Measuring Economic Capital: Value at Risk, Expected Tail Loss and Copula Approach

Transcription

1 Measuring Economic Capital: Value at Risk, Expected Tail Loss and Copula Approach by Jeungbo Shim, Seung-Hwan Lee, and Richard MacMinn August 19, 2009 Please address correspondence to: Jeungbo Shim Department of Business Administration Illinois Wesleyan University P.O. Box 2900 Bloomington, IL Jeungbo Shim Assistant Professor Illinois Wesleyan University Bloomington, IL Tel: Seung-Hwan Lee Assistant Professor Illinois Wesleyan University Bloomington, IL Tel: Richard MacMinn Professor Illinois State University Normal, IL Tel:

2 Measuring Economic Capital: Value at Risk, Expected Tail Loss and Copula Approach Abstract Investigating dependence structure between variables is important when modeling multivariate data in the study of finance and insurance risks. In this paper, we explore these practical applications for assessing the economic capital measured by Value at Risk and Expected Tail Loss. The procedures are based on copula functions that provide a flexible methodology for modeling multivariate dependence. Using copulas, we examine the impact of dependence structure between market risk and underwriting risk of insurance portfolio on the insurer s economic capital, particularly for the sample of the U.S. property liability insurance industry. Special attention is given to the issue of choosing the copula that best fits our application data. We also assess diversification benefit by comparing the difference between the diversified VaR and the undiversified VaR. Numerical studies show that the insurer s capital requirements and the level of diversification benefit may vary depending on the choice of copulas. The result suggests that t copula is preferred to Gaussian copula in estimating economic capital for our given data. Keywords: Copulas; Value at Risk; Dependence; Diversification; Property-liability insurers 1. Introduction Economic capital is taking on growing importance along with the increasing natural and man-made catastrophes within the insurance industry. One survey revealed that 65% of all insurance respondents calculate a form of economic capital as part of their business practice and an additional 19% are considering calculating economic capital. 1 Despite its increasing significance, there is currently no consensus as to how to define and calculate economic capital. We present a model for assessing the economic capital in an insurance context. Capital is the financial cushion that insurance companies use to absorb adverse consequences due to catastrophic claims costs or unfavorable asset returns. 2 The meaning of capital varies depending on the viewpoint. Although not always precisely defined, economic capital is distinct from regulatory capital in the sense that 1 The survey implemented by Tillinghast included responses from insurance executives in over 200 global companies, including 32 respondents from North American life insurers. See the details in the 2006 Tillinghast ERM survey at 2 Capital is used interchangeably with policyholder surplus in the insurance industry. 2

3 Figure 1 VaR and Economic Capital regulatory capital is the mandatory minimum capital required by the regulators while economic capital serves to underpin all the actual risks borne by the insurer. Conceptually, economic capital can be expressed as protection needed to secure survival in a worst case scenario. More specifically, economic capital can be defined as the difference between the expected loss and a worst tolerable loss at a selected confidence level. Expected loss is the anticipated average loss over a defined period of time and generally imputed as a cost of doing business. Expected loss does not contribute to the capital charge since funds equivalent to the expected loss are held in the insurer s reserves. Because economic capital is the level of capital that the firm should hold to maintain its probability of default below a certain threshold (1 c %), economic capital can be viewed as the Value at Risk (VaR) at a given confidence level. Economic capital is based on a probabilistic assessment of potential future losses and therefore itself random. The definition of VaR and economic capital is presented graphically in Figure 1. Because asset-side risks and insurance underwriting risks are aggregated to determine the insurer s total economic capital, the distribution is based on the aggregate return. Thus, VaR is at left tail where negative returns imply the losses of the company portfolio value. 3

4 To estimate total economic capital for an insurance company, standalone economic capital calculated separately for each risk type must be aggregated. The conventional approach for aggregation is simply to add up standalone risk capital. This simple additive approach may be easy to implement but overlooks potentially important diversification effects. The risks of insurance company that should be covered under the economic capital approach can be categorized into primary components of asset risk, liability risk, and operational risk. 3 Because these risks are less than perfectly correlated, the potential for diversification suggests that the whole risk will be less than the simple sum of standalone risk. In modeling multivariate risks, the commonly used measure of dependence structure is the linear correlation. Linear correlation is a natural scalar measure of dependence under the assumption of elliptical distributions such as the multivariate normal. However, the joint distributions of the real world in finance and insurance are not always the case of the multivariate normal. Embrechts et al. (2002) point out that the presence of non-linear derivative products and skewed heavy-tailed insurance claims invalidates the assumption of multivariate normal distribution. Therefore, the use of linear correlation under the assumptions of normality can mislead the level of real dependence between different risks (Wirch, 1999). Furthermore, linear correlation is unable to fully capture the degree of dependence in the tail of the underlying distribution where extreme events are highly dependent. Thus, linear correlation may not be an appropriate dependence measure for heavy-tailed distributions (Embrechts et al., 2002). 3 Asset risk is generally decomposed of credit risk (e.g. when debt obligations are not met) and market risk (e.g. a fall in the value of an insurer's investments). Liability risk refers to insurance underwriting risk (e.g. the risk that arises from under-estimating the liabilities from business written or inadequate pricing). Operational risk includes a risk of system failure or malpractice. 4

