# The Rule of 72

The rule of 72 is an arithmetic trick that helps you understand compound growth.  It will not make you intuitive but it will help you to understand when to look deeper.

Here’s how it works.  The time until money doubles under a constant rate of interest (growth) is approximately 72 divided by the interest rate.  It is a reasonable approximation for rates between 3% and 12% and that will pick up most investment situations.  Similarly the rate of return is 72 divided by the number of doubles.

T=72/i or i=72/T

A \$25,000 painting on The Road Show cost \$100, 50 years ago.  250 times the investment.  What is the rate of return?  8 doubles is 256.  So each one took 50/8 = 6.25 years so the rate of return is about 72 divided by 6.25 equals 11.5%.  The right answer is about 11.67%

Looking from the other end, suppose you think you can get 7%. You can estimate that the time to double your money is about 10.29 years.  Actual is about 10.25 years.  Plenty close enough once we realize that strategic decisions are seldom improved by precision.  The 2 week difference will not affect anything.  We are looking for shape and direction in these cases.

By the same reasoning in about 20 years you would have 4 times, in 30 years about 8 times and so on.  Learn the doubles list, 2, 4, 8, 16, 32, 64, 128, 256, …

Right away you can see that in going from 30 years to 60, the last three periods are important.  In 30 years, about 3 doubling periods you get 8 times your money, but if you increase the time to 60 years, you get 6 doubles = 64.  If you started with \$1,000 the first 3 doubles returned \$7,000 new dollars (for a total of \$8,000.)  But the last 3 doubles brought home \$56,000 more. (64,000 less the 8,000 there at 30 years.)  Shortening the time to 50 years means you miss the last double.

If you want to shorten the overall time to get a given amount of money, you must increase your investment or increase the rate so that the time to double is shorter.

If you have only 40 years to get the \$64,000 then at 7% you need to do it in 4 doubles.  (40 divided by the time to double at 7%  = 40/10.28  about 4.)

Four doubles is 16 times your money so you must start with \$4,000 to have \$64,000 in 40 years.  You have to “buy up” the first two doubles.

Alternatively you could invest at a higher rate.  If you need 6 doubles in 40 years then the doubling period needs to be about 6.7 years.  If you know the time to double you can get the rate the same way.  72 divided by time = rate.  In this case about 10.7%  The precise answer is higher.  About 10.95% and again precision is not valuable.  Investments that return 10.95% over 40 years look the same as ones that can earn 10.7%.  Nobody could sort them out at the beginning.  Avoid getting caught up in the details.

So how do we apply this?

Time is a very good defense to risk.  Rates that are higher include inherent risk.  You will not get 10% with the same underlying investments characteristics as those that will present at 7%.    You don’t need higher rates if you have longer.

Avoid costs that reduce the rate.  Taxes are dreadful.  If you lose taxes at 30% along the way the 7% yield becomes 4.9%.  The \$64,000 end hoped for becomes \$17,600.  Of the potential \$63,000 new dollars you only keep \$16,600.  You lose 74% of the potential income to taxes.  See Einstein was wrong.

Pay attention to investment costs.  At 6.5% return you will only reach about \$44,000 in 40 years.  Investment managers need to add value not cost.  Investment managers and investment advisers have different jobs.  Do not confuse them.  Understand and look for investment alpha.

Learn to protect time.  Losing money matters but losing time matters more.  If you start with \$1,000 expecting 7% but instead breakeven with it for 10 years, you will need to throw in another \$1,000 or wait 10 years longer to reach the same goal.  Whatever time you lose you will have to buy back with capital.  To make money, the first thing to do is to avoid losing money.

The last double earns more than all the ones prior to it.  Start early.  If you start late it is the last double that you will miss.

If you have a pension plan be wary.  The last double is often the reason employees are terminated around 55 or so.  If your pension base is growing, the employer will need to buy all the doubles they missed earlier when you were earning much less.  An employer’s pension contribution in that last period before retirement can easily exceed the salary they pay you.  So – Thanks for your service.  Goodbye!

Play around with this for a while.  Draw some pictures.  There is no free lunch.  To reach a  future amount for a given rate, you need to contribute capital and time.  If you want to increase the rate you need to contribute risk tolerance.  It is a simple calculation and the Rule of 72 will help you understand how the variables relate to each other.

Don Shaughnessy is a retired partner in an international accounting firm and is presently with The Protectors Group, a large personal insurance, employee benefits and investment agency in Peterborough Ontario. don.s@protectorsgroup.com

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