A Mathematical View of Our World

A Mathematical View A Mathematical View of Our Worldof Our World

11stst ed. ed.

Parks, Musser, Trimpe, Parks, Musser, Trimpe, Maurer, and MaurerMaurer, and Maurer

Chapter 13Chapter 13

Consumer Mathematics:Consumer Mathematics:

Buying and SavingBuying and Saving

Section 13.1Section 13.1

Simple and Compound InterestSimple and Compound Interest• GoalsGoals

• Study simple interestStudy simple interest• Calculate interestCalculate interest• Calculate future valueCalculate future value

• Study compound interestStudy compound interest• Calculate future valueCalculate future value

• Compare interest ratesCompare interest rates• Calculate effective annual rateCalculate effective annual rate

13.1 Initial Problem13.1 Initial Problem• Suppose you discover you are the only Suppose you discover you are the only

direct descendant of a man who loaned the direct descendant of a man who loaned the Continental Congress $1000 in 1777 and Continental Congress $1000 in 1777 and was never repaid.was never repaid.

• Using an interest rate of 6% and a Using an interest rate of 6% and a compounding period of 3 months, how much compounding period of 3 months, how much should you demand from the government?should you demand from the government?• The solution will be given at the end of the The solution will be given at the end of the

section.section.

Simple InterestSimple Interest

• If If PP represents the principal, represents the principal, rr the the annual interest rate expressed as a annual interest rate expressed as a decimal, and decimal, and tt the time in years, then the time in years, then the amount of simple interest is:the amount of simple interest is:

I Prt

Example 1Example 1

• Find the interest on a loan of $100 at Find the interest on a loan of $100 at 6% simple interest for time periods of:6% simple interest for time periods of:

a)a) 1 year1 year

b)b) 2 years2 years

c)c) 2.5 years2.5 years

Example 1, cont’dExample 1, cont’d

• Solution: We have Solution: We have PP = 100 and = 100 and rr = 0.06. = 0.06.

a)a) For For t t = 1 year, the calculation is:= 1 year, the calculation is:

$100 0.06 1 $6I Prt


• Solution, cont’d: We have Solution, cont’d: We have PP = 100 and = 100 and rr = 0.06. = 0.06.

b)b) For For t t = 2 years, the calculation is:= 2 years, the calculation is:

c)c) For For t t = 2.5 years, the calculation is:= 2.5 years, the calculation is:

$100 0.06 2 $12I Prt

$100 0.06 2.5 $15I Prt

Future ValueFuture Value

• For a simple interest loan, the For a simple interest loan, the future valuefuture value of of the loan is the principal plus the interest.the loan is the principal plus the interest.

• If If PP represents the principal, represents the principal, I I the interest, the interest, rr the annual interest rate, and the annual interest rate, and tt the time in the time in years, then the future value is:years, then the future value is:

1F P I P Prt P rt

Example 2Example 2

• Find the future value of a loan of $400 Find the future value of a loan of $400 at 7% simple interest for 3 years.at 7% simple interest for 3 years.


• Solution: Use the future value formula Solution: Use the future value formula with with PP = 400, = 400, rr = 0.07, and = 0.07, and tt = 3. = 3.•

1

400 1 0.07 3 $484

F P rt

Example 3Example 3• In 2004, Regular Canada Savings Bonds In 2004, Regular Canada Savings Bonds

paid 1.25% simple interest on the face paid 1.25% simple interest on the face value of bonds held for 1 year. value of bonds held for 1 year.

• If the bond is cashed early, the investor If the bond is cashed early, the investor receives the face value plus interest for receives the face value plus interest for every full month.every full month.

• Suppose a bond was purchased for $8000 Suppose a bond was purchased for $8000 on November 1, 2004.on November 1, 2004.


a)a) What was the value of the bond if it What was the value of the bond if it was redeemed on November 1, was redeemed on November 1, 2005?2005?

b)b) What was the value of the bond if it What was the value of the bond if it was redeemed on July 10, 2004?was redeemed on July 10, 2004?


a)a) Solution: If the bond was redeemed Solution: If the bond was redeemed on November 1, 2005, it had been on November 1, 2005, it had been held for 1 year.held for 1 year.

• The future value of the bond after 1 year The future value of the bond after 1 year is:is:

1

8000 1 0.0125 1 $8100

F P rt


b)b) Solution: If the bond was redeemed Solution: If the bond was redeemed on July 10, 2004, it had been held for on July 10, 2004, it had been held for 7 full months.7 full months.

• The future value of the bond after 7/12 of The future value of the bond after 7/12 of a year is: a year is:

1

78000 1 0.0125 $8058.3312

F P rt

Example 4Example 4

• What is the simple interest on a $500 What is the simple interest on a $500 loan at 12% from June 6 through loan at 12% from June 6 through October 12 in a non-leap year?October 12 in a non-leap year?

Example 4, cont’dExample 4, cont’d• Solution: The time must be converted Solution: The time must be converted

to years.to years.• (30 - 6) + 31 + 31 + 30 + 12 = 128 days(30 - 6) + 31 + 31 + 30 + 12 = 128 days

• A non-leap year has 365 days.A non-leap year has 365 days.

