Reflection statement 2
Points 25
Reflection statement 2 – Write a one (full) page paper, single spaced, 12 pt. font., Calibri or
Arial, stating your thoughts on the assigned topic.
Please put your name, course description FIN 348, date, and Reflection # at top left-hand side of
each reflection statement.
Topic for Reflection statement 2:
Why is knowing the PV (present value) or FV (future value) of an investment important
when making financial decisions to invest or not to invest?
Principles of Managerial Finance
Fifteenth Edition
Chapter 5
Time Value of Money
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Learning Goals (1 of 2)
LG 1 Discuss the role of time value in finance, the use of
computational tools, and the basic patterns of cash
flow.
LG 2 Understand the concepts of future value and present
value, their calculation for single cash flow amounts,
and the relationship between them.
LG 3 Find the future value and the present value of both an
ordinary annuity and an annuity due, and find the
present value of a perpetuity.
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Learning Goals (2 of 2)
LG 4 Calculate both the future value and the present value
of a mixed stream of cash flows.
LG 5 Understand the effect that compounding interest more
frequently than annually has on future value and on the
effective annual rate of interest.
LG 6 Describe the procedures involved in (1) determining
deposits needed to accumulate a future sum, (2) loan
amortization, (3) finding interest or growth rates, and
(4) finding an unknown number of periods.
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5.1 The Role of Time Value in Finance
(1 of 5)
• Time Value of Money
– Refers to the observation that it is better to receive money
sooner than later
• Future Value Versus Present Value
– Suppose that a firm has an opportunity to spend $15,000
today on some investment that will produce $17,000 spread
out over the next 5 years as follows:
Year
1
2
3
4
5
Cash flow
$?4,400
$ 5,000
$ 4,000
$ 3,000
$ 2,000
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5.1 The Role of Time Value in Finance
(2 of 5)
• Future Value Versus Present Value
– Is this investment a wise one?
– Timeline
? A horizontal line on which time zero appears at the leftmost
end and future periods are marked from left to right; can be
used to depict investment cash flows
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Figure 5.1 Timeline
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5.1 The Role of Time Value in Finance
(3 of 5)
• Future Value Versus Present Value
– To make the correct investment decision, managers must
compare the cash flows depicted in Figure 5.1 at a single
point in time
– Compounding
? Used to find the future value of each cash flow at the end of an
investment’s life
– Discounting
? Used to find the present value of each cash flow at time zero
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Figure 5.2 Compounding and Discounting
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5.1 The Role of Time Value in Finance
(4 of 5)
• Computational Tools
– Financial Calculators
– Electronic Spreadsheets
– Cash Flow Signs
? To provide a correct answer, financial calculators and
electronic spreadsheets require that a calculation’s relevant
cash flows be entered accurately as cash inflows or cash
outflows
? Cash inflows are indicated by entering positive values
? Cash outflows are indicated by entering negative values
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Figure 5.3 Calculator Keys
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5.1 The Role of Time Value in Finance
(5 of 5)
• Basic Patterns of Cash Flow
– Single Amount
? A lump-sum amount either
currently held or expected at
some future date
– Annuity
? A level periodic stream of cash
flows
– Mixed Stream
Mixed Cash Flow Stream
Year
A
B
0
?$3,000
?$ 50
1
100
50
2
800
?100
3
1,200
280
4
1,200
?60
5
1,400
Blank
6
300
Blank
? A stream of cash flows that is
not an annuity
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5.2 Single Amounts (1 of 7)
• Future Value of a Single Amount
– The Concept of Future Value
? Future Value
– The value on some future date of money that you invest
today
? Compound Interest
– Interest that is earned on a given deposit and has become
part of the principal at the end of a specified period
? Principal
– The amount of money on which interest is paid
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Personal Finance Example 5.1 (1 of 2)
If Fred Moreno places $100 in an account paying 8% interest
compounded annually (i.e., interest is added to the $100
principal 1 time per year), after 1 year he will have $108 in
the account. That’s just the initial principal of $100 plus 8%
($8) in interest. The future value at the end of the first year is
Future value at end of year 1 = $100 × (1 + 0.08) = $108
If Fred were to leave this money in the account for another
year, he would be paid interest at the rate of 8% on the new
principal of $108. After 2 years there would be $116.64 in the
account. This amount would represent the principal after the
first year ($108) plus 8% of the $108 ($8.64) in interest.
