Thursday

December 8, 2016
Posted by **jenny** on Monday, April 30, 2007 at 10:55pm.

Compound Interest

With compound interest, the interest due and paid at the end of the interest compounding period is added to the initial starting principal to form a new principal, and this new principal becomes the amount on which the interest for the next interest period is based. The original principal is said to be compounded, and the difference between the the final total, the compound amount, accumulated at the end of the specified interest periods, and the original amount, is called the compound interest.

In its most basic use, if P is an amount deposited into an account paying a periodic interest, then S is the final compounded amount accumulated where S = P(1+i)^n, i is the periodic interest rate in decimal form = %Int./(100m), n is the number of interest bearing periods, and m is the number of interest paying periods per year. For example, the compound amount and the compound interest on $5000.00 resulting from the accumulation of interest at 6% annual interest compounded monthly for 10 years is as follows: Since m = 12, i = .06/12 = .005. Since we are dealing with a total of 10 years with 12 interest periods per year, n = 10 x 12 = 120. From this we get S = $5000(1+.005)^120 = $5000(1.8194) = $9097. Consequently, the compound interest realized is $9097 - $5000 = $4097. Of course the compound interest can be calculated directly from the simple expression I = P[(1+i)^n - 1].

The interest associated with annuities, loans, mortgages, savings accounts, IRA's, etc. are all based on the compound interest principle. Some additional applications of the use of compound interest are offered below.

1--What will R dollars (your contribution and your copany matching contribution) deposited in a bank account at the end of each of n periods, and earning interest at I%, compounded n times per year, amount to in N years?

This is called an ordinary annuity, differeing from an annuity due. An ordinary annuity consists of a definite number of deposits made at the ENDS of equal intervals of time. An annuity due consists of a definite number of deposits made at the BEGINNING of equal intervals of time.

For an ordinary annuity over n payment periods, n deposits are made at the end of each period but interest is paid only on (n - 1) of the payments, the last deposit drawing no interest, obviously. In the annuity due, over the same n periods, interest accrues on all n payments and there is no payment made at the end of the nth period.

The formula for determining the accumulation of a series of periodic deposits, made at the end of each period, over a given time span is

S(n) = R[(1 + i)^n - 1]/i

where S(n) = the accumulation over the period of n inter, P = the periodic deposit, n = the number of interest paying periods, and i = the annual interest % divided by 100 divided by the number of interest paying periods per year. This is known as an ordinary annuity.

When an annuity is cumputed on the basis of the payments being made at the beginning of each period, an annuity due, the total accumulation is based on one more period minus the last payment. Thus, the total accumulation becomes

S(n+1) = R[(1 + i)^(n+1) - 1]/i - R = R[{(1 + i)^(n + 1) - 1}/i - 1]

Simple example: $200 deposited annually for 5 years at 12% annual interest compounded annually. Therefore, R = 200, n = 5, and i = .12.

Ordinary Annuity

..................................Deposit.......Interest.......Balance

Beginning of month 1........0................0.................0

End of month..........1.....200...............0...............200

Beg. of month.........2.......0.................0...............200

End of month..........2.....200..............24...............424

Beg. of month.........3.......0.................0................424

End of month..........3.....200............50.88..........674.88

Beg. of month.........4.......0.................0.............674.88

End of month..........4.....200............80.98..........955.86

Beg. of month.........5.......0.................0.............955.86

End of month..........5.....200...........114.70........1270.56

S = R[(1 + i)^n - 1]/i = 200[(1.12)^5 - 1]/.12 = $1270.56

Annuity Due

..................................Deposit.......Interest.......Balance

Beginning of month 1......200..............0................200

End of month..........1.......0...............24................224

Beg. of month.........2.....200..............0..................424

End of month..........2.......0.............50.88...........474.88

Beg. of month.........3.....200..............0...............674.88

End of month..........3.......0.............80.98...........755.86

Beg. of month.........4.....200..............0..............955.86

End of month..........4.......0...........114.70..........1070.56

Beg. of month.........5.....200..............0..............1270.56

End of month..........5.......0...........152.47..........1423.03

S = [R[(1 + i)^(n +1) - 1]/i - R] = 200[(1.12)^6 - 1]/.12 - 200 = $1,423.03