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A Mathematical View of Our World

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A Mathematical View of Our World

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A Mathematical View of Our World

1st ed.

Parks, Musser, Trimpe, Maurer, and Maurer

Chapter 13

Consumer Mathematics:

Buying and Saving

- Goals
- Study simple interest
- Calculate interest
- Calculate future value

- Study compound interest
- Calculate future value

- Compare interest rates
- Calculate effective annual rate

- Study simple interest

- Suppose you discover you are the only direct descendant of a man who loaned the Continental Congress $1000 in 1777 and was never repaid.
- Using an interest rate of 6% and a compounding period of 3 months, how much should you demand from the government?
- The solution will be given at the end of the section.

- If P represents the principal, r the annual interest rate expressed as a decimal, and t the time in years, then the amount of simple interest is:

- Find the interest on a loan of $100 at 6% simple interest for time periods of:
- 1 year
- 2 years
- 2.5 years

- Solution: We have P = 100 and r = 0.06.
- For t = 1 year, the calculation is:

- Solution, cont’d: We have P = 100 and r = 0.06.
- For t = 2 years, the calculation is:
- For t = 2.5 years, the calculation is:

- For a simple interest loan, the future value of the loan is the principal plus the interest.
- If P represents the principal, I the interest, r the annual interest rate, and t the time in years, then the future value is:

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

- Solution: Use the future value formula with P = 400, r = 0.07, and t = 3.

- In 2004, Regular Canada Savings Bonds paid 1.25% simple interest on the face value of bonds held for 1 year.
- If the bond is cashed early, the investor receives the face value plus interest for every full month.
- Suppose a bond was purchased for $8000 on November 1, 2004.

- What was the value of the bond if it was redeemed on November 1, 2005?
- What was the value of the bond if it was redeemed on July 10, 2004?

- Solution: If the bond was redeemed on November 1, 2005, it had been held for 1 year.
- The future value of the bond after 1 year is:

- Solution: If the bond was redeemed on July 10, 2004, it had been held for 7 full months.
- The future value of the bond after 7/12 of a year is:

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

- Solution: The time must be converted to years.
- (30 - 6) + 31 + 31 + 30 + 12 = 128 days
- A non-leap year has 365 days.

- The interest will be:

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

a. $65.32

b. $653.15

c. $651.37

d. $65.14

- Ordinary interest simplifies calculations by using 2 conventions:
- Each month is assumed to have 30 days.
- Each year is assumed to have 360 days.

- A homeowner owes $190,000 on a 4.8% home loan with an interest-only option.
- An interest-only option allows the borrower to pay only the ordinary interest, not the principal, for the first year.

- What is the monthly payment for the first year?

- Solution: Use the simple interest formula, measuring time according to ordinary interest conventions.
- The monthly payments are:

- Reinvesting the interest, called compounding, makes the balance grow faster.
- To calculate compound interest, you need the same information as for simple interest plus you need to know how often the interest is compounded.

- Suppose a principal of $1000 is invested at 6% interest per year and the interest is compounded annually.
- Find the balance in the account after 3 years.

- Solution: We must calculate the interest at the end of each year and then add that interest to the principal.
- After 1 year:
- The interest is:
- The new balance is $1060.00
- We could also have used the future value formula.

- Solution, cont’d:
- After 2 years the new balance is:
- After 3 years the new balance is:

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

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

- If P represents the principal, r the annual interest rate expressed as a decimal, m the number of equal compounding periods per year, and t the time in years, then the future value of the account is:

- Find the future value of each account at the end of 3 years if the initial balance is $2457 and the account earns:
- 4.5% simple interest.
- 4.5% compounded annually.
- 4.5% compounded every 4 months.
- 4.5% compounded monthly.
- 4.5% compounded daily.

- Solution: We have P = 2457 and t = 3.
- We have r = 0.045 with simple interest.
- We have r = 0.045 compounded annually.

- Solution, cont’d: We have r = 0.045
- Compounded every 4 months:
- Compounded monthly:
- Compounded daily:

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

- Find the future value of each account at the end of 100 years if the initial balance is $1000 and the account earns:
- 7.5% simple interest.
- 7.5% compounded annually.

