Compound Interest
What is the Compound Interest?
Compound interest is calculated on the principal plus all interest already earned, not just the original principal — which is what makes it grow faster than simple interest over time. Each compounding period, the balance is multiplied by (1 + r/n), and that new, larger balance is what the next period's interest is calculated on.
How often interest compounds matters: for the same nominal annual rate, compounding quarterly earns more than compounding annually, and compounding monthly earns more still, though the difference shrinks as n gets very large.
What Each Variable Means
Units
| Quantity | Symbol | Unit |
|---|---|---|
| Principal / Final amount | P, A | currency unit (e.g. $) |
| Rate | r | decimal |
| Compounding frequency | n | times per year |
| Time | t | years |
When to Use It
- Modeling how a savings account or investment grows with periodic compounding
- Comparing loan or investment options that compound at different frequencies
- Understanding why compounding more frequently increases returns
Where This Formula Comes From
After one period, the balance is the principal plus one period's interest on it.
P + P(r/n) = P(1 + r/n)The new balance, not the original principal, earns the next period's interest.
P(1 + r/n)(1 + r/n) = P(1 + r/n)²Over t years with n compounding periods per year, that's n×t total periods, each multiplying the balance by the same factor.
A = P(1 + r/n)^(nt)Step-by-Step Example
Problem: Find the value of $2,000 invested at 5% annual interest, compounded quarterly, after 3 years.
Identify P, r, n, and t. The rate is a decimal here.
P = 2000, r = 0.05, n = 4, t = 3Divide the annual rate by the number of periods per year.
r/n = 0.05 / 4 = 0.0125Multiply the compounding frequency by the number of years.
nt = 4 × 3 = 12Compute (1 + r/n) raised to the nt power.
(1.0125)^12 ≈ 1.16075This gives the final amount.
A = 2000 × 1.16075 ≈ 2321.51Interactive Calculator
Solving for Other Variables
P = A / (1 + r/n)^(nt)Solve for the principal needed today to reach a target amount A.Common Mistakes
Mistake: Entering the interest rate as a whole-number percentage instead of a decimal.
Fix: In the formula itself, r must be a decimal — 5% is 0.05, not 5. (The calculator on this page accepts a percentage and converts it for you.)
Mistake: Using the number of years instead of the total number of compounding periods in the exponent.
Fix: The exponent is n×t, the total number of compounding periods — not just t. For quarterly compounding over 3 years, that's 12, not 3.
Practice Questions
Find the value of $1,000 at 6% compounded annually (n = 1) for 5 years.
Hint: With n = 1, A = P(1 + r)^t.
Which grows faster over 10 years: $500 at 4% compounded monthly, or $500 at 4% compounded annually?
Hint: Compare (1 + 0.04/12)^120 to (1.04)^10.
Frequently Asked Questions
How is this different from simple interest?
Simple interest is calculated only on the original principal every period. Compound interest is calculated on the principal plus all previously earned interest, so it grows faster over time — see the Simple Interest page for a side-by-side example.
What happens as the compounding frequency n gets very large?
As n approaches continuous compounding, the formula approaches A = Pe^(rt), where e is Euler's number — the theoretical maximum growth for a given nominal rate.
Does a higher n always mean significantly more interest?
It increases the total, but with diminishing returns. The jump from annual to monthly compounding matters more than the jump from monthly to daily.
References
- OpenStax Contemporary Mathematics — Compound Interest