Answer: 5.5%

## 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 --- ## Step-by-Step Calculation ### Given Data: | Parameter | Value | |-----------|-------| | Initial Expected Return, E(R) | 10% | | GDP Consensus Forecast | 6% | | Interest Rate Consensus Forecast | 3% | | GDP Factor Beta (β₁) | 1.5 | | Interest Rate Factor Beta (β₂) | -1.00 | | Actual GDP Growth | 5% | | Actual Interest Rate | 4% | | Firm-Specific Return (ε) | -2% | ### 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 $$R = E(R) + \text{GDP Contribution} + \text{Interest Rate Contribution} + \epsilon$$ $$R = 10\% + (-1.5\%) + (-1\%) + (-2\%)$$ $$R = 10\% - 1.5\% - 1\% - 2\% = \boxed{5.5\%}$$ --- ## 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: | Component | Calculation | Result | |-----------|-------------|--------| | Initial Expected Return | E(R) | +10.0% | | GDP Surprise Effect | 1.5 × (5% - 6%) | -1.5% | | Interest Rate Surprise Effect | -1.0 × (4% - 3%) | -1.0% | | Firm-Specific Return | ε | -2.0% | | **Total Return** | | **5.5%** | ### 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. --- ## 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. --- ## Reference Answer **Answer: C (5.5%)** The updated expected return for BBC is calculated as: $$R = 10\% + 1.5 \times (-1\%) + (-1.0) \times (+1\%) + (-2\%) = 5.5\%$$