When VaR Can Be Used to Infer Standard Deviation

Summary

The document asks whether standard deviation can be recovered from a value-at-risk estimate, an expected return, and a normal z-score. It presents an algebraic rearrangement of the normal VaR relationship, then questions a negative result obtained from the supplied inputs.

The answer explains that this inference relies on returns following a normal distribution, where the z-score describes a normal quantile. A lognormal example shows why the rearranged expression does not generally equal standard deviation: the VaR quantile and the distribution’s standard deviation follow different formulas. The discussion does not resolve whether the original negative value reflects a sign convention or inconsistent inputs, and it offers no broader estimation procedure. Its main lesson is that a normal quantile formula cannot be applied indiscriminately across distributions.

Key ideas

• Inferring standard deviation from VaR and expected return requires a distributional assumption.
• The z-score formula applies when returns follow a normal distribution and the VaR sign convention is consistent.
• For a lognormal variable, the VaR quantile and standard deviation have different expressions.
• A negative inferred value signals that the inputs or formula application need review.

Tags

# Calculate standard deviation from the value at risk


# 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?

## Answer by Raskolnikov (score 2)

https://quant.stackexchange.com/a/37266

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

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.