Amortization Formulas

1
Amortization Formulas

• Section 5.4

2
Periodic Payment

• A loan is essentially a loan purchased by the
  lending institution in which the borrower agrees
  to make regular periodic payments.
• So to find the payment amount for a loan we can
  use the formula to find the payment for an
  annuity

3
Periodic Payment, example

• Find the monthly payment for a 7200 loan at
  10.2 annual interest compounded monthly to be
  repaid over 4 years.
• We can use the formula with A 7200, n 412,
  and r .102/12.

4
Outstanding Principal

• We can always find the outstanding principal
  remaining at the beginning of any period by
  filling out an amortization table up to that
  point.
• But suppose we want to find the principal due at
  the beginning of the 20th period, that can take a
  lot of work.
• So well try to find an easier way of computing
  this.

5

• Lets look back at the time value equation of the
  loan.
• The amount of the loan is equal to the present
  value of all the payments.

A
R(1r)-1
R(1r)-2
R(1r)-3
R(1r)-(n-2)
R(1r)-(n-1)
R(1r)-n
So A R(1r)-1 R(1r)-2 R(1r)-3 . . .
R(1r)-(n-2) R(1r)-(n-1) R(1r)-n
6

• Now lets look at this equation at the beginning
  of the kth period.

A
A(1r)k-1
R
R(1r)1
R(1r)k-3
R(1r)k-2
R(1r)-1
R(1r)-(n-k)
R(1r)-(n-k1)
The values in red are the present values of the
payments yet to be made.
The sum of these gives us the principal at the
beginning of the kth payment period.
7

• But this is just the present value of an ordinary
  annuity.
• We had n payments and have already made k - 1, so
  we have n - (k - 1) n - k 1 payments left.
• So we end up with the following formula for the
  outstanding principal at the beginning of the kth
  payment period

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Outstanding Principal, example

• Find the outstanding principal at the beginning
  of the 16th year of a 30-year, 200,000 home loan
  at 6.24 compounded monthly with monthly
  payments.
• First we need to find the monthly payment.
• A 200,000, n 3012, r .0624/12
• Using k 1512 1, (so n - (k - 1) 180) we
  get

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Interest and Principal in kth payment

• Now that we can find the balance at the beginning
  of any period, it is easy to find the interest
  and principal in any payment.
• To find the interest in any payment we multiply
  the outstanding balance by the periodic interest
  rate r.
• So the interest in the kth payment is given by
• To get the principal in the kth payment we simply
  subtract the interest from the payment amount

10
Interest and Principal in kth payment

• Find the interest and principal paid in the
  payment at the beginning of the 16th year for the
  loan above.
• We have R 1230.13 (R 1230.13393), so

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Total Interest Paid

• We have one last formula, the total interest paid
  during the term of a loan.
• We get this by subtracting the amount of the loan
  from the total amount paid out.
• This is simply nR - A.

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• Jared takes out a 6-year loan to purchase a
  17,000 car. If the interest on the loan is 9.3
  compounded monthly, find
• The monthly payment.
• The outstanding balance at the beginning of the
  5th year.
• The interest in the first payment of the 5th
  year.
• The principal in the first payment of the 5th
  year.
• a)
• b)
• c) Multiply the answer in b) by .00775 I49
  6742.79(.00775) 52.26
• d) Subtract the answer in c) from the payment
  P49 308.97 - 52.26 256.71

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Key Suggested Problems

• Sec. 5.4 12, 13, 16, 22
• Final exam, Monday, March 13, 530-718 pm
• Mills - UH 014
• Nikolov - MQ 264