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Compound interest

Interest that earns interest, and why the years matter more than the rate looks.

FinanceMankiw ch. 14About 5 min: 2 reading, 3 practiceUpdated 27 October 2026

General education, not financial advice.

After this lesson you can

Work out a future value with amount × (1 + r)ᴺ
Tell compound from simple interest
Estimate doubling time with the rule of 70

Money over time: lesson 1 of 5

Mankiw ch. 14
Compound interest 5 min
Present value 5 min
Mankiw ch. 11
Measuring inflation with a price index 4 min
Real and nominal interest rates 3 min
Bogle ch. 4
Costs and the market return 3 min

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Pairs with Mankiw, Principles of Macroeconomics (6th ed.), chapter 14, "The Basic Tools of Finance": present value and the FYI box on compounding

What it is

When interest is left in the account, next year's interest is worked out on the bigger balance. That is compounding: your money earns interest, and the interest earns interest too.

After one year, an amount grows by the factor (1 + r), where r is the yearly rate as a decimal (5% is 0.05). After N years it has been multiplied by that factor N times:

future value = amount × (1 + r)ᴺ

Quick check

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100 at 10% a year, compounded. What is it after 2 years?
Show answer
121. 100 → 110 → 121; the second year earns interest on 110.

Why it matters

Small differences in a rate turn into large differences over long periods. That works for you on savings and against you on debt, where unpaid interest can start charging interest of its own.

A handy check is the rule of 70: something growing at x% a year doubles in about 70 ÷ x years. At 7% a year, money doubles roughly every 10 years. (Some books use 72 instead of 70; both are rough guides.)

Worked example

You put 1,000 in an account paying 6% a year, added once a year, and leave it for 10 years.

Year	Calculation	Balance
1	1,000 × 1.06	1,060.00
2	1,060 × 1.06	1,123.60
3	1,123.60 × 1.06	1,191.02
10	1,000 × 1.06¹⁰	1,790.85

With simple interest (60 a year on the original 1,000, never on interest) you would have 1,600. The extra 190.85 is interest earned on interest. By the rule of 70, money at 6% doubles in about 70 ÷ 6 ≈ 12 years.

Interest rate: 6%

1,000 over 30 years: compound against simple interest. Move the slider to change the rate.

Text description

Line chart over 30 years starting from 1,000 at 6% a year. Simple interest grows in a straight line to 2,800. Compound interest curves upward, slowly at first and faster later, to 5,743.49.

Common mistake

Multiplying the rate by the number of years. "6% for 10 years is 60%" is simple interest. Compounding gives about 79% here, and the gap grows every year.

Key terms

future value: what an amount grows to after interest
compounding: earning interest on earlier interest
simple interest: interest on the original amount only
rule of 70: doubling time ≈ 70 ÷ growth rate (%)

Practise the key terms

Tap a card to turn it over. 4 cards.

Front	Back
future value	what an amount grows to after interest
compounding	earning interest on earlier interest
simple interest	interest on the original amount only
rule of 70	doubling time ≈ 70 ÷ growth rate (%)

Practice

Try 30 years instead of 10, then change the rate by just one point.

Quick quiz

Tap an answer to check it.

200 earns 10% a year, compounded yearly. What is it after 2 years?
Show answer
242. Year 1 gives 220; year 2 adds 10% of 220, which is 22.
By the rule of 70, about how long does money take to double at 2% a year?
Show answer
35 years. 70 ÷ 2 = 35.

Work it through

You save 2,000 at 4% a year, compounded yearly. Work out the balance step by step.

Balance after year 1
Interest earned in year 2
Balance after year 2
Balance after year 3

Practice problems

Basic

500 is saved at 8% a year, compounded yearly. What is it worth after 2 years?
Answer and full solution
Answer: 583.20
1. Year 1: 500 × 1.08 = 540.
2. Year 2: 540 × 1.08 = 583.20.
3. Or in one go: 500 × 1.08² = 583.20.

Intermediate

Using the rule of 70, about how long does money take to double at 3.5% a year?
Answer and full solution
Answer: About 20 years
1. Rule of 70: doubling time ≈ 70 ÷ growth rate.
2. 70 ÷ 3.5 = 20 years.

Challenge

1,000 earns 12% a year, but interest is added monthly (1% a month). What is it after one year, and how does that compare with yearly compounding?
Answer and full solution
Answer: About 1,126.83, which is 6.83 more than the 1,120 from yearly compounding
1. Monthly rate: 12% ÷ 12 = 1% = 0.01.
2. Twelve monthly steps: 1,000 × 1.01¹² ≈ 1,126.83.
3. Yearly compounding gives 1,000 × 1.12 = 1,120.
4. Adding interest more often lets it earn interest sooner, so the balance ends a little higher.
5. So a quoted "12% a year, added monthly" is a nominal rate: the true yearly growth is about 12.68%.

In short

Compounding: interest earns interest.
Future value = amount × (1 + r)ᴺ, with r as a decimal.
Simple interest only ever uses the original amount.
Rule of 70: doubling time ≈ 70 ÷ rate.
Small rate differences become large over long periods.

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Check yourself

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In the formula amount × (1 + r)ᴺ, what does N stand for?
Show answer
The number of years (or periods) the money is left to grow.
Why does compound interest beat simple interest at the same rate?
Show answer
Each year's interest is worked out on a larger balance, because earlier interest has been added in.
Something grows at 5% a year. Roughly how long until it doubles?
Show answer
About 14 years (70 ÷ 5).