Chapter 2 Finance Matters

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1 Chapter 2 Finance Matters Chapter 2 Finance Matters 2.1 Pe r c e n t s 2.2 Simple and Compound Interest 2.3 Credit Cards 2.4 Annuities and Loans Chapter Summary Chapter Review Chapter Test Handling personal finances is a responsibility of an adult in today s society. So that you are better able to handle this responsibility, this chapter introduces you to the mathematics that is essential to understanding different types of financial dealings. You will learn how to work with percentages, determine the amount of interest earned in a savings account or certificate of deposit, calculate the finance charges on a credit card, find the value of an annuity, and compute the amount of a loan payment. If you like to shop, have a credit card, plan to buy a car or house, or save for retirement, this chapter will pique your interest. page 63

2 Research Projects Chapter 2 Finance Matters 1. The Loan Process: Contact a loan company and obtain all the necessary information so that you can explain the loan application process, the required fees, and how the calculations are made. Also, include all of the forms used for the application process. 2. PMI: Obtain information regarding Private Mortgage Insurance (PMI). Under what circumstances will a lender require PMI? How much does PMI cost? How can you eliminate PMI from your mortgage? 3. Escrow: Obtain information about the purpose and use of escrow accounts. Find out how, when, and why escrow accounts are used. Discuss the use of escrow accounts both during the loan application process and during the lifetime of the loan. Math Projects 1. Car of My Dreams: Use the Internet to find the cost of the car of your dreams. Include the cost of all the options you desire. Do further research to find the terms and rates for financing such a car. Perform the calculations to determine the monthly payments to purchase the car. Make sure you include sales tax and license fees, if these apply in your state, and add 3% to the cost for dealer preparation. How much do you actually pay for the car? How much interest do you pay? 2. Amortization: An amortization schedule is a table listing the amount of principal and interest that are included in each loan payment and the balance of the loan that remains after each payment. Such tables are often included with the annual mortgage statement sent by mortgage companies to their customers. If you have a $1200 loan at 13.2%, compounded quarterly, repaid in 1 year with quarterly payments of $325.15, the amortization table would look as follows. Payment Interest Repaid Principal Repaid Loan Balance $1, $ $39.60 $ $ $ $30.18 $ $ $ $20.44 $ $ $ $10.39 $ $ 0.00 a) Write an amortization schedule for an 18-month car loan of $18,000. The loan has a 9.84% annual interest, compounded monthly. b) Suppose a person decides to add an extra $50 each month. Recalculate the amortization schedule. 3. Credit Card Accounts: Get a copy of two actual billing statements from Master Card or VISA accounts. Check calculation of daily rates, finance charges from purchases, cash advances, and balance transfers, and determination of the account balance with purchases and payments included. Clearly show how all components of the statement make sense. See Section 2.3 for the mathematics to do this. page 64

3 2.1 Percents Section 2.1 Percents Percents are frequently used in business transactions. For example, the sales tax rate might be 7.5%, the interest rate on a 30-year loan might be 9.16%, a department store might be offering a 25% discount on all items, or inflation might be increasing at a rate of 1.3% per year. The ability to work with percents is essential to an investigation of the mathematics of finance. What is a Percent? The word percent means out of one hundred. A score of 93% on a test means that if the test was worth 100 points you scored 93 points. Percents give us a standard way of comparing data. For example, to compare three quizzes in which your scores were 22 out of 25 points, 27 out of 30 points, and 43 out of 50 points, we represent each score as an amount out of outof 25 = = = 088. = 88% outof 30 = = = 090. = 90% outof 50 = = = 086. = 86% Since each score is represented as an amount out of 100, we can see that a score of 27 out of 30 was the highest mark of the three quizzes. The three quiz scores, when written as an amount out of 100, can be represented by a fraction, a decimal, or a percent. Since a percent means out of 100, to change a percent to a decimal, you drop the percent sign and divide by 100. This can be quickly accomplished by moving the decimal point two places to the left and dropping the percent sign. To change a decimal to a percent, use the reverse process. Instead of moving the decimal point two places to the left, move the decimal point two places to the right. Instead of dropping the percent sign, attach the percent sign. To change a fraction to a percent, first change the fraction to a decimal by dividing the denominator into the numerator and then change the decimal to a percent. page 65

4 2.1 Percents Example 1 Change 33.65% to a decimal. Solution: To change 33.65% to a decimal, move the decimal point two places to the left and drop the percent sign. Example 2 Change into a percent rounded off to the nearest tenth of a percent. Solution: To change into a percent, divide 73 by 121, move the decimal point two places to the right, and round off the result to the nearest tenth. Finding the Percent of a Number Percents are often given as a rate that is used to find a portion of a number. A sales tax rate of 7.5% means that the tax is 7.5% of the price. A 25% discount means that the amount of the discount is 25% of the price. When a percent is used as a rate in this context, you must find the percent of a number. A percent of a number is found by multiplying the percent and the number. You must remember to convert the percent to a decimal before performing the multiplication. Example 3 Find 7.5% of 360. Solution: 7.5% is changed to a decimal and multiplied by 360. Example 4 7.5% of 360 = = 27 If the sales tax rate is 8%, how much sales tax is charged on a $99.49 coat and how much is actually paid for the coat? Solution: The sales tax is 8% of the cost of the coat. 8% of $ = $ = $ $. 796 page 66

