Table of Contents Other books by Moshe A. Milevsky Title Page Copyright Dedication Introduction: An Equation Can’t Predict Your Future … But It Can Help You Plan for It Reality Check Chapter 1: How Long Will My Number Last? Equation #1: Leonardo Fibonacci (1170–1250) The Spending Rate: A Burning Question Fibonacci’s Fabulous Flash of Finance Manipulating the First Equation Is It Really His? Can We Really Know Interest Rates? Back to Fibonacci’s Life Story Okay, Here Come the Rabbits He Retired Wealthy Chapter 2: How Long Will I Spend in Retirement? Equation #2: Benjamin Gompertz (1779–1865) Simplistic Retirement Planning Gompertz’s Big Discovery So How Long Should I Plan for in Retirement? Back to Benjamin Gompertz Chapter 3: Is a Pension Annuity Worth It? Equation #3: Edmond Halley (1656–1742) Take the Pension or a Lump Sum? Halley’s History, Part I Pension Value: Age and Interest Back to Halley’s History, Part II The Great Comet Chapter 4: What is a Proper Spending Rate? Equation #4: Irving Fisher (1867–1947) Retirement Inflation: It’s Getting Personal Sustainable Spending Rates: Maybe The Fourth Equation, Explained Fisher’s Optimal Retirement Plan, Applied In Fisher’s Words Confirming the Numbers Irving Fisher’s Rise, Fall and Rise Again The 1929 Crash Chapter 5: How Much in Risky Stocks versus Safe Cash? Equation #5: Paul Samuelson (1915–2009) Time Is on Your Side? Human Capital: You Are Wealthier than You Think The Equation, Explained The Equation, Applied: Case Studies and Examples Retirees Should Protect Their Equities Paul Samuelson’s Life and Impact Chapter 6: What Is Your Financial Legacy Today? Equation #6: Solomon S. Huebner (1882–1964) Human Life Value: Raison d’être of Life Insurance What’s the Most Effective Way to Create a Legacy? Life Insurance: The Formula More Examples: Different Ages If You Try This at Home He Wasn’t a Quant Secondary Markets and Life Settlements Term versus Whole Life But Sunny Sol Was a Fan of Annuities Too The Historical Huebner versus Fisher Chapter 7: Is My Current Plan Sustainable? Equation #7: Andrei N. Kolmogorov (1903–1987) Will Your Retirement Plan Work Out? A Purely Imaginary Case to Help From Monte Carlo to Netherlands, Sweden and then Russia The Equation Itself Is Only Partial Detailed Example Verification that Kolmogorov’s Equation Is Satisfied Physics, Not Finance Back to Andrei Nikolaevich Kolmogorov Conclusion: Controversies, Omissions and Concluding Thoughts What Equations Are Missing? Appendix: Crash Course on Natural and Unnatural Logarithms References and Sources Acknowledgments About the Author Short Poem by Maya Milevsky (age 11) Index Other books by Moshe A. Milevsky Strategic Financial Planning over the Lifecycle: A Conceptual Approach to Personal Risk Management with N. Charupat and H. Huang (Cambridge, 2012) Your Money Milestones (FT Press, 2010) Pensionize™ Your Nest Egg with A. Macqueen (Wiley, 2010) Are You a Stock or a Bond? (FT Press, 2009) Lifetime Financial Advice: Human Capital, Asset Allocation and Insurance with R. Ibbotson, P. Chen and K. Zhu (CFA Institute, 2007) The Calculus of Retirement Income (Cambridge, 2006) Wealth Logic (Captus, 2002) Insurance Logic with A. Gottesman (Stoddart, 2002) Money Logic with M. Posner (Stoddart, 1999) To the memory of my grandfather, Rav Hillel Mannes, PhDc; a German gentleman, scholar and orthodox Jew. Introduction An Equation Can’t Predict Your Future … But It Can Help You Plan for It Most books about retirement planning are written as guides, instruction manuals or “how-to” books. The authors tell you what to do, when to do it, and what to expect. I know this quite well because I have authored many such tomes myself. Rest assured, this is not one of those books. This book tells stories which I hope will lead into conversations. It is a narrative involving seven people, their discoveries and the conceptual innovations that made it possible for you to stop working and enjoy the money you have accumulated, one day. These protagonists—or scientific heroes— didn’t achieve their breakthroughs while hunched over a laboratory workbench, peering through a microscope or trekking through jungles. They made their discoveries sitting in front of a blank sheet of paper, but while thinking very carefully about life and money. And, like the greatest thinker of them all, Albert Einstein, they too expressed their discoveries using a very beautiful language called mathematics. Alas, the seven equations profiled in this book aren’t as 2 famous or as elegant as the simplicity of E = MC , but they are far more practical for your retirement. You see, time is running out. North American baby boomers are getting within shouting distance of their golden years. Most have finally grasped that—despite the dreamy commercials and brochures—retirement isn’t a long vacation that begins at the mythical age of 65. It’s a gradual winding-down process, possibly involuntary, with austere financial implications. The timeworn questions about proper savings rates, the best mutual funds or the ideal size of your nest egg number have been pushed aside by a pragmatic economic reality: “This is what I have, give or take a few more years of saving. How do I make it last?” In other words, it is time to have some conversations about retirement income planning, also known as de-accumulation planning. The stories in this book should lead you into those conversations. These conversations should take place with your family and loved ones and possibly with a professional financial advisor as well (whether you love them or not). Reality Check Today, there are two cold, hard facts about aging consumers heading into the second decade of the 21st century: i) most are not saving enough money to maintain their current standard of living, and ii) many are financially illiterate. Alas, study after published study continues to document a shocking lack of knowledge about basic financial affairs and a correspondingly miniscule financial cushion for retirement. These two problems are obviously linked. Yes, I know from many years of teaching experience that financial conversations are often dry and humorless. So I promise to do my best to lighten up the topic by keeping the technicalities to a minimum and focusing on the art. “Art,” you say? Yes. In my mind, famous equations are like beautiful Picassos. Even if I don’t quite understand the painting or the mathematics I can certainly appreciate the beauty and genius behind it. The seven equations presented in this book typify, at least for me, the conciseness, elegance and beauty that the best of the best equations demonstrate. By the end of this book, if you’re not already inclined to appreciate mathematical equations for what they are, I hope you’ll agree about the beauty. Here’s a brief outline of what’s ahead. In the first chapter, I will start by asking a simple question: How long will your nest egg last, if you were to stop contributing today, and instead withdraw a fixed amount each year while earning a fixed interest rate each year for the rest of your life? Although neither of these fixes are realistic in practice, this sort of analysis provides a quick and sobering assessment of whether you can maintain your standard of living, or when the money will run out if you can’t. The underlying mathematics of this first equation is rather simple, and does not require any complex algorithms. Along the way, you’ll learn about the 13th- century Italian educator and mathematician Leonardo Fibonacci and his remarkable technique for solving such financial problems. In the second chapter, I’ll address the length of life itself. Here is the motivating question: Given your family history, current lifestyle and recent