# PM: "Mean Variance Frontier" vs. "Minimum Variance Frontier", What's the difference?

“Minimum Variance Frontier” makes sense…lowest amount of risk (st. dev.) given a desired E®, but why do they use the term “Mean Variance Frontier”? From the readings, it seems like they use these terms interchangeably without any explanation between the difference of the two.

Ain’t a bit of difference between the two…

Yeah, the minimum variance frontier is the frontier that you get from mean variance optimization, and it’s also the efficient frontier, so I guess they were just trying to vary the language.

Typically the Y axis is the mean return. You could also construct the e.g. median-variance frontier, but no one uses that.

Would the median variance frontier look any different? I thought MVO assumes that returns are normally distributed, so that the distribution is fully described by mean and variance, and, more importantly, is symmetric, so mean=median. I’m not sure how you would compute the median analytically from the kind of data that goes into MVO, although I think it would be something useful?

C’mon it was just a quip…

DarienHacker Wrote: ------------------------------------------------------- > Typically the Y axis is the mean return. You > could also construct the e.g. median-variance > frontier, but no one uses that. Ah, thats great. I get it now. Thanks Darien Hacker, not bad for a hockey thug! so it’s “minimum” variance frontier for minimum acceptable risk for given return, and it’s “mean” variance frontier for mean return for a given level of risk. That makes sense, Thanks!

bchadwick: I don’t know any restriction or assumption on shape of distribution for constructing a minimum-variance frontier. It’s simple: for a given y value, find the minimum x (variance) value. Repeat. Y axis could be mean return, or maximum return, or median return, or average cost, or almost any measure. If what you’re measuring is more valuable when risk is minimized, then go ahead and construct the frontier. Also, you made a mistake earlier when equating the minimum-variance and efficient frontier. Efficient frontier is generally a subset of the min-var frontier. In portfolio management, you start with the min-var frontier and then pick the “top” side of it: for a given volatility, the client would obviously prefer a higher return. So the efficient frontier is the portion of the min-var frontier where slope is positive. UAECFA: I’ve never heard of constructing a frontier that isn’t for minimum variance. So all efficient frontiers (that I’ve encountered) are minimum variance frontiers. If what’s on the Y axis is the mean of something, you have a mean-variance plot, and the minimum-variance frontier is also called the mean-variance frontier. (You could also construct the maximum-variance frontier, but people aren’t really interested in that.) — Bonus questions: 1. While we’re hashing out usage: what if my X axis is VaR (or some other risk/dispersion metric)? Would I still call it an efficient frontier? Would it be a mean-variance frontier? 2. Instead of return, I need to plot my funding cost vs risk. How do I find the optimal portfolio?

> 2. Instead of return, I need to plot my funding > cost vs risk. How do I find the optimal > portfolio? There is a LOS regarding this. We are expected to plot the surplus in an ALM company (i.e. insurance co.). It was one of the approaches in the Asset Allocation readings. St. dev. was on the x axis and the surplus was on the y axis. The effecient frontier appeared to be the same. With respect to cost, I think you would want the most efficient cost for a given level of risk so the mean-variance frontier would look the same. However, you would want the LOWEST cost so the effiecient frontier would be the bottom portion of the curve, not the top portion. I could be totally wrong though.

UAECFA Wrote: ------------------------------------------------------- > DarienHacker Wrote: > -------------------------------------------------- > ----- > > Typically the Y axis is the mean return. You > > could also construct the e.g. median-variance > > frontier, but no one uses that. > > > Ah, thats great. I get it now. Thanks Darien > Hacker, not bad for a hockey thug! > > so it’s “minimum” variance frontier for minimum > acceptable risk for given return, and it’s “mean” > variance frontier for mean return for a given > level of risk. That makes sense, Thanks! Perhaps it makes sense but it isn’t true. The frontier is always the boundary of some, um, convex hull. I think Darien’s point about efficient frontier being only part of the mean-variance frontier is right in some very orthodox kinda way but generally all these terms are about the set of portfolios that has the minimum variance for a given level of return. Also "It’s simple: for a given y value, find the minimum x (variance) value. Repeat. " is one terrible algorithm for calculating an efficient frontier. First, it requires another step of interpolating and then you only get some approximation of the frontier. Next, is that you don’t get corner points and you will likely interpolate right through them unless you make a very fine grid. Implementing this isn’t so easy either as once you pick a return, finding the set of portfolios that give you that return is not so easy and then you still have the task of finding the minimum variance portfolio. The usual quadratic programming solution is several orders of magnitude faster and gives you an exact result. Now if you believed that the median return was different from the mean return for your portfolio finding an algorithm for calculating the frontier is not so easy because you lose that linearity of the mean, i.e., the median of the mixture distribution doesn’t equal the weighted sum of the medians. So we would need to do some math. The reason that you don’t see that is that for portfolios that contain some reasonable number of securities, under really broad assumptions you have normality from some central limit theorem or other (the iid CLT of the CFA exams is nothing like the only CLT and you can relax both the i’s). That means that for most of the frontier the mean and median are the same and we have some fine tools for mean-variance optimization.