Measuring Profitability of Life Insurance Products under Solvency II · 2020. 7. 28. · As life...

Measuring Profitability of Life Insurance Products under

Solvency II

Karen Tanja Rödel*‡

[email protected]

Stefan Graf‡

[email protected]

Alexander Kling‡

[email protected]

Andreas Reuß‡

[email protected]

This version: 27th

of July, 2020

In this paper, we propose a novel method for the measurement of profitability of life

insurance products. In contrast to most of the existing literature, we consider the

development of the insurance contracts over their entire lifetime under the real-world

probability measure and distinguish between different sources of capital. We study the

pathwise realization of random variables describing shareholder profitability to obtain and

analyze their distribution. These distributions are more versatile than single statistics such

as expected values since they additionally allow for the analysis of extreme outcomes.

Moreover, we specifically consider the strain on shareholders arising from the solvency

capital requirement under Solvency II. We use a cost of capital approach based on the

explicit computation of the solvency capital requirement and the interrelated capital

required from shareholders for each year of the projection period.

To demonstrate the feasibility of our profit measures, we provide a concrete application to

products with interest rate guarantees including an internal model approach for market

risks under Solvency II. Our numerical application shows that our proposed profit

measures are particularly suitable for revealing the profitability of different life insurance

products in today’s regulatory environment.

Keywords: Life Insurance, Profitability, Solvency II, Shareholder Perspective, Interest

Rate Guarantees

* Corresponding author

‡ ifa (Institute for Finance and Actuarial Sciences), Lise-Meitner-Straße 14, 89081 Ulm, Germany

mailto:[email protected]




Measuring Profitability of Life Insurance Products under Solvency II

1

1 Introduction

Understanding the profitability of insurance business is essential in order to manage the

business successfully and to satisfy the shareholders who provide the capital required to

run the business. Therefore, it is not surprising that a lot of research has been conducted on

this topic. Measuring profitability of long-term life insurance business is particularly

challenging as the insurance contracts are represented by random future cash flows

stretching out over many years and even decades. Furthermore, for life insurance contracts

with a considerable savings component, premiums are invested over a long time period.

Abkemeier and Vodrazka (2002) provide a good overview of the profit measures

frequently used in the life insurance sector. They discuss the internal rate of return (IRR),

which is easily understood but only a single number that cannot capture all aspects of the

business. Moreover, they address the return on equity (ROE) and the return on assets

(ROA), which are usually based on local GAAP quantities. The profit margin is defined by

the present value of all future profits divided by the present value of premiums, and the

embedded value is the value of future shareholder cash flows discounted at the cost of

capital.

In the last few decades, the focus shifted from traditional measurements based on local

GAAP accounting methods to an economic perspective based on market values. The aim is

to perform a market-consistent valuation and consider all types of risks including non-

market risks. Hancock et al. (2001) introduce an economic value approach which considers

the present value of expected future cash flows including the cost of risk. The authors point

out the differences between their approach and the embedded value as well as the risk-

adjusted return on capital (RAROC). The latter divides the economic profit by the risk

capital and thus assumes capital to be scarce. Goh and Wang (2013), Junus et al. (2012)

and Lebel (2009) discuss the market-consistent embedded value (MCEV) and the market-

consistent value of new business (VNB). These measures allow for all types of risks, i.e.

hedgeable market risk through risk-neutral valuation as well as non-hedgeable risks

through an additional cost. The VNB is defined as the expected present value of future

profits net of these costs. De Mey (2009) studies financial reporting of life insurers and

points out how it should be adjusted to better suit shareholders and investors. Since the

VNB relies on a forward-looking view, it fixes one of the faults discussed in De Mey

(2009). The author also mentions that ranges of possible outcomes and worst cases would

be of interest rather than just best estimates.

For an in-depth analysis of the profitability of life insurance business, various factors have

to be considered. We want to especially point out regulatory requirements that oblige


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insurance companies to hold certain amounts of risk capital, e.g. Solvency II in the

European Union. These capital requirements have a material influence on how much

shareholder capital has to be injected and on how long it has to be retained in the company.

As shareholders expect an adequate remuneration for the capital they provide, there is an

associated cost for retaining capital. Scotti (2005) summarizes the theory of cost of equity

capital and defines it as the amount of capital supporting the business multiplied by the

cost of capital rate. The author also discusses the estimation of the cost of capital and

provides some empirical results on value creation including the cost of capital.

Krvavych and Sherris (2006), Nirmalendran et al. (2012) and Braun et al. (2018) study

profit measures within a one-year horizon including capital requirements. Krvavych and

Sherris (2006) seek to maximize the shareholder value under solvency constraints on the

basis of the return on risk capital. Nirmalendran et al. (2012) consider the economic value

added which they define as the expected present value of profits to shareholders based on

market values. Moreover, the authors include an adjustment for the cost of capital and set a

target solvency level. Braun et al. (2018) compute the return on risk-adjusted capital

(RORAC) of a life insurer under Solvency II and study its implications on the asset

management. They define RORAC as the expected change in equity divided by the capital

requirement for market risk derived from the standard formula.

As life insurance business is based on long-term commitments, an analysis of the

profitability across several years appears more suitable than on a one-year horizon.

Blackburn et al. (2017) evaluate a life annuity business based on an economic valuation

approach and the MCEV. The authors’ aim is to demonstrate the effect of risk management

and in particular risk transfer on solvency and shareholder value. Their model includes

frictional costs on shareholder capital and a dividend / recapitalization strategy designed to

satisfy the solvency capital requirement under Solvency II. For the calculation of the

capital requirement, Blackburn et al. (2017) focus on longevity risk and apply the Solvency

II standard formula.

Wilson (2015) and Wilson (2016) consider the VNB and the new business margin (NBM),

which is simply the VNB divided by the present value of the premiums. The author further

presents two possibilities to improve these measures: First, Wilson (2016) criticizes that

the role of the required capital is not prominent enough for the capital-intensive nature of

life insurance business. The author solves this issue by rearranging the VNB to the return

on capital or risk-adjusted performance measure (RAPM), which sets the adjusted earnings

in relation to the allocated capital. However, Wilson (2016) points out that both VNB and

RAPM should be considered since the RAPM could be large simply owing to a small

denominator. Second, Wilson (2016) criticizes that the VNB fails to capture risk and

returns from financial market positions. The author hence includes real-world investment


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returns and the associated financial market risk to the RAPM. Essentially, the author

changes from an expected value under the risk-neutral measure to an expected value under

the real-world measure.

In this paper, we propose a novel method for the measurement of profitability of life

insurance products from the perspective of shareholders. The profit measures are based on

the measures in Wilson (2016). However, we do not just study expected values but instead

the pathwise realization of the underlying random variables to obtain and analyze their

complete probability distribution. Further, our proposed profit measures are based on the

real-world development of insurance contracts over the entire lifetime of the contracts and

reflect different sources of capital. We specifically consider the strain on shareholders

arising from the solvency capital requirement under Solvency II via an explicit calculation

of the cost of capital for the entire projection period. To demonstrate the feasibility of our

profit measures in a concrete application, we introduce a specific model framework that

includes an internal model approach for Solvency II capital requirements concerning

market risk. This allows us to compare the profitability of different types of interest rate

guarantees. We consider a product with a cliquet-style guarantee, a product with a maturity

guarantee and a unit-linked product without any guarantee. In the course of the analysis,

we assess how product characteristics such as the inclusion of guarantees or the path-

dependence of the cliquet guarantee are reflected in our profit measures.

This paper is structured as follows. We present the general company setup and the

computation of the solvency capital requirement in Section 2. In Section 3, we introduce

our profit measures. Section 4 contains the model framework including the setup of the

insurance products for our numerical results following in Section 5. The results include an

evaluation of our profit measures as well as some sensitivity analyses. Finally, we

conclude in Section 6.

