Compound Interest

When you borrow money from a bank, you pay interest. Interest is really a fee charged for borrowing the money, it is a percentage charged on the principle amount for a period of a year - usually. If you want to know how much interest you will earn on your investment or if you want to know how much you will pay above the cost of the principal amount on a loan or mortgage, you will need to understand how compound interest works. * Compound interest is paid on the original principal and on the accumulated past interest. .

Conversion periods are:

Semi-Annually = (m = 2)

Annually = m = 1 / P × (1 + r) = (annual compounding)

Quarterly = m = 4 / P (1 + r/4)4 = (quarterly compounding) Monthly = m = 12 / P (1 + r/12)12 = (monthly compounding)

The fundamental formula for compound amount is:

F = P(1 + i)n

F = compound amount

P = original principal

i = interest per conversion period which is equal to nominal rate (j) divided by the conversion period (m) n = total number of conversions period for the whole term; (m * t)

Illustrative Examples:

1.) Find the compound interest on P1,000 at the end of 8 ½ years at 8% compounded quarterly. P = P1,000 j = 8%

m = 4 (quarterly) t = 8 ½ years

i = 8%/3 = 2% = .02 n = 34 interest periods

F = P(1 + i)n

F = 1,000 (1 + .02)34

= 1,000 (1.960676)

= P1,960.68 --> compound amount { By Scientific Calculator: Press 1.02, xy , 34, =, x, 1000, = } P1,960.68 – 1,000 = P960.68 -> compound interest

2.) Accumulate P5,000 for 25 years at 9% compounded monthly. P = P5,000 i = 9/12 = 3/4% = 7.5% = .0075 j = 9% t = 25 years

m = 12 (monthly) n = 25 x 12 = 300

F = 5,000 (1 + .0075)300

= P5000 (1 + .0075)200 (1 + .0075)100

= 5000 (4.456675) (2.111084) { By Scientific Calculator, Press: = P47,042,08 1.0075, xy, 300, =, x, 5000, = }

3.) Accumulate P4000 at 12% compounded semi-annually for 4 years and 10 months. Interest rate per conversion period

i = 12%/2 = 6% = .06

Total number of conversion periods

n = 2(4 10/12) = 9 2/3

In the compound interest formula, we assumed that the time would be an integral number of conversion periods. When n is not an integer, we use the approximate accumulation method.

The total time in the above example is 9 whole periods and 4 months left over.

a.) To obtain an amount of F1, on that date, use F = P (1 + i)n

F1 = 4000 (1 + .06)9

= 4000 (1.68947896)

= P6,757.9158

b.) The simple interest for the remaining 4 months is: use I = P r t / I = F1 r t

I = F1 r t

= 6,757.9158 x .12 x 4/12

= P270.3166

c.) Add F1 and I to obtain the value of the approximate compound amount F.

F = 6,757.9158 + 270.3166

= P7,028.23 --> approximate compound amount

NOTE: The actual amount is:

F = 4000 (1 + .06)9 2/3 = 4000 (1 + .06)9 4000 (1 + .06)1/3 4000 (1 + .06)1/3 = 4000 (1.68947896) (1.01961282) (1.01961282)

= P7,025.60

{ By Scientific Calculator, press: 1.06, xy, 9, a b/c, 2, a b/c, 3, =, x,4000, = }

Present Value of F and Compound Discount

The present value at compound interest of a given sum of money is the principal (P), which, if invested now at the given rate, would amount to (F) after (n) interest periods.

Therefore, to discount an amount F for n conversions periods means to find its present value P on a day which is n periods before F is due.

Present value of P -> P = F/(1 + i)n / F (1 + i)-n

Discount on F -> D = F – P, where

P = principal or present value

F = amount...