A Performance-Perspective Under Solvency II

Maximizing the Return on Risk-Adjusted Capital:
A Performance-Perspective Under Solvency II
Alexander Braun, Hato Schmeiser, and Florian Schreiber∗
Extended Proposal (March, 2015): Please do not cite nor redistribute.
Abstract
We draw on historical time series data for the last 25 years and construct a large number of
asset portfolios of a stylized European insurance company that calculates its Solvency II market risk
capital requirements by means of the standard formula. Then, given a fixed underwriting portfolio,
the insurer’s one-year profit can be determined. Finally, for each asset allocation, we derive the return
on risk-adjusted capital (RoRAC), i.e., the expected profit over the Solvency II capital charge. Our
preliminary results indicate that portfolios with a low level of market risk and therefore, low capital
requirements, lead to the highest RoRAC values. More diversified portfolios that also provide a higher
return, however, result in relatively low RoRAC figures. Hence, it can be concluded that with a higher
level of market risk, the capital charges increase disproportionately compared to the realized profit.
Particularly in the initial phase after Solvency II’s introduction, this could cause additional risks for
insurance companies that base their business management on risk-adjusted performance measures
such as the RoRAC. Moreover, systemic risk of the whole insurance sector is likely to increase, too.
Keywords: Asset Management, Solvency II, Performance Measurement, RoRAC
1
Introduction
At the beginning of January 2016, the regulatory landscape for European insurance companies will change
substantially. After more than ten years development time and a total of five quantitative impact studies (QIS), the European Union (EU) is expected to start the implementation of its new risk-based capital
standards, termed Solvency II. In contrast to its predecessor Solvency I, the new framework applies to
all EU member states and aims at providing a more comprehensive assessment of the various risks that
are associated with the insurance business. In accordance with the regulatory regime for the banking
sector, Basel II, Solvency II has been designed as a three pillar approach as well (see, e.g., EC, 2014).
The first pillar contains quantitative requirements that prescribe how the insurer has to evaluate its assets and liabilities, while the second pillar focuses on more qualitative requirements such as the insurer’s
governance and risk management system. Moreover, the latter further outlines the Supervisory Review
Process (SPR). Finally, the requirements with respect to transparency and disclosure are included in
∗ Alexander
Braun ([email protected]), Hato Schmeiser ([email protected]), and Florian Schreiber
([email protected]) are from the Institute of Insurance Economics, University of St. Gallen, Tannenstrasse 19,
CH-9010 St. Gallen.
the third pillar. However, due to its complexity, particularly smaller and medium-sized insurers are
faced with high obstacles regarding the proper implementation of Solvency II. In this regard, the first
pillar imposes the major challenge, since the determination of the technical provisions as well as the
so-called solvency capital requirement (SCR), the key magnitude of Solvency II, demand a critical degree
of risk management know-how and capacity. In order to facilitate this calculation process, the regulator offers a standard formula that is divided into several modules covering various risk categories. For
European life insurance companies, market risk accounts for almost 70 percent of the overall SCR (see,
e.g., Fitch Ratings, 2011). Besides a classic internal model that has been directly tailored to the specific
situation of the insurance company, however, the market risk module of the standard formula relies on a
simple stress factor approach. For more volatile asset classes such as stocks, these stress factors result in
high capital charges, which, in turn, may cause the portfolio to be inadmissible. Consequently, it must be
expected that Solvency II restricts the insurance industry with respect to the chosen investment decision
and therefore, entails severe consequences for the pricing and demand of several asset classes as well as
the world’s capital markets in general (see Fitch Ratings, 2011).
Although the general framework of Solvency II and in particular the standard formula have been intensively analyzed and discussed both among academics (see, e.g., Linder and Ronkainen, 2004; Eling et al.,
2007; Doff, 2008; Steffen, 2008) and practitioners (see, e.g., Fitch Ratings, 2011; Ernst & Young, 2011,
2012b,a), only a few articles focus on the possible asset management constraints that insurers could face
under the new regulatory regime. Fischer and Schluetter (2014) provide an in-depth analysis of the equity risk submodule and and its impact on the investment strategy of an shareholder-value maximizing
insurer within an option-pricing framework. Their results show that the standard formula exerts a strong
influence on both the capital and investment strategy. Similarly, the work of Braun et al. (2015a) highlights that the market risk standard formula suffers from several severe shortcomings, which may create
incentives to invest in less-diversified portfolios associated with an increased default risk from a proper
asset-liability perspective. These thoughts have been extended and further developed by Braun et al.
