Life Insurance Mathematics

by Gerber, Hans U.; Neuhaus, Walther

• ISBN13: 9783540622420
• ISBN10: 354062242X
• Edition: 3rd
• Format: Hardcover
• Copyright: 1997-03-01
• Publisher: Springer Verlag

Summary

From the reviews: "The highly esteemed 1990 first edition of this book now appears in a much expanded second edition. The difference between the first two English editions is entirely due to the addition of numerous exercises. The result is a truly excellent book, balancing ideally between theory and practice. ....As already hinted at above, this book provides the ideal bridge between the classical (deterministic) life insurance theory and the emerging dynamic models based on stochastic processes and the modern theory of finance. The structure of the bridge is very solid, though at the same time pleasant to walk along. I have no doubt that Gerber's book will become the standard text for many years to come. Metrika, 44, 1996, 2

Table of Contents

1 The Mathematics of Compound Interest	1	(14)
1.1 Mathematical Bases of Life Contingencies	1	(1)
1.2 Effective Interest Rates	1	(1)
1.3 Nominal Interest Rates	2	(1)
1.4 Continuous Payments	3	(1)
1.5 Interest in Advance	4	(2)
1.6 Perpetuities	6	(3)
1.7 Annuities	9	(2)
1.8 Repayment of a Debt	11	(2)
1.9 Internal Rate of Return	13	(2)
2 The Future Lifetime of a Life Aged x	15	(8)
2.1 The Model	15	(1)
2.2 The Force of Mortality	16	(1)
2.3 Analytical Distributions of T	17	(1)
2.4 The Curtate Future Lifetime of (x)	18	(2)
2.5 Life Tables	20	(1)
2.6 Probabilities of Death for Fractions of a Year	21	(2)
3 Life Insurance	23	(12)
3.1 Introduction	23	(1)
3.2 Elementary Insurance Types	23	(3)
3.2.1 Whole Life and Term Insurance	23	(1)
3.2.2 Pure Endowments	24	(1)
3.2.3 Endowments	25	(1)
3.3 Insurances Payable at the Moment of Death	26	(1)
3.4 General Types of Life Insurance	27	(2)
3.5 Standard Types of Variable Life Insurance	29	(2)
3.6 Recursive Formulae	31	(34)
4 Life Annuities	35	(14)
4.1 Introduction	35	(1)
4.2 Elementary Life Annuities	35	(2)
4.3 Payments made more Frequently than Once a Year	37	(2)
4.4 Variable Life Annuities	39	(2)
4.5 Standard Types of Life Annuity	41	(1)
4.6 Recursion Formulae	42	(1)
4.7 Inequalities	43	(3)
4.8 Payments Starting at Non-integral Ages	46	(3)
5 Net Premiums	49	(10)
5.1 Introduction	49	(1)
5.2 An Example	49	(3)
5.3 Elementary Forms of Insurance	52	(2)
5.3.1 Whole Life and Term Insurance	52	(1)
5.3.2 Pure Endowments	53	(1)
5.3.3 Endowments	54	(1)
5.3.4 Deferred Life Annuities	54	(1)
5.4 Premiums Paid m Times a Year	54	(1)
5.5 A General Type of Life Insurance	55	(1)
5.6 Policies with Premium Refund	56	(1)
5.7 Stochastic Interest	56	(3)
6 Net Premium Reserves	59	(16)
6.1 Introduction	59	(1)
6.2 Two Examples	59	(2)
6.3 Recursive Considerations	61	(2)
6.4 The Survival Risk	63	(1)
6.5 The Net Premium Reserve of a Whole Life Insurance	63	(1)
6.6 Net Premium Reserves at Fractional Durations	64	(1)
6.7 Allocation of the Overall Loss to Policy Years	65	(3)
6.8 Conversion of an Insurance	68	(1)
6.9 Technical Gain	69	(1)
6.10 Procedure for Pure Endowments	70	(1)
6.11 The Continuous Model	71	(4)
7 Multiple Decrements	75	(8)
7.1 The Model	75	(1)
7.2 Forces of Decrement	76	(1)
7.3 The Curtate Lifetime of (x)	76	(1)
7.4 A General Type of Insurance	77	(1)
7.5 The Net Premium Reserve	78	(2)
7.6 The Continuous Model	80	(3)
8 Multiple Life Insurance	83	(10)
8.1 Introduction	83	(1)
8.2 The Joint-Life Status	83	(1)
8.3 Simplifications	84	(1)
8.4 The Last-Survivor Status	85	(2)
8.5 The General Symmetric Status	87	(2)
8.6 The Schuette-Nesbitt Formula	89	(1)
8.7 Asymmetric Annuities	90	(1)
8.8 Asymmetric Insurances	91	(2)
