Ch 4

Report
Chapter 4
Time Value
of Money:
Valuing Cash
Flow
Streams
Copyright © 2012 Pearson Prentice Hall. All rights reserved.
Chapter Outline
4.1
4.2
4.3
4.4
4.5
Valuing a Stream of Cash Flows
Perpetuities
Annuities
Growing Cash Flows
Solving for Variables Other Than Present Value
or Future Value
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4-2
Learning Objectives
• Value a series of many cash flows
• Value a perpetual series of regular cash flows called a
perpetuity
• Value a common set of regular cash flows called an annuity
• Value both perpetuities and annuities when the cash flows
grow at a constant rate
• Compute the number of periods, cash flow, or rate of return
in a loan or investment
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4-3
4.1 Valuing a Stream of Cash Flows
• Rules developed in Chapter 3:
– Rule 1: Only values at the same point in time
can be compared or combined.
– Rule 2: To calculate a cash flow’s future value,
we must compound it.
– Rule 3: To calculate the present value of a
future cash flow, we must discount it.
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4-4
4.1 Valuing a Stream of Cash Flows
Applying the Rules of Valuing Cash Flows
• Suppose we plan to save $1,000 today, and
$1,000 at the end of each of the next two years.
• If we earn a fixed 10% interest rate on our
savings, how much will we have three years from
today?
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4-5
4.1 Valuing a Stream of Cash Flows
• We can do this in several ways.
• First, take the deposit at date 0 and move it
forward to date 1.
• Combine those two amounts and move the
combined total forward to date 2.
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4-6
4.1 Valuing a Stream of Cash Flows
• Continuing in the same fashion, we can solve the
problem as follows:
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4-7
4.1 Valuing a Stream of Cash Flows
• Another approach is to compute the future value
in year 3 of each cash flow separately.
• Once all amounts are in year 3 dollars, combine
them.
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4-8
4.1 Valuing a Stream of Cash Flows
• Consider a stream of cash flows: C0 at
date 0, C1 at date 1, and so on, up to CN at
date N.
• We compute the present value of this cash
flow stream in two steps.
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4-9
4.1 Valuing a Stream of Cash Flows
• First, compute the present value of each cash
flow.
• Then combine the present values.
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4-10
Example 4.1
Present Value of a Stream of Cash Flows
Problem:
• You have just graduated and need money to buy a new car.
• Your rich Uncle Henry will lend you the money so long as
you agree to pay him back within four years.
• You offer to pay him the rate of interest that he would
otherwise get by putting his money in a savings account.
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4-11
Example 4.1
Present Value of a Stream of Cash Flows
(cont’d)
Problem:
• Based on your earnings and living expenses, you think you
will be able to pay him $5000 in one year, and then $8000
each year for the next three years.
• If Uncle Henry would otherwise earn 6% per year on his
savings, how much can you borrow from him?
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4-12
Example 4.1
Present Value of a Stream of Cash Flows
(cont’d)
Solution:
Plan:
• The cash flows you can promise Uncle Henry are as follows:
• Uncle Henry should be willing to give you an amount equal
to these payments in present value terms.
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4-13
Example 4.1
Present Value of a Stream of Cash Flows
(cont’d)
Plan:
• We will:
– Solve the problem using equation 4.1
– Verify our answer by calculating the future value of this
amount.
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4-14
Example 4.1
Present Value of a Stream of Cash Flows
(cont’d)
Execute:
• We can calculate the PV as follows:
5000 8000 8000 8000
PV 



2
3
1.06 1.06 1.06 1.064
 4716.98  7119.97  6716.95  6336.75
 $24,890.65
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4-15
Example 4.1
Present Value of a Stream of Cash Flows
(cont’d)
Execute:
• Now, suppose that Uncle Henry gives you the money, and
then deposits your payments in the bank each year.
• How much will he have four years from now?
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4-16
Example 4.1
Present Value of a Stream of Cash Flows
(cont’d)
Execute:
• We need to compute the future value of the annual
deposits.
• One way is to compute the bank balance each year.
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4-17
Example 4.1
Present Value of a Stream of Cash Flows
(cont’d)
Execute:
• To verify our answer, suppose your uncle kept his
$24,890.65 in the bank today earning 6% interest.
• In four years he would have:
FV= $24,890.65×(1.06)4=$31,423.87 in 4 years
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4-18
Example 4.1
Present Value of a Stream of Cash Flows
(cont’d)
Evaluate:
• Thus, Uncle Henry should be willing to lend you $24,890.65
in exchange for your promised payments.
• This amount is less than the total you will pay him
($5000+$8000+$8000+$8000=$29,000) due to the time
value of money.
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4-19
Example 4.1a
Present Value of a Stream of Cash Flows
Problem:
• You have just graduated and need money to pay
the deposit on an apartment.
• Your rich aunt will lend you the money so long as
you agree to pay her back within six months.
• You offer to pay her the rate of interest that she
would otherwise get by putting her money in a
savings account.
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4-20
Example 4.1a
Present Value of a Stream of Cash Flows
(cont’d)
Problem:
• Based on your earnings and living expenses, you think you
will be able to pay her $70 next month, $85 in each of the
next two months, and then $900 each month for months 4
through 6.
• If your aunt would otherwise earn 6% per year on her
savings, how much can you borrow from her?
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4-21
Example 4.1a
Present Value of a Stream of Cash Flows
(cont’d)
Solution:
Plan:
• The cash flows you can promise your aunt are as follows:
• She should be willing to give you an amount equal to these
payments in present value terms.
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4-22
Example 4.1a
Present Value of a Stream of Cash Flows
(cont’d)
Plan:
• We will:
– Solve the problem using equation 4.1
– Verify our answer by calculating the future value of this
amount.
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4-23
Example 4.1a Present Value of a Stream
of Cash Flows (cont’d)
Execute:
• We can calculate the PV as follows:
70
85
85
90
90
90
PV 





