Solvency II SCR based on Expected Shortfall

Transcription

1 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: [email protected] Date: September 27, 2014 Supervisor: Dr. T.J. Boonen Second reader: Prof. dr. ir. M.H. Vellekoop Supervisor: Dr. J.C. Gielen

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3 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 modules 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 estimate 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 allocation of SCR(ES θ ) is very close to SCR(VaR α ). This can be explained by noticing that the VaR 99.5% 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

4 Contents Preface vi 1 Introduction 1 2 An appropriate risk measure to estimate sufficient capital Coherent Risk Measures Short overview of well known risk measures Risk measure estimation methods Historical Simulation The delta-normal method Monte Carlo simulation Non-normality of financial risks Advantages and disadvantages of risk measures Standard deviation Value-at-Risk Expected Shortfall Other regulatory frameworks The Swiss Solvency Test Basel III Appropriate risk measure Solvency II Three pillars Valuation Solvency Capital Requirement Overview of the main risks for a life annuity insurance company Market risk Life risk Calculation of the SCR Market SCR Life SCR Total SCR Methodology Calibrating the SCR stress scenarios The calibration of equity risk The calibration of interest rate risk The calibration of longevity risk Comparing the SCR calibrated on VaR with the SCR calibrated on ES The fictitious life annuity insurance company Estimating the SCR Matching the total SCR value iv

5 Solvency II SCR based on Expected Shortfall Wouter Alblas v 5 Results The SCR stress scenarios Equity risk Interest rate risk Longevity risk Comparing the SCR(VaR α ) with the SCR(ES θ ) for the fictitious life annuity insurance company The changes in allocation of the total SCR The changes in allocation of the total SCR with empirical stress scenarios Intuitive explanation of the changes in allocation of the total SCR Sensitivity analysis Changing the asset portfolio Changing the liability portfolio Conclusion Summary of the findings Discussion References 35 Appendices 37

6 Preface 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 research 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 Management. Finally, I would like to thank my parents, girlfriend and friends for supporting and distracting me during this adventure. vi

7 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 supervisory framework is scheduled for January 1 st Solvency II is the successor of Solvency I, which was introduced in 1973, and its main objective is to ensure that insurance 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 measure is the Expected Shortfall. This risk measure 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 regulatory 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 1

8 2 Wouter Alblas Solvency II SCR based on Expected Shortfall 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.

9 Chapter 2 An appropriate risk measure to estimate sufficient capital Risk measures are, among other things, used to estimate the amount of sufficient economic 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) Coherent Risk Measures Coherent risk measurers were introduced by Artzner, et al. (1999). They discuss desirable 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 X 1 and X 2, ρ(x 1 + X 2 ) ρ(x 1 ) + ρ(x 2 ). 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 X 1 can be neutralized by a positive performance of X 2 and therefore total risk of the combined portfolio decreases. 3

10 4 Wouter Alblas Solvency II SCR based on Expected Shortfall 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 = E((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 VaR α (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 VaR α (X). The Expected Shortfall 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 ρ(x) = µ ES α (X) = µ E(X X VaR α (X)).

11 Solvency II SCR based on Expected Shortfall Wouter Alblas 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 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 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 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 distributed, 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 normality, 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.

12 6 Wouter Alblas Solvency II SCR based on Expected Shortfall 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 Standard deviation When modern portfolio theory was introduced by Markowitz (1952), the standard deviation 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 standard 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 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 nonnormally 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.

13 Solvency II SCR based on Expected Shortfall Wouter Alblas 7 Especially when we deal with non-normal distributed risk, the VaR can really underestimate 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 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, subadditivity, 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 potential 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 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%.

14 8 Wouter Alblas Solvency II SCR based on Expected Shortfall Basel III The Basel III framework is a global regulatory framework for banks which is planned to be implemented in 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 capital. 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 calculation. 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.

