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?

Question

BlackTom AI · Accepted Answer

First, lay out what we know: the portfolio consists of a stock with beta 1.56 and a risk-free asset with beta 0. The portfolio beta is a weighted average of the components' betas, so if w is the weight in the stock and (1 - w) is the weight in the risk-free asset, the portfolio beta is: Beta_P = w*(1.56) + (1 - w)*0 = 1.56w. The problem states Beta_P = 1.06, so 1.56w = 1.06. Solving for w gives w = 1.06 / 1.56 ≈ 0.6795, which rounds to about 0.68. The remaining weight goes to the risk-free asset: 1 - 0.68 ≈ 0.32.

Option 1: Stock weight = 0.032; risk-free weight = 0.68. Here the stock weight is far too small (0.032 instead of ~0.68), and the weights do not sum to 1 (0.032 + 0.68 = 0.712). Even if we compute the portfolio beta with 0.032 in stock, Beta_P ≈ 0.032*1.56 ≈ 0.0499, which is far from 1.06. This option is inconsistent.

Option 2: Stock weight = 0.44; risk-free weight = 0.56. The beta of this portfolio would be 0.44*1.56 ≈ 0.6864, which does not equal 1.06. The weights do sum to 1, but the resulting beta is incorrect, so this is not the right mix.

Option 3: Stock weight = 0.28; risk-free weight = 0.72. The resulting beta would be 0.28*1.56 ≈ 0.4368, again not matching the required 1.06. The mix is not consistent with the target beta.

Option 4: Stock weight = 0.68; risk-free weight = 0.32. This satisfies the weights summing to 1 and yields Beta_P ≈ 0.68*1.56 ≈ 1.0608, which is effectively the target 1.06 (allowing for rounding). Numerically, this is the only option that aligns with the given portfolio beta and proper weight balance.

In summary, the correct approach is to set 1.56w = 1.06, solve for w ≈ 0.68, and then allocate the remaining ~0.32 to the risk-free asset. The other options fail because they either misallocate weights (not summing to 1) or produce a portfolio beta that does not match the target 1.06.

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? 单项选择题

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