Class notes
ActSci 650
Actuarial Maths I
Lecture Notes ActSci650 Fall24 Actuarial
Maths I 2025 | The Ultimate Guide for Yr.
(2026/2027)
Dr. Daniel Bauer
Department of Risk and Insurance
Wisconsin School of Business
University of Wisconsin-Madison
975 University Avenue, Madison, WI 53706
Based on Bowers et al. (1997) as well as the lecture notes by Ru¨diger Kiesel (Ulm University), Ragnar Norberg (London
School of Economics), Eric Ulm (Georgia State University), Shaun Wang (Georgia State University), and Hans-Joachim
Zwiesler (Ulm University).
Fall 2024
,Contents
Contents 1
1 Introduction 1
1.1 Example: The Grumpy Old Actuary . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Review of Basic Concepts From Probability . . . . . . . . . . . . . . . . . . . . . . 3
2 Survival Distributions and Life Tables 4
2.1 Basic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Analytic Mortality Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Curtate Future Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Life Tables ........................................ 16
2.5 Fractional Ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Select and Ultimate Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Non-parametric Survival Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7.1 Background: Construction of a survival function from population counts . . 33
2.7.2 The Kaplan-Meier Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7.3 The Nelson-Aalen Estimator .......................... 35
3 Life Insurance 38
3.1 Insurances Payable at the Moment of Death . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Insurances Payable at the End of the Year of Death ................. 48
3.3 Varying Benefit Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Life Annuities 67
4.1 Continuous Life Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Discrete Life Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Life Annuities with m-thly Payments . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Benefit Premiums 87
5.1 Premiums by the Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Percentile Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Gross Premium Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
, 1
CONTENTS 2
6 Benefit Reserves 107
6.1 Fully Continuous Benefit Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Fully Discrete Benefit Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3 Retrospective formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4 Benefit Reserves for General Insurances . . . . . . . . . . . . . . . . . . . . . . . . 112
6.5 Recursion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.6 Benefit Reserves with Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Chapter 1
Introduction
1.1 Example: The Grumpy Old Actuary
Mr. (55) is an actuary – a grumpy old actuary: He dislikes all his heirs, so when he dies he “will
consider” his remaining savings wasted. He just celebrated his 55th birthday and decides to invest
S0 = $100,000 to secure his future financial situation; he picks his 70th birthday as the pay-out year.
Since he has taken all actuarial exams, he knows how to calculate his expected payoff.
His first thought is to put all his money in a savings account, where he earns an annual interest
rate of i = 4.5%. Thus, after 15 years he would have
S15 = S0 (1 + i)15 = $100,000 · 1.04515 = $193,528.24.
The probability to survive until the age of 70 when being 55 now is approximatively 0.75, so the
expected amount at his disposal is
palive S15 + pdead 0 = palive S15 + (1 − palive)0
= 0.75 · $193,528.24 + 0.25 · 0
= $145,146.18.
On his way to the bank he meets his colleagues (55)∗ and (55)∗∗ who are in the same situation as
him. While sitting together for a coffee, they come up with the following idea: “How about putting
all our savings of 3S0 = $300,000 in a shared account and those who survive share the accumulated
capital 3S15 = $580,584.73 equally.” (55) excuses himself and runs to his car, where he has a
calculator, to decide whether this scheme would be beneficial for him. The expected amount at his
disposal is now
p(55) is alive pay-out + p(55) is dead 0
, = p(55) is alive and (55)∗∗ died died but (55)∗∗ is alive
3S15
+p (55)∗∗ died but (55)∗ is alive + pboth, (55)∗ and (55)∗∗, are alive
2
1
CHAPTER 1. INTRODUCTION 2
But instead of returning to his colleagues, he wonders now how good he could do if he finds 4,
5 or even more persons to join the scheme. Say he finds l55 people who are in the same situation as
him and of which l70 until the age of 70. The expected amount at his disposal will then be
p(55) is alive (55) is dead
and by the strong law of large numbers we have
l 55
l70 1 X
= 1 {Person k is alive}{Person k is alive}
l55 l55 k =1
| {z }
iid Bernoulli random variables
= Pr(Person 1 is alive) = 0.75.
Therefore,
,
and hence the expected discounted amount at his disposal is
.
That is insurance, and clearly (55) should have known that.
This type of contract is called pure endowment, and a combination of several endowment
contracts is called life annuity. In contrast, contracts that pay upon death if the insured has heirs he
wants to secure are called life insurance contacts. In the example, S0 is
• the amount the insured pays initially, i.e. the (single upfront) premium;
• the expected discounted amount at the insured’s disposal.
