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Lecture notes

Life Assurance Mathematics

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This book consists largely of material written for Parts A2 and D1 of the U.K. actuarial examinations (old system). It is hoped that the material given here will prove useful for much of Subjects 104 and 105 of the new examinations and certain similar examinations at universities, and that it may...

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  • July 19, 2024
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1




LIFE ASSURANCE MATHEMATICS


W.F.Scott




c
°1999 W.F.Scott
Department of Mathematical Sciences
King’s College
University of Aberdeen
Aberdeen
AB24 3UE
U.K.

,2

Preface
This book consists largely of material written for Parts A2 and D1 of the U.K. actuarial exami-
nations (old system). It is hoped that the material given here will prove useful for much of Subjects
104 and 105 of the new examinations and certain similar examinations at universities, and that it
may also be useful as a general reference work on life assurance mathematics.


William F. Scott. November 10, 1999

,Contents

1 NON-SELECT LIFE TABLES 9
1.1 Survivorship functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Probabilities of death and survival . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 The force of mortality, µx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 The expectation of life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 The assumption of a uniform distribution of deaths . . . . . . . . . . . . . . . . . . . 17
1.6 Central death rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Laws of mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 SELECT LIFE TABLES 29
2.1 What is selection? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Construction of select tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 The construction of A1967-70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Some formulae for the force of mortality. . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Select tables used in examinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 ASSURANCES 37
3.1 A general introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Whole life assurances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Commutation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 The variance of the present value of benefits . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Assurances payable at the end of the year of death. . . . . . . . . . . . . . . . . . . . 41
3.6 Assurances payable at the end of the 1/m of a year of death. . . . . . . . . . . . . . 44
3.7 Temporary and deferred assurances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.8 Pure endowments and endowment assurances . . . . . . . . . . . . . . . . . . . . . . 46
3.9 Varying assurances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.10 Valuing the benefits under with profits policies . . . . . . . . . . . . . . . . . . . . . 50
3.11 Guaranteed bonus policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.13 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 ANNUITIES 61
4.1 Annuities payable continuously . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Annuities payable annually . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Temporary annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Deferred annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3

, 4 CONTENTS

4.5 Annuities payable m times per annum . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.6 Complete annuities (or “annuities with final proportion”) . . . . . . . . . . . . . . . 70
4.7 Varying annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5 PREMIUMS 81
5.1 Principles of premium calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Notation for premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 The variance of the present value of the profit on a policy. . . . . . . . . . . . . . . . 83
5.4 Premiums allowing for expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.5 Premiums for with profits policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.6 Return of premium problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.7 Annuities with guarantees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.8 Family income benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 RESERVES 95
6.1 What are reserves? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Prospective reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.3 Net premium reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4 Retrospective reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.5 Gross premium valuations and asset shares. . . . . . . . . . . . . . . . . . . . . . . . 102
6.6 The variance of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.7 Zillmerised reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.8 Full preliminary term reserves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.9 Reserves for with-profits policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.11 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7 APPLICATIONS OF RESERVES 119
7.1 Surrender values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2 Paid-up policy values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.3 Alterations and conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.4 The actual and expected death strains . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.5 Mortality profit and loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.6 Other sources of profit and loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8 EXTRA RISKS 137
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.2 A constant addition to the force of mortality . . . . . . . . . . . . . . . . . . . . . . 137
8.3 A variable addition to the force of mortality . . . . . . . . . . . . . . . . . . . . . . . 139
8.4 Rating up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.5 Debts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

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