Transcript Document

Time value of money
Some important concepts
Financial management: Lecture 3
Today’s agenda
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Review of what we have learned in the last lecture
Discuss the concept of the time value of money
• present value (PV)
• discount rate (r)
• net present value (NPV)
Learn how to draw cash flows of projects
Learn how to calculate the present value of annuities
Learn how to calculate the present value of
perpetuities
Inflation, real interest rates and nominal interest
rates, and their relationship
Financial management: Lecture 3
What have we learned in the
last lecture
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Three types of business organization and their
characteristics
Three types of financial managers
The goal of corporation and agency problem
The motivation for the study of the financial
market
The seven functions of a financial market
The cost of capital
Financial management: Lecture 3
Financial choices
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Which would you rather receive today?
• TRL 1,000,000,000 ( one billion Turkish
lira )
• USD 850 ( U.S. dollars )
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Both payments are absolutely
guaranteed.
What do we do?
Financial management: Lecture 3
Financial choices
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We need to compare “apples to apples” this means we need to get the TRL:USD
exchange rate
From www.bloomberg.com we can see:
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Therefore TRL 1bn = USD 627
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• USD 1 = TRL 1,594,500
Financial management: Lecture 3
Financial choices with time
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Which would you rather receive?
• $1000 today
• $1200 in one year
Both payments have no risk, that is,
• there is 100% probability that you will be paid
• there is 0% probability that you won’t be paid
Financial management: Lecture 3
Financial choices with time (2)
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Why is it hard to compare ?
•
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$1000 today
$1200 in one year
This is not an “apples to apples” comparison.
They have different units
$100 today is different from $100 in one year
Why?
•
A cash flow is time-dated money
• It has a money unit such as USD or TRL
• It has a date indicating when to receive money
Financial management: Lecture 3
Present value
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In order to have an “apple to apple”
comparison, we convert future payments to the
present values
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•
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this is like converting money in TRL to money in USD
Certainly, we can also convert the present value to the
future value to compare payments we get today with
payments we get in the future.
Although these two ways are theoretically the same,
but the present value way is more important and has
more applications, as to be shown in stock and bond
valuations.
Financial management: Lecture 3
Present value (2)
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The formula for converting future cash flows or
payments:
Ct
PV0 
(1  rt )t
PV0
= present value at time zero
Ct
= cash flow in the future (in year t)
= discount rate for the cash flow in year t
rt
Ct i
Financial management: Lecture 3
Example 1
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What is the present value of $100
received in one year (next year) if the
discount rate is 7%?
• Step 1: draw the cash flow diagram
• Step 2: think !
$100
•
PV<?> $100 PV=?
Step 3: PV=100/(1.07)1=
Year one
Financial management: Lecture 3
Example 2
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What is the present value of $100
received in year 5 if the discount rate is
7%?
• Step 1: draw the cash flow diagram
• Step 2: think !
$100
•
PV<?> $ 100
PV=?
Step 3: PV=100/(1.07)5 =
Financial management: Lecture 3
Year 5
Example 3
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What is the present value of $100
received in year 20 if the discount rate is
7%?
• Step 1: draw the cash flow diagram
• Step 2: think !
PV<?> $ 100
PV=?
• Step 3: PV=100/(1.07)20 =
Financial management: Lecture 3
$100
Year 20
Present value of multiple cash
flows
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For a cash flow received in year one and a
cash flow received in year two, different
discount rates must be used.
The present value of these two cash flows is
the sum of the present value of each cash flow,
since two present value have the same unit:
time zero USD.
PVt 0 (C1, C2 )  PVt 0 (C1 )  PVt 0 (C2 )
 C1 (1  r1 )1  C2 (1  r2 ) 2
Financial management: Lecture 3
Example 4
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John is given the following set of cash flows and
discount rates. What is the PV?
$100
$100
C1  100
r1  10%
C2  100
r2  9%
•
•
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PV=?
Year one
Step 1: draw the cash flow diagram
Step 2: think !
PV<?> $200
Step 3: PV=100/(1.1)1 + 100/(1.09)2 =
Financial management: Lecture 3
Year two
Example 5
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John is given the following set of cash flows and
discount rates. What is the PV?
$100 $200
$50
C1  100
r1  0.1
C2  200
C3  50
r2  0.09
•
•
•
PV=?
Yr 1
Yr 2
Yr 3
r3  0.07
Step 1: draw the cash flow diagram
Step 2: think !
PV<?> $350