9 The Total Claim Amount in a Portfolio	93	(10)
9.1 Introduction	93	(1)
9.2 The Normal Approximation	93	(1)
9.3 Exact Calculation of the Total Claim Amount Distribution	94	(2)
9.4 The Compound Poisson Approximation	96	(2)
9.5 Recursive Calculation of the Compound Poisson Distribution	98	(2)
9.6 Reinsurance	100	(1)
9.7 Stop-Loss Reinsurance	101	(2)
10 Expense Loadings	103	(6)
10.1 Introduction	103	(1)
10.2 The Expense-Loaded Premium	104	(1)
10.3 Expense-Loaded Premium Reserves	105	(4)
11 Estimating Probabilities of Death	109	(10)
11.1 Problem Description	109	(1)
11.2 The Classical Method	110	(1)
11.3 Alternative Solution	111	(1)
11.4 The Maximum Likelihood Method	112	(1)
11.5 Statistical Inference	112	(4)
11.6 The Bayesian Approach	116	(1)
11.7 Multiple Causes of Decrement	116	(2)
11.8 Interpretation of Results	118	(1)
Appendix A. Commutation Functions	119	(6)
A.1 Introduction	119	(1)
A.2 The Deterministic Model	119	(1)
A.3 Life Annuities	120	(1)
A.4 Life Insurance	121	(1)
A.5 Net Annual Premiums and Premium Reserves	122	(3)
Appendix B. Simple Interest	125	(3)
Appendix C. Exercises	128	(40)
C.0 Introduction	128	(1)
C.1 Mathematics of Compound Interest: Exercises	129	(4)
C.1.1 Theory Exercises	129	(1)
C.1.2 Spreadsheet Exercises	130	(3)
C.2 The Future Lifetime of a Life Aged x: Exercises	133	(5)
C.2.1 Theory Exercises	134	(2)
C.2.2 Spreadsheet Exercises	136	(2)
C.3 Life Insurance	138	(4)
C.3.1 Theory Exercises	138	(3)
C.3.2 Spreadsheet Exercises	141	(1)
C.4 Life Annuities	142	(4)
C.4.1 Theory Exercises	142	(2)
C.4.2 Spreadsheet Exercises	144	(2)
C.5 Net Premiums	146	(6)
C.5.1 Notes	146	(1)
C.5.2 Theory Exercises	146	(4)
C.5.3 Spreadsheet Exercises	150	(2)
C.6 Net Premium Reserves	152	(4)
C.6.1 Theory Exercises	152	(3)
C.6.2 Spreadsheet Exercises	155	(1)
C.7 Multiple Decrements: Exercises	156	(2)
C.7.1 Theory Exercises	156	(2)
C.8 Multiple Life Insurance: Exercises	158	(3)
C.8.1 Theory Exercises	158	(2)
C.8.2 Spreadsheet Exercises	160	(1)
C.9 The Total Claim Amount in a Portfolio	161	(2)
C.9.1 Theory Exercises	161	(2)
C.10 Expense Loadings	163	(2)
C.10.1 Theory Exercises	163	(1)
C.10.2 Spreadsheet Exercises	164	(1)
C.11 Estimating Probabilities of Death	165	(3)
C.11.1 Theory Exercises	165	(3)
Appendix D. Solutions	168	(39)
D.0 Introduction	168	(1)
D.1 Mathematics of Compound Interest	169	(3)
D.1.1 Solutions to Theory Exercises	169	(1)
D.1.2 Solutions to Spreadsheet Exercises	170	(2)
D.2 The Future Lifetime of a Life Aged x	172	(3)
D.2.1 Solutions to Theory Exercise	172	(2)
D.2.2 Solutions to Spreadsheet Exercises	174	(1)
D.3 Life Insurance	175	(4)
D.3.1 Solutions to Theory Exercises	175	(3)
D.3.2 Solution to Spreadsheet Exercises	178	(1)
D.4 Life Annuities	179	(4)
D.4.1 Solutions to Theory Exercises	179	(2)
D.4.2 Solutions to Spreadsheet Exercises	181	(2)
D.5 Net Premiums: Solutions	183	(6)
D.5.1 Theory Exercises	183	(4)
D.5.2 Solutions to Spreadsheet Exercises	187	(2)
D.6 Net Premium Reserves: Solutions	189	(3)
D.6.1 Theory Exercises	189	(1)
D.6.2 Solutions to Spreadsheet Exercises	190	(2)
D.7 Multiple Decrements: Solutions	192	(2)
D.7.1 Theory Exercises	192	(2)
D.8 Multiple Life Insurance: Solutions	194	(4)
D.8.1 Theory Exercises	194	(3)
D.8.2 Solutions to Spreadsheet Exercises	197	(1)
D.9 The Total Claim Amount in a Portfolio	198	(2)
D.9.1 Theory Exercises	198
D.10 Expense Loadings	200	(3)
D.10.1 Theory Exercises	200	(1)
D.10.2 Spreadsheet Exercises	201	(2)
D.11 Estimating Probabilities of Death	203	(4)
D.11.1 Theory Exercises	203	(4)
Appendix E. Tables	207	(6)
E.0 Illustrative Life Tables	207	(2)
E.1 Commutation Columns	209	(2)
E.2 Multiple Decrement Tables	211	(2)
References	213	(2)
Index	215