The Rule of 72 is a general mathematical guideline, in financial planning, that determines how long an investment portfolio will take to double. The Rule assumes a fixed rate of return (ROR), and calculates how long (years) it will take a portfolio to double in size, given that fixed ROR. This is an important concept to understand, for both retirees, and even active savers, who depend on fixed-rate investments to deliver the lion’s share of the returns from their nest’s egg.

## Great Minds Have Spoken

Albert Einstein, the noted mathematician, and the most influential physicists of many generations, reportedly attributed compound interest as being one of the more remarkable “inventions” of the world. A quote, credited to Einstein, goes like this:

**“Compound interest is the eighth wonder of the world. He who understands it, earns it…he who doesn’t…pays it!”**

For retirees, and those nearing retirement, as well as millions of working people who will eventually join the ranks of retirees, a deep understanding of compound interest is critical. The connotation here is simple: As your savings grow with interest, that interest attracts further interest, spurring the growth of your retirement nest egg. It is this virtuous cycle, of growth attracting even more growth, that has led many analysts to describe the phenomenon of compound interest as “Interest on interest”.

Compound interest is a relatively well-known concept, used by financial institutions that help you save and borrow. However, the world of finance also harbors a lesser-known, yet equally important piece of math: The Rule of 72.

## The Rule of 72 Unveiled: How doubling works

At its simplest, the Rule of 72 (the Rule) is a mathematical calculation, with compound interest at its heart. The Rule provides a quick way for anyone to estimate how long it will take for a sum of money to double (or to halve – if we’re looking at inflation’s impact on savings). To understand the application of this “quick method”, let’s first look at an example using the traditional method of computing compound interest.

We’ll assume that a saver has $10,000 to invest, and that the current investment opportunity yields a 12% rate of return. If this investment were to return simple interest, over a 6-year time horizon, it would deliver a steady stream of interest ($1,200) each year, to produce a total return of $17,200 – the original principal of $10,000 + $7,200 interest each year – by the end of year 6.

However, what if we threw the power of compounding into the mix, to work its magic?

The most striking difference between the two investment scenarios, that retirees will immediately pickup on, is the total return they enjoy with compound interest. Over the investment horizon, they’ll enjoy $2,538 ($9,738 minus $7,200) more interest through compounding, than they will through simple interest. But that’s not the only striking feature that compounding delivers to an investment portfolio.

Notice the rose-colored column in Table 2. The ending balance in year 6 is almost double the initial principal of $10,000. We arrived at that conclusion through a series of six iterative calculations. However, if this retiree wanted a quick answer to the question:

**“How long will it take for my nest egg to double?”**

…thanks to the Rule of 72, we could provide them the answer in short order!

The answer is: Approximately 6-years, and we calculate it by dividing the constant 72 by the interest rate. In this example, 72 divided by 12 = 6, which approximates the result we achieved after six iterations of calculations in Table 2. Later in our discussions, we’ll see how to use the Rule in conjunction with inflation, which has the impact of diminishing our savings.

## The Mechanics of the Rule

The Rule, as illustrated in the above examples, seems rather straightforward and simple to understand: 72 divided by a compound interest rate. The more mathematical-minded amongst us, however, would resort to a more intricate formula involving a natural Logarithm calculation. Here’s the spreadsheet (Microsoft Excel) equivalent of the Rule using Logs:

Applying this formula to the variables in Table 2, we get the following result:

…which is a more accurate answer to the retiree’s question. We’ll very briefly revisit the Log formula later in this discussion. However, when financial calculators and spreadsheets aren’t readily available, the Rule seems to provide us a relatively close approximation (6-years).

## Variations to the Rule

There are several variations of the Rule that retirees can use to forecast the doubling (and halving) effect of interest (and inflation) on their nest egg. Although the difference in results, produced from these variations, is negligible, they may be meaningful to some. These variations are a good spin on the original if you wish to “personalize” your forecasts.

In general, the “base” denominator of the Rule of 72 appears to be 8% (more on this later). To produce a “variant” formula, one must adjust the numerator (72) by 1 (either up or down), for every 3-point difference in rates from the “base” denominator (i.e., 8%).

In Table 3, because “5%” is one 3-point deviation down from 8%, we subtract 1 from 72, to get a variant numerator – i.e., 71. And because “11%” is one 3-point deviation up from 8%, we add 1 to 72, to get a variant numerator – i.e., 73.

As you can see from the calculations in Table 3 above, there is a slight difference between the doubling calculated under the Rule of 72 (e.g., 14.40 years @5%), and those performed by the variant rules (e.g., 14.20 years @5% under the “rule of 71”). However, where the retirement portfolio contains many individual investments, or if this is a sizable portfolio spanning decades, then those differences could add-up to build wealth for you and your family.

## Limitations and Exceptions to the Rule

So, is the Rule of 72 a useful tool, and does it work? At its core, the Rule of 72 (we’ll ignore some of its variations for this discussion, but the same logic applies to those variants too) represents a relationship between two numbers – a constant numerator (72), and a denominator (which can represent one of several elements – more on that later). This comparison works well to highlight a mathematical relationship between those two numbers – that’s basic math. However, there are limitations and exceptions to the Rule that retirees and investors shouldn’t discount.

