Group 1: Armijo, Hairston, Pitrone
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Practical Exercise
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Learning Objectives
Introduction:
Concept of Value
History
Features
Theories
Measures
Significance
Time Value of Money
Basics
Opportunity Cost
Present vs Future Value
Use in Capital Budgeting
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Concept of Value
History
• Historically, based on an underlying
commodity, such as gold, and the price of gold
regulated the value of money, keeping it
stable.
• Today, the Federal Reserve regulates interest
rates and it is this power that underlies the
value of money.
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Concept of Value
•
•
•
•
Features
The price of money is based on its purchasing power.
This may or may not have anything to do with the
actual prime rate, which is based on the whim of the
banks.
Rationally, the value of money, independent of the
Fed, is based on yesterday's purchasing power.
Therefore, money is based on "pre-existing prices."
Since money is based on what it can buy, a preexisting price must exist for money to have value.
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Concept of Value
Measures
• M1 : the amount of cash available in bank deposits.
• M2 : (M1) + all savings deposits of any kind, as well
as all available liquid money in money markets.
• M3 : (M1 + M2) + all forms of certificates and large
liquid cash investments in dollars worldwide)
broadest conception,
• M1 + M2 +M3 estimate of how much readily
available cash exists and thus gives a rough idea of
the dollar's value.
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Concept of Value
Significance
• Since money is based on what it can buy, a preexisting price must exist for money to have value.
The real significance here is that the value of money
dictates how much your savings, earnings and debt is
worth. If the value of money is low, then your salary
is low, but your debts are equally low.
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Time Value of Money
 Money, like all things, exists in time. Most
people understand that over time, the value of
$1 can change. If you can invest your dollar,
even at a very low rate, it will be worth more in
the future than it is now because it will earn
interest. This change is known as the time
value of money and is an important concept to
understand as you make financial goals and
plans
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Time Value of Money
Basics
 The time value of money is an economic concept that
accounts for the difference in value certain sum of
money has based on the time involved in gaining or
losing it. In essence, the time value of money is a way
of acknowledging the difference between being paid
today and being paid at some future time, requiring a
wait. For most people, waiting for money is much less
desirable than having it immediately. This is because
waiting involves the potential for an opportunity cost.
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Time Value of Money
Opportunity Cost
 An opportunity cost is a loss that results from a missed
opportunity. Opportunity cost is conceptually related to
the time value of money. If a person receives money
sooner rather than later, they can invest or spend it and
enjoy the money’s value. If they must wait, however, the
money is of less value to them because they will miss out
on any opportunities between the present and the time
they receive the money. By determining the value of this
opportunity cost, it is possible to compare the difference
in the value of money lost due to waiting. Capital
budgeting decisions necessarily involve the choice of one
opportunity cost over another.
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Time Value of Money
Present Vs. Future Value
 The time value of money is usually expressed as the
difference between the present value of a sum of
money and that same sum’s future value. The present
value is usually the outright value of the money, if
paid immediately, while the future value is the
amount of money plus interest. This is because
receiving the same amount money in the future
means the loss of an opportunity to earn interest.
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Time Value of Money
Use in Capital Budgeting

Present and future values of money are critical to capital budgeting.

Budgeting requires decisions to allocate or invest money.

By opting to place money into an investment, an individual or a business will be
denying themselves the use of that money until the investment pays off.
 If the investment’s value at maturity exceeds the calculated future value of the
investment’s principal, this could be an excellent choice.
 But if the future value of the money exceeds the value of the investment, it
might be better to choose another investment or keep the money in cash.