(2015b), who draw on a partial internal model in order to provide estimates for the actual ruin probabilities of insurance companies under Solvency II. The alarming results indicate that the admissible
portfolios under the standard formula exhibit tremendously high ruin probabilities. Hence, insurers using the standard formula face both asset management restrictions and higher ruin probabilities.
In this paper, however, we take the regulation as laid out by the European Insurance and Occupational
Pensions Authority (EIOPA) as given and analyze the impact of an insurer’s asset allocation on a riskadjusted performance measure. That is, we resort to historical time series data for the three major asset
classes stocks, government bonds, as well as real estate over the last 20 years in order to construct a large
number of asset portfolios.1 Then, the capital requirements under the market risk standard formula as
well as the insurer’s profits associated with these portfolios are calculated. In this regard, we draw on the
assumption that the insurer’s underwriting portfolio is given and cannot be changed within the one-year
time period. By taking the ratio between the expected profit and the solvency capital requirement for
1 In
later versions of the paper, further asset classes such as corporate bonds, alternatives etc. will be included.
2
market risk, the return on risk-adjusted capital (RoRAC) is obtained. Now, for an insurance company
aiming to maximize its RoRAC, the optimality of each asset portfolio under the Solvency II standard
formula can be assessed. Our preliminary results show that the lower-risk portfolios exhibiting lower
capital charges lead to the highest RoRAC figures. With an increasing level of market risk, however, the
Solvency II capital requirements increase disproportionately compared to the expected return. As a consequence, a drop in the RoRAC figures can be observed. Since many insurers rely on these risk-adjusted
performance measure for their business strategy, substantial shifts in the asset portfolios can be expected.
The rest of the paper is organized as follows. In Section 2 both the design of Solvency II’s market risk
standard formula as well as its calibration as undertaken by the Committee of European Insurance and
Occupational Pensions Supervisors (CEIOPS) and EIOPA are presented. The underlying assumptions
for the stylized insurance company are presented in Section 3. Moreover, the definition of the RoRAC
is included in the section, too. A short explanation of the underlying historical data can be found in
Section 4. Finally, the preliminary results are outlined in Section 5.
2
Solvency II Market Risk Standard Formula
Design of the Standard Formula
The market risk standard formula has been calibrated on the basis of historical data and relies on the value
at risk (VaR) with a confidence level of 99.5 percent and a one-year time horizon (see, e.g., EIOPA, 2014a).
In its documentation, the regulator defines the difference between the insurer’s assets and liabilities as
Basic Own Funds (BOF ). The overall market risk capital charge (SCRMkt ) is composed as total of the
individual capital charges by the six sub-modules, that are contained in the market risk module, and
the correlation between them. Within each sub-module, on the other hand, the determination of the
capital requirements is based on a specific scenario that has an impact on the level of the BOF (see
EIOPA, 2014a). Thereby, each scenario is used to measure the influence of exogenous shocks from the
capital markets, as reflected by the stress factors, on the BOF . This resulting change is denoted as
∆BOF . In order to keep the analysis as simple as possible, we only consider interest rate risk, equity
risk, as well as property risk. The former category spans all assets such as fixed-income instruments and
liabilities that react sensitively to interest rate changes (see, e.g., EIOPA, 2014a). Since both upward as
well as downward changes in the term structure influence the BOF , the interest rate risk sub-module
is divided into two pre-defined scenarios. Thus, the capital requirement for interest rate risk, Mktint , is
calculated as (see EIOPA, 2014a):
p
MktU
int
=
∆BOF |up ,
(1)
MktDown
int
=
∆BOF |down .
