You own a portfolio equally invested in a risk-free asset and two stocks. If one of the stocks has a beta of 1.16 and the total portfolio is exactly as risky as the market, what must the beta be for the other stock in your portfolio?
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If the portfolio is equally invested in a risk-free asset and two stocks, and the total portfolio is as risky as the market, the beta of the entire portfolio can be used to determine the beta of the other stock in the portfolio.
Given that one of the stocks has a beta of 1.16 and the portfolio is equally invested in this stock and another stock:
The formula for calculating the beta of a portfolio is a weighted average of the betas of its individual components:
Portfolio Beta = Weight of Stock 1 × Beta of Stock 1 + Weight of Stock 2 × Beta of Stock 2
Since the portfolio consists of equal investments in the risk-free asset and two stocks, the weight of each stock in the portfolio would be 0.5 (or 50%).
Given that the beta of one stock (Stock 1) is 1.16 and the portfolio has the same risk as the market (which is usually represented by a beta of 1), we can rearrange the formula to solve for the beta of the other stock (Stock 2):
1 = 0.5 × 1.16 + 0.5 × Beta of Stock 2
1 = 0.58 + 0.5 × Beta of Stock 2
Subtracting 0.58 from both sides:
0.42 = 0.5 × Beta of Stock 2
Beta of Stock 2 = 0.42 / 0.5
Beta of Stock 2=0.84
Therefore, the beta of the other stock in the portfolio must be 0.84 to make the entire portfolio as risky as the market.