Morningstar Advisor - June/July 2010 - 49

Reminiscence of Jimmie Savage: A Son’s Perspective
by Sam Savage The work of my father, Leonard J. Savage, deals with the very definition of probability. For hundreds of years, the probability of an event, E, was defined as the frequency with which E would occur if you repeatedly preformed identical experiments. For example, if you roll a die thousands of times, you will determine the percentage of 1s, 2s, etc. This is known as the “frequentist” approach to probability. So, how about estimating the probability that the Golden Gate Bridge will fall down tomorrow? In cases like this, which are like most of life’s uncertainties, my father strongly advocated an approach known as “subjective probability.” Here the probability of event E is defined as the gamble you would be willing to make on its occurrence. For example, how much would you pay for a contract that paid you $100 if the number 3 appears on the roll of a die? How about a contract that pays $100 if the Golden Gate Bridge falls down tomorrow? If you were risk-neutral, you would pay $1/6 for the first contract, while different people would pay very different amounts for the second contract, but at least you can cover both kinds of events. The subjective approach makes use of Bayesian analysis and was also championed by the Italian statistician Bruno de Finetti and English statistician Dennis Lindley. My view is that of course the probability of a bridge falling down is subjective. And how would I get a first cut at this probability? By running a Monte Carlo simulation of the physics of the bridge and finding the frequency of bridge failures. This estimate would be continually updated as I learned more about the situation. So, I see these two approaches as complementary, rather than conflicting. But back in the day, there was a tremendously intellectual, and sometimes emotionally heated, debate between the two schools of thought. My brother, at about age 10, was present at a social gathering of academicians when he was approached by a frequentist statistician, who advised him that our father was a deeply deluded man. In the long run, subjective probability has been the foundation of much economic theory, along with the recent rise of prediction markets, such as Intrade. More recently, behavioral economics has questioned the degree to which humans can in fact estimate probabilities in the first place, and, clearly, recent events have shown that not everybody can.

any particular assumption about the shape of the distribution. You can assume normal. You can assume Student’s t. You can assume many, many distributions, and you don’t have to assume that the covariance matrix is the best way to characterize the relationships between these securities. Now that we have the computational power, what is your view of applying this scenario approach to the portfolio-construction problem that you first worked on in 1952?
Markowitz: I have no objection to what you and Sam do with oil and gas. I might very well do the very same thing. But when it comes to whether David Swensen, in running the Yale endowment, ought to be using scenario analysis instead of doing a mean-variance analysis at the asset-class

level and then implementing it himself—I think David is doing just fine.
Kaplan: Well, Harry, let me give some more context to this. In your 1959 book, you mentioned semi-variance as an alternative risk measure. Later, in a paper you cowrote in 1993, you solved the mean-semivariance optimization problem—using the scenario approach.2 You made a list of scenarios, and under each scenario, there’s a return for each security. As I recall, you set it up as a conventional quadratic optimization problem, except it had lots and lots more variables than the typical meanvariance problem because there had to be like at least one additional variable for each scenario. Do you recall the paper I’m referring to?

Markowitz: Yes. It was a paper about how to compute mean semi-deviation frontiers using a mean-variance optimizer. It’s a much slicker algorithm for solving that problem than what I published in 1959.

I tell my class that one of the problems with mean semi-variance is that you can’t just take a matrix full of semi-co-variances and trace out a frontier. There are semico-variances, but they’re local to the portfolio and its neighboring portfolios that have the same pattern of “profitable years” versus “non-profitable years.” So, you have to use finite samples. You can use historical returns on securities, or if you have some kind of model of covariance, you have to sample synthetic years and use these in the algorithm. You’re perfectly right

2 Harry Markowitz, Peter Todd, Ganlin Xu, and Yuji Yamane, “Computation of Mean-Semivariance Efficient Sets by the Critical Line Algorithm,” Annals of Operations Research, December 1993.

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