Risk Measures

Let’s consider that there are different types (i.e. distributions) of assets, all with the same volatility and mean. The standard mean-variance analysis indicates that all these assets are equally risky. In reality market, however, participants view the risk in them differently.

In practice, the problem of risk comparisons is difficult because the underlying distribution of market prices and returns of various assets is unknown. One can try

  • to identify the distribution by maximum likelihood methods
  • test the distributions against other distributions by using methods such as the Kolmogorov-Smirnov test

Practically, it is impossible to accurately identify the distribution of financial returns.

The most common approach to the problem of comparing the risk of assets having different distributions is to employ a risk measure that represents the risk of an asset as a single number that is comparable across assets.

Three risk measures: Volatility, Value-at-Risk, Expected Shortfall


It is sufficient as a risk measure only when financial returns are normally distributed.


VaR is a single summary statistical measure of risk. It is distribution independent.

The three steps in VaR calculations:

  1. to specify the probability, p, of losses exceeding VaR: 1% (the most common); 0.1% for applications like economic capital or long-run risk analysis for pension funds
  2. to specify the holding period: usually one day
  3. to specify the probability distribution of the P/L of the portfolio: by using past observations and a statistical model.

There are three main issues that arise in the implementation of VaR:

  • VaR is only a quantile on the P/L distribution.
  • VaR is not a coherent risk measure: not subadditivity. It is subadditive in the special case of normally distributed returns.
  • VaR is easy to manipulate

Expected Shortfall

Also be known as tail VaR or conditional VaR (CVaR). It measures the expected loss when losses exceed VaR.

The ES is the negative expected value of P/L over the tail density f_{\textrm{VaR}}\left(\cdot\right)

    \begin{eqnarray*} \textrm{ES} & = & -\left[Q\mid Q\leqslant-\textrm{VaR}\left(p\right)\right]\\ & = & -\int_{-\infty}^{-\textrm{VaR}\left(p\right)}xf_{\textrm{VaR}}\left(x\right)dx \end{eqnarray*}

If the P/L distribution is standard normal, then

    \[ \textrm{ES}=-\frac{\phi\left(\Phi^{-1}\left(p\right)\right)}{p} \]

where \phi and \Phi are the normal density and distribution respectively.

Here is a sample R code

Advantages of using ES:

  1. Any bank that has a VaR-based risk management system could implement ES without much additional effort.
  2. ES is subadditive while VaR is not.

However, in practice the vast majority of financial institutions employ VaR and not ES. The reasons may be:

  1. ES is measured with more uncertainty than VaR. The first step in ES estimation is ascertaining the VaR and the second step is obtaining the expectation of tail observations. This means that there are at least two sources of error in ES.
  2. More importantly, ES is much harder to backtest than VaR because the ES procedure requires estimates of the tail expectation to compare with the ES forecast. Therefore, in backtesting, ES can only be compared with the output from a model while VaR can be compared with actual observations.

Holding Periods

In practice, the most common holding period is daily, but many other holding periods are also employed: e.g. hourly (or every 20/10-min) 90% VaR is used on the trading floor.

Basel Accords require financial institutions to model risk using 10-day holding periods. The majority of risk managers employ scaling laws to obtain such risk levels.

Square-root-of-time scaling \sqrt{T}

It supposes the observed random variables \left\{X_{t}\right\} are IID with variance \sigma^2 over time. The variance of sum of T consecutive Xs is then

    \[ \textrm{Var}\left(X_{t}+X_{t+1}+\ldots+X_{t+T}\right)=\textrm{Var}\left(X_{t}\right)+\textrm{Var}\left(X_{t+1}\right)+\ldots+\textrm{Var}\left(X_{t+T}\right)=T\sigma^{2} \]

This implies that volatility scales up by \sqrt{T}.

The square-root-of-time scaling rule does not apply to VaR unless we assume the returns are normal. It should not be considered to obtain multi-day VaR forecasts by scaling up daily VaR using \sqrt{T}, although the 1996 amendment of Basel Accords explicitly recommends to do so.

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