What Is Compound Interest? Formula, Calculation and Examples

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Compound interest is interest calculated on the original principal and on the interest already added to the balance. Each time interest is credited, the base for the next calculation grows, so the amount earned (or owed) rises every period even when the rate stays the same. The standard compound interest formula is:

A = P × (1 + r/n)^(n × t)

where A is the ending balance, P is the starting principal, r is the annual interest rate as a decimal, n is the number of compounding periods per year, and t is the number of years. Subtract P from A and you have the compound interest itself.

What Is Compound Interest?

Interest is the price of money over time. With simple interest, that price is always charged on the original principal. With compound interest, it is charged on a balance that keeps growing because past interest has been added to it.

A short comparison shows the difference. Put $1,000 into a hypothetical account paying 5% a year:

• Simple interest: $50 every year, forever. After five years, you have $1,250.
• Compound interest (annual): $50 in year one, then 5% of $1,050 ($52.50) in year two, then 5% of $1,102.50 in year three, and so on. After five years, you have $1,276.28.

The $26.28 gap is interest earned on earlier interest. It looks small at first, but the gap widens every year because the base keeps getting larger.

For someone new to saving, the plainest way to put it is this: money left in an account that compounds earns a return, and then that return starts earning a return too. For a borrower, the same idea runs in reverse. Interest that isn’t paid becomes part of what you owe, and you are charged interest on it.

The concept also extends beyond bank accounts. “Compound growth” describes any situation where gains are reinvested and begin generating their own gains, such as reinvested dividends in a stock fund. The arithmetic is the same, although the returns are not fixed the way a savings rate is.

How Compound Interest Works

Calculating Compound Interest

The full formula has two common forms:

• Ending balance: A = P(1 + r/n)^(nt)
• Interest only: CI = P[(1 + r/n)^(nt) − 1]

When interest compounds once a year (n = 1), the formula simplifies to the version many textbooks start with:

A = P(1 + r)^t

That annual version is the “equation for annually compounded interest.” The term inside the parentheses, (1 + r), is the growth factor for one period. Raising it to a power applies that growth once per period, each time to the new, larger balance.

Worked example (hypothetical): a $10,000 loan at 5%, compounded annually for 3 years

1. Growth factor per year: 1 + 0.05 = 1.05
2. Number of periods: 3
3. 1.05³ = 1.157625
4. Ending balance: $10,000 × 1.157625 = $11,576.25
5. Total interest: $11,576.25 − $10,000 = $1,576.25

Broken out year by year, assuming no payments are made and all interest is added to the balance:

Under simple interest, each year would cost exactly $500, for $1,500 in total. The extra $76.25 comes entirely from charging interest on interest.

Converting From Yearly to Daily (or Any Other Period)

Rates are quoted annually, but compounding often happens more frequently. To use the formula with any frequency, divide the annual rate by the number of periods and multiply the years by the same number:

• Monthly: periodic rate = r ÷ 12; periods = 12 × t
• Quarterly: periodic rate = r ÷ 4; periods = 4 × t
• Daily: periodic rate = r ÷ 365; periods = 365 × t (some institutions use a 360-day year, which slightly raises the daily rate)

So a 6% annual rate compounded monthly becomes 0.5% per month, applied 12 times a year.

Continuous Compounding

If compounding happened at every instant, the formula becomes:

A = Pe^(rt)

where e is the mathematical constant (about 2.71828). Continuous compounding is the theoretical upper limit. In practice, it gives almost the same result as daily compounding.

Adding Regular Deposits

Most savers don’t make one deposit and walk away; they add money every month. The future value of a series of equal deposits at the end of each period is:

FV = PMT × [(1 + i)^N − 1] ÷ i

where PMT is the deposit, i is the periodic rate (r/n), and N is the total number of deposits. To get the total balance, add this to the compounded value of any starting principal. This is the formula behind most “compound interest with monthly contributions” calculators.

Interpreting Compound Interest

The ending balance on its own hides what drove it. Three questions help read a compound interest result:

1. How much came from deposits, and how much from interest? Subtract total contributions from the ending balance.
2. What is the effective annual rate? Convert the stated rate to an annual equivalent so different compounding schedules can be compared (see APR vs. APY below).
3. How sensitive is the result to time? Because time sits in the exponent, adding years changes the result far more than adding a few tenths of a percent of rate over a short horizon.

The Effect of Compounding Frequency

The more often interest is credited, the sooner each slice of interest starts earning interest. That benefits savers and lenders and costs borrowers.

Hypothetical example: $10,000 at a 10% annual rate for 10 years, no payments or withdrawals

Two patterns stand out. The jump from simple to annual compounding is large: nearly $6,000. Each step after that adds less, and moving from daily to continuous adds only a few dollars. Frequency matters, but the biggest factors are the rate and the length of time.