• The interest will be:The interest will be:

128500 0.12 365$21.04

I Prt

Question:Question:

What is the simple interest on a What is the simple interest on a $2000 loan at 8% from March 19th $2000 loan at 8% from March 19th through August 15th in a leap year?through August 15th in a leap year?

a. $65.32a. $65.32

b. $653.15b. $653.15

c. $651.37c. $651.37

d. $65.14 d. $65.14

Ordinary InterestOrdinary Interest

• Ordinary interest simplifies calculations Ordinary interest simplifies calculations by using 2 conventions:by using 2 conventions:• Each month is assumed to have 30 days.Each month is assumed to have 30 days.

• Each year is assumed to have 360 days.Each year is assumed to have 360 days.

Example 5Example 5

• A homeowner owes $190,000 on a A homeowner owes $190,000 on a 4.8% home loan with an interest-only 4.8% home loan with an interest-only option.option.• An interest-only option allows the An interest-only option allows the

borrower to pay only the ordinary interest, borrower to pay only the ordinary interest, not the principal, for the first year.not the principal, for the first year.

• What is the monthly payment for the What is the monthly payment for the first year?first year?


• Solution: Use the simple interest Solution: Use the simple interest formula, measuring time according to formula, measuring time according to ordinary interest conventions.ordinary interest conventions.

• The monthly payments are:The monthly payments are:

1190000 0.048

12

$760

I Prt

Compound Interest Compound Interest

• Reinvesting the interest, called Reinvesting the interest, called compoundingcompounding, makes the balance grow , makes the balance grow faster.faster.

• To calculate compound interest, you To calculate compound interest, you need the same information as for need the same information as for simple interest plus you need to know simple interest plus you need to know how often the interest is compounded.how often the interest is compounded.

Example 6Example 6

• Suppose a principal of $1000 is Suppose a principal of $1000 is invested at 6% interest per year and invested at 6% interest per year and the interest is compounded annually.the interest is compounded annually.

• Find the balance in the account after 3 Find the balance in the account after 3 years.years.


• Solution: We must calculate the Solution: We must calculate the interest at the end of each year and interest at the end of each year and then add that interest to the principal.then add that interest to the principal.

• After 1 year:After 1 year:• The interest is:The interest is:

• The new balance is $1060.00The new balance is $1060.00• We could also have used the future value We could also have used the future value

formula.formula.

1000 0.06 1 $60I


• Solution, cont’d:Solution, cont’d:• After 2 years the new balance is:After 2 years the new balance is:

• After 3 years the new balance is:After 3 years the new balance is:

1

1060 1 0.06 1 $1123.60

F P rt

1

1123.6 1 0.06 1 $1191.02

F P rt


• Solution, cont’d: The interest earned each Solution, cont’d: The interest earned each year increases because of the increasing year increases because of the increasing principal.principal.


• Solution, cont’d: The following table shows Solution, cont’d: The following table shows the pattern in the calculations for the pattern in the calculations for subsequent years.subsequent years.

Compound Interest, cont’d Compound Interest, cont’d

• If If PP represents the principal, represents the principal, rr the the annual interest rate expressed as a annual interest rate expressed as a decimal, decimal, mm the number of equal the number of equal compounding periods per year, and compounding periods per year, and tt the time in years, then the future value the time in years, then the future value of the account is:of the account is:

1mt

rF P

m

Example 7Example 7

• Find the future value of each account at the Find the future value of each account at the end of 3 years if the initial balance is $2457 end of 3 years if the initial balance is $2457 and the account earns:and the account earns:

a)a) 4.5% simple interest. 4.5% simple interest.

b)b) 4.5% compounded annually.4.5% compounded annually.

c)c) 4.5% compounded every 4 months.4.5% compounded every 4 months.

d)d) 4.5% compounded monthly.4.5% compounded monthly.

e)e) 4.5% compounded daily.4.5% compounded daily.


• Solution: We have Solution: We have PP = 2457 and = 2457 and tt = 3. = 3.

a)a) We have We have r r = 0.045 with simple interest.= 0.045 with simple interest.•

b)b) We have We have r r = 0.045 compounded = 0.045 compounded annually.annually.

•

2457 1 0.045 3 $2788.70F

1 31 2457 1 0.045 1

$2803.85

mtF P r m

Example 7, cont’dExample 7, cont’d• Solution, cont’d: We have Solution, cont’d: We have rr = 0.045 = 0.045

c)c) Compounded every 4 months:Compounded every 4 months:•

d)d) Compounded monthly:Compounded monthly:•

e)e) Compounded daily:Compounded daily:•

12 32457 1 0.045 12 $2811.42F

3 32457 1 0.045 3 $2809.31F

365 32457 1 0.045 365 $2812.10F


• Solution, cont’d: The results are Solution, cont’d: The results are summarized below.summarized below.

Example 8Example 8

• Find the future value of each account Find the future value of each account at the end of 100 years if the initial at the end of 100 years if the initial balance is $1000 and the account balance is $1000 and the account earns:earns:

a)a) 7.5% simple interest. 7.5% simple interest.

b)b) 7.5% compounded annually.7.5% compounded annually.


