Solvency II SCR based on Expected
Shortfall
Wouter Alblas
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics Author: Wouter Alblas Student nr: 5886635
Email: wouteralblas@hotmail.com Date: September 27, 2014
Supervisor: Dr. T.J. Boonen
Second reader: Prof. dr. ir. M.H. Vellekoop Supervisor: Dr. J.C. Gielen
Solvency II SCR based on Expected Shortfall — Wouter Alblas iii
Abstract
This thesis examines the consequences for an average life annuity insurance company if the Solvency II SCR estimation is based on a more appropriate risk measure instead of Value-at-Risk (VaR). The allocation of SCR based on VaR over the three risk mod-ules equity risk, interest rate risk and longevity risk is compared with the allocation based on Expected Shortfall (ES), a risk measure we feel is more appropriate to esti-mate sufficient capital. Following the guidelines set by EIOPA we calibrated the stress scenarios based on VaR and ES and applied them to a fictitious life annuity insurance company. We found that for EIOPA’s current confidence interval α of 99.5%, the allo-cation of SCR(ESθ) is very close to SCR(VaRα). This can be explained by noticing that
the VaR99.5% is already captured in the ”unexpected” long tail for the current data. Might EIOPA choose to calibrate the stress scenarios on a smaller confidence interval, underestimation of longevity risk and overestimation of equity risk will take place if the current data is used. This difference in allocation is maximized for the confidence interval α of 98.5%.
Keywords Solvency II, Expected Shortfall, Value-at-Risk, Solvency Capital Requirement, Interest rate risk, Equity risk, Longevity risk
Preface vi
1 Introduction 1
2 An appropriate risk measure to estimate sufficient capital 3
2.1 Coherent Risk Measures . . . 3
2.2 Short overview of well known risk measures . . . 4
2.3 Risk measure estimation methods . . . 5
2.3.1 Historical Simulation . . . 5
2.3.2 The delta-normal method . . . 5
2.3.3 Monte Carlo simulation . . . 5
2.4 Non-normality of financial risks . . . 5
2.5 Advantages and disadvantages of risk measures . . . 6
2.5.1 Standard deviation . . . 6
2.5.2 Value-at-Risk . . . 6
2.5.3 Expected Shortfall . . . 7
2.6 Other regulatory frameworks . . . 7
2.6.1 The Swiss Solvency Test . . . 7
2.6.2 Basel III . . . 8
2.7 Appropriate risk measure . . . 8
3 Solvency II 9 3.1 Three pillars . . . 9
3.2 Valuation . . . 9
3.3 Solvency Capital Requirement . . . 10
3.4 Overview of the main risks for a life annuity insurance company . . . 10
3.4.1 Market risk . . . 11 3.4.2 Life risk . . . 11 3.5 Calculation of the SCR . . . 12 3.5.1 Market SCR . . . 12 3.5.2 Life SCR . . . 14 3.5.3 Total SCR . . . 14 4 Methodology 15 4.1 Calibrating the SCR stress scenarios . . . 15
4.1.1 The calibration of equity risk . . . 15
4.1.2 The calibration of interest rate risk . . . 16
4.1.3 The calibration of longevity risk . . . 18
4.2 Comparing the SCR calibrated on VaR with the SCR calibrated on ES . 19 4.2.1 The fictitious life annuity insurance company . . . 19
4.2.2 Estimating the SCR . . . 19
4.2.3 Matching the total SCR value . . . 20 iv
Solvency II SCR based on Expected Shortfall — Wouter Alblas v
5 Results 21
5.1 The SCR stress scenarios . . . 21
5.1.1 Equity risk . . . 21
5.1.2 Interest rate risk . . . 22
5.1.3 Longevity risk . . . 22
5.2 Comparing the SCR(VaRα) with the SCR(ESθ) for the fictitious life an-nuity insurance company . . . 24
5.2.1 The changes in allocation of the total SCR . . . 24
5.2.2 The changes in allocation of the total SCR with empirical stress scenarios . . . 24
5.2.3 Intuitive explanation of the changes in allocation of the total SCR 25 5.3 Sensitivity analysis . . . 26
5.3.1 Changing the asset portfolio . . . 26
5.3.2 Changing the liability portfolio . . . 30
6 Conclusion 33 6.1 Summary of the findings . . . 33
6.2 Discussion . . . 34
References 35
Writing a Master’s Thesis is a challenging mission which cannot be completed without the help of others. First and foremost, I would like to thank my supervisor at KPMG, Jeroen Gielen, for guiding me through the process of writing this thesis. Always asking the right questions and challenging me to think intuitively about the complex matter. Secondly, I would like to thank my supervisor at the University of Amsterdam, Tim Boonen, for supporting during this half year. Giving me the freedom to take my re-search in the direction that I wanted, for supplying me with interesting papers and for the various discussions we had in his office. A special thanks to KPMG for giving me the opportunity to use all their resources and guidance throughout this mission. I am grateful for the assistance and interest of various colleagues at Financial Risk Manage-ment. Finally, I would like to thank my parents, girlfriend and friends for supporting and distracting me during this adventure.
Chapter 1
Introduction
Solvency II is the new supervisory framework for insurers and reinsurers in Europe and puts new and stricter demands on the required economic capital, risk management and reporting standards of insurance companies. The application date of this new su-pervisory framework is scheduled for January 1st 2016. Solvency II is the successor of Solvency I, which was introduced in 1973, and its main objective is to ensure that in-surance companies hold sufficient economic capital to avoid bankruptcy and therefore it protects the policyholder. Or to put it differently, Solvency II aims to reduce the risk that an insurance company is unable to meet its financial claims. To make sure that insurers hold sufficient economic capital, they will have to meet new quantitative requirements under the new Solvency II framework. This capital requirement is called the Solvency Capital Requirement (SCR) and covers all the risks that an insurer faces. The SCR is described as follows by EIOPA (2010): “The SCR should correspond to the Value-at-Risk of the basic own funds of an insurance or reinsurance undertaking subject to a confidence level of 99.5% over a one-year period”. The Value-at-Risk is thus the signature risk measure for the SCR within the Solvency II framework.
The Value-at-Risk (VaR) is a widely used risk measure and can be described as the maximum loss within a certain confidence level. In the case of the SCR, the confidence level of 99.5% tells us that the firm can expect to lose no more than VaR in the next year with 99.5% confidence, so only once every 200 years the VaR loss level will be exceeded. Mathematically, VaR is a quantile. The Value-at-Risk is however being criticized for not being subadditive, which is shown by Artzner et al. (1999), this simply means that this risk measure not always rewards diversification. Besides not being subadditive, the VaR does also not consider the shape of the tail beyond the confidence level. This means that the VaR does not take in to account what happens beyond the confidence level, so it does not consider the very worst case scenarios. This is thoroughly disussed by Yamai and Yoshiba (2004). David Einhorn, an American hedge fund manager, described the VaR as “An airbag that works all the time, except when you have a car accident”. The fact that VaR is not subadditive and that it does not consider the tail beyond the confidence level makes it not a very suitable risk measure to use for a capital requirement calculation.
There are alternative risk measures for VaR which are subadditive and consider the shape of the tail beyond the confidence level, these risk measurers are called coherent risk measurers. Coherent risk measurers are defined by Artzner et al. (1999) as: “A risk measure satisfying the four axioms of translation invariance, subadditivity, positive homogeneity, and monotonicity is called coherent”. The most famous coherent risk mea-sure is the Expected Shortfall. This risk meamea-sure is equal to the expected value of the loss, given that the loss is greater than the VaR, therefore the Expected Shortfall also depends on the confidence interval. In other words, the Expected Shortfall accounts for the expected loss given that we are in the very worst case scenario. Two other regu-latory frameworks for financial institutions, the Swiss Solvency Test and the Basel III framework both use the Expected Shortfall as their signature risk measure. There might
also be other non coherent risk measures which are at first sight more appropriate to estimate sufficient economic capital than the VaR.
