Compound Interest and the Time Value of Money Explained

Money today is worth more than the same amount in the future because it can earn interest — the time value of money. Compound interest is the engine behind this: you earn interest not only on your principal but also on previously earned interest. This guide explains the formula with worked examples.

Calculate growth on any investment with the free Compound Interest Calculator.

Simple vs. Compound Interest

Simple interest is earned only on the original principal: I = P·r·t.
Compound interest is earned on principal plus accumulated interest, so it grows faster over time.

The Compound Interest Formula

📐 Formula

FV = P · (1 + r/n)^(n·t) P = principal, r = annual rate (decimal) n = compounding periods per year, t = years FV = future value

Worked Example

Invest $1,000 at 10% annual interest, compounded annually, for 2 years.

📐 Formula

FV = 1000 · (1 + 0.10/1)^(1×2) = 1000 · (1.10)² = 1000 · 1.21 = $1,210 Interest earned = 1210 − 1000 = $210

With simple interest you would earn only 1000 × 0.10 × 2 = $200 — compounding adds the extra $10 (interest on the first year's interest).

Compounding Frequency Matters

The more often interest compounds, the more you earn. For $1,000 at 10% over one year:

📐 Formula

Annually (n=1): 1000 × 1.10 = 1,100.00 Quarterly (n=4): 1000 × 1.025⁴ ≈ 1,103.81 Monthly (n=12): 1000 × (1.00833)¹² ≈ 1,104.71

Effective Annual Rate (EAR)

The EAR expresses the true annual return once compounding is included, making rates with different frequencies comparable:

📐 Formula

EAR = (1 + r/n)ⁿ − 1 At 10% compounded monthly: EAR = (1 + 0.10/12)¹² − 1 ≈ 10.47%

The Rule of 72

A quick shortcut: an investment doubles in roughly 72 ÷ interest rate years. At 10%, money doubles in about 7.2 years.

Common Mistakes

Using the annual rate without dividing by n — the periodic rate is r/n.
Confusing the nominal rate with the EAR — always compare using EAR.
Mismatching n and t units — both must be consistent (per year).

Try It Yourself

Enter your principal, rate, time, and compounding frequency into the Compound Interest Calculator to get the future value, interest earned, EAR, and a period-by-period breakdown. For regular repayments instead, use the Loan / EMI Calculator; to discount future cash flows, the NPV & IRR Calculator.

Key Takeaways

Compound interest earns interest on interest: FV = P(1 + r/n)^(nt).
$1,000 at 10% for 2 years compounded annually grows to $1,210.
More frequent compounding raises the effective annual rate (EAR).
The Rule of 72 estimates doubling time as 72 ÷ rate.