The calculus of retirement income: financial models for pension annuities and life insurance
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| Format: | Book |
| Language: | English |
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Cambridge
Cambridge Univ. Press
2006
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| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Publisher description Table of contents only Contributor biographical information Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke Includes bibliographical references (p. 301-307) and index |
| Physical Description: | XIV, 321 S. graph. Darst. 24 cm |
| ISBN: | 0521842581 9780521842587 |
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| 100 | 1 | |a Milevsky, Moshe Arye |d 1967- |e Verfasser |0 (DE-588)171757416 |4 aut | |
| 245 | 1 | 0 | |a The calculus of retirement income |b financial models for pension annuities and life insurance |c Moshe A. Milevsky |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge |b Cambridge Univ. Press |c 2006 | |
| 300 | |a XIV, 321 S. |b graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 500 | |a Includes bibliographical references (p. 301-307) and index | ||
| 650 | 7 | |a Actuariële wetenschappen |2 gtt | |
| 650 | 4 | |a Annuités - Modèles mathématiques | |
| 650 | 7 | |a Levensverzekering |2 gtt | |
| 650 | 7 | |a Pensioen |2 gtt | |
| 650 | 4 | |a Pensions de vieillesse - Modèles mathématiques | |
| 650 | 4 | |a Revenu de retraite - Modèles mathématiques | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Old age pensions |x Mathematical models | |
| 650 | 4 | |a Annuities |x Mathematical models | |
| 650 | 4 | |a Retirement income |x Mathematical models | |
| 650 | 0 | 7 | |a Altersversorgung |0 (DE-588)4001479-4 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Mathematisches Modell |0 (DE-588)4114528-8 |D s |
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| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0633/2005029455-t.html |3 Table of contents only | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0733/2005029455-b.html |3 Contributor biographical information | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016223697&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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| _version_ | 1804137238638886912 |
|---|---|
| adam_text | Contents
List of Figures and Tables page x
I MODELS OF ACTUARIAL FINANCE
1 Introduction and Motivation 3
1.1 The Drunk Gambler Problem 3
1.2 The Demographic Picture 5
1.3 The Ideal Audience 9
1.4 Learning Objectives 10
1.5 Acknowledgments 12
1.6 Appendix: Drunk Gambler Solution 14
2 Modeling the Human Life Cycle 17
2.1 The Next Sixty Years of Your Life 17
2.2 Future Value of Savings 18
2.3 Present Value of Consumption 20
2.4 Exchange Rate between Savings and Consumption 22
2.5 A Neutral Replacement Rate 26
2.6 Discounted Value of a Life Cycle Plan 27
2.7 Real vs. Nominal Planning with Inflation 28
2.8 Changing Investment Rates over Time 30
2.9 Further Reading 32
2.10 Problems 33
3 Models of Human Mortality 34
3.1 Mortality Tables and Rates 34
3.2 Conditional Probability of Survival 35
3.3 Remaining Lifetime Random Variable 37
3.4 Instantaneous Force of Mortality 38
3.5 The ODE Relationship 39
3.6 Moments in Your Life 41
3.7 Median vs. Expected Remaining Lifetime 44
3.8 Exponential Law of Mortality 45
3.9 Gompertz Makeham Law of Mortality 46
3.10 Fitting Discrete Tables to Continuous Laws 49
3.11 General Hazard Rates 51
3.12 Modeling Joint Lifetimes 53
3.13 Period vs. Cohort Tables 55
3.14 Further Reading 59
3.15 Notation 60
3.16 Problems 60
3.17 Technical Note: Incomplete Gamma Function in Excel 61
3.18 Appendix: Normal Distribution and Calculus Refresher 62
4 Valuation Models of Deterministic Interest 64
4.1 Continuously Compounded Interest Rates? 64
4.2 Discount Factors 66
4.3 How Accurate Is the Rule of 72? 67
4.4 Zero Bonds and Coupon Bonds 68
4.5 Arbitrage: Linking Value and Market Price 70
4.6 Term Structure of Interest Rates 72
4.7 Bonds: Nonflat Term Structure 73
4.8 Bonds: Nonconstant Coupons 74
4.9 Taylor s Approximation 75
4.10 Explicit Values for Duration and Convexity 76
4.11 Numerical Examples of Duration and Convexity 78
4.12 Another Look at Duration and Convexity 80
4.13 Further Reading 81