2 Company setup, solvency capital requirement (SCR) and solvency

ratio

In this section, we introduce the general company setup by illustration of a simplified

economic balance sheet under Solvency II in Fig 1.


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Assets Liabilities

shareholder fund

𝐴𝑆𝐻

shareholder capital

𝐸𝑆𝐻

retained profits

𝐸𝑅

policyholder fund

𝐴𝑃𝐻

present value of future profits

𝑃𝑉𝐹𝑃

best estimate of liabilities

𝐿

Fig 1 Simplified economic balance sheet under Solvency II

Under Solvency II, the companies need to report the balance sheet positions at market

value as instructed by the directive of the European Parliament and the Council (2009). For

the assets 𝐴 = 𝐴𝑆𝐻 + 𝐴𝑃𝐻, we distinguish between a shareholder fund 𝐴𝑆𝐻 covering

statutory shareholder capital 𝐸𝑆𝐻 and retained profits 𝐸𝑅 (i.e. realized statutory profits not

paid out to shareholders, as shown in the statutory balance sheet), and a policyholder fund

𝐴𝑃𝐻. The policyholder fund covers the best estimate of liabilities (BEL) 𝐿 and the present

value of future profits (PVFP), which is equal to the expected future statutory profits

(before tax). The former is defined as expected present value of future cash flows required

to settle the insurance obligations. The difference between the market value of assets and

the best estimate of liabilities defines the company’s own funds 𝑂𝐹 (in gray), i.e. 𝑂𝐹 =

𝐴 − 𝐿. The own funds can also be explicitly calculated by adding up the statutory

shareholder capital 𝐸𝑆𝐻, the retained profits 𝐸𝑅 and the 𝑃𝑉𝐹𝑃. A risk margin as described

in article 77 of the directive of the European Parliament and the Council (2009) is not

included as we focus on market risk and seek to keep the presentation simple. (Deferred)

taxes are ignored as well. The development of the shareholder fund reflects capital

injections (cash inflows from shareholders) as well as cash outflows to shareholders

(dividends and capital withdrawals). Note that expected future profits, i.e. the 𝑃𝑉𝐹𝑃,

cannot be paid out to shareholders until profits are realized under statutory accounting

rules.

𝑂𝐹

own funds


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Article 101 of the Solvency II directive of the European Parliament and the Council (2009)

defines the SCR as the one-year Value-at-Risk of the basic own funds with a confidence

level of 99.5%. Thus, insurance companies are required to possess sufficient own funds to

survive an extremely negative year that statistically only occurs once every 200 years. In

order to determine the 99.5% Value-at-Risk, companies have to model and evaluate all risk

factors they are exposed to. While isolated evaluations and aggregations using a standard

formula are accepted by regulators, we focus on the use of an internal model analyzing all

relevant risk factors at once. As in Christiansen and Niemeyer (2014), we define the SCR

at time 𝑡 by the quantile

𝑆𝐶𝑅𝑡 = 𝑞99.5% (𝑂𝐹𝑡 − 𝑒−∫ 𝑟(𝑠)𝑑𝑠

𝑡+1𝑡 𝑂𝐹𝑡+1) ,

where 𝑂𝐹𝑡 are the own funds at time 𝑡 and 𝑟(𝑠) is the risk-free interest rate.

The solvency ratio 𝑆𝑅𝑡 at time 𝑡 is the quotient of own funds 𝑂𝐹𝑡 and solvency capital

requirement 𝑆𝐶𝑅𝑡.

3 Profit measures

In this section, we introduce measures for assessing the profitability of insurance products

from the shareholders’ perspective. The profit measures are based on shareholder cash

flows (Δ𝑡)𝑡=0,…,𝑇, where 𝑇 is the (maximum) remaining term of the considered insurance

contracts. The value of Δ𝑡 is negative in case of a cash inflow from the shareholders

(capital injection) and positive when funds are paid out to the shareholders (dividends and

capital withdrawals). Remaining capital that is not needed for the policyholders’ maturity

benefits at time 𝑇 is paid out to shareholders and thus included in Δ𝑇.

All the cash flows we consider for measuring profitability emerge from a projection under

the real-world probability measure. Our profit measures and their interpretation resemble

those of Wilson (2016). However, in contrast to Wilson (2016), we do not only study

expected values but the considered random variables and their distribution as a whole.

3.1 Excess profit and excess profit margin

First, we consider the present value of the shareholder cash flows

𝐶𝐹𝑆𝐻 =∑𝑒−∫ 𝑟(𝑠)𝑑𝑠𝑡0 Δ𝑡

𝑇

𝑡=0

.

Typically, Δ0 is negative (capital injection), whereas subsequent Δ𝑡 are positive (in case of

dividend payments or capital withdrawals) or negative (in case of additional capital


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injections). In order to conclude how profitable a business is for shareholders, it is not

sufficient to merely consider these cash flows. We need to additionally consider that the

company’s capital requirement under Solvency II is (at least partly) covered by shareholder

capital and that shareholders expect an adequate return on this capital.

Our valuation is based on a cost of capital approach as described in article 37 of the

commission delegated regulation of the European Commission (2015) in the context of the

risk margin. We adapt the given formula by modifying the discounting to its continuous

version and setting the shareholders’ capital in the company as reference value. We thus

obtain the present value of the cost of shareholder capital

𝐶𝑜𝐶𝑆𝐻 =∑𝑒−∫ 𝑟(𝑠)𝑑𝑠𝑡+10 (𝑒𝑐𝑜𝑐 − 1) 𝐸𝑡

𝑆𝐻

𝑇−1

𝑡=0

,

where 𝐸𝑡𝑆𝐻 is the shareholders’ capital shown in the balance sheet in Fig 1, and 𝑐𝑜𝑐

denotes the cost of capital rate. It can be interpreted as the return that shareholders expect

above the risk-free interest rate. Note that no cost of capital is applied to retained profits

𝐸𝑡𝑅 as this component of own funds represents realized profits from the considered

insurance contracts and is not provided by shareholders. This is a key difference to profit

measures proposed in the current scientific literature that typically do not explicitly

consider this source of capital.

Based on these definitions, we obtain the shareholders’ excess profit 𝐸𝑃 as the difference

between shareholder cash flows and cost of capital, i.e.

𝐸𝑃 = 𝐶𝐹𝑆𝐻 − 𝐶𝑜𝐶𝑆𝐻 .

Additionally, we define the excess profit margin 𝐸𝑃𝑀 to be the absolute excess profit in

relation to the (present value of the) premiums paid by policyholders, i.e.

𝐸𝑃𝑀 =𝐸𝑃

𝑃𝑉(𝑝𝑟𝑒𝑚𝑖𝑢𝑚𝑠) .

3.2 Shareholder account

In order to obtain a more dynamic view, we additionally introduce the projection of a

shareholder account (𝑆𝐴𝑡)𝑡=0,…,𝑇. We let the shareholder account begin with an initial

balance of zero. If shareholders need to make an initial capital injection, the account is

decreased accordingly, i.e. 𝑆𝐴0 = Δ0 = −𝐸0𝑆𝐻. Then, we add upcoming cash flows to the

current account balance and deduct the cost of capital year by year. Additionally, the

shareholder account earns the risk-free interest rate. For 𝑡 = 1,… , 𝑇, we then have


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𝑆𝐴𝑡 = 𝑆𝐴𝑡−1 ⋅ 𝑒∫ 𝑟(𝑠)𝑑𝑠𝑡𝑡−1 +Δ𝑡 − (𝑒

𝑐𝑜𝑐 − 1)𝐸𝑡−1𝑆𝐻 .