• Solution: We have Solution: We have PP = 1000, = 1000, tt = 100, = 100, and and r r = 0.075.= 0.075.

a)a) With simple interest, the future value is:With simple interest, the future value is:•

b)b) With annually compounded interest, the With annually compounded interest, the future value is:future value is:

•

1000 1 0.075 100 $8500F

11000.0751000 1 $1,383,0001F

Question:Question:

If you loan your friend $100 at 3% If you loan your friend $100 at 3% interest compounded daily, how much interest compounded daily, how much will she owe you at the end of 1 year? will she owe you at the end of 1 year?

a. $103.05a. $103.05

b. $103.00b. $103.00

c. $103.33c. $103.33

d. $103.02d. $103.02

Interest, cont’dInterest, cont’d

• Simple interest exhibits arithmetic Simple interest exhibits arithmetic growth.growth.

• The same amount is added each year.The same amount is added each year.

• Compound interest exhibits Compound interest exhibits exponential growth.exponential growth.

• The same amount is multiplied each The same amount is multiplied each year.year.

Example 9Example 9

• How much money should be How much money should be invested at 4% interest compounded invested at 4% interest compounded monthly in order to have $25,000 monthly in order to have $25,000 eighteen years later?eighteen years later?


• Solution: We know Solution: We know FF = 25,000, = 25,000,

rr = 0.04, = 0.04, mm = 12 and = 12 and tt = 18. = 18.• Solve for Solve for PP, the necessary principal., the necessary principal.

•

1218

216

25000 1 0.04 12

25000$12,183.39

1 0.04 12

P

P

Effective Annual RateEffective Annual Rate

• The The effective annual rate (EAReffective annual rate (EAR) or ) or annual annual percentage yield (APY)percentage yield (APY) is the simple interest is the simple interest that would give the same result in 1 year.that would give the same result in 1 year.• The stated rate is called the The stated rate is called the nominal ratenominal rate..

• This provides a basis for comparing different This provides a basis for comparing different savings plans.savings plans.• APY is used only for savings accounts.APY is used only for savings accounts.

• EAR is used in any context.EAR is used in any context.

Example 10Example 10

• Find the effective annual rate by Find the effective annual rate by computing what happens to $100 computing what happens to $100 over 1 year at 12% annual interest over 1 year at 12% annual interest compounded every 3 months.compounded every 3 months.


• Solution: The balance at the end of Solution: The balance at the end of 1 year is:1 year is:

• Since the account increased by Since the account increased by $12.55 in 1 year, the EAR is $12.55 in 1 year, the EAR is 12.55%.12.55%.•

41100 1 0.12 4 $112.55F

100 1 0.1255 1 $112.55F

Effective Annual Rate, cont’dEffective Annual Rate, cont’d• If If rr represents the annual interest rate represents the annual interest rate

expressed as a decimal and expressed as a decimal and mm is the number is the number of equal compounding periods per year, then of equal compounding periods per year, then the effective annual rate is:the effective annual rate is:

• Note: The same formula is used for APY.Note: The same formula is used for APY.

1 1m

rEAR

m

Effective Annual Rate, cont’dEffective Annual Rate, cont’d

Example 11Example 11• A bank offers a savings account with an A bank offers a savings account with an

interest rate of 0.25% compounded daily, interest rate of 0.25% compounded daily, with a minimum deposit of $100.with a minimum deposit of $100.

• The same bank offers an 18-month CD with The same bank offers an 18-month CD with an interest rate of 2.13% compounded an interest rate of 2.13% compounded monthly, with deposits less than $10,000.monthly, with deposits less than $10,000.

• Find the effective annual rate for each Find the effective annual rate for each option.option.

Example 11, cont’dExample 11, cont’d• Solution, cont’d: The effective annual rate for Solution, cont’d: The effective annual rate for

the savings account is:the savings account is:

• The EAR for the account is about 0.2503%.The EAR for the account is about 0.2503%.

365

1 1

1 0.0025 365 1

0.00250312

mEAR r m

Example 11, cont’dExample 11, cont’d• Solution: The effective annual rate for the Solution: The effective annual rate for the

CD is:CD is:

• The EAR for the CD is about 2.1509%.The EAR for the CD is about 2.1509%.

12

1 1

1 0.0213 12 1

0.02150918

mEAR r m

13.1 Initial Problem Solution13.1 Initial Problem Solution

• Suppose you discover you are the only direct Suppose you discover you are the only direct descendant of a man who loaned the descendant of a man who loaned the Continental Congress $1000 in 1777 and was Continental Congress $1000 in 1777 and was never repaid.never repaid.

• Using an interest rate of 6% and a Using an interest rate of 6% and a compounding period of 3 months, how much compounding period of 3 months, how much should you demand from the government?should you demand from the government?

Initial Problem Solution, cont’dInitial Problem Solution, cont’d

• We have We have PP = $1000, = $1000, rr = 0.06, = 0.06, mm = 4 = 4 and and tt = 223. = 223.

