Valuation of Cash Flows
Chapter 3.1-3.3
Chapter 4
outline
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•
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Financial costs and benefits
The Time Value of Money
Present value and future value
Valuation of streams of cash flows
Valuation of Annuities and Perpetuities
Further Examples
Financial Costs and Benefits
Identifying Costs and Benefits
Can we identify the costs and benefits of the
following projects?
• Advertising campaign
• Lowering production cost
• Management training program
What information is required in order to measure
or value the associated costs and benefits?
The Important Role of Market Prices
Suppose you could trade 10 ounces of platinum
and receive 20 ounces of gold, would you trade?
• The price of one ounce of gold is $250
• The price of one ounce of platinum is $550
What if the jeweler thought that the price of
gold was too low?
Market Prices as Cash Value
Suppose the jeweler can produce $10,000 worth of jewelry
from 20 ounces of gold but only $6,000 worth of jewelry from
10 ounces of platinum.
The Jeweler should still not trade 10 ounces of gold for 20
ounces of platinum – why?
When a good is trades in a “competitive market” (which
means that you are free to buy and sell the good at the same
price), price determines the cash value of the good
Non-Competitive Markets
When a good is not traded in a competitive market
(which means that you buy and sell the good at
different prices) its value is determined by your
own personal preferences (or benefit) from
obtaining the good.
Suppose that just before the holidays your boss
surprises you by offering all employees the choice
between a brand new gardening tool-kit and $375.
A brand-new tool-kit costs $450 in stores and can
be sold for $400 through one of the internet
websites. What would you choose?
Non-Competitive Markets
If you have a garden at home and you were
planning on buying the kit anyway then you would
value this gift at $450. Alternatively, if you were not
planning on buying the kit in the first place then
you would value the gift at $375.
When the market for the good is not competitive,
the cash value of the economic opportunity depends
on your preferences (and market prices)
The Time Value of Money
Timing of Payments
For most decisions in life (and also financial decisions) the
costs and benefits occur at different points in time.
• MBA
• Marriage
• Mortgage
Consider the investment opportunity
• $100,000 today
• $105,000 in one year
Are we “comparing apples to oranges”?
How do we compare payments in different times?
Interest Rates
Suppose that the current annual interest rate is 7%.
This means that on a current deposit of $100 we
will earn a total of $7 of interest over the course of
the year leading to a total of $107 in our bank
account.
We refer to this as the risk-free interest rate and
we denote this rate by rf = 7% or 1+rf=1.07
This means that we can exchange (without any risk)
$1 today (or at time “0”) for $1.07 received in one
year (or at time “1”)
Comparing Payments Across Time
Back to our investment opportunity.
We know that the benefit is $105,000 in one year. Can we
calculate the cost of this investment in dollars paid in one year
instead of dollars paid today in order to compare the two?
By using the risk-free interest rate of 7%:
Costs = $100, 000 today
= $100, 000 ´ (1+ rf ) = $107, 000 in one year
The net value of the investment in terms of “dollars in one year”
$105,000-107,000= - $2,000
Comparing Payments Across Time
Back to our investment opportunity (again…). We will lose $2000 “dollars
in one year” if we choose this investment opportunity, but how much
will we loose in terms of “dollars today”?
We know that the cost is $100,000 today. The benefit of this investment
in dollars paid today instead of dollars in one year is
By using the risk-free interest rate of 7%:
Benefits = $105, 000 in one year
$100, 000
=
= $98,130.84 today
1+ rf
The net value of the investment in terms of “dollars today” is
$98,130.84-100,000= - $1,869.16
Comparing Payments Across Time
We valued the investment opportunity in terms of “dollars today” and
“dollars in one year” and both led us to the conclusion not to invest in
the project since it leads to a net loss.
Is it always true that both approaches lead to the same conclusion?
As long as we convert costs and benefits to the same point in time, we
can compare them to make a decision
Not surprisingly
The net value of the investment in terms of “dollars today” and its value
in terms of “dollars in one year” satisfy
- $1,869.16 (1.07) = $2,000
The Net Present Value Decision Rule
The term Present Value or PV refers to the fact that value is calculated in
today’s dollars
The Net Present Value (NPV) of a project or investment opportunity is the
difference between the present value of benefits and the present value
of costs
Net Present Value (NPV)
NPV = PV(Benefits)- PV(Costs)
If we use positive cash-flows to represent benefits and negative cash
flows to represent costs then
NPV = PV(Benefits -Costs)
We accept projects with positive NPV
Present Value & Future Value
of Cash Flows
The Time Line
The first (and very important) tool to develop is
the ability to represent the stream of cash flows
of an economic opportunity on a time line.
The Time Line
Problem: Suppose you must pay tuition of $10,000
per year for the next two years. Your tuition
payments must be made in equal installments at the
start of each semester. What is the timeline of your
tuition payments?