Step 3: PV=100/(1.1)1 + 200/(1.09)2 + 50/(1.07)3 =
Financial management: Lecture 3
Projects
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A “project” is a term that is used to describe
the following activity:
• spend some money today
• receive cash flows in the future
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A stylized way to draw project cash flows is
Expected cash flows
Expected cash flows
as follows:
in year one (probably positive)
in year two (probably positive)
Initial investment
(negative cash flows)
Financial management: Lecture 3
Examples of projects
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An entrepreneur starts a company:
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initial investment is negative cash outflow.
future net revenue is cash inflow .
An investor buys a share of IBM stock
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cost is cash outflow; dividends are future cash inflows.
A lottery ticket:
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investment cost: cash outflow of $1
jackpot: cash inflow of $20,000,000 (with some very small
probability…)
Thus projects can range from real investments, to
financial investments, to gambles (the lottery ticket).
Financial management: Lecture 3
Firms or companies
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A firm or company can be regarded as a
set of projects.
• capital budgeting is about choosing the best
projects in real asset investments.
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How do we know one project is worth
taking?
Financial management: Lecture 3
Net present value
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A net present value (NPV) is the sum of
the initial investment (usually made at
time zero) and the PV of expected future
cash flows.
NPV  C0  PV (C1  CT )
T
 C0  
Ct
t 1(1  rt )
t
Financial management: Lecture 3
NPV rule
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If NPV > 0, the manager should go
ahead to take the project; otherwise, the
manager should not.
Financial management: Lecture 3
Example 6
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Given the data for project A, what is the
NPV?
$50
$10
-$50
C0  50
C1  50
r1  7.5%
C2  10
r2  8.0%
Yr 1
• Step1: draw the cash flow graph
• Step 2: think! NPV<?>10
• Step 3: NPV=-50+50/(1.075)+10/(1.08)2 =
Yr 0
Financial management: Lecture 3
Yr 2
Example 1
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John got his MBA from SFSU. When he was interviewed by a
big firm, the interviewer asked him the following question:
A project costs 10 m and produces future cash flows, as shown
in the next slide, where cash flows depend on the state of the
economy.
In a “boom economy” payoffs will be high
•
over the next three years, there is a 20% chance of a boom
•
over the next three years, there is a 50% chance of normal
•
over the next 3 years, there is a 30% chance of a recession
• In a “normal economy” payoffs will be medium
In a “recession” payoffs will be low
In all three states, the discount rate is 8% over all time
horizons.
Tell me whether to take the project or not
Financial management: Lecture 3
Cash flows diagram in each
state
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Boom economy
Normal economy
-$10 m
$8 m
$3 m
$3 m
$7 m
$2 m
$1.5 m
$1 m
$0.9 m
-$10 m
$6 m
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Recession
-$10 m
Financial management: Lecture 3
Example 1 (continues)
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The interviewer then asked John:
• Before you tell me the final decision, how do
you calculate the NPV?
• Should you calculate the NPV at each economy or
take the average first and then calculate NPV
• Can your conclusion be generalized to any
situations?
Financial management: Lecture 3
Calculate the NPV at each
economy
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In the boom economy, the NPV is
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In the average economy, the NPV is
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In the bust economy, the NPV is
• -10+ 8/1.08 + 3/1.082 + 3/1.083=$2.36
• -10+ 7/1.08 + 2/1.082 + 1.5/1.083=-$0.613
• -10+ 6/1.08 + 1/1.082 + 0.9/1.083 =-$2.87
The expected NPV is
0.2*2.36+0.5*(-.613)+0.3*(-2.87)=-$0.696
Financial management: Lecture 3
Calculate the expected cash
flows at each time
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At period 1, the expected cash flow is
•
C1=0.2*8+0.5*7+0.3*6=$6.9
At period 2, the expected cash flow is
•
C2=0.2*3+0.5*2+0.3*1=$1.9
At period 3, the expected cash flows is
•
C3=0.2*3+0.5*1.5+0.3*0.9=$1.62
The NPV is
•
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NPV=-10+6.9/1.08+1.9/1.082+1.62/1.083
=-$0.696
Financial management: Lecture 3
Perpetuities
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We are going to look at the PV of a perpetuity starting one year from
now.
Definition: if a project makes a level, periodic payment into perpetuity,
it is called a perpetuity.
Let’s suppose your friend promises to pay you $1 every year, starting
in one year. His future family will continue to pay you and your future
family forever. The discount rate is assumed to be constant at 8.5%.
How much is this promise worth?
C
C
C
C
C
C
PV
???
Yr1
Yr2
Yr3 Yr4
Yr5
Financial management: Lecture 3
Time=infinity
Perpetuities (continue)
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Calculating the PV of the perpetuity could be hard
PV 
C
(1  r )1

