Mean Variance Optimization

The Thinking Man's Ouija Board

Make no mistake about it, portfolio optimizers are big business. Open
almost any journal aimed at investment professionals and you'll find full
page ads trumpeting the "Nobel Prize winning algorithm" inherent
in the $2000 chunk of software being touted. Never mind that the basic
code occupies no more than 50-200K on the single floppy that eventually
comes in the mail. Advisors and clients alike are dazzled by the upwardly
convex curves and precise portfolio compositions flashing on the monitor
and lasered into spiffy folders.

I've gotten quite a bit of correspondence about the availability and
usefulness of optimization techniques for small investors from readers
of ** Efficient Frontier** and

For those of you new to the asset allocation game, here's the plot:
In 1951 Harry Markowitz published a mathematical technique for finding
the precise portfolio compositions which yield the best combinations of
portfolio risk and return. These portfolios form the upper left border
of the risk/return plot of all possible portfolios from a given group of
assets, and are said to form the *efficient frontier* of the plot
-- hence the name of this site. This site's logo symbolizes this concept:

The vertical axis of the logo represents return, the horizontal axis risk. The multicolored sail like object symbolizes all of the possible portfolio combinations which can be formed from the assets. The upwardly curving part of the sail on its upper left is the "efficient frontier." Portfolios lying on this efficient frontier have the highest return for a given degree of risk. Looked at from another perspective, they have the lowest risk for a given return. Obviously, the efficient frontier is the place to be.

The inputs to the formula are remarkably simple -- the return and standard
deviation for each asset, as well as the correlations between each asset.
For example, a simple 3 asset portfolio has 9 inputs; 3 returns, 3 SDs,
and 3 correlations. A 10 asset portfolio would have 65 inputs (10 returns,
10 SDs, and 45 correlations). The advent of the microcomputer put this
technology on the desktop, Professor Markowitz won a well deserved Nobel
Prize for his work, and financial analysts become intoxicated with the
technique's allure.

However, there is a large and ugly fly in the ointment -- the technique
works only in retrospect. It turns out that the outputted portfolio compositions
are exquisitely sensitive to even very small changes in the input data.
Change a few pieces of the input data slightly and the resultant portfolio
compositions change drastically. Since the required input returns, SDs,
and correlations are known with precision only in retrospect, mean variance
optimization is worthless as a predictor of *future* optimal portfolios.
This is because it is impossible to predict with anywhere near the required
accuracy the returns, SDs, and correlations.

As a simple example, I fed the following inputs into my MVO for a simple 5 asset model:

Asset | Return | Standard Deviation |

A | 12.5% | 25% |

B | 11.5% | 25% |

C | 10.5% | 15% |

D | 9.5% | 15% |

E | 5% | 1% |

Assets A and B represent high return/risk assets, such as emerging markets stocks, C and D lower return/risk assets such as US and European stocks, and E cash. All of the stock assets had typical mutual correlations of 0.5 with each other and 0.0 with cash. The optimizer calculated the following optimal "corner portfolios." (Corner portfolios are the basic output of the MVO algorithm. To find the efficient frontier composition between any two corner portfolios they are combined in appropriate amounts.)

Asset | Portfolio 1 | Portfolio 2 | Portfolio 3 | Portfolio 4 | Portfolio 5 |

A % | 0 | 15 | 32 | 80 | 100 |

B % | 0 | 4 | 11 | 20 | 0 |

C % | 1 | 56 | 57 | 0 | 0 |

D % | 0 | 25 | 0 | 0 | 0 |

E % | 99 | 0 | 0 | 0 | 0 |

Return% | 5.06 | 10.59 | 11.24 | 12.3 | 12.5 |

Std% | 1.00 | 13.88 | 16.03 | 22.91 | 25.00 |

Note that the "maximum return" portfolio 5 consists of 100%
asset A. In the real world of rebalanced portfolios two assets with SD=25%
and correlation= 0.5 will have a rebalanced return of about 0.8% over their
averaged returns, so in actuality the maximum return portfolio will turn
out to be a nearly 50/50 mix of A and B, with a return of about 12.8% (the
average of 11.5% and 12.5%, plus the "rebalancing factor" of
0.8%) and an SD of 22.1%. In other words, the maximum return portfolio
often turns out to be a mixture of 2 or more assets, but MVO will always
assign the maximum return portfolio as 100% of the highest performing asset.

Next, you'll recognize that at the lower return/risk region (portfolios
2, 3, and 4) the output favored A over B, and C over D by very wide margins.
This is simply the result of A having a 1% higher return than B, with the
same true of C and D. If your asset return forecast is off by even 1% then
that asset's allocation may be off by several fold. The Lord Almighty Herself
cannot make returns forecasts with that sort of accuracy. The same sorts
of errors occur with SD and correlation aberrations. Consider a 10 asset
model. What do you think the odds are of correctly predicting with any
accuracy all 65 required input parameters?

In addition, the tendency for asset returns to mean revert introduces
a perverse bias into optimizer results. If you are using returns over the
past 5, 10, or even 20 years you are likely to overestimate the returns
of the higher performing assets and vice versa. This will result in the
optimizer overweighting precisely those assets which are likely to underperform
in the future. Some optimizers actually allow the financial analyst to
feed raw mutual fund data directly into the algorithm! This is a recipe
for disaster.

It is becoming increasingly obvious that naive portfolio allocations,
followed with discipline, will beat most active allocation strategies with
great regularity. So why do financial analysts use MVO? For starters, the
Nobel Prize does get your attention. It's major attraction, however, is
its flashiness. Human nature favors the complex over the simple. James
P. O'Shaugnnessy, in ** What Works on Wall Street** cites a psychological
study in which two subjects, Smith and Jones, are asked to identify whether
cell samples shown on a screen are healthy or sick. Smith and Jones are
given a few simple but effective rules for making this decision. Smith
is given the correct feedback about whether he is right or wrong, and after
a while is right 80% of the time. Jones is given incorrect feedback, and
does much more poorly. Smith and Jones are then asked to explain to each
other how they made their decisions. Smith's, as expected, is short and
simple. Jones' is complex and convoluted, and much more impressive than
Smith's. Next, they are asked to look at a new set of cell samples. So
impressed is Smith with Jones' complex (but less effective) method that
he begins to use it.

The same is true of mean variance optimization versus fixed balanced
portfolio allocations. Financial analysts and investors have been conned
by MVO's complexity and elegance. It's failure is reminscent of
communism's. Marx's system fails because of the flaws inherent
in human nature: Markowitz' system fails because of the flaws inherent in economic
forecasting.

*copyright (c) 1997, William J. Bernstein*