You own a portfolio equally invested in a risk-free asset and two shares. If one of the shares has a beta of 1.04 and the total portfolio is equally as risky as the market, what must the beta be for the other share in your portfolio?

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BlackTom AI · Accepted Answer

We start by identifying how the portfolio beta is formed. Since the portfolio is equally invested in a risk-free asset and two shares, the weights are w = 1/3 for each asset. The risk-free asset has beta 0, and the two shares have betas b1 = 1.04 and b2 = x (the unknown). The portfolio beta is the weighted average of the individual betas: beta_p = (1/3)*0 + (1/3)*1.04 + (1/3)*x = (1/3)*(1.04 + x).

Next, we use the condition given: the total portfolio is equally as risky as the market, which means beta_p = 1. Setting (1/3)*(1.04 + x) = 1 leads to 1.04 + x = 3, so x = 3 - 1.04 = 1.96.

Now evaluate the provided options:
- Option 1.37: If x = 1.37, beta_p = (1/3)*(1.04 + 1.37) = (1/3)*2.41 ≈ 0.8033, which is clearly less than 1, so the portfolio would be less risky than the market, not equally risky.
- Option 2.30: If x = 2.30, beta_p = (1/3)*(1.04 + 2.30) = (1/3)*3.34 ≈ 1.1133, which is above 1, making the portfolio more risky than the market.
- Option 2.97: If x = 2.97, beta_p = (1/3)*(1.04 + 2.97) = (1/3)*4.01 ≈ 1.3367, still greater than 1, not equal to the market beta.
- Option 1.96: If x = 1.96, beta_p = (1/3)*(1.04 + 1.96) = (1/3)*3.00 = 1.00, exactly matching the market beta as required.
- Option 1.54: If x = 1.54, beta_p = (1/3)*(1.04 + 1.54) = (1/3)*2.58 ≈ 0.86, which is also not equal to 1.

Thus, only the value 1.96 yields a portfolio beta of 1, satisfying the condition that the portfolio is as risky as the market. The other options fail because they produce betas either below or above 1, rather than equal to 1.

BFIN011 Week 8: Practice Quiz

You own a portfolio equally invested in a risk-free asset and two shares. If one of the shares has a beta of 1.04 and the total portfolio is equally as risky as the market, what must the beta be for the other share in your portfolio?

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