3.3 Return on capital

Alternatively to the profit in absolute terms, we directly consider the return on capital

𝑅𝑂𝐶. We calculate it as

𝑅𝑂𝐶 =𝐶𝐹𝑆𝐻

∑ 𝑒−∫ 𝑟(𝑠)𝑑𝑠𝑡+10 𝐸𝑡

𝑆𝐻𝑇−1𝑡=0

=𝐶𝐹𝑆𝐻

𝐶𝑜𝐶𝑆𝐻/(𝑒𝑐𝑜𝑐 − 1) .

Clearly, the shareholders’ return needs to exceed (𝑒𝑐𝑜𝑐 − 1) for it to be high enough to

compensate for the provision of shareholder capital. A return of (𝑒𝑐𝑜𝑐 − 1) is equivalent to

an excess profit of zero. Nevertheless, the return offers some additional information by

putting more weight on the amount of capital shareholders have to provide. Hence, it

illustrates how capital efficient the investment is for shareholders. For a more detailed

discussion on the importance of focusing on capital for capital-intensive business such as

life insurance, see Wilson (2016).

As all of the profit measures introduced above are random variables, we will analyze their

distributions and derive suitable key statistics in Section 5.1.

4 Model framework

In this section, we describe the model framework used for the numerical application of our

profit measures. After specifying the dynamics of the financial market, we continue with

the design of the insurance products and the corresponding model companies. Finally, we

discuss management rules underlying the projection of shareholder cash flows.

4.1 Financial market

For the financial market, we adopt the setting used in Rödel et al. (2020) which includes

three asset classes: a money market account, a portfolio of zero-coupon bonds and a stock

index. In what follows, we briefly summarize the most important dynamics and refer to

Rödel et al. (2020) for more details.

The money market account evolves according to the risk-free interest rate 𝑟(𝑡), that is

𝑑𝑀(𝑡) = 𝑟(𝑡)𝑀(𝑡)𝑑𝑡 ,

where the risk-free interest rate is stochastic and follows the Hull-White model (cf. Hull

and White (1990)). Its dynamics under the risk-neutral measure 𝒬 and the real-world

measure 𝒫 are given by


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𝑑𝑟(𝑡) = (𝜃(𝑡) − 𝑎𝑟(𝑡))𝑑𝑡 + 𝜎𝑟𝑑𝑊1(𝑡) 𝑢𝑛𝑑𝑒𝑟 𝒬,

𝑑𝑟(𝑡) = (𝜃(𝑡) + 𝜆𝑟 − 𝑎𝑟(𝑡))𝑑𝑡 + 𝜎𝑟𝑑�̃�1(𝑡) 𝑢𝑛𝑑𝑒𝑟 𝒫.

(𝑊1(𝑡))𝑡≥0 and (�̃�1(𝑡))𝑡≥0

are Wiener processes, 𝜆𝑟 is the risk premium, 𝑎 is the mean

reversion rate and 𝜎𝑟 the standard deviation of the risk-free interest rate. In order to match

the term structure of interest rates observed in the market, we set 𝜃(𝑡) following Brigo and

Mercurio (2006) as

𝜃(𝑡) =𝜕𝑓𝑀(0, 𝑡)

𝜕𝑇+ 𝑎𝑓𝑀(0, 𝑡) +

𝜎𝑟2

2𝑎(1 − 𝑒−2𝑎𝑡) .

𝑓𝑀(0, 𝑡) is the forward rate for time 𝑡 observed in the market, and 𝜕𝑓𝑀/𝜕𝑇 is its partial

derivative with respect to the second argument.

For this setting, the time 𝑡 price of a zero-coupon bond with maturity 𝑇 ≥ 𝑡 is explicitly

given in Brigo and Mercurio (2006) as

𝑃(𝑡, 𝑇) = 𝔼𝒬 [𝑒−∫ 𝑟(𝑠)𝑑𝑠𝑇𝑡 |ℱ𝑡] = 𝐴(𝑡, 𝑇)𝑒

−𝐵(𝑡,𝑇)𝑟(𝑡) ,

where

𝐵(𝑡, 𝑇) =1

𝑎(1 − 𝑒−𝑎(𝑇−𝑡)) ,

𝐴(𝑡, 𝑇) =𝑃𝑀(0, 𝑇)

𝑃𝑀(0, 𝑡)exp {𝐵(𝑡, 𝑇)𝑓𝑀(0, 𝑡) −

𝜎𝑟2

4𝑎(1 − 𝑒−2𝑎𝑡)𝐵2(𝑡, 𝑇)} ,

and 𝑃𝑀(0, 𝑡) is the observed time zero market price of a zero-coupon bond with maturity 𝑡.

The bond portfolio consists of zero-coupon bonds with different times to maturity as in

Graf et al. (2011) and Barbarin and Devolder (2005). It is self-financing and admits yearly

trading times 𝑖 = 0, … , 𝑇 − 1 at which the portfolio composition can be changed. We

assume that the market offers bonds with times to maturity 𝑗 of one up to some 𝑇∗ ∈ ℕ. 𝑥𝑖𝑗

gives the proportion of all the money invested in the bond portfolio that is invested in

bonds with maturity 𝑖 + 𝑗 during the time period [𝑖, 𝑖 + 1). Continuous rebalancing

throughout the year is applied such that the proportions are constant between 𝑖 and 𝑖 + 1.

The same split is used for all projection paths. For the dynamics of zero-coupon bonds with

maturity 𝑖 + 𝑗, the introduced dynamics of the risk-free interest rate imply

𝑑𝑃(𝑡, 𝑖 + 𝑗) = 𝑃(𝑡, 𝑖 + 𝑗) (𝑟(𝑡)𝑑𝑡 − 𝐵(𝑡, 𝑖 + 𝑗)𝜎𝑟 𝑑𝑊1(𝑡)) 𝑢𝑛𝑑𝑒𝑟 𝒬 ,

𝑑𝑃(𝑡, 𝑖 + 𝑗) = 𝑃(𝑡, 𝑖 + 𝑗)[ (𝑟(𝑡) − 𝐵(𝑡, 𝑖 + 𝑗)𝜆𝑟)𝑑𝑡 − 𝐵(𝑡, 𝑖 + 𝑗)𝜎𝑟 𝑑�̃�1(𝑡)] 𝑢𝑛𝑑𝑒𝑟 𝒫 .

Lastly, the stock index follows a modified geometric Brownian motion based on Black and

Scholes (1973) with dynamics


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𝑑𝑆(𝑡) = 𝑟(𝑡)𝑆(𝑡)𝑑𝑡 + 𝜎𝑆𝑆(𝑡) (𝜌𝑑𝑊1(𝑡) + √1 − 𝜌2𝑑𝑊2(𝑡)) 𝑢𝑛𝑑𝑒𝑟 𝒬,

𝑑𝑆(𝑡) = (𝑟(𝑡) + 𝜆𝑆)𝑆(𝑡)𝑑𝑡 + 𝜎𝑆𝑆(𝑡) (𝜌𝑑�̃�1(𝑡) + √1 − 𝜌2𝑑�̃�2(𝑡)) 𝑢𝑛𝑑𝑒𝑟 𝒫.

(𝑊2(𝑡))𝑡≥0 and (�̃�2(𝑡))𝑡≥0

are Wiener processes under the measures 𝒬 and 𝒫,

respectively. The processes 𝑊1 and 𝑊2 are independent of each other, just like �̃�1 and �̃�2.

𝜌 is the parameter by which we set the instantaneous correlation between interest rates and

stock index. Furthermore, 𝜎𝑆 is the standard deviation of the stocks, and 𝜆𝑆 is the risk

premium.

In order to describe how the companies allocate their assets to the three investment

opportunities, we recall the shareholder fund and policyholder fund introduced in Fig 1.

