What is Compounding?

Compounding is like a snowball rolling down a hill…

You start with just a handful of the powder packed in your hands. When you put it down at the top of a hill, it doesn’t seem like much. But, to your surprise, you may end up with a much larger mass of snow by the time you get to the bottom of the hill. As the little snowball starts rolling, it can pick up additional snow as it rolls through. Then, if it keeps going, it has more surface area from which to pick up even more. The process of snow building on snow leads to compounding growth in its size.

Compound interest works by charging or earning interest on other interest. If you’re the depositor, compound interest works by leaving the interest that you make in the investment.

From there, you’ll make money on your original investment, plus earn interest on the funds you reinvested. Each period, the balance of your investment gets bigger and bigger — which means, all things being equal, that the amount of your earnings should also get larger and larger over time.

Unfortunately, the same process happens on money that you owe. If you don’t pay off the interest that accrues in one month, that interest adds to your balance. Then, when your lender calculates the amount you owe the next month, you end up paying interest on the interest you didn’t pay off.

There are plenty of compound interest calculators out there, but let’s walk through an example of annual compounding to explain how it works. Assume you hypothetically start with $100 in a CD and let it grow at a 5% annual rate of return. The critical thing to remember is that you must not take that interest out in order for compounding to work.

At the beginning of the first year, you have $100. Then, at the end of the year, you earn 5% interest on that $100. So, at the end of year one, you have $105 – the initial investment plus the interest for the year using a ‘simple interest’ calculation.

At the end of year 2, you earn 5% interest again. But, that interest applies to the entire balance, not just the initial investment. So, interest in year 2 is $5.25 ($105 x 5%). That is, you get another $5 in interest on the first $100 plus an extra quarter from the $5 of reinvested earnings. The balance in your account is now:

Since we already have a formula for the term “Year 1 Ending Value,” we can plug that equation in place of the variable. So, the expanded formula would look like this:

Which you can simplify to:

This same process works for any number of years. All you have to do is turn the exponent into whatever year’s balance you’re trying to calculate.

For example, if you want to know the balance after ten years, the equation would be:

Or, the more general annual compounding formula of:

Where:

FV = future value PV = Present Value i = annual interest rate n = number of years

Of course, the answer includes the original investment.

If you only want to know how much interest you’ve earned, you need to subtract the money you started with. That means the compound interest formula can be generalized to:

Compounding can happen at any interval. Annual compounding is common, but so are quarterly and monthly compounding. In some cases, compounding can be continuous. The closest mental image we can get to continuous compounding is to imagine calculating interest every second of every day.

Let’s step through the process. First, let’s say you want to calculate monthly compounding. You start with a hypothetical $100 and earn an annual interest rate of 10% on your account. Because there are 12 months in a year, you need to divide the yearly interest rate by 12 to get the monthly interest rate. In this case, the monthly rate is 0.83% (0.10 / 12).

Now we can solve for however many months we’re interested in. All you have to do is plug the number of months into the formula rather than the number of years:

By compounding more frequently, you end up with a slightly larger number. You can see this by solving for the value at the end of year 2, which is also the end of month 24.

Daily compounding would result in a daily interest rate of 0.027% per day (0.10 / 365 days). At the end of year 2, you would have 730 days of compounding interest.

The continuous compounding formula looks like this:

Where;

FV = future value P = Principal e = is the mathematical constant for the base of the natural logarithm, which is approximately 2.7183 r = interest rate t = time between the start and end value, expressed as the number of periods of the same length as the interest rate

For example, with continuous compounding after two years at an annual interest rate of 10%, we get:

You may notice the result of continuous compounding is the same as the outcome of daily compounding in our examples. That’s not always the case, but with smaller numbers and rounding, it happens to work out that way.

It’s said that Albert Einstein once called compounding the eighth wonder of the world. The ability for things to multiply exponentially is incredible.

Consider a cell that divides every day. On the second day, there are two cells. On the third day, there are four cells. After a month, there are 1,073,741,824 cells. Within three months, there would be twice as many cells as stars in the universe.

Now, while exponential growth is a remarkable thing, you won’t be able to take a penny and turn it into $10,000,000 in three months. With the power of compound interest, you can save more for retirement money than you earn in your lifetime. And, if you’re not careful, compounding debts can bankrupt you faster than you realize.

In investing, compounding almost always refers to using investment earnings to buy more assets. This process can include using capital gains to purchase more investments, or it could mean reinvesting dividends in more of the company stock.

Compounding in investing allows investors to increase the number of shares in their portfolio thus increasing its growth potential when these additional shares produce their own dividend and potential growth.

Imagine a person saving $100 per month from the time of their 20th birthday until they retire at 65. That’s 540 months, which means they deposited $54,000 into their savings account. Now, assume they earn a hypothetical 5% interest on those savings each year.

The total value of those 540 deposits on the person’s 65th birthday will be $201,703. In other words, the small monthly contributions turn into something much more meaningful thanks to compounding.

Compounding in finance can turn into a headache if you’re not careful. In general, financing refers to borrowing money to pay for something, and then managing debts. When structuring a deal, project finance is how you fund the project. If you’re not careful, compounding can turn a project from a cash cow into a money pit.

Consider borrowing $1M as a revolving line of credit to build a factory. You estimate it will take three years to get the plant up and running. From there, you’ll have positive cash flows of $240,000 per year with which you can pay off the loan.

Your lender will charge you a 12% annual interest rate, compounded monthly. If you dedicate all of your positive cash flow, how long do you think it will take to pay off that line of credit?

By the time you have revenues flowing, you’ll owe $1,430,769 due to compounding interest, and your interest payments in your first year of operation will be more than $180,000. If everything goes well, you’ll end up paying every penny of profits to your lender for over 10 years before you make anything for yourself.

Examples of compounding are for illustrative purposes only and do not represent any financial product or investment return available.

Compounding for investment (non-bank) products must take into account that actual returns will vary and can be negative over certain periods of time.