Let 
be the principal (initial investment), 
be the annual compounded rate, 
the "nominal rate," 
be the number of times interest
is compounded per year (i.e., the year is divided into 
conversion periods),
and 
be the number of years (the "term"). The interest
rate per conversion period is then
 |
(1)
|
If interest is compounded 
times at an annual rate of 
(where, for example, 10% corresponds to 
), then the effective rate over 
the time (what an investor would earn if he did not redeposit
his interest after each compounding) is
 |
(2)
|
The total amount of holdings 
after a time 
when interest is re-invested is then
 |
(3)
|
Note that even if interest is compounded continuously, the return is still finite since
 |
(4)
|
where e is the base of the natural logarithm.
The time required for a given principal to double (assuming 
conversion period) is given by solving
 |
(5)
|
or
 |
(6)
|
where ln is the natural logarithm. This function can be approximated by the so-called rule of 72:
 |
(7)
|