Using the VaR for a capital requirement calcultation, a risk measure that at first sight does not seem to be a very suitable risk measure for this task, might result in overestimating or underestimating the capital requirements for certain risks. The results of Stevens et al. (2010) already suggested that using the simplified approach of Solvency II leads to an overestimation of the capital requirements and we would like to know if this is due to using VaR. In this paper we will answer the question: Is there a more appropriate risk measure to estimate sufficient economic capital than Value-at-Risk, and if so, what happens to the allocation of the total SCR for an average life annuity insurance company if the Solvency II SCR estimation is based on a more appropriate risk measure instead of Value-at-Risk?
First there will be examined if there is a more appropriate risk measure to estimate sufficient economic capital than VaR and for that purpose an overview of the advantages and disadvantages of VaR and other relevant risk measures will be given. Secondly, thorough analysis of the Solvency II framework and the estimation of the SCR is needed. After that the stress scenarios to estimate the SCR will be calibrated based on a more appropriate risk measure. The SCR for a standard life annuity insurance company will then be estimated by using the “old” and the “new” stress scenarios and the different results will be thoroughly discussed.
Chapter 2
An appropriate risk measure to
estimate sufficient capital
Risk measures are, among other things, used to estimate the amount of sufficient eco-nomic capital to be kept in reserve in order to protect an asset, company, portfolio or liability for any negative financial impacts that may arise in the future. A risk measure is a function ρ of random variable X.“The risk ρ(X) of a given financial position X is specified as the minimal capital that should be added to the position in order to make that position acceptable” (Follmer and Knispel, 2013, p. 1). A position X is acceptable if ρ(X) ≤ 0.
2.1
Coherent Risk Measures
Coherent risk measurers were introduced by Artzner, et al. (1999). They discuss de-sirable properties of risk measures and define coherent risk measures as follows: “A risk measure satisfying the four axioms of translation invariance, subadditivity, positive homogeneity, and monotonicity is called coherent”.
Translation invariance
For all risks X and all real numbers α, we have ρ(X + α) = ρ(X) − α.
The real number α can be seen as cash added to the portfolio. Thus adding an amount of cash α to the portfolio will decrease the amount of economic capital that is required to hold with α. This axiom ensures the desirable property that adding the sufficient amount of cash to the portfolio, leads to the result that then no extra capital to hold is needed. In an equation this comes down to ρ(X + ρ(X)) = 0.
Subadditivity For all X1 and X2,
ρ(X1+ X2) ≤ ρ(X1) + ρ(X2).
This property says that a merger of portfolios does not increase capital requirements. This is a desirable property because it captures the meaning of possible diversification. Diversification means that the total risk of a combined portfolio is reduced when the two individual portfolios are not perfectly correlated, because a negative performance of X1 can be neutralized by a positive performance of X2 and therefore total risk of the
combined portfolio decreases.
Positive homogeneity For all λ ≥ 0 and all X,
ρ(λX) = λρ(X).
This can be interpreted as: When you double your portfolio, then you double your risk. It is quite straightforward that this is a desirable property. This axiom also implies that ρ(0) = 0.
Monotonicity
For all X and Y with X ≤ Y , we have
ρ(X) ≥ ρ(Y ).
This axiom is desirable since it tells us that when Y generates more than or as much profit as X, then the capital requirement for Y should be lower or equal to the capital requirement for X.
2.2
Short overview of well known risk measures
In this paper we will discuss three well known risk measures: The standard deviation, the Value-at-Risk and the Expected Shortfall. These three risk measures are most used in practice and therefore relevant to discuss for estimating the SCR within the Solvency II framework. These three risk measures are discussed by Hull (2012).
The standard deviation σ of a portfolio is given by the square root of the variance of the portfolio. Let X be a continious random variable with µ as mean, which represents a financial postion, then the standard deviation is defined as:
ρ(X) = σ =pE((X − µ)2.
The Value-at-Risk (VaR) can be described as the worst case value of a financial position within a certain confidence level. Mathematically, VaR is a quantile. To put it differently, the VaR of a portfolio is given by the smallest possible x such that the probability that the value of the portfolio X < x does not exceed 1 − α, where α is the confidence interval. Since ρ(X) is specified as the minimal capital that should be added to the position in order to make that position acceptable, the VaR should be substracted from the mean µ if it represents the amount of significant economic capital. We will however still refer to this as VaR. This results in the following mathematical definition:
ρ(X) = µ − VaRα(X) = µ − inf(x ∈ R : P (X < x) ≤ (1 − α)).
The Expected Shortfall (ES) is the expected value of the financial position, given that the value of the financial position is smaller than VaRα(X). The Expected
Short-fall therefore also depends on the confidence interval α. In other words, the Expected Shortfall accounts for the expected value of the financial position given that we are in the worst case scenario. Since ρ(X) is specified as the minimal capital that should be added to the position in order to make that position acceptable, the ES should be substracted from the mean µ if it represents the amount of significant economic capital. We will however still refer to this as ES. Mathematically, this equals
Solvency II SCR based on Expected Shortfall — Wouter Alblas 5
2.3
Risk measure estimation methods
There are different calculation methods to compute the different risk measures. In this paper we will discuss three common approaches: The historical simulation method, the delta-normal method and the Monte Carlo simulation method. In this Section the three named methods will be briefly explained and discussed.
2.3.1 Historical Simulation
Historical simulation is the estimation method which is most used in practice and is discussed by Li et al. (2012). Historical simulation is a simple resampling method since is does not assume any distribution about the underlying process of financial returns. It is a non-parametric model which assumes that the distribution of past returns is a perfect representation of the expected future returns. The fact that the model does not make any assumptions about the probability distribution is an advantage. A disadvantage is that risk estimates can become biased when the past returns turn out to be not representative for the future returns. Logically, historical simulation is very dependent of its length of the sample and the weights given to the past returns. In the latter we speak of weighted historical simulation, then often decreasing weights are given to observed returns that are further from the present.
2.3.2 The delta-normal method
The delta-normal method, sometimes called the variance-covariance method, is based on the assumption that the returns follow a normal distribution. It uses the historical returns to compute variances and correlations. The main advantage of the delta-normal method is its simplicity, but the main drawback is that it may underestimate the risk being faced if the returns are not normally distributed.
2.3.3 Monte Carlo simulation
The Monte Carlo simulation method is a more sophisticated approach than the two methods named above. “Monte Carlo simulation uses random samples from known populations of simulated data to track a statistic’s behavior” (Li et al., 2012, p.3). This model allows any assumption about the stochastic process to be made. Based on this assumption it will sample many different paths to build up a probability distribution. It is a fairly easy and flexible simulation method and it can model instruments with non-linear and path- dependence payoff functions. The main disadvantage is that the method can be time consuming, since a large number of samples is needed.
2.4
Non-normality of financial risks
When the financial position distribution, or profit-loss distribution is normally dis-tributed, the same information is given by the standard deviation, Value-at-Risk and Expected Shortfall. This is named by Yamai and Yoshiba (2004). In the case of normal-ity, VaR and Expected Shortfall are multiples of the standard deviation. “For example, VaR at 99% confidence level is 2.33 times the standard deviation, while expected shortfall at the same confidence level is 2.67 times the standard deviation” (Yamai and Yoshiba, 2004, p. 4).