5 An alternative and theoretically more robust method to capture a wide of range of dependence structure amongst different risks is to use copulas. Copulas formulate a joint multivariate distribution by combining univariate marginal distributions in such a way that various types of dependence can be represented. Because a copula embodies all the information about the dependence between the components of random vector, it has several advantages as a dependence measure. For example, copulas identify both linear and non-linear dependence between random variables. More importantly, copulas well capture the upper tail or the lower tail behavior. Thus, copulas can be a useful technique to analyze dependence structure in an insurance setting where extreme events appear to occur simultaneously and asset risks and liability risks diversify against one another to some degree. There are several studies that model dependence structure with copulas in the insurance literature. Among them, Frees et al. (1996) examine the effects of dependent mortality models on annuity valuation with data from a Canadian insurer. They estimate the probability of joint survivorship of two annuitants using a bivariate survivorship function called Frank copula. Their estimation results provide evidence that strong positive dependence exists between joint lives and annuity values decrease when dependent mortality models are used. Tang and Valdez (2006) examine the differences of the insurer s capital requirement under different copula assumptions. They perform simulation using historical loss ratios of five lines of business for the aggregated Australian industry to account for possible dependencies between insurance risks. They find that the choice of copula has a significant impact on the multi-line insurer s capital requirement and diversification benefit. As they point out, one limitation of their analysis is focusing merely on insurance underwriting risks, not considering other sources of risks such as market risk. In reality, market risk is one of 5

6 dominant risks for property-liability insurers, accounting for 30% of the total economic capital of a typical insurer, while insurance underwriting risk accounts for 50% of the capital requirement (Oliver, Wyman & Company, 2001). 4 We contribute to the literature in two main aspects. First, we incorporate two major categories of risks- market risk (asset-side risk) as well as insurance underwriting risk (liability-side risk) in modeling the insurer s total economic capital. According to the new solvency requirements developing across the EU, insurers are required to hold capital against market risk, credit risk and operational risk, which are not covered by contemporary practice. 5 The current EU regime concentrates mainly on the insurance underwriting risks. Our study will provide practitioners and regulators with important insights in determining total capital requirements where both asset-side risks and liability-side risks and their interactions are considered. Second, we attempt to identify which copula is the best one for the given application data. The selection of the copula that best fits the data is critical since the estimates of capital requirements can be significantly different according to the dependence structure chosen. In addition, we quantify diversification benefit for each copula model by comparing the difference between the diversified VaR and the undiversified VaR on the aggregated portfolio returns. In this paper, we focus on how different dependence structure between market risk and underwriting risk has a substantial impact on the insurer s total required 4 Credit risk and operational risk account for 5% and 15% of the total economic capital requirement of a typical property liability insurer on average, a modest amount in comparison to the 55% and 25%, respectively for a bank (Oliver, Wyman & Company, 2001). 5 The new Solvency II rules will introduce economic risk-based solvency requirements and be expected to operate in 2012 across all EU Member States. The new regime would establish more risk-sensitive and harmonized solvency requirements. 6

7 Figure 2 Process of Risk Aggregation and Economic Capital Calculation Market Risk Underwriting Risk Stock Marginal Mortgage Marginal HO Marginal WC Marginal Rank Correlations beteen Risks Monte Carlo Simulation (Random returns/loss) Elliptical Copulas (Gaussian, t, Cauchy) VaR economic capital using the sample of U.S. property-liability insurance industry. 6 We include six business lines to represent insurer s underwriting risk and five categories of assets are chosen as proxies for market risks. We first choose the best fitting marginal distribution for each market risk class and business line based on historical quarterly returns. Using estimated parameters from selected marginal distributions and Spearman s rank correlation coefficients as inputs in the simulation algorithm, we then perform Monte Carlo simulation to generate random return series for each market and underwriting risk under several different copulas. To estimate the insurer s economic capital, we calculate both the Value at Risk (VaR) and the Expected Tail Loss (ETL) from the aggregated return distribution. Figure 2 illustrates conceptually risk aggregation process that various marginal distributions of market 6 The recent report by the International Actuarial Association (IAA) and Solvency II rules generally emphasize the importance of recognizing and modeling interdependencies of multiple risks. We do not include credit risk and operational risk due to data limitations in this paper. 7

8 and underwriting risk components are combined to generate a single return distribution- and economic capital- for the insurer. This paper is organized as follows. Section 2 describes a brief introduction to copula theory, and risk measures such as VaR and ETL are discussed in Section 3. Section 4 illustrates how to measure the difference between diversified VaR and undiversified VaR. Section 5 provides numerical examples and conclusions for the assessment of economic capital are presented in Section Copula Functions as a Dependence Measure The statistical properties and applications of copulas have been extensively studied in the recent works (e.g.; Frees and Valdez, 1998; Breymann et al., 2003; Demarta and McNeil, 2003; Trivedi and Zimmer, 2005). Copula functions are used to fully capture real dependence amongst different risks. We first discuss several approaches of dependence measures and then present basic properties of copulas, their relationships to measures of dependence, and technical algorithms that generate random vectors. 2.1 Dependence measures The choice of the appropriate dependence measure is essential in determining capital requirements. Consider an insurer with k classes of asset portfolios and n k lines of insurance business facing return random vector, ( X1,..., X,..., X n). The insurer s total return is given by the weighted sum of X i s. We assume that is a random return variable and has a marginal distribution, F ( x) = P( X x). The dependence between random variables, X,...,, 1 X n is determined by their joint distribution function 1 n 1 1 n n i k i T X i F ( x,..., x ) = P( X x,..., X x ). (1) 8

