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
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
1
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
2
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
3
? 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.
4
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.
5
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