Future Value & Compounding
How money grows when interest earns interest
If you invest $1,000 today at 6% a year, why is it worth more than the $1,600 that simple interest would give 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 (today's 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.
- Identify inputs: \(PV = 1000\), \(r = 0.06\), \(n = 10\).
- Growth factor: \((1.06)^{10} = 1.790847\).
- 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.
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