As a concept, time value of money provides a means to analyze the opportunity
costs of capital budgeting decisions. Using the time value of money allows these
decisions to take place with a better understanding of whether or not a particular
choice in allocating money is better or worse than other available choices.
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Video
Easy Quiz
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Time Value of Money Pre-Test

1. Diane invests $500 today in an account earning 7%. How much will it be worth in 5 years? _____ 10 years?
_____ 20 years? _____

2. Same facts as #1, except Diane finds an account earning 10%. How much will it be worth in 5 years? _____ 10
years? _____20 years? _____

3. Elaine needs to save up $4,000 in 4 years. If she can set aside $1,000 today, what rate of return does she need
on her account? _____

4. Same facts as in #3, except now Elaine can set aside $50 per month.

What rate of return does she need on her account? _____

5. Frank wants to buy a $10,000 car. The car dealer offers him financing of 60 payments at 9% interest.

What will his payments be? _____

6. Same facts as #5, except the dealer also offers 48 payments at 8%.

Now what will Frank’s payments be? _____

7. Gayle has a credit card with a $500 balance on it and a 19% interest rate. He wants to pay off his card in two
years.

What will his monthly payments be? _____ How much interest will he pay? _____
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Time Value of Money Pre-Test
Diane invests $500 today in an account earning 7%. How much will it be worth in 5 years?
$701
10 years? $984
20 years? $1,935
2. Same facts as #1, except Diane finds an account earning 10%. How much will it be worth in 5 years? $805
10 years? $1,297
20 years? $3,364
3. Elaine needs to save up $4,000 in 4 years. If she can set aside $1,000 today, what rate of return does she
need on her account? 41%
4. Same facts as in #3, except now Elaine can set aside $50 per month. What rate of return does she need on
her account?
24%
5. Frank wants to buy a $10,000 car. The car dealer offers him financing of 60 payments at 9% interest. What
will his payments be?
$208
6. Same facts as #5, except the dealer also offers 48 payments at 8%. Now what will Frank’s payments be?
$244
7. Gayle has a credit card with a $500 balance on it and a 19% interest rate. If he wants to pay off his card in
two years, what will his monthly payments be? $25.20
How much interest will he pay?
$104.80 ($25.20 x 24 = $604.80, less the $500 original balance = $104.80)
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Where to Start: The Timeline
 A timeline is a linear representation of the timing of
potential cash flows.
 Drawing a timeline of the cash flows will help you
visualize the financial problem.
 As simplistic as it sounds…it’ll keep you out of trouble
down the road.
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The Time Line Basics
0
1
2
3
CF1
CF2
CF3
I%
CF0
• Tick marks designate ends of periods.
• Time 0 is the starting point (the beginning of Period 1);
Time 1 is the end of Period 1 (the beginning of Period
2); and so on.
• Inflows are positive cash flows.
• Outflows are negative cash flows identified via a “-”
(minus) sign.
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Three Rules of Time Travel
Financial decisions often require combining cash flows or comparing values.
Three rules govern these processes.
The Three Rules of Time Travel
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The 1st Rule of Time Travel
A dollar today and a dollar in one year are not equivalent.
Why?
Thus, it is only possible to compare or combine values at the
same point in time.
 Example: Which would you prefer: A gift of $1,000 today or
$1,210 at a later date?
 To answer this, you will have to compare the alternatives to
decide which is worth more.
 What factors do we need to consider?
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The 2nd Rule of Time Travel
To move a cash flow forward in time, you must compound it.
Suppose you have a choice between receiving $1,000 today
or $1,210 in two years. You believe you can earn 10% on
the $1,000 today, but want to know what the $1,000 will be
worth in two years. The time line looks like this:
We would technically be indifferent!
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The 2nd Rule of Time Travel
Future Value of a Cash Flow
FVn  C  (1  r )  (1  r ) 
 (1  r )  C  (1  r ) n
n times
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The Composition of Interest Over Time
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Example
Problem: Suppose you have a choice between
receiving $5,000 today or $10,000 in five years.
Which is the better deal (all things equal)?
You believe you can earn 10% on the $5,000
today, but want to know what the $5,000 will
be worth in five years.
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Solution
Solution: The time line looks like this:
1
0
$5,000
x 1.10
$5, 500
2
x 1.10
$6,050
3
x 1.10
$6,655
4
x 1.10
$7,321
5
x 1.10
$8,053
Thus in five years, the $5,000 is expected to grow to:
$5,000 × (1.10)5 = $8,053, so we say the future value of $5,000 at 10% for
five years is $8,053.
Decision: You would be better off forgoing the gift of $5,000 today and
taking the $10,000 in five years.
What might alter this decision?
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The 3rd Rule of Time Travel
 To move a cash flow backward in time, we must
discount it.
 Present Value of a Cash Flow:
PV  C  (1  r )
n
C