- Solution: We have P = 1000, t = 100, and r = 0.075.
- With simple interest, the future value is:
- With annually compounded interest, the future value is:

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

a. $103.05

b. $103.00

c. $103.33

d. $103.02

- Simple interest exhibits arithmetic growth.
- The same amount is added each year.

- Compound interest exhibits exponential growth.
- The same amount is multiplied each year.

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

- Solution: We know F = 25,000,
r = 0.04, m = 12 and t = 18.

- Solve for P, the necessary principal.

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

- This provides a basis for comparing different savings plans.
- APY is used only for savings accounts.
- EAR is used in any context.

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

- Solution: The balance at the end of 1 year is:
- Since the account increased by $12.55 in 1 year, the EAR is 12.55%.

- If r represents the annual interest rate expressed as a decimal and m is the number of equal compounding periods per year, then the effective annual rate is:
- Note: The same formula is used for APY.

- A bank offers a savings account with an interest rate of 0.25% compounded daily, with a minimum deposit of $100.
- The same bank offers an 18-month CD with an interest rate of 2.13% compounded monthly, with deposits less than $10,000.
- Find the effective annual rate for each option.

- Solution, cont’d: The effective annual rate for the savings account is:
- The EAR for the account is about 0.2503%.

- Solution: The effective annual rate for the CD is:
- The EAR for the CD is about 2.1509%.

- Suppose you discover you are the only direct descendant of a man who loaned the Continental Congress $1000 in 1777 and was never repaid.
- Using an interest rate of 6% and a compounding period of 3 months, how much should you demand from the government?

- We have P = $1000, r = 0.06, m = 4 and t = 223.
- The value of your ancestor’s loan is:

- Goals
- Study amortized loans
- Use an amortization table
- Use the amortization formula

- Study rent-to-own

- Study amortized loans

- Home mortgage rates have decreased and Howard plans to refinance his home.
- He will refinance $85,000 at either 5.25% for 15 years or 5.875% for 30 years.
- In each case, what is his monthly payment and how much interest will he pay?
- The solution will be given at the end of the section.

- The interest on a simple interest loan is simple interest on the amount currently owed.
- The simple interest each month is called the finance charge.
- Finance charges are calculated using an average daily balance and a daily interest rate.

- Assuming the billing period is June 10 through July 9, determine each of the following:
- The average daily balance
- The daily percentage rate
- The finance charge
- The new balance

- Solution: The daily balances are shown below.

- Solution, cont’d: The average daily balance is:

- Solution: The daily percentage rate is:
- Solution: The finance charge is the simple interest on the average daily balance at the daily rate:

- Solution: The new balance is the sum of the previous balance, any new charges, and the finance charge, minus any payments:
287.84 + 144.10 + 4.33 – 150.00 = 286.27

- The new balance is $286.27.

- Amortized loans are simple interest loans with equal periodic payments over the length of the loan.
- The important variables for an amortized loan are:
- Principal
- Interest rate
- Term (length) of the loan
- Monthly payment

- Each payment includes the interest due since the last payment and an amount paid toward the balance.
- The amount paid each month is constant, but the split between principal and interest varies.
- The amount of the last payment may be slightly more or less than usual.

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

a. increase as time goes by.

b. decrease as time goes by.

c. remain the same every month.

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

- Solution: Monthly payment #1:
- The interest owed is
- The payment toward the principal is
$340 - $10 = $330

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

- Solution, cont’d: Monthly payment #2:
- The interest owed is
- The payment toward the principal is
$340 - $6.70 = $333.30

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

- Solution, cont’d: Monthly payment #3:
- The interest owed is
- The remaining balance plus the interest is: $336.70 + $3.37 = $340.07.
- The third and final payment is $340.07.

- Solution, cont’d: The amortization schedule for this loan is shown below.

- The monthly payments for an amortized loan can be determined in one of three ways:
- Using an amortization table.
- Using a formula.
- Using financial software or an online calculator.

- An amortization table gives pre-calculated monthly payments for common loan rates and terms.
- An example of a table is shown on the next slide.

