You want your portfolio beta to be 1.10. Currently, your portfolio consists of $3,000 invested in stock A with a beta of 1.65 and $2,000 in stock B with a beta of.72. You have another $5,000 to invest and want to divide it between an asset with a beta of 1.48 and a risk-free asset. How much should you invest in the risk-free asset?
A) $0
B) $775
C) $1,885
D) $3,115
E) $5,000

Question

Quizplus · Accepted Answer

To achieve a portfolio beta of 1.10, we first calculate the weighted average beta of the current portfolio and then determine the required investment in the risk-free asset and the asset with a beta of 1.48 to achieve the target beta. The current portfolio beta is calculated as follows:$$ \text{Current Portfolio Beta} = \left(\frac{3000}{3000+2000}\right) \times 1.65 + \left(\frac{2000}{3000+2000}\right) \times 0.72 = 1.386 $$With an additional $5,000 to invest, the total portfolio value will be $10,000. To achieve a portfolio beta of 1.10, we use the formula:$$ \text{Target Portfolio Beta} = \left(\frac{\text{Current Investment} \times \text{Current Beta} + \text{New Investment in Asset} \times \text{Beta of Asset} + \text{New Investment in Risk-Free Asset} \times 0}{\text{Total Portfolio Value}}\right) $$Given that the beta of the risk-free asset is 0, we rearrange the formula to solve for the new investments. Let $ x $ be the amount invested in the risk-free asset and $ 5000 - x $ be the amount invested in the asset with a beta of 1.48. The equation to solve is:$$ 1.10 = \frac{(3000 \times 1.65 + 2000 \times 0.72 + (5000 - x) \times 1.48)}{10000} $$Solving this equation gives us the value of $ x $, the amount to invest in the risk-free asset, which is $1,885.

You Want Your Portfolio Beta to Be 1