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Personal Finance Example 5.1 (2 of 2)
The future value after 2 years is
Future value after 2 years = $108 × (1 + 0.08)
= $116.64
Substituting the expression $100 × (1 + 0.08) from the firstyear calculation for the $108 value in the second-year
calculation gives us
Future value after 2 years = $100 × (1 + 0.08) × (1 + 0.08)
= $100 × (1 + 0.08)2
= $116.64
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5.2 Single Amounts (2 of 7)
• Future Value of a Single Amount
– The Equation for Future Value
?
?
?
?
FVn = future value after n periods
PV0 = initial principal, or present value when time = 0
r = annual rate of interest
n = number of periods (typically years) that the money remains
invested
FVn = PV0 ? (1 + r ) n
(5.1)
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Personal Finance Example 5.2
Jane Farber places $800 in a savings account paying 3%
interest compounded annually. She wants to know how
much money will be in the account after 5 years. Substituting
PV0 = $800, r = 0.03, and n = 5 into Equation 5.1 gives the
future value after 5 years:
FV5 = $800 × (1 + 0.03)5 = $800 × (1.15927) = $927.42
We can depict this situation on a timeline as follows:
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Personal Finance Example 5.3 (1 of 5)
In Personal Finance Example 5.2, Jane Farber places $800 in
her savings account at 3% interest compounded annually and
wishes to find out how much will be in the account after 5 years.
Calculator use We can use a financial
calculator to find the future value directly.
First enter ?800 and depress PV; next
enter 5 and depress N; then enter 3 and
depress I/Y (which is equivalent to “r” in our
notation); finally, to calculate the future
value, depress CPT and then FV. The
future value of $927.42 should appear on
the calculator display as shown at the left.
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Personal Finance Example 5.3 (2 of 5)
Remember that the calculator differentiates inflows from outflows
by preceding the outflows with a negative sign. For example, in
the problem just demonstrated, the $800 present value (PV),
because we entered it as a negative number, is considered an
outflow. Therefore, the calculator shows the future value (FV) of
$927.42 as a positive number to indicate that it is the resulting
inflow. Had we entered $800 present value as a positive number,
the calculator would show the future value of $927.42 as a
negative number. Simply stated, the cash flows—present value
(PV) and future value (FV)—will have opposite signs.
(Note: In future examples of calculator use, we will use only a
display similar to that shown on the previous slide. If you need a
reminder of the procedures involved, review the previous slide.)
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Personal Finance Example 5.3 (3 of 5)
Spreadsheet use Excel offers a mathematical function that
makes the calculation of future values easy. The format of
that function is FV(rate,nper,pmt,pv,type). The terms inside
the parentheses are inputs that Excel requires to calculate
the future value. The terms rate and nper refer to the interest
rate and the number of time periods, respectively. The term
pv represents the lump sum (or present value) that you are
investing today.
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Personal Finance Example 5.3 (4 of 5)
For now, we will ignore the other two inputs, pmt and type,
and enter a value of zero for each. The following Excel
spreadsheet shows how to use this function to calculate the
future value.
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Personal Finance Example 5.3 (5 of 5)
Changing any of the values in cells B2, B3, or B4
automatically changes the result shown in cell B5 because
the formula in that cell links back to the others. As with the
calculator, Excel reports cash inflows as positive numbers
and cash outflows as negative numbers. In the example
here, we have entered the $800 present value as a negative
number, which causes Excel to report the future value as a
positive number. Logically, Excel treats the $800 present
value as a cash outflow, as if you are paying for the
investment you are making, and it treats the future value as
a cash inflow when you reap the benefits of your investment
5 years later.