2
3
4
5
1.005 1.005 1.005 1.005 1.005 1.0056
 $69.65  $84.16  $83.74  $88.22  $87.78  $87.35
 $500.90
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4-24
Example 4.1a
Present Value of a Stream of Cash Flows
(cont’d)
Execute:
• Now, suppose that your aunt gives you the money, and
then deposits your payments in the bank each month.
• How much will she have six months from now?
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4-25
Example 4.1a
Present Value of a Stream of Cash Flows
(cont’d)
Execute:
• We need to compute the future value of the monthly
deposits.
• One way is to compute the bank balance each month.
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4-26
Example 4.1a
Present Value of a Stream of Cash Flows
(cont’d)
Execute:
• To verify our answer, suppose your aunt kept her $500.90
in the bank today earning 6% interest.
• In six months she would have:
FV= $500.90×(1.005)6=$516.11 in 6 months
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4-27
Example 4.1a
Present Value of a Stream of Cash Flows
(cont’d)
Evaluate:
• Thus, your aunt should be willing to lend you $500.90 in
exchange for your promised payments.
• This amount is less than the total you will pay her
($70+$85+$85+$90+$90+$90=$510) due to the time
value of money.
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4-28
4.1 Valuing a Stream of Cash Flows
Using a Financial Calculator: Solving for Present
and Future Values
• Financial calculators and spreadsheets have the
formulas pre-programmed to quicken the process.
• There are five variables used most often:
–
–
–
–
–
N
PV
PMT
FV
I/Y
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4-29
4.1 Valuing a Stream of Cash Flows
Example 1:
• Suppose you plan to invest $20,000 in an account
paying 8% interest.
• How much will you have in the account in 15
years?
• To compute the solution, we enter the four
variables we know and solve for the one we want
to determine, FV.
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4-30
4.1 Valuing a Stream of Cash Flows
Example 1:
• For the HP-10BII or the TI-BAII Plus calculators:
–
–
–
–
–
Enter
Enter
Enter
Enter
Press
15 and press the N key.
8 and press the I/Y key (I/YR for the HP)
-20,000 and press the PV key.
0 and press the PMT key.
the FV key (for the TI, press “CPT” and then “FV”).
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4-31
4.1 Valuing a Stream of Cash Flows
Given:
15
8
-20,000
Solve for:
0
63,443
Excel Formula: = FV(0.08,15,0,-20000)
Notice that we entered PV (the amount we’re putting in to the
bank) as a negative number and FV is shown as a positive number
(the amount we take out of the bank).
It is important to enter the signs correctly to indicate the direction
the funds are flowing.
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4-32
Example 4.2
Computing the Future Value
Problem:
• Let’s revisit the savings plan we considered earlier. We plan
to save $1000 today and at the end of each of the next two
years.
• At a fixed 10% interest rate, how much will we have in the
bank three years from today?
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4-33
Example 4.2 Computing the Future
Value (cont’d)
Solution:
Plan:
• We’ll start with the timeline for this savings plan:
• Let’s solve this in a different way than we did in
the text, while still following the rules.
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4-34
Example 4.2 Computing the Future
Value (cont’d)
Plan:
• First we’ll compute the present value of the cash flows.
• Then we’ll compute its value three years later (its future
value).
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4-35
Example 4.2 Computing the Future
Value (cont’d)
Execute:
• There are several ways to calculate the present value of the
cash flows.
• Here, we treat each cash flow separately an then combine
the present values.
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4-36
Example 4.2 Computing the Future
Value (cont’d)
Execute:
• Saving $2735.54 today is equivalent to saving $1000 per
year for three years.
• Now let’s compute future value in year 3 of that $2735.54:
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4-37
Example 4.2 Computing the Future
Value (cont’d)
Evaluate:
• The answer of $3641 is precisely the same result we found
earlier.
• As long as we apply the three rules of valuing cash flows,
we will always get the correct answer.
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4-38
4.2 Perpetuities
• The formulas we have developed so far allow us
to compute the present or future value of any
cash flow stream.
• Now we will consider two types of cash flow
streams:
– Perpetuities
– Annuities
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4-39
4.2 Perpetuities
Perpetuities
– A perpetuity is a stream of equal cash flows
that occur at regular intervals and last forever.
– Here is the timeline for a perpetuity:
– the first cash flow does not occur immediately;
it arrives at the end of the first period
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4-40
4.2 Perpetuities
• Using the formula for present value, the
present value of a perpetuity with
payment C and interest rate r is given by:
C
C
C
PV=


