These questions are taken from the 2015 (April and October) IFoA papers, with additional parts

in some cases to give them a flavour of the UEA exams

The marks available are based on the IFoA papers being 100 marks for a 3 hour paper.

April 2015 Q2
2) The mortality of a rare form of flying beetle is being studied. It has been discovered that beetles kept in
a protected environment have a constant force of mortality, μ, but that those in the wild have a force of
mortality which is 50% higher. It has been proven that the beetles revert immediately to the higher rate of
mortality if they are released from the protected environment.

A beetle born and always living in the wild has a 58% chance of living for eight days.

Calculate the probability of living the same length of time for:

(a) a beetle born and reared in the protected environment.

(b) a beetle born in the protected environment which is scheduled to be released into the wild after
six days. [4]

April 2015 Q5
5) (i) State the principle of correspondence as it applies to death rates. [1]

A nightclub opens at 10.00 p.m. and closes at 2.00 a.m. It admits only people aged over 21 years on the
production of an identity card giving date of birth.

The table below shows the number of people entering in various intervals between 10.00 p.m. and 2.00
a.m. on 30 June 2013. No-one was admitted after 1.00 a.m., and you may assume that all those who enter
the premises stay until 2.00 a.m.

Year of birth 10.00–11.30pm 11.30–12.00 pm 12.00 p.m.–1.00 a.m.

.
1989 100 300 200
1990 200 400 350
1991 150 400 300
1992 100 250 200

During the period of opening, 40 people aged 22 last birthday required medical attention for heat
exhaustion.

(ii) Calculate the rate per person-hour at which those attending the night club aged 22 last birthday
required medical attention for heat exhaustion, stating any assumptions you make. [6]
[Total 7]
April 2015 Q7 (changed)
7) i) Describe what is meant by a Markov chain. [2]

A simplified model of the internet consists of the following websites with links between the websites as
shown in the diagram below. You may assume transition intensities from one state to another are all
constant

N(ile) B(anan
a)

C(hee H(andbook
p) )

BN
Let t p x be the probability of a browser being on page Banana at time x being on page Nile at time
x+t etc
Let  x
BN
be the transition intensity from state Banana to state Nile at time x.
 BB
ii) Using consistent notation derive a relationship for t px [6]
t

[Total 8]
April 2015 Q12
12) A study was made of a group of people seeking jobs. 700 people who were just starting to look for
work were followed for a period of eight months in a series of interviews after exactly one month, two
months, etc. If the job seeker found a job during a month, the job was assumed to have started at the end
of the month.

Unfortunately, the study was unable to maintain contact with all the job seekers.

The data from the study are shown in the table below:

Months since Found employment Contact lost

start of study
1 100 50
2 70 0
3 50 20
4 40 20
5 20 30
6 20 60
7 12 38
8 6 0

(i) (a) Describe two types of censoring present in the investigation.

(b) Describe an example of a person to whom each type applies. [3]

(ii) Calculate the Kaplan-Meier estimate of the function for “remaining without employment”. [6]

A Weibull distribution with a rate h(t) given by the formula h(t )     t  1 was fitted to these data.
The estimated value of λ was 0.18 and the estimated value of β was 0.3.

(iii) Test the goodness-of-fit of the data to this Weibull distribution. [6]
[Total 15]
September 2015 Q6

6) (i) Describe what is meant by a proportional hazards model. [3]

A pharmaceutical company is interested in testing a new treatment for a debilitating but non-fatal
condition in cows. A randomised trial was carried out in which a sample of cows with the
condition was assigned to either the new treatment or the previous treatment. The event of
interest was the recovery of a cow from the condition. The results were analysed using a Cox
regression model.

The final model estimated the hazard, h(t, x) as:

h(t , x)  h0 (t ) exp( 0 z   1 x   2 xz )
where:

h0 (t ) is the baseline hazard;

z is a covariate taking the value 1 if the cow was assigned the new treatment and 0 if the cow was
assigned the previous treatment;

x is a covariate denoting the length of time (in days) for which the cow had been suffering from
the condition when treatment was started;

and t is the number of days since treatment started.

 0 ,  1 and  2 are parameters. Their estimated values were  0 = 0.8,  1 = 0.4

and  2 = - 0.1.

(ii) Determine the characteristics of the baseline cow. [1]

For a particular cow, the new treatment and the previous treatment have exactly the same hazard.

