Markowitz’s Portfolio Selection

Harry Markowitz is often referred to as the father of Modern Portfolio Theory–a collection of mathematical models that quantify the behavior of assets and portfolios of assets. Harry’s work specifically addresses the latter and examines how assets may be combined to reduce volatility and (potentially) increase returns. I briefly alluded to Harry’s work in an earlier article on portfolio construction, but wanted to cover the major points in greater detail.

A quick word of caution: the book reads like a Ph.D. thesis because, well, it is. The math is dense and requires more than basic arithmetic to get through it. That being said, if you’ve got some background in linear algebra and statistics, and are looking for a math challenge then have at it. I suggest tackling it in chunks, preferably with a cup of strong coffee to help keep yourself alert.

The first few chapters mainly lay down a lot of the mathematical groundwork and cover a lot of basic statistical models and formulations. Markowitz demonstrates that the average return of a portfolio is the weighted sum of asset returns. On the other hand, the standard deviation (volatility) of a portfolio is not simply the weighted sum of standard deviations, but relies also on the correlation between assets.

Consider two assets, A and B, for which we know the expected return, the expected variance/standard deviation, and the expected correlation. The average return and standard deviation of a portfolio composed of these two assets is calculated as follows

Markowitz Portfolio Equations

Equations 2 and 3 above clearly show how correlation between assets is related to the standard deviation of a portfolio. All things equal, increasing the correlation between assets increases portfolio volatility and vice versa. Within the assumptions that Markowitz is working with this is not subjective opinion, it’s a mathematical fact. Therefore, a diversified portfolio does not simply mean owning different assets, but owning uncorrelated assets (as quantified by the correlation coefficient).

However, the above formulations should come with a warning label when applied to real world asset management. Like any other mathematical model they are a representation of reality, and consequently are built upon assumptions. The equations above only apply over a single holding period and do not account for any sort of external behavior such as cash flow or rebalancing. Furthermore, they require inputs–specifically inputs for expected returns, expected standard deviations/variances, and expected correlations. Thus they are useful for understanding the interplay among assets and how a given asset may impact a portfolio. In no way does the calculated result forecast/predict/guarantee a specific outcome.

Markowitz turns the mathiness up a few more notches and develops what has become his pièce de résistance: the derivation of efficient portfolios. The procedure works as an optimization routine to determine how much of a given asset should be held in a portfolio to deliver the highest possible return for a given level of volatility. He defines an efficient portfolio in an indirect manner by describing what is inefficient

A portfolio is inefficient if it is possible to obtain higher expected (or average) return with no greater variability of return, or obtain greater certainty of return with no less average or expected return. [1a]

At the core of all the complicated math is the relationship between return and volatility/variability. As a mathematical exercise it’s both incredibly smart and extremely elegant. However, optimization of any sort is probably one of the most dangerous endeavors to undertake. The concept of optimization implies that the inputs regarding future behavior are known and the resulting future outcome can be computed precisely. This is a laughable concept. In reality all of the inputs are at the very least unknown, and at the very best imprecise. That’s the thing with models, the results they generate are only as good as the inputs provided.

This does not imply that Markowitz’s work is a failure in any way. The major problem is the misunderstanding and misapplication of his work by practitioners on how to properly use his models. The concept of the optimized efficient portfolio teaches us to diversify among uncorrelated assets, but it is up to us as investors to understand the expected behavior of those assets and the risks that we are taking. Markowitz’s work thus becomes less of a panacea, and more of a philosophical framework for portfolio construction–a brilliant one at that.

Here’s the ultimate question: How did Markowitz apply this knowledge to his own portfolio? Jason Zweig provides some insight

In the 1950s, a young researcher at the RAND Corporation was pondering how much of his retirement fund to allocate to stocks and how much to bonds. An expert in linear programming, he knew that “I should have computed the historical co-variances of the asset classes and drawn an efficient frontier. Instead, I visualized my grief if the stock market went way up and I wasn’t in it–or if it went way down and I was completely in it. My intention was to minimize my future regret. So I split my contributions 50/50 between bonds and equities.” The researcher’s name was Harry M. Markowitz. Several years earlier, he had written an article called “Portfolio Selection” for the Journal of Finance showing exactly how to calculate the tradeoff between risk and return. In 1990, Markowitz shared the Nobel Prize in economics, largely for the mathematical breakthrough that he had been incapable of applying to his own portfolio. [2]

When it comes to models, remember: garbage in, garbage out.

Bonus! An hour long conversation with Harry Markowitz courtesy of

1. Markowitz, Harry M. Portfolio Selection. Second Edition. Blackwell Publishing. 2007.
(a) p. 129
2. Zweig, Jason. Your Money & Your Brain. Simon & Schuster. New York, NY. 2007. p. 4.

NOTES – Portfolio Selection.pdf

What I’m Reading
Warren Buffett Loses A Bit of Ground in His ‘Million-Dollar Bet’ (Carol Loomis)
Uncertainty is a fact of life, so get used to it (Caroline Baum)
The Portfolio Management Assumptions that Harm Clients (Scott MacKillop)
Ignorance: The World’s Invisible Enemy (Kent Thune)