Compound Interest Formula – A = P(1 + r/n)^(nt)

The complete compound interest formula with variable definitions, compounding frequency table, APY derivation, and three fully worked examples.

Formula
Compound interest is computed on the principal plus all previously accumulated interest. The final balance A equals the principal P multiplied by the growth factor (1 + r/n) raised to the total number of compounding periods nt.
Variables
SymbolNameDescriptionUnit
AFinal AmountTotal balance after t years, including all interest$
PPrincipalInitial deposit or loan amount$
rAnnual Rate (decimal)Annual interest rate divided by 100: r = R% / 100—
nCompounding frequencyNumber of times interest is compounded per yearper year
tTimeDuration of the investment or loanyears
CICompound InterestInterest earned: CI = A − P$
How to Use
  1. Convert the annual percentage rate to decimal: r = R / 100.
  2. Determine n (compounding frequency): 1 annually, 12 monthly, 365 daily, etc.
  3. Compute the per-period rate: r / n.
  4. Compute total periods: n × t.
  5. Apply the formula: A = P × (1 + r/n)^(n×t).
  6. Compound interest = A − P.
Examples
1. P = $5,000, R = 8%, t = 5 years, compounded monthly

r = 0.08, n = 12, total periods = 60

ℹSimple interest on the same terms = (5000 × 8 × 5)/100 = $2,000. Compounding adds an extra $449.23.
2. Compare compounding frequencies: P = $10,000, R = 6%, t = 10 years
FrequencynFormula resultFinal Amount
Annually110000 × (1.06)^10$17,908.48
Quarterly410000 × (1.015)^40$18,140.18
Monthly1210000 × (1.005)^120$18,193.97
Daily36510000 × (1.000164)^3650$18,220.40
💡The difference between annual and daily compounding is $311.92 — smaller than most people expect. Rate and time matter far more than compounding frequency.
3. Find the principal needed to reach $20,000 in 8 years at 5% compounded annually

Rearrange A = P(1 + r)^t for P: P = A / (1 + r)^t

ℹYou need to deposit $13,536.76 today at 5% compounded annually to have $20,000 in 8 years. This is called the present value.
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