MATH 166 Spring 2016
F.4
c
Wen
Liu
F.4 Present Value of Annuities and Amortization
Present Value of an Ordinary Annuity: The present value P V of an ordinary annuity of n = mt
payments of P M T dollars each made at the end of the period with interest compounded at the rate
of r/m per period is
1 − (1 + r/m)−mt
P V = P MT
r/m
Definition: The process of paying the debt is called amortizing (or killing) the debt.
Amortization Formula: The periodic payment P M T to be made at the end of each period on a
loan of P V dollars that is to be amortized over n = mt periods with interest at the rate of r/m per
period is
r/m
P MT = P V
1 − (1 + r/m)−mt
Definition: Outstanding principal (or equity) is the amount paid towards principal.
Examples:
1. Find the amount needed to deposit into an account today that will yield pension payments of
$25000 at the end of each of the next 23 years if the account earns interest at a rate of 5.1%/yr
compounded annually.
2. Two oil wells are for sale. The first will yield payments of $11500 at the end of each of the next
13 years, while the second will yield $6200 at the end of each of the next 28 years. Interest
rates are assumed to hold steady at 4.5% per year over the next 28 years. Which has the higher
present value?
3. Carl is the beneficiary of a $20000 trust fund set up for him by his grandparents. Under the
terms of the trust, he is to receive equal installments from this fund at the end of each year
over a 7-year period. If the fund earns interest at the rate of 10%/year compounded annually,
what amount will he receive each year?
Page 1 of 4
MATH 166 Spring 2016
F.4
c
Wen
Liu
4. A group of private investors purchased a condominium complex for $4 million. They made
an initial down payment of 12% and obtained financing for the balance. The loan is to be
amortized over 13 years at an interest rate of 10% per year compounded quarterly.
(a) What is the required quarterly payment?
(b) How much total interest will be paid on the loan?
5. Lupe made a down payment of $2100 toward the purchase of a new car. To pay the balance of
the purchase price, she has secured a loan from her bank at the rate of 12%/year compounded
monthly. Under the terms of her finance agreement she is required to make payments of
$220/month for 24 months. What is the cash price of the car?
6. Five years ago, Diane secured a bank loan of $360000 to help finance the purchase of a loft
in the San Francisco Bay area. The term of the mortgage was 30 years, and the interest rate
was 10% per year compounded monthly on the unpaid balance. Because the interest rate for
a conventional 30-year home mortgage has now dropped to 5% per year compounded monthly,
Diane is thinking of refinancing her property.
(a) What is Diane’s current monthly mortgage payment?
(b) What is Diane’s current outstanding balance?
Page 2 of 4
MATH 166 Spring 2016
F.4
c
Wen
Liu
(c) If Diane decides to refinance her property by securing a 30-year home mortgage loan in the
amount of the current outstanding principal at the prevailing interest rate of 5% per year
compounded monthly, what will be her monthly mortgage payment? Use the rounded
outstanding balance.
(d) How much less would Diane’s monthly mortgage payment be if she refinances? Use the
rounded values from parts (a)-(c).
7. The Martinezes are planning to refinance their home (assuming that there are no additional
finance charges). The outstanding balance on their original loan is $200000. Their finance
company has offered them two options:
Option A: A fixed-rate mortgage at an interest rate of 6.5% per year compounded monthly,
payable over a 25-year period in 300 equal monthly installments.
Option B: A fixed-rate mortgage at an interest rate of 6.25% per year compounded monthly,
payable over a 15-year period in 180 equal monthly installments.
(a) Find the monthly payment required to amortize each of these loans over the life of the
loan.
(b) How much interest would the Martinezes save if they chose the 15-year mortgage instead
of the 25-year mortgage? Use the rounded monthly payment values from part (a).
8. The Turners have purchased a house for $140000. They made an initial down payment of
$20000 and secured a mortgage with interest charged at the rate of 10%/year compounded
monthly on the unpaid balance. The loan is to be amortized over 30 yr.
(a) What monthly payment will the Turners be required to make?
Page 3 of 4
MATH 166 Spring 2016
F.4
c
Wen
Liu
(b) How much total interest will they pay on the loan?
(c) What will be their equity after 10 years?
9. Find the amortization table for a $8000 loan amortized in five annual payments if the interest
rate is 6.5% per year compounded annually.
End of
Period
0
1
2
3
4
5
Payment
Made
Payment Toward Payment Toward Outstanding
Interest
Principal
Principal
8000
Page 4 of 4