# Calculate standard deviation from the value at risk

I have the following data:

• VaR

• VaR%

• Expected return

Am I right to think that I would be able to derive standard deviation from this?

Using the formula: VaR%= ER-(zscore*SD), I should be able to calculate SD= (ER-VaR%)/zscore

Given:

• VaR%= 0.5%,

• ER= 0.3%,

• zscore= 1.65(95% confidence)

then, SD= (0.3-0.5)/1.65= -0.12

But SD can never be negative. I the derivation wrong or anything else wrong? I cant figure it out. Can someone help please?

• The sign convention for VaR is ambiguous, in some books they will write VaR%= 0.5% when they refer to a 1/2 percent loss and in others VaR%= -0.5%. Some sign issue like this is likely the cause of your problem. – noob2 Dec 7 '17 at 9:52
• In other words SD=(0.3-(-0.5))/1.65 – noob2 Dec 7 '17 at 13:57

This is only correct if the expected returns are normally distributed. Remember that z-score is in essence the quantile function or the VaR of the normal distribution. If you try to apply this to any other distribution, you are going to be sorry.

Take a lognormal distributed variable$\sim \text{logN}(\mu,\sigma^2)$, the VaR in that case is

$$\text{VaR}_{95\%}=\exp(\mu-\sigma 1.65) \; .$$

With your formula, you would deduce that SD is equal to

$$\text{SD}=(\text{ER}-\text{VaR}_{95\%})/1.65 = (\exp(\mu+\sigma^2/2)-\exp(\mu-\sigma 1.65))/1.65$$

whereas the standard deviation of a lognormal variable is

$$\text{SD} = \exp(\mu+\sigma^2/2)\sqrt{\exp(\sigma^2)-1} \; .$$

• And just to point out the obvious, real world returns do not follow the normal distribution. Real world returns exhibit excess kurtosis (i.e. fat tails). You can easily reject the hypothesis of normality by conducting a studentized range test, classic normality tests like $\chi^2$ etc... or by simply looking at a QQ plot. – Matthew Gunn Dec 7 '17 at 16:19