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MATH3063/6129 Actuarial Mathematics I

Practical Assignment 2022/23

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

Completed work should be submitted via Blackboard before 23:59 on Monday, 12 Dec

2022. The deadline is strict and penalties for late work will be applied in accordance with the

University’s late work policy.

Your submission should include a written report and the Excel spreadsheet of your calculations.

Please note that your last attempt will be marked. Note that all the answers must be

presented in your written report. Therefore please avoid expressions such as ”Please see the

spreadsheet” in the report. The Excel spreadsheet is submitted to prove that the work is done

using Excel and also to check the accuracy of the answers presented in the report.

There is a strict limit of four 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 Integrity Guidance)

To submit your report and Excel spreadsheet go to the Blackboard page of MATH3063/6129,

under the Assignments tab there is an assignment called ”Practical Assignment Submission”. In

your submission please attach the following two files:

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

(2) The Excel spreadsheet called spreadsheet-ID.xls, where ID is your student ID number.

Where necessary, use the repeated Simpson’s Rule with a step size of 0.25 years.

You are given the following survival model:

Ultimate rates: Makeham’s law given by

μx = A+B c

x

Select rates: 2 year select period, q[x] = 0.70qx and q[x]+1 = 0.80qx+1

Choose the following :

A between 1× 104 and 2× 104

B between 2× 104 and 3.5× 104

c between 1.075 and 1.08

Also choose a rate of interest i between 3% and 6% per annum effective.

State these values clearly in your report and use the same values throughout the

assignment.

Assume ω = 110.

(a) (i) [20 marks] Calculate tpx and tp[x] for x = 30, 45, 60 and t = 0, 1, 2, . . . , 110?x. [Hint: For

this part, you can present results in the report by completing a table similar to below.]

t tp30 tp[30] tp45 tp[45] tp60 tp[60]

0 1.000000 1.000000 1.0000000 1.000000 1.000000 1.000000

(ii) [10 marks] Plot the survival probabilities tpx and tp[x] for x = 30, 45, 60 against time (on

the same graph in Excel) and briefly comment on this plot. Make sure the lines and the axes

are clearly labelled in Excel.

(b) (i) [5 marks] Calculate ex and e[x] for x = 30, 45, 60.

(ii) [10 marks] Using numerical integration, calculate e?x and sd(Tx) for x = 30, 45, 60.

(iii) [10 marks] Comment briefly on your findings in parts (b)(i) and (b)(ii).

(c) Santa is aged exactly 60 years and has made a proposal to a life office for a whole life

insurance. Premiums P are payable annually in advance for a maximum of 15 years. The

sum insured is payable at the end of the year of death, and is ￡500,000 on death during the

first 15 years, and ￡100,000 thereafter.

(i) [35 marks] Calculate P using the select survival model above and the interest rate i per

year. Assume that the expenses will be

20% of the first year’s premium and 5% of all premiums after the first year

on each premium date an additional expense starting at ￡20 and increasing with infla-

tion from that date at a (compound) rate of i

2

% per year.

(ii) [10 marks] The underwriters of the life office consider Santa to have a higher than normal

mortality risk because of excessive consumption of mince pie and his dangerous occupation of

driving an overloaded sleigh at heights. Accordingly, they consider Santa’s risk of mortality

is equivalent to a constant addition of 0.002416 to the normal force of mortality at all ages.

Using select survival model calculate the curtate life expectancy of Santa. Compare this

value with your finding in part (b). Briefly comment on this result.

[Total 100 marks]

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