We assume that the shareholder fund 𝐴𝑆𝐻 is entirely invested in the money market account,

whereas the policyholder fund 𝐴𝑃𝐻 is invested in all three asset classes (with a fraction of

money market investments 𝑥𝑀, a fraction of bond portfolio 𝑥𝐵 and a stock ratio 𝑥𝑆). The

asset allocation is not dynamic (i.e. not path-dependent), and a continuous rebalancing is

applied to achieve the desired allocation. For 𝑡 ∈ [𝑖, 𝑖 + 1), we obtain the following

dynamics of the policyholder fund:

𝑑𝐴𝑃𝐻(𝑡)

𝐴𝑃𝐻(𝑡)= 𝑥𝑀

𝑑𝑀(𝑡)

𝑀(𝑡)+ 𝑥𝑆

𝑑𝑆(𝑡)

𝑆(𝑡)+ 𝑥𝐵∑ 𝑥𝑖𝑗

𝑑𝑃(𝑡, 𝑖 + 𝑗)

𝑃(𝑡, 𝑖 + 𝑗)

𝑇∗

𝑗=1


𝐴𝑃𝐻(𝑡)= 𝑟(𝑡)𝑑𝑡 + (𝑥𝑆𝜎𝑆𝜌 − 𝑥𝐵𝜎𝑟∑ 𝑥𝑖𝑗𝐵(𝑡, 𝑖 + 𝑗)

𝑇∗

𝑗=1)𝑑𝑊1(𝑡)

+ 𝑥𝑆𝜎𝑆√1 − 𝜌2𝑑𝑊2(𝑡) 𝑢𝑛𝑑𝑒𝑟 𝒬


𝐴𝑃𝐻(𝑡)= (𝑟(𝑡) + 𝑥𝑆𝜆𝑆 − 𝑥𝐵𝜆𝑟∑ 𝑥𝑖𝑗𝐵(𝑡, 𝑖 + 𝑗)

𝑇∗

𝑗=1)𝑑𝑡

+ (𝑥𝑆𝜎𝑆𝜌 − 𝑥𝐵𝜎𝑟∑ 𝑥𝑖𝑗𝐵(𝑡, 𝑖 + 𝑗)𝑇∗

𝑗=1)𝑑�̃�1(𝑡)

+ 𝑥𝑆𝜎𝑆√1 − 𝜌2𝑑�̃�2(𝑡) 𝑢𝑛𝑑𝑒𝑟 𝒫

Conveniently, the investment of shareholder fund assets in the money market account

implies that a cash flow from or to shareholders does not affect the computation of the

SCR. Thus, shareholders can raise the solvency ratio by injecting additional capital without

affecting the SCR.


10

4.2 Insurance products

The profit measures introduced in Section 3 are applied to three different types of

insurance products. There are two types of traditional products with embedded interest rate

guarantees, a cliquet-style guarantee and a maturity guarantee, as well as a unit-linked

product without any guarantee. In order to compare these products, we consider three

separate model companies each selling one of the products only and starting with the same

balance sheet and investment strategy. In Section 5.2.2, we also provide a sensitivity

calculation with a higher stock ratio for the model company selling unit-linked business.

4.2.1 Cliquet company

The cliquet guarantee follows the design of Miltersen and Persson (2003). An interest rate

guarantee 𝑔 and an additional bonus are granted on an annual basis. The issuing company

guarantees that not only the single premium 𝑃 but also already credited bonus payments

earn at least the guaranteed rate 𝑔. The payoff to policyholders is thus

𝑃𝑒∑ (𝑔+𝛿𝑐(𝜁𝑖+1−𝑔)+)𝑇−1

𝑖=0 ,

where 𝜁𝑖+1 = ln (𝐴𝑖+1𝑃𝐻

𝐴𝑖𝑃𝐻) is the logarithmic return of the policyholder fund from time 𝑖 to

𝑖 + 1. The participation rate 𝛿𝑐 determines to what extent policyholders participate in

returns above the guaranteed rate. For the market-consistent valuation of this product, we

adjust the efficient simulation scheme of Kijima and Wong (2007) as explained in Rödel et

al. (2020). A change of measure to the forward measure 𝒬𝑇 via

𝑑𝑊1𝑇(𝑡) = 𝑑𝑊1(𝑡) + 𝐵(𝑡, 𝑇)𝜎𝑟 𝑑𝑡 ,

𝑑𝑊2𝑇(𝑡) = 𝑑𝑊2(𝑡)

allows us to express the value of the liabilities as

𝐿𝑡 = 𝔼𝒬 [𝑒−∫ 𝑟(𝑠)𝑑𝑠

𝑇𝑡 𝑃𝑒∑ (𝑔+𝛿𝑐(𝜁𝑖+1−𝑔)

+)𝑇−1𝑖=0 |ℱ𝑡]

= 𝑃(𝑡, 𝑇)𝑃𝑒∑ (𝑔+𝛿𝑐(𝜁𝑖+1−𝑔)+)𝑡−1

𝑖=0 𝔼𝒬𝑇[∏𝑒𝑔+𝛿𝑐(𝜁𝑖+1−𝑔)

+

𝑇−1

𝑖=𝑡

|ℱ𝑡] .

We approximate the conditional expectation under the forward measure by Monte Carlo

simulation of (𝜁𝑖+1)𝑖=𝑡,…,𝑇−1 as shown in Rödel et al. (2020).


11

4.2.2 Maturity company

We design the maturity guarantee following Briys and De Varenne (1997) and Grosen and

Jørgensen (2002). At maturity, policyholders receive a guaranteed interest rate 𝑔 on their

single premium 𝑃 as well as a terminal bonus. The payoff at maturity equals

𝑃𝑒𝑔𝑇 + 𝛿𝑚𝑃 (𝑒∑ 𝜁𝑖+1𝑇−1𝑖=0 − 𝑒𝑔𝑇)

+

,

where 𝛿𝑚 is the participation rate. As in Rödel et al. (2020), the market value of the

liabilities at time 𝑡 is


𝑇𝑡 (𝑃𝑒𝑔𝑇 + 𝛿𝑚𝑃 (𝑒

∑ 𝜁𝑖+1𝑇−1𝑖=0 − 𝑒𝑔𝑇)

+

) |ℱ𝑡]

= 𝔼𝒬 [𝑒−∫ 𝑟(𝑠)𝑑𝑠𝑇𝑡 𝑃𝑒𝑔𝑇|ℱ𝑡] + δ𝑚𝔼

𝒬 [𝑒−∫ 𝑟(𝑠)𝑑𝑠𝑇𝑡 (𝑃𝑒∑ 𝜁𝑖+1

𝑇−1𝑖=0 − 𝑃𝑒𝑔𝑇)

+

|ℱ𝑡]

= 𝑃(𝑡, 𝑇)𝑃𝑒𝑔𝑇 + 𝛿𝑚𝐶𝑡 (𝑃𝑒∑ 𝜁𝑖+1𝑡−1𝑖=0 , 𝑃𝑒𝑔𝑇) ,

where 𝐶𝑡 is the time 𝑡 price of a European call on the asset 𝑃𝑒∑ 𝜁𝑖+1𝑡−1𝑖=0 with maturity 𝑇 and

strike price 𝑃𝑒𝑔𝑇 as given in Rödel et al. (2020).

4.2.3 Unit-linked company

The unit-linked product is essentially the cliquet-style traditional product without any

guarantee. Therefore, the payoff to policyholders is

𝑃𝑒𝛿𝑢 ∑ 𝜁𝑖+1𝑇−1𝑖=0

with a participation rate 𝛿𝑢. We obtain the market value of the liabilities at time 𝑡 by

straightforward calculation of the conditional expectation of the discounted payoff under

the risk-neutral measure 𝒬, i.e.