(1  r )n
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Example
Problem: Suppose you are offered an investment that
pays $10,000 in five years. If you expect to earn a 10%
return, what is the value of this investment today?
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Solution
Solution: The $10,000 is worth:
$10,000 ÷ (1.10)5 = $6,209
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Applying the Rules of Time Travel
The Three Rules of Time Travel
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Valuing a Stream of Cash Flows
Based on the first rule of time travel we can derive a general formula for
valuing a stream of cash flows: if we want to find the present value of a
stream of cash flows, we simply add up the present values of each.
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Valuing a Stream of Cash Flows
 Present Value of a Cash Flow Stream
PV 
N
 PV (C )
n  0
n

N

n  0
Cn
(1  r ) n
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Future Value of Cash Flow Stream
 Future Value of a Cash Flow Stream with a Present Value of PV
FVn  PV  (1  r ) n
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The Easy Way
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Three Methods to Work TVM Problems
 Solve the FV equation using a regular
calculator.
 Use a financial calculator — that is, one
with financial functions.
 Use a computer with a spreadsheet
program such as Microsoft Excel®.
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Regular Calculator Solution
0
1
2
3
10%
-$100
$110.00
$121.00
$133.10
$100 x 1.10 x 1.10 x 1.10 = $133.10
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Spreadsheet Solution
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What is the PV of $100 due
in three years if I = 10%?
Solve FVN = PV x (1 + I )N for PV
PV = FVN / (1 + I )N
PV = $100 / (1.10)3
= $100(0.7513) = $75.13
If I offer you $75.13 today or $100 three years from
now, which would you prefer?
Spreadsheet Solution
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Solving for I
Assume that a bank offers an account that will pay
$200 after five years on each $75 invested. What
is the implied interest rate?
Spreadsheet Solutions
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Solving for N
Assume an investment earns 20 percent per year.
How long will it take for the investment to double?
Spreadsheet Solutions
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Annuities
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Types of Annuities
Three Year Ordinary Annuity
0
I%
1
2
3
PMT
PMT
PMT
Three Year Annuity Due
0
1
2
PMT
PMT
3
I%
PMT
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What is the FV of a three-year ordinary annuity of
$100 invested at 10%?
0
1
2
$100
$100
3
10%
$100
110
121
FV = $331
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Spreadsheet Solutions
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What is the PV of the annuity?
0
1
2
3
$100
$100
$100
10%
$90.91
82.65
75.13
$248.69 = PV
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Spreadsheet Solutions
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What is the FV and PV if the
annuity were an annuity due?
0
1
2
3
10%
$100
?
$100
$100
?
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What is the FV of the annuity due?
0
1
2
$100
$100
3
10%
$100
110
121
133
FV = $364
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What is the PV of the annuity due?
0
1
2
3
10%
$100
$100.00
$90.91
82.64
$273.55 = PV
$100
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Present Value & the Decision Rule
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PV & NPV Decision Rule
The net present value (NPV) of a project or investment is the difference
between the present value of its benefits and the present value of its costs.
NPV  PV (Benefits)  PV (Costs)
NPV  PV (All project cash flows)
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Calculating the Net Present Value
 Calculating the NPV of future cash flows allows us to evaluate an
investment decision.
 Net Present Value compares the present value of cash inflows (benefits) to
the present value of cash outflows (costs).
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The NPV Decision Rule
When making an investment decision,
take the alternative with the highest NPV.
Choosing this alternative is equivalent to
receiving its NPV in cash today.
Should this ALWAYS be true?
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The NPV Decision Rule (cont'd)
Accepting or Rejecting a Project
 Accept those projects with positive NPV because accepting
them is equivalent to receiving their NPV in cash today.
 Reject those projects with negative NPV because accepting
them would reduce the wealth of investors.
