You own a portfolio equally invested in a risk-free asset and two stocks. If one of the stocks 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 stock in your portfolio?

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BlackTom AI · Accepted Answer

We start by parsing the setup: the portfolio is equally invested in a risk-free asset and two stocks, so each component has equal weight in the overall portfolio.
Since the risk-free asset has a beta of 0, its contribution to the portfolio beta is zero regardless of its weight, but because we’re told the portfolio is equally risky as the market, the overall portfolio beta must equal 1.
With three equal-weighted components (risk-free asset, Stock A, Stock B), the portfolio beta is the average of the betas of the two stocks (the risk-free asset contributes 0). Mathematically: Beta_portfolio = (beta_A + beta_B) / 3.
We know beta_A = 1.04, and Beta_portfolio must be 1. Therefore (1.04 + beta_B) / 3 = 1. Solving gives 1.04 + beta_B = 3, so beta_B = 3 - 1.04 = 1.96.
Now evaluate the provided options to see which one matches this required beta_B:
- If beta_B = 1.37, then (1.04 + 1.37) / 3 = 2.41 / 3 ≈ 0.803, which is below 1, so this option cannot yield a portfolio with market-like risk.
- If beta_B = 1.54, then (1.04 + 1.54) / 3 = 2.58 / 3 ≈ 0.86, still below 1.
- If beta_B = 1.96, then (1.04 + 1.96) / 3 = 3.00 / 3 = 1.0, which matches the market beta exactly.
- If beta_B = 2.30, then (1.04 + 2.30) / 3 = 3.34 / 3 ≈ 1.113, which is above 1.
- If beta_B = 2.97, then (1.04 + 2.97) / 3 = 4.01 / 3 ≈ 1.337, well above 1.
Thus, only the beta of 1.96 yields a portfolio beta equal to 1, aligning the portfolio’s overall risk with the market.

FNBSLW 344-23 Chapter 11 Practice Quiz (not graded)

You own a portfolio equally invested in a risk-free asset and two stocks. If one of the stocks 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 stock in your portfolio?

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