𝑇𝑡 𝑃𝑒𝛿𝑢∑ 𝜁𝑖+1

𝑇−1𝑖=0 |ℱ𝑡]

= 𝑃𝑒𝛿𝑢∑ 𝜁𝑖+1𝑡−1𝑖=0 𝔼𝒬 [𝑒−∫ 𝑟(𝑠)𝑑𝑠

𝑇𝑡 𝑒𝛿𝑢∑ 𝜁𝑖+1

𝑇−1𝑖=𝑡⏟

=: 𝑒𝑋

|ℱ𝑡]

= 𝑃𝑒𝛿𝑢∑ 𝜁𝑖+1𝑡−1𝑖=0 𝑒𝔼

𝒬[𝑋|ℱ𝑡]+12𝑉𝑎𝑟𝒬(𝑋|ℱ𝑡) ,

where 𝑋 is normally distributed. Its conditional mean and variance are easily computed by

standard techniques.


12

4.3 Management rules

Next, we explain the projection of the balance sheet in Fig 1 from time 𝑡 to time 𝑡 + 1

including the management rules that ultimately determine the shareholder cash flows

(Δ𝑡)𝑡=0,…,𝑇.

In order to strictly distinguish between retained profits and future profits contained in the

PVFP, we introduce a policyholder account (𝑃𝐴𝑡)𝑡=0,…,𝑇 based on statutory book values.

The policyholder account is updated annually to reflect the capital growth of the initially

deposited single premium. Its update is therefore determined retrospectively but also

reflects a prospective view based on typical statutory reserving requirements with locked-

in discount rates. The value of the policyholder account at time 𝑡 for the different

companies is

𝑃𝐴𝑡 =

{

𝑃𝑒∑ (𝑔+𝛿𝑐(𝜁𝑖+1−𝑔)

+)𝑡−1𝑖=0 , cliquet

𝑃𝑒𝑔𝑡 + 𝛿𝑚𝑃 (𝑒∑ 𝜁𝑖+1𝑡−1𝑖=0 − 𝑒𝑔𝑡)

+

𝑃𝑒𝛿𝑢∑ 𝜁𝑖+1𝑡−1𝑖=0 , unit-linked

, maturity .

As the policyholder account is assumed to correspond to the statutory reserves of the

contracts, it matches the sum of PVFP and BEL.

On the asset side, shareholder fund and policyholder fund evolve according to the

stochastic differential equations given in Section 4.1, i.e. we obtain

𝐴𝑡+1−𝑆𝐻 = 𝐴𝑡

𝑆𝐻 ⋅ 𝑒∫ 𝑟(𝑠)𝑑𝑠𝑡+1𝑡 ,

𝐴𝑡+1−𝑃𝐻 = 𝐴𝑡

𝑃𝐻 ⋅ 𝑒𝜁𝑡+1 ,

the split second before any further adjustments occur. Subsequently, the realization of

profits is reflected by a shift between shareholder fund and policyholder fund. The

policyholder fund is matched to the policyholder account, which results in an adjustment

term of 𝐴𝑡+1−𝑃𝐻 − 𝑃𝐴𝑡+1, the realized statutory profit, that is transferred from the

policyholder fund to the shareholder fund. In a last step, the shareholder fund is adjusted

by cash flows to/from shareholders. We thus obtain

𝐴𝑡+1𝑆𝐻 = 𝐴𝑡+1−

𝑆𝐻 + (𝐴𝑡+1−𝑃𝐻 − 𝑃𝐴𝑡+1) − Δ𝑡+1 ,

𝐴𝑡+1𝑃𝐻 = 𝐴𝑡+1−

𝑃𝐻 − (𝐴𝑡+1−𝑃𝐻 − 𝑃𝐴𝑡+1) = 𝑃𝐴𝑡+1 .

On the liability side, we compute the BEL at time 𝑡 + 1 depending on the current market

situation as given in the formulas of Section 4.2. The PVFP at time 𝑡 + 1 is calculated

indirectly as the difference between policyholder account and BEL, i.e. 𝑃𝑉𝐹𝑃𝑡+1 =

𝑃𝐴𝑡+1 − 𝐿𝑡+1.


13

The assets backing shareholder capital earn the risk-free interest rate. We assume that cash

flows to shareholders are financed through shareholder capital and subsequently through

retained profits, and also ensure that the shareholder capital does not become negative, i.e.

𝐸𝑡+1𝑆𝐻 = max (𝐸𝑡

𝑆𝐻 ⋅ 𝑒∫ 𝑟(𝑠)𝑑𝑠𝑡+1𝑡 − Δ𝑡+1 , 0) .

This implies that the capital received from shareholders grows at risk-free rates and is

reimbursed as fast as possible.

The assets backing retained profits also earn the risk-free interest rate. Additionally, profits

realized from time 𝑡 to time 𝑡 + 1 are added, and profits paid out to shareholders are

deducted, i.e.

𝐸𝑡+1𝑅 = 𝐸𝑡

𝑅 ⋅ 𝑒∫ 𝑟(𝑠)𝑑𝑠𝑡+1𝑡 + (𝐴𝑡+1−

𝑃𝐻 − 𝑃𝐴𝑡+1) + min (𝐸𝑡𝑆𝐻 ⋅ 𝑒∫ 𝑟(𝑠)𝑑𝑠

𝑡+1𝑡 − Δ𝑡+1 , 0 ) .

It remains to explain how the exact amount of Δ𝑡+1 is determined. The shareholder capital

before any injections 𝐸𝑡+1−𝑆𝐻 , the retained profits before injections 𝐸𝑡+1−

𝑅 and 𝑃𝑉𝐹𝑃𝑡+1 are

compared with the solvency capital requirement 𝑆𝐶𝑅𝑡+1 to check whether the target

solvency ratio 𝑇𝑆𝑅 is reached. If the own funds are not sufficient, shareholders have to

inject additional capital such that the target ratio is achieved. Conversely, if the solvency

ratio is higher than the target ratio, shareholders receive payments subject to the accounting

restriction that the statutory shareholders’ equity consisting of shareholder capital and

retained profits has to be non-negative at all times. As a consequence, the actual solvency

ratio may exceed the target solvency ratio. In summary, Δ𝑡+1 is determined by

Δ𝑡+1 =

{

min(−(𝑇𝑆𝑅 ⋅ 𝑆𝐶𝑅𝑡+1 − 𝑂𝐹𝑡+1−) , 𝐸𝑡+1−𝑆𝐻 + 𝐸𝑡+1−

𝑅 ) , 𝑖𝑓 𝑂𝐹𝑡+1−𝑆𝐶𝑅𝑡+1

< 𝑇𝑆𝑅

min(𝑂𝐹𝑡+1− − 𝑇𝑆𝑅 ⋅ 𝑆𝐶𝑅𝑡+1 , 𝐸𝑡+1−𝑆𝐻 + 𝐸𝑡+1−

𝑅 ) , 𝑖𝑓 𝑂𝐹𝑡+1−𝑆𝐶𝑅𝑡+1

≥ 𝑇𝑆𝑅

,

where 𝑂𝐹𝑡+1− = 𝐸𝑡+1−𝑆𝐻 + 𝐸𝑡+1−

𝑅 + 𝑃𝑉𝐹𝑃𝑡+1 are the own funds at time 𝑡 + 1 before any

capital injections.

As we do not consider other business, remaining funds are distributed at maturity of the

contracts. This means that the value of the insurance contracts 𝐿𝑇 is paid out to

policyholders and the remaining capital to shareholders such that Δ𝑇 = 𝐸𝑇−𝑆𝐻 + 𝐸𝑇−

𝑅 .

5 Numerical results

Tab 1 lists all relevant parameters applied in our numerical analyses.


