Question

You have $1,000 to invest and are considering buying some combination of the shares of two companies, Donkey Inc and Elephant Inc. Shares of Donkey Inc will pay a 10 percent return if the Democrats are elected, an event you believe to have a 60 percent probability; otherwise the shares pay a zero return. Shares of Elephant Inc will pay 8 percent if the Republicans are elected (a 40 percent probability), zero otherwise. Either the Democrats or the Republicans will be elected.

a)What is your expected return if you invest $500 in each stock? (Hint: Consider what your return will be if the Democrats win and if the Republicans win, then weight each outcome by the probability that event occurs.)

B) Devise an investment strategy that is riskless, that is, one in which the return on your $1,000 does not depend at all on which party wins.

 Invest $ ______in ElephantInc and $ ______in DonkeyInc.

Answer 1

Ok we're calculating expected returns. So first we need to know the formula for expected value:
Prob A x ($ amount invested) x (% return) = expected value of A
Prob B x ($ amount invested) x (% return) = expected value of B
Then you sum the expected value of the two in order to get the expected returns.

A) 0.6 x $500 x 0.10 = $30 expected value of A
0.4 x $500 x 0.08 = $16 expected value of B
Expected Value of A + expected value of B = expected return so $30 + $16 = $46

B) In order to solve for the strategy that gives us equal returns we need to set two equations equal to each other, create an additional equation, and solve for the size of the two investments necessary. First we're going to set the expected value formulas equal to each other:

(Prob A 0.6) x 0.1 x (X dollars) = (Prob B 0.4) x .08 x (Y dollars)
By multiplying the decimals on both sides we get:
.06X = 0.032Y

Now remember X+Y has to equal $1000 because that's the amount we can invest. So the equation we will create will be X + Y = 1000
By solving for Y, we can rearrange that equation to be Y = 1000 - X
Now plug that into the other equation:

0.06X = 0.032 (1000 - X)
0.06X = 32 - 0.032 X
0.092X = 32
X = $347.83 ( Donkey inc.)
Y = 1000 - $347.83 = $652.17 (Elephant Inc.)

If you plug those numbers (X and Y investments) back into the expected value formulas to calculate your expected return. You'll find that the two equally contribute to your return, thus exempting you from who the winner of the race is.