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Modelling Longevity Dynamics for Pensions and Annuity Business

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Mortality improvements, uncertainty in future mortality trends and the relevant impact on life annuities and pension plans constitute important topics in the field of actuarial mathematics and life insurance techniques. In particular, actuarial calculations concerning pensions, life annuities and other living benefits (provided, for example, by long-term care insurance products and whole life sickness covers) are based on survival probabilities which necessarily extend over a long time horizon. In order to avoid underestimation of the related liabilities, the insurance company (or the pension plan) must adopt an appropriate forecast of future mortality.

Great attention is currently being devoted to the management of life annuity portfolios, both from a theoretical and a practical point of view, because of the growing importance of annuity benefits paid by private pension schemes. In particular, the progressive shift from defined benefit to defined contribution pension schemes has increased the interest in life annuities with a guaranteed annual amount.

This book provides a comprehensive and detailed description of methods for projecting mortality, and an extensive introduction to some important issues concerning longevity risk in the area of life annuities and pension benefits. It relies on research work carried out by the authors, as well as on a wide teaching experience and in CPD (Continuing Professional Development) initiatives. The following topics are dealt with: life annuities in the framework of post-retirement income strategies; the basic mortality model; recent mortality trends that have been experienced; general features of projection models; discussion of stochastic projection models, with numerical illustrations; measuring and managing longevity risk.

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Product Details

  • ISBN-13: 9780199547272
  • Publisher: Oxford University Press, USA
  • Publication date: 4/15/2009
  • Pages: 400
  • Product dimensions: 6.30 (w) x 9.30 (h) x 1.10 (d)

Meet the Author

Ermanno Pitacco is full professor of Actuarial Mathematics in the Faculty of Economics, University of Trieste, Academic Director of the Master in Insurance and Risk Management at the MIB School of Management in Trieste, Full member of the Istituto Italiano degli Attuari (Italy), Affiliate member of the Institute of Actuaries (UK). He has authored 90 papers and textbooks in the field of actuarial techniques. Michel Denuit is Professor of Statistics and Actuarial Mathematics, Université Catholique de Louvain. He was a founding member of the Belgian Actuarial Bulletin. He is also Proceedings Editor for Insurance: Mathematics and Economics, Editor for ASTIN Bulletin, Associate Editor, Methodology and Computing in Applied Probability, Member of the Advisory Board for the Wiley Encyclopedia of Actuarial Science, Member of the Advisory Board for the Wiley Encyclopedia of Quantitative Risk Analysis and Assessment, and Associate Editor of the Australian and New Zealand Journal of Statistics. Steven Haberman is Professor of Actuarial Science in Cass Business School, City University, Fellow of the Institute of Actuaries (UK), Associate of Society of Actuaries (US) and Honorary Member of the Istituto Italiano degli Attuari (Italy), He is author of about 150 papers and textbooks in the field of actuarial mathematics and actuarial techniques and was a founding editor of Founding Editor of the international journal "Journal of Pension Economics and Finance ". Annamaria Olivieri is Full professor of Mathematical Methods for Economics, Actuarial Science and Finance, Faculty of Economics, University of Parma and a full member of the Istituto Italiano degli Attuari (Italy). She has been the author of many papers and textbooks in the field of actuarial mathematics and actuarial techniques.