The assumption that financial risk is normally distributed is often criticized and it is said that the normality assumption does lead to an underestimation of the risks being faced. It is an observed fact that asset returns are fat-tailed and asymmetric and therefore profit- loss distributions tend to be non-normal. Yamai and Yoshiba (2004) state: “Non-normality of the profit-loss distribution is caused by non-linearity of the portfolio position or non-normality of the underlying asset prices”.
In this paper, the distribution of financial risk will not be further examined, but both distributions, normal and non-normal, will be considered when we search for an appropriate risk measure to estimate sufficient economic capital.
2.5
Advantages and disadvantages of risk measures
2.5.1 Standard deviation
When modern portfolio theory was introduced by Markowitz (1952), the standard devi-ation was introduced as a risk measure to the great public. The modern portfolio theory maximizes portfolio return for a given amount of risk, which is quantified by the stan-dard deviation, and where the financial returns are assumed to be normally distributed. The standard deviation was therefore the first measure that was used to quantify risk. There are some clear advantages of using the standard deviation as a risk measure: It does satisfy the axioms of subadditivity and positive homogeneity, it gives a good indication of the variability of the risk, it always exits and it is easy to understand and communicate, but over the years quantile based downside risk measures have become more popular in risk management. One of the reasons is that the standard deviation does not satisfy the axioms of translation invariance and monotonicity. Other reasons are named by Zhu (2010). The most important is that standard deviation is a dispersion measure that does not discriminate between upside and downside risks, since it does not deal with asymmetry. The standard deviation is therefore often called a deviation risk measure since it is a quantifier of financial risk, but not always of quantifier of downside risk. The standard deviation could underestimate risks for negative skewed distributions, and thus not consider the tail risk. In general, the standard deviation fails to capture non-normality. Therefore, and the fact that the standard deviation does not satisfy the axioms of translation invariance and monotonicity, the standard deviation is not a suitable risk measure to estimate sufficient economic capital.
2.5.2 Value-at-Risk
In the last twenty years the Value-at-Risk has become the most widely used risk measure in the financial sector, and has often been called the standard risk measure for financial risk management. “Value-at-Risk is an attempt to provide a single number summarizing the total risk in a portfolio of financial assets. It has become widely used by corporate treasurers and fund managers as well as by financial institutions” (Hull, 2012, p. 471). Value-at-Risk can simply be described as the maximum loss within a certain confidence level, for a given portfolio, probability and time horizon. In contrast to the standard deviation, the Value-at-Risk focuses on which worst case scenario can happen up to a certain confidence level and therefore only deals with the relevant downside risk. VaR captures risk in one single number, of course dependent upon its confidence interval. One of the other advantages of VaR is that it is easy to understand, since it answers the simple question: How bad can things get within a certain confidence interval? Next to that VaR is relatively easy to estimate robustly and it always exits.
When the risk is assumed to be normally distributed, VaR satisfies the four axioms of translation invariance, subadditivity, positive homogeneity, and monotonicity and can then be called coherent. This is not the case when the risk is assumed to be non-normally distributed, the VaR then does not satisfy the subadditivity and cannot be called coherent. Therefore, VaR as a whole is not a coherent risk measure. It is shown by Artzner et al. (1999) that Value-at-Risk fails to satisfy the subadditivity property. This means the VaR of a combined portfolio can be larger than the sum of the VaR of its stand-alone risks, therefore the VaR does not encourage diversification.
Another disadvantage of the Value-at-Risk is that it does not consider what happens beyond the VaR level, and therefore it does not consider the very worst case scenarios.
Solvency II SCR based on Expected Shortfall — Wouter Alblas 7
Especially when we deal with non-normal distributed risk, the VaR can really underes-timate the impact of the possible extreme losses and therefore VaR can be misleading. This is also why VaR should not be used as risk measure in the optimal portfolio choice. This will result in so called gambling portfolios. Then a portfolio will be constructed which will incur relatively large losses whenever they occur. This latter is however not relevant when the VaR is used to calculate sufficient economic capital.
The fact that VaR is not subadditive and that it does not consider the tail beyond the confidence level makes it not a very suitable risk measure to use for a capital requirement calculation.
2.5.3 Expected Shortfall
Expected Shortfall was proposed as an alternative to Value-at-Risk in the late 1990s. The Expected Shortfall is the expected value of the loss, given that the loss is greater than the VaR, therefore the ES also depends on the confidence interval. In other words, the Expected Shortfall accounts for the expected loss given that we are in the very worst case scenario. The Expected Shortfall is also called: Conditional VaR (CVaR), Tail VaR (TVaR) or Expected Tail Loss (ETL).
The Expected Shortfall satisfies the four axioms of translation invariance, subad-ditivity, positive homogeneity, and monotonicity under any risk distribution and can therefore be called a coherent risk measure. So in contrast to the Value-at-Risk, the Expected Shortfall is subadditive and does reward for diversification.
Next to that it considers the tail beyond the VaR level and thus takes into account what happens in the very worst case scenarios and considers the consequences of po-tential default. The Expected Shortfall reflects the also losses if the risk distribution features a fatter tail than the normal distribution.
In 2006 EIOPA, at the time called CEIOPS, generally acknowledged the theoretical advantages of using the Expected Shortfall to calculate the SCR. However there were concerns about the practicality of an SCR measurement based on Expected Shortfall, since there was seen to be scarcity of data about the tails, which can easily lead to an increase in modeling errors. Yamai and Yoshiba (2004) argue that the estimation errors of Expected Shortfall are much greater than those of VaR. This is a potential drawback of the Expected Shortfall. These estimation errors can however be reduced by increasing the sample size of the simulation or by making assumptions on the shape of the tail. Another disadvantage is that the Expected Shortfall does not exist for distributions with an infinite mean, these distributions are however seldom associated with profit loss distributions and therefore this disadvantage is not relevant in this case.
2.6
Other regulatory frameworks
To give some perspective, we will shortly discuss which risk measure is used in two other regulatory frameworks for financial institutions: The Swiss Solvency Test and the Basel III framework.
2.6.1 The Swiss Solvency Test
The Swiss Solvency Test (SST) is a regulatory framework for insurance companies in Switzerland. The SST is quite similar to the Solvency II framework, except for the important difference that the stress scenarios for the SST are calibrated by using the Expected Shortfall with confidence level 99.0%.
2.6.2 Basel III
The Basel III framework is a global regulatory framework for banks which is planned to be implemented in 2018. Basel III was set up in a different manner then Solvency II since it is a regulatory framework for a different part of the financial industry, the framework does however also use stress scenarios to see the impact of shocks on certain risk drivers. The current Basel II framework uses stress scenarios calibrated on the VaR, but the Basel III framework will be using Expected Shortfall to calibrate the stress scenarios.
2.7
Appropriate risk measure
This Section dealt with finding an appropriate risk measure to estimate sufficient cap-ital. Since Value-at-Risk does, among other things, not consider the tail beyond the confidence level, it is not a very suitable risk measure for a capital requirement calcu-lation. The Expected Shortfall does not only consider the tail beyond the confidence level, it also satisfies the four axioms of translation invariance, subadditivity, positive homogeneity and monotonicity and can therefore be called a coherent risk measure. In this research we therefore chose to calibrate the SCR estimations on Expected Shortfall instead of Value-at-Risk and compare the differences. Before we can do this, we first need to have an insight in the Solvency II framework and the current estimation of the SCR.
Chapter 3
Solvency II
Solvency II is a new regulatory framework for the European insurance industry and puts new and stricter demands on the required economic capital, risk management and reporting standards of insurance companies. Solvency II is the successor of Solvency I, which was introduced in 1973, and its underlying idea “is that insurers should hold an amount of capital that enables them to absorb unexpected losses and meet the obligations towards policy-holder at a high level of equitableness” (Stevens et al., 2010, p.5).
