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published in my 1959 book [Portfolio Selection: Efficient Diversification of Investments]. I never ever assumed that probability distributions were normal. I never justified meanvariance analysis in terms of probability distributions being normal. My basic assumption is that you act under uncertainty to maximize expected utility. In fact, I was a student of Sam’s father, and he convinced me, as well as half the rest of the world, that the way to act under uncertainty is to maximize expected utility using probability beliefs where there weren’t any objective probabilities. So, how do I get off peddling mean-variance analysis when I believe in expected utility? An illustration of what I believe is given in Chapter six of my 1959 book, where I recommend maximizing expected logarithm of one-plus-return if you’re in for the long run. I have a table showing various levels of return, the log utility of that return, and the value of a simple quadratic approximation. Between a 30% or 40% loss and a 40% or 50% gain on the portfolio as a whole, the quadratic approximation is very close to the log utility. So, the expected value of the one has to be close to the expected value of the other, as long as we’re talking about portfolios that rarely lose much more than 30% or 40% or gain much more than 40% or 50%. Of course, the expected value of the quadratic is a function of mean and variance. So, for the nine or 10 securities that I had available for my illustrative mean-variance analysis, I show that not only does the approximation look good in the table, but, in fact, if you knew the mean and variance, you could guess the expected utility quite well. What Levy and I did was to do that same experiment using historical returns on investment companies and a variety of utility functions. We confirmed that if your probability distribution is not too spread out,

Markowitz, Mean Variance Optimization, and Expected Utility
Harry Markowitz’s revolutionary 1952 paper “Portfolio Selection” [Journal of Finance, Vol. 7, Issue 1, pp. 77–91] introduced a now commonplace concept: Investors can construct an “efficient portfolio” by investing in diverse securities combined to maximize expected returns while minimizing expected volatility. Markowitz demonstrated how mean variance optimization can be used to engineer the best combination of securities based on their mean (or expected return), their variance (the square of standard deviation of return), and their correlation with each other. Critics contend that MVO is limited by its simplified use of mean and variance to describe the distribution of an asset’s returns. The result is typically plotted as a classic, symmetrical bell curve, but return distributions don’t always neatly follow that outline. A normal, or bell-curve, distribution may underestimate the likelihood of returns at either extreme. (See “Déjà Vu All Over Again,” Morningstar Advisor, February/March 2009.) However, expected utility theory, which is the foundation of the economics of investing, does not assume that returns follow a bell-curve distribution. According to the theory, an investor ranks alternative portfolios by their “expected utility,” which is the probabilityweighted average of the “utility” of every one of the portfolio’s possible returns. The utility of each return depends on the investor’s attitude toward risk. The important point here is that expected utility can be calculated for a wide range of distributions. There is no presumption that returns follow a normal distribution. It is a simple, elegant approach to taking risk and investors’ risk attitudes into account when assessing alternative portfolios. In fact, the Morningstar Rating for mutual funds (star rating) is based on the expected utility theory. But Markowitz had been “indoctrinated at point-blank range in expected utility theory” by Leonard J. Savage. So why did he base his model of portfolio selection on mean and variance? Markowitz first addressed this question in his landmark 1959 book, Portfolio Selection: Efficient Diversification of Investments, and elaborated on it in a 1979 paper with Haim Levy, “Approximating Expected Utility by a Function of Mean and Variance” [The American Economic Review, Vol. 69, No. 3 (June, 1979), pp. 308–317] These works demonstrate that if the probability distribution of the return on the portfolio as a whole is not too wide, mean and variance can be used to form a good estimate of expected utility even though returns do not follow a normal distribution. Levy and Markowitz further found that the distributions in annual returns of investment companies were not too wide for MVO to be a useful approach to construct portfolios.

and the annual returns on investment companies turn out to be not too spread out, then if you know mean and variance, you can guess expected utility quite well.

Savage: Not at all, that’s just a fluke. I had

heard of Harry and the Markowitz optimization model but had not paid much attention to it. I was not in that line of work. But in 1989, I got a phone call from a guy with a Texas drawl thick enough to cut with a knife. He asked me an interesting question. He said that when oil companies choose portfolios of exploration sites, they sort these sites from best to worst on something like expected net present value. Then, they

‘It’s Jimmie Savage’s Son!’
Kaplan: Sam, as Harry just mentioned, he was a student of your father. Is that how you developed an interest in portfolio theory?

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Morningstar Advisor - June/July 2010

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