Compound Interest Formula
Calculate compound interest with A = P(1+r/n)^nt.
See how principal, rate, and compounding frequency grow your money over time.
The Formula
Compound interest is interest calculated on both the initial principal and the accumulated interest from previous periods. This is what makes savings grow exponentially over time.
Variables
| Symbol | Meaning |
|---|---|
| A | Final amount (principal + interest) |
| P | Principal (initial investment or loan amount) |
| r | Annual interest rate (as a decimal, e.g. 5% = 0.05) |
| n | Number of times interest is compounded per year |
| t | Time in years |
Compounding Frequencies
| Frequency | n value |
|---|---|
| Annually | 1 |
| Semi-annually | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Daily | 365 |
| Continuously | A = P × e^(r×t) |
Example 1
You invest $10,000 at 6% annual interest, compounded monthly, for 5 years. What is the final amount?
P = $10,000, r = 0.06, n = 12, t = 5
A = 10,000 × (1 + 0.06/12)^(12×5)
A = 10,000 × (1.005)^60
A = 10,000 × 1.34885
A = $13,488.50 (you earned $3,488.50 in interest)
Example 2
How much do you need to invest today at 8% compounded annually to have $50,000 in 10 years?
Rearrange: P = A / (1 + r/n)^(n×t)
P = 50,000 / (1 + 0.08)^10
P = 50,000 / 2.15892
P = $23,159.67 (invest this amount today)
When to Use It
Use the compound interest formula for financial planning:
- Projecting savings account or investment growth over time
- Comparing different compounding frequencies (monthly vs. annually)
- Calculating the total cost of a loan with compound interest
- Planning retirement savings goals
Key Notes
- Formula: A = P(1 + r/n)^(nt): A is the final amount, P is the principal, r is the nominal annual interest rate (decimal), n is compounding periods per year, and t is time in years. More frequent compounding increases the final amount, but with diminishing returns as n increases.
- Compound vs simple interest: Simple interest: A = P(1 + rt) — grows linearly. Compound interest grows exponentially. For a 10% rate over 30 years: simple gives 4× principal; compound (annual) gives 17.4× principal. The gap widens dramatically over time.
- Continuous compounding limit: A = Pe^(rt): As n → ∞, (1 + r/n)^(nt) → e^(rt). For r = 10% and t = 30 years: Pe^(3) ≈ 20.1P vs annual compounding's 17.4P — the difference from continuous vs annual compounding is real but small compared to the rate and time effects.
- Real vs nominal return: The formula gives nominal return in future dollars. Real return adjusts for inflation: r_real ≈ r_nominal − inflation (Fisher equation). A 7% nominal return with 3% inflation yields approximately 4% real purchasing power growth.
- Applications: Compound interest calculations are used in savings and investment projections, loan amortization, bond pricing, population and economic growth models, and any scenario where a quantity grows proportionally to its current size.