Being Warren Buffett [A Classroom Simulation of Risk And Wealth When Investing In The Stock Market].pdf

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Being Warren Buffett: A Classroom Simulation of
Risk and Wealth when Investing in the Stock Market
Dean P. Foster & Robert A Stine
Statistics Department, The Wharton School
University of Pennsylvania
Philadelphia, PA 19104-6340
September 19, 2005
Students who are new to Statistics and its role in modern Finance have a hard time
making the connection between variance and risk. To link these, we developed a classroom
simulation in which groups of students roll dice that simulate the success of three investments.
The simulated investments behave quite differently: one remains almost constant, another drifts
slowly upward, and the third climbs to extremes or plummets. As the simulation proceeds,
some groups have great success with this last investment – they become the “Warren Buffetts”
of the class, accumulating far greater wealth than their classmates. For most groups, however,
this last investment leads to ruin because of its volatility, the variance in its returns. The
marked difference in outcomes surprises students who discover how hard it is to separate luck
from skill. The simulation also demonstrates how portfolios, weighted combinations of
investments, reduce the variance. Students discover that a mixture of two poor investments
emerges as a surprising performer. After this experience, our students immediately associate
financial volatility with variance. This lesson also introduces students to the history of the stock
market in the US. We calibrated the returns on two simulated investments to mimic returns on
US Treasury Bills and stocks.
Modified 9/20/05
Being Warren Buffet
1. Introduction
The definition of variance as the expected squared deviation from the mean often strikes
students as capricious. When shown a histogram, our students seldom suggest measuring
spread by finding the average squared deviation from the mean – unless they have read the text
or met the definition in other courses. Those who have seen boxplots pick the interquartile
range and others suggest the average absolute deviation. Students often ask us “Why average
squared deviations from the mean?” When teaching an introductory course, we cannot appeal
to efficiency arguments that assume normality to justify variance as a “natural” measure of
When dealing with money, however, the definition of variance is just right. In Finance,
the risk of an investment is precisely the variance of its returns. Rather than link these through
definitions, we have found it more engaging and memorable to let students experience the
effects of variance first-hand in a simulation. In this experiment, students roll dice that
determine the value of several investments and reveal the role of variance. The discussion of
this simulation requires only basic properties of means and variances, with the most
sophisticated property being that the variance of a sum of independent quantities is the sum of
the variances.
We have used this dice simulation successfully for over 10 years in courses taught at
three levels. Because the exercise requires relatively little background, it can be used early in
the curriculum before normality and standard error. The simulation has become a standard
component of the introductory undergraduate course in Statistics at Wharton. Students need
only have been introduced to histograms and their connection to the mean, standard deviation,
and variance. The idea of a discrete random variable is useful (for that is what the students will
be simulating) but this is not necessary. We also regularly use this simulation in the required
MBA course. MBA students generally have a better sense of the economics of investing, but
many are nonetheless surprised to discover the rich connection between Statistics and Finance.
In more advanced courses, such as undergraduate courses in mathematical statistics or
probability, we use the dice simulation to illustrate discrete random variables. The simulation
and ensuing discussion consume a full hour and 20-minute class; it also works well divided into
2 one-hour classes.
Being Warren Buffet
The investments simulated by the dice in this exercise mimic actual investments. One
investment resembles a conservative money-market fund whose interest has been adjusted for
the effects of inflation. At the other extreme, a second investment matches our intuitive
definition of a risky stock. It resembles the performance of many of the high-flying tech stocks
in the late 1990s during the dot-com bubble. A third lies between these extremes and performs
like the overall stock market.
We have students simulate the value of these investments by rolling three differently
colored dice. We label the three investments Red , White, and Green because it is easy to find
dice in these colors. We have, on occasion, tried to save class time by using a computer to run
the simulation; it is easy to program the simulation in Excel, say. End-of-the-term course
evaluations have shown, however, that students who have done the experiment “by hand” more
often mention this lesson as one that was particularly effective. Students not only see the
importance of variance in Statistics, but they also discover the relevance of Statistics in the real
world. After this simulation, everyone appreciates the importance of variance when looking at
data. As our nation discusses privatizing Social Security and shifting retirement investments
into stocks, it would be useful if more citizens understood these lessons.
