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R: Assignment 2

Background

? This assignment is worth 50% of your course grade. You can complete

it yourself, or in groups of two or three people (each group member will

receive the same grade).

? The due date will be advised via Blackboard.

? In answering the questions properly, you will need to consult the Part 2

notes for the course, use the R help file and think carefully. Do not trawl the

internet looking for inspiration ¨C that is a waste of time.

? There are three main sections: 1, 2 and 3. Answer all questions.

? Section 4 is a ¡°substitute question¡±. If you choose this option you can do

only two questions from 1, 2 or 3, rather than three.

? Each question will be marked out of 10. The score will be determined

holistically and subjectively (¨¤ la marking an essay), so there are no pre-set

levels of marks for any particular parts of a question.

? Do all calculations and computer code with R.

Instructions

? Be creative. Don¡¯t spend too long doing this assignment.

? No inter-group copying or extensive answer sharing.

? Ask me for help if you get muddled, but not before

you have carefully read the assignment and discussed problems with your

group members.

? Your answers should include a number of Word (or PDF) files and R code

.R files. Submit all files in one .zip file.

? Email me your answers directly; in the subject line

put ¡°LND¡± followed by your student IDs (e.g. LND 2011213, 2011215,

2011214). Repeat your ids and full names in email text.

? I do not want to see spreadsheets or CSV files. For your data outputs

please send them to .CSV files from R and format them there as you wish

in Excel. Then integrate your formatted data outputs into a Word (or PDF)

file with the related text.

1

R: Assignment 2

? Provide one Word (or PDF) file that explains who your team members are

(student IDs and names) and outlines which questions you have selected.

Call this file ¡°overview.doc¡± or something else obvious.

? Provide a separate Word (or PDF) file for each question you answer. Call

them something logical and obvious e.g. ¡°answer1.doc¡±. In each file, briefly

outline how you went about answering the question, including references to

attached R scripts and functions. This is also where you should display the

following: any tables of numbers (fully annotated and nicely formatted); any

general text relating to your analysis; and your specific answers to different

parts of the questions.

? Don¡¯t forget to include all of your R scripts and functions in the ¡°.zip¡± file.

? I have deliberately left the specific requirements of each question vague.

Decide for yourself how to express your answers to me. Obviously, if the

question simply asks you do something on R, then all you need to do is give

me your scripts and/or functions and in your answer document say that the

code does what the question asks. Where you need to make substantive

conclusions from results or data summaries, show me a table of numbers

and/or a graph, describe in words what is being displayed and say why you

conclude what you do (and also give me the R code).

Assignment Files

? I have included a set of auxiliary computer files to be used in conjunction

with this assignment. I will refer to these as the ¡°Assignment Files¡±.

Assignment Files

Functions

fsumstats Like from the notes (adjusts for NA values)

fmoment Called by fsumstats

freg Unfinished regression function

fregquick Same as freg but with R2

Data Spreadsheets

assetsq2.csv Asset descriptions in CSV format.

pricesq2.csv Prices in CSV format.

? You can install all the files in your R working directory if you so wish (but my

strong suggestion is to use a simple structure in which you have ¡°functions¡±

and ¡°data¡± as separate folders, with the folder paths appropriately added

in your R code).

2

R: Assignment 2

1 Returns, Portfolios and Summary Statistics

The questions in this section relate to the type of analysis introduced in chapters

1 and 2 of the Part 2 notes.

1.1 Statistical Summaries

Extend the statistical summary function ¡°fsumstats¡± found in the Assignment

Files. Make at least the following additions.

a. Summaries for the minimum, maximum and quantiles corresponding to

.025 and .5 and .975 (make sure you give each row an appropriate name).

b. Autocorrelation estimates - use several lags that you think are practical. For

the autocorrelation function, follow the details for ¦Ñ?(k) from section 2.12 in

the Part 2 notes.

c. P-values for the autocorrelation estimates. Assume that if the true autocorrelation

is zero then ¡Ì

(T)?¦Ñ(k) ¡« N(0, 1) where T is the number of observations.

Required: Make the p-values simply the value in the CDF of the

test statistic under the null (like the ones for kurtosis and skewness) and

assume that the underlying null hypothesis is that the true autocorrelation

is zero.

1.2 Practical Regression

For this question use the function ¡°fregquick¡± from the Assignment Files. For

the base data use the ¡°pricesq2.csv¡± and ¡°assetsq2.csv¡± files, also in the Assignment

Files.

Part A

a. First convert all the equity prices (total return indices that include capital

changes and dividends) and the FRCAC40 index (including dividends) to log

returns. And then calculate excess returns relative to TRBD3MT (a proxy

for the risk free rate).Required: Dynamically select the columns for the

various series you require, using the details in ¡°assetsq2.csv¡± i.e. your code

should work even if the column structure was different (assets and indices

in a different order, for example) or more or less time periods were included.

Hint: Don¡¯t forget to transform TRBD3MT to the correct number of decimal

places.

b. Run regressions for the excess returns of every French stock (y-variable)

3

R: Assignment 2

versus the excess return for the CAC40 (x-variable), over the entire period

contained in the data. Required: Store the estimated parameters (intercept

and slope) and R2

values for each asset.

c. Do the same regressions using what you think are interesting subsets of the

data through time based on when the VSTOXXI index is high (you decide

what ¡°high¡± is). Required: Store the estimated parameters (intercept and

slope) and R2

values for each asset.

d. Do the same regressions using what you think are interesting subsets of

the data through time based on when the ¡°US SMOOTHED US RECESSION

PROBABILITIES NADJ¡± index is high (you decide what ¡°high¡± is). Required:

Store the estimated parameters (intercept and slope) and R2

values

for each asset.

e. Calculate, report and comment upon, the summary statistics of the estimated

parameters and R2

in each case. Required: Use your version of

¡°fsumstats¡±.

