Understanding the Question

This question tests your understanding of multi-factor models for asset returns. The 2-factor model decomposes the actual return of an asset into its expected return plus surprises from systematic factors and firm-specific events.

The 2-Factor Model Formula

The actual return using a multi-factor model is calculated as:

$R = E(R) + \beta_1 \times (F_1 - E(F_1)) + \beta_2 \times (F_2 - E(F_2)) + \epsilon$

Where:

• E(R) = Initial expected return
• β₁, β₂ = Factor betas (sensitivities)
• F₁, F₂ = Actual factor values
• E(F₁), E(F₂) = Expected factor values (consensus forecasts)
• ε = Firm-specific (idiosyncratic) return

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Step-by-Step Calculation

Given Data:

Step 1: Calculate Factor Surprises

GDP Surprise: $F_1 - E(F_1) = 5\% - 6\% = -1\%$

GDP grew 1% less than expected (negative surprise)

Interest Rate Surprise: $F_2 - E(F_2) = 4\% - 3\% = +1\%$

Interest rates rose 1% more than expected (positive surprise)

Step 2: Calculate Factor Contributions

GDP Contribution: $\beta_1 \times (F_1 - E(F_1)) = 1.5 \times (-1\%) = -1.5\%$

Since BBC has a positive GDP beta (1.5), lower-than-expected GDP growth reduces returns

Interest Rate Contribution: $\beta_2 \times (F_2 - E(F_2)) = -1.00 \times (+1\%) = -1\%$

Since BBC has a negative interest rate beta (-1.00), higher-than-expected rates reduce returns

Step 3: Calculate Total Updated Return

$R = E(R) + \text{GDP Contribution} + \text{Interest Rate Contribution} + \epsilon$

$R = 10\% + (-1.5\%) + (-1\%) + (-2\%)$

$R = 10\% - 1.5\% - 1\% - 2\% = \boxed{5.5\%}$

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Option Analysis

Option A: 1.5% INCORRECT

This answer likely results from making multiple calculation errors:

• Possibly adding all negative contributions incorrectly: -1.5% + (-1%) + (-2%) = -4.5%, then 10% - 8.5% = 1.5%
• Or incorrectly treating the interest rate contribution as positive (+1% instead of -1%)

This demonstrates a fundamental misunderstanding of how negative betas work with factor surprises.

Option B: 3.5% INCORRECT

This answer might come from:

• Miscalculating: 10% - 1% - 1% - 2% = 6% (using wrong GDP beta of 1 instead of 1.5), then further errors
• Or forgetting to include one of the factor contributions
• Possibly: 10% - 1.5% - 1.5% - 3.5% = 3.5% (arithmetic error)

This suggests incomplete application of the factor model formula.

Option C: 5.5% CORRECT

This is the correct answer derived from the proper application of the 2-factor model:

Option D: 6.5% INCORRECT

This answer likely results from:

• Omitting the firm-specific return: 10% - 1.5% - 1% = 7.5% (close but not exact)
• Or: 10% - 1% - 2% - 0.5% = 6.5% (errors in beta calculations)
• Possibly forgetting to account for the negative interest rate beta

This indicates partial understanding but missing key components of the model.

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Key Concepts to Remember

1. Factor Surprises Matter, Not Factor Levels: The model uses unexpected changes (surprises), not the absolute levels of factors.

2. Beta Determines Direction of Impact:

   • Positive beta + negative surprise → negative impact on returns
   • Negative beta + positive surprise → negative impact on returns

3. Firm-Specific Returns Are Idiosyncratic: These are company-specific events not captured by systematic factors.

4. All Components Must Be Included: Initial expected return, all factor contributions, AND firm-specific return.

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Reference Answer

Answer: C (5.5%)

The updated expected return for BBC is calculated as: $R = 10\% + 1.5 \times (-1\%) + (-1.0) \times (+1\%) + (-2\%) = 5.5\%$