What Is Compound Interest? How Interest Earns Interest, With Worked Examples

Compound interest is interest paid on the original amount and also on the interest already added to it, so each period's interest is larger than the last.

Compounding is the reason a savings balance, a loan or an investment return can grow faster than a simple percentage suggests, and the reason a loss followed by a gain of the same size does not bring a balance back to where it started. This page compares simple and compound interest, works through examples by hand, explains compounding frequency and the annual percentage yield, covers the rule of 72, and shows how the same arithmetic applies to investment returns. It is descriptive throughout.

Simple interest and compound interest

Simple interest is paid on the original amount only, called the principal. Compound interest is paid on the principal plus the interest earned so far. The SEC's investor education site, Investor.gov, defines compound interest in the same terms: interest paid on principal and on interest that has built up.

Take $10,000 at 5 percent a year for ten years.

• Simple interest pays 5 percent of $10,000, which is $500, every year. After ten years the balance is $10,000 plus 10 x $500, or $15,000.
• Compound interest, credited once a year, pays 5 percent of whatever the balance is at the time. The formula is the principal times (1 + rate) raised to the number of years: $10,000 x 1.05^10 = $16,288.95.

The difference, $1,288.95, is interest earned on earlier interest.

A worked example, year by year

The compound balance grows by a larger dollar amount each year, even though the rate never changes.

In year 2 the extra $25 is 5 percent of the $500 earned in year 1. By year 10 the interest earned in a single year is about 55 percent larger than in year 1.

Time matters far more than frequency. At the same 5 percent rate:

Doubling the time from 10 to 20 years makes the gap about five times larger, and at 30 years the compound balance is roughly 1.7 times the simple one. The SEC runs a free compound interest calculator on Investor.gov that repeats these sums for other amounts, rates and periods.

How often interest is added: compounding frequency

Interest can be credited yearly, quarterly, monthly or daily. When the quoted annual rate is split into smaller, more frequent pieces, each piece starts earning sooner, so the result is a little higher. The formula divides the rate by the number of periods per year, n, and raises it to the total number of periods: principal x (1 + rate / n)^(n x years).

Going from yearly to monthly adds about $181 over ten years. Going from monthly to daily adds only about $17, and continuous compounding, the mathematical limit, adds just 56 cents more. Frequency matters, but far less than the rate and the time.

APR, APY and the effective rate

Because frequency changes the result, a quoted rate alone does not say how much a balance grows in a year. Two terms separate the two ideas:

• The stated or nominal rate is the yearly rate before compounding: 5 percent in the examples above. On US loans the quoted APR does not count compounding within the year either, but it also folds in certain fees, so it can sit above the loan's interest rate.
• The annual percentage yield (APY), also called the effective annual rate, is the growth over one full year once compounding is counted. A 5 percent rate compounded monthly gives (1 + 0.05 / 12)^12 minus 1, an APY of 5.12 percent. Compounded daily it gives 5.13 percent.

In the United States, Regulation DD, which implements the Truth in Savings Act, sets the formula that banks and the other depository institutions it covers use for the APY in deposit account disclosures and advertising. Its stated purpose is to let consumers make meaningful comparisons between institutions, so two accounts that compound differently can be compared on the same footing.

The rule of 72

The rule of 72 is a shortcut for doubling time: divide 72 by the yearly rate in percent. At 5 percent, 72 / 5 gives 14.4 years. The exact answer, from logarithms, is 14.21 years, so the shortcut is close. At 8 percent it gives 9 years, against an exact 9.01. At 2 percent it gives 36 years, against an exact 35.0. The rule works best for rates in the middle range and drifts further from the exact answer at very high or very low rates.

The same arithmetic describes inflation. At 3 percent inflation a year, prices double in about 24 years by the rule (exact 23.4), which is the same as the spending power of money halving. After ten years, $1 of today's spending power is worth about 74 cents (1 / 1.03^10).

Compounding in investment returns

Investment returns compound too, but they change every period, which leads to three results that surprise beginners.

• Returns multiply, they do not add. A gain of 1 percent in each of twelve months compounds to 1.01^12 minus 1, or 12.68 percent for the year, not 12 percent.
• A loss needs a larger gain to recover. A fall of 50 percent followed by a rise of 50 percent leaves a balance 25 percent lower (0.5 x 1.5 = 0.75). Getting back to the start after a 50 percent fall takes a 100 percent gain; after a 20 percent fall it takes 25 percent.
• The average return overstates the growth. A year of plus 20 percent and a year of minus 20 percent average 0 percent, yet the balance ends 4 percent lower (1.2 x 0.8 = 0.96). The steady yearly rate that produces the same ending value, the compound annual growth rate, is minus 2.02 percent a year.

Kresmion's portfolio analytics chain returns in the same way. The performance line is a time-weighted return that chains each day's return onto the last. Deposits and withdrawals are taken out, so they do not count as gains or losses. In the monthly returns grid, the yearly total is compounded from the months shown rather than added up. Both sit on the Analytics tab of the portfolio, which needs a Kresmion account and runs its calculations in the browser.

Honest limitations

The worked examples use a fixed rate, no taxes, no fees and no withdrawals, which real accounts rarely have. Fees and taxes compound as well, in the opposite direction: a yearly charge comes out of a balance that would otherwise have kept growing. Inflation reduces what a compounded balance can buy, so a nominal balance overstates growth in spending power. Investment returns are not fixed, so a compound growth rate describes a path that already happened; it is not a rate the future has to follow. The rule of 72 is an approximation, as the figures above show.

Key takeaways

Frequently asked questions

What is the difference between simple and compound interest?

Simple interest is calculated on the original amount only, so it adds the same sum every period. Compound interest is calculated on the original amount plus the interest already credited, so each period adds a little more than the last. On $10,000 at 5 percent for ten years the two give $15,000 and $16,288.95.

Does compounding daily make a big difference?

Less than the label suggests. On $10,000 at 5 percent for ten years, daily compounding ends about $198 higher than yearly compounding, and only about $17 higher than monthly. The rate and the length of time matter far more than the frequency.

What is the difference between APR and APY?

APR, the figure quoted on US loans, is a yearly rate that does not count compounding within the year and includes certain loan fees. APY, the figure quoted on US savings accounts, is the growth over a full year once compounding is counted. For the same nominal rate with no fees, APY is equal to or higher than APR: a 5 percent rate compounded monthly has an APY of 5.12 percent.

Does compound interest work against borrowers?

It can. On card balances and on many loans, interest that is not paid is added to the balance and then charged interest itself, by the same arithmetic as the savings examples on this page. Some loans, such as many simple-interest auto loans, do not charge interest on unpaid interest. The rate and how often it compounds are set out in the loan or card terms.

Is the rule of 72 exact?

No. It is a shortcut that is close for moderate rates: 14.4 years at 5 percent against an exact 14.21 years. It drifts further from the exact figure at very high or very low rates.

This page is information, not investment advice.

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Source: Investor.gov (US Securities and Exchange Commission), glossary entry "Compound Interest", https://www.investor.gov/introduction-investing/investing-basics/glossary/compound-interest ; Investor.gov Compound Interest Calculator, https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator ; Regulation DD (Truth in Savings), 12 CFR 1030.1 (purpose and coverage) and Appendix A to part 1030, Annual Percentage Yield Calculation, https://www.ecfr.gov/current/title-12/chapter-X/part-1030/appendix-Appendix%20A%20to%20Part%201030 ; all balances, yields and doubling times on this page computed by Kresmion for hypothetical amounts ; Kresmion portfolio analytics (time-weighted return and monthly returns grid, account required).

Kresmion Research.