9 The traditional approach that measures the dependence is to use Pearson s linear correlation coefficient, assuming that random variables have jointly a multivariate normal distribution. If ( X 1,..., X n ) has a multivariate normal distribution and more generally an elliptical distribution, it is unproblematic to employ the correlation matrix as a measurement of dependence structure. However, most real world risks such as credit or operational risks are not likely to have an elliptical distribution (Shaw, 1997). In addition, Pearson s correlation coefficient is influenced by extreme values, and not invariant under non-linear monotonic increasing transformations of random variables, resulting in incomplete description of dependence structure (Kumar and Shoukri, 2008). Thus, the use of linear correlation in the jointly non-normal distributions may have serious deficiency in measuring real dependency. Two widespread concordance (global trend) measures are the Spearman s rank and Kendall s rank correlations. They are non-parametric statistics used to measure the strength of association between random variables in rank-ordered scale. It is wellknown that both Spearman s and Kendall s correlations are invariant under any monotonic transformations. However, they are unlikely to fully capture extreme tail dependence which is a major concern in quantifying capital charge. It is also very complicated to estimate the multivariate joint probability distribution with these rank correlations, especially when a large number of variables are involved. An alternative to measure the dependence structure is copulas that overcome the aforementioned limitations of the correlations. Copulas allow for any choice of a marginal for each risk type and facilitate a more precise description of extreme outcomes. Copulas provide a way of isolating the peculiar behavior of individual risk s marginal distribution from the description of dependence structure. Thus, T 9

10 copula modeling fits well into an insurance framework, where each asset and liability risk of the insurance company has a different distributional characteristic. 2.2 Copula Definition 7 From a statistical point of view, a copula is a function that combines univariate marginal distributions to construct a joint distribution with a specific dependence structure. Thus it provides a way to create probability distributions to model dependent multivariate components. n To elaborate, a copula is a function, C :[0,1] [0,1], such that 1. For ui [0,1], i = 1,..., n, C(0,...,0, u i,0,...,0) = 0 and C(1,...,1, ui,1,...,1) = ui. n 2. For all ( u11,..., u1n ),( u21,..., u2n) [0,1], with u, 1i u2 i i {1,, n}, 2 2 i1 + + in componentwise, ( 1) C( u, u ) 0. i1 = 1 in = 1 The important aspect of copulas is summarized by Sklar s theorem (1959), which shows that the univariate marginal behavior and the multivariable dependence structure can be separated from the joint multivariate distributions. Theorem (Sklar, 1959) If F is a joint distribution function with marginal distributions, F,...,, 1 F then there exists a copula C such that n 1i1 nin F ( x,..., x ) = C( F ( x ),..., F ( x )). (2) 1 n 1 1 n n Using the standard transformation method, it can be shown that F ( X ), i = 1,..., n, has a uniform distribution defined on the interval[0,1]. Therefore, the copula can be viewed as a multivariate function with standard uniform marginal distribution. If the marginal distributions, F,...,, 1 F are continuous, then F has a unique copula. However, n if one or more marginals are not continuous, then copula for F is no longer unique. i i 7 The notion of copula and its must-read literature are well discussed in Embrechts (2009). 10

11 Let ui be the observed value of Fi ( X i ). Then for continuous univariate marginals, the unique copula function is given by C( u,..., u ) = F( F ( x ),..., F ( x )), (3) n 1 1 n n where F1,..., Fn 1 1 denote the quantile functions of the univariate marginals, F,..., 1 F n. From Sklar s theorem, it is noted that any marginals can be used, since a copula links marginals to their multivariate distribution, and it is independent of marginals as a measure of dependence. Hence, such a copula is constructed under the assumption that marginal distributions are known and can be consistently estimated from a given data. A special category of copulas used in our numerical analysis are presented in the following section. 2.3 Elliptical Copulas The class of elliptical copulas such as Gaussian, Student-t, and Cauchy copula is flexible since elliptical copulas allow for both positive and negative dependence. Elliptical copulas are particularly suitable in modeling dependence structures with multi-dimensions. Indeed, the introduction of a correlation matrix as a multidimensional parameter allows a great flexibility in terms of structure s shape and simulation procedures. However, a special class of copulas termed as Archimedean copulas limits the complex nature of the dependence structure because they represent dependence structure with a few parameters (Nelsen, 2006). Copulas of the Archimedean family such as Gumbel and Clayton copulas cannot account for negative dependence. 8 In this paper, we focus on elliptical copulas which are best suited to our problem. 8 The class of Archimedean copulas is well discussed in Nelsen (2006). Gumbel is an appropriate modeling choice when variables are strongly correlated in the upper tail of the joint distribution while Clayton is suitable for strong lower tail dependence. 11

