covariance and statistical bias

In the book, it calculates the covariance of a 12 month data set. It divides it by 11. The explanation for this is: “we…divide the values by (n-1) rather than by n to avoid statistical bias” ------------------------------- What is this statistical bias? I don’t understand why they do this. pg. 234 in the Corporate Finance and Portfolio Management Book CFAI

Think of it like a margin of safety. Dividing by n-1 instead of n increases variance/covariance, so it’s more conservative.

It has to do with the notion of ‘degrees of freedom’, look it up. Basically just think of it this way: disregarding the covariance with itself, with how many variables can you find covariance with any given one? For January, you can find covariance for the 11 following months Feb-Dec with January. Hope that helps.

First, the question is beyond the scope. Second, statistical bias is the tendency for an estimator to produces estimates that are either systematically greater than or less than the thing they are trying to estimate. In the case of the covariance, you pick up bias by using the sample means instead of the population means. All the data are closer to their sample mean than to their population mean so the products of (X[i] - X-bar)(Y[i] - Y-bar) are all systematically too small. If we then average them by dividing by n, the estimator is biased downwards for the population covariance E(X - muX)(Y - muY). Through some math, you can show that dividing by n-1 instead of n removes this bias.