Amortized Loans

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

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Example using Table 1

• Amortization tables have been created so that
  people dont 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, thats because rounding error was
  introduced into the table.

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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.

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Table 1 - Amortization Table at 10
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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.
• Lets 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.

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Table 2 - Amortization Table for 1000 Loan