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Mathematics · Reviewed 2026-10-11

Compound interest formula

A = P(1 + r/n)^(nt) projects the future value (A) of a starting principal (P) at a nominal annual rate (r), compounded (n) times per year for (t) years. This guide explains each variable, works through examples, and shows how the formula changes for monthly deposits and continuous compounding.

Test your own starting balance, deposits and rate in the compound interest calculator after reviewing the assumptions below.

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The formula
A = P × (1 + r/n)^(n × t)

A = future value (final balance)
P = principal (starting balance)
r = nominal annual rate (decimal, e.g. 0.07 for 7%)
n = compounding periods per year
    annual = 1, monthly = 12, daily = 365
t = number of years

This formula expects a nominal annual rate that is divided by the compounding frequency. An APY is already an effective annual yield: for a lump sum over whole years, you can apply the APY once per year. To represent the same APY with n periods, convert it to a nominal annual rate: r = n × ((1 + APY)^(1/n) − 1), with APY written as a decimal. For deposit accounts, check the account disclosure because the frequency of compounding and the frequency of crediting can differ. See the CFPB APY calculation rules.

For a side-by-side tool comparison, see the best savings calculators guide.

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Worked example: $10,000 at 7% for 30 years

Apply the formula step by step with monthly compounding (n = 12):

  1. P = $10,000, r = 0.07, n = 12, t = 30
  2. r/n = 0.07 / 12 = 0.005833
  3. (1 + r/n) = 1.005833
  4. n × t = 12 × 30 = 360
  5. (1 + r/n)^(nt) is approximately 8.1165
  6. A = $10,000 × (1 + 0.07/12)^(12×30) = $81,165

Compare to simple interest at the same rate: A = P(1 + r×t) = $10,000 × (1 + 0.07×30) = $31,000. The difference is the effect of interest earning interest, year after year.

Compound interest formula with monthly contributions

With recurring deposits, the result depends on deposit timing and how the annual rate is converted to a monthly factor. The equation below describes Snowballr's projection convention: deposits arrive at each month-end and grow at an equivalent monthly rate. A bank's actual crediting schedule can differ; check its account disclosure for a product-specific result.

q = (1 + r/n)^(n/12) − 1

A = P(1 + q)^(12 × t) + PMT × [((1 + q)^(12 × t) − 1) / q]

PMT = contribution at each month-end
q = equivalent monthly growth factor in this model
At q = 0, A = P + PMT × 12 × t

Worked example: $10,000 starting + $500/month at 7% for 30 years (monthly compounding):

  • Principal portion: $81,165
  • Contributions: $500 at each month-end for 360 months
  • Total: $691,150

Continue with the monthly compound interest guide for the rate and deposit-timing convention, or compare six transparent scenarios in the compound interest case studies.

Try your own assumptions below, or open the full compound interest calculator.

Try the formula with your numbers

The starting values match the worked example: $10,000 initially, $500 at each month-end, a fixed 7% nominal annual rate, monthly compounding, and 30 years. The projection excludes taxes, fees, and inflation. Change the inputs to compare scenarios.

Daily compound interest formula

For a projection that assumes daily compounding, set n = 365:

A = P × (1 + r/365)^(365 × t)

At a nominal 5% rate compounded daily, $10,000 for one year becomes $10,513. Product terms vary: verify the disclosed interest rate, APY, compounding frequency, and crediting frequency before using an account-specific projection.

Use our daily compound interest calculator for any rate and timeframe. Investor.gov's compound interest calculator also lets you compare annual, monthly, and daily compounding frequencies.

Continuous compound interest formula

As compounding frequency approaches infinity (continuous compounding), the formula becomes:

A = P × e^(r × t)

e = Euler's number ≈ 2.71828

$10,000 at 7% for 30 years with continuous compounding is $81,662. This is a mathematical limit, not a typical deposit-crediting schedule.