The following section describes the dice simulation. The third section describes the
origins of the simulated investments and explains how portfolios improve investments by
reducing variation. This section also introduces the notion of volatility drag to quantify the
effects of variation. The concluding section returns to the theme of distinguishing luck from
2. The Dice Simulation
2.1 Getting Started
Before describing the simulation, we divide the class into teams. Teams of 3 or 4 students seem
about right. One person on each team plays the role of nature (or the market) and rolls the dice.
Another keeps track of the dice and reads off their values, and a third records the outcomes.
Others can help out by retrieving the dice and checking the calculations.
Once we have divided the class into teams, we pose the following question. We’ve
found it useful to elicit a written preference from each team before starting the simulation. This
Being Warren Buffet
gets them talking about the simulation and avoids too many “Monday morning quarterbacks” in
the subsequent discussion. If a team has chosen an investment before starting the simulation,
the team members seem more interested in following their choice as the simulation evolves.
Question 1. Which of the three investments summarized in the following
table seems the most attractive to the members of your group?
Expected Annual
Percentage Change
SD of Annual
Percentage Change
Table 1. Expected value and standard deviation of the annual percentage change in the value of
three investments.
We describe the information in Table 1 using examples such as the following:
Suppose that you invest $1000 in one of these choices, say Red . Table 1
tells you that you can expect the value of your investment to be 71% larger at
the end of the first year, up to $1,710.
Similarly, if we start with $1000 in each of these, we’d expect to have $1,083 in Green and
$1,008 in White after one year. Because the expected value of a product of independent random
variables is the product of expectations, we can find the expectations for each investment over a
longer horizon given this assumption. Over 20 years, the initial investment of $1000 in Red
grows in expectation to an astonishing $1000 × (1.71) 20 = $45,700,000. By comparison, the
initial investment in Green grows to $4,927 and White creeps up to $1,173.
Students find such calculations of expected values quite reasonable, but have little
intuition for how to anticipate the importance of the standard deviation – other than to recognize
that the presence of a large standard deviation means that the results are not guaranteed. The
massive standard deviation leads some students to question the wisdom of investing in Red , but
most find it difficult to see how to trade off its large average return for the variation. The annual
return on Red is about 9 times that of Green , but its standard deviation is also 6.5 times larger.
Few students appreciate the bumpy ride promised by Red .
Being Warren Buffet
At this point in the discussion, many teams find the large average return of Red quite
appealing. Regardless of the level of the class, we have found the following example useful as a
means of suggesting the impact of variation on the long-run behavior of an investment.
Suppose that a graduate lands a good job that pays $100,000 per year. In the first year, the
company does well and her salary grows by 10% to $110,000. The next year is leaner, and she
has to take 10% cut in pay, reducing her salary down to $99,000. The average percentage
change in her salary is zero, but the net effect is a loss of 1% of the starting salary over the two
years. Figured at an annual rate, that’s a loss of 0.5% per year. It turns out that this simple
example is a special case of a more general property that captures how variance eventually
wipes out investments in Red .
2.2 Running the Simulation
After this introduction, we pass out three dice to each team along with a data-collection
form similar to that suggested in Figure 1. (A full-page version of this form suitable for use in
class is available at This form organizes the results of the
simulation in a format useful in later steps. The unused last column saves space to compute the
returns on a portfolio later in the exercise. We collect these sheets at the end of the simulation
so that we can review the results in the next class. We have found that students keep better
records when we tell them in advance that we will collect these forms at the end of class.
Starting value
Return 1
Value 1
Return 2
Value 2
Return 3
Figure 1. Initial rows of the data-collection form used to record the value of the three
investments simulated by rolling a red die, a white die, and a green die.
Before the class begins the simulation, we carefully explain how the dice determine the
values of the investments. Each roll of all three dice simulates a year in the market, and the
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