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 with the
future value (FV) function discussed on page 197. The
syntax of the NPV function is NPV(rate, value1, value2,
value 3, . . .). The rate argument is the interest rate, and
value1, value2, value3, . . . represent a stream of cash flows.
The NPV function assumes that the first payment in the
stream arrives 1 year in the future and that all subsequent
payments arrive at 1-year intervals.
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Example 5.13 (6 of 7)
To find the future value of a mixed stream, the trick is to use
the NPV function to first find the present value of the mixed
stream and then find the future of this present value lump
sum amount. The Excel spreadsheet below illustrates this
approach (notice that the NPV appears as an outflow
because it represents the net present value of the stream of
investment costs).
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Example 5.13 (7 of 7)
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5.4 Mixed Streams (2 of 2)
• Present Value of a Mixed Stream
– To determine the present value of a mixed stream of cash
flows, compute the present value of each cash flow and
then add all the individual present values to find the total
present value
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Example 5.14 (1 of 3)
Frey Company, a shoe manufacturer, has the opportunity to receive
the following mixed stream of cash flows over the next 5 years.
If the firm must earn at least 9% on its
investments, what is the most it should
pay for this opportunity? This situation is
depicted on the following timeline.
Time
0
1
2
3
4
5
Cash flow
$ 0
400
800
500
400
300
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Example 5.14 (2 of 3)
Calculator use You can use the NPV
function on your financial calculator to find
the present value of the mixed cash flow
stream. Recall that to accomplish this task
you must first enter the mixed stream of
cash flows into your financial calculator’s
cash flow register by using the CF key. For
Frey enter the following into your calculator’s
cash flow register: CF0 = 0, CF1 = 400, CF2
= 800, CF3 = 500, CF4 = 400, CF5 = 300.
Next enter the interest rate of 9% and then
solve for the NPV. The present value of Frey
Company’s mixed cash flow stream found
using a calculator is $1,904.76.
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Example 5.14 (3 of 3)
Spreadsheet use To calculate the present value of a mixed
stream in Excel, we will use the NPV function. The present
value of the mixed stream of future cash flows can be
calculated as shown on the following Excel spreadsheet.
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5.5 Compounding Interest More
Frequently Than Annually (1 of 7)
• Semiannual Compounding
– Compounding of interest over two periods within the year
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Personal Finance Example 5.15
Fred Moreno has decided to invest $100 in a savings account
paying 8% interest compounded semiannually. If he leaves
his money in the account for 24 months (2 years), he will
receive 4% interest compounded over four periods, each of
which is 6 months long. Table 5.4 shows that after 12 months
(1 year) with 8% semiannual compounding, Fred will have
$108.16; after 24 months (2 years), he will have $116.99.
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Table 5.4 Future Value from Investing
$100 at 8% Interest Compounded
Semiannually over 24 Months (2 Years)
Period
6 months
Beginning
principal
$100.00
12 months
104.00
$104.00 × (1 + 0.04) =
$108.16
18 months
108.16
$108.16 × (1 + 0.04) =
$112.49
24 months
112.49
$112.49 × (1 + 0.04) =
$116.99
Future value calculation
$100.00 × (1 + 0.04) =
Future value at end
of period
$104.00
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5.5 Compounding Interest More
Frequently Than Annually (2 of 7)
• Quarterly Compounding
– Compounding of interest over four periods within the year
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Personal Finance Example 5.16
Fred Moreno has found an institution that will pay him 8%
interest compounded quarterly. If he leaves his money in
this account for 24 months (2 years), he will receive 2%
interest compounded over eight periods, each of which is 3
months long. Table 5.5 shows the amount Fred will have at
the end of each period. After 12 months (1 year), with 8%
quarterly compounding, Fred will have $108.24; after 24
months (2 years), he will have $117.17.