3.1
Three pillars
Similar to the Basel II framework, the Solvency II framework consist of 3 pillars. These pillars are described by De Nederlandsche Bank (2009).
The first pillar prescribes the quantitative requirements which an insurer must meet. It covers both the valuation of the assets and the liabilities as the capital requirements. The valuations are to be done in a market-consistent manner. The capital requirement can be fulfilled by using the standard formula or an internal model. These quantitative requirements are supported by the so-called Quantitative Impact Studies (QIS).
In the second pillar requirements on standards of risk management and governance are given. The third pillar focuses on transparency for supervisors and the public and reporting standards. For the purpose of this paper we will only focus on the quantitative requirements prescribed in pillar I.
3.2
Valuation
EIOPA (2010) describes how the assets and liabilities of an insurance or reinsurance undertaking should be valued: Assets should be valued at the amount for which they could be exchanged between knowledgeable willing parties in an arm’s length transac-tion. Liabilities should be valued at the amount for which they could be transferred, or settled, between knowledgeable willing parties in an arm’s length transaction. Valuing assets on a market-consistent basis is not that complicated for an insurance company, valuing liabilities can become a little more complicated. If the expected cash flows of the liabilities can be perfectly replicated using a portfolio of assets, then the market price of that portfolio can be used to value the liabilities. Often, perfect replication is not possible for the liabilities of an insurance company. In that case QIS5, which is the most recent Quantitative Impact Study, prescribes to use the Best Estimate and the Risk Margin to value the liabilities.
The Best Estimate of the liabilities corresponds to the probability weighted average of discounted future cash flows, or in other words, the present value of the expected future liability payments. The Risk Margin is calculated by determining the cost of holding an amount of own funds equal to the SCR over the lifetime of the insurance
obligation. To determine the cost of holding that amount of own funds a so-called Cost-of-Capital rate is used. The sum of the Best Estimate and the Risk Margin is known as the Technical Provisions. The Risk Margin is a part of the Technical Provisions in order to ensure that the value of the Technical Provisions is equivalent to the amount that insurance companies would be expected to require in order to take over and meet the insurance obligations.
3.3
Solvency Capital Requirement
To make sure that insurers hold sufficient economic capital, they will have to meet new quantitative requirements under the new Solvency II framework. This capital require-ment is called the Solvency Capital Requirerequire-ment (SCR) and covers all the risks that an insurer faces. There are three ways to estimate the SCR: By using and internal model, by using the standard formula or by using a combination of the two. Whether the stan-dard formula or an internal model is used to estimate the SCR a main requirement is prescribed and this follows from EIOPA (2010): “The SCR should correspond to the Value-at-Risk of the basic own funds of an insurance or reinsurance undertaking subject to a confidence level of 99.5% over a one-year period”. Building a full internal model is complicated, this results in a lot of companies using (part of) the standard formula. The principle of the standard formula is to apply a set of shocks to certain risk drivers and calculate the impact on the value of the assets and liabilities for various risks. These shock are calibrated to the 99.5% VaR level. Predefined correlation matrices are used to aggregate to total SCR for all risks together. The standard formula SCR is divided into modules as shown in Figure 3.1.
Figure 3.1: The different modules of the standard formula SCR
3.4
Overview of the main risks for a life annuity insurance
company
In this paper we will only discuss the main risks relevant for a life annuity insurance company. As shown in Figure 3.2 market risk and life risk account together for 91.1%
Solvency II SCR based on Expected Shortfall — Wouter Alblas 11
of the diversified BSCR of a life insurance undertaking (EIOPA, 2011, graph 35) and are therefore the two main risks relevant for a life annuity insurance company.
Figure 3.2: The diversified average BSCR structure following from research done by EIOPA on life insurance undertakings (EIOPA, 2011, graph 35)
3.4.1 Market risk
Market risk is defined by EIOPA (2010) as follows: “Market risk arises from the level or volatility of market prices of financial instruments. Exposure to market risk is measured by the impact of movements in the level of financial variables such as stock prices, in-terest rates, real estate prices and exchange rates”. Market risk is the largest component of the SCR and for life insurance undertakings market risk accounts for 67.4% of the diversified BSCR (Figure 3.2). The largest components within this module are equity risk and interest rate risk.
Equity risk
“Equity risk arises from the level or volatility of market prices for equities. Exposure to equity risk refers to all assets and liabilities whose value is sensitive to changes in equity prices” (EIOPA, 2010, p. 112). A separation is made between “Global” equity and “Other” equity. The first category includes equities listed in EEA or OECD countries. The “Other” equities include equities listed in other than EEA or OECD countries, hedge funds, private equities and other alternative investments.
Interest rate risk
“Interest rate risk exists for all assets and liabilities for which the net asset value is sensitive to changes in the term structure of interest rates or interest rate volatility. This applies to both real and nominal term structures” (EIOPA, 2010, p. 110). Interest rate risk is determined by applying a set of shocks, both upward and downward, to the current yield curve to test the sensitivity of the asset values to interest rate changes. 3.4.2 Life risk
Life risk is defined by EIOPA (2010) as follows: “Life risk covers the risk arising from the underwriting of life insurance, associated with both the perils covered and the processes followed in the conduct of the business”. Life risk is the second largest module and it
accounts for 23.7% of the diversified BSCR for life insurance undertakings (Figure 3.2). The most important component for a life annuity insurance company is longevity risk.
Longevity risk
“Longevity risk is associated with (re)insurance obligations (such as annuities) where a (re)insurance undertaking guarantees to make recurring series of payments until the death of the policyholder and where a decrease in mortality rates leads to an increase in the technical provisions, or with (re)insurance obligations (such as pure endowments) where a (re)insurance undertaking guarantees to make a single payment in the event of the survival of the policyholder for the duration of the policy term” (EIOPA, 2010, p. 151). Longevity risk basically comes down to the risk which is associated to higher than expected pay-outs because of increasing life expectancy.
3.5
Calculation of the SCR
In this research we will only consider equity risk, interest rate risk and longevity risk. In Figure 3.3 an overview of our SCR calculation and its different modules is given.
Figure 3.3: The different modules of our SCR calculation
3.5.1 Market SCR
The market SCR is a combination of the different market sub-risks, in this case equity risk and interest rate risk. This is done by using predefined correlation matrices and the following formulas:
SCRmkt = max(SCRmktup; SCRmktdown),
SCRmktup =
s X
rxc
CorrMktUpr,c∗ Mktup,r∗ Mktup,c,
SCRmktdown =
s X
rxc
CorrMktDownr,c∗ Mktdown,r∗ Mktdown,c,
Solvency II SCR based on Expected Shortfall — Wouter Alblas 13
CorrMktUpr,c = The entries of the correlation matrix CorrMktUp, Mktup,r , Mktup,c = Capital requirements for the individual market risks
un-der the interest rate up stress,
CorrMktDownr,c = The entries of the correlation matrix CorrMktDown,
Mktdown,r , Mktdown,c = Capital requirements for the individual market risks
un-der the interest rate down stress,
and the correlation matrices CorrMktUp and CorrMktDown are defined as:
CorrMktUp Interest Equity
Interest 1 0
Equity 0 1
CorrMktDown Interest Equity
Interest 1 0.5
Equity 0.5 1
Equity SCR
The capital requirement for equity risk is determined by the decrease of the net value of assets minus liabilities (NAV ) after a negative shock has been given to the equity. This negative shock implies that the value of equity will decrease with a certain percentage. The shock differs for “Global” equity and “Other” equity. Mathematically, for each category (“Global” and “Other”) i this comes down to:
Mkteq,i= max(4NAV |equity shocki; 0).
