1. Amortized Loans Section 5.4

2. Introduction The word amortize comes from the Latin word admoritz which means “bring to death”. What we are saying is that we want to bring the debt to death! More gently it is retiring the debt. The important factors related to an amortized loan are the principal, annual interest rate, the length of the loan and the monthly payment. If we know any 3 of the above factors, the fourth can be found.

3. Charting the history of a loan Chart the history of an amortized loan of $1000 for three months at 12% interest, with a monthly payment of $340. When the 1 st payment is made 1/12 of a year has gone by, so the interest is $1000 x.12 x 1/12 = $10. The payment first goes toward paying the interest, then the rest is applied to the unpaid balance. The net payment is $340 - $10 = $330. The new balance is $1000 - $330 = $670. Now we calculate the interest on the remaining balance. $670 x.12 x 1/12 = $6.70. The net payment is $340 - $6.70 = $333.30. The new balance is $670 - $333.30 = $336.70. Once again we calculate the interest on the remaining balance. $336.70 x.12 x 1/12 = $3.37. Thus the last payment has to cover the interest and the remaining balance. This is $3.37 + $336.70 = $340.07. Thus the last payment is $340.07

4. A table of the previous example Beginning balance $1000 PaymentInterestNet Payment New Balance $340.00$10.00$330.00$670.00 $340.00$6.70$333.30$336.70 $340.07$3.37$336.70$0.00

5. Finding a monthly payment Many times we know the length of a loan, the annual interest rate and the amount of the loan. Can we afford to make the monthly payment??? This question is very important when considering a mortgage. The monthly payment formula is basically derived from the equation future value of annuity = future value of loan amount.

6. Payment formula Let P be present value or full amount of loan, r is the annual interest rate, t is the length of the loan and PMT is the monthly payment.

7. Example What is the monthly payment for a loan of $29,000 for 5 years at an annual interest rate of 5%. The monthly payment is $547.27 Note: If you follow this schedule, you will make 60 payments of $547.27 which in total is $32836.20. The amount of interest paid to the lender is $32836.20 - $29000 = $3836.20

8. Example using Table 1 Amortization tables have been created so that people don’t need to use the complicated payment formula. For example, find the monthly payment for a $10000 loan at 10% annual interest for 5 years. Looking at Table 1, this corresponds to the entry of $212.48. Verify using the PMT formula. You may be off by a cent or two, that’s because rounding error was introduced into the table.

9. Another example using table 1 What would be the payment on a loan of $58,000 at 10% annual interest for 30 years? $58000 = $50000 + 4 x $2000 We will use the entries for $50000 at 30 years and $2000 at 30 years. The PMT = $438.79 + 4 x $17.56 = $509.03 Verify using the PMT formula. Rounding error has been introduced.

10. Table 1 - Amortization Table at 10% Amount5 years 10 years 15 years 20 years 25 years 30 years 35 years 40 years 1002.131.331.080.970.910.880.860.85 2004.252.652.151.941.821.761.721.70 50010.636.615.384.834.554.394.304.25 100021.2513.2210.759.669.098.788.608.50 200042.5026.4421.5019.3118.1817.5617.2016.99 5000106.2466.0853.7448.2645.4443.8842.9942.46 10000212.48132.16107.4796.5190.8887.7685.9784.92 20000424.95264.31214.93193.01181.75175.52171.94169.83 500001062.36660.76537.31482.52454.36438.79429.84424.58 1000002124.711321.511074.61965.03908.71877.58859.68849.15

11. Example Using Table 2 Recall that we calculated the monthly payment of a $29000 loan for 5 years at 5% annual interest to be $547.27. Let’s use table 2. The entry that corresponds to 5% for 5 years is $18.871234. Since this is a $1000 table, and the loan amount is for $29000, we multiply the $18.871234 by 29 to get a monthly payment of $547.265786 or properly $547.27. The same as we computed using the formula.

12. Table 2 - Amortization Table for $1000 Loan Percent5 years10 years15 years20 years25 years30 years 518.87123410.605527.9079366.5995575.8459005.368216 619.33280211.1020508.4385687.1643116.4430145.995505 719.80119911.6108488.9882837.7529897.0677926.653025 820.27639412.1327599.5565218.3644017.7181627.337646 920.75835512.66757710.1426668.9972608.3919648.046226 1021.24704513.21507410.7460519.6502169.0870078.775716 1121.74242313.77500111.36596910.3218849.8011319.523234 1222.24444814.34709512.00168111.01086110.53224110.286126 1322.75307314.93107412.65242211.71575711.27835311.061995 1423.26825115.52664413.31741412.43520812.03761011.848718 1523.78993016.13349613.99587113.16789612.80830612.644440