4.14 Notation 82
4.15 Problems 82
5 Models of Risky Financial Investments 83
5.1 Recent Stock Market History 83
5.2 Arithmetic Average Return versus Geometric Average
Return 86
5.3 A Long Term Model for Risk 88
5.4 Introducing Brownian Motion 91
5.5 Index Averages and Index Medians 97
5.6 The Probability of Regret 98
5.7 Focusing on the Rate of Change 100
5.8 How to Simulate a Diffusion Process 101
5.9 Asset Allocation and Portfolio Construction 102
5.10 Space Time Diversification 104
5.11 Further Reading 107
5.12 Notation 108
5.13 Problems 108
Contents vii
6 Models of Pension Life Annuities 110
6.1 Motivation and Agenda 110
6.2 Market Prices of Pension Annuities 110
6.3 Valuation of Pension Annuities: General 114
6.4 Valuation of Pension Annuities: Exponential 115
6.5 The Wrong Way to Value Pension Annuities 115
6.6 Valuation of Pension Annuities: Gompertz Makeham 116
6.7 How Is the Annuity s Income Taxed? 119
6.8 Deferred Annuities: Variation on a Theme 121
6.9 Period Certain versus Term Certain 123
6.10 Valuation of Joint and Survivor Pension Annuities 125
6.11 Duration of a Pension Annuity 128
6.12 Variable vs. Fixed Pension Annuities 130
6.13 Further Reading 134
6.14 Notation 136
6.15 Problems 136
7 Models of Life Insurance 138
7.1 A Free (Last) Supper? 138
7.2 Market Prices of Life Insurance 138
7.3 The Impact of Health Status 139
7.4 How Much Life Insurance Do You Need? 140
7.5 Other Kinds of Life Insurance 142
7.6 Value of Life Insurance: Net Single Premium 143
7.7 Valuing Life Insurance Using Pension Annuities 145
7.8 Arbitrage Relationship 147
7.9 Tax Arbitrage Relationship 148
7.10 Value of Life Insurance: Exponential Mortality 149
7.11 Value of Life Insurance: GoMa Mortality 149
7.12 Life Insurance Paid by Installments 150
7.13 NSP: Delayed and Term Insurance 150
7.14 Variations on Life Insurance 151
7.15 What If You Stop Paying Premiums? 154
7.16 Duration of Life Insurance 157
7.17 Following a Group of Policies 159
7.18 The Next Generation: Universal Life Insurance 160
7.19 Further Reading 162
7.20 Notation 162
7.21 Problems 162
8 Models of DB vs. DC Pensions 164
8.1 A Choice of Pension Plans 164
8.2 The Core of Denned Contribution Pensions 165
8.3 The Core of Denned Benefit Pensions 169
viii Contents
8.4 What Is the Value of a DB Pension Promise? 172
8.5 Pension Funding and Accounting 176
8.6 Further Reading 180 j
8.7 Notation 181
8.8 Problems 182
II WEALTH MANAGEMENT: *
APPLICATIONS AND IMPLICATIONS !
9 Sustainable Spending at Retirement 185
9.1 Living in Retirement 185
9.2 Stochastic Present Value 187
9.3 Analytic Formula: Sustainable Retirement Income 190
9.4 The Main Result: Exponential Reciprocal Gamma 192
9.5 Case Study and Numerical Examples 193
9.6 Increased Sustainable Spending without More Risk? 202
9.7 Conclusion 206
9.8 Further Reading 208
9.9 Problems 208
9.10 Appendix: Derivation of the Formula 209
10 Longevity Insurance Revisited 215
10.1 To Annuitize or Not To Annuitize? 215
10.2 Five 95 Year Olds Playing Bridge 216
10.3 The Algebra of Fixed and Variable Tontines 218
10.4 Asset Allocation with Tontines 220
10.5 A First Look at Self Annuitization 225
10.6 The Implied Longevity Yield 226
10.7 Advanced Life Delayed Annuities 234
10.8 Who Incurs Mortality Risk and Investment Rate Risk? 241
10.9 Further Reading 244
10.10 Notation 245
10.11 Problems 245
III ADVANCED TOPICS
11 Options within Variable Annuities 249
11.1 To Live and Die in VA 249
11.2 The Value of Paying by Installments 252
11.3 A Simple Guaranteed Minimum Accumulation Benefit 257
11.4 The Guaranteed Minimum Death Benefit 258
11.5 Special Case: Exponential Mortality 259
11.6 The Guaranteed Minimum Withdrawal Benefit 262
11.7 Further Reading 268 j
11.8 Notation 269 j
Contents ix
12 The Utility of Annuitization 270
12.1 What Is the Protection Worth? 270
12.2 Models of Utility, Value, and Price 271
12.3 The Utility Function and Insurance 272
12.4 Utility of Consumption and Lifetime Uncertainty 274
12.5 Utility and Annuity Asset Allocation 278
12.6 The Optimal Timing of Annuitization 281
12.7 The Real Option to Defer Annuitization 282
12.8 Advanced RODA Model 287
12.9 Subjective vs. Objective Mortality 289
12.10 Variable vs. Fixed Payout Annuities 290
12.11 Further Reading 291
12.12 Notation 292
13 Final Words 293
14 Appendix 295
Bibliography 301
Index 309
Figures and Tables
