Time Value Of Money

Time Value of Money

Core Concepts

Future Value (FV)

The value of a present sum after earning interest for n periods at rate r per period.

$$FV = PV \times (1 + r)^n$$

Future value grows exponentially with time, which is the mathematical basis of compound interest.

Present Value (PV)

The current worth of a future sum, discounted back at rate r for n periods. This is the inverse of future value.

$$PV = \frac{FV}{(1 + r)^n}$$

Present value is the cornerstone of all valuation: a dollar today is worth more than a dollar tomorrow because of the opportunity cost of capital.

Compounding Conventions

Interest can compound at different frequencies. The nominal annual rate r_nom compounded m times per year produces different effective yields.

Discrete compounding (m times per year):

$$FV = PV \times \left(1 + \frac{r_{nom}}{m}\right)^{m \times t}$$

Continuous compounding:

$$FV = PV \times e^{r \times t}$$

Effective Annual Rate (EAR):

$$EAR = \left(1 + \frac{r_{nom}}{m}\right)^m - 1$$

For continuous compounding: EAR = e^(r_nom) - 1