Annualizing Volatility Implied by a VaR Limit
Summary
The document explains how to interpret a VaR limit specified over a fraction of a year and convert the volatility implied over that horizon into annualized volatility. For a horizon represented as one of m equal yearly intervals, first identify the VaR horizon, the corresponding risk-free rate, and the volatility implied for that period. Annualization then scales the period volatility by the square root of the number of periods in a year.
The answer illustrates both monthly and daily conventions and flags a notation mix-up between the number of observations in the VaR calculation and the annualization frequency. It also states the key assumption behind square-root-of-time scaling: returns across periods are uncorrelated and have the same volatility. The exchange excerpt is brief and its correction distinguishes trading-day horizons from month-based horizons; it does not lay out the full VaR formula or discuss departures from the scaling assumptions.
Key ideas
- A VaR limit implies volatility over the horizon used to calculate that VaR.
- Annualize period volatility by multiplying by the square root of the number of periods per year.
- Distinguish the number of observations in a VaR horizon from the annualization frequency.
- Square-root-of-time scaling assumes equal period volatility and uncorrelated returns.
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# What does this formula (to derive annualized volatility from VaR) mean?
# What does this formula (to derive annualized volatility from VaR) mean?
I'm faced with the formula shown in the image below, which I just don't understand, in part because I've no grounding in stats, and in part because I don't even understand the notation:
What's going on here? Is this showing in the first line how to compute the VaR, and then in the second line how to derive the annualized volatility from some of the variables used in the first computation?
I don't even understand the "T time intervals of 1/m years" part. What's an "mth" of a year? And what does $\sigma_{\frac{1}{m}}$ mean? Is that the volatility for one time period? If so, then don't we need to aggregate the values for all the discrete time periods somehow? I'd expect to see a "sum from 1 to m" somewhere...
I'm totally confused, and any help would be very gratefully received.
## Answer by Richi Wa (score 2)
https://quant.stackexchange.com/a/4157
I guess you want to calculate vola pa for SRRI. The logic is the following:
- If you have a VaR Limit for $1/m$ th of a year (e.g. if $m = 12$ then for one month which is equivalent to $20$ banking days) and the risk free interest rate for a $1/m$ th of a year, this is $rf_{1/m}$ in the formula (e.g. $1$ month LIBOR), then $T=20$ and you can calculate the volatility for $20$ days, this is $\sigma_{1/m}$ ,which is implied by your VaR Limit.
- You calculate a vola pa from it. If you calculate with $20$ days in step 1 then you have $1/m = 1/12$ and you can calculate your annual volatility by multiplying the $20$-days volatility by $\sqrt{12}$ because the year has $12$ months. Take care: this assumes uncorrelated monthly returns all having the same volatility.
EDIT: There is mix up in this answer between $T$ and $m$. Calculating the VaR with daily data we need $T=20$ and then we annualize the volatility implied by the VaR-Limit by scaling with $\sqrt{250}$.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.