3. Assume the economy consisted of three types of people. 50% are fad followers, 45% are passive investors (they have read this book and so hold the market portfolio), and 5% are informed traders. The portfolio consisting of all the informed traders has a beta of 1.5 and an expected return of 15%. The market expected return is 11%. The risk-free rate is 5%.

a. What alpha do the informed traders make?
b. What is the alpha of the passive investors?
c. What is the expected return of the fad followers?
d. What alpha do the fad followers make

a. Alpha of the informed traders = 15% - (1.5 * 11%) - 5% = 2.5%

b. Alpha of the passive investors = 11% - 5% = 6%
c. Expected return of the fad followers = 11%
d. Alpha of the fad followers = 0%

To answer these questions, we need to understand the concepts of alpha and beta. Alpha represents the excess return that an investment generates above or below a benchmark or index. Beta measures the sensitivity of an investment's returns to the overall market's returns.

a. To calculate the alpha of the informed traders, we need the expected return of their portfolio, the risk-free rate, and the beta. The formula to calculate alpha is as follows:

Alpha = Expected Return - (Risk-free Rate + Beta * (Market Expected Return - Risk-free Rate))

Given that the expected return of the informed traders' portfolio is 15%, the market expected return is 11%, and the risk-free rate is 5%, and the beta is 1.5, the calculation would be:

Alpha = 15% - (5% + 1.5 * (11% - 5%))
Alpha = 15% - (5% + 1.5 * 6%)
Alpha = 15% - (5% + 9%)
Alpha = 15% - 14%
Alpha = 1%

Therefore, the informed traders make an alpha of 1%.

b. The alpha of the passive investors can be calculated using the same formula as above, with the difference that their beta is equal to 1 since they hold the market portfolio:

Alpha = Expected Return - (Risk-free Rate + Beta * (Market Expected Return - Risk-free Rate))

Given that the expected return of the market is 11%, and the risk-free rate is 5%, the calculation for alpha of the passive investors would be:

Alpha = 11% - (5% + 1 * (11% - 5%))
Alpha = 11% - (5% + 1 * 6%)
Alpha = 11% - (5% + 6%)
Alpha = 11% - 11%
Alpha = 0%

Therefore, the passive investors have an alpha of 0%.

c. To calculate the expected return of the fad followers, we need to consider that they do not actively manage their portfolio and hold the market portfolio. Since they hold the market portfolio, their expected return would be the same as the market expected return, which is 11%.

Therefore, the expected return of the fad followers is 11%.

d. To calculate the alpha of the fad followers, we need to use the same formula as above, but since their beta is equal to 1 (as they hold the market portfolio), their alpha would also be 0%.

Alpha = Expected Return - (Risk-free Rate + Beta * (Market Expected Return - Risk-free Rate))
Alpha = 11% - (5% + 1 * (11% - 5%))
Alpha = 11% - (5% + 1 * 6%)
Alpha = 11% - (5% + 6%)
Alpha = 11% - 11%
Alpha = 0%

Therefore, the fad followers have an alpha of 0%.

a. To calculate the alpha of the informed traders, we need to use the formula:

Alpha = Expected Return - (Risk-Free Rate + Beta * (Market Expected Return - Risk-Free Rate))

Given that the expected return of the informed traders is 15%, the beta is 1.5, the market expected return is 11%, and the risk-free rate is 5%, we can plug these values into the formula:

Alpha = 15% - (5% + 1.5 * (11% - 5%))
Alpha = 15% - (5% + 1.5 * 6%)
Alpha = 15% - (5% + 9%)
Alpha = 15% - 14%
Alpha = 1%

Therefore, the informed traders make an alpha of 1%.

b. The alpha of the passive investors can be calculated using the formula:

Alpha = Expected Return - (Risk-Free Rate + Beta * (Market Expected Return - Risk-Free Rate))

Since passive investors hold the market portfolio, their beta is equal to 1. We are given that the market expected return is 11% and the risk-free rate is 5%, so we can substitute these values into the formula:

Alpha = 11% - (5% + 1 * (11% - 5%))
Alpha = 11% - (5% + 1 * 6%)
Alpha = 11% - (5% + 6%)
Alpha = 11% - 11%
Alpha = 0%

Therefore, the passive investors have an alpha of 0%.

c. The fad followers do not actively invest or make investment decisions based on analysis or information. Therefore, their expected return will be the same as the market return. Given that the market expected return is 11%, the expected return of the fad followers is also 11%.

d. The formula for calculating alpha is the same for all investors:

Alpha = Expected Return - (Risk-Free Rate + Beta * (Market Expected Return - Risk-Free Rate))

Since the fad followers do not actively select investments or have a beta value, we can assume their beta is 0. Plugging in the values, we have:

Alpha = 11% - (5% + 0 * (11% - 5%))
Alpha = 11% - (5% + 0 * 6%)
Alpha = 11% - (5% + 0%)
Alpha = 11% - 5%
Alpha = 6%

Therefore, the fad followers make an alpha of 6%.