15 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 transaction. 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 9

16 10 Wouter Alblas Solvency II SCR based on Expected Shortfall obligation. To determine the cost of holding that amount of own funds a so-called Costof-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 requirement is called the Solvency Capital Requirement (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 standard 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%

17 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) 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, interest 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 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

18 12 Wouter Alblas Solvency II SCR based on Expected Shortfall 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 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: where SCR mkt = max(scr mktup ; SCR mktdown ), SCR mktup = CorrMktUp r,c Mkt up,r Mkt up,c, rxc SCR mktdown = CorrMktDown r,c Mkt down,r Mkt down,c, rxc

19 Solvency II SCR based on Expected Shortfall Wouter Alblas 13 CorrMktUp r,c = The entries of the correlation matrix CorrMktUp, Mkt up,r, Mkt up,c = Capital requirements for the individual market risks under the interest rate up stress, CorrMktDown r,c = The entries of the correlation matrix CorrMktDown, Mkt down,r, Mkt down,c = Capital requirements for the individual market risks under 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 Equity 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: Mkt eq,i = max( NAV equity shock i ; 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: where Mkt eq = CorrIndex r,c Mkt r Mkt c, rxc CorrIndex r,c = The entries of the correlation matrix CorrIndex, Mkt r, Mkt c = Capital requirements for equity risk per individual category, and the correlation matrix CorrIndex is defined as: CorrIndex Global Other Global Other

20 14 Wouter Alblas Solvency II SCR based on Expected Shortfall 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 + s up ) and (1 + s down ). In Appendix B the upward shocks s up and downward shocks s down 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: Life SCR Up Mktint = NAV up, Down Mktint = NAV down. 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 Total SCR Life long = ( NAV longevity shock). 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 = Corr i,j SCR i SCR j, where Corr i,j = The entries of the correlation matrix Corr, ij SCR i, SCR j = 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 Life

21 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 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 assumptions about the precise calibration method for interest rate risk. To transform the principal components and eigenvectors into VaR and ES (with a certain confidence level) based interests rate scenarios, the method described in Frye (1997) was followed The calibration of equity risk The calibration method for equity risk is discussed by CEIOPS (2010). The distribution 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 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. 15

22 16 Wouter Alblas Solvency II SCR based on Expected Shortfall The calibration of interest rate risk The calibration method for interest rate risk is discussed by CEIOPS (2010). The following 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 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 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 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: X t = b 1t U b mt U m where X t is the absolute change in the yield curve at time t, U k is a time independent m x 1 vector and b kt is a time dependent scalar. PCA transforms the data into a new coordinate system such that the first coordinate, or principal component b 1t, 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: X t = b 1t U b Kt U K + E t where E t 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 U k, which are the eigenvectors of the covariance matrix of the original data, are sorted in order of decreasing eigenvalue. These eigenvectors U k 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.

23 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 represents the term to maturity and the vertical axis represents the eigenvector level. The principal components are derived via a matrix multiplication: b kt = U k X t 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 estimate, but the downside of the method is that it tends to overstate the VaR. In the original method of Frye each principal component vector b k is assumed to be normally distributed to compute the VaR γ (b k ) and ES γ (b k ), instead we use the empirical distribution of each principal component vector to compute the VaR γ (b k ) and ES γ (b k ) 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 U k. VaR γ (b k ) U k ES γ (b k ) U k for k = 1, 2,..., K and for γ = α, (1 α); 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. VaR γ1 (b 1 ) U 1 + VaR γ2 (b 2 ) U VaR γk (b K ) U K ES γ1 (b 1 ) U 1 + ES γ2 (b 2 ) U ES γk (b K ) U K for γ 1,γ 2,...,γ K = α, (1 α); for γ 1,γ 2,...,γ K = 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, b 1 and b 2, are sufficient to explain most of the variance, and we assume that

24 18 Wouter Alblas Solvency II SCR based on Expected Shortfall 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 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, U 1 and U 2. 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 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 distributions, 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.

25 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 distribution. 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 empirical VaR or ES. The empirical VaR or empirical ES will serve as the empirical 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 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 1 st 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 to e15,000 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 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 2 nd Duration matching will be used to assign the amount invested in the 5-year and 30-year UK Gilts 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 26 th 2014 will be used. We will use the mortality table of the Dutch Actuarial Society: AG 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 31 st 2014.

26 20 Wouter Alblas Solvency II SCR based on Expected Shortfall Matching the total SCR value The SCR based on VaR stress scenarios, SCR(VaR α ), will be computed for different confidence intervals between 99.75% and 95%. To compare the SCR(VaR α ) properly with the SCR(ES θ ), 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(VaR α ) = SCR(ES θ ). 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.

