Present Value of an Annuity; Amortization

Present Value of an Annuity; Amortization

Section 3-4

Prof. Nathan Wodarz Math 109 - Fall 2008

Contents

1 Present Value of an Annuity

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1.1 Present Value of an Ordinary Annuity . . . . . . . . . . . . . . . 2

1.2 Problem Solving Strategy . . . . . . . . . . . . . . . . . . . . . . 3

2 Amortization

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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

PV = PMT 1 - (1 + i)-n 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:

PMT

=

PV

1

-

i (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

PMT

=

PV

1

-

i (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 balance 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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