Consider the single-index model. The alpha of a stock is 3.75%. The return on the market index is 11%. The risk-free rate of return is 2.5%. During a sample period, the stock earns a return that exceeds the risk-free rate by 13%, and there is no firm-specific residual affecting the stocks return (ie e = 0). The β of the stock is

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Consider the single-index model. The alpha of a stock is 3.75%. The return on the market index is 11%. The risk-free rate of return is 2.5%. During a sample period, the stock earns a return that exceeds the risk-free rate by 13%, and there is no firm-specific residual affecting the stocks return (ie e = 0). The β of the stock is

BlackTom AI · Accepted Answer

First, restate what we know from the problem to ensure clarity: we are in a single-index model where the stock's return can be expressed in excess terms as r_i − rf = α_i + β_i (r_m − rf) + e_i. Given data: α_i = 3.75% (0.0375), r_m = 11% (0.11), rf = 2.5% (0.025), and the realized excess return r_i − rf = 13% (0.13) with e_i = 0. So we should use the excess-return form to solve for β.

Option-by-option analysis:

Option a: None of the options is correct. This would be true only if none of the numerical values matches the computed β. Since we will compute a β that does appear among the options (1.088), this option will be incorrect, but we must show the reasoning that leads to the numeric result before deeming correctness.

Option b: 1.143. If we plugged β = 1.143 into the excess-return equation r_i − rf = α + β(r_m − rf), we would have the right-hand side = 0.0375 + 1.143 × (0.11 − 0.025) = 0.0375 + 1.143 × 0.085 ≈ 0.0375 + 0.097155 ≈ 0.134655, which would imply r_i − rf ≈ 13.47%, not 13%. Therefore this β does not satisfy the given data exactly.

Option c: 1.538. Using β = 1.538, the excess-return equation gives 0.0375 + 1.538 × 0.085 ≈ 0.0375 + 0.13073 ≈ 0.16823, i.e., r_i − rf ≈ 16.82%, which is far from the specified 13%. This is clearly inconsistent with the provided excess return.

Option d: 0.692. With β = 0.692, the excess term yields 0.0375 + 0.692 × 0.085 ≈ 0.0375 + 0.05882 ≈ 0.09632, so r_i − rf ≈ 9.63%, again not matching 13%. This occurs due to β being too small to generate the observed excess return.

Option e: 1.088. Here, compute β from the equation: 0.13 = 0.0375 + β × 0.085. Subtract 0.0375 from both sides to get 0.0925 = β × 0.085. Divide: β = 0.0925 / 0.085 ≈ 1.088235..., which matches 1.088 when rounded to three decimals. This exactly satisfies the given excess-return condition with e = 0, so this β is consistent with all the provided information.

Summary of reasoning: we must work with excess returns r_i − rf, not raw returns, in the single-index model formula as specified. Solving 0.13 = 0.0375 + β(0.11 − 0.025) leads to β = (0.13 − 0.0375) / 0.085 = 0.0925 / 0.085 ≈ 1.088, which aligns with option e. The other options fail because they produce excess returns that are either too high or too low when plugged back into the excess-return equation, demonstrating their inconsistency with the given data.

FINS2624-Portfolio Mgmt - T1 2025

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