Compound Interest. Contents. 1 Mathematics of Finance. 2 Compound Interest

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1 Compound Interest Contents 1 Mathematics of Finance 1 2 Compound Interest 1 3 Compound Interest Computations 3 4 The Effective Rate 5 5 Document License CC BY-ND 4.0) License Links Mathematics of Finance At some point in their lives, most people in the developed world must borrow money to pay for a car, a house, an education, or some other expensive item, service, or learning opportunity. Many also have credit cards or similar financial instruments that allow them to instantly borrow money to pay for everyday items. Borrowing money via a loan or credit card is not free the borrower must pay a fee which we call interest. The goal of this lesson is to explain some of the mathematics behind the most common type of interest compound interest), to define the concepts that are used when money is borrowed, and to show how to use the formulas that are associated with compound interest and those concepts. Following lessons will examine common types of loans that use compound interest. People don t only borrow money, they also invest it. An investment is simply a loan to a borrower who must eventually pay the investor the amount borrowed and an extra fee interest). Following lessons will examine some methods of investment that use compound interest. 2 Compound Interest A certificate of deposit CD) is a promissory note that a bank gives to someone who agrees to deposit some money for a predetermined amount of time ending on the maturity date. No money may be withdrawn from a CD before its maturity date unless a penalty is paid). Suppose an investor opens a CD account at a bank. The amount of money deposited into the account is called the principal. The fee paid by the bank to the investor is called the interest. Interest is earned gradually an investor will earn more interest in two months than they will in one month. The amount of money in the account at any given time is called the balance or the compound amount. 1

2 The account has an interest rate r. The interest rate is an annual rate unless stated otherwise. Interest is compounded m times per year. One period is 1/m th of a year. The CD earns r/m interest at the end of each period. The amount of interest earned is r/m) F where F is the balance at the end of the previous period. The interest earned at the end of a period is added to the balance. This new larger balance is the amount that earns interest during the next period. When interest is added to the balance, then that interest begins to earn interest. In effect, the interest has been reinvested into the CD instead of being paid out to the investor) to create a larger principal, which then earns a larger amount of interest during the next period. When interest is reinvested so that it will earn interest, we say that we are compounding interest. The interest rate r is properly called a nominal interest rate. However, we will not use that term in this or following lessons. Example: Suppose and investor deposits $200 into a CD that earns 4% compounded quarterly and matures after one year. What is the balance when the CD matures? The interest rate is 4% so r = This is the annual interest rate since no other time period for the interest rate was specified. Since interest is compounded quarterly and quarterly means 4 times per year every 3 months), we have m = 4. This means the CD will earn r/m = 4%/4 = 1% = 0.01 interest every 3 months. The principal is $200, so the balance after 0 months is $200. Let F represent the balance. After 0 months: F = $200 After 3 months: F = = $202 Note: The CD earned $2 interest during the first quarter. After 6 months: F = = $ Note: The CD earned $2.02 interest during the second quarter. After 9 months: F = = $ Note: The CD earned $2.04 interest during the third quarter. After 1 year: F = = $ Note: The CD earned $2.06 interest during the fourth quarter. We conclude that the balance will be $ when the CD matures. Notice that the amount of interest earned during each quarter increases during the year $2 during the first quarter, and $2.06 during the last quarter). This is caused by the interest being compounded. It is very important to understand that 4% compounded quarterly does not mean that 4% interest is earned at the end of each quarter. Instead, it means that the annual) interest rate is 4%, interest is compounded 4 times per year, and 4%/4 = 1% = 0.01 interest is earned at 2

3 the end of each quarter period). Similarly, if the interest is 6% compounded monthly, then the interest rate is 6%, interest is compounded 12 times per year, and 6%/12 = 0.5% = interest is earned at the end of each period month). 3 Compound Interest Computations The previous example required four computations. While the computations were not difficult, they were repetitive. It would be very tedious to do a problem like the above example if the interest was compounded daily requiring 365 computations), and it is very likely that the final result would wrong. Fortunately, formulas that eliminate the need for multiple computations may be derived using some basic mathematical facts. We will not derive these formulas, we will just use them. The formulas used for compound interest computations are given below: r = the annual interest rate m = the number of interest payments per year n = the total number of interest payments P = the principal F = the compound amount balance) F = P P = 1 + r ) n m F ) 1 + r n m The above formulas are also found in the Finance Formulas document. This document contains all of the finance formulas that will be used in this course. You should use this document when you are working through the homework problems or taking a quiz. The best way to understand the formulas is by working through some examples. Example: If $10, 000 is deposited into a certificate of deposit account that earns 5.2% compounded daily, what is the balance after 15 years? The interest rate is 5.2%, so r = Interest is compounded daily, so m = 365. Interest is earned for 15 years and there are 365 interest payments per year, so n = = The principal initial deposit) is $10, 000, so P = 10, 000. The quantity we wish to find is F. Using the above values and the formula for F, we get the following: F = ) The only task remaining is to compute F using a calculator. First, you need to know a few things about your calculator. The key is used to raise a quantity to a power. The following table shows what appears on the screen of the calculator when various keys are pressed. 3

4 Key Pressed Screen ^ ) / * Use the above information to get the following on the screen of your calculator: /365)^5475 Once the above is on the screen of your calculator, press ENTER to compute the result. The screen should show Since F is a dollar amount, we conclude that F = $21, The following is the work that you should write when you solve this problem: F = ) = $21, Example: What amount should be deposited into a certificate of deposit account that earns 6.8% compounded monthly if the account needs to be worth $10, 000 when it matures in 8 years? The interest rate is 6.8%, so r = Interest is compounded monthly, so m = 12. Interest is earned for 8 years and there are 12 interest payments per year, so n = 8 12 = 96. The balance is $10, 000, so F = 10, 000. The quantity we wish to find is P the principal). Using the above values and the formula for P, we get the following: P = = $ The value of P may be computed by evaluating the following expression on your calculator: ) / /12)^96 4

5 4 The Effective Rate Suppose account A has an interest rate r that is compounded m times per year. Suppose account B has the same principal as A, but has an interest rate r 1 that is compounded annually. If the balances of A and B are equal after one year, then r 1 is called the effective rate of A and is denoted r eff. Note: Actually, A and B will produce the same balance after t years for any t.) The effective rate is the actual or true annual interest rate of account A. It is similar to the annual percentage rate APR) that is used by consumers to compare interest rates, and the effective rate may be used to compare interest rates too. One difference between the effective rate and the APR is that the APR may include one-time fees such as loan origination fees) paid by a borrower, while the effective rate does not take such fees into consideration. The effective rate is a mathematical entity only, while the APR is more of a legal entity. The formulas used to compute the effective rate is given below. It may also be found on the PDF document containing the finance formulas. r = the annual interest rate m = the number of interest payments per year r eff = the effective rate r eff = 1 + r m) m 1 Example: Is it better to take out a loan that charges 4.52% compounded daily or a loan that charges 4.55% compounded quarterly? Since the periods of the two loans are different daily and quarterly), the only way to compare them is to compute their effective rates since an effective rate is compoundly annually. The loan with the smaller effective rate will be the better loan. 4.52% compounded daily r = m = 365 r eff = ) = = 4.62% The value of r eff may be computed by evaluating the following expression on your calculator: /365)^

6 4.55% compounded quarterly r = m = 4 r eff = ) = = 4.63% The value of r eff may be computed by evaluating the following expression on your calculator: /4)^4-1 We now see that the loan which charges 4.52% compounded daily is the better loan for a borrower. 6

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