Scaling Value at Risk Across Time Horizons
Summary
The document reviews ways to convert a one-day value-at-risk estimate to a longer horizon. For models where VaR is proportional to volatility, such as a normal model or a suitably specified Student-t model, volatility and VaR are often scaled by the square root of time. If the VaR expression also includes an expected-return term, that term scales linearly with time while the volatility component scales with the square root of time. One answer gives the common illustration of using the square root of 250 trading days for an annual horizon.
This shortcut depends on assumptions, especially that returns do not exhibit relevant autocorrelation and that the model’s distributional form remains appropriate over the longer period. Mean reversion can reduce scaling, while positive feedback can increase it. The responses caution that simple scaling can be inappropriate and recommend modeling the target horizon directly, for example through a longer-horizon simulation. The discussion offers general guidance, not a universal annualization rule.
Key ideas
- When VaR is a constant multiple of volatility, it commonly scales with the square root of time.
- An expected-return component scales with time, while the volatility component scales with square root of time.
- Using the square root of the number of trading days assumes no material autocorrelation.
- Mean reversion or positive feedback can make simple time scaling inaccurate.
- Re-running the risk model at the target horizon can be preferable to annualizing daily VaR mechanically.
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Full text
# How to extrapolate VaR?
# 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?
## Answer by Richi Wa (score 4, accepted)
https://quant.stackexchange.com/a/15749
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.
## Answer by RossFabricant (score 3)
https://quant.stackexchange.com/a/15745
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.
## Answer by dg_risk (score 1)
https://quant.stackexchange.com/a/15747
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.
## Answer by Onyxx (score 0)
https://quant.stackexchange.com/a/15786
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.