14

Company-specific

𝐿0 surcharge 𝑇 𝑇𝑆𝑅 𝑐𝑜𝑐

100 5% 20 100% 6%

𝑥𝑀 𝑥𝑆 𝑥𝐵

5% 10% 85%

Product-specific

𝑔 𝛿𝑐 𝛿𝑚 𝛿𝑢

0.5% 16% 65% 81%

Short rate dynamics

𝑎 𝑟0 𝜎𝑟 𝜆𝑟

0.1 -0.334% 1.48% 0%

Stock dynamics

𝜎𝑆 𝜆𝑆 𝜌

16.95% 4% 20%

Tab 1 Parameter set

The contracts have a lifetime of 𝑇 = 20 years, during which both the cliquet company and

the maturity company guarantee an annual interest rate of 𝑔 = 0.5%. We assume an

additional premium surcharge of 5%, which means that policyholders are asked to pay a

premium of 105 for an initial best estimate of liabilities 𝐿0 of 100.

In order to arrive at a premium surcharge of 5%, we need to fix the participation rates to

𝛿𝑐 = 16% for the cliquet company, 𝛿𝑚 = 65% for the maturity company and 𝛿𝑢 = 81%

for the unit-linked company. Note that smoothing mechanisms (e.g. resulting from

amortized cost accounting for bonds) have not been modeled explicitly. Therefore, these

participation rates (applied to market value returns) cannot be compared to minimum legal

requirements that are applied to book value returns.

All three model companies target a solvency ratio of 𝑇𝑆𝑅 = 100%, which is in accordance

with the regulatory minimum requirement. The cost of capital rate is set to 𝑐𝑜𝑐 = 6% as

for the calculation of the risk margin in article 39 of the commission delegated regulation

of the European Commission (2015).

For the investment strategy of the policyholder fund, we assume that all companies equally

invest 𝑥𝑀 = 5% in the money market account, 𝑥𝑆 = 10% in stocks and the remaining


15

𝑥𝐵 = 85% in zero-coupon bonds. This allocation is maintained by continuous rebalancing.

The bond strategy follows a buy and hold strategy, that is bonds are initially bought with

maturity 𝑇 and are then simply held to match the lifetime of the insurance contracts.

For the initial term structure of interest rates in our numerical analyses, we use the term

structure given by EIOPA as of December 31st 2018. Furthermore, we base the choice of

the model parameters on the quarterly published calibration of the German Association of

Actuaries specified in DAV (2015) and DAV (2018).

The risk premium 𝜆𝑟 for the short rate dynamics is set to 𝜆𝑟 = 0%. For the stock

dynamics, we choose 𝜆𝑆 = 4% as in Korn and Wagner (2018).

For our analyses, we consider 6,000 real-world paths of the capital market for the next

𝑇 = 20 years. We compute 𝑆𝐶𝑅𝑡 for each of the 20 years by simulation of another 10,000

one-year paths under the real-world measure 𝒫. As the liabilities of the cliquet company

cannot be valued in closed form, further simulation paths under the pricing measure 𝒬 are

required. The market-consistent valuation before stress is performed for each of the 20

years using 5,000 risk-neutral paths while the valuation at the end of the 10,000 one-year

paths of the SCR computation is reduced to 100 paths to avoid overstraining the nested

simulations. An additional analysis showed that the Monte Carlo error is negligible in view

of the available accuracy in the computation of the SCR.

5.1 Profit measures

In this section, we evaluate and compare the model companies by applying the profit

measures introduced in Section 3. We begin with the excess profit margin 𝐸𝑃𝑀 =𝐸𝑃

𝑃 and

subsequently discuss its components shareholder cash flows 𝐶𝐹𝑆𝐻 and cost of capital

𝐶𝑜𝐶𝑆𝐻. Then, we give some advice on how the excess profit margins of different

companies may be compared. Finally, we analyze the shareholder account (𝑆𝐴𝑡)𝑡=0,…,𝑇,

which highlights the time component, and the return on capital 𝑅𝑂𝐶.

5.1.1 Excess profit and excess profit margin

Fig 2 illustrates the probability distribution of the shareholders’ excess profit margin

𝐸𝑃𝑀 =𝐶𝐹𝑆𝐻−𝐶𝑜𝐶𝑆𝐻

𝑃.


16

Fig 2 Quantile plots of the excess profit margin 𝐸𝑃𝑀

The quantile plots in Fig 2 show that the variance of this profit measure differs

significantly between the companies. The measure varies most for the cliquet company

showing an especially widespread quantile plot, followed by the maturity company and

finally the unit-linked company with the smallest variance. Moreover, the downside

dominates in the uncertainty of the excess profit for the cliquet as well as the maturity

company because shareholders are fully hit by negative events but only partially benefit

from positive events (asymmetry of the liabilities due to the guarantee). The mean excess

profit margin is the lowest for the maturity company followed by the cliquet and the unit-

linked company (see numbers in Tab 2).

These results reflect the nature of the liabilities of the three different types of life insurance

products for the shareholders: Unit-linked business performs best considering this profit

measure since it provides a low level of uncertainty for the shareholders and at the same

time proves to be more profitable in expectation. Products with guarantees show a much

higher uncertainty for the shareholders. At this point, it is unclear whether the cliquet

company should be “ranked” above or below the maturity company. While the cliquet

company offers a higher mean and median, it is also exposed to more uncertainty. We will

discuss this matter further at the end of this section.

Shareholder cash flows and cost of capital


17

In order to better understand the characteristics of the profit measure 𝐸𝑃𝑀, we now study

its first component, the shareholder cash flows 𝐶𝐹𝑆𝐻, in detail. Except for the initial cash

flow at time zero, which is fixed by the model parameters, the further development of

(Δ𝑡)𝑡=1,…,𝑇 is random and depends on how the capital market evolves on a specific path. At

time 𝑡 = 0, the 𝑃𝑉𝐹𝑃 is given by 5 since policyholders pay a premium surcharge of 5% on

their contract value of 𝐿0 = 100. The maturity company and the unit-linked company

reach solvency ratios of 𝑆𝑅0 = 231% and 𝑆𝑅0 = 549%, respectively, which are both far

above the required 100% without additional capital injections. As the cliquet company

only reaches a solvency ratio of 65% before capital injections, shareholders need to

provide −Δ0 = 2.71 right away to reach the target of 𝑆𝑅0 = 100%. The development of

the shareholder cash flows (Δ𝑡)𝑡=0,…,𝑇 is illustrated in Fig 3.


18

Fig 3 Quantile plots of the development of the shareholder cash flows (Δ𝑡)𝑡=0,…,𝑇

Considering the medians at each point in time, shareholders are typically required to inject

capital in the first few years of the contracts and then tend to receive positive cash flows

over time. The payout to shareholders is particularly large at the end of the projection

horizon 𝑇 = 20.

The maturity company and the cliquet company show a fairly similar pattern and

uncertainty in shareholder cash flows, in particular during the first 10 years. For both

companies, the uncertainty is quite large in the first years and then decreases towards

maturity. The decline is more pronounced for the maturity company than for the cliquet

company. This illustrates that the maturity guarantee is more predictable towards the end

of the contract compared to the cliquet guarantee as the former is not path-dependent. For

the unit-linked company, the uncertainty is much lower over the entire projection horizon

because it does not include a guarantee.

Next, we study the second component of the excess profit measure, the cost of capital

𝐶𝑜𝐶𝑆𝐻. It captures that different amounts of shareholder capital have to be injected and

kept over different time periods depending on the specific company. Having a closer look

at the development of the shareholder capital in the companies in Fig 4, we find that the

amount increases at the beginning. It then decreases after some years as retained profits

have been accumulated and capital requirements decrease. As for the cash flows shown in

Fig 3, the uncertainty is by far the smallest for the unit-linked company, where the 95%-

quantile reaches values up to about 5. In contrast, the 95%-quantiles of the companies with

guarantees reach levels of 30 to 35.


