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Mathematics · Reviewed 2026-06-08

Compound interest formula

A = P(1 + r/n)^(nt) is the standard compound interest formula. It computes the future value (A) of a principal (P) growing at annual rate (r) compounded (n) times per year over (t) years. This guide walks through the formula, every variant (monthly contributions, daily, continuous), and provides worked numerical examples.

By Snowballr Research Team
Last reviewed June 8, 2026Fact-checked against primary sourcesEditorial standards
Coverage: Compound interest · Retirement · FIRE · Debt payoff · Mortgages · Fraud prevention
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The formula
A = P × (1 + r/n)^(n × t)

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

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) = 1.005833^360 = 8.1165
  6. A = $10,000 × 8.1165 = $81,165

Compare to simple interest at the same rate: A = P(1 + r×t) = $10,000 × (1 + 0.07×30) = $31,000. The $50,165 gap is pure compounding — interest earning interest, year after year.

Compound interest formula with monthly contributions

Real investment plans usually add contributions over time. The full formula has two parts: the principal's compounding plus the future value of every contribution (an annuity).

A = P(1 + r/n)^(nt) + PMT × [((1 + r/n)^(nt) − 1) / (r/n)]

PMT = contribution per compounding period
    (e.g., for monthly contributions, PMT = monthly amount)

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

  • Principal portion: $10,000 × 1.005833^360 = $81,165
  • Contributions portion: $500 × [(1.005833^360 − 1) / 0.005833] = $612,000
  • Total: ~$693,165

Run any combination instantly in our free compound interest calculator.

Daily compound interest formula

For high-yield savings accounts (HYSAs), CDs, and money market accounts that compound daily, use n = 365:

A = P × (1 + r/365)^(365 × t)

$10,000 at 5% APY compounded daily for 1 year: A = $10,000 × (1 + 0.05/365)^365 = $10,513. That's $13 more than annual compounding ($10,500) — daily compounding produces a small but real edge.

Use our daily compound interest calculator for any rate and timeframe.

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 continuous: A = $10,000 × e^(0.07×30) = $10,000 × 8.166 = $81,660. Only $495 (0.6%) more than monthly compounding. Continuous compounding is mathematically interesting but practically irrelevant — monthly captures 99%+ of the benefit.

How compounding frequency changes results

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

FrequencynFuture valuevs annual
Annual1$76,123—
Quarterly4$80,232+$4,109
Monthly12$81,165+$5,042
Daily365$81,635+$5,512
Continuous∞$81,660+$5,537

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's the formula. The exponential structure is what makes compound interest grow so much faster than simple interest at long horizons — each period builds on every prior period's interest.

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)  →  ~$693,165

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.

Frequently asked questions

What is the compound interest formula?

A = P(1 + r/n)^(nt) where A is the future amount, P is principal, r is annual rate as a decimal, n is compoundings per year (12 for monthly, 365 for daily), and t is time in years. For $10,000 at 7% compounded monthly over 30 years: A = 10000 × (1 + 0.07/12)^(12×30) = $81,165.

What is the compound interest formula with monthly contributions?

Two-part formula. Principal compounds via A = P(1 + r/n)^(nt). Monthly contributions compound via future-value-of-annuity: FV_PMT = PMT × [((1 + r/n)^(nt) − 1) / (r/n)]. Total: A_total = A_principal + FV_PMT. For $500/month at 7% for 30 years: $500 × [((1.00583)^360 − 1) / 0.00583] = $612,000.

What is the daily compound interest formula?

A = P × (1 + r/365)^(365 × t). At 5% APY on $10,000 for 1 year: $10,000 × (1 + 0.05/365)^365 = $10,513. Daily compounding produces a tiny edge vs annual (0.13% more) but is standard for high-yield savings accounts and CDs.

What is the continuous compound interest formula?

A = P × e^(rt), where e ≈ 2.71828. This is the mathematical limit as compounding frequency approaches infinity. At 7% for 30 years: A = $10,000 × e^(0.07×30) = $10,000 × 8.166 = $81,660. Only marginally higher than monthly compounding ($81,165 — a 0.6% difference).

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?

Use the FV function: =FV(rate/n, n×t, -pmt, -principal). Example for $10K at 7% monthly for 30 years: =FV(0.07/12, 12*30, 0, -10000) returns $81,165. For monthly contributions of $500, change pmt to 500: =FV(0.07/12, 12*30, -500, -10000) returns ~$693,165.

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

Simple interest: A = P × (1 + r × t). Linear growth — interest is only on the principal. Compound interest: A = P × (1 + r/n)^(nt). Exponential growth — interest also earns interest. $10,000 at 7% for 30 years: simple = $31,000; compound monthly = $81,165. The 2.6× gap is pure compounding.

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