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5.2 Single Amounts (3 of 7)
• Future Value of a Single Amount
– A Graphical View of Future Value
? Figure 5.4 illustrates how the future value of $1 depends on
the interest rate and the number of periods that money is
invested
? It shows that (1) the higher the interest rate, the higher the
future value, and (2) the longer the money remains invested,
the higher the future value
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Figure 5.4 Future Value Relationship
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5.2 Single Amounts (4 of 7)
• Future Value of a Single Amount
– Compound Interest versus Simple Interest
? Simple Interest
– Interest that is earned only on an investment’s original
principal and not on interest that accumulates over time
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Table 5.1 Simple Interest versus Compound
Interest
Blank
Time (year)
0 (initial deposit)
Account Balance
Simple Interest
Compound Interest
$1,000
$1,000.00
1
1,050
1,050.00
2
1,100
1,102.50
3
1,150
1,157.62
4
1,200
1,215.51
5
1,250
1,276.28
10
1,500
1,628.89
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5.2 Single Amounts (5 of 7)
• Present Value of a Single Amount
– The Concept of Present Value
? Present Value
– The value in today’s dollars of some future cash flow
? Discounting Cash Flows
– The process of finding present values; the inverse of
compounding interest
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Personal Finance Example 5.4 (1 of 2)
Paul Shorter has an opportunity to receive $300 one year
from now. What is the most that Paul should pay now for this
opportunity? The answer depends in part on what Paul’s
current investment opportunities are (i.e., what his
opportunity cost is). Suppose Paul can earn a return of 2%
on money that he has on hand today. To determine how
much he’d be willing to pay for the right to receive $300 one
year from now, Paul can think about how much of his own
money he’d have to set aside right now to earn $300 by next
year. Letting PV0 equal this unknown amount and using the
same notation as in the future value discussion, we have
PV0 × (1 + 0.02) = $300
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Personal Finance Example 5.4 (2 of 2)
Solving for PV0 gives us
$300
(1 + 0.02)
= $294.12
PV0 =
The value today (“present value”) of $300 received 1 year
from today, given an interest rate of 2%, is $294.12. That is,
investing $294.12 today at 2% would result in $300 in 1 year.
Given his opportunity cost (or his required return) of 2%,
Paul should not pay more than $294.12 for this investment.
Doing so would mean that he would earn a return of less
than 2% on this investment. That’s unwise if he has other
similar investment opportunities that pay 2%. However, if
Paul could buy this investment for less than $294.12, he
would earn a return greater than his 2% opportunity cost.
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5.2 Single Amounts (6 of 7)
• Present Value of a Single Amount
– The Equation for Present Value
?
?
?
?
FVn = future value after n periods
PV0 = initial principal, or present value when time = 0
r = annual rate of interest
n = number of periods (typically years) that the money remains
invested
FVn
PV0 =
(1 + r ) n
(5.2)
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Personal Finance Example 5.5 (1 of 4)
Pam Valenti has been offered an investment opportunity that
will pay her $1,700 eight years from now. Pam has other
investment opportunities available to her that pay 4%, so she
will require a 4% return on this opportunity. How much
should Pam pay for this opportunity? In other words, what is
the present value of $1,700 that comes in 8 years if the
opportunity cost is 4%? Substituting FV8 = $1,700, n = 8, and
r = 0.04 into Equation 5.2 yields
$1, 700
$1, 700
PV0 =
=
= $1, 242.17
8
(1 + 0.04) 1.36857
The following timeline shows this analysis.
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Personal Finance Example 5.5 (2 of 4)
Calculator use Using the calculator’s
financial functions and the inputs shown at
the left, you should find the present value
to be $1,242.17. Notice that the calculator
result is represented as a negative value
to indicate that the present value is a cash
outflow (i.e., the investment’s cost).
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Personal Finance Example 5.5 (3 of 4)
Spreadsheet use The format of Excel’s present value
function is very similar to the future value function covered
earlier. The appropriate syntax is PV(rate,nper,pmt,fv,type).
The input list inside the parentheses is the same as in
Excel’s future value function with one exception. The present
value function contains the term fv, which represents the
future lump sum payment (or receipt) whose present value
you are trying to calculate. The following Excel spreadsheet
illustrates how to use this function to calculate the present
value.
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Personal Finance Example 5.5 (4 of 4)
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5.2 Single Amounts (7 of 7)
• Present Value of a Single Amount
– A Graphical View of Present Value
? Figure 5.5 illustrates how the present value of $1 depends on
the interest rate and the number of periods an investor must
wait to receive $1
? The figure shows that, everything else being equal, (1) the
higher the discount rate, the lower the present value; and (2)
the longer the waiting period, the lower the present value
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Figure 5.5 Present Value Relationship
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5.3 Annuities (1 of 8)
• Types of Annuities
– Annuity
? A stream of equal periodic cash flows over a specified time
period
? These cash flows can be inflows or outflows of funds
– Ordinary Annuity
? An annuity for which the cash flow occurs at the end of each
period
– Annuity Due
? An annuity for which the cash flow occurs at the beginning of
each period
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Table 5.2 Comparison of Ordinary Annuity
and Annuity Due Cash Flows ($1,000, 5
Years)
blank
Year
0
Annual cash flows
Annuity A (ordinary)
$
Annuity B (annuity due)
0
$1,000
1
1,000
1,000
2
1,000
1,000
3
1,000
1,000
4
1,000
1,000
5
1,000
0
Totals
$5,000
$5,000
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Personal Finance Example 5.6 (1 of 2)
Fran Abrams is evaluating two annuities. Both annuities pay
$1,000 per year, but annuity A is an ordinary annuity, while
annuity B is an annuity due. To better understand the
difference between these annuities, she has listed their cash
flows in Table 5.2. The two annuities differ only in the timing
of their cash flows: The cash flows occur sooner with the
annuity due than with the ordinary annuity.
