Students in business and other areas who are new to Statistics have a hard time making the connection between variance and risk. To convey the connection, we developed a classroom simulation. In the simulation, groups of students roll three colored dice that determine the success of three “investments”. The simulated investments behave quite differently. The value of 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. For most groups, however, this last investment leads to ruin because of variance in its returns. The marked difference in outcomes shows students how hard it is to separate luck from skill. The simulation also demonstrates how portfolios, weighted combinations of investments, reduce the variance. In the simulation, a mixture of two poor investments is surprisingly good. Rather than use arbitrary properties, we calibrated the returns on two simulated investments to mimic returns on US Treasury Bills and stocks.
The definition of variance as the expected squared deviation from the mean often strikes students as capricious. Why square the deviations from the mean rather than use the absolute value? Why average the values? Without the machinery of maximum likelihood or concepts of asymptotic efficiency, one is left to vague, heuristic explanations. When dealing with money, however, the definition of variance is just right. Rather than make this connection with formulas and theorems, we have found it more useful and memorable to let students experience the effects of variance first-hand. After defining means and variances with some basic examples, we use this ‘dice game’ to show the importance of these concepts. The discussion of the 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.
The three investments in this simulation have different characteristics. One investment resembles an old-fashioned savings account whose interest has been adjusted for the effects of inflation. At the other extreme, a second investment matches our intuitive definition of being very risky. A third lies between these extremes.
We have students simulate the changing 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. Though we have tried to save class time by letting a computer roll the dice (it’s easy to program the simulation in Excel, say), we have found that students find the results more impressive when they roll the dice themselves. Although the simulation and ensuing discussion consume only an hour and 20-minute class (it also works well divided into 2 one-hour classes), the lessons of the simulation are among those that students take away from our course. After this simulation, everyone appreciates the importance of variance when looking at data.
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Being Warren Buffett: A classroom simulation of financial risk