- A couple is buying a vehicle for $20,995.
- They pay $7000 down and finance the remainder at an annual interest rate of 4.5% for 48 months.
- Use the amortization table to determine their monthly payment.

- Solution: The amount being financed is $20,995 – $7000 = $13,995.
- In the table, find the row corresponding to 4.5% and the column corresponding to 4 years.
- This entry is highlighted on the next slide.

- Solution, cont’d: The value 22.803486 indicates the couple will pay $22.803486 for each $1000 they borrowed.
- They will pay $319.14 per month.

- The couple in the previous example borrowed $13,995 to buy a car and will pay the loan over 4 years.
- If their payments are $340.02, what interest rate are they being charged?

- Solution: They are paying $340.02 a month for $13,995, or approximately $24.295820 per $1000.
- Look at the amortization table to see to which interest rate this payment amount corresponds.

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

- If P is the amount of the loan, r is the annual interest rate expressed as a decimal, and t is the length of the loan in years, then the monthly payment for an amortized loan is:

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

- Solution: We have P = 13,995, r = 0.045, and
t = 4.

- Suppose a student accumulated $7800 in student loans which she must now pay over 10 years.
- Determine her monthly payment amount using an interest rate of 3.37%

- Solution: Solution: We have P = 7800,
r = 0.0337, and t = 10.

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

a. $527.38

b. $124.00

c. $722.46

d. $899.13

- Suppose you can afford car payments of $250 per month.
- If a 3-year loan at 4% interest is available, how much can you finance?

- Solution: We know PMT = 250,
r = 0.04, and t = 3.

- Solution, cont’d: Solve for P.
- You can borrow $8468.

- In a rent-to-own transaction, you rent the item at a monthly rate, but after a contracted number of payments, the item becomes yours.
- The difference between the retail price of the item and the total of your monthly payments is the interest.

- Suppose you can rent-to-own a $500 television for 24 monthly payments of $30.
- What amount of interest would you pay for the rent-to-own television?
- What annual rate of simple interest on $500 for 24 months yields the same amount of interest found in part (a)?

- Solution:
- The total of your monthly payments will be 24($30) = $720.
- You will pay $720 - $500 = $220 in interest over the 2 years.

- Solution:
- Solve the simple interest formula for r:
- The equivalent simple interest rate is:

- Home mortgage rates have decreased and Howard plans to refinance his home. He will refinance $85,000 at either 5.25% for 15 years or 5.875% for 30 years.
- In each case, what is his monthly payment and how much interest will he pay?

- The 15-year loan has an interest rate of 5.25%.
- According to the amortization table, the monthly payment per $1000 would be $8.038777.
- Under this loan, Howard’s monthly payment would be $8.038777(85) which is approximately $683.30.

- For the 15-year loan, Howard will pay a total of ($683.30)(12)(15) = $122,994.
- The amount spent on interest is $122,994 - $85,000 = $37,994.

- The 30-year loan has an interest rate of 5.875%, which is not found in the table.
- Using the amortization formula, we find a monthly payment amount of $502.81.

- For the 30-year loan, Howard will pay a total of ($502.81)(12)(30) = $181,011.60.
- The amount spent on interest is $181,011.60 - $85,000 = $96,011.60

- Goals
- Study affordability guidelines
- Study mortgages
- Interest rates and closing costs
- Annual percentage rates
- Down payments

- Suppose you have saved $15,000 toward a down payment on a house and your total yearly income is $45,000. What is the most you could afford to pay for a house?
- Assume you pay 0.5% of the value for insurance, you pay 1.5% of the value for taxes, your closing costs will be $2000, and you can obtain a fixed-rate mortgage for 30 years at 6% interest.
- The solution will be given at the end of the section.

- The 2 most common guidelines for buying a house are:
- The maximum house price is 3 times your annual gross income.
- Your maximum monthly housing expenses should be 25% of your gross monthly income.

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

- Solution:
- The purchase price should be no more than 3($60,000) = $180,000.
- The monthly expenses for mortgage payments, property taxes, and homeowner’s insurance should be no more than

- Some lenders allow monthly expenses up to 38% of the buyer’s monthly income.
- We call the 25% level the low maximum monthly housing expense estimate.
- We call the 38% level the high maximum monthly housing expense estimate.