27 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(VaR α ) with the allocation of SCR(ES θ ) for a fictitious life annuity insurance company. 5.1 The SCR stress scenarios In this paper the SCR stress scenarios calibrated on VaR 99.5%, ES 98.76%, VaR 98.5% and ES 97.10% will be shown, these specific values for θ, are chosen because, SCR(VaR 99.5% ) = SCR(ES 98.76% ), SCR(VaR 98.5% ) = SCR(ES 97.10% ). 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(VaR α ) is compared with the SCR(ES θ ) 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 Equity risk Similar to the method named in the QIS 5 Technical Specifications we will use a symmetric adjustment of 9%, which will lower the equity risk shock. With this symmetric adjustment included, the stress rates for equity risk are: α VaR α ES θ 99.5% % % 98.5% % % 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. 21

28 22 Wouter Alblas Solvency II SCR based on Expected Shortfall 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 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 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 improvements:

29 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.

30 24 Wouter Alblas Solvency II SCR based on Expected Shortfall α VaR α ES θ 99.5% % % 98.5% % % 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(VaR α ) with the SCR(ES θ ) for the fictitious life annuity insurance company. Since θ is chosen such that the total SCR(VaR α ) equals SCR(ES θ ), 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 allocations 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 intervals α used for VaR 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 scenarios, 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 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.

31 Solvency II SCR based on Expected Shortfall Wouter Alblas 25 Figure 5.5: Comparing the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ) 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(ES θ ) when compared with SCR(VaR α ). 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 different 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. Besides that the changes in interest rate SCR differ greatly when empirical stress scenarios are used compared to when EIOPA stress scenarios are used 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

32 26 Wouter Alblas Solvency II SCR based on Expected Shortfall Figure 5.6: Comparing the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ) 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 VaR 98.5% 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 VaR 98.5% 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 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(VaR α ) with the allocation of SCR(ES θ ).

33 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.

34 28 Wouter Alblas Solvency II SCR based on Expected Shortfall We will compare the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ) 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 concluded 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.

35 Solvency II SCR based on Expected Shortfall Wouter Alblas 29 Figure 5.9: Comparing the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ) for the fictitious life annuity insurance company, with an asset portfolio consisting of 100% stocks. 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 5.10: Comparing the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ) for the fictitious life annuity insurance company, with an asset portfolio consisting of 50% stocks and 50% government bonds. 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.

36 30 Wouter Alblas Solvency II SCR based on Expected Shortfall Figure 5.11: Comparing the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ) for the fictitious life annuity insurance company, with an asset portfolio consisting of 10% stocks and 90% government bonds. 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 5.12: Comparing the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ) for the fictitious life annuity insurance company, with an asset portfolio consisting of 100% government bonds. 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 Changing the liability portfolio To test if our results are sensitive to the composition of the liability portfolio we will vary with the distribution of the insured members and compare the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ). We will compare the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ) for the following liability portfolios: Young life annuity insurance company: Insured members with an average age of 35; Average life annuity insurance company: Insured members with an average age of 50 (base scenario); Old life annuity insurance company: Insured members with an average age of 65. The annuity that will be paid does not differ from the annuity described in Chapter 4. All other named assumptions made in Chapter 4 do still hold. For a precise description of the liability portfolios see appendices C, E and G. The number of insured members

37 Solvency II SCR based on Expected Shortfall Wouter Alblas 31 Figure 5.13: Comparing the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ) for the fictitious young life annuity insurance company, where the insured members have an average age of 35. 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. per liability portfolio is chosen based on matching the Best Estimate of the liabilities to the Best Estimate of the liabilities of the base scenario. The asset portfolio will consist of 25% stocks and 75% government bonds, similar to the base scenario. Duration matching is used to assign the amounts invested to the 5- year and 30-year UK Gilts. To match the duration in the case of the young life annuity insurance company Canadian 50-year government bonds are added to the portfolio, these are 2.96% coupon bonds with a remaining time to maturity of 50 years. Figures 5.13 and 5.14 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 liability portfolios. To see how sensitive our results are to the composition of the liability portfolio we can compare these figures to each other and the base scenario in Figure 5.5. Figure 5.5, 5.13 and 5.14 have a most things in common. Namely, 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%, down to 97.5% the difference grows smaller again. We see that for all liability portfolios that longevity risk is underestimated and equity risk overestimated when VaR is used. For a confidence interval of 98.5%, longevity SCR would grow with over 4% and equity SCR would decline with close to 2% if ES was used instead of VaR for all three different liability portfolios. The only thing that differs for the different liability portfolios is interest rate SCR, if you compare Figure 5.13 with 5.14 you can clearly see that interest rate SCR changes are dependent on the composition of the liability portfolio. We can conclude that for different liability 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. Also can be concluded that interest rate SCR is dependent on the composition of the liability portfolio.

