer or a com-
bination of layers from a limited number of op-
tions. A simple method is to calculate the risk-
adjusted profit for the candidate layers using
Equation (9) and select a layer associated with
the highest score. Layer ($610 million, $1,030
million) is the best of the six quoted options. In
this case, actuaries do not need to fit a nonlin-
ear curve on reinsurance prices and to solve the
complicated optimization problem.
The underwriting performance may impact the
reinsurance selection from two perspectives: (1)
the more profitable the business, the more risk
the insurer can retain, and the less reinsurance the
insurer may be willing to buy; (2) the more prof-
itable the business, the more capital can be de-
ployed for reinsurance, and the more reinsurance
the insurer is able to buy. The optimal reinsur-
ance layer, assuming a 3.93% gross underwrit-
ing profit rate with μ = 22:28, is ($680 million,
$1,390 million). If the company could make 2%
more underwriting profit by lowering its non-
catastrophe loss ratio, the reinsurance optimiza-
tions could be formularized by the following pa-
rameters:26
1. The benchmark point for minimum accept-
able profit may increase to 2%. In this case, the
semivariance will not be impacted by a profitabil-
ity change; the optimal layer remains the same as
($680 million, $1,390 million).
2. The minimum acceptable profit rate re-
mains at 0%. In this case, the semivariance re-
duces with improved profitability. The semivari-
ance decreases from 0.07% to 0.05% and down-
side deviation from 2.652% to 2.240%. This is
26None of the three parameterizations (1, 2a, or 2b) can fully re-
flect risk perception and risk aversion associated with improved
profitability. The true parameters are probably somewhere among
1, 2a, and 2b.
because smaller catastrophe events no longer
produce underwriting losses, and larger events
would produce 2% less loss.
(a) If the penalty on the semivariance remains
at 22.28, the optimal layer becomes ($740 mil-
lion, $1,420 million). The insurer would max-
imize its risk-adjusted profit by retaining more
loss from relatively small events (higher reten-
tion). This is because the reinsured has more
underwriting profit to cover smaller catastrophe
events. And by increasing the retention level, it
could have additional capital to buy more protec-
tion from a higher layer ($1,390 million, $1,420
million). The limit is reduced from $710 mil-
lion to $680 million because the downside risk
is smaller.
(b) If the reinsured would like to use the same
level of profit, or 1.567% of gross premium, to
fully hedge downside risk, μ would be 31.22. In
this scenario, the optimal layer is ($630 million,
$1,555 million). The reinsured would like to buy
a wider layer with a lower retention due to in-
creased risk-aversion (willing to pay the same
amount of price to hedge a semivariance that is
28.6% smaller than before).
4. ConclusionsWhen selecting reinsurance layers for catastro-
phe loss, the reinsured weighs two dimensions
in the decision making process: profit and risk.
The reinsured would give up too much of its
underwriting profit if purchasing excessive rein-
surance. On the other hand, the reinsured would
still be under the risk of large catastrophe losses
if carrying little reinsurance. This study explores
the determination of the optimal reinsurance
layer for catastrophe loss. The reinsured is as-
sumed to be risk-averse and chooses the rein-
surance layer that maximizes the underwriting
profit net of reinsurance adjusted for downside
risk. It provides a theoretical and practical model
under classical mean-variance framework to esti-
mate the optimal reinsurance layer. Theoretically,
204 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Optimal Layers for Catastrophe Reinsurance
the paper improves previous studies by utilizing
both catastrophe and noncatastrophe loss infor-
mation simultaneously and using the lower par-
tial moment to measure risk. Practically, the op-
timal layer is determined numerically by the risk
appetite of the reinsured, the reinsurance price
quotes by layer, and the loss (frequency and
severity) distributions of the business written by
the reinsured. The proposed approach uses three
parameters to reflect the insurer’s risk percep-
tion and risk aversion. T is the minimum accept-
able profit and the benchmark point to define
“downside.” The moment that represents one’s
risk perception of larger losses is k; the higher
the k, the greater the fear of severe losses. μ
is the risk-penalty coefficient which represents
one’s risk aversion. The higher the μ, the greater
the risk aversion to downside risks. The DRAP
(downside-risk-adjusted profit) framework pro-
vides a flexible approach to capture the insurer’s
risk appetite comprehensively and precisely. All
the information required by the model should be
readily available from catastrophe modeling ven-
dors and the actuarial database of an insurance
company.
