# 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.)

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

• Thanks. Just testing: would this give exactly zero vol for a steady $x\%$ per month (with the last month $(x\%+1)^{(21/31)}-1$)? (I'm also somewhat unsure about the $21/31$ factor before the $R_{25}$ in the weighted mean.) – Řídící Jan 25 '16 at 20:42
• Thank you for putting me on what might be the right track. +1 However, I'm fairly sure that $\mu$ top right should be $25/31~\mu$ and $21/31~R_{25}$ bottom right should be $R_{25}$. At least than the steady growth would have zero variance. – Řídící Jan 26 '16 at 16:22
• @KinnisalMountainChicken thanks indeed I hadn't rescaled the last returns to a monthly return. I've edited my answer. But I'm not sure I understand why the top right $\mu$ should be $25/31 \mu$ and the $21/31 R_{25}$ should be $R_{25}$. When calculating the mean and the variance you are basically doing a weighted sum (of the returns, and of the squared differences to the mean respectively). The $21/31$ comes from your confidence on the last value which is lower proportional to the fewer days over which you have observed it: first 24 months would have a weight of $1$, and the last one $21/31$. – Borja Jan 27 '16 at 7:53
• Which is why in the denominator you have $24 + 21/31$ instead of $25$. This should also yield a volatility of 0 for a steady return of $x$% – Borja Jan 27 '16 at 8:01
• Because I would tend to think that you shouldn't rescale the last month's return, but rather the last months' '$\mu$'. (And, also, but this is unrelated the previous point, I think that the $R$'s should be replaced by, or defined as, $\text{Ln}(1+R)$.) – Řídící Jan 27 '16 at 18:47