5.1 Simple and Compound Interest

Transcription

1 5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest? Businesses operate with borrowed money. When a business needs order inventory or expand, it may borrow the money needed for the expansion. The borrower will be charged interest for the opportunity to use the money. The interest on the loan is typically charged at some percentage of the amount borrowed called the interest rate. Not only do businesses borrow, but banks may also borrow money from other banks or even individuals. For instance, you may invest money with a bank and receive interest from the bank. Most consumers borrow money regularly using credit cards. If the balance is not paid off when the lending period is through, you must pay interest to the credit card company for the privilege of borrowing the money. In this section, we ll examine several type of interest. Simple interest is interest where a fixed amount is paid based on the amount borrowed and the length of time the money is borrowed. In compound interest, interest accumulates according to the amount borrowed over time and any interest that has accumulated during that period of time. Both types of interest are used extensively in business and finance. 1

2 Question 1: What is simple interest? In business, individuals or companies often borrow money or assets. The lender charges a fee for the use of the assets. Interest is the fee the lender charges for the use of the money. The amount borrowed is the principal or present value of the loan. Simple interest is interest computed on the original principal only. If the present value PV, in dollars, earns interest at a rate of r for t years, then the interest is I PV rt The future value (also called the accumulated amount or maturity value) is the sum of the principal and the interest. This is the amount the present value grows to after the present value and interest are added. Simple Interest The future value FV at a simple interest rate r per year is FV PV PV rt PV 1rt where PV is the present value that is deposited for t years. The interest rate r is the decimal form of the interest rate written as a percentage. This means an interest rate of 4% per year is equivalent to r In this text, we use the variable names commonly used in finance textbooks. Instead of writing the present value as the single letter P, we use two letters, PV. Be very careful to interpret this as a single variable and not a product of P and V. Similarly, the future value is written FV. This set of letters represents a single quantity, not a product of F and V. This allows us to use groups of letters to represent quantities that suggest their meaning. 2

3 If we know two of the quantities in this formula, we can solve for the other quantity. This formula is also used to calculate simple interest paid on investments or deposits at a bank. In these cases, we think of the deposits or investment as a loan to the bank with the interest paid to the depositor. Example 1 Simple Interest An investment pays simple interest of 4% per year. An investor deposits $500 in this investment and makes no withdrawals for 5 years. a. How much interest does the investment earn over the five-year period? Solution Use I PV rt to compute the interest, I Set PV = 500, r = 0.04, and t = b. What is the future value of the investment in 5 years? Solution The future value is computed using FV PV 1 rt, A Set PV = 500, r = 0.04, and t = c. Find an expression for the future value if the deposit accumulates interest for t years. Assume no withdrawals over the period. Solution In this part, the time t is variable, FV t t 3

4 This relationship corresponds to a linear function of t. The vertical intercept is 500 and the slope is 20. This tells us that the initial investment is $500 and the accumulated amount increases by $20 per year. Figure 1 The linear function describing the accumulated amount in Example 1c. Example 2 Simple Interest A small payday loan company offers a simple interest loan to a customer. They will loan the customer $750. The customer promises to repay the company $808 in two weeks. What is the annual interest rate for this loan? Solution Since there are 52 weeks in a year, the length of this loan is 2 52 years. Use the information in the problem in the simple interest formula, FV PV 1 rt, to solve for the rate r: 4

5 r r r r 2 52 Set FV = 808, PV = 750 and Divide both sides by 750 Subtract 1 or Multiply both sides by t 52 from both sides r 2.01 This decimal corresponds to an interest rate of 201% per year. Because of such high rates, many states are passing legislation to limit the interest rates that pay day loan companies charge. 5

6 Question 2: What is compound interest? In a loan or investment earning compound interest, interest is periodically added to the present value. This additional amount earns interest. In other words, the interest earns interest. Let us illustrate this process with a concrete example. Suppose we deposit $500 in an account that earns interest at a rate of 4% compounded annually. This rate is the nominal or stated rate. By saying that interest is compounding annually, we mean that interest is added to the principal at the end of each year. For instance, we use the simple interest formula, FV PV 1 rt value at the end of the first year, FV 500( ) , to compute the future To find the future value at the end of the second year, we let the present value be the future value from the end of the first year in the simple interest formula, A future value from first year Since the present value in this amount includes the interest from the first year, the interest from the first year is earning interest. This is the effect of compounding. To find the future value at the end of the third year, we let the future value at the end of the second year be the present value in the simple interest formula, 2 6

