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Chapter 13 Consumer Mathematics: Buying and Saving
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Section 13.1 Simple and Compound Interest <ul><li>Goals </li></ul><ul><ul><li>Study simple interest </li></ul></ul><ul><ul><ul><li>Calculate interest </li></ul></ul></ul><ul><ul><ul><li>Calculate future value </li></ul></ul></ul><ul><ul><li>Study compound interest </li></ul></ul><ul><ul><ul><li>Calculate future value </li></ul></ul></ul>
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Simple Interest <ul><li>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: </li></ul>
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Example 1 <ul><li>Find the interest on a loan of $100 at 6% simple interest for time periods of: </li></ul><ul><ul><li>1 year </li></ul></ul><ul><ul><li>2 years </li></ul></ul><ul><ul><li>2.5 years </li></ul></ul>
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Example 1, cont’d <ul><li>Solution: We have P = 100 and r = 0.06. </li></ul><ul><ul><li>For t = 1 year, the calculation is: </li></ul></ul>
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Example 1, cont’d <ul><li>Solution, cont’d: We have P = 100 and r = 0.06. </li></ul><ul><ul><li>For t = 2 years, the calculation is: </li></ul></ul><ul><ul><li>For t = 2.5 years, the calculation is: </li></ul></ul>
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Future Value <ul><li>For a simple interest loan, the future value of the loan is the principal plus the interest. </li></ul><ul><li>If P represents the principal, I the interest, r the annual interest rate, and t the time in years, then the future value is: </li></ul>
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Example 2 <ul><li>Find the future value of a loan of $400 at 7% simple interest for 3 years. </li></ul>
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Example 2, cont’d <ul><li>Solution: Use the future value formula with P = 400, r = 0.07, and t = 3. </li></ul>
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Example 3 <ul><li>In 2004, Regular Canada Savings Bonds paid 1.25% simple interest on the face value of bonds held for 1 year. </li></ul><ul><li>If the bond is cashed early, the investor receives the face value plus interest for every full month. </li></ul><ul><li>Suppose a bond was purchased for $8000 on November 1, 2004. </li></ul>
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Example 3, cont’d <ul><li>What was the value of the bond if it was redeemed on November 1, 2005? </li></ul><ul><li>What was the value of the bond if it was redeemed on July 10, 2004? </li></ul>
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Example 3, cont’d <ul><li>Solution: If the bond was redeemed on November 1, 2005, it had been held for 1 year. </li></ul><ul><ul><li>The future value of the bond after 1 year is: </li></ul></ul>
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Example 3, cont’d <ul><li>Solution: If the bond was redeemed on July 10, 2004, it had been held for 7 full months. </li></ul><ul><ul><li>The future value of the bond after 7/12 of a year is: </li></ul></ul>
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Ordinary Interest <ul><li>Ordinary interest simplifies calculations by using 2 conventions: </li></ul><ul><ul><li>Each month is assumed to have 30 days. </li></ul></ul><ul><ul><li>Each year is assumed to have 360 days. </li></ul></ul>
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Example 5 <ul><li>A homeowner owes $190,000 on a 4.8% home loan with an interest-only option. </li></ul><ul><ul><li>An interest-only option allows the borrower to pay only the ordinary interest, not the principal, for the first year. </li></ul></ul><ul><li>What is the monthly payment for the first year? </li></ul>
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Example 5, cont’d <ul><li>Solution: Use the simple interest formula. </li></ul><ul><li>The monthly payments are: </li></ul>
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Compound Interest <ul><li>Reinvesting the interest, called compounding , makes the balance grow faster. </li></ul><ul><li>To calculate compound interest, you need the same information as for simple interest plus you need to know how often the interest is compounded. </li></ul>
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Example 6 <ul><li>Suppose a principal of $1000 is invested at 6% interest per year and the interest is compounded annually. </li></ul><ul><li>Find the balance in the account after 3 years. </li></ul>