Moving Cash Flows Across Time
When comparing cash flows obtained in different points in
time we need to first convert all cash flows to one point in
time so that we do not “compare apples to oranges”
Rule 1
It is only possible to compare or combine
values at the same point in time
Rule 2
To move a cash flow forward in time, we
must compound it
Rule 3
To move a cash flow back in time, we must
discount it
Compounding Cash Flows
Suppose we have $1,000 today and we want to determine
the equivalent amount in one year’s time. If the annual riskfree interest rate is 10% then we can calculate:
($1000 today)(1.10) = $1100 in one year
What if we were interested in the equivalent amount in two
year’s time? Given that the annual risk-free rate for two
years is also 10% then we can calculate:
Future Value
The value of a cash flow that is moved forward in time is known as
its future value
When interest rate is 10% then,
$1210 is the future value of $1000 two years from today
$1100 is the future value of $1000 one year from today
Future Value
If we were interested in the future value of $1000 three
years from today we would simply multiply one more period
FV3 = $1000 ´(1.10)´(1.10)´(1.10)
Future Value of a Cash Flow “$C”
“n” years from today, given interest “r”
FVn = $C ´ (1+ r)n
The future value is increasing in the interest rate “r” and the
number of periods “n”.
Discounting Cash Flows
Suppose we anticipate having $1,000 in one year and we want to
compute the value today of this payment. If the annual risk-free
interest rate is 10% then we can calculate:
($1000 in one year)/(1.10) = $909.09 today
Suppose we need the money now…does our preference for money
today matter for this decision between $909.09 today or $1000
tomorrow?
Suppose you anticipate receiving $1000 in two years. Given that the
annual risk-free rate for two years is also 10% then we can calculate:
$1000
PV =
= 826.45
2
1.10
Present Value
If we were interested in the present value of $1000 received
three years from today we would simply multiply one more
period
$1000
PV =
= 751.31
3
1.10
Present Value of a Cash Flow “$C”
Received “n” years from today, given interest “r”
$C
PV =
(1+ r)n
The present value is decreasing in the interest rate “r” and
the number of periods “n”
Savings problem
Suppose you plan to save $1000 annually in the next
three years starting from today.
Time line of deposits:
If the interest rate on your savings account is 10%, then
how much money will you have in your account three
years from now?
Savings problem
We are asked to find the future value of the stream of cash
flows.
The future value at time 3 of the payment of $1000 received
at time 0:
The future value at time 3 of the payment of $1000 received
at time 1:
The future value at time 3 of the payment of $1000 received
at time 2:
The future value of the stream of cash flows:
Savings problem
An alternative way to find the future value of the stream of cash
flows is to follow the balance on our saving account.
At time 0 we have …………. in our bank account.
At time 1 we have ………… in our bank account.
At time 2 we have …………. in our bank account.
At time 3 we have accumulated …………….
Savings problem
Now suppose that you wish to invest all the money
now instead of $1000 each year. What size
investment is required in order to obtain the exact
same amount of accumulated funds three years from
now?
You are asked to find the present value of the stream
of cash flows.
The present value of the stream of cash flows is
……………. P V
N


n  0
N
P V (C n ) 

n  0
Cn
(1  r )
n
Stream of Cash Flows
Valuation
Present Value of Stream of Cash Flows
The present value of a
stream of “n” cash flows
of arbitrary amounts is
the sum of the present
value of all cash flows.
N
PV


n  0
N
P V (C n ) 

n  0
Cn
(1  r )
n
Car Loan
Suppose that you need money to purchase a new car.
Your uncle will lend you the money so long as you agree
to pay him back within four years, and you offer to pay
him the rate of interest that he would otherwise get by
putting his money in a savings account.
Based on your yearnings and living expenses, you think
you will be able to pay him $5000 in one year (time
t=1), and then $8000 each year for the next three years.
If your uncle would otherwise earn 6% per year on his
savings, how much can you borrow from him?
Car Loan
You are asked about the present value of the stream of
cash flows you plan to pay your uncle.
The present value of the stream of cash flows is the
sum of present values of all individual cash flows
$5000 $8000 $8000 $8000
PV =
+
+
+
2
3
4
1.06
1.06
1.06
1.06
= $24,890.65
The Future Value Present Value Link
So you are asking you uncle to give up $24,890.65 now
and in return receive a stream of cash flows. Your
calculation indicated that your uncle should be
indifferent between the two alternatives.
To verify lets compare the future wealth of your uncle
if he had invested instead of leant to you $24,890.65.
FV4 = $24,890.65´1.06 4 = $31, 423.87
What shall we compare this to?
The Future Value Present Value Link
Future Value of a Cash Flow Stream with Present Value of PV
FVn = PV ´ (1+ r)
n
We can alternatively calculate the balance on the saving
account of your uncle if he was to save your payments for the
next four years by calculating the sum of future values of all
individual installments:
FV4 = $5000(1.06)3 + $8000(1.06)2 + $8000(1.06) + $8000
= $31,423.88
The Net Present Value
Net Present Value (NPV)
NPV = PV(Benefits)- PV(Costs) = PV(Benefits - Costs)
Consider the following investment opportunity:
If you invest $1000 today , you will receive $500 at the end
of each of the next three years. If you could otherwise earn
10% per year on your money, should you undertake the
investment opportunity?
$500 $500 $500
NPV = -$1000 +
+
+
2
1.1 1.1
1.13
Special Stream of Cash Flows
Valuation
Annuities and Perpetuities
Several of real-life situations involve streams of cash flows
with “standard” structure.
