You own a portfolio equally invested in a risk-free asset and two stocks. One of the stocks has a beta of 1.31 and the total portfolio is equally 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 equally as risky as the market, it means the portfolio's beta equals the market beta (which is 1, since the market's beta is always considered 1).
Given that one stock has a beta of 1.31, and the portfolio is equally as risky as the market, we can use the beta of the risk-free asset and the two stocks to calculate the beta of the other stock.
Let the beta of the other stock be β.
The portfolio consists of:
Risk-free asset (beta = 0),
Stock with beta = 1.31, and
The other stock with an unknown beta = β.
The portfolio beta is a weighted average of the betas of its components:
Portfolio beta = Weight of asset 1 × Beta of asset 1 + Weight of asset 2 × Beta of asset 2 +
Weight of asset 3×Beta of asset 3
Since the portfolio is equally invested:
Portfolio beta = 1/3 × 0 + 1/3 × 1.31 + 1/3 × β.
Given that the portfolio beta equals the market beta (which is 1):
1 = 1/3 × 1.31 + 1/3 × β.
Solving for β:
1 = 0.437 + 1/3 × β
2/3 = 1/3 × β
β = 2
Therefore, the beta for the other stock in the portfolio must be β=2.