i
Figures
2.1 The human financial life cycle: Savings, wealth
consumption (constant investment rate) page 25
2.2 The human financial life cycle: Savings, wealth
consumption (varying investment rate) 32
3.1 RP2000 mortality table used for pensions 36
3.2 Relationships between mortality descriptions 40
3.3 The CDF versus the PDF of a normal remaining lifetime R.V. 42
3.4 The hazard rate for the normal distribution 42
3.5 The CDF versus the PDF of an exponential remaining
lifetime R.V. 47 ;
3.6 RP2000 (unisex pension) mortality table vs. best Gompertz fit
vs. exponential approximation 50
4.1 Evolution of the bond price over time 69
4.2 Model bond value vs. valuation rate 71
4.3 The term structure of interest rates 73
4.4 Taylor s D as maturity gets closer 77
4.5 How good is the approximation? 81
5.1 Visualizing the stochastic growth rate 89
5.2 Sample path of Brownian motion over 40 years 92
5.3 Another sample path of Brownian motion over 40 years 93
5.4 Sample paths: BM vs. nsBM vs. GBM 94
5.5 What is the Probability of Regret (PoR)? 99
5.6 Space time diversification 107
6.1 Pension annuity quotes: Relationship between credit rating and
average payout (income) 113
6.2 One sample path Three outcomes depending on h 135
8.1 Pension systems 165
8.2 Salary/wage profile vs. weighting scheme: Modeling pension ¦
vesting career averages 169 j
i
x j
Figures and Tables xi
8.3 ABO vs. PBO vs. RBO 174
9.1 The retirement triangle 186
9.2 Stochastic present value (SPV) of retirement consumption 189
9.3 Minimum wealth required at various ages to maintain a fixed
retirement ruin probability 200
9.4 Probability given spending rate is not sustainable 201
9.5 Expected wealth: 65 year old consumes $5 per year but
protects portfolio with 5% out of the money puts 204
9.6 Ruin probability conditional on returns 205
10.1 I want a lifetime income 228
10.2 Advanced life delayed annuity 235
11.1 Three types of puts 250
11.2 Titanic vs. vanilla put 260
12.1 Expected loss 271
Tables
1.1 Old age dependency ratio around the world 6
1.2 Expected number of years spent in retirement around the
world 7
2.1 Financial exchange rate between $1 saved annually over 30
working years and dollar consumption during retirement 23
2.2 Government sponsored pension plans: How generous are they? 26
2.3 Discounted value of life cycle plan = $0,241 under first
sequence of varying returns 31
2.4 Discounted value of life cycle plan = $0,615 under second
sequence of varying returns 31
3.1 Mortality table for healthy members of a pension plan 35
3.2 Mortality odds when life is normally distributed 41
3.3 Life expectancy at birth in 2005 43
3.4 Increase since 1950 in life expectancy at birth E[T0] 44
3.5 Mortality odds when life is exponentially distributed 46
3.6 Example of fitting Gompertz Makeham law to a group
mortality table—Female 49
3.7 Example of fitting Gompertz Makeham law to a group
mortality table—Male 49
3.8 How good is a continuous law of mortality?—Gompertz vs.
exponential vs. RP2000 50
3.9 Working with the instantaneous hazard rate 52
3.10 Survival probabilities at age 65 54
3.11 Change in mortality patterns over time—Female 56
3.12 Change in mortality patterns over time—Male 57
4.1 Year end value of $1 under infrequent compounding 65
4.2 Year end value of $1 under frequent compounding 65
4.3 Years required to double or triple $1 invested at various
interest rates 67
4.4 Valuation of 5 year bonds as a fraction of face value 70
4.5 Valuation of 10 year bonds as a fraction of face value 70
4.6 Estimated vs. actual value of ,000 bond after change in
valuation rates 80
5.1 Nominal investment returns over 10 years 84
5.2 Growth rates during different investment periods 85
5.3 After inflation (real) returns over 10 years 86
5.4 Geometric mean returns 87
5.5 Probability of losing money in a diversified portfolio 90
5.6 SDE simulation of GBM using the Euler method 102
6.1 Monthly income from ,000 premium single life pension
annuity 111
6.2 A quick comparison with the bond market 112
6.3 Monthly income from ,000 premium joint life pension
annuity 112
6.4 IPAF ax: Price of lifetime $1 annual income 118
6.5 Taxable portion of income flow from $l for life annuity
purchased with non tax sheltered funds 121
6.6 DPAFua45: Price of lifetime $1 annual income for 45 year old 123
