ï»¿ 70085
WORLD BANK WORKING PAPER
GCMNB
Solvency Measures for Insurance Companies:
Is There a Room for Improvement?
DRAFT
Eugene N. Gurenko
Alexander Itigin
June 26, 2009
Abstract
Over the last few years, there has been a considerable debate in the industry and in the academic
community about the accuracy and reliability of risk-based solvency measures in approximating
the level of risk retained by insurance and reinsurance entities. This topic was further
reinvigorated by the recent enactment of Solvency II requirements by the European Parliament 1,
which should replace the current Solvency I regulations in 2012. Despite divergence of opinions
about the adequacy of specific risk based measures of required solvency capital, there seems to
be a general consensus that in principle, well-designed risk based capital (RBC) or solvency
requirements can help achieve an efficient reduction in the expected costs of insolvencies by
helping regulators to identify weak insurers and intervene well before capital falls below
specified levels. 2
The main objective of this paper is to revisit both issues â€“ the accuracy and reliability of most
common risk-based measures of insurersâ€™ solvency and the practicability of risk-based measures
of solvency for corrective regulatory action. The paper also touches upon the role of financial
incentives in developing the RBC estimates.
Introduction
In this paper, we investigate the practical value of the RBC approaches for insurance regulators
and equity investors by applying two most well-known measures of risk based solvency
(Solvency II and Swiss Solvency Test) to a portfolio of catastrophe risk for a mono-line
catastrophe risk insurer. We begin by providing a brief overview of the existing literature on the
main limitations of most common risk-based solvency measures. Using 15 different scenarios,
we demonstrate how the RBC measures obtained for a chosen risk portfolio can be significantly
reduced by exploiting common data and modeling uncertainties, which in some cases can be
used to manipulate the RBC values. We then discuss how the existing system of incentives in the
industry may affect insurersâ€™ decisions with regard to determination of the level of RBC. The
paper concludes by offering several concrete recommendations on how to make the RBC
approaches to insurersâ€™ solvency less prone to adverse incentives and potential data
manipulation.
In general, insurance solvency risk can be defined as a future random value of an insurerâ€™s
financial net worth, e.g. the difference between the companyâ€™s equity capital and the overall
1
The text of final Solvency II Directive was adopted by the European Parliament on April 22, 2009.
2
Cummings et al (1995)
1
value of insurance claims arising from the existing insurance risk portfolio at any future point in
time 3. Solvency risk measures enable to assess and quantify the risk 4. In the simplest and most
common case, a risk measure reduces the complex characteristics of underlying risk exposure to
a simple number, thus enabling to quantify it.
Over the last decade, risk based measures of solvency have gained world-wide recognition
through the implementation of the Basel II Agreement in banking. However, risk-based measures
of solvency have been much less common in insurance, 5 where until recently the very notion of
risk-based supervision has been mainly academic, except for a few countries. The situation
however changed for the better with the recent passage of the Solvency II regulatory framework
by the EU Parliament, which sets a concrete implementation frame-work for the implementation
of RBC solvency requirements all EU countries. In addition to Solvency II, there is a variety of
other standard and special risk measures 6. In this paper we will concentrate on Value at Risk
(VaR) and Expected Shortfall (ES) solvency risk measures as being most common due to their
implementation requirement under the Solvency II and the Swiss Solvency Test frameworks.
The VaR is one of the most popular risk measures, mainly due to its simplicity, wide
applicability and universality7. It is the most widely used risk measure in financial institutions
for market risk and credit risk. It is used in Basel II as well as in life insurance; it is also
proposed as a risk measure under the Solvency II framework. However, the Value at Risk has
some serious limitations. Most criticisms zoom in on the fact that as a risk measure the Value at
Risk is not sub-additive in general. 8 Sub-additivity is the mathematical equivalent of the
diversification effect. In a nutshell, it means that the measure of the sum of two risks should not
exceed the sum of measures of two risks. Hence, the concept of sub-additivity implies that risk
can be reduced through diversification, a concept widely used in economic theory. For a sub-
additive risk measure, portfolio diversification always leads to risk reduction, while for a non-
sub-additive risk measure it is not the case in general. If a non-sub-additive risk measure is used,
it may happen that the diversified portfolio requires more solvency capital than the original one,
which is counter-intuitive and highly undesirable from the regulatory point of view 9.
