04 // Interactive Formula Compendium // FINANCEA = P · (1 + r / n)^(n · t)
Compound Interest & Continuous Compounding Formula Sheet
The compound interest formula A = P(1 + r/n)^(nt) computes the future value A of an initial principal P earning a nominal annual rate r compounded n times per year for t years.
Formula Reference
Main Formula
A = P · (1 + r / n)^(n · t)
Rearranged Forms
A_continuous = P · e^(r · t) (Continuous Compounding Limit, n → ∞)
APY = (1 + r / n)ⁿ − 1 (Effective Annual Percentage Yield)
t_double = ln(2) / [ n · ln(1 + r / n) ] ≈ 72 / (100 · r) (Doubling Time)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| A | Future Compound Value (A) | USD |
| P | Initial Principal (P) | USD |
| r | Nominal Annual Interest Rate (decimal) | 1/yr |
| n | Compounding Periods per Year (n) | 1/yr |
| t | Investment Horizon (t) | yr |
Inputs & Parameters
[USD]
[1/yr]
[1/yr]
[yr]
Calculated Result (Future Compound Value (A))
A =20,096.61USD
Formula Notes & Background
- Each compounding period applies a periodic interest rate i = r / n, multiplying the balance by (1 + r/n).
- Over t years there are N = n · t total compounding periods, so P is multiplied by (1 + r/n) exactly nt times: A = P(1 + r/n)^(nt).
- Taking the limit as n → ∞ using lim_{k→∞} (1 + x/k)^k = e^x yields the continuous compounding law A = P e^(rt).
Step-by-Step Examples
#1: Invest $10,000 at 7% APR compounded monthly (n = 12) for 10 years.
P = $10,000r = 0.07n = 12t = 10
A = 10000 × (1 + 0.07 / 12)^(12 × 10) = 10000 × (1.0058333)¹²⁰
→ A = $20,096.61 (Total interest earned = $10,096.61, APY = 7.229%)
#2: Compare the same $10,000 at 7% for 10 years under continuous compounding.
P = $10,000r = 0.07t = 10
A = 10000 × e^(0.07 × 10) = 10000 × e^0.7
→ A = $20,137.53 ($40.92 higher than monthly compounding)
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Frequently Asked Questions
What is the difference between APR and APY?+
APR (r) is the stated nominal annual rate without compounding, whereas APY = (1 + r/n)ⁿ − 1 includes intra-year compounding to show the true effective annual return.
Why does the Rule of 72 use 72 instead of 69.3 (100 × ln 2)?+
While continuous compounding gives ln(2) ≈ 0.6931, discrete annual compounding at typical 6%–10% rates makes 72 both more accurate and easily divisible by 2, 3, 4, 6, 8, 9, and 12.
How do you solve for the required initial principal P to hit a target future value A?+
Multiply the target A by the discount factor: P = A × (1 + r/n)^(−nt).