• The value of your ancestor’s loan is:The value of your ancestor’s loan is:

4 2231000 1 0.06 4

$585,746,479

F


LoansLoans• GoalsGoals

• Study amortized loansStudy amortized loans• Use an amortization tableUse an amortization table• Use the amortization formulaUse the amortization formula

• Study rent-to-ownStudy rent-to-own

13.2 Initial Problem13.2 Initial Problem

• Home mortgage rates have decreased and Home mortgage rates have decreased and Howard plans to refinance his home.Howard plans to refinance his home.

• He will refinance $85,000 at either 5.25% He will refinance $85,000 at either 5.25% for 15 years or 5.875% for 30 years.for 15 years or 5.875% for 30 years.

• In each case, what is his monthly payment In each case, what is his monthly payment and how much interest will he pay?and how much interest will he pay?• The solution will be given at the end of the section.The solution will be given at the end of the section.

Simple Interest LoansSimple Interest Loans

• The interest on a The interest on a simple interest loansimple interest loan is simple interest on the amount is simple interest on the amount currently owed.currently owed.

• The simple interest each month is The simple interest each month is called the called the finance chargefinance charge..• Finance charges are calculated using an Finance charges are calculated using an

average daily balanceaverage daily balance and a and a daily interest daily interest raterate..

Example 1Example 1


• Assuming the billing period is June 10 Assuming the billing period is June 10 through July 9, determine each of the through July 9, determine each of the following:following:

a)a) The average daily balanceThe average daily balance

b)b) The daily percentage rateThe daily percentage rate

c)c) The finance chargeThe finance charge

d)d) The new balanceThe new balance


a)a) Solution: The daily balances are shown Solution: The daily balances are shown below.below.


a)a) Solution, cont’d: The average daily Solution, cont’d: The average daily balance is:balance is:

2 287.84 6 333.44 4 183.44 11 203.44 7 281.94

2 6 4 11 7

7521.50$250.72

30


b)b) Solution: The daily percentage rate is:Solution: The daily percentage rate is:

c)c) Solution: The finance charge is the Solution: The finance charge is the simple interest on the average daily simple interest on the average daily balance at the daily rate: balance at the daily rate:

21%0.057534%

365

250.72 0.21 365 30 $4.33I


d)d) Solution: The new balance is the sum Solution: The new balance is the sum of the previous balance, any new of the previous balance, any new charges, and the finance charge, charges, and the finance charge, minus any payments:minus any payments:

287.84 + 144.10 + 4.33 – 150.00 = 287.84 + 144.10 + 4.33 – 150.00 = 286.27286.27

• The new balance is $286.27.The new balance is $286.27.


Amortized LoansAmortized Loans• Amortized loansAmortized loans are simple interest loans are simple interest loans

with equal periodic payments over the with equal periodic payments over the length of the loan.length of the loan.

• The important variables for an amortized The important variables for an amortized loan are:loan are:

• PrincipalPrincipal

• Interest rateInterest rate

• Term (length) of the loanTerm (length) of the loan

• Monthly paymentMonthly payment

Amortized Loans, cont’dAmortized Loans, cont’d

• Each payment includes the interest Each payment includes the interest due since the last payment and an due since the last payment and an amount paid toward the balance.amount paid toward the balance.

• The amount paid each month is The amount paid each month is constant, but the split between principal constant, but the split between principal and interest varies.and interest varies.

• The amount of the last payment may be The amount of the last payment may be slightly more or less than usual.slightly more or less than usual.

Question:Question:

When paying off an amortized loan, When paying off an amortized loan, the percent of the monthly payment the percent of the monthly payment going toward the interest will: going toward the interest will:

a. increase as time goes by.a. increase as time goes by.

b. decrease as time goes by.b. decrease as time goes by.

c. remain the same every month.c. remain the same every month.

Example 2Example 2

• Chart the history of an amortized loan Chart the history of an amortized loan of $1000 for 3 months at 12% interest of $1000 for 3 months at 12% interest with monthly payments of $340.with monthly payments of $340.


• Solution: Monthly payment #1:Solution: Monthly payment #1:• The interest owed isThe interest owed is

• The payment toward the principal is The payment toward the principal is

$340 - $10 = $330 $340 - $10 = $330

• The new balance is $1000 - $330 = $670.The new balance is $1000 - $330 = $670.

11000 0.12 $1012I

Example 2, cont’dExample 2, cont’d• Solution, cont’d: Monthly payment #2:Solution, cont’d: Monthly payment #2:

• The interest owed isThe interest owed is

• The payment toward the principal is The payment toward the principal is

$340 - $6.70 = $333.30 $340 - $6.70 = $333.30

• The new balance is $670 - $333.30 = The new balance is $670 - $333.30 = $336.70$336.70

1670 0.12 $6.7012I

Example 2, cont’dExample 2, cont’d• Solution, cont’d: Monthly payment #3:Solution, cont’d: Monthly payment #3:

• The interest owed isThe interest owed is

• The remaining balance plus the interest The remaining balance plus the interest is: is: $336.70 + $3.37 = $340.07. $336.70 + $3.37 = $340.07.

• The third and final payment is $340.07.The third and final payment is $340.07.

