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Chris Gilbert
Chris Gilbert
Articles  | Author's Website |

The Magic of Compound Interest

The meaning and the equations behind the greatest mathematical force

May 20, 2016

So you want financial independence? You want your money to work for you? But how? Well, investing is certainly a great place to start. More specifically, value investing. But how do we really grow wealth into a sum substantial enough to claim financial independence? The answer – by utilitizing compound interest.

Talking points

  • What is compound interest and why is it important to investors?
  • Breaking down compound interest into equations and examples.
  • The magic of compound interest.

What is compound interest?

The basic and boring definition of compound interest is the interest on the initial principal plus the accumulated interest of previous periods. Or, in our case, an investment that has earned interest (or dividends) that we have reinvested into the original principle. It can also be thought of as the interest on interest.

However you want to view it, compound interest will make your investment grow much quicker than simple interest, interest calculated on just the principal. We'll take a look at an example later in this post.

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The concept of compound interest has been around a while. Albert Einstein is believed to have called it "the greatest mathematical discovery of all time" and "the eighth wonder of the world." Compound interest can transform a small amount of money into a high-powered, income-generating machine. However, it cannot work without two crucial components: reinvested earnings and time.

For instance, let's say you have $10,000 earning 10% interest a year. After one year, you will have $11,000 ($10,000 * 1.10). If you reinvest the $1,000 you made off the interest rather than withdrawing it, your $11,000 grows to $12,100 ($11,000 * 1.10). Notice if you reinvest your earnings you make $100 more over two years than if you would have withdrawn them ($12,100-$12,000). This is important for two reasons. One, you made more money using compound interest. And, two, you didn't have to work at all; your money worked for you. While $100 might not seem like much, after many years, as I will demonstrate later, utilizing compound interest will cause your investment to balloon compared to just using simple interest.


Breaking down compound interest

Before we continue, let's briefly look at the mathematics behind this seemingly magical concept. It's important to understand the formula so you can use it in a variety of ways. It also allows you to change certain aspects of the formula to fit what you're calculating. I have a free spreadsheet that has multiple financial calculators available at the click of a button here.

Let's start with the basic annual compound interest formula:

FV = P * (1 + r/n) ^ (t * n)

  • FV = Future value of the investment
  • P = Principal amount (initial deposit)
  • r = Annual interest rate
  • t = Number of years
  • n = Number of times compounded per year (12 = monthly; 1 = yearly)

For example, if you invest $10,000 and earn a 15% annual return for 10 years you will calculate:

  • FV = 10,000 * (1 + 0.15 / 12) ^ (10 * 12)
  • FV = 10,000 * (1.0125) ^ 120
  • FV = 10,000 * (4.44)
  • FV = 44,402.13

The investment balance after 10 years would be$44,402.13.

Next, let's look at the same calculation, but this time we'll add in monthly deposits. This solution actually calls for two equations; however, one of them we've already seen: FV = P * (1 + r / n) ^ (t * n). The other equation is for the monthly addition aspect. This equation is:

(DEP * (((1 + r / n) ^ (t * n) - 1)) / (r / n))

  • DEP = Deposit
  • r = Annual interest rate
  • t = Number of years
  • n = Number of times compounded per year (12 = monthly; 1 = yearly)

So the final formula for compound interest with monthly deposits is:

FV = [P * (1 + r) ^ (t * n)] + [DEP * (((1 + r / n) ^ (t * n) - 1) / (r / n))]]

I know, your eyes are glossed over by now. But let's go through a quick example and then we'll wrap this section up. If you invest the same $10,000, earn the same 15% a year for 10 years and deposit $100 a month for the whole 10 years, the formula would look like this:

  • FV = [10,000 * (1 + 0.15 / 12) ^ (10 * 12)] + [100 * (((1 + 0.15 / 12) ^ (10 * 12) - 1) / (0.15 / 12))]
  • FV = 44,402.13 + [100 * ((1.0125 ^ 120 - 1) / 0.0125)]
  • FV = 44,402.13 + [100 * ((4.44 - 1) / 0.0125)]
  • FV = 44,402.13 + [100 * (3.44 / 0.0125)]
  • FV = 44,402.13 + [100 * 275.22]
  • FV = 44,402.13 +27,521.71
  • FV = $71,923.84

I love this formula because you can really see the power of compound interest as well as the magic of adding monthly deposits. A $100 a month deposit amounts to almost a $30,000 difference over 10 years.

The magic of compound interest

Compound interest can transform your investments, but as you can see, monthly deposits and time really affect the way it works. Let's take a look at two examples,

First, let's say Red invests $10,000 and earns 10% annual returns for 20 years. Now let's say his buddy Blue invests the same amount and also earns 10% annual returns for 20 years. The only difference being Blue also deposits $100 a month for the entire time. The graph below shows the difference these monthly deposits can make.

Notice how Blue's investment more than doubles Red's after 20 years. What a difference $100 a month can make! Also notice how the curves show a significant increase in the latter half of the graph. We'll look further into that next.

Now, let's say Red and Blue decide to invest the same amount ($10,000) and get the same annual returns (10%), only this time Red decides to invest 10 years after Blue does. In this example both Red and Blue contribute $100 a month as well. The graph below demonstrates the importance of the time component in compound interest.

By only starting 10 years sooner than Red, Blue's investment more than tripled in value. Blue seems like a smart fellow. And these graphs exhibit the importance of monthly deposits and starting early when it comes to investing.

Summary

  • Albert Einstein called compound interest "the greatest mathematical discovery of all time" and "the eighth wonder of the world."
  • But to make the most of it, you need three crucial components: reinvested earnings, monthly deposits and time.
  • As shown with the graphs above, these components can increase your investments by two to three times in as little as 10 years.

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About the author:

Chris Gilbert
I am a full-time pharmacist as well as an avid investor. I discovered the art of value investing by reading the works of Benjamin Graham. After that I began to drift towards Peter Lynch, Warren Buffett, and Charlie Munger. My investing philosophy now is simply to find great companies at a discount while using a common sense, value oriented strategy, and maintaining a long-term outlook. I am determined to educate the individual investor that investing is not hard, not time consuming, and not expensive. Through the use of value investing and the YourPortfolio Software, my goal is to help you obtain better results in a fraction of the time.

Visit Chris Gilbert's Website


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