A stock has a beta of 1.56 and an expected return of 17.3 percent. A risk-free asset currently earns 5.1 percent. If a portfolio of the two assets has a beta of 1.06, what are the portfolio weights?

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BlackTom AI · Accepted Answer

To determine the portfolio weights, we use the portfolio beta formula: Beta_p = w_stock * Beta_stock + w_rf * Beta_rf.
Since the risk-free asset has a beta of 0, Beta_p = w_stock * Beta_stock.
Given Beta_p = 1.06 and Beta_stock = 1.56, we solve for w_stock: w_stock = 1.06 / 1.56 ≈ 0.6795.
The remaining weight goes to the risk-free asset, so w_rf = 1 - w_stock ≈ 1 - 0.6795 = 0.3205.
Now evaluate the options:
Option 1: Stock weight 0.28; risk-free 0.72. This would yield a portfolio beta of 0.28 * 1.56 ≈ 0.4368, far from 1.06, so it’s incorrect.
Option 2: Stock weight 0.032; risk-free 0.68. This gives Beta_p ≈ 0.032 * 1.56 ≈ 0.0499, which is far too low; clearly incorrect.
Option 3: Stock weight 0.44; risk-free 0.56. Beta_p ≈ 0.44 * 1.56 ≈ 0.6864, still short of 1.06; not correct.
Option 4: Stock weight 0.68; risk-free 0.32. Beta_p ≈ 0.68 * 1.56 ≈ 1.0608, which matches the given portfolio beta very closely; this aligns with the calculation and rounding, so this is the correct construction.
Option 5: Stock weight 0.72; risk-free 0.28. Beta_p ≈ 0.72 * 1.56 ≈ 1.1232, which is higher than 1.06, so this is not correct.
In summary, only the fourth option yields a portfolio beta that closely equals the target 1.06 when using the given stock beta and the zero beta of the risk-free asset, and the weights sum to 1 as required.

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

A stock has a beta of 1.56 and an expected return of 17.3 percent. A risk-free asset currently earns 5.1 percent. If a portfolio of the two assets has a beta of 1.06, what are the portfolio weights?

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