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## Chapter 10

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**Chapter 10**Consumer Mathematics START EXIT**Chapter Outline**10.1 Credit Cards 10.2 Mortgages 10.3 Installment Plans 10.4 Leasing Chapter Summary Chapter Exercises**10.1 Credit Cards**• It’s no secret that credit cards can be useful to both businesses and individual consumers. • It’s also no secret that letting things get out of control with credit cards can be disastrous to your financial well-being. • Given these facts, it’s worthwhile to have a solid understanding of the mathematics involved in their use.**10.1 Credit Cards**• Since using a credit card means borrowing money, it also means interest. • One common complaint about credit cards is that their interest rates are often very high compared to the rates on other loans, such as car loans or home loans. • With those sorts of loans, the lender has collateral. This means that if you do not repay the loan as promised, the lender has the right to take the property for which you borrowed the money. • Credit cards, on the other hand, are normally unsecured loans, meaning they are issued without any collateral.**10.1 Credit Cards**• Debit cards present an alternative for those who really don’t want to borrow. • Purchases made with a debit card are paid out of a checking, savings, or a similar account immediately, so you are not borrowing any money.**10.1 Credit Cards**• Another similar type of card is sometimes known as a travel and entertainment card (T&E). • Payments for purchases made with these cards are handled in much the same way as with credit cards. • However, while a credit card allows you flexibility in when you pay the money back, with a T&E card you are normally required to pay off the charges in full each month. Therefore, there is usually no interest charged on these cards.**10.1 Credit Cards**• The calculation of credit card interest poses a bit of a challenge. On the one hand, since statements are produced and payments are due monthly, it makes sense that interest should be computed and charged to the account monthly. • On the other hand, since the balance changes from day to day, it seems that interest should be calculated daily.**10.1 Credit Cards**• The most common method of calculating interest on the credit card is known as the average daily balance (ADB) method. • The interest is computed and added to the account monthly.**10.1 Credit Cards**Example 10.1.1 • Problem • Joanna has a credit card whose billing period begins on the 17th day of each month. On July 17 her balance was $815.49. She made a $250 payment on July 28. She also made new charges of $27.55 on July 21, $129.99 on August 5, and $74.45 on August 8. • Find her average daily balance. • Solution • See table on the next slide. • ADB = $23,253.97/31 = $750.13**10.1 Credit Cards**Example 10.1.1 Cont. • Solution**10.1 Credit Cards**Example 10.1.2 • Problem • Suppose that Joanna’s credit card carries an interest rate of 15.99%. How much interest would she owe for the billing month from Example 10.1.1? • Solution I = PRT I = $750.13 x 15.99% x 31/365 I = $10.19**10.1 Credit Cards**Example 10.1.3 • Problem • What will the total balance be on Joanna’s monthly statement? • Solution $797.48 + $10.19 = $807.67**10.1 Credit Cards**• Most credit cards offer a feature known as a grace period, which adds an interesting wrinkle to the matter of interest. • It’s a period of time, typically 20 to 25 days, beginning on the card’s billing date. If you pay the entire balance within a grace period, and if you paid your previous month’s balance off in full, you pay no interest at all.**10.1 Credit Cards**• Although the main source of profit for credit card issuers is interest, two other sources are annual fees and commissions. • An annual fee is a fee paid by the cardholder simply for having the credit card. • Commissions are not paid by the cardholder but by a merchant who accepts credit card payments. These fees may be a percent of the amount charged, a flat amount per transaction, or a combination of the two.**10.1 Credit Cards**Example 10.1.5 • Problem • Travis bought a pair of shoes for $107.79 and charged them to his credit card. The credit card company charges the shoe store 45 cents for each transaction, plus 1.25% of the amount charged. How much will the credit card company pay to the shoe store? • Solution $107.79 x 1.25% = $1.35 $1.35 + $0.45 = $1.80 $107.79 -- $1.80 = $105.99**10.1 Credit Cards**• The credit card industry is highly competitive, with literally hundreds of different card issuers competing for each potential card holder. • People who just take whatever offer is first presented to them often overlook or miss out on opportunities to pay significantly less for their credit card use. • Ideally, as a consumer, you would want to choose the card that has both the lowest interest rate and the lowest annual fee. • What if, though, the card with the lowest annual fee carries a higher interest rate, while the card with the lowest interest rate has a low annual fee?**10.1 Credit Cards**Example 10.1.6 • Problem • Jerome expects that he will normally carry a credit card balance of around $800. Which of the three options in the table would be the lowest-cost option for him?