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## 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

- Probability
- Return
- Profit and loss
- Portfolio value

### Value-at-Risk, VaR

The loss on a trading portfolio such that there is a probability of losses equalling or exceeding in a given trading period and a probability of losses being lower than VaR.

We may write it as , 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 on an investment portfolio by the random variable , with a particular realization indicated by . If we hold one unit of an asset, the P/L would be indicated by:
More generally, if the portfolio value is :
i.e. the P/L is the portfolio value times returns. The density of the P/L is denoted by . 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 , 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}