Consider the single-index model. The alpha of a stock is 2%. The return on the market index is 10% and the risk-free rate of return is 3%. The stock earns a return 14%, and there are no firm-specific events affecting the stock performance. The β of the stock is

Question

BlackTom AI · Accepted Answer

Jensen’s alpha equals the stock’s actual excess return over its CAPM-predicted return, so alpha = R_i − [R_f + β(R_m − R_f)].

1) Relevant knowledge points and concepts
- CAPM predicted return: R_expected = R_f + β(R_m − R_f).
- Jensen’s alpha (α) is defined as α = R_actual − R_expected.
- Rearranging gives β = (R_actual − R_f − α) / (R_m − R_f).
- The statement “no firm-specific events” implies the observed alpha is the systematic abnormal return captured by the model, so the relationship above applies directly.

2) Step-by-step reasoning and calculations
Given:
- α = 2% (0.02)
- R_m = 10% (0.10)
- R_f = 3% (0.03)
- R_actual = 14% (0.14)

Start from the rearranged formula for β:
β = (R_actual − R_f − α) / (R_m − R_f).

Compute numerator:
R_actual − R_f − α = 0.14 − 0.03 − 0.02 = 0.09 (9%).

Compute denominator:
R_m − R_f = 0.10 − 0.03 = 0.07 (7%).

Compute β:
β = 0.09 / 0.07 = 1.285714... ≈ 1.29.

3) Why the correct answer is correct
This β makes the CAPM-predicted return R_f + β(R_m − R_f) = 0.03 + 1.2857×0.07 = 0.12 (12%). Adding the given alpha of 2% yields the observed 14% return, matching the data exactly. Thus β ≈ 1.29 is the value consistent with the definitions.

4) Why other options are incorrect (briefly)
- 0.64 or 0.75: produce much lower expected returns (≈7.48% and 8.25%), so adding α=2% gives actual ≈9.48% and 10.25%, not 14%.
- 1.17: gives expected ≈11.19%, so with α=2% actual ≈13.19%, not 14%.
- 1.50: gives expected ≈13.50%, so with α=2% actual ≈15.50%, not 14%.

Conclusion: The calculated β ≈ 1.29, which matches the provided correct answer 1.29.

Consider the single-index model. The alpha of a stock is 2%. The return on the market index is 10% and the risk-free rate of return is 3%. The stock earns a return 14%, and there are no firm-specific events affecting the stock performance. The β of the stock is单项选择题

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