6.7 Value V(r, T) of term certain annuity factor vs. immediate
pension annuity factor 124
6.8 Duration value D (in years) of immediate pension annuity
factor 129
6.9 Pension annuity factor at age x = 50 when r = 5% 131
6.10 Annuity payout at age x = 65 ( ,000 premium) 134
7.1 U.S. monthly premiums for a ,000 death benefit 139
7.2 U.S. monthly premiums for a ,000 death benefit—
50 year old nonsmoker 140
7.3 Net single premium for ,000 of life insurance protection 150
7.4 Net periodic premium for ,000 of life insurance protection 151
7.5 Model results: ,000 life insurance—Monthly premiums
for 50 year old by health status 153
7.6 ,000 life insurance—Monthly premiums for 50 year old
by lapse rate 156
7.7 Duration value D (in years) of NSP for life insurance 158
7.8 Modeling a book of insurance policies over time 159
8.1 DC pension retirement income 171
8.2 DC pension: Income replacement rate 171
8.3 DB pension retirement income 172
8.4 DB pension: Income replacement rate 173
8.5 Current value of sample retirement pension by valuation rate
and by type of benefit obligation 175
Figures and Tables xiii
8.6 Change in value (from age 45 to 46) of sample retirement
pension by valuation rate and by type of benefit obligation 177
8.7 Change in pension value at various ages assuming r = 5%
valuation rate 177
8.8 Change in PBO from prior year 178
8.9 Change in ABO from prior year 178
9.1 Probability of retirement ruin given (arithmetic mean)
return i of 7% with volatility a of 20% 195
9.2 Probability of retirement ruin given fi of 5% with a of 20% 197
9.3 Probability of retirement ruin given fi of 5% with a of 10% 197
9.4(a) Maximum annual spending given tolerance for 5%
probability of ruin 198
9.4(b) Maximum annual spending given tolerance for 10%
probability of ruin 198
9.4(c) Maximum annual spending given tolerance for 25%
probability of ruin 199
9.5 Probability of ruin for 65 year old male given collared
portfolio under a fixed spending rate 202
9.6 Probability of ruin for 65 year old female given collared
portfolio under a fixed spending rate 203
10.1 Algebra of fixed tontine vs. nontontine investment 218
10.2 Investment returns from fixed tontines given survival to
year s end 219
10.3 Algebra of variable tontine vs. nontontine investment 220
10.4 Optimal portfolio mix of stocks and safe cash 224
10.5 Monthly income from immediate annuity ( ,000
premium) 231
10.6 Cost for male of monthly from immediate annuity 231
10.7 Cost for female of monthly from immediate annuity 232
10.8 Should an 80 year old annuitize? 232
10.9 ALDA: Net single premium (uax) required at age x to
produce $1 of income starting at age x + u 236
10.10 ALDA income multiple: Dollars received during retirement
per dollar paid today 239
10.11 Lapse adjusted ALDA income multiple 240
10.12 Profit spread (in basis points) from sale of ALDA given
mortality misestimate of 20% 244
11.1 BSM put option value as a function of spot price and
maturity—Strike price = 252
11.2 Discounted value of fees 256
11.3 Annual fee (in basis points) needed to hedge the death
benefit—Female 258
11 4 Annual fee (in basis points) needed to hedge the death
benefit—Male 259
xiv Figures and Tables
11.5 Value of exponential Titanic option 262
11.6 GMWB payoff and the probability of ruin within 14.28
years 265
11.7 Impact of GMWB rate and subaccount volatility on
required fee k 268
12.1 Relationship between risk aversion y and subjective
insurance premium Iy 275
12.2 When should you annuitize in order to maximize your
utility of wealth? 288
12.3 Real option to delay annuitization for a 60 year old male
who disagrees with insurance company s estimate of his
mortality 289
12.4 When should you annuitize?—Given the choice of fixed
and variable annuities 291
14.1(a) RP2000 healthy (static) annuitant mortality table—Ages
50 89 296
14.1(b) RP2000 healthy (static) annuitant mortality table—Ages
90 120 296
14.2 International comparison (year 2000) of mortality rates qx
at age 65 297
14.3(a) 2001 CSO (ultimate) insurance mortality table—Ages
50 89 298
14.3(b) 2001 CSO (ultimate) insurance mortality table—Ages
90 120 298
14.4 Cumulative distribution function for a normal random
variable 299
14.5 Cumulative distribution function for a reciprocal Gamma
random variable 299
|
| adam_txt |
Contents
List of Figures and Tables page x
I MODELS OF ACTUARIAL FINANCE
