Topic 1 - Life Table Functions

Question 1

Welcome to Actuarial Mathematics 2

Pre-requisites: Actuarial Mathematics 1 (FIN1013)

MODULE DESCRIPTION
This module builds on the functions and techniques introduced in Actuarial Mathematics 1 (FIN1013), with the introduction of cash flows dependent on death, survival or other uncertainties.

CONTACT HOURS
Lectures times
Monday: 1pm –2pm – 6UQ/0G/006
Monday: 3pm -4pm - PFC/02/018

Tutorial Times & Location
Monday: 9am-10am – 14UQ/01/007
Monday: 10am-11am - 14UQ/01/007
Monday: 11am-12pm – 14UQ/01/007

MODULE AIMS
The aims of this module are to:
1. Introduce students to the concept of uncertainty in cash flows via decrements
2. Introduce life assurance and annuity functions and the impact of decrements on these functions
3. Describe and calculate net and gross premiums and reserves.

LEARNING OUTCOMES
At the conclusion of this module students will be able to:
1. Understand further actuarial functions allowing for decrements used and the mathematical techniques
employed by an actuary
2. Demonstrate the relationship between simple annuity and assurance functions
3. Solve equations of value to determine premium levels or reserves.

MODULE ASSESSMENT
Assessment type
Class Test (10%): The class test will occur midway through the semester and will test concepts
already introduced.
Assignment (15%): The assignment will consist of two parts:
i) Excel based project and
ii) Group presentation. The assignment is worth 15% of the overall mark for the module
Exam (75%): A two-hour end-of-term written exam worth 75% of the overall mark for the module. Exam questions will be mainly based on the lecture material and tutorial work.

READING LIST
ActEd Actuarial Mathematics 1 (CM1) notes

Students should ensure that they have a copy of the “Formulae and Tables for Examinations” booklet from the Institute and Faculty of Actuaries in advance of commencing this module.

INTERNATIONAL DIMENSION
This module introduces elements which focus on the influences of life expectancy of lives, including different
factors, such as age, gender, location, education etc. They are aware that life expectancy is a function of location which is a function of diet, climate, access to medical care, education etc

CONNECTIONS WITH PRACTICE
The assignment was borne out of experience of dealing with students in a professional environment. It is essential that students develop key excel and softer skills as well as technical model building skills to enhance their contribution at the start of their placement. The module co-ordinator is a Fellow of the Institute and Faculty of Actuaries and has incorporated professional expertise into the material. Students are taught the principles of building basic actuarial models and are also aware of the bridge between the principles taught within the module and how they can be used to build actuarial models and apply them in a professional environment.
Building on Actuarial Mathematics 1, students pull in more realistic assumptions such as mortality to understand the key influences on a pension scheme. Such as interest rates, inflation and mortality.

ETHICS, REPONSIBILITY AND SUSTAINABILITY
This module introduces student to some of the key legislation applying to pension schemes in the UK (Pensions Act 2004). They are made aware of the requirements that a scheme actuary (holder of a scheme actuary certificate and deemed to have sufficient experience and knowledge) is able to give advice to the pension scheme and sign off figures. Students are aware that there are a number of statutory roles performed by specific actuaries with specific certificates.
Students are also required to undertake an assignment with the objective to calculate a set of figures and deliver a presentation to a Trustee of a pension scheme. Students are encourage to develop their softer skills, such as team working, time management, communication (verbal, written and listening), integrity and professionalism.

TEACHING OUTLINE
Topic 1
Life Table Functions
- Introduction to survival probabilities
- Determine expressions for survival probabilities and life table functions
- Life table functions at non-integer ages

Overview of Course
- Actuarial Mathematics 2 introduces the concept of decrements and actuarial functions dependent on an eventuality
- This module builds on the principles of Actuarial Mathematics 1
- This module is analogous to the Profession’s Core Principle subject Actuarial Mathematics (CM1)
- This module covers syllabus items 4-6 are covered in Actuarial Mathematics 2
- Copies of the CM1 notes are held in the McClay library
- Additional reading –Actuarial Mathematics –Bowers, N.L.; Gerber, H.U. et al

Topic 1 Overview
- Recap
- Introduction to survival probabilities
- Probabilities of survival and mortality
- Survival probabilities and life table functions
- Life table functions at non-integer ages

Answer 1

Introduction

Question 2

Recap
In AM1 we looked at i and various relationships

• Express 1+i in terms of i(p)
• Express v in terms of i
• Express d in terms of v and/or i
• Express 1+i in terms of δ

Answer 2

1+i = (1+i(p)/p)^p
v = 1/1+i
d = (1-v) = i/1+i = iv
1+i = e^δ

Question 3

Introduction to survival probabilities

Why will references to annuities in this module differ from definitions met in AM1?

