Question

Stock X has an expected return of 11% and the standard deviation of the expected return is 12%. Stock Z has an expected return of 9% and the standard deviation of the expected return is 18%. The correlation between the returns of the two stocks is +0.2. These are the only two stocks in a hypothetical world.

A.What is the expected return and the standard deviation of a portfolio consisting of 90% Stock X and 10% Stock Z? Will any rational investor hold this portfolio (in this hypothetical two stock world)? Explain why or why not.

B.What is the expected return and the standard deviation of a portfolio consisting of 10% Stock X and 90% Stock Z? Will any rational investor hold this portfolio (in this hypothetical two stock world)? Explain why or why not. (You might want to do Part C first).

C.What is the maximum amount of Stock Z a rational investor will hold in his or her portfolio? What is the expected return and the standard deviation of this portfolio? The maximum amount is a percentage between 0% and 100%, and to receive full credit your answer should be within 0.2 percentage points of the correct answer. (Hint: Set up Excel to calculate the portfolio expected return and standard deviation as a function of the portfolio weights, which must sum to 100%. You can find the correct answer to this part by manually changing the portfolio weights, or by using the Solver function on Excel).

D.Explain why different rational investors might hold different portfolios of these two stocks. Identify the range of portfolios a rational investor might hold. Your answer should take this form: A rational investor will hold a maximum of ___% in Stock X (with ___% in Z), or a minimum of _____% in Stock X (with _____ in Z). The set of feasible portfolios will fall within the range defined by these two end points.

Answer 1

Solution:

The data is:

Stocks	Expected Return	Standard Deviation	Variance
X	11%	12%	1.440%
Z	9%	18%	3.240%

r = 0.2

When stock X=90% and Z= 10%, the expected return and standard deviation are as follows:

Expected Return:	10.800%
Variance:	1.27656%
Sd:	11.2985%

Yes an investor can think of creating such a portfolio in order to minimize the risk. However this is not the optimum portfolio proportions.

When stock X=10% and Z= 90%, the expected return and standard deviation are as follows:

Expected Return:	9.200%
Variance:	2.71656%
Sd:	16.4820%

               No, an investor will not think of creating such a portfolio because the risk (sd) is so high and the return is not sufficient as compared to the above portfolio where taking risk (sd) of 11.29% will yield return of 10.8% whereas in this portfolio, taking risk of 16.4% only yields 9.2% return.

          A rational investor will hold only 26% of stock Z in his/her portfolio. The expected return and standard deviation is 10.48% and 10.834% respectively.

         Different investors might hold different portfolios of these 2 stocks since the utility function may differ from investor to investor. A rational investor should hold maximum of 26% in stock Z and rest in stock X.

Note: here expected return and standard deviations of the portfolio are calculated as:

Expected return: Proportion in stock X * 11% + Proportion in stock Z * 9%

Standard deviation: square root(Proportion in stock X^2 * 12%^2 + Proportion in stock Z^2*18%^2 + 2*Proportion in stock X *Proportion in stock Y * 0.2 * 12% *18%)

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