Life Insurance Theory

Transcription

1 Life Insurance Theory

2 Life Insurance Theory Actuarial Perspectives by F. Etienne De Vylder SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

3 A C.I.P. Catalogue record for this book is available from the Library of Congress ISBN ISBN (ebook) DOI / Printed on acid-free paper All Rights Reserved 1997 Springer Science+ Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

4 TABLE OF CONTENTS List offigures and Tables Preface A Guide to Terminology and Notation Chapter 1. Financial Models 1.1. Units 1.2.' Constant interest rate 1.3. Variable interest rates 1.4. Deterministic time-capitals 1.5. Stochastic time-capitals 1.6. Annuities-certain Stochastic interests Chapter 2. Mortality Models 2.1. Life tables 2.2. Future lifetime X 2.3. Force ofmortality 2.4. Decease in the middle ofthe year 2.5. Expected future lifetime 2.6. Analytic life tables 2.7. Restricted life tables 2.8. Selected life tables 2.9. Commutation functions Chapter 3. Construction oflife Tables 3.1. Problem description 3.2. National tables 3.3. Private tables 3.4. Analytic least-squares graduation 3.5. Maximum likelihood graduation 3.6. Determination ofinitial parameters in the Makeham case xi xiii xv

5 vi Life Insurance Theory Chapter 4. Basic Concepts oflife Insurance Mathematics 4.1. Life insurance models 4.2. Contracts 4.3. Ruin problems in portfolios 4.4. Validity level ofrelations 4.5. Approximations 4.6. Null events Chapter 5. Life Annuities (One Life) 5.1. Deferred life capital 5.2. Constant life annuities 5.3. Partitioned life annuities 5.4. General variable life annuities 5.5. Classical variable life annuities 5.6. Annuities on status xl 5.7. Variable interest rates Chapter 6. Lire Insurances (One Life) 6.1. Constant life insurances 6.2. General variable life insurances 6.3. Classical variable life insurances 6.4. Endowments 6.5. Insurance ofa remaining debt at death 6.6. Variable interest rates Chapter 7. Relations Between Life Annuities and Life Insurances (One Life) 7.1. Constant annuities and insuranees. Price level 7.2. Constant annuities and insuranees. P-resent value level 7.3. Variable annuities and insuranees. General discrete case 7.4. Variable annuities and insuranees. General continuous case 7.5. Classical variable annuities and insurances Chapter 8. Decompositions oftime-capitais (One Life) 8.1. Reserves ofa time-capital 8.2. The decomposition formula 8.3. Evaluation of areserve at a non-integer instant 8.4. Fouret's formula 8.5. Thiele's formula 8.6. Insurances payable in the middle ofthe year of death

6 Table of Contents vii Chapter 9. Life Insurance Contracts (One Life) Life insurance contracts Reserves of a contract Practical constraints on contracts Contracts with partitioned prerniums Risk and savings prerniums Illustration: general endowment insurance Positive reserves (analytic proofs) Variation ofprices with interest rate i Variation ofreserves with interest rate i Variation ofreserves with time t Transformation of a contract Expense loadings 80 Chapter 10. Ruin Probability of a Life Insurance Company True interest rate and true mortality Profit of a contract Profit of a c10sed portfolio Probability of ruin in a c10sed portfolio Solvency parameter of a portfolio Merger of c10sed portfolios considered at the same moment Merger of c10sed portfolios considered at different moments Probability of ruin in an open portfolio Open portfolio with constant entries Open portfolio with exponential growth Open portfolio with linear growth Evaluation ofvariances. General methodology Deferred life capital Generaliife insurance Life annuities Variance ofreserves 94 Chapter 11.lnsurances on a Status (Several Lives) Definition of a status Probabilities on a status Deferred capitals on a status Life annuities on a status Life insurances on a status Alternative notations 103

7 viii Life Insurance Theory Chapter 12. Decomposition of Time-Capitals (Several Lives) Reserves of a time-capital on two lives Time-capitals vanishing at first decease The decomposition formula Iterative formulas Evaluation of areserve at a non-integer instant Fouret's formula Thiele's formula 109 Chapter 13. Life Insurance Contracts (Several Lives) Life insurance contracts on severallives Reserves of a contract Practical constraints on contracts Contracts with partitioned premiums 113 Chapter 14. Multiple Decrement Models Extinction graphs of a group oflives Other graphs Events and probabilities on a graph Annuities on states of a graph Transition capitals on a graph Transition Theorem for time-capitals (price level) Illustrations in case ofgraph Gr4(X,y) Transition Theorem for time-capitals (present value level) 128 Chapter 15. Variances (Several Lives) Ruin problems Evaluation ofvariances. General methodology Deferred life capitals Generallife insurances Life annuities Variance ofreserves 138 Chapter 16. Population Groups on a Graph Closed graph model Open graph model Estimation ofinstantaneous transition rates Estimations in a graph with two states Estimations in agraph with three states Estimations in a graph with four states Evaluation of state probabilities 152

8 Table of Contents State probabilities in a graph with two states State probabilities in a graph with three states State probabilities in a graph with four states Mortality Estimations Appendix A. Summation by Parts Appendix B. Linear Interpolations Appendix C. Probability Theory Appendix D. A Differential Equation Appendix E. Inversion of apower Series Appendix F. Summary of Formulas References Notation Index Subject Index IX