• Mortgages
• Car loans
• Retirement savings
• Corporate and treasury bonds
• stocks
We will study the valuation of
• Annuity: periodic stream of cash flows with finite life
(Growing annuity has growing cash flows)
• Perpetuity: annuity with infinite life
(Growing perpetuity has growing cash flows)
Perpetuity
The PV of a Perpetuity equals the amount of funds
required today in order to generate this perpetuity.
How much is required today in order to generate an
annual payment of $1 starting one year from now and
continuing forever if the interest rate is 8%?
The present value of a perpetuity is:
C
PV (C in perpetuity) =
r
Perpetuities
Perpetuities
Annuities
The PV of an Annuity equals the amount of funds
required today in order to generate this annuity.
How much is required today in order to generate an
annual payment of $1 starting one year from now and
continuing for 10 years if the interest rate is 8%?
The present value of an “n” year annuity is:
æ
Cç
1
PV (C in annuity) =
1ç
rç
1+ r
è
(
)
ö
÷
n÷
÷
ø
Annuities
Annuities
Annuities
Annuities
Growing Perpetuity
A growing perpetuity with growth rate “g” :
The Present Value of a Growing Perpetuity
C
PV (growing perpetuity) =
r-g
Growing Perpetuities
Growing Perpetuities
Growing Annuities
The present value of an “n” year growing
annuity is:
nö
æ æ
C ç
1+ g ö ÷
PV (growing annuity) =
1- ç
÷ ÷
ç
r - g è è 1+ r ø ø
Growing Annuities
Growing Annuities
Further examples
Loan Payment
Loan Payment
Loan Term
Loan Term
Mortgage Example
Q21(2nd edition):
When you purchased your house, you took our a 30-year annualpayment mortgage with an interest rate of 6% per year. The annual
payment of the mortgage is $12,000. You have just made a payment
and have now decided to pay the mortgage off by repaying the
outstanding balance. What is the payoff amount if:
a.
b.
c.
You have lived in the house for 12 years (there are 18 years left
on the mortgage)
You have lived in the house for 20 years (there are 10 years left
on the mortgage)
You have lived in the house for 12 years (there are 18 years left
on the mortgage) and you decide to pay off the mortgage
immediately before the twelfth payment is due?
Amortization Schedule
a. The PV(18 remaining payments)=
$12, 000 æ
1 ö
PV =
= $129, 931
ç118 ÷
0.06 è 1.06 ø
b. The PV(10 remaining payments)=
$12, 000 æ
1 ö
PV =
= $88, 321
ç110 ÷
0.06 è 1.06 ø
c. The PV(remaining payments just before 12th payment)=
PV = $12, 000 +$129, 931= $141, 931
Cost of schooling
Q27(2nd edition): [modified]
Your oldest daughter is about to start kindergarten at a private
school. Tuition is $10,000 per year, payable at the beginning of the
school year. You expect to keep your daughter in private school
through high school. You expect tuition to increase at a rate of 5%
per year over the 13 years of her schooling.
What is the present value of the tuition payments if the interest rate
is 6% per year? How much would you need to have in the bank now
to fund all 13 years of tuition?
What would your answers to the above questions be if interest rate
is 5% per year?
Cost of Schooling
When the interest rate is 6% you would need to
have in the bank:
12
$10, 500 æ æ 1.05 ö ö
çç1- ç
PV = $10, 000 +
÷ ÷÷ = $122,890
6% - 5% è è 1.06 ø ø
If interest rate is 5% we cannot
use the standard formula
PV =13´ $10, 000
Growing annuity when growth rate is larger than
interest rate
The standard formula is applicable for
all cases in which the interest rate is
either larger or smaller than the growth
rate
nö
æ
æ
ö
CF1
1+ g
çç1- ç
PV =
÷ ÷÷
r - g è è 1+ r ø ø
Investment in Art
Q33(2nd edition):
You are thinking about buying a piece of art that costs $50,000. The
art dealer is proposing the following deal: He will lend you the
money, and you will repay the loan by making the same payment
every two years for the next 20 years (i.e., a total of 10 payments).
If the interest rate is 4%, how much will you have to pay every two
years?
The interest rate over a period of 2 years is:
1.042 =1.0816
Calculating biennial payment:
X æ
1 ö
$50, 000 =
Þ X = $7, 505.3
ç110 ÷
0.0816 è 1.0816 ø
Retirement
Q35(2nd edition):
You are saving for retirement. To live comfortably, you decide you will need
to save $2 million by the time you are 65. Today is your 30th birthday, and
you decide, starting today and continuing on every birthday up to and
including your 65th birthday, that you will put the same amount into a
savings account.
If the interest rate is 5%, how much must you set aside each year to make
sure that you will have $2 million in the account on your 65th birthday?
The present value of $2 million received at age 65
FV = $2, 000, 000 Þ PV =
Calculating annual installment:
2000000
= $462, 754.9
30
1.05
X æ
1 ö
$462, 754.9 =
Þ X = $30,102.9
ç130 ÷
0.05 è 1.05 ø