Value at Risk, or VaR for short, is a measure of the amount that could be lost from a position, portfolio, desk, bank, etc. VaR is generally understood to mean the maximum loss an investment could incur at a given confidence level over a specified time horizon. There are other risk measures used in practice but this is the simplest and most common.

Example

An equity derivatives hedge fund estimates that its Value at Risk over one day at the 95% confidence level is $500,000. This is interpreted as one day out of 20 the fund expects to lose more than half a million dollars.

Long answer

VaR calculations often assume that returns are normally distributed over the time horizon of interest. Inputs for a VaR calculation will include details of the portfolio composition, the time horizon, and parameters governing the distribution of the underlyings. The latter set of parameters includes average growth rate, standard deviations (volatilities) and correlations. (If the time horizon is short you can ignore the growth rate, as it will only have a small effect on the final calculation.)

With the assumption of normality, VaR is calculated by a simple formula if you have a simple portfolio, or by simulations if you have a more complicated portfolio. The difference between simple and complicated is essentially the difference between portfolios without derivatives and those with. If your portfolio only contains linear instruments then calculations involving normal distributions, standard deviations, etc., can all be done analytically. This is also the case if the time horizon is short so that derivatives can be approximated by a position of delta in the underlying.

The simulations can be quite straightforward, albeit rather time consuming. Simulate many realizations of all of the underlyings up to the time horizon using traditional Monte Carlo methods. For each realization calculate the portfolio s value. This will give you a distribution of portfolio values at the time horizon. Now look at where the tail of the distribution begins, the left-hand 5% tail if you want 95% confidence, or the 1% tail if you are working to 99%, etc.

If you are working entirely with normal distributions then going from one confidence level to another is just a matter of looking at a table of numbers for the standardized normal distribution (see Table 2.1). As long as your time horizon is sufficiently short for the growth to be unimportant you can use the square-root rule to go from one time horizon to another. (The VaR will scale with the square root of the time horizon; this assumes that the portfolio return is also normally distributed.)

Table 2.1: Degree of confidence and the relationship with deviation from the mean.

Degree of confidence

Number of standard deviations from the mean

99%

2.326342

98%

2.053748

97%

1.88079

96%

1.750686

95%

1.644853

90%

1.281551

An alternative to using a parameterized model for the underlyings is to simulate straight from historical data, bypassing the normal distribution assumption altogether.

VaR is a very useful concept in practice for the following reasons:

• VaR is easily calculated for individual instruments, entire portfolios, or at any level right up to an entire bank or fund

• You can adjust the time horizon depending on your trading style. If you hedge every day you may want a one-day horizon; if you buy and hold for many months, then a longer horizon would be relevant

• It can be broken down into components, so you can examine different classes of risk, or you can look at the marginal risk of adding new positions to your book

• It can be used to constrain positions of individual traders or entire hedge funds

• It is easily understood, by management, by investors, by people who are perhaps not that technically sophisticated

Of course, there are also valid criticisms as well:

• It does not tell you what the loss will be beyond the VaR value

• VaR is concerned with typical market conditions, not the extreme events

• It uses historical data, 'like driving a car by looking in the rear-view mirror only'

• Within the time horizon positions could change dramatically (due to normal trading or due to hedging or expiration of derivatives).

A common criticism of traditional VaR has been that it does not satisfy all of certain commonsense criteria. Artzner et al. (1997) specify criteria that make a risk measure coherent. And VaR as described above is not coherent.

Prudence would suggest that other risk-measurement methods are used in conjunction with VaR including, but not limited to, stress testing under different real and hypothetical scenarios, including the stressing of volatility especially for portfolios containing derivatives.