12 Gaussian Copula The Gaussian copula is derived from the multivariate Gaussian distribution. The dependence structure among the margins is described by a copula function, C, such that C ( u,..., u ) = G( Φ ( u ),..., Φ ( u )), (4) G 1 1 R 1 n 1 n where G denotes the joint distribution function of the multivariate standard normal distribution function with linear correlation matrix R, and 1 Φ is the inverse of the univariate standard normal distribution function. In the case of bivariate distribution, a bivariate copula function can be written as Φ ( u1 ) Φ ( u2) G 1 ( s1 2 R12 s1s2 + s2 ) CR ( u1, u2) = exp 2 1/ 2 ds 2 1ds2, 2 π (1 R12 ) 2(1 R12 ) (5) where R12 is the linear correlation coefficient of the corresponding bivariate normal distribution. Note that R = ( R ij ) with R ij = ij ii jj for i, j = 1,, n, where is ij the ( i, j) th component of the covariance matrix ( ) of the distribution. The most interesting property of the copula in our study is about tail dependence. The Gaussian copula does not allow for extreme events to be dependent. To illustrate the concept of tail dependence with bivariate variables, let X1 and X 2 be continuous random variables with marginal distribution functions F 1 and F 2. The coefficients of tail dependence are asymptotic measures of the dependence in the tails of the bivariate distribution of X 1 and X 2. The coefficients of lower and upper tail dependences of X 1 and X 2 are defined as λ = lim P( X < F ( u) X < F ( u)), (6) L u

13 λ = lim P( X > F ( u) X > F ( u)), (7) U u respectively, provided that the limit exists in the interval [0,1]. Hence the tail dependence states the probabilities of having a high (low) extreme value of X 2 given that a high (low) extreme value of X1 occurs. The dependence structures are of a great variety from one copula to another. If λ U = 0( λ L = 0 ), for example, random variables X 1 and X 2 are said to be asymptotically independent in the upper (lower) tail, i.e., no tail dependence, which is the case of the Gaussian copula. On the contrary, if λu (0,1] ( λl (0,1] ), X 1 and X 2 are asymptotically dependent in the upper (lower) tail. 9 We now describe the algorithm that provides random vector generation from the Gaussian copulac. Let be positive definite such that = G R T AA for some n n matrix A. If Z,..., 1 Zn denote a random sample from N (0,1) and µ + AZ is a normal distribution with mean, µ and covariance,, where Z = ( Z1,, Z ) T n, Lower triangular matrix A can be obtained using Cholesky decomposition method. Construct the lower triangular matrix A so that the covariance matrix = T AA using Cholesky decomposition. Then, 1. Simulate n independent standard normal random variates Z = ( Z1,..., Z ) T n from N (0,1). 2. Take the matrix product of A and Z, i.e., Y = AZ. 3. Set Ui = Φ ( Yi ) for i = 1,..., n., where Φ is the distribution function of the standard normal. 4. Set X = F 1 ( U ) for i = 1,..., n. i i i 5. ( X1,..., X ) T n is the random vector with marginals, F,...,, 1 Fn and Gaussian G copula C. R 9 See Embrechts, McNeil, and Straumann (1999) for a proof. 13

14 Student s t-copula Student s t-copula (or simply t-copula) is implied by a multivariate Student s t distribution. The t-copula with v degrees of freedom can be written as C ( u,..., u ) = t ( t ( u ),..., t ( u )), (8) t 1 1 v, R 1 n v, R v 1 v n where t v, R denotes the multivariate t -distribution function, and t v is the marginal distribution function of t, bivariate case, t-copula has the following form v R (i.e. the usual univariate t-distribution function). For the ( v+ 2) / t ( u1 ) t ( u2 ) t v v 1 ( s1 2 R12 s1s2 + s2 ) v, R 1 2 = 2 1/ π (1 R12 ) v(1 R12 ) (9) C ( u, u ) 1 ds ds, where R 12 is the linear correlation coefficient of the corresponding bivariate tv distribution, if v > 2. Unlike the Gaussian copula, t-copula generates joint extreme movements regardless of the marginal behavior of random variables. The t-copula has both lower and upper tail dependence, expressing dependence between extreme events. In order to quantify such dependences with various degrees of freedom and different correlations, we calculate coefficients of upper tail dependence ( λ U ) for the t-copula based on the property of the equation (7). It is noted in Table1 that the coefficient of Table 1 Coefficients of Tail Dependence ( λ U ) for t-copulas a v \ R R denotes the linear correlation coefficient and v is degrees of freedom a 12 14