1336.70 0.12 $3.3712I

Example 2, cont’dExample 2, cont’d• Solution, cont’d: The amortization Solution, cont’d: The amortization

schedule for this loan is shown below.schedule for this loan is shown below.

Monthly PaymentsMonthly Payments

• The monthly payments for an The monthly payments for an amortized loan can be determined in amortized loan can be determined in one of three ways:one of three ways:• Using an amortization table.Using an amortization table.

• Using a formula.Using a formula.

• Using financial software or an online Using financial software or an online calculator.calculator.

Amortization TableAmortization Table

• An An amortization tableamortization table gives pre- gives pre-calculated monthly payments for calculated monthly payments for common loan rates and terms.common loan rates and terms.

• An example of a table is shown on the An example of a table is shown on the next slide. next slide.

Amortization Table, cont’dAmortization Table, cont’d

Example 3Example 3

• A couple is buying a vehicle for A couple is buying a vehicle for $20,995.$20,995.

• They pay $7000 down and finance the They pay $7000 down and finance the remainder at an annual interest rate of remainder at an annual interest rate of 4.5% for 48 months.4.5% for 48 months.

• Use the amortization table to Use the amortization table to determine their monthly payment.determine their monthly payment.


• Solution: The amount being financed is Solution: The amount being financed is $20,995 – $7000 = $13,995.$20,995 – $7000 = $13,995.

• In the table, find the row corresponding In the table, find the row corresponding to 4.5% and the column corresponding to 4.5% and the column corresponding to 4 years.to 4 years.• This entry is highlighted on the next slide.This entry is highlighted on the next slide.



• Solution, cont’d: The value 22.803486 Solution, cont’d: The value 22.803486 indicates the couple will pay indicates the couple will pay $22.803486 for each $1000 they $22.803486 for each $1000 they borrowed.borrowed.•

• They will pay $319.14 per month.They will pay $319.14 per month.

22.803486 13.995 319.13479

Example 4Example 4

• The couple in the previous example The couple in the previous example borrowed $13,995 to buy a car and will borrowed $13,995 to buy a car and will pay the loan over 4 years.pay the loan over 4 years.

• If their payments are $340.02, what If their payments are $340.02, what interest rate are they being charged?interest rate are they being charged?


• Solution: They are paying $340.02 a Solution: They are paying $340.02 a month for $13,995, or approximately month for $13,995, or approximately $24.295820 per $1000.$24.295820 per $1000.• Look at the amortization table to see to Look at the amortization table to see to

which interest rate this payment amount which interest rate this payment amount corresponds.corresponds.



• Solution, cont’d: The interest rate for Solution, cont’d: The interest rate for the amortized loan, according to the the amortized loan, according to the table, is approximately 7.75%.table, is approximately 7.75%.

Amortization FormulaAmortization Formula• If If PP is the amount of the loan, is the amount of the loan, rr is the annual is the annual

interest rate expressed as a decimal, and interest rate expressed as a decimal, and tt is the length of the loan in years, then the is the length of the loan in years, then the monthly payment for an amortized loan is:monthly payment for an amortized loan is:

12

12

112 12

1 112

t

t

r rP

PMTr

Example 5Example 5

• Use the monthly payment formula to Use the monthly payment formula to determine the monthly car payment for determine the monthly car payment for a loan of $13,995 at 4.5% annual a loan of $13,995 at 4.5% annual interest for 48 months.interest for 48 months.

Example 5, cont’dExample 5, cont’d• Solution: We have Solution: We have PP = 13,995, = 13,995, rr = 0.045, and = 0.045, and

tt = 4. = 4.

12 4

12 4

0.045 0.04513995 1

12 12

0.0451 1

12

$319.14

PMT

Example 6Example 6

• Suppose a student accumulated $7800 Suppose a student accumulated $7800 in student loans which she must now in student loans which she must now pay over 10 years.pay over 10 years.

• Determine her monthly payment Determine her monthly payment amount using an interest rate of 3.37%amount using an interest rate of 3.37%


• Solution: Solution: We have Solution: Solution: We have PP = 7800, = 7800,

rr = 0.0337, and = 0.0337, and tt = 10. = 10.

1210

1210

0.0337 0.03377800 1

12 12$76.66

0.03371 1

12

PMT

Question:Question:

Use the amortization formula to Use the amortization formula to determine the amount of the monthly determine the amount of the monthly payment for a loan of $30,000 at 5% for payment for a loan of $30,000 at 5% for 3 years. 3 years.

a. $527.38a. $527.38

b. $124.00b. $124.00

c. $722.46c. $722.46

d. $899.13d. $899.13

Example 7Example 7

• Suppose you can afford car payments Suppose you can afford car payments of $250 per month.of $250 per month.

• If a 3-year loan at 4% interest is If a 3-year loan at 4% interest is available, how much can you finance?available, how much can you finance?


• Solution: We know Solution: We know PMTPMT = 250, = 250,

rr = 0.04, and = 0.04, and tt = 3. = 3.

12 3

12 3

0.04 0.041

12 12250

0.041 1

12

P

Example 7, cont’dExample 7, cont’d• Solution, cont’d: Solve for Solution, cont’d: Solve for PP..

