Notation and Terminology

Notation and Terminology#

Basic Notation and Terminology

$P =$ Principal (i.e., value of initial deposit)
$A =$ Accumulated amount (i.e., sum of the principal and interest)
$r =$ Nominal interest rate
$t =$ Term of investment (in years)
$m =$ Number of conversion periods per year, (a conversion period is the interval of time between successive interest payments)

Annually	Semiannually	Quarterly	Monthly	Weekly	Daily
$m = 1$	$m = 2$	$m = 4$	$m = 12$	$m = 52$	$m = 365$

Simple Interest

Interest is always computed based on the original principal.

Interest Earned

$I = P r t$

Accumulated Amount

$A = P (1 + r t)$

Present Value

The present value of an investment is the principal investment, $P$ , that yields a given accumulated amount, $A$ , in the future.

Discrete Compound Interest

Interest payments are added to the principal at the end of each conversion period and earn interest during future conversion periods.

Accumulated Amount

$A = P {(1 + \frac{r}{m})}^{m t}$

Present Value Formula

$P = A {(1 + \frac{r}{m})}^{- m t}$

Continuous Compound Interest

Continuous compounding of interest is equivalent to a discrete compounding of interest where $m$ , the number of conversion periods per year, goes to infinity.

Accumulated Amount

$A = P e^{r t}$

Present Value Formula

$P = A e^{- r t}$

Effective Rate of Interest

The effective interest rate, $r_{eff}$ , is the simple interest rate that produces the same accumulated amount in 1 year as the nominal rate, $r$ , compounded $m$ times a year.

r_{eff} = {(1 + \frac{r}{m})}^{m} - 1