Download Being Warren Buffett: a classroom simulation of the stock market

Being Warren Buffett: a classroom
and computer simulation of the
stock market
June 30, 2010
Nicholas J. Horton
Department of Mathematics and Statistics
Smith College, Northampton, MA
[email protected]
http://www.math.smith.edu/~nhorton
Acknowledgements and references
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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), 60:53-60.
More information, the handout form and copy of 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
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Overview
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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
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Being Warren Buffett
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Objectives
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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
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Background information
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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:
Investment
Green
Red
White
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Expected annual return
8.3%
SD(annual return)
20%
71%
132%
0.8%
4%
Question: Which of the three investments seems the
most attractive to the members of your group?
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Dice outcomes
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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
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Example:
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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)
Horton - MOSAIC
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
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Repeat the process for 20 years
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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 Buffett’s” of the class
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Group results form
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Usually, red doesn’t do as well as green
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But occasionally it wins big!
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Expected returns for 20 years
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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
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We’d always want to pick RED, no?
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Observed returns (using simulation)
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Used R to simulate 5000 20-year histories, available as
“res.csv”
Observed Q1, median, Q3
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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)
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GREEN: 5.9%
RED:
72.7%
WHITE: 25.0%
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Another strategy (“pink”)
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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
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How to implement PINK
3
$3000
0.05
$150
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Being Warren Buffett
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$1000
1
$1000
Pink
$1000
2
$2000
0.525
$1050
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Implementation in R
# Becoming Warren Buffett simulator (Foster et al TAS)
# Nicholas Horton, [email protected]
# $Id: buffett.R,v 1.2 2010/06/29 13:01:17 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 = years*numsims
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Being Warren Buffett
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Implementation in R (cont.)
library(Hmisc)
process = function(color) {
xmat = matrix(rMultinom(matrix(rep(1/6,6), 1 , 6), n),
nrow=numsims)
res = rep(1000,numsims) # starting investment
for (i in 1:years) {
res = res*color[xmat[,i]]
}
return(res)
}
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Implementation in R (cont.)
pink = function(col1,col2) {
xmat1 = matrix(rMultinom(matrix(rep(1/6,6), 1 , 6), n),
nrow=numsims)
xmat2 = matrix(rMultinom(matrix(rep(1/6,6), 1 , 6), n),
nrow=numsims)
res = rep(500,numsims)
for (i in 1:years) {
redtmp = res*col1[xmat1[,i]]
whitetmp = res*col2[xmat2[,i]]
res = (redtmp+whitetmp)/2
}
return(res*2)
}
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Implementation in R (cont.)
greenres = process(green)
redres = process(red)
whiteres = process(white)
pinkres = pink(red,white)
boxplot(greenres,redres,whiteres,pinkres)
boxplot(greenres,redres[redres<50000],whiteres,pinkres[pi
nkres<50000])
# plotting on a log scale may be worth investigating
results = data.frame(greenres,redres,whiteres,pinkres)
write.csv(results,"res.csv")
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Connections to reality and thoughts on “pink”
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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 don’t know
of any investment that performs like RED. If you know of
one, please tell us so we can make PINK!”
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Boxplots of results (needs rescaling)
$80 billion!
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Boxplots of results (where returns <=$50,000)
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Implementation in SAGE (kudos to Randy P.)
http://www.sagenb.org/home/pub/2184
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Teaching materials and checklist
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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)
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Extensions and assessment
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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
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Conclusions
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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)
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Being Warren Buffett: a classroom
and computer simulation of the
stock market
June 30, 2010
Nicholas J. Horton
Department of Mathematics and Statistics
Smith College, Northampton, MA
[email protected]
http://www.math.smith.edu/~nhorton