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Personal Finance Example 5.6 (2 of 2)
Although the cash flows of both annuities in Table 5.2 total
$5,000, the annuity due would have a higher future value
than the ordinary annuity because each of its five annual
cash flows can earn interest for 1 year more than each of the
ordinary annuity’s cash flows. In general, as we will
demonstrate later in this chapter, the value (present or
future) of an annuity due is always greater than the value of
an otherwise identical ordinary annuity.
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5.3 Annuities (2 of 8)
• Finding the Future Value of an Ordinary Annuity
?? (1 + r )n ? 1 ??
FVn = CF1 ? ?
? (5.3)
r
??
??
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Personal Finance Example 5.7 (1 of 5)
Fran Abrams wishes to determine how much money she will
have after 5 years if she chooses annuity A, the ordinary
annuity. She will deposit the $1,000 annual payments that
the annuity provides at the end of each of the next 5 years
into a savings account paying 7% annual interest. This
situation is depicted on the following timeline.
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Personal Finance Example 5.7 (2 of 5)
As the figure shows, after 5 years, Fran will have $5,750.74
in her account. Note that because she makes deposits at the
end of the year, the first deposit will earn interest for 4 years,
the second for 3 years, and so on. Plugging the relevant
values into Equation 5.3, we have
?[(1 + 0.07)5 ? 1] ?
FV5 = $1, 000 ? ?
? = $5, 750.74
0.07
?
?
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Personal Finance Example 5.7 (3 of 5)
Calculator use Using the calculator inputs
shown at the left, you can confirm that the
future value of the ordinary annuity equals
$5,750.74. In this example, we enter the
$1,000 annuity payment as a negative value,
which in turn causes the calculator to report
the resulting future value as a positive value.
You can think of each $1,000 deposit that
Fran makes into her investment account as a
payment into the account or a cash outflow,
and after 5 years the future value is the
balance in the account, or the cash inflow
that Fran receives as a reward for investing.
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Personal Finance Example 5.7 (4 of 5)
Spreadsheet use To calculate the future value of an
annuity in Excel, we will use the same future value function
that we used to calculate the future value of a lump sum,
but we will add two new input values. Recall that the future
value function’s syntax is FV(rate,nper,pmt,pv,type). We
have already explained the terms rate, nper, and pv in this
function. The term pmt refers to the annual payment the
annuity offers. The term type is an input that lets Excel
know whether the annuity being valued is an ordinary
annuity (in which case the input value for type is 0 or
omitted) or an annuity due (in which case the correct input
value for type is 1).
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Personal Finance Example 5.7 (5 of 5)
In this particular problem, the input value for pv is 0 because
there is no up-front money received that is separate from the
annuity. The only cash flows are those that are part of the
annuity stream. The following Excel spreadsheet demonstrates
how to calculate the future value of the ordinary annuity.
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5.3 Annuities (3 of 8)
• Finding the Present Value of an Ordinary Annuity
1 ??
? CF1 ? ??
PV0 = ?
(5.4)
? ? ?1 ?
n ?
? r ? ?? (1 + r ) ??
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Example 5.8 (1 of 3)
Braden Company, a small producer of plastic toys, wants to
determine the most it should pay for a particular ordinary
annuity. The annuity consists of cash inflows of $700 at the
end of each year for 5 years. The firm requires the annuity to
provide a minimum return of 4%. The following timeline
depicts this situation.
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Example 5.8 (2 of 3)
Table 5.3 shows that one way to find the present value of the annuity
is to simply calculate the present values of all the cash payments
using the present value equation (Equation 5.2) and sum them. This
procedure yields a present value of $3,116.28. Calculators and
spreadsheets offer streamlined methods for arriving at this figure.