(2)
3
∆BOF |up and ∆BOF |down indicate the change in the insurer’s basic own funds that is caused by an
upward and downward movement in the current term structure. The change of the latter is given by applying the interest rate stresses to the basic risk-free rate for maturity t (rt ) as follows (see EIOPA, 2014a):
∆rtup
∆rtdown
=
=
rt · (1 + sup
t ) − rt
rt · (1 +
sdown
)
t
∀t, in the upward scenario,
− rt
∀t, in the downward scenario,
(3)
(4)
down
with sup
being the interest rate shocks for the two scenarios. Finally, the interest rate sensit and st
tivity of the insurer’s assets and liabilities is measured by the duration (see Braun et al., 2015a) and the
shocks are translated into ∆BOF -values.
The second sub-module covers equity risk that arises from changes in equity prices (see EIOPA, 2014a).
In order to take into account specific characteristics of different equity investments, the capital charges
for equity risk, Mkteq , is split in two categories. Equities that are listed in a member state of the European Economic Area (EEA) or the Organisation for Economic Cooperation and Development (OECD)
are called Type 1 equities, while all remaining equities are assigned to the Type 2 equities class (see
EIOPA, 2014a). These are hedge funds, non-listed equities, commodities, as well as other alternative
investments. The overall Mkteq is then calculated by aggregating the individual capital charges of the
two categories by taking into account the given correlation matrix. The former are calculated as follows
(see EIOPA, 2014a):
Mkteq,i = max (∆BOF | equity shocki ; 0) ,
(5)
where equityshocki denotes the stress factor for equity category i. With CorrIndexeq being the correlation
coefficient between Type 1 and Type 2 equities and i, j denoting the two categories, the aggregation
formula is given by (see EIOPA, 2014a):
Mkteq =
sX
CorrIndexij · Mkteq,i · Mkteq,j .
(6)
ij
The assets, liabilities, and financial investments that react sensitively to changes in real estate prices
are further contained in the property risk sub-module (see EIOPA, 2014a). The capital requirements for
this specific risk class, Mktprop , can be calculated as:
Mktprop = max (∆BOF | property shock; 0) .
4
(7)
Finally, the overall capital charges for market risk (SCRMkt ) are derived by aggregating the results
from the individual sub-modules by using the correlation matrix as given by the regulator (see, e.g.,
EIOPA, 2014a):
SCRMkt = max

 sX X

i
up
up
CorrMktup
ij · Mkti · Mktj ;
j
sX X
i
· Mktdown
· Mktdown
CorrMktdown
j
i
ij


,

j
with i, j ∈ {int; eq; prop} (see Braun et al., 2015a). As mentioned before, the upward and downward
scenarios contained in the interest rate risk sub-module are labeled by the superscripts. Moreover, for
each scenario, a different correlation matrices, are needed. The latter are denoted as CorrMktup and
CorrMktdown , respectively.
Calibration of the Standard Formula
Similar to the articles of Braun et al. (2015a,b), we draw on the former Solvency II directives by CEIOPS
(see CEIOPS, 2010a, CEIOPS, 2010b, CEIOPS, 2010c, and CEIOPS, 2010d) and the recent technical
specification documents by EIOPA (see EIOPA, 2014a and EIOPA, 2014b). Additionally, we further
resort to the latest errata document published by EIOPA (see EIOPA, 2014c). The stress factors for the
interest rate risk sub-module have been derived based on EUR and GBP government zero bond yields
as well as EUR and GBP LIBOR rates (see CEIOPS, 2010b). However, we assume a flat term structure
and that the European insurance company under consideration invests in EUR-denominated assets only
(see Braun et al., 2015a). Therefore, foreign exchange (FX) risks can be neglected. The basic risk-free
rate, that is subjected to the interest rate stress, is proxied by the mean of the AAA-rated Eurozone zero
bond spot yield curve for maturities ranging from 1 to 30 years. On December 31, 2012, a risk-free rate of
0.92 percent results. Both the upward as well as downward stresses are obtained by averaging the given
stress factors for all maturities. By doing so, the shock in the upward scenario amounts to +43 percent,
while the downward shock equals –37 percent. However, according to the technical specifications, the
absolute increase in the upward scenario should be at least one percentage point (see EIOPA, 2014a).2
Since the asset and liability durations (DA and DL ) act as a link between the sensitivity of the insurer’s
bond portfolio on the one hand, and its insurance liabilities on the other hand, we calculate ∆BOF as
follows (see Braun et al., 2015a).
