Time Value Of Money -Finance

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THE BASIS OFTHE BASIS OF
FINANCEFINANCE
Time Value ofTime Value of
MoneyMoney
© Pearson Education Limited 2004
Fundamentals of Financial Management, 12/e
Created by: Gregory A. Kuhlemeyer, Ph.D.
Carroll College, Waukesha, WI

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After studying this part,After studying this part,
you should be able to:you should be able to:
1. Understand what is meant by "the time value of money."
2. Understand the relationship between present and future value.
3. Describe how the interest rate can be used to adjust the value of
cash flows – both forward and backward – to a single point in
time.
4. Calculate both the future and present value of: (a) an amount
invested today; (b) a stream of equal cash flows (an annuity);
and (c) a stream of mixed cash flows.
5. Distinguish between an “ordinary annuity” and an “annuity due.”
6. Use interest factor tables and understand how they provide a
shortcut to calculating present and future values.
7. Use interest factor tables to find an unknown interest rate or
growth rate when the number of time periods and future and
present values are known.
8. Build an “amortization schedule” for an installment-style loan.

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The Time Value of MoneyThe Time Value of Money
The Interest Rate
Simple Interest
Compound Interest
Amortizing a Loan
Compounding More Than
Once per Year

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Obviously, $10,000 today$10,000 today.
You already recognize that there is
TIME VALUE TO MONEYTIME VALUE TO MONEY!!
The Interest RateThe Interest Rate
Which would you prefer -- $10,000$10,000
todaytoday or $10,000 in 5 years$10,000 in 5 years?

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TIMETIME allows you the opportunity to
postpone consumption and earn
INTERESTINTEREST.
Why TIME?Why TIME?
Why is TIMETIME such an important
element in your decision?

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Types of InterestTypes of Interest
Compound InterestCompound Interest
Interest paid (earned) on any previous
interest earned, as well as on the
principal borrowed (lent).
Simple InterestSimple Interest
Interest paid (earned) on only the original
amount, or principal, borrowed (lent).

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Simple Interest FormulaSimple Interest Formula
FormulaFormula SI = P0(i)(n)
SI: Simple Interest
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods

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SI = P0(i)(n)
= $1,000(.07)(2)
= $140$140
Simple Interest ExampleSimple Interest Example
Assume that you deposit $1,000 in an
account earning 7% simple interest for
2 years. What is the accumulated
interest at the end of the 2nd year?

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Class work 1:Simple InterestClass work 1:Simple Interest
interest at the end of the 3rd
year?

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interest at the end of the 5th
year?

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interest at the end of the 7th
year?

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FVFV = P0 + SI
= $1,000 + $140
= $1,140$1,140
Future ValueFuture Value is the value at some future
time of a present amount of money, or a
series of payments, evaluated at a given
interest rate.
Simple Interest (FV)Simple Interest (FV)
What is the Future ValueFuture Value (FVFV) of the
deposit?

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The Present Value is simply the
$1,000 you originally deposited.
That is the value today!
Present ValuePresent Value is the current value of a
future amount of money, or a series of
payments, evaluated at a given interest
rate.
Simple Interest (PV)Simple Interest (PV)
What is the Present ValuePresent Value (PVPV) of the
previous problem?

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0
5000
10000
15000
20000
1st Year 10th
Year
20th
Year
30th
Year
Future Value of a Single $1,000 Deposit
10% Simple
Interest
7% Compound
Interest
10% Compound
Interest
Why Compound Interest?Why Compound Interest?
FutureValue(U.S.Dollars)

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Assume that you deposit $1,000$1,000 at
a compound interest rate of 7% for
2 years2 years.
Future ValueFuture Value
Single Deposit (Graphic)Single Deposit (Graphic)
0 1 22
$1,000$1,000
FVFV22
7%

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FVFV11 = PP00 (1+i)1
= $1,000$1,000 (1.07)
= $1,070$1,070
Compound Interest
You earned $70 interest on your $1,000
deposit over the first year.
This is the same amount of interest you
would earn under simple interest.
Single Deposit (Formula)Single Deposit (Formula)

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FVFV11 = PP00 (1+i)1
= $1,000$1,000 (1.07)
= $1,070$1,070
FVFV22 = FV1 (1+i)1
= PP00 (1+i)(1+i) = $1,000$1,000(1.07)(1.07)
= PP00 (1+i)2
= $1,000$1,000(1.07)2
= $1,144.90$1,144.90
You earned an EXTRA $4.90$4.90 in Year 2 with
compound over simple interest.

