The Compound Interest Equation
P = C (1 + r/n)^{ nt}
where
P = future value
C = initial deposit
r = interest rate (expressed as a fraction: eg. 0.06)
n = # of times per year interest in compounded
t = number of years invested
Simplified Compound Interest Equation
When interest is only compounded once per yer (n=1), the equation simplifies to:
P = C (1 + r)^{ t}
Continuous Compound Interest
When interest is compounded continually (i.e. n >
), the compound interest equation takes the form:
P = C e ^{rt}
Demonstration of Various Compounding
The following table shows the final principal (P), after t = 1 year and t = 10 years, of an account initally with C = $10000, at 6% interest rate, with the given compounding (n). As is shown, the method of compounding (n) has an effect that is initially small but becomes more significant over time (t).
n  P (t=1)  P (t=10) 
1 (yearly)  $ 10600.00  17908.48 
2 (semianually)  $ 10609.00  18061.11 
4 (quarterly)  $ 10613.64  18140.18 
12 (monthly)  $ 10616.78  18193.97 
52 (weekly)  $ 10618.00  18214.89 
365 (daily)  $ 10618.31  18220.27 
continuous  $ 10618.37  18221.19 
Loan Balance
Situation: A person initially borrows an amount
A and
in return agrees to make
n repayments per year,
each of an amount
P.
While the person is repaying the loan, interest is accumulating
at an annual percentage rate of
r, and this interest
is compounded
n times a year (along with each payment). Therefore, the person
must continue paying these installments of amount
P until
the original amount and any accumulated interest is repayed.
This equation gives the amount
B that the person still
needs to repay after
t years.
B = A (1 + r/n)^{nt}  P

(1 + r/n)^{nt}  1
(1 + r/n)  1

where
B = balance after t years
A = amount borrowed
n = number of payments per year
P = amount paid per payment
r = annual percentage rate (APR)