(iii) Calculate the number of days for which that cow had the condition before the initiation of
treatment. [2]

Under the previous treatment, cows whose treatment began after they had been suffering from
the condition for three days had a median recovery time of 14 days once treatment had started.

(iv) Calculate the proportion of these cows which would still have had the condition after 14
days if they had been given the new treatment. [4]
[Total 10]
April 2015 Q9

9) (i) Describe an example of a situation when graduation by parametric formula would be used. [1]

(ii) State two advantages and two disadvantages of graduation by parametric formula. [4]

(iii) (a) Explain why the χ2 test is different when considering the goodness of fit of graduated data
compared with when considering the similarity of two sets of data.

(b) Describe how this is dealt with when the graduation has been carried out by parametric formula. [4]
[Total 9]

April 2015 Q10

10) In a computer game a player starts with three lives. Events in the game which cause the player to lose
a life occur with a probability dt  o(dt ) in a small time interval dt.
However the player can also find extra lives. The probability of finding an extra life in a small time
interval dt is dt  o(dt ) . The game ends when a player runs out of lives.
(i) Outline the state space for the process which describes the number of lives a player has. [1]

(ii) Draw a transition graph for the process, including the relevant transition rates. [3]

(iii) Determine the generator matrix for the process. [2]

(iv) Explain what is meant by a Markov jump chain. [1]

(v) Determine the transition matrix for the jump chain associated with the process. [2]

(vi) Determine the probability that a game ends without the player finding an extra life. [1]
[Total 10]
April 2015 Q11
11 A new disease has been discovered which is transmitted by an airborne virus.

Anyone who contracts the disease suffers a high fever and then in 60% of cases dies within an hour and in
40% of cases recovers. Having suffered from the disease once, a person builds up antibodies to the
disease and thereafter is immune.

(i) Draw a multiple state diagram illustrating the process, labeling the states and possible transitions
between states. [2]

(ii) Express the likelihood of the process in terms of the transition intensities and other observable
quantities, defining all the terms you use [4]

(iii) Derive the maximum likelihood estimator of the rate of first time sickness. [2]

Three years ago medical students visited the island where the disease was first discovered and found that
of the population of 2,500 people, 860 had suffered from the disease but recovered. They asked the
leaders of the island to keep records of the occurrence and the outcome of each incidence of the disease.
The students intended to return exactly three years later to collect the information.

(iv) Derive an expression (in terms of the transition intensities) for the probability that an islander who
has never suffered from the disease will still be alive in three years’ time. [4]

(v) Set out the information which the students would need when they returned three years later in order to
calculate the rate of sickness from the disease. [2]
[Total 14]
April 2015 Q3

3) (i) Explain what is meant by a proportional hazards model. [3]

(ii) Outline three reasons why the Cox proportional hazards model is widely used in empirical work. [3]
[Total 6]

April 2015 Q8
8) (i) State why it is important to divide data into homogeneous classes when undertaking mortality
investigations. [2]

(ii) List four factors, apart from smoking behaviour, by which mortality data are often classified by life
insurance companies. [2]

In a particular life insurance market, it has for many years been the practice for all companies to charge
smokers higher premiums than non-smokers for the same term assurance policy. Suppose one company
decides to switch to charging smokers and non-smokers the same premiums for term assurance policies.

The other companies retain differential pricing for smokers and non-smokers.

(iii) Discuss the likely implications for the company making the switch. [4]
[Total 8]

September 2015 Q1

1 List four factors, other than age and sex, by which mortality statistics are often
subdivided. [2]

September 2015 Q2

2 Describe the differences between a stochastic and a deterministic model. [4]

September 2015 Q3

3 (i) Define how the following forms of censoring arise in a survival investigation:
 right censoring
 type I censoring
 random censoring [3]
An experience analysis is conducted where the event of interest is the lapse of a term assurance
policy.

(ii) Explain whether each form of censoring listed in part (i) occurs in each of the following
situations. If it is not possible to state whether a form of censoring occurs, explain why this is the
case.

(a) A policyholder dies.

(b) A subset of the policies is migrated to a new administration system and no data are provided
from the new system to the experience analysis team.

(c) A policy reaches its maturity date. [4]

[Total 7]
September 2015 Q4
4) Company A and Company B are two small insurance companies which have recently merged
to form Company C. Company C is reviewing its premium rates for a whole of life product and
so is conducting an analysis of mortality rates experienced.

