Question

(i) Which security has greater total risk? Which has greater systematic risk? Which has greater unsystematic risk? Which security will have a higher risk premium?

(ii) Construct a two-asset equally weighted portfolio is minimising the overall risk. What is the portfolio's Beta? What is the standard deviation of the portfolio?

(iii) Calculate the Sharpe ratios for the three securities and the equally weighted portfolio in part ii. Is it possible to build a two-asset equally weighted portfolio with a higher Sharpe ratio than the one in part ii?

Required:

Answer 1

(i).

The total risk is indicated by standard deviation of the security. Since, security A has highest standard deviation, it has the greatest total risk. Systematic risk is indicated by Beta. Since, security B has highest beta of 1.6, it has the highest systematic risk. The unsystematic risk is indicated by the residual variances which are not provided in the statement. Risk Premium referes the the difference between security return and risk free rate of return. Thus, Security A has the highest risk premium of 12%.

 

(ii).

The Overall risk of Security A = 28% * 1.2 = 33.6%

The Overall risk of Security B = 12% * 1.6 = 19.2%

The Overall risk of Security C = 8% * 1.1 = 8.8%

 

To minimize the overall risk, we will select select Security B and C for the two-asset equally eighted portfolio.

Portfolio Beta, βP = wB βB + wC βC = (0.5)(1.6) + (0.5)(1.1) = 1.35

Porfolio Standard Deviation, σP = √ [wB2σB2 +  wC2σC2 + 2wBwCσBσCρB,C]

σP = √ [(0.5)2(12%)2 + (0.5)2(8%)2 + 2(0.5)(0.5)(12%)(8%)(0.5)] = √ (76) = 8.72%

 

(iii). 

The formula is 

Sharpe Ratio, S = [E(R) - RF] / σ

For Security A, SA = (14% - 2%) / 28% = 0.43

For Security B, SB = (10% - 2%) / 12% = 0.67

For Security C, SC = (8% - 2%) / 8% = 0.75

Expected Return on Portfolio, E(RP) = wB E(R)B + wC E(R)C= (0.5)(10%) + (0.5)(8%) = 9%

For Security P, SP = (9% - 2%) / 8.72% = 0.803

 

Yes, By changing weights of securities in the portfolio (taking more of Security C and less of Security B), we can actually increase the Sharpe's Ratio.

Yes, By changing weights of securities in the portfolio (taking more of Security C and less of Security B), we can actually increase the Sharpe's Ratio.