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Example 6, cont’d <ul><li>Solution: We must calculate the interest at the end of each year and then add that interest to the principal. </li></ul><ul><li>After 1 year: </li></ul><ul><ul><li>The interest is: </li></ul></ul><ul><ul><li>The new balance is $1060.00 </li></ul></ul><ul><ul><ul><li>We could also have used the future value formula. </li></ul></ul></ul>
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Example 6, cont’d <ul><li>Solution, cont’d: </li></ul><ul><li>After 2 years the new balance is: </li></ul><ul><li>After 3 years the new balance is: </li></ul>
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Example 6, cont’d <ul><li>Solution, cont’d: The interest earned each year increases because of the increasing principal. </li></ul>
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Example 6, cont’d <ul><li>Solution, cont’d: The following table shows the pattern in the calculations for subsequent years. </li></ul>
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Compound Interest <ul><li>Shortcut formula rather than calculating for each year: If </li></ul><ul><ul><ul><li>P represents the principal </li></ul></ul></ul><ul><ul><ul><li>r the annual interest rate expressed as a decimal, </li></ul></ul></ul><ul><ul><ul><li>m the number of equal compounding periods per year </li></ul></ul></ul><ul><ul><ul><li>t the time in years </li></ul></ul></ul><ul><ul><ul><li>then the future value of the account is: </li></ul></ul></ul>
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Example 7 <ul><li>Find the future value of each account at the end of 3 years if the initial balance is $2457 and the account earns: </li></ul><ul><ul><li>4.5% simple interest. </li></ul></ul><ul><ul><li>4.5% compounded annually. </li></ul></ul><ul><ul><li>4.5% compounded every 4 months. </li></ul></ul><ul><ul><li>4.5% compounded monthly. </li></ul></ul><ul><ul><li>4.5% compounded daily. </li></ul></ul>
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Example 7, cont’d <ul><li>Solution: We have P = 2457 and t = 3. </li></ul><ul><ul><li>We have r = 0.045 with simple interest. </li></ul></ul><ul><ul><li>We have r = 0.045 compounded annually. </li></ul></ul>
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Example 7, cont’d <ul><li>Solution, cont’d: We have r = 0.045 </li></ul><ul><ul><li>Compounded every 4 months: </li></ul></ul><ul><ul><li>Compounded monthly: </li></ul></ul><ul><ul><li>Compounded daily: </li></ul></ul>
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Example 7, cont’d <ul><li>Solution, cont’d: The results are summarized below. </li></ul>
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Section 13.2 Loans <ul><li>Goals </li></ul><ul><ul><li>Study amortized loans </li></ul></ul><ul><ul><ul><li>Use an amortization table </li></ul></ul></ul><ul><ul><ul><li>Use the amortization formula </li></ul></ul></ul><ul><ul><li>Study rent-to-own </li></ul></ul>
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13.2 Initial Problem <ul><li>Home mortgage rates have decreased and Howard plans to refinance his home. </li></ul><ul><li>He will refinance $85,000 at either 5.25% for 15 years or 5.875% for 30 years. </li></ul><ul><li>In each case, what is his monthly payment and how much interest will he pay? </li></ul><ul><ul><li>The solution will be given at the end of the section. </li></ul></ul>
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Simple Interest Loans <ul><li>The interest on a simple interest loan is simple interest on the amount currently owed. </li></ul><ul><li>The simple interest each month is called the finance charge . </li></ul><ul><ul><li>Finance charges are calculated using an average daily balance and a daily interest rate . </li></ul></ul>
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Example 1, cont’d <ul><li>Assuming the billing period is June 10 through July 9, determine each of the following: </li></ul><ul><li>The average daily balance </li></ul><ul><li>The daily percentage rate </li></ul><ul><li>The finance charge </li></ul><ul><li>The new balance </li></ul>
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Example 1, cont’d <ul><li>Solution: The daily balances are shown below. </li></ul>
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Example 1, cont’d <ul><li>Solution, cont’d: The average daily balance is: </li></ul>
