# Game-theoretic approaches to optimal risk sharing

This dissertation addresses topics in optimal risk sharing and capital allocation.

First, allocating the buffer is important for performance measurement. It is not uncommon that the managers of the divisions are evaluated on the basis of the return earned on the amount of risk capital to be withheld for their portfolio. This requires an allocation of buffer to divisions that is perceived as “fair” by the managers. Second, the allocation is important for decisions regarding whether to increase or decrease the engagement in the operational activities of certain divisions. The attractiveness of a specific risk (e.g., a specific financial investment) is typically evaluated by means of a risk-return trade-off. Evaluating the performance of a division’s activities in isolation, however, can be very misleading. So, the allocation indicates the riskiness of a division’s portfolio for the total firm.

The second chapter of this dissertation proposes to use the Aumann-Shapley value as a solution of this problem. This allocation rule received considerable attention in the literature (see, e.g., Tasche, 1999; Denault, 2001; Tsanakas and Barnett, 2003; Kalkbrener, 2005) and is given by a gradient of a specific function. I derive a generalization of the Aumann-Shapley value that is also well-defined if this gradient does not exist. Aumann and Shapley (1972) show in their seminal book that a very general asymptotic approach leads towards the Aumann-Shapley value for many functions, but not the one that corresponds to the risk capital allocation problem. As a solution, I propose a weaker asymptotic approach. I, however, show existence in case of the risk capital allocation problem. Moreover, the approach that I use to characterize the allocation rule allows us to give an explicit formula for the corresponding capital allocation. The specific formula has a geometric interpretation. It still satisfies some properties that were known to be valid for the Aumann-Shapley value if it exists.

The third and fourth chapter analyze optimal risk sharing for firms. First, I study risk sharing in the context of longevity risk. Longevity risk is the risk of populations living longer or shorter than expected, for example, through medical advances or declining health risks such as smoking. It is the systematic risk involved in life-contingent liabilities that arises from the fact that death rates change in an unpredictable way. This risk is a major concern for Defined Benefit pension funds, since they typically promise their participants a retirement payment until death. An increase in longevity leads to an increase in the present value of these pension liabilities. On the other hand, death benefit insurers offer a pay-off in case of (early) death. If people live longer than expected, this leads to either later payment, or a larger chance of not paying out this insurance. This implies that an increase of longevity leads to a decrease of the present value of the death benefit insurer’s liabilities. Exposure to longevity risk can be rather substantial for pension funds and death benefit insurers (see, e.g., Jones, 2013).