Question: What if the project is a ‘loser’ today, but creates
goodwill in the community and/or preserves NFP status?
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Choosing Among Alternatives
We can also use the NPV decision rule to
choose among projects. To do so, we must
compute the NPV of each alternative, and then
select the one with the highest NPV. This
alternative is the one which will lead to the
largest increase in the value of the firm.
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Uneven Cash Flow Streams: Setup
0
10%
1
$100
2
$300
3
4
$300
-$50
$ 90.91
247.93
225.40
-34.15
$530.09 = PV
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Spreadsheet Solutions
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Return on Investment (ROI)
 The financial performance of an investment is
measured by its return on investment.
 Time value analysis is used to calculate investment
returns.
 Returns can be measured either in dollar terms or
in rate of return terms.
 Assume that a hospital is evaluating a new MRI. The
project’s expected cash flows are given on the next
slide.
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MRI Investment Expected Cash Flows
($$$ in K’s)
0
1
-$1,500
$310
2
$400
3
4
$500
$750
Where do these numbers come from?
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Simple Dollar Return
0
1
2
3
-$1,500 $310 $400
$500
310
400
500
750
$ 460 = Simple dollar return
4
$750
 Is this a good measure?
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Discounted Cash Flow (DCF) Dollar Return
0
8%
1
2
3
-$1,500 $310 $400
$500
287
343
397
551
$ 78 = net present value (NPV)
4
$750
 Is this a better measure?
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Spreadsheet Solution
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Opportunity Cost Rate
 To find an investment’s dollar return (NPV), we need to apply a discount
rate.
 The discount rate is the opportunity cost rate or the rate that could be
earned on alternative investments of similar risk.
 Typically firms apply the WACC (more later)
 It does NOT depend on the source of the investment funds.
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Opportunity Cost Rate (Cont.)
 When calculating NPV, the discounting process automatically recognizes
the opportunity cost of capital. Thus,
 A positive NPV means that the investment is expected to create value for the investor.
 A negative NPV means that the investment is expected to lose value for the investor.
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Rate of (Percentage) Return
0
10%
1
2
3
-$1,500
$310 $400
$500
282
331
376
511
$ 0.00 = NPV, so E(R) = 10.0% .
4
$750
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Spreadsheet Solution
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Rate of Return (Cont.)
 In capital investment analyses, the rate of return often is called internal
rate of return (IRR) or the percentage return expected on the investment.
 To interpret the rate of return, it must be compared to the cost of capital.
In this example, 10 % IRR > 8% WACC, so the project is a winner.
 The IRR & NPV should both indicate whether a project will produce a
positive result (or not).
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Intra-Year Compounding
• Thus far, all examples have assumed
annual compounding.
• When compounding occurs intra-year, the
following occurs:
– Interest is earned on interest during the year
(more frequently).
– The future value of an investment is larger than
under annual compounding.
– The present value of an investment is smaller
than under annual compounding.
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0
1
2
3
10%
-100
133.10
Annual: FV3 = 100 x (1.10)3 = 133.10
0
0
1
1
2
3
2
4
5
3
6
5%
-100
134.01
Semiannual: FV6 = 100 x (1.05)6 = 134.01
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Amortization
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Amortization
Construct an amortization schedule for a
$1,000, 10% annual rate loan with three
equal payments.
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Step 1: Find the required payments
0
1
2
3
PMT
PMT
PMT
10%
-$1,000
Step 2: Find interest charge for Year 1.
Step 3: Find repayment of principal in
Year 1.
Step 4: Find ending balance at end of
Year 1.
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Finally:
Repeat these steps for Years 2 and 3
to complete the amortization table
YR
1
2
3
TOTAL
BEG
BAL
$1,000
698
366
PMT
INT
$402
402
402
$1,206
$100
70
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$206
PRIN
PMT
END
BAL
$302 $698
332
366
366
0
$1,000
Note that annual interest declines over time while the principal payment
increases.
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Questions?
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