Compound Interest Formula
The compound interest formula is A = P(1 + r/n)^(nt). It is the single equation that drives savings accounts, fixed deposits, mutual fund projections and almost every long-term investment calculator on this site. This guide unpacks what every symbol means, walks through worked examples, and shows how to rearrange the formula for the principal, rate or time. If you are new to the concept, start with what compound interest is.
The formula and what each symbol means
A = P (1 + r/n)^(n × t)- A — the future amount (principal + compound interest)
- P — the principal: the starting amount you invest or borrow
- r — the annual interest rate as a decimal (12% = 0.12)
- n — how many times interest is compounded per year (1 yearly, 4 quarterly, 12 monthly)
- t — the time in years
The compound interest earned is simply A − P. Notice that r/n spreads the
annual rate across each period, while n × t counts the total number of compounding periods.
Worked example — yearly compounding
Invest $10,000 at 8% p.a., compounded yearly, for 5 years:
- Convert the rate: r = 8% = 0.08, n = 1.
- Growth factor per year: 1 + 0.08/1 = 1.08.
- Number of periods: n × t = 1 × 5 = 5.
- A = 10,000 × 1.08⁵ = 10,000 × 1.46933 = $14,693.
- Interest earned = 14,693 − 10,000 = $4,693.
Why n matters — the same rate, four ways
The nominal rate stays 8%, but compounding more often raises the effective rate, because each period's interest joins the balance and earns its own interest sooner. Same $10,000, same 8% p.a., same 10 years:
| Compounding | n | Future Value | Effective Annual Rate |
|---|---|---|---|
| Yearly | 1 | $21,589 | 8.00% |
| Half-yearly | 2 | $21,912 | 8.16% |
| Quarterly | 4 | $22,080 | 8.24% |
| Monthly | 12 | $22,196 | 8.30% |
The gap looks small over a decade — about $600 between yearly and monthly on $10,000. Over 30 years on $100,000 it is tens of thousands of dollars. The Lump Sum Calculator lets you switch frequencies and see the effect on your own numbers.
The effective annual rate
Effective Rate = (1 + r/n)^n − 1This is the honest "what did I actually earn per year" figure that banks quote as APY. It is always at least the nominal rate, and the gap grows with both the rate and the compounding frequency. When comparing two products, always compare effective rates, never nominal ones.
Rearranging the formula
Because the formula is one equation, you can solve it for any variable when the others are known:
- Find the principal you need to start with:
P = A / (1 + r/n)^(nt) - Find the time required:
t = ln(A/P) / (n × ln(1 + r/n)) - Find the annual rate:
r = n × ((A/P)^(1/(nt)) − 1)— this is exactly how a CAGR is computed.
The time equation is why the Rule of 72 works: dividing 72 by the annual rate estimates doubling time, because ln(2) ≈ 0.693.
Frequently asked questions
Jump straight into a calculator when you are ready: the multi-phase Timeline Simulator handles compounding plus monthly contributions, the SIP Calculator models regular deposits, and the Inflation Calculator runs the same math backwards on prices.