# Amortization Schedule Mortgage Mathematical Formula

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```					 Chapter 5

The Time Value
Of Money
We know that receiving \$1 today is worth
more than \$1 in the future. This is due
to OPPORTUNITY COSTS.
The opportunity cost of receiving \$1 in
the future is the interest we could have
Today                              Future
Measuring Opportunity Cost
   Translate \$1 today into its equivalent in
the future (COMPOUNDING).
Today                                      Future

?
   Translate \$1 in the future into its
equivalent today (DISCOUNTING).
Today                                      Future

?
Simple and Compound
Interest
   Simple Interest
– Interest paid on the principal sum only
– Example: Bond Coupon Payments
   Compound Interest
– Interest paid on the principal and on prior
interest which has not been paid or withdrawn
– Example: Certificates of Deposit (CD’s)
Types of Time Value
Problems
 Single Sum: One dollar amount
 Annuity: A series of equal cash flows for a
specified number of periods at a constant
interest rate
– Ordinary annuity: cash flows occur at the end
of each period
– Annuity due: cash flows occur at the beginning
of each period
Ordinary Annuity and Annuity Due

7
Mathematical Formulas
   FV = PV ( 1 + i ) n

   PV = FV       1
( 1 + i )n

 FVIFA = ( 1 + i ) n - 1
i
1
 PVIFA = 1 - ( 1+ i ) n
i
Solving Time Value Problems
   Mathematical solutions are the most accurate
– Formulas in the text book
– Complex for annuity problems
   We will be utilizing the Tables in the Text:
– Table I: Future Value          Table III: FV Annuity
– Table II: Present Value        Table IV: PV Annuity
   Each problem will have four variables, three will
be given and the fourth will need to be calculated
Decision Tree Approach
1) Decide between Single Sum or Annuity
2) Decide between Present Value or Future Value
– Present Value: Bring future amounts back to present
– Future Value: Value of present amounts in future
– Be careful of wording to questions (land example)
3) Apply appropriate formula/Use appropriate Table
Time Value Decision Tree
Number of
Payments?

Single Sum                                                              Annuity
(One Payment)                                                     (More than One Payment)

Present Value                     Future Value             Ordinary Annuity                                          Annuity Due
PV=FV(PVIF i,n)                   FV=PV(FVIF i,n)            (End of Period)                                      (Beginning of Period)
Use Table II                      Use Table I

Present Value              Future Value               Present Value              Future Value
PVAN=PMT(PVIFA i,n)        FVAN=PMT(FVIFA i,n)      PVAND=PMT(PVIFA i,n)(1+i) FVAND=PMT(FVIFA i,n)(1+i)
Use Table IV               Use Table III              Use Table IV               Use Table III
Am I Using the Right Table?
   Single Sum Problems:
PV        – Factor is always < 1
FV        – Factor is always > 1
   Annuity Problems:
PVAN      – Factor is always < n
FVAN      – Factor is always > n
Single Sum Formulas
Future Value:        FV = PV ( FVIFi,n )
Present Value:       PV = FV ( PVIFi,n )

FV       Future Value
PV       Present Value
FVIF     Future Value Interest Factor (Table I)
PVIF     Present Value Interest Factor (Table II)
n        Number of Periods
i        Interest Rate
Future Value - single sums
If you deposit \$100 in an account earning 6%,
how much would you have in the account
after 5 years?
PV = 100                        FV =    ?