∆BOF |up
∆BOF |down
=
=
(−A0 · DA · ∆rup ) − (−L0 · DL · ∆rup ),
(−A0 · DA · ∆r
down
) − (−L0 · DL · ∆r
down
(8)
).
(9)
Since the equity risk sub-module has been split into two classes, the regulator ran several calibration scenarios (see CEIOPS, 2010a). For the “Type 1 equities”, historical returns of the MSCI World
Developed Equity Index have been analyzed and a stress factor of 39 percent (see EIOPA, 2014a) has
2 Please
note that in former technical documentations (see, e.g., EIOPA, 2012b; EIOPA, 2012a), both the upward and
downward scenario had to be manually adjusted to achieve an absolute change of at least one percentage point.
5
been determined. The “Type 2 equities” stress factor, on the other hand, has been derived by analyzing a benchmark index for each asset class contained in this category. More in detail, they drew on
the LPX50 Total Return Index for private equity, the S&P GSCI Total Return Index for commodities,
the HFRX Global Hedge Fund Index for hedge funds, as well as the MSCI Emerging Markets BRIC
Index that represents the emerging markets (see CEIOPS, 2010a). Since the obtained stresses of these
asset classes on a single basis may underestimate or overestimate the appropriate stress factor applicable
to the overall “Type 2 equities”, the regulator proposed a single stress factor amounting to 49 percent
(see EIOPA, 2014a).3
Finally, the Investment Property Databank (IPD) index for the United Kingdom has been employed
to calibrate the stress factor for the property risk sub-module (see CEIOPS, 2010c). CEIOPS chose the
IPD index since all main property sectors such as retail, office, industrial and residential are covered.
The analysis revealed that the variations of these market segments result in almost identical historical
VaR figures (99.5 percent confident level) for the period from 1987 to the end of 2008. Therefore, a single
property stress of 25 percent has been chosen (see CEIOPS, 2010c and EIOPA, 2014a). An overview is
presented in Table 1.
Submodule
Shock %
Interest Rate Risk
Type 1 Equities
Type 2 Equities
Property Risk
–37.00/+43.00
–39.00
–49.00
–25.00
Table 1: Input Data for the Solvency II Market Risk Standard Formula
This table shows the stress factors as suggested by the regulator (see EIOPA, 2014a for the Solvency II market risk
standard formula. Detailed information how the stresses have been derived can be found in the CEIOPS directives
(see CEIOPS, 2010a, CEIOPS, 2010b, and CEIOPS, 2010c).
3
Basic Model Assumptions
Stylized Insurance Company
To be in line with the standard formula, we analyze the situation of an exemplary insurance companies
at two points in time. The design of our model is close to the one presented by Braun et al. (2011) who
combine a so-called traffic light approach with a stochastic pension fund model.4 The deterministic assets
at the beginning of the period are given by:
A0 = EC0 + Π0 ,
3 Please
(10)
note that the 39 percent and the 49 percent equal the base levels of the equity stresses without taking into account
the so-called symmetric adjustment (for further information refer to CEIOPS, 2010a and EIOPA, 2014a).
4 A relatively similar model has been presented by Kahane and Nye (1975) who simultaneously optimize both the investment
and insurance portfolios of the property-liability insurance sector.
6
where Π0 denotes the premium income and EC0 the available equity capital. In the short term, however,
we assume that the underwriting portfolio is given (premiums as well as stochastic liabilities) and thus,
is no decision variable of the insurer. Consequently, the stochastic assets at the end of the period depend
exclusively on the portfolio return (r̃p ):
Ã1 = A0 · (1 + r̃P ).
(11)
The aggregated portfolio return r̃P can be calculated by drawing on the portfolio weights wi and discrete
returns ri for each asset class i (i = 1, ..., n):
r̃P = w1 ,
w2 ,
···
 
r̃1
 

 r̃2 

′

wn 
 ..  = w R,
 ., 
 
(12)
r̃n
with w being the vector of portfolio weights and R the random vector of asset class returns.
