Based on current dividend yields and expected capital gains, the expected rates

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 and 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 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?

Explanation / Answer

To answer simply,

a. S&P index beta is 1. So, 12% = 6% + 1*(Market risk premium) from CAPM => market risk premium = 6%

Using this to find the cost of equity for A, cost of equity = 6% + .8*6% = 10.8% and

for B, cost of equity = 6% + 1.5*6% = 15%.

But, you can see that A is expected to earn more than the cost of equity while B less. So, I would choose to add A to my portfolio.

b. Bills are expected to be completely risk-free. So, their standard deviation is taken to be 0. When you are not talking with respect to a market portfolio, we just use the standard deviations as measures of risk instead of betas.

Hence, the portfolio with the highest sharpe ratio wins.

Sharpe ratio A = (rp - rf)/sigma p = (11% - 6%)/10% = 0.5

Sharpe ratio B = (14% - 6%)/31% = .258

Shapre ratio index = (12% - 6%)/20% = 0.3

I would still invest in A as it has the highest risk adjusted returns.

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