A value-at-risk model basically works like this. You have some stuff, which is worth X today. Tomorrow it will be worth X + Y, where Y ranges from more or less negative infinity to positive infinity. Y is a function of a bunch of correlated random variables, rates and credit and stock prices and general whatnot. You look at a distribution of moves in those variables and take (usually) a 2-standard deviation daily move; if 95% of the time rates move by -10 to +10 basis points, your VaR model will assume a -10bp or +10bp move, whichever is bad for you. You take the 95%-worst-case, taking into account correlation etc., and tot up how much you’d lose in that case. Then you write that number down and feel a bit better, since you’ve sort of implicitly replaced “we have $X today and will have some number between negative and positive infinity tomorrow” with “we have $X today and will have some number between ($X – VaR) and positive infinity tomorrow,” though of course the first statement is true but unhelpful and the second is not true and also unhelpful.
But that aside! You get your VaR from a distribution of your variables, but the obvious question is what distribution. A good answer would be like “the distribution of those variables over the next three months,” say, for quarterly reporting, but of course that is only a good answer because it begs the question; if you knew what would happen over the next three months you would, one assume, always end those three months with more than $X and this VaR thing would be moot or moot-ish.1
So instead you look at things that you think will allow you to predict that future distribution as accurately as possible, which is epistemically troubling since VaR is a measure of how inaccurate your predictions might turn out to be. Anyway! You pick a distribution of variables based on the sort of stuff that you always use to estimate future distributions in your future-distribution-estimating business, which could mean distributions implied by market prices (e.g. option implied vol) but which seems to mostly mean historical distributions. You look at the last N days of data and assume that the world will be similarly distributed in the following M days, because really what else is there to do.
Picking the number of days to use is hard because, one, this is in some strict sense a nonsense endeavor, but also two, the world changes over time, so looking back one year is for instance rather different from looking back four years. Here is how different: Read more »









