Apr 10

Relative (short horizon) versus absolute (long horizon) value at risk (VaR) - 10 minute screencast

by David Harper, CFA, FRM, CIPM

FRM | CFA |

absoluteVaRthumb

For FRM candidates, Paul Wilmott shares two formulas for value at risk (VaR). The first is what Jorion has elsewhere called relative VaR (as in, relative to the expected value). When the time horizon is short, this is the same as the next version because the assumed, expected return is equal to zero. So, this VaR formula...

var_relative 

...is a special case of this formula (below) which Jorion has elsewhere called Absolute VaR (as in, absolute loss relative to zero):

var_absolute

The difference is that the latter ("absolute VaR") includes the expected return: the loss is mitigated by expected gains. (Wilmott's notation +/- sign is a bit confusing and, in my opinion, inconsistent. It is better to say: absolute VaR = - return + [scaled volatility]. That way you don't run the risk of adding the gain and loss together, but no matter).

So, the practical difference is that you can use the first (the simpler that excludes expected return) when the horizon is short enough such that assuming expected return = 0 is an acceptable assumption. On an equally practical level, many people point out that scaling VaR over long horizons is an unfortunate use of the model. That is, VaR was designed for daily trading distributions but it's extension into monthly/yearly periods is flawed. And, mathematically, they are correct because the "square root rule" presumes i.i.d. in the return volatility. Empirically, we know that isn't the case!

Here is the screencast:

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