Quantitative Finance

Exercise 1: CAPM model [50 marks]
You will find on Blackboard a data file named “2020-21_Fama French5Factors_XXX.xlsx” for you to use (more details will be posted on Blackboard early in the semester). It contains data on the CAPM and the Fama-French tree- and five-factor models, which have been very successful in finance. The main variable of interest is the return on a portfolio of stocks (r_ portfolio) , which has been assigned to you to study. The returns on a risk-free asset (r_rf) and the market portfolio (r_mktport), which is composed of the stocks in NYSE, AMEX, and NASDAQ), can be found in the same file. It also includes the other four factors SMB, HML, RMW and CMA. In solving this exercise, you should to show understanding of these three models. If they are new to you, spend some time reviewing them.

Tasks:
•    Our main interest is to evaluate the performance of your stock. Obtain the scatter plot of this stock’s risk premium against the market risk premium. Comment on it.
•    Estimate the Capital Asset Pricing Model and interpret the coefficients. Is the stock aggressive or defensive? Test the significance of the coefficients at the 5% level.
•    From the remaining variables, add two factors to the model in order to estimate the so called Fama-French 3-factor model. Are these additional factors statistically significant?
•    Add two more factors to the model in order to estimate the so-called Fama-French 5-factor model. Are these additional factors statistically significant?
•    Report the coefficient of determination (??) and comment on the goodness of fit of each regression. Test the residuals of the regressions for violation of the CLRM assumptions and report any violations. Visit https://en.wikipedia.org/wiki/Quantitative_analysis_(finance) for more details.

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Exercise 2: ARMA model [25 marks]
The file “2020-09_univariatemodels_XXX.xlsx” is available on blackboard. It contains a financial time series, which we will refer to as yt.

Tasks:
•    Test whether there is a unit root in the time series yt.
•    Estimate the AR(p) model for p=1,2,3,4, as well as the ARMA(1,1), ARMA(2,1), ARMA(1,2) and ARMA(2,2) models. Report the results in a table with one column for each model. For each model, include in the table the Akaike and Bayes Information Criteria (IC), as well as the Ljung-Box test of white noise of the first four autocorrelations.
•    Compare the performance of the different models. Using both the Akaike and Schwarz IC, select the best of these AR(p) and ARMA(p,q) models that describes variable yt. What is the conclusion of the Ljung-Box test?
•    At this point, you may also want to consider whether to estimate any alternative model.

Exercise 3: GARCH Model Selection [25 marks]
Make use of the same datafile than in Exercise 2.

Tasks:
•    Estimate the ARCH(1), GARCH(1,1), EGARCH(1,1) and ARCH-M(1,1) models. Assume that the mean equation is best modelled as having an intercept only.
•    Summarise the results, including IC, in a table. Select the best model based on the IC and any relevant hypothesis test.
•    At this point, you can propose other model(s)/specification(s), which, in your view, would fit the data better that the ones you have estimated before.

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