Simple Interest Formulas The Compound Interest Equation

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Simple Interest Formulas
The Compound Interest Equation
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
P = C (1 + r/n) nt
Simplified Compound Interest Equation P = C (1 + r) t
When interest is only compounded once per year (n=1), the equation simplifies to:
Continuous Compound Interest
When interest is compounded continually (i.e. n -->
P=Ce
rt
), the compound interest equation takes the form:
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).
(n) Number of Times
Compounded
(P) Final Principal
at the end of one year
1 (yearly)
$ 10600.00
2 (semiannually)
$ 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 repaid. This equation gives the amount B that the person still needs to repay after t
years.
nt
r⎞
⎛
B = A⎜ 1 + ⎟
⎝
n⎠
nt
r⎞
⎛
⎜ 1+ ⎟ − 1
⎝
n⎠
−P
r⎞
⎛
⎜ 1+ ⎟ − 1
⎝
n⎠
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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