Life Insurance Mathematics
Complete Course Summary
Survival Models, Life Tables & Selection, Insurance Benefits,
Life Annuities, Premiums, and Policy Values
Lecture Notes, Step-by-Step Solution Guides &
Worked Exam Exercises
This document is an independent study summary and is not affiliated with, endorsed by, or a substitute for
any university’s official course materials.
,Life Insurance Mathematics — Complete Course Summary 1
Contents
I Lecture Notes & Worked Exercises 3
1 Survival Models 3
1.1 The Future Lifetime Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Force of Mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Actuarial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Mean and Standard Deviation of Tx . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Curtate Future Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Life Tables, Fractional Ages, and Selection 5
2.1 Life Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Fractional Age Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Select Mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Insurance Benefits 7
3.1 Valuation Functions and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Whole Life Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Term Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Pure Endowment and Endowment Insurance . . . . . . . . . . . . . . . . . . . . . . 8
3.5 Deferred Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
(m)
3.6 Relating Āx , Ax and Ax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Life Annuities 9
4.1 Review of Annuities-Certain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Annual Life Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Continuous Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.4 Annuities Payable m-thly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.5 Deferred and Guaranteed Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 Premiums 11
5.1 Premium Calculation Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2 Gross Premiums and Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.3 Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.4 Portfolio Percentile Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Policy Values 13
6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.2 Recursive Formula for Policy Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
II Quick-Reference Step-by-Step Guides 15
7 Guide: Survival Probabilities & Life Tables 15
8 Guide: EPV of Insurance Benefits 16
9 Guide: EPV of Annuities 16
10 Guide: Premium Calculation 17
11 Guide: Policy Values and Recursions 17
,Life Insurance Mathematics — Complete Course Summary 2
12 Guide: Profit and the Portfolio Percentile Premium (Extended) 18
III Worked Exam Exercises 19
13 Survival Models and Life Tables 19
14 Insurance Benefits 20
15 Life Annuities 20
16 Premiums 21
17 Policy Values 22
IV Practice Questions 24
,Life Insurance Mathematics — Complete Course Summary 3
Part I Lecture Notes & Worked Ex-
ercises
1 Survival Models
1.1 The Future Lifetime Random Variable
Consider a life aged x, denoted (x), where x ≥ 0. The future lifetime of (x) is a positive, continuous
random variable Tx . Then x + Tx represents the age at death of (x).
Let Fx be the lifetime distribution from age x, i.e. the cdf of Tx :
Fx (t) = Pr[Tx ≤ t] (probability that (x) dies before age x + t),
and let Sx be the corresponding survival function:
Sx (t) = 1 − Fx (t) = Pr[Tx > t] (probability that (x) survives beyond age x + t).
Now consider the collection of random variables {Tx }x≥0 , and let T0 and Tx refer to one and the
same individual, currently aged x. If this person dies within t years, then Tx ≤ t and T0 ≤ x + t.
Since the events {Tx ≤ t} and {T0 ≤ x + t | T0 > x} describe the same thing (given that the
individual has already survived to age x),
Pr[x < T0 ≤ x + t] F0 (x + t) − F0 (x)
Fx (t) = Pr[Tx ≤ t] = Pr[T0 ≤ x + t | T0 > x] = = ,
Pr[T0 > x] S0 (x)
which implies
S0 (x) − S0 (x + t) S0 (x + t)
Fx (t) = , Sx (t) = .
S0 (x) S0 (x)
This gives the multiplication rule, one of the most important identities in the course:
S0 (x + t) = S0 (x) · Sx (t)
where S0 (x + t) is the probability of survival from birth to age x + t, S0 (x) is the probability of
survival from birth to age x, and Sx (t) is the probability, having survived to age x, of further
surviving to age x + t.
More generally, for any u, t ≥ 0:
Sx (t + u) = Sx (t) · Sx+t (u).
1.2 Force of Mortality
For a very small interval dx, the quantity µx dx is (approximately) the probability that a life who
has attained age x dies before attaining age x + dx:
µx dx ≈ Pr[T0 ≤ x + dx | T0 > x].
The force of mortality (hazard rate) is
f0 (x) fx (t)
µx = (with 0 fixed, x variable), µx+t = (with x fixed, t variable).
S0 (x) Sx (t)
,Life Insurance Mathematics — Complete Course Summary 4
Derivation. By definition,
1 S0 (x + dx) . −1 d d
µx = lim Pr[T0 ≤ x+dx | T0 > x] = lim 1 − dx = S0 (x) = − ln S0 (x).
dx→0 dx dx→0 S0 (x) S0 (x) dx dx
Analogously, fx (t) = µx+t · t px .