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FVFV11 = P0(1+i)1
FVFV22 = P0(1+i)2
General Future ValueFuture Value Formula:
FVFVnn = P0 (1+i)n
or FVFVnn = P0 (FVIFFVIFi,n) -- See Table ISee Table I
General FutureGeneral Future
Value FormulaValue Formula
etc.

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FVIFFVIFi,n is found on Table I
at the end of the book.
Valuation Using Table IValuation Using Table I
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469

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FVFV22 = $1,000 (FVIFFVIF7%,2)
= $1,000 (1.145)
= $1,145$1,145 [Due to Rounding]
Using Future Value TablesUsing Future Value Tables
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469

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TVM on the CalculatorTVM on the Calculator
Use the highlighted row
of keys for solving any
of the FV, PV, FVA,
PVA, FVAD, and PVAD
problems
N: Number of periods
I/Y: Interest rate per period
PV: Present value
PMT: Payment per period
FV: Future value
CLR TVM: Clears all of the inputs
into the above TVM keys

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Using The TI BAII+ CalculatorUsing The TI BAII+ Calculator
N I/Y PV PMT FV
Inputs
Compute
Focus on 3Focus on 3rdrd
Row of keys (will beRow of keys (will be
displayed in slides as shown above)displayed in slides as shown above)

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Entering the FV ProblemEntering the FV Problem
Press:
2nd
CLR TVM
2 N
7 I/Y
-1000 PV
0 PMT
CPT FV

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N: 2 Periods (enter as 2)
I/Y: 7% interest rate per period (enter as 7 NOT .07)
PV: $1,000 (enter as negative as you have “less”)
PMT: Not relevant in this situation (enter as 0)
FV: Compute (Resulting answer is positive)
Solving the FV ProblemSolving the FV Problem
N I/Y PV PMT FV
Inputs
Compute
2 7 -1,000 0
1,144.90

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Julie Miller wants to know how large her deposit
of $10,000$10,000 today will become at a compound
annual interest rate of 10% for 5 years5 years.
Story Problem ExampleStory Problem Example
0 1 2 3 4 55
$10,000$10,000
FVFV55
10%

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Calculation based on Table I:
FVFV55 = $10,000 (FVIFFVIF10%, 5)
= $10,000 (1.611)
= $16,110$16,110 [Due to Rounding]
Story Problem SolutionStory Problem Solution
Calculation based on general formula:
FVFVnn = P0 (1+i)n
FVFV55 = $10,000 (1+ 0.10)5
= $16,105.10$16,105.10

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Class work 4: FVClass work 4: FV
0 1 2 3 4 55
$10,000$10,000
FVFV55
12%

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0 1 2 3 4 55
$10,000$10,000
FVFV55
24%

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semi-annual interest rate of 24% for 5 years5 years.
0 1 2 3 4 55
$10,000$10,000
FVFV55
24%

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(quarterly) interest rate of 24% for 5 years5 years.
0 1 2 3 4 55
$10,000$10,000
FVFV55
24%

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(monthly) interest rate of 24% for 5 years5 years.
0 1 2 3 4 55
$10,000$10,000
FVFV55
24%

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Entering the FV ProblemEntering the FV Problem
Press:
2nd
CLR TVM
5 N
10 I/Y
-10000 PV
0 PMT
CPT FV

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The result indicates that a $10,000
investment that earns 10% annually
for 5 years will result in a future value
of $16,105.10.
Solving the FV ProblemSolving the FV Problem
N I/Y PV PMT FV
Inputs
Compute
5 10 -10,000 0
16,105.10

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We will use the ““Rule-of-72Rule-of-72””..
Double Your Money!!!Double Your Money!!!
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?

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Approx. Years to Double = 7272 / i%
7272 / 12% = 6 Years6 Years
[Actual Time is 6.12 Years]
The “Rule-of-72”The “Rule-of-72”
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?

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How long does it take to double
$5,000 at a compound rate of 10% per
year (approx.)?
How long does it take to double
$5,000 at a compound rate of % per
year (approx.)?
Class work 9: Doubling ?Class work 9: Doubling ?

