Quantitative Methods in Finance, Regression and Probability Regression

Question 1: Regression and Stopping Distance

Instructions

There are four questions each worth 25 marks. Answer all four questions. In the light of the emergency coronavirus measures this assignment will constitute 100% of the marks awarded for this module and replaces the final exam.

Question 1 [Regression]

Expalin Quantitative Methods in Finance?

The data in Table 1 below links the stopping distance (in metres) of a car y to the speed of a car x (speed measured in mph).

Table 1: Data for Question 1

(a) Enter this data into R and list the commands used

(b) Fit a suitable regression model to this data

(c) Using the model fitted in part (b) estimate the stopping distance for speeds of 10, 80, 90 and 100 mph.

(d) For the estimates in part (c) use R to construct a 95% prediction interval for each of these four speeds: 10, 80, 90 and 100mph. What is the interpretation of the prediction intervals in this case?

(e) How is the process of choosing a regression model in part (b) different to other regression examples you might see in accounting and finance?

Question 2 [Probability regression]

(a) List the assumptions underpinning the classical regression model.

The data in Table 2 below link the probability Y that a sales voucher is redeemed compared to the size of the discount X offered.

Table 2: Data for Question 2

(b) Enter the data in Table 2 into R and list the commands used.

(c) Fit the regression model .

(d) What is the regression model in part (c) called?

(e) Show that the linear regression model in part (c) cannot satisfy the classical linear regression assumptions listed in part (a).

(f) If X�is equal to 31.63 estimate the probability that a voucher is redeemed in each case.

(g) Suggest a better model than the model in part (c). Use this model to re-estimate the probability in part (f).

Question 3 [Financial price data]

(a) Download price data for a financial asset of your choice. [Hint: possibilities here include stock price data from yahoo finance or cryptocurrency data from coinmarketcap.com]. Explain why the asset or index you have chosen is interesting and give the precise dates for which you have collected the data.

(b) Download the data into R and calculate the log-returns. List the R commands used.

(c) Fit an ARCH(1) model to the log-returns in part (b) and test for the presence of an ARCH effect within this data. List the R commands and R packages used.

(c) Fit a GARCH(1, 1) model to the log-returns in part (b) and test for the presence of ARCH and GARCH effects within this data. List the R commands and R packages used.

(e) List the stylised empirical facts of financial time series.

(f) [Independent research required]. Explain how more advanced family GARCH models can be fitted to the data in part (b) using R.

Question 4 [Basic hypothesis tests]

The data in Table 3 below lists the corporation tax rates for selected EU and non-EU countries.

Table 3: Data on Corporation Tax rates for Question 4

(a) Read this data into R and list the R commands used.

(b) Test for a difference in the mean corporation tax rate between EU and non-EU countries.

(c) Test for a difference in the variance of the corporation tax rate between EU and non-EU countries.

(d) Based on the findings in part (c-d) what are the implications of Brexit upon the rate of UK corporation tax?

(e) What implications might the current coronavirus pandemic have upon the data in Table 3?

Some of the commonly used quantitative methods in finance include:

1. Time series analysis: This involves analyzing historical data to identify patterns and trends over time. Time series analysis is commonly used in finance to forecast future prices of financial assets.

2. Statistical inference: This involves using statistical methods to make inferences about a population based on a sample of data. Statistical inference is used in finance to estimate the parameters of financial models and test hypotheses.

3. Optimization: This involves finding the optimal values of a set of decision variables subject to constraints. Optimization is used in finance to develop portfolio optimization models, which aim to find the best combination of assets to maximize returns while minimizing risk.