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1. Given the market model $R_{i t}=\alpha_{i}+\beta_{i} R_{M t}+\varepsilon_{i ...
Aug 4, 2024
Solution
a
Given the market model Rit=αi+βiRMt+εitR_{i t}=\alpha_{i}+\beta_{i} R_{M t}+\varepsilon_{i t}, where Rit=log⁡(PitPit−1)R_{i t}=\log \left(\frac{P_{i t}}{P_{i t-1}}\right), PitP_{i t} represents the price of the ii-th stock at time tt, for t=1,…,Tt=1, \ldots, T, RMtR_{M t} represents the logarithm of time tt returns to the market, and εit\varepsilon_{i t} represents the errors also at time tt
b
Suppose αi=βi=0\alpha_{i}=\beta_{i}=0 in the market model for i=1,…,Ni=1, \ldots, N. This simplifies the model to Rit=εitR_{i t}=\varepsilon_{i t}
c
Cumulative abnormal returns (CAR) over the time period are calculated as the sum of the abnormal returns, which in this case are the error terms εit\varepsilon_{i t}
d
Therefore, the cumulative abnormal returns represent the sum of the error terms over the time period
Answer
(a) Cumulative abnormal returns represent the sum of the squared error terms over the time period
Key Concept
Cumulative Abnormal Returns (CAR)
Explanation
In the given market model, with αi=βi=0\alpha_{i}=\beta_{i}=0, the returns RitR_{i t} are solely represented by the error terms εit\varepsilon_{i t}. Therefore, the cumulative abnormal returns over the time period are the sum of these error terms.
Solution
a
Event Study Analysis: When the error term εt\varepsilon_{t} in the market model Rt=α+βRMt+εtR_{t}=\alpha+\beta R_{M t}+\varepsilon_{t} is not normally distributed, event study analysis must appeal to large sample theory. This is done by averaging over a large number of common events rather than relying on a single event
Answer
(b) It appeals to large sample theory by averaging over a large number of common events rather than use only one event
Key Concept
Large Sample Theory
Explanation
When the error term in the market model is not normally distributed, event study analysis uses large sample theory to ensure the robustness of its statistical analysis. This involves averaging over many events to mitigate the impact of non-normality.
Solution
a
Definition of CAPM: The Capital Asset Pricing Model (CAPM) is a financial theory that describes the relationship between systematic risk and expected return for assets, particularly stocks
b
Options Analysis: - (a) The Capital Pricing Model: This is a common misnomer; the correct term is the Capital Asset Pricing Model. - (b) The Arbitrage Pricing Model: This is a different model used in financial theory. - (c) The Global Minimum Variance portfolio: This is a portfolio with the lowest possible risk for a given return. - (d) All of (a) to (c) are correct: This is incorrect as only (a) is related to CAPM. - (e) None of (a) to (c) are correct: This is incorrect as (a) is related to CAPM
Answer
(a) The Capital Pricing Model
Key Concept
Capital Asset Pricing Model (CAPM)
Explanation
The CAPM is a model that describes the relationship between the expected return of an asset and its risk, measured by beta. It is used to determine a theoretically appropriate required rate of return of an asset, considering the asset's non-diversifiable risk.
Solution
a
Definition of MLE: Maximum Likelihood Estimation (MLE) is a method used to estimate the parameters of a statistical model. It finds the parameter values that maximize the likelihood function, given the observed data
b
Consistency of MLE: Even if the normality assumption of stock returns is not valid, MLE can still produce consistent estimates of the coefficients as long as the model is correctly specified. Consistency means that as the sample size increases, the estimates converge to the true parameter values
c
Normality Assumption: The assumption that stock returns are normally distributed is often used for simplicity, but empirical evidence suggests that stock returns may not follow a normal distribution. Despite this, MLE can still be effective if the model is correctly specified
Answer
(b) If the normality assumption of stock returns is not valid, then MLE will still produce consistent estimates of the coefficients as long as the model is correctly specified.
Key Concept
Consistency of MLE
Explanation
MLE can still produce consistent estimates even if the normality assumption is not valid, provided the model is correctly specified. This means that the estimates will converge to the true parameter values as the sample size increases.