Answer 3

This module introduces decrements
- Death
- Withdrawal
- Retirement

Therefore references to annuities in this module differ from definitions met in AM1
- AM1 – annuities certain i.e. an¬ = (1−𝑣^𝑛)/𝑖
- AM2 –annuities will allow for mortality (q𝑥) a𝑥
- This is more realistic

Question 4

Give the standard notation for the probability a life aged x survives the next t years (i.e. survives to age x+t)

Answer 4

𝑡p𝑥

Question 5

What does the survival probability 𝑡p𝑥 stand for?

Answer 5

The probability a life aged x survives the next t years (i.e. survives to age x+t)

Question 6

Give the standard notation for the probability a life aged x survives the next t years (i.e. survives to age x+t) in terms of the force of mortality

Answer 6

𝑡p𝑥 = exp(-∫0,t 𝜇,𝑥+𝑠 ds) = e^(-∫0,t 𝜇,𝑥+𝑠 ds)

Question 7

What is 𝜇𝑥?

Answer 7

𝜇𝑥 is the force of mortality - instantaneous rate of mortality at x

Question 8

Give the standard notation for the probability a life aged x dies within the next t years (i.e. dies before age x+t)

Answer 8

𝑡q𝑥

Question 9

What does the survival probability 𝑡q𝑥 stand for?

Answer 9

The probability a life aged x dies within the next t years (i.e. dies before age x+t)

Question 10

State the formula for calculating the probability a life aged x dies within the next t years (i.e. dies before age x+t)

Answer 10

𝑡q𝑥 = ∫0,t 𝑠,𝑝,𝑥 . 𝜇,𝑥+𝑠 ds

Question 11

State the relationship between 𝑡p𝑥 and 𝑡q𝑥

Answer 11

𝑡p𝑥 = 1 - 𝑡q𝑥

Question 12

How does the 𝑡p𝑥 and 𝑡q𝑥 change if t=1?

Answer 12

p𝑥 and q𝑥

Question 13

What survival probabilities are included in the IFoA tables?

Answer 13

q𝑥 and 𝜇𝑥

Question 14

Explain and illustrate the principle of consistency

Answer 14

𝑡+𝑠,p,𝑥 = 𝑡,p,𝑥 x 𝑠,p,𝑥+𝑡 = 𝑠,p,𝑥 x 𝑡,p,𝑥+𝑠

• Probability life aged x survives to age x+t+s
• Relationship known as principle of consistency
• For example, set x = 40, t = 10 and s = 5
• 15p40 = 10p40 x 5p50 = 5p40 x 10p45
• Diagram page 15 week 1

Question 15

Give the standard notation and formula for calculating the probability a life aged x survives to age x+n then dies in the next m years (between age x+n and x+n+m)

Answer 15

n|m,q𝑥 = np𝑥 x mq,𝑥+n

Question 16

Give the standard notation and formula for calculating the probability a life aged 50 survives to age 65 then dies in the next 10 years

Answer 16

Set x = 50, n = 15 and m = 10

15|10,q50 = 15p50 x 10q65

Question 17

Give the standard notation and formula for calculating the probability a life aged x survives to age x+n then dies within a year

Answer 17

n|q𝑥 = np𝑥 x q,𝑥+n

Question 18

Life table functions

Give the standard notation for the following variables associated with life tables:
- starting age of life table
- number of lives at age α i.e. at start
- number of lives at age 𝑥
- limiting age of life table – max age
- number of deaths between 𝑥 and 𝑥+1

Answer 18

α
𝑙,α
𝑙,𝑥
ω
d,𝑥

Question 19

Life table functions

Give a definition for the following standard notation representing variables associated with the life tables:
- α
- 𝑙,α
- 𝑙,𝑥
- ω
- d,𝑥

Answer 19

• starting age of life table
• number of lives at age α i.e. at start (Known as radix)
• number of lives at age 𝑥
• limiting age of life table – max age
• number of deaths between 𝑥 and 𝑥+1