Common Compounding Schedules

Compounding frequency depends on the product and its terms. Typical arrangements include:

• Savings and money market accounts: frequently compounded daily, though credited monthly or quarterly at many banks.
• Certificates of deposit (CDs): commonly daily or monthly compounding.
• U.S. Series I savings bonds: interest compounds semiannually.
• Credit cards: interest is often calculated on a daily basis, which compounds unpaid interest quickly.
• Loans: many compound monthly. On student loans, unpaid interest added to the principal is called interest capitalization.

Accrual vs. Crediting

Interest can accrue (be calculated) daily but only be credited (added to the balance) monthly. Under U.S. Regulation DD, which implements the Truth in Savings Act, banks must disclose both their compounding and crediting policies. Interest begins earning more interest only after it is credited. If you close an account between crediting dates, the account terms decide whether accrued interest is paid.

Why Is Compound Interest Important?

The Power of Compounding Over Time

Because the exponent in the formula is time, compounding rewards patience more than anything else.

Hypothetical example: one deposit of $5,000 at 6%, compounded annually

• After 20 years: $5,000 × 1.06^20 = $16,035.68
• After 40 years: $5,000 × 1.06^40 = $51,428.59

Doubling the time more than triples the balance. In the second 20 years, the account earns about $35,400, compared with about $11,000 in the first 20, because it is compounding on a much larger base.

Estimating Doubling Time

The Rule of 72 gives a quick estimate of how long money takes to double: divide 72 by the annual rate (as a whole number). At 4%, money doubles in roughly 72 ÷ 4 = 18 years; the exact figure from the formula is about 17.7 years. The shortcut works best for rates in the mid-single digits to low double digits. The Rule of 72 explainer covers its variations and where the estimate drifts.

When Interest Overtakes Your Deposits

For a regular saver, there is a point where cumulative interest exceeds cumulative contributions. Under the assumptions of $200 deposited at the end of each month, 6% annual interest compounded monthly, and no withdrawals, that crossover arrives around year 21. After that, the account has earned more from interest than from new money. At a lower rate the crossover comes later, and at a higher rate it comes sooner.

Example of Compound Interest: Starting Early vs. Saving More Later

This hypothetical compares two savers who put in exactly the same total amount. Both earn 6% a year compounded monthly, and both deposit at the end of each month with no withdrawals.

Saver A deposits $200 a month for 30 years.

• Total deposits: $72,000
• Ending balance: $200 × [(1.005^360 − 1) ÷ 0.005] ≈ $200,903
• Interest earned: about $128,900

Saver B waits 15 years, then deposits $400 a month for 15 years.

• Total deposits: $72,000
• Ending balance: $400 × [(1.005^180 − 1) ÷ 0.005] ≈ $116,327
• Interest earned: about $44,300

Both savers contributed the same $72,000 and finished on the same date. Saver A ends with roughly $84,600 more, purely because the early deposits had an extra 15 years to compound. The same logic applies to tax-advantaged retirement accounts such as IRAs, 401(k) and 403(b) plans, and the federal Thrift Savings Plan, where starting contributions early gives compounding the longest runway.

Compound Interest on Debt

The formula has no preference between saver and borrower. When interest is charged on unpaid interest, debt grows the same way savings do.

Hypothetical example: a $3,000 credit card balance at a 24% APR, compounded daily, with no payments and no fees for one year

• Daily rate: 0.24 ÷ 365 ≈ 0.0658%
• Ending balance: $3,000 × (1 + 0.24/365)^365 ≈ $3,813
• Effective annual cost: about 27.1%, higher than the 24% stated rate

Real card statements typically use an average daily balance method and require minimum payments, so actual figures differ. The mechanism holds, though: if minimum payments barely exceed the interest charged, the principal shrinks slowly and total interest over time can be large. Paying more than the minimum reverses the effect, because every extra dollar removes a dollar that would otherwise have been compounding against you.

Amortizing Loans and Reducing Balances

Not all loan interest compounds in practice. On a standard amortizing loan (most mortgages and auto loans), each scheduled payment first covers the interest due for that period, and the rest reduces principal. Interest is calculated on the diminishing (reducing) balance, but because it is paid off every month, it doesn’t pile onto the balance. Compounding takes hold when payments are missed, deferred, or too small to cover the interest, as with student loans in deferment where unpaid interest capitalizes.

Compound Interest in Investing

Investments don’t pay a fixed interest rate, but they can compound when earnings are reinvested.

