January 28, 2003
Volume 2, Issue 2

Integrating the Outliers
Two Lessons from the St. Petersburg Paradox
The risk-reducing formulas behind portfolio theory rely on a number of demanding andultimately unfounded premises. First, they suggest that price changes are statisticallyindependent from one another . . . The second assumption is that price changes are distributed
in a pattern that conforms to a standard bell curve.
Do financial data neatly conform to such assumptions? Of course, they never do.
Benoit B. Mandelbrot
1
A Multifractal Walk down Wall Street
The very fact that the Petersburg Problem has not yielded a unique and generally acceptable
solution to more than 200 years of attack by some of the world’s great intellects suggests,
indeed, that the growth-stock problem offers no hope of a satisfactory solution.
David Durand2
Growth Stocks and the Petersburg Paradox

Bernoulli’s Challenge
Competent investors take great pride in their ability to place an appropriate value on a financial
claim. This ability is the core of investing: Markets are just vehicles to trade cash for future
claims, and vice versa.
O.K. Here’s a cash flow stream for you to value: Say the house flips a fair coin. If it lands onheads, you receive $2 and the game ends. If it lands on tails, the house flips again. If thesecond flip lands on heads, you get $4; if it lands on tails, the game continues. For eachsuccessive round, the payoff for landing on heads doubles (i.e., $2, $4, $8, $16, etc.) and you
progress to the next round until you land heads. How much would you pay to play this game?

Michael J. Mauboussin
212-325-3108
michael.mauboussin@csfb.com

Kristen Bartholdson
212-325-2788
kristen.bartholdson@csfb.com

Daniel Bernoulli, one of a family of distinguished mathematicians, first presented this problem
3
to the Imperial Academy of Sciences in 1738. Bernoulli’s game, known as the St. Petersburg
Paradox, challenges classical theory, which says that a player should be willing to pay thegame’s expected value to participate. The expected value of this game is infinite. Each roundnn
has a payoff of $1 (probability of 1/2 and a payoff of $2 , or ½ x $2, ¼ x $4, ⅛ x $8, etc.) So,
Expected value = 1 + 1 +1 + 1 . . . = ∞Naturally, very few people would be willing to pay even $20 to play the game. Bernoulli tried to
explain the paradox with the marginal utility of money. He argued that the amount you’d bewilling to pay is a function of your resources—the greater your resources, the more you’d bewilling to pay. Still, Bernoulli’s explanation is not altogether satisfactory. The St. Petersburg
Paradox has kept philosophers, mathematicians, and economists thinking for over two-and-a4
half centuries.
Philosophical discourse aside, the St. Petersburg Paradox illuminates two very concrete ideas
for investors. The first is that the distribution of stock market returns does not follow the patternthat standard finance theory assumes. This deviation from theory is important for risk
management, market efficiency, and individual stock selection.

The second idea relates to valuing growth stocks. What do you pay today for a business with a low
probability of an extraordinarily high payoff? This question is more pressing than ever in a world with violent
value migrations and increasing returns.

What’s Normal?
Asset price distributions are of great practical significance for portfolio managers. Standard finance theory
assumes that asset price changes follow a normal distribution—the well-known bell curve. That thisassumption is roughly accurate most of the time allows analysts to use very robust probability statistics. For
example, for a sample that follows a normal distribution, you can identify the population average andcharacterize the likelihood of variance from that average.
5

However, much of nature—including the man-made stock market—is not normal. Many natural systems
have two defining characteristics: an ever-larger number of smaller pieces and similar-looking pieces acrossthe different size scales. For example, a tree has a large trunk and a number of ever-smaller branches, andthe small branches resemble the big branches. These systems are fractal. Unlike a normal distribution, no
average value adequately characterizes a fractal system. Exhibit 1 contrasts normal and fractal systems
6
visually and shows the probability functions that represent the data. Fractal systems follow a power law.
Exhibit 1: Probability Density Functions for Normal and Fractal Systems

NormalProbability

SamplePayoffFractalLog (Probability)

SampleLog (Payoff)

Source: Larry S. Liebovitch and Daniela Scheurle, “Two Lessons from Fractals and Chaos,” Complexity, Vol. 5, 4, 2000.