C 

C
(1  r ) 2

1
i 1(1  r )
i
Financial management: Lecture 3
C
(1  r ) 
Perpetuities (continue)
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To calculate the PV of perpetuities, we can have
some math exercise as follows:
1

1
1
(1  r )
S    2    
S   2   3     
  S  S

1 /(1  r )
1
S


1   1  1 /(1  r ) r
Financial management: Lecture 3
Perpetuities (continue)
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Calculating the PV of the perpetuity could also be
easy if you ask George
C
C
C
PV 


(1  r )1 (1  r ) 2
(1  r ) 

1

C
C 
 C.    C.S 
i
r
i 1(1  r )
i 1
i
Financial management: Lecture 3
Calculate the PV of the
perpetuity
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Consider the perpetuity of one dollar
every period your friend promises to pay
you. The interest rate or discount rate is
8.5%.
Then PV =1/0.085=$11.765, not a big
gift.
Financial management: Lecture 3
Perpetuity (continue)
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What is the PV of a perpetuity of paying $C
every year, starting from year t +1, with a
constant discount rate of r ?
PV 
C
(1  r )
t 1

C
(1  r )
C
Yr0
t+1
t 2
C
t+2

C
C
C
(1  r ) 
C
t+3 t+4 T+5
Financial management: Lecture 3
C
Time=t+inf
Perpetuity (continue)
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What is the PV of a perpetuity of paying $C
every year, starting from year t +1, with a
constant discount rate of r ?
PV 
C
(1  r )t 1

C
(1  r )t  2

C
(1  r ) 
 1
1
1 



t
1
2

(1  r )  (1  r ) (1  r )
(1  r ) 
C
C

1
C
1
C


. 

t
i
t r
(1  r ) i 1(1  r ) (1  r )
(1  r )t r
Financial management: Lecture 3
Perpetuity (alternative method)
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What is the PV of a perpetuity that pays $C
every year, starting in year t+1, at constant
discount rate “r”?
•
Alternative method: we can think of PV of a perpetuity
starting year t+1. The normal formula gives us the
value AS OF year “t”. We then need to discount this
value to account for periods “1 to t”
Vt  C
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That is
r
PV 
Vt
(1  r )
t

C
(1  r )t r
Financial management: Lecture 3
Annuities
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Well, a project might not pay you forever.
Instead, consider a project that promises to
pay you $C every year, for the next “T” years.
This is called an annuity.
Can you think of examples of annuities in the
real world?
C
C C C
C
C
PV
???
Yr1
Yr2
Yr3 Yr4
Yr5
Financial management: Lecture 3
Time=T
Value the annuity
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Think of it as the difference between two
perpetuities
•
•
add the value of a perpetuity starting in yr 1
subtract the value of perpetuity starting in yr
T+1
1

C
C
1

PV  
 C 
 r (1  r )T r 
r (1  r )T r


Financial management: Lecture 3
Example for annuities
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you win the million dollar lottery! but wait,
you will actually get paid $50,000 per
year for the next 20 years if the discount
rate is a constant 7% and the first
payment will be in one year, how much
have you actually won (in PV-terms) ?
Financial management: Lecture 3
My solution
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Using the formula for the annuity
 1

1
PV  50,000 * 


 0.07 1.07 20 * 0.07 
 $529,700 .71
Financial management: Lecture 3
Example
You agree to lease a car for 4 years at $300
per month. You are not required to pay any
money up front or at the end of your
agreement. If your opportunity cost of
capital is 0.5% per month, what is the cost of
the lease?
Financial management: Lecture 3
Solution
 1