The Equity SCR is a combination of the capital requirements for “Global” equity and “Other” equity, this is done by using a predefined correlation matrix and the following formula: Mkteq= s X rxc CorrIndexr,c∗ Mktr∗ Mktc, where
CorrIndexr,c = The entries of the correlation matrix CorrIndex,
Mktr , Mktc = Capital requirements for equity risk per individual
cat-egory,
and the correlation matrix CorrIndex is defined as:
CorrIndex Global Other
Global 1 0.75
Interest rate SCR
To estimate the capital requirement for interest rate risk, an upward and a downward shock is given to the interest term structure. The altered term structures are derived by multiplying the current interest rate curve by (1 + sup) and (1 + sdown). In Appendix B the upward shocks sup and downward shocks sdown are given for calibrating with ES and VaR and different confidence intervals. Using these altered term structures will off course result in a change of value of interest sensitive assets and liabilities. The capital requirement for the downward and upward shock is determined by the changes in net value of assets minus liabilities when the shocked interest rate curve is used instead of the normal term structure. This leads to the following definitions:
MktintU p = 4NAV |up,
MktintDown = 4NAV |down.
3.5.2 Life SCR
In this research we are only considering longevity risk as a life sub-risk. Therefore the life SCR equals the longevity SCR.
Longevity SCR
The longevity SCR is estimated by the change in net value of assets minus liabilities after a permanent percentage decrease in mortality rates for each age has took place. This decrease means that people will live longer and will lead to a change of the value of the insurance products and therefore also to a change in the value of NAV. The definition of longevity SCR is therefore
Lifelong = (4NAV |longevity shock).
3.5.3 Total SCR
The different risk modules, in our case market risk and life risk, are combined to estimate the SCR. In this paper we will not take into account the SCR for operational risk and the Adjustment for the risk absorbing effect of technical provisions and deferred taxes, therefore BSCR equals SCR. The total SCR is calculated by using a predefined correlation matrix and the following formula:
SCR = s X ij Corri,j ∗ SCRi∗ SCRj, where
Corri,j = The entries of the correlation matrix Corr,
SCRi , SCRj = Capital requirements for the individual SCR risks according
to the rows and columns of the correlation matrix Corr, and the correlation matrix Corr is defined as:
Corr Market Life
Market 1 0.25
Chapter 4
Methodology
This research consists of two parts: The first part will focus on calibrating the SCR stress scenarios for certain main risks based on either the Value-at-Risk or the Expected Shortfall following the calibration methods used in the calibration of Solvency II. The second part consists of comparing the estimated SCR calibrated on Value-at-Risk with the estimated SCR calibrated on Expected Shortfall for a fictitious life annuity insurance company.
4.1
Calibrating the SCR stress scenarios
As mentioned before, in this paper we will focus on the main risks relevant for a life annuity insurance company. The stress scenarios will be calibrated for: Equity risk, interest rate risk and longevity risk. Different confidence levels will be used for the Value-at-Risk and Expected Shortfall to fully show the difference between calibrating with Value- at-Risk and Expected Shortfall.
The derived stress scenarios based on the Value-at-Risk with 99.5% confidence level will probably not be equal to the stress scenarios as noted in by EIOPA (2010). This has two reasons:
• The data used in this research is similar to the data used for the QIS 5 stress scenarios calibration, as named by CEIOPS (2010), except for the fact that the time span of the data differs. Due to lack of availability of older data we will decrease time spans for the data sets compared to the data time spans used by EIOPA. To “compensate” for this missing of some older data, and to make our conclusions more “up-to-date”, all the data will be updated to 2014, instead of data updated to 2009.
• The methods followed by EIOPA to calibrate the stress scenarios are described vaguely in the official literature and therefore we had to make a small assump-tions about the precise calibration method for interest rate risk. To transform the principal components and eigenvectors into VaR and ES (with a certain confi-dence level) based interests rate scenarios, the method described in Frye (1997) was followed.
4.1.1 The calibration of equity risk
The calibration method for equity risk is discussed by CEIOPS (2010). The distri-bution of annual holding period returns derived from the MSCI World Development index is used to calculate the empirical VaR and empirical ES. The data, sourced from Bloomberg, spans a daily period of 41 years, starting in June 1973 until June 2014. The empirical VaR and empirical ES will serve as the stress rate for “Global” equity. Since we assume that the fictitious life annuity insurance company does not hold “Other” equity it is not relevant to calibrate the stress scenarios for “Other” equity.
4.1.2 The calibration of interest rate risk
The calibration method for interest rate risk is discussed by CEIOPS (2010). The fol-lowing 4 datasets are used:
• Euro area government bond yield curve, with maturities from 1 year to 15 years, spaced out in annual intervals. The daily data spans a period of approximately 10 years and runs from September 2004 to July 2014. The data is sourced from the website of the European Central Bank.
• The UK government liability curve. The data is daily and sourced from the website of the Bank of England. The data covers a period from January 1998 to June 2014, and contains rates of maturities starting from 1 year up until 25 year whilst the in between data points are spaced on annual intervals.
• Euro vs Euribor IR swap rates. The daily data is downloaded from Datastream and covers a period from 1999 to 2014. The data contains the 1 to 10 year rates spaced out in one year intervals, as well as the 12 year, 15 year, 20 year, 25 year and 30 year rates.
• UK (GBP) 6m IRS swap rates. The daily data is downloaded from Datastream and covers a period from 1999 to 2014. The data contains the 1 to 10 year rates spaced out in one year intervals, as well as the 12 year, 15 year, 20 year, 25 year and 30 year rates.
In this paper we will calibrate the stress interest rate scenarios by using Principal Component Analysis (PCA) as prescribed by EIOPA. To give some perspective we will also calibrate empirical stress scenarios.
Interest rate risk stress scenarios by using PCA
Principal Component Analysis can be used to describe movements of the yield curve and is explained by Barber and Copper (2010). PCA is mathematically defined as an orthogonal linear transformation that converts data of possibly correlated variables into a set of values of linearly uncorrelated variables. A yield curve change, for m maturities, can then be represented exactly as a linear combination of m vectors:
Xt= b1tU1+ ... + bmtUm
where Xt is the absolute change in the yield curve at time t, Uk is a time independent
m x 1 vector and bkt is a time dependent scalar.
PCA transforms the data into a new coordinate system such that the first coordinate, or principal component b1t, has the greatest variance by any projection of the data, the
second coordinate has the second greatest variance, etcetera. The objective of PCA is to determine a small set of components that best explain the total variance of the data. The number of components is then small K m, but with high explanatory power:
Xt= b1tU1+ ... + bKtUK+ Et
where Et is the error term.
When PCA is used to describe interest rate movements, the first six components pick up between 99.2% and 99.5% of the variance with a 90% confidence interval (Barber and Copper, 2010, p.15). The derived factors Uk, which are the eigenvectors of the
covariance matrix of the original data, are sorted in order of decreasing eigenvalue. These eigenvectors Uk describe the different yield curve movements and the first three
are interpreted as the shift, twist and butterfly moves of the yield curve (Novosyolov and Satchkov, 2009). Figure 4.1 shows the first three eigenvectors of the Euro vs Euribor IR swap rates, the three named yield curve movements can be observed here as well.
Solvency II SCR based on Expected Shortfall — Wouter Alblas 17
Figure 4.1: The shift, twist and butterfly moves of the yield curve. The horizontal axis repre-sents the term to maturity and the vertical axis reprerepre-sents the eigenvector level.