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investment that earns 12% annually
will double to $2,000 in 6.12 years.
Note: 72/12% = approx. 6 years
Solving the Period ProblemSolving the Period Problem
N I/Y PV PMT FV
Inputs
Compute
12 -1,000 0 +2,000
6.12 years

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Assume that you need $1,000$1,000 in 2 years.2 years.
Let’s examine the process to determine
how much you need to deposit today at a
discount rate of 7% compounded
annually.
0 1 22
$1,000$1,000
7%
PV1PVPV00
Present ValuePresent Value
Single Deposit (Graphic)Single Deposit (Graphic)

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PVPV00 = FVFV22 / (1+i)2
= $1,000$1,000 / (1.07)2
= FVFV22 / (1+i)2
= $873.44$873.44
Present ValuePresent Value
0 1 22
$1,000$1,000
7%
PVPV00

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PVPV00 = FVFV11 / (1+i)1
PVPV00 = FVFV22 / (1+i)2
General Present ValuePresent Value Formula:
PVPV00 = FVFVnn / (1+i)n
or PVPV00 = FVFVnn (PVIFPVIFi,n) -- See Table IISee Table II
General PresentGeneral Present
Value FormulaValue Formula
etc.

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PVIFPVIFi,n is found on Table II
at the end of the book.
Valuation Using Table IIValuation Using Table II
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681

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PVPV22 = $1,000$1,000 (PVIF7%,2)
= $1,000$1,000 (.873)
= $873$873 [Due to Rounding]
Using Present Value TablesUsing Present Value Tables
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681

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N: 2 Periods (enter as 2)
PV: Compute (Resulting answer is negative “deposit”)
PMT: Not relevant in this situation (enter as 0)
FV: $1,000 (enter as positive as you “receive $”)
Solving the PV ProblemSolving the PV Problem
N I/Y PV PMT FV
Inputs
Compute
2 7 0 +1,000
-873.44

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Julie Miller wants to know how large of a
deposit to make so that the money will
grow to $10,000$10,000 in 5 years5 years at a discount
rate of 10%.
Story Problem Class work 10Story Problem Class work 10
0 1 2 3 4 55
$10,000$10,000
PVPV00
10%

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Calculation based on general formula:
PVPV00 = FVFVnn / (1+i)n
PVPV00 = $10,000$10,000 / (1+ 0.10)5
= $6,209.21$6,209.21
Calculation based on Table I:
PVPV00 = $10,000$10,000 (PVIFPVIF10%, 5)
= $10,000$10,000 (.621)
= $6,210.00$6,210.00 [Due to Rounding]
Story Problem SolutionStory Problem Solution

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Solving the PV ProblemSolving the PV Problem
N I/Y PV PMT FV
Inputs
Compute
5 10 0 +10,000
-6,209.21
future value that will earn 10%
annually for 5 years requires a
$6,209.21 deposit today (present
value).

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Class work 11: PV of FVClass work 11: PV of FV
rate of 12%.
0 1 2 3 4 55
$20,000$20,000
PVPV00
12%

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rate of 18%.
0 1 2 3 4 55
$40,000$40,000
PVPV00
18%

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rate of 20%.
0 1 2 3 4 55
$60,000$60,000
PVPV00
20%

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rate of 25%.
0 1 2 3 4 55
$80,000$80,000
PVPV00
25%

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Types of AnnuitiesTypes of Annuities
Ordinary AnnuityOrdinary Annuity: Payments or receipts
occur at the end of each period.
Annuity DueAnnuity Due: Payments or receipts
occur at the beginning of each period.
An AnnuityAn Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.

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Examples of AnnuitiesExamples of Annuities
Student Loan Payments
Car Loan Payments
Insurance Premiums
Mortgage Payments
Retirement Savings

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Parts of an AnnuityParts of an Annuity
0 1 2 3
$100 $100 $100
(Ordinary Annuity)
EndEnd of
Period 1
EndEnd of
Period 2
Today EqualEqual Cash Flows
Each 1 Period Apart
EndEnd of
Period 3

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Parts of an AnnuityParts of an Annuity
0 1 2 3
$100 $100 $100
(Annuity Due)
BeginningBeginning of
Period 1
Period 2
Today EqualEqual Cash Flows
Each 1 Period Apart
Period 3

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FVAFVAnn = R(1+i)n-1
+ R(1+i)n-2
+
... + R(1+i)1
+ R(1+i)0
Overview of anOverview of an
Ordinary Annuity -- FVAOrdinary Annuity -- FVA
R R R
0 1 2 nn n+1
FVAFVAnn
R = Periodic
Cash Flow
Cash flows occur at the end of the period
i% . . .

