Interest and Exponential Growth

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, of an account initally with C = $10000, at 6% interest rate, with the given compounding (n). As is shown, the method of compounding has little effect.

n	P
1 (yearly)	$ 10600.00
2 (semi-anually)	$ 10609.00
4 (quarterly)	$ 10613.64
12 (monthly)	$ 10616.78
52 (weekly)	$ 10618.00
365 (daily)	$ 10618.31
continuous	$ 10618.37

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)