5 Example Percents Note: Dollar amounts should be rounded off to the nearest cent. The sales tax is added to the cost of the coat to get the actual amount paid. Cost including sales tax = $ $7.96 = $ A television set retails for $ During a sale, a store gives a 25% discount. What is the sale price? Solution: The discount is 25% of the $ retail price. Example 6 Discount = 0.25 $ $ The discount is subtracted from the price of the TV to get the final price. Sale price = $ $ = $ The amount paid for gasoline at a service station was $ If this amount included 28.5% in various taxes, what is the cost of the gasoline without tax? Solution: The $17.99 amount is the sum of the cost of the gasoline and the 28.5% tax. Let x = the cost of the gasoline without the tax. The tax is 28.5% of x or 0.285x. Since the total cost of the gasoline is $17.99, we have the following equation. x x= $ x = x = x = $ Markups and Markdowns Percents are also used as a basis for marking up (increasing) or marking down (decreasing) the price of an item. For example, a retailer might mark up the wholesale cost of an item 40% before it is sold in the store or mark down the retail price of an item 20% during a sale. (Note: The wholesale cost is the price a store or retailer pays for an item.) Example 7 To make a profit, a college bookstore marks up the wholesale cost of books 40% to obtain the retail price for books sold in the bookstore. a) If an algebra book has a wholesale cost of $24.00, what is the retail price? b) If the retail price marked in a biology book is $49.49, what is the wholesale cost of the book? Solution: a) The amount of the markup is obtained by multiplying the percent of the markup by the cost. The amount of the markup is then added to the wholesale cost. page 67

6 The retail price = Cost+ Markup = $ ( ) = $ $. 960 = $ Percents Example 8 b) Let x = the wholesale cost of the book. The retail price = Cost+ Markup $ = x x $ = 14. x $ x = $ = x The retail price of a dress was $ However, when the dress was placed on the sale rack, it was marked down 20%. Then, at a Moonlight Sale, the dress was marked down an additional 30%. a) What was the cost of the dress after the final 30% markdown? b) Is the final sale price equal to a 50% discount on the original price of the dress? Solution: a) The amount of the markdown is obtained by multiplying the percent of the markdown by the price. The amount of the markdown is then subtracted from the price. To find the final sale price in this problem, find the price after a 20% markdown. Then mark down that price by an additional 30%. Price after the 20 % markdown= Price Markdown = $ ($ ) = $ $ = $ Price after the 30% markdown = $ ($ ) = $ $ = $ b) To answer this question, compare the price after a 50% discount and the $43.68 sale price obtained previously. Price after the 50 % markdown= Price Markdown = $ ($ ) = $ $ = $ The sale price after a 50% discount is less than the price after successive 20% and 30% markdowns. page 68

7 2.1 Percents Percent Increase and Decrease Percents are also used as a standard way to describe changes in prices, costs, profits, and other quantities. You might read that the cost of living increased 1.2% or the sales for a company decreased 12.6%. How are such percents determined? They are calculated by taking the amount of the increase or decrease (A), dividing it by the starting or base amount (B), and changing the resulting decimal into a percent by moving the decimal point two places to the right. Example 9 If the price of a dozen eggs increased from $1.25 to $1.40, what was the percent increase in the price of eggs? Solution: The amount of the increase is $1.40 $1.25 = $0.15. The base (starting) amount is $1.25. A Percent increase = B 015. = = 012. = 12% 125. Example 10 In 2006, the sales for a company were $2,475,000; in 2007, the sales dropped to $1,950,000. What was the percent decrease in sales? Solution: The amount of the decrease was $2,475,000 $1,950,000 = $525,000. The base (starting) amount is $2,475,000. A Percent decrease = B 525, 000 = % 2, 475, explain Apply Explore Explain 1. What is a percent? 2. How do you change a decimal to a percent? 3. How do you change a percent to a decimal? 4. How do you change a fraction to a percent? 5. If the sales tax rate in your county is 5.2%, how is the actual amount you must pay for a taxable item determined? 6. If a store is giving a 30% discount, how is the discount price for each item determined? 7. What is the difference between a markup and a markdown? How is each calculated? page 69