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FVAFVA33 = $1,000(1.07)2
+
$1,000(1.07)1
+ $1,000(1.07)0
= $1,145 + $1,070 + $1,000
= $3,215$3,215
Example of anExample of an
Ordinary Annuity -- FVAOrdinary Annuity -- FVA
$1,000 $1,000 $1,000
0 1 2 33 4
$3,215 = FVA$3,215 = FVA33
7%
$1,070
$1,145

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FVAFVA33 = ?
Class work 15 -- FVAClass work 15 -- FVA
$1,000 $1,000 $1,000
0 1 2 33 4
8%

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FVAFVA33 = ?
$1,000 $1,000 $1,000
0 1 2 33 4
9%

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FVAFVA33 = ?
$1,000 $1,000 $1,000
0 1 2 33 4
11%

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Hint on Annuity ValuationHint on Annuity Valuation
The future value of an ordinary
annuity can be viewed as
occurring at the endend of the last
cash flow period, whereas the
future value of an annuity due
can be viewed as occurring at
the beginningbeginning of the last cash
flow period.

61
FVAFVAnn = R (FVIFAi%,n)
FVAFVA33 = $1,000 (FVIFA7%,3)
= $1,000 (3.215) = $3,215$3,215
Valuation Using Table IIIValuation Using Table III
Period 6% 7% 8%
1 1.000 1.000 1.000
2 2.060 2.070 2.080
3 3.184 3.215 3.246
4 4.375 4.440 4.506
5 5.637 5.751 5.867

62
N: 3 Periods (enter as 3 year-end deposits)
PV: Not relevant in this situation (no beg value)
PMT: $1,000 (negative as you deposit annually)
FV: Compute (Resulting answer is positive)
Solving the FVA ProblemSolving the FVA Problem
N I/Y PV PMT FV
Inputs
Compute
3 7 0 -1,000
3,214.90

63
FVADFVADnn = R(1+i)n
+ R(1+i)n-1
+
... + R(1+i)2
+ R(1+i)1
= FVAFVAnn (1+i)
Overview View of anOverview View of an
Annuity Due -- FVADAnnuity Due -- FVAD
R R R R R
0 1 2 3 n-1n-1 n
FVADFVADnn
i% . . .
Cash flows occur at the beginning of the period

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FVADFVAD33 = $1,000(1.07)3
+
$1,000(1.07)2
+ $1,000(1.07)1
= $1,225 + $1,145 + $1,070
= $3,440$3,440
$1,000 $1,000 $1,000 $1,070
0 1 2 33 4
$3,440 = FVAD$3,440 = FVAD33
7%
$1,225
$1,145

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FVADFVAD33 = ?
Class work 18Class work 18
$1,000 $1,000 $1,000
0 1 2 33 4
8%

66
FVADFVAD33 = ?
$1,000 $1,000 $1,000
0 1 2 33 4
9%

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FVADFVAD33 = ?
$1,000 $1,000 $1,000
0 1 2 33 4
11%

68
FVADFVADnn = R (FVIFAi%,n)(1+i)
FVADFVAD33 = $1,000 (FVIFA7%,3)(1.07)
= $1,000 (3.215)(1.07) = $3,440$3,440
Valuation Using Table IIIValuation Using Table III
Period 6% 7% 8%
1 1.000 1.000 1.000
2 2.060 2.070 2.080
3 3.184 3.215 3.246
4 4.375 4.440 4.506
5 5.637 5.751 5.867

69
Solving the FVAD ProblemSolving the FVAD Problem
N I/Y PV PMT FV
Inputs
Compute
3 7 0 -1,000
3,439.94
Complete the problem the same as an “ordinary annuity”
problem, except you must change the calculator setting
to “BGN” first. Don’t forget to change back!
Step 1: Press 2nd
BGN keys
Step 2: Press 2nd
SET keys
Step 3: Press 2nd
QUIT keys

70
PVAPVAnn = R/(1+i)1
+ R/(1+i)2
+ ... + R/(1+i)n
Ordinary Annuity -- PVAOrdinary Annuity -- PVA
R R R
0 1 2 nn n+1
PVAPVAnn
R = Periodic
Cash Flow
i% . . .

