MATH2040/6131 Financial Mathematics
Assignment 2024/25
This Assignment counts as 20% of your overall mark for this module.
Completed work should be submitted on Blackboard before 23:59 on Monday 16 December 2024. 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 MATH2040 Blackboard page, where you will find an assignment called Coursework submission 2024/25. [This applies to MATH2040 and MATH6131 students.] Please submit the following two files:
1. A report in a file named report-ID.pdf, where ID is your student ID number;
2. 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, so that it is coherent and self-contained, 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 well presented.
The spreadsheet should be clearly set out, with the calculations and simulations relating to different parts of the questions clearly identified.
1. A local government authority is considering how to finance a project to build a road bridge that will connect a large Scottish island to the mainland. It is expected that construction of the bridge will take two years, at a cost of £5 million per year, payable quarterly in advance. Once constructed, the bridge will incur operating and maintenance costs totalling £0.75 million per year, payable continuously, and will generate revenue from atoll (charge) of £T per vehicle for each single crossing of the bridge. It is estimated that there will be 1.5 million single crossings of the bridge in each year of operation, uniformly distributed during the year. The toll is expected to be reduced to £U after the first ten years of operation of the bridge, a level that will generate revenue equal to twice the continuing operating and maintenance costs each year.
(a) One possible method of financing the project is to borrow the construction costs from a bank. The bank will charge interest at a rate of 4% per annum effective and will allow full or partial repayment of the loan at any time.
(i) Obtain the present value of the construction costs at the start of the project, using the bank borrowing rate. [4]
(ii) Find the value of T such that the discounted payback period (DPP) of the project is twelve years. [5]
(iii) Obtain the value of U. [1]
(iv) Obtain the present value of the profit (revenue less costs) at the start of the project, using the bank borrowing rate, assuming that the DPP is indeed twelve years and that the term of the project is infinite. [5]
(b) An alternative method of financing the project has been proposed: form a special purpose company to issue ten million ordinary shares, each of price £P, to raise capital equal to the present value of the construction costs in part (a)(i). Dividends on these shares are expected to be paid annually in arrear and to grow at a compound annual rate of 2% in perpetuity.
(i) Obtain the value of P. [2]
(ii) If shareholders are to achieve an expected annual yield of 8% per annum effective, obtain the amount of the first dividend per share. [3]
(c) The above model for the project, and 代寫 MATH2040/6131 Financial Mathematics the suggested methods of financing, involve various assump- tions, expectations, and limitations. Identify, discuss, and investigate these, using a spreadsheet to explore the impact of changing these aspects in order to make the model and financing more realistic, and take note of the corresponding risks. [30]
[Total 50]
Note: Parts (a) and (b) of this question are based on Q5 of the 2021/22 final exam. For part (c), think about how the stated assumptions or expectations may change, think about what is missing and may be included, and think about how appropriate the suggested financing methods may be.
2. An equity investment fund manager models the future performance of the fund, as follows: 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 j = 0.10 and s2 = (0.08)2 , respectively. [Here, log means the natural logarithm.]
The fund offers investors a choice of two five-year investment products, with the following cashflows:
• A single investment of £1 million made at the start of the five-year period. The accumulated value of this single investment, X, is returned to the investor at the end of the fiveyear period. • An annual investment of £1 million made at the start of each of the five years. The accumulated value of this total investment, Y , is returned to the investor athe end of the five-year perio(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 probability that X will be (i) less than £1.5 million, and (ii) more than £2 million. [8]
(e) Using simulation, estimate the probability that X will be (i) less than £1.5 million, and (ii) more than £2 million. Also, give 95% confidence limits for these probabilities. [Use 10,000 simulations of X, and use the confidence interval for a proportion.] Compare with your exact results obtained in part (d). [12]
(f) Using simulation, estimate the probability that Y will be (i) less than £6 million, and (ii) more than £8 million. Also, give 95% confidence limits for these probabilities. [Use 10,000 simulations of Y , and use the confidence interval for a proportion.] Explain why, unlike in part (d) for X , these probabilities cannot be calculated exactly, but must be estimated by using simulation. [12] (g) In return for payment of an initial fee by any investor who chooses the annual investment product, the fund offers a guarantee that the value of the investment returned to the investor at the end of the five-year period will be at least equal to the value of the annual investments of £1 million accumulated at the (constant) effective annual risk-free interest rate of 2%. Using simulation, estimate the fair value of this fee. [Use 10,000 simulations of Y.] [5]
(h) Discuss your results. [5][Total 50]
Note: For parts (a) and (b), see the statements of Theorems 10.1 and 10.2 of the lecture notes (not the corresponding proofs). For parts (c) and (d), see Examples 10.2 and 10.3 of the lecture notes. For the confidence intervals in parts (e) and (f), you may find Example 12.4.4 of Garrett useful. For the simulations in parts (e), (f), and (g) you may wish to use the Excel Data Analysis add-in. Use a fixed seed for your simulations, and state the seed you used, so that the random numbers generated are reproducible.