## Risk measures

While there approximately an infinite number of market risk measures have been proposed, by and large only two are in general use, Value-at-Risk (VaR) and Expected Shortfall (ES). Both of these date back to the 1990s.

The information below comes from Financial Risk Forecasting.

### Notation

$\DeclareMathOperator{\VaR}{VaR}$ $\DeclareMathOperator{\ES}{ES}$ $\DeclareMathOperator{\E}{E}$
• $p$ Probability
• $Y,y$ Return
• $Q$ Profit and loss
• $\vartheta$ Portfolio value

### Value-at-Risk, VaR

The loss on a trading portfolio such that there is a probability $p$ of losses equalling or exceeding $\VaR$ in a given trading period and a $(1-p)$ probability of losses being lower than VaR.

We may write it as $\VaR(p)$, to make the dependence on probability explicit, e.g. VaR(0.05). The most common probability levels are 1% or 5%, but numbers higher and lower that that are often used in practice.

VaR is a quantile on the distribution of profit and loss profit and loss (P/L). We indicate the profit and loss $P/L$ on an investment portfolio by the random variable $Q$, with a particular realization indicated by $q$. If we hold one unit of an asset, the P/L would be indicated by: $Q=P_t-P_{t-1}.$ More generally, if the portfolio value is $\vartheta$: $Q=\vartheta Y$ i.e. the P/L is the portfolio value times returns. The density of the P/L is denoted by $f_q\left(\cdot\right)$. VaR is then given by: \begin{equation} \Pr \left[ Q\leq -\VaR(p)\right] =p \end{equation} or \begin{equation} p=\int_{-\infty }^{-\VaR(p)}f_q\left( x\right) dx \end{equation}

\begin{equation} p=\int^{-\VaR}_{\infty} f(x)dx \end{equation}

or \begin{equation} \Pr[x \le -\VaR]=p \end{equation}

We use a minus sign because VaR is a positive number, and we are dealing with losses — that is probability of losses being larger (more negative) than negative VaR.

### Expected Shortfall, ES

Expected loss conditional on VaR being violated, i.e. the expected profit/loss of $Q$, when it is lower than the negative VaR: \begin{equation} \ES=-\E [Q | Q \le -\VaR(p)]. \end{equation} or \begin{equation} \ES=\int^{-\VaR}_\infty x f(x)dx \end{equation}