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interest and loans

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    nossi ch 13 updated nossi ch 13 updated Presentation Transcript

    • Chapter 13 Consumer Mathematics: Buying and Saving
    • Section 13.1 Simple and Compound Interest
      • Goals
        • Study simple interest
          • Calculate interest
          • Calculate future value
        • Study compound interest
          • Calculate future value
    • Simple Interest
      • 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:
    • Example 1
      • Find the interest on a loan of $100 at 6% simple interest for time periods of:
        • 1 year
        • 2 years
        • 2.5 years
    • Example 1, cont’d
      • Solution: We have P = 100 and r = 0.06.
        • For t = 1 year, the calculation is:
    • Example 1, cont’d
      • 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:
    • Future Value
      • 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:
    • Example 2
      • Find the future value of a loan of $400 at 7% simple interest for 3 years.
    • Example 2, cont’d
      • Solution: Use the future value formula with P = 400, r = 0.07, and t = 3.
    • Example 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.
    • Example 3, cont’d
      • 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?
    • Example 3, cont’d
      • 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:
    • Example 3, cont’d
      • 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:
    • Ordinary Interest
      • Ordinary interest simplifies calculations by using 2 conventions:
        • Each month is assumed to have 30 days.
        • Each year is assumed to have 360 days.
    • Example 5
      • 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?
    • Example 5, cont’d
      • Solution: Use the simple interest formula.
      • The monthly payments are:
    • Compound Interest
      • 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.
    • Example 6
      • 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.
    • Example 6, cont’d
      • 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.
    • Example 6, cont’d
      • Solution, cont’d:
      • After 2 years the new balance is:
      • After 3 years the new balance is:
    • Example 6, cont’d
      • Solution, cont’d: The interest earned each year increases because of the increasing principal.
    • Example 6, cont’d
      • Solution, cont’d: The following table shows the pattern in the calculations for subsequent years.
    • Compound Interest
      • Shortcut formula rather than calculating for each year: If
          • P represents the principal
          • r the annual interest rate expressed as a decimal,
          • m the number of equal compounding periods per year
          • t the time in years
          • then the future value of the account is:
    • Example 7
      • 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.
    • Example 7, cont’d
      • Solution: We have P = 2457 and t = 3.
        • We have r = 0.045 with simple interest.
        • We have r = 0.045 compounded annually.
    • Example 7, cont’d
      • Solution, cont’d: We have r = 0.045
        • Compounded every 4 months:
        • Compounded monthly:
        • Compounded daily:
    • Example 7, cont’d
      • Solution, cont’d: The results are summarized below.
    • Section 13.2 Loans
      • Goals
        • Study amortized loans
          • Use an amortization table
          • Use the amortization formula
        • Study rent-to-own
    • 13.2 Initial Problem
      • 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.
    • Simple Interest Loans
      • 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 .
    • Example 1
    • Example 1, cont’d
      • 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
    • Example 1, cont’d
      • Solution: The daily balances are shown below.
    • Example 1, cont’d
      • Solution, cont’d: The average daily balance is:
    • Example 1, cont’d
      • Solution: The daily percentage rate is:
      • Solution: The finance charge is the simple interest on the average daily balance at the daily rate:
    • Example 1, cont’d
      • 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.
    • Example 1, cont’d
    • Amortized Loans
      • 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
    • Amortized Loans, cont’d
      • 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.
    • Example 2
      • Chart the history of an amortized loan of $1000 for 3 months at 12% interest with monthly payments of $340.
    • Example 2, cont’d
      • Solution: Monthly payment #1:
        • The interest owed is
        • The payment toward the principal is
        • $340 - $10 = $330
        • The new balance is $1000 - $330 = $670.
    • Example 2, cont’d
      • 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
    • Example 2, cont’d
      • 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.
    • Example 2, cont’d
      • Solution, cont’d: The amortization schedule for this loan is shown below.
    • Example 3
      • 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. (For any assignment you may use an amortization calculator on the internet)
    • Example 3, cont’d
      • 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.
    • Example 3, cont’d
    • Example 3, cont’d
      • 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.
    • Rent-to-Own
      • 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.
    • Example 8
      • 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)?
    • Example 8, cont’d
      • Solution:
        • The total of your monthly payments will be 24($30) = $720.
        • You will pay $720 - $500 = $220 in interest over the 2 years.
    • Example 8, cont’d
      • Solution:
        • Solve the simple interest formula for r :
        • The equivalent simple interest rate is:
    • 13.2 Initial Problem Solution
      • 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?
    • Initial Problem Solution, cont’d
      • 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.
    • Initial Problem Solution, cont’d
      • 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.
    • Initial Problem Solution, cont’d
      • 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.
    • Initial Problem Solution, cont’d
      • 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
    • Section 13.3 Buying a House
      • Goals
        • Study affordability guidelines
        • Study mortgages
          • Interest rates and closing costs
          • Annual percentage rates
          • Down payments
    • 13.3 Initial Problem
      • 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.
    • Affordability Guidelines
      • 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. (housing expenses include mortgage payment, insurance and property taxes)
    • Example 1
      • 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?
    • Example 1, cont’d
      • Solution:
        • The purchase price should be no more than 3($60,000) = $180,000.
        • The monthly (multiply by 1/12) expenses for mortgage payments, property taxes, and homeowner’s insurance should be no more than
    • Affordability Guidelines, cont’d
      • 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.
    • Example 2
      • 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?
    • Example 2, cont’d
      • Solution: The low estimate is 25% of the total monthly income.
      • The high estimate is 38% of the total monthly income.
    • Mortgages
      • 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.
    • Mortgages, cont’d
      • 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 .
    • Example 3
      • 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.
    • Down Payment
      • 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.
      • 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.
    • 13.3 Initial Problem Solution
      • 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.
    • Initial Problem Solution, cont’d
      • 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.
    • Initial Problem Solution, cont’d
      • The first affordability guideline says you can spend at most 3($45,000) = $135,000 on a house.
      • $135,000 - 13,000 = $122,000 to finance.
      • Next, consider your monthly expenses:
        • You would be financing $122,000 at 6% for 30 years.
        • The monthly mortgage payments would be 122($5.995505) = $732. (from the table or website calculator.)
    • Initial Problem Solution, cont’d
      • 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 $732 + $225 = $957.
      • According to the second affordability guideline you can only afford monthly expenses of at most $938.
      • ($45,000/12=$3750 and then .25 of $3750=$938)
        • The monthly expenses for this house are above your maximum. You cannot afford it.
    • Initial Problem Solution, cont’d
      • 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.
    • Dave Ramsey’s website
      • http://www.daveramsey.com/
    • Chapter 13 Assignment due Tues August 12
      • Section 13.1 pg 812 Show use of a formula on each problem (1, 3, 13a, 27, 28 and find the most recently released CPI - I think it was in the news last week.)
      • Section 13.2 pg 829(3, 9, 11, 15, 21 *** for 15 and 21 use an amortization website calculator )
      • Section 13.3 pg 842 Show work on each problem (1, 5, 7, 9, 13, 14, 23)
      • NOTE: You may use an amortization website to calculate any amortization rather than the textbook table .