What is the compound interest formula?

The balance after n periods is K = K₀ · (1 + p/100)ⁿ, where K₀ is the starting capital and p is the interest rate for one period as a percentage. The bracket is the multiplier for one period and the exponent says how many times that multiplier has acted.

What does compounding mean?

It is the moment the interest is added to the capital. With monthly compounding the annual rate is divided by 12 and the number of periods multiplied by 12: K = K₀ · (1 + p/1200)^(12n). The more frequent the compounding, the larger the final balance, though the differences are small — for 1000 at 5% over 2 years it comes to 2.44.

What is the difference between a percentage point and a percent?

A percentage point is the plain difference between two figures given as percentages. A rate rising from 4% to 5% rises by 1 percentage point and by 25% at the same time, because 1 is 25% of 4. A change in a quantity that is itself a percentage is reported in percentage points precisely so that the two numbers are not confused.

Why does my statement differ from the formula?

Usually because of rounding and tax. A bank rounds the interest to the smallest coin after every compounding rather than only at the end, and it withholds tax on investment income. The compound interest formula gives the amount before both corrections, so a discrepancy of a few cents is normal.

Intermediate level

Compound interest and percentage points

Q: How does compound interest differ from simple interest?

With simple interest the interest is always computed on the starting amount, so the same sum is added each period. With compound interest the interest is added to the capital and the next period is computed on the enlarged sum. For 1000 at 10% over 3 years simple interest gives 1300 and compound interest 1331.

Compound interest is interest charged on an amount that has already grown — which is why capital rises as a power rather than in equal steps. Learn the formula for the final balance, compounding more often than once a year, compound decline, and the difference between a percent and a percentage point.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Where this is used

Real situations where you count exactly the way this lesson teaches:

Savings account
A deposit of 1000 at 5% a year compounded monthly comes to 1000 · (1 + 0.05/12)²⁴ ≈ 1104.94 after two years, against 1102.50 with annual compounding. Those 2.44 of difference are the entire meaning of the word compounding in the terms and conditions.
Credit card
A debt of 2000 charged 1.5% a month grows in a year to 2000 · 1.015¹² ≈ 2391.24, that is by 19.6% and not by 12 · 1.5% = 18%. Compound interest treats debt exactly as it treats savings — only against you.
Economic analysis
With inflation at 8% a year, prices after three years are multiplied by 1.08³ ≈ 1.2597. So 1000 kept in a drawer then buys what 1000 : 1.2597 ≈ 794 used to — a real loss of 20.6%, not of 24%.
Banking and the press
Raising a reference rate from 5.75% to 6.25% is a rise of 0.5 percentage points and at the same time a rise of 8.7%, since 0.5 : 5.75 ≈ 0.087. A headline quoting one of those figures instead of the other says something entirely different about the same decision.

All formulas

Balance after n periods
$K_n = K_0 \left(1 + \frac{p}{100}\right)^{n}$
one period multiplier raised to the power n
Compounding m times a year
$K_n = K_0 \left(1 + \frac{p}{100 \cdot m}\right)^{m \cdot n}$
the rate is divided by m and the number of periods multiplied by m
Compound decline
$K_n = K_0 \left(1 - \frac{p}{100}\right)^{n}$
the same formula with a minus — depreciation, loss of value
Interest earned
$O = K_n - K_0$
what the bank added, that is the difference of the balances
Percentage points
$\text{p.p.} = p_2 - p_1$
the plain difference between two figures given as percentages

The lesson on percentages computed a percentage of one number. Here a percentage acts repeatedly — and that changes the arithmetic completely, because the second time round it is charged on an amount that has already grown.

Simple against compound

Take 1000 at $10\%$ a year for three years.

With simple interest the interest is computed each year on the starting amount, so 100 is added every year: 1300 in total.

With compound interest the interest is added to the capital, and the next year is computed on the new sum:

Each year the amount added is larger than the last, because the interest is charged on the current balance.

After three years there is 1331 rather than 1300. That difference of 31 is interest on the interest — and it is what makes the two methods diverge dramatically over longer periods.