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Example 1, cont’d <ul><li>Solution: The daily percentage rate is: </li></ul><ul><li>Solution: The finance charge is the simple interest on the average daily balance at the daily rate: </li></ul>
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Example 1, cont’d <ul><li>Solution: The new balance is the sum of the previous balance, any new charges, and the finance charge, minus any payments: </li></ul><ul><li>287.84 + 144.10 + 4.33 – 150.00 = 286.27 </li></ul><ul><ul><li>The new balance is $286.27. </li></ul></ul>
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Amortized Loans <ul><li>Amortized loans are simple interest loans with equal periodic payments over the length of the loan. </li></ul><ul><li>The important variables for an amortized loan are: </li></ul><ul><ul><li>Principal </li></ul></ul><ul><ul><li>Interest rate </li></ul></ul><ul><ul><li>Term (length) of the loan </li></ul></ul><ul><ul><li>Monthly payment </li></ul></ul>
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Amortized Loans, cont’d <ul><li>Each payment includes the interest due since the last payment and an amount paid toward the balance. </li></ul><ul><ul><li>The amount paid each month is constant, but the split between principal and interest varies. </li></ul></ul><ul><ul><li>The amount of the last payment may be slightly more or less than usual. </li></ul></ul>
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Example 2 <ul><li>Chart the history of an amortized loan of $1000 for 3 months at 12% interest with monthly payments of $340. </li></ul>
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Example 2, cont’d <ul><li>Solution: Monthly payment #1: </li></ul><ul><ul><li>The interest owed is </li></ul></ul><ul><ul><li>The payment toward the principal is </li></ul></ul><ul><ul><li>$340 - $10 = $330 </li></ul></ul><ul><ul><li>The new balance is $1000 - $330 = $670. </li></ul></ul>
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Example 2, cont’d <ul><li>Solution, cont’d: Monthly payment #2: </li></ul><ul><ul><li>The interest owed is </li></ul></ul><ul><ul><li>The payment toward the principal is </li></ul></ul><ul><ul><li>$340 - $6.70 = $333.30 </li></ul></ul><ul><ul><li>The new balance is $670 - $333.30 = $336.70 </li></ul></ul>
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Example 2, cont’d <ul><li>Solution, cont’d: Monthly payment #3: </li></ul><ul><ul><li>The interest owed is </li></ul></ul><ul><ul><li>The remaining balance plus the interest is: $336.70 + $3.37 = $340.07. </li></ul></ul><ul><ul><li>The third and final payment is $340.07. </li></ul></ul>
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Example 2, cont’d <ul><li>Solution, cont’d: The amortization schedule for this loan is shown below. </li></ul>
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Example 3 <ul><li>A couple is buying a vehicle for $20,995. </li></ul><ul><li>They pay $7000 down and finance the remainder at an annual interest rate of 4.5% for 48 months. </li></ul><ul><li>Use the amortization table to determine their monthly payment. (For any assignment you may use an amortization calculator on the internet) </li></ul>
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Example 3, cont’d <ul><li>Solution: The amount being financed is $20,995 – $7000 = $13,995. </li></ul><ul><li>In the table, find the row corresponding to 4.5% and the column corresponding to 4 years. </li></ul><ul><ul><li>This entry is highlighted on the next slide. </li></ul></ul>
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Example 3, cont’d <ul><li>Solution, cont’d: The value 22.803486 indicates the couple will pay $22.803486 for each $1000 they borrowed. </li></ul><ul><li>They will pay $319.14 per month. </li></ul>
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Rent-to-Own <ul><li>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. </li></ul><ul><li>The difference between the retail price of the item and the total of your monthly payments is the interest. </li></ul>
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Example 8 <ul><li>Suppose you can rent-to-own a $500 television for 24 monthly payments of $30. </li></ul><ul><li>What amount of interest would you pay for the rent-to-own television? </li></ul><ul><li>What annual rate of simple interest on $500 for 24 months yields the same amount of interest found in part (a)? </li></ul>
