Scaling Multi-Period Variance for Simple Returns

Summary

The document clarifies a variance formula used to scale returns across multiple periods. It interprets the expression as applying the square-root-of-time idea to the variance of compounded simple returns, where the multi-period gross return is the product of each period’s gross return. The calculation starts from the single-period return’s expected value and variance.

The answer points out that this setup concerns simple returns rather than log returns and that the formula’s notation can be confusing. It also illustrates how interpreting the inputs as a mean return of 0.9% and volatility of 2.35% leads to an annualized volatility near 9%. That example depends on the assumed meaning and scale of the inputs; the document does not provide enough detail to assess dependence assumptions or establish that the scaling is appropriate in every setting.

Key ideas

• The formula concerns variance of compounded simple returns across multiple periods.
• Multi-period gross returns are formed by multiplying each period’s gross return.
• The calculation uses the single-period return’s expectation and variance.
• The formula should not be confused with scaling log-return variance, and input units matter.

Tags

# Dorfleitner's Standard Deviation


# Dorfleitner's Standard Deviation












Can someone please advise how to compute the following (as my results go into thousands):

E.g. I have used

and the result (for T=12) = 580103.7261

Thanks

## Answer by Enrico Schumann (score 1, accepted)

https://quant.stackexchange.com/a/46740

The formula looks like the square-root-of-$t$ rule for scaling return-variance, but for simple multi-period returns, not log-returns. That is, the formula shows you how to compute $$\mbox{var}\bigg(1+R^{(\tau)}\bigg) = \mbox{var}\bigg(\prod_{t=1}^{\tau} 1+R_t\bigg)$$ starting from the expectation and variance of $R_t$.

In which case $r^d$ would be a single-period return. (The notation is unfortunate: writing $\sigma^{\tau}_{i}$ when $\tau$ is an integer.) What exactly do the values of $\mu$ and $\sigma$ show? If I take them to mean 0.9% and 2.35%, i.e. 0.009 and 0.0235, then I get an annualised vol of about 9%, which seems reasonable.

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.