MATH2040/6131 Financial Mathematics

Assignment 2025/26

This Assignment counts as 20% of your overall mark for this module.

Completed work should be submitted on Blackboard before 23:59 on Monday 15 December 2025. This deadline is strict and the standard University penalty for late submission of work will apply.

To submit your work, go to the Assignments tab in the Blackboard page, where you will find an assignment called Coursework submission 2025/26. Please submit the following two files:

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

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

Note that all your results, explanations, and discussion must be presented in your report, and all calculations and simulations must be done in your Excel spreadsheet, without use of Macros/VBA. Therefore please avoid using expressions such as “Please see the spreadsheet” in the report.

The strict page limit of the report is 4 A4 pages, and a font size of no smaller than 11pt should be used. The report should be written in a coherent manner, with careful explanation, and should be self contained and well presented. [It may help to think of yourself as a graduate trainee, with the report to be submitted to your line manager.]

The spreadsheet should be clearly set out, with the calculations and simulations relating to different parts of the questions clearly identified.

Your work should be entirely your own, and in accordance with the University Academic Integrity Guidance.

1. An electricity generation company is considering an investment project involving the construction, operation, and eventual decommissioning of a coal-fired power station. Construction is expected to take three years, and to cost £420 million, payable in equal instalments at the start of each year. Once constructed, the power station is expected to enjoy 40 years of operational life, to produce an annual output of 3 billion kWh of energy, and to incur an annual operating cost of £120 million, both of these annual figures being spread uniformly throughout the year. At the end of its operational life, it will be necessary to decommission the power station, a process that is expected to take two years and to cost £100 million, payable in equal instalments at the start of each year. The company has no spare funds to finance the project, but may borrow the construction costs from its bankers, who charge an effective annual rate of interest of 5% on borrowings and pay an effective annual rate of interest of 3 1 2% on deposits.

(a) Determine the minimum price (in pence per kWh) at which electricity must be sold if this project is to just break even, and find the length of time that must elapse before the company will have repaid its bank indebtedness at this minimum price, assuming that borrowings may be reduced by repayment at any time. [15]

(b) If the price at which electricity may be sold is twice that found in part (a), find the accumulated profit achieved by the company on this investment project at the end of the project term. [5]

(c) The above model for the project involves various assumptions and expectations. Identify, discuss, and investigate these, using a spreadsheet to explore the impact of changing these aspects in order to make the model more realistic and/or to test its robustness to such changes, and take note of the corresponding risks. [30]

[Total 50]

Note: You may find Example 5.5 in the lecture notes helpful for part (a).

2. An equity fund manager at an investment firm models the future performance of the fund, as follows: it is assumed that, in each year, t, the corresponding annual effective yield, it , is independent of that in any other year, and is such that the corresponding accumulation factor, 1 + it , is lognormally distributed, with (constant) parameters µ and σ2 , so that log(1 + it) ∼ N(µ, σ2). The mean and variance of it are assumed to be j = 0.09 and s2 = (0.05)2 , respectively. [Here, log means the natural logarithm.]

The fund offers investors a choice of two three-year investment products, with the following cashflows:

• A single investment of £1 million made at the start of the three-year period. The accumulated value, X, of this single investment is returned to the investor at the end of the three-year period.

• An annual investment of £1 million made at the start of each of the three years. The accumu-lated value, Y , of this total investment is returned to the investor at the end of the three-year period.

(a) Calculate the mean and standard deviation of X. [2]

(b) Calculate the mean and standard deviation of Y . [4]

(c) Calculate the values of µ and σ 2 . [2]

(d) Calculate the 95% confidence limits for X. [4]

(e) Using simulation, estimate the mean, the standard deviation, and the 95% confidence limits for X, and compare with your exact results obtained in parts (a) and (d). [Use 10,000 simulations of X.] [8]

(f) Using simulation, estimate the mean, the standard deviation, and the 95% confidence limits for Y . [Use 10,000 simulations of Y .] [8]

Another fund manager at the investment firm has questioned the validity of the modelling assumption that the annual effective yield, it , in year t is independent of that in that in any other year. He has suggested that the model be generalised to allow for a degree of dependence between the annual effective yields in successive years, by letting log(1 + it) ∼ N(µt , σ2), where µ1 = µ and µt = µ + k [log(1 + it−1) − µ] for t ≥ 2. Here, the parameter k satisfies 0 ≤ k ≤ 1 and controls the degree of dependence, the case k = 0 corresponding to the assumption of independence used by the first fund manager.

(g) Using simulation, estimate the mean, the standard deviation, and the 95% confidence limits for X under this generalised model, for both k = 1/2 and k = 1. [Use 10,000 simulations of X.] [6]

(h) Using simulation, estimate the mean, the standard deviation, and the 95% confidence limits for Y under this generalised model, for both k = 1/2 and k = 1. [Use 10,000 simulations of Y .] [6]

(i) Discuss all of your results. [10]

[Total 50]

Note: Use a fixed seed for your simulations, and state the seed you use, so that the random numbers generated are reproducible.