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 »