Using the statistics of normal distributions to characterize a fractal system like financial markets is potentially7very hazardous. Yet theoreticians and practitioners do it daily. The distinction between the two systems
boils down to probabilities and payoffs. Fractal systems have few, very large observations that fall outside
the normal distribution. The classic example is the crash of 1987. The probability (assuming a normal
distribution) of the market’s 20%-plus plunge in one day was so infinitesimally low it was practically zero.
And still the losses were a staggering $2 trillion-plus.
A comparison of a normal coin toss game and the St. Petersburg game illustrates the point. Assume thatyou toss a coin and receive $2 if it lands heads and nothing if it lands tails. The expected value of the gameis $1, the amount you would be willing to pay to play the game in a fair casino. We simulated 1 million
rounds of 100 tosses each, and plotted the payoffs in Exhibit 2. Just as you would expect, we got a well8
defined normal distribution.

Page 2

Exhibit 2: Standard Coin Toss Game
9.0%
8.0%
7.0%

Probability

6.0%
5.0%
4.0%
3.0%
2.0%
1.0%
0.0%
$0.40
-1.0%

$0.60

$0.80

$1.00

$1.20

$1.40

$1.60

PayoffSource: CSFB analysis.

We then simulated the St. Petersburg game 1 million times, and plotted that distribution (see Exhibit 3).
While the underlying process is stochastic, the outcome is a power law. For example, half the time the gameonly pays $2, and three-quarters of the time it pays $4 or less. However, a run of 30 provides a $1.1 billion
payoff, but is only a 1-in-1.1 billion probability. Lots of small events and a few very large events characterizea fractal system. Further, the average winnings per game is unstable with the St. Petersburg game, so noaverage accurately describes the game’s long-term outcome.
Exhibit 3: Fractal Coin Toss Game0-5

Log (Probability)

-10
-15
-20
-25
-30
-35
0

5

10

15
Log (Payoff)

20

24

29

Source: CSFB analysis.

Are stock market returns fractal? Benoit Mandelbrot shows that by lengthening or shortening the horizontal
axis of a price series—effectively speeding up or slowing down time—prices series are indeed fractal. Not
only are rare large changes interspersed with lots of smaller one, the price changes look similar at various
scales (e.g., daily, weekly, and monthly returns). Mandelbrot calls financial time series multifractal, adding
the prefix “multi” to capture the time adjustment.

Page 3In an important and fascinating book, Why Stock Markets Crash, geophysicist Didier Sornette argues thatstock market distributions comprise two different populations, the body (which you can model with standard
theory) and the tail (which relies on completely different mechanisms). Sornette’s analysis of market
drawdowns convincingly dismisses the assumption that stock returns are independent, a key pillar of
9classical finance theory. His work provides fresh and thorough evidence of finance theory’s shortcomings.

St. Petersburg and Growth Stock Investing
10

The St. Petersburg Paradox also provides insight for growth stock valuation. What should you be willing to
pay for a very small probability that a company can grow its cash flows by a very significant amount11
forever?
David Durand took up this question in his classic 1957 article, “Growth Stocks and the Petersburg
12
Paradox.” He encourages a good deal of caution, emphasizing reversion-to-the-mean thinking andmodeling. But if anything, the challenge to value the low probability of significant value is even morepressing today than it was when Durand took on the challenge 45 years ago.
Consider, for example, that of the nearly 2,000 technology initial public offerings since 1980, only 5%
13account for over 100% of the $2-trillion-plus in wealth creation. And even within this small wealthgenerating group, only a handful delivered the bulk of the huge payoffs. Given the winner-take-mostcharacteristics of many growth markets, there’s little reason to anticipate a more normal wealth-creation
distribution in the future.
In addition, the data show that the distribution of economic return on investment is wider in corporate
14
America today than it was in the past. So the spoils awaiting the wealth creators, given their outsized
returns, are greater than ever before. Like the St. Petersburg game, the majority of the payoffs from future
deals are likely to be modest, but some will be huge. What’s the expected value? What should you be willingto pay to play?