The principal components are derived via a matrix multiplication: bkt= U’kXt for k = 1, 2, ..., K
In the case when PCA is used to describe yield curve movements, the annual absolute interest rate changes are used as input data X. The columns of matrix U represent the eigenvectors of the covariance matrix of the imput data, which are sorted in order of decreasing eigenvalue.
To transform the principal components and eigenvectors into VaR and ES (with a certain confidence level) based interests rate stress scenarios, the method described in Frye (1997) is mostly followed. This method provides an inuitive and rapid VaR esti-mate, but the downside of the method is that it tends to overstate the VaR. In the original method of Frye each principal component vector bk is assumed to be normally
distributed to compute the VaRγ(bk) and ESγ(bk), instead we use the empirical
dis-tribution of each principal component vector to compute the VaRγ(bk) and ESγ(bk) in
an empirical way. So we derive a down risk VaR or ES and an up risk VaR or ES and multiply both separately with the corresponding eigenvector Uk.
VaRγ(bk) ∗ Uk for k = 1, 2, ..., K and for γ = α, (1 − α);
ESγ(bk) ∗ Uk for k = 1, 2, ..., K and for γ = θ, (1 − θ).
These “VaR/ES eigenvectors” are then combined with each other to create VaR or ES level absolute interest rate stress scenarios. The number of combinations that is possible depends on the number of eigenvectors that are chosen.
VaRγ1(b1) ∗ U1+ VaRγ2(b2) ∗ U2+ ... + VaRγK(bK) ∗ UK for γ1,γ2,...,γK = α, (1 − α);
ESγ1(b1) ∗ U1+ ESγ2(b2) ∗ U2+ ... + ESγK(bK) ∗ UK for γ1,γ2,...,γK = θ, (1 − θ);
If you decided that K eigenvectors are sufficient to explain most of the variance, then the number of combinations that are possible is equal to 2K. The up (down) absolute interest rate stress scenarios are then the maximum (minimum) of the combinations for each maturity taken together. This vector of absolute interest rate stress scenarios is then converted to percentage interest rate stress scenarios by dividing it by the average interest rates for each maturity.
Let us look at an example to make it clearer. We assume that in this example two eigenvectors, b1and b2, are sufficient to explain most of the variance, and we assume that
both principal component vectors are normally distributed and that we want the 99.5% VaR interest stress rates.The principal components VaR’s are approximately equal to -2.33 and 2.33 (99.5% quantile and 0.5% quantile of a standard normal distribution) times the standard deviation of the principal component vector. To create a shift up, shift down, twist up and twist down these principal component VaR’s are multiplied with both eigenvectors, U1and U2. By adding the different shift and twist movements we
can create the four scenarios: UpUp, UpDown, DownUp and DownDown. The maximum (minimum) absolute interest rate changes of these 4 scenarios per maturity will then form the absolute up (down) stress vector. The percentage interest rate stress vectors are then computed by dividing it by the average interest rates for each maturity.
Since we have 4 datasets, we will also have 4 different up and down shock vectors. The swap rates are not defined for all maturities between 1 year and 25 years, and therefore linear interpolation is used to fill in shocks for this maturities. For the Euro area government bond yield curve is no extrapolation performed. The mean result of the 4 different up and down shock vectors has been taken to arrive at a generalized up and down shock vector.
Interest rate risk empirical stress scenarios
The annual absolute interest rate changes of the 4 datasets are used to compute the empirical stress scenarios. Per dataset for each maturity the empirical VaR and ES will be calculated and these lead to a vector of absolute up and down shocks. The percentage interest rate stress vectors per dataset are then computed by dividing it by the average interest rates for each maturity. Since the swap rates are not defined for all maturities between 1 year and 25 years, linear interpolation is used to fill in shocks for this maturities. For the Euro area government bond yield curve is no extrapolation performed. The mean result of the 4 different up and down shock vectors has been taken to arrive at a generalized up and down shock vector.
4.1.3 The calibration of longevity risk
The calibration method for Longevity risk is discussed by CEIOPS (2010). The unisex mortality tables of nine countries, namely: Denmark, France, United Kingdom, Estonia, Italy, Sweden, Poland, Hungary and Czech Republic, from 1992 till 2009 are used as data. In this data age bands of five years are used. This data is sourced from the Human Mortality Database. In this paper we will calibrate the longevity stress scenarios by assuming that annual mortality improvements follow a normal distribution as prescribed by EIOPA, to give some perspective we will also calibrate empirical stress scenarios. In the Solvency II framework the same shock in mortality rates is used for all different ages. This is mainly done because of simplicity of calculations. In this research we follow the guidelines set by EIOPA and therefore we also use one mortality shock for all different ages.
Longevity risk stress scenarios by assuming a normal distribution
Annual mortality rate changes are calculated per country, per age band and per year based on the data from the Human Mortality Database. Then the means and standard deviations of the annual mortality improvements are computed, and it is assumed that all follow a normal distribution. In this case we are talking about 198 normal distribu-tions, since we have 9 countries and 22 different age bands. This results in 198 different VaR’s or ES’s. The average of these VaR’s or ES’s will be the one mortality shock for all different ages.
Solvency II SCR based on Expected Shortfall — Wouter Alblas 19
Longevity risk empirical stress scenarios
In this case is assumed that all annual mortality rate changes follow the same distribu-tion. This is assumption is made out of practicality, since it makes it possible to have enough data about the tails of the distribution and we thus can calculate a robust em-pirical VaR or ES. The emem-pirical VaR or emem-pirical ES will serve as the emem-pirical stress rate for longevity risk.
4.2
Comparing the SCR calibrated on VaR with the SCR
calibrated on ES
The second part of our research consists of comparing the estimated SCR calibrated on Value-at-Risk with the estimated SCR calibrated on Expected Shortfall for a fictitious life annuity insurance company.
4.2.1 The fictitious life annuity insurance company
The liability portfolio of this company consists of life annuity products only. For the liability portfolio there is no new accrual assumed, and therefore no incoming cash flows because of premium.
The portfolio consist of 98,758 insured male members. The insured members in this portfolio have a mean age of 50 years, the youngest group is 21 years old and the oldest group has an age of 79 years. Discrete age intervals of 2 years between age 21 and 79 are used. See appendices C and D for a precise overview of the liability portfolio and its expected payouts. We further assume that all members have their birthday on the 1st of January.
The annuity will be paid at the end of every year starting from age 65 if still alive. Since there is no new accrual assumed, the amount been received differs per starting age. For starting age 21 the amount equals e1,000 and this linearly increases over the starting ages up toe15,000 for age 65 and above. Indexation is ignored in this portfolio. We assume that this life annuity insurance company has a funding ratio of 115% and therefore the asset portfolio equals the amount of the liability portfolio times 1.15. The asset portfolio consists for 25% out of stocks and for 75% out of government bonds. Since their AAA credit rating and high yield to maturity compared to other European government bonds, we picked the 5-year and 30-year UK Gilt, which are bonds issued by the British government. The 5-year UK Gilt is a 2.13% coupon bond with a remaining time to maturity of 5 years and the 30-year UK Gilt is a 3.45% coupon bond with a remaining time to maturity of 30 years. The coupon rates are based on the market data of July 2nd 2014. Duration matching will be used to assign the amount invested in the 5-year and 30-year UK Gilts.
4.2.2 Estimating the SCR
The SCR’s for Equity risk, Interest rate risk and Longevity risk will be estimated as described in Chapter 3. For each risk the net value of the assets minus liabilities must be computed after a shock has been given. This will be compared to the net value of the assets minus liabilities in a “shock-free” situation. To discount cash flows the Euro area central government bonds yield curve of June 26th 2014 will be used. We will use the
mortality table of the Dutch Actuarial Society: AG 2012-2062. This is the most recent prognosis of the Dutch Actuarial Society and assembled by a broad delegation of Dutch insurers with actuarial practice in mind. The valuation date is December 31st 2014.