How compounding frequency changes results

$10,000 at 7% for 30 years under different compounding frequencies:

FrequencynFuture valuevs annual
Annual1$76,123—
Quarterly4$80,192+$4,069
Monthly12$81,165+$5,042
Daily365$81,645+$5,523
Continuous (limit)∞$81,662+$5,539

The jump from annual to monthly captures 91% of the maximum possible benefit. Beyond monthly, the gains are tiny — don't optimize for compounding frequency; optimize for rate, fees, and time horizon.

How to derive the compound interest formula

Start with one compounding period. If you have P dollars and earn rate i per period:

  • After 1 period: P × (1 + i)
  • After 2 periods: [P × (1 + i)] × (1 + i) = P × (1 + i)^2
  • After k periods: P × (1 + i)^k

With n compoundings per year for t years: total periods k = n×t, and per-period rate i = r/n. Substituting:

A = P × (1 + r/n)^(n × t)

That equation is for a lump sum. For monthly deposits, this calculator converts the selected fixed nominal annual rate to an equivalent monthly growth factor, then adds each contribution at month-end. This is a projection convention; an account that credits interest only on a separate schedule can produce a different result. The exponential structure is what makes compound interest grow faster than simple interest at long horizons.

Compound interest formula in Excel

Excel's built-in FV() function applies the formula automatically:

=FV(rate/n, n*t, -PMT, -P)

Example ($10K at 7% monthly for 30 yrs, no contributions):
=FV(0.07/12, 12*30, 0, -10000)  →  $81,165

With $500/month contributions:
=FV(0.07/12, 12*30, -500, -10000)  →  $691,150

The negative signs on PMT and P represent money you're putting in. Excel returns the FV as a positive number representing what comes back out.

Sources for formulas and account terms

Frequently asked questions

What is the formula for compound interest?

A = P(1 + r/n)^(nt), where A is the future amount, P is principal, r is the nominal annual rate as a decimal, n is the number of compounding periods per year and t is time in years. For $10,000 at 7% compounded monthly over 30 years, the result is $81,165.

What is the compound interest formula with monthly contributions?

For Snowballr's projection convention, deposits are made at each month-end and grow at equivalent monthly factor q = (1 + r/n)^(n/12) − 1. Then future value is P(1 + q)^(12t) + PMT × [((1 + q)^(12t) − 1) / q]; when q = 0, use P + 12t × PMT. This convention may differ from a bank's actual interest-crediting schedule. For $10,000 plus $500 monthly at 7% over 30 years, the shared engine returns $691,150.

What is the daily compound interest formula?

For a nominal 5% rate credited daily, use A = P × (1 + r/365)^(365t). On $10,000 for one year this gives $10,513. If a bank quotes APY rather than a nominal APR, use the effective annual APY directly or convert it before applying a daily formula; do not count compounding twice.

What is the continuous compound interest formula?

A = P × e^(rt), where e ≈ 2.71828. This is the mathematical limit as frequency approaches infinity. At 7% for 30 years, $10,000 becomes $81,662; it is a mathematical comparison, not how most deposit products credit interest.

How do I derive the compound interest formula?

After one period, balance = P × (1 + i) where i = r/n. After two periods: P × (1 + i)^2. After (n×t) periods: P × (1 + i)^(nt) = P × (1 + r/n)^(nt). The exponential grows because each period's interest also earns interest going forward — that's the 'compound' part.

Compound interest formula in Excel?

Excel's FV function is =FV(rate/n, n×t, -PMT, -principal) when contributions match the compounding period. For $10,000 at 7% monthly for 30 years it returns $81,165; with $500 paid at each month-end it returns $691,150. Set type to 0 for end-of-period payments.

What's the difference between simple and compound interest formulas?

Simple interest is A = P × (1 + rt), while compound interest is A = P × (1 + r/n)^(nt). For $10,000 at 7% over 30 years, simple interest gives $31,000 and monthly compounding gives $81,165.

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