38 32 Wouter Alblas Solvency II SCR based on Expected Shortfall Figure 5.14: Comparing the allocation of SCR(VaR α ) with the allocation of SCR(ES θ ) for the fictitious old life annuity insurance company, where the insured members have an average age of 65. 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.

39 Chapter 6 Conclusion This research paper 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. This Chapter consists of a summary of the findings and a discussion. 6.1 Summary of the findings In the first part of this paper a literature study is undertaken which deals with finding an appropriate risk measure to estimate sufficient capital. 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 calculation. 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. The quantitative research of this paper consist of two parts: The first focusses on calibrating the SCR 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 SCR(VaR α ) with the SCR(ES θ ) for a fictitious life annuity insurance company. Since θ is chosen such that the total SCR(VaR α ) equals SCR(ES θ ), we focused on the allocation of the total SCR over the three risk modules: Equity SCR, interest rate SCR and longevity SCR. By comparing these allocations for using VaR and ES, we saw if the current method, which uses VaR, underestimates or overestimates certain risks. This was done for different confidence intervals α used for VaR. For the confidence interval α used in Solvency II, namely 99.5%, the difference in allocation is very small between SCR(VaR α ) with the SCR(ES θ ). When the confidence interval decreases down to 98.5%, the differences are maximized. Then it is shown that by using the VaR to determine the shocks, longevity risk is significantly underestimated and equity risk is overestimated. This conclusion can be drawn because when ES is used to determine the shock, the longevity SCR would grow and the equity SCR would decline if compared to VaR is used. Down to the confidence interval of 97.5% the differences become smaller again. These results are very solid in terms of variation with the composition of the asset and liability portfolio of the life annuity insurance company. If the percentage of stocks in the portfolio declines, the wight given to the overestimation of equity risk increases and the weight given to the underestimation of longevity risk declines. Interest rate SCR is very sensitive to changes in the liability portfolio and that is one of the reasons that we can not draw one uniform conclusion about the behaviour of interest rate SCR. To test 33

40 34 Wouter Alblas Solvency II SCR based on Expected Shortfall the dependence of the results on the estimation methods used by EIOPA, we compared the results with the differences in allocation when the stress scenarios were determined in an empirical way. There we found that when empirical stress scenarios were used, the difference in allocation is also maximized for α = 98.5%, and compared to estimation methods used by EIOPA the difference in allocation increased, meaning longevity risk is more underestimated and equity risk is more overestimated when empirical stress scenarios are used. Overall, we can conclude that for EIOPA s current confidence interval α of 99.5%, the allocation of SCR(ES θ ) is very close to SCR(VaR α ). This can be explained by noticing that the VaR 99.5% 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 will most likely take place if the current data is used. This under- and overestimation is maximized for α = 98.5%. If EIOPA chooses to use different calibration data then is used now, then a similar research like this will be able to show if any underestimation might take place. 6.2 Discussion There are some limitations to this research paper. First, the literature of EIOPA about the calibration methods is very vague, therefore we had to make an assumption about the calibration method for interest rate risk. There is assumed that the method of Frye (1997) is used to transform principal components and eigenvectors into VaR and ES based interest rate stress scenarios. The downside of this method is that it tends to overstate the VaR. Secondly, in this research we use a fictitious life annuity insurance company with one annuity product, this is a simplification since most life annuity insurance companies have more complex liability portfolios. The conclusions could be made more relevant to the current market if market data of a few life annuity insurance companies was taken to see the different in allocation between the allocation of SCR(ES θ ) and SCR(VaR α ). Finally, this research is very dependent on the calibration data which is used. We use similar data as prescribed by EIOPA, except for using different time spans, but we feel that the conclusions are relevant for the current Solvency II framework. If the data is replaced this might result in different conclusions, but this is the case for various economic research papers.

