The portfolio optimization higher-moment credo

The question of skewness and kurtosis in portfolio optimization.


Problem 4 of “The top 7 portfolio optimization problems” concerns the use of higher moments.

“Further adventures with higher moments” is the most recent in a series of posts on the efficacy of higher moments in optimization.  This set includes the observation that “trade selection” is a better term than “portfolio optimization”.


For any particular optimization problem there are a few possibilities of what you might believe about higher moments:

  1. The returns are normally distributed
  2. Adding higher moments to the optimization has no benefit
  3. The effect of adding higher moments is unknown
  4. Can’t be bothered to add higher moments even though they would help
  5. Use higher moments

1) Normally distributed returns

If you believe this, then you might miss the satire in “The distribution of financial returns made simple”.

In this case higher moments are known, and of no use.

2) No benefit of higher moments

There is persistent misinformation that this possibility doesn’t exist.  The commonly repeated message is that mean-variance optimization implies that you believe returns are normal.

That is not the case.  It has things backwards: if returns are normal, then any optimization is mean-variance (though it could be done inefficiently by adding higher moments).

Doing mean-variance optimization is perfectly okay when the returns have symmetric distributions.  It is the best thing to do if there is no predictability of higher moments.

3) Unknown effect

This is appropriate, for instance, when it is clear that there is some predictability of skewness and kurtosis, but it is not clear that the predictability will translate into better results.

4) Can’t be bothered

Incorporating higher moments would improve the utility of the portfolio, but seemingly not the overall utility that includes the process of managing the portfolio.

5) Include higher moments

Higher moments are put into the optimization — either explicitly, or implicitly via a scenario optimization.


For most assets other than equities, my bias would be to do scenario optimization.  High-level asset allocation would generally be an exception.

For equities I am somewhere among beliefs 2 through 4.  Before I started looking at the stability of skewness and kurtosis among random portfolios, I was leaning towards 4.  Since I have done that experimenting, I am much closer to 2.


I believe the stars keep shining all through the night.
I believe if we just keep trying it will be all right.

from “I Believe” by Chris Isaak

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