8 2.1 Percents 8. In determining the percent increase in sales for a company, what information is needed and how is the percent increase calculated? 9. In determining the percent decrease in enrollment at a college, what information is needed and how is the percent decrease determined? Apply 10. Change the following percents to decimals. a) 9% b) 0.7% c) 17.5% d) 120% 11. Change the following percents to decimals. a) 5% b) 9.125% c) 234% d) 0.03% 12. Change the following decimals to percents. a) 0.02 b) c) d) Change the following decimals to percents. a) 0.35 b) 0.06 c) d) Change the following fractions to exact percents. a) 3/4 b) 7/8 c) 21/32 d) 39/ Change the following fractions to percents accurate to the nearest tenth of a percent. a) 5/7 b) 2/3 c) 16/29 d) 5/ If the sales tax rate is 4.5%, find the amount of tax on items that are priced at a) $34.00 b) $ c) $ If the sales tax rate is 7%, find the amount of tax on items that are priced at a) $34.00 b) $ c) $ A sporting goods store marks up the wholesale price of fishing lures 60%. Find the retail price for a lure that wholesales for a) $1.79 b) $2.99 c) $ A toy store marks up the wholesale price of toys 35%. Find the retail price of a toy that wholesales for a) $20.00 b) $45.60 c) $ A jewelry store marks up the cost of a ring 40%. Find the cost of a ring that retails for a) $79.95 b) $ c) $ A hardware store marks up the cost of paint 25%. Find the cost of a gallon of paint that retails for a) $17.00 b) $25.50 c) $ Find the percent increase in price. a) $5.00 to $5.75 b) $32.50 to $34.58 c) $25,000 to $27, Find the percent decrease in price. a) $9.00 to $7.79 b) $432 to $ c) $4999 to $3000 Explore 24. If the inflation rate is predicted to be 2.3% a year, how much will $ in groceries cost a year later? 25. A compact disc player now sells for $ If the cost for consumer goods decreased 0.3% in the past month, what was the cost of the compact disc player one month ago? 26. If a radio that costs $ is marked up 35% and later the marked-up price is marked down 35%, how much below cost is the radio being sold for? page 70

9 2.1 Percents 27. The manufacturer s selling price on a pair of shoes is $ a) If a shoe outlet marks up that price 40%, what is the retail price of the pair of shoes? b) If a shoe outlet marks down that retail price by 40%, how much of a loss is the shoe outlet taking on the pair of shoes? 28. According to the IRS Tax Rate Schedule for the year 2004, if you are single with taxable income over $70,350 but not over $146,750, your tax is $14,325 plus 28% of the amount over $70,350. If your taxable income is $80,000, how much tax do you owe? 29. According to the IRS Tax Rate Schedule for the year 2004, if you are married and filing a joint return with taxable income over $29,050 but not over $70,350, your tax is $4000 plus 25% of the amount over $29,050. If your taxable income is $65,000, how much tax do you owe? 30. Which stock showed the greater percent increase in selling price: a) A share of Netco stock that sold for $54.50 on Monday and $57.25 on Friday or b) A share of Trapco stock that sold for $20.75 on Monday and $22.25 on Friday? 31. Which stock showed the greater percent decrease in selling price: a) A share of Mitek stock that sold for $ and $29.5 one year later or b) A share of Yotek stock that sold for $8.625 and $6.25 one year later? 32. A bank made $234,567,000 in loans during the second quarter of the year. If 62% of that amount was made for home loans and 85% of the amount of home loans was made for single-family dwellings, how much money was loaned for single-family dwellings? page 71

10 2.2 Simple and Compound Interest Section 2.2 Simple and Compound Interest Interest is the fee charged for the use of money. If we deposit money in a bank, the bank may use the money to provide loans for other customers. In return for the use of the money, the bank will pay a certain percentage of the amount invested. In a similar manner, if we borrow money from a bank, we will be required to pay interest to the bank in return for the privilege of using the money. One way to calculate interest is to use simple interest. Simple interest indicates that the interest is earned only at the end of the specified time and is earned only on the amount deposited. The formula used to calculate simple interest is given by the following. Example 1 If you deposit $1500 in a bank for three years at an annual rate of 9%, find the amount of simple interest you will earn. Solution: Since I = Prt, I = (1500)(0.09)(3) = $ Example 2 If you deposit $1500 in the bank for three years and the bank is paying simple interest of 1.5% each month, find the amount of interest earned. Solution: Because the time and the interest rate are not given in the same units of time, you cannot merely substitute the numbers into the formula as we did in Example 1. Since the rate is per month, change the time of 3 years into months by multiplying by 12. This gives t = 3 12 = 36 months. We now have the interest rate per month and the time in months, so we can substitute P = $1500, r = 0.015, and t = 36 months into I = Prt, giving I = (1500)(0.015)(36) = $810. Example 3 If you earned $500 on a $12,000 investment that earned simple interest for 18 months, what is the annual interest rate? Solution: To find the annual interest rate, first convert the time into years. This gives t = = 1.5 years. Now substituting P = $12,000, I = $500, and t = 1.5 into the formula I = Prt gives the following. page 72