The formula

Every period multiplies the capital by the same change multiplier $\left(1 + \tfrac{p}{100}\right)$ . Repeated $n$ times it becomes a power:

K_n = K_0 \left(1 + \frac{p}{100}\right)^{n}

where $K_0$ is the starting capital, $p$ is the rate for one period as a percentage, and $n$ is the number of periods. The interest itself is the difference of the balances:

O = K_n - K_0

For our example $K_3 = 1000 \cdot 1.1^3 = 1000 \cdot 1.331 = 1331$ , so the interest is 331.

Compounding more often than once a year

Banks add interest not once a year but monthly or quarterly. The moment it is added is called compounding. The annual rate is then divided by the number of compounding periods $m$ , and the number of periods multiplied by the same number:

K_n = K_0 \left(1 + \frac{p}{100 \cdot m}\right)^{m \cdot n}

The more frequent the compounding, the larger the final balance — because the interest starts working sooner.

A deposit of 1000 at 5% a year for 2 years: what does annual compounding give, and what does monthly?

Compound decline

The same formula with a minus describes a repeated fall:

K_n = K_0 \left(1 - \frac{p}{100}\right)^{n}

A car bought for $80\,000$ and losing $15\%$ of its value each year is worth, after four years,

80\,000 \cdot 0.85^4 = 80\,000 \cdot 0.522 \approx 41\,760

rather than $80\,000 - 4 \cdot 12\,000 = 32\,000$ . A drop of $15\%$ four times over is not a drop of $60\%$ but of just under $48\%$ — because each time it is subtracted from a smaller amount.

Percentage points

When the quantity that changes is itself a percentage, there are two different ways to report it. A rate rises from $4\%$ to $5\%$ :

the difference is $1$ percentage point (p.p.) — plain subtraction, $5\% - 4\%$ ,
but the rise is $25\%$ , because $1$ is a quarter of $4$ .

One change described by two numbers: a difference of 1 percentage point is a rise of 25%.

Both numbers are true and describe the same event — which is why it always has to be clear which one is meant. A percentage point is a difference, a percent is a ratio.

Unemployment fell from 8% to 6%. By how much did it fall?

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

Final balance ($) — principal, annual rate, years: 10000, 10%, 2

Common mistakes

Multiplying the rate by the number of years — $10\%$ over $3$ years is a multiplier of $1.1^3 = 1.331$ , not $1.3$ . Percentages do not add.
Reading four drops of $15\%$ as a drop of $60\%$ — each successive drop is taken from a smaller amount, so the total is less.
Confusing a percentage point with a percent — a rise from $4\%$ to $5\%$ is $1$ p.p. but $25\%$ .
Forgetting to divide the rate when compounding — with monthly compounding the annual rate is divided by $12$ ; changing only the exponent inflates the answer several times over.
Expecting agreement to the last cent — a bank rounds the interest after every compounding and withholds tax, so the formula's result can be a few cents out.

Formula card

Topic: Compound interest

Balance after n periods
$K_n = K_0 \left(1 + \frac{p}{100}\right)^{n}$
one period multiplier raised to the power n
Compounding m times a year
$K_n = K_0 \left(1 + \frac{p}{100 \cdot m}\right)^{m \cdot n}$
the rate is divided by m and the number of periods multiplied by m
Compound decline
$K_n = K_0 \left(1 - \frac{p}{100}\right)^{n}$
the same formula with a minus — depreciation, loss of value
Interest earned
$O = K_n - K_0$
what the bank added, that is the difference of the balances
Percentage points
$\text{p.p.} = p_2 - p_1$
the plain difference between two figures given as percentages

A deposit of 1000 at 10%: each year the amount added is larger, because the interest is charged on the current balance.

A rate rising from 4% to 5% is one percentage point — and at the same time a rise of 25%, since 1 is a quarter of 4.

Frequently asked questions

$A fraction bar and a percent sign rendered as geometric forms, with a circle split into equal orbital slices, against a cosmic nebula in violet and magenta$

Numbers

A percentage is a fraction in disguise — three ways to write one number

1/2, 0.5 and 50% are not three different things but three ways of writing the same number. Here is where the fraction bar, the decimal point and the % sign came from, why 1/3 never ends, why +20% and −20% do not return you to the start, and why fractions are the first serious wall in school mathematics.

11 min