The insurance company’s liabilities at t = 0 equal the (discounted) future payments to the policyholders. Consequently, the stochastic value at the end of the period can be expressed as:
L̃1 = L0 · (1 + r̃L )
(13)
where r̃L denotes the stochastic growth rate of the liabilities. When taking the limited liability of the
shareholders into account, the insurer’s stochastic equity capital at time t = 1 is given by the following
expression:
˜ 1 = (Ã1 − L̃1 , 0)+
EC
(14)
Finally, the stochastic profit (P̃ ) over the considered period can be obtained by determining the
change in the equity capital. Hence,
P̃
=
˜ 1 − EC0
EC
=
Ã1 − L̃1 − (A0 − L0 )
=
A0 · r̃P − L0 · r̃L
=
˜ 1 − EC0 , −EC0 )+ .
(EC
(15)
In order to calculate both the SCR as demanded by the Solvency II market risk standard formula
and the insurer’s stochastic profit, a few more assumptions with respect to the balance sheet needs to be
made. Firstly, the total sum of the balance sheet is fixed to EUR 10 bn, while the equity quote is set
to 12 percent (see, e.g., Braun et al., 2015a,b). Consequently, the liabilities account for 88 percent. The
7
duration of the liabilities is assumed to be 10 (see, e.g., Steinmann, 2006), and the mean of the liability
growth rate has been set to 1.75 percent (see Federal Financial Supervisory Authority (BaFin), 2012).5
Risk-Adjusted Performance Measurement
Classic performance measures such as the return on equity (ROE) (see, e.g., Modigliani and Miller, 1958)
or the return on investment (ROI) (see, e.g., Phillips and Phillips, 2009) are widely accepted and applied
by a large number of companies from a large variety of industries. However, the also exhibit several
disadvantages that may lead to distortions with respect to the economic perspective on performance.
One well-known shortcoming is that these ratios measure the company’s success on the basis of book
values. By doing so, all cash flows resulting from different investments are implicitly considered as riskfree. That is, the inherent and unequal risks between different projects are neglected. In case of an
insurance company, however, all business activities are associated with risks. With the beginning of
2016, these risks need to be underpinned by a certain amount of equity capital that most insurers will
calculate by the standard formula of Solvency II. Obviously, investments and business activities with a
greater level of risk demand higher capital resources than less risky activities. On the other hand, it
is expected that they also result in higher net profits. In order to take all these aspects into account,
so-called risk-adjusted performance measures such as the return on risk-adjusted capital (RoRAC) have
been developed. The latter can be calculated as (see, e.g., Matten, 1996):
RoRAC =
E(P̃ )
,
SCR0
(16)
with SCR0 being the solvency capital requirement at the beginning of the period and E(P̃ ) the expected
profit. Hence, the RoRAC measures the performance by the relationship between the expected profit
and the risk capital necessary to achieve this profit. In other words, the SCR in the denominator causes
an implicit risk-adjustment of the expected profit. From the RoRAC perspective, an asset portfolio is
better than another asset portfolio if the expected profit can be achieved by a lower level of risk capital.
The formal design of Equation (16) may be misinterpreted in a way that low capital charges result in
artificially high and distorted RoRAC values. However, one needs to take into account that at the same
time, a lower SCR is only achieved if the composition of the investment portfolio and the interaction
with the insurance liabilities are not very risky. Since a less risky portfolio is also likely to result in a
lower expected profit, the RoRAC is unaffected. The interesting point, on the other hand, is whether
the nominator (expected profit) or the denominator (SCR) rise more sharply in case the level of market
risk is increased. From that point of view, an insurance company can select its optimal asset allocation.
5 Please
note that this figure equals the technical interest rate in Germany. However, with the beginning of January 2015,
the latter has been decreased to 1.25 percent, which will be taken into account in the next version of this draft.