Company A recorded the number of policies in force every 1 January using a definition of age
next birthday whereas Company B recorded the number of policies in force every 1 April using
an age definition of age last birthday. Both companies recorded deaths as they happened using an
age definition of age last birthday.

These are the data for the most recent years.

Company A
Age next Number of Number of Number of
birthday policies policies policies
1 Jan. 2012 1 Jan. 2013 1 Jan. 2014
51 8,192 6,421 8,118
52 7,684 8,298 7,187
53 9,421 8,016 9,026

Company B
Age last Number of Number of Number of
birthday policies policies policies
1 Jan. 2012 1 Jan. 2013 1 Jan. 2014
51 4,496 3,817 4,872
52 5,281 5,218 3,812
53 4,992 5,076 5,076

In the calendar year 2013 Company A recorded 28 deaths of those aged 52 last birthday and
Company B recorded 17 deaths of those aged 52 last birthday.

(i) Estimate the force of mortality for the combined company for age 52 last birthday, stating all
assumptions that you make. [6]

(ii) Explain the exact age to which your estimate applies. [1]
[Total 7]
September 2015 Q7

7) A school offers a one year course in a foreign language as an evening class. This is divided
into three terms of 13 weeks each with one lesson per week. At the end of each lesson all the
students sit a test and any that pass are awarded a qualification, and no longer attend the course.

Last year 33 students started the course. Of these 13 dropped out before completing the year, and
16 passed the test before the end of the year. The last lesson attended by the students who did not
stay for the whole 39 lessons is shown in the table below along with their reason for leaving.
Number of Last lesson Reason for
Students attended leaving

5 1 Dropped out
1 6 Dropped out
2 7 Passed test
2 13 Dropped out
5 14 Passed test
6 27 Passed test
4 28 Dropped out
1 30 Dropped out
3 36 Passed test

(i) Calculate the Nelson-Aalen estimate of the survival function. [5]

(ii) Sketch a graph of the Nelson-Aalen estimate of the survival function, labeling the axes. [2]

(iii) Determine the probability that a student who starts the course passes by the end of the year.
[1]

Since only four students had not passed by the end of the year and a total of 16 had passed, the
school claims in its publicity that 80% of students are awarded the qualification by the end of the
year.

(iv) Comment on the school’s claim in light of your answer to part (iii). [2]
[Total 10]
September 2015 Q9
9) Doctors at a health centre are carrying out an investigation to see if obesity affects the
likelihood of dying from heart disease. They propose to use a model with four states:

1. Obese
2. Not obese
3. Dead due to heart disease, and
4. Dead due to other causes

(i) Write down, defining all the terms you use, the likelihood for the transition intensities. [3]

(ii) Derive the maximum likelihood estimator of the force of mortality from heart disease for
Obese people. [3]

The investigation has followed several thousand people aged 50–59 years for five years and has
the following data:

Waiting time in state Obese (in person-years) 14,392

Waiting time in state Not obese (in person-years) 18,109
Number of deaths due to heart disease for those persons who are Obese 178
Number of deaths due to heart disease for those persons who are Not obese 190
Number of deaths due to other causes for those persons who are Obese 89
Number of deaths due to other causes for those persons who are Not obese 53

The doctors want to promote healthy living and therefore wish to claim that Obese people have a
much higher chance, statistically, of dying from heart disease than do people who are Not obese.

(iii) Test whether this claim is true at the 90% confidence level. [5]
[Total 11]
September 2015 Q8
8 (i) Define a Markov Jump Process. [1]

A company provides phones on contracts under which it is responsible for repairing or replacing
any phones which break down.

When a customer reports a fault with a phone, it is immediately taken to the company’s repair
shop and it is assessed whether it can be fixed (meaning fixable at reasonable cost). Based on
previous experience, it is estimated that the probability of a phone being fixable is 0.75. If a
phone is not fixable it is discarded and the customer is provided with a new phone.

If a repaired phone breaks again the company, in line with its customer charter, will not attempt
to repair it again, and so discards the phone and replaces it with a new one.

The status of a phone is to be modeled as a Markov Jump Process with state space
{Never Broken (NB), Repaired (R), Discarded (D)}.

