Abstract
AbstractAllocating risk properly to subunits is crucial for performance evaluation and internal capital allocation of portfolios held by banks, insurance companies, investment funds and other entities subject to financial risk. We show that by using coherent measures of risk it is impossible to allocate risk satisfying simultaneously the natural game theoretical requirements of Core Compatibility and Strong Monotonicity. To obtain the result we characterize the Shapley value on the class of totally balanced games and also on the class of exact games as being the only risk allocation method satisfying Strong Monotonicity, Equal Treatment Property and Efficiency. Moreover, we clarify and interpret the related game theoretical requirements that have appeared in the literature so far and have been applied to risk allocation.
Highlights
If a firm consists of divisions, is it important to measure properly the risk of the firm, and to allocate the risk capital of the firm to the divisions
Drehmann and Tarashev (2013) consider Shapley’s axiomatization of the Shapley value for systemic risk allocation games but it has the same shortcomings as Denault (2001) and they even interpret Additivity as Efficiency
Those requirements are natural, they are not related to the properties of Core Compatibility (CC), Equal Treatment Property (ETP) or Strong Monotonicity (SM)
Summary
If a firm (financial enterprise, bank, insurance company, investment fund, portfolio, etc.) consists of divisions (individuals, products, subportfolios, risk factors, etc.), is it important to measure properly the risk of the firm, and to allocate the risk capital of the firm to the divisions. In a TU game using the values (the negative of the risk) of the coalitions (subsets) of the players (divisions) a solution concept (a risk allocation rule) determines how to share the value of the grand coalition (the firm). Csoka et al (2009) showed that the class of risk allocation games (using coherent measures of risk) coincides with the class of totally balanced games. We will prove (Proposition 3.5) that on the class of totally balanced (exact) games if a risk allocation rule meets CC and SM together, there does exist a risk allocation rule which satisfies CC, SM and ETP together. In Example 3.8 we will illustrate that our impossibility result (Theorem 3.7) can be made stronger, because in all practical applications SM can be replaced by the following requirement: if a division’s stand-alone performance increases, its allocated risk should not increase.
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