Either work with someone else on this assignement, or work by yourself. Type up your answers to the questions (two or three pages max). Print out two copies--one to give me and one to keep for class discussion. Each group with two or more people gets THREE dice. If you are doing it alone, you get one and will have to pretend that you have three dice.

For this homework exercise, you will use three dice to simulate the uncertainty in financial markets. There are three basic instruments in this simplified market, each associated with a single die.

Color Die | Annual Return | Variability (SD) |
---|---|---|

green | 7.5% | 20% |

Red | 71% | 130% |

White | 0% | 5% |

- Before starting the simulation, which of the three investments looks most appealing to you? Does the group agree (have a consensus) or is it not so clear? Explain your choice briefly.
- The rest of this assignment deals with a small simulation of these three financial instruments represented by the dice described above. For each "year" of the game, you will roll all three dice, and use the outcomes to determine what has happened to the value of each investment. Each of the three investments starts off with an initial value of $1000. Run the game out for 25 years. Which investment wins? Is it the same one that you picked in Question 1? Try to explain any differences or surprises.
- The "volatility corrected growth rate" is the E(R) - Var(R)/2, where R is the annual return. The idea of the correction is simple. If you have a 10% gain followed by a 10% loss, you end up at .99 of where you started. So this standard deviation of 10% has lowered your return by 1% every two rounds--namely the variance/2. Work out the volatility corrected growth rate for the first three investments. This might explain what happened to your Red investment. :-)
- Now considers the performance of a hybrid investment, one which mixes the outcomes of "Red"; and "White". To compute the value of this investment, roll both the red and white dice for each round. It’s easiest to describe what to do with an example. Before doing any simulating, what do you think of this hybrid?
- Again, simulate the hybrid. What happens? How does it compare to the previous instruments?
- Work out the mean and variance for the Pink investment. From these compute the volatility corrected growth rate for pink. (Hint: Note that the return for Pink is simply the return for red + white/2. So you can use linearity of expectation! Also note that returns on red and white are independent, so the covariance is zero.)
- For the first three investment, compute the expected log return, and the variance of the log return. Is the expected log return close to your growth rate you computed above? Can you see why? (Think about Taylor's expansion for the log.)
- Which of the 4 investments do you like the best now?

The following table shows how the rolls of the dice affect the values of the three investments.

Roll |
Green |
Red |
White |

1 |
0.8 |
0.06 |
0.9 |

2 |
0.9 |
0.2 |
1 |

3 |
1.05 |
1 |
1 |

4 |
1.1 |
3 |
1 |

5 |
1.2 |
3 |
1 |

6 |
1.4 |
3 |
1.1 |

For example, suppose that on the first roll of all three dice, you obtain

Then the values of the investments after the first year become

Green: | $1000 * 0.9 | = $900 |

Red: | $1000 * 3 | = $3000 |

White: | $1000 * 1 | = $1000 |

For the next roll, the values are compounded from these. Suppose that
on the second roll of all three dice, you obtain

then the values of the three investments after two years are

Green: | $900 * 1.1 | = $990 |

Red: | $3000 * .2 | = $600 |

White: | $1000 * 1.1 | = $1100 |

For the first round, using the same dice rolls as above (Green 2),
(Red 5) and (White 3), the value of this "Pink"; investment is