SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT
BAS 300 – ACTUARIAL MATHEMATICS I
END OF SEMESTER FINAL EXAMINATION
MONDAY 7th JUNE, 2021
09:00 – 12:00 HOURS
TIME ALLOWED: 3 HOURS PLUS 5 MINUTES READING TIME.
INSTRUCTIONS TO CANDIDATES:
1. Read the instructions very carefully.
2. Check that you have the correct examination paper in front of you.
3. There are FIVE (05) questions in this paper. ANSWER ANY FOUR (04)
4. Write down the number of questions that you have answered on the cover of
the examination answer booklet provided.
5. Begin answering each question on a new page in the answer booklet
provided only.
6. No books, files or mechanical/ electronic aids are permitted in the
examination room. Use of non-programmable calculators is allowed.
Actuarial tables 2002 are a requirement for this examination.
7. There shall be NO communication among students during the examination.
Any students caught doing this will be disqualified.
DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO.
Page 1 of 5
QUESTION ONE
A. Describe the role of sensitivity analysis.
[5 Marks]
B. In assessing the suitability of a model for a particular exercise, what are the important
features to consider?
[7 Marks]
C. A UNILUS student wishes to model the size of insurance claims of a certain insurance
company. List the key steps the student will carry out in the modelling process.
[8 Marks]
D. Distinguish between a stochastic and deterministic model with the help of an example.
[5 Marks]
[TOTAL: 25 MARKS]
QUESTION TWO
A. For the force of mortality,  x that is known to follow Gompertz Law, calculate the
parameters B and c if 50  0.017609 and 55  0.028359
[5 Marks]
B. If x  0.01908  0.001 x  70 for x  55 , calculate 5 q60
[5 Marks]
C. Mortality of a group of lives is assumed to follow Gompertz Law. Calculate  x for a 30year old and a 70-year old, given that  x is 0.003 for a 50-year old and 0.01 for a 60year old.
[6 Marks]
D. Using ELT15 (Males) mortality, approximate the curtate expectation of life for
i.
A 1 year old baby and 7 year old boy.
[3 Marks]
ii.
A 16 year old UNILUS student and 20 year old actuarial intern.
[3 Marks]
iii.
A 35 year old actuarial analyst and 50 year old medical officer.
[3 Marks]
[TOTAL: 25 MARKS]
QUESTION THREE
A. Explain why lives might be ‘lost to the investigation’ if we carry out:
i.
A national investigation into the rate of death from natural causes.
[3 Marks]
ii.
A study of the mortality of life insurance policyholders.
[3 Marks]
B. An investigation is carried out into the mortality rates of married male accountants. A
group of 10,000 married male accountants is selected at random on 1 January 2016.
Page 2 of 5
Each member of the sample group supplies detailed personal information as at 1
January 2016 including name, address and date of birth. The same information is
collected as at each 1 January in the years 2017, 2018, 2019 and 2020. The
investigation closes in 2020. Describe the ways in which the available data for this
investigation may be censored.
[7 Marks]
C. Given that the likelihood function can be written as:
L    j j B 1   j 
k
d
n j d j
j 1
Show that the maximum likelihood estimate of  j is
dj
nj
for j  1,2,..., k
[6 Marks]
D. Butterflies of a certain species have short lives. After hatching, each butterfly
experiences a lifetime defined by the following probability distribution:
Lifetime (days)
Probability
1
0.10
2
0.30
3
0.25
4
0.20
5
0.15
Calculate  j for j  1,2,...,5 (to 3 decimal places)
[6 Marks]
[TOTAL: 25 MARKS]
QUESTION FOUR
A. A clinical trial has been carried out the effectiveness of a new drug. Sixty patients were
involved in a trial, which followed them for 2 years from the start of their treatment.
The following data show the period in complete months from the start of treatment to
the end of observation for those patients who died or withdrew from the trial before
the end of the 2-year period.
Deaths
8
10
10
16
20
Withdrawals
2
6
9
16
18
i.
22
Calculate the Kaplan-Meire estimate of the survival function [5 Marks]
Page 3 of 5
22
ii.
Construct an approximate 95% confidence interval for the probability that a
patient survives for at least 18 months after the start of the drug treatment.
[4 Marks]
B. A life insurance company has carried out a mortality investigation. It followed a sample
of independent policyholders aged between 50 and 55 years. Policyholders were
followed from their 50th birthday until they died, withdrew from the investigation while
still alive, or reached their 55th birthday (whichever of these events occurred first).
Describe the types of censoring that are present in this investigation.
[4 Marks]
C. 10,000 school children have been selected to take part in a one year medical study.
If the initial annual rate of mortality is 0.00025 for each child and deaths are expected
to occur independently, calculate the probability that 2 or more of the participants will
die before the end of the study.
[6 Marks]
D. A large computer company always maintains a workforce of exactly 5,000 young
workers, immediately replacing any worker who leaves. Calculate the probability that
there will be fewer than 3 deaths during any 6 month period, assuming that all workers
experience a constant force of mortality of 0.0008 per annum.
[6 Marks]
[TOTAL: 25 MARKS]
QUESTION FIVE
A. If mortality follows Gompertz Law such that  x  0.00003  1.1x calculate the values of
80 , 81, 82 , 83 (to 7 decimal places) and the first, second and third differences derived
from the quantities.
[6 Marks]
B. State the characteristics of a good graduation
[5 Marks]
C. Explain why it is necessary to graduate crude rates of mortality for practical use.
[6 Marks]
D. A study of causes of death in elderly men in the 1988s showed the proportions given
in the table below.
Cause of death
Cancer
Proportion of deaths
Number of deaths in
in 1988
2021
8%
286
Page 4 of 5
Heart disease
22%
805
Other
40%
1548
Respiratory disease
19%
755
Other causes
11%
464
circulatory
disease
Carry out a chi square test to determine whether these percentages can still be
considered to provide an accurate description of causes of death in 2021.
[8 Marks]
[TOTAL: 25 MARKS]
END OF EXAMINATION PAPER
Page 5 of 5