11 2.2 Simple and Compound Interest 500 = ( 12, 000)( 15. ) r 500 = 18, 000r 500 r = = 278. % 18, 000 If interest is left in an account along with the principal, the amount in the account is the total of the principal and the interest. Therefore, the amount (A) is given by the following A= P+ I Substituting I = Prt gives A= P+ Prt ( ) Factoring Pout of the righthand side gives A= P 1+ rt Example 4 If you deposit $1500 in a bank for three years and the bank pays 9% simple interest per year, find the amount in the account after the interest has been added to the account. Solution: Substituting P = $1500, r = 0.09, and t = 3 into A = P(1 + rt), we have A = 1500( )= 1500(1.27) = $1905. Example 5 Suppose a bank pays Bill 6% simple interest each year on the amount in the account for the entire year. He deposits $1000 on January 1. If Bill lets the interest accumulate in the account, how much is in the account after 1 year? If the simple interest for the second year is determined from the amount in the account after the first year s interest is added, how much is in the account after two years? Repeat this process to determine the amount in the account after three years. Solution: Using the formula A = P(1 + rt), with P = $1000, r = 0.06, and t = 1 year, we have A = 1000( ) = $ Thus, during the second year, $1060 is the principal in the account. The amount in the account at the end of the second year is A = 1060( ) = $ Similarly, during the third year, $ is the principal in the account, so the amount at the end of the third year is A = ( ) = $ Compound Interest In Example 5, interest was paid on both the principal and the previously earned interest. This method of earning interest on the interest is called compounded interest. At each step of the calculations in Example 5, we multiplied by an additional factor of (1 + r), giving the formula page 73

12 for compound interest. 2.2 Simple and Compound Interest Now let s try Example 5 again. Example 6 Suppose a bank pays Bill 6% interest, compounded annually. He deposits $1000 on January 1. If he lets the interest accumulate in the account, how much does he have after three years? Solution: Using the formula A = P(1 + r) n, after three years at 6%, compounded annually, we get A = 1000( ) 3 = $ Note that the result of the calculations are the same as in Example 5. More important, we can immediately find the result, without computing the amount from previous years. When using this formula, the interest is added to the account at the end of every period rather than at the end of the year. For example, if a problem states that interest is compounded monthly, the interest is added each month. If the annual interest rate is 6%, the monthly interest rate is 6% 12 = = A period is the time interval between successive additions of interest to an account. For example, if interest is compounded monthly, the number of periods per year is 12. If the interest is compounded daily, the number of periods per year is 365. Example 7 Carol is depositing $1500 into an account earning 9%, compounded monthly. How much money will be in the account after 25 years? Solution: Use the formula A = P(1 + r) n, P = $1500, r = = , and n = = 300. Example 8 A = 1500( ) 300 = 1500( ) = $14, If $1500 is deposited into an account earning 9%, compounded daily, how much money will be in the account after 25 years? Solution: Use the formula A = P(1 + r) n, with r = = and n = = A = 1500( ) 9125 = 1500( ) = $14, page 74

13 2.2 Simple and Compound Interest (Note: If more digits are used in the value for r, A = $14, ) Compare the results of Examples 7 and 8. Notice that by compounding more frequently, the amount of interest earned has increased. Starting with the compound interest formula, we can solve for the amount in the account (A), or the present value (P). The next problem involves solving for P, the present value of the account. Example 9 How much money must be deposited into an account that earns 6%, compounded monthly, so that $20,000 can be withdrawn in seven years? Solution: Use the formula A = P(1 + r) n, with r = = and n = 7 12 = , 000 = P( ) 20, 000 = P( ) P = 20, = $ 13, explain Apply Explore Explain 1. What is meant by principal? 2. How is simple interest is calculated? 3. If you know the simple interest rate, the amount of interest earned, and the principal, how can you determine the length of time of the investment? 4. If the interest is given as an annual rate and time is given in months, what step(s) must you take to determine the simple interest? 5. What information is needed to determine the amount in an account earning compound interest? 6. What information is needed to determine the present value of an account earning compound interest? 7. Why does an account earning compound interest accumulate money faster than an account earning simple interest? 8. When compound interest is computed, why is the annual interest rate divided by the number of periods per year? Apply In Problems 9-16, use the simple interest formula I = Prt and A = P(1 + rt) and the given information to find the indicated value. 9. P = $2000 5% annually t = 4 years Find I. 10. P = $ % monthly t = 3 months Find I. 11. P = $ % monthly t = 6 months Find I. 12. P = $3000 4% annually t = 6 years Find A. 13. P = $2000 5% annually t = 4 years Find A. page 75