4

R: Assignment 2

2 Regression and Optimisation

The questions in this section relates to chapters 3 and 4 of the Part 2 notes.

2.1 Technical Regression

.

# create same model as before

n<-100 ; set.seed(999) ; x<-cbind(rep(1,n),rnorm(n,.1,.4))

e<-rnorm(n,0,.2) ; b<-rbind(0,1) ; y<-x%*%b+e

The code above repeats the fake data that we used in the Part 2 notes for

checking regression estimates. Use this in your answers to the questions below.

Part A

Let the following formula define an objective function to minimize the sum of the

absolute value of errors from a linear regression.

min?b[(|y ? X?b|)¡ä¦É)]

(1)

where ¦É is a vector of ones and the absolute value operation is for each element

of the vector within it.

For the avoidance of doubt, a simple regression model based on equation (1)

can be re-written as

min¦Â?0,¦Â?1¡ÆTi=1|yi ? ¦Â?0 ? ¦Â?1xi| (2)

a. Find the ¡°optimal¡± betas for a regression model that has the objective function

in formula (2), using the fake data defined above.

Hint: Follow the example for minimising the sum of squared errors in the Part

2 notes.

Part B

Add the following components to the output structure variable in the regression

function ¡°freg¡± from the Assignment Files. The section references refer to the

formulas given in the Part 2 notes.

a. yhat from the y? in section 3.3.

b. e from estimated errors in section 3.3.

5

R: Assignment 2

c. rsq from R2

in section 3.9.

d. rsqadj from R¡¥2

in section 3.9.

e. covbeta from cov(?b) in section 3.9.

f. tbeta from ti, for every ?bi, from section 3.9.

Hint: Start with the list of variables contained in ¡°freg¡± and then add out$yhat

and so on.

Part C

The following assumptions describe a maximum likelihood set-up for a simple

linear regression that assumes the errors are IID Normal.

.

Table 1: Assumptions for a Linear Regression MLE

1. The data are y and x from the fake data defined above.

2. The linear regression equation is yi = ¦Â0 + ¦Â1xi + ?i.

3. The errors, ?i, are independently and identically Normally distributed.

4. We have ?i = yi ? ¦Â0 ? ¦Â1xi.

5. The applicable Normal distribution is f(?i) = 1

6. The likelihood function for IID Normals is L =¡Çni=1 f(ei)

7. We do not know ¦Ò, ¦Â0 or ¦Â1: we seek to estimate them such

that L is maximised.

a. Solve the maximum likelihood equation (¡°MLE¡±) for the regression model

defined by the assumptions in Table 1 with a numerical optimisation.

Hint: Follow the MLE example in the Part 2 notes, changing only the pieces

that you need to.

6

R: Assignment 2

3 Simulation

The background material for simulation was covered in Chapter 5.

3.1 Option Valuation via Simulation

Consider the option to sell an asset at time T to someone at the maximum

price of the asset between 0 and T. This is a so-called Asian put option which

depends on the price path of the asset (as discussed in class).

Assume that the underlying asset follows geometric Brownian motion, as per

chapter 5 of the Part 2 notes.

Let the risk-neutral valuation formula (Campbell, Lo and MacKinlay (1997), section

9.4) of an Asian put option be

H(0) = e?¦ÌrfTE?[max0¡Üt¡ÜTP(t)]? P(0), (3)

where H is the option, P is the price of the asset, ¦Ìrf is the risk free rate, the

¡°max¡± function applies to any price between the start (t = 0) and end (t = T) of

the option and the expectation is with respect to ¡°risk neutral probabilities¡±.

The analytic solution (Goldman, Sosin and Gatto, 1979) to equation (3) is

H(0) = P(0)e¨C¦ÌrfT ?(?¦ÁT¦Ò¡ÌT) [1 ?¦Ò22¦Ìrf]? P(0)+P(0) (1 +¦Ò22¦Ìrf) [1 ? ?(?(¦Á+¦Ò2)T¦Ò¡ÌT)] (4)

where ?(.) is a function that represents the cumulative N(0, 1) distribution, ¦Ò is

the asset¡¯s standard deviation and ¦Á = ¦Ìrf ? ¦Ò

2/2.

Let the option for this question be defined by the assumptions in Table 2.

Part A

a. Value an Asian put option with equation (4). Use the assumptions in Table 2.

Part B

7

R: Assignment 2

.

Table 2: Assumptions for an Asian Put Option

1. Starting price of €15.

2. Period length of 3 (three years).

3. Risk free rate of 0.5% per year.

4. Asset standard deviation of 25% per year.

5. Strike price set by the maximum of the price over the 3 years.

a. Value the Asian put option, as above, by using price path simulations. Use

10,000 simulations. Required: Use equation (3) to define your valuation.

Use n = 100 for the number of steps. Hint: Follow the code from chapter

5 of the Part 2 notes, changing only what you need to.

b. Increase the number of steps (n) and the number of simulations until you

are sure that your answer is converging to the analytic results given by

equation (4). Required: Report your simulated option values for different

levels of n and simulations.

c. Explain why increasing the number steps brings the simulated values closer

to the analytic solution.

8

R: Assignment 2

4 Substitute Question

You are welcome to substitute one of the questions from 1, 2 or 3 for your one

of your own creation ¡ª subject to my prior approval!

If you want to take this option then please talk to me first. The idea is that you

can base your substitute question on a part of the notes that you find interesting

but isn¡¯t covered in the other questions. Good topics include: searching for

momentum and reversal effects in asset returns, non-parametric analysis, nonlinear

regression and bootstrapping. Or maybe you have a different dataset that

you wish to analyse.

9


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