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Example 8, cont’d <ul><li>Solution: </li></ul><ul><ul><li>The total of your monthly payments will be 24($30) = $720. </li></ul></ul><ul><ul><li>You will pay $720 - $500 = $220 in interest over the 2 years. </li></ul></ul>
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Example 8, cont’d <ul><li>Solution: </li></ul><ul><ul><li>Solve the simple interest formula for r : </li></ul></ul><ul><ul><li>The equivalent simple interest rate is: </li></ul></ul>
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13.2 Initial Problem Solution <ul><li>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. </li></ul><ul><li>In each case, what is his monthly payment and how much interest will he pay? </li></ul>
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Initial Problem Solution, cont’d <ul><li>The 15-year loan has an interest rate of 5.25%. </li></ul><ul><li>According to the amortization table, the monthly payment per $1000 would be $8.038777. </li></ul><ul><li>Under this loan, Howard’s monthly payment would be $8.038777(85) which is approximately $683.30. </li></ul>
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Initial Problem Solution, cont’d <ul><li>For the 15-year loan, Howard will pay a total of ($683.30)(12)(15) = $122,994. </li></ul><ul><li>The amount spent on interest is $122,994 - $85,000 = $37,994. </li></ul>
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Initial Problem Solution, cont’d <ul><li>The 30-year loan has an interest rate of 5.875%, which is not found in the table. </li></ul><ul><li>Using the amortization formula, we find a monthly payment amount of $502.81. </li></ul>
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Initial Problem Solution, cont’d <ul><li>For the 30-year loan, Howard will pay a total of ($502.81)(12)(30) = $181,011.60. </li></ul><ul><li>The amount spent on interest is $181,011.60 - $85,000 = $96,011.60 </li></ul>
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Section 13.3 Buying a House <ul><li>Goals </li></ul><ul><ul><li>Study affordability guidelines </li></ul></ul><ul><ul><li>Study mortgages </li></ul></ul><ul><ul><ul><li>Interest rates and closing costs </li></ul></ul></ul><ul><ul><ul><li>Annual percentage rates </li></ul></ul></ul><ul><ul><ul><li>Down payments </li></ul></ul></ul>
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13.3 Initial Problem <ul><li>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? </li></ul><ul><li>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. </li></ul><ul><ul><li>The solution will be given at the end of the section. </li></ul></ul>
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Affordability Guidelines <ul><li>The 2 most common guidelines for buying a house are: </li></ul><ul><ul><li>The maximum house price is 3 times your annual gross income. </li></ul></ul><ul><ul><li>Your maximum monthly housing expenses should be 25% of your gross monthly income. (housing expenses include mortgage payment, insurance and property taxes) </li></ul></ul>
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Example 1 <ul><li>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? </li></ul>
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Example 1, cont’d <ul><li>Solution: </li></ul><ul><ul><li>The purchase price should be no more than 3($60,000) = $180,000. </li></ul></ul><ul><ul><li>The monthly (multiply by 1/12) expenses for mortgage payments, property taxes, and homeowner’s insurance should be no more than </li></ul></ul>
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Affordability Guidelines, cont’d <ul><li>Some lenders allow monthly expenses up to 38% of the buyer’s monthly income. </li></ul><ul><ul><li>We call the 25% level the low maximum monthly housing expense estimate. </li></ul></ul><ul><ul><li>We call the 38% level the high maximum monthly housing expense estimate. </li></ul></ul>
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Example 2 <ul><li>Suppose Andrew and Barbara both have jobs, each earning $24,000 a year, and they have no debts. </li></ul><ul><li>What are the low and high estimates of how much they can afford to pay for monthly housing expenses? </li></ul>
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Example 2, cont’d <ul><li>Solution: The low estimate is 25% of the total monthly income. </li></ul><ul><li>The high estimate is 38% of the total monthly income. </li></ul>
