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Using CAPM to calculate Expected Return

Thread starter #1
I am having trouble understanding Slide 34 in Chapter 13 (Elton).

The question is: What is the expected return on an asset with a Beta of 2.0?

I understand you use the CAPM formula but I do not understand how you get the answer.

12% = Rf + 1.5(Rm-Rf) - I understand
6% = Rf + .5(Rm-Rf) - I understand
I do not understand that from here, you get to 6%=(Rm-Rf) -> rF = 3%

So to get 6, you do 12-6, but how does that get you to the rF rate?
 

David Harper CFA FRM

David Harper CFA FRM
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#2
Hi @omar72787 Given (Rm-Rf) = 6%, we can plug that back into 6% = Rf + .5*(Rm-Rf):
Rf + .5*(Rm-Rf), but we know Rm-Rf = 6% so that
6% = Rf + .5*6% -->
6% = Rf + 3% -->
Rf = 6% - 3%.

This is solving for two variables with two equations, so there is more than one way to approach it. For example, if we wanted to solve for Rf first (instead of the equity risk premium first), we could do this be seeking to eliminate (Rm_Rf) in a subtraction. Noticing that the second .5(Rm-Rf) is one-third the first 1.5(Rm-Rf), would could triple it as follows:
  • 12% = Rf + 1.5(Rm-Rf)
  • [6% = Rf + .5(Rm-Rf)]*3; i.e., multiply both sides by three with the goal of equalizing the (Rm-Rf).
which gives us:
  • 12% = Rf + 1.5(Rm-Rf)
  • 18% = 3*Rf + 1.5(Rm-Rf)
swap them so the second is first and vice versa
  • 18% = 3*Rf + 1.5(Rm-Rf)
  • 12% = Rf + 1.5(Rm-Rf)
and then subtract the second from the first:

  • 18% = 3*Rf + 1.5(Rm-Rf)
  • -[12% = Rf + 1.5(Rm-Rf)]
  • 6% = 2*Rf; so now we can see Rf = 3%, which then can be used to solve (Rm-Rf). I hope that help!
 
Thread starter #3
Thank you for the adequate response. I think once you tell me how I know that (Rm-Rf) = 6%, I'll be able to figure out this simple problem. We don't know anything about the risk-free rate except that it is an unknown. The only values that are given are the expected returns with their associated beta's.
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#4
Hi @omar72787 Solving for (Rm-Rf) is easier than solving for Rf (which I did above). We only need to subtract the second equation from the first:

12% = Rf + 1.5(Rm-Rf)
-[6% = Rf + .5(Rm-Rf)]
=6% = 0 + (Rm -Rf)

This is just:
X = Rf + 1.5P, where P = ERP
-[Y = Rf + 0.5P]
=(X-Y) = (Rf-Rf) + (1.5P - 0.5P); notice the Rf cancels out and the difference in P is conveniently 1.0
= (X-Y) = 0 + 1.0P. I hope that helps!
 
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