Integrating the force of mortality recovers the survival function — the key link between the two:
Z y Z y Z y
d
µx dx = − ln S0 (x) dx = − ln S0 (y) − ln S0 (0) =⇒ S0 (y) = exp − µx dx .
0 0 dx 0
Rt
The same relation holds locally: Sx (t) = exp{− 0 µx+s ds}.
1.3 Actuarial Notation
t px = Sx (t) = Pr[Tx > t] = Pr[T0 > x + t | T0 > x] (survives t years)
t qx = Fx (t) = Pr[Tx ≤ t] = 1 − t px (dies within t years)
u|t qx = Fx (u + t) − Fx (u) = Pr[u < Tx ≤ u + t] = Sx (u) − Sx (u + t) = u px − u+t px
The quantity u|t qx is the probability that (x) survives u years and then dies within the following
t years, i.e. dies between ages x + u and x + u + t. It decomposes as
u|t qx = u px · t qx+u .
When the subscript before the vertical bar or the leading subscript equals 1, it is dropped: px := 1 px ,
qx := 1 qx .
A key multiplicative rule, used constantly when splitting a survival period into pieces:
t+u px = t px · u px+t .
Since µx+t = fx (t)/t px , we also have fx (t) = t px · µx+t , and therefore
Z t Z t
Fx (t) = t qx = fx (s) ds = s px · µx+s ds.
0 0
Exercise (Tutorial 1): a die-before probability
Calculate the probability
√ that a life aged 0 will die between ages 18 and 36, given the survival
1
function S0 (x) = 10 100 − x for 0 ≤ x ≤ 100.
Solution. We want 18|18 q0 = Pr[18 < T0 ≤ 36] = S0 (18) − S0 (36):
√ √ √ √
1 1 1 1 1
S0 (18) − S0 (36) = 10 100 − 18 − 10 100 − 36 = 10 82 − 64 = 10 (9 − 8) = 10 .
1.4 Mean and Standard Deviation of Tx
d
The complete life expectancy of a life aged x is e̊x = E[Tx ]. Using fx (t) = t px µx+t = − dt t px
and integration by parts,
Z ∞ Z ∞ Z ∞
e̊x = t · fx (t) dt = t px dt = Sx (t) dt.
0 0 0
Rn
For a finite term: e̊x:n| = 0 t px dt.
R∞
Similarly, E[Tx2 ] = 2 0 t · t px dt, so that Var[Tx ] = E[Tx2 ] − (e̊x )2 .
Complete Course Summary
Survival Models, Life Tables & Selection, Insurance Benefits,
Life Annuities, Premiums, and Policy Values
Lecture Notes, Step-by-Step Solution Guides &
Worked Exam Exercises
This document is an independent study summary and is not affiliated with, endorsed by, or a substitute for
any university’s official course materials.
,Life Insurance Mathematics — Complete Course Summary 1
Contents
I Lecture Notes & Worked Exercises 3
1 Survival Models 3
1.1 The Future Lifetime Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Force of Mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Actuarial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Mean and Standard Deviation of Tx . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Curtate Future Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Life Tables, Fractional Ages, and Selection 5
2.1 Life Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Fractional Age Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Select Mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Insurance Benefits 7
3.1 Valuation Functions and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Whole Life Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Term Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Pure Endowment and Endowment Insurance . . . . . . . . . . . . . . . . . . . . . . 8
3.5 Deferred Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
(m)
3.6 Relating Āx , Ax and Ax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Life Annuities 9
4.1 Review of Annuities-Certain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Annual Life Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Continuous Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.4 Annuities Payable m-thly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.5 Deferred and Guaranteed Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 Premiums 11
5.1 Premium Calculation Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2 Gross Premiums and Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.3 Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.4 Portfolio Percentile Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Policy Values 13
6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.2 Recursive Formula for Policy Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
II Quick-Reference Step-by-Step Guides 15
7 Guide: Survival Probabilities & Life Tables 15
8 Guide: EPV of Insurance Benefits 16
9 Guide: EPV of Annuities 16
10 Guide: Premium Calculation 17
11 Guide: Policy Values and Recursions 17
,Life Insurance Mathematics — Complete Course Summary 2
12 Guide: Profit and the Portfolio Percentile Premium (Extended) 18
III Worked Exam Exercises 19
13 Survival Models and Life Tables 19
14 Insurance Benefits 20
15 Life Annuities 20
16 Premiums 21
17 Policy Values 22
IV Practice Questions 24
,Life Insurance Mathematics — Complete Course Summary 3
Part I Lecture Notes & Worked Ex-
ercises
1 Survival Models
1.1 The Future Lifetime Random Variable
Consider a life aged x, denoted (x), where x ≥ 0. The future lifetime of (x) is a positive, continuous
random variable Tx . Then x + Tx represents the age at death of (x).