- As discussed previously, retirees must consider the impact of inflation when using the Rule as a meaningful resource. While rates of return increase a nest eggs’ value, inflation erodes it
- The Rule works well when used with certain denominators, including 2, 4, 6, 8, 9, and 12 (be they percentages or years). That’s because 72 is equally divisibly by them
- The Rule produces its most accurate result at 8%. As interest rates increase or decrease above and below that threshold, however, slight deviations in the results, produced by the Rule versus the more accurate Log formula (discussed in The Mechanics) creep in.
- Do those minor variations discredit the Rule as an effective quick-forecast tool? Absolutely not! Retirees and investors may also use one of the variations of the Rule to customize the results for their unique situations – but even those variants are bound by the same general principles governing the Rule of 72
- Most significantly, the Rule is a powerful ally when dealing with fixed-rate investments, such as fixed annuities and certificates of deposits (CDs). That’s because the Rule factors a single denominator, and is therefore unsuitable to account for variable rate annuities

With a slew of variables impacting the future growth of an investment, the Rule is but one tool – albeit a simple and powerful one – to quickly * forecast* growth (doubling) and erosion (halving) of an investment, based on the single denominator used. It cannot, however, act as a financial

**modelling tool.**

*prediction*## Broader Applications

As a retiree, an employee considering their impending retirement plans, or even as a cautious investor, the Rule does provide you with a convenient, back-of-the-napkin tool to predict when your savings will double. It gives you some mental relaxation by helping you avoid doing some onerous math. Perhaps, instead of firing-up that calculator, or building a spreadsheet, you can even use this handy graphic, courtesy of the Federal Reserve Bank of St. Louis, to do a quick look-up when pressed for time.

But the Rule, which involves a simple, one-step division exercise, has broader application than simply predicting when your nest egg will double in value. While in the same vein as “doubling”, here are some broader useful applications of the Rule:

**1) Credit Card and Other Debt:**

Most lenders (especially credit card issuers!) encourage borrowers to “just let the debt roll on…don’t focus on repaying it!”. Instead, they encourage borrowers to focus on enjoying that new car, beautiful home renovation, or much-deserved retirement vacation. Let’s see what sanity check the Rule provides us:

Assume you charged $5,000 to your credit card for that home reno project, and your lender charges you a “very competitive” 12% interest rate. If you don’t start chipping-away at that debt, gradually and systematically, within six years (72 divided by 12 = 6), you’ll owe $10,000 on your credit card. The Rule quickly tells you that within 6-years, you’ve racked-up as much interest on that loan as the amount of principal you originally borrowed!

Entering retirement with any amount of debt is risky. But owing twice as much as you initially borrowed, just as you plan on hanging up your gloves and calling it a day, is downright irresponsible.

**2) Inflationary Impact:**

Inflation has an inverse relationship on your retirement nest egg compared to interest. But essentially, the application of the Rule is the same. While the numerator remains 72, now you substitute the rate of inflation as the denominator. And our interpretation of the result changes -from doubling to halving.

Let’s suppose someone you trust (so there’s no risk to your investment) approaches you and asks for a loan of $5,000 for a 12-year term. They promise to pay you at a healthy 12% annual rate, with the principal and interest paid at maturity. On the face of it, this looks like a great opportunity – give them $5,000 today, and 12-years later collect – risk free – nearly $19,500 ($19,479.88 to be precise!). What’s not to love? Well, let’s introduce you to the party spoiler – inflation. Assume inflation runs at a steady 6% over the duration of the term.

If you do some quick math using the Rule of 72, you’ll see that inflation will halve your principal in 12 years (72 divided by 6 = 12). In effect, instead of receiving $19,479.88 at maturity, you’ll only receive $16,979.88 ($19,479.88 minus $2,500) – in real terms. These are somewhat simplistic calculations. In real terms, however, you’ll receive much less than $16,979.88 because inflation will also erode accumulated interest (…but’s a discussion for an Advanced Financial Math class!).

**3) Estimating Expected Rate of Returns:**

Finally, as a retiree, you’ll often be tempted to jump in with both feet when slick investment advisors make compelling pitches “Double your money in no time with this once-in-a-lifetime opportunity”. Can the Rule help you make an informed decision? Absolutely!

If you wanted to double your investment over a specified time-horizon, what would it take to make that happen? Let’s assume your same trusted source pitches you an idea: Give me $5,000 for 8-years, and I’ll guarantee you an annual rate of return (ROR) of 7.5%. We’ll park our party-spoiling inflation outside the door for now, and use the Rule to assess whether you’ll manage to double your investment with that pitch.

Because it’s the rate we’re looking to calculate, we’ll need to re-jig the formula we’ve used so far, to now use the investment time as the denominator (instead of the usual rate parameter).

The result: If you wish to lock-in your money for 8-years, in the hopes of doubling it, then a 7.5% ROR just won’t cut it. Thanks to a slightly re-worked Rule of 72, you’ll quickly ascertain that you’ll need at least a 9% (72 divided by 8) ROR to achieve your goal of doubling what you invested.

Although we’ve deliberately kept the examples here relatively simple, they still serve to underline the core principles of the Rule – that compounding cuts both ways. As Einstein noted, whether it’s earning it or paying it, the Rule is a quick-n-dirty formula to use for judging the impact that compounding (interest and inflation) has on a retirement nest egg.