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Mortgages <ul><li>A mortgage is a loan that is guaranteed by real estate. </li></ul><ul><li>The interest rate of a fixed-rate mortgage is set for the entire term. </li></ul><ul><li>The interest rate of an adjustable-rate mortgage (ARM) can change. </li></ul>
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Mortgages, cont’d <ul><li>The finalizing of a house purchase is called the closing . </li></ul><ul><li>Points are fees paid to the lender at the time of the closing. </li></ul><ul><ul><li>Loan origination fees </li></ul></ul><ul><ul><li>Discount charges </li></ul></ul><ul><li>Points and any other expenses paid at the time of the closing are called closing costs . </li></ul>
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Example 3 <ul><li>Suppose you will borrow $80,000 for a home at 6.5% interest on a 30-year fixed-rate mortgage. </li></ul><ul><li>The loan involves a one-point loan origination fee and a one-point discount charge. What are your added costs? </li></ul><ul><ul><li>Note: One point is equal to 1 percent of the loan amount. </li></ul></ul><ul><li>Solution: Each fee will cost you 1% of $80,000, or $800. </li></ul><ul><li>Your total added fees are $1600. </li></ul>
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Down Payment <ul><li>A down payment on a house is the amount of cash the buyer pays at closing, minus any points and fees. </li></ul><ul><ul><li>Traditionally a down payment is 20% of the value, but can be lower. </li></ul></ul><ul><li>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? </li></ul>
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<ul><li>Solution: The maximum price you can afford to pay is your down payment amount divided by 20%. </li></ul><ul><ul><li>The most expensive house you can afford is one that is selling for $125,000. </li></ul></ul>
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13.3 Initial Problem Solution <ul><li>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? </li></ul><ul><li>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. </li></ul>
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Initial Problem Solution, cont’d <ul><li>Your total income is $45,000 </li></ul><ul><li>You have $15,000 saved for the purchase </li></ul><ul><ul><li>$2000 will be used for closing costs. </li></ul></ul><ul><ul><li>This leaves $13,000 for a down payment. </li></ul></ul>
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Initial Problem Solution, cont’d <ul><li>The first affordability guideline says you can spend at most 3($45,000) = $135,000 on a house. </li></ul><ul><li>$135,000 - 13,000 = $122,000 to finance. </li></ul><ul><li>Next, consider your monthly expenses: </li></ul><ul><ul><li>You would be financing $122,000 at 6% for 30 years. </li></ul></ul><ul><ul><li>The monthly mortgage payments would be 122($5.995505) = $732. (from the table or website calculator.) </li></ul></ul>
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Initial Problem Solution, cont’d <ul><li>The insurance and taxes are 2% of the home’s value annually. </li></ul><ul><ul><li>This adds $225 to the monthly expenses, for a total monthly expense of $732 + $225 = $957. </li></ul></ul><ul><li>According to the second affordability guideline you can only afford monthly expenses of at most $938. </li></ul><ul><li>($45,000/12=$3750 and then .25 of $3750=$938) </li></ul><ul><ul><li>The monthly expenses for this house are above your maximum. You cannot afford it. </li></ul></ul>
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Initial Problem Solution, cont’d <ul><li>A house priced $135,000 is slightly out of your reach, so your options are: </li></ul><ul><ul><li>Wait for interest rates to fall. </li></ul></ul><ul><ul><li>Increase your income. </li></ul></ul><ul><ul><li>Come up with a larger down payment. </li></ul></ul><ul><ul><li>Choose a less expensive house. </li></ul></ul>
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Dave Ramsey’s website <ul><li>http://www.daveramsey.com/ </li></ul>
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Chapter 13 Assignment due Tues August 12 <ul><li>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.) </li></ul><ul><li>Section 13.2 pg 829(3, 9, 11, 15, 21 *** for 15 and 21 use an amortization website calculator ) </li></ul><ul><li>Section 13.3 pg 842 Show work on each problem (1, 5, 7, 9, 13, 14, 23) </li></ul><ul><li>NOTE: You may use an amortization website to calculate any amortization rather than the textbook table . </li></ul>
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