0                               5
Mathematical Solution:
FV = PV (FVIF i, n )
Present Value - single sums
If you will receive \$100 5 years from now,
what is the PV of that \$100 if your opportunity
cost is 6%?
PV =   ?                         FV = 100

0                                 5

Mathematical Solution:
PV = FV (PVIF i, n )
Annuity Formulas
FV Ordinary Annuity: FVAN = PMT ( FVIFAi,n )
PV Ordinary Annuity: PVAN = PMT ( PVIFAi,n )
FV Annuity Due: FVAND = PMT ( FVIFAi,n ) (1+i)
PV Annuity Due: PVAND = PMT ( PVIFAi,n ) (1+i)

FVAN     Future Value of an Ordinary Annuity
PVAN     Present Value of an Ordinary Annuity
FVAND    Future Value of an Annuity Due
PVAND    Present Value of an Annuity Due
FVIFA    Future Value Interest Factor – Annuity (Table III)
PVIFA    Present Value Interest Factor – Annuity (Table IV)
n        Number of Periods
i        Interest Rate
PMT      Payment
Examples of Annuities

equal coupon interest payments over
the life of the bond.
 If you borrow money to buy a house
or a car, you will pay a stream of
equal payments.
– Amortization Schedule (Chapter 19)
Future Value – Ordinary Annuity
If you invest \$1,000 at the end of the next 3
years, at 8%, how much would you have after
3 years?
1000          1000    1000

0          1            2        3

Mathematical Solution:
FVAN = PMT (FVIFA i, n )
Present Value – Ordinary Annuity
What is the PV of \$1,000 at the end of each of
the next 3 years, discounted at 8%?

1000          1000    1000

0          1            2        3

Mathematical Solution:
PVAN = PMT (PVIFA i, n )
Ordinary Annuity Calculations
1000        1000        1000

0         1            2          3
Using an interest rate of 8%, we
find that:
 The Future Value (at 3) is
\$3,246.00.
 The Present Value (at 0) is
\$2,577.00.
What Happens Here?
1000       1000          1000
0           1             2           3
 Same time line and \$1000 cash flows
 Cash flows occur at the beginning of each
year, rather than at the end of each year,
making this an “annuity due.”
 What does this change? i, n, PMT, PVIFA?
 Difference is 1 more year of interest (1 + i)
Ordinary Annuity and Annuity Due

22
Future Value - annuity due
If you invest \$1,000 at the beginning of each
of the next 3 years at 8%, how much would
you have at the end of year 3?

Mathematical Solution:       Simply compound the FV of the
ordinary annuity one more period:
FVAND = PMT (FVIFA i, n ) (1 + i)
Present Value – Annuity Due
What is the PV of \$1,000 at the beginning of
each of the next 3 years, discounted at 8%?

Mathematical Solution:       Simply compound the FV of the
ordinary annuity one more period:
PVAND = PMT (PVIFA i, n ) (1 + i)
Solving for i, n, or PMT
   Solve using simple algebra
– You have three “givens”, solve for the fourth
   For interest (i) and number of periods (n), you will
be solving for the factor. Using appropriate Table:
– n given: Go to n on table and scan across to find i
– i given: Go to i on table and scan down to find n
   For payment (PMT), you will find the factor the
same way as if you are solving for PV or FV.
– Make sure you are using the right formula
Solving for Interest (i)
If you deposit \$100 in an account and have
\$133.82 at the end of 5 years, what was the

Mathematical Solution (Can Use Either Formula):
FV    = PV (FVIF i, n )
133.82 = 100 (FVIF i, 5 )  (divide both sides by 100)
1.3382 = FVIF i, 5        (use FVIF Table I)
i      = 6%
Solving for Number of Periods (n)
If you invested \$76,060 in an account today
bearing 10% interest, how many end of year
withdraws of \$10,000 could you make before
all of the money is gone?

Which Formula Should We Use?
PVAN = PMT (PVIFA i, n )
FVAN = PMT (FVIFA i, n )
Other Time Value Problems
   Perpetuity
– Security that pays an equal amount each period forever
(example is preferred stock)
– PVPER0 = PMT/i
   Uneven Cash Flows
– Treated like Single Sum
   Deferred Annuity
– Payout does not start for a number of years
– Combines elements of annuity and single sum problem
   Amortization Schedule (Chapter 19)
– Used for Mortgage, Car and Other Loans
– Repay portion of principal with each payment
Unequal Payments

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