Furthermore, the VaR measure does not consider the risks at the tail of distributions, which in
3
For simplification reasons, this definition of solvency risk disregards potential claims on the companyâ€™s solvency
capital that may arise out of poor operational performance (e.g. fraud, failure of systems, court awards against the
company, etc.), and loss in the value of companyâ€™s assets.
4
See ARTZNER et al. (1999) or ACERBI AND TASCHE (2002)
5
See for instance, the Swiss Solvency Test (e.g. WHITE PAPER ON SST (2004)).
6
See TASCHE (2002), ALBRECHT (2003), ACERBI (2004), GRÃœNDL AND WINTER (2005), MCNEIL, FREY AND
EMBRECHTS (2006), LANGMANN (2005)
7
STRASSBURGER (2006)
8
See the discussion in EMBRECHTS, MCNEIL AND STRAUMANN (2002) and in MCNEIL, FREY AND EMBRECHTS
(2006).
9
Numerous illustrations of this problem can be found in LANGMANN (2005).
2
general results in underestimation of risk 10. Thus, the VaR does not address the question of â€œhow
bad is badâ€? 11. Another disadvantage is the absence of continuity in the VaR measure with respect
to the confidence level â€“ it may happen that for slightly different confidence levels one obtains
considerably different values of the VaR 12.
In contrast to the VaR, our second considered risk measure the Expected Shortfall (ES) posses
the property of sub-additivity, which is an important advantage due to the proper representation
of the risk diversification principle 13. In the pertinent literature, the ES is also often known as
Conditional Value at Risk (CVaR), Tail Value at Risk (TVaR) or Average Value at Risk â€“ all of
which refer to the same risk measure.
Beside the sub-additivity, another key advantage of the ES over the VaR measure is that the ES
answers the question â€œhow bad is badâ€? as it not only describes but also quantifies the fact of
insolvency. Furthermore, the ES is continuous with respect to the confidence level as little
modifications of confidence level result in commensurately small resulting changes of its value.
The ES measure is not without flaws though. It only reflects losses above the chosen confidence
level and therefore is â€œsensitive mainly to extreme eventsâ€? 14. The ES will react sensitively to an
adjustment of probabilistic distributions in the tail zone, which poses a problem as often there is
no sufficient data for extreme events to clearly define the shape of the tail. Therefore, the
effectiveness of the ES depends on the stability of estimates for probabilistic distributions. This
is particularly the case for low-frequency and high-severity risks (e.g. natural hazards) and risks
with unknown dependence structures. 15 All the above mentioned limitations may lead to
significant estimation errors in applying the Expected Shortfall as a risk measure. Furthermore,
there has been some criticism on the fact that pretending to cover the whole tail, the Expected
Shortfall calculates only an expected value (in practice, an average) of all â€œworstâ€? losses, hence
ignoring the fact that every unlikely event will happen sooner or later given enough time 16.
Therefore, even though the ES provides more conservative estimates of the RBC for a given
portfolio of risk than the VAR (e.g. specifically for catastrophic events with longer return
periods), the real life portfolio losses will eventually exceed the ES projections as well.
Nevertheless, as we discuss below, despite the above mentioned limitations, the ES measure of
portfolio risk is perhaps one of the best currently available to insurance practioners for the
purposes of determining the RBC requirements for highly volatile business lines with a high risk
exposure in the tail of the probabilistic distribution.
10
(e.g. ACERBI (2004), page 155 f., ARTZNER et al. (2002), page 169, KORYCIORZ (2004), page 70, LANGMANN
(2005), page 36 ff.).
11
(ARTZNER et al. (2002), page 169 or DHAENE et al. (2004), page 5)
12
STRASSBURGER (2006).
13
ACERBI AND TASCHE (2002)
14
MEYERS (2002), page 15.
15
WANG (2001), page 4.
16
ROOTZÃ‰N AND KLÃœPPELBERG, 1999, page 5.
3
Sensitivity of Risk-Based Solvency Estimates to Data and Assumptions
In this Section we demonstrate the sensitivity of different risk-based solvency measures to
correlation assumptions, treatment of extreme data points and the chosen levels of confidence for
the computations 17. This exercise is reduced to thirteen different but rather common scenarios
demonstrating the sensitivity of risk-based solvency measures to data and model uncertainty for
the same portfolio of catastrophe insurance risk. We demonstrate that each of these scenarios
produces different, often highly diverging values of the RBC. A description of the actuarial
work done for this purpose follows.
