# A numerical approach to the risk capital allocation problem

## Henryk Gzyl and Silvia Mayoral

#### Need to know

• We provide a numerical procedure to allocate capital in ranges provided by experts.
• We provide a numerical procedure to find a distortion function from a collection of risk prices.
• That provides a risk measure consistent with a collection of risk prices as standalone risks.
• That risk measure can be further used to further compute the price of other risks.

#### Abstract

Capital risk allocation is an important problem in corporate, financial and insurance risk management. There are two theoretical aspects to this problem. The first aspect consists of choosing a risk measure, and the second consists of choosing a capital allocation rule subordinated to the risk measure. When these two complementary aspects are settled, the problem is reduced to a computational problem. No universal prescription for choosing a risk measure exists, let alone a subordinated capital allocation rule. However, expert opinion may be expressed as a range of capital to be allocated to each risk. For example, the managers of each line of activity of a corporation or bank may have a way to estimate a minimum and maximum losses with high confidence. In that case, one is confronted with two interesting numerical problems. First, to devise a method to allocate risk capital within the bounds provided by the expert. Second, to use the upper bounds as stand-alone risk prices to determine a risk measure, which can then be used to compute the prices of other risks. The risk measure could also be used to compute a range for the capital allocations of a new collection of risks, to which our numerical procedure to allocate risk capital can be applied. The aim of this paper is to use a model-free, nonparametric approach based on the method of maximum entropy in the mean to solve both problems and to examine how they are intertwined. In this way we circumvent the problem of choosing between risk valuation and risk pricing methods, and between risk capital allocation rules consistent with them. Also, the maximum-entropy-based methodology provides us with a method to determine coherent risk measures consistent with the bounds provided by expert opinion or management. The risk measure provides us with a method to determine either risk prices or bounds to be used for capital allocation for a different collection of risks.

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