Assume the economy consisted of three types of people. 50% are fad followers, 45% are passive investors who 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?
Answers
a. 1%
b. 0
c. 10.6%
d. –0.1%
How do you get to the answers?
. 1%
b. 0
c. 10.6%
d. –0.1%
To calculate the answers to these questions, we need to understand the concepts of beta, expected return, risk-free rate, and alpha.
Beta is a measure of an asset's sensitivity to movements in the overall market. A beta of 1 indicates that an asset's price will move in line with the market, while a beta greater than 1 suggests that the asset will be more volatile than the market, and a beta less than 1 suggests the asset will be less volatile than the market.
Expected return is the estimated return on an investment based on its predicted performance. The risk-free rate represents the return on an investment with zero risk, typically defined as the return on government bonds.
Alpha measures the excess return of an investment or portfolio compared to its expected return based on its beta. It provides an indication of the asset manager's skill in delivering returns above or below the market's expectations.
Now let's calculate the answers to the given questions:
a. To find the alpha of the informed traders, we first need to calculate their expected return based on their beta and the market's expected return:
Informed traders' expected return = risk-free rate + beta * (market expected return - risk-free rate)
= 5% + 1.5 * (11% - 5%)
= 5% + 1.5 * 6%
= 5% + 9%
= 14%
Alpha of the informed traders = Informed traders' expected return - Market expected return
= 14% - 11%
= 3%
Therefore, the alpha of the informed traders is 3%.
b. The passive investors hold the market portfolio, which means they have the same beta as the market. Since their beta is equal to 1, their expected return will be the same as the market's expected return.
Alpha of the passive investors = 0 (Since their expected return is the same as the market's)
c. To find the expected return of the fad followers, we need to calculate their weighted return based on the market and informed traders:
Fad followers' expected return = market expected return * (1 - percentage of informed traders) + informed traders' expected return * percentage of informed traders
= 11% * (1 - 5%) + 14% * 5%
= 11% * 0.95 + 14% * 0.05
= 10.45% + 0.7%
= 11.15%
Therefore, the expected return of the fad followers is 10.15%.
d. To find the alpha of the fad followers, we need to calculate the difference between their expected return and the market expected return.
Alpha of the fad followers = Fad followers' expected return - Market expected return
= 10.15% - 11%
= -0.85%
Therefore, the alpha of the fad followers is -0.85%.
In summary:
a. The informed traders make an alpha of 3%.
b. The passive investors have an alpha of 0.
c. The expected return of the fad followers is 10.15%.
d. The fad followers make an alpha of -0.85%.
To calculate the answers, we need to use the formula for the capital asset pricing model (CAPM):
Expected Return = Risk-Free Rate + Beta * (Market Expected Return - Risk-Free Rate) + Alpha
Where:
- Risk-Free Rate is 5%
- Expected Return of the market is 11%
- Beta of the portfolio consisting of all informed traders is 1.5
- Expected Return of the portfolio consisting of all informed traders is 15%
Now let's calculate the answers step by step:
a. To find the alpha for the informed traders, we will use the formula mentioned above:
15% = 5% + 1.5 * (11% - 5%) + Alpha
Solving for Alpha:
15% = 5% + 1.5 * 6% + Alpha
Alpha = 15% - 5% - 1.5 * 6%
Alpha = 1%
Therefore, the alpha for the informed traders is 1%.
b. Passive investors who hold the market portfolio have a beta of 1 (since they hold the market portfolio), which means they have no alpha. So, the alpha for passive investors is 0.
c. To find the expected return of the fad followers, we can use the formula without considering the alpha since fad followers are not informed traders:
Expected Return of fad followers = Risk-Free Rate + Beta * (Market Expected Return - Risk-Free Rate)
= 5% + 1 * (11% - 5%)
= 10.6%
Therefore, the expected return for fad followers is 10.6%.
d. Fad followers do not have any alpha since they only follow trends and do not have any information advantage. So, the alpha for fad followers is 0 - Expected Return of the fad followers, we know that the expected return for fad followers is 10.6% from the previous step:
Alpha = 0 - 10.6%
Alpha = -0.1%
Therefore, the alpha for the fad followers is -0.1%.