The Calculus of Retirement Income: Financial Models for Pension Annuities and Life Insurance

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ISBN: 9780521842587 | 0521842581
Cover: Hardcover
Copyright: 3/13/2006

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The book introduces and develops the basic actuarial models and underlying pricing of life-contingent pension annuities and life insurance from a unique financial perspective. The ideas and techniques are then applied to the real-world problem of generating sustainable retirement income towards the end of the human life-cycle. The role of lifetime income, longevity insurance, and systematic withdrawal plans are investigated in a parsimonious framework. The underlying technology and terminology of the book are based on continuous-time financial economics by merging analytic laws of mortality with the dynamics of equity markets and interest rates. Nonetheless, the book requires a minimal background in mathematics and emphasizes applications and examples more than proofs and theorems. It can serve as an ideal textbook for an applied course on wealth management and retirement planning in addition to being a reference for quantitatively-inclined financial planners.

List of Figures and Tables

I MODELS OF ACTUARIAL FINANCE

1 Introduction and Motivation

(14)

1.1 The Drunk Gambler Problem

(2)

1.2 The Demographic Picture

(4)

1.3 The Ideal Audience

(1)

1.4 Learning Objectives

(2)

1.5 Acknowledgments

(2)

1.6 Appendix: Drunk Gambler Solution

(3)

2 Modeling the Human Life Cycle

(17)

2.1 The Next Sixty Years of Your Life

(1)

2.2 Future Value of Savings

(2)

2.3 Present Value of Consumption

(2)

2.4 Exchange Rate between Savings and Consumption

(4)

2.5 A Neutral Replacement Rate

(1)

2.6 Discounted Value of a Life-Cycle Plan

(1)

2.7 Real vs. Nominal Planning with Inflation

(2)

2.8 Changing Investment Rates over Time

(2)

2.9 Further Reading

(1)

2.10 Problems

(1)

3 Models of Human Mortality

(30)

3.1 Mortality Tables and Rates

(1)

3.2 Conditional Probability of Survival

(2)

3.3 Remaining Lifetime Random Variable

(1)

3.4 Instantaneous Force of Mortality

(1)

3.5 The ODE Relationship

(2)

3.6 Moments in Your Life

(3)

3.7 Median vs. Expected Remaining Lifetime

(1)

3.8 Exponential Law of Mortality

(1)

3.9 Gompertz-Makeham Law of Mortality

(3)

3.10 Fitting Discrete Tables to Continuous Laws

(2)

3.11 General Hazard Rates

(2)

3.12 Modeling Joint Lifetimes

(2)

3.13 Period vs. Cohort Tables

(4)

3.14 Further Reading

(1)

3.15 Notation

(1)

3.16 Problems

(1)

3.17 Technical Note: Incomplete Gamma Function in Excel

(1)

3.18 Appendix: Normal Distribution and Calculus Refresher

(2)

4 Valuation Models of Deterministic Interest

(19)

4.1 Continuously Compounded Interest Rates?

(2)

4.2 Discount Factors

(1)

4.3 How Accurate Is the Rule of 72?

(1)

4.4 Zero Bonds and Coupon Bonds

(2)

4.5 Arbitrage: Linking Value and Market Price

(2)

4.6 Term Structure of Interest Rates

(1)

4.7 Bonds: Nonflat Term Structure

(1)

4.8 Bonds: Nonconstant Coupons

(1)

4.9 Taylor's Approximation

(1)

4.10 Explicit Values for Duration and Convexity

(2)

4.11 Numerical Examples of Duration and Convexity

(2)

4.12 Another Look at Duration and Convexity

(1)

4.13 Further Reading

(1)

4.14 Notation

(1)

4.15 Problems

(1)

5 Models of Risky Financial Investments

(27)

5.1 Recent Stock Market History

(3)

5.2 Arithmetic Average Return versus Geometric Average Return

(2)

5.3 A Long-Term Model for Risk

(3)

5.4 Introducing Brownian Motion

(6)

5.5 Index Averages and Index Medians

(1)

5.6 The Probability of Regret

(2)

5.7 Focusing on the Rate of Change

100

(1)

5.8 How to Simulate a Diffusion Process

101

(1)

5.9 Asset Allocation and Portfolio Construction

102

(2)

5.10 Space Time Diversification

104

(3)

5.11 Further Reading

107

(1)

5.12 Notation

108

(1)

5.13 Problems

108

(2)

6 Models of Pension Life Annuities

110

(28)

6.1 Motivation and Agenda

110

(1)

6.2 Market Prices of Pension Annuities

110

(4)

6.3 Valuation of Pension Annuities: General

114

(1)

6.4 Valuation of Pension Annuities: Exponential

115

(1)

6.5 The Wrong Way to Value Pension Annuities

115

(1)

6.6 Valuation of Pension Annuities: Gompertz-Makeham

116

(3)

6.7 How Is the Annuity's Income Taxed?

119

(2)

6.8 Deferred Annuities: Variation on a Theme

121

(2)

6.9 Period Certain versus Term Certain

123

(2)

