18.3 Estimators for Risk Measurements

The value at risk discussed in the previous chapter is not the single measure of the market risk. In this section we introduce an alternative risk measure. In addition we discuss how to estimate the measure given extremely high loss.

Definition 18.12 (Value-at-Risk and Expected Shortfall)  
Let $ 0 < q < 1,$ and let $ F$ be the distribution of the loss $ X$ of a financial investment within a given time period, for example, one day or 10 trading days. Typical values for $ q$ are $ q = 0.95$ and $ q = 0.99.$
a) The Value-at-Risk (VaR) is the $ q$-quantile

$\displaystyle VaR _q (X) = x_q = F^{-1} (q).$

b) The expected shortfall is defined as

$\displaystyle S_q = {\mathop{\text{\rm\sf E}}} \{ X\vert X > x_q\}. $

Value-at-Risk is today still the most commonly used measurement, which can quantify the market risk. It can be assumed, however, that in the future the expected shortfall will play at least an equal role.

Definition 18.13 (Coherent Risk Measure)  
A coherent risk measure is a real-valued function $ \rho:
\mathbb{R} \to \mathbb{R}$ of real-valued random variables, which model the losses, with the following characteristics:
(A1)     $ X \ge Y $ a.s. $ \Longrightarrow \rho (X) \ge \rho (Y)$ (Monotonicity)
(A2)      $ \rho (X+Y) \le \rho (X) + \rho (Y) $ (Subadditivity)
(A3)     $ \rho (\lambda X) = \lambda \rho (X)$ for $ \lambda \ge 0$ (Positive homogeneity)
(A4)     $ \rho (X+a) = \rho (X) + a $ (Translation equivariance)
   

These conditions correspond to intuitive obvious requirements of a market risk measurement:
(A1) When the loss from investment $ X$ is always larger than that from investment $ Y$, then the risk from investment $ X$ is also larger.
(A2) The risk of a portfolio consisting of investments in $ X$ and $ Y$ is at most as large as the sum of the individual risks (diversification of the risk).
(A3) When an investment is multiplied, then the risk is also multiplied accordingly.
(A4) By adding a risk free investment, i.e., a non-random investment with known losses $ a$ ($ a<0$, when the investment has fixed payments), to a portfolio, the risk changes by exactly $ a$.

The VaR does not meet condition (A2) in certain situations. Let $ X$ and $ Y$, for example, be i.i.d. and both can take on the value 0 or 100 with probability $ {\P}(X=0) = {\P}(Y=0) = p$ and $ {\P}(X=100) =
{\P}(Y=100) = 1-p.$ Then $ X+Y$ can be 0, 100 and 200 with probability $ {\P}(X+Y=0) = p^2$, $ {\P}(X+Y=100) = 2p(1-p)$ and $ {\P}(X+Y=200) =
(1-p)^2$ respectively. For $ p^2 < q < p$ and $ q < 1-(1-p)^2$, for example, for $ p=0.96, q=0.95$, it holds that

$\displaystyle VaR _q (X) = VaR _q (Y) = 0, \; \;$   but$\displaystyle \; \; VaR _q (X+Y) = 100. $

The expected shortfall, on the other hand, is a coherent risk measure that always fulfills all four conditions. It also gives a more intuitive view of the actual risk of extreme losses than the Value-at-Risk. The VaR only depends on the probability of losses above the $ q$-quantile $ x_q$, but it doesn't say anything about whether these losses are always just a little above the threshold $ x_q$ or whether there are also losses that are much larger than $ x_q$ that need to be taken into account. In contrast the expected shortfall is the expected value of the potential losses from $ x_q$ and depends on the actual size of the losses.

The Value-at-Risk is simply a quantile and can be, for example, estimated as a sample quantile $ \hat{F}_n^{-1}(q)$, where $ \hat{F}_n(x)$ is the empirical distribution of a sample of negative values, i.e., losses, from the past. As was discussed at the beginning of the chapter, this particular estimator of $ q
\approx 1$, which is for the typical VaR-level of 0.95 and 0.99, is often too optimistic. An alternative VaR estimator, which has the possibility of reflecting extreme losses better, is the POT or the Hill quantile estimator.

Analogous estimators for the expected shortfall are easy to derive. This risk measure is closely related to the average excess function when $ u = x_q$, as immediately can be seen from the definition:

$\displaystyle S_q = e(x_q) + x_q. $

Here we only consider the POT estimator for $ S_q$. Since $ F_u (x) \approx W_{\gamma, \beta} (x)$ for a sufficiently large threshold $ u$, it holds from Theorem 17.5, b) with $ \alpha = 1/\gamma$

$\displaystyle e(v) \approx \frac{\beta + (v - u)\gamma}{1-\gamma} \; \;$   for$\displaystyle \; \; v > u.$

Therefore, for $ x_q > u$ we have

$\displaystyle \frac{S_q}{x_q} = 1 + \frac{e(x_q)}{x_q} \approx \frac{1}{1-\gamma} + \frac{\beta - \gamma u}{x_q (1-\gamma)}. $

The POT estimator for the expected shortfall $ S_q$ is thus

$\displaystyle \hat{S}_{q,u} = \frac{\hat{x}_q}{1-\hat{\gamma}} + \frac{\hat{\beta} -
\hat{\gamma}u}{1-\hat{\gamma}}, $

where $ \hat{x}_q$ is the POT quantile estimator.