# 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.

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looks like the EWMA/VaR model –  pyCthon Sep 20 '12 at 3:31

I guess you want to calculate vola pa for SRRI. The logic is the following:

1. 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.
2. 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 $\sqrt{250}$.

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Reading it again I see that there is a mix up. In your screenshot and in my answer. The best is if you read the correct formula in The Esma publication on page 14. –  Richard Sep 20 '12 at 10:51
In the ESMA publication above VaR is calculated using weekly data. Then $T=4$ for a monthly (e.g. $20$ days) VaR and then you see the scaling by $\sqrt{52}$ because the year has $52$ weeks- –  Richard Sep 20 '12 at 10:57
Thanks. You're edging me towards enlightenment, but I still don't understand a couple of points: (a) If $\sigma_{\frac{1}{m}}$ is a single value, then presumably the rf bit is also a single value? But the text seems to imply that it's several values: "the risk free rate for each of the T intervals of 1/m years"? Or am I reading that wrong, and it just means that we should use the same value for each period? –  Sloucher Sep 20 '12 at 21:53
(b) if the formula only deals with single values, that makes it easier, and I can see how you can get the VaR based on the three inputs (volatility, risk-free rate and T). But how can that be solved for volatility given a particular value for VaR? –  Sloucher Sep 20 '12 at 21:56
To your a) this is the risk free rate for the holding period that your VaR has. E.g. if you have the 20 days VaR then this should be your 20 days (i.e. 1 month) risk free rate - one number for this one time period. The phrase which you quote is strange ... and I find in in the ESM text too. Nevertheless I am of the opinion that this is one number on a given day. –  Richard Sep 23 '12 at 19:29