Tail conditional expectation for multivariate distributions: A game theory approach

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Abstract

This paper proposes using the Shapley values in allocating the total tail conditional expectation (TCE) to each business line (Xj, j=1,,n) when there are n correlated business lines. The joint distributions of Xj and S (S=X1+X2++Xn) are needed in the existing methods, but they are not required in the proposed method.

Introduction

A risk measure is defined as a mapping from the set of random variables representing the risk exposure to a real number. The well-known risk measures in the literature are value at risk (VaR), tail conditional expectation (TCE), and shortfall expectation (SE). Let X denote the possible loss of a portfolio at a given time horizon. Then V aRX(1α) is the size of loss for which there is a small probability α for exceeding that loss (also shown by x1α or ξ1α); therefore, V aRX(1α) is defined as the smallest value x satisfying Pr(X>x)=α. The mathematical form of the value at risk, V aRX(1α), is given by V aRX(1α)=inf{x|Pr(X>x)α}.

The tail conditional expectation, TCEX(1α), is the mean of worse losses, given that the loss will exceed a particular value x1α. It is expressed by TCEX(1α)=E[X|X>V aRX(1α)]=E(X|X>x1α).

Finally, the shortfall expectation, SEX(1α), is defined as SEX(1α)=TCEX(1α)+x1α(1αPr(xx1α)).

1α is called the confidence level, and in practice it is often set to 0.95 or 0.99. It follows from the definitions that SEX(1α)TCEX(1α)V aRX(1α). When X is a continuous random variable, then Pr(XV aRX(1α))=1α and SEX(1α) is equal to TCEX(1α). When compared to the VaR measure, the TCE provides a more conservative measure of risk for the same degree of confidence level, and it provides an effective tool for analyzing the tail of the loss distribution. In multivariate cases, assume that a company manages n lines of business and that the risk managers of that company estimate the aggregated risk of all business lines and are interested to know how much risk is concealed in each business line. Let Xj denote the jth loss variable (j=1,,n). If ζS(1α) indicates the risk measure for S, where S=X1+X2++Xn, we would like to determine ζXj(1α) as the risk measure for Xj such that ζS(1α)=i=1nζXj(1α).

In recent years, attention has turned to coherent risk measurements. A risk measure (ζ) is called a coherent risk measure if, and only if, it satisfies all of the following four axioms (Artzner et al., 1999).

  • Sub-additivity: This means that the risk of two, or more, portfolios together cannot get any worse than adding the two, or more, risks separately; this is the diversification principle.

  • Positive homogeneity: ζ(λX)=λζ(X) for λ0.

  • Translation invariance: ζ(X+a)=ζ(X)+a for any aR.

  • Monotonicity: If X1X2, then ζ(X1)ζ(X2). This means that if portfolio X2 always has better values than portfolio X1 under all scenarios, then the risk of X2 should be less than the risk of X1.

It is well known that the VaR fails to satisfy the coherency principle. In general, the VaR is not a coherent risk measure, as it violates the sub-additivity principle and often underestimates the tail risk. An immediate consequence is that the VaR might discourage diversification (Artzner et al., 1999). Zhu and Li (2012) studied the asymptotic relation between the TCE and the VaR and showed that, for a large class of continuous heavy-tailed risks, the TCE is asymptotically proportional to the VaR of aggregation, given that the aggregated risk exceeds a large threshold. It is proven that the SE is a coherent risk measure; therefore, the TCE is a coherent measure for continuous distributions.

In this study, we consider the TCE since it exhibits properties that are considered desirable and applicable in a variety of situations. To find the risk concealed in each individual variable in multivariate environments, we use the cooperative game theory concept, and apply the Shapley value decomposition to calculate the TCE for each variable. In existing approaches to estimate the risk share for each variable from the total risk, the joint distribution of Xj(j=1,,n) and sum of all variables (S) is required, where estimating the joint distribution is not a straightforward task. The proposed method uses Shapley values in a cooperative game theory approach to allocate the total TCE fairly to its constituents without the need to fit any joint distributions.

The remainder of the paper is organized as follows. Section  2 presents the existing approaches in estimating risk measures in multivariate environments. Section  3 reviews the concepts of the cooperative game theory and Shapley values. Section  4 discusses the concept of Shapley values in risk allocation and describes the proposed method. Several numerical examples for multivariate normal and non-normal distributions are illustrated in Section  5. Finally, Section  6 concludes.

Section snippets

TCE for multivariate distributions

In multivariate cases, where we have multiple lines of correlated business (Xj,j=1,,n), the total TCE is calculated from TCES(1α)=E(S=i=1nXi|S>s1α).

Then, the risk contribution of each business line (Xj,j=1,,n) in the total risk should be determined. In the approach proposed by Panjer (2002), the contribution of the jth line of business is defined as TCEXi|S(1α)=E[Xj|S>S1α].

The formula above is based on the additivity property of expected values. We call Panjer method the decomposition

Cooperative game theory and Shapley values

A cooperative game is where players can encourage cooperative behavior and make coalitions. In the game, based on each player’s contribution, the total gain (utility/cost) by the coalition will be divided among the coalition members Driessen (1988). In a cooperative game, let ν(H) denote the characteristic function, or the gain value of coalition H which is a subset of the set Ω={1,2,,N} in which N is the total number of players. In the cooperative game theory formulation, a characteristic

Formulating the TCE allocation by the Shapley values

In the multivariate environments, we have a total risk and are interested in subdividing it between the variables. The share of each variable from the total risk should be determined based on its contribution to the total risk. The problem can be viewed as a cooperative game such that variables act like the players and the total gain relates to the total risk. Since the Shapley value is a fair allocation strategy and has several desirable properties, we use the Shapley values to decompose the

Numerical examples

Here, several examples are presented, and the results of the game theory approach are compared with those of the decomposition approach. In the first and second examples, the data have a multivariate normal distribution. The third example deals with the case where the data follow a skew-normal distribution.

Example 1 Three-Dimensional Normal Distribution with Positive Correlations

Assume that we have three loss random variables, X=(X1,X2,X3), and XMV N(μ,), where μ=(578)and=(21.521.530.820.81).

Let α=0.05,Ω={X1,X2,X3}, and S=SΩ=X1+X2+X3. Then TCES(0.95) is 27.881.3

An empirical example

Here, we consider a real case, where data are borrowed from an insurance company. In this example, we have three variables (liability, disablement, and driving insurance), and 300 data points are available for each variable. The descriptive statistics of the 300 pieces of data are shown in Table 4.

By using the best fit distribution method, the distributions of all coalitions are summarized in Table 5. The Weibull distribution is well fitted to most of the cases in this example. The following

Conclusion

We applied the Shapley values to find the share of each business line from the total tail conditional expectation (TCE) risk in multivariate situations. In this method, the joint distributions of each random variable and the sum of all random variables are not required, and this is an advantage of using the proposed approach compared to the existing approaches. The numerical results showed that the common existing method (the decomposition method) sometimes provides inaccurate estimations. The

Acknowledgments

The authors are grateful to the anonymous referees for their numerous constructive and thoughtful comments that led to this improved version.

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