Compound interest: the interest formula

When we talk about interest, we are actually talking about a type of function called an exponential function. Those get their own guide, but here is a little introduction to what they are and what they can be used for, through the one example everybody meets: money in a bank account.

When do I use this?

If we have some money in a bank and we get a certain rate of interest per year, or per month or the like, we can use the interest formula to find how much is in the account after a number of years. It is a percentage increase, from percentage increase and decrease, repeated once for every year.

The formula

$$K_n = K_0 \times (1 + r)^n$$

In this formula:

• $K_0$ is our start amount, the money we start with in the account.
• $r$ is how many per cent interest we get, written as a decimal.
• $n$ is how long we get interest for. That can be years, or months, for example.
• $K_n$ is the amount we have in the account after that time, with that rate and that start amount. $K_n$ is the value we are trying to work out.

There are quite a few letters here, so let's look at what each part of the formula does. The rate has to be a decimal: if we have 2% interest, then $r = 0.02$. The reason the rate is added to 1 is that we are not finding a percentage of the amount, we are adding a percentage on top. If we just wrote

$$K_0 \times 0.02$$

we would find 2% of $K_0$. But when we write

$$K_0 \times (1 + 0.02)$$

we add 2% on top of $K_0$. The $(1 + r)$ is called the growth factor, and the power $n$ multiplies by it once for every period, so the interest in year two is worked out on what was in the account after year one. That is what "compound" means.

One more thing to keep in mind: $n$ and $r$ have to fit together. We cannot measure the rate per month and then count $n$ in years. They have to be in the same unit.

Worked examples

Example 1: £1000 at 2% for 2 years. Let's say we had £1000 in the account with 2% interest per year. How much money would we have after 2 years? We can use the formula. In our example

$$K_0 = 1000, \quad r = 2\% = 0.02, \quad n = 2$$

We have to remember that the rate goes in as a decimal:

$$K_n = 1000 \times (1 + 0.02)^2 = 1040.4$$

With 2% interest for 2 years we have £1040.40. The reason we multiply by 1.02 is that we add the percentage on top of our start amount. When we multiply by 1.02, we add 2% on top of the original amount, and we do it twice, once for each year.

Example 2: £2000 at 3% for 5 years. Let's say we started with £2000 in the account ($K_0$), a rate of 3% per year ($r$), and left the money there for 5 years ($n$). We put the values in the places of $K_0$, $r$ and $n$:

$$K_n = 2000 \times (1 + 0.03)^5 \approx 2318.55$$

So after 5 years there is about £2318.55 in the account. As we can see, we have got £318.55 more than we started with. Working out $(1.03)^5$ is a job for the calculator; the indices glossary has the background on what a power like that means.

Where the formula comes from

This interest formula actually comes from the exponential function, which has the equation

$$f(x) = b \times a^x$$

In this equation $b$ is the start value, just like $K_0$, and $a$ is what we call the growth factor, which is the percentage increase $(1 + r)$. The same function also describes things that shrink by a percentage every year, where the growth factor is smaller than 1. All of that is in the guide about exponential growth and decay.

Common mistakes

• Multiplying by 0.02 instead of 1.02. That finds 2% of the money instead of adding 2% to it. The 1 keeps what you already have, the 0.02 is the interest.
• Putting the rate in as 2 instead of 0.02. The rate has to be a decimal. $1000 \times (1 + 2)^2$ is 9000, which is not what a bank pays.
• Thinking the interest is the same every year. It is not, because each year's interest is worked out on the new balance, which includes the previous years' interest. The power in the formula is what takes care of that.
• Mixing units. A rate per year goes with $n$ in years, a rate per month with $n$ in months.

Related

• What is a percentage?
• Percentage increase and decrease with a multiplier
• Exponential growth and decay
• What are indices (powers) and roots?