Additionally, we would like to make the fol-
lowing concluding remarks:
1. From the perspective of enterprise risk
management (ERM), catastrophe insurance is a
risk management tool to mitigate one of many
risks faced by insurance companies. Catastro-
phe reinsurance should not be arranged or eval-
uated solely upon the information of catastrophe
losses. Instead, its arrangement should be viewed
concurrently with other types of risks, such as
noncatastrophe underwriting risk and various in-
vestment risks. In this paper we consider both
catastrophe and noncatastrophe losses and the
analysis (simulation) is carried out with both sets
of variables operating concurrently. This particu-
lar path views the catastrophe reinsurance cover
as a part of the ERM process. We believe the
decision reached by viewing the transaction as a
step in the ERM process to be a superior deci-
sion. This idea may be extended to all other el-
ements of reinsurance considered and/or utilized
by an insurer. In effect, our suggestion is that
the reinsurance decision, for catastrophe reinsur-
ance and otherwise, is an important element of
the total ERM process.
2. The reinsurance purchase decision is sel-
dom, if ever, guided solely by the dry mechanics
of pricing a layer above a particular attachment
point to pay a certain percentage of the covered
layer. An aspect of the transaction that goes be-
yond the mechanical factors deals with “who”
the prospective reinsurer is. This is an important
input item, but it is always an intangible. The size
of the reinsurer, the size of its surplus, the finan-
cial rating of the reinsurer, the length and quality
of the relationship with the reinsurer, how much
of the reinsurance is retained for its own account,
and so forth form important intangibles that are
impossible to factor into any simulation. All the
same, these factors do operate and they can in-
fluence the final decision.
3. Another aspect of the reinsurance decision
is the way the ultimate decision maker may be
able to use the outputs of modeling such as those
proposed in this paper. The models and their out-
put in effect provide the ultimate decision maker
with some absolute points of reference that can
be factored into the final decision. For exam-
ple, if the model results show a clearly economi-
cally advantageous reinsurance proposition is be-
ing offered, the ultimate decision maker now has
some “elbow room” to fully capitalize on the ad-
vantage that is being offered: he may seek to ex-
pand layers of coverage, extend the terms of cov-
erage, add additional reinstatement provisions,
and so on. On the other hand, if the proposed
reinsurance is particularly disadvantageous, the
ultimate decision maker also is well-armed to
seek alternatives that are consistent with his ap-
petite for risk: change the point of attachment,
change the size of the reinsured layer, seek out-
VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 205
Variance Advancing the Science of Risk
Table 4. Regression statistics
Variable Coefficients Standard Error t Stat
x2¡ x1 1.2300 0.0995 12.37x2
2¡ x21 1:2978 ¤10¡4 6:6023 ¤10¡6 19.66
x32¡ x3
1 ¡1:3077 ¤10¡8 6:0976 ¤10¡10 ¡21:45x2 log(x2)¡ x1 log(x1) ¡0:1835 0.0135 ¡13:56log(x2)¡ log(x1) 45.4067 3.935 11.54
side bids for the same coverage, and so on. In
all cases, the knowledge that is imparted from
these simulations to the ultimate decision maker
enhances his level of comfort with what is being
offered as well as with any final decision.
4. The downside risk measure and utility func-
tion (downside-risk-adjusted profit) in this study
can be adapted to analyze whether an insurance
company should write catastrophe exposures and
the design of the catastrophe reinsurance pro-
gram would be one component of such an analy-
sis. For example, if the profit in a property line is
not high enough to cover reasonable reinsurance
costs, or even negative, it is better to not write
that property line. The risk-adjusted profit of
Table 5. Fitted vs. actual prices¤
Upper Bound ofRetention Layer Reinsurance Limit Reinsurance Price Rate-on-line Fitted Price Fitted Rate-on-line
305 420 115 20.8 18.09% 20.84 18.12%420 610 190 21.7 11.42% 21.69 11.41%610 915 305 19.8 6.50% 19.87 6.51%610 1,030 420 25.2 5.99% 25.18 6.00%
1,030 1,800 770 28.7 3.72% 28.73 3.73%1,800 3,050 1,250 39.1 3.13% 39.10 3.13%
305 610 305 42.5 13.93% 42.52 13.94%305 915 610 62.3 10.22% 62.39 10.23%305 1,030 725 67.7 9.33% 67.70 9.34%305 1,800 1,495 96.5 6.45% 96.43 6.45%305 3,050 2,745 135.6 4.94% 135.53 4.94%420 915 495 41.5 8.39% 41.55 8.39%420 1,030 610 46.9 7.68% 46.87 7.68%420 1,800 1,380 75.6 5.47% 75.60 5.48%420 3,050 2,630 114.7 4.36% 114.69 4.36%610 1,800 1,190 53.9 4.53% 53.91 4.53%610 3,050 2,440 93.0 3.81% 93.01 3.81%915 1,030 115 5.3 4.64% 5.32 4.62%915 1,800 885 34.0 3.85% 34.04 3.85%915 3,050 2,135 73.1 3.42% 73.14 3.43%
1,030 3,050 2,020 67.8 3.36% 67.83 3.36%
*The actual prices below layer (1800, 3050) are derived by combining the six layers with known prices.