Event data set
To carry out our analyses, we used a stochastic data set of insured property losses in South East
Europe. The data was provided by RMSI (2008), a risk modeling consultancy. The data set
approximates losses that would be sustained by an insurance portfolio consisting exclusively of
earthquake risk related liabilities, e.g. a portfolio of stand-alone earthquake insurance covers for
residential property in four countries â€“ Croatia, Bosnia and Herzegovina, Macedonia and
Albania. A deductible of 3 percent was assumed for each insured loss. For each country, the
RMSI provided an estimate of insured residential loss in the capital city and in the rest of
country. Insurance penetration was assumed to be 8 percent in capital cities and 6 percent
outside. The probability of occurrence for each loss event and its severity were estimated by
RMSI based on the underlying physical hazard and building vulnerability data.
We assumed that the event frequency is distributed according to the Poisson distribution. From
the original event data set described above, we derived the expected frequency (Poisson lambda)
of 1.79. A customized distribution for loss severity results directly from the original event data
set. Through a Monte-Carlo simulation for these frequency-severity parameters, we produced
100,000 data records of annual residential earthquake insured losses. Our sensitivity analysis was
based on this derived data set. The data set specifies both (i) the total annual loss for the total
residential portfolio in four countries and (ii) the total annual loss for each country separately.
Further, for each country the loss in the capital city and outside is specified. Each of the 100,000
annual insured losses can occur with the entry probability of 1/100,000 = 0.001 percent.
Measures of required solvency capital
For the above described portfolio of insurance risk we performed the calculations of required
solvency capital for each of three most common solvency capital measures: Solvency I, Solvency
II and the Swiss Solvency Test (SST). Although the Solvency I measure cannot be viewed as a
true risk-based measure of solvency, as we demonstrate below under certain circumstances it can
17
Following the regulatory definitions of VAR and ES under the Solvency II and Swiss Solvency Test approaches, we
define the confidence levels for these solvency measures at 99.5 and 99 percent, respectively.
4
exhibit some mild properties of a risk-based capital measure. For this reason and for the sake of
analytical completeness the Solvency I measure is included in our analysis. The mathematical
formulas used for these calculations are described below.
Solvency I
Required Solvency Capital = max (Premium Index, Claims Index)
Premium Index =
Retention Rate * (0.18 * min (Gross Premium, â‚¬50m) + 0.16 * max (Gross Premium - â‚¬50m,
0))
Claims Index =
Retention Rate * (+ 0.26 * min (Mean Gross Loss last 3 yrs, â‚¬35m)+ 0.23 * max (Mean
Gross Loss last 3 yrs - â‚¬35m, 0))
Where
Retention Rate = Total Retained Loss for last 3 yrs / Total Gross Loss for last 3 yrs
Gross Premium = + Expected Annual Loss + RoRaC + Admin Expenses
Where RoRaC denotes Return on Risk Adjusted Capital and is calculated as follows
RoRaC = Risk Adjusted Capital * Target Rate of Return
The Target Rate of Return was chosen to be 15 percent. The Risk Adjusted Capital was
calculated as
Risk Adjusted Capital = + Annual Loss VaR (99.5%) â€“ Expected Annual Loss
Hence,
RoRaC = (+ Annual Loss VaR (99.5%) â€“ Expected Annual Loss) *15%
Please notice that for the Risk Adjusted Capital was chosen for the confidence level of
99.5%, which is in line with the most common industry practice and regulatory requirements
for investment-grade rated insurance companies. However, other choices are also possible
depending on the companyâ€™s rating and solvency requirements, the extent of risk aversion,
and the quality of risk management.
5
Finally, we assumed the Administrative Expenses to be 10 percent of Gross Premium
Written.
Admin Expenses = 10% of Gross Premium
Further, we assumed
(i) Retention Ratio = 1 (no reinsurance)
(ii) Claims Index = 0 (no claims for last 3 yrs)
With the hindsight of the formula for the Required Solvency capital under the Solvency II (see
below), we can then rewrite the above presented RaRoC formula for Solvency I as follows:
RoRaC =(Required Solvency Capital II) *15%,
where Required Solvency Capital II denotes the Required Solvency Capital under the
Solvency II.