1
Lease Cost  300  

48 
 .005 .0051  .005 
Cost  $12,774.10
Financial management: Lecture 3
Lottery example
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Paper reports: Today’s JACKPOT =
$20mm !!
• paid in 20 annual equal installments.
• payment are tax-free.
• odds of winning the lottery is 13mm:1
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Should you invest $1 for a ticket?
• assume the risk-adjusted discount rate is 8%
Financial management: Lecture 3
My solution
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Should you invest ?
Step1: calculate the PV
1.0mm 1.0mm
1.0mm
PV 


2
(1.08) (1.08)
(1.08) 20

 $9.818 mm
Step 2: get the expectation of the PV
1
1
E[ PV ] 
* 9.818 mm  (1 
)*0
13mm
13mm
 $0.76  $1

Pass up this this wonderful opportunity
Financial management: Lecture 3
Mortgage-style loans

Suppose you take a $20,000 3-yr car loan with
“mortgage style payments”
•
•
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annual payments
interest rate is 7.5%
“Mortgage style” loans have two main
features:
•
•
They require the borrower to make the same payment
every period (in this case, every year)
The are fully amortizing (the loan is completely paid off
by the end of the last period)
Financial management: Lecture 3
Mortgage-style loans
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The best way to deal with mortgage-style loans
is to make a “loan amortization schedule”
The schedule tells both the borrower and
lender exactly:
•
•
•

what the loan balance is each period (in this case year)
how much interest is due each year ? ( 7.5% )
what the total payment is each period (year)
Can you use what you have learned to figure
out this schedule?
Financial management: Lecture 3
My solution
year
Beginning
balance
Interest
payment
Principle
payment
Total
payment
Ending
balance
0
1
$20,000
$1,500
$6,191
$7,691
$13,809
2
13,809
1,036
6,655
7,691
7,154
3
7,154
537
7,154
7,691
Financial management: Lecture 3
0
Future value

The formula for converting the present value to
future value:
FVt i  PVt 0  (1  rt i )i
PVt 0 = present value at time zero
FVt i = future value in year i
rt i = discount rate during the i years
Ct i
Financial management: Lecture 3
Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1629.
Was this a good deal? Suppose the interest rate is 8%.
Financial management: Lecture 3
Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1629.
Was this a good deal?
To answer, determine $24 is worth in the year 2003,
compounded at 8%.
FV  $24  (1  .08)
 $75.979 trillion
374
FYI - The value of Manhattan Island land is
well below this figure.
Financial management: Lecture 3
Inflation
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What is inflation?
What is the real interest rate?
What is the nominal interest rate?
Financial management: Lecture 3
Inflation rule
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Be consistent in how you handle inflation!!
Use nominal interest rates to discount
nominal cash flows.
Use real interest rates to discount real
cash flows.
You will get the same results, whether you
use nominal or real figures
Financial management: Lecture 3
Example
You own a lease that will cost you $8,000 next
year, increasing at 3% a year (the forecasted
inflation rate) for 3 additional years (4 years
total). If discount rates are 10% what is the
present value cost of the lease?
1  real interest rate =
1+ nominal interest rate
1+inflation rate
Financial management: Lecture 3
Inflation
Example - nominal figures
Year
1
2
3
4
Cash Flow
8000
8000x1.03 = 8240
8000x1.03 2 = 8487.20
8000x1.03 3 = 8741.82
PV @ 10%
8000
1.10  7272.73
8240
 6809.92
1.102
8487.20
 6376.56
1.103
8741.82
 5970.78
1.104
$26,429.99
Financial management: Lecture 3
Inflation
Example - real figures
Year
1
2
3
4
Cash Flow
8000
1.03 = 7766.99
8240
= 7766.99
1.032
8487.20
= 7766.99
1.033
8741.82
=
7766.99
4
1.03
[email protected]%
7766.99
1.068  7272.73
7766.99
 6809.92
1.0682
7766.99
 6376.56
1.0683
7766.99
4  5970.78
1.068
= $26,429.99
Financial management: Lecture 3