4.2.3 Matching the total SCR value
The SCR based on VaR stress scenarios, SCR(VaRα), will be computed for different
confidence intervals between 99.75% and 95%. To compare the SCR(VaRα) properly
with the SCR(ESθ), the confidence interval for the ES, θ, will be chosen based on
matching the total SCR value. In an equation this comes down to, SCR(VaRα) = SCR(ESθ).
By using this criteria we can properly compare the allocations of the SCR over the different risk modules and clearly see the differences between using VaR stress scenarios and ES stress scenarios.
Chapter 5
Results
In this chapter the results of the research are presented. The first part shows the stress scenarios for equity risk, interest rate risk and longevity risk based on Value-at-Risk and the stress scenarios based on Expected Shortfall. The second part consists of comparing the allocation of SCR(VaRα) with the allocation of SCR(ESθ) for a fictitious life annuity
insurance company.
5.1
The SCR stress scenarios
In this paper the SCR stress scenarios calibrated on VaR99.5%, ES98.76%, VaR98.5% and ES97.10% will be shown, these specific values for θ, are chosen because,
SCR(VaR99.5%) = SCR(ES98.76%),
SCR(VaR98.5%) = SCR(ES97.10%).
These equations are true for the portfolio of the fictitious life annuity insurance company that we use in this research. If another portfolio is used, the values for θ change, but only by a very little amount, therefore we feel that using these values θ is appropriate to give a hint of the differences between the stress rates calibrated on VaR and ES. The stress scenarios calibrated on other confidence intervals will be used in the second part of the research when the SCR(VaRα) is compared with the SCR(ESθ) for a fictitious
life annuity insurance company.
Since the interesting part of this paper is comparing the resulting SCR values and not comparing stress scenarios, we will mainly state stress scenarios and not spend a long time discussing the differences.
5.1.1 Equity risk
Similar to the method named in the QIS 5 Technical Specifications we will use a sym-metric adjustment of 9%, which will lower the equity risk shock. With this symsym-metric adjustment included, the stress rates for equity risk are:
α VaRα ESθ
99.5% -35.16% -35.14% 98.5% -30.25% -29.72%
For α equals 99.5% the difference between using VaR and ES is very small. When a smaller confidence interval of 98.5% is chosen, using VaR leads to a greater shock.
Figure 5.1: Interest rate up shocks for α = 99.5%. The horizontal axis represents the term to maturity and the vertical axis represents up shock percentage. The blue line represents the shocks calibrated on VaR and the red line represents the shocks calibrated on ES.
Figure 5.2: Interest rate down shocks for α = 99.5%. The horizontal axis represents the term to maturity and the vertical axis represents down shock percentage. The blue line represents the shocks calibrated on VaR and the red line represents the shocks calibrated on ES.
5.1.2 Interest rate risk
For the calculation of interest rate stress scenarios a maximum up shift of 100% and a maximum down shift of -90% was used. The Figures 5.1, 5.2, 5.3 and 5.4 show the resulting stress scenarios by using Principal Component Analysis.
Similar as for equity risk, for α equals 99.5% the difference between using VaR and ES is very small, but although the differences are small, calibrating with VaR always leads to greater stress rates. When a smaller confidence interval of 98.5% is chosen, the difference is bigger, but then calibrating with ES leads to greater stress rates.
5.1.3 Longevity risk
The following table displays the stress scenarios for longevity risk, while using the same method as EIOPA, that is by assuming normally distributed mortality rate improve-ments:
Solvency II SCR based on Expected Shortfall — Wouter Alblas 23
Figure 5.3: Interest rate up shocks for α = 98.5%. The horizontal axis represents the term to maturity and the vertical axis represents up shock percentage. The blue line represents the shocks calibrated on VaR and the red line represents the shocks calibrated on ES.
Figure 5.4: Interest rate down shocks for α = 98.5%. The horizontal axis represents the term to maturity and the vertical axis represents down shock percentage. The blue line represents the shocks calibrated on VaR and the red line represents the shocks calibrated on ES.
α VaRα ESθ
99.5% -18.78% -18.87% 98.5% -16.19% -16.90%
For both confidence intervals, calibrating with ES leads to bigger shock rates. The effect is greater when a confidence interval of 98.5% is used.
5.2
Comparing the SCR(VaR
α) with the SCR(ES
θ) for the
fictitious life annuity insurance company
This Section is focused on comparing the SCR(VaRα) with the SCR(ESθ) for the
ficti-tious life annuity insurance company. Since θ is chosen such that the total SCR(VaRα)
equals SCR(ESθ), we will focus on the allocation of the total SCR over the three risk
modules: Equity SCR, interest rate SCR and longevity SCR. By comparing these al-locations for using VaR and ES, we can see if the current method, which uses VaR, underestimates or overestimates certain risks. This is done for different confidence in-tervals α used for VaR.
5.2.1 The changes in allocation of the total SCR
Figure 5.5 shows the change in allocation of the total SCR over the three different risk modules when ES is used to calibrate the shock scenarios instead of VaR. The stress scenarios used to estimate the SCR are as prescribed by EIOPA, that means that the interest rate stress scenarios are estimated by using PCA and the longevity stress scenarios are estimated by assuming Normal distributions.
Further in this research we will vary with the estimation method of the stress scenar-ios, the asset portfolio and the liability portfolio of the fictitious life annuity insurance company. The scenario which uses the stress scenarios estimation methods as prescribed by EIOPA and the portfolios of the fictitious life annuity insurance company as described in chapter 4, will be referred to as the base scenario. Figure 5.5. therefore shows the change in allocation of the total SCR for the base scenario.
As the figure shows, for the confidence interval α used in Solvency II, namely 99.5%, the difference in allocation is very small. When the confidence interval decreases down to 98.5%, the differences become more significant. Then we can see that by using the VaR to determine the shocks, longevity risk and interest rate risk are underestimated and equity risk is overestimated. We can draw this conclusion because the figure shows that when we would use ES to determine the shock the longevity SCR would grow with 4.82%, interest rate SCR would grow with 2.27% and equity SCR would decline with 1.77%, when a confidence interval α of 98.5% is chosen.
These changes do not add up to 0 since the amounts of SCR for the different risk measures differ. Therefore a decline of 2% for equity SCR leads to a greater change in the actual amount of SCR, than an increase of 2% for interest rate risk, because the amount of equity SCR is greater than the amount of interest rate SCR.
When the confidence interval is smaller than 98.5%, the differences become smaller down to 97.5%. In this paper we will ignore confidence intervals lower than 97.5%, since we do not think that these “small” confidence intervals will ever be used in practice.
5.2.2 The changes in allocation of the total SCR with empirical stress scenarios
In order to check how dependent these results are of the estimation methods used by EIOPA, we will now look at the differences in allocation when the stress scenarios for the three risk modules are determined in an empirical way.
Solvency II SCR based on Expected Shortfall — Wouter Alblas 25
Figure 5.5: Comparing the allocation of SCR(VaRα) with the allocation of SCR(ESθ) for the fictitious life annuity insurance company, when the stress scenarios are determined in a similar way as EIOPA prescribes (base scenario). The horizontal axis represents the confidence interval α used for VaR in the calibration method. The vertical axis represents the change in allocation of the SCR’s when the stress scenarios are calibrated on ES instead of VaR.
