Bessel's Correction and Unbiased Variance

If you know a return series' true population mean exactly, computing its variance is simple: average the squared deviations from that known mean. In practice you never know the true mean — you have to estimate it from the same sample, using the sample average $\bar{x}$. That reuse is the problem: $\bar{x}$ is, by construction, the number that makes the sum of squared deviations smallest for this particular sample, so deviations measured from $\bar{x}$ are systematically a bit smaller than deviations from the true (unknown) mean would be. Dividing by the sample size $n$ therefore slightly underestimates the true variance on average.

Bessel's correction fixes this by dividing by $n - 1$ instead of $n$:

$$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 .$$

In plain English: shrinking the denominator by one inflates the estimate just enough to offset the bias introduced by estimating the mean from the same data, making $s^2$ an unbiased estimator of the true population variance. The "$n-1$" is often described as the "degrees of freedom" lost to estimating the mean.

For a small sample of five daily returns with sample mean $\bar{x}$ and sum of squared deviations equal to 20, dividing by $n = 5$ gives a variance of 4, while dividing by $n - 1 = 4$ gives 5 — a 25% difference that matters a lot at small sample sizes and shrinks toward negligible as $n$ grows large.