1. Being Warren Buffett: a classroom simulation of the stock market March 24, 2009 Nicholas J. Horton Department of Mathematics and Statistics Smith College, Northampton, MA nhorton@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), 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 – Causeweb - 2009 Being Warren Buffett 2

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 Can be implemented using dice (and calculators) in a classroom setting Computer generation of results complements the analytic and hand simulations Horton – Causeweb - 2009 Being Warren Buffett 3

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 Horton – Causeweb - 2009 Being Warren Buffett 4

5. Background information Imagine that you have $1000 to invest in the stock market, for 20 years Three investment possibilities are presented to students in groups of 2 or 3: Question: Which of the three investments seems the most attractive to the members of your group? Horton – Causeweb - 2009 Being Warren Buffett 5 InvestmentExpected annual returnSD(annual return) Green 8.3% 20% Red71%132% White 0.8% 4%

6. Dice outcomes The investments rise or fall based on the outcomes of a 6-sided die: Horton – Causeweb - 2009 Being Warren Buffett 6 OutcomeGreenRedWhite 10.80.050.95 20.90.21 31.111 4 31 51.231 61.431.1

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) Horton – Causeweb - 2009 Being Warren Buffett 7 RoundGreenRedWhite Start$1000 Return 10.931 Value 1$ 900$3000$1000 Return 21.10.21.1 Value 2$ 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 Buffett’s” of the class Horton – Causeweb - 2009 Being Warren Buffett 8

9. Group results form Horton – Causeweb - 2009 Being Warren Buffett 9

10. Usually, red doesn’t do as well as green Horton – Causeweb - 2009 Being Warren Buffett 10

11. But occasionally it wins big! Horton – Causeweb - 2009 Being Warren Buffett 11

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 We’d always want to pick RED, no? Horton – Causeweb - 2009 Being Warren Buffett 12

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% Horton – Causeweb - 2009 Being Warren Buffett 13

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 Horton – Causeweb - 2009 Being Warren Buffett 14

15. How to implement PINK Horton – Causeweb - 2009 Being Warren Buffett 15 Pink $1000 31 $3000 $1000 $2000 2 0.0510.525 $150$1000 $1050

16. 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!” Horton – Causeweb - 2009 Being Warren Buffett 16

17. Boxplots of results (needs rescaling) Horton – Causeweb - 2009 Being Warren Buffett 17 $80 billion!

18. Boxplots of results (where returns <=$50,000) Horton – Causeweb - 2009 Being Warren Buffett 18

19. 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) Horton – Causeweb - 2009 Being Warren Buffett 19

20. 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 Horton – Causeweb - 2009 Being Warren Buffett 20

21. 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) Horton – Causeweb - 2009 Being Warren Buffett 21

22. Being Warren Buffett: a classroom simulation of the stock market March 24, 2009 Nicholas J. Horton Department of Mathematics and Statistics Smith College, Northampton, MA nhorton@smith.edu http://www.math.smith.edu/~nhorton