15 tail dependence is increasing in R 12 and decreasing in v (see Embrechts et al., 2002 for a proof). Thus, t-copula is more suitable than Gaussian copula to model the case where extreme events occur simultaneously. Furthermore, the coefficient of upper tail dependence tends to zero as degrees of freedom ( v ) tend to infinity for R 12 < 1, implying that a t-copula converges to a Gaussian copula as v tends to infinity. To simulate random vector ( 1,..., ) T t X X n from the t-copula, C, algorithm can be used (Embrechts et al., 2001): v R, the following 1. Construct the lower triangular matrix A so that the covariance matrix = T AA using Cholesky decomposition. 2. Simulate n independent standard normal random variates Z = ( Z1,..., Z ) T n from N (0,1). 3. Simulate a chi-squared random variates S with v degrees of freedom, T independent of Z = ( Z1,..., Z n ). 4. Take the matrix product of A and Z, i.e., Y = AZ. v 5. Set Ti = Yi for i = 1,..., n. S 6. Set U = t ( T ) for i = 1,..., n. 7. Set i v i X = F 1 ( U ) for i = 1,..., n. i i i 6. ( X1,..., X ) T n is the random vector with marginals,,...,, t F1 Fn and t-copula, C v, R. Cauchy Copula The Cauchy copula is a special case of the t-copula where the degrees of freedom ( v ) is equal to one. The copula generated by a multivariate Cauchy distribution is C ( u,..., u ) = t ( t ( u ),..., t ( u )), (10) c 1 1 1, R 1 n 1, R n where t 1,R is the joint distribution function of a standard Cauchy random vector. Similar to t-copula, the bivariate Cauchy copula is (3/ 2) t 1 ( u1 ) t 1 ( u2 ) c 1 ( s1 2 R12 s1s2 + s2 ) 1, R 1 2 = 2 1/ π (1 R12 ) (1 R12 ) (11) C ( u, u ) 1 ds ds, 15

16 where t 1 ( ) 1 is the inverse of a standard Cauchy distribution with 2 z 1 1 t1( z) = dw. The Cauchy copula also yields tail dependence because of π 1+ w its relationship with the t-copulas. To simulate a random vector, ( X1,..., X ) T n from the Cauchy copula, we follow the t-copula algorithm with v = 1. Independence Copula The independence copula can be expressed as ind C ( u,, u ) = u u u. (12) 1 n 1 2 n From Sklar s theorem, it is clear that the dependence structure of the continuous random variables is given by the above (12) if and only if they are independent. 3. Risk Measures The calculation of economic capital is a process that begins with the quantification of the risks that any given company faces over a given time period. Risk measures provide a magnitude of the severity of a potential loss in a portfolio and present meaningful amounts to hold to support the risk. Thus, the capital required for the company can be determined by the risk measure. Value-at-Risk (VaR) and Expected Tail Loss (ETL), most prominent risk measures among regulators and practitioners, are used to evaluate risks and to determine the capital requirements in this paper. 3.1 Value-at-Risk The Value-at-Risk (VaR) is a statistical-based risk measurement that summarizes the expected maximum loss over a target horizon period, at a given level of confidence. Consider a real-valued random variable X that represents the loss of some risky portfolio in a specified time period, with the distribution, F( x) = P{ X x}, measuring the severity of the risk of holding portfolio. Then, the 16

17 maximum loss is simply infinity since F is unbounded in most models. Thus, the maximum possible loss is given by inf{ x : F( x ) = 1} on the real line, where inf denotes a greatest lower bound. For eachα, 0 < α < 1, the VaR is given by the largest number x such that the probability that the loss X does not exceed x is at least α. Formally, the VaR with given level of α is defined as 10 VaR ( X ) = inf{ x : F( x) α}. (13) α Assuming that the risk profile of the institution remains constant over the horizon, VaR can be derived by reading the quantile off the actual empirical distribution. The VaR is a commonly used risk measure in financial institutions to assess the risk associated with any type of portfolio taking account of correlations between risk factors. 11 VaR presents a probability statement about the potential change in the value of a portfolio resulting from a change in market factors over a given period of time (Crouhy et al., 2001). One of advantages of VaR is that it aggregates all of the risks in a portfolio into a single number suitable for reporting to regulators or disclosure in an annual report. Because VaR measures the worst amount the firm is likely to lose, it can be very useful to determine capital requirements at the level of the firm. Dhaene et al. (2008) find that when substituting the subadditivity condition by the regulator s condition, VaR is the most efficient capital requirement in the sense that it reduces some reasonable cost function. However, VaR has some limitations as a risk measure. For example, VaR does not capture statistical properties of the significant loss beyond the quantile point of interest. Thus, even if the severity of losses doubles beyond quantile point, the VaR 10 VaR can be expressed as VaR ( X ) = inf{ x : S( x) 1 α} with S( x) = 1 F ( x). See α McNeil et al., (2005). 11 VaR information can be used to implement portfolio-wide hedging strategies that are otherwise rarely impossible (Kuruc and Lee, 1998; Dowd, 1999). Properties of VaR have been discussed extensively in the literature(e. g., Dowd and Blake, 2006). 17

