# How to extrapolate VaR?

I have a model predicting 1-day VaR.

How does 1-year VaR follow from it?

Shall I just multiply by 365 or another method?

• Square root of time. – SmallChess Dec 3 '14 at 23:43
• as @Richard pointed out, the scaling rule depends on the distribution. Value at Risk is a distribution quantile. The Quantile of a Normal Distribution with $\mu = 0$ scales with $\sqrt{T}$, in general it does not! – vanguard2k Dec 4 '14 at 14:14

It depends on the method by which you calculate VaR. Some models (t-distributuion, normal) lead to a form of VaR such that it is just scaled volatility: $$VaR = c \sigma$$ with some proper $c$ (e.g. $q_{\alpha}$ in the case of normal, bit more complicated for the t-distribution). Then as $\sigma$ scales with square-root-of-time so does VaR.

If VaR is modelled with some expectaion of the form $$VaR = -\mu+c \sigma$$ then $\mu$, the expectation, scales with time and volatility as above.

Furthermore: it depends on the setting but usually it is just nonsense to calculate an annual VaR from a daily one.

The standard approach is to multiply by the square root of the number of trading days in a year. If you assume there are 250 trading days in the year, you multiply by $\sqrt{250}$.

Investopedia is one source explaining this approach.

• The link recommends to multiply the standard deviation by $\sqrt{250}$, what about the mean $\mu\cdot 250$? – emcor Dec 4 '14 at 8:40

The most commonly used approach is multiplication by the square-root of T, 19.1 in this case.

This assumes no autocorrelation, however (Markov process). Interest rates tend to show a mean reversion, so the number would be smaller than 19.1. Other cases could show the oppoite effect if there are positive feedbacks. In both of these cases, a simple time scaling is not possible and the model should be re-run for the new time horizon.

The most common approach is to multiply by sqrt of 250. This is the standard. Although very basic. A much better solution is to make your monte carlo simulation on a 1 year time period using scaled parameters over 1 year.