The Rule of 72

How long to double your money at a given interest rate.

What is the rule of 72?

In short:

The quickest way to figure out how long it will take your investment to double with annual compounding interest is to use “the rule of 72”. This is a quick and relatively accurate way of calculating how fast your investment will double at a certain interest rate.

The rule of 72

In-depth:

The rule of 72 seeks to provide an estimate for doubling time in the context of compounding interest. Compound interest is the interest (or rate of return) on both the initial investment and the interest the initial investment generates. Compound interest is a powerful tool because you get paid interest on a larger and larger amount every compounding period.

To use the rule of 72, simply divide 72 by your interest rate, ex. 72/ i, the result is the number of years it will take to double your investment with compounding interest.

An example of this, say you believe you can obtain a rate of return (interest) of 10%, 72/10 = 7.2, it will take you 7.2 years to double your investment. What is a good interest rate for me to use? If you are invested in the stock market with a portfolio similar to the S&P 500 then you can expect to make an average return of 10-11% per year. This is the average return for the S&P 500 over many decades. That said, the stock market performs differently every year so one year might be low and another might be very high.

Why the rule of 72 works?

The rule of 72 is an estimation to solve for the time variable when the compound interest formula is rearranged. Note that the rule of 72 only applies to annually compounding interest scenarios. The compound interest formula below:

Final = Initial Investment(1 + (interest rate/compounding times per year))^(compounding times * length of investment)

We can use algebra to rearrange the compound interest formula but will encounter the need to use logarithms to solve for the exact time. Generally, most people find using logarithms in their head difficult so a proxy solution can be found by solving for how long does it take 1 to double (i.e. 1 is the initial investment).

Concept 1:

Solving without the rule of 72:

Since we are using annual compounding interest we can adjust the formula from above:

Final = Initial Investment * (1 + (interest rate))^( length of investment)

Stating we want to double:

2 = Initial Investment * (1 + (interest rate))^( length of investment)

Dealing with exponent:

            This is where logs come in.

Length of investment = log_10² /log_{10^{interest rate}}

Concept 2:

To show the math behind the rule of 72 we can once again rearrange the formula where we dealt with the exponent by using natural logs.

Length of investment = ln(2) / ln( 1 + (interest rate/100))

Solving for this will get us close to:

Length of investment = 72 / interest rate

*To understand why there is a “time to double” you need to understand what the equation is qualitatively saying and the theory it is relying upon, that is the time value of money.

Time Value of Money:

The core tenet of compounding interest is the time value of money. The time value of money is the idea that money has more value today than it does tomorrow.

Look at a logical example to help make more sense of this. If someone could choose between receiving $1,000 today and $1,000 in a year, the vast majority would pick to receive the money today, (this is economically rational). Now, if the scenario changes to receiving $1,000 today or $1,100 in a year, the number of people who choose today vs a year change. More people will choose a year from now rather than today so that they can receive the extra $100, how many changes depends on the other opportunities available at the time to use that $1,000.

The extra $100 or 10% represents the Time Value of Money. How much does it cost for a large number of people to forego having money today for money sometime in the future. This depends on the other opportunities available at the time.