Question

 

1. The (annual) expected return and standard deviation of returns for 2 assets are as follows: Asset A Asset B E[r] 10% 20% SD[r] 30% 50% The correlation between the returns is 0.15. a. Calculate the expected returns and standard deviations of the following portfolios: (i) 80% in A, 20% in B (ii) 50% in A, 50% in B (iii) 20% in A, 80% in B

b. Find the weights for a portfolio with an expected return of 25%? What is the standard deviation of this portfolio?

2. In addition to the information in Q.1, assume that the (annual) risk-free (T-bill) rate is 4%.

a. Calculate the expected returns and standard deviations of the following portfolios: (i) 75% in the risk-free asset, 25% in B (ii) 25% in the risk-free asset, 75% in B (iii) 50% in the risk-free asset, 50% in the portfolio in Q.1a(ii)

b. Calculate the Sharpe ratios of (i) asset A (ii) asset B (iii) the portfolio in Q.1a(i) (iv) the portfolio in Q.1a(ii) (v) the portfolio in Q.1a(iii) (vi) the portfolio in Q.1a(iii)

Answer 1

ER (A) = 10%; ER(B) = 20%; SD (A) = 30%; SD (B) = 50%; Correlation_A,B = 0.15

At the portfolio level, when we combine these two assets with respective weights W_A and W_B - the expected return shall be : ER(P) = W_A * ER(A) + W_B * ER(B) and the portfolio standard deviation shall be given by : SD(P) =

Now we can calculate the ER & SD for various combination of Asset A & B:

(i) W_A = 80% and W_B = 20%

ER(P) = 80% * 10% + 20% * 20% = 12%

SD (P) = [ (80%)²*(30%)² + (20%)²*(50%)² + 2*80%*20%*0.15*30%*50%]^1/2 = 27.35%

(ii) W_A = 50% and W_B = 50%

ER(P) = 50% * 10% + 50% * 20% = 15%

SD (P) = [ (50%)²*(30%)² + (50%)²*(50%)² + 2*50%*50%*0.15*30%*50%]^1/2 = 31.02%

(iii) W_A = 20% and W_B = 80%

ER(P) = 20% * 10% + 80% * 20% = 18%

SD (P) = [ (20%)²*(30%)² + (80%)²*(50%)² + 2*20%*80%*0.15*30%*50%]^1/2 = 41.33%

b. If the ER(P) is 25%, then let us assume that the W_A is x and then W_B will be (1-x) and ER(P) will be :

x * 10% + (1-x) * 20% = 25%; solving we get x = -50% which is W_A and hence W_B is 150%

At these weights, the SD(P) will be:

SD (P) = [ (-50%)²*(30%)² + (150%)²*(50%)² + 2*-50%*150%*0.15*30%*50%]^1/2 = 74.25%

The risk free asset return is given at 4% and its SD is going to be zero. Hence we can calculate as below:

(i) W_r (weight in risk free asset) = 75% and W_B = 25%

ER(P) = 75% * 4% + 25% * 20% = 8%

SD (P) = [ (75%)²*(0%)² + (25%)²*(50%)²]^1/2 = 12.50% (note that the third term will be zero since SD(risk free asset) is zero)

(ii) W_r = 25% and W_B = 75%

ER(P) = 25% * 4% + 75% * 20% = 16%

SD (P) = [ (25%)²*(0%)² + (75%)²*(50%)²]^1/2 = 37.50%

(iii) W_r = 50% and Portfolio of 50% A and 50% B

ER(P) = 50% * 4% + 50% * 15% = 9.50%

SD (P) = [ (50%)²*(0%)² + (50%)²*(31.02%)²]^1/2 = 15.51%

Sharpe ratio = (Expected Return - Risk free rate)/Standard Deviation

Sharpe ratio Asset A = (10% - 4%)/30% = 20%

Sharpe ratio Asset B = (20% - 40%)/50% = 32%

Sharpe ratio Portfolio (80% A and 20% B) = (12% - 4%)/27.35% = 29.25%

Sharpe ratio Portfolio (50% A and 50% B) = (15%-4%)/31.02% = 35.46%

Sharpe ratio Portfolio (18%-4%)/41.33% = 33.87%