Empirical Methods in Finance HEC Paris
Assignment 2
• Please hand in only one pdf file with your answers and one m file. Name the file as ”Assign- ment 2 – NAME1 NAME2 NAME3 NAME4.pdf” and ”Assignment2_NAME1_NAME2_NAME3_NAME4.m”, respectively, using your last names.
• Send your pdf to [email protected] with the following EXACT object: EMF ASSIGN- MENT 2 by January 2nd, 2020.
• You can work in a group of up to four students.
• Your m file for should work, i.e., it should produce the values you use in your answers on the pdf when it is run.
Question 1
You have just been hired as a trader in a large investment bank. Your boss is convinced that a short-term trend following strategy easily produces good returns. For example, you can use a regression of this type,
rt = β1 + β2rt−1 + εt,
where rt is the return on a stock index for day t, β1 is the coefficient for a constant, β2 is the parameter for the daily return lagged by one day rt−1, and εt is an error term. If the coefficient β2 is significantly positive, then it means that positive returns tend to be followed by positive returns, so it is profitable to follow the trend.
Your boss asks for some analysis on this type of strategy using the daily returns on the S&P 500 Index from 1950 to 2017 that you can find in the file Data.xlsx. Use a 5% significance level for making inferences.
a)
Run this regression over the full sample period1 assuming that the errors εt have equal variance and have zero covariance between themselves. What is the coefficient for β2 and its t-ratio? Is it
1Of course, you lose one observation for the dependent variable because you use the lagged return as a regressor. 1
significantly positive (provide the threshold you used)? Is the short-term trend-following strategy profitable based on your analysis (yes or no)? Compute the adjusted R2 for this regression. Is this regression a good model to explain the daily returns on this stock index?
b)
You are now worried that the errors might have different variances and be correlated between them- selves. Use the function fctNeweyWestCovarianceMatrix to compute the Newey-West covariance matrix of the regression coefficients and use a maximum lag of 252 (i.e., one year of daily returns). What is the t-ratio for the β2 coefficient? Is it significantly positive (provide the threshold you used)?
c)
Cut your sample period in half (use t = 1 to t = 8529 for the first half and t = 8530 to the end for the second half). Estimate the regression for both periods. Based on the results for the earlier period, report the β2 and its t-ratio (still using the Newey-West covariance matrix estimator). Did the short-term trend following strategy work in the first period (justify your answer)? Report the β2 and its t-ratio for the second period. If, based on your analysis on the first period, you had decided to follow the short-term trend following strategy in the second period, would it have worked (justify your answer)?
d)
To assess the risk of being invested in the stock market each day, you now compute the Value-at- Risk at the 5% level (VaR(5%)) of the S&P 500 Index. Use the function fctEstimateGARCH2 to estimate the model
rt = β1+ε, ε∼N(0,σt2)
σ2 = (1 − θ − α)σ ̄2 + θσ2 + αε2 .
t t−1 t−1
Then, produce a graph with the daily returns and the VaR(5%) on the vertical axis and the
dates on the horizontal axis. Copy and paste the figure on your report. Compute the proportion of daily returns that exceed the VaR(5%).
e)
Given the difficulty in estimating the average daily return, it is often suggested to set it equal to 0 when computing the daily VaR(5%). Compute the daily VaR(5%) assuming the average daily
2IMPORTANT: Use the function provided with this assignment.
2
return is 0 and compute the proportion of daily returns that exceed the VaR(5%). Which proportion is closer to the chosen level of 5%?
f)
The leverage effect in financial markets refers to the fact that variance tends to increase more after a negative shock than after a positive shock. In a GARCH model like the one in part d), the leverage effect is not captured because the shock to variance is ε2t−1 (i.e., a -2% shock has the same effect as a +2% effect). Modify the function fctEstimateGARCH.m to estimate the model
rt = β1+ε, ε∼N(0,σt2) ( α+α)
σ2=1−θ−1 2σ ̄2+θσ2+αε2I +αε2I . t 2 t−1 1 t−1 εt−1<=0 2 t−1 εt−1>0
where Iεt−1<=0 is an indicator variable equal to 1 when εt−1 <= 0 and 0 otherwise. The term in front of σ ̄2 has been changed such that the long-run variance is equal to σ ̄2. Save your new function as fctEstimateGARCHLeverage.m (and send it with your assignment). Here are a few hints:• In the GARCH model in part d), we need the following constraints on the parameters: α > 0, θ > 0, and θ + α < 1.• I impose these constraints by taking the exponential of the parameters for α and θ (see lines 46-47 in fctEstimateGARCH.m) to make sure they are positive and stopping the iteration whenever I have θ + α > 1 (see lines 50-54).
• In the GARCH with the leverage effect, I need the following constraints on the parameters: α1 >0,α2 >0,θ>0,andθ+α1+α2 <1.2If the data exhibit the leverage effect, then α1 should be higher than α2. What are your estimated values for α1 and α2? To see whether this is a better model for time-varying variance, compute the proportion of daily returns that break the new VaR(5%) measure to see if it is closer to 5%.Question 2The file G7 Stock Markets.xlsx contains monthly returns on the stock markets of G7 countries. Do an PCA on the covariance matrix of these monthly returns (you can use the function eig on MATLAB, type help eig in the command window for more information). What is the proportion of variance explained by the first principal component (i.e., the one with λmax)? What is the proportion of variance explained by the first two principal components? Does it suggest a strong factor structure in G7 stock market indexes? 3
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