Present Value of an Annuity; Amortization

Document Sample
Present Value of an Annuity; Amortization Powered By Docstoc
					    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
                                        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




                                       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
                                                  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.




                                        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 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.

                                      5
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


                                       6

				
DOCUMENT INFO