**10.1 Credit Cards**Example 10.1.6 Cont • Solution • Bank A I = PRT I = $800 x 9% x 1 = $72 Total = $72 + $80 = $152 • Bank B I = $800 x 15% x 1 = $120 Total = $120 + $25 = $145 • Bank C I = $800 x 23.99% = $191.92**Problem 1**• Rudy has a credit card with a billing period that begins on the 10th day of each month. On May 10th, his balance was $1,045. He made a $250 payment on May 20th and charged $100 on May 22nd and $175 on May 24th. Find his average daily balance. CHECK YOUR ANSWER**Solution 1**• On May 10th, his balance was $1,045. He made a $250 payment on May 20th and charged $100 on May 22nd and $175 on May 24th. Find his average daily balance. • ADB = $39,744.32/31 = $1,282.07 BACK TO GAME BOARD**Problem 2**• Suppose that Rudy’s card carries 3.99% interest. How much interest would he owe for the billing month from a previous problem? CHECK YOUR ANSWER**Solution 2**• Suppose that Rudy’s card carries 3.99% interest. How much interest would he owe for the billing month from a previous problem? • I = PRT • I = $1,282.07 x 3.99% x 31/365 • I = $4.34 BACK TO GAME BOARD**Problem 3**• Nicky bought a lamp for $45.99 and charged it to her credit card. The credit card company charges the department store 20 cents for each transaction plus 2% of the amount charged. How much will the credit card company pay to the department store? CHECK YOUR ANSWER**Solution 3**• Nicky bought a lamp for $45.99 and charged it to her credit card. The credit card company charges the department store 20 cents for each transaction plus 2% of the amount charged. How much will the credit card company pay to the department store? • Fee = ($45.99 x 2%) + $0.20 = $1.12 • $45.99 -- $1.12 = $44.87 BACK TO GAME BOARD**Problem 4**• Elaine expects that she will normally carrya credit card balanceof around $2,000. Which of the threeoptions in the tablewould be the lowestcost option for her? CHECK YOUR ANSWER**Solution 4**• I = PRT • I = $2,000 x 9% x 1 = $180 $180 + $80 = $260 • I = $2,000 x 15% x 1 = $300 $300 + $25 = $325 • I = $2,000 x 23.99% x 1 = $479.80 + $0 = $479.80 BACK TO GAME BOARD**10.2 Mortgages**• A mortgage is a loan that is secured by real estate. • If the borrower fails to pay back the loan as promised, the lender has the right to take the real estate through a legal process known as foreclosure. • Another way of saying that the lender has the right to do this is to say that the lender has a lien on the property. • With few exceptions, the amount borrowed with a mortgage loan must be less than the value of the property. This is sometimes called the maximum loan to value percentage. • The difference between the value of a property and the amount that is owed against it is called the homeowner’s equity.**10.2 Mortgages**Example 10.2.1 • Problem • Les and Rhonda own a house worth $194,825. The balance they owe on their first mortgage is $118,548. They want to take out a second mortgage and all of the lenders they have spoken with require a minimum equity of 5%. • Find (a) their equity now, (b) the maximum they can borrow with the second mortgage, and (c) their equity if they borrow the maximum.**10.2 Mortgages**Example 10.2.1 Cont. • Solution (a) $194,825 -- $118,548 = $76,277 (b) The maximum they can owe: 95% x $194,825 = $185,084 Since they already owe $118,548, they can borrow $185,084 -- $118,548 = $66,536 (c) If they borrow the maximum, they will have only 5% equity left or 5% x $194,825 = $9,741**10.2 Mortgages**• However, second mortgages are not to be confused with a similar financial product called a home equity line of credit. • The difference between the two is that, with a home equity line, instead of being lent a set amount up front, the borrowers are given a checkbook or debit card that they can use to borrow money as needed against their home equity up to a set limit. • Occasionally, some finance companies will offer mortgage loans for more than the value of the property. In those cases, it’s actually possible to have negative equity.**10.2 Mortgages**• A fixed (or fixed-rate) mortgage has an interest rate that is set at the beginning and never changes over the entire term of the loan. • If mortgage rates drop, a borrower can often take an advantage of this by refinancing. • An adjustable-rate mortgage (or ARM) is a more complicated type of a loan. It will usually have some initial period of time for which the rate is fixed. After that, the rate can change. • Adjustable-rate mortgages will usually carry better interest rates up front.**10.2 Mortgages**Example 10.2.3 • Problem • Chantal took out a $142,000 mortgage with a 30-year, fixed-rate loan at 7.2%. Find her monthly mortgage payment • Solution • PV = PMTan/i $142,000 = PMTa360/0.006 PMT = $963.88**10.2 Mortgages**• Real property taxes are taxes that you must pay on the basis of the value of real estate you own. • Mortgage lenders do not always require an escrow account. However, monthly escrow payments are easier to swallow than one large annual tax payment. • Escrow may also be required for homeowners’ insurance premium. • One additional expense that may be included with your monthly payment is private mortgage insurance (PMI). It protects the mortgage lender financially in the event that you do not pay and are forced to foreclose.