18 figure may not be influenced and gives us no indication of how much (i.e., a loss in excess of VaR) that might be. 3.2 Expected Tail Loss as a Coherent Risk Measure Expected tail loss (ETL) is a coherent risk measure in the sense of Artzner et al (1997, 1999). 12 ETL is also known as the expected shortfall by Tasche (2001) and Acerbi and Tasche (2002). 13 ETL calculates the conditional expected loss beyond VaR. For a givenα, ETL is defined by 1 1 ETL ( X ) = α VaRu ( X ) du. 1 α (14) α ETLα ( X ) can be expressed as a linear combination of the corresponding quantile and its expected shortfall (e.g., Denuit et al., 2005) P{ X > VaRα ( X )} ETLα ( X ) = VaRα ( X ) + E{ X VaRα ( X ) X > VaRα ( X )}. 1 α (15) ETL overcomes some drawbacks of VaR because ETL tells us what we can expect to lose if a tail event occurs. Note that ETL is more conservative than VaR since it averages VaR over all levels u distribution and ETL α α, looking further into the tail of the loss > VaR. Dhaene et al. (2008) showed that ETL is the optimal α capital requirement at a given confidence level when the regulator s condition is imposed to the class of concave distortion risk measures. 4. The Measure of Diversification Benefit An important implication of diversification effect is to mitigate the riskiness of the firm by pooling uncorrelated risks. The risk of the portfolio of businesses will be less than the sum of the stand-alone risks of the businesses if the risks of business lines are not perfectly correlated with one another (Merton and Perold, 1993). Myers 12 See e.g., Dowd and Blake (2006) and Dhaene et al., (2008) for more discussions about the properties of coherent risk measure. 13 There is no consensus of terminology. Expected tail loss (ETL) is also called conditional VaR, tail VaR and conditional tail expectation. 18

19 and Read (2001) show that if each line of business is assumed to be organized as a stand-alone firm, total capital requirements for those lines increase because of loss of diversification. Perold (2001) argues that diversification across business segments diminish the firm s deadweight cost of risk capital. If we do not consider the effect of diversification when allocating capital, we may overestimate firm risk, leading to overcharging capital requirements. In general, the diversification benefits can be quantified by three key factors: the number of risk positions (or assets), the concentration of risk factors, and the cross-risk correlations. For the illustration purpose, we take only two assets (asset a and asset b ). Then, the diversified portfolio variance can be expressed in terms of covariance of weighted assets a and b as follows: σ = ( w σ ) + ( w σ ) + 2 w w ρ σ σ, (16) p a a b b a b ab a b where σ a and σ b represent standard deviation of asset a and asset b, respectively, wa and w b are respective weights, such that wa + wb = 1, and ρab is correlation coefficient between asset a and asset b. The equation (16) shows that portfolio variance accounts for the risk of the individual asset ( σ andσ ) and correlation ( ρ ab ) between them. If we assume the portfolio return is normally distributed, the portfolio variance can be translated into a portfolio VaR measure in the delta-normal model. The portfolio VaR is then 2 a 2 b VaR = ασ W = α w σ + w σ + w w ρ σ σ W (17) 2 2 p p ( a a) ( b b) 2 a b ab a b, where α is the confidence level and W represents the initial portfolio value. If the correlation ( ρ ) between two assets is assumed to be zero, the portfolio VaR reduces to ab 19

20 VaR = αw σ W + αw σ W = VaR + VaR (18) p ( a a ) ( b b ) a b. The result shows that the portfolio risk is lower than the sum of individual stand alone risk ( VaRp < VaRa + VaRb ). 14 This reflects the fact that a portfolio of two assets will be less risky than a portfolio consisting of either asset on its own if returns are independent of each other. If the correlation reaches its maximum value of 1, and both w a and wb are positive, equation (17) reduces to VaR = VaR + VaR + 2 VaR VaR = VaR + VaR. (19) 2 2 p a b a b a b If two assets are perfectly correlated, the portfolio VaR is equal to the sum of individual stand alone VaR. In other words, there is no benefit of risk diversification when all correlations are unity. This is a special case because correlations are generally not perfect. The results provide us with important insights that diversification benefit can be measured by the difference between the diversified VaR and the undiversified VaR. The undiversified VaR is defined as the sum of individual VaR when all correlation are exactly unit and the diversified VaR is defined as the portfolio VaR in all circumstances except for the special case where the correlation is one (Jorion, 2007) Numerical Studies We present numerical examples to explore the influence of dependence structure on the insurer s economic capital charge and the level of diversification benefit under different copulas. We first discuss the process of aggregating risks considered for this exercise. The correlation matrix and the marginal distributions which are essential inputs for the specification of copulas are presented. We then 14 This shows that VaR is a coherent risk measure for normal and elliptical distributions in general (Jorion, 2007) 15 We also assume that there is no short position in our portfolio. 20

21 discuss our simulation results. We also discuss the choice of the best copula for the given data in this section. 5.1 Aggregating Insurance Underwriting Risk and Market Risk We measure insurance underwriting risk using quarterly industry-wide underwriting returns by line of insurance business. Following the literature (e.g., Doherty and Kang, 1988; Gron, 1994; Doherty and Garven, 1995), the underwriting return is defined as premiums written net of the present value of incurred losses for each quarter divided by premiums written for the quarter. The quarterly data on losses and premiums by line is provided by the National Association of Insurance Commissioners (NAIC). We include six business lines Homeowners (HO), Auto Physical Damage (APD), Auto Liability (AL), Commercial Multiple Peril (CMP), Workers Compensation (WC), and Other Liability (OL) that account for about 85% of total underwriting risk in terms of premium written. Each business line has a different risk characteristic depending on the payout patterns and product coverage. Long-tail lines of business like AL, WC, and OL are considered to be more risky and complex since losses of long-tail lines are not paid until long after the accident has occurred due to several factors such as loss adjustment procedures and litigated claims. Because the products of short-tail lines such as HO, APD, and CMP are more standardized and their losses are fully developed or ultimately paid in one or two years, short-tail lines are considered to be less risky. Five categories of assets Stocks, Government Bonds (GBond), Corporate Bonds (CBond), Real Estate, and Mortgages are selected to represent the insurer s market risk (or asset-side risk). As proxies for market risks of asset portfolio, the quarterly time series of returns are obtained from the standard rate of return series: Stocks- the total return on the Standard & Poor s 500 stock index; Government Bond 21