7 2 A future value from first two years Let us summarize these amounts in a table. 3 End of the Calculation for the Future Value Future Value First Year 500(1.04) 520 Second Year Third Year The middle column establishes a simple pattern. At the end of each year, the future value is equal to the present value times several factors of These factors correspond to the compounding of interest. In general, if interest is compounded annually, then the future value is FV PV 1 r where PV is the principal, r is the nominal rate and t is the time in years. If interests compounds more than once a year, finding the future value is more challenging. It is more likely that interest is compounded quarterly (4 times a year), monthly (12 times a year) or daily (365 times a year). The length of time between which interest is earned is the conversion period. The length of time over which the loan or investment earns interest is the term. To account for compounding over shorter conversion periods, we need more factors in the expression for the future value. However, in each of these factors we only earn a fraction of the interest rate. t 7

8 For instance, suppose deposit $500 in an account earning 4% compounded quarterly. To calculate the future value, we multiply the principal by a factor corresponding to onefourth of the interest rate each quarter. The future value after one quarter is FV 500( ) After two quarters, the future value contains two factors corresponding to one percent interest per quarter, FV Continue this pattern for twelve conversion periods (twelve quarters or three years) gives FV If we compare this expression to the expression for compounding quarterly, A , we note several differences. When we compound quarterly, we get four times as many factors in the future value. This is due to the fact compounding quarterly means we need four times as many factors. When we compound quarterly, each factor utilizes a rate that is one-fourth the rate for compounding annually. 8

9 Compound Interest The future value FV of the present value PV compounded over n conversion periods at an interest rate of i per period is FV PV 1 i n where r nominal rate i, m number of conversion periods in a year and number of conversion periods in a year term in years nmt. r You may also see compound interest computed from the formula A P1 m mt. This is the exact same formula as the one above except the present value is called the principal P and the future value is called the accumulated amount A. Example 3 Compound Interest A customer deposits $5000 in an account that earns 1% annual interest compounded monthly. If the customer makes no further deposits or withdrawals from the account, how much will be in the account in five years? Solution To utilize the compound interest formula, FV PV 1 i, we must find the present value PV, the interest rate per conversion period i, and the number of conversion periods n. The present value or principal n 9

10 is the amount of the original deposit so P The account earns 1% annual interest, compounded monthly. This means the account earns i percent per month over each conversion period. Since the interest is compounded monthly over 5 years, there are n 12 5 or 60 conversion periods during the time this money is deposited. The future value is FV dollars 12 If the future value, interest rate, and number of conversion periods is known, we can solve for the present value in FV PV 1 i n. In problems like this, we want to know what amount should we start with to grow to a known future value. Example 4 Present Value A couple needs $25,000 for a large purchase in five years. How much must be deposited now in an account earning 2% annual interest compounded quarterly to accumulate this amount? Assume no further deposits or withdrawals during this time period. Solution To find the amount needed today, we must find the present value of $25,000. The interest for each conversion period is i 0.02 percent per period 4 The account earns interest over a total of 45 or 20 conversion periods. Substitute these values into the compound interest formula, n FV PV 1 i, and solve for PV: 10

11 25000 PV Substitute FV 25, 000, i 0.005,and n PV Divide both sides by PV We round the present value in the last step to two decimal places. This ensures the value is accurate to the nearest cent. If the couple invests $22, for five years, it will grow to $25,000 at this interest rate. 11

12 Question 3: What is an effective interest rate? The amount of interest compounded depends on several factors. The nominal rate r and the number of conversion periods m both influence the future value over a predetermined time period. A savings account earning a higher nominal rate over fewer conversion periods might have the same future value as another savings account with a lower nominal rate and a higher number of conversion periods. To help us compare nominal interest rates, we use the effective interest rate. The effective interest rate is the simple interest rate that leads to the same future value in one year as the nominal interest rate compounded m times per year. The effective interest rate is m r re 1 1 m where r is the nominal interest rate, and m is the number of conversion periods per year. Another name for the effective interest rate is the annual percentage yield or APR. Example 5 Best Investment An investor has the opportunity to invest in one of two opportunities. The first opportunity is a certificate of deposit (CD) earning 1.140% compounded daily. The second opportunity is an investment yielding a dividend of 1.141% compounded quarterly. Which investment is best? Solution The better investment is the one with the higher effective interest rate. The nominal rate for the CD is r Interest is earned on a daily basis so m 365. This gives an effective rate of 12

