Future Value & Compounding
How money grows when interest earns interest
If you invest $1,000 today at 6% a year, why is it worth far more than $1,600 after ten years?
Why a dollar today beats a dollar tomorrow
A dollar you hold today can be invested to earn a return, so it becomes more than a dollar in the future. That difference is the time value of money. The rate at which money grows per period is the interest rate $r$.
Simple vs. compound growth
Under simple interest you earn a return only on the original principal. Under compound interest you earn a return on the principal and on interest already credited. Almost all of finance uses compounding, where the future value after $n$ periods is:
Reading the formula
$PV$ is the present value (todays sum), $r$ is the interest rate per period written as a decimal, and $n$ is the number of periods. The factor $(1+r)^n$ is the growth factor: it is greater than 1 whenever $r>0$, and it grows faster and faster as $n$ increases.
Because the exponent $n$ sits above the base $(1+r)$, doubling the time more than doubles the growth. This curvature is the whole point of the time value of money.
- 1. Identify inputs: $PV = 1000$, $r = 0.06$, $n = 10$.
- 2. Growth factor: $(1.06)^{10} = 1.790847$.
- 3. Multiply: $FV = 1000 \times 1.790847 = 1790.85$.
💡 Hint
Check your understanding
- The time value of money means a sum today is worth more than the same sum later.
- Compound future value is $FV = PV(1+r)^n$; the growth factor $(1+r)^n$ drives everything.
- Because time enters as an exponent, compounding makes growth accelerate.