Tail conditional expectation for multivariate distributions: A game theory approach
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 denote the possible loss of a portfolio at a given time horizon. Then is the size of loss for which there is a small probability for exceeding that loss (also shown by or ); therefore, is defined as the smallest value satisfying . The mathematical form of the value at risk, , is given by
The tail conditional expectation, , is the mean of worse losses, given that the loss will exceed a particular value . It is expressed by
Finally, the shortfall expectation, , is defined as
is called the confidence level, and in practice it is often set to 0.95 or 0.99. It follows from the definitions that . When is a continuous random variable, then and is equal to . 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 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 denote the th loss variable (). If indicates the risk measure for , where , we would like to determine as the risk measure for such that
In recent years, attention has turned to coherent risk measurements. A risk measure () is called a coherent risk measure if, and only if, it satisfies all of the following four axioms (Artzner et al., 1999).
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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.
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Positive homogeneity: for .
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Translation invariance: for any .
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Monotonicity: If , then . This means that if portfolio always has better values than portfolio under all scenarios, then the risk of should be less than the risk of .
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 and sum of all variables () 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 (), the total TCE is calculated from
Then, the risk contribution of each business line () in the total risk should be determined. In the approach proposed by Panjer (2002), the contribution of the th line of business is defined as
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 denote the characteristic function, or the gain value of coalition which is a subset of the set in which 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, , and , where Let , and . Then 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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2020, Journal of Multivariate AnalysisCitation Excerpt :The corresponding results for other special cases such as TESN, TESSGH along with truncated ESL (TESL) and truncated EST (TEST) distributions are also available and can be obtained from the authors upon request. Over the two past decades, several works have been carried out to extend the concept of tail risk measures from the univariate case to the multivariate case; one may refer to [2,10,14,17,22]. Here, we present two alternate extensions of the classical univariate TCE to a multivariate set-up.
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