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Table 5.5 Future Value from Investing $100
at 8% Interest Compounded Quarterly over
24 Months (2 Years)
Period
Beginning principal
Future value calculation
Future value at end of period
3 months
$100.00
$100.00 × (1 + 0.02) =
$102.00
6 months
102.00
$102.00 × (1 + 0.02) =
$104.04
9 months
104.04
$104.04 × (1 + 0.02) =
$106.12
12 months
106.12
$106.12 × (1 + 0.02) =
$108.24
15 months
108.24
$108.24 × (1 + 0.02) =
$110.41
18 months
110.41
$110.41 × (1 + 0.02) =
$112.62
21 months
112.62
$112.62 × (1 + 0.02) =
$114.87
24 months
114.87
$114.87 × (1 + 0.02) =
$117.17
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Table 5.6 Future Value at the End of Years
1 and 2 from Investing $100 at 8% Interest,
Given Various Compounding Periods
blank
End of year
Compounding period
Annual
Semiannual
Quarterly
1
$108.00
$108.16
$108.24
2
116.64
116.99
117.17
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5.5 Compounding Interest More
Frequently Than Annually (3 of 7)
• A General Equation for Compounding
r
FVn = PV0 1 +
m
mn
(5.9)
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Personal Finance Example 5.17 (1 of 2)
The preceding examples calculated the amount that Fred
Moreno would have after 2 years if he deposited $100 at 8%
interest compounded semiannually or quarterly. For
semiannual compounding, m would equal 2 in Equation 5.9;
for quarterly compounding, m would equal 4. Substituting the
appropriate values for semiannual and quarterly
compounding into Equation 5.9, we find that
1. For semiannual compounding:
0.08
FV2 = $100 1 +
2
22
= $100 (1 + 0.04) 4 = $116.99
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Personal Finance Example 5.17 (2 of 2)
2. For quarterly compounding:
0.08
FV2 = $100 1 +
4
42
= $100 (1 + 0.02)8 = $117.17
These results agree with the values for FV2 in Tables 5.4 and
5.5.
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5.5 Compounding Interest More
Frequently Than Annually (4 of 7)
• We can simplify the computation process by using a
calculator or spreadsheet program
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Personal Finance Example 5.18 (1 of 3)
Fred Moreno wished to find the future value of
$100 invested at 8% interest compounded
both semiannually and quarterly for 2 years.
Calculator use1 If the calculator were used for
the semiannual compounding calculation, the
number of periods would be 4, and the interest
rate would be 4%. The future value of $116.99
will appear on the calculator display as shown
to the left.
1Many
calculators allow the user to set the number of payments per year. Most of these calculators are preset for monthly
payments, or 12 payments per year. Because we work primarily with annual payments—one payment per year—it is
important to be sure that your calculator is set for one payment per year. Although most calculators are preset to recognize
that all payments occur at the end of the period, it is also important to make sure that your calculator is correctly set on the
END mode. To avoid including previous data in current calculations, always clear all registers of your calculator before
inputting values and making each computation. You can punch the known values into the calculator in any order; the order
specified in this as well as other demonstrations of calculator use included in this text merely reflects convenience and
personal preference.
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Personal Finance Example 5.18 (2 of 3)
For the quarterly compounding case, the
number of periods would be 8 and the interest
rate would be 2%. The future value of $117.17
will appear on the calculator display as shown
to the left.
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Personal Finance Example 5.18 (3 of 3)
Spreadsheet use The future value of the single amount with
semiannual and quarterly compounding also can be
calculated as shown on the following Excel spreadsheet.
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5.5 Compounding Interest More
Frequently Than Annually (5 of 7)
• Continuous Compounding
– Compounding of interest, literally, all the time
– Equivalent to compounding interest an infinite number of
times per year
FVn = PV0 e rn
(5.10)
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Personal Finance Example 5.19 (1 of 3)
To find the value after 2 years (n = 2) of Fred Moreno’s $100
deposit (PV0 = $100) in an account paying 8% annual
interest (r = 0.08) compounded continuously, we can
substitute into Equation 5.10:
FV2 ( continuous compounding ) = $100 e0.082
= $100 2.71830.16
= $100 1.1735 = $117.35
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Personal Finance Example 5.19 (2 of 3)
Calculator use To find this value using the
calculator, you must first find the value of
e0.16 by punching in 0.16 and then pressing
2nd and then ex to get 1.1735. Next
multiply this value by $100 to obtain the
future value of $117.35, as shown at the
left. (Note: On some calculators, you may
not have to press 2nd before pressing ex.)