 ......
2
3
(1  r) (1  r) (1  r)
• Notice that all the cash flows are the
same.
• Also, the first cash flow starts at time 1.
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4-41
4.2 Perpetuities
• Let’s derive a shortcut by creating our own
perpetuity.
• Suppose you can invest $100 in a bank
account paying 5% interest per year
forever.
• At the end of the year you’ll have $105 in
the bank – your original $100 plus $5 in
interest.
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4-42
4.2 Perpetuities
• Suppose you withdraw the $5 and reinvest
the $100 for another year.
• By doing this year after year, you can
withdraw $5 every year in perpetuity:
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4-43
4.2 Perpetuities
• To generalize, suppose we invest an
amount P at an interest rate r.
• Every year we can withdraw the interest
we earned, C=r × P, leaving P in the bank.
• Because the cost to create the perpetuity
is the investment of principal, P, the value
of receiving C in perpetuity is the upfront
cost, P.
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4-44
4.2 Perpetuities
Present Value of a Perpetuity
C
PV (C in perpetuity) 
r
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(Eq. 4.4)
4-45
Example 4.3
Endowing a Perpetuity
Problem:
• You want to endow an annual graduation party at your alma
mater. You want the event to be a memorable one, so you
budget $30,000 per year forever for the party.
• If the university earns 8% per year on its investments, and
if the first party is in one year’s time, how much will you
need to donate to endow the party?
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4-46
Example 4.3
Endowing a Perpetuity (cont’d)
Solution:
Plan:
• The timeline of the cash flows you want to
provide is:
• This is a standard perpetuity of $30,000 per
year. The funding you would need to give the
university in perpetuity is the present value of
this cash flow stream
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4-47
Example 4.3
Endowing a Perpetuity (cont’d)
Execute:
• From the formula for a perpetuity,
PV  C / r  $30, 000/ 0.08  $375,000 today
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4-48
Example 4.3
Endowing a Perpetuity (cont’d)
Evaluate:
• If you donate $375,000 today, and if the university invests
it at 8% per year forever, then the graduates will have
$30,000 every year for their graduation party.
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4-49
Example 4.3a
Endowing a Perpetuity
Problem:
• You just won the lottery, and you want to endow a
professorship at your alma mater.
• You are willing to donate $4 million of your winnings for this
purpose.
• If the university earns 5% per year on its investments, and
the professor will be receiving her first payment in one year,
how much will the endowment pay her each year?
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4-50
Example 4.3a
Endowing a Perpetuity (cont’d)
Solution:
Plan:
• The timeline of the cash flows you want to
provide is:
• This is a standard perpetuity. The amount she can
withdraw each year and keep the principal intact
is the cash flow when solving equation 4.4.
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4-51
Example 4.3a
Endowing a Perpetuity (cont’d)
Execute:
• From the formula for a perpetuity,
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4-52
Example 4.3a
Endowing a Perpetuity (cont’d)
Evaluate:
• If you donate $4,000,000 today, and if the university
invests it at 5% per year forever, then the chosen professor
will receive $200,000 every year.
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4-53
4.3 Annuities
• Annuities
– An annuity is a stream of N equal cash flows
paid at regular intervals.
– The difference between an annuity and
a perpetuity is that an annuity ends after
some fixed number of payments
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4-54
4.3 Annuities
Present Value of An Annuity
• Note that, just as with the perpetuity, we
assume the first payment takes place one
period from today.
C
C
C
C
PV=


 ......
2
3
(1  r) (1  r) (1  r)
(1  r) N
• To find a simpler formula, use the same
approach as we did with a perpetuity:
create your own annuity.
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4-55
4.3 Annuities
• With an initial $100 investment at 5%
interest, you can create a 20-year annuity
of $5 per year, plus you will receive an
extra $100 when you close the account at
the end of 20 years:
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4-56
4.3 Annuities
• The Law of One Price tells us that because
it only took an initial investment of $100
to create the cash flows on the timeline,
the present value of these cash flows is
$100:
$100  PV( 20 - year annuity of $5 per year )  PV( $100 in 20 years )
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4-57
4-57
4.3 Annuities
• Rearranging:
PV( 20 - year annuity of $5 per year )  $100  PV( $100 in 20 years )
=$100 -
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$100
 $100  $37.69  $62.31
20
(1.05)
4-58
4.3 Annuities
• We usually want to know the PV as a
function of C, r, and N.
• Since C can be written as $100(0.05)=$5,
we can further re-arrange:
$5
$5
$5 
1 
PV( 20 - year annuity of $5 per year )=
- 0.0520 
1



0.05 (1.05)
0.05  1.0520 
1 
1 
 $5 
1



(0.05)  1.0520 
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4-59
4.3 Annuities
• In general:
1
1
PV (annuity of C for N periods with interest rate r)  C  1 
r  (1  r)N
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


