Terminology. Compound Interest Computing Basic Values. More Terminology. Basic Method of Computing Compound Interest

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1 Terminology Compound Interest Computing Basic Values The Compound interest method is generally used in long-term borrowing. There is usually more than one period for computing interest during the borrowing time. The interest for each period is added (compounded or converted) to the principal before the interest for the next period is computed. The final sum at the end of the period of borrowing is called the compound amount. The process of accumulating a principal to obtain a compound amount is called compound accumulation. Compound interest is the difference between the original principal and the compound amount. More Terminology The period for computing interest, usually at regular stated intervals such as annually, semi-annually, quarterly, or monthly, is called the conversion period or the interest period. The interest rate per conversion period is equal to the stated amount interest rate divided by the number of conversion periods in one year. The stated annual interest rate is called the nominal annual rate, or simply the nominal rate. Thus, if the nominal rate is 12%, the interest rate for the annual conversion period is also 12%; but if the conversion period is semiannual, the interest rate for the period of six months is 12%/2, or 6%. Basic Method of Computing Compound Interest The basic method of computing compound interest for each conversion period is the same as the method of computing simple interest. Thus, if there is only one conversion period, compound interest is the same as simple interest. Example: What are the compound amount and the compound interest at the end of nine months if 10,000 is borrowed at 4% compounded quarterly? Finding the Compound Amount by Formula To find the compound amount by formula, we shall use the following symbols; Let P = original principal i = interest rate per conversion period n = number of conversion periods S = compound amount, or principal at the end of the n th period The compound amount formula is: Example 1: What is the compound amount at the end of nine months if 10,000 is borrowed at 4% compounded quarterly? S=P(1+i) n 1

2 Example 2: Example 3: Find the amount if 1500 invested at 12% compounded quarterly and due at the end of 4.5 years. A note having a face value of 1000 and bearing interest at 16% compounded quarterly will mature in 10.5 years. What is the maturity value? The Interest Rate changes during Compound Accumulation If the interest rate changes during compound accumulation, the compound amount is the product of the principal and the several accumulation factors that are at different interest rates for respective given periods. Example 4 If the principal is 500 and the interest rate is 6% compounded semiannually for the first five years and 8% compounded quarterly for the next six years, what is the compound amount at the end of the 11 th year? The Conversion Periods include a Fractional Part Method A: Use the formula S=P(1+i) n, where n is a whole number representing the total number of whole conversion periods. The computed value, S, is further compounded by the simple interest method for the remaining fractional period. Method B: Use the formula S=P(1+i) n, where n is a mixed number that equals the entire time. Example 5: Find the compound amount and the compound interest when 1000 is invested for three years and two months at 6% compounded semiannually. Finding the Present Value and Compound Discount The difference between the value of the compound amount and its present value is called compound discount. Therefore the compound discount on the compound amount is the same as the compound interest on the present value. Example 1 Find the present value of 2,479.27, due at the end of 4¼ years if money is worth 12% compounded quarterly. 2

3 Compound Discount on Notes Example 2 If 1,000 is due five years from now and money is worth 4% compounded quarterly, find its present value and the compound discount. For interest bearing notes: Step 1: Find the compound amount of the face value of the note according to the rate and the time stipulated on the note. The compound amount is the maturity value. Step 2: Find the present value (on the date of discount) of the maturity value, according to the rate and the time of discount. Example 3: A note of 1000 dated January 1, 1995, at 6% compounded quarterly for ten years, was discounted on January 1, What are the proceeds and the compounded discount if the note was discounted at 8% compounded semiannually? The Discount Time Includes a Fraction of a Conversion Period When the discount time includes a fraction of a conversion period, the proceeds may be computed by either of the following methods: Method A: Use the formula P=S(1+i) -n, where n is a whole number representing the total number of whole conversion periods plus 1. The computed value P is then accumulated by the simple interest method for the extra fractional period included in n to obtain the proceeds. Method B: Use the formula P=S(1+i) n, where n is a mixed number that equals the entire discount time. Example 4: A non-interesting bearing note of 1000 is discounted at 6% compounded semiannually for three years and two months. Find the proceed and the compound discount. Method B is more reasonable than the other because the compound discount method is used throughout. However, Method A is widely employed in practice because its computation is less complicated than that of Method B. 3

4 Example 1: Finding the Interest Rate If 1000 will accumulate to in 17 years, what is the interest rate compounded annually? Example 2: At what nominal interest rate compounded semiannually for ten years will 300 accumulate to 890? Effective Annual Interest Rate The effective annual interest rate is commonly abbreviated as the effective rate. If the principal is 1, the value of the compound interest for a one-year period is the effective rate. In general, the effective rate is the ration of the compound interest earned for a one-year period to the principal as shown below: Example 3 If 1 is invested at 6% compounded quarterly for one year, what is the effective rate? Compound interest for a one - year period Effective Rate= Principal Example 4: Find the effective rate if money is worth 6% compounded quarterly on the investment market. 4

5 Example 5: Bank A offers its depositors an interest rate of 6% compounded monthly, while Bank B gives its depositors an interest rate of 6½% compounded semiannually. Which of the two banks makes the better offer? Finding the number of conversion periods How long will it take 1000 to accumulate to the amount of 1,105 at 24% compounded monthly? 5