19

Fig 4 Quantile plots of the development of shareholder capital (𝐸𝑡𝑆𝐻)𝑡=0,…,𝑇 in the companies

Fig 5 shows a comparison of the shareholder cash flows and the cost of capital on a present

value basis as defined in Section 3.1. We note that the acceptance of more uncertainty is

rewarded with a higher expected present value of the cash flows. Regarding the cost of

capital, the unit-linked company clearly requires the least capital from its shareholders.

Fig 5 Quantile plots of the present value of the shareholder cash flows 𝐶𝐹𝑆𝐻 and the cost of capital

𝐶𝑜𝐶𝑆𝐻


20

Profitability comparison

It is not always obvious how the companies should be ranked according to their excess

profit margins. We have seen that higher uncertainty may be linked to higher expected

values. As an additional view, we therefore list shortfall probabilities and the ratio of mean

to standard deviation of the excess profit margin in Tab 2.

Cliquet Maturity Unit-linked

ℙ(𝐸𝑃𝑀 < 0) 31.3% 27.1% 2.8%

𝔼[𝐸𝑃𝑀] 3.3% 2.6% 5.2%

√𝑉𝑎𝑟(𝐸𝑃𝑀) 16.7% 8.3% 2.7%

𝔼[𝐸𝑃𝑀]

√𝑉𝑎𝑟(𝐸𝑃𝑀) 0.20 0.31 1.91

Tab 2 Key figures of the excess profit margin 𝐸𝑃𝑀

Shortfall is the case in which the excess profit falls short of zero, i.e. the shareholder cash

flows are not sufficient to cover the cost of capital. It is equivalent to a final shareholder

account balance below zero. The shortfall probability is slightly lower for the maturity

company compared to the cliquet company.

We consider mean over standard deviation to interrelate the two quantities in a simple way.

This key indicator rewards high means and punishes high standard deviations. It resembles

the well-known Sharpe ratio of Sharpe (1966) and the related information ratio. Instead of

the excess return over a risk-free return or a risky index as benchmark, we consider the

excess return over the cost of capital. Of course, there are other ways to interrelate

opportunity and risk depending on the shareholders’ specific attitude to risk. When

considering mean over standard deviation, the maturity company performs better than the

cliquet company. Apparently, the higher expected values of the cliquet company cannot

justify the higher risk. The unit-linked company has by far the lowest shortfall probability

and highest mean over standard deviation of all three companies.

5.1.2 Shareholder account

Fig 6 shows the quantile plots of the development of the shareholder account, which allow

us to analyze the aggregated cash flows and resulting costs of capital. For each path, we

can determine if and when the overall outcome becomes positive for shareholders.


21

Fig 6 Quantile plots of the development of the shareholder account (𝑆𝐴𝑡)𝑡=0,…,𝑇

We notice that the quantile plot of the unit-linked company is the fastest to grow towards

positive values followed by maturity and cliquet. In terms of a payback period for the

medians, the account of the unit-linked company becomes positive after five years,

whereas it takes 14 years for the maturity company and 18 years for the cliquet company.

Therefore, the shareholders of the cliquet company tend to have to wait the longest until

their initial investments pay off, if ever. The width of the quantile plots shows the

uncertainty in the value of the shareholder account over the years. As observed for

previous quantities, the uncertainty is smallest for the unit-linked company followed by the

maturity company and the cliquet company.


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5.1.3 Return on capital

The return on the capital provided by shareholders is shown in Fig 7. The advantage of this

measure is that it is independent of the choice of the cost of capital rate 𝑐𝑜𝑐.

Fig 7 Quantile plots of the return on shareholder capital 𝑅𝑂𝐶

The dashed line in Fig 7 shows the minimum target for a cost of capital rate of 6%. Note

that the location of the dashed line reveals the same shortfall probabilities we have already

seen for the excess profit margin in Tab 2 and the shareholder account in Fig 6. With

regard to all quantiles and the mean, the unit-linked company performs the best ahead of

maturity and finally cliquet. We can observe mean returns of 46% for unit-linked, 20% for

maturity and 16% for cliquet. Although the cliquet company admits the least uncertainty in

the value of the return, this is not of advantage. In this case, uncertainty mainly represents

beneficial upside potential, which the cliquet company lacks compared to the other two

companies due to its high capital requirements.

5.2 Sensitivity analysis

We performed a number of sensitivity analyses and show some selected results in this

section. We start with some sensitivity on the term structure of interest rates and then

proceed to changing the stock ratio and the target solvency ratio. Finally, we close with a

summary of the expected excess profit margin and risk for all our sensitivities.


23

5.2.1 Term structure of interest rates

We start our sensitivity analyses with a change in the term structure of interest rates by

assuming that the interest rate level changes immediately after the contract has been sold.

This implies that the pricing of the contract is not affected by the changed interest level,

i.e. guaranteed rates as well as participation rates remain unchanged compared to the base

case.

In Fig 8, we show quantile plots and, in Tab 3, we show the key figures of the excess profit

margin for a shift of spot rates by +/- 50 basis points (bp).

Fig 8 Quantile plots of the excess profit margin for a decrease of spot rates by 50 bp (-50bp), the base

case (BC) and an increase of spot rates by 50 bp (+50bp)


-50bp BC +50bp -50bp BC +50bp -50bp BC +50bp

ℙ(𝐸𝑃𝑀 < 0) 74.4% 31.3% 11.6% 55.9% 27.1% 14.2% 8.8% 2.8% 0.8%

𝔼[𝐸𝑃𝑀] -12.8% 3.3% 13.3% -3.8% 2.6% 6.3% 3.3% 5.2% 7.2%

√𝑉𝑎𝑟(𝐸𝑃𝑀) 22.2% 16.7% 13.6% 11.6% 8.3% 7.1% 2.9% 2.7% 2.7%

𝔼[𝐸𝑃𝑀]

√𝑉𝑎𝑟(𝐸𝑃𝑀) -0.57 0.20 0.98 -0.33 0.31 0.89 1.12 1.91 2.69

Tab 3 Key figures of the excess profit margin for a decrease of spot rates by 50 bp (-50bp), the base case

(BC) and an increase of spot rates by 50 bp (+50bp)


24

It is obvious that a change in interest rates has a significant impact on the profitability for

all three model companies. The higher the interest rate, the lower is the probability of a

negative excess profit and the higher is the expected excess profit of the company. The

effect is especially pronounced for the cliquet company where e.g. the probability of a

negative excess profit is 74% if interest rates decrease by 50 bp and roughly 12% if interest

rates increase by 50 bp. The expected excess profit margin of the company can become

significantly negative for decreasing interest rates or can increase from 3.3% to 13.3% and

even surpass the maturity and the unit-linked company for increasing interest rates. The

main reason for this rather high sensitivity of the results of the cliquet company with

respect to interest rates is the fact that both risk and upside potential are mainly taken by

the shareholders.

For all three companies, a higher interest rate level also means a reduced uncertainty in the

company’s profit as the guarantees become less relevant. This effect is however less

distinct than the effect on the expected value.

We also performed sensitivity analyses where we changed the level of interest rates before

the pricing of the contracts. Under this assumption, a change in interest rates hardly affects

the results at all. This shows that insurance companies can practically avoid negative

expected profits by an adjustment of the pricing of new business. However, if interest rates

change after the products have been sold, a proper management of interest rate risk is

important, in particular for companies with a cliquet-style guarantee.

5.2.2 Stock ratio

In this section, we analyze the influence of the company’s stock ratio on its profitability. In

Fig 9, we show quantile plots and, in Tab 4, we show the key figures of the excess profit

margin for an increase in the stock ratio from 𝑥𝑆 = 10% in the base case to 𝑥𝑆 = 20% for

all companies. As an additional sensitivity, we increase the stock ratio of the unit-linked

company to 𝑥𝑆 = 100%. Stock ratios up to 100% are realistic in the market for companies

offering unit-linked contracts, unlike for companies offering cliquet or maturity guarantees.