8
4
Empirical Data and Portfolio Construction
As mentioned before, our analysis is based on three major asset classes of European insurance companies, i.e., stocks, government bonds, and real estate investments.6 Each of the latter is proxied by a
representable benchmark index, for which we analyzed historical time series data over the last 25 years
from January, 1989 until December, 2014.7 With a 25-year time horizon, the calibration contains both
a wide variety of interest rate environments with high interest rates at the beginning of the 90’s as well
as even negative interest rates at the end of 2014. Moreover, with the dot-com bubble at the end of the
late 90’s, the financial crisis between 2007-2008, and the government debt crisis since the end of 2009
several different business cycles are covered as well. Given its broad range and wide scope, the stocks
portfolio is represented by the EURO STOXX 50 Index, in which 50 blue-chip firms from more than
ten Eurozone countries are contained (see Braun et al., 2015a). The German REX Performance Index
(REXP) is taken to estimate the expected returns that are achieved by the insurer’s government bonds
subportfolio. In this index, 30 German Bunds with different maturities (ranging from one to ten years)
as well as three different coupon types are listed. Although this index includes debt securities issued by
Germany only, we deem it to be an appropriate proxy for the government bonds portfolio of a European
insurance company (see Braun et al., 2015a). Finally, we draw on the open-end and actively managed
Grundbesitz Europa Fund that represents the insurer’s investments in the real estate category. Given the
scarce database on real estate indices, the Grundbesitz Europa has the advantage that investments both
in commercial as well as residential property all across Europe are undertaken. The descriptive statistics
for the selected benchmark indices representing the insurer’s investments in the three asset classes stocks,
government bonds, and real estate can be found in Table 2.
No.
Asset Class
Benchmark Index
1
2
3
Stocks
Government Bonds
Real Estate
EURO STOXX 50 (TR)
REX Performance Index
Grundbesitz Europa Fund (TR)
µi
σi
Duration
9.21%
5.96%
4.81%
19.26%
3.34%
1.76%
–
4.92
–
Table 2: Descriptive Statistics for Asset Categories
This table shows the mean (µi ), the standard deviation (σi ), as well as the corresponding duration (as of 12/31/2012)
for each benchmark index representing the insurer’s investments in stocks, government bonds, and real estate from
01/01/1993 until 12/31/2012. All indices measure the total return (TR), i.e., both dividends and coupons are included.
The figures are shown on an annual basis.
Based on the asset classes shown in Table 2, we systematically select a large number of stylized
portfolios. More in detail, we open up a tree in order to find every possible combination of the three asset
categories by taking into account the current legal investment limits in Germany (see BMJ, 2011). In
the latter, stocks are limited to account for 20 percent of the asset portfolio at most, while investments
in real estate must not exceed 25 percent.8 Holdings of government bonds issued by a member state of
6 Similar
to the work of Braun et al. (2015a), we refrain from constructing the portfolios based on individual securities.
Instead, we resort to whole asset classes to model the insurer’s asset composition.
7 All figures have been obtained from Bloomberg.
8 Please note that for reasons of simplicity, we do not differentiate between the insurer’s so-called free and restricted assets.
For more information, please refer to Braun et al. (2015a,b), among others.
9
6.6
6.4
6.2
6
5.6
5.8
µA (in percent)
2.5
3
3.5
σA (in percent)
4
Figure 1: Portfolio Space
This figure shows the constructed portfolios (total of 12,726) in the mean-standard deviation space.
the European Union, on the other hand, are not restricted. Consequently, since the total investments in
stocks and real estate are restricted to a maximum of 45 percent, government bonds represent at least
55 percent of each asset allocation. In the selection process, the step size has been set to 0.002. In case
of stocks, for instance, a total of 101 possible portfolio weights results ranging from 0 to 0.2 (as given by
the investment regulation). The only further side condition is that the portfolio weights need to add up
to 100 percent.9 Finally, we end up with a total of 12,726 different portfolios that can be displayed in a
mean-standard deviation space as shown in Figure 1. From the latter, it can be seen that both efficient
as well as inefficient portfolios have been constructed. Given the historical time series of the asset classes
as contained in Table 2, the expected asset return (µA ) ranges from approximately 5.6 to 6.6 percent,
while the minimum (maximum) standard deviation amounts to roughly 2.4 percent (4.2 percent). In the
figure, the characteristic notch located at an expected return of 6.1 percent and a standard deviation of
3.1 percent can be explained by the investment limits for the asset categories stocks and real estate.