4-60
Example 4.4
Present Value of a Lottery Prize
Annuity
Problem:
• You are the lucky winner of the $30 million state lottery.
• You can take your prize money either as (a) 30 payments of
$1 million per year (starting today), or (b) $15 million paid
today.
• If the interest rate is 8%, which option should you take?
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4-61
Example 4.4
Present Value of a Lottery Prize
Annuity (cont’d)
Solution:
Plan:
• Option (a) provides $30 million in prize money but paid over
time. To evaluate it correctly, we must convert it to a
present value. Here is the timeline:
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4-62
Example 4.4
Present Value of a Lottery Prize
Annuity (cont’d)
Plan (cont’d):
• Because the first payment starts today, the last payment
will occur in 29 years (for a total of 30 payments).
• The $1 million at date 0 is already stated in present value
terms, but we need to compute the present value of the
remaining payments.
• Fortunately, this case looks like a 29-year annuity of $1
million per year, so we can use the annuity formula.
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4-63
Example 4.4
Present Value of a Lottery Prize
Annuity (cont’d)
Execute:
• From the formula for an annuity,
PV( 29-year annuity of $1million)  $1 million 
1 
1 
1

0.08  1.0829 
 $1 million  11.16
 $11.16 million today
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4-64
Example 4.4
Present Value of a Lottery Prize
Annuity (cont’d)
Execute (cont’d):
• Thus, the total present value of the cash flows is $1 million
+ $11.16 million = $12.16 million. In timeline form:
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4-65
Example 4.4
Present Value of a Lottery Prize
Annuity (cont’d)
Execute (cont’d):
• Financial calculators or Excel can handle annuities easily—
just enter the cash flow in the annuity as the PMT:
Given:
29
Solve for:
8.0
1,000,000
0
-11,158,406
Excel Formula: =PV(RATE,NPER, PMT, FV) = PV(0.08,29,1000000,0)
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4-66
Example 4.4
Present Value of a Lottery Prize Annuity
(cont’d)
Evaluate:
• The reason for the difference is the time value of money.
• If you have the $15 million today, you can use $1 million
immediately and invest the remaining $14 million at an 8%
interest rate.
• This strategy will give you $14 million  8% = $1.12 million
per year in perpetuity!
• Alternatively, you can spend $15 million – $11.16 million =
$3.84 million today, and invest the remaining $11.16
million, which will still allow you to withdraw $1 million each
year for the next 29 years before your account is depleted.
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4-67
Example 4.4a
Present Value of an Annuity
Problem:
• Your parents have made you an offer you can’t refuse.
• They’re planning to give you part of your inheritance
early.
• They’ve given you a choice.
• They’ll pay you $10,000 per year for each of the next
seven years (beginning today) or they’ll give you their
2007 BMW M6 Convertible, which you can sell for
$61,000 (guaranteed) today.
• If you can earn 7% annually on your investments,
which should you choose?
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4-68
Example 4.4a
Present Value of an Annuity (cont’d)
Solution:
Plan:
• Option (a) provides $10,000 paid over time. To evaluate it
correctly, we must convert it to a present value. Here is the
timeline:
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4-69
Example 4.4a
Present Value of an Annuity (cont’d)
Plan (cont’d):
• The $10,000 at date 0 is already stated in present value
terms, but we need to compute the present value of the
remaining payments.
• Fortunately, this case looks like a 6-year annuity of $10,000
per year, so we can use the annuity formula.
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4-70
Example 4.4a
Present Value of an Annuity (cont’d)
Execute:
• From the formula for a annuity,
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4-71
Example 4.4a
Present Value of an Annuity (cont’d)
Execute (cont’d):
• Thus, the total present value of the cash flows is $10,000 +
$47,665. In timeline form:
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4-72
Example 4.4a
Present Value of an Annuity (cont’d)
Execute (cont’d):
• Financial calculators or Excel can handle annuities easily—
just enter the cash flow in the annuity as the PMT:
Given:
6
Solve for:
7
10000
0
-47,665
Excel Formula: =PV(RATE,NPER, PMT, FV) = PV(0.07,6,10000,0)
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4-73
Example 4.4a
Present Value of an Annuity (cont’d)
Evaluate:
• Lucky you!
• Even if you don’t want to keep it, the fact that you can sell
it for more than the annuity is worth means you’re better off
taking the BMW.
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4-74
4.3 Annuities
Future Value of an Annuity
FV (annuity)  PV  (1  r)N


1
N
 (1  r )
1 
N 

(1  r ) 