1 Introduction and Motivation 3
1.1 The Drunk Gambler Problem 3
1.2 The Demographic Picture 5
1.3 The Ideal Audience 9
1.4 Learning Objectives 10
1.5 Acknowledgments 12
1.6 Appendix: Drunk Gambler Solution 14
2 Modeling the Human Life Cycle 17
2.1 The Next Sixty Years of Your Life 17
2.2 Future Value of Savings 18
2.3 Present Value of Consumption 20
2.4 Exchange Rate between Savings and Consumption 22
2.5 A Neutral Replacement Rate 26
2.6 Discounted Value of a Life Cycle Plan 27
2.7 Real vs. Nominal Planning with Inflation 28
2.8 Changing Investment Rates over Time 30
2.9 Further Reading 32
2.10 Problems 33
3 Models of Human Mortality 34
3.1 Mortality Tables and Rates 34
3.2 Conditional Probability of Survival 35
3.3 Remaining Lifetime Random Variable 37
3.4 Instantaneous Force of Mortality 38
3.5 The ODE Relationship 39
3.6 Moments in Your Life 41
3.7 Median vs. Expected Remaining Lifetime 44
3.8 Exponential Law of Mortality 45
3.9 Gompertz Makeham Law of Mortality 46
3.10 Fitting Discrete Tables to Continuous Laws 49
3.11 General Hazard Rates 51
3.12 Modeling Joint Lifetimes 53
3.13 Period vs. Cohort Tables 55
3.14 Further Reading 59
3.15 Notation 60
3.16 Problems 60
3.17 Technical Note: Incomplete Gamma Function in Excel 61
3.18 Appendix: Normal Distribution and Calculus Refresher 62
4 Valuation Models of Deterministic Interest 64
4.1 Continuously Compounded Interest Rates? 64
4.2 Discount Factors 66
4.3 How Accurate Is the Rule of 72? 67
4.4 Zero Bonds and Coupon Bonds 68
4.5 Arbitrage: Linking Value and Market Price 70
4.6 Term Structure of Interest Rates 72
4.7 Bonds: Nonflat Term Structure 73
4.8 Bonds: Nonconstant Coupons 74
4.9 Taylor's Approximation 75
4.10 Explicit Values for Duration and Convexity 76
4.11 Numerical Examples of Duration and Convexity 78
4.12 Another Look at Duration and Convexity 80
4.13 Further Reading 81
4.14 Notation 82
4.15 Problems 82
5 Models of Risky Financial Investments 83
5.1 Recent Stock Market History 83
5.2 Arithmetic Average Return versus Geometric Average
Return 86
5.3 A Long Term Model for Risk 88
5.4 Introducing Brownian Motion 91
5.5 Index Averages and Index Medians 97
5.6 The Probability of Regret 98
5.7 Focusing on the Rate of Change 100
5.8 How to Simulate a Diffusion Process 101
5.9 Asset Allocation and Portfolio Construction 102
5.10 Space Time Diversification 104
5.11 Further Reading 107
5.12 Notation 108
5.13 Problems 108
Contents vii
6 Models of Pension Life Annuities 110
6.1 Motivation and Agenda 110
6.2 Market Prices of Pension Annuities 110
6.3 Valuation of Pension Annuities: General 114
6.4 Valuation of Pension Annuities: Exponential 115
6.5 The Wrong Way to Value Pension Annuities 115
6.6 Valuation of Pension Annuities: Gompertz Makeham 116
6.7 How Is the Annuity's Income Taxed? 119
6.8 Deferred Annuities: Variation on a Theme 121
6.9 Period Certain versus Term Certain 123
6.10 Valuation of Joint and Survivor Pension Annuities 125
6.11 Duration of a Pension Annuity 128
6.12 Variable vs. Fixed Pension Annuities 130
6.13 Further Reading 134
6.14 Notation 136
6.15 Problems 136
7 Models of Life Insurance 138
7.1 A Free (Last) Supper? 138
7.2 Market Prices of Life Insurance 138
7.3 The Impact of Health Status 139
7.4 How Much Life Insurance Do You Need? 140
7.5 Other Kinds of Life Insurance 142
7.6 Value of Life Insurance: Net Single Premium 143
7.7 Valuing Life Insurance Using Pension Annuities 145
7.8 Arbitrage Relationship 147
7.9 Tax Arbitrage Relationship 148
7.10 Value of Life Insurance: Exponential Mortality 149
7.11 Value of Life Insurance: GoMa Mortality 149
7.12 Life Insurance Paid by Installments 150
7.13 NSP: Delayed and Term Insurance 150
7.14 Variations on Life Insurance 151
7.15 What If You Stop Paying Premiums? 154
7.16 Duration of Life Insurance 157
7.17 Following a Group of Policies 159
7.18 The Next Generation: Universal Life Insurance 160
7.19 Further Reading 162
7.20 Notation 162
7.21 Problems 162
8 Models of DB vs. DC Pensions 164
8.1 A Choice of Pension Plans 164
8.2 The Core of Denned Contribution Pensions 165
8.3 The Core of Denned Benefit Pensions 169
viii Contents
8.4 What Is the Value of a DB Pension Promise? 172
8.5 Pension Funding and Accounting 176
8.6 Further Reading 180 j
8.7 Notation 181
8.8 Problems 182
II WEALTH MANAGEMENT: *
APPLICATIONS AND IMPLICATIONS !
