Question

Use the information for securities X, Y and Z in the table below to answer parts a and b:

	Security X	Security Y	Security Z
Expected return	8%	8%	17%
Beta	0.7	1.3	2.5

The risk-free rate is 2% and the expected return of the market portfolio is 8%.

1. According to the CAPM, are these securities overpriced, fairly priced or underpriced?
2. Using the CAPM, calculate the abnormal return of a portfolio that takes a long position in security X by $35,000, a short position in security Y by $15,000, and a long position in security Z by $5,000.
3. You have $100,000 and you want to set up a portfolio of security X and security Y with an abnormal return of 2.7% based on the CAPM. Specify the position to be taken (long or short) and the dollar amount to be invested in each security.

Answer 1

Formula of CAPM :

Expected Return = RiskFree Rate + beta*(Market's return - RiskFree Rate)

X = 2% + 0.7*(8%-2%) = 6.2%

X = 2% + 1.3*(8%-2%) = 9.8%

X = 2% + 2.5*(8%-2%) = 17%

By this formula we can get the following table:

	BETA	Return Given	Return by CAPM
X	0.7	8%	6.20%
Y	1.3	8%	9.80%
Z	2.5	17%	17%

A.) If the Original Return(Return Given) is more than the CAPM return, the stock is underpriced and vice versa.

So, Here X is Underpriced, Y is Overpriced and Z is at fair value.

B.)

	Expected Return(CAPM)	Amount	%in portfolio	Return in portfolio
X	6.2%	35,000	63.64%	6.2%*63.64% = 3.95%
Y	9.8%	15,000	27.27%	-9.8%*27.27% = -2.67
Z	17%	5,000	9.09%	17%*9.09% = 1.55
		55,000		2.82%*55,000 = 1550

Here we have short Y, therefore negative return should be considered. Hence the abnormal return of the portfolio will be $1550.

C.) let us assume, we have taken long positions in both X and Y with x% of X, thus (100-x)% of Y. So, we can write the equation of the required return as: x*(6.2%) + (1-x)*(9.8%) = 2.7%, by solving it we get x = 1.97, which is not possible as x should be less than 1. So, now if we short Y than we have required return as: x*(6.2%) - (1-x)*(9.8%) = 2.7%, by solving this we get x = 0.78.

So, we have long position in X of 0.78*100,000 = $78,000. And Short position in Y of 0.22*100,000 = $22,000.

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