B
Key Concept
Black's modification to the market model
Explanation
Black (1972) set α=0\alpha=0 in the market model, simplifying the equation to Rt=βRMt+εtR_{t}=\beta R_{M t}+\varepsilon_{t}. This adjustment implies that the expected return on the asset is solely dependent on the market return and the asset's sensitivity to the market (beta), without any additional constant term.
Solution
a
Market Model: The market model is given by Rit=αi+βiRMt+εitR_{i t}=\alpha_{i}+\beta_{i} R_{M t}+\varepsilon_{i t} where Rit=log⁡(PitPit−1)R_{i t}=\log \left(\frac{P_{i t}}{P_{i t-1}}\right), PitP_{i t} represents the price of the iith stock at time tt, RMtR_{M t} represents the logarithm of time tt returns to the market, and εit\varepsilon_{i t} represents the errors at time tt
b
Assumptions: Given αi=0\alpha_{i}=0 and βi=0\beta_{i}=0 for i=1,…,Ni=1, \ldots, N, the market model simplifies to Rit=εitR_{i t}=\varepsilon_{i t}
c
Cumulative Abnormal Returns: Cumulative abnormal returns (CAR) over a time period are the sum of the abnormal returns over that period. Since Rit=εitR_{i t}=\varepsilon_{i t}, the CAR is the sum of the error terms ∑t=1Tεit\sum_{t=1}^{T} \varepsilon_{i t}
d
Interpretation: - Option (a): CAR represents the sum of the squared error terms over the time period. This is incorrect because CAR is the sum of the error terms, not their squares. - Option (b): CAR represents the logarithm of the price increase over the time period. This is incorrect because CAR is the sum of the error terms, not the logarithm of the price increase. - Option (c): CAR are normally distributed. This is incorrect because the distribution of CAR depends on the distribution of the error terms εit\varepsilon_{i t}. - Option (d): All of (a) to (c) are correct. This is incorrect as none of the individual options are correct. - Option (e): None of (a) to (c) are correct. This is correct as none of the individual options accurately describe CAR
Answer
(e) None of (a) to (c) are correct
Key Concept
Cumulative Abnormal Returns (CAR)
Explanation
CAR is the sum of the abnormal returns over a time period, which in this case simplifies to the sum of the error terms ∑t=1Tεit\sum_{t=1}^{T} \varepsilon_{i t}. None of the given options (a) to (c) correctly describe this.
Solution
a
Solnik's Measure of Diversification: Solnik's measure of diversification is defined as the variance of randomly selected portfolios consisting of a large number of assets with weights corresponding to its market value
Answer
(c) It is the variance of randomly selected portfolios consisting of a large number of assets with weights corresponding to its market value.
Key Concept
Solnik's Measure of Diversification
Explanation
Solnik's measure of diversification focuses on the variance of portfolios that are randomly selected and consist of a large number of assets. The weights of these assets correspond to their market value, which helps in understanding the level of diversification achieved in a portfolio.
Solution
a
Aggregation of Returns: When aggregating weekly returns to monthly returns by summing them, the skewness of the distribution of returns is affected
b
Skewness Formula: The skewness of a distribution is given by E[(r−E[r])3]σ3/2\frac{\mathbb{E}\left[(r - \mathbb{E}[r])^3\right]}{\sigma^{3/2}}
c
Effect on Skewness: Aggregating returns over time tends to reduce the skewness of the distribution. This is because the aggregation process averages out extreme values, leading to a more symmetric distribution
d
Monthly Skewness: Given that the skewness of weekly returns is reduced when aggregated to monthly returns, the skewness of monthly returns will not be the same as the skewness of weekly returns
e
Conclusion: Therefore, the correct answer is (e) None of (a) to (c) are correct
Answer
(e) None of (a) to (c) are correct
Key Concept
Aggregation of returns affects skewness
Explanation
When weekly returns are aggregated to monthly returns, the skewness of the distribution tends to decrease, leading to a more symmetric distribution. Therefore, none of the provided options (a) to (c) correctly describe the skewness of monthly returns.