9 Sustainable Spending at Retirement 185
9.1 Living in Retirement 185
9.2 Stochastic Present Value 187
9.3 Analytic Formula: Sustainable Retirement Income 190
9.4 The Main Result: Exponential Reciprocal Gamma 192
9.5 Case Study and Numerical Examples 193
9.6 Increased Sustainable Spending without More Risk? 202
9.7 Conclusion 206
9.8 Further Reading 208
9.9 Problems 208
9.10 Appendix: Derivation of the Formula 209
10 Longevity Insurance Revisited 215
10.1 To Annuitize or Not To Annuitize? 215
10.2 Five 95 Year Olds Playing Bridge 216
10.3 The Algebra of Fixed and Variable Tontines 218
10.4 Asset Allocation with Tontines 220
10.5 A First Look at Self Annuitization 225
10.6 The Implied Longevity Yield 226
10.7 Advanced Life Delayed Annuities 234
10.8 Who Incurs Mortality Risk and Investment Rate Risk? 241
10.9 Further Reading 244
10.10 Notation 245
10.11 Problems 245
III ADVANCED TOPICS
11 Options within Variable Annuities 249
11.1 To Live and Die in VA 249
11.2 The Value of Paying by Installments 252
11.3 A Simple Guaranteed Minimum Accumulation Benefit 257
11.4 The Guaranteed Minimum Death Benefit 258
11.5 Special Case: Exponential Mortality 259
11.6 The Guaranteed Minimum Withdrawal Benefit 262
11.7 Further Reading 268 j
11.8 Notation 269 j
Contents ix
12 The Utility of Annuitization 270
12.1 What Is the Protection Worth? 270
12.2 Models of Utility, Value, and Price 271
12.3 The Utility Function and Insurance 272
12.4 Utility of Consumption and Lifetime Uncertainty 274
12.5 Utility and Annuity Asset Allocation 278
12.6 The Optimal Timing of Annuitization 281
12.7 The Real Option to Defer Annuitization 282
12.8 Advanced RODA Model 287
12.9 Subjective vs. Objective Mortality 289
12.10 Variable vs. Fixed Payout Annuities 290
12.11 Further Reading 291
12.12 Notation 292
13 Final Words 293
14 Appendix 295
Bibliography 301
Index 309
Figures and Tables '
i
Figures
2.1 The human financial life cycle: Savings, wealth
consumption (constant investment rate) page 25
2.2 The human financial life cycle: Savings, wealth
consumption (varying investment rate) 32
3.1 RP2000 mortality table used for pensions 36
3.2 Relationships between mortality descriptions 40
3.3 The CDF versus the PDF of a "normal" remaining lifetime R.V. 42
3.4 The hazard rate for the normal distribution 42
3.5 The CDF versus the PDF of an "exponential" remaining
lifetime R.V. 47 ;
