II. Module 7 Homework 3 [final]: Bayesian Portfolio Theory

This problem will introduce some ideas from Bayesian portfolio theory. 

Here is the problem: our portfolio theory so far has a hole in it. We solve a portfolio theory assuming that investors know means and variances  and  perfectly. We then estimate  and , and plug those estimates in to the portfolio problem. Doesn't the estimation error matter?

In fact, as you saw when we calculated sample mean-variance frontiers, the estimation error is huge, and drives all the problems of mean-variance theory. The mean-variance optimizer jumps on small differences in mean returns due to sampling error, and assets that appear too correlated due to sampling error, and recommends huge long-short positions. Perhaps incorporating the sampling uncertainty of the parameters will solve this "wacky weight" issue.

Now, your first instinct might be, no, parameter uncertainty will just give standard errors for our portfolio weights -- a good thing -- but sampling uncertainty should not bias the weights either way. That instinct is wrong, because portfolio weights are not a linear function of the data.

The bottom line answer is, parameter uncertainty is a real risk to the investor, you should include parameter uncertainty in the portfolio calculation, and this consideration generally recommends less aggressive exploitation of statistical models.