• You can borrow $8468.You can borrow $8468.

36

36

0.04250 1 1

12$8468

0.04 0.041

12 12

P

Rent-to-OwnRent-to-Own

• In a rent-to-own transaction, you rent In a rent-to-own transaction, you rent the item at a monthly rate, but after a the item at a monthly rate, but after a contracted number of payments, the contracted number of payments, the item becomes yours.item becomes yours.

• The difference between the retail price The difference between the retail price of the item and the total of your of the item and the total of your monthly payments is the interest.monthly payments is the interest.

Example 8Example 8

• Suppose you can rent-to-own a $500 Suppose you can rent-to-own a $500 television for 24 monthly payments of $30.television for 24 monthly payments of $30.

a)a) What amount of interest would you pay for What amount of interest would you pay for the rent-to-own television?the rent-to-own television?

b)b) What annual rate of simple interest on What annual rate of simple interest on $500 for 24 months yields the same $500 for 24 months yields the same amount of interest found in part (a)?amount of interest found in part (a)?


a)a) Solution: Solution: • The total of your monthly payments will The total of your monthly payments will

be 24($30) = $720.be 24($30) = $720.

• You will pay $720 - $500 = $220 in You will pay $720 - $500 = $220 in interest over the 2 years.interest over the 2 years.


b)b) Solution: Solution: • Solve the simple interest formula for Solve the simple interest formula for rr::

• The equivalent simple interest rate is:The equivalent simple interest rate is:

Ir

Pt

220

0.22 22%500 2

r

13.2 Initial Problem Solution13.2 Initial Problem Solution• Home mortgage rates have decreased Home mortgage rates have decreased

and Howard plans to refinance his and Howard plans to refinance his home. He will refinance $85,000 at home. He will refinance $85,000 at either 5.25% for 15 years or 5.875% either 5.25% for 15 years or 5.875% for 30 years.for 30 years.

• In each case, what is his monthly In each case, what is his monthly payment and how much interest will payment and how much interest will he pay?he pay?


• The 15-year loan has an interest rate of The 15-year loan has an interest rate of 5.25%.5.25%.

• According to the amortization table, the According to the amortization table, the monthly payment per $1000 would be monthly payment per $1000 would be $8.038777.$8.038777.

• Under this loan, Howard’s monthly Under this loan, Howard’s monthly payment would be $8.038777(85) payment would be $8.038777(85) which is approximately $683.30.which is approximately $683.30.


• For the 15-year loan, Howard will pay a For the 15-year loan, Howard will pay a total of ($683.30)(12)(15) = $122,994.total of ($683.30)(12)(15) = $122,994.

• The amount spent on interest is The amount spent on interest is $122,994 - $85,000 = $37,994.$122,994 - $85,000 = $37,994.


• The 30-year loan has an interest rate of The 30-year loan has an interest rate of 5.875%, which is not found in the table.5.875%, which is not found in the table.

• Using the amortization formula, we find Using the amortization formula, we find a monthly payment amount of $502.81.a monthly payment amount of $502.81.


• For the 30-year loan, Howard will pay a For the 30-year loan, Howard will pay a total of ($502.81)(12)(30) = total of ($502.81)(12)(30) = $181,011.60.$181,011.60.

• The amount spent on interest is The amount spent on interest is $181,011.60 - $85,000 = $96,011.60$181,011.60 - $85,000 = $96,011.60


Buying a HouseBuying a House

• GoalsGoals

• Study affordability guidelinesStudy affordability guidelines

• Study mortgagesStudy mortgages• Interest rates and closing costsInterest rates and closing costs• Annual percentage ratesAnnual percentage rates• Down paymentsDown payments

13.3 Initial Problem13.3 Initial Problem

• Suppose you have saved $15,000 toward a down Suppose you have saved $15,000 toward a down payment on a house and your total yearly income is payment on a house and your total yearly income is $45,000. What is the most you could afford to pay $45,000. What is the most you could afford to pay for a house?for a house?

• Assume you pay 0.5% of the value for insurance, Assume you pay 0.5% of the value for insurance, you pay 1.5% of the value for taxes, your closing you pay 1.5% of the value for taxes, your closing costs will be $2000, and you can obtain a fixed-rate costs will be $2000, and you can obtain a fixed-rate mortgage for 30 years at 6% interest. mortgage for 30 years at 6% interest. • The solution will be given at the end of the section.The solution will be given at the end of the section.

Affordability GuidelinesAffordability Guidelines

• The 2 most common guidelines for The 2 most common guidelines for buying a house are:buying a house are:• The maximum house price is 3 times your The maximum house price is 3 times your

annual gross income.annual gross income.

• Your maximum monthly housing expenses Your maximum monthly housing expenses should be 25% of your gross monthly should be 25% of your gross monthly income. income.

Example 1Example 1

• If your annual gross income is $60,000, If your annual gross income is $60,000, what do the guidelines tell you about what do the guidelines tell you about purchase price and monthly expenses purchase price and monthly expenses for your potential home purchase?for your potential home purchase?


