Introduction to Actuarial Science
Summary
EBB827A05
Semester II A
Wouter Voskuilen
S4916344
Lecture notes by L. Spierdijk,
Theory from Actuarial Mathematics for Life Contingent Risks, 3rd Edition, 2020
1
,Wouter Voskuilen Introduction to Actuarial Science
Contents
1 Week 1 3
1.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 𝑇𝑥 : the remaining lifetime of (𝑥) . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Important results for 𝑇𝑥 (with IAN) . . . . . . . . . . . . . . . . . . . 4
2 Week 2 7
2.1 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Week 3 9
3.1 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 Whole life insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.2 1/𝑚-thly benefit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.3 𝑛-year term insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.4 Pure endowment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.5 𝑛-year endowment insurance . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.6 Deferred insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.7 Non-standard insurance products . . . . . . . . . . . . . . . . . . . . . 12
4 Week 4 13
5 Week 5 14
5.1 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1.1 Revision: Annuity certain . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1.2 Life annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Week 6 17
6.1 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.1.1 Premium principle #1: the equivalence principle . . . . . . . . . . . . 17
6.1.2 Premium principle #2: the portfolio percentile principle . . . . . . . . 17
7 Week 7 19
7.1 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2
, Wouter Voskuilen Introduction to Actuarial Science
1 Week 1
1.1 Chapter 2
1.1.1 𝑇𝑥 : the remaining lifetime of (𝑥)
There is uncertainty about the policyholder’s lifetime, the insurance company does not know
the policyholder’s date of death.
Discounting is done to account for the time value of money.
The uncertainty in the remaining lifetime of a policyholder is dealt with by viewing the re-
maining lifetime as a random variable, the present value of payment upon the policyholder’s
death becomes a random variable.
Then, the insurer is typically interested in the expected value of the present value, called the
actuarial present value, and the variance of the present value.
The key random variable in life insurance is 𝑇𝑥 , the remaining lifetime of somebody aged
𝑥, often referred to as (𝑥).
We know 𝐹𝑥 (·) as the CDF of 𝑇𝑥 , defined by
𝐹𝑥 (𝑡) = ℙ(𝑇𝑥 ≤ 𝑡).
Actuaries sometimes prefer to use the survival distribution, which is defined as
𝑆 𝑥 (𝑡) = 1 − 𝐹𝑥 (𝑡) = ℙ(𝑇𝑥 > 𝑡).
𝐹𝑥 (𝑡) and 𝑆 𝑥 (𝑡) are conditional distributions:
𝐹𝑥 (𝑡) = ℙ(𝑇𝑥 ≤ 𝑡) = ℙ(𝑇0 ≤ 𝑥 + 𝑡|𝑇0 > 𝑥).
and
𝑆0 (𝑥 + 𝑡)
𝑆 𝑥 (𝑡) = ℙ(𝑇𝑥 > 𝑡) = ℙ(𝑇0 > 𝑥 + 𝑡 |𝑇0 > 𝑥) = ,
𝑆0 (𝑥)
using the definition of conditional probability, ℙ(𝐴|𝐵) = ℙ(𝐴∩𝐵)ℙ(𝐵)
.
Furthermore, we can write
𝑆0 (𝑥 + 𝑇) = 𝑆0 (𝑥)𝑆 𝑥 (𝑡).
Hence, the probability that (0) survives until age 𝑥 + 𝑡 equals the probability that (0) survives
until age 𝑥 times the probability that (𝑥) survives until age 𝑥 + 𝑡.
A survival function is valid if it satisfies the following 3 conditions:
Condition 1: 𝑆 𝑥 (0) = 1;
Condition 2: lim𝑡→∞ 𝑆 𝑥 (𝑡) = 0;
3
Summary
EBB827A05
Semester II A
Wouter Voskuilen
S4916344
Lecture notes by L. Spierdijk,
Theory from Actuarial Mathematics for Life Contingent Risks, 3rd Edition, 2020
1
,Wouter Voskuilen Introduction to Actuarial Science
Contents
1 Week 1 3
1.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 𝑇𝑥 : the remaining lifetime of (𝑥) . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Important results for 𝑇𝑥 (with IAN) . . . . . . . . . . . . . . . . . . . 4
2 Week 2 7
2.1 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Week 3 9
3.1 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 Whole life insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.2 1/𝑚-thly benefit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.3 𝑛-year term insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.4 Pure endowment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.5 𝑛-year endowment insurance . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.6 Deferred insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.7 Non-standard insurance products . . . . . . . . . . . . . . . . . . . . . 12
4 Week 4 13
5 Week 5 14
5.1 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1.1 Revision: Annuity certain . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1.2 Life annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Week 6 17
6.1 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.1.1 Premium principle #1: the equivalence principle . . . . . . . . . . . . 17
6.1.2 Premium principle #2: the portfolio percentile principle . . . . . . . . 17
7 Week 7 19
7.1 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2
, Wouter Voskuilen Introduction to Actuarial Science
1 Week 1
1.1 Chapter 2
1.1.1 𝑇𝑥 : the remaining lifetime of (𝑥)
There is uncertainty about the policyholder’s lifetime, the insurance company does not know
the policyholder’s date of death.
Discounting is done to account for the time value of money.
The uncertainty in the remaining lifetime of a policyholder is dealt with by viewing the re-
maining lifetime as a random variable, the present value of payment upon the policyholder’s
death becomes a random variable.
Then, the insurer is typically interested in the expected value of the present value, called the
actuarial present value, and the variance of the present value.
The key random variable in life insurance is 𝑇𝑥 , the remaining lifetime of somebody aged
𝑥, often referred to as (𝑥).
We know 𝐹𝑥 (·) as the CDF of 𝑇𝑥 , defined by
𝐹𝑥 (𝑡) = ℙ(𝑇𝑥 ≤ 𝑡).
Actuaries sometimes prefer to use the survival distribution, which is defined as
𝑆 𝑥 (𝑡) = 1 − 𝐹𝑥 (𝑡) = ℙ(𝑇𝑥 > 𝑡).
𝐹𝑥 (𝑡) and 𝑆 𝑥 (𝑡) are conditional distributions:
𝐹𝑥 (𝑡) = ℙ(𝑇𝑥 ≤ 𝑡) = ℙ(𝑇0 ≤ 𝑥 + 𝑡|𝑇0 > 𝑥).
and
𝑆0 (𝑥 + 𝑡)
𝑆 𝑥 (𝑡) = ℙ(𝑇𝑥 > 𝑡) = ℙ(𝑇0 > 𝑥 + 𝑡 |𝑇0 > 𝑥) = ,
𝑆0 (𝑥)
using the definition of conditional probability, ℙ(𝐴|𝐵) = ℙ(𝐴∩𝐵)ℙ(𝐵)
.
Furthermore, we can write
𝑆0 (𝑥 + 𝑇) = 𝑆0 (𝑥)𝑆 𝑥 (𝑡).
Hence, the probability that (0) survives until age 𝑥 + 𝑡 equals the probability that (0) survives
until age 𝑥 times the probability that (𝑥) survives until age 𝑥 + 𝑡.
A survival function is valid if it satisfies the following 3 conditions:
Condition 1: 𝑆 𝑥 (0) = 1;
Condition 2: lim𝑡→∞ 𝑆 𝑥 (𝑡) = 0;
3
