Finance Class Homework Question - help please?!?

Question

Ok, the question reads:

You have been managing a $5 million portfolio that has a beta of 1.25 and a required rate of return of 12%. The current risk-free rate is 5.25%. Assume that you have another $500000. If you invest the money in a stock with a beta of .75, what will be the required return on your $5.5 million portfolio?

I came up with the new beta for the portfolio as:

b = 1.25+(500000/5500000)*.75 = 1.32

and

to get the required rate of return on the market (Km) I got:

12=5.25+1.25(Km-5.25)

Km=10.65

Kp (required rate of return for portfolio) = 5.25+1.32(10.65 - 5.25) = 12.378

However, my professor's answer is Kp=11.75....anyone know where I went wrong??

Thanks!!!!

Answer 1

car,

I worked through your problem and here's my assessment of your answer.  First, it looks like you calculated Km correctly using the CAPM formula.

The point that you went wrong was in calculating the beta of the new $5.5M portfolio.

Ignoring the calculations for a moment and thinking about what you're doing with your portfolio: you begin with a portfolio of $5.0M with a beta 1.25.  You're then adding in an additional $500K of securities that have a lower beta than your original portfolio.  Effectively, what you're doing is "diluting" or diminishing your original portfolio's beta.  As a consequence, your new $5.5M portfolio should have a LOWER beta than the original.  In your calculation, you're showing a beta of 1.32, which is actually a slight increase in the portfolio's beta.  This should be a tip-off that the calculation isn't agreeing with the intuition I just described.

The beta for the new $5.5M portfolio will be a weighted average of the beta for the old $5.0M portion and the newer $500K portion:

b = (1.25)*(5,000,000/5,500,000) + (0.75)*(500,000/5,500,000)

   = 0.068 + 1.136

   = 1.204

Using a beta of 1.204 will get you your professor's answer of 11.75:

Kp = 5.25 + (1.204)*(10.65 - 5.25) = 11.75

I hope this helps.