We will see that this is not a coincidence. However, note that this in general not the present value of
the pay-out. In fact, in our example (55) either gets
ActSci 650
Actuarial Maths I
Lecture Notes ActSci650 Fall24 Actuarial
Maths I 2025 | The Ultimate Guide for Yr.
(2026/2027)
Dr. Daniel Bauer
Department of Risk and Insurance
Wisconsin School of Business
University of Wisconsin-Madison
975 University Avenue, Madison, WI 53706
Based on Bowers et al. (1997) as well as the lecture notes by Ru¨diger Kiesel (Ulm University), Ragnar Norberg (London
School of Economics), Eric Ulm (Georgia State University), Shaun Wang (Georgia State University), and Hans-Joachim
Zwiesler (Ulm University).
Fall 2024
,Contents
Contents 1
1 Introduction 1
1.1 Example: The Grumpy Old Actuary . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Review of Basic Concepts From Probability . . . . . . . . . . . . . . . . . . . . . . 3
2 Survival Distributions and Life Tables 4
2.1 Basic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Analytic Mortality Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Curtate Future Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Life Tables ........................................ 16
2.5 Fractional Ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Select and Ultimate Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Non-parametric Survival Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7.1 Background: Construction of a survival function from population counts . . 33
2.7.2 The Kaplan-Meier Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7.3 The Nelson-Aalen Estimator .......................... 35
3 Life Insurance 38
3.1 Insurances Payable at the Moment of Death . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Insurances Payable at the End of the Year of Death ................. 48
3.3 Varying Benefit Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Life Annuities 67
4.1 Continuous Life Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Discrete Life Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Life Annuities with m-thly Payments . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Benefit Premiums 87
5.1 Premiums by the Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Percentile Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Gross Premium Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
, 1
CONTENTS 2
6 Benefit Reserves 107
6.1 Fully Continuous Benefit Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Fully Discrete Benefit Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3 Retrospective formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4 Benefit Reserves for General Insurances . . . . . . . . . . . . . . . . . . . . . . . . 112
6.5 Recursion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.6 Benefit Reserves with Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Chapter 1
Introduction
1.1 Example: The Grumpy Old Actuary
Mr. (55) is an actuary – a grumpy old actuary: He dislikes all his heirs, so when he dies he “will
consider” his remaining savings wasted. He just celebrated his 55th birthday and decides to invest
S0 = $100,000 to secure his future financial situation; he picks his 70th birthday as the pay-out year.
Since he has taken all actuarial exams, he knows how to calculate his expected payoff.
His first thought is to put all his money in a savings account, where he earns an annual interest
rate of i = 4.5%. Thus, after 15 years he would have
S15 = S0 (1 + i)15 = $100,000 · 1.04515 = $193,528.24.
The probability to survive until the age of 70 when being 55 now is approximatively 0.75, so the
expected amount at his disposal is
palive S15 + pdead 0 = palive S15 + (1 − palive)0
= 0.75 · $193,528.24 + 0.25 · 0
= $145,146.18.
On his way to the bank he meets his colleagues (55)∗ and (55)∗∗ who are in the same situation as
him. While sitting together for a coffee, they come up with the following idea: “How about putting
all our savings of 3S0 = $300,000 in a shared account and those who survive share the accumulated
capital 3S15 = $580,584.73 equally.” (55) excuses himself and runs to his car, where he has a
calculator, to decide whether this scheme would be beneficial for him. The expected amount at his
disposal is now
p(55) is alive pay-out + p(55) is dead 0
, = p(55) is alive and (55)∗∗ died died but (55)∗∗ is alive
3S15
+p (55)∗∗ died but (55)∗ is alive + pboth, (55)∗ and (55)∗∗, are alive
2
1
CHAPTER 1. INTRODUCTION 2
But instead of returning to his colleagues, he wonders now how good he could do if he finds 4,
5 or even more persons to join the scheme. Say he finds l55 people who are in the same situation as
him and of which l70 until the age of 70. The expected amount at his disposal will then be
p(55) is alive (55) is dead
and by the strong law of large numbers we have
l 55
l70 1 X
= 1 {Person k is alive}{Person k is alive}
l55 l55 k =1
| {z }
iid Bernoulli random variables
= Pr(Person 1 is alive) = 0.75.
Therefore,
,
and hence the expected discounted amount at his disposal is
.
That is insurance, and clearly (55) should have known that.
This type of contract is called pure endowment, and a combination of several endowment
contracts is called life annuity. In contrast, contracts that pay upon death if the insured has heirs he
wants to secure are called life insurance contacts. In the example, S0 is
• the amount the insured pays initially, i.e. the (single upfront) premium;
• the expected discounted amount at the insured’s disposal.
We will see that this is not a coincidence. However, note that this in general not the present value of
the pay-out. In fact, in our example (55) either gets