Solution
a
Aggregation of Returns: When aggregating weekly returns to monthly returns, the monthly return RMonthly,tR_{\text{Monthly}, t} is the sum of the four weekly returns. Mathematically, this can be expressed as: RMonthly,t=∑i=14RWeekly,ti R_{\text{Monthly}, t} = \sum_{i=1}^{4} R_{\text{Weekly}, t_i}
b
Market Model for Weekly Returns: The given market model for weekly returns is: RWeekly,t=α+βRMarket,t+εt R_{\text{Weekly}, t} = \alpha + \beta R_{\text{Market}, t} + \varepsilon_t where εt\varepsilon_t has a mean of zero and a positive and finite variance σ2\sigma^2
c
Market Model for Monthly Returns: By summing the weekly returns, the monthly return model can be derived. Since the errors εt\varepsilon_t are independent and identically distributed, the monthly return model will be: RMonthly,t=4α+β∑i=14RMarket,ti+∑i=14εti R_{\text{Monthly}, t} = 4\alpha + \beta \sum_{i=1}^{4} R_{\text{Market}, t_i} + \sum_{i=1}^{4} \varepsilon_{t_i} The variance of the error term for the monthly model will be the sum of the variances of the weekly errors: Var(∑i=14εti)=4σ2 \text{Var}(\sum_{i=1}^{4} \varepsilon_{t_i}) = 4\sigma^2 Thus, the variance of the error term in the monthly model is four times the variance of the error term in the weekly model
d
Alpha in Monthly Model: The alpha term in the monthly model is the sum of the four weekly alphas: 4α 4\alpha Therefore, the alpha in the monthly model is four times the alpha in the weekly model
Answer
The correct option is (c): The variance of the error term of the market model that applies to monthly returns is four times larger than the variance of the error in the market model that corresponds to weekly returns and the alpha that corresponds to the market model of monthly return is four times larger than alpha in the market model that corresponds to weekly returns.
Key Concept
Aggregation of returns and its impact on the market model
Explanation
When aggregating weekly returns to monthly returns, the monthly return model retains the structure of the weekly return model, but the variance of the error term and the alpha value are scaled by the number of weeks in a month. Specifically, the variance of the error term becomes four times larger, and the alpha value becomes four times larger.
Solution
a
Ordinary Least Squares (OLS): OLS is a method for estimating the unknown parameters in a linear regression model. It minimizes the sum of the squared differences between the observed and predicted values. In the context of the market model Rit=αit+βiRmt+εitR_{i t}=\alpha_{i t}+\beta_{i} R_{m t}+\varepsilon_{i t}, OLS can be used to estimate αi\alpha_{i} and βi\beta_{i} consistently if the error terms εit\varepsilon_{i t} are uncorrelated and homoscedastic
b
Maximum Likelihood Estimates (MLE): MLE is a method of estimating the parameters of a statistical model by maximizing the likelihood function. For the market model, if the error terms εit\varepsilon_{i t} are normally distributed, the MLE of the coefficients will be algebraically equivalent to the OLS estimates
c
Generalized Least Squares (GLS): GLS is a technique used to estimate the parameters of a linear regression model when there is heteroscedasticity or autocorrelation in the error terms. In the market model, if the error terms εit\varepsilon_{i t} are heteroscedastic or autocorrelated, GLS will produce consistent estimates of the coefficients
Answer
(d) All of (a) to (c) are correct
Key Concept
Estimation Methods for Market Model Coefficients
Explanation
In the context of the market model used in event study analysis, Ordinary Least Squares (OLS), Maximum Likelihood Estimates (MLE), and Generalized Least Squares (GLS) can all produce consistent estimates of the coefficients αi\alpha_{i} and βi\beta_{i} under different conditions. OLS is consistent if the error terms are uncorrelated and homoscedastic, MLE is equivalent to OLS if the errors are normally distributed, and GLS is consistent if there is heteroscedasticity or autocorrelation in the error terms.