Let Fx be the lifetime distribution from age x, i.e. the cdf of Tx :
Fx (t) = Pr[Tx ≤ t] (probability that (x) dies before age x + t),
and let Sx be the corresponding survival function:
Sx (t) = 1 − Fx (t) = Pr[Tx > t] (probability that (x) survives beyond age x + t).
Now consider the collection of random variables {Tx }x≥0 , and let T0 and Tx refer to one and the
same individual, currently aged x. If this person dies within t years, then Tx ≤ t and T0 ≤ x + t.
Since the events {Tx ≤ t} and {T0 ≤ x + t | T0 > x} describe the same thing (given that the
individual has already survived to age x),
Pr[x < T0 ≤ x + t] F0 (x + t) − F0 (x)
Fx (t) = Pr[Tx ≤ t] = Pr[T0 ≤ x + t | T0 > x] = = ,
Pr[T0 > x] S0 (x)
which implies
S0 (x) − S0 (x + t) S0 (x + t)
Fx (t) = , Sx (t) = .
S0 (x) S0 (x)
This gives the multiplication rule, one of the most important identities in the course:
S0 (x + t) = S0 (x) · Sx (t)
where S0 (x + t) is the probability of survival from birth to age x + t, S0 (x) is the probability of
survival from birth to age x, and Sx (t) is the probability, having survived to age x, of further
surviving to age x + t.
More generally, for any u, t ≥ 0:
Sx (t + u) = Sx (t) · Sx+t (u).
1.2 Force of Mortality
For a very small interval dx, the quantity µx dx is (approximately) the probability that a life who
has attained age x dies before attaining age x + dx:
µx dx ≈ Pr[T0 ≤ x + dx | T0 > x].
The force of mortality (hazard rate) is
f0 (x) fx (t)
µx = (with 0 fixed, x variable), µx+t = (with x fixed, t variable).
S0 (x) Sx (t)
,Life Insurance Mathematics — Complete Course Summary 4
Derivation. By definition,
1 S0 (x + dx) . −1 d d
µx = lim Pr[T0 ≤ x+dx | T0 > x] = lim 1 − dx = S0 (x) = − ln S0 (x).
dx→0 dx dx→0 S0 (x) S0 (x) dx dx
Analogously, fx (t) = µx+t · t px .
Integrating the force of mortality recovers the survival function — the key link between the two:
Z y Z y Z y
d
µx dx = − ln S0 (x) dx = − ln S0 (y) − ln S0 (0) =⇒ S0 (y) = exp − µx dx .
0 0 dx 0
Rt
The same relation holds locally: Sx (t) = exp{− 0 µx+s ds}.
1.3 Actuarial Notation
t px = Sx (t) = Pr[Tx > t] = Pr[T0 > x + t | T0 > x] (survives t years)
t qx = Fx (t) = Pr[Tx ≤ t] = 1 − t px (dies within t years)
u|t qx = Fx (u + t) − Fx (u) = Pr[u < Tx ≤ u + t] = Sx (u) − Sx (u + t) = u px − u+t px
The quantity u|t qx is the probability that (x) survives u years and then dies within the following
t years, i.e. dies between ages x + u and x + u + t. It decomposes as
u|t qx = u px · t qx+u .
When the subscript before the vertical bar or the leading subscript equals 1, it is dropped: px := 1 px ,
qx := 1 qx .
A key multiplicative rule, used constantly when splitting a survival period into pieces:
t+u px = t px · u px+t .
Since µx+t = fx (t)/t px , we also have fx (t) = t px · µx+t , and therefore
Z t Z t
Fx (t) = t qx = fx (s) ds = s px · µx+s ds.
0 0
Exercise (Tutorial 1): a die-before probability
Calculate the probability
√ that a life aged 0 will die between ages 18 and 36, given the survival
1
function S0 (x) = 10 100 − x for 0 ≤ x ≤ 100.
Solution. We want 18|18 q0 = Pr[18 < T0 ≤ 36] = S0 (18) − S0 (36):
√ √ √ √
1 1 1 1 1
S0 (18) − S0 (36) = 10 100 − 18 − 10 100 − 36 = 10 82 − 64 = 10 (9 − 8) = 10 .
1.4 Mean and Standard Deviation of Tx
d
The complete life expectancy of a life aged x is e̊x = E[Tx ]. Using fx (t) = t px µx+t = − dt t px
and integration by parts,
Z ∞ Z ∞ Z ∞
e̊x = t · fx (t) dt = t px dt = Sx (t) dt.
0 0 0
Rn
For a finite term: e̊x:n| = 0 t px dt.
R∞
Similarly, E[Tx2 ] = 2 0 t · t px dt, so that Var[Tx ] = E[Tx2 ] − (e̊x )2 .
