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Amortized Loans. Section 5.4. 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.

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amortized loans

Amortized Loans

Section 5.4

introduction
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.
charting the history of a loan
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
a table of the previous example
A table of the previous example
  • Beginning balance $1000
finding a monthly payment
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.
payment formula
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.
example
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
example using table 1
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.
another example using table 1
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.
example using table 2
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.