By substituting RoRaC with the above produced right side of equation, we can then rewrite the
formula for Gross Premium as follows:
GP = Expected Loss + 15% *(Required Solvency Capital II) + Admin Expenses
The Solvency I premium-based formula can then be rewritten as follows:
Premium Index =
Retention Rate * ((0.18 * min (Expected Loss + 15% *(Required Solvency Capital II) + Admin
Expenses), â‚¬50m) + 0.16 * max ((Expected Loss + 15% *(Required Solvency Capital II) +
Admin Expenses), - â‚¬50m, 0))
By expressing the premium-based index of Solvency I in terms of Solvency II, we have
turned it into a risk-based capital measure. The problem with the Solvency I measure however
is that it accounts only for a very small fraction of Solvency II capital requirements, which
can be traced throughout our modeling scenarios below.
Solvency II
Required Solvency Capital = + Annual Loss VaR (99.5%) â€“ Expected Annual Loss
where Annual Loss VaR (99.5%) denotes the annual VAR for the confidence level of 99.5
percent - the 99.5 percentile of the annual loss distribution or the value of the annual loss
which, with probability of 99.5%, will not be exceeded. This value can be estimated directly
from the annual loss data as described above.
6
Swiss Solvency Test
Required Solvency Capital = + Annual Loss TVaR(99.0%) - Expected Annual Loss
where Annual Loss TVaR (99.0%) denotes the Annual Loss Tail Value at Risk (also often
called the Expected Shortfall) at the confidence level of 99.0 percent. This is the average
annual loss calculated from the data set values that exceed those in excess of 99.0 percentile
of the annual loss distribution. In other words, itâ€™s an average calculated for the losses from
the data set which can only be exceeded with probabilities lower than 1.0 percent. To put it
differently, TVaR is the risk measure which approximates the whole tail of the distribution
whereas VaR looks at losses up to a predefined percentile, thus disregarding the tail of
distributions.
In Figure 1, we provide an graphic illustration of the differences among the above described
measures of RBC.
Figure 1
7
Return on Equity
For the purposes of this paper, we assumed that the required solvency capital is financed by
shareholdersâ€™ equity. Usually, companyâ€™s shareholders would require some adequate return on
their investment in the company, which we assumed to be 15 percent. Then the Required Return
on Equity (RoE) can be calculated as follows:
Required RoE = Solvency Capital * Required Rate of Return on Equity
Notice that one would obtain different RoEs for different solvency capital calculation
frameworks.
Sensitivity Analysis
The objective of the sensitivity analysis is to investigate the sensitivity of the risk-based solvency
requirements as well as the RoE to the uncertainties in the underlying stochastic model for the
annual insured loss.
To assess this sensitivity, we first considered several different realistic data uncertainty
scenarios. Each of these scenarios differs in some way from the original (â€œbenchmarkâ€?)
modeling scenario used to calculate the annual insured loss for the above presented measures of
the risk. As a result, each scenario generates an annual maximum loss estimate which
incorporates the corresponding uncertainty of the data. We then calculated the required solvency
capital and RoE for each of the scenarios and compared them with the ones calculated for the
original annual insured loss model.
All in all, we have considered the following 15 data uncertainty scenarios, including the original
â€œbenchmarkâ€? scenario and a subset of Scenario 8. A brief description of the selected scenarios
and produced results follows 18:
Scenario 0 (â€œBenchmarkâ€?)
Takes all loss values in the stochastic data set as is for the purposes of calculating solvency
measures.
Required
Solvency
Capital Required RoE
Solvency I 8,139,722.94 1,220,958.44
Solvency II 202,952,499.20 30,442,874.88
Swiss Solvency Test 216,493,498.73 32,474,024.81
18
See Aanex I for the results of all scenarios.
8
The first clear observation one can make from the benchmark scenario is that a very significant
gap exists between the estimates of solvency obtained through the existing and most commonly
applied Solvency I regime and those obtained by applying the risk-based solvency measures (e.g.
Solvency II and Swiss Solvency Test). The second observation demonstrates that the Swiss
Solvency Test is a more conservative (to the tune of 15 percent) estimate of risk than Solvency
II. Finally, one can clearly see that due to a huge difference in solvency capital requirements
mandated by different solvency regimes, an insurer with a portfolio of catastrophic risk would
receive a significant boost to its bottom line simply by staying with the Solvency I regime.
Scenario 1
Drops the largest single loss from 100,000 annual losses in the model.