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Table of Contents

Preface v

1 Life annuities 1

1.1 Introduction 1

1.2 Annuities-certain versus life annuities 2

1.2.1 Withdrawing from a fund 2

1.2.2 Avoiding early fund exhaustion 5

1.2.3 Risks in annuities-certain and in life annuities 6

1.3 Evaluating life annuities: deterministic approach 8

1.3.1 The life annuity as a financial transaction 8

1.3.2 Actuarial values 9

1.3.3 Technical bases 12

1.4 Cross-subsidy in life annuities 14

1.4.1 Mutuality 14

1.4.2 Solidarity 16

1.4.3 'Tontine' annuities 18

1.5 Evaluating life annuities: stochastic approach 20

1.5.1 The random present value of a life annuity 20

1.5.2 Focussing on portfolio results 21

1.5.3 A first insight into risk and solvency 24

1.5.4 Allowing for uncertainty in mortality assumptions 27

1.6 Types of life annuities 31

1.6.1 Immediate annuities versus deferred annuities 31

1.6.2 The accumulation period 33

1.6.3 The decumulation period 36

1.6.4 The payment profile 38

1.6.5 About annuity rates 40

1.6.6 Variable annuities and GMxB features 41

1.7 References and suggestions for further reading 43

2 The basic mortality model 45

2.1 Introduction 45

2.2 Life tables 46

2.2.1 Cohort tables and period tables 46

2.2.2 'Population' tables versus 'market' tables 47

2.2.3 The life table as a probabilistic model 48

2.2.4 Select mortality 49

2.3 Moving to an age-continuous context 51

2.3.1 The survival function 51

2.3.2 Other related functions 53

2.3.3 The force of mortality 55

2.3.4 The central death rate 57

2.3.5 Assumptions for non-integer ages 57

2.4 Summarizing the lifetime probability distribution 58

2.4.1 The life expectancy 59

2.4.2 Other markers 60

2.4.3Markers under a dynamic perspective 62

2.5 Mortality laws 63

2.5.1 Laws for the force of mortality 64

2.5.2 Laws for the annual probability of death 66

2.5.3 Mortality by causes 67

2.6 Non-parametric graduation 67

2.6.1 Some preliminary ideas 67

2.6.2 The Whittaker-Henderson model 68

2.6.3 Splines 69

2.7 Some transforms of the survival function 73

2.8 Mortality at very old ages 74

2.8.1 Some preliminary ideas 74

2.8.2 Models for mortality at highest ages 75

2.9 Heterogeneity in mortality models 77

2.9.1 Observable heterogeneity factors 77

2.9.2 Models for differential mortality 78

2.9.3 Unobservable heterogeneity factors. The frality 80

2.9.4 Frailty models 83

2.9.5 Combining mortality laws with frailty models 85

2.10 References and suggestions for further reading 87

3 Mortality trends during the 20th century 89

3.1 Introduction 89

3.2 Data sources 90

3.2.1 Statistics Belgium 91

3.2.2 Federal Planning Bureau 91

3.2.3 Human mortality database 92

3.2.4 Banking, Finance, and Insurance Commission 92

3.3 Mortality trends in the general population 93

3.3.1 Age-period life tables 93

3.3.2 Exposure-to-risk 95

3.3.3 Death rates 96

3.3.4 Mortality surfaces 101

3.3.5 Closure of life tables 101

3.3.6 Rectangularization and expansion 105

3.3.7 Life expectancies 111

3.3.8 Variability 113

3.3.9 Heterogeneity 115

3.4 Life insurance market 116

3.4.1 Observed death rates 116

3.4.2 Smoothed death rates 118

3.4.3 Life expectancies 122

3.4.4 Relational models 123

3.4.5 Age shifts 127

3.5 Mortality trends throughout EU 129

3.6 Conclusions 135

4 Forecasting mortality: an introduction 137

4.1 Introduction 137

4.2 A dynamic approach to mortality modelling 139

4.2.1 Representing mortality dynamics: single-figures versus age-specific functions 139

4.2.2 A discrete, age-specific setting 140

4.3 Projection by extrapolation of annual probabilities of death 141

4.3.1 Some preliminary ideas 141

4.3.2 Reduction factors 144

4.3.3 The exponential formula 145

4.3.4 An alternative approach to the exponential extrapolation 146

4.3.5 Generalizing the exponential formula 147

4.3.6 Implementing the exponential formula 148

4.3.7 A general exponential formula 149

4.3.8 Some exponential formulae used in actuarial practice 149

4.3.9 Other projection formulae 151

4.4 Using a projected table 152

4.4.1 The cohort tables in a projected table 152

4.4.2 From a double-entry to a single-entry projected table 153

