Question

Asset 1 has an expected return of 10% and a standard deviation of 20%. Asset 2 has an expected return of 20% and a standard deviation of 50%. The correlation coefficient between the two assets is 0.0. Calculate the expected return and standard deviation for each of the following portfolios, and plot them on a graph. SHOW ALL WORK!

Portfolio	% Invested in Asset 1	% Invested in Asset 2
A	100	0
B	75	25
C	50	50
D	25	75
E	0	100

Now, repeat these calculations after changing just one assumption: suppose the standard deviation of asset 1 equals zero. In other words, asset 1 pays a risk-free (because it never varies) return of 10%. How does the graph of the expected return and standard deviation for various portfolios change in this case?

Answer 1

(A) Expected Portfolio Return = w1 x r1 + w2 x r2, where r1 and r2 are the returns of Asset 1 and Asset 2 respectively and w1 and w2 are the weights of Asset 1 and Asset 2 respectively. Further, w1 + w2 = 1. Expected Standard Deviation = [{w1 x s1}^(2) + {w2 x s2}^(2) + 2 x w1 x w2 x s1 x s2 x Correlation Coefficient]^(1/2) where s1 and s2 are the standard deviations of Asset 1 and Asset 2 respectively.

Coefficient of Correlation = 0

s1 = 20 % and s2 = 50 %

r1 = 10 % and r2 = 20 %

(B) If the standard deviation of Asset 1 becomes zero, Asset 1 is rendered risk-free and therefore, Expected Portfolio Return = r1 + y x (r2 -r1) where y is the weight of the risky asset 2 and r1 and r2 are the returns of Asset 1 and Asset 2 respectively. Similarly, Expected Standard Deviation = y x s2 where s2 is the standard deviation of Asset 2.

As is observable the graph representing the risk-return profile becomes a straight line when the standard deviation of Asset 1 becomes zero. This happens because the equations for both expected portfolio return portfolio standard deviation become equations linear in nature.