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Dr. Savage’s Tutorial on Understanding Uncertainty Through Simulation

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Definition of the Flaw of Averages

The Flaw of Averages is the term I use to describe the fallacies that arise when single numbers (usually averages) are used to represent uncertain outcomes. And surprisingly even most graduates of statistics courses have trouble with the basic concepts of dealing with uncertainty. Worse yet, Generally Accepted Accounting Principles (GAAP) is strewn with these fallacies in many areas. This, however, has not prevented new laws requiring CEOs to certify their inevitably flawed financial statements (see Accounting for Uncertainty, Sam Savage and Marc van Allen, Journal of Portfolio Management, Fall 2002). Although the Flaw of Averages takes on many forms, they fall into two primary categories of misunderstanding: portfolio effects and nonlinearities.

Portfolio Effects

This form of the flaw of averages results from ignoring the effects of diversification and statistical dependence. This is the problem addressed by Modern portfolio theory, whose foundation was laid by Harry Markowitz and Bill Sharpe. However, its implications are largely ignored beyond Wall Street. For example, in capital budgeting, projects are often ranked from “best” to “worst”, whereupon funds are allocated from the top of the list down until the budget is exhausted. Although some firms refer to this as portfolio management, it ignores the true portfolio effects (see Holistic vs. Hole-istic Exploration and Production Strategies, Ben C. Ball and Sam L. Savage, the Journal of Petroleum Technology. Sept 1999 and its accompanying notes and spreadsheet model.

Even the most basic effect of diversification across assets is difficult to understand intuitively. To counter this problem, Rick Medress, president of Cineval LLC, chose to simulate portfolios for his investors instead of describing performance with a single average outcome. Cineval is a media-consulting firm in Los Angeles that provides valuations of film investments. Starting with the historical box office receipts of action films, Medress dramatically demonstrated the effects of diversification across different films with a Blitzogram (interactive histogram) as shown below.

Note how quickly the probability of loss (the height of the leftmost bar) decreases as the number of films in the portfolio is increased, and how the histogram approaches the Normal distribution due to the central limit theorem.


This form of the flaw of averages is based on something known by mathematicians as Jensen’s Inequality, and with a name like that, no wonder it has not entered the general vocabulary. It states that when nonlinear relationships are involved, the value based on average assumptions is not the average value of the entity. For example, the value of an “out of the money” call option, based on average stock price is zero, but zero is not its market value. Pursuit of this topic led Fischer Black and Myron Scholes to their well known option pricing formula.

For a more sobering example, consider the state of a drunk, wandering back and forth on a busy highway. His average position is the centerline. Therefore:

(Illustration from "INSIGHT.xla: Business Analysis Software for Microsoft Excel," 1st edition, by S.L. Savage, copyright 1998.Reproduced with permission of Brooks/Cole, an imprint of the Wadsworth Group, a division of Thomson Learning.)

This form of the Flaw of Averages underlies retirement planning. The chance that a retirement fund will survive given the average rate of return may be 100%, but the chance of survival averaged over all possible market moves may be quite small.

The example below shows the cash remaining in a retirement account assuming "Average" returns (left) vs. simulated returns (right).

To correct this problem, Monte Carlo Simulation is now increasingly being used to illuminate the potential future behavior of retirement accounts. American Express, Bessemer Trust, and Financial Engines, co-founded by Nobel laureate William Sharpe are three examples of this latest trend. “Our clients require a proper perception of risk" says Bessemer Managing Director, Andrew Parker. "One of them, when shown a simulation of how his expenses might impact his portfolio value over a decade, dramatically curtailed his spending. Now, it's much more likely that his assets can stand the test of time."