Spreadsheet use The following Excel
spreadsheet shows how to calculate the
future value of Fred’s deposit with
continuous compounding.
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Personal Finance Example 5.19 (3 of 3)
As expected, Fred’s deposit grows more with continuous
compounding than it does with semiannual ($116.99) or
quarterly ($117.17) compounding. In fact, continuous
compounding produces a greater future value than any other
compounding frequency.
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5.5 Compounding Interest More
Frequently Than Annually (6 of 7)
• Nominal and Effective Annual Rates of Interest
– Nominal (Stated) Annual Rate
▪ Contractual annual rate of interest charged by a lender or
promised by a borrower
– Effective (True) Annual Rate (EAR)
▪ The annual rate of interest actually paid or earned
m
r
EAR = 1 + − 1 (5.11)
m
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5.5 Compounding Interest More
Frequently Than Annually (7 of 7)
• Nominal and Effective Annual Rates of Interest
– Annual Percentage Rate (APR)
▪ The nominal annual rate of interest, found by multiplying the
periodic rate by the number of periods in one year, that must
be disclosed to consumers on credit cards and loans as a
result of “truth-in-lending laws.”
– Annual Percentage Yield (APY)
▪ The effective annual rate of interest that must be disclosed to
consumers by banks on their savings products as a result of
“truth-in-savings laws.”
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Personal Finance Example 5.20 (1 of 6)
Fred Moreno wishes to find the effective annual rate
associated with an 8% nominal annual rate (r = 0.08) when
interest is compounded (1) annually (m = 1), (2) semiannually
(m = 2), and (3) quarterly (m = 4). Substituting these values
into Equation 5.11, we get
1. For annual compounding:
1
0.08
1
EAR = 1 +
−
1
=
(1
+
0.08)
− 1 = 1 + 0.08 − 1 = 0.08 = 8%
1
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Personal Finance Example 5.20 (2 of 6)
2. For semiannual compounding:
2
0.08
2
EAR = 1 +
−
1
=
(1
+
0.04)
− 1 = 1.0816 − 1 = 0.0816 = 8.16%
2
3. For quarterly compounding:
4
0.08
4
EAR = 1 +
−
1
=
(1
+
0.02)
− 1 = 1.0824 − 1 = 0.0824 = 8.24%
4
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Personal Finance Example 5.20 (3 of 6)
Calculator use To find the EAR using the calculator, you
first need to enter the nominal annual rate and the
compounding frequency per year. Most financial calculators
have a NOM key for entering the nominal rate and either a
P/Y or C/Y key for entering the compounding frequency per
year. Once you enter these inputs, depress the EFF or CPT
key to display the corresponding effective annual rate.
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Personal Finance Example 5.20 (4 of 6)
Spreadsheet use You can convert nominal interest rates to
effective rates (or vice versa) using Excel’s EFFECT and
NOMINAL functions. To find the EAR, the EFFECT function
asks you to input the nominal annual rate and the
compounding frequency. If you input an EAR and the
compounding frequency, the NOMINAL function provides the
nominal annual rate or the annual percentage rate (APR).
Interest rate conversions from the 8% APR to the
semiannual EAR and from the quarterly EAR back to the 8%
APR are shown on the following Excel spreadsheet.
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Personal Finance Example 5.20 (5 of 6)
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Personal Finance Example 5.20 (6 of 6)
These examples demonstrate two important points. First, the
nominal rate equals the effective rate if compounding occurs
annually. Second, the effective annual rate increases with
increasing compounding frequency, up to a limit that occurs
with continuous compounding.
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5.6 Special Applications of Time Value
(1 of 4)
• Determining Deposits Needed to Accumulate a Future
Sum
(1 + r )n − 1
(5.12)
CF1 = FVn
r
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Personal Finance Example 5.21 (1 of 3)
You want to determine the equal
annual end-of-year deposits required
to accumulate $30,000 after 5 years,
given an interest rate of 6%.
Calculator use Using the calculator
inputs shown at the left, you will find
the annual deposit amount to be
$5,321.89. Thus, if $5,321.89 is
deposited at the end of each year for
5 years at 6% interest, there will be
$30,000 in the account after 5 years.
