This method is used mainly for linear products – spot and futures products. Linear VAR cannot be used to quantify the risk of derivatives as the model cannot take into account higher order risks such as gamma and vega. This method is also referred to as the variance-covariance method.
The formula for calculating analytical VaR is given by :
The alpha value can be calculated by using the excel function NORMSINV. So for a 99% confidence interval VaR we would have NORMSINV(0.01) = -2.33
We can apply the above formula to following:
Mean : 11.5%
Standard Deviation : 18.78%
The first step is to convert these figures which are annualized , to a daily figure (or for the required number of days VaR is being calculated for).
The mean is very simply : 1/ 252 * 11.5% = 0.046%
The standard deviation is the square root of (1/252) * 18.78% = 1.18%
Note that we take 252 days due to the number of trading days in a year.
When calculating standard deviations we always apply the square root of time rule we talked about earlier.
Now we can calculate the VaR of the portfolio for a 1 day time horizon at 99% confidence interval:
VaR (1,99%) = -(0.046% -2.33 * 1.18%) = 2.70%
One final thing to look at would be to convert the VaR to a monetary figure as this is more meaningful. So if we had £20 million invested in the portfolio the VaR would be:
VaR (1,99%) = 20,000,000 * 2.70% = £540,680
If the above VaR was for 95% we would quite easily do the following :
VaR(1,95%) = -(0.046% -1.64 * 1.18%) = 1.89%*20,000,000 = £377,840
If we wanted to convert the VaR to a 10 day VaR, we would apply the square root of time rule :
VaR(10,95%) = 3.16228 * £377,840 = £1,194,835
VaR(10,99%) = 3.16228 * £540,680 = £1,709,781
The good thing about Analytic VaR is that it is very easy to implement. The data needs to be gathered and historic volatilities and correlations calculated. The historic statistics can also be bought from external vendors or even gotten straight from a Bloomberg Terminal.
However there is a huge drawback with analytic risk in that it assumes that distributions are normal. This can lead to inaccurate VaR estimates (overestimated at lower confidence intervals and underestimated at higher confidence intervals).
Analytic VaR ignores fat tails due to assumption that distribution is normal. This can lead to risk being underestimated as a model based on the normal distribution would underestimate the proportion of outlying observations and therefore the true value of risk.
In summary while analytic VaR is a good starting point, it is wholly inappropriate for portfolios that have a significant investments in derivatives, where the payoff structure is non-linear and discontinuous.