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.
- 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 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
(Green 2) (Red 5) (White 3)
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
(Green 4) (Red 2) (White 6)
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 |
- 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?
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
Pink: $1000 * (3 + 1)/2 = $2000
and compounded in the second round which had (Green 4), (Red 2), and (White 6)
Pink: $2000 * (0.2 + 1.1)/2 = $1300
- 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?
| |