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Personal Finance Example 5.21 (2 of 3)
Spreadsheet use In Excel, solving for the annual cash flow
that helps you reach the $30,000 means using the payment
function. Its syntax is PMT (rate,nper,pv,fv,type). We have
previously discussed all the inputs in this function. The
following Excel spreadsheet illustrates how to use this
function to find the annual payment required to save
$30,000.
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Personal Finance Example 5.21 (3 of 3)
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5.6 Special Applications of Time Value
(2 of 4)
• Loan Amortization
– The determination of the equal periodic loan payments
necessary to provide a lender with a specified interest return
and to repay the loan principal over a specified period
– Loan Amortization Schedule
▪ A schedule of equal payments to repay a loan
▪ It shows the allocation of each loan payment to interest and
principal
1
CF1 = ( PV0 r ) 1 −
(5.13)
n
(1 + r )
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Personal Finance Example 5.22 (1 of 4)
Alex May borrows $6,000 from a bank. The bank requires Alex to
repay the loan fully in 4 years by making four end-of-year payments.
The interest rate on the loan is 10%. What is the loan payment that
Alex will have to make each year? Plugging the appropriate values
into Equation 5.13, we have
1
CF1 = ($6, 000 0.10) 1 −
= $600 0.316987 = $1,892.82
4
(1 + 0.10)
Calculator use Using the calculator inputs shown
at the left, you verify that Alex’s annual payment
will be $1,892.82. Thus, to repay the interest and
principal on a $6,000, 10%, 4-year loan, equal
annual end-of-year payments of $1,892.82 are
necessary.
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Personal Finance Example 5.22 (2 of 4)
Table 5.7 provides a loan amortization schedule that shows
the principal and interest components of each payment. The
portion of each payment that represents interest (column 3)
declines over time, and the portion going to principal
repayment (column 4) increases. Every amortizing loan
displays this pattern; as each payment reduces the principal,
the interest component declines, leaving a larger portion of
each subsequent loan payment to repay principal. Notice
that after Alex makes the fourth payment, the remaining loan
balance is zero.
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Personal Finance Example 5.22 (3 of 4)
Spreadsheet use The first spreadsheet below shows how to
calculate the annual loan payment, and the second
spreadsheet illustrates the construction of an amortization
schedule.
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Personal Finance Example 5.22 (4 of 4)
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Table 5.7 Loan Amortization Schedule
($6,000 Principal, 10% Interest, 4-Year
Repayment Period)
blank
Payments
blank
Beginning-ofyear principal
(1)
Loan
payment
(2)
Interest
[0.10 × (1)]
(3)
Principal
[(2) − (3)]
(4)
1
$6,000.00
$1,892.82
$600.00
$1,292.82
$4,707.18
2
4,707.18
1,892.82
470.72
1,422.10
3,285.08
3
3,285.08
1,892.82
328.51
1,564.31
1,720.77
4
1,720.77
1,892.82
172.08
1,720.74
______a
End-of-year
End-of-year
principal [(1) − (4)]
(5)
aBecause
of rounding, a slight difference ($0.03) exists between the beginning-of-year-4 principal
(in column 1) and the year-4 principal payment (in column 4).
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5.6 Special Applications of Time Value
(3 of 4)
• Finding Interest or Growth Rates
1/ n
FVn
r =
PV
0
− 1 (5.14)
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Personal Finance Example 5.23 (1 of 4)
Consumers across the United States are familiar with Dollar Tree
stores, which offer a vast array of items that cost just $1. Most
shoppers at Dollar Tree probably do not know that the company’s
stock was one of the best-performing stocks during the decade
that ended in 2016. An investor who purchased a $10 share of
Dollar Tree stock at the end of 2006 saw the firm’s stock price
grow to $70 by 2016’s close. What compound annual growth rate
does that increase represent? Or, equivalently, what average
annual rate of interest did shareholders earn over that period? Let
the initial $10 price represent the stock’s present value in 2006,
and let $70 represent the stock’s future value 10 years later.
