Time value of money

compound interestpresent valuesimple vs compound interestfuture value

Simple interest, periodic and continuous compounding, and discounting a future amount back to what it is worth today.

Simple Interest

I = P \, r \, t

Interest earned on a principal at a flat rate per period, without compounding.

Compound Interest (Periodic)

A = P \left( 1 + \frac{r}{n} \right)^{n t}

Amount after t periods when interest compounds n times per period at rate r.

Continuous Compounding

A = P \, e^{r t}

Amount after t periods when interest compounds continuously at rate r.

Present Value

\mathit{PV} = \frac{\mathit{FV}}{(1 + r)^{t}}

Today's value of a future amount, discounted at rate r per period.

How they fit together

Simple interest pays only on the original principal; compound interest pays on the interest too, and the gap between them widens without limit as the term lengthens. Increasing the compounding frequency helps, but with sharply diminishing returns — going from annual to monthly matters, monthly to daily barely does, and the limit as the periods become infinitely short is continuous compounding, Pe^(rt).

Present value is the same machinery run backwards, and it is the one that changes decisions: money arriving in ten years is not worth its face value today. The near-universal error is a rate–period mismatch, feeding an annual rate into a calculation counting monthly periods. The rate and the period must always describe the same interval. The rule of 72 is the useful sanity check — 72 divided by the percentage rate gives the doubling time to within a few percent for ordinary rates.