14 14. P = $ % daily t = 1 year Find A. 15. A = $3100 4% annually t = 6 years Find P. 16. A = $6000 5% annually t = 4 years Find P. 2.2 Simple and Compound Interest In Problems 17-22, use the compound interest formula A = P(1 + r) n and the given information to determine the value of the specified variable. In all cases, the interest rates are given as annual rates. 17. P = $2000 6% compounded monthly t = 4 years Find A. 18. P = $6000 5% compounded daily t = 3 years Find A. 19. P = $ % compounded daily t = 4 years Find A. 20. A = $6000 6% compounded monthly t = 2 years Find P. 21. A = $7500 6% compounded quarterly t = 5 years Find P. 22. A = $ % compounded quarterly t = 10 years Find P. Explore 23. Dylan deposits $4700 into a savings account earning simple interest at 4.6% annually. He intends to leave the money in the bank for six months. How much money, including both principal and interest, can he withdraw at the end of this time? 24. You win $4700 in a charity drawing and decide to deposit it into a savings account earning simple interest at 5.51% annually. You intend to leave the money in the bank until the account is worth $5000. How long must you wait? 25. Ed purchased 500 shares of stock for $23.63 per share. After holding the stock for 18 months, he sells the stock for $26.37 per share. Assuming that there are no commissions on either the purchase or sale of the stock, what was the annual simple interest rate earned on the investment? 26. Angelina purchased 2000 shares of stock for $87.88 per share. After holding the stock for 18 months, she sells the stock for $93.12 per share. Assuming that there are no commissions on either the purchase or sale of the stock, what was the annual simple interest rate earned on the investment? 27. In 1996, Salvador purchased a 10-year bond with a face value of $25,000. The purchase price was $14, If the bond is redeemed for its face value in 2006, what is the simple interest rate on the bond? 28. Phuong Lan purchased a tax-free bond with a face value of $50,000. The purchase price was $31,250. If the bond is redeemed for its face value in ten years, what is the simple interest rate on the bond? 29. Suppose you have a savings account that earns 5% interest annually and you let the interest accumulate in your account at the end of each year. If the account initially has a balance of $10,000, use the procedure given in Example 6 to find how much is in the account at the end of each of the next three years. 30. Antonio deposits $15,000 into a certificate of deposit that guarantees 6.6% annual interest rate, compounded quarterly. How much will be in the account at the end of five years? 31. The Lee family has decided to invest $12,000 in an account that pays 7.3% interest, page 76

15 compounded daily. What is the value of the account after 12 years? 2.2 Simple and Compound Interest 32. A business has decided to place a certain amount of current profits into a bank account earning 9%, compounded monthly, for a period of five years. At the end of that time, the business will use the money to purchase $15,000 worth of new equipment. How much should the business deposit into the account? 33. A relative has decided to establish a bank account for your newborn daughter that will pay for some of her future college expenses. It is intended that the amount be worth $10,000 eighteen years from now. Assuming that the account will earn 7.6%, compounded quarterly, how much money should be deposited into the account. 34. After extensive negotiations, the Indians of Manhattan Island agree with the banks of Amsterdam to invest $24 at 5% interest, compounded daily. After keeping this investment for 400 years, the Indians decide to cash in the account. What is the account balance? page 77

16 2.3 Credit Cards Section 2.3 Credit Cards Credit cards give consumers a convenient way to make purchases and keep track of the amounts spent in various areas. However, not understanding the language and mathematics of credit cards can lead to frustration and debt. The intent of this section is to clarify credit card terminology and calculations so that intelligent credit card decisions can be made. Annual percentage rate (APR) The yearly interest rate that is used to calculate finance charges on unpaid balances of credit card accounts. Lenders are required by law to disclose the APR used on credit card accounts. Average daily balance The average daily balance is determined by taking the sum of the ending balance each day and dividing that sum by the number of days in a billing cycle. A daily balance is affected by purchases, payments, interest, and fees. Credit limit The credit limit is the maximum amount you may charge on a credit card. This includes purchases, cash advances, balance transfers, finance charges, and fees. Fees Fees are the amounts lenders charge for certain credit card activities. Fee amounts are in addition to finance charges. Listed below are some common credit cards fees. 1. an annual fee just for having the card. 2. a cash-advance fee if you use the card to get a cash. 3. a balance-transfer fee if you transfer a balance from another credit card. 4. a late-payment fee if your payment is received after the date due. 5. an over-the-limit fee if you charge over your credit limit. These fees can either be a specified amount (flat fee) or a percentage of a transfer or advance. Finance charge The dollar amount you pay for using a credit card. Finance charges occur when an account has an unpaid balance. Most credit cards have a minimum finance charge and several methods are used in determining finance charges. We will examine one of the most common methods which uses the APR and average daily balance. Grace period The grace period is the number of days you have to pay your bill in full without accruing finance charges. The standard grace period is from 20 to 30 days from the statement date on a bill. Typically, no grace period is given on cash advances, balance transfers, or accounts that have balances carried over from previous billing periods. Minimum payment The minimum amount a cardholder must pay for a billing cycle. Most credit cards require a minimum payment of 2% of the unpaid balance rounded off to the nearest dollar. The disclosure statement that comes with a credit card will explain how the credit card issuer determines its APR, finance charges, and fees. A section of a typical credit card billing statement is shown. The examples that follow show you how to determine finance charges from such a statement. page 78

