Introduction to Valuation: The
Time Value of Money
Chapter 4
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Prepare for Capital Budgeting
Part 2: Understand financial statement and cash flow
C2-Identify cash flow from financial statement
C3-Financial statement and comparison
Part 3: Valuation of future cash flow
C4-Basic concepts
C5-More exercise
Part 4: Valuing stocks and bonds
C6-Bond
C7-Stock
Part 5: Capital budgeting
4.1
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Chapter Outline
1.
2.
3.
4.
5.
Future Values: Definitions and Formula
Present Values
PV – Important Relationship
Calculate Rates and Number of Periods
Calculator Keys
4.2
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1. Example
Suppose you put $5000 in bank for one year at 3%
interest rate per year. What is the value of your
money in one year?
 Interest = 5000(.03) = 150
 Value in one year = principal + interest
= 5000+5000(.03) = 5000 + 150 = 5150
= 5000(1 + .03) = 5150
 Suppose you leave the money in for another year.
How much will you have two years from now?
 FV = [5000(1.03)](1.03)

= 5000(1.03)2 = 5304.5
4.3
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Example (cont..)

1 year: Value = 5000(1+.03)

2 years: Value = [5000(1+.03)](1+.03)

3 years:
Value = {[5000(1+.03)](1+.03)} (1+.03)

4 years: Value =
{{[5000(1+.03)](1+.03)} (1+.03)} (1+.03)
4.4
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Example (cont..)
1 year: Value = 5000(1+.03)
= 5000(1+.03)1
 2 years: Value = [5000(1+.03)](1+.03)
= 5000(1+.03)2
 3 years:
Value = {[5000(1+.03)](1+.03)} (1+.03)
= 5000(1+.03)3
 4 years: Value =
{{[5000(1+.03)](1+.03)} (1+.03)} (1+.03)
= 5000(1+.03)4

4.5
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Basic Definitions
FV = PV(1 + r)t
 Present Value – earlier money on a time line
 Future Value – later money on a time line
 Interest rate – “exchange rate” between earlier money
and later money





Discount rate
Cost of capital
Opportunity cost of capital
Required return
Time value of money: A dollar in hand today is worth
more than a dollar promised at some time in the future.
4.6
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Future Values: General Formula
 FV
= PV(1 + r)t
FV = future value
 PV = present value
 r = period interest rate
 T = number of periods

 Future
value factor = (1 + r)t
4.7
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Effects of Compounding
 Compounding:
the process of accumulating
interest over time to earn more interest.
 Compound interest: interest earned on both the
initial principal and the interest earned from
prior periods.
 Compound interest (total interest)
=Simple interest+Interest on interest
 Simple interest: interest on principal
4.8
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Illustration on Compounding
2004
2005
2006
2007
$5000
$5150
$5304.5
Interest earned (2004-2005)=$150
$150
Interest earned (2004-2006)=150+(150+4.5)=$304.5
$150
$4.5
$150
4.9
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Illustration on Compounding (cont..)
2004
2005
$5000
$5150
2006
$5304.5
2007
$5463.6
Interest earned (04-06)=150+(150+4.5)=$304.5
Interest earned(04-07)=150+(150+4.5)+(150+9.1)=$463.6
$150
$4.5
$9.1
$150
$150
Compounding effect=Total interest earned - simple interest
=463.6 – 150 – 150 – 150 = 463.6 - 3(150) = 13.6
4.10
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Figure 4.1
4.11
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Future Values – Example
Suppose you had a relative deposit $10 at 5.5% interest
200 years ago. How much would the investment be
worth today?
 FV = 10(1.055)200 = 447,189.84
What is the effect of compounding?
 Total interest = FV- PV = 447, 179.84
 Simple interest = 200[(10)(.055)] = 110
 Compounding effect = Total interest – simple interest
= 447,179.84 -110= 447,069.84
Compounding has added $447,069.84 to the value of the
investment.
4.12
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2. Present Values

How much do I have to invest today to have some
amount in the future?
FV = PV(1 + r)t
 Rearrange to solve for PV = FV / (1 + r)t

Present Value factor (Discount factor)= 1 / (1 + r)t
 When we talk about discounting, we mean finding the
present value of some future amount.
 When we just say the “value” of something, we are
talking about the present value unless we specifically
indicate that we want the future value.

4.13
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Example
If you want to have $5000 in your account this year,
how much you should put in the bank last year
given 3% interest rate per year?

