Conditional tail expectations

In a recent posting I looked at the calculation of percentiles and quantiles, which underpin many calculations for ICA and Solvency II. Simply put, an \(\alpha\)-quantile is the value which is not expected to be exceeded \(\alpha\times 100\)% of the time. This value is denoted \(Q_{\alpha}\). Mathematically, for a continuous random variable, \(X\), and a given probability level \(\alpha\) we have:

$$\Pr(X\leq Q_\alpha)=\alpha$$

Thus, ICA and Solvency II work is about 99.5%-quantiles or \(Q_{99.5\%}\). However, quantiles and percentiles are not universally used for determining regulatory capital. In North America, for example, the conditional tail expectation (CTE) is widely used. The CTE is the expected value given that an extreme event has actually occurred. Mathematically, we have:

$${\rm CTE}_{\alpha}=E[X|X>Q_\alpha]$$

which shows that quantiles and CTEs are related. One immediate result from the equation above is that \(CTE_\alpha\geq Q_\alpha\). Thus, a regulatory or reporting environment based around the CTE may use a lower value for \(\alpha\) than 99.5%, but this does not mean that the CTE-based regulatory environment is weaker.

A useful reference guide is Hardy (2006), which gives two advantages of the CTE over quantile methods. The first advantage of the CTE is that:

The above quotation references a basic statistical phenomenon: there is less variance for the mean of the largest \(n\) numbers in a set than there is over a single order statistic. The second advantage of the CTE is that it is coherent, while a quantile approach is not (or not always). For two risks \(X\) and \(Y\), for example, it is always the case that:

$$CTE_\alpha(X+Y)\leq CTE_\alpha(X) + CTE_\alpha(Y)$$

which is known as subadditivity — diversification cannot make the total risk greater, but it might make the overall risk smaller if the risks are not perfectly correlated. For quantile methods, however, it is possible to construct examples whereby:

$$Q_\alpha(X+Y)> Q_\alpha(X) + Q_\alpha(Y)$$

which violates an intuitive principle: it should not be possible to reduce the capital requirement for a risk by splitting it into constituent parts. The above feature of quantiles is one reason why supervisory authorities in quantile-regulated territories require capital to be identified for each risk separately before being aggregated into a company-wide capital requirement. This way, the regulator can ensure that the failure of quantiles to be subadditive is not being used to game the capital requirements.