4.4.3 Age shifting 155

4.5 Projecting mortality in a parametric context 156

4.5.1 Mortality laws and projections 156

4.5.2 Expressing mortality trends via Weibull's parameters 160

4.5.3 Some remarks 162

4.5.4 Mortality graduation over age and time 163

4.6 Other approaches to mortality projections 165

4.6.1 Interpolation versus extrapolation: the limit table 165

4.6.2 Model tables 166

4.6.3 Projecting transforms of life table functions 167

4.7 The Lee-Carter method: an introduction 169

4.7.1 Some preliminary ideas 169

4.7.2 The LC model 171

4.7.3 From LC to the Poisson log-bilinear model 172

4.7.4 The LC method and model tables 173

4.8 Further issues 173

4.8.1 Cohort approach versus period approach. APC models 173

4.8.2 Projections and scenarios. Mortality by causes 175

4.9 References and suggestions for further reading 175

4.9.1 Landmarks in mortality projections 175

4.9.2 Further references 178

5 Forecasting mortality: applications and examples of age-period models 181

5.1 Introduction 181

5.2 Lee-Carter mortality projection model 186

5.2.1 Specification 186

5.2.2 Calibration 188

5.2.3 Application to Belgian mortality statistics 200

5.3 Cairns-Blake-Dowd mortality projection model 203

5.3.1 Specification 203

5.3.2 Calibration 206

5.3.3 Application to Belgian mortality Statistics 207

5.4 Smoothing 209

5.4.1 Motivation 209

5.4.2 P-splines approach 210

5.4.3 Smoothing in the Lee-Carter model 212

5.4.4 Application to Belgian mortality statistics 213

5.5 Selection of an optimal calibration period 214

5.5.1 Motivation 214

5.5.2 Selection procedure 216

5.5.3 Application to Belgian Mortality statistics 217

5.6 Analysis of residuals 218

5.6.1 Deviance and Pearson residuals 218

5.6.2 Application to Belgian mortality statistics 220

5.7 Mortality projection 221

5.7.1 Time series modelling for the time indices 221

5.7.2 Modelling of the Lee-Carter time index 223

5.7.3 Modelling the Cairns-Blake-Dowd time indices 228

5.8 Prediction intervals 229

5.8.1 Why bootstrapping? 229

5.8.2 Bootstrap percentiles confidence intervals 230

5.8.3 Application to Belgian mortality statistics 232

5.9 Forecasting life expectancies 234

5.9.1 Official projections performed by the Belgian Federal Planning Bureau (FPB) 235

5.9.2 Andreev-Vaupel projections 235

5.9.3 Application to Belgian mortality statistics 237

5.9.4 Longevity fan charts 240

5.9.5 Back testing 240

6 Forecasting mortality: applications and examples of age-period-cohort models 243

6.1 Introduction 243

6.2 LC age-period-cohort mortality projection model 246

6.2.1 Model Structure 246

6.2.2 Error structure and model fitting 248

6.2.3 Mortality rate projections 253

6.2.4 Discussion 253

6.3 Application to United Kingdom mortality data 254

6.4 Cairns-Blake-Dowd mortality projection model: allowing for cohort effects 263

6.5 P-splines model: allowing for cohort effects 265

7 The longevity risk: actuarial perspectives 267

7.1 Introduction 267

7.2 The longevity risk 268

7.2.1 Mortality risks 268

7.2.2 Representing longevity risk: stochastic modelling issues 270

7.2.3 Representing longevity risk: some examples 273

7.2.4 Measuring longevity risk in a static framework 276

7.3 Managing the longevity risk 293

7.3.1 A risk management perspective 293

7.3.2 Natural hedging 299

7.3.3 Solvency issues 303

7.3.4 Reinsurance arrangements 318

7.4 Alternative risk transfers 330

7.4.1 Life insurance securitization 330

7.4.2 Mortality-linked securities 332

7.4.3 Hedging life annuity liabilities through longevity bonds 337

7.5 Life annuities and longevity risk 343

7.5.1 The location of mortality risks in traditional life annuity products 343

7.5.2 GAO and GAR 346

7.5.3 Adding flexibility to GAR products 347

7.6 Allowing for longevity risk in pricing 350

7.7 Financing post-retirement income 354

7.7.1 Comparing life annuity prices 354

7.7.2 Life annuities versus income drawdown 356

7.7.3 The 'mortality drag' 359

7.7.4 Flexibility in financing post-retirement income 363

7.8 References and suggestions for further reading 369

References 373

Index 389

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