Calculator use Using the calculator’s
inputs shown at the left, you will find the
present value of the ordinary annuity to be
$3,116.28. Because the present value in
this example is a cash outflow
representing what Braden Company is
willing to pay for the annuity, we show it as
a negative value in the calculator display.
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Example 5.8 (3 of 3)
Spreadsheet use The following spreadsheet shows how to
calculate present value of the ordinary annuity.
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Table 5.3 Long Method for Finding the
Present Value of an Ordinary Annuity
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5.3 Annuities (4 of 8)
• Finding the Future Value of an Annuity Due
? ?(1 + r )n ? 1? ?
?
? ? ? (1 + r ) (5.5)
FVn = CF0 ? ? ?
?
r
??
??
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Personal Finance Example 5.9 (1 of 4)
Recall from an earlier example, illustrated in Table 5.2, that
Fran Abrams wanted to choose between an ordinary annuity
and an annuity due, both offering similar terms except for the
timing of cash flows. We calculated the future value of the
ordinary annuity in Example 5.7, but we now want to calculate
the future value of the annuity due. The timeline on the next
slide depicts this situation. Take care to notice on the timeline
that when we use Equation 5.5 (or any of the shortcuts that
follow) we are calculating the future value of Fran’s annuity due
after 5 years even though the fifth and final payment in the
annuity due comes after 4 years (which is equivalent to the
beginning of year 5). We can calculate the future value of an
annuity due using a calculator or a spreadsheet.
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Personal Finance Example 5.9 (2 of 4)
Calculator use Before using your
calculator to find the future value of an
annuity due, you must either switch it
to BEGIN mode or use the DUE key,
depending on the specific calculator.
Then, using the inputs shown at the
left, you will find the future value of
the annuity due to be $6,153.29.
(Note: Because we nearly always
assume end-of-period cash flows, be
sure to switch your calculator back to
END mode when you have completed
your annuity-due calculations.)
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Personal Finance Example 5.9 (3 of 4)
Spreadsheet use The following Excel spreadsheet
illustrates how to calculate the future value of the annuity
due. Remember that for an annuity due the type input value
must be set to 1, and we must also specify the pv input value
as 0 because there is no upfront cash other than what is part
of the annuity stream.
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Personal Finance Example 5.9 (4 of 4)
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5.3 Annuities (5 of 8)
• Finding the Future Value of an Annuity Due
– Comparison of an Annuity Due with an Ordinary Annuity
Future Value
? The future value of an annuity due is always greater than the
future value of an otherwise identical ordinary annuity
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5.3 Annuities (6 of 8)
• Finding the Present Value of an Annuity Due
1 ??
? CF0 ? ??
PV0 = ?
? (1 + r ) (5.6)
? ? ?1 ?
n ?
? r ? ?? (1 + r ) ??
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Example 5.10 (1 of 3)
In Example 5.8 involving Braden Company, we found the
present value of Braden’s $700, 5-year ordinary annuity
discounted at 4% to be $3,116.28. We now assume that
Braden’s $700 annual cash inflow occurs at the start of each
year and is thereby an annuity due. The following timeline
illustrates the new situation.
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Example 5.10 (2 of 3)
We can calculate its present value using a calculator or a
spreadsheet.
Calculator use Before using your calculator to
find the present value of an annuity due, you
must either switch it to BEGIN mode or use
the DUE key, depending on the specifics of
your calculator. Then, using the inputs shown
at the left, you will find the present value of the
annuity due to be $3,240.93 (Note: Because
we nearly always assume end-of-period cash
flows, be sure to switch your calculator back to
END mode when you have completed your
annuity-due calculations.)
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Example 5.10 (3 of 3)
Spreadsheet use The following spreadsheet shows how to
calculate the present value of the annuity due.
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5.3 Annuities (7 of 8)
• Finding the Present Value of an Annuity Due
– Comparison of an Annuity Due with an Ordinary Annuity
Present Value
? The present value of an annuity due is always greater than the
present value of an otherwise identical ordinary annuity
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5.3 Annuities (8 of 8)
• Finding the Present Value of a Perpetuity
– Perpetuity
? An annuity with an infinite life, providing continual annual cash
flow
PV0 = CF1 ? r (5.7)
– Growing Perpetuity
? An annuity with an infinite life, providing continual annual cash
flow, with the cash flow growing at a constant annual rate
? CF1 ?
PV0 = ?
? (5.8)
?r?g?
