MATH3510 Assignment 2023
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MATH3510 Assignment 2023
Important Information - Read this first!
This assignment forms the coursework proportion of MATH3510. You are required to use an Excel spreadsheet for all calculations. You must submit this Excel spreadsheet (as described below) and provide selected answers/outputs in Gradescope (either by typing your answer as indicated or uploading a file containing your answer, PDF format is recommended in the latter case). Marks available for each question are shown in Gradescope.
Questions will become available at 11am on Thursday 16 November 2023. The submission deadline is 2pm on Thursday 23 November 2023. Late submissions will be accepted until 2pm Thursday 30 November, but late submission will result in a penalty of 15% of the marks awarded. (If you need more time due to mitigating circumstances apply for an extension or exemption in the usual way.)
The spreadsheet you submit for marking should be based on the template file you can download from Minerva. Download and save a copy of the template file.
The first thing you must do is enter your student number in cell B1 of the first worksheet. This will create a set of parameter values for you to use (different students will have different parameters and therefore different answers to questions).
If you believe there are problems with your parameters, or the parameters do not generate, notify the module leader immediately.
You are required to submit both the spreadsheet and your answers in Gradescope for your work to be marked.
The Excel file should be submitted via Turnitin. There is a link in the Assignment folder on Minerva. Please follow the naming convention below for your spreadsheet file:
last name__student ID__spreadsheet.xlsx
where your “last name” should be consistent with your University records.
Academic Integrity
You must complete the academic integrity statement in Gradescope.
As this work contributes towards the award of actuarial exemptions (through your MATH3510 module grade), it is important you prepare your own spreadsheet and summary document inde- pendently.
Students submitting highly similar documents may be excluded from eligibility from exemptions at the discretion of the independent examiner and be subject to the University’s own misconduct procedures.
A guide to the marking scheme and some advice is included at the end of the questions.
Answer Submissions
In Gradescope you will be prompted to input or upload parts of your work. For some short answer questions you can simply type the answer. In other cases you should upload a file containing the information requested.
Where you have to produce tables it is recommended that you first do so in LaTeX or Word and then upload a PDF file of the table.
Where you have to provide longer mathematical explanations as part of your answer LaTeX is recommended, but legible handwritten work is also acceptable. If you have an image of handwritten work, you should first convert this to PDF format before uploading.
One option is to use www.smallpdf. com for creating pdf documents.
Questions
All numerical values in the report should be quoted to THREE decimal places. Failure to do so may result in a loss of marks.
Question 1.
Consider a population of 100,000 people at the age of 20. You should carry out a comparison of the Gompertz and De Moivre models for survival. In particular, you should:
(i) In your spreadsheet, calculate S0 (从) for each integer age from age 从 = 20 up to and including age 从 = 100 for each model (the limiting age for De Moivre’s model should be taken as w = 100).
(ii) Create a chart in your spreadsheet which compares S0 (从) for the two survival models.
(iii) Using your calculations from part (i), calculate l从 in your spreadsheet for each integer age from age 20 up to and including age 100 for both models.
(iv) Create a completed version of the table below using values from your spreadsheet. S0 (从)G
and l①(G) refer to the Gompertz model while S0 (从)D and l①(D) refer to De Moivre’s model.
Age(从) S0(G)(从) l①(G) S0(D)(从) l①(D) |
30 40 50 60 70 |
Your spreadsheet contains data from two life tables sourced from the UK office for national statis- tics (ONS) one for newborn lives in the period 1981-83 and in the period 2014-2016.
(v) In your spreadsheet, calculate p① for integer ages between 20 and 99 for each of the life tables provided. Now calculate p① for the same range of ages for the Gompertz model and De Moivre’s model and create a chart comparing the four data series.
(vi) Suppose you must select either the Gompertz model with the parameters provided or De Moivre’s model with limiting age w = 100 as the mortality model for life insurance provision for a new insurance company. Use your analysis from parts (i)-(v) to explain in a few sentences which model you would select.
Question 2
You should now use the ONS lifetable for newborn lives in 2014-16 provided. In your spreadsheet calculate the following for each integer age from 20 to 100: e①∶a|, A① , A , A(12b| and ① . The values for a and b are given in the parameter section of your spreadsheet. You may assume l101 = 0.
(i) Create a completed version of the following table.
Age(从) |
e①∶a| |
A① |
A(12) |
A(12) 1 ①∶b| |
① |
30 40 50 60 70 |
(ii) Write a mathematical explanation of your working to reach the values for A① and A(12b| (this explanation should not include discussion of Excel functions used or your spreadsheet).
Question 3
In this question you must use the interest rate provided for Q3 and not the rate for Q1.
You will investigate the profit or loss arising from experienced mortality. This is a topic some of you will study further in MATH3520 Actuarial Mathematics 2. Some concepts we have not yet discussed in lectures. There is no need to read ahead to complete this question.
(i) Your spreadsheet contains an extract from the standard actuarial tables. Use this extract
to calculate A①(1)∶n| and A①∶n| and where ① is the Age parameter for Q3 and n is the Term
parameter for Q3. Only calculate values for ages where you have the required information in the table.
(ii) Use the formulae given below to calculate the premium for an n-year term insurance policy and an n-year endowment insurance policy issued to a life initially aged ① with benefit B. The formulae assume that the benefit B is paid at the end of year of death (or at the end of the policy in the case of the endowment insurance) and premiums are paid at the start of each year within the term provided the policyholder is alive.
Term insurance premium =
Endowment insurance premium =
(iii) In the final chapter of the module we will introduce reserving. The “reserve” is the amount of money the insurer must hold at any given time to meet the policy benefits and not become insolvent.
Use the formula below to calculate the reserve, V, at time t = 1 for the term insurance or endowment insurance policy. You should select which policy based on the Type parameter.
V = Premium ⋅ (1 + i) − Bq① .
p①
Now assume we have a portfolio of N independent and identical policies which are either term insurance or endowment insurance according to your Type parameter.
(iv) The mortality profit arising in the first year for the portfolio described above is:
(expected number of deaths - actual number of deaths) × (B − V).
Using the actual number of deaths stated in your parameter values and the extract from the standard tables provided, calculate the mortality profit in the first year.
(v) Comment on this answer in one or two sentences.
Marking Scheme and Guidance
The assignment will be assessed in the categories shown in the table below. Marks are obtained by: accurately carrying out calculations; preparing a spreadsheet which is easy for others to understand; and providing good reasoning/explanations of concepts.
In a professional actuarial environment, these skills are just as important as your mathematical skills.
Category |
Criteria |
Numerical accuracy |
Marks are obtained by obtaining correct numerical answers and correct charts/tables. |
Presentation of Spreadsheet |
The contents of the spreadsheet should be clearly labelled and well organised. There should be a minimum of “hard coding’ ’ (i.e. typing numbers directly into formulas). |
Reasoning/Explanations |
For some questions some judgment, comparison or independent research may be required. You will gain marks for concise (i.e. not overly long) and relevant comments and justifications. |
The marks available for each question are shown in Gradescope. The majority of marks are in the first two categories.
Feedback
It is intended that the final marks for your submissions are released two weeks after the submission deadline.
In addition to your individual total mark, Gradescope comments will be provided on your individ- ual submission.
Moreover, group feedback will be provided to the class on any areas which proved to be more difficult.
2023-11-19