The index model for stock B has been estimated with the following result: RB,t = 0.01 + 1.1RM,t + eB,t. If σM = 0.20 and R2B= 0.50, the standard deviation of the return on stock B is

Question

BlackTom AI · Accepted Answer

Systematic variance = β^2 σ_M^2, so start by computing the systematic component of stock B’s variance.

1) Relevant concepts
- Index (single-factor) model: RB = α + βRM + e. Total variance of RB, σ_B^2 = β^2 σ_M^2 + σ_e^2 (systematic + residual).
- R^2 for the regression = (systematic variance) / (total variance). Thus R^2 = (β^2 σ_M^2) / σ_B^2.
- Standard deviation σ_B is the square root of total variance σ_B^2.

2) Step-by-step reasoning and calculations
- Given β = 1.1 and σ_M = 0.20.
- Compute systematic variance: β^2 σ_M^2 = (1.1)^2 × (0.20)^2.
  - (1.1)^2 = 1.21
  - (0.20)^2 = 0.04
  - Systematic variance = 1.21 × 0.04 = 0.0484
- Given R^2 = 0.50, use R^2 = systematic variance / total variance:
  - 0.50 = 0.0484 / σ_B^2
  - Solve for total variance: σ_B^2 = 0.0484 / 0.50 = 0.0968
- Take square root to get standard deviation:
  - σ_B = sqrt(0.0968) ≈ 0.3111

3) Intermediate numerical steps were shown above (β^2, σ_M^2, multiplication, division by R^2, square root).

4) Why this answer is correct
- The R^2 definition directly links the known systematic variance to the unknown total variance; solving that equation yields σ_B^2 = 0.0968, and the standard deviation is its square root ≈ 0.3111. This is the correct transformation from variance to standard deviation.

5) Why the other choices are incorrect
- 0.0968 is the total variance, not the standard deviation. 
- 0.2111 is close to βσ_M (1.1×0.2 = 0.22) but that is the systematic standard deviation, not the total standard deviation when R^2 ≠ 1.
- 0.4111 and 0.0484 are not consistent with the variance decomposition: 0.0484 is the systematic variance (not a standard deviation), and 0.4111 is too large relative to the computed total variance.

Conclusion: The correct standard deviation of stock B’s return is 0.3111, which matches the provided answer.

The index model for stock B has been estimated with the following result: RB,t = 0.01 + 1.1RM,t + eB,t. If σM = 0.20 and R2B= 0.50, the standard deviation of the return on stock B is单项选择题

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