MTH1004 summative coursework: report
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MTH1004 summative coursework: report
Date set: Monday 6 March 2023
Submission date: Friday 31 March 2023 by noon
Return date: Friday 5 May 2023
Instructions
This coursework comprises 15% of the overall module assessment. This is an individual assessment, so you must not discuss this assessment with other people. See our guidelines on collusion and plagiarism on the following webpage.
http://intranetexeteruk/emps/studentinfo/subjects/mathematics/assessment/academicmisco nduct
Conduct the two investigations outlined below. For each investigation, write a short report of your findings, addressing each part of the investigation in a separate section. The data and R functions
needed for the two investigations are contained in the file MTH1004T2CW .RData, which you should download from ELE at https://vleexeteruk/course/view.
Each report should comprise at most two sides of word-processed text (e.g. written with Word or LaTeX) followed by as many pages of figures or tables that you feel is appropriate and then an appendix containing the R code (with explanatory comments but without the R output) that you used to obtain your reported results. No credit will be awarded to additional pages of text. (You do not need to write two sides of text to achieve full marks—the model solutions contain only one side of text.) You must use the page size A4 and a font size of at least 11 points, lines must be single-spaced and all margins must be at least one inch wide. You must combine your reports and save them as a single pdf file before uploading it to BART at https://bartexeteruk by the date and time shown above.
Assessment criteria
Approximately 50 marks are available for each investigation. Of these 50 marks, approximately 20 are awarded for your method and numerical results, 20 for your discussion of your results, and 10 for the quality of your presentation of your report.
Investigations
1. If Xt represents the price of a stock market index on day t then the “log return”for day t is Yt = log(Xt/Xt − 1 ). (Here, log denotes the natural logarithm.) The “value at risk” is the value, v, such that ( −v, ∞) is a 99% prediction interval for the log return. The file MTH1004T2CW .RData includes a data frame called finance which contains the log returns for a stock market index on 1859 consecutive trading days (which excludes weekends and holidays). The file also contains some R functions which are described at the end of this document and which you may use to answer this question.
(a) A colleague suggests modelling the log returns as realisations of independent and identi- cally distributed random variables with a Normal distribution, N(µ, σ2 ). Use the method of moments to estimate the parameters of this model and then assess the realism of the model. Include in your report the formulae for your parameter estimates, the numerical values of these estimates, a discussion of the model’s realism and appropriate graphical evidence. You do not need to include the derivation of your formulae.
(b) Now consider modelling the log returns as realisations of independent and identically distributed random variables with a Student T distribution, Stu(α, β, γ), where β > 0 and γ > 4. This is a generalisation of the standard Student T distribution and has mean α, variance β2 γ/(γ − 2) and kurtosis (3γ − 6)/(γ − 4). (The kurtosis is the standardised fourth moment, which we use here instead of the third moment because the third moment equals zero for the Student T distribution.) Use these moments and the R functions described below to repeat part (a) for this model and then explain which of the two models you prefer.
(c) Use your preferred model to compute the following predictions and briefly state your opinion about their trustworthiness:
(i) the value at risk;
(ii) the probability that the price Xt reduces by at least 3% of its value in one day.
2. A study sought to determine the effectiveness of two treatments for arthritis for adults in
the UK. All adult patients in Devon who were referred for treatment for arthritis during a particular month were included in the study. The patients were randomly allocated to either a control group (who received the standard treatment) or an intervention group (who received a new treatment) in such a way that patients were twice as likely to be allocated to the intervention group. For each patient, the degrees of flexion in their joint was measured before and after receiving three weeks of treatment. An increase in degrees of flexion represents an improvement in a patient’s condition. The data are contained in the data frame arthritis in the file MTH1004T2CW .RData.
(a) Comment on the strengths and weaknesses of the design of the study.
(b) Use point estimates and confidence intervals to assess the effects of the two treatments, and whether their effects differ. Include in your report the formulae for your point estimates and intervals, as well as their numerical values.
R functions
The file MTH1004T2CW .RData contains functions dstudent, pstudent and qstudent to compute the pdf, cdf and quantiles of Stu(α, β, γ) distributions. In particular, dstudent(y,a,b,g) will compute the pdf at y = y for α = a, β = b and γ = g; pstudent(y,a,b,g) will compute the cdf at y = y for α = a, β = b and γ = g; and qstudent(p,a,b,g) will compute the p-quantile for α = a, β = b and γ = g. The file also contains a function, kurtosis, to compute the sample kurtosis. In particular, kurtosis(y) will compute the sample kurtosis for data in a vector y.
2023-03-31