# Present Value of an Annuity; Amortization by Levone

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```									    Present Value of an Annuity; Amortization
Section 3-4

Prof. Nathan Wodarz
Math 109 - Fall 2008

Contents
1   Present Value of an Annuity                                                  2
1.1 Present Value of an Ordinary Annuity . . . . . . . . . . . . . . .       2
1.2 Problem Solving Strategy . . . . . . . . . . . . . . . . . . . . . .     3

2   Amortization                                                                 4
2.1 Amortization . . . . . . . . . . . . . . . . . . . . . . . . . . . .     4
2.2 Equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   6

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1     Present Value of an Annuity
1.1    Present Value of an Ordinary Annuity
Present Value of an Ordinary Annuity

• Last section: Paid into an account gradually, accumulated savings

• This section: One lump sum deposited at beginning, slowly paid back

– Loans
– Annuities (as an insurance product)

Present Value of an Ordinary Annuity
1 − (1 + i)−n
PV = PMT
i
• PV = present value (amount) (often denoted S )

• PMT = periodic payment (at end of each period) (often denoted R)

• i = rate per period

• n = number of payments (periods)

• To solve for the payment:
i
PMT = PV
1 − (1 + i)−n

• We will not solve for i

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Present Value of an Ordinary Annuity

Problem 1. Find the present value of the ordinary annuity, with payments of \$50
made quarterly for 10 years at 8% interest compounded quarterly.

A. \$490.90

B. \$1345.13

C. \$1367.77

D. \$1376.77

E. None of the above.

Present Value of an Ordinary Annuity

Problem 2. Tammy borrowed \$10,000 to purchase a new car at an annual interest
rate of 11%. She is to pay it back in equal monthly payments over a 5-year period.
How much total interest will be paid over the period of the loan? Round to the
nearest dollar.

A. \$92

B. \$1435

C. \$3045

D. \$3630

E. None of the above.

1.2   Problem Solving Strategy
Problem Solving Strategy

• In general, single payments will be simple or compound interest

– Look for hints as to whether simple or compound interest is used
– Shorter time periods are often (but not always) simple interest

• Continuing payments involve annuities

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– If account is increasing in value - future value problem
– If account is decreasing in value - present value problem
– Amortization problems (below) are always present value problems

2     Amortization
2.1    Amortization
Amortization

• Borrow money from bank

• Repay it in equal installments

• View as bank buying annuity from you

• After last payment back to bank, loan is amortized (literally “killed oﬀ”)

• Payments determined by earlier formula
i
PMT = PV
1 − (1 + i)−n

Amortization

Problem 3. Find the payment necessary to amortize a loan of \$10,100 at 12%
compounded monthly, if there are to be 48 monthly payments.

A. \$261.74

B. \$265.97

C. \$266.16

D. \$1217.28

E. None of the above.

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Amortization

Problem 4. The monthly payments on a \$73,000 loan at 13% annual interest are
\$807.38. How much of the ﬁrst monthly payment will go toward the principal?

A. \$16.55

B. \$104.96

C. \$702.42

D. \$790.83

E. None of the above.

Amortization Schedules

• How can we compute outstanding loan balances?

• Not as simple as just subtracting payments

– This ignores interest

• Suppose there are n payments left. Outstanding balance is present value
of an annuity with same payments as before, but with the fewer number of
payments.

Amortization Schedules

Problem 5. A \$7,000 debt is to be amortized in 15 equal monthly payments of
\$504.87 at 12% annual interest on the unpaid balance. What is the unpaid bal-
ance after the second payment?

A. \$5,990.26

B. \$6,860.00

C. \$6,971.87

D. \$8,126.78

E. None of the above.

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2.2   Equity
Equity
• For an asset that you take out a loan on, equity measures how much of that
asset you actually “own”
• Equity = (current net market value) - (unpaid loan balance)
• A home equity loan is a loan taken out using the equity in your house as
collateral
– Essentially, taking out a second mortgage
• Equity can be negative - you owe more than the asset is worth

Equity
Problem 6. A home was purchased 14 years ago for \$70,000. The home was
ﬁnanced by paying a 20% down payment and signing a 25 year mortgage at 8.5%
compounded monthly on the unpaid balance. The market value is now \$100,000.
The owner wishes to sell the house. How much equity (to the nearest dollar) does
the owner have in the house after making 168 monthly payments?
A. \$22,913
B. \$51,127
C. \$51,768
D. \$61,414
E. None of the above.

Summary
Summary
You should be able to:
• Calculate present value of an annuity
• Understand amortization
• Be able to compute unpaid balances and equities

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