Question

Veronica decides to purchase stocks of two firms: A and B using her $200,000 cash. The market price for stock A is $80 and for B is $50. The expected return for A is 15% (25% for B) and the standard deviation for A is 20% (35% for B). The correlation between A and B is zero and a T-bill with maturity of 120 days is traded for 97.8 (face value 100). Assume that the markets rate will remain constant over time and that the bank will not allow Veronica to borrow more than $300,000.

a. Veronica decides to invest in one risky (only A or only B) and one risk-free security.

1. What will be the standard deviation of a portfolio with expected return 20%?

2. If Veronica is risk neutral, How many securities from type A or B she buy?

b. Veronica decides to invest in both A and B equally. What will be the expected return of a portfolio with standard deviation equals 0.10078

Answer 1

Part a 1) Only A
Return on T-bill = (Face value-Current value)*360days/(face value*Maturity period) = (100-97.8)*360/(100*120) = 2.2*360/12000 = 6.6% per annum
Expected return from the portfolio (With A & T-bill) = (Return from A*Weight on A)+(Return from T-bill*Weight on T-bill) = 20%
(15%*Weight on A)+(6.6%*[1-Weight on A]) = 20%
15%Weight on A+6.6%-6.6%Weight on A = 20%
8.4%Weight on A = 13.4%
Weight on A = 13.4%/8.4% = 1.6
Weight on T-bill - 1- weight on A = 1-1.6 = -0.6
To earn 20% return, invest ($200,000*1.6) $320,000 in security A & borrow or short sell $120,000 T-bills
Standard deviation of the portfolio (A & T-bill) = Weight on A*Standard deviation of A = 1.6*20% = 32%

Part a 1) Only B
Expected return from the portfolio (With B & T-bill) = (Return from B*Weight on B)+(Return from T-bill*Weight on T-bill) = 20%
(25%*Weight on B)+(6.6%*[1-Weight on B]) = 20%
25%Weight on B+6.6%-6.6%Weight on B = 20%
18.4%Weight on B = 13.4%
Weight on B = 13.4%/18.4% = 0.73
Weight on T-bill - 1- weight on B = 1-0.73 = 0.27
To earn 20% return, invest ($200,000*0.73) $146,000 in security B & invest $54,000 ($200,000*0.27) in T-bills
Standard deviation of the portfolio (B & T-bill) = Weight on b*Standard deviation of B = 0.73*35% = 25.55%

Part a 2)
Security from firm A = Funds available/Market price = $200,000/$80 = 2,500
Security from firm B = Funds available/Market price = $200,000/$50 = 4,000

Part b)
Weight on A=0.25, Weight on B = 0.25, Weight on T-bill=0.5
Expected return on the portfolio = (Weight on A*return from A)+(Weight on B*Return from B)+(Weight on T-bill*Return from T-bill) = (0.25*15%)(0.25*25%)+(0.5*6.6%) = 3.75%+6.25%+3.3% = 13.3%