COMPOUND INTEREST

Definition of Terms:
Compound Interest – whenever at stipulated intervals during the term of an investment, the interest due is added to the principal and thereafter earns interest, the sum which represents the increase in the original principal at the end of the term is called compound interest.
Compound Amount – it is the total amount due which is the sum of the original principal and the compound interest.
Conversion Period – it is the time between successive conversions of interest into principal. Conversion periods are the following: m = 12 if conversion period is monthly m = 2 if conversion period is semi-annually m = 4 if conversion period is quarterly m = 1 if conversion period is annually

Compound Interest Formula: F = P ( 1 + I )n
Where:

F = compound amount or Future Value or Final Amount P = original principal or Present Value i = interest rate per conversion period which is equal to the nominal rate (j) divided by the conversion period (m) n = total number of conversion periods for the whole term; number of conversion periods per year (m) multiplied by the time or term (t).

EXAMPLE: 1. Determine the compound amount and interest if P5,000 is invested at 10% compounded quarterly for 4 years.

Solution:

Interest rate per conversion period: Total number of conversion periods:

i = j/m = 10%/4 = 2.5% or .025 n =mt = 4(4) = 16

Compound Amount: Compound Interest:
F = P ( 1 + i )n = P5,000 ( 1.48451 ) I = F - P = P5,000 ( 1 + .025 )16 = P7,422.55 = P7,422.55 – P5,000 = P2,422.25 PRESENT VALUE OF F AND COMPOUND DISCOUNT

To discount an amount F for n conversion periods means to find its present value P on a day which n periods before F is due. From F = P ( 1 + i )n , the present value P is obtained by: P = F ( 1 + i ) -n or P = F / ( 1 + i )n

The discount on F can be solved by: D = F – P

EXAMPLES:

1. To provide for the purchase of a gift worth P150,000 to his son when he graduates from college 4 years from now, how much should Papa M invest today at 10% compounded semi-annually?

Solution:

P = F ( 1 + i ) –n or P = F / ( 1 + i )n = P150,000 ( 1 + .05 ) -8 = P150,000 (.67684 ) = P101,525.90

2. If money is worth 7.5% compounded monthly, find the discount if P100,000 is discounted for 1 year.

Solution:

Interest rate per conversion period: i = 7.5% / 12 = .625% or .00625

Total Conversion Periods: n = (12)(1) = 12

Present Value P: P = F ( 1 + i ) n = P100,000 ( 1 + .00625 ) -12 =P92,796.06

NOMINAL AND EFFECTIVE RATES

The rate which is used to describe a variety of compound interest is called the nominal rate j. With a given nominal rate j compounded m times per year, the corresponding effective rate w is the rate which, if compounded annually, is equivalent to j.

Effective rate corresponding to a nominal rate is obtained by:

w = ( 1 + j/m )m – 1 or w = ( 1 + i )m – 1

To find the nominal rate corresponding to an effective rate, use:

J = m [ ( 1 + w)1/m

EXAMPLES:

1. Determine the effective rate corresponding to the rate 9% compounded (a) Annually (b) Monthly
Solution:
a) w = ( 1 + j/m )m – 1 b) w = ( 1 + 9 / 12 ) 12 - 1 = ( 1 + .09/1 )1 – 1 = ( 1 + .0075 )12 - 1 = 1 + .09 – 1 = .09 or 9% = ( 1.09380690 ) – 1 = .0938 or 9.38% 2. What nominal rate compounded quarterly will yield the effective rate 8%?
Solution:
J = m [ ( 1 + w )1/m – 1] = 4 [ ( 1 + 8% )1/4 – 1] = 4 [ 1.01942655 ) – 1 ] = 4 ( .01942655) = .077706 or 7.77%

3. Which rate of interest gives the better annual yield, 5% compounded monthly or 5.5% compounded semi-annually.
Solution:
The two nominal rates can be compared by computing their corresponding effective rates. w of ( 5%. M = 12) w of ( 5.5%, m = 2 ) w = ( 1 + j/m ) m – 1 w = ( 1 + j/m )m - 1 = ( 1 + 5/12 )12 – 1 = ( 1 + 5.5/2 )2 - 1 = ( 1.05116190 ) – 1 = ( 1 + 2.75 ) 2 - 1 = .0511619 or 5.12% = ( 1.05575625 ) – 1 = .05575625 or 5.58% Since 5.58% is greater than 5.12%, therefore, (5.5%, m = 2 ) gives more annual yield than ( 5%, m = 12 )

PROBLEM SET # 3: COMPOUND INTEREST

Solve the following problems:

1. Find the compound amount and the compound interest of:

a. P5,000 for 7 years at (7%. m=4)

b. P4,800 for 4 years at (18%, m=12)

c. P8,800 for 2 yeas at (14%, m=2)

2. Find the present value of P25,500 due in 7 years if money is worth 12% compounded annually.

3. What principal is needed to accumulate to P12,600 in 2 years at 11% compounded quarterly

4. What nominal rate compounded quarterly is equivalent to 16% compounded quarterly?

5. If the nominal rate is 12% converted once each year, find the effective rate.