Question

If the economy enters a boom, the stock of Company E will return 20% and the stock of Company F will return 40%. On the other hand, if the economy enters a recession, the stock of Company E will return -10% and the stock of Company F will return -25%. The boom state is one-and-one-half times as likely as the recession state. The risk-free rate in the market is 2%, while the risk premium of the market portfolio is 6%. You are planning to set up a portfolio of these two securities with a beta of 1.6. Assuming that these two securities are fairly priced according to the CAPM, what should be their weights in the portfolio?

Answer 1

Hi

here portfolio beta = 1.6

risk free rate rf = 2%

risk premium of market rmf = 6%

So return of the portfolio as per CAPM = rf + beta*rmf

=2 + 1.6*6

   = 2 + 9.6 = 11.6%

Lets say weight ot security E = w

then weight of security F = 1-w

boom state is one and half time as likely as recession state.

So lets say recession state probability = p

   so boom state probability = 1.5p

p+1.5p = 1

p = 1/2.5 = 0.4

hence recesion state probability = 0.4

and boom state probaility = 0.6

So expected return = (0.6*0.2+0.4*(-0.1))* w + (0.6*0.4 +0.4*(-0.25))*(1-w)

= (0.12 - 0.04)*w + (0.24-0.1)*(1-w)

= 0.08w + 0.14 - 0.14w

0.116-0.14 = -0.06w

w = 0.024/0.06

w = 0.4

1-w = 0.6

Hence weight of security E = 40%

weight of security F = 60%

Thanks