Question

You own a portfolio equally invested in a risk-free asset and two stocks (If one of the stocks has a beta of 0.97 and the total portfolio is equally as risky as the market, what must the beta be for the other stock in your portfolio? (Hint: Remember that the market has a Beta=1; also remember that equally invested means that each asset has the same weight- since there are 3 assets, each asset's weight is 1/3 or 0.3333). Enter the answer with 4 decimals (e.g. 1.1234)

Answer 1

The portfolio consists of 3 assets: risk-free asses, stock 1 and stock 2

The portfolio is an equally invested portfolio

Weights

Weight of risk-free asset in the portfolio = w_F = 1/3

Weight of Stock 1 in the portfolio = w₁ = 1/3

Weight of stock 2 in the portfolio = w₂ = 1/3

Betas

Beta of risk-free asset is zero

Beta of risk-free asset = β_F = 0

Beta of stock 1 = β₁ = 0.97

Beta of stock 2 = β₂

Beta of the total portfolio

It is given that the total portfolio is equally as risky as the market. This means that the beta of the total portfolio is equal to 1

β_P = 1, we need to calculate beta of the stock 2

Beta of the total portfolio is calculated using the formula:

β_P = w_F*β_F + w₁*β₁ + w₂*β₂

where, w_F = 1/3, w₁ = 1/3, w₂ = 1/3, β_P = 1, β_F = 0, β₁ = 0.97

1 = (1/3)*0 + (1/3)*0.97 + (1/3)*β₂

(1/3)*0.97 + (1/3)*β₂ = 1

0.97 + β₂ = 3

β₂ = 3 - 0.97 = 2.03

Answer -> Beta of other stock = 2.03