Solvenc Delta to orig. Delta to orig. Delta RoE
y Capital event set RoE event set abs
Solvency I 8,133,534.23 -0.08% 1,220,030.13 -0.08% -928.31
Solvency II 202,780,919 -0.08% 30,417,137 -0.08% -25,736
Swiss Solvency Test 216,132,656 -0.17% 32,419,898 -0.17% -54,126
As can be seen from the table, by eliminating the largest single outlier in the data series we have
achieved almost no effect on the risk-based solvency measures.
Scenario 2
Drops two largest loses out of 100,000 annual losses in the model (e.g., annual losses above the
return period of 50,000 yrs).
Delta to
Solvency orig. event Delta to orig Delta RoE
Capital set RoE event set abs
Solvency I 8,101,527.67 -0.47% 1,215,229.15 -0.47% -5,729.29
Solvency II 201,745,962 -0.59% 30,261,894 -0.59% -180,980
Swiss Solvency Test 215,813,117 -0.31% 32,371,967 -0.31% -102,057
Similar to Scenario 2, by dropping 2 largest annual losses from the data set we have observed
only a slight decline (well under 1 percent) in the estimates of solvency capital for all three
measures.
Scenario 6
Drops ten largest losses out of 100,000 annual losses in the model (annual losses above the
return period of 10,000 yrs).
Solvency Delta to orig. Delta to orig. Delta RoE
Capital event set RoE event set abs
9
Solvency I 8,046,913.55 -1.14% 1,207,037.03 -1.14% -13,921.41
Solvency II 200,155,238 -1.38% 30,023,285 -1.38% -419,589
Swiss Solvency Test 213,635,878 -1.32% 32,045,381 -1.32% -428,643
As can be seen from the table above, the two measures of risk-based solvency react almost
similar to the omission of 10 largest losses from the distribution (-1.38 and -1.32 percent,
respectively).
Scenario 9
Drops hundred largest losses out of 100,000 annual losses in the model (annual losses above the
return period of 1,000 yrs).
Delta to
Solvency orig. Delta to orig. Delta RoE
Capital event set RoE event set abs
Solvency I 7,541,847.08 -7.35% 1,131,277.06 -7.35% -89,681.38
Solvency II 185,418,312 -8.64% 27,812,746 -8.64% -2,630,127
Swiss Solvency Test 196,072,237 -9.43% 29,410,835 -9.43% -3,063,189
Scenario 9 demonstrates however that ultimately the quantity translates into quality â€“ all three
measures of solvency have shown about same reaction to the elimination of 100 largest loss
events from the data set, by registering a drop of -7.35, -8.64 and -9.43 percent for Solvency I,
Solvency II and Swiss Solvency Test, respectively. In the case of our risk portfolio, it means that
by eliminating all losses in the data series in excess of 1000 year return period, which in our case
amounts to only 0.1 percent of all loss events, an insurer can reduce its required solvency capital
by about 10 percent. The simulation also shows that the Swiss Solvency Test is the most
sensitive to this change.
Scenario 10
In this scenario, we error on the conservative side by introducing a 30 percent upward adjustment
of losses occurring from events with a return period in excess of 1000 years. The scenario
effectively tries to compensate for considerable uncertainty involved in modeling losses from
events with such long return periods.
Delta to
Solvency orig. event Delta to orig. Delta RoE
Capital set RoE event set abs
Solvency I 8,158,044.76 0.23% 1,223,706.71 0.23% 2,748.27
Solvency II 202,844,723 -0.05% 30,426,708 -0.05% -16,166
Swiss Solvency Test 227,163,262 4.93% 34,074,489 4.93% 1,600,464
The outcomes of this simulation by and large conform with the criticism of the VAR based
measures of risk capital discussed in the introduction to this paper. As can be seen from the result
10
table for this scenario, despite the appearance of a noticeably heavier tail in the underlying loss
distribution, Solvency II estimate of solvency remains virtually the same. At the same time, the
TVAR based-measure of risk has shown a considerable sensitivity to this adjustment by
increasing by about 5 percent. A slight change in the Solvency I capital requirement can be
attributed to the increase in gross premium (stemming from increased standard deviation), which
in the absence of losses over the last 3 years, is used as a basis for calculating the solvency
requirement.
Scenario 12
Introduces a 100 percent upward adjustment for losses with a return period of 1000 years.
