Question

1. 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?

Please show all work.

Answer 1

probability of boom + probability of recession = 1

= 1.5* probability of recession +probability of recession = 1

=> probability of recession = 0.4

probability of boom =0.6

formulas for Expected return and Expected Standard Deviation

  

where pi represents the individual probabilities in different scenarios

Ri represents the corresponding returns in different scenarios and

represents the expected return calculated as above.

Expected return of Company E = 0.6*20%+0.4* (-10%) =8%

Expected return of Company F = 0.6*40%+0.4* (-20%) =14%

As the stocks are fairly priced as per CAPM

Expected Return  = Risk free rate + beta of stock *Market risk premium

So for Company E

=> 8% = 2% + Beta of Company E * 6%

=> Beta of Company E = 1

and for Company F

=> 14% = 2% + Beta of Company F * 6%

=> Beta of Company E = 12%/6% = 2

As the beta of a portfolio is the weighted average beta of constituent stocks

weight of E* 1 + weight of F*2 = 1.6

and weight of F = 1- weight of E

=> weight of E* 1 + (1- weight of E) *2 = 1.6

=> weight of E = 2-1.6 = 0.4

and weight of F = 1-0.4 = 0.6

So, the weight of company E in the portfolio should be 0.4 or 40%

and the weight of company F in the portfolio should be 0.6 or 60%