# Analytical analysis of VaR vs. ES and 99% vs. 97.5%

# An evaluation of the Basel III market risk proposals

### Analytical analysis of VaR vs. ES and 99% vs. 97.5%

The data analysis finds that GARCH ES(97.5%) and VaR(99%) are almost exactly the same. Below, this is verified analytically for both the conditional normal and conditional student-t(3) distributions. Furthermore, if one goes deep into the tails, a similar result holds asymptotically.

We assume, without loss of generality, that the portfolio value is one.

### Conditionally normal models

For a conditionally normal model (like GARCH and EWMA), VaR is \begin{equation} VaR_t(p)= \sigma_t \Phi^{-1}(p) \end{equation} where $$\Phi$$ is the normal distribution. In that case, ES is \begin{equation}ES_t(p)= \sigma_t \frac{\phi\left({\Phi^{-1}(p)}\right)}{1-p}\end{equation} For VaR(99%) and ES(97.5%), $$\Phi^{-1}(p)$$=2.3263479 and $$\phi\left({\Phi^{-1}(p)}\right)/p$$=2.6652142 \begin{equation}\frac{ES_t}{VaR_t}=1.1456645\end{equation}

If we evaluate VaR at 99% and ES at 97.5% we get

\begin{equation}\frac{ES_t(0.975)}{VaR_t(0.99)}=\frac{2.3378028}{2.3263479}=1.004924\end{equation}

So, VaR(99%)=ES(97.5%). That will apply to any conditional normal models, like members of the GARCH family, EWMA, MA, etc.

### Conditionally fat (Student-t) models

Denote the density of the Student-t by $$\delta()$$, the distribution by $$\Delta()$$ and the degrees of freedom by $$\nu$$. Then the VaR is

\begin{equation}VaR_t(p)= \sigma_t \Delta^{-1}(p)\end{equation}

and the ES is

\begin{equation}ES_t(p)= \sigma_t \frac{\delta_\nu\left({\Delta_\nu^{-1}(p)}\right)}{1-p} \left( \frac{\nu + \left(\Delta_\nu^{-1}(p)\right)^2}{\nu-1} \right)\end{equation}

If we evaluate VaR at 99% and ES at 97.5%, where $$\nu=3$$ we get

\begin{equation}\frac{ES_t(0.975)}{VaR_t(0.99)}=\frac{5.0395831}{4.5407029}=1.1098685\end{equation}

So, for the Student-t(3) the VaR(99%) is 90% of the ES(97.5%). That will apply to any conditional Student-t(3) models, like members of the t-GARCH family.

In the special case of Student-t GARCH, the degrees of freedom parameter is likely to be higher, because the GARCH equation contributes to fat tails independently of the conditional, so the difference between the two would be smaller.

For example, if the degrees of freedom equal 6, a common outcome for Student-t GARCH models, we get:

\begin{equation}\frac{ES_t(0.975)}{VaR_t(0.99)}=\frac{3.2561511}{3.1426684}=1.0361103\end{equation}

### Results for the extremes

For regularly varying distributions at infinity, where $$\alpha$$ is the tail index ( in the special case of Student-t, the tail index is equal to the degrees of freedom), the following holds:

\begin{equation}ES = \frac{\alpha}{\alpha-1} VaR\end{equation}

This suggests that ES will be quite close to VaR with a constant difference between them.