Solvency Delta to orig. Delta to orig event Delta RoE
Capital event set RoE set abs
Solvency I 8,200,795.66 0.75% 1,230,119.35 0.75% 9,160.91
Solvency II 202,593,247 -0.18% 30,388,987 -0.18% -53,887
Swiss Solvency
Test 252,059,378 16.43% 37,808,906 16.43% 5,334,882
The results presented in the table only confirm our previous conclusion about the lack of
sensitivity of VAR type measures as well as the Solvency I index to across the board changes in
the tail of risk distributions and at the same time a considerable sensitivity of TVAR measures to
the tail risk. The differences are quite illustrative â€“ when the TVAR goes up by 16.43 percent the
values of Solvency I and II measures remains almost same.
Scenario 14
Disregards correlations between annual losses in eight sub-portfolios corresponding to capital
city and rest or country for each county under consideration.
Delta to
Solvency orig. Delta to orig. Delta RoE
Capital event set RoE event set abs
Solvency I 5,982,594.31 -26.50% 897,389.15 -26.50% -323,569.29
Solvency II 131,048,211 -35.43% 19,657,231 -35.43% -10,785,643
Swiss Solvency Test 146,867,953 -32.16% 22,030,193 -32.16% -10,443,831
In this scenario, an insurer disregards correlations that may exist between different sub-portfolios
of risk in its aggregate portfolio â€“ a scenario quite typical for many insurance entities involved in
underwriting catastrophe risk often without sufficient information about potential risk
11
accumulations. It is interesting to observe that in this case all three measures of solvency drop by
approximately the same amount â€“ about 30 percent.
Conclusions
Based on the above presented sensitivity analysis we can make several important observations.
When utilizing Solvency I and Solvency II measures, the regulators should be mindful of the fact
that these measures are inherently (e.g. by their design) insensitive to the risk exposure in the tail
of abnormal risk distributions and hence can be rather misleading if used to determine the
solvency requirements for insurance-based entities involved in underwriting volatile classes of
insurance risk with a substantial exposure in the tail of the distribution (e.g. marine, aviation,
commercial liability and natural disasters.
The Swiss Solvency Test is the measure of risk-based solvency best suited for â€œtail dominatedâ€?
business lines due to its considerable sensitivity to risk exposure in the tail of abnormal
distributions.
1. Regulators however should be aware of the Swiss Solvency Test limitations. Our
computational work demonstrates that the TVAR based approaches to determining insurersâ€™
solvency requirements should be used with caution due to its considerable sensitivity to
modeling uncertainties in the tail of abnormal distributions, as we have shown in the scenarios
9, 10, 11, and 12. In practical terms, by requiring TVAR based measures in the absence of
proper oversight, the regulators may create ample space for potential data manipulation in
cases when insurers are under pressure to reduce their capital costs. In modeling natural
hazards, uncertainties of this type and range are quite typical for return periods of 200 year or
longer.
ï‚§
The VaR based Solvency II measure shows substantially lower sensitivity to model and data
uncertainties in the tail of the distribution in scenarios that increase the weight of the
distribution tail, as demonstrated by scenarios 10, 11, 12 that. However, in the cases where we
have eliminated the data outliers in excess of a 200-year return period the Solvency II
measure has also shown substantial sensitivity due to a shift of the whole distribution to the
left, as is in Scenario 9. We therefore conclude that while being reasonably sensitive to the
reduced uncertainty, the Solvency II measure equally fails to record increased uncertainty in
the long-run.
We also show a very significant impact of correlation assumptions on all three measures of
solvency, which underscores the importance of accumulation control procedures and up-to-
date information about correlation between different risks in insurersâ€˜ portfolios.
12
2. Finally, the empirical analysis shows a considerable actuarial discretion allowed by different
measures of solvency and assumptions that go into them, which is particularly the case for the
Swiss Solvency Test. We demonstrate that the material financial outcomes of such actuarial
discretion in calculating the insurerâ€™s solvency may invite management and shareholdersâ€™
pressure to reduce the estimates of RBC for a given risk portfolio. In the extreme cases, the
choice of the RBC measure and of modeling approach can either add or subtract millions of
dollars to/from the companyâ€™s bottom line. Given the prevailing structure of management
compensation and short-term profit maximizing incentives of shareholders in most public
insurance companies, this finding not only underscores the importance of the actuarial
profession, but also raises the question about its professional independence and ability to
effectively resist pressure from the management and shareholders whose myopic bottom-line
objectives may be at odds with the goal of achieving long-term solvency 19.
