You are trying to decide how to allocate your money over the next year between o

Question

You are trying to decide how to allocate your money over the next year between one-year T-Bills and a stock index. Suppose that T-Bills are going to return 2.5% for sure over the next year, while you forecast that the expected stock index return is 4.0% and the standard deviation of the stock index return is 15.0%.

(a) Suppose that w is the fraction of your wealth that you allocate to the stock index. For w = 0%, 1%, 2%, . . . , 99%, 100%, (i.e. for 101 different values of w), calculate the expected return (E[rw]) and standard deviation of return (w) for the strategy that invests a fraction w in the stock market and (1w) in T-Bills. (You don’t need to present any of these numbers). Plot out first E[rw] against w, and then w against w. (Again, you don’t need to present these plots). What shape do these graphs have? Why?

(b) Now, for each of these 101 different strategies, calculate the utility of the strategy (Uw) for an investor with the utility function U(E[r], ) = E[r]2 2 . Plot out Uw against w. (Please do include this graph in your write-up). What does your graph suggest is the right allocation to stocks for this investor? Does this agree with the formula for the optimal allocation?

Explanation / Answer

(a)

Expected return is calculated as the weighted average of the likely profits of the assets in the portfolio, weighted by the likely profits of each asset class. Expected return is calculated by using the following formula

E(R) = w1R1+w2R2+w3R3+.................+wnRn

A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation. Standard deviation is calculated as the square root of variance

Standard deviation = [ w2A*2(RA) + w2B*2(RB) + 2*(wA)*(wB)*Cov(RA, RB) ] 1/2

In this case, as in every case involving a riskless and a risky asset, the relationship is linear.

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