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Title: Being Warren Buffett: a classroom and computer simulation of the stock market

1
Being Warren Buffett a classroom and computer
simulation of the stock market
• September 24, 2010
• Nicholas J. Horton
• Department of Mathematics and Statistics
• Smith College, Northampton, MA
• nhorton_at_smith.edu
• http//www.math.smith.edu/nhorton

2
Acknowledgements and references
• Activity developed by Robert Stine and Dean
Foster (Wharton School, University of
Pennsylvania)
• Published paper Being Warren Buffett A
classroom simulation of risk and wealth when
investing in the stock market, The American
Statistician (2006), 6053-60.
the TAS paper can be found at
• http//www-stat.wharton.upenn.edu/stine
• A copy of these notes plus the R code to run the
simulation and results from 5000 simulations can
be found at
• http//www.math.smith.edu/nhorton/buffett

3
Overview
• The concepts of expected value and variance are
challenging for students
• A hands-on simulation can help to fix these
ideas, in the context of the stock market
• Allows students to experience variance
first-hand, in a setting where long tails exist
• Can be implemented using dice (and calculators)
in a classroom setting
• Computer generation of results complements and
extends the analytic and hand simulations

4
Objectives
• Understanding discrete random variables to model
stock market returns
• Calculate and interpret expectations for return
from a given investment strategy
• Calculate and interpret standard deviations of
returns from a given investment strategy
• Compare the risk and return for these strategies
• Spark thinking about diversification and
rebalancing of investments
• Build complementary empirical and analytic
problem solving skills

5
Background information
• Imagine that you have 1000 to invest in the
stock market, for 20 years
• Three investment possibilities are presented to
students working in groups of 2 or 3
• Question Which of the three investments seems
the most attractive to the members of your group?

Investment Expected annual return SD(annual return)
Green 8.3 20
Red 71 132
White 0.8 4
6
Dice outcomes
• The investments rise or fall based on the
outcomes of a 6-sided die

Outcome Green Red White
1 0.8 0.05 0.95
2 0.9 0.2 1
3 1.1 1 1
4 1.1 3 1
5 1.2 3 1
6 1.4 3 1.1
7
Example
• Suppose on the first roll your team gets the
following outcomes (Green 2) (Red 5) (White 5),
then on the second roll, you get (Green 4) (Red
2) (White 6)

Round Green Red White
Start 1000 1000 1000
Return 1 0.9 3 1
Value 1 900 3000 1000
Return 2 1.1 0.2 1.1
Value 2 990 600 1100
8
Repeat the process for 20 years
• 1 student to roll the dice (green, red and white)
• 1 student to determine the return and calculate
the new value on the results handout
• 1 student to supervise and catch errant dice
• At the end of class, each team enters their
results on the classroom computer
• Find out who are the Warren Buffetts of the
class

9
Group results form
10
Usually, red doesnt do as well as green
11
But occasionally it wins big!
12
Expected returns for 20 years
• Use property that the expectation of a product is
the product of the expectation
• GREEN 1000(1.083)20 4,927
• RED 1000(1.710)20 45,700,632
• WHITE 1000(1.008)20 1,173
• Wed always want to pick RED, no?

13
Observed returns (using simulation)
• Used R to simulate 5000 20-year histories,
available as res.csv
• Observed Q1, median, Q3
• GREEN 2,058 3,621 6,269
• RED 0 16 1,993
• WHITE 1,011 1,141 1,321
• Percentage ending with less than initial
investment (1000)
• GREEN 5.9
• RED 72.7
• WHITE 25.0

14
Another strategy (pink)
• Consider a strategy where you balance investments
between RED (dangerous) and WHITE (boring) each
year
• Call this PINK
• Smaller average returns, but far less variable
• Can be calculated using existing rolls (average
returns), using space on the results form

15
How to implement PINK
Pink
1000
3
1
2
3000
1000
2000
0.05
1
0.525
150
1000
1050
16
Implementation in R
• Becoming Warren Buffett simulator (Foster et al
TAS)
• Nicholas Horton, nhorton_at_smith.edu
• Id buffett.R,v 1.2 2010/06/29 130117
nhorton Exp
• green c(0.8, 0.9, 1.1, 1.1, 1.2, 1.4)
• red c(0.05, 0.2, 1, 3, 3, 3)
• white c(0.95, 1, 1, 1, 1, 1.1)
• years 20 numsims 5000
• n yearsnumsims

17
Implementation in R (cont.)
• library(Hmisc)
• process function(color)
• xmat matrix(rMultinom(matrix(rep(1/6,6), 1 ,
6), n),
• nrownumsims)
• res rep(1000,numsims) starting investment
• for (i in 1years)
• res rescolorxmat,i
• return(res)

18
Implementation in R (cont.)
• pink function(col1,col2)
• xmat1 matrix(rMultinom(matrix(rep(1/6,6), 1
, 6), n), nrownumsims)
• xmat2 matrix(rMultinom(matrix(rep(1/6,6), 1
, 6), n), nrownumsims)
• res rep(500,numsims)
• for (i in 1years)
• redtmp rescol1xmat1,i
• whitetmp rescol2xmat2,i
• res (redtmpwhitetmp)/2
• return(res2)

19
Implementation in R (cont.)
• greenres process(green)
• redres process(red)
• whiteres process(white)
• pinkres pink(red,white)
• results data.frame(greenres,redres,whiteres,pink
res)
• write.csv(results,"res.csv")

20
Connections to reality and thoughts on pink
• GREEN performs like the US stock market (adjusted
for inflation)
• WHITE represents the (inflation adjusted)
performance of US Treasury Bills
• Quote from authors We made up RED. We dont
know of any investment that performs like RED. If
you know of one, please tell us so we can make
PINK!

21
Boxplots of results (needs rescaling)
22
Boxplots of results (where returns lt50,000)
23
log of results (returnslt1 bumped to 1)
24
Implementation in SAGE (kudos to Randy Pruim)
• http//www.sagenb.org/home/pub/2184

25
Teaching materials and checklist
• Copies of handout describing the simulation (one
per student)
• Copies of results sheet (one per group)
• Set of three die (though one will work in a
pinch, one set per group)
• Remind students to bring calculators (or run this
in a lab rather than lecture)
• Time requirements between 50 and 80 minutes
(depending in part on whether you calculate
expected values, motivate the simulation
parameters in terms of historical inflation and
stock returns and whether pink is introduced)

26
Extensions and assessment
• The activity was developed for use in both an MBA
and PhD program
• The paper introduces concepts of volatility
drag and volatility adjusted return as more
advanced topics (potentially applicable as a
project at the end of a undergraduate probability
class), as well as connections to calculus
• Verifying the expected value and standard
deviation of one of the investment strategies is
a straightforward homework assignment (other
assessments possible)
• Students without formal exposure to expectations
of discrete random variables can still fully
participate in the simulation

27
Conclusions
• Hands-on activity is popular with students
• Helps to reinforce important but often confused
concepts in the context of a real world
application
• Small group work helps to address questions as
they arise
• Students turn in results to allow review of
results (in addition to immediate display of
summary and graphical statistics)

28
Being Warren Buffett a classroom and computer
simulation of the stock market
• September 24, 2010
• Nicholas J. Horton
• Department of Mathematics and Statistics
• Smith College, Northampton, MA
• nhorton_at_smith.edu
• http//www.math.smith.edu/nhorton