Solution
a
Aggregating Weekly Returns to Monthly Returns: When aggregating weekly stock returns to monthly returns, the skewness of the monthly returns can be influenced by the distribution of the weekly returns. If the weekly returns are normally distributed, the monthly returns will also tend to be normally distributed due to the Central Limit Theorem. However, if the weekly returns are skewed, the monthly returns may also exhibit skewness
b
Market Model for Monthly Returns: The market model for monthly returns when weekly returns are aggregated can be expressed as Rit=αi+βiRmt+εitR_{i t} = \alpha_i + \beta_i R_{m t} + \varepsilon_{i t}, where RitR_{i t} is the return on the asset, RmtR_{m t} is the return on the market portfolio, and εit\varepsilon_{i t} is the error term. This model assumes that the relationship between the asset returns and the market returns remains consistent over the aggregation period
c
Test Statistic for Black (1972) CAPM: To test the validity of the Black (1972) version of the Capital Asset Pricing Model, the appropriate test statistic is given by option (a): T−N−K2N(1+μ^KTΩ^K−1μ^K)−1α^TΩε−1α^\frac{T-N-K_{2}}{N}\left(1+\widehat{\mu}_{K}^{T} \widehat{\Omega}_{K}^{-1} \widehat{\mu}_{K}\right)^{-1} \widehat{\alpha}^{T} \Omega_{\varepsilon}^{-1} \widehat{\alpha} where μ^K=T−1∑t=1TZKt\widehat{\mu}_{K}=T^{-1} \sum_{t=1}^{T} Z_{K t}, Ω^K=T−1∑t=1T(ZKt−μ^K)(ZKt−μ^K)T\widehat{\Omega}_{K}=T^{-1} \sum_{t=1}^{T}\left(Z_{K t}-\widehat{\mu}_{K}\right)\left(Z_{K t}-\widehat{\mu}_{K}\right)^{T}, and Ω^ε=T−1∑t=1Tε^tε^tT\widehat{\Omega}_{\varepsilon}=T^{-1} \sum_{t=1}^{T} \widehat{\varepsilon}_{t} \widehat{\varepsilon}_{t}^{T}. This statistic tests the null hypothesis that the intercepts αi\alpha_i are jointly zero, which is a key implication of the CAPM
Answer
(a) T−N−K2N(1+μ^KTΩ^K−1μ^K)−1α^TΩε−1α^\frac{T-N-K_{2}}{N}\left(1+\widehat{\mu}_{K}^{T} \widehat{\Omega}_{K}^{-1} \widehat{\mu}_{K}\right)^{-1} \widehat{\alpha}^{T} \Omega_{\varepsilon}^{-1} \widehat{\alpha}
Key Concept
Test Statistic for CAPM
Explanation
The test statistic provided in option (a) is used to test the validity of the Black (1972) version of the Capital Asset Pricing Model by examining whether the intercepts αi\alpha_i are jointly zero.
Solution
a
Adjusting for Risk-Free Rate: The multifactor pricing model needs to account for the risk-free rate of return. This is typically done by subtracting the risk-free rate from the returns on traded portfolios. The adjusted model can be written as: Zt−RfiN=B1(f1t−RfiK1)+B2f2t+εt Z_{t} - R_{f} i_{N} = B_{1} (f_{1 t} - R_{f} i_{K_{1}}) + B_{2} f_{2 t} + \varepsilon_{t} where iN i_{N} and iK1 i_{K_{1}} are vectors of ones
b
Incorporating Macroeconomic Factors: The macroeconomic factors f2t f_{2 t} are included in the model without adjustment for the risk-free rate. This is because these factors are not returns on traded portfolios but rather represent economic variables
c
Model Specification: The correct specification of the multifactor pricing model, considering the risk-free rate and the nature of the factors, is: Zt−RfiN=B1(f1t−RfiK1)+B2f2t+εt Z_{t} - R_{f} i_{N} = B_{1} (f_{1 t} - R_{f} i_{K_{1}}) + B_{2} f_{2 t} + \varepsilon_{t} This ensures that the returns on traded portfolios are adjusted for the risk-free rate, while the macroeconomic factors are included as they are
Answer
(b) Zt−RfiN=B2γ2+B1(f1t−RfiK1)+B2f2t+εtZ_{t}-R_{f} i_{N}=B_{2} \gamma_{2}+B_{1}\left(f_{1 t}-R_{f} i_{K_{1}}\right)+B_{2} f_{2 t}+\varepsilon_{t}, where iNi_{N} and iK1i_{K_{1}} are N×1N \times 1 and K1×1K_{1} \times 1 vectors containing only the number 1 for each element of the vectors
Key Concept
Adjusting for Risk-Free Rate
Explanation
The multifactor pricing model must adjust the returns on traded portfolios by subtracting the risk-free rate, while macroeconomic factors are included without such adjustment. This ensures the model accurately reflects the different nature of the factors involved.