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42 36 Wouter Alblas Solvency II SCR based on Expected Shortfall Markowitz, H. (1952). Portfolio Selection, The Journal of Finance, 7, Novosyolov, A. and D. Satchkov (2008) Global term structure modelling using principal component analysis, Journal of Asset Management, 9, R Development Core Team (2012). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL ISBN Stevens, R., A. De Waegenaere and B. Melenberg (2010). Calculating capital requirements for longevity risk in life insurance products using an internal model in line with Solvency II. 20ralph%20stevens.pdf, consulted on 5 June Yamai, J. and T. Yoshiba (2004). Value-at-risk versus expected shortfall: A practical perspective, Journal of Banking and Finance, 29, Zhu, M. (2010). Portfolio optimization with tail risk measures and non-normal returns /16268/PhD%20Dissertation%20Minfeng%20Zhu% pdf?sequence=1, consulted on June 24, 2014.

43 Appendices Appendix A: Zero curve The Euro area central government bonds yield curve of June 26 th

44 38 Wouter Alblas Solvency II SCR based on Expected Shortfall Appendix B: Interest rate stress scenarios SCR(VaR 99.5% ) SCR(ES 98.76% ) SCR(VaR 98.5% ) SCR(ES 97.10% ) Maturity Up Down Up Down Up Down Up Down % -90.0% 75.6% -90.0% 69.4% -90.0% 70.8% -90.0% % -90.0% 76.4% -90.0% 66.8% -90.0% 68.9% -90.0% % -89.5% 67.7% -89.4% 59.2% -87.6% 61.1% -88.0% % -86.9% 63.5% -86.8% 55.9% -83.5% 57.6% -84.3% % -81.0% 61.4% -80.9% 54.5% -74.4% 56.1% -76.1% % -72.8% 57.4% -72.5% 51.0% -65.8% 52.5% -67.4% % -63.6% 51.5% -63.3% 45.8% -57.2% 47.2% -58.7% % -57.9% 48.2% -57.6% 42.9% -51.9% 44.2% -53.3% % -57.9% 49.5% -57.6% 44.2% -52.0% 45.4% -53.4% % -56.7% 48.7% -56.4% 43.5% -51.0% 44.6% -52.4% % -55.9% 48.1% -55.7% 42.9% -50.4% 44.0% -51.7% % -56.2% 48.3% -55.9% 43.1% -50.8% 44.2% -52.0% % -55.6% 47.8% -55.3% 42.6% -50.3% 43.7% -51.4% % -54.7% 47.0% -54.4% 41.8% -49.4% 42.9% -50.6% % -54.1% 46.6% -53.8% 41.5% -48.9% 42.5% -50.0% % -52.5% 43.8% -52.3% 39.1% -48.4% 40.0% -49.1% % -53.0% 44.6% -52.7% 39.8% -48.8% 40.7% -49.5% % -53.3% 45.2% -53.1% 40.5% -49.1% 41.4% -49.8% % -53.6% 45.8% -53.4% 41.1% -49.4% 42.0% -50.0% % -53.8% 46.4% -53.6% 41.6% -49.6% 42.5% -50.2% % -54.6% 47.5% -54.4% 42.7% -50.3% 43.6% -51.0% % -55.3% 48.7% -55.1% 43.8% -51.0% 44.7% -51.7% % -56.1% 49.8% -55.9% 44.9% -51.7% 45.8% -52.4% % -56.7% 50.8% -56.5% 45.9% -52.3% 46.8% -53.0% % -57.4% 51.9% -57.2% 46.9% -52.9% 47.8% -53.6%

45 Solvency II SCR based on Expected Shortfall Wouter Alblas 39 Appendix C: Liability portfolio of the average life annuity insurance company (base scenario) Group Number of policies Age Annuity paid yearly starting at , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Appendix D: Expected payouts for the average life annuity insurance company (base scenario)

46 40 Wouter Alblas Solvency II SCR based on Expected Shortfall Appendix E: Liability portfolio of the young life annuity insurance company Group Number of policies Age Annuity paid yearly starting at , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Appendix F: Expected payouts for the young life annuity insurance company

47 Solvency II SCR based on Expected Shortfall Wouter Alblas 41 Appendix G: Liability portfolio of the old life annuity insurance company Group Number of policies Age Annuity paid yearly starting at , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Appendix H: Expected payouts for the old life annuity insurance company