Compound Interest

Most people have a rough idea of how interest works, but the particulars are often fuzzy. For example, let's say that you invest $100 in a savings account with a yearly interest rate of 6%. Six percent of 100 is 6, so after the first year, you would have $106 (100 + 6). After the second year, you would have $112 (106 + 6), and so on. Pretty simple, right? You just get 6% of your original principal amount every year.

That structure is so simple that it's actually called simple interest, but many real-world applications don't use it. Instead, you'll see compound interest, which is calculated based on how much money you currently have, not what you initially put in. For instance, you'll still have $106 after the first year in the example above, but the second year will add 6% of $106 instead of the original $100, giving you about $112.36 after year 2. The calculation for year 3 then uses $112.36 as the base figure, meaning that your 6% increases are bigger every year.

Compound interest is a much better investment for people collecting interest, but the math is considerably more challenging. This article will explore two different formulas you might need to calculate compound interest and get into some practical applications. Without further ado, let's get started!

The yearly compound interest formula

If you put P dollars in a savings account with an annual interest rate r, and the interest is compounded yearly, the amount A you have after t years can be calculated using the following formula:

$A = P {(1 + r)}^{t}$

That probably looks a little abstract, so let's try working with an example. Suppose you invest $4,000 at 7% interest, compounded yearly. How much money will you have after 5 years?

We're trying to solve for A, with P = 4000, r = 7%, and t = 5 years. Remember to turn the percentage into a decimal before doing any calculations. If we plug all of these numbers into our formula, we get:

$A = 4000 {(1 + 0.07)}^{5}$

$A = 4000 (1.40255)$

$A = $ 5610.20$

Compound interest problems almost always involve working with money, so you can assume that you should round to the hundredths place even if you aren't explicitly told to do so. After all, there's no coin worth 0.001 in most money systems. The exception to this rule would be working with a monetary system that supports more decimals, such as many cryptocurrencies.

You may also see problems that expect you to use the formula above to find a variable other than A. For instance, you might be given the amount you have after a set period of time and asked to find the original deposit, how long interest has accrued, or what the annual rate is. You solve all of these problems by subbing the numbers you have into the formula you know. Just make sure to put all of the numbers you have in the correct spot!

The general compound interest formula

There's no rule that says compound interest must be compounded annually, meaning that the formula above won't always work. If interest is compounded more frequently than once a year, you need to divide the interest rate by the number of compounding periods to accurately calculate how much money you will have in the future. We use n for this number, giving us a new variable to add to our formula:

$A = P {(1 + \frac{r}{n})}^{n t}$

This formula is derived from the one above. When interest is compounded annually, we would have the fraction r/1 and multiply t by 1 since it only compounds once per year. Neither changes any values, so we don't have to use the n variable at all.

The other variables mean the same thing they did above, but it's worth calling attention to t. Even though the time in the problem will be expressed in something other than years since we're using the general compound interest formula instead of the yearly one, t is still expressed in years.

For example, let's say that you've invested $1,000 at 9% interest compounded monthly. How much money would you have after 18 months?

In this problem, we're solving for A, P is 1,000, r is 9% (or 0.09), and n is 12 (since that's how many times the interest will compound in a year). The 18 months are represented by setting t to 1.5 since 18 months is a year and a half. Plugging these numbers into the formula above, we get:

$A = 1000 {(1 + \frac{0.09}{12})}^{12 \times 1.5}$

$A = 1000 (1.143960)$

$A = $ 1143.96$

We've rounded to the nearest cent. Essentially, compound interest questions require you to memorize the relevant formula and work out the math. As long as you take your time and avoid making mistakes, it is simpler than it seems on the surface.

Again, you may see problems asking you to solve for P, r, or t instead of A. The process is identical except that you would be solving for a different variable and manipulating the formula appropriately. Of course, this also means that you need to pay attention to the information provided to you and not simply assume that you're trying to solve for A.

Where might I see compound interest?

You might see compound interest described as a "better deal" than simple interest, but that distinction is completely dependent on context. If you're collecting interest on money deposited in a savings account, compound interest is a better deal since you'll end up with more money. However, if you take on debt to finance a significant purchase such as a house or car and get charged compound interest, you'll end up paying more money over time. That is not a better deal at all.

That said, compound interest is the standard in finance and economics, so you probably couldn't choose a loan with simple interest even if it would benefit you. The best thing to do is to understand what you're paying and try to pay off the loan entirely before the interest compounds too much.

If you're looking for examples beyond a savings account, compound interest is common for Canadian mortgage loans (but not American mortgages), interest paid on corporate and government bonds, and in valuing financial products such as derivatives.

The most important number in calculus, e, was also discovered by a mathematician trying to solve a compound interest problem. The concept is more important in the broader scheme of mathematics than you might expect.

Important compound interest vocabulary

It's impossible to fairly compare financial products with different compounding frequencies, so most governments (including the United States) have legislation requiring lenders to express the price of their products in a standardized manner; usually, equivalency to an annual rate. This nominal rate goes by many names, including the effective annual percentage rate (EAPR), annual equivalent rate (AER), effective interest rate, effective annual rate, and annual percentage yield.

Similarly, compound interest doesn't always compound yearly or monthly. Other common terms include quarterly, monthly, weekly, daily, half-yearly, and continuously. Some instruments don't have compound interest at all until they reach maturity, at which point it's retroactively applied. Furthermore, an uncommon term can exist. For example, you'd be hard-pressed to find a financial instrument where interest compounds bi-annually, but it's theoretically possible and the general compound interest formula would work.

Compound interest practice questions

a. If you invest $5000 at 2.5% interest compounded annually, how much money will you have after five years?

$A = 5000 {(1 + \frac{0.025}{1})}^{1 \times 5}$

$A = 5000 {(1.025)}^{5}$

$A = 5000 (1.131408)$

$$ 5657.04$

b. Mark invests $2000 at an annual rate of 4% compounded annually. How much money will he have in 10 years?

$A = 2000 {(1 + \frac{0.04}{1})}^{1 \times 10}$

$A = 2000 (1.480244)$

$$ 2960.49$

c. Gwen invests $12000 at an annual rate of 8% compounded quarterly. How much money will she have in 4 years?

$A = 12000 {(1 + \frac{0.08}{4})}^{4 \times 4}$

$A = 12000 {(1.02)}^{16}$

$A = 12000 (1.372786)$

$$ 16473.43$

d. Chris invests $1500 at an annual rate of 4.2% compounded monthly. How much money will he have in 3 years?

$A = 1500 {(1 + \frac{0.042}{12})}^{12 \times 3}$

$A = $ 1500 (1.144602)$

$$ 1701.05$

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