Value-at-Risk

From Traderpedia

Definition:
A measure of how much the value of an asset or portfolio is likely to change (generally thought of in negative terms) over a certain time period at a given level of confidence.


For example, the 1-day Value-at-Risk (VaR) for a specific portfolio could be $1000 at a 95% confidence.

VaR is often used by medium and long term investment managers (e.g. systematic hedge funds, investment banks) to put a value on the amount of money (or percentage of capital) they determine is "at risk" (i.e. could be lost based on the model they are using). As stated above, a VaR figure must be accompanied by a time frame (usually 1day or 1month) and a confidence level (usually 95% or 99%). A value at 95% confidence means that we expect the return to exceed the stated value 95% of the time, or alternatively 5% of the time we will lose more than that.

The actual calculation of VaR is subjective and therefore often includes several assumptions. It can be calculated "ex-ante" (looking forward - using a model to predict future returns) or "ex-post" (looking back - basing the calculation purely on observed past returns).

A simple demonstration of both calcualtion types follows. The example assumes a simple portfolio with a constant 1 lot long FTSE future from Jan-2008 to Dec-2008, paying £1 per point. The intial capital in the portfolio is £10,000.


Contents

[edit] Ex-Ante

This method uses an assumption that portfolio returns will be normally distributed in the future.

To perform the calculation we need to know the mean daily return and the standard deviation of returns, easily done with excel using the AVERAGE and STDEV functions on our daily returns. In the example above, the mean daily return is -£7.81 and the std. dev. is £112.88. Convert these to percentage of capital: -0.08% and 1.13% respectively.

To calculate the daily VaR, we assume that future returns are normally distributed, with a mean of -0.08 and std dev of 1.13. We want to know where the 5% value occurs in the normal distribution, so in Excel we can use the following formula: NORMINV(0.05,-0.08,1.13) which gives the result -1.93. This would be interpretted as a VaR of 1.93% at a confidence interval of 95%, which means "we would expect to see a daily return exceeding -1.93% of capital 19 days out of 20".

A 99% confidence number could be obtained by substituting 0.05 with 0.01 above (-2.71% in the example portfolio)


[edit] Ex-Post

This assumes that futures returns will be distributed in a similar way to past returns.

This method simply lines up all previously observed returns in ascending order and then takes the number which is the 5% largest value as our observed VaR. This can be easily caclulated in Excel using the formual:

SMALL(B2:B255,ROUND(5*(COUNT(B2:B255)/100),0)) (where B2:B255 is the range of daily returns).

For the portfolio above, the vaue is -202 which was the 13th smallest return of the portfolio (The FTSE lost 202 pips 18-Nov to 19-Nov 2008). This represents -2.02% of capital and would be interpretted as a VaR of 2.02% at a confidence interval of 95%, which means "we would expect to see a daily return exceeding -2.02% of capital 19 days out of 20".

A 99% confidence number could be obtained by substituting 5 with 1 above (-3.24% in the example portfolio)


[edit] Summary

It should be clear that different methodologies can produce vastly different results, so any published figures should be treated with caution. The Ex-Ante model above normally underestimates VaR (as we see from the example) as the fat tail/black swan phenomenon means that returns are not normally distributed, especially at the extremeties. For this reason most ex-ante VaR models use a more sophisticated model than the simple normal distribution.

For portfolios consisting of multiple securities, appropriate consideration should also be taken with correlation between returns.



[edit] See also