### Compare and contrast: VaR v. CTE

#### (Mar 9, 2019)

Insurance reserving in many countries looks at extreme scenarios over a single year.  The idea is that the insurer faces an uncertain liability, as represented by a random variable, $$X$$ say.  The question is then how to set a reserve for this liability, $$V_\alpha(X)$$?  In this blog we consider two common regulatory approaches to setting $$V_\alpha(X)$$:

1. Value-at-risk (VaR).  This is a quantile-based approach, i.e. $$V_\alpha(X)=Q_\alpha(X)$$, where $$Q_\alpha(X)$$ is the $$\alpha$$-quantile of $$X$$, i.e. $$\Pr[X\leq Q_\alpha(X)]=\alpha$$.  The quantile approach to reserving underlies Solvency II in the European Union.  The reserve $$V_\alpha(X)$$ is thus set so that the liability $$X$$…

### 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…