Time Value of Money
Present value of future cash
flows or payments
Fin 351: lectures 3-4
This week’s plan




Some administrative issues
Understand the concept of the time value of money
Learn how to compare:
•
•
Cash flows or payments you get today
Cash flows or payments you get in the future
• Understand the following terms:
•
•
•
•
•
present value (PV)
discount rate (r)
net present value (NPV)
annuity
perpetuity
Fin 351: lectures 3-4
Today’s plan (2)



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
Fin 351: lectures 3-4
Financial choices

Which would you rather receive today?
• TRL 1,000,000,000 ( one billion Turkish
lira )
• USD 652.72 ( U.S. dollars )


Both payments are absolutely
guaranteed.
What do we do?
Fin 351: lectures 3-4
Financial choices

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:

Therefore TRL 1bn = USD 610.64

• USD 1 = TRL 1,637,600
Fin 351: lectures 3-4
Financial choices with time


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
Fin 351: lectures 3-4
Financial choices with time (2)




Why is it hard to compare ?
•
•
$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
Fin 351: lectures 3-4
Present value

In order to have an “apple to apple”
comparison, we convert future payments to the
present values
•
•
•
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.
Fin 351: lectures 3-4
Present value (2)

The formula for converting future cash flows or
payments:
PVt 0  Ct i 
1
(1  rt i )i
PVt 0
= present value at time zero
Ct i
rt i
= cash flow in the future (in year i)
= discount rate for the cash flow in year i
Ct i
Fin 351: lectures 3-4
rt i
Example 1

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
Fin 351: lectures 3-4
Example 2

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<?> $
PV=?
Step 3: PV=100/(1.07)5 =
Fin 351: lectures 3-4
Year 5
Example 3

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<?> $
PV=?
• Step 3: PV=100/(1.07)20 =
Fin 351: lectures 3-4
$100
Year 20
Present value of multiple cash
flows


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
Fin 351: lectures 3-4
Example 4

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%
•
•
•
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 =
Fin 351: lectures 3-4
Year two
Example 5

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 =
Fin 351: lectures 3-4
Projects

A “project” is a term that is used to describe
the following activity:
• spend some money today
• receive cash flows in the future

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)
Fin 351: lectures 3-4
Examples of projects




An entrepreneur starts a company:
•
•
initial investment is negative cash outflow.
future net revenue is cash inflow .
An investor buys a share of IBM stock
•
cost is cash outflow; dividends are future cash inflows.
A lottery ticket:
•
•
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).
Fin 351: lectures 3-4
Firms or companies

A firm or company can be regarded as a
set of projects.
• capital budgeting is about choosing the best
projects in real asset investments.

How do we know one project is worth
taking?
Fin 351: lectures 3-4
Net present value

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 )
Fin 351: lectures 3-4
t
NPV rule

If NPV > 0, the manager should go
ahead to take the project; otherwise, the
manager should not.
Fin 351: lectures 3-4
Example 6

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
Fin 351: lectures 3-4
Yr 2
Example 7







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
Fin 351: lectures 3-4
Cash flows diagram in each
state


Boom economy
-$10 m
Normal economy
$8 m
$3 m
$3 m
$7 m
$2 m
$1.5 m
$1 m
$0.9 m
-$10 m
$6 m

Recession
-$10 m
Fin 351: lectures 3-4
Example 7 (continues)

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?
Fin 351: lectures 3-4
Calculate the NPV at each
economy

In the boom economy, the NPV is

In the average economy, the NPV is

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
Fin 351: lectures 3-4
Calculate the expected cash
flows at each time




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
•
•
NPV=-10+6.9/1.08+1.9/1.082+1.62/1.083
=-$0.696
Fin 351: lectures 3-4
Perpetuities



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?
$1
$1 $1
$1
$1
$1
PV
???
Yr1
Yr2
Yr3 Yr4
Yr5
Fin 351: lectures 3-4
Time=infinity
Perpetuities (continue)

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
Fin 351: lectures 3-4
C
(1  r ) 
Perpetuities (continue)

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
Fin 351: lectures 3-4
Perpetuities (continue)

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
Fin 351: lectures 3-4
Calculate the PV of the
perpetuity


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.
Fin 351: lectures 3-4
Perpetuity (continue)

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 )
$1
Yr0
t+1
t 2

$1 $1
t+2
$1
C
(1  r ) 
$1
t+3 t+4 T+5
Fin 351: lectures 3-4
$1
Time=t+inf
Perpetuity (continue)

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
Fin 351: lectures 3-4
Perpetuity (alternative method)

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

That is
r
PV 
Vt
(1  r )
t
Fin 351: lectures 3-4

C
(1  r )t r
Annuities


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?
$1 $1 $1 $1 $1
$1
PV
???
Yr1
Yr2
Yr3 Yr4
Yr5
Fin 351: lectures 3-4
Time=T
Value the annuity

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


Fin 351: lectures 3-4
Example for annuities

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) ?
Fin 351: lectures 3-4
My solution

Using the formula for the annuity
 1

1
PV  50,000 * 


 0.07 1.07 20 * 0.07 
 $529 ,700 .71
Fin 351: lectures 3-4
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?
Fin 351: lectures 3-4
Solution
 1

1
Lease Cost  300  

48 
 .005 .0051  .005 
Cost  $12,774.10
Fin 351: lectures 3-4
Lottery example

Paper reports: Today’s JACKPOT =
$20mm !!
• paid in 20 annual equal installments.
• payment are tax-free.
• odds of winning the lottery is 13mm:1

Should you invest $1 for a ticket?
• assume the risk-adjusted discount rate is 8%
Fin 351: lectures 3-4
My solution


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
Fin 351: lectures 3-4
Mortgage-style loans

Suppose you take a $20,000 3-yr car loan with
“mortgage style payments”
•
•

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)
Fin 351: lectures 3-4
Mortgage-style loans


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?
Fin 351: lectures 3-4
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
Fin 351: lectures 3-4
7,691
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
Fin 351: lectures 3-4
Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1629.
Was this a good deal? Suppose the interest rate is 8%.
Fin 351: lectures 3-4
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)
374
 $75.979 trillion
FYI - The value of Manhattan Island land is
well below this figure.
Fin 351: lectures 3-4