1
 C  ((1  r ) N  1)
r
C

r
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(Eq. 4.6)
4-75
Example 4.5
Retirement Savings Plan Annuity
Problem:
• Ellen is 35 years old, and she has decided it is time to plan
seriously for her retirement.
• At the end of each year until she is 65, she will save
$10,000 in a retirement account.
• If the account earns 10% per year, how much will Ellen
have saved at age 65?
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4-76
Example 4.5
Retirement Savings Plan Annuity
(cont’d)
Solution
Plan:
• As always, we begin with a timeline. In this case, it is helpful
to keep track of both the dates and Ellen’s age:
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4-77
Example 4.5
Retirement Savings Plan Annuity
(cont’d)
Plan (cont’d):
• Ellen’s savings plan looks like an annuity of
$10,000 per year for 30 years.
• (Hint: It is easy to become confused when you
just look at age, rather than at both dates and
age. A common error is to think there are only
65-36= 29 payments. Writing down both dates
and age avoids this problem.)
• To determine the amount Ellen will have in the
bank at age 65, we’ll need to compute the future
value of this annuity.
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4-78
Example 4.5
Retirement Savings Plan Annuity
Execute:
1
FV  $10,000 
(1.1030  1)
0.10
 $10,000  164.49
 $1.645 million at age 65
Using Financial calculators or Excel:
Given:
30
Solve for:
10.0
0
-10,000
-1,644,940
Excel Formula: =FV(RATE,NPER, PMT, PV) = FV(0.10,30,10000,0)
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4-79
Example 4.5
Retirement Savings Plan Annuity
Evaluate:
• By investing $10,000 per year for 30 years (a total of
$300,000) and earning interest on those investments, the
compounding will allow her to retire with $1.645 million.
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4-80
Example 4.5a
Retirement Savings Plan Annuity
Problem:
• Adam is 25 years old, and he has decided it is time to plan
seriously for his retirement.
• He will save $10,000 in a retirement account at the end of
each year until he is 45.
• At that time, he will stop paying into the account, though he
does not plan to retire until he is 65.
• If the account earns 10% per year, how much will Adam
have saved at age 65?
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4-81
Example 4.5a
Retirement Savings Plan Annuity
Solution
Plan:
• As always, we begin with a timeline. In this case, it is
helpful to keep track of both the dates and Adam’s age:
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4-82
Example 4.5a
Retirement Savings Plan Annuity
• Adam’s savings plan looks like an annuity of $10,000 per
year for 20 years.
• The money will then remain in the account until Adam is 65
– 20 more years.
• To determine the amount Adam will have in the bank at age
45, we’ll need to compute the future value of this annuity.
• Then we’ll compound the future value into the future 20
more years to see how much he’ll have at 65.
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4-83
Example 4.5a
Retirement Savings Plan Annuity
Execute:
Using Financial calculators or Excel:
Given:
20
Solve for:
10.0
0
-10,000
$572,750
Excel Formula: =FV(RATE,NPER, PMT, PV) = FV(0.10,20,10000,0)
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4-84
Example 4.5a
Retirement Savings Plan Annuity
Execute:
Using Financial calculators or Excel:
Given:
20
10.0
-$572,750
0
Solve for:
$3,853,175
Excel Formula: =FV(RATE,NPER, PMT, PV) = FV(0.10,20,0,-572750)
Copyright © 2012 Pearson Prentice Hall. All rights reserved.
4-85
Example 4.5a
Retirement Savings Plan Annuity
Evaluate:
• By investing $10,000 per year for 20 years (a
total of $200,000) and earning interest on those
investments, the compounding will allow him to
retire with $3.85 million.
• Even though he invested for 10 fewer years than
Ellen did, Adam will end up with more than twice
as much money because he’s starting his
retirement plan ten years earlier than she will.
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4-86
4.4 Growing Cash Flows
• A growing perpetuity is a stream of cash
flows that occur at regular intervals and
grow at a constant rate forever.
• For example, a growing perpetuity with a
first payment of $100 that grows at a rate
of 3% has the following timeline:
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4-87
4.4 Growing Cash Flows
Present Value of a Growing
Perpetuity
C
PV (growing perpetuity) 
rg
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(Eq. 4.7)
4-88
Example 4.6
Endowing a Growing Perpetuity
Problem:
• In Example 4.3, you planned to donate money to your alma
mater to fund an annual $30,000 graduation party.
• Given an interest rate of 8% per year, the required donation
was the present value of PV=$30,000/0.08=$375,000.
• Before accepting the money, however, the student
association has asked that you increase the donation to
account for the effect of inflation on the cost of the party in
future years.
• Although $30,000 is adequate for next year’s party, the
students estimate that the party’s cost will rise by 4% per
year thereafter.
• To satisfy their request, how much do you need to donate
now?
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4-89
Example 4.6
Endowing a Growing Perpetuity
(cont’d)
Solution:
Plan:
• The cost of the party next year is $30,000, and the cost
then increases 4% per year forever. From the timeline, we
recognize the form of a growing perpetuity and can value it
that way.
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4-90
Example 4.6
Endowing a Growing Perpetuity
(cont’d)
Execute:
• To finance the growing cost, you need to
provide the present value today of:
PV  $30,000 / (0.08  0.04)  $750,000 today
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4-91
Example 4.6
Endowing a Growing Perpetuity
(cont’d)
Evaluate:
• You need to double the size of your gift!
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4-92
Example 4.6a
Endowing a Growing Perpetuity
Problem:
• In Example 4.3a, you planned to donate $4 million to your
alma mater to fund an endowed professorship.
• Given an interest rate of 7% per year, the professor would
be able to collect $200,000 per year from your generosity.
• The inflation rate is expected to be 2% per year.
• How much can the professor be paid in the first year in
order to allow her annual salary to increase by 2% each
year and keep the principal intact?
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4-93
Example 4.6a
Endowing a Growing Perpetuity
(cont’d)
Solution:
Plan:
• The salary needs to increase 2% per year forever. From the
timeline, we recognize the form of a growing perpetuity and
can value it that way.
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4-94
Example 4.6a
Endowing a Growing Perpetuity
(cont’d)
Evaluate:
• She can only withdraw $120,000 in her first year.
• In the second year, her payment will be $120,000
X 1.02 = $122,400 and the payments will
continue to increase each year.
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4-95
4.4 Growing Cash Flows
Present Value of a Growing Annuity
• A growing annuity is a stream of N
growing cash flows, paid at regular
intervals
• It is a growing perpetuity that eventually
comes to an end.
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4-96
4.4 Growing Cash Flows
• The following timeline shows a growing
annuity with initial cash flow C, growing at
a rate of g every period until period N:
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4-97
4.4 Growing Cash Flows
• Present Value of a Growing Annuity:
1  1 g  
PV= C 
1  
 
