Suppose an investor invests $100 in a stock that in each period can either double or half with equal probability. The returns in each period are uncorrelated.

a. Calculate the variance of the investors dollar position after one period.

b. Calculate the variance of the investors dollar position after holding the stock for two periods. Report the ratio of variance after two periods to the variance after one period (in (a)).

c. Comment on the ratio in (b) and why you might or might not have expected it to be 0.5 (a case if one shares their wealth equally between two uncorrelated assets for one period).

d. Suppose now in (b) that the investor only invested $36.8229 in the stock. Now calculate and comment the variance after two periods and ratio of variance after two periods to the variance of investing $100 after one period (in (a)).