MATH2040/6131 Financial Mathematics

This assignment is worth 20% of the overall mark for the course.

Completed work should be submitted on Blackboard before 16.00 on Monday, 5 December 2022. The deadline is strict and penalties for late work will be applied in accordance with the University’s late work policy.

To submit your report and Excel spreadsheet go to the Blackboard page of MATH2040/6131, under the Class tests, Exam Papers and Courseworks tab there is an assignment called 2022/2023 - Coursework Assignment.  In your submission please attach the following two files:

• The report in a file called report-ID.pdf, where ID is your student ID number;

• The Excel spreadsheet in a file called spreadsheet-ID.xls, where ID is your student ID number.

Note that all your results must be presented in your written report and all your calculations should be done entirely in Excel, without use of Macros/VBA.  Therefore please avoid expressions such as ”Please see the spread- sheet” in the report. There is a strict limit of five A4 pages for the written report, which is easily sufficient to receive full credit.  Font sizes of at least 11pt must be used.  Careful explanation and clear presentation are important.  All coursework must be carried out and written up independently (see University’s Academic In- tegrity Guidance)

Where necessary, use fixed values for your simulations in Excel.

1. A company is considering setting up a branch.  The company is aware that access to capital may become difficult in twelve years time.  It therefore has two decision criteria.  The following cashflows are generated at an effective interest rate if 9% per annum:

• Outflows :  Between the present time and the opening of the branch in three years time the company will spend £1.5 million per annum on re- search and development of products. This outlay is a constant continuous payment stream. The rent on the branch building will be £0.3 million per annum paid quarterly in advance for twelve years starting in three years time. Staff costs are assumed to be £1 million in the first year, £1.05m in the second year, rising by 5% per annum each year thereafter.  Staff costs are assumed to be incurred at the beginning of each year starting in three years time and assumed to be incurred for 12 years.

• Inflows :  The company expects the sale of products to produce a net income at a rate of £1 million per annum for the first three years after the branch opens rising to £1.9 million per annum in the next three years and to £2.5 million for the following six years. This net income is assumed to be received continuously throughout each year. The company expects to be able to sell the branch operation 15 years from the present time for £8m.

(a) Calculate the present values of the outflows and inflows.                    [5]

(b) Plot the net present value of the project, NPV (i) for i = 1%, 2%, .., 9%. Comment on the figure.                                                                    [10]

(c) Calculate the internal rate of return. Comment on your answer.         [5]

(d) If the company decides to consider an effective interest rate of 7%, is the discounted payback period less than 12 years ? Justify and comment on your answer.                                                                                    [10]

(e) If the company decides to sell the branch after 10 years, repeat (a), (b), (c) and (d).                                                                                      [10]

(f) Compare your results with the ones in (b), (c) and (d). Maybe plot some figures.                                                                                             [10]

[50]

• Question 2

A company holds a large amount of capital and wishes to distribute some of it to its policyholders by way of two options:

• Option A. £1000 for each policyholder will be put into a fund from which the expected annual effective rate of return from the investments will be 5.5% and the standard deviation of annual returns 7%.  The annual effective rates of return will be independent and (1 + it ) is lognormally distributed, where it  is the rate of return in year t.  The policy holder will receive the accumulated investment at the end of ten years.

• Option B. £1000 will be invested for each policyholder for five years at a rate of return of 6% per annum effective. After five years, the accumulated sum will be invested for a further five years at another rate. This rate will be 1% per annum effective with probability 0.2, 3% per annum effective with probability 0.3, 6% per annum effective with probability 0.2, and 8% per annum effective with probability 0.3. The policyholder will receive the accumulated investment at the end of ten years.

Effective interest rates will be independent of that in any other such one-year period.

(a) For both options, calculate the mean accumulated value and standard devia- tion of the asset after 10 years.                                                                  [8]

(b) Using simulation, estimate the probability that the accumulated value is less than £1, 300 and give the 95% confidence limits for this probability for both options.  [In each case, use 10,000 simulations and use the confidence interval for a proportion.]                                                                                     [15]

(c) Assuming that investments of £1, 000 are made at the start of each year, calculate the mean and standard deviation of the accumulated amount at the end of 10 years for both options.                                                                [8]

(d) Using simulation, estimate the probability that the accumulated value is less than £15, 000 and give the 95% confidence limits for this probability for both options.  [In each case, use 10,000 simulations and use the confidence interval for a proportion.]