Formulas: Time Value of Money

Module: M02

Future Value — Lump Sum

Annual compounding: $F V = P V \times (1 + r)^{N}$

Periodic compounding ( $m$ times/year): $F V = P V \times (1 + \frac{r}{m})^{N \times m}$

Continuous compounding: $F V = P V \times e^{r \times N}$

Annual: $P V = \frac{F V}{( 1 + r ) ^{N}}$

Periodic: $P V = \frac{F V}{( 1 + \frac{r}{m} ) ^{N \times m}}$

Continuous: $P V = \frac{F V}{e ^{r \times N}}$

Future Value: $F V = P M T \times \frac{( 1 + r ) ^{N} - 1}{r}$

Present Value: $P V = P M T \times \frac{1 - ( 1 + r ) ^{- N}}{r}$

$F V_{due} = F V_{ordinary} \times (1 + r)$

$P V_{due} = P V_{ordinary} \times (1 + r)$

Standard (first payment at $t = 1$ ): $P V_{0} = \frac{P M T _{1}}{r}$

Deferred (first payment at year $n$ ): $P V_{0} = \frac{P M T}{r} \times \frac{1}{( 1 + r ) ^{n - 1}}$

$P V_{0} = \frac{P M T _{1}}{r - g}$

where $g$ = constant growth rate, requires $r > g$

From periodic rate: $E A R = (1 + \frac{r}{m})^{m} - 1$

From continuous rate: $E A R = e^{r} - 1$

$r_{cc} = ln (1 + H P R) = ln (\frac{P _{1}}{P _{0}})$