THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF RISK AND ACTUARIAL STUDIES
TERM 1 2025
ACTL3151
Assignment
Due: Friday 18 April 2025 5pm
This is an individual assignment. The total mark is 100 and is 20% of total
course mark.
Assignment Tasks
As a life actuary you are tasked with the following two tasks.
Task I (25 marks)
Create life tables that list the values for Ax, a¨x,
2Ax, (IA)x, and (Ia¨)x assuming an
annual interest rate of 3% and the following Makeham-Beard (1932) mortality law
for the hazard function:
µx (t) =
eϵ + eα+β(x+t)
1 + eα+ρ+β(x+t)
hence, ∫ t
0
µx (s) ds = te
ϵ +
e−ρ − eϵ
β
ln
(
1 + eα+ρ+β(x+t)
1 + eα+ρ+βx
)
with parameters:
α = −17.1472
β = 0.174283
ϵ = −5.96492
ρ = 0.247414
(Instructions: You’ll need to use software, like Excel, for the calculations.
However, you’re not limited to any specific software choice.)
Please arrange the values into columns, following the same order and structure as
AM92 of the Actuarial Tables. Please provide only the final numerical values here,
rounded at the third decimal at most, without including the calculation process and
Excel spreadsheets in the main document. If you choose to include calculation meth-
ods and Excel spreadsheets, add them to an appendix, which won’t be evaluated.
There are no specific guidelines for the appendix.
Page limit: 4 A4 pages maximum.
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Task II (75 marks)
Consider unit-linked endowment policies that offer a guaranteed minimum death
benefit with the following key features:
• The policies have a term of 20 years.
• There is a guaranteed minimum death benefit of $80, 000.
• On death of the policyholder, there is a benefit payable at the end of the policy
year of death of the basic sum insured or the bid value of the units allocated
to the policy, whichever is greater.
• The policyholder may surrender the policy only at the end of a policy year.
The surrender benefit is the same as the bid value of the units.
• Level monthly premiums are payable in advance until maturity or earlier death,
and 85% of each year’s premium is invested in units at the offer price.
• The units are subject to a bid-offer spread of 5%.
• The units are subject to an annual management charge of $50 plus 1% of the
bid value of the units. The management charge will be deducted at the end
of each policy year and before the death and surrender benefits are paid.
There are four options for the maturity benefit:
• Policy A: At maturity, 100% of the full bid value of the units is payable to the
policyholder if the policyholder is still alive.
• Policy B: At maturity, 105% of the full bid value of the units is payable to the
policyholder if the policyholder is still alive.
• Policy C: At maturity, 110% of the full bid value of the units is payable to the
policyholder if the policyholder is still alive.
• Policy D: At maturity, 120% of the full bid value of the units is payable to the
policyholder if the policyholder is still alive.
Calculate the monthly level premiums for the mentioned policies for a 45-year-
old, based on the following information:
• Initial Expenses: $500;
• Renewal Expenses: $70 at the start of each year starting from the second year;
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• Initial Commission: 20% of the first premium at the start of the first year;
• Renewal Commission: 2% of the premium at the start of each year starting
from the second year; No-renewal commission will be incurred to the company
during the years of suspension.
• Risk discount rate: 8% per annum;
• The company holds unit provisions/reserves equal to the full bid value of the
units. It sets up non-unit provisions (reserves) to zeroise any negative non-unit
fund cashflows.
• The company is targeting a profit margin of 12%.
To determine the premiums, you further make the following assumptions on the
following variables: mortality, rate of surrender, unit fund growth rate, and non-unit
fund interest rate.
Assumptions
• Mortality: Makeham-Beard hazard function with the same parameters as in
Task I;
• Independent rate of surrender: 5% pa for the standard policy;
• Unit fund growth rate: 7% per annum;
• Non-unit fund interest rate: 3% per annum;
• Since surrender is only allowed to happen at the end of the year, it is gener-
ally not worthwhile for a policyholder to surrender in the final year. So you
assume that there are no surrenders in the last year of the term (that is, the
independent rate of surrender in final year is 0).
(i) Provide the final numerical results for the annual premiums for the above poli-
cies. The premiums should be rounded to the nearest integer.
(Instructions: You’ll need to use software, like Excel, for the calculations. How-
ever, you’re not limited to any specific software choice.)
Page limit: 1/2 A4 pages maximum.
Font size: greater than Times New Roman font 12pt.
(ii) Outline the algorithm of your calculation in (i).
(This part is for the marker to decide whether the procedure you used in de-
termining the premiums is reasonable. You should include all the steps involved in
your calculations while describing the algorithm, but try to avoid paragraphs and
long sentences. Use bullet points and/or numbers to list the steps. You can use the
standard notations used in lecture notes and formulas. )
Page limit: 2 A4 pages maximum.
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Font size: greater than Times New Roman font 12pt.
(iii) Analyze the sensitivity of the profit margin to changes in the unit fund’s growth
rate. You should first recalculate the profit margin for each policy, using the (integer)
premium initially determined in (i) and adjusting the unit fund’s growth rate by +2%
and −2%. Then, recalculate the profit margin for each policy considering mortality
shifts where death rates are subject to shocks of +10% and +50% (i.e. qx×1.1 for a
shock of 10% at all ages), while maintaining all other factors constant, and provide
a brief discussion of the outcomes.
Only the numerical results for the profit margin adjustments and a discussion
should be provided here. Detailed calculations and methodologies should not be
included here.
Submission
• The font size of the main body of your assignment must be at least 12pt.
• The assignment you submit must have a title, your name and zID.
• Assignments must be submitted via the Turnitin Submission Inbox that is
available on the course website (in the Assignment section). As long as the
due date has not been reached, you can resubmit your work: the previous
version of your assignment will be replaced by the new version.
• All parts of your assignment must be uploaded as a unique pdf document.
• Please check Course Outlines for the School policy on late assignments.
• Students are reminded of the risk that technical issues may delay or even
prevent their submission (such as internet connection and/or computer break-
downs). Students should then allow enough time (at least 24 hours is recom-
mended) between their submission and the due time. The Turnitin module
will not let you submit a late report. No paper copy will be either accepted
or graded.
• Programming code or spreadsheets should not appear in the main body. They
must be provided as appendices at the end of your assignment.
• The font size requirement does not apply to the appendices.
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