9 FIGURES AND TABLES Figores Figure 1.1. Present values (constant interest rate) Figure 1.2. Discount factors (variable interests) Figure 3.1. Lexis diagram Figure 8.1. Decomposition of a time-capital Figure 9.1. Variation of reserves with time t Figure Extinction graph Gr2(x) Figure Complete extinction graph Grs(x.y) Figure Extinction graph Gr4(x,y) with amalga.meted states Figure Extinction graph Gr3(X.y) with amalga.meted states Figure Complete extinction graph Grl~x,y,Z) Figure Complete active-disabled graph Figure Active-disabled graph with amalga.meted states Figure Aecidental death graph Figure Healthy-siekgraph with returns Figure Open population graph with two states Figure Open population graph with three states Figure Open population graph with four states Figure Closed probability graph with two states Figure Closed probability graph with three states Figure Closed probability graph with four states Figure Open population graph with two states Figure A.l. Geometrie proof of interpolation formulas Figure C.l. Representation of double events Tables Table Systematie verifieation of relations between insuranees on statuses 103 Table C.l. Verifieation of a double event relation 165 Table C.2. Verifieation of a tripie event relation 166 Table C.3. Symmetrieal tripie events 166

10 PREFACE This book is different from all other books on Life Insurance by at least one of the following characteristics The treatment of life insurances at three different levels: time-capital, present value and price level. We call time-capital any distribution of a capital over time: is the time-capital with amounts Cl, ~,..., CN at moments Tl, T2,.., TN resp. For instance, let (x) be a life at instant 0 with future lifetime X. Then the whole life insurance Ax oo is the time-capital (I,X). The whole life annuity äx oo is the time-capital (1,0) + (1,1) + (1,2) (I,'X), where 'X is the integer part ofx. The present value at 0 of time-capital (*) is the random variable In particular, the present value ofa x 00 and ä x 00 is (*) Cl V T1 + ~ v T, CNV TN. (**) Ax 0 = ~ and ä x 0 = 1 + v + v v'x resp. The price (or premium) of a time-capital is the expectation of its present value. In particular, the price ofa x 00 and äx 00 is Ax = E(~) and äx = E(I + v + v v'x) resp. It is important to know at what level relations are valid. For instance, the c1assical relation Ax = 1-8 ax is in fact valid at present value level: Ax 0 = 1-8 ax o. The latter relation implies that Var(A x 0) = 82 Var( a x 0). 2. The introduction of general capital-functions c, (r~o). For instance, the variable whole life insurance with capital-function c, is the time-capital Ai c,)oo = (Cx,X) with present value Ai c,)o = cx~ and price Ai c,) = E(cx~). Similarly, general variable annuities with capital-function c, are defined. Then all c1assical relations (between life insurances and life annuities, between yearly annuities and partitioned annuities,... ) can be resumed in simple general relations with general capital-functions.

11 xiv Preface 3. The introduction of extinction graphs of a group of lives. For instance, the evolution ofthe group oftwo lives (x) and (y) can be visualized by a graph with 4 states xy (x and y alive), xly (x dead and y alive), ylx (y dead and x alive) and xlyl (x and y both dead). Annuities can be defined on the states and insurances can be connected with a transition from one state to another state. Very general relations at present value level can be proved on graphs. Models with decrements due to other causes than death (sickness, disablement,... ) can be treated by graphs. 4. The consideration and evaluation of long term ruin probabilities of a life insurance company, under assumptions on true mortality, true interest rate and evolution of the volume of the company. Acknowledgments I am grateful to Kluwer Academic Publishers, in particular to Mr. Allard Winterink, Acquisitions Editor, to have accepted the publication ofthis book. Mr. Winterink remained in permanent pleasant contact during the complete submission and publication process. Professor Helene Cossette (Lava!, Quebec) and Professor Etienne Marceau (Laval, Quebec) have developed computer programs for the numerical evaluation of ruin probabilities, based on Chapter 10, in a portfolio of a large Canadian life insurance company. Thanks, Helene and Etienne, for this agreeable collaboration. The first version of the book has been used in a life insurance course at a Belgian university. Thanks to Mr. Frederic Kint, one of my students, numerous misprints in the initial text have been detected and eliminated. Finally, I want to thank Mrs. Anne Cocriamont for reading my manuscript, checking for language and spelling mistakes. Ghent, June 1997, F. Etienne De Vylder

12 A GUIDE TO TERMINOLOGY AND NOTATION The notations :=. =: :<=> <=>: are used for definitions. The defined symbol. or property. is on the side of the double point. The symbol == connects identical quantities expressed in different notations. or it indicates that a function has a constant value on some domain R := ] oo[ == (-00. +(0). ~:= [0. oo[ == [0. (0). We call :::; the inequality symbol and < the strict inequality symbol and we use a corresponding consistent terminology. Examples xis stictly positive :<=> x> O. x is larger than y :<=> x ~ y. The letters j. k. m. n. r and the Greek letter v denote positive integers avb := max(a.b). 3I\b:= min(a.b). Hf is an abbreviation of if and only if. In definitions. iff is not used, but there the meaning of if is iff in all cases. 't is the largest integer smaller than t (i.e. the integer part of t). t' is the smallest integer larger than t and t := ('t+t')/2. If t is an integer then t = 't = ~ = t'. Cf course. sometimes t'. t"... are any numbers without connexion with t and accents are also used for derivatives. Anyway. the meaning of the accents is clear from the context. We denote by e a number between 0 and 1. f(a,b) == fa b. The indicator function IProp of the proposition Prop equals 1 if Prop is true and o if Prop is false. We recall that multiplications and divisions must be performed first, then additions and substractions. if no brackects are used. Hence. k+ 1/2 == k+(1/2). The international notation system for Life Insurance is adopted throughout the book. Unfortunately. tbis system ignores stochastics. and in contradiction with Risk Theory notation. capital letters never represent random variables. For the latter. the superscript 0 is used systematically in the monograph.