### Conditional tail expectations

#### (Oct 10, 2014)

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…