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Measuring Profitability of Life Insurance Products under
Solvency II
Karen Tanja Rödel*‡
Stefan Graf‡
Alexander Kling‡
Andreas Reu߇
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
Measuring Profitability of Life Insurance Products under Solvency II
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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
Measuring Profitability of Life Insurance Products under Solvency II
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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
Measuring Profitability of Life Insurance Products under Solvency II
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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.
Measuring Profitability of Life Insurance Products under Solvency II
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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
Measuring Profitability of Life Insurance Products under Solvency II
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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
Measuring Profitability of Life Insurance Products under Solvency II
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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
Measuring Profitability of Life Insurance Products under Solvency II
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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
Measuring Profitability of Life Insurance Products under Solvency II
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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
Measuring Profitability of Life Insurance Products under Solvency II
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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
𝑑𝐴𝑃𝐻(𝑡)
𝐴𝑃𝐻(𝑡)= 𝑟(𝑡)𝑑𝑡 + (𝑥𝑆𝜎𝑆𝜌 − 𝑥𝐵𝜎𝑟∑ 𝑥𝑖𝑗𝐵(𝑡, 𝑖 + 𝑗)
𝑇∗
𝑗=1)𝑑𝑊1(𝑡)
+ 𝑥𝑆𝜎𝑆√1 − 𝜌2𝑑𝑊2(𝑡) 𝑢𝑛𝑑𝑒𝑟 𝒬
𝑑𝐴𝑃𝐻(𝑡)
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.
Measuring Profitability of Life Insurance Products under Solvency II
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).
Measuring Profitability of Life Insurance Products under Solvency II
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
𝐿𝑡 = 𝔼𝒬 [𝑒−∫ 𝑟(𝑠)𝑑𝑠
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.
𝐿𝑡 = 𝔼𝒬 [𝑒−∫ 𝑟(𝑠)𝑑𝑠
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.
Measuring Profitability of Life Insurance Products under Solvency II
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.
Measuring Profitability of Life Insurance Products under Solvency II
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.
Measuring Profitability of Life Insurance Products under Solvency II
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
Measuring Profitability of Life Insurance Products under Solvency II
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.
Measuring Profitability of Life Insurance Products under Solvency II
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
Measuring Profitability of Life Insurance Products under Solvency II
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.
Measuring Profitability of Life Insurance Products under Solvency II
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.
Measuring Profitability of Life Insurance Products under Solvency II
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
Measuring Profitability of Life Insurance Products under Solvency II
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.
Measuring Profitability of Life Insurance Products under Solvency II
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.
Measuring Profitability of Life Insurance Products under Solvency II
22
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.
Measuring Profitability of Life Insurance Products under Solvency II
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)
Cliquet Maturity Unit-linked
-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)
Measuring Profitability of Life Insurance Products under Solvency II
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.
Measuring Profitability of Life Insurance Products under Solvency II
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
Cliquet Maturity Unit-linked
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.
Measuring Profitability of Life Insurance Products under Solvency II
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%
Measuring Profitability of Life Insurance Products under Solvency II
27
Cliquet Maturity Unit-linked
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.
Measuring Profitability of Life Insurance Products under Solvency II
28
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
Measuring Profitability of Life Insurance Products under Solvency II
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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