William J. Bernstein
The Appropriate Use of the Mean Variance Optimizer
Few investment tools are more seductive than the Markowitz mean variance optimizer ("MVO"). For those of you unfamiliar with this beast, it will take the basic characteristics of the portfolio component assets (expected return, standard deviation of returns, and the correlations among all of the assets) and ground out the "efficient frontier" of portfolios -- a spectrum of asset allocations which are maximally efficient. In other words, each portfolio has the maximum return for a given degree of risk, or the minimum risk for a given return. Pretty heady stuff.
In fact, these things are so flashy that most of the commercially available ones are basically gee-whizzes for financial advisors to impress clients with, with relatively rudimentary analytical capabilities.
Only problem is, in order to obtain the future efficient frontier portfolios, you have to be able to predict future asset returns, SDs, and correlations. And, hey, if you can do that, you don't need an optimizer -- you need the ability to handle the tsuris which comes with being the world's wealthiest human.
What about simply using historical data? That's probably the dumbest thing you can do, for the simple reason that most national stock markets have a fairly strong tendency to mean revert. All other things being equal, the optimizer tends to pick those assets with the highest assigned returns. If you use historical returns you will wind up with a portfolio of the previously best performing assets, which are liable to be the future worst performing assets. Not cool. As a simple example, if you had optimized your portfolio at the end of 1989 using historical data, the outputted portfolios would have been very heavy weighted towards Japanese equity. Needless to say, had you actually done this you would be by now so far at the back of the class that your best chance of survival would be a search party.
So, what use is the thing? Well, first and foremost an MVO is a superb teaching tool. Play around with one for a few hours and you will begin to acquire a grasp of the rather counterintuitive way in which real portfolios behave. The process can be likened to one's first few hours of flight instruction -- "Golly, if I bank the airplane enough I gain airspeed and lose altitude. If I increase power the houses get smaller, and we go slower," becomes "Geez, willya lookit that -- add in a few percent of Iranian equity and my portfolio becomes less risky."
Can you use an MVO to help you shape your portfolio? Yes, but you've got to be very careful. An MVO is like a chainsaw. Used appropriately, it is a powerful tool for clearing your backyard. Used inappropriately it will send your local surgeon's kids to college. Same thing with MVOs. Want to wind up in the financial version of intensive care? Just throw in some historical (or even plausible) returns and believe what comes out the other end.
What's the right way? In order to answer that question, you have to realize that the chances of your allocation, no matter how skillfully chosen, winding up exactly on the future efficient frontier are zero. In fact, your chances of winding up even within 1% of return of the EF are about the same as your chances of winning the Miss America contest. In order to do so, after all, you have to be able to accurately predict almost all of the MVO inputs. For a 10 asset portfolio, that's 65 parameters. Rotsa ruck.
Rather, the most rational way to use an MVO is to find a "reasonable" allocation (hereafter known as the "coward's portfolio," or "CP") which does fairly well under a wide range of scenarios. In other words, pick an allocation, and then figure out as many ways as you can of blowing that allocation up with adverse inputs.
In order to do this, I've taken the following 12 assets, and used some "baseline" returns/SD assumptions:
Asset Return SD S&P 500 .07 .15 US Small Stocks .09 .25 Latin American Stocks .07 .40 Pacific Large Stocks .07 .25 Pacific Small Stocks .09 .30 European Large Stocks .07 .20 European Small Stocks .09 .25 REIT Stocks .06 .20 Precious Metals Stocks .03 .40 Natural Resources Stocks .05 .20 Non US Bonds .04 .09 Long Term US Bonds .04 .09 One Year Corporate Bonds .03 .015
(All returns/SDs are inflation adjusted, and decimailzed. E.g., the return of .07 for the S&P 500 denotes an inflation adjusted return of 7%. The European small and large asset classes are 2/3 continental European and 1/3 British, the Pacific small and large asset classes are equal parts Japan and EAFE-PACXJ.)