3.6 RP2000 (unisex pension) mortality table vs. best Gompertz fit
vs. exponential approximation 50
4.1 Evolution of the bond price over time 69
4.2 Model bond value vs. valuation rate 71
4.3 The term structure of interest rates 73
4.4 "Taylor's D" as maturity gets closer 77
4.5 How good is the approximation? 81
5.1 Visualizing the stochastic growth rate 89
5.2 Sample path of Brownian motion over 40 years 92
5.3 Another sample path of Brownian motion over 40 years 93
5.4 Sample paths: BM vs. nsBM vs. GBM 94
5.5 What is the Probability of Regret (PoR)? 99
5.6 Space time diversification 107
6.1 Pension annuity quotes: Relationship between credit rating and
average payout (income) 113
6.2 One sample path Three outcomes depending on h 135
8.1 Pension systems 165
8.2 Salary/wage profile vs. weighting scheme: Modeling pension ¦
vesting career averages 169 j
i
x j
Figures and Tables xi
8.3 ABO vs. PBO vs. RBO 174
9.1 The retirement triangle 186
9.2 Stochastic present value (SPV) of retirement consumption 189
9.3 Minimum wealth required at various ages to maintain a fixed
retirement ruin probability 200
9.4 Probability given spending rate is not sustainable 201
9.5 Expected wealth: 65 year old consumes \$5 per year but
protects portfolio with 5% out of the money puts 204
9.6 Ruin probability conditional on returns 205
10.1 I want a lifetime income 228
10.2 Advanced life delayed annuity 235
11.1 Three types of puts 250
11.2 Titanic vs. vanilla put 260
12.1 Expected loss 271
Tables
1.1 Old age dependency ratio around the world 6
1.2 Expected number of years spent in retirement around the
world 7
2.1 Financial exchange rate between \$1 saved annually over 30
working years and dollar consumption during retirement 23
2.2 Government sponsored pension plans: How generous are they? 26
2.3 Discounted value of life cycle plan = \$0,241 under first
sequence of varying returns 31
2.4 Discounted value of life cycle plan = \$0,615 under second
sequence of varying returns 31
3.1 Mortality table for healthy members of a pension plan 35
3.2 Mortality odds when life is normally distributed 41
3.3 Life expectancy at birth in 2005 43
3.4 Increase since 1950 in life expectancy at birth E[T0] 44
3.5 Mortality odds when life is exponentially distributed 46
3.6 Example of fitting Gompertz Makeham law to a group
mortality table—Female 49
3.7 Example of fitting Gompertz Makeham law to a group
mortality table—Male 49
3.8 How good is a continuous law of mortality?—Gompertz vs.
exponential vs. RP2000 50
3.9 Working with the instantaneous hazard rate 52
3.10 Survival probabilities at age 65 54
3.11 Change in mortality patterns over time—Female 56
3.12 Change in mortality patterns over time—Male 57
4.1 Year end value of \$1 under infrequent compounding 65
4.2 Year end value of \$1 under frequent compounding 65
4.3 Years required to double or triple \$1 invested at various
interest rates 67
4.4 Valuation of 5 year bonds as a fraction of face value 70
4.5 Valuation of 10 year bonds as a fraction of face value 70
4.6 Estimated vs. actual value of \,000 bond after change in
valuation rates 80
5.1 Nominal investment returns over 10 years 84
5.2 Growth rates during different investment periods 85
5.3 After inflation (real) returns over 10 years 86
5.4 Geometric mean returns 87
5.5 Probability of losing money in a diversified portfolio 90
5.6 SDE simulation of GBM using the Euler method 102
6.1 Monthly income from \,000 premium single life pension
annuity 111
6.2 A quick comparison with the bond market 112
6.3 Monthly income from \,000 premium joint life pension
annuity 112
6.4 IPAF ax: Price of lifetime \$1 annual income 118
6.5 Taxable portion of income flow from \$l for life annuity
purchased with non tax sheltered funds 121
6.6 DPAFua45: Price of lifetime \$1 annual income for 45 year old 123
6.7 Value V(r, T) of term certain annuity factor vs. immediate
pension annuity factor 124
6.8 Duration value D (in years) of immediate pension annuity
factor 129
6.9 Pension annuity factor at age x = 50 when r = 5% 131
6.10 Annuity payout at age x = 65 (\,000 premium) 134
7.1 U.S. monthly premiums for a \,000 death benefit 139
7.2 U.S. monthly premiums for a \,000 death benefit—
50 year old nonsmoker 140
7.3 Net single premium for \,000 of life insurance protection 150
7.4 Net periodic premium for \,000 of life insurance protection 151
7.5 Model results: \,000 life insurance—Monthly premiums
for 50 year old by health status 153
7.6 \,000 life insurance—Monthly premiums for 50 year old
by lapse rate 156
7.7 Duration value D (in years) of NSP for life insurance 158
7.8 Modeling a book of insurance policies over time 159
8.1 DC pension retirement income 171
8.2 DC pension: Income replacement rate 171
8.3 DB pension retirement income 172
8.4 DB pension: Income replacement rate 173
8.5 Current value of sample retirement pension by valuation rate
and by type of benefit obligation 175
Figures and Tables xiii
8.6 Change in value (from age 45 to 46) of sample retirement
pension by valuation rate and by type of benefit obligation 177
8.7 Change in pension value at various ages assuming r = 5%
valuation rate 177
8.8 Change in PBO from prior year 178
8.9 Change in ABO from prior year 178
9.1 Probability of retirement ruin given (arithmetic mean)
return \i of 7% with volatility a of 20% 195
9.2 Probability of retirement ruin given fi of 5% with a of 20% 197
9.3 Probability of retirement ruin given fi of 5% with a of 10% 197
9.4(a) Maximum annual spending given tolerance for 5%
probability of ruin 198
9.4(b) Maximum annual spending given tolerance for 10%
probability of ruin 198
9.4(c) Maximum annual spending given tolerance for 25%
probability of ruin 199
9.5 Probability of ruin for 65 year old male given collared
portfolio under a fixed spending rate 202
9.6 Probability of ruin for 65 year old female given collared
portfolio under a fixed spending rate 203
10.1 Algebra of fixed tontine vs. nontontine investment 218
10.2 Investment returns from fixed tontines given survival to
year's end 219
10.3 Algebra of variable tontine vs. nontontine investment 220
10.4 Optimal portfolio mix of stocks and safe cash 224
10.5 Monthly income from immediate annuity (\,000
premium) 231
10.6 Cost for male of \ monthly from immediate annuity 231
10.7 Cost for female of \ monthly from immediate annuity 232
10.8 Should an 80 year old annuitize? 232
10.9 ALDA: Net single premium (uax) required at age x to
produce \$1 of income starting at age x + u 236
10.10 ALDA income multiple: Dollars received during retirement
per dollar paid today 239
10.11 Lapse adjusted ALDA income multiple 240
10.12 Profit spread (in basis points) from sale of ALDA given
mortality misestimate of 20% 244
11.1 BSM put option value as a function of spot price and
maturity—Strike price = \ 252
