Understanding Compound Interest and Its Formula

What is compound interest?

Compound interest is often called "interest on interest." Unlike simple interest, which only calculates growth on the starting amount, compound interest continually calculates growth on a new, larger total at the end of each period.

Calculating the interest on a new total requires finding the percent of a number that includes the previous interest. This means the amount of interest earned grows larger every year, creating a curved, accelerating path rather than a straight line.

After mastering the basics of this growth model, you can explore the exact mathematical differences in simple interest vs compound interest.

See why the amount compounds

To see why the final amount compounds, track a starting principal of $1{,}000$ dollars earning $10\%$ interest each year. This compounding effect is a type of consecutive percent change applied to a growing base.

• Year 1: The starting principal is $1{,}000$ dollars. Ten percent of $1{,}000$ is $100$. The new total is $1{,}100$ dollars.
• Year 2: The new starting principal is $1{,}100$ dollars. Ten percent of $1{,}100$ is $110$. The new total is $1{,}210$ dollars.
• Year 3: The starting principal is now $1{,}210$ dollars. Ten percent of $1{,}210$ is $121$. The new total is $1{,}331$ dollars.

Notice that the interest earned in the third year ($121$ dollars) is larger than the interest earned in the first year ($100$ dollars). Calculating this year by year takes a long time, which is why a formula is used.

Use the compound-interest formula

The standard formula calculates the total final amount directly, rather than calculating the interest for each year separately.

The standard compound interest formula uses an annual rate expressed as a decimal.

The formula for annual compounding is:

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

Here is what each variable represents:

• $A$ is the final amount, including the original principal and all accumulated interest.
• $P$ is the principal, the starting amount of money.
• $r$ is the annual interest rate expressed as a decimal, not a whole number.
• $t$ is the time the money is invested or borrowed, measured in years.

Handle compounding frequency

Interest is not always calculated once a year. When interest is added more frequently, you must adjust the formula to account for the number of compounding periods in a single year.

The modified formula is:

$A = P\left(1 + \dfrac{r}{n}\right)^{nt}$

The new variable $n$ represents the number of times the interest is compounded per year.

• Annually: Compounded once a year, so $n = 1$.
• Semi-annually: Compounded twice a year, so $n = 2$.
• Quarterly: Compounded four times a year, so $n = 4$.
• Monthly: Compounded twelve times a year, so $n = 12$.

Because interest is applied more often, the total amount grows slightly faster with frequent compounding than with annual compounding at the same interest rate.

Find the interest from the final amount

The standard formula gives you the final amount $A$, which includes the original principal. It does not isolate the interest earned.

To find the amount of compound interest ($CI$) by itself, subtract the starting principal from the final amount:

$CI = A - P$

You can also write this entirely in terms of the formula:

$CI = P\left(1 + \dfrac{r}{n}\right)^{nt} - P$

Always check whether a question asks for the final amount or just the interest earned.

Worked examples

Review these examples to see how the formulas are applied to different compounding frequencies.

Example 1: Calculating annual compound interest

Question: An investment of $5{,}000$ dollars earns $6\%$ interest compounded annually for $3$ years. What is the total compound interest earned?

Method:

1. Identify the given values: $P = 5{,}000$, $r = 0.06$, $n = 1$, $t = 3$.
2. Substitute the values into the formula to find the final amount:

$A = 5{,}000(1 + 0.06)^3$

$A = 5{,}000(1.06)^3$

3. Calculate the power and multiply:

$A = 5{,}000 \times 1.191016$

$A = 5{,}955.08$

4. Subtract the principal to find the interest:

$CI = 5{,}955.08 - 5{,}000$

Answer: The compound interest earned is $955.08$ dollars.

Check: A simple interest estimate ($5{,}000 \times 0.06 \times 3 = 900$) confirms the answer is reasonable, as compound interest should be slightly higher.

Example 2: Calculating quarterly compounding

Question: What is the final amount if $2{,}000$ dollars is invested at $8\%$ compounded quarterly for $2$ years?

Method:

1. Identify the given values: $P = 2{,}000$, $r = 0.08$, $n = 4$ (quarterly), $t = 2$.
2. Substitute into the frequency formula:

$A = 2{,}000\left(1 + \dfrac{0.08}{4}\right)^{4 \times 2}$

3. Simplify the rate per period and the total number of periods:

$A = 2{,}000(1 + 0.02)^8$

$A = 2{,}000(1.02)^8$

4. Calculate the result:

$A = 2{,}000 \times 1.171659...$

$A \approx 2{,}343.32$

Answer: The final amount is $2{,}343.32$ dollars.

Example 3: Setting up monthly compounding

Question: Write the expression for the final amount when $800$ dollars is borrowed at $12\%$ interest compounded monthly for $5$ years.

Method:

1. Identify the variables: $P = 800$, $r = 0.12$, $n = 12$ (monthly), $t = 5$.
2. Substitute them into $A = P\left(1 + \dfrac{r}{n}\right)^{nt}$.
3. $A = 800\left(1 + \dfrac{0.12}{12}\right)^{12 \times 5}$
4. Simplify the internal fraction and the exponent.

$A = 800(1 + 0.01)^{60}$

Answer: $A = 800(1.01)^{60}$.

Common mistakes

When applying the formula, avoid these frequent errors:

• Using a whole number instead of a decimal: If the rate is $5\%$, the value of $r$ must be $0.05$. Using $5$ in the formula will calculate a $500\%$ interest rate.
• Forgetting to multiply the exponent by $n$: When interest is compounded monthly for $3$ years, the total number of periods is $12 \times 3 = 36$. It is a mistake to leave the exponent as $3$.
• Confusing the final amount with the interest: The formula $A = P(1+r)^t$ returns the total balance. If a question asks for the interest, you must subtract the original principal.

Frequently asked questions

Does compound growth apply to things other than money?

Yes. The same mathematical concept models population growth, bacterial reproduction, and radioactive decay. Any system where growth is based on the current size, rather than the starting size, uses a compounding formula.

What happens if the time is not a whole number?

The formula works exactly the same way. If you invest money for $2.5$ years with annual compounding, you substitute $t = 2.5$ directly into the formula.

Is continuous compounding different?

Yes. As the compounding frequency $n$ becomes infinitely large, the standard formula shifts to a special continuous compounding formula: $A = Pe^{rt}$, where $e$ is a mathematical constant.