Book contents
- Frontmatter
- Epigraph
- Contents
- Preface: In Memoriam for Jared
- Acknowledgments
- 1 King Billy, Protestant Hero of england
- 2 Tontine's Economic Origins: Cheaper Debt
- 3 A Most Curious Will(iam) and Older Than You Think
- 4 The Million Act to Fight a War against France
- 5 Don't Englishmen Die? Anti-Selection vs. Fraud
- 6 Is Your Tontine a Stock or a Bond?
- 7 Optimal Tontine: Hedging (Some) Longevity Risk
- 8 Conclusion: Tontines for the Twenty-First Century
- Epilogue: What Did William Really Know?
- Appendix A The List of Nominees
- Appendix B The Gompertz-Makeham Law of Mortality
- Appendix C 14% for One, 12% for Two, or 10% for Three?
- Source Notes and Guide to Further Reading
- References
- Index
Appendix B - The Gompertz-Makeham Law of Mortality
Published online by Cambridge University Press: 05 May 2015
- Frontmatter
- Epigraph
- Contents
- Preface: In Memoriam for Jared
- Acknowledgments
- 1 King Billy, Protestant Hero of england
- 2 Tontine's Economic Origins: Cheaper Debt
- 3 A Most Curious Will(iam) and Older Than You Think
- 4 The Million Act to Fight a War against France
- 5 Don't Englishmen Die? Anti-Selection vs. Fraud
- 6 Is Your Tontine a Stock or a Bond?
- 7 Optimal Tontine: Hedging (Some) Longevity Risk
- 8 Conclusion: Tontines for the Twenty-First Century
- Epilogue: What Did William Really Know?
- Appendix A The List of Nominees
- Appendix B The Gompertz-Makeham Law of Mortality
- Appendix C 14% for One, 12% for Two, or 10% for Three?
- Source Notes and Guide to Further Reading
- References
- Index
Summary
In a number of earlier chapters I made reference to the Gompertz-Makeham (or just plain Gompertz) Law of Mortality. In particular, I used this law of mortality in Chapter 7 when I introduced and described Jared's tontine payout rate and in Chapter 2 when I discussed the probability density function of the “present value” of the tontine versus annuity cash-flow payout. Well, in this appendix I offer a brief explanation of this well-known law, as well as the analytic representation I used plus a bit of information about the person after whom it is named. For those interested the technical details, see Milevsky (2012), which is a popular book covering the most important equations in the field of retirement income planning.
For starters, age-dependent mortality rates for adults – for example, those displayed in Table 5.3 but continued to older ages – seem rather arbitrary at first, but there is actually an underlying pattern to them. In particular, for people between the ages of twenty and ninety – mortality rates not only increase consistently every year with age, they actually increase by approximately 9% every year.
In mathematical symbols, if the starting mortality or death rate per year was q percent at age y, (for example, q = 2% at age fifty) then it is q(1 + z) percent in year (y + 1) and then q(1 + z)2 percent in year (y + 2), and then q(1 + z)3 percent in year (y + 3), and so on, where z is approximately 9%. Human adult mortality rates – regardless of what particular group of humans or population you select and whether it is in the seventeenth or nineteenth or twenty-first century – are an exponentially increasing function of age with a (growth rate) parameter of 9%.
What this also means (mathematically) is that if you take the logarithms of these annual mortality rates denoted by q, they can be approximated quite nicely by a straight line and determined by a slope and an intercept.
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- King William's TontineWhy the Retirement Annuity of the Future Should Resemble its Past, pp. 226 - 229Publisher: Cambridge University PressPrint publication year: 2015
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