The company considers the rate at which phones break down to vary according to whether a
phone has previously been repaired as follows:

Status Probability of break down in small

interval of time, dt
Never Broken 0.1dt + o(dt)
Repaired 0.2dt + o(dt)

(ii) Draw a transition diagram for the possible transitions between the states, including the
associated transition rates. [2]

Let PNB (t ) , PR (t ) and PD (t ) be the probabilities that a phone is in each state after time t
since it was provided as a new phone.

(iii) Determine Kolmogorov’s forward equations in component form for PNB (t ) , PR (t ) and
PD (t ) . [2]

(iv) Solve the equations in part (iii) to obtain PNB (t ) , PR (t ) . [4]

(v) Calculate the probability that a phone has not been discarded by time t. [1]
[Total 10]
September 2015 Q11
11 (i) Describe why an insurance company might want to compare the results of a mortality
investigation with previous experience. [2]

A large life insurance company has undertaken an investigation of the mortality of its
policyholders. Currently it assumes that mortality at age x,  x , is equal to a standard table. The
company wishes to use the results from the investigation to see whether the standard table is still
appropriate. Below are shown some data from the
investigation.

Age x Number of Actual death Expected death

policies in force claims claims from
standard table
70 1,000 13 23.74
71 1,200 28 31.80
72 1,100 31 32.50
73 1,100 34 36.20
74 1,000 39 36.63
75 1,000 41 40.73
76 950 41 42.99
77 900 40 45.20
78 850 46 47.34
79 800 48 49.35

(ii) Perform an overall test of the hypothesis that the underlying mortality of the company’s
policyholders is, over this range of ages, represented by the standard table. [6]

(iii) Evaluate the suitability of the standard table for use in the company’s financial modelling by
performing two additional tests for different possible inconsistencies between the actual death rates
and those represented by the standard table. [6]

The company discovers that at age 70 years, one individual owns 25 of the policies in the
investigation, the remaining policies each being owned by different individuals.

(iv) Assess the impact of this on the variance of the number of claims at age 70 years. [4]
[Total 18]
April 2011 Q1
1 Give three advantages of the two-state model over the Binomial model for estimating transition
intensities where exact dates of entry into and exit from observation are known. [3]

April 2011 Q3
3 Describe the ways in which the design of a model used to project over only a short time frame may
differ from one used to project over fifty years. [4]

April 2011 Q5
5 (i) Explain why a mortality experience would need to be graduated. [3]

An actuary has conducted investigations into the mortality of the following classes of lives:

(a) the female members of a medium-sized pension scheme

(b) the male population of a large industrial country
(c) the population of a particular species of reptile in the zoological collections of the southern
hemisphere

The actuary wishes to graduate the crude rates.

(ii) State an appropriate method of graduation for each of the three classes of lives and, for each
class, briefly explain your choice. [3]
[Total 6]

April 2011 Q6
6 A study of the mortality of a certain species of insect reveals that for the first 30 days of life,
the insects are subject to a constant force of mortality of 0.05. After 30 days, the force of
mortality increases according to the formula:

 30  x  0.05 exp(0.01x )

where x is the number of days after day 30.

(i) Calculate the probability that a newly born insect will survive for at least 10 days. [1]

(ii) Calculate the probability that an insect aged 10 days will survive for at least a further 30
days. [3]

(iii) Calculate the age in days by which 90 per cent of insects are expected to have died. [4]
[Total 8]
April 2011 Q8
8 (i) Explain the difference between the central and the initial exposed to risk, in the context of
mortality investigations. [2]

An investigation studied the mortality of infants aged less than 1 year. The following table gives
details of 10 lives involved in the investigation. Infants with no date of death given were still
alive on their first birthday.

Life Date of birth Date of death

1 1 August 2008 -
2 1 September 2008 -
3 1 December 2008 1 February 2009
4 1 January 2009 -
5 1 February 2009 -
6 1 March 2009 1 December 2009
7 1 June 2009 -
8 1 July 2009 -
9 1 September 2009 -
10 1 November 2009 1 December 2009

(ii) Calculate the maximum likelihood estimate of the force of mortality, using a two-state model
and assuming that the force is constant. [3]

(iii) Hence estimate the infant mortality rate, q 0 . [1]

(iv) Estimate the infant mortality rate, q 0 ,using the initial exposed to risk. [1]

(v) Explain the difference between the two estimates. [2]

[Total 9]
April 2011 Q9
9 (i) Define a Markov jump process. [2]

A study of a tropical disease used a three-state Markov process model with states:

1. Not suffering from the disease

2. Suffering from the disease
3. Dead

The disease can be fatal, but most sufferers recover. Let t p xij be the probability that a person in
state i at age x is in state j at age x+t. Let  xt be the transition intensity from state i to state j at
ij

age x+t.

