16.2 Backtesting with Expected Shortfall

In the following we consider the expected shortfall from $ L_{t+1}$ as an alternative to the VaR and develop a backtesting method for this risk measurement. The expected shortfall, also called the Tail-VaR, is in the Delta-Normal Model, i.e. under the assumptions from (15.7) and (15.8), defined by

$\displaystyle E(L_{t+1} \ \vert \ L_{t+1} > VaR_t)$ $\displaystyle =$ $\displaystyle E(L_{t+1} \mid L_{t+1} > z_\alpha \ \sigma_t)$  
  $\displaystyle =$ $\displaystyle \sigma_t \ E(L_{t+1} / \sigma_t \mid L_{t+1} / \sigma_t > z_\alpha
).$ (16.13)

Here $ z_\alpha = \Phi^{-1}(\alpha)$ represents the $ \alpha$ quantile of the standard normal distribution, where $ \Phi $ is the standard normal distribution function.

Under this model (15.7) and (15.8) $ Z_{t+1} =
L_{t+1}/\sigma_t$ has a standard normal distribution. For a defined threshold value $ u$ we obtain

$\displaystyle \vartheta = E(Z_{t+1} \ \vert \ Z_{t+1} > u)$ $\displaystyle =$ $\displaystyle \frac{\varphi(u)}{1-\Phi(u)}$ (16.14)
$\displaystyle \varsigma^2 = Var(Z_{t+1} \ \vert \ Z_{t+1} > u)$ $\displaystyle =$ $\displaystyle 1+ u \cdot \vartheta - \vartheta^2 ,$ (16.15)

where $ \varphi$ is the standard normal density. For given observations from a forecast distribution and its realizations $ (\hat{F}_{t+1}(\cdot / \hat{\sigma}_t) , L_{t+1} / \hat{\sigma}_t)
$ we consider (15.14) as the parameter of interest. Replacing the expected value with a sample mean and the unobservable $ Z_{t+1}$ with

$\displaystyle \hat{Z}_{t+1} = \frac{L_{t+1}}{\hat{\sigma}_t} ,$ (16.16)

where $ \sigma_t$ in (15.8) is estimated with (15.9) or (15.10), we obtain an estimator for $ \vartheta$

$\displaystyle \hat{\vartheta} = \frac{1}{N(u)} \sum_{t=0}^n \hat{Z}_{t+1} \ { \boldsymbol{1}} (\hat{Z}_{t+1}>u).$ (16.17)

$ N(u)$ is the random number of times that the threshold value $ u$ is exceeded:

$\displaystyle N(u) = \sum_{t=1}^n \boldsymbol{1}(\hat{Z}_{t+1} > u) .$

Inferencing on the expected shortfall, i.e., on the difference $ \hat{\vartheta} - \vartheta $, we obtain the following asymptotical result:

$\displaystyle \sqrt{N(u)}\Big(\frac{\hat{\vartheta} - \vartheta}{\hat{\varsigma}} \Big) \stackrel{\cal{L}}{\longrightarrow}$   N$\displaystyle (0,1)$ (16.18)

(15.18) can be used to check the adequacy of the Delta-Normal model.