r - g   1 r  
N
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4-98
Example 4.7
Retirement Savings with a Growing
Annuity
Problem:
• In Example 4.5, Ellen considered saving $10,000 per year
for her retirement. Although $10,000 is the most she can
save in the first year, she expects her salary to increase
each year so that she will be able to increase her savings by
5% per year. With this plan, if she earns 10% per year on
her savings, how much will Ellen have saved at age 65?
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4-99
Example 4.7
Retirement Savings with a Growing
Annuity (cont’d)
Solution:
Plan:
Her new savings plan is represented by the following
timeline:
This example involves a 30-year growing annuity with a growth
rate of 5% and an initial cash flow of $10,000. We can use Eq.
4.8 to solve for the present value of a growing annuity.
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4-100
Example 4.7
Retirement Savings with a Growing
Annuity (cont’d)
Execute:
The present value of Ellen’s growing annuity is
given by:
  1.05 30 
1
PV= $10,000 
1  
 
0.10 - 0.05   1.10  
 $10,000  15.0463
 $150, 463today
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4-101
Example 4.7
Retirement Savings with a Growing
Annuity (cont’d)
Execute:
•Ellen’s proposed savings plan is equivalent to
having $150,463 in the bank today. To determine
the amount she will have at age 65, we need to
move this amount forward 30 years:
FV= $150, 463  1.1030
 $2.625 million in 30 years
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4-102
Example 4.7
Retirement Savings with a Growing
Annuity (cont’d)
Evaluate:
•Ellen will have saved $2.625 million at age 65 using
the new savings plan. This sum is almost $1 million
more than she had without the additional annual
increases in savings.
•Because she is increasing her savings amount each
year and the interest on the cumulative increases
continues to compound, her final savings is much
greater.
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4-103
Example 4.7a
Retirement Savings with a Growing
Annuity
Problem:
• In Example 4.5a, Adam considered saving $10,000 per year
for his retirement. Although $10,000 is the most he can
save in the first year, he expects his salary to increase each
year so that he will be able to increase his savings by 4%
per year. With this plan, if he earns 10% per year on his
savings, how much will Adam have saved at age 65?
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4-104
Example 4.7a
Retirement Savings with a Growing
Annuity (cont’d)
Solution:
Plan:
His new savings plan is represented by the following
timeline:
This example involves a 20-year growing annuity with a growth
rate of 4% and an initial cash flow of $10,000. We can use Eq.
4.8 to solve for the present value of a growing annuity.
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4-105
Example 4.7a
Retirement Savings with a Growing
Annuity (cont’d)
Execute:
The present value of Adam’s growing annuity is
given by:
  1.04  20 
1
PV= $10,000 
1  
 
0.10 - 0.04   1.10  
 $10,000  11.2384
 $112,384 today
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4-106
Example 4.7a
Retirement Savings with a Growing
Annuity (cont’d)
Execute:
•Adam’s proposed savings plan is equivalent to
having $112,384 in the bank today. To determine
the amount he will have at age 65, we need to move
this amount forward 40 years:
FV= $112,384  1.10
40
 $5,086, 416 in 40 years
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4-107
Example 4.7a
Retirement Savings with a Growing
Annuity (cont’d)
Evaluate:
•Adam will have saved $5.086 million at age 65
using the new savings plan. This sum is over $1
million more than he had without the additional
annual increases in savings.
•Because he is increasing his savings amount each
year and the interest on the cumulative increases
continues to compound, his final savings is much
greater.
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4-108
4.5 Solving for Variables Other Than
Present Value or Future Value
• In some situations, we use the present
and/or future values as inputs, and solve
for the variable we are interested in.
• We examine several special cases in this
section.
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4-109
4.5 Solving for Variables Other Than
Present Value or Future Value
Solving for Cash Flows
C
P
1
1 
1
N
r
(1  r) 
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(Eq. 4.8)
4-110
Example 4.8
Computing a Loan Payment
Problem:
• Your firm plans to buy a warehouse for $100,000.
• The bank offers you a 30-year loan with equal annual
payments and an interest rate of 8% per year.
• The bank requires that your firm pay 20% of the purchase
price as a down payment, so you can borrow only $80,000.
• What is the annual loan payment?
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4-111
Example 4.8
Computing a Loan Payment (cont’d)
Solution:
Plan:
• We start with the timeline (from the bank’s
perspective):
• Using Eq. 4.8, we can solve for the loan payment,
C, given N=30, r = 8% (0.08) and P=$80,000
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4-112
Example 4.8
Computing a Loan Payment (cont’d)
Execute:
• Eq. 4.8 gives the payment (cash flow) as follows:
C
P
1
1 
 1
N 
r
(1 r) 