**10.2 Mortgages**• Since there are other expenses other principal and interest, such as insurance and taxes, looking at the monthly payment we calculated so far is somewhat misleading. • The total payment is referred to as the PITI: • Principal • Interest • Taxes • Insurance**10.2 Mortgages**Example 10.2.6 • Problem • Suppose that Chantal’s annual property taxes are expected to be $3,300 and her homeowners’ insurance premium is $750. PMI is $45 per month. Her principal and interest portion of the payment is $963.88 (calculated in a previous example). Find her total PITI. • Solution Annual Escrow = $3,300 + $750 = $4,050 Every month, Chantal will pay $4,050/12 = $337.50 in order to accumulate the full amount. PITI = $963.88 + $45 + $337.50 = $1,346.38**10.2 Mortgages**• When deciding whether to approve someone for a mortgage loan, the lenders consider the following: • Your Credit History • Employment Stability • Income • How does a lender determine whether or not your income is adequate for a loan? • Most lenders decide this by using ratio tests, comparing the total monthly PITI for the loan to a percentage of your gross monthly income.**10.2 Mortgages**The 28% Rule Total PITI cannot exceed 28% of gross monthly income The 36% Rule Total PITI and all other long-term debt payments cannot exceed 36% of gross monthly income.**10.2 Mortgages**Example 10.2.7 • Problem • Antoine and Maria earn a combined annual income of $76,500. They are trying to qualify for a mortgage on a house for which the total monthly payment (PITI) would be $1,494.57. Do they pass the 28% test? • Solution • Their gross monthly income is $76,500/12 = $6,375 • 28% of this would be 28% x $6,375 = $1,785 • Since their PITI is only $1,494.57, they pass the test.**10.2 Mortgages**Example 10.2.8 • Problem • Suppose that Antoine and Maria have two car loans. The first carries a monthly payment of $309.15 and has 2 years and 7 months left to go. The second has 6 months left to go and the monthly payment is $175.14. They have student loan payments of $109.15 per month and the minimum monthly payments on their credit cards total $75. Do they pass the 36% test? • Solution • 36% of their gross monthly income is 36% x $6,375 = $2,295 • Their total debt payments are $1,494.57 + $309.15 + $109.15 + $75 = $1,987.87 • This total is less than $2,295, so they pass the 36% test as well.**10.2 Mortgages**• In addition to the monthly PITI payments, buying real estate usually involves a significant cash outlay up front. Some of the many expenses involved include: • Down payment • Legal fees • Appraisal • Title search and insurance • Inspections • Flood check • Recording fees • Mortgage taxes • Application and origination fees**10.2 Mortgages**Example 10.2.9 • Problem • Drew and Joanne are buying a house for $128,550. They will make a minimum 3% down payment, and closing costs will total $2,100. Annual property taxes are $2,894 and homeowners’ insurance is $757 annually. How much money will they need up front? • Solution • Their down payment will be 3% x $128,550 = $3,856.50 Total = $3,856.50 + $2,100 + $2,894 + $757 = $9,607.50**10.2 Mortgages**• Lenders often offer an opportunity to “buy” a lowerinterest rate by paying points. • Points a fee paid to the lenderup front, in exchange for alower interest rate. One pointis equal to 1% of the amount ofthe loan.**10.2 Mortgages**Example 10.2.10 • Problem • Suppose that Drew and Joanne are offered the mortgage choices in the table shown on a previous slide. • How much more would they have to come up with if they chose to pay the points? • What would their monthly mortgage payment be with each of the options offered? • How much would they save over the 30-year life of the loan if they paid the points.**10.2 Mortgages**Example 10.2.10 Cont. • Solution • The price of the house is $128,550. Drew and Joanne are making a $3,856.50 down payment, so they will be borrowing $128,550 -- $3,856.50 = $124,693.50. • Since each point is 1% of the loan, 2.5 points is 2.5% or 2.5% x $124,693.50 = $3,117.34 • Using the present value annuity formula with PV = $124,693.50 and a 7.5% interest rate for 30 years gives that the monthly payment with no points would be $871.88. • Using the 6.25% rate gives a monthly payment of $767.76 • So the total payments are: No points: 360 x $871.88 = $313,876.80 Points: 360 x $767.76 = $276,393.60 Therefore, they would save $37,483.20**Problem 1**• Nancy and Drew are considering a $150,000 mortgage with a 20-year fixed-rate loan at 8%. Find their monthly mortgage payment. CHECK YOUR ANSWER**Solution 1**• Nancy and Drew are considering a $150,000 mortgage with a 20-year fixed-rate loan at 8%. Find their monthly mortgage payment. • PV = PMT x an/i • $150,000 = PMT x an/i • $150,000 = PMT x 111.86428 • PMT = $1,251.42 BACK TO GAME BOARD**Problem 2**• Suppose that Nancy and Drew’s annual property taxes are expected to be $3,200 and their homeowners’ insurance annual premium is $1,900. Find their total PITI. CHECK YOUR ANSWER**Solution 2**• Suppose that Nancy and Drew’s annual property taxes are expected to be $3,200 and their homeowners’ insurance annual premium is $1,900. Find their total PITI. • PI = $1,251.42 • TI = ($3,200 + $1,900)/12 = $425.00 • PITI = $1,251.42 + $425.00 = $1,676.42 BACK TO GAME BOARD