Plugging the appropriate values into Equation 5.13, we find that
Dollar Tree stock increased almost 21.5% per year over this
decade.
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Personal Finance Example 5.23 (2 of 4)
r = ($70 $10)(1/10) –1 = 0.2148 = 21.48%
Calculator use Using the calculator to find the
interest or growth rate, we treat the earliest value as
a present value, PV, and the latest value as a future
value, FV. (Note: Most calculators require either the
PV or the FV value to be input as a negative value
to calculate an unknown interest or growth rate.) If
we think of an investor buying Dollar Tree stock for
$10 at the end of 2016, we treat that $10 payment
as a cash outflow. Then the $70 future value
represents a cash inflow, as if the investor sold the
stock in 2016 and received cash. The calculator
screenshot confirms that the growth rate in Dollar
Tree stock over this period was 21.48%.
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Personal Finance Example 5.23 (3 of 4)
Spreadsheet use The following spreadsheet shows how to
find Dollar Tree’s growth rate using Excel’s RATE function. The
syntax of that function is RATE(nper,pmt,pv,fv,type,guess). We
have encountered the function’s arguments nper, pmt, pv, fv,
and type previously. In this problem, $10 is the present value,
and $70 is the future value. We set the arguments pmt and
type to zero because those arguments are needed to work with
annuities, but we are calculating the growth rate by comparing
two lump sums. The new argument in this function is guess,
which in nearly all applications you can set to zero.
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Personal Finance Example 5.23 (4 of 4)
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Personal Finance Example 5.24 (1 of 4)
Jan Jacobs can borrow $2,000 today, and she must repay the
loan in equal end-of-year payments of $482.57 over 5 years.
Notice that Jan’s payments will total $2,412.85 (i.e., $482.57
per year × 5 years). That’s more than she borrowed, so she is
clearly paying interest on this loan, as we’d expect. The
question is, what annual interest rate is Jan paying? You
could calculate the percentage difference between what Jan
borrowed and what she repaid as follows:
$2, 412.85 − $2, 000
= 0.206 = 20.6%
$2, 000
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Personal Finance Example 5.24 (2 of 4)
Unfortunately, for two reasons this calculation does not tell us
what interest rate Jan is paying. First, this calculation sums Jan’s
payments over 5 years, so it does not reveal the interest rate on
her loan per year. Second, because each of Jan’s payments
comes at a different time, it is not valid to simply add them up.
Time-value-of-money principles tell us that even though each
payment is for $482.57, the payments have different values
because they occur at different times. The key idea in this problem
is that there is some interest rate at which the present value of the
loan payments is equal to the loan principal. It’s this interest rate
that equates the loan principal to the present value of payments
that we want to find. Solving for that algebraically is very difficult,
so we rely on a calculator or spreadsheet to find the solution.
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Personal Finance Example 5.24 (3 of 4)
Calculator use (Note: Most calculators
require you to input either the PMT or
the PV value as a negative number to
calculate an unknown interest rate on
an equal-payment loan. We take the
approach of treating PMT as a cash
outflow with a negative number.) Using
the inputs shown at the left, you will
find that the interest rate on this loan is
6.6%.
Spreadsheet use You can also calculate the interest on this
loan as shown on the following Excel spreadsheet.
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Personal Finance Example 5.24 (4 of 4)
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5.6 Special Applications of Time Value
(4 of 4)
• Finding an Unknown Number of Periods
FVn
log
PV0
n=
log (1 + r )
(5.15)
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Personal Finance Example 5.25 (1 of 3)
Ann Bates wishes to determine how long it will take for her
initial $1,000 deposit, earning 8% annual interest, to grow to
$2,500. Applying Equation 5.15, at an 8% annual rate of
interest, how many years, n, will it take for Ann’s $1,000,
PV0, to grow to $2,500, FVn?
$2,500
log
0.39794
$1,
000
=
n=
= 11.9
log(1.08)
0.03342
Ann will have to wait almost 12 years to reach her savings
goal of $2,500.
