Estimating Volatility from a Partial Final Monthly Return

Summary

The document considers how to estimate volatility when a return series contains complete monthly observations plus a final observation covering only part of a month. It proposes a weighted variance calculation: first scale the partial-period return to an estimated full-month return by compounding it over the fraction of the month observed, then give that observation a weight equal to the observed fraction of a month. The remaining complete monthly returns receive full weight. The mean is computed with the same weighting scheme, and volatility is the square root of the resulting variance.

The method is offered as a possible approach when only monthly data are available; the response says daily returns would be preferable for annualizing volatility. The partial-month scaling assumes the observed return can be compounded to represent a full month, and the example does not discuss how that estimate behaves under changing volatility or calendar effects. The proposed formula also defines a variance estimate but does not provide a separate annualization step.

Key ideas

• A partial month can be assigned a fractional weight in a variance estimate.
• Scale the partial-period return to an estimated full-month return before including it.
• Use the same fractional weighting when calculating the mean and variance.
• Daily returns are preferable when available for estimating and annualizing volatility.

Tags

# Volatility of monthly performances, where the last month is short


# Volatility of monthly performances, where the last month is short












I'd like to calculate the vol of a return series of, say, 25 months. However, the last of those months is not completed yet. The last data point only refers to the first 21 days of the month (say, January). (All the others refer to whole months.)

Is it as simple as $\text{Vol}=\text{StDev}(\text{Ln}(1+R))\times \sqrt(12 \times 25 / (24 + 21/31) )$?

(I'm guessing not.)

## Answer by Borja (score 4)

https://quant.stackexchange.com/a/22918

Ideally you'd want to use daily returns and just annualise it, but if you only have monthly returns then calculating the weighted variance in the following way might do it:

$$ Var = \frac{\sum_{i=0}^{24}(R_i - \mu)^2}{24 + \frac{21}{31}} + \frac{\frac{21}{31} (R_{25}' - \mu)^2}{24 + \frac{21}{31}} $$

$$ Vol = \sqrt{Var} $$

Where $R_i$ is the returns of your $i^{th}$ month, and $R_{25}'$ is the returns of the 25th month (only up to its 21st day), compounded to a month (as you wrote in your comment):

$$ R_{25}' = (1 + R_{25}) ^ {\frac{31}{21}} - 1 $$

$\mu$ is the weighted mean: $$ \mu = \frac{\sum_{i=0}^{24} R_i }{24 + \frac{21}{31}} + \frac{\frac{21}{31} R_{25}'}{24 + \frac{21}{31}} $$

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.