6.10 Valuation of Joint and Survivor Pension Annuities

125

(3)

6.11 Duration of a Pension Annuity

128

(2)

6.12 Variable vs. Fixed Pension Annuities

130

(4)

6.13 Further Reading

134

(2)

6.14 Notation

136

(1)

6.15 Problems

136

(2)

7 Models of Life Insurance

138

(26)

7.1 A Free (Last) Supper?

138

(1)

7.2 Market Prices of Life Insurance

138

(1)

7.3 The Impact of Health Status

139

(1)

7.4 How Much Life Insurance Do You Need?

140

(2)

7.5 Other Kinds of Life Insurance

142

(1)

7.6 Value of Life Insurance: Net Single Premium

143

(2)

7.7 Valuing Life Insurance Using Pension Annuities

145

(2)

7.8 Arbitrage Relationship

147

(1)

7.9 Tax Arbitrage Relationship

148

(1)

7.10 Value of Life Insurance: Exponential Mortality

149

(1)

7.11 Value of Life Insurance: GoMa Mortality

149

(1)

7.12 Life Insurance Paid by Installments

150

(1)

7.13 NSP: Delayed and Term Insurance

150

(1)

7.14 Variations on Life Insurance

151

(3)

7.15 What If You Stop Paying Premiums?

154

(3)

7.16 Duration of Life Insurance

157

(2)

7.17 Following a Group of Policies

159

(1)

7.18 The Next Generation: Universal Life Insurance

160

(2)

7.19 Further Reading

162

(1)

7.20 Notation

162

(1)

7.21 Problems

162

(2)

8 Models of DB vs. DC Pensions

164

(21)

8.1 A Choice of Pension Plans

164

(1)

8.2 The Core of Defined Contribution Pensions

165

(4)

8.3 The Core of Defined Benefit Pensions

169

(3)

8.4 What Is the Value of a DB Pension Promise?

172

(4)

8.5 Pension Funding and Accounting

176

(4)

8.6 Further Reading

180

(1)

8.7 Notation

181

(1)

8.8 Problems

182

(3)

II WEALTH MANAGEMENT: APPLICATIONS AND IMPLICATIONS

9 Sustainable Spending at Retirement

185

(30)

9.1 Living in Retirement

185

(2)

9.2 Stochastic Present Value

187

(3)

9.3 Analytic Formula: Sustainable Retirement Income

190

(2)

9.4 The Main Result: Exponential Reciprocal Gamma

192

(1)

9.5 Case Study and Numerical Examples

193

(9)

9.6 Increased Sustainable Spending without More Risk?

202

(4)

9.7 Conclusion

206

(2)

9.8 Further Reading

208

(1)

9.9 Problems

208

(1)

9.10 Appendix: Derivation of the Formula

209

(6)

10 Longevity Insurance Revisited

215

(34)

10.1 To Annuitize or Not To Annuitize?

215

(1)

10.2 Five 95-Year-Olds Playing Bridge

216

(2)

10.3 The Algebra of Fixed and Variable Tontines

218

(2)

10.4 Asset Allocation with Tontines

220

(5)

10.5 A First Look at Self-Annuitization

225

(1)

10.6 The Implied Longevity Yield

226

(8)

10.7 Advanced-Life Delayed Annuities

234

(7)

10.8 Who Incurs Mortality Risk and Investment Rate Risk?

241

(3)

10.9 Further Reading

244

(1)

10.10 Notation

245

(1)

10.11 Problems

245

(4)

III ADVANCED TOPICS

11 Options within Variable Annuities

249

(21)

11.1 To Live and Die in VA

249

(3)

11.2 The Value of Paying by Installments

252

(5)

11.3 A Simple Guaranteed Minimum Accumulation Benefit

257

(1)

11.4 The Guaranteed Minimum Death Benefit

258

(1)

11.5 Special Case: Exponential Mortality

259

(3)

11.6 The Guaranteed Minimum Withdrawal Benefit

262

(6)

11.7 Further Reading

268

(1)

11.8 Notation

269

(1)

12 The Utility of Annuitization

270

(23)

12.1 What Is the Protection Worth?

270

(1)

12.2 Models of Utility, Value, and Price

271

(1)

12.3 The Utility Function and Insurance

272

(2)

12.4 Utility of Consumption and Lifetime Uncertainty

274

(4)

12.5 Utility and Annuity Asset Allocation

278

(3)

12.6 The Optimal Timing of Annuitization

281

(1)

12.7 The Real Option to Defer Annuitization

282

(5)

12.8 Advanced RODA Model

287

(2)

12.9 Subjective vs. Objective Mortality

289

(1)

12.10 Variable vs. Fixed Payout Annuities

290

(1)

12.11 Further Reading

291

(1)

12.12 Notation

292

(1)

13 Final Words

293

(2)

14 Appendix

295

(6)

Bibliography

301

(8)

Index

309

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The Calculus of Retirement Income: Financial Models for Pension Annuities and Life Insurance

Summary

Table of Contents

Supplemental Materials