(ii) Show from first principles that:

d
t px t px
13 11
 13
x t  t p x
12
 x23t
dt

The study revealed that sufferers who contract the disease a second or subsequent time are more
likely to die, and less likely to recover, than first-time sufferers.

(iii) Draw a diagram showing the states and possible transitions of a model which allows for this
effect yet retains the Markov property. [3]
[Total 9]
April 2011 Q10
10 At Miracle Cure hospital a pioneering new surgery was tested to replace human lungs
with synthetic implants. Operations were carried out throughout June 2010. Patients
who underwent the surgery were monitored daily until the end of August 2010, or
until they died or left hospital if sooner. The results are shown below. Where no date
is given, the patient was alive and still in hospital at the end of August.
Patient Date of surgery Date of leaving Reason for
Observation leaving
observation
A June 1 June 3 Died
B June 3 July 2 Left Hospital
C June 5
D June 8
E June 9 July 11 Died
F June 12
G June 16 June 21 Died
H June 17 Aug 12 Left Hospital
I June 22
J June 24 June 29 Died
K June 25 Aug 20 Died
L June 26
M June 29 Aug 6 Left Hospital
N June 30

(i) Explain whether each of the following types of censoring is present and for those present
explain where they occur:

• right censoring
• left censoring
• informative censoring [3]

(ii) Calculate the Kaplan-Meier estimate of the survival function for these patients, stating all
assumptions that you make. [6]

(iii) Sketch, on a suitably labelled graph, the Kaplan-Meier estimate of the survival function. [2]

(iv) Estimate the probability that a patient will die within four weeks of surgery. [1]
[Total 12]
April 2011 Q11
11 An historian has investigated the force of mortality from tuberculosis in a particular town in a
developed country in the 1860s using a sample of records from a cemetery.

He wishes to test whether the underlying mortality from tuberculosis in the town is the same as
the national force of mortality from this cause of death, as reported in death registration data.

The data are shown in the table below.

Age-group Deaths in Central exposed to National force

Sample risk in sample of mortality
5–14 13 3,685 0.0051
15–24 47 2,540 0.0199
25–34 52 1,938 0.0309
35–44 50 1,687 0.0316
45–54 33 1,386 0.0286
55–64 23 1,018 0.0230
65–74 13 663 0.0202
75–84 3 260 0.0070

(i) Carry out an overall test of the null hypothesis that the underlying mortality from tuberculosis
in the town is the same as the national force of mortality, and state your conclusion. [6]

(ii) (a) Identify two differences between the experience of the sample and the national experience
which the test you performed in (i) might not detect.

(b) Carry out a test for each of the differences in (ii)(a). [7]

(iii) Comment on the results from all the tests carried out in (i) and (ii). [1]
[Total 14]
April 2014 Q4 (already done as class example)

4 (i) State the principle of correspondence as it relates to mortality investigations.[1]

Two small countries conduct population censuses on an annual basis. Country A records its
population on 1 February every year based on an age definition of age last birthday. Country B
records its population on every 1 August using a definition of age nearest birthday. Each country
records deaths as they happen based on age next birthday.

Below are some data from the last few years.

Country A

Age last Population Population Population

Birthday 1 February 2011 1 February 2012 1 February 2013

44 382,000 394,000 401,000

45 374,000 381,000 385,000
46 354,000 372,000 375,000

Country B
Age nearest Population Population Population
Birthday 1 August 2011 1 August 2012 1 August 2013

44 382,000 394,000 401,000

45 374,000 381,000 385,000
46 354,000 372,000 375,000

In the combined lands of Countries A and B in the calendar year 2012 there were 4,800 deaths of
those aged 46 next birthday and 4,500 deaths of those aged 45 next birthday.

The two countries decide to form an economic union, after which it will be mandatory to offer
the same rates for life insurance to residents of each country.

(ii) Estimate the death rate at age 45 years last birthday for the two countries
combined. [6]

(iii) Explain the exact age to which your estimate relates. [1]
[Total 8]