Integrating the Outliers
The St. Petersburg Paradox may be centuries old, but its lessons are as fresh as ever. One of the major
challenges in investing is how to capture (or avoid) low-probability, high-impact events. Unfortunately,
standard finance theory has little to say about the subject.

____________________________
1

Benoit B. Mandelbrot, “A Multifractal Walk down Wall Street,” Scientific American, February 1999, 70-73.
David Durand, “Growth Stocks and the Petersburg Paradox,” Journal of Finance, 12, September 1957,
348-363.
3
Daniel Bernoulli, “Exposition of a New Theory on the Measurement of Risk,” Econometrica, 22, January
1954, 23-36. Originally published in 1738. Daniel’s cousin, Nicolaus, initially proposed the game.
4
See http://plato.stanford.edu/entries/paradox-stpetersburg/.
5
Much of this section relies on Larry S. Liebovitch and Daniela Scheurle, “Two Lessons from Fractals andChaos,” Complexity, Vol. 5, 4, 2000, 34-43. See http://www.ccs.fau.edu/~liebovitch/complexity-20.html.
6Michael J. Mauboussin and Kristen Bartholdson, “More Power to You: Power Laws and What They Meanfor Investors,” The Consilient Observer, September 24, 2002.
7
If you assume that you flipped a coin nonstop 16 hours a day (estimating 8 hours of sleep), and if each coinflip takes three seconds, it would take 14.3 years to complete 100 million coin tosses.
8
Mandelbrot. Also, Benoit B. Mandelbrot, Fractals and Scaling in Finance (New York: Springer Verlag,
1997).
9
Didier Sornette, Why Stock Markets Crash: Critical Events in Complex Financial Systems (Princeton:
Princeton University Press, 2003). See http://www.ess.ucla.edu/faculty/sornette/.
10
See another classic article: Peter L. Bernstein, “Growth Companies vs. Growth Stocks, ” Harvard
Business Review, September-October 1956.
11
Peter L. Bernstein, Against the Gods: The Remarkable Story of Risk (New York: John Wiley & Sons,
1996), 107-109.
12
Durand.
13
Stephen R. Waite, Quantum Investing (New York: Texere, 2003), 129.
14
Michael J. Mauboussin, Bob Hiler, and Patrick J. McCarthy, “The (Fat) Tail that Wags the Dog,” Credit
Suisse First Boston Equity Research, February 4, 1999.
2

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AMSTERDAM............. 31 20 5754 890
ATLANTA ................... 1 404 897 2800
BALTIMORE............... 1 410 659 8800
BANGKOK....................... 62 614 6000
BEIJING.................... 86 10 6410 6611
BOSTON..................... 1 617 556 5500
BUDAPEST .................. 36 1 202 2188
BUENOS AIRES....... 54 11 4394 3100
CHICAGO ................... 1 312 750 3000
FRANKFURT ................. 49 69 75 38 0
HOUSTON .................. 1 713 890 6700
HONG KONG............... 852 2101 6000
JOHANNESBURG
27 11 343 2200

KUALA LUMPUR.........603 2143 0366
LONDON ...................44 20 7888 8888
MADRID .....................34 91 423 16 00
MELBOURNE .............61 3 9280 1888
MEXICO CITY ..............52 5 283 89 00
MILAN .............................39 02 7702 1
MOSCOW....................7 501 967 8200
MUMBAI......................91 22 230 6333
NEW YORK.................1 212 325 2000
PALO ALTO................1 650 614 5000
PARIS........................33 1 53 75 85 00
PHILADELPHIA ..........1 215 851 1000

SAN FRANCISCO...... 1 415 836 7600
SÃO PAULO ............ 55 11 3841 6000
SEOUL ....................... 82 2 3707 3700
SINGAPORE ................. 65 6212 2000
SYDNEY ..................... 61 2 8205 4433
TAIPEI ...................... 886 2 2715 6388
TOKYO ....................... 81 3 5404 9000
TORONTO.................. 1 416 352 4500
WARSAW................... 48 22 695 0050
WASHINGTON........... 1 202 354 2600
ZURICH ....................... 41 1 333 55 55

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