(last year) vs.
(this year)
PV= 5000/(1+3%) = $4854
4.14
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PV –Example
 Suppose
your grandmother know you need
$2500 in one year for car down payment. If you
can earn 3% quarterly interest when put the
money in the bank, how much does she need to
give you today?
 PV = 2500 / (1.03)4 = 2221.2
4.15
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3. PV – Important Relationship I

For a given interest rate and future value – the longer
the time period, the lower the present value
 What is the present value of $500 to be received in
5 years? 10 years? The discount rate is 10%
 5 years: PV = 500 / (1.1)5 = 310.46
 10 years: PV = 500 / (1.1)10 = 192.77
Future Value=Present Value+Interest Earned
PV = FV / (1 + r)t
4.16
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PV – Important Relationship II

For a given time period and future value – the higher
the interest rate, the smaller the present value
 What is the present value of $500 received in 5
years if the interest rate is 10%? 15%?
 Rate = 10%: PV = 500 / (1.1)5 = 310.46
 Rate = 15%; PV = 500 / (1.15)5 = 248.58
Future Value=Present Value+Interest Earned
PV = FV / (1 + r)t
4.17
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4. The Basic PV Equation Refresher
= FV / (1 + r)t
 There are four parts to this equation
 PV
 PV,
FV, r and t
 If we know any three, we can solve for the
fourth
4.18
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Discount Rate
 Often
we will want to know what the implied
interest rate is in an investment
 Rearrange the basic PV equation and solve for r
FV = PV(1 + r)t
 r = (FV / PV)1/t – 1

 If
you are using formulas, you will want to
make use of both the yx and the 1/x keys
4.19
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Discount Rate – Example
 You
are looking at an investment that will pay
$1200 in 5 years if you invest $1000 today.
What is the implied rate of interest?
 r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%
4.20
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6. Calculator Keys
 Texas
Instruments BA-II Plus
FV = future value
 PV = present value
 I/Y = period interest rate


Interest is entered as a percent, not a decimal
N = number of periods
 Remember to clear the registers (CLR TVM) after
each problem
 Other calculators are similar in format

4.21
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Calculator Settings (Appendix D)
 Compounding
frequency
Set P/Y=1: Press [2 nd] [I/Y] (P/Y), show {P/Y},
[1] [ENTER]
 Set C/Y=1: [ ] [1] [ENTER] [2 nd] [CPT] (QUIT)

 End
mode and annuities due
Start with [2 nd] [PMT] (BGN)
 Switch between END and BGN using [2 nd]
[ENTER] (SET)
 End with [2 nd] [CPT] (QUIT)

4.22
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Future Values – Example 1
Suppose you invest $1000 for 5 years with 5% interest
rate. How much would you have at the end of 5th year?
 FV = 1000(1.05)5 = 1276.28
 Calculator:
N = 5; I/Y = 5; PV = 1000;
CPT FV = -1276.28

4.23
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Present Value – Example 1

Suppose your grandmother know you need $2500 in
one year for car down payment. If you can earn 3%
quarterly when put the money in the bank, how much
does she need to give you today?


PV = 2500 / (1.03)4 = 2221.2
Calculator
N=4
 I/Y=3
 FV=2500
 CPT PV = -2221.2

4.24
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Using Financial Calculator
 When
using a financial calculator, be sure and
remember the sign convention or you will
receive an error when solving for r or t
4.25
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Discount Rate – Example 1
 You
are looking at an investment that will pay
$1200 in 5 years if you invest $1000 today.
What is the implied rate of interest?
r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%
 Calculator – the sign convention matters!!!





N=5
PV = -1000 (you pay 1000 today)
FV = 1200 (you receive 1200 in 5 years)
CPT I/Y = 3.714 (%)
4.26
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Number of Periods – Example 1
 You
want to purchase a new car and you are
willing to pay $20,000. If you can invest at 10%
per year and you currently have $15,000, how
long will it be before you have enough money
to pay cash for the car?

Calculator:
I/Y = 10; FV = 20,000; PV = -15,000;
CPT N = 3.02 years
4.27
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Review Questions
1.
 Know how to calculate the future value, present
value, and rate of return of an investment.
 What is the difference between simple interest
and compound interest? How to calculate
compounding effect?
 What is a compounding process and what is a
discounting process?
4.28
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Review Questions (cont..)
3.
 How will discount factor and future value factor
change with the interest rate and length of time?
 As you increase the length of time involved,
what happens to FV for a given PV and rate?
What happens to PV for a given FV and rate?
If you increase the rate, what happens to FV for a
given PV and time length? What happens to PV
for a given FV and time length?
4.29
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