Question

Based on current dividend yields and expected capital gains, the expected rate of
returns on portfolio A and B are 10% and 11% respectively. The beta of portfolio
A is 0.80 while that of B is 1.4. risk-free rate is 6%, while expected return of the
market portfolio is 10%. The standard deviation of portfolio A is 25% , while that
of B is 22%, and that of the stock market 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 the risk-free asset and one of these alternative
portfolios, which would you choose?

Answer 1

Solution:

(A)
The market index portfolio is a diversified portfolio where only systematic risk is rewarded. So, we can add an portfolio that is earning more than its market risk. Therefore, excess return (alpha) needs to be calculated and positive alpha portfolio should be added to the existing market index portfolio.
	Portfolio A	Portfolio B
Expected Return (given)	10%	11%
	Portfolio A	Portfolio B
Risk free Rate (given)	6%	6%
Beta (given)	0.8	1.4
Expected Market Return (given)	10%	10%
CAPM Formula: Required Return = Risk free Return + Beta *(Expected Market Return-Risk free Return)	6% + 0.8*(10% - 6%)	6% + 1.4*(10% - 6%)
	= 9.200%	= 11.600%
Alpha = Expected Return - Required CAPM return	10%-9.2%	11%-11.6%
	= 0.80%	= -0.60%
As portfolio A has a positive alpha of 0.8% so it should be added to the holdings.

(B)
For this situation we should consider full risk and reward function. So, we need to choose the portfolio with the higher sharpe ratio that means the portfolio that is generating higher return on per unit of risk taken.
	Portfolio A	Portfolio B
Standard deviation	25%	22%
Sharpe Ratio = (Expected Return - Risk Free Rate)/Standard deviation	(10% - 6%) / 25%	(11% - 6%) / 22%
	= 0.16	= 0.23
As portfolio B has a higher sharpe ratio of 0.23, that is higher reward to risk ratio. So, we need to need to choose portfolio B along with risk free asset