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Amortized Loans

Amortized Loans

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Amortized Loans

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  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 1st 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

  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%

  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