• Dividend reinvestment: a dividend reinvestment plan (DRIP) uses each dividend to buy more shares, which then pay their own dividends. Funds tracking a broad index such as the S&P 500 report “total return” figures that assume dividends were reinvested.
• Zero-coupon bonds: these pay no periodic interest. They are bought at a discount to face value and rise toward full value at maturity, and the price path reflects interest compounding inside the bond. A traditional coupon bond pays interest out to the holder, so that interest doesn’t compound unless the investor reinvests it.
• Reinvested interest in general: any interest that is withdrawn and spent stops compounding. Whether returns compound depends on whether they stay invested.

With variable returns, the formula still applies, but r becomes an average. The figure that fits is the compound annual growth rate (CAGR): (Ending value ÷ Starting value)^(1/years) − 1. It describes the smoothed rate that would have produced the same result, not what happened in any single year.

Calculating Compound Interest With Tools

A compound interest calculator automates the formula, but checking a result by hand or in a spreadsheet is straightforward. Using the earlier example of $1,000 at 5% compounded annually for five years:

Method 1: Year-by-year multiplication

1. Put the starting balance (1000) in cell B2.
2. In B3, enter =B2*1.05.
3. Copy that formula down to B7 (five rows of growth).
4. B7 shows $1,276.28; subtract $1,000 for $276.28 of interest.

Method 2: Direct formula

1. Enter principal in B1 (1000), rate in B2 (0.05), and periods in B3 (5).
2. In B4, enter =B1*(1+B2)^B3-B1.
3. The result is $276.28.

Method 3: Built-in future value function In Excel or Google Sheets, =FV(0.05, 5, 0, -1000) returns $1,276.28. The function also handles regular deposits: =FV(0.06/12, 360, -200, 0) returns about $200,903, matching Saver A above. The negative signs mark money paid in.

All three methods should agree. If a calculator’s output differs from your own check, the usual causes are a different compounding frequency, deposit timing (start vs. end of period), or a 360-day vs. 365-day year.

Distinctions Readers Often Confuse

Simple Interest vs. Compound Interest

Simple interest = P × r × t, always on the original principal. Compound interest uses an exponent, so the base grows. Over one period at the same rate, they’re identical; the gap opens only from the second period onward.

APR vs. APY

The annual percentage rate (APR) is the stated yearly rate before compounding. The annual percentage yield (APY) includes compounding:

APY = (1 + r/n)^n − 1

A 5% rate compounded monthly gives an APY of about 5.12%; compounded daily, about 5.13%. The Truth in Savings Act requires deposit accounts to disclose APY, which makes it the figure to use when comparing savings products. On the borrowing side, the Truth in Lending Act requires lenders to disclose loan terms such as the finance charge and APR, and the loan agreement states how interest is calculated.

Compound Interest vs. Cumulative Interest

Cumulative interest is simply the total of all interest paid or earned over a period. Compound interest describes how that interest was calculated. A loan can have large cumulative interest without compounding, as in a long amortizing mortgage.

Nominal vs. Real Growth

A balance that compounds at 4% while prices rise at 3% gains purchasing power at only about 1% a year: (1.04 ÷ 1.03) − 1 ≈ 0.97%. Historical figures on U.S. inflation by year show how much that drag has varied.

Limitations of Compound Interest

• Taxes reduce the effective rate. In the U.S., interest in a taxable account is generally taxed as ordinary income, which leaves less to compound. Tax-advantaged retirement accounts defer or exempt that tax.
• Inflation erodes purchasing power. Compounding helps offset rising costs only if the rate exceeds inflation after taxes.
• Rates change. Savings rates are often variable, so a projection at today’s rate is an estimate, not a promise.
• Investment returns are not guaranteed. Applying a steady rate to stocks produces a smooth curve that real markets don’t follow, and losses compound too: a 50% drop needs a 100% gain to recover.
• Fees compound against you. An annual fee taken as a percentage of assets reduces the base each year, so its cost grows with the balance.
• The formula assumes no withdrawals. Any money taken out stops compounding from that point.

Who Benefits From Compound Interest?

Anyone on the receiving end of interest: savers with deposit accounts and CDs, bondholders, and long-term investors who reinvest. Banks benefit as lenders, earning interest on loans and putting that income back to work. Borrowers who carry balances are on the paying side. For a household, compounding shows up on both sides of net worth: growing assets and growing debts.

FAQs

The Bottom Line

Compound interest matters to savers, investors who reinvest, and anyone carrying a balance on a card or loan. Time drives results more than compounding frequency, so starting deposits early and leaving earnings invested builds the most growth. On the debt side, paying more than the minimum stops unpaid interest from compounding, and comparing APY rather than APR shows the true yield on savings products.