9 Since
the weights for each asset class start from zero, we also implicitly exclude short-sales.
10
5
Preliminary Results
For each asset portfolio as shown in Figure 1, we now calculate the market risk capital charges under the
Solvency II standard formula and derive the insurer’s expected profit. Then, by means of Equation (16),
the RoRAC can be computed. Figure 2 shows how the RoRAC depends on the insurer’s expected profit
(subfigure 2a) and the insurer’s solvency capital (subfigure 2b), respectively. Generally, a RoRAC above
one implies that the expected profit exceeds the insurer’s risk capital (in our context: SCR). In other
words, per Euro risk capital a profit above one Euro has been realized. In Figure 2, this threshold is
indicated by the dashed horizontal line. At first glance, it can be seen from Figures 2a and b that only a
few portfolios lead to RoRAC figures greater than one. On average, a RoRAC amounting to 0.59 results.
That is, per Euro risk capital, a profit of EUR 0.59 is achieved. Moreover, Figure 2a indicates that a
linear relationship is not existent, implying that the maximum expected profit (approx. EUR 507 bn) is
not associated with the maximum RoRAC (1.14). In contrast, the maximum profit results in a RoRAC
of 0.55, which is even lower than the average across all portfolios. Hence, it must be concluded that
the denominator in Equation (16) exerts a stronger influence on the RoRAC than the expected profit in
the numerator. In this respect, Figure 2(b) shows a decreasing (almost) linear relationship between the
RoRAC and the SCR. The portfolios exhibiting a low level of market risk and thus, associated with a
relatively low capital charge under Solvency II, yield the highest RoRAC values. Portfolios with higher
shares of the risky asset classes stocks and real estate, on the other hand, face stricter capital requirements. Hence, the latter have a dampening effect on the resulting RoRAC. The interaction between the
expected profit and the SCR are shown in Figure 3. So far, to sum up, we conclude that the RoRAC
of a portfolio strongly depends on the underlying solvency capital requirement. Therefore, it must be
assumed that the standard formula will have a considerable influence on the chosen asset allocation of
an insurance company.
In later versions of this research proposal, we will extend the total number of asset classes to five by
including corporate bonds and an alternative investment category.10 Moreover, the time horizon will be
expanded to span 25 years from the beginning of 1989 until the end of 2014. Additionally, the technical
interest rate will be adjusted to 1.25 percent. Then, all portfolios with the associated SCR, expected
profits, and the resulting RoRAC figures will be analyzed more in detail. With the introduction of
Solvency II at hand, we believe that the results are of great interest for practitioners who are involved in
the development of investment strategies with the beginning of 2016.
10 It
might also be the case that one or even two more asset classes such as money market instruments, private equity etc.
will be included in order to provide a broad range of stylized portfolios.
11
Return on Risk−Adjusted Capital
0.44
0.46
0.48
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Return on Risk−Adjusted Capital
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.42
0.5
0.4
Expected Profit (in EUR bn)
0.6
0.8
1
1.2
1.4
Solvency Capital Requirement (in EUR bn)
(a) RoRAC vs. Expected Profit
(b) RoRAC vs. SCR
Figure 2: RoRAC vs. Expected Profit vs. SCR
0.5
0.48
0.46
0.44
0.42
0.4
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
Solvency Capital Requirement (in EUR bn)
0.2
Return on Risk−Adjusted Capital
0.52
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Expected Profit (in EUR bn)
This figure shows the return on risk-adjusted capital (RoRAC) depending on the insurer’s expected profit (subfigure 2a)
and the insurer’s solvency capital (subfigure 2), respectively. The dashed horizontal line Both the expected profit as
well as the SCR are denoted in EUR bn.
Figure 3: Expected Profit vs. SCR vs. RoRAC
This figure shows the relationship between the expected profit, the underlying solvency capital requirement, and the
resulting return on risk-adjusted capital (RoRAC).
12
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