11.2 Discounted value of fees 256
11.3 Annual fee (in basis points) needed to hedge the death
benefit—Female 258
11 4 Annual fee (in basis points) needed to hedge the death
benefit—Male 259
xiv Figures and Tables
11.5 Value of exponential Titanic option 262
11.6 GMWB payoff and the probability of ruin within 14.28
years 265
11.7 Impact of GMWB rate and subaccount volatility on
required fee k 268
12.1 Relationship between risk aversion y and subjective
insurance premium Iy 275
12.2 When should you annuitize in order to maximize your
utility of wealth? 288
12.3 Real option to delay annuitization for a 60 year old male
who disagrees with insurance company's estimate of his
mortality 289
12.4 When should you annuitize?—Given the choice of fixed
and variable annuities 291
14.1(a) RP2000 healthy (static) annuitant mortality table—Ages
50 89 296
14.1(b) RP2000 healthy (static) annuitant mortality table—Ages
90 120 296
14.2 International comparison (year 2000) of mortality rates qx
at age 65 297
14.3(a) 2001 CSO (ultimate) insurance mortality table—Ages
50 89 298
14.3(b) 2001 CSO (ultimate) insurance mortality table—Ages
90 120 298
14.4 Cumulative distribution function for a normal random
variable 299
14.5 Cumulative distribution function for a reciprocal Gamma
random variable 299 |
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| author | Milevsky, Moshe Arye 1967- |
| author_GND | (DE-588)171757416 |
| author_facet | Milevsky, Moshe Arye 1967- |
| author_role | aut |
| author_sort | Milevsky, Moshe Arye 1967- |
| author_variant | m a m ma mam |
| building | Verbundindex |
| bvnumber | BV023019590 |
| callnumber-first | H - Social Science |
| callnumber-label | HD7105 |
| callnumber-raw | HD7105.3 |
| callnumber-search | HD7105.3 |
| callnumber-sort | HD 47105.3 |
| callnumber-subject | HD - Industries, Land Use, Labor |
| classification_rvk | QX 400 |
| classification_tum | WIR 190f |
| ctrlnum | (OCoLC)62109936 (DE-599)BVBBV023019590 |
| dewey-full | 368.3/701 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 368 - Insurance |
| dewey-raw | 368.3/701 |
| dewey-search | 368.3/701 |
| dewey-sort | 3368.3 3701 |
| dewey-tens | 360 - Social problems and services; associations |
| discipline | Wirtschaftswissenschaften |
| discipline_str_mv | Wirtschaftswissenschaften |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV023019590 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:12:32Z |
| indexdate | 2024-07-09T21:09:07Z |
| institution | BVB |
| isbn | 0521842581 9780521842587 |
| language | English |
| lccn | 2005029455 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016223697 |
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| physical | XIV, 321 S. graph. Darst. 24 cm |
| publishDate | 2006 |
| publishDateSearch | 2006 |
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| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Milevsky, Moshe Arye 1967- Verfasser (DE-588)171757416 aut The calculus of retirement income financial models for pension annuities and life insurance Moshe A. Milevsky 1. publ. Cambridge Cambridge Univ. Press 2006 XIV, 321 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Includes bibliographical references (p. 301-307) and index Actuariële wetenschappen gtt Annuités - Modèles mathématiques Levensverzekering gtt Pensioen gtt Pensions de vieillesse - Modèles mathématiques Revenu de retraite - Modèles mathématiques Mathematisches Modell Old age pensions Mathematical models Annuities Mathematical models Retirement income Mathematical models Altersversorgung (DE-588)4001479-4 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Altersversorgung (DE-588)4001479-4 s Mathematisches Modell (DE-588)4114528-8 s DE-604 http://www.loc.gov/catdir/enhancements/fy0633/2005029455-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0633/2005029455-t.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0733/2005029455-b.html Contributor biographical information HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016223697&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Milevsky, Moshe Arye 1967- The calculus of retirement income financial models for pension annuities and life insurance Actuariële wetenschappen gtt Annuités - Modèles mathématiques Levensverzekering gtt Pensioen gtt Pensions de vieillesse - Modèles mathématiques Revenu de retraite - Modèles mathématiques Mathematisches Modell Old age pensions Mathematical models Annuities Mathematical models Retirement income Mathematical models Altersversorgung (DE-588)4001479-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| subject_GND | (DE-588)4001479-4 (DE-588)4114528-8 |
| title | The calculus of retirement income financial models for pension annuities and life insurance |
| title_auth | The calculus of retirement income financial models for pension annuities and life insurance |
| title_exact_search | The calculus of retirement income financial models for pension annuities and life insurance |
| title_exact_search_txtP | The calculus of retirement income financial models for pension annuities and life insurance |
| title_full | The calculus of retirement income financial models for pension annuities and life insurance Moshe A. Milevsky |
| title_fullStr | The calculus of retirement income financial models for pension annuities and life insurance Moshe A. Milevsky |
| title_full_unstemmed | The calculus of retirement income financial models for pension annuities and life insurance Moshe A. Milevsky |
| title_short | The calculus of retirement income |
| title_sort | the calculus of retirement income financial models for pension annuities and life insurance |
| title_sub | financial models for pension annuities and life insurance |
| topic | Actuariële wetenschappen gtt Annuités - Modèles mathématiques Levensverzekering gtt Pensioen gtt Pensions de vieillesse - Modèles mathématiques Revenu de retraite - Modèles mathématiques Mathematisches Modell Old age pensions Mathematical models Annuities Mathematical models Retirement income Mathematical models Altersversorgung (DE-588)4001479-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Actuariële wetenschappen Annuités - Modèles mathématiques Levensverzekering Pensioen Pensions de vieillesse - Modèles mathématiques Revenu de retraite - Modèles mathématiques Mathematisches Modell Old age pensions Mathematical models Annuities Mathematical models Retirement income Mathematical models Altersversorgung |
| url | http://www.loc.gov/catdir/enhancements/fy0633/2005029455-d.html http://www.loc.gov/catdir/enhancements/fy0633/2005029455-t.html http://www.loc.gov/catdir/enhancements/fy0733/2005029455-b.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016223697&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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