The Debt Retirees Owe to Fibonacci

Lenny had a problem. A close friend of his invested some money a few years prior, in a local Italian bank — in Pisa actually — that had promised him steady interest of 4 percent per month. Rather than sitting by and letting the money rapidly grow and compound over time, Lenny’s friend started withdrawing large and irregular sums of money from the account every few months. These sums were soon exceeding the interest he was earning and the whole process was eating heavily into his capital. To make a long story short, Lenny was approached by this friend and asked how long the money would last, if he kept up these withdrawals. Reasonable question, no?

Now, if I were Lenny, I would have pulled out my handy HP business calculator, entered the cash flows, pushed the relevant buttons and quickly obtained the answer. In fact, with any calculator these sorts of questions can be answered quite easily using the technique known as “present value analysis” — which is something all business school professors teach their finance students on the first day of class.

Unfortunately, Lenny didn’t have access to an HP business calculator that performed the necessary compound interest calculations. To be honest, Lenny didn’t have a calculator at all because they hadn’t been invented yet. You see, Lenny was asked this question over 800 years ago, in the early part of the 13th century. But to answer it — which he certainly did — he invented a technique that today is called present value analysis. Yes. The one I mentioned we teach business students.

You might have heard of Lenny by his more formal name: Leonardo Pisano filius (“family,” in Latin) Bonacci, a.k.a. Fibonacci (1170-1250), probably the most famous mathematician of the Middle Ages.

In fact, Fibonacci helped solve his friend’s problem, wrote the first commercial mathematics textbook Liber Abaci and introduced the methodology for solving complicated questions involving interest rates. Let me repeat. His technique is still used and taught with slight refinements to college students 800 years later. Now that is academic immortality! (He published and his name hasn’t perished yet.)

The Spending Rate: A Burning Question

Let’s translate Fibonacci’s mostly hypothetical 800 year-old puzzles into a problem with more recent implications. Imagine that you are thinking about retiring and to date have managed to save a total of $300,000 in your retirement account. For now, I’ll stay away from a discussion about taxes or the account type. Allow me to further assume that you are entitled to a retirement pension income of $25,000 per year. This is the sum total of your Social Security and corporate pension plans — but the $25,000 is not enough. You need at least a total of $55,000 per year, to maintain your current standard of living. This leaves a gap of approximately $30,000 per year, which you hope to fill with your $300,000 nest egg. The pertinent question, then, becomes: Is the $300,000 enough to fill the budget deficit of $30,000 per year? And if not, how long will the money last?

As you probably suspected intuitively by now your $300,000 nest egg is not enough. Think about it this way: The ratio of $30,000 per year (the income you want to generate) divided by the original $300,000 (your nest egg) is 10 percent. There is no financial instrument that I am aware of that can generate a consistent and reliable 10 percent per year. If you don’t want to risk any of your hard earned nest egg in today’s volatile environment, the best you can hope for is about 3 percent after inflation is accounted for, and even that is pushing it. Sure, you might think that you are earning 5 percent guaranteed by a bank, or 5 percent in dividends or 5 percent in bond coupons, but an inflation rate of 2 percent will erode the true return to a mere 3 percent. Needless to say, 3 percent will only generate $9,000 per year in interest from your $300,000 nest egg. That is a far cry from (actually $21,000 short of) the extra $30,000 you wanted to extract from the nest egg.

You have no choice. In retirement you will have to eat into your principal.