Question

Based on current dividend yields and expected capital gains, the expected rates of return on portfolios A and B are 11% and 14%, respectively. The beta of A is .8, while that of B is 1.5. The T-bill rate is currently 6%, while the expected rate of return of the S&P 500 index is 12%. The standard deviation of portfolio A is 10% annually, while that of B is 31%, and that of the index is 20%.

a. If you currently hold a market-index portfolio, would you choose to add either of these portfolios to your holdings? Explain.

b. If instead you could invest only in bills and one of these portfolios, which would you choose?

Short answer

a.Taking a long position in Portfolio A and a short position in portfolio B would be advisable.

b. Portfolio A appears to be the preferred choice

Step-by-step solution

Calculation of alphas of two portfolios

	E ( r )	δ	β
Portfolio A	11%	10%	0.8
Portfolio B	14%	31%	1.5
Market index	12%	20%	1.00
Risk-free asset	6%	0%	0.00

α_A= E(r_A) – required return (predicted by CAPM)

= -.11 - [.06 + 8 x (.12 - .06)

= .02%

α_B= E(r_B) – required return (predicted by CAPM)

= -.11 - [.06 + 1.5 x (.12 - .06)

= - 1.0%

This implies that taking a long position in Portfolio A and a short position in portfolio B would be advisable.

Explanation on investment in portfolios

Sharpe measure would be an appropriate criterion, in case only one of the two portfolios is held.

In this case,

S_A= E(r_A) - r_f/ A = .11 - .06 / .10 = .5

S_B= E(r_B) - r_f/ B = .14 - .06 / .31 = .26

From the above, Portfolio A appears to be the preferred choice.