13 r e For the other investment, r and m 4. The effective rate for this investment is r e The effective rate for the CD, %, is higher than the effective rate for the investment, %. Because of this, the CD is the better investment. By law, the effective rate of interest is shown in all transactions involving interest charges. The APR is always prevalent in advertisements, such as the one below for five-year CD rates from Bankrate.com on December 29, Institution APR Rate Minimum Deposit Bank of America % 1.19 Compounded monthly $1000 We can also use the APR to compute accumulated amounts. Suppose we want to compute the future value from depositing $1000 in the Bank of America five year CD. We could calculate the future value using the rate, FV Alternatively, we compute the future value using the APR and compound annually, 13

14 FV This gives us another way of computing accumulated amounts. The future value FV compounded at an effective interest rate (APR) of r e is FV PV 1 r e t where PV is the present vvalue or principal, and t is the term in years. Since the APR is always shown in financial transactions, this formula allows us to compute accumulated amounts from the APR. We can also use the compound interest formula to find the rate at which an amount grows. In this case, we think of PV as the original amount and FV as the amount it grows to. Example 6 Growth of Ticket Prices In 2000, the average price of a movies theater ticket was $5.39. In 2010, the average price increased to $7.89. At what annual percentage rate did prices increase over the period from 2000 to 2010 on average? Source: National Association of Theater Owners Solution The original price in 2000 is $ This price grows in ten years to $7.89. Use these values in 1 rate r e : t FV PV r e to find the effective 14

15 r re e 10 To solve for r e, remove the tenth power by raising both sides of the equation to the one-tenth power re Multiply exponents, r e Subtract 1 from both sides r e r e Over this period, the price of tickets increased by an average of 3.88% per year. 15

16 Question 4: What is continuous compound interest? As the frequency of compounding increases, the effective interest rate also increases. We can see this by computing the effective interest rate at a specific nominal rate, say r 0.1. Frequency Number of conversion periods per year m Effective interest rate 0.1 m 1 m 1 annually semiannually quarterly monthly daily hourly every minute 525, As the number of conversion periods per year increases, the effective interest rate gets closer and closer to In fact, it is possible to show that the effective interest rate gets closer and closer to the value e as the frequency of computing increases. If this is done at a nominal rate of r 0.1, the accumulated amount is A P 1r 0.1 t 1 e 1 P P e Pe t e t t Set re e Simplify using the fact that a m n a mn 16

17 In general, as the frequency of compounding increases, the effective interest rate gets r closer and closer to e 1. We can express this symbolically by writing, m r r 1 1 e 1 as m m Think of the symbol as meaning approaches. Larger and larger values of m mean that we are compounding interest more and more frequently. When this happens, we say that the interest is term compounded continuously. The future value FV of the present value PV compounded continuously at a nominal interest rate of r per period is FV PV e rt where t is the time in years. Like the compound interest formula, this formula may also be written in several equivalent forms. In a biological context, the size of a population P with an initial amount of P 0 growing at a continuous rate of r % per year over t years grows according to P P e 0 rt In some business applications, an original amount of money or principal P grows to an accumulated amount A at a continuous rate of r % per year over t years according to A Pe rt In each of these applications, some quantity is growing at a continuous rate r. The original amount of the quantity is multiplied by a factor of quantity at some later time. rt e to yield the amount of the 17

18 Example 7 Continuous Interest Third Federal Savings and Loan offers a CD that earns 1.79% compounded quarterly (on February 3, 2012). If $5000 is invested in the CD, how much more money would be in the account in 5 years if the interest is compounded continuously versus quarterly? Solution The future value with interest compounded quarterly is FV The future value with interest compounded continuously is FV e The future value with continuous interest is greater than the future value with interest compounded quarterly by In general, compounding some amount continuously will always yield a larger amount than compounding the same amount at the same rate a finite number of times per year. The greater number of times the amount is compounded in a year, the closer the future value will be to the future value compounded continuously. 18