being warren buffett: a classroom and computer simulation of the stock market june 30, 2010 nicholas...

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

marketJune 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 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

Horton - MOSAIC Being Warren Buffett 2

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


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


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%

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

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

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 Buffett’s” of the class


Group results form


Usually, red doesn’t do as well as green


But occasionally it wins big!


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

We’d always want to pick RED, no?


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%


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


How to implement PINK


Pink$1000

3 1$3000 $1000 $2000

2

0.05 1 0.525$150 $1000 $1050

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

Horton – Causeweb - 2009 Being Warren Buffett 16

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)

}


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)

}Horton – Causeweb - 2009 Being Warren Buffett 18

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[pinkres<50000])

# plotting on a log scale may be worth investigating

results = data.frame(greenres,redres,whiteres,pinkres)

write.csv(results,"res.csv")


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 don’t know

of any investment that performs like RED. If you know of one, please tell us so we can make PINK!”


Boxplots of results (needs rescaling)

Horton – MOSAIC Being Warren Buffett 21

$80 billion!

Boxplots of results (where returns <=$50,000)


Implementation in SAGE (kudos to Randy P.)http://www.sagenb.org/home/pub/2184


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)


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


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)


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

marketJune 30, 2010

Nicholas J. Horton

Department of Mathematics and Statistics

Smith College, Northampton, MA

[email protected]

http://www.math.smith.edu/~nhorton

being warren buffett: a classroom and computer simulation of the stock market june 30, 2010 nicholas...

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