Figure 5.6 shows the change in allocation of the total SCR over the three different risk modules when ES is used to calibrate the shock scenarios instead of VaR. The stress scenarios used to estimate the SCR are this time determined in an empirical way. Similar as when EIOPA stress scenarios are used, when α = 99.5% the difference in allocation is very small. Once again, the differences are maximized when α equals 98.5%. At that particular confidence interval, the differences are considerably greater if compared to the base scenario, since the longevity SCR is 9.82% higher and the equity SCR is 5.65% lower for SCR(ESθ) when compared with SCR(VaRα). Down to 97.5% we see the same
trend, the differences are declining. The changes in interest rate SCR differ greatly when empirical stress scenarios are used compared to when EIOPA stress scenarios are used. Comparing the scenarios in figure 5.5 and 5.6 there can be concluded that for differ-ent stress scenario estimation methods it still holds that the difference in SCR allocation is maximized at α = 98.5% and that longevity risk is underestimated and equity risk overestimated when VaR is used. The effect of over- and underestimation is however considerably greater when the stress scenarios are determined in an empirical way. Be-sides that the changes in interest rate SCR differ greatly when empirical stress scenarios are used compared to when EIOPA stress scenarios are used.
5.2.3 Intuitive explanation of the changes in allocation of the total SCR
For both using EIOPA stress scenarios and empirical stress scenarios we see that the changes in allocation of the total SCR are very small for the confidence interval α used in Solvency II, namely 99.5%. For that particular confidence interval, there is not a great difference in using VaR or ES to calibrate the stress scenarios. When the confidence interval decreases, this difference becomes more significant and is maximized for α = 98.5%. We see that by using the VaR to determine the shocks, longevity risk is underestimated and equity risk is overestimated, especially at the specific confidence interval of 98.5%. When we decrease the confidence interval even more down to 97.5% the differences grow smaller.
By comparing the empirical distribution of the tail of the data used for equity risk and longevity risk we will intuitively explain why the differences are maximized at 98.5%. The left tail of the data used for equity risk is shown in figure 5.7 and the left tail of the data used for longevity risk is shown in figure 5.8. Both figures have two horizontal
Figure 5.6: Comparing the allocation of SCR(VaRα) with the allocation of SCR(ESθ) for the fictitious life annuity insurance company, when the stress scenarios are determined in an empiral way. The horizontal axis represents the confidence interval α used for VaR in the calibration method. The vertical axis represents the change in allocation of the SCR’s when the stress scenarios are calibrated on ES instead of VaR.
axis. The first horizontal axis represents the annual holding period return for equity risk and the annual mortality changes for longevity risk. On the second horizontal axis is in both cases the (1 - cumulative percentage) displayed.
One of the main disadvantages of VaR is that it does not consider the shape of the tail. When the tail is shaped like the tail of a Normal distribution, this does not lead to problems, but when the distribution has a long shaped tail, then this can lead to an underestimation of the risk. The underestimation is maximized when the ”unexpected” long tail is takes place beyond the VaR level. When the VaR level is captured in the middle of the ”unexpected” long tail, no underestimation will take place.
If you compare Figures 5.7 and 5.8 you see that the distribution of longevity risk has a very long shaped tail, the distribution of equity risk is probably not close to Normal either, but is tail is definitely not as ”unexpectedly” long as the tail of the distribution of longevity risk. This already explains a little why longevity risk is underestimated and equity risk is overestimated. To understand why the difference is maximized at α = 98.5% you need to focus on the second horizontal axis. For equity risk the VaR98.5%
is situated in the middle of the unexpected long tail, therefore there will not be any underestimation of the risk when VaR is used. For longevity risk however, the VaR98.5% is situated just before the start of the unexpected long tail, and this of course leads to a big underestimation of the risk in comparison with using the ES.
5.3
Sensitivity analysis
In this Section we will test the sensitivity and solidness of our results. This will be done by changing the composition of the asset portfolio and the liability portfolio of the fictitious life annuity insurance company.
5.3.1 Changing the asset portfolio
As described in Chapter 4, the fictitious life annuity insurance company which is used in this research has an asset portfolio which consists for 25% out of stocks and for the remainder of government bonds. To test if our results are sensitive to the composition of the asset portfolio we will vary with the percentage of stocks in the portfolio and compare the allocation of SCR(VaRα) with the allocation of SCR(ESθ).
Solvency II SCR based on Expected Shortfall — Wouter Alblas 27
Figure 5.7: The left tail of the empirical distribution of the calibration data used in this research for equity risk. The first horizontal axis represents the annual holding period return, the second horizontal axis displays the (1 - cumulative percentage) and the vertical axis represents the frequency.
Figure 5.8: The left tail of the empirical distribution of the calibration data used in this research for longevity risk. The first horizontal axis represents the annual mortality changes, the second horizontal axis displays the (1 - cumulative percentage) and the vertical axis represents the frequency.
We will compare the allocation of SCR(VaRα) with the allocation of SCR(ESθ) for
the following asset portfolios: • 100% stocks;
• 50% stocks and 50% government bonds;
• 25% stocks and 75% government bonds (base scenario); • 10% stocks and 90% government bonds;
• 100% government bonds.
When the asset portfolio contains government bonds, the amounts invested to the 5-year and 30-year UK Gilts are assigned by using duration matching.
Figures 5.9, 5.10, 5.11 and 5.12 show the change in allocation of the total SCR over the three different risk modules when ES is used to calibrate the shock scenarios instead of VaR for the different asset portfolios. To see how sensitive our results are to the composition of the asset portfolio we can compare these figures to each other and the base scenario in figure 5.5.
When the percentage of stocks in the portfolio increases, (figure 5.9 and 5.10) we see the same trend as in the base scenario. The changes in allocation of the total SCR are very small for a confidence interval α of 99.5%, which is used in Solvency II. For that particular confidence interval, there is not a great difference in using VaR or ES to calibrate the stress scenarios. When the confidence interval decreases, this difference becomes more significant and is maximized for α = 98.5%. For this confidence interval, using the VaR to determine the shocks leads to longevity risk and interest rate risk being underestimated and equity risk being overestimated. We can draw this conclusion because the figures show that when we would use ES to determine the shock the longevity SCR would grow, the interest rate SCR would grow and the equity SCR would decline. When we decrease the confidence interval even more down to 97.5% the differences grow smaller.
Figure 5.11 displays a similar trend as the base scenario, except for one difference. The similarity with the base scenario is that the change in allocation of the total SCR is very small for a confidence interval α of 99.5%, this difference becomes more significant and is maximized for α = 98.5%, and down to 97.5% the difference grows smaller again. This figure differs from the base scenario if you focus on the actual percentage change per risk module SCR. When an asset portfolio of 10% stocks and 90% government bonds is used, the biggest percentage change comes from equity risk SCR. This is different for portfolios with a greater percentage of stocks in their portfolio, since in those cases the biggest percentage change comes from longevity risk. Since the change in equity risk grows in comparison with the base scenario, this implies that the change in longevity SCR and interest rate SCR declines in comparison with the base scenario. In the overall we can however still conclude that longevity risk is underestimated and equity risk overestimated when VaR is used.
Figure 5.12 looks somewhat different, but this makes sense. In this case, the asset portfolio consists for 100% of government bonds, meaning there is no equity risk. We do see however that longevity risk is underestimated for all confidence intervals. We have to keep in mind that asset portfolios consisting for 100% out of bonds or stocks are not very realistic.
Comparing the “realistic” scenarios in figure 5.5, 5.10 and 5.11 there can be con-cluded that for different asset portfolios it still holds that the difference in SCR allocation is maximized at α = 98.5% and that longevity risk is underestimated and equity risk overestimated when VaR is used. The actual percentage change per risk module changes if the percentage of stocks decreases in the portfolio, but the main trends do not change that much.
