Question

The table below presents the state-based returns of securities A and B, the risk-free security and the market portfolio, where p is the probability of each state. Use the information therein to answer parts a and b.

State	p	Security A	Security B	Risk-free security	Market portfolio
Recession	0.5	-4%	44%	2%	-6%
Normal	0.4	10%	-10%	2%	20%
Boom	0.1	40%	-30%	2%	30%

1. Calculate the expected return and its standard deviation of securities A and B.
2. Suppose you borrow $30,000 at the risk-free rate and along with the $60,000 you have, you invest $15,000 in security A and $75,000 in security B. Calculate the expected return, its standard deviation and the CAPM-implied beta of this portfolio.

Please give detail

Answer 1

a.

= 0.5*(-0.04) + 0.4*0.1+0.1*0.4 = 0.06 = 6%

= 0.5 * 0.44 + .4*(-0.1)+0.1*(-0.3) = .15 = 15%

= 0.5 * (-0.04 - 0.06)2   + 0.4*(0.1-0.06)2 +0.1*(0.4-0.06)2 = 0.0172

= 0.131 = 13.1%

0.5*(0.44-0.15)2 +0.4*(-0.1 - 0.15)2 +0.1(-0.3-0.15)2 =0.0873

= 0.2955 = 29.55 %

b.

Total investment = 30000+60000 = 90000

Weightage of A in the portfolio = 15000/90000 = .1667

= 1-0.1667 = 0.8333

So, the A is 16.67 % in out portfolio and B is 83.33%

= 0.1667* 0.06 + 0.8333*0.15 = 0.134997 = 13.5%

= 0.16672 *0.0172 + 0.83332 * 0.0873 = 0.061098

= 0.2472

CAPM Model states that :

where Rf is the rsk free rate, Rm is the return of market portfolio

= (0.5 * -0.06 )+0.4*0.2+0.1*0.3 = 0.08 = 8%

= (0.134997 - 0.02)/(0.08-0.02) = 1.92