• Solution: Solution: • The purchase price should be no more The purchase price should be no more

than 3($60,000) = $180,000.than 3($60,000) = $180,000.

• The monthly expenses for mortgage The monthly expenses for mortgage payments, property taxes, and payments, property taxes, and homeowner’s insurance should be no homeowner’s insurance should be no more than more than 0.25 1 12 60000 $1250.

Affordability Guidelines, cont’dAffordability Guidelines, cont’d

• Some lenders allow monthly expenses Some lenders allow monthly expenses up to 38% of the buyer’s monthly up to 38% of the buyer’s monthly income.income.• We call the 25% level the low maximum We call the 25% level the low maximum

monthly housing expense estimate.monthly housing expense estimate.

• We call the 38% level the high maximum We call the 38% level the high maximum monthly housing expense estimate.monthly housing expense estimate.

Example 2Example 2

• Suppose Andrew and Barbara both Suppose Andrew and Barbara both have jobs, each earning $24,000 a year, have jobs, each earning $24,000 a year, and they have no debts.and they have no debts.

• What are the low and high estimates of What are the low and high estimates of how much they can afford to pay for how much they can afford to pay for monthly housing expenses?monthly housing expenses?


• Solution: The low estimate is 25% of the Solution: The low estimate is 25% of the total monthly income.total monthly income.•

• The high estimate is 38% of the total The high estimate is 38% of the total monthly income.monthly income.•

48,0000.25 $1000

12

48,0000.38 $1520

12

Question:Question:

If you make $28,000 per year, can you If you make $28,000 per year, can you afford to buy a house for $83,000 with afford to buy a house for $83,000 with monthly housing expenses of $650? monthly housing expenses of $650?

a. Yes, according to the low maximum a. Yes, according to the low maximum guideline.guideline.

b. Yes, according to the high maximum b. Yes, according to the high maximum guideline.guideline.

c. No c. No

MortgagesMortgages

• A A mortgagemortgage is a loan that is guaranteed is a loan that is guaranteed by real estate.by real estate.

• The interest rate of a The interest rate of a fixed-rate fixed-rate mortgagemortgage is set for the entire term. is set for the entire term.

• The interest rate of an The interest rate of an adjustable-rate adjustable-rate mortgage (ARM)mortgage (ARM) can change. can change.

Mortgages, cont’dMortgages, cont’d

• The finalizing of a house purchase is called The finalizing of a house purchase is called the the closingclosing..

• PointsPoints are fees paid to the lender at the time are fees paid to the lender at the time of the closing.of the closing.• Loan origination feesLoan origination fees

• Discount chargesDiscount charges

• Points and any other expenses paid at the Points and any other expenses paid at the time of the closing are called time of the closing are called closing costsclosing costs..

Example 3Example 3

• Suppose you will borrow $80,000 for a Suppose you will borrow $80,000 for a home at 6.5% interest on a 30-year home at 6.5% interest on a 30-year fixed-rate mortgage.fixed-rate mortgage.

• The loan involves a one-point loan The loan involves a one-point loan origination fee and a one-point discount origination fee and a one-point discount charge. What are your added costs?charge. What are your added costs?• Note: One point is equal to 1 percent of the Note: One point is equal to 1 percent of the

loan amount.loan amount.


• Solution: Each fee will cost you 1% of Solution: Each fee will cost you 1% of $80,000, or $800.$80,000, or $800.

• Your total added fees are $1600.Your total added fees are $1600.

Annual Percentage RateAnnual Percentage Rate

• The The annual percentage rate (APR)annual percentage rate (APR) helps borrowers compare the true cost helps borrowers compare the true cost of a loan.of a loan.

• The APR includes the annual interest The APR includes the annual interest rate, any points, and any other loan rate, any points, and any other loan processing or private mortgage processing or private mortgage insurance fees.insurance fees.

Example 4Example 4

• Suppose you plan to borrow $80,000 for Suppose you plan to borrow $80,000 for a home at 6.5% interest on a 30-year a home at 6.5% interest on a 30-year fixed-rate mortgage.fixed-rate mortgage.

• Determine the loan’s APR if there is a 1-Determine the loan’s APR if there is a 1-point loan origination fee and a 1-point point loan origination fee and a 1-point discount charge.discount charge.


• Solution: The loan is $80,000 plus Solution: The loan is $80,000 plus points totaling $1600, for a total of points totaling $1600, for a total of $81,600.$81,600.

• According to the amortization table, the According to the amortization table, the total monthly payment will be total monthly payment will be 81.6($6.320680) or about $515.77, if 81.6($6.320680) or about $515.77, if the fees were paid monthly instead of at the fees were paid monthly instead of at the time of closing.the time of closing.


• Solution, cont’d: The interest rate that Solution, cont’d: The interest rate that corresponds to a loan of $80,000 at corresponds to a loan of $80,000 at 6.5% for 30 years with a monthly 6.5% for 30 years with a monthly payment of $515.77 is found:payment of $515.77 is found:• Divide $515.77 by 80 to find the amount Divide $515.77 by 80 to find the amount

per $1000, which is $6.44713.per $1000, which is $6.44713.

