Investment
How To Calculate Compound Interest
Compound interest is interest that earns interest. Once interest is added to your balance, that added amount starts earning too — which is why long horizons grow faster than people expect.
Quick Answer
A = P(1 + r/n)^(nt)
- A
- Final amount (principal + interest)
- P
- Starting principal
- r
- Annual nominal interest rate as a decimal (8% = 0.08)
- n
- Number of times interest compounds per year
- t
- Time in years
Compound interest is calculated by adding 1 to the periodic rate (annual rate divided by compounding periods per year), raising it to the total number of periods, and multiplying by the starting principal. Interest earned is A − P.
What Is Compound Interest?
Compound interest is the interest you earn not just on your original money, but on the interest that money has already produced. Put $1,000 in an account paying 8% a year and after twelve months you have $1,080. Leave it alone and the second year you earn 8% on $1,080 rather than on $1,000 — so you add $86.40 instead of $80. The extra $6.40 looks trivial in year two, but the same mechanic is what turns decades of retirement contributions into most of the final balance.
Simple interest, by contrast, pays only on the principal and never reinvests itself. Over one year the two are identical; over twenty they are dramatically different. This page uses the lump-sum compounding formula, which assumes a single starting deposit and no contributions afterward. Savings plans with regular deposits need a different formula (an annuity) because every contribution has a different amount of time to grow.
Two numbers in the formula do most of the work: the rate per period (r/n) and the total number of periods (nt). Anything that raises the periodic rate or increases how often interest is credited raises the result. This is why a 3.50% account compounding daily can beat a 3.55% account compounding annually, and why comparing accounts by headline rate alone is misleading.
The standard way to compare two offers with different schedules is the effective annual rate, also called APY in consumer banking. It collapses any compounding frequency into the equivalent once-a-year figure. Once both offers are expressed as effective annual rates, the higher number is unambiguously better.
The same formula describes obligations as well as investments. Credit card balances, student loans and mortgages all compound — interest accrues on an unpaid balance that already includes previously assessed interest. The math is identical; only the sign of who benefits changes. That symmetry is why carrying a balance at 22% while holding savings at 4% loses money at an accelerating pace.
One boundary worth knowing up front: the lump-sum formula ignores contributions, taxes, fees and inflation. Real balances grow more slowly once fees are deducted and investment returns are taxed, and purchasing power grows more slowly still after inflation. Use this formula to understand the mechanism and compare offers; use it with those adjustments before trusting it as a forecast.
Formula
A = P (1 + r/n)^(nt)
The general form. Divide the annual rate by compounding periods per year, add 1, raise it to the number of years times periods per year, multiply by principal. Interest earned equals A − P.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P | Starting principal | currency | The amount deposited or invested at the start. Assumes no additional contributions. |
| r | Annual nominal interest rate as a decimal | decimal | Divide the quoted percentage by 100. An 8% rate is 0.08 in the formula, not 8. |
| n | Compounding periods per year | count/year | 1 annually, 2 semiannually, 4 quarterly, 12 monthly, 365 daily. |
| t | Time in years | years | Must be in years when r is annual. Convert months by dividing by 12. |
| A | Final amount | currency | Principal plus all accumulated interest. |
A = P · e^(rt)
The limiting case as the compounding frequency grows without bound. It gives the maximum possible result for a given nominal rate, and is used in derivatives pricing and theoretical finance rather than consumer accounts.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| e | Euler's number | constant | Approximately 2.71828. |
How To Calculate Compound Interest
- 1
Convert the percentage rate to a decimal
Divide the quoted annual rate by 100. An 8% rate becomes 0.08. Skipping this step is the single most common error and inflates the result enormously, because the formula treats the rate as a factor rather than a percentage.
- 2
Determine the compounding frequency
Find n from the account terms: 1 for annual, 2 semiannual, 4 quarterly, 12 monthly, 365 daily. If the terms do not state it, assume monthly for savings products and daily for most cards, but confirm before relying on the number.
- 3
Compute the periodic rate
Divide the annual decimal rate by the number of periods: r ÷ n. With r = 0.08 compounding monthly, the periodic rate is 0.08 ÷ 12 = 0.0066667, or about 0.667% per month.
- 4
Count the total number of periods
Multiply periods per year by the number of years: n × t. Twenty years of monthly compounding is 12 × 20 = 240 periods. This is where the term in years matters — entering months here while r stays annual produces nonsense.
- 5
Apply the growth factor
Add 1 to the periodic rate and raise it to the total period count: (1 + r/n)^(nt). For annual compounding at 8% over 20 years that is (1.08)^20 = 4.660957.