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PVAPVA33 = $1,000/(1.07)1
+
$1,000/(1.07)2
+
$1,000/(1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32$2,624.32
$1,000 $1,000 $1,000
0 1 2 33 4
$2,624.32 = PVA$2,624.32 = PVA33
7%
$934.58
$873.44
$816.30

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PVAPVA33 = ?
Class work 21:Class work 21:
$1,000 $1,000 $1,000
0 1 2 33 4
8%

73
PVAPVA33 = ?
$1,000 $1,000 $1,000
0 1 2 33 4
9%

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PVAPVA33 = ?
$1,000 $1,000 $1,000
0 1 2 33 4
11%

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Hint on Annuity ValuationHint on Annuity Valuation
The present value of an ordinary
annuity can be viewed as
occurring at the beginningbeginning of the
first cash flow period, whereas
the future value of an annuity
due can be viewed as occurring
at the endend of the first cash flow
period.

76
PVAPVAnn = R (PVIFAi%,n)
PVAPVA33 = $1,000 (PVIFA7%,3)
= $1,000 (2.624) = $2,624$2,624
Valuation Using Table IVValuation Using Table IV
Period 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993

77
N: 3 Periods (enter as 3 year-end deposits)
PV: Compute (Resulting answer is positive)
PMT: $1,000 (negative as you deposit annually)
FV: Not relevant in this situation (no ending value)
Solving the PVA ProblemSolving the PVA Problem
N I/Y PV PMT FV
Inputs
Compute
3 7 -1,000 0
2,624.32

78
PVADPVADnn = R/(1+i)0
+ R/(1+i)1
+ ... + R/(1+i)n-1
= PVAPVAnn (1+i)
Annuity Due -- PVADAnnuity Due -- PVAD
R R R R
0 1 2 n-1n-1 n
PVADPVADnn
R: Periodic
Cash Flow
i% . . .

79
PVADPVADnn = $1,000/(1.07)0
+ $1,000/(1.07)1
+
$1,000/(1.07)2
= $2,808.02$2,808.02
$1,000.00 $1,000 $1,000
0 1 2 33 4
$2,808.02$2,808.02 = PVADPVADnn
7%
$ 934.58
$ 873.44

80
PVADPVADnn = ?
$1,000.00 $1,000 $1,000
0 1 2 33 4
8%

81
PVADPVADnn = ?
$1,000.00 $1,000 $1,000
0 1 2 33 4
9%

82
PVADPVADnn = ?
$1,000.00 $1,000 $1,000
0 1 2 33 4
11%

83
PVADPVADnn = R (PVIFAi%,n)(1+i)
PVADPVAD33 = $1,000 (PVIFA7%,3)(1.07)
= $1,000 (2.624)(1.07) = $2,808$2,808
Valuation Using Table IVValuation Using Table IV
Period 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993

84
Solving the PVAD ProblemSolving the PVAD Problem
N I/Y PV PMT FV
Inputs
Compute
3 7 -1,000 0
2,808.02
Complete the problem the same as an “ordinary annuity”
problem, except you must change the calculator setting
to “BGN” first. Don’t forget to change back!
Step 1: Press 2nd
BGN keys
Step 2: Press 2nd
SET keys
Step 3: Press 2nd
QUIT keys

85
1. Read problem thoroughly
2. Create a time line
3. Put cash flows and arrows on time line
4. Determine if it is a PV or FV problem
5. Determine if solution involves a single
CF, annuity stream(s), or mixed flow
6. Write the formula
7. Solve the problem
8. Check with financial calculator (optional)
Steps to Solve Time ValueSteps to Solve Time Value
of Money Problemsof Money Problems

86
Julie Miller will receive the set of cash
flows below. What is the Present ValuePresent Value
at a discount rate of 10%10%.
Mixed Flows ExampleMixed Flows Example
0 1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $100
PVPV00
10%10%

87
1. Solve a “piece-at-a-timepiece-at-a-time” by
discounting each piecepiece back to t=0.
2. Solve a “group-at-a-timegroup-at-a-time” by first
breaking problem into groups
of annuity streams and any single
cash flow groups. Then discount
each groupgroup back to t=0.
How to Solve?How to Solve?