• In the table, this value corresponds to a In the table, this value corresponds to a rate between 6.5% and 6.75%.rate between 6.5% and 6.75%.

Example 4, cont’dExample 4, cont’d• Solution, cont’d: Use linear interpolation to Solution, cont’d: Use linear interpolation to

determine the APR. determine the APR. •

• The payment is 76.5% of the way from the The payment is 76.5% of the way from the payment for 6.5% to the payment for 6.75%.payment for 6.5% to the payment for 6.75%.

•

• The APR is about 6.691%The APR is about 6.691%

6.44713 6.320680100% 76.5%

6.485981 6.320680

6.5 07.65 6.75 6.5 6.691%

APR, cont’dAPR, cont’d• Note that the APR is always greater than or Note that the APR is always greater than or

equal to the stated annual interest rate.equal to the stated annual interest rate.

Down PaymentDown Payment• A A down paymentdown payment on a house is the amount of on a house is the amount of

cash the buyer pays at closing, minus any cash the buyer pays at closing, minus any points and fees.points and fees.

• Traditionally a down payment is 20% of the Traditionally a down payment is 20% of the value, but can be lower. value, but can be lower.

• The The loan to value ratioloan to value ratio of a mortgage is the of a mortgage is the percent of the home’s value that is not paid for percent of the home’s value that is not paid for by the down payment.by the down payment.

• For example, a down payment of 20% results in For example, a down payment of 20% results in an 80% loan to value ratio.an 80% loan to value ratio.

Down Payment, cont’dDown Payment, cont’d

• Another guideline for the maximum Another guideline for the maximum price you can afford when buying a price you can afford when buying a home is to find your maximum price home is to find your maximum price by dividing the amount you have for by dividing the amount you have for a down payment by the percent of a down payment by the percent of the value of the house that amount the value of the house that amount represents.represents.

Example 5Example 5

• If you have $25,000 for a down If you have $25,000 for a down payment, what is the highest-priced payment, what is the highest-priced home you can afford if a 20% down home you can afford if a 20% down payment is required?payment is required?

Example 5, cont’dExample 5, cont’d• Solution: The maximum price you can Solution: The maximum price you can

afford to pay is your down payment afford to pay is your down payment amount divided by 20%.amount divided by 20%.

•

• The most expensive house you can afford The most expensive house you can afford is one that is selling for $125,000. is one that is selling for $125,000.

25,000$125,000

0.20

Question:Question:

Suppose you have $5000 for a down Suppose you have $5000 for a down payment on a house that is selling payment on a house that is selling for $82,000. If the lender requires a for $82,000. If the lender requires a 5% down payment for first-time 5% down payment for first-time homebuyers, is this house within homebuyers, is this house within your price range? your price range?

a. Yesa. Yes b. Nob. No

13.3 Initial Problem Solution13.3 Initial Problem Solution• Suppose you have saved $15,000 toward a Suppose you have saved $15,000 toward a

down payment on a house and your total down payment on a house and your total yearly income is $45,000. What is the most yearly income is $45,000. What is the most you could afford to pay for a house?you could afford to pay for a house?

• Assume you pay 0.5% of the value for Assume you pay 0.5% of the value for insurance, you pay 1.5% of the value for insurance, you pay 1.5% of the value for taxes, your closing costs will be $2000, and taxes, your closing costs will be $2000, and you can obtain a fixed-rate mortgage for 30 you can obtain a fixed-rate mortgage for 30 years at 6% interest.years at 6% interest.


• Your total income is $45,000Your total income is $45,000• You have $15,000 saved for the You have $15,000 saved for the

purchasepurchase• $2000 will be used for closing costs.$2000 will be used for closing costs.

• This leaves $13,000 for a down This leaves $13,000 for a down payment.payment.


• The first affordability guideline says you The first affordability guideline says you can spend at most 3($45,000) = $135,000 can spend at most 3($45,000) = $135,000 on a house.on a house.

• Next, consider your monthly expenses:Next, consider your monthly expenses:• You would be financing $122,000 at 6% for 30 You would be financing $122,000 at 6% for 30

years.years.

• The monthly mortgage payments, from the The monthly mortgage payments, from the table, would be 122($5.995505) = $732.table, would be 122($5.995505) = $732.


• The insurance and taxes are 2% of the The insurance and taxes are 2% of the home’s value annually.home’s value annually.• This adds $225 to the monthly expenses, for a This adds $225 to the monthly expenses, for a

total monthly expense of $957.total monthly expense of $957.

• According to the second affordability According to the second affordability guideline you can only afford monthly guideline you can only afford monthly expenses of at most $938.expenses of at most $938.• The monthly expenses for this house are The monthly expenses for this house are

above your maximum. You cannot afford it.above your maximum. You cannot afford it.


• A house priced $135,000 is slightly A house priced $135,000 is slightly out of your reach, so your options are:out of your reach, so your options are:• Wait for interest rates to fall.Wait for interest rates to fall.

• Increase your income.Increase your income.

• Come up with a larger down payment.Come up with a larger down payment.

• Choose a less expensive house.Choose a less expensive house.

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A Mathematical View of Our World