# Compound Interest

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Compound Interest

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### MATH INVS – Compound Interest:

MATH INVS – Compound Interest Arnie Dris

### Definition of Compound Interest:

Definition of Compound Interest If, at stated intervals during the term of an investment, the interest due is added to the principal and thereafter earns interest, the sum by which the original principal has increased by the end of the term of the investment is called compound interest .

### Definition of Compound Amount:

Definition of Compound Amount At the end of the term, the total amount due, which consists of the original principal plus the compound interest, is called the compound amount .

### Definition of Conversion Period:

Definition of Conversion Period We speak of interest being compounded , or payable , or converted into principal. The time between successive conversions of interest into principal is called the conversion period . In a compound interest transaction we must know the conversion period; and the rate at which interest is earned during a conversion period

### Example 1:

Example 1 Thus, if the rate is 6% compounded quarterly, the conversion period is 3 months and interest is earned at the rate 6% per year during each period, or at the rate 1.5% per conversion period.

### Example 2:

Example 2 Find the compound amount after 1 year if PhP 100.00 is invested at the rate 8%, compounded quarterly. Solution: The rate per conversion period is 0.02. Original principal is PhP 100.00. At end of 3 mo. PhP 2.000 interest is due; new principal is PhP 102.000. At end of 6 mo. PhP 2.040 interest is due; new principal is PhP 104.040.

### Example 2 (continued):

Example 2 (continued) The rate per conversion period is 0.02. Original principal is PhP 100.00. At end of 9 mo. PhP 2.081 interest is due; new principal is PhP 106.121. At end of 1 yr. PhP 2.122 interest is due; new principal is PhP 108.243. The compound interest earned in 1 year is PhP 8.243. The rate of increase of principal per year is 8.243/100 = 0.08243, or 8.243%.

### Exercise 1:

Exercise 1 By the method of Example 2 find the compound amount after 1 year if PhP 100 is invested at the rate 6%, payable quarterly. What was the compound amount after 6 months? At what rate per year does principal increase in this case? Find the annual rate of growth of principal under the rate 0.04, converted quarterly.

### The compound interest formula:

The compound interest formula Let the interest rate per conversion period be r , expressed as a decimal. Let P be the original principal and let A be the compound amount to which P accumulates by the end of k conversion periods. Then we shall prove that A = P (1 + r ) k . The method of Example 2, applies in establishing the last equation.

### The compound interest formula (continued):

The compound interest formula (continued) Original principal invested is P . Interest due at end of 1 st period is Pr . New principal at end of 1 st period is P + Pr = P (1 + r ). Interest due at end of 2 nd period is P (1 + r ) r . New principal at end of 2 nd period is P (1 + r ) + P (1 + r ) r = P (1 + r )(1 + r ) = P (1 + r ) 2 .

### The compound interest formula (continued – part 2):

The compound interest formula (continued – part 2) By the end of each period, the principal on hand at the beginning of the period has been multiplied by (1 + r ). Hence, by the end of k periods, the original principal P has been multiplied k successive times by (1 + r ) or by (1 + r ) k .

### Definition of present value:

Definition of present value If money can be invested at the rate r per period, the sums P and A , connected by the equation A = P (1 + r ) k , are equally desirable, because if P is invested now it will grow to the value A by the end of k periods. We shall call P the present value of A , due at the end of k periods.

### Fundamental problems under compound interest:

Fundamental problems under compound interest The accumulation problem – This is the determination of the amount A when we know the principal P , the interest rate, and the time for which P is invested. To accumulate P , means to find the compound amount A resulting from the investment of P .

### Fundamental problems under compound interest (continued):

Fundamental problems under compound interest (continued) The discount problem – This is the determination of the present value P of a known amount A , when we know the interest rate and the date on which A is due. To discount A means to find its present value P . The discount on A is ( A – P ).

### Example 3:

Example 3 Find the compound amount after 9 years and 3 months on a principal P = 3000, if the rate is 6%, compounded quarterly. Solution: The rate per period is r = 0.015; the number of periods is k = 4(9.25) = 37. A = 3000(1.015) 37 = 3000(1.73477663) = 5204.33 The compound interest earned is 5204.33 – 3000 = 2204.33.

### Example 4:

Example 4 Find the present value of PhP 5000, due at the end of 4 years and 6 months, if money earns 4%, converted semi-annually. Solution: Rate per period is r = 0.02; number of periods is k = 2(4.5) = 9. P = 5000(1.02) -9 = 5000(0.83675527) = 4183.78 The discount on A is PhP 5000 – PhP 4183.78 = PhP 816.22.

### Definition of accumulation and discount factors:

Definition of accumulation and discount factors The quantity (1 + r ) is sometimes called the accumulation factor , while the quantity 1/(1 + r ) = (1 + r ) -1 , is called the discount factor . In many books the letter v is used to denote the discount factor, or v = (1 + r ) -1 . Thus, for example, at the rate 7%, v 4 = (1.07) -4 .

### Exercise 2:

Exercise 2 (a) If the rate is i , compounded annually, and if the original principal is P , derive the formula for the compound amount after 10 years. (b) After n years. If PhP 100 had been invested in the year 1800 A.D. at 3% compounded annually, what would be the compound amount now? (a) If the rate is j , compounded m times per year, derive a formula for the compound amount of a principal P after 10 years. (b) After n years. 