80, 000
1 
1 
1 
30 
0.08 
(1.08) 
 $7106.19
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4-113
Example 4.8
Computing a Loan Payment (cont’d)
Execute (cont’d):
• Using a financial calculator or Excel:
Given:
30
8.0
-80,000
Solve for:
0
7106.19
Excel Formula: =PMT(RATE,NPER, PV, FV) =
PMT(0.08,30,-80000,0)
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4-114
Example 4.8
Computing a Loan Payment (cont’d)
Evaluate:
• Your firm will need to pay $7,106.19 each year to repay the
loan.
• The bank is willing to accept these payments because the
PV of 30 annual payments of $7,106.19 at 8% interest rate
per year is exactly equal to the $80,000 it is giving you
today.
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4-115
Example 4.8a
Computing a Loan Payment
Problem:
• Suppose you accept your parents’ offer of a 2007 BMW M6
convertible, but that’s not the kind of car you want.
• Instead, you sell the car for $61,000, spend $11,000 on a
used Corolla, and use the remaining $50,000 as a down
payment for a house.
• The bank offers you a 30-year loan with equal monthly
payments and an interest rate of 6% per year, and requires
a 20% down payment.
• How much can you borrow, and what will be the payment
on the loan?
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4-116
Example 4.8a
Computing a Loan Payment (cont’d)
Solution:
Plan:
• To calculate the amount we can borrow, we need to find out
what amount $50,000 is 20% of:
$50,000 = .2 X Value
Value = $50,000/.2 = $250,000
• Because you’ll be putting $50,000 down, your loan amount
will be $250,000 - $50,000 = $200,000.
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4-117
Example 4.8a
Computing a Loan Payment (cont’d)
Solution:
Plan:
• We start with the timeline:
• Note, we need to use the monthly interest rate. Since the
quoted rate is an APR, we can just divide the annual rate by
12:
r = .06/12 = .005
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4-118
Example 4.8a
Computing a Loan Payment (cont’d)
Execute:
• Eq. 4.8 gives the payment (cash flow) as follows:
= $1,199.10
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4-119
Example 4.8a
Computing a Loan Payment (cont’d)
Execute (cont’d):
• Using a financial calculator or Excel:
Given:
360
0.5
200,000
Solve for:
0
-1199.10
Excel Formula: =PMT(RATE,NPER, PV, FV) =
PMT(0.005,360,200000,0)
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4-120
4.5 Solving for Variables Other Than
Present Value or Future Value
• Rate of Return
– The rate of return is the rate at which the
present value of the benefits exactly offsets the
cost.
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4-121
4.5 Solving for Variables Other Than
Present Value or Future Value
• Suppose you have an investment
opportunity that requires a $1000
investment today and will pay $2000 in six
years.
• What interest rate, r, would you need so
that the present value of what you get is
exactly equal to the present value of what
you give up?
2000
1000 
6
(1  r)
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4-122
4.5 Solving for Variables Other Than
Present Value or Future Value
• Rearranging:
1000  (1  r)  2000
6
1
6
 2000 
1 r  

