Question

Assume that the economy has three types of people. 15% are fad followers, 75% are passive investors, and 10% are informed traders. The portfolio consisting of all informed traders has a beta of 1.3 and an alpha of 2.21%. The market has an expected return of 12% and the risk-free rate is 4 %. What is the alpha for the fad followers? Enter your answer as a percentage to two decimal places (i.e. 0.12% rather than 0.0012; the percent sign is not necessary).

Answer 1

According to the CAPM,

Expected Return = Risk-free rate * [Beta * (Expected Market Return - Risk-free rate)]

Expected Return_{Informed Trader} = 4% + [1.3% * (12% - 4%)]

= 4% + [1.3 * 8%] = 4% + 10.4% = 14.4%

Return_{Informed Trader} = Alpha + Expected Return_{Informed Trader}

= 2.21% + 14.4% = 16.61%

Market's Return = [W_Fad * Return_Fad] + [W_Passive * Return_Passive] + [W_{Informed Trader} * Return_{Informed Trader}]

12% = [0.15 * Return_Fad] + [0.75 * 12%] + [0.10 * 16.61%]

12% = [0.15 * Return_Fad] + 9% + 1.661%

12% = [0.15 * Return_Fad] + 10.661%

0.15 * Return_Fad = 12% - 10.661%

Return_Fad = 1.339% / 0.15 = 8.93%

Market's beta = [W_Fad * beta_Fad] + [W_Passive * beta_Passive] + [W_{Informed Trader} * beta_{Informed Trader}]

1 = [0.15 * beta_Fad] + [0.75 * 1] + [0.10 * 1.3]

1 = [0.15 * beta_Fad] + 0.75 + 0.13

1 = [0.15 * beta_Fad] + 0.88

0.15 * beta_Fad = 1 - 0.88

beta_Fad = 0.12 / 0.15 = 0.8

Alpha_Fad = Return_Fad - Expected Return_Fad by CAPM

= 8.93% - [4% + 0.8 * (12% - 4%)]

= 8.93% - [4% + 6.4%]

= 8.93% - 10.4% = 1.47%