C
Key Concept
Fama and French Factors
Explanation
In their 1993 paper, Fama and French introduced three factors into their multifactor pricing model: the market factor (Rm), the High minus Low (HML) factor, and the Small minus Big (SMB) factor. The Momentum (MOM) factor was not part of their original model.
Solution
a
Definition of Abnormal Returns: Abnormal returns are the difference between the actual returns and the expected returns based on a market model
b
Market Model: The one-factor market model is given by Rt=α+β(Rmt−Rft)+εtR_{t} = \alpha + \beta (R_{mt} - R_{ft}) + \varepsilon_{t}, where RtR_{t} is the return on an asset, RmtR_{mt} is the return on the market portfolio, RftR_{ft} is the risk-free rate of return, and εt\varepsilon_{t} is the error term
c
Zero Abnormal Returns by Construction: When it is stated that abnormal returns before the event are zero by construction, it means that the sum of the estimated errors (residuals) from the market model, when estimated on data before the event, will be zero. This is because the model is typically estimated with an intercept, ensuring that the average of the residuals is zero
Answer
(c) It means that the abnormal returns before the event are zero because the estimated errors obtained by estimating the market model on data before the event occurs will have a sum of zero as long as there is an intercept in the model.
Key Concept
Zero Abnormal Returns by Construction
Explanation
In the context of the one-factor market model, abnormal returns before the event are zero by construction because the model is estimated with an intercept, ensuring that the average of the residuals (errors) is zero.
Solution
a
Assumption of Zero Mean: The errors εit\varepsilon_{it} are often assumed to have an expected value of zero given the market returns, i.e., E[εit∣RMt]=0\mathbb{E}[\varepsilon_{it} \mid R_{Mt}] = 0
b
Assumption of No Autocorrelation: The errors εit\varepsilon_{it} are assumed to be uncorrelated over time, i.e., E[εitεis∣RM1,…,RMT]=0\mathbb{E}[\varepsilon_{it} \varepsilon_{is} \mid R_{M1}, \ldots, R_{MT}] = 0 for t≠st \neq s
c
Assumption of Normal Distribution: The errors εit\varepsilon_{it} are typically assumed to be normally distributed, not uniformly distributed
Answer
None of (a) to (c) are correct
Key Concept
Assumptions about errors in event study analysis
Explanation
In event study analysis, the common assumptions about the errors εit\varepsilon_{it} include that they have an expected value of zero given the market returns, are uncorrelated over time, and are normally distributed. The options provided in the question do not align with these typical assumptions.
Solution
a
Definition of Cumulative Abnormal Returns (CAR): Cumulative Abnormal Returns (CAR) measure the sum of the differences between the expected return and the actual return over a specific period
b
Interpretation of the Graph: The line graph shows the average CAR of Exxon over five stock splits from 1940 to 2017. The graph starts with a downward trend, then moves upwards, peaking at around 0.04 at 100 days since the split, and then fluctuates before trending downwards
c
Analysis of the Trend: The upward movement of the CAR line before and after the split date suggests a positive market reaction to the stock splits. This indicates that investors might perceive stock splits as a positive signal, leading to an increase in stock returns
d
Conclusion: Based on the graph, the correct interpretation is that there is a fairly strong trend increase during the event window starting somewhat before the split date
Answer
(a) There is a fairly strong trend increase during the event window starting somewhat before the split date.
Key Concept
Cumulative Abnormal Returns (CAR)
Explanation
The graph shows that the average CAR of Exxon increases significantly around the stock split dates, indicating a positive market reaction to the stock splits.
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