 1000 
 1.1225,or
r  12.25%
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4-123
4.5 Solving for Variables Other Than
Present Value or Future Value
• Suppose your firm needs to purchase a
new forklift.
• The dealer gives you two options:
– A price for the forklift if you pay cash
($40,000)
– The annual payments if you take out a loan
from the dealer (no money down and four
annual payments of $15,000).
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4-124
4.5 Solving for Variables Other Than
Present Value or Future Value
• Setting the present value of the cash flows
equal to zero requires that the present
value of the payments equals the purchase
price:
1
1 
40,000  15,000  1 
r  (1  r) 4 
• The solution for r is the interest rate
charged by the dealer, which you can
compare to the rate charged by your bank.
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4-125
4.5 Solving for Variables Other Than
Present Value or Future Value
• There is no simple way to solve for the
interest rate.
• The only way to solve this equation is to
guess at values for r until you find the
right one.
• An easier solution is to use a financial
calculator or a spreadsheet.
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4-126
4.5 Solving for Variables Other Than
Present Value or Future Value
Given:
Solve for:
4
40,000
-15,000
0
18.45
Excel Formula: =RATE(NPER,PMT,PV,FV)=Rate(4,-25000,40000,0)
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4-127
Example 4.9
Computing the Rate of Return with a
Financial Calculator
Problem:
• Let’s return to the lottery example (Example 4.4).
• How high of a rate of return do you need to earn investing
on your own in order to prefer the $15 million payout?
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4-128
Example 4.9
Computing the Rate of Return with a
Financial Calculator
Solution:
Plan:
• We need to solve for the rate of return that
makes the two offers equivalent.
• Anything above that rate of return would make
the present value of the annuity lower than the
$15 million lump sum payment and
• anything below that rate of return would make it
greater than the $15 million.
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4-129
Example 4.9
Computing the Rate of Return with a
Financial Calculator
Execute:
Given:
14,000,000
29
Solve for:
1,000,000
0
5.72
Excel Formula: =RATE(NPER, PMT, PV,FV) =
RATE(29,1000000,-14000000,0)
The rate equating the two options is 5.72%.
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4-130
Example 4.9
Computing the Rate of Return with a
Financial Calculator
Evaluate:
• 5.72% is the rate of return that makes giving up the $15
million payment and taking the 30 installments of $1
million exactly a zero NPV action.
• If you could earn more than 5.72% investing on your own,
then you could take the $15 million, invest it and generate
thirty installments that are each more than $1 million.
• If you could not earn at least 5.72% on your investments,
you would be unable to replicate the $1 million
installments on your own and would be better off taking the
installment plan.
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4-131
Example 4.9a
Computing the Internal Rate of Return
with a Financial Calculator
Problem:
• Let’s return to the BMW example (Example 4.4a).
• What rate of return would make you indifferent
between the car and the $10,000 per year payout
(even if the car is your favorite color and has HD
radio)?
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4-132
Example 4.9a
Computing the Internal Rate of Return
with a Financial Calculator
Solution:
Plan:
• We need to solve for the rate of return that
makes the two offers equivalent.
• Anything above that rate of return would make
the present value of the annuity lower than the
$61,000 car and
• anything below that rate of return would make it
greater than the $61,000.
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4-133
Example 4.9a
Computing the Internal Rate of Return
with a Financial Calculator
Execute:
Given:
Solve for:
6
-51,000
10,000
0
4.85%
Excel Formula: =RATE(NPER, PMT, PV,FV) = RATE(6,10000,-61000,0)
The rate equating the two options is 4.85%.
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4-134
Example 4.9a
Computing the Internal Rate of Return
with a Financial Calculator
Evaluate:
• 4.85% is the rate of return that makes giving up the
$61,000 car and taking the 7 installments of $10,000
exactly a zero NPV action.
• If you can earn more than 4.85% investing on your own,
then you can take the $61,000, invest it and generate
seven installments that are each more than $10,000.
• If you can not earn at least 4.85% on your investments,
you would be unable to replicate the $10,000 installments
on your own and would be better off taking the generous
payments your parents have offered.
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4-135
4.6 Solving for Variables Other Than
Present Value or Future Value
• Solving for the Number of Periods
– In addition to solving for cash flows or the
interest rate, we can solve for the amount of
time it will take a sum of money to grow to a
known value.
– In this case, the interest rate, present value,
and future value are all known.
– We need to compute how long it will take for
the present value to grow to the future value.
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4-136
Example 4.10
Solving for the Number of Periods in a
Savings Plan
Problem:
• Let’s return to saving for a down payment on a
house.
• Imagine that some time has passed and you have
$10,050 saved already, and you can now afford
to save $5,000 per year at the end of each year.
• Also, interest rates have increased so that you
now earn 7.25% per year on your savings.
• How long will it take you to get to your goal of
$60,000?
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4-137
Example 4.10
Solving for the Number of Periods in a
Savings Plan
Solution:
Plan:
• The timeline for this problem is
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4-138
Example 4.10
Solving for the Number of Periods in a
Savings Plan
Plan (cont’d):
• We need to find N so that the future value of our current
savings plus the future value of our planned additional
savings (which is an annuity) equals our desired amount.
• There are two contributors to the future value: the initial
lump sum $10,050 that will continue to earn interest, and
the annuity contributions of $5,000 per year that will earn
interest as they are contributed.
• Thus, we need to find the future value of the lump sum plus
the future value of the annuity
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4-139
Example 4.10
Solving for the Number of Periods in a
Savings Plan
Execute:
Given:
Solve for:
7.25
-10,050
-5,000
60,000
7
Excel Formula: =NPER(RATE,PMT, PV, FV) =
NPER(0.0725,-5000,-10050,60000)
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4-140
Example 4.10
Solving for the Number of Periods in a
Savings Plan
Evaluate:
• It will take seven years to save the down payment.
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4-141
Example 4.10a
Solving for the Number of Periods in a
Savings Plan
Problem:
• Let’s return to Ellen and Adam.
• Suppose Ellen decides she will continue working
until she has as much at retirement as her
brother, Adam, will have when he retires.
• She will continue to contribute $10,000 each year
to her retirement account.
• How much longer will she need to work to tie the
competition with her brother?
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Example 4.10a
Solving for the Number of Periods in a
Savings Plan
Solution:
Plan:
• The timeline for this problem is
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Example 4.10a
Solving for the Number of Periods in a
Savings Plan
Plan (cont’d):
• We need to find N so that the FV of the $1,645,000 she’ll
have at age 65 plus the $10,000 she’ll contribute each year
is equal to $3,850,000.
• Remember, she’s earning 10% on her investments.
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Example 4.10a
Solving for the Number of Periods in a
Savings Plan
Execute:
Given:
Solve for:
10
-1645000
-10,000
3850000
8.57
Excel Formula: =NPER(RATE,PMT, PV, FV) =
NPER(0.10,-10000,-1645000,3850000)
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Example 4.10a
Solving for the Number of Periods in a
Savings Plan
Evaluate:
• Ellen will have to work until she’s 73 ½ years old.
• (Here’s hoping she really loves her job!)
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Chapter Quiz
1.
How do you calculate the present value of a cash flow
stream?
2.
What is the intuition behind the fact that an infinite
stream of cash flows has a finite present value?
3.
What are some examples of annuities?
4.
What is the difference between an annuity and a
perpetuity?
5.
What is an example of a growing perpetuity?
6.
How do you calculate the cash flow of an annuity?
7.
How do you calculate the rate of return on an investment?
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