For more of Jeff Danziger’s cartoons, visit
www.danzigercartoons.com
Now a book from John Wiley & Sons, published June 2009
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. Holeistic 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 mediaconsulting 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.
Nonlinearities
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, cofounded 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."