- Suppose Andrew and Barbara both have jobs, each earning $24,000 a year, and they have no debts.
- What are the low and high estimates of how much they can afford to pay for monthly housing expenses?

- Solution: The low estimate is 25% of the total monthly income.
- The high estimate is 38% of the total monthly income.

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

a. Yes, according to the low maximum guideline.

b. Yes, according to the high maximum guideline.

c. No

- A mortgage is a loan that is guaranteed by real estate.
- The interest rate of a fixed-rate mortgage is set for the entire term.
- The interest rate of an adjustable-rate mortgage (ARM) can change.

- The finalizing of a house purchase is called the closing.
- Points are fees paid to the lender at the time of the closing.
- Loan origination fees
- Discount charges

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

- Suppose you will borrow $80,000 for a home at 6.5% interest on a 30-year fixed-rate mortgage.
- The loan involves a one-point loan origination fee and a one-point discount charge. What are your added costs?
- Note: One point is equal to 1 percent of the loan amount.

- Solution: Each fee will cost you 1% of $80,000, or $800.
- Your total added fees are $1600.

- The annual percentage rate (APR) helps borrowers compare the true cost of a loan.
- The APR includes the annual interest rate, any points, and any other loan processing or private mortgage insurance fees.

- Suppose you plan to borrow $80,000 for a home at 6.5% interest on a 30-year fixed-rate mortgage.
- Determine the loan’s APR if there is a 1-point loan origination fee and a 1-point discount charge.

- Solution: The loan is $80,000 plus points totaling $1600, for a total of $81,600.
- According to the amortization table, the total monthly payment will be 81.6($6.320680) or about $515.77, if the fees were paid monthly instead of at the time of closing.

- Solution, cont’d: The interest rate that corresponds to a loan of $80,000 at 6.5% for 30 years with a monthly payment of $515.77 is found:
- Divide $515.77 by 80 to find the amount per $1000, which is $6.44713.
- In the table, this value corresponds to a rate between 6.5% and 6.75%.

- Solution, cont’d: Use linear interpolation to determine the APR.
- The payment is 76.5% of the way from the payment for 6.5% to the payment for 6.75%.
- The APR is about 6.691%

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

- Note that the APR is always greater than or equal to the stated annual interest rate.

- A down payment on a house is the amount of cash the buyer pays at closing, minus any points and fees.
- Traditionally a down payment is 20% of the value, but can be lower.

- The loan to value ratio of a mortgage is the percent of the home’s value that is not paid for by the down payment.
- For example, a down payment of 20% results in an 80% loan to value ratio.

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

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

- Solution: The maximum price you can afford to pay is your down payment amount divided by 20%.
- The most expensive house you can afford is one that is selling for $125,000.

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

a. Yesb. No

- Suppose you have saved $15,000 toward a down payment on a house and your total yearly income is $45,000. What is the most you could afford to pay for a house?
- Assume you pay 0.5% of the value for insurance, you pay 1.5% of the value for taxes, your closing costs will be $2000, and you can obtain a fixed-rate mortgage for 30 years at 6% interest.

- Your total income is $45,000
- You have $15,000 saved for the purchase
- $2000 will be used for closing costs.
- This leaves $13,000 for a down payment.

- The first affordability guideline says you can spend at most 3($45,000) = $135,000 on a house.
- Next, consider your monthly expenses:
- You would be financing $122,000 at 6% for 30 years.
- The monthly mortgage payments, from the table, would be 122($5.995505) = $732.

- The insurance and taxes are 2% of the home’s value annually.
- This adds $225 to the monthly expenses, for a total monthly expense of $957.

- According to the second affordability guideline you can only afford monthly expenses of at most $938.
- The monthly expenses for this house are above your maximum. You cannot afford it.

- A house priced $135,000 is slightly out of your reach, so your options are:
- Wait for interest rates to fall.
- Increase your income.
- Come up with a larger down payment.
- Choose a less expensive house.