Policy Recommendations
Based on the above presented conclusions we would like to make the following three
recommendations:
1. Given the inherent lack of sensitivity of the Solvency II measure to the risk exposure in the
tail of abnormal distributions, regulators may consider supplementing it with other risk-based
solvency measures (e.g. such as TVaR) when it comes to determining solvency requirements
for insurance entities with heavy exposure to volatile â€œtail dominatedâ€? business lines (e.g.
aviation, marine, mortgage insurers, natcat and commercial liability).
This recommendation however should not be interpreted as an argument for using instead of the
VaR the expected shortfall based approaches (TVAR), as the latter suffer from considerable
sensitivity to modeling uncertainties in the tail of abnormal distributions.
2. Due to the sensitivity of the risk-based solvency measures (e.g. mainly TVaR and less so
VaR) to underlying data and modeling assumptions in determining the risk-based solvency,
insurance regulators may consider requiring their full disclosure. To avoid potential misuse of
modeling and data uncertainties by some insurance entities in calculating the RBC
requirements, it may be advisable to establish standard universal modeling and data quality
and verification requirements for the most critical building blocks of the risk modeling
process (e.g. the loss return period, treatment of outliers beyond the given return period
threshold, suggested shape of the tail in abnormal distributions in the absence of sufficient
19
For a comprehensive discussion of management incentives to take excessive risks in a highly leveraged liability
business see â€œRegulating Bankersâ€™ Payâ€? by Lucian Bebchuk and Holger Spamann, Harvard Law and Economics
Discussion Paper No. 641, 2009.
13
data, as well as correlation assumptions used to model individual classes of risk as well as
aggregate portfolios).
3. We also would like to recommend strengthening the independence of actuaries from potential
management pressures to adjust the estimates of risk-based capital downwards for the sake of
short-term profitability. This can be achieved by requiring companies to subject their
estimates to external actuarial audits as well as to introduce standard solvency estimation
models approved by the insurance regulators.
Insurers disagreeing with the estimates of their risk-based solvency produced by such
standard models should then be able to justify to the regulator (which would require a
modeling audit by an independent modeling provider) why their own custom-tailored
solvency model is superior to that proposed by the regulator for the entire market.
14
References
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16
17
ANNEX I
Complete list of Solvency Sensitivity Tests
Scenario 1
Drop the largest loss of 100,000 annual losses in the model
Scenario 2
Drop two largest losses of 100,000 annual losses in the model (annual losses above the return
period of 50,000 yrs)
Scenario 3
Drop three largest losses of 100,000 annual losses in the model (annual losses above the return
period of 33,000 yrs)
Scenario 4
Drop four largest losses of 100,000 annual losses in the model (annual losses above the return
period of 25,000 yrs)
Scenario 5
Drop five largest losses of 100,000 annual losses in the model (annual losses above the return period
of 20,000 yrs)
Scenario 6
Drop ten largest losses of 100,000 annual losses in the model (annual losses above the return period
of 10,000 yrs)
Scenario 7
Drop fifteen largest losses of 100,000 annual losses in the model (annual losses above the return
period of 7,000 yrs)
Scenario 8
Drop twenty largest losses of 100,000 annual losses in the model (annual losses above the return
period of 5,000 yrs)
Scenario 9
Drop hundred largest losses of 100,000 annual losses in the model (annual losses above the return
period of 1,000 yrs)
18
Scenario 10
Underestimate of annual losses with return periods above 1000yrs in the original model. 30%
adjustment of these losses is required.
Scenario 11
Underestimate of annual losses with return periods above 1000yrs in the original model. 50%
adjustment of these losses is required.
Scenario 12
Underestimate of annual losses with return periods above 1000yrs in the original model. 100%
adjustment of these losses is required.
Scenario 13
Disregard correlation between annual losses in four sub-portfolios corresponding to four countries
under consideration.
Scenario 14
Disregard correlation between annual losses in eight sub-portfolios corresponding to capital city and
rest or country for each county under consideation.
