I have recently downloaded a file related to VaR. I know that VaR at 100*p% level is the value of x in the following equation: Pr(Loss in a t-day period >= x) = p. However, the file has a different calculation method of finding VaR. According to the file: For example, if mark-to-market = $100M standard deviation of annual return = 15% default holding period = 10 days credibility level = 99%, then VaR = 100M*15%*(10/252)^0.5*2.33 = 7M. I don't know how we can get this formula. Can someone explain this? Thanks!
It probably helps to know the context, I've never seen a VaR calc like that before. It's just a guess, but I think this VaR calculation looks much more like one that would be used in e.g. a bank, where their VaRs are calculated for much shorter timescales (they will calculate 1 day VaRs). The reason I think that is because of the short holding period of their assets, the fact that they are using business days (10/252), and the high standard deviation of returns. Regardless, you shouldn't need to replicate this sort of calculation in an actuarial exam.
It looks like they have assumed a distribution of returns and then simplified the P(X>x)=p formula to something numerical. The 100M*15% gives the absolute standard deviation, the 2.33 is giving you the upper percentile, and the root 10/252 term gives you the time dependence. It might be a useful little exercise if you have the time to think about how you would create a little VaR calculator to give to one of your non-actuarial colleagues. You would probably end up with a formula not unlike this at some point.
Yes, banks are required to calculate the VaR over a 10-day time period when determining their risk capital for solvency purposes (Basel II I think.) 100m is the amount of capital at risk. 15% is the annual standard deviation (SD). The (10/252)^0.5 term is used to scale the annual SD down to the corresponding 1-day SD. This is based on the lognormal distribution, which assumes that the variance is proportional to the length of the time interval and so the SD is proportional to the square root of the length of the time interval. 2.33 is then the 1% one tail probability of the normal distribution. Hope this helps.
