### 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…

### Quantiles and percentiles

#### (Aug 20, 2014)

Quantiles are points taken at regular intervals from the cumulative distribution function of a random variable. They are generally described as q-quantiles, where q specifies the number of intervals which are separated by q−1 points. For example, the 2-quantile is the median, i.e. the point where values of a distribution are equally likely to be above or below this point.

A percentile is the name given to a 100-quantile.  In Solvency II work we most commonly look for the 99.5th percentile, i.e. the point at which the probability that a random event exceeds this value is 0.5%.  The simplest approach to estimating the 99.5th percentile might be to simulate 1,000 times and take the 995th or 996th largest…

Tags: quantile, percentile, Solvency II, Excel, R