17 2.3 Credit Cards Joseph R. Citizen Rate Information Type of Balance APR Daily Rate Average Daily Balance Purchase(s) 10.10% % $ Cash Advance(s) 20.49% % $ Balance Transfer(s) 6.49% % $ Days in Billing Cycle 30 (Note: the daily rate is determined by dividing the APR by 365.) Interest charges are actually added to an account daily and calculated using compound interest. However, lenders give information in this form so that you can compute finance charges on the account using the simple interest formula. Example 1 What is the total finance charges on this credit card account? Solution: Since the daily rate is given for each type of balance and the number of days in the billing cycle is 30, we use the simple interest formula, I = Prt, to calculate interest charges. Remember to convert the daily percentage rates to a decimal by moving the decimal point two places to the left, i.e % = , before doing calculations. Example 2 Purchases: I = ($845.75)( )(30) = $7.02 Cash Advances: I = ($276.09)( )(30) = $4.65 Balance Transfers: I = ($575.50)( )(30) = $3.07 Total Finance Charges = $14.74 What is the overall account APR for the billing period? Solution: The total interest was $ The total daily balance was $ $ $ = $ The time for the billing cycle was 30 days. We can use the simple interest formula with I = $14.74, P = $ , t = 30 and solve for the overall daily rate. I = Prt = ()( r 30) = r r = = r page 79

18 Example 3 To get the overall annual rate we multiply that daily rate by 365. APR = 365( ) = 10.57% 2.3 Credit Cards If the balance on the account in Example 2 before the finance charges was $ and the credit card company requires the standard payment of 2% of the total balance rounded to the nearest dollar, what is the minimum payment on the account? Solution: The total balance on the account is the sum of the $ and the finance charges. Example 4 Total Balance = $ $14.74 = $ The minimum payment is 2% of that balance rounded to the nearest dollar. Payment: ($ )(0.02) = $ $29.00 To encourage on-time credit card payments, a lender may apply a penalty rate if a customer makes two late payments is a six month period. Compare the finance charge on an average daily balance of $2300 for a 31 day billing cycle at the normal APR of 9.9% versus a penalty APR of 22.9%. Solution: We calculate the finance charge by finding the daily rate for each percent and use the simple interest formula, I = Prt. At the 9.9% APR: Daily rate = 9.9% 365 = % = I = ($2300)( )(31) = $19.34 At the 22.9% APR: Daily rate = 22.9% 365 = % = I = ($2300)( )(31) = $44.73 You will notice that the finance charge has more than doubled at the penalty rate. Further, the lender will continue to use the penalty rate in future billing cycles until the account is paid in full. With the information supplied on a credit card statement you will be able to check on the finance charges and minimum payment that was determined by the credit card company. Large unpaid balances, high APR s, fees, late payments, and making only minimum payments could make a credit card very expensive. The best way to keep finance changes at a minimum is to pay off all balances in full within the grace period allowed by your card. (Note: Finance charges are determined using the simple interest formula I = Prt.) page 80

19 2.3 Credit Cards 2.3 explain Apply Explore Explain 1. What is an APR? 2. How are average daily balances computed? 3. What is a grace period? 4. What is a credit limit? 5. What are minimum payments and how are they typically calculated? 6. How is the daily rate determined from an APR? 7. What is the best way to minimize the finance charges on a credit card account? Apply Joseph R. Citizen Rate Information Type of Balance APR Daily Rate Average Daily Balance Purchase(s) 9.9% % $ Cash Advance(s) 18.5% % $ Balance Transfer(s) 4.9% % $ Days in Billing Cycle Verify that the daily rates are correct. 9. Find the total finance charge (for the month shown) of Joseph R. Citizen s credit card account. 10. If his account balance was $ before the finance charges in Problem 9, what is the new balance on the account? If the credit card company requires a 2% minimum payment on the total balance, to the nearest dollar what is the minimum payment on Joseph R. Citizen s account? 11. What is the overall account APR for the billing cycle? Sally R. Citizen Rate Information Type of Balance APR Daily Rate Average Daily Balance Purchase(s) 6.9% % $45.25 Cash Advance(s) 15.9% % $ Balance Transfer(s) 9.9% % $ Days in Billing Cycle Verify that the daily rates are correct. 13. Find the total finance charge of Sally R. Citizen s credit card account. 14. What is the overall account APR for the billing cycle? 15. If her account balance was $ before the finance charges in Problem 13, what is the new balance on the account? If the credit card company requires a 2% minimum page 81