88
““Piece-At-A-Time”Piece-At-A-Time”
0 1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $100
10%
$545.45$545.45
$495.87$495.87
$300.53$300.53
$273.21$273.21
$ 62.09$ 62.09
$1677.15$1677.15 == PVPV00 of the Mixed Flowof the Mixed Flow

89
““Group-At-A-Time” (#1)Group-At-A-Time” (#1)
0 1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $100
10%
$1,041.60$1,041.60
$ 573.57$ 573.57
$ 62.10$ 62.10
$1,677.27$1,677.27 == PVPV00 of Mixed Flowof Mixed Flow [Using Tables][Using Tables]
$600(PVIFA10%,2) = $600(1.736) = $1,041.60
$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57
$100 (PVIF10%,5) = $100 (0.621) = $62.10

90
““Group-At-A-Time” (#2)Group-At-A-Time” (#2)
0 1 2 3 4
$400 $400 $400 $400$400 $400 $400 $400
PVPV00 equals
$1677.30.$1677.30.
0 1 2
$200 $200$200 $200
0 1 2 3 4 5
$100$100
$1,268.00$1,268.00
$347.20$347.20
$62.10$62.10
PlusPlus
PlusPlus

91
Use the highlighted
key for starting the
process of solving a
mixed cash flow
problem
Press the CF key
and down arrow key
through a few of the
keys as you look at
the definitions on
the next slide
Solving the Mixed FlowsSolving the Mixed Flows
Problem using CF RegistryProblem using CF Registry

92
Defining the calculator variables:
For CF0: This is ALWAYS the cash flow occurring
at time t=0 (usually 0 for these problems)
For Cnn:* This is the cash flow SIZE of the nth
group of cash flows. Note that a “group” may only
contain a single cash flow (e.g., $351.76).
For Fnn:* This is the cash flow FREQUENCY of the
nth group of cash flows. Note that this is always a
positive whole number (e.g., 1, 2, 20, etc.).
* nn represents the nth cash flow or frequency. Thus, the
first cash flow is C01, while the tenth cash flow is C10.

93
Steps in the Process
Step 1: Press CF key
Step 2: Press 2nd
CLR Work keys
Step 3: For CF0 Press 0 Enter ↓ keys
Step 4: For C01 Press 600 Enter ↓ keys
Step 5: For F01 Press 2 Enter ↓ keys

94
Steps in the Process
Step 10: Press ↓ ↓ keys
Step 11: Press NPV key
Step 12: For I=, Enter 10 Enter ↓ keys
Step 13: Press CPT key
Result: Present Value = $1,677.15

95
General Formula:
FVn = PVPV00(1 + [i/m])mn
n: Number of Years
m: Compounding Periods per Year
i: Annual Interest Rate
FVn,m: FV at the end of Year n
PVPV00: PV of the Cash Flow today
Frequency ofFrequency of
CompoundingCompounding

96
Julie Miller has $1,000$1,000 to invest for 2
Years at an annual interest rate of
12%.
Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2)
= 1,254.401,254.40
Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2)
= 1,262.481,262.48
Impact of FrequencyImpact of Frequency

97
Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2)
= 1,266.771,266.77
Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2)
= 1,269.731,269.73
Daily FV2 = 1,0001,000(1+[.12/365])(365)(2)
= 1,271.201,271.20
Impact of FrequencyImpact of Frequency

98
investment that earns a 12% annual
rate compounded quarterly for 2
years will earn a future value of
$1,266.77.
Solving the FrequencySolving the Frequency
Problem (Quarterly)Problem (Quarterly)
N I/Y PV PMT FV
Inputs
Compute
2(4) 12/4 -1,000 0
1266.77

99
Problem (Quarterly Altern.)Problem (Quarterly Altern.)
Press:
2nd
P/Y 4 ENTER
2nd
QUIT
12 I/Y
-1000 PV
0 PMT
2 2nd
xP/Y N
CPT FV

100
investment that earns a 12% annual
rate compounded daily for 2 years will
earn a future value of $1,271.20.
Problem (Daily)Problem (Daily)
N I/Y PV PMT FV
Inputs
Compute
2(365) 12/365 -1,000 0
1271.20

