Q:

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.)

A:

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

