“Even the youngest of us may be wrong sometimes.” [George Bernard Shaw]
Abstract
Value at Risk (VaR) is the most commonly used measure of the amount that could be lost from a position or a portfolio.
VaR is understood to be the maximum loss which could occur at a given confidence level over a specified time horizon. More precisely, it is the threshold value such that the probability that the mark-to-market loss on the portfolio over the given time horizon exceeds this value is the given probability level. We assume normal markets and no trading in the portfolio.
VaR calculations often assume that returns are normally distributed over the time horizon. Common parameters are 1% and 5% probabilities and a one day or 10 days (2 weeks) horizon. The inputs for VaR calculations include details of the portfolio, the time horizon and the parameters which drive the distribution of the underlyings (the average growth rate, volatilities and correlations). For short time horizons the growth rate is negligible.
Confidence Level
If you are working with standard deviations only then you can go from one confidence level to another by adding the corresponding difference of their distances of standard deviations from the mean:
| Degree of Confidence | # of Standard Deviations from Mean |
|---|---|
| 95% | 1.64485362695147 |
| 99% | 2.32634787404084 |
| 99.9% | 3.09023230616781 |
You can move from one time horizon to another by multiplying or dividing with the square root of time. This is only true if the difference between time horizons is short enough to ignore the growth rate.
Different Approaches
There are three different basic approaches to calculate VaR:
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Variance / Covariance Method
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Historical Simulation
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Monte Carlo Simulation
The variance / covariance method requires only the two factors average return and standard deviation. Therefore it is also called the parametric method. It assumes that returns are normally distributed over the time horizon.
The historical simulation requires more computational data and produces a more accurate VaR. It also assumes that history repeats itself.
The Monte Carlo simulation is more complex to implement. Its advantage is that you can design future patterns to differ from historical patterns.
(Dis-)Advantages of Using VaR
| Advantages of Using VaR | Disadvantages of Using VaR |
|---|---|
| Easy to calculate for single assets, portfolios and whole companies | No knowledge of loss beyond VaR |
| Easy to adjust to trading style: If you hedge daily take 1 day, if on average you buy and hold for 2 weeks take 10 days | VaR is developed for normal markets, not for extreme events |
| Easy to break down into components or to analyze marginal risk (add or take away an asset) | [In case of historical simulation] It uses historical data (history is assumed to be repeated) |
| Good limit measure for single traders, desks or whole companies | Positions can change dramatically during time horizon (trading, hedging, expiration) |
| Easy to understand | Does not satisfy commonsense criteria like (external link!) coherence |
Because of its drawbacks VaR is suggested to be used only in conjunction with stress testing and with backtesting.
Further Reading
John C. Hull “Options, Futures and Other Derivatives”
Paul Wilmott “Frequently asked Questions in Quantitative Finance”, Jon Wiley & Sons, Ltd