25

Fig 9 Quantile plots of the excess profit margin for an increase in the stock ratio from 𝑥𝑆 = 10% in the

base case (BC) to 𝑥𝑆 = 20% for all companies and 𝑥𝑆 = 100% for the unit-linked company


BC 𝑥𝑆 = 20% BC 𝑥𝑆 = 20% BC 𝑥𝑆 = 20% 𝑥𝑆 = 100%

ℙ(𝐸𝑃𝑀 < 0) 31.3% 25.9% 27.1% 20.1% 2.8% 4.7% 20.1%

𝔼[𝐸𝑃𝑀] 3.3% 9.3% 2.6% 6.7% 5.2% 7.2% 14.4%

√𝑉𝑎𝑟(𝐸𝑃𝑀) 16.7% 25.8% 8.3% 14.0% 2.7% 4.5% 19.6%

𝔼[𝐸𝑃𝑀]

√𝑉𝑎𝑟(𝐸𝑃𝑀) 0.20 0.36 0.31 0.48 1.91 1.60 0.73

Tab 4 Key figures of the excess profit margin for an increase in the stock ratio from 𝑥𝑆 = 10% in the

base case (BC) to 𝑥𝑆 = 20% for all companies and 𝑥𝑆 = 100% for the unit-linked company

An increase in the company’s stock ratio has a significant impact on the company’s

profitability. As expected, the volatility of the company’s excess profit increases with an

increasing stock ratio, in our example by more than 50% for all three model companies. If

the unit-linked company offers a stock ratio of 100%, the company’s profitability will

reach a similar (even slightly higher) level of uncertainty as the profitability of the other

two companies in the base case.


26

An increase in the stock ratio also results in a higher expected excess profit for all three

companies. This effect may appear less obvious than the increase in volatility but can be

explained by the real-world approach. A higher stock ratio results in a higher expected

return of the company’s asset portfolio. This leads to higher expected benefits for the

policyholders as well as higher expected profits for the shareholders. Interestingly, the

effect on the expected excess profit of the two companies selling guarantees is even higher

than the effect on the volatility in this case. This results in an increase of the quotient

𝔼[𝐸𝑃𝑀] ⁄ √𝑉𝑎𝑟(𝐸𝑃𝑀) for the cliquet company and the maturity company. For the unit-

linked company, however, this quotient is decreasing for an increasing stock ratio. In the

case of a stock ratio of 100%, it reaches a level that is below 1 and is much closer to the

values observed for the other two companies.

5.2.3 Target solvency ratio

In this section, we analyze the influence of the target solvency ratio on the company’s

profit. In Fig 10, we show quantile plots and, in Tab 5, we show the key figures of the

excess profit margin for an increase in the target solvency ratio from 𝑇𝑆𝑅 = 100% in the

base case to 𝑇𝑆𝑅 = 200%.

Fig 10 Quantile plots of the excess profit margin for an increase in the target solvency ratio from

𝑇𝑆𝑅 = 100% in the base case (BC) to 𝑇𝑆𝑅 = 200%


27


BC

𝑇𝑆𝑅

200% BC

𝑇𝑆𝑅

200% BC

𝑇𝑆𝑅

200%

ℙ(𝐸𝑃𝑀 < 0) 31.3% 39.9% 27.1% 28.4% 2.8% 3.2%

𝔼[𝐸𝑃𝑀] 3.3% -0.3% 2.6% 2.2% 5.2% 5.2%

√𝑉𝑎𝑟(𝐸𝑃𝑀) 16.7% 19.6% 8.3% 8.9% 2.7% 2.9%

𝔼[𝐸𝑃𝑀]

√𝑉𝑎𝑟(𝐸𝑃𝑀) 0.20 -0.01 0.31 0.25 1.91 1.82

Tab 5 Key figures of the excess profit margin for an increase in the target solvency ratio from

𝑇𝑆𝑅 = 100% in the base case (BC) to 𝑇𝑆𝑅 = 200%

For the unit-linked company and the maturity company, an increase in the target solvency

ratio only has a small impact on the results. The reason for this is that both companies

already reach solvency ratios above 200% in most scenarios without any additional

shareholder capital.

For the cliquet company, we observe a much higher impact of the target solvency ratio. A

target solvency ratio of 200% even leads to a negative expected excess profit. The reason

for this is a much higher cost of capital because more shareholder capital is needed than in

the base case. The present value of shareholder cash flows is not affected by an increase in

the target solvency ratio since shareholder capital in the company accumulates at the risk-

free rate. As a consequence, the probability of a negative excess profit also increases. Note

that, from a competitive point of view, it might be reasonable for the insurer to target a

solvency ratio above the minimum requirement for reasons of perceived trust and stability.

This proves to be most costly for the cliquet company.

The effect of the target solvency ratio on the volatility of the excess profit is of minor

relevance. Since negative paths are affected stronger than positive paths, the volatility

slightly increases.

5.2.4 Overview

To summarize our sensitivity analyses, we provide an overview of all sensitivities shown

by displaying the corresponding expectation and variation of the excess profit margin in

Fig 11.


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Fig 11 Expectation and variation of the excess profit margin for all sensitivities shown

We can see from this overview that some of the sensitivities have a similar impact on all

three considered companies (e.g. interest rates (IR) up and down) while other sensitivities

mainly affect certain companies (e.g. target solvency ratio). In addition, this overview

shows that no company clearly outperforms the others. Even if the unit-linked company

“dominates” the other companies in the base case, there are possible parameter

combinations where the unit-linked company faces a higher risk for the shareholders than

the other companies.

6 Conclusion

In this paper, we introduced new profit measures to analyze the profitability of life

insurance products from the perspective of shareholders. In contrast to existing literature,

we consider the real-world development of the insurance contracts over their entire lifetime

and distinguish between different sources of capital. Moreover, we include the impact of

Solvency II capital requirements on the capital provided by shareholders in a cost of capital

approach. Our profit measures are based on the explicit computation of the SCR for every

year of the projection period in an internal model approach. We thus arrive at full

distributions of random variables describing shareholder profitability. These distributions

are more versatile than single statistics such as expected values as they additionally allow

for the analysis of extreme outcomes. To demonstrate the feasibility of our theoretical


29

proposition, we provide a concrete application of our profit measures to products with

interest rate guarantees and discuss ways to compare and interpret the results.

In our numerical application, we compare a cliquet company (offering traditional products

with a cliquet-style interest rate guarantee) with a maturity company (offering traditional

products with a maturity guarantee) and a unit-linked company (offering products without

any guarantee). We confirm that life insurance products including interest rate guarantees

are capital-intensive under Solvency II. This is reflected in lower expected excess profits,

lower expected returns on capital and higher shortfall probabilities compared to a unit-

linked product. The commitment to guarantees also leads to a material uncertainty in the

excess profit. This observation is more pronounced for the cliquet company than for the

maturity company due to the path-dependence of the benefits. Moreover, this high

uncertainty results in a high sensitivity to a change in the interest rates, in the stock ratio

and in the target solvency ratio. Our numerical application shows that our proposed profit

measures are particularly suitable for revealing the differences in the profitability of

various types of life insurance products. In particular, our profit measures are more suitable

than traditional measures which cannot as adequately account for the specific risk of the

products in today’s regulatory requirements.

For further research, we may study the application of our proposed profit measures in an

extended model framework including other risk factors than equity and interest rate risk.

Moreover, we may analyze other and more complex insurance products or the more

complete view of an insurance entity writing different types of the considered products

simultaneously.

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