is the minimum variance portfolio equation taught anywhere in CFA?

CFA General
#1

is the minimum variance portfolio equation x = (σb²-ρabσaσb) / (σa² + σb² – 2ρabσaσb) taught anywhere in CFA?

I went through portfolio management topics in both level 1 and 2 and could not find any mention of this formula in the currimulum.

#2

Don’t think so.

#3

That’s weird, how a can program that mostly useful in wealth management not have this formula?

#4

Normally you need the Excel Solver or better to construct the efficient frontier and since it is too technical the curriculum just describes it rather than calculate. It seems easy for two a two asset portfolio but after that it gets out of hand…

#5

Back in the day, I recall this formula being somewhere in the curriculum. Can’t remember if it was L3 or not.

But I have to say, learning the two-asset case does very little for people who are genuinely interested in using mean-variance optimization in a case of three or more constituent assets. In the realm of realistic portfolio construction with many assets, the pressing topics are:

  • Handling the computational explosion with three or more assets, where the number of equation terms rapidly balloons to accommodate all of the possible asset correlation pairs,
  • What algorithm/program to use in order to actually solve the frontier points if not manually solving these formulas, generally an iterative process similar to a generalized reduced gradient (GRG) algorithm which can be found with solutions like Excel Solver (previously mentioned, but the free version suffers from a 200-variable limit, insufficient for very large portfolios),
  • Handling the adjustments to inputs, such as what happens if you tweak any correlations in your matrix (ensuring that the modified matrix is positive semidefinite, i.e., internally consistent and does not imply any negative variances, through eigendecomposition analysis and any number of matrix adjustment procedures),
  • Determining the “correct” data frequency and sample length for your initial starting estimates, unless you are an all-seeing guru who has a very good idea of what the returns and other inputs would be, which is the ridiculous idea prescribed by Harry Markowitz, especially in his most recent writings (which begs the question, if you know that already, why optimize and combine when concentrating on the better-performing assets would make more sense?),
  • Determining whether you should even be using mean-variance optimization as the correct objective function, as opposed to another parametric or non-parametric target to construct your frontier,
  • Countless other considerations that have everything to do with determining and defining “proper” constraints to other myriad input considerations.

None of the above would be in the scope of the CFA program. You’d be looking at a PhD or MSQF for most of it, and even then, you’d likely be forced into having to do a lot of outside reading to connect all of the dots, especially with regard to non-Gaussian approaches.