Use the following information to answer the question(s)below.
Suppose that the market portfolio is equally likely to increase by 24% or decrease by 8%.Security "X" goes up on average by 29% when the market goes up and goes down by 11% when the market goes down.Security "Y" goes down on average by 16% when the market goes up and goes up by 16% when the market goes down.Security "Z" goes up on average by 4% when the market goes up and goes up by 4% when the market goes down.
-The risk-free rate is closest to:
A)0%.
B)4%.
C)8%.
D)16%.

Question

Quizplus · Accepted Answer

To find the risk-free rate, we need to use the Capital Asset Pricing Model (CAPM):
Expected Return = Risk-Free Rate + Beta*(Expected Market Return - Risk-Free Rate)
The market portfolio has two outcomes: 24% or -8%, each with a probability of 0.5. Therefore, the expected market return is:
Expected Market Return = 0.5*(24%) + 0.5*(-8%) = 8%
Now we can find the beta for each security:
Beta X = (0.29-(-0.11))/(0.24-(-0.08)) = 2.5
Beta Y = (0.16-(-0.16))/(-0.08-0.24) = -1
Beta Z = (0.04-0.04)/(0.24-(-0.08)) = 0
Using the CAPM:
Expected Return X = Risk-Free Rate + 2.5*(8% - Risk-Free Rate)
Expected Return Y = Risk-Free Rate - 1*(8% - Risk-Free Rate)
Expected Return Z = Risk-Free Rate + 0*(8% - Risk-Free Rate)
We know that the expected return for a risk-free asset is simply the risk-free rate. To find the risk-free rate, we can set the expected return of one of the securities equal to the risk-free rate, and solve for it. Let's use security Y:
16% = Risk-Free Rate - 0.08 + Risk-Free Rate
Risk-Free Rate = 12%/2 = 6%
Therefore, the closest choice is B) 4%.

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