What is the Portfolio Beta?

Corporate Finance: If you are given a beta, how do you determine portfolio weights?

• FOR EXAMPLE: Using CAPM, a stock has a beta of 1.1 and an expected return of 15 percent. A risk-free asset currently earns 5 percent. What is the expected return on a portfolio that is equally invested in the two assets? If a portfolio of the two assets has a beta of .6, what are the portfolio weights? If a portfolio of the two assets has an expected return of 9 percent, what is its beta? If a portfolio of the two assets has a beta of 2.20, what are the portfolio weights? How do you interpret the weights for the two assets in this case?

• Answer:

  Despite sounding an awful lot like a homework question, I'll bite... Firstly, the CAPM formula and the info we have: E[R_i] = R_f + \beta_{i}(E[R_m] - R_f) \beta_{i} = 1.1, \beta_{m} = 1, \beta_{rf} = 0 E[R_i] = 0.15, E[R_m] =\ ?, R_f = 0.05 What is the expected return on a portfolio that is equally invested in the two assets? Expected return of a portfolio is simply the weighted average of the individual asset expected returns. 0.5[E(R_i)] + 0.5(R_f) = E[R_p] Substitute in values above. If a portfolio of the two assets has a beta of .6, what are the portfolio weights? Likewise, the beta of a portfolio is the weighted average of the individual asset betas. (x)(\beta_{i}) + (1-x)(\beta_{rf}) = \beta_p = 0.6 Substitute in values above, solve for x. If a portfolio of the two assets has an expected return of 9 percent, what is its beta? The CAPM equation holds for portfolios as well as individual assets. If you knew the expected return of the market you could use the original CAPM equation, substituting in E(R_i) = 0.09 and solving for \beta. Instead you'll need to take a different approach. Find the portfolio weights of the two assets that give E[R_p] = 0.09 (similar to first question). Use these weights to find the beta of the portfolio (similar to second question). If a portfolio of the two assets has a beta of 2.20, what are the portfolio weights? Use the same approach as in question 2. How do you interpret the weights for the two assets in this case? Weights will be 2 for the risky asset and -1 for the risk free. The interpretation is left as an exercise to the reader — I can't do all the work for you :)