20 2.3 Credit Cards payment on the total balance, to the nearest dollar what is the minimum payment on Sally R. Citizen s account? Explore 16. A credit card has an APR on purchases of 14.9% with a penalty rate of 23.9% if you make two late payments in a six month period. Suppose that in a billing cycle of 30 days you have an average daily balance on purchases of $ a) What is the finance change at the 14.9% APR? b) What is the finance charge at the penalty rate of 23.9%? 17. A credit card has an APR on cash advances of 12.9% with a penalty rate of 22.9% if you make two late payments in a six month period. Suppose that in a billing cycle of 31 days you have an average daily balance on cash advances of $ a) What is the finance change at the 12.9% APR? b) What is the finance charge at the penalty rate of 22.9%? 18. A credit card has a cash-advance fee of 3% of the amount advanced. If you take a $800 cash advance and it accrues finance charges for 20 days of the 30 day billing cycle, how much is the $800 advance costing you if the cash-advance APR is 11.9%? 19. A credit card has a balance-transfer fee of 2% of the amount advanced. If you make a $1500 balance transfer and it accrues finance charges for 25 days of the 31 day billing cycle, how much is the $1500 transfer costing you if the cash-advance APR is 8.9%? page 82

21 2.4 Annuities and Loans Section 2.4 Annuities and Loans Annuities As you were doing the problems in Section 2.2, the situations might have seemed a little beyond your current financial status. For many, the idea of depositing $10,000 is not a realistic situation. Most people are more likely to save a little money at the end of every month or every week rather than to make one large deposit. An account in which money is deposited at the end of each period is called an ordinary annuity. Suppose you deposit $100 into an account at the end of every month for four months and interest is compounded monthly at an annual rate of 12%. How much is in the account at the end of four months? To solve this problem, consider the compound interest earned by each deposit using the formula discussed in the previous section, A = P(1 + r) n The first deposit is in the bank for three months, so its value will be $100( ) 3. The second deposit is in the bank for two months, so its value will be $100( ) 2. The third deposit is in the bank for one month, so its value will be $100( ). The fourth deposit has a value of $100 (since it has not earned any interest). Adding these four terms gives (1.01) + 100(1.01) (1.01) 3 = $ This method certainly solved the problem. However, if these monthly deposits continued for 30 years, this would mean 360 deposits. This process would soon become very tiresome. There is a formula that is used to determine the sum of all the payments in an annuity without doing all the individual computations. We will be using PMT (which stands for payment) as a reminder that an annuity is different from an account with one deposit. For an ordinary annuity, you make deposits into the account every period. Example 1 Suppose you deposit $100 every month for 20 years into an account earning 6% compounded monthly, a) What is the value of the account after 20 years? b) What were your total deposits? c) How much interest was earned? page 83

22 Solution: a) Using PMT = 100, r = = 0.005, and n = = 240 gives the following. 2.4 Annuities and Loans Example 2 b) The total deposits were 240 payments of $100 each, which gives = $24, c) The amount of interest earned is the difference between the amount in the annuity and the amount deposited. Therefore, the interest is $46, $24, = $22, Suppose that when Hector retires, he has an annuity worth $120,000. The annuity was created by using equal monthly payments over a period of ten years. If the account had an interest rate of 8.1%, compounded monthly, how much were the monthly payments? Solution: Using S = 120,000, r = = , and n = = 120 gives the following. Loans Borrowing money from a bank is the most common way of buying a house or a car. In this section, we look at the mathematics of fixed-rate loans, ones in which the interest rates do not change. Although we will not show the derivation, the formula for loans is as follows. Loan Formula L= PMT 1 ( 1+ r) r n where L = amount of the loan PMT = amount of each payment r = periodic interest rate n = the number of payments This formula can be used in several ways. First, if the amount of the payments, the length of time remaining on the loan, and the interest rates are known, the formula can be used to determine the amount of the loan. page 84

23 Example Annuities and Loans Maria has a car loan with monthly payments of $ The loan is for three years and has an annual interest rate of 12%. What is the amount of her car loan? Solution: Use PMT = , r = = 0.01, and n = 3 12 = 36. Another use of the loan formula is to determine the loan payments if you know the amount of money being borrowed. Example 4 Suppose you are buying a used car and take out a loan for $9000 and make payments for four years. If the annual interest rate is 6%, compounded monthly, what are the monthly payments? Solution: We know L = 9000, r = = 0.005, and n = 4 12 = 48 and must determine the payments, PMT. Substituting these values, we get the following.. We can use the loan formula to find out the total amount paid on a loan, the total interest paid on a loan, and the balance of a loan at any point during the life of the loan. Example 5 When buying a $180,000 home in Moscow, Idaho, Theresa made a down payment of $40,000 and took out a loan for the remaining $140,000. The loan has a 30-year term with monthly payments and an annual rate of 6.6%. a) What is the monthly payment? b) What is the total of the payments over the 30 years? c) How much interest will be paid on the loan? d) What is the balance of the loan after ten years? Solution: a) Use L = 140,000, r = = , and n = = 360. page 85

24 2.4 Annuities and Loans b) 360 payments of $ give = $321, c) The interest is the difference between the total payments and the value of the loan. $321, $140,000 = $181, d) To determine the balance of the loan after ten years, we calculate the amount of a loan (L) if payments of $ are made for the remaining 20 years of the loan at the same rate (r = 0.009). The problems that follow will give you practice in working with annuities and loans. 2.4 explain Apply Explore Explain 1. What is an ordinary annuity? 2. What is the difference between an ordinary annuity and a compund interest account? 3. How can you determine the total amount of interest earned by an ordinary annuity? 4. How can you determine the total amount you deposit into an ordinary annuity? 5. In a fixed-rate loan, the payments remain the same each month. Does the amount of interest paid each month remain the same? Explain. 6. In a fixed-rate loan, the payments remain the same each month. Does the amount of principal repaid each month remain the same? Explain. page 86

