MATH3510: Actuarial Mathematics 1 (Year 2022/23)
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MATH3510: Actuarial Mathematics 1 (Year 2022/23)
MATH3510 Coursework Assignment
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 de- scribed below) and provide selected answers/outputs in Gradescope (either by typing your answer or uploading a file containing your answer, PDF format is recommended in the latter case) . Marks available for each question are shown in Gradescope.
The submission deadline is 9am on Monday 28 November 2022. Late submissions will be accepted until 9am Monday 5 December, but late submission will result in a penalty of 15% of the marks awarded. (If you need more time due to mitigating circumstances you can apply for an extension in the usual way.)
To begin, download the Excel template file from Minerva (in the Assignment folder of Learning Resources) . Enter your student number in cell B1 of the Parameters worksheet and save the file. The template will create a set of parameter values for you to use (differ- ent students will have different parameters and your answers will be different to those of other students) . If you believe there are problems with your parameters, or the parameters do not generate, notify the module leader immediately.
You will see that some cells on the template are locked. You cannot edit these parts.
You should be able to edit all other parts of the file freely.
You are required to submit both your completed spreadsheet and your Gradescope answers to be eligible for any credit. 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.
As this work may contribute towards the award of actuarial exam exemptions (through your MATH3510 module grade), it is important that you prepare your own spreadsheet and summary document independently. You must enter your initials on the academic integrity statement and upload this as instructed in Gradescope. Students suspected of collusion may be excluded from eligibility from exemptions at the discretion of the independent examiner and be subject to the University’s own misconduct procedures.
All numerical values in your answers should be quoted to three decimal places.
1. In the Weibull survival model the force of mortality, µ北 , is
µ北 = kxn ,
where x > 0, n > 1 and k > 0. Using your values of the parameters k and n calculate the following in your spreadsheet.
(i) S0 (x) for all integer ages from 0 to 111 inclusive.
(ii) p北 for integer ages 0 to 110 inclusive.
(iii) Create and complete a version of the table below to submit in Gradescope.
Age(x) |
S0 (x) |
北 |
30 40 50 60 70 |
|
|
(In this, and other questions, you do not need to present the data for ages other than those shown in the table.)
(iv) In your spreadsheet, create an appropriate chart showing S0 (x) for the Weibull model. Add to your chart the survival function S0(M)(x) (for integer ages x from
0 to 111 inclusive) if we assume mortality follows Makeham’s law, S0(M)(x) = exp :-Ax - (c北 - 1)、,
where A = 0.0001, B = 4.25 × 10 −6 and c = 1.125. Your chart should be submitted as a PDF in Gradescope.
(v) You are required to select one of the survival functions to model human mortality for a life insurance application. State your choice briefly discuss your reasons. Your discussion should include a brief comparison of the two models.
2. In the template spreadsheet you are provided with two life tables sourced from the office for national statistics. One is for the UK population born between 1981 and
1983 and the other is for the UK population born between 2014 and 2016.
(i) Using the 1981-1983 life table, calculate q北 for integer ages between 20 and 100 inclusive in your spreadsheet. Then calculate e北 for all integer ages between 30 and 50. You may assume there are no lives aged 101.
In Gradescope, provide a mathematical description of your method. You may upload a file with a handwritten or typed answer. (Hint: Your explanation should not feature Excel formulas and references to your spreadsheet. Instead explain the results from the notes you use.)
(ii) Repeat the calculations for the 2014-16 table (again assuming there are no lives aged 101) and create and complete a version of the table below to submit in Gradescope.
Age(x) |
1981-83 q北 e北 |
2014-16 q北 e北 |
||
30 40 50 |
|
|
|
|
(iii) Comment on the differences. Include in your answer two possible reasons for the changes observed. You may need to do a little independent research to find possible reasons.
3. Using the 1981-83 life table from question 2 and your parameter values, calculate the following quantities for integer ages from 20: A北:m and . In each case consider carefully the highest age you can make this calculation based on the data you have.
You may assume there are no lives ages 101 and that deaths are uniformly distributed in each year of age. Create and complete a version of the table below to submit in Gradescope.
Age(x) |
北 |
A1 |
A北:m |
|
北 |
北:u |
|
|
30 40 50 60 70 |
|
|
|
|
|
|
4. A life insurance policy is structured as follows. The policy is issued to a life aged x years old. In this question x is one of your parameters. The policy holder will receive a benefit of ↔B1 at the end of year of death, provided death occurs within 35 years, i.e. K北 < 34. If the policyholder survives for at least 35 years, then the benefit is ↔B2 at the end of year of death. Use the values of B1 and B2 from your parameters worksheet.
To answer the following you may use the 1981-83 lifetable from question 2, your interest rate parameter and your calculations from Q3 as needed. You may also assume that deaths are uniformly distributed in each year of age.
(i) Calculate the expected present value of benefits, EPV (Benefits), in your spread- sheet.
(ii) The policy holder pays insurance premiums at the start of each month. The monthly premium amount is P . In your spreadsheet, evaluate the expected
present value of the premiums paid by the policy holder, EPV (Premiums), for
values of P between ↔0 and ↔100 (in steps of ↔5. For each value of P , calculate
EPV (Benefits) - EPV (Premiums).
Create a chart of this information to upload to Gradescope.
(iii) Estimate the value of P where EPV (Benefits) - EPV (Premiums) = 0 and include this point on your chart. (Hint: You may wish to use the “goal seek” function within Excel to help you do this. You may also use other basic methods to estimate this.)
(iv) In order to make a profit, the actual insurance premiums are higher than the level of the net premium, this is known as “profit loading” .
Suppose a particular policy has monthly premium £A (payable at the start of each month). The profit loading means that these premiums are increased by £B, i.e. the total monthly premium is £(A + B).
Produce a chart showing the expected present value of the total monthly premi- ums (i.e. including the profit loading) in the first two years. Your chart should also show how the expected present value is split between the profit loading and the premium.
Outline 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; presenting charts which are easy to read and providing good reasoning where required. In a professional actuarial environment, these skills are just as important as your mathematical skills.
Numerical accuracy: Marks are obtained by obtaining cor-
rect 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 you use).
Presentation and reasoning: Presentation marks are awarded based
on absence of typos and easy to read
charts. For some questions some judg-
ment, comparison or independent re-
search is 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.
2022-11-25