Question 20

Give the formula for calculating the number of lives at age 𝑥

Answer 20

For α ≤ 𝑥 ≤ ω and t ≥ 0

𝑙,𝑥 = 𝑙,α . 𝑥-α,pα

Question 21

Given that, for α ≤ 𝑥 ≤ ω and t ≥ 0,

𝑙,𝑥 = 𝑙,α . 𝑥-α,pα

Using independence and the above formula derive a function for tp𝑥 in terms of 𝑙

Answer 21

Step 1 (manipulate formula provided)
If 𝑙,𝑥 = 𝑙,α . 𝑥-α,pα
then 𝑥-α,pα = 𝑙,𝑥/𝑙,α

Step 2
tp𝑥 . 𝑥-α,pα = 𝑡+𝑥−α,pα (independence)
Therefore, tp𝑥 = (𝑡+𝑥−α,pα)/𝑥-α,pα

Using 𝑥-α,pα = 𝑙,𝑥/𝑙,α and 𝑡+𝑥−α,pα = 𝑙,𝑥+t/𝑙,α

tp𝑥 = (𝑙,𝑥+t/𝑙,α)/(𝑙,𝑥/𝑙,α)
tp𝑥 = (𝑙,𝑥+t/𝑙,α) . 𝑙,α/𝑙,𝑥 (𝑙,α cancels out on top and bottom)
tp𝑥 = 𝑙,𝑥+t/𝑙,𝑥

Question 22

Using tp𝑥 = 𝑙,𝑥+t/𝑙,𝑥 derive a function for tq𝑥

Answer 22

Remember tp𝑥 = 1 - tq𝑥
tq𝑥 = 1 - tp𝑥
tq𝑥 = 1 - 𝑙,𝑥+t/𝑙,𝑥
tq𝑥 = 𝑙,𝑥/𝑙,𝑥 - 𝑙,𝑥+t/𝑙,𝑥
tq𝑥 = (𝑙,𝑥 - 𝑙,𝑥+t)/𝑙,𝑥

Question 23

Recap
State the formula for valuing active member in the assignment

Answer 23

P x [(1+s)/(1+ipre)]^(nra-x) x 𝑎n¬(12) @ i’

Question 24

Remember the formula for valuing active
member in the assignment:
P x [(1+s)/(1+ipre)]^(nra-x) x 𝑎n¬(12) @ i’

What two key simplifications does this formula make?

Answer 24

• Assumed everyone survives to NRA
• Assumed everyone survives post NRA for specific period

Question 25

P x [(1+s)/(1+ipre)]^(nra-x) x 𝑎n¬(12) @ i’

Two key simplifications:
- Assumed everyone survives to NRA
- Assumed everyone survives post NRA for specific period

State a new formula which relaxes these assumptions

Answer 25

P x [(1+s)/(1+ipre)]^(nra-x) x l,nra/l,𝑥 x 𝑎,nra

Question 26

For α≤ x ≤ ω-1

State the formula for calculating d,𝑥 and q,𝑥

Answer 26

number of deaths between 𝑥 and 𝑥+1 = number of lives at age 𝑥 - number of lives at age 𝑥+1
d,𝑥 = 𝑙𝑥 −𝑙𝑥+1

q,𝑥 = (𝑙𝑥 −𝑙𝑥+1)/𝑙𝑥 = d,𝑥/𝑙𝑥

Question 27

What life table values are included in the IFoA tables?

Answer 27

𝑙𝑥 and d𝑥 included in the tables
(Look at page 74 of the tables)

Question 28

Using the useful relationships and formulae learned so far regarding survival probabilities & life table functions, show the different ways of expressing 2p60

Answer 28

2p60 = p60 x p61 = 𝑙61/𝑙60 x 𝑙62/𝑙61 = 𝑙62/𝑙60

Question 29

Survival probabilities & life table functions

Example
Using the AM 92 Ultimate tables calculate the following (pg 74-79):
p40
10p50
5q55

Answer 29

p40 = 1-q40 = 1 - 0.000937 = 0.999063

10p50 = 𝑙60/𝑙50= 9287.2164/9712.0728 = 0.95625

5q55 = (𝑙55 –𝑙60)/l55 = (9557.8179-9287.2164)/9557.8179 = 0.028312

Question 30

Survival probabilities & life table functions

Derive a formula for n|m,q𝑥 = np𝑥 . mq,𝑥+n in terms
of 𝑙x

Answer 30

n|m,q𝑥 = np𝑥 x mq,𝑥+n
n|m,q𝑥 = 𝑙,x+n/𝑙x . (𝑙,x+n - 𝑙,x+n+m)/𝑙,x+n
n|m,q𝑥 = (𝑙,x+n - 𝑙,x+n+m)/𝑙,x

Question 31

Survival probabilities & life table functions

Question
Using the AM 92 Ultimate tables calculate the
following:
p35
10q50
d55
5|10q40

Answer 31

p35 = 1-q35 = 1-0.000689 = 0.999311

10q50 = 1- 10p50= 1- 0.95625 = 0.04375

d55 = (𝑙55 – 𝑙56) = (9557.8179-9515.104) = 42.7139

5|10q40 = (𝑙45 –𝑙55)/𝑙40
= (9801.3123-9557.8179)/9856.2863 = 0.0247

Question 32

𝑡p𝑥 = exp(-∫0,t 𝜇,𝑥+𝑠 ds) = e^(-∫0,t 𝜇,𝑥+𝑠 ds)

How does this formula change if 𝜇 is a constant force of mortality

Answer 32

If μ is a constant force of mortality then this
formula becomes:
tp𝑥 = 𝑒^(−∫0,t 𝜇,𝑥+𝑠 ds) = e^(-tμ)

Question 33

Survival probabilities & life table functions

Example
A population is subject to a constant force of
mortality of 0.01, calculate 5p35 and 2q50.

