Quickly Calculating the Magic of Compound Interest

I taught Algebra I and Algebra II, so I saw firsthand how the actual compound interest formula completely befuddles students trying to understand how a bunch of variables fit together.  For anyone that needs a reminder from their high school math class, this is the compound interest formula:

Thank you Money.com for having quick high school math resources on hand so I did not have to figure out how to type the exponents in the compound interest formula.

Being years removed from teaching, I think having students interact with it while learning basic algebra is a bit of a disservice to their financial futures.  Math curriculums decide that students know how to work with multiple variables and have learned their order of operations, so they can do this math.  That is true.  Students can do the math once they have basic algebra skills, but they do not fully understand how the variables work together at this stage of their math education.  This lack of understanding of why this formula works prevents students from really understanding the magic of compound interest.

If high school curriculum prioritized teaching students how to support themselves rather than teaching them how to become the best employees to be stuck in a future 9-to-5, we would probably teach students the quick formulas to fully understand compound interest first.  With an understanding of the magic of compound interest, we could then introduce the more complicated compound interest formula above.

The magic of compound interest is the path to make students interested in the formula because compound interest can make you wealthy without doing any work.  Every high school student would like that, as would any adult.

The Rule of 72

To make compound interest easy to understand, start with the Rule of 72.  A much friendlier looking formula, the Rule of 72 says when you divide 72 by an interest rate, the answer will tell you the number of years until your money doubles.  In other words:

Using the formula makes it easier to understand.  If the money in your high-yield savings account (HYSA) has a 4% interest rate, just take 72 ÷ 4 to determine how many years it will take for your money to double at that rate.  You will find that with a 4% interest rate, your money will take 18 years to double.  

That is a pretty long time, but it is also remarkable that your money just magically doubles by sitting there without you contributing anything to it.  If you have $10,000 in a HYSA with a 4% interest rate, you will have $20,000 in 18 years!  What is more remarkable is that if you continue to leave that money alone for the duration of your career, you will have $40,000 in 36 years by doing absolutely nothing!  That $10,000 becomes $40,000 in 36 years because it has time to double twice.  When your doubled money doubles, you start feeling like your money grows money with ease.

This gets even more fun with higher interest rates.  The average annual return of the S&P 500 over the last 30 years has been about 10%.  The S&P 500 is the basis for a lot of index funds, including Vanguard’s VOO, meaning you will track the S&P 500’s performance if you invest in that index fund.  When you account for inflation, the returns are closer to 7%.  That is still fantastic, but we will use 10% since we are dealing with actual dollar amounts for the compound interest equation.

• Keep in mind that 10% is the average annual growth over that period and is not guaranteed.  In real life, the growth is not linear—some years it will be more, and some years it will be less (and could even be negative).  Over the long haul, though, the year-to-year fluctuations become less important.  

To calculate how long it takes your money to double when invested in VOO or a similar total stock market index fund tied to the S&P 500, divide 72 by 10.  Your money will double in 7.2 years.  If you invest $10,000 in VOO, and take no further action, it will become $20,000 in 7.2 years.  In 14.4 years, it will be $40,000.  In 21.6 years, it will be $80,000.  In 28.8 years, it will be $160,000.  In 36 years, it will be $320,000.

That math is remarkable for wealth building!  To emphasize:

• Invest $10,000 in a high-yield savings account with a 4% interest rate for 36 years, and it will become $40,000.

• Invest $10,000 in a total stock market index fund with a 10% rate of return for 36 years, and it will become $320,000.

By investing in a total stock market index fund, your money grows to become $280,000 larger than if you just left it in a high-yield savings account.  Higher interest rates mean faster doubling.  When your doubled money doubles, it grows beyond what you can imagine!  The $10,000 invested in the total stock market index fund doubled five times while the $10,000 invested in a HYSA doubled only twice.  Five doublings versus two doublings meant $280,000.

That is why compound interest is magical.

The Rule of 114

Similar to the Rule of 72, the Rule of 114 tells you how long it will take your money to triple.  Most investors use the Rule of 72 to test out investments and extrapolate earnings into the future, while the Rule of 114 is mostly used as a fun way to see the magic of compound interest.  We can use the same investments as before to play with the numbers.

If we invest $10,000 in a HYSA with a 4% interest rate and want to know how long it will take to triple in value, we can take 114 ÷ 4 to find our answer.  It will take 28.5 years to triple.  In other words, the HYSA will have $30,000 in it in 28.5 years without contributing another cent.

If we invest $10,000 into a total stock market index with 10% average returns, we can divide 114 by 10 to learn our investment will triple in 11.4 years.  It takes only 11.4 years for the $10,000 investment to become $30,000.  In 22.8 years, the investment will become $90,000.  That is already triple the HYSA investment years ahead of time.  If we play it out further, the difference is starker:

• The money in the HYSA at a 4% interest rate takes 57 years to become $90,000.

• The money in the total stock market index with 10% average returns will be $2,430,000 in 57 years.

Again, over 57 years, a 10% interest rate allowed the money to triple five times, while the 4% interest rate only tripled twice.  That is a life-changing difference.

Both the Rule of 72 and the Rule of 114 can be used for any rate.  Many investments involve 7–9%  rates, and many individuals preparing for early retirement work out the hypothetical growth of their investments with rates from 7–9% to account for diverse portfolios that are not completely invested in total stock market index funds.  It is easy to play around with the numbers since you just substitute the interest rate of your choice into the denominator and calculate how long it takes your money to double or triple!

Revisiting Algebra Class

Seeing how quickly money doubled would have made me way more excited about compound interest than algebra class did.  The concept of compound interest did not click for me when I initially learned it in Algebra I.  (I get to blame my dad because he was the teacher of my eighth grade Algebra I class!)  I could do the math to insert numbers into the equation, but the compounding growth was not a tangible or finance-altering fact for me at that point.  My ah-ha moment hit in an elective Economics class in high school because I entered each class primed to think about finances.  That financial scope makes the power of compound interest more pronounced than an algebra class.

Now that we are thinking about the huge impact of higher interest rates on how quickly our money doubles and triples, we can reconsider the equation from algebra class:

The big precision point in the algebra class formula is “n,” the amount of times interest is compounded per time period.  This gets down to whether interest compounds on a daily, weekly, monthly, or annual basis—for example, n = 12 when interest is compounded monthly.  From a wealth building standpoint, this nuance is irrelevant.  The market is much too volatile for precise calculations like this, and we base our predictions on annual returns rather than parsing out fluctuations over shorter durations, like a month.

So why does this equation matter?  The nuance matters when you borrow money.  If interest compounds daily, you can owe a lot of money quickly because the rate of compounding is so much faster.  While loans like car loans use simple interest that is predictable, the interest that compounds quickly is interest on credit card debt.  Credit card debt interest compounds daily, so it grows as quickly as possible.

If you avoid debt, invest, and grow your net worth, you can forget about the algebra class formula forever.  Use the Rule of 72 to estimate how your net worth will grow over time.  If you get into debt, algebra class will show you why you want to get out of debt as soon as possible.  Compound interest is magical when it grows your wealth, but it hurts you rapidly when it grows your debt.  Let your money work for you.  Grow your wealth using compound interest.