Math Tables: Interest and Exponential Growth


The Compound Interest Equation

P = C (1 + r/n) nt
    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 = $10000, at 6% interest rate, with the given compounding (n). As is shown, the method of compounding (n) has an effect that is initially small but becomes more significant over time (t).
nP (t=1)P (t=10)
1 (yearly)$ 10600.0017908.48
2 (semi-anually)$ 10609.0018061.11
4 (quarterly)$ 10613.6418140.18
12 (monthly)$ 10616.7818193.97
52 (weekly)$ 10618.0018214.89
365 (daily)$ 10618.3118220.27
continuous$ 10618.3718221.19

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