Morningstar Advisor - June/July 2010 - 47

go back and bring the Markowitz-Usmen research, on shifting beliefs among different probability distributions, up to date. While he was at it, he found out how you should shift your belief between the stable Paretian distribution and the Student’s t-distribution. We concluded that you should massively shifted against the stable Paretian distribution in favor of the Student’s t-distribution. So, I’m a Bayesian, and I am not worried about Mandelbrot’s stable Paretian distributions. For daily returns, Student’s t-distribution with four-and-a-half degrees of freedom, plus or minus a half degree, is a good model of stock returns. When you aggregate them up to lower frequencies, things start looking normal. So, I go back to the Levy-Markowitz paper: As long as your probability distributions aren’t too spread out, mean variance is a good approximation to the expected utility.
Kaplan: Sam, you mentioned that when you

Fat Tails and Bayesian Inference
When a normal (or Gaussian) distribution is plotted, it forms the familiar bell-shaped curve. The bulk of results cluster symmetrically around the mean: 68% of the data will fall within one standard deviation, and 95% will fall within two standard deviations. The outliers— or the “tails” on either side of the bell—trail off. The Pearson family of distributions includes the normal distribution, as well as the Student’s t. The form of a particular t-distribution is determined by its degrees of freedom. A t-distribution is symmetrical, like the normal, but it has “fatter” tails. That is, it allows for a higher incidence of outliers. Mathematician Benoît Mandelbrot urges the use of stable Paretian distributions to model returns of securities because they are so erratic. A stable Paretian distribution has tails that are much fatter than those of a t-distribution. In fact, the tails of stable Paretian distributions are so “fat” that standard deviation is meaningless—and thus so is MVO. Markowitz answers Mandelbrot using Bayesian inference. Bayesians determine the probability of a hypothesis by gathering evidence that supports or undermines it. As the evidence accumulates, Bayesians shift their belief against or in favor of the hypothesis. Markowitz contends that the weight of the evidence supports a conclusion that the Student’s t-distribution is a better model of stock returns than either the normal distribution or the stable Paretian. The Student’s t-distribution is not inconsistent with MVO.

Fat Tails, Skinny Tails
Historical Frequency (Months) 256 128 64 32 16 8 4 2 1 Stable Paretian Student’s-t Normal Historical

started working with the oil industry and doing optimization models to form diversified portfolios, you used the scenario approach. Explain the scenario approach and how that squares with this whole debate about stable distributions versus t-distributions versus normal distributions.
Savage: The beauty of the scenario approach

is that it’s completely agnostic. Suppose I’m trying to analyze the outcomes of the roll of a die. If you study the physics, you’ll say that you can have any of the faces show up with equal probability. But instead of that, you roll the die a thousand times and store those thousand die rolls. Those would be scenarios. Now, let’s do an example that you couldn’t easily solve with theory. Suppose I had two dice that were statistically related by having magnets imbedded in them. I could roll the dice together and the number that appeared on one die would, in fact, have implications for the number on the other die. I could store these numbers in two

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Monthly Return (%)

columns of thousands of rows in Excel. The scenarios in the two columns capture the joint distribution of the dice. In his 1959 book, Harry does something similar in a beautiful way. He has spinners with concentric rings of numbers to model joint

distributions of statistically dependent things. That is, there is an inner ring of numbers representing one uncertain asset and an outer ring of numbers representing a statistically related asset. That actually is very much in the spirit of the scenarios of dice I just described. I must point out that when I first

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Morningstar Advisor - June/July 2010

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