101
Problem (Daily Alternative)Problem (Daily Alternative)
Press:
2nd
P/Y 365 ENTER
2nd
QUIT
12 I/Y
-1000 PV
0 PMT
2 2nd
xP/Y N
CPT FV

102
Effective Annual Interest Rate
The actual rate of interest earned
(paid) after adjusting the nominal
rate for factors such as the number
of compounding periods per year.
(1 + [ i / m ] )m
- 1
Effective AnnualEffective Annual
Interest RateInterest Rate

103
Basket Wonders (BW) has a $1,000
CD at the bank. The interest rate
is 6% compounded quarterly for 1
year. What is the Effective Annual
Interest Rate (EAREAR)?
EAREAR = ( 1 + 6% / 4 )4
- 1
= 1.0614 - 1 = .0614 or 6.14%!6.14%!
BWs EffectiveBWs Effective
Annual Interest RateAnnual Interest Rate

104
is 8% compounded quarterly for 1
EAREAR = ?
Class work 27: EffectiveClass work 27: Effective

105
is 8% compounded monthly for 1
EAREAR = ?

106
is 8% compounded weekly for 1
EAREAR = ?

107
is 8% compounded daily for 1
EAREAR = ?

108
Converting to an EARConverting to an EAR
Press:
2nd
I Conv
6 ENTER
↓ ↓
4 ENTER
↑ CPT
2nd
QUIT

109
1. Calculate the payment per period.
2. Determine the interest in Period t.
(Loan Balance at t-1) x (i% / m)
3. Compute principal paymentprincipal payment in Period t.
(Payment - Interest from Step 2)
4. Determine ending balance in Period t.
(Balance - principal paymentprincipal payment from Step 3)
5. Start again at Step 2 and repeat.
Steps to Amortizing a LoanSteps to Amortizing a Loan

110
Julie Miller is borrowing $10,000$10,000 at a
compound annual interest rate of 12%.
Amortize the loan if annual payments are
made for 5 years.
Step 1: Payment
PVPV00 = R (PVIFA i%,n)
$10,000$10,000 = R (PVIFA 12%,5)
$10,000$10,000 = R (3.605)
RR = $10,000$10,000 / 3.605 = $2,774$2,774
Amortizing a Loan ExampleAmortizing a Loan Example

111
Amortizing a Loan ExampleAmortizing a Loan Example
End of
Year
Payment Interest Principal Ending
Balance
0 --- --- --- $10,000
1 $2,774 $1,200 $1,574 8,426
2 2,774 1,011 1,763 6,663
3 2,774 800 1,974 4,689
4 2,774 563 2,211 2,478
5 2,775 297 2,478 0
$13,871 $3,871 $10,000
[Last Payment Slightly Higher Due to Rounding]

112
The result indicates that a $10,000 loan
that costs 12% annually for 5 years and
will be completely paid off at that time
will require $2,774.10 annual payments.
Solving for the PaymentSolving for the Payment
N I/Y PV PMT FV
Inputs
Compute
5 12 10,000 0
-2774.10

113
Using the AmortizationUsing the Amortization
Functions of the CalculatorFunctions of the Calculator
Press:
2nd
Amort
1 ENTER
1 ENTER
Results:
BAL = 8,425.90* ↓
PRN = -1,574.10* ↓
INT = -1,200.00* ↓
Year 1 information only
*Note: Compare to 3-82

114
Press:
2nd
Amort
2 ENTER
2 ENTER
Results:
BAL = 6,662.91* ↓
PRN = -1,763.99* ↓
INT = -1,011.11* ↓
Year 2 information only
*Note: Compare to 3-82

115
Press:
2nd
Amort
1 ENTER
5 ENTER
Results:
BAL = 0.00 ↓
PRN =-10,000.00 ↓
INT = -3,870.49 ↓
Entire 5 Years of loan information
(see the total line of 3-82)

116
Usefulness of AmortizationUsefulness of Amortization
2.2. Calculate Debt OutstandingCalculate Debt Outstanding --
The quantity of outstanding
debt may be used in financing
the day-to-day activities of the
firm.
1.1. Determine Interest ExpenseDetermine Interest Expense --
Interest expenses may reduce
taxable income of the firm.

Time Value Of Money -Finance

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