22 -the Lehman Brothers intermediate term total return; Corporate Bond -Moody s corporate bond total return; Real Estate-the National Association of Real Estate Investment Trusts (NAREIT) total return; and Mortgages-the Merrill Lynch mortgage backed securities total return. 16 To produce a distribution of overall portfolio returns, we aggregate underwriting return series of insurance business lines and market return series of asset classes using their respective weights. The weight for each asset class is given by the relative amounts invested in each asset class, while the weight for each business line is specified according to the proportion of earned premium by line of business. Weights are constructed to sum to unity by scaling the dollar amount in each asset class and business line by the sum of all amounts of invested asset and earned premium considered. Table 2 lists descriptive statistics for the return series of asset classes and insurance business lines used in the numerical analysis and presents their weights used in aggregating portfolio returns. Several interesting patterns are observable. The mean return ranges from 1.8% to 2.5% for asset classes while it varies from 26.5% to 37.2% for liability lines. Stocks among asset classes show the highest volatility, 7.9%, consistent with common perception. Homeowners line is more volatile than other business lines. One important pattern to notice is that minimum returns for all asset classes are negative. Furthermore, the last column of Table 2 shows that assets variables account for a considerable proportion of total weights. These results imply that the aggregated portfolio return rate will be significantly influenced by the return rate of asset classes. 16 Employing external indices as proxies for the behavior of particular risks is common among the banking institutions. For example, CreditMetrics TM of J.P. Morgan opted equity returns as a proxy for asset returns since internal asset returns are not directly observable (Basel Committee on Banking Supervision, 2003; Morone, et al., 2007). 22

23 Table 2 Descriptive Statistics and Weights for Asset Classes and Liability Lines Variables Mean SD Min Max Weight Stock GBond CBond RealEstate Mortgage HO APD AL CMP WC OL Variables: GBond (Government Bond), CBond (Corporate Bond), HO (Homeowners), APD (Auto Physical Damage), AL (Auto Liability), CMP (Commercial Multiple Peril), WC (Workers Compensation), and OL (Other Liability). 5.2 Correlations The critical parameter that needs to be estimated for the simulation algorithm using copulas is the correlation matrix between market risks and underwriting risks. The industry-wide correlation matrix is estimated by the historical quarterly time series of returns from each asset class and business line over the sample period Table 3 provides the Spearman s rank correlation coefficients for the return series of five asset classes and six insurance lines. Rank correlation is invariant since the simulated numbers replicate the same rank correlations that are used to generate them. The returns of government bonds and mortgages are strongly positively related each other, 93 percent correlations. Of the pair-wise correlations between insurance underwriting risks, the pair of workers compensation (WC) and auto liability (AL) shows the highest positive correlations with 85 percent. High correlations between them may be justified since both WC and AL have similar long-term payout patterns. The correlation between APD and WC is negative 40 percent. It can be argued that 23

24 Table 3 Correlation Matrix for Asset and Liability Return Series Variable Stock GBond CBond RealEstate Mortgage HO APD AL CMP WC OL Stock GBond CBond RealEstate Mortgage HO APD AL CMP WC OL Variables: GBond (Government Bond), CBond (Corporate Bond), HO (Homeowners), APD (Auto Physical Damage), AL (Auto Liability), CMP (Commercial Multiple Peril), WC (Workers Compensation), and OL (Other Liability). APD line that covers damage to auto physical is less likely to be correlated with WC line that provides coverage for bodily injuries due to job-related accidents. The returns of government bonds, corporate bonds, real estate and mortgages are negatively correlated with those of several business lines of insurance, suggesting that insurer s investment in these types of assets provides effective hedges against some losses in insurance liability lines. This view is consistent with the findings of recent empirical studies that the insurer s asset returns are correlated with liability risks to some degree (e.g., Cummins et al., 2006; Ren, et al., 2008) Fitting Marginal distributions Determining the appropriate marginal distributions with a given data set is important along with the choice of copula because capital requirement measured by VaR and ETL is largely dependent on the tail behavior of distribution of each risk. Various techniques including graphical or numerical method can be used to model marginal distribution. The choice of distributions for each asset class and liability line is detailed. 24

25 Let F( x ) be the fitted distribution function of a random variable X and then the empirical distribution function for X,..., 1 X n is defined as K ˆ 1 Fi ( x ) = I ( X ji x ), (20) K j = 1 where i = 1,, n, and K is the number of observations for the i th random variable. The fit of the marginal distributions can be assessed with a graphical comparison of the specified theoretical distribution function against its empirical version, as demonstrated in Figure 3. For an illustration, we present here the fit of the univariate marginals using lognormal distribution for Auto Liability line. Figure 3 (left) shows that the plot of the empirical distribution function, F ˆ ( x ), is fairly close to the smooth curve of the theoretical distribution, F( x ). Hence we can conclude that the lognormal distribution seems a suitable model for AL line. Figure 3 Graphical Method to Model Marginal Distribution An alternative method is to use the probability-probability plot (P-P plot) that helps to determine how well a specified distribution fits a given data set. Figure 3 (right) illustrates P-P plot method using the sample data of AL line. The adoption of lognormal distribution for AL appears reasonable since there is no significant discrepancy between the dotted curve of empirical distribution function and the diagonal reference line. 25