Following results were obtained
Original Model
Solvency Capital RoE
Solvency I 8,139,722.94 1,220,958.44
Solvency II 202,952,499.20 30,442,874.88
Swiss Solvency Test 216,493,498.73 32,474,024.81
Scenario 1
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,133,534.23 -0.08% 1,220,030.13 -0.08% -928.31
Solvency I
Solvency II 202,780,919.69 -0.08% 30,417,137.95 -0.08% -25,736.93
Swiss Solvency Test 216,132,656.83 -0.17% 32,419,898.52 -0.17% -54,126.29
Scenario 2
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,101,527.67 -0.47% 1,215,229.15 -0.47% -5,729.29
Solvency I
Solvency II 201,745,962.29 -0.59% 30,261,894.34 -0.59% -180,980.54
Swiss Solvency Test 215,813,117.00 -0.31% 32,371,967.55 -0.31% -102,057.26
19
Scenario 3
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,097,918.30 -0.51% 1,214,687.75 -0.51% -6,270.70
Solvency I
Solvency II 201,655,771.50 -0.64% 30,248,365.72 -0.64% -194,509.15
Swiss Solvency Test 215,520,385.85 -0.45% 32,328,057.88 -0.45% -145,966.93
Scenario 4
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,083,386.27 -0.69% 1,212,507.94 -0.69% -8,450.50
Solvency I
Solvency II 201,201,331.01 -0.86% 30,180,199.65 -0.86% -262,675.23
Swiss Solvency Test 215,230,036.28 -0.58% 32,284,505.44 -0.58% -189,519.37
Scenario 5
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,081,955.98 -0.71% 1,212,293.40 -0.71% -8,665.04
Solvency I
Solvency II 201,182,647.41 -0.87% 30,177,397.11 -0.87% -265,477.77
Swiss Solvency Test 214,953,992.68 -0.71% 32,243,098.90 -0.71% -230,925.91
Scenario 6
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,046,913.55 -1.14% 1,207,037.03 -1.14% -13,921.41
Solvency I
Solvency II 200,155,238.85 -1.38% 30,023,285.83 -1.38% -419,589.05
Swiss Solvency Test 213,635,878.49 -1.32% 32,045,381.77 -1.32% -428,643.04
Scenario 7
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,005,048.94 -1.65% 1,200,757.34 -1.65% -20,201.10
Solvency I
Solvency II 198,890,030.89 -2.00% 29,833,504.63 -2.00% -609,370.25
Swiss Solvency Test 212,677,587.49 -1.76% 31,901,638.12 -1.76% -572,386.69
Scenario 8
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
7,971,196.92 -2.07% 1,195,679.54 -2.07% -25,278.90
Solvency I
Solvency II 197,888,232.03 -2.50% 29,683,234.80 -2.50% -759,640.07
Swiss Solvency Test 211,358,070.87 -2.37% 31,703,710.63 -2.37% -770,314.18
Scenario 9
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
7,541,847.08 -7.35% 1,131,277.06 -7.35% -89,681.38
Solvency I
Solvency II 185,418,312.81 -8.64% 27,812,746.92 -8.64% -2,630,127.96
20
Swiss Solvency Test 196,072,237.73 -9.43% 29,410,835.66 -9.43% -3,063,189.15
Scenario 10
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,158,044.76 0.23% 1,223,706.71 0.23% 2,748.27
Solvency I
Solvency II 202,844,723.80 -0.05% 30,426,708.57 -0.05% -16,166.31
Swiss Solvency Test 227,163,262.78 4.93% 34,074,489.42 4.93% 1,600,464.61
Scenario 11
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,170,259.30 0.38% 1,225,538.89 0.38% 4,580.45
Solvency I
Solvency II 202,772,873.54 -0.09% 30,415,931.03 -0.09% -26,943.85
Swiss Solvency Test 234,276,438.81 8.21% 35,141,465.82 8.21% 2,667,441.01
Scenario 12
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,200,795.66 0.75% 1,230,119.35 0.75% 9,160.91
Solvency I
Solvency II 202,593,247.88 -0.18% 30,388,987.18 -0.18% -53,887.70
Swiss Solvency Test 252,059,378.88 16.43% 37,808,906.83 16.43% 5,334,882.02
Scenario 13
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
8,054,991.29 -1.04% 1,208,248.69 -1.04% -12,709.75
Solvency I
Solvency II 200,128,110.95 -1.39% 30,019,216.64 -1.39% -423,658.24
Swiss Solvency Test 213,953,653.42 -1.17% 32,093,048.01 -1.17% -380,976.80
Scenario 14
Solvency Capital Delta to orig event set RoE Delta to orig event set Delta RoE abs
5,982,594.31 -26.50% 897,389.15 -26.50% -323,569.29
Solvency I
Solvency II 131,048,211.53 -35.43% 19,657,231.73 -35.43% -10,785,643.15
Swiss Solvency Test 146,867,953.30 -32.16% 22,030,193.00 -32.16% -10,443,831.81
21