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 2.62%. 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

Expected return on market = 12%, Risk free rate = 4%

Beta of informed traders = 1.3

Expected Return of informed investors according to CAPM = Risk free rate + Beta(Expected market return - risk free rate) = 4% + 1.3(12% - 4%) = 4% + 1.3 x 8% = 4% + 10.4% = 14.40%

Alpha of informed investors = Expected return of informed investors - Expected return of informed investors according to CAPM

2.62% = Expected return of informed investors = 14.40%

Expected return of informed investors = 14.40% + 2.62% = 17.02%

As we know that passive investors invest in the market portfolio, therefore Expected return of passive investors = Expected return of market = 12%. We also know that Beta of market portfolio is equal to 1, hence Beta of passive investor = 1

Economy consists of various types of investors. Market return denotes a return earned by an average investor. Average return earned by an investor in economy will be equal to expected return of the market. Hence return of market can be used evaluate return earned by an average investor in the economy.

Expected return of an investor in economy = Expected return of market = 12%

Expected return of an investor in economy = Weight of fad followers x expected return of fad followers + Weight of passive investors x expected return of passive investors + Weight of informed traders x Expected return of informed traders

12% = 15% x expected return of fad followers + 75% x 12% + 10% x 17.02%

12% = 15% x expected return of fad followers + 9.00% + 1.702%

12% - 10.702% = 15% x expected return of fad followers

1.298% = 15% x expected return of fad followers

Expected return of fad followers = 1.298% / 15% = 8.6533%

As average return earned by an investor in economy is equal to expected return of market. We know that Beta of market is equal to 1. Therefore beta of portfolio of average investor in economy will also be 1

beta of portfolio of average investor in economy = Weight of fad followers x Beta of fad followers + Weight of passive investors x Beta of passive investors + Weight of informed traders x Beta of informed traders

1 = 15% x Beta of fad followers + 75% x 1 + 10% x 1.3

1 = 15% x Beta of fad followers + 75% + 13%

15% x Beta of fad followers = 1 - 88% = 12%

Beta of fad followers = 12% / 15% = 0.80

Expected return of fad followers according to CAPM = Risk free rate + Beta (expected return of market - risk free rate) = 4% + 0.80(12% - 4%) = 4% + 0.80 x 8% = 4% + 6.4% = 10.40%

Alpha of fad followers = Expected return of fad followers - Expected return of fad followers according to CAPM = 8.6533% - 10.40% = -1.7467% = -1.75% (rounded to two decimal places)

Hence Alpha of fad followers = -1.75%