26 As numerical measures, Skewness and kurtosis can be used to derive a fitting distribution. Skewness is a measure of the asymmetry of the probability distribution where positive (negative) skewness indicates a long right (left) tail, while zero skewness reveals symmetry around the mean. Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high Kurtosis portray a distinct peak near the mean and tend to have heavy tails, whereas data sets with low Kurtosis show a flat top with skinny tails. Using the identical historical quarterly returns used in the estimation of correlations and following all procedures discussed above, we present the best fitting distributions for asset classes and business lines in Table Table 4 Best Fitting Distributions and Parameter Estimates Variables Distribution Skewness Kurtosis Parameter1 c Parameter2 d Stock Gumbel(Min) a GBond Normal CBond Logistic RealEstate Logistic Mortgage Gumbel(Max) b HO Gumbel(Min) a APD Gamma AL Lognormal CMP Normal WC Lognormal OL Gamma Variables: GBond (Government Bond), CBond (Corporate Bond), HO (Homeowners), APD (Auto Physical Damage), AL (Auto Liability), CMP (Commercial Multiple Peril), WC (Workers Compensation), and OL (Other Liability). a Gumbel(Min) denotes Gumbel minimum that highlights the left tail. b Gumbel(Max) denotes Gumbel maximum highlights the right tail. c d Parameter1 and Parameter 2 indicate the estimates of the parameters in the probability density function The variables for Stock, Mortgage, and Homeowners (HO) display the extreme value distributions where Gumbel minimum highlights the left tail (i.e., thin right tail and thick left tail) while Gumbel maximum highlights the right tail (i.e., thin left tail and 17 We also employed the commonly used goodness-of-fit tests such as Anderson-Darling test, Kolmogorov-Smirnov test and Chi-Square test. 26

27 thick right tail). Both Auto Liability (AL) and Workers Compensation (WC) best fit the Log-normal distribution which exhibits a relatively heavier tail than Gamma distributions of Auto Physical Damage (APD) and Other Liability (OL) lines. We estimate the parameters for each of the selected distributions using the maximum likelihood method that often leads to the unbiased estimates with minimum variance. The results are provided in Table Simulation Results We perform Monte Carlo simulation to generate random return series for each asset class and business line by applying the procedure described in the previous section. The scatter plots between return series and economic capital measured by VaR and ETL are presented for each copula model. We also discuss the results of diversification benefit in this section Scatter Plot of Simulated Return Series We present scatter plots of the return series to observe graphically the dependence level between asset classes and insurance business lines for the following copulas: Gaussian copula, t-copula with degrees of freedom v =10 and v =2, Cauchy copula and Independence copula. We simulate 5000 random values of each asset class and business line for each of copula models considered using correlations and parameter estimates of the marginal distributions in Table 3 and Table 4. The scatter plots of five selected classes -Stock, GBond, HO, AL and WC are presented to conserve the space. Each dot stands for a joint observation for each pair. The scatter plots show that different copulas impose distinctive tail behaviors, leading to different level of risk measures. Even with the identical rank correlations, the level of dependence structure can be differentiated by copula types. In general, we observe that the Gaussian copula (Figure 4) demonstrates a modest linear relationship 27

28 across returns in each asset and liability class and observations tend to cluster around the center of the distribution. It is important to realize that dependencies may be different according to the correlation parameters initially determined by historical data. For example, the scatter plot between AL and WC shows a relatively strong linear relationship, compared to that of other pairs. This positively sloping linear pattern is consistent with high rank correlation (85%) between AL and WC as shown in Table 3. However, neither upper nor lower tail dependence is detected across all pairs in the Gaussian copula. This is justifiable because the degree of tail dependence for the Gaussian copula is zero if the correlation of any pair is less than unity. Little evidence of tail dependence implies that extreme events appear to occur independently under Gaussian copula model regardless of how high a correlation we choose between asset and liability classes. In contrast, t-copula differs from Gaussian copula, even when their components have the same correlation coefficients. The scatter plots of t-copula with v =10 (Figure 5) shows that observations are more scattered toward upper or lower tail distribution than those of Gaussian copula. This is reasonable given that t-copula permits symmetric tail dependence. The joint observations in Figure 5, however, appear somewhat similar to the case of Gaussian copula even if simulated from t- copula. As we noted in Section 2, this phenomenon results from the asymptotic characteristic of the t-copula that the tail behavior of t-copula resembles that of Gaussian copula as the degrees of freedom increase. Although 10 degrees of freedom may not be sufficient enough for asymptotic behavior, we still observe some similitude with the Gaussian case. As expected for these forms of copulas, the t-copula with v =2 in Figure 6 and Cauchy copula in Figure 7 display more pronounced non-linear dependence between 28