# Chapter SixRisk Measures and Capital Allocation

OpRisk is a significant risk exposure to most firms and therefore requires effective risk management. Thus, these risks should be modeled, measured, and capital should be held so that a bank can withstand extreme losses. A risk measure is a single number quantifying an exposure to the risk. In particular, risk managers and regulators are interested in assessing the probability that extreme losses may occur and this can be represented through the quantile of the loss distribution (over a specified time horizon). This led to the regulatory requirement for risk capital to be measured as a Value-at-Risk (VaR), which is just a quantile of the loss distribution at some high confidence level (i.e., quantile of the loss distribution at the 0.999 confidence level over a 1-year horizon for OpRisk).

Since VaR had come into widespread use in the financial markets for quantifying market risk, adacemics began to undertake theoretical studies of the properties of such a risk measure and they began to notice that using VaR as a risk measure could sometimes have a poor outcome. Specifically, the diversification principle may fail in some circumstances. The wisdom in choosing the 0.999 VaR as a risk measure for capital is highly contested (see, e.g., Daniélsson *et al.* 2001). Using economic reasoning, a list of axiomatic properties for a good (*coherent*) risk measure was suggested in the seminal paper by Artzner *et al.* (1999). In particular, an alternative ...

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