Answer 33

5p35 = e^(-5x0.01) = 0.95123
2q50 = 1 - e^(-2x0.01) = 1 –0.9802 = 0.0198

Question 34

Question
A population is subject to a constant force of
mortality. If the probability of a life currently aged 30 surviving until age 40 is 0.8825, calculate μ.

Answer 34

10p30 = e(-10μ) = 0.8825
e^(-10μ) = 0.8825
-10μ = ln(0.8825)
μ = -ln(0.8825)/10 = 0.0125

Question 35

Non-integer ages

• So far only looked at integer ages
• Sometimes need to calculate 5p35.5

What is the first step required when calculating 5p35.5?

Answer 35

Break it down into the following
0.5p35.5 x 4p36 x 0.5p40

Question 36

What are the two methods to calculate non-integer age values?

Answer 36

• Uniform distribution of deaths (UDD)
• Constant force of mortality (CFM)

Question 37

What is the UDD method of calculating non-integer survival probabilities?

Answer 37

Remember sq𝑥 =∫0,𝑠 𝑡𝑝𝑥.𝜇,𝑥+𝑡 dt
Assume for x and 0 ≤t ≤1, 𝑡𝑝𝑥.𝜇,𝑥+𝑡 is constant
sq𝑥 =∫0,𝑠 𝑡𝑝𝑥.𝜇,𝑥+𝑡 dt = ∫0,𝑠 q𝑥 dt = s.q𝑥
tp𝑥 = sp𝑥 x t-s,p,𝑥+s for 0 ≤ s < t ≤1
t-s,q,𝑥+s = 1 - t-s,p,𝑥+s = 1 - tp𝑥 /sp𝑥 = 1 - (1-tq𝑥)/(1-sq𝑥)
1 - (1-t.q𝑥)/(1-s.q𝑥)
t-s,q,𝑥+s = (t-s)q𝑥 /(1-s.q𝑥)

Question 38

What is the CFM method of calculating non-integer survival probabilities?

Answer 38

Remember tp𝑥 =𝑒^(−∫0,𝑡 𝜇,𝑥+𝑟 dr)
Assume for 𝑥 and 0 ≤ t ≤ 1, 𝜇,𝑥+𝑡 = 𝜇 is constant
t-s,p,𝑥+s = 𝑒^(−∫s,𝑡 𝜇,𝑥+𝑟 dr) = 𝑒^-(t-s)μ for 0 ≤ s < t ≤1
Using p𝑥 = e^-μ
t-s,p,𝑥+s = (p𝑥)^t-s

Question 39

Non-integer ages - UDD

Calculate 5p35.5 using AM92 Ult

Answer 39

0.5p35.5 x 4p36 x 0.5p40

4p36 = 𝑙40/𝑙36 = 9856.2863/9887.6126
= 0.99683

0.5p40 = 1 - 0.5q40 = 1 - 0.5q40 = 1 - 0.5(0.000937)
= 0.99953

0.5p35.5 = 1 - 0.5q35.5 = 1 - [(0.5)q35/(1-0.5.q35)]
= 1 - [0.5(0.000689)/(1-0.5x0.000689)] = 0.999655

5p35.5 = 0.999655x0.99683x0.99953 = 0.996

Question 40

Non-integer ages - CFM

Calculate 5p35.5 using AM92 Ult

Answer 40

0.5p35.5 x 4p36 x 0.5p40

4p36 = 𝑙40/𝑙36 = 9856.2863/9887.6126
= 0.99683

0.5p40 =(p40)^0.5 = (1 - q40)^0.5
= (1 - 0.000937)^0.5 = 0.99953

0.5p35.5 = (p35)^0.5 =(1 - q35)^0.5
=(1 - 0.000689)^0.5 = 0.999655

5p35.5 = 0.999655x0.99683x0.99953 = 0.996

Question 41

Summary
- Introduced tp𝑥 and tq𝑥
- Looked at life table functions and
- The relationships between life table functions
and survival probabilities.
- Looked at life table functions at non-integer
ages.

Answer 41

Summary