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Personal Finance Example 5.11 (1 of 2)
Ross Clark wishes to endow a chair in finance at his alma
mater. In other words, Ross wants to make a lump sum
donation today that will provide an annual stream of cash
flows to the university forever. The university indicated that
the annual cash flow required to support an endowed chair is
$400,000 and that it will invest money Ross donates today in
assets earning a 5% return. If Ross wants to give money
today so that the university will begin receiving annual cash
flows next year, how large must his contribution be? To
determine the amount Ross must give the university to fund
the chair, we must calculate the present value of a $400,000
perpetuity discounted at 5%.
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Personal Finance Example 5.11 (2 of 2)
Using Equation 5.7, we can determine that this present value
is $8 million when the interest rate is 5%:
PV0 = $400,000 ÷ 0.05 = $8,000,000
In other words, to generate $400,000 every year for an
indefinite period requires $8,000,000 today if Ross Clark’s
alma mater can earn 5% on its investments. If the university
earns 5% interest annually on the $8,000,000, it can
withdraw $400,000 per year indefinitely without ever
touching the original $8,000,000 donation.
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Personal Finance Example 5.12 (1 of 2)
Suppose, after consulting with his alma mater, Ross Clark
learns that the university requires the endowment to provide
a $400,000 cash flow next year, but subsequent annual cash
flows must grow by 2% per year to keep up with inflation.
How much does Ross need to donate today to cover this
requirement? Plugging the relevant values into Equation 5.8,
we have:
$400, 000
PV0 =
= $13,333,333
0.05 ? 0.02
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Personal Finance Example 5.12 (2 of 2)
Compared to the level perpetuity providing $400,000 per
year, the growing perpetuity requires Ross to make a much
larger initial donation, $13.3 million versus $8 million.
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5.4 Mixed Streams (1 of 2)
• Mixed Stream
– A stream of unequal periodic cash flows that reflect no
particular pattern
• Future Value of a Mixed Stream
– To determine the future value of a mixed stream of cash
flows, compute the future value of each cash flow at the
specified future date and then add all the individual future
values to find the total future value
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Example 5.13 (1 of 7)
Shrell Industries, a cabinet manufacturer, expects to receive the
following mixed stream of cash flows over the next 5 years from
one of its small customers.
Time
Cash flow
If Shrell expects to earn 8% on its
0
$
0
1
11,500
investments, how much will it accumulate
2
14,000
after 5 years if it immediately invests these
3
12,900
cash flows when they are received? This
4
16,000
situation is depicted on the following timeline. 5
18,000
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Example 5.13 (2 of 7)
Calculator use Most financial calculators do not have a
built-in function for finding the future value of a mixed stream
of cash flows, but most of them have a function for finding
the present value. Once you have the present value of the
mixed stream, you can move it forward in time to find the
future value. To accomplish this task you must first enter the
mixed stream of cash flows into your financial calculator’s
cash flow register, usually denoted by the CF key, starting
with the cash flow at time zero. Be sure to enter cash flows
correctly as either cash inflows or outflows.
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Example 5.13 (3 of 7)
Once you enter the cash flows, you will need to use the
calculator’s net present value (NPV) function to find the
present value of the cash flows. For Shrell, enter the
following into your calculator’s cash flow register: CF0 = 0,
CF1 = 11,500, CF2 = 14,000, CF3 = 12,900, CF4 = 16,000,
CF5 = 18,000. Next enter the interest rate of 8% and then
solve for the NPV, which is the present value of the mixed
stream of cash flows at time zero. The present value of the
mixed stream of cash flows is $56,902.30, so you need to
move this amount forward to the end of year 5 to find the
future of the mixed stream. Enter ?56,902.30 as the PV, 5 for
N, 8 for I/Y, and then compute FV.
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Example 5.13 (4 of 7)
You will find that the future value at the end of year 5 of
Shrell’s mixed cash flows is $83,608.15. An alternative
approach to using the calculator’s cash flow register and
NPV function is to find the future value at time 5 of each
cash flow and then sum the individual future values to find
the future value of Shrell’s mixed stream. Finding the future
value of a single cash flow was demonstrated earlier (in
Personal Finance Example 5.3). As you have already
discovered, summing the individual future values of Shrell
Industries’ mixed cash flow stream results in a future value of
$83,608.15 after 5 years.
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Example 5.13 (5 of 7)
Spreadsheet use A relatively simple way to use Excel to
calculate the future value of a mixed stream is to use the
Excel net present value (NPV) function combined w