Interest & 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 is compounded

t = number of years invested

Simplified Compound Interest Equation

When interest is only compounded once per year (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 initially with C = $10000, at 6% interest rate, with the given compounding (n). As is shown, the method of compounding has little effect.

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 ofr, 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 repaid. 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)

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