- 6
Multiply by principal, then subtract to isolate interest
A = P × the growth factor gives the final amount. Subtract the original principal to get interest earned. Any difference between two accounts shows up entirely inside the growth factor.
Examples
Example 1: Annual compounding — $10,000 at 8% for 20 years
- P
- $10,000
- r
- 8%
- n
- 1 (annual)
- t
- 20 years
| Step | Calculation | Result |
|---|---|---|
| Convert the rate | 8% ÷ 100 | 0.08 |
| Periodic rate | 0.08 ÷ 1 | 0.08 |
| Total periods | 1 × 20 | 20 |
| Growth factor | (1 + 0.08)^20 | 4.660957 |
| Final amount | $10,000 × 4.660957 | $46,609.57 |
| Interest earned | $46,609.57 − $10,000 | $36,609.57 |
Result: $46,609.57 (interest earned $36,609.57)
Example 2: Monthly compounding — same $10,000 at 8% for 20 years
- P
- $10,000
- r
- 8%
- n
- 12 (monthly)
- t
- 20 years
| Step | Calculation | Result |
|---|---|---|
| Periodic rate | 0.08 ÷ 12 | 0.0066667 |
| Total periods | 12 × 20 | 240 |
| Growth factor | (1 + 0.0066667)^240 | 4.926803 |
| Final amount | $10,000 × 4.926803 | $49,268.03 |
| Interest earned | $49,268.03 − $10,000 | $39,268.03 |
Result: $49,268.03 (interest earned $39,268.03)
Calculator
Final amount
$49,268.03
- Interest earned
- $39,268.03
- Starting principal
- $10,000.00
- Effective annual rate (APY)
- 8.300%
A = P(1 + r/n)^(nt). Excludes contributions, taxes and fees.
Prefer a full-width tool? Open the Compound Interest calculator page.
Common Mistakes
Entering the rate as a percentage instead of a decimal
Using 8 rather than 0.08 multiplies the result by an enormous factor. Always divide by 100 first, and sanity-check: the final amount should never exceed the principal by more than a few multiples at ordinary rates.
Comparing headline rates without matching compounding frequency
A 3.50% account compounding daily outgrows a 3.55% account compounding annually. Convert both to effective annual rate (APY) before comparing — the frequency difference can silently reverse which offer is better.
Mixing time units
When the rate is annual, time must be in years. Using months for t while keeping r annual compounds the error across every period. Divide months by 12, or convert the rate to monthly at the same time — never change only one of them.
Assuming the formula includes regular contributions
This formula models a single lump sum. Monthly deposits each compound for a different duration, so a savings plan needs the annuity formula instead. Using the lump-sum version badly understates what a contributing saver ends up with.
Treating the result as spendable real money
Nominal growth ignores inflation, taxes and fees. A $46,610 balance twenty years from now buys appreciably less than $46,610 today. For planning decisions, deflate the result or use a real (inflation-adjusted) rate.
FAQ
What is the difference between simple and compound interest?
Simple interest pays only on the original principal and produces the same interest amount every period. Compound interest adds each period's interest to the balance so subsequent periods earn on a larger amount. They are identical for a single period and diverge increasingly after that.
How often does interest typically compound?
Savings accounts and certificates of deposit usually compound monthly or daily. Credit cards generally compound daily on the average daily balance. Bonds typically pay semiannually. Check the account agreement, because the frequency is stated there and materially changes the result.
What is the effective annual rate and why does it matter?
The effective annual rate converts any compounding frequency into the equivalent once-a-year rate. It lets you compare two offers that compound differently on a single scale. In consumer banking it is published as APY, while the nominal rate is the APR.
Does this formula work for loans?
The growth mechanism is identical, whether the balance is an investment or a debt. What differs is that loan balances usually decrease with scheduled payments, so amortization — not pure compounding — determines what you actually owe each month.
How do I handle contributions made over time?
Each contribution compounds for a different length of time, so the lump-sum formula does not apply. Use the future value of an annuity for regular equal deposits, or compute each deposit separately with this formula and sum the results.
References
- [1]U.S. Securities and Exchange Commission, Office of Investor Education (Investor.gov), Compound Interest Calculator — https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator
- [2]OpenStax, Principles of Finance (open textbook), 2024 — https://openstax.org/details/books/principles-finance
- [3]Consumer Financial Protection Bureau, Truth in Savings Act — Regulation DD, Annual Percentage Yield calculation — https://www.consumerfinance.gov/rules-policy/regulations/1030/