the primary insurer without the property line will
be higher than that by adding the line and buy-
ing the optimal catastrophe reinsurance cover. In
reality, a property line may not be profitable and
it may not be a viable option to completely exit
or even shrink the line. Under the scenario of un-
profitable property lines with predetermined ex-
posures, the proposed method can still help the
primary insurer to find an optimal reinsurance
solution and to mitigate severe downside pains
from property lines by giving up a portion of
profit from other profitable lines.
5. Uncertainty in modeling and estimating net
underwriting profit is an important consideration.
Model and parameter risks inherent in catastro-
phe loss simulations can influence actual vs. per-
ceived costs as well as the optimal amount of
capacity companies choose to buy.
Finally, there is no question that, when all is
said and done, the ultimate decision maker has to
weigh many things, both objective and subject-
ive, on the way to finalizing the reinsurance de-
206 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
Optimal Layers for Catastrophe Reinsurance
cision. Having the results of the simulations pre-
sented in this paper serves to improve the quality
of decision making.
Appendix: Fitting the ReinsurancePrice CurveIn calculating catastrophe reinsurance rates,
the premium and rate-on-line are associated with
two values: the starting and end points of a layer.
When the layer is infinitesimal, rate-on-line can
be thought as a function of the midpoint of the
layer. In other words, rate-on-line in a continu-
ous setting is f(x), a function with a single vari-
able. Let p(x1,x2) be the reinsurance rate with
retention at x1 and upper limit of the layer at x2,
then
p(x1,x2) =
Z x2
x1f(x)dx: (A.1)
Because the reinsurance rate is expressed as
the integral, it meets the addable requirement
of reinsurance pricing: p(x1,x3) = p(x1,x2)+
p(x2,x3). From actual quotes, it is clear that f(x)
is a decreasing nonlinear function of x. To cap-
ture the nonlinearity, we assume that f(x) con-
tains a quadratic term x2, and logarithm log(x),
and inverse term 1=x in addition to a linear
term x.
f(x) = ¯0 +¯1x+¯2x2 +¯3 log(x)+¯4x
¡1:
(A.2)27
Combining A.1 and A.2, the reinsurance rate is
p(x1,x2) = ¯0(x2¡ x1)+ 12¯1(x
22¡ x21)
+ 13¯2(x
32¡ x31)
+¯3(x2 log(x2)¡ x1 log(x1))
+¯4(log(x2)¡ log(x1)): (A.3)
By Equation (A.3), one could fit a linear re-
gression with quotes as observations of the de-
pendent variable, p, and x2¡ x1, x22¡ x21, x32¡x31, x2 log(x2)¡ x1 log(x1), and log(x2)¡ log(x1)
27One could also consider adding other polynomial terms.
as corresponding observations of the indepen-
dent variables.28 There are 21 observations in Ta-
ble 5 and five possible variables in the regression.
To avoid over-fitting problem associated with re-
gression, we selected the model with the lowest
value of BIC (Bayesian information criterion).
Another way to minimize over-fitting is to obtain
more quotes to increase the number of observa-
tions. Finally, actuaries should review the shape
of fitted reinsurance curve to check its reason-
ableness. The regression results are reported in
Table 4.
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208 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2
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2010121 Bootstrap Estimation of the Predictive
Distributions of Reserves Using Paid and Incurred Claims by Huijuan Liu and Richard Verrall
136 Robustifying Reserving by Gary G. Venter and Dumaria R. Tampubolon
155 Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach by Jackie Li
170 The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium by Stephen P. D’Arcy, Richard A. Derrig, and Herbert I. Weisberg
191 Optimal Layers for Catastrophe Reinsurance by Luyang Fu and C. K. “Stan” Khury
VOLUME 04 ISSUE 02