25 Apply 2.4 Annuities and Loans In problems 7-12, use the annuity formula and the given information to solve the following problems. In all cases, interest rates are given as annual rates. 7. PMT = $200 6% compounded monthly t = 5 years Find S. 8. PMT = $100 4% compounded quarterly t = 2 years Find S. 9. PMT = $150 6% compounded quarterly t = 25 years Find S. 10. S = $35,000 12% compounded monthly t = 15 years Find PMT. 11. S = $200,000 6% compounded monthly t = 20 years Find PMT. 12. S = $435,000 9% compounded quarterly t = 30 years Find PMT. In Problems 13-18, use the loan formula and the given information to find the indicated value. In all cases, interest rates are given as annual rates. 13. PMT = $200 6% compounded monthly t = 5 years Find L. 14. PMT = $100 4% compounded quarterly t = 2 years Find L. 15. PMT = $250 8% compounded quarterly t = 5 years Find L. 16. L = $35,000 12% compounded monthly t = 15 years Find PMT. 17. L = $120,000 8% compounded quarterly t = 20 years Find PMT. 18. L = $235, % compounded monthly t = 30 years Find PMT. Explore 19. Sanjay has decided to invest $500 each quarter into a retirement account that has annual earnings of 9.3%, compounded quarterly. If Sanjay continues his investments for 25 years, how much money will he have in the retirement account? 20. Antonio will be retiring in 15 years. With his children grown and finished with college, he can save money toward his retirement. He decides to deposit $800 per month into a mutual fund that is earning 6.72%, compounded monthly. How much will Antonio have in the account when he retires? page 87

26 2.4 Annuities and Loans 21. George listened when his banker told him to start saving money in an Individual Retirement Account (IRA). Beginning when he was 22, George deposited $100 every month into an account earning 9% compounded monthly. a) How much will be in the account when George retires at age 70? b) How much of this money did George deposit? c) How much of this money is interest? George s brother Skippy decided to spend the first ten years buying himself toys. He reasoned that he could accumulate more money than George if he deposited $200 every month starting at age 32. Skippy also plans to retire at age 70 and to use the same 9% account as George. d) How much money will be in Skippy s account? e) How much of this money did Skippy deposit? f) How much of this money is interest? 22. When Lisa was first hired by the Environmental Protection Agency as a research scientist, she had sufficient income to deposit $600 each quarter into an IRA paying 10% interest, compounded quarterly. The quarterly deposits lasted for 12 years. a) How much was in the account at the end of 12 years? b) Because of her parents nursing home costs, Lisa was not able to continue these deposits. Instead, she deposited the entire IRA account into a 30-year certificate of deposit earning 12% compounded monthly. What was the value of the account when it matured? 23. So You Want to Be a Millionaire: You have decided to make monthly deposits into an account earning 6.3% interest, compounded monthly. If you want to have $1,000,000 in 20 years, how much should you deposit each month? 24. Grandpa has decided to set up a college fund for his newborn grandson. How much should he deposit every month into an account paying 7.5% interest compounded monthly so that the account will be worth $30,000 by the time his grandson is 18? 25. Juanita is making quarterly deposits into an annuity that will be worth $175,000 in 35 years. The annuity earns 8.4% annual interest, compounded quarterly. a) What are the quarterly payments into the annuity? b) What is the value of the annuity after 25 years? 26. Akiko is making monthly deposits into an annuity that will be worth $200,000 in 30 years. The annuity earns 7.8% annual interest, compounded monthly. a) What are the monthly payments into the annuity? b) What is the value of the annuity after 15 years? 27. You have decided to buy a Toyota, using your savings and an $8000 loan. If the loan is at 13.2%, compounded monthly, and has monthly payments for four years, find the a) monthly payment. b) total paid over four years. c) total interest paid. page 88

27 2.4 Annuities and Loans 28. You have decided to buy a Ford Escape, using a $26,000 loan. If the loan is at 9%, compounded monthly, and has monthly payments for five years, find the a) monthly payment. b) total paid over four years. c) total interest paid. 29. Olivia s Visa(t m) card has a balance of $ She plans to pay it off in three years, using equal monthly payments. The interest rate is 20.4%, compounded monthly. Assuming that no additional charges are made to the account, find the following. a) monthly payment b) total paid over three years c) total interest paid 30. Phil and Angelika are refinancing a mortgage. The existing loan is a 30-year mortgage for $175,000 at 9.6%, compounded monthly. They have made payments on the loan for nine years. Find the remaining balance on the loan. page 89

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