The correlation grid used was for quarterly returns from July 1988 to September 1997. Three parameters were varied:
1) Inflation could be "normal," "high," or "low." For high/low inflation the returns of precious metals equity are raised/lowered by 0.1, and for natural resources by 0.05. One year bond and REIT are not changed under any of these scenarios. All other stock and bond assets are lowered/raised by 0.04 for high/low inflation.
2) Foreign dominance. Under the high/low scenarios the returns of all foreign stock and bond assets are raised/lowered by 0.04.
3) Small/large dominance. Under high/low scenarios, the returns of all foreign and domestic small cap stock assets are raised/lowered by 0.04.
Each scenario is denoted by 3 letters. For example, scenario "lhn" corresponds to low inflation, high foreign returns, and a normal small cap premium. In this scenario the S&P would return 0.11 (0.07 + 0.04 for low inflation), and European and Pacific large caps 0.15 (0.07 + 0.04 for low inflation + 0.04 for foreign dominance).
There are thus 27 different scenarios, which would seem to cover most, but not all, economic and financial environments. For example, the current scenario is lll, the late 70s to early 80s hhh, the mid 80s nhn, the mid 60s lnh. The period of the great depression is not describable in this schema, as there was negative inflation and very low stock returns.
These 27 scenarios were tested against the following "reasonable" allocation:
Asset Allocation S&P 500 15% US Small Stocks 15% Latin American Stocks 5% Pacific Large Stocks 5% Pacific Small Stocks 6% European Large Stocks 6% European Small Stocks 6% REITs 5% Precious Metals Stocks 5% Natural Resources Stocks 2% Non US Bonds 10% Long Term US Bonds 0% One Year Corporate Bonds 20%
The return of this allocation was plotted against the unconstrained efficient frontier for all 27 scenarios, along with the S&P 500. The MVO used is "MvoPlus," a proprietary optimizer which closely approximates the returns of a rebalanced portfolio.
The plot below represents the nnn case -- that is, no adjustments to the "normal" scenario. This is a screenshot from MvoPlus, modified slightly to show the return/SD of the SP and S&P 500. .
I've plotted the returns for the above "coward's portfolio" ("CP") as well as the S&P. Note that the CP is pretty close to the EF curve, and that it is considerably more efficient than the S&P 500. The "efficiency" of a portfolio is defined as the distance below the EF curve. The closer to the curve, the more efficient it is.
I've also plotted the same screenshots for the other 26 scenarios below. You can simultaneously view all 27 scenarios by clicking here, or view them individually by clicking on the individual scenarios below.
It's a Low, Low, Low World
It turns out that of the 27 possible scenarios, the CP was more efficient than the S&P 500 20 times, less efficient 5 times (lhl, lll, lln, lhl, and nll), and about the same 2 times (llh and lnn).
The "worst case" for the CP is the lll portfolio -- with low inflation/high overall stock returns, but low foreign and small cap dominance. Buy the S&P, and you wind up almost smack dab on the sacred curve. That, unfortunately, is precisely the universe we've been living in for the past decade, and has encouraged the skeptics to proclaim the "Death of Diversification."
Maybe I'm just whistling past the graveyard, but it seems to me that it's pretty unlikely that the next decade will look anything like the last one. In fact, if the returns of global equity assets mean revert then over long enough time horizons the return scenario should look most like the nnn world, in which the CP beats the pants off of a domestic only strategy.
Anyway, the purpose of this piece is not to predict which of the above scenarios will best predict future returns, but rather how to properly use an MVO. VisualMvo, a freestanding Windows based MVO, will be available soon. If you are interested contact David Wilkinson via email. He may also be reached via snailmail or telephone at:
311 Ned's Mountain Road
Ridgefield CT 06877
1 203 778 1632
The MVO used in this piece, MvoPlus, with its mutliperiod rebalanced return formulation, will also be available from David sometime in early Spring. Happy Portfolio Hunting!
copyright (c) 1998, Wm. Bernstein