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Personal Finance Example 5.25 (2 of 3)
Calculator use Using the calculator, we treat
the initial value as the present value, PV, and
the latest value as the future value, FV. (Note:
Most calculators require either the PV or the
FV value to be input as a negative number to
calculate an unknown number of periods. We
treat Ann’s $1,000 initial deposit as a cash flow
and give it a negative number.) Using the
inputs shown at the left, we verify that it will
take Ann 11.9 years to reach her $2,500 goal.
Spreadsheet use You can calculate the number of years for
the present value to grow to a specified future value using
Excel’s NPER function, as shown below.
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Personal Finance Example 5.25 (3 of 3)
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Personal Finance Example 5.26 (1 of 2)
Bill Smart can borrow $25,000 at a 7.25% annual interest rate. The
lender requires Bill to make equal, end-of-year payments of
$3,878.07. Bill wishes to determine how long it will take to fully
repay the loan. The algebraic solution to this problem is a bit
tedious, so we will find the answer with a calculator or spreadsheet.
Calculator use (Note: Most calculators require either
the PV or the PMT value to be input as a negative
number to calculate an unknown number of periods.
We treat the loan payments as cash outflows here
and show them with a negative number.) Using the
inputs at the left, you will find the number of periods to
be 9 years. So, after making 9 payments of
$3,878.07, Bill will have a zero outstanding balance.
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Personal Finance Example 5.26 (2 of 2)
Spreadsheet use The number of years to pay off the loan
also can be calculated as shown on the following Excel
spreadsheet.
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Review of Learning Goals (1 of 11)
• LG 1
– Discuss the role of time value in finance, the use of
computational tools, and the basic patterns of cash flow.
▪ Financial managers and investors use time-value-of-money
techniques when assessing the value of expected cash flow
streams
▪ Alternatives can be assessed by either compounding to find
future value or discounting to find present value
▪ Financial managers rely primarily on present-value techniques
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Review of Learning Goals (2 of 11)
• LG 1 (Cont.)
– Discuss the role of time value in finance, the use of
computational tools, and the basic patterns of cash flow.
▪ Financial calculators and electronic spreadsheets streamline
the application of time-value techniques
▪ Cash flow patterns are of three types: a single amount or lump
sum, an annuity, or a mixed stream
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Review of Learning Goals (3 of 11)
• LG 2
– Understand the concepts of future value and present value,
their calculation for single amounts, and the relationship
between them.
▪ Future value (FV) relies on compound interest to translate
current dollars into future dollars
▪ The initial principal or deposit in one period, along with the
interest earned on it, becomes the beginning principal of the
following period
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Review of Learning Goals (4 of 11)
• LG 2 (Cont.)
– Understand the concepts of future value and present value,
their calculation for single amounts, and the relationship
between them.
▪ The present value (PV) of a future amount is the amount of
money today that is equivalent to the given future amount,
considering the return that can be earned
▪ Present value is the inverse of future value
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Review of Learning Goals (5 of 11)
• 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.
▪ An annuity is a pattern of equal periodic cash flows
▪ For an ordinary annuity, the cash flows occur at the end of the
period
▪ For an annuity due, cash flows occur at the beginning of the
period
▪ The future or present value of an ordinary annuity can be
found by using algebraic equations, a financial calculator, or a
spreadsheet program
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Review of Learning Goals (6 of 11)
• LG 3 (Cont.)
– 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.
▪ The value of an annuity due is always r% greater than the
value of an identical annuity
▪ The present value of a perpetuity—an infinite-lived annuity—
equals the annual cash payment divided by the discount rate
▪ The present value of a growing perpetuity equals the initial
cash payment divided by the difference between the discount
rate and the growth rate
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Review of Learning Goals (7 of 11)
• LG 4
– Calculate both the future value and the present value of a
mixed stream of cash flows.
▪ A mixed stream of cash flows consists of unequal periodic
cash flows that reflect no particular pattern
▪ The future value of a mixed stream of cash flows is the sum of
the future values of each cash flow
▪ Similarly, the present value of a mixed stream of cash flows is
the sum of the present values of the individual cash flows
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Review of Learning Goals (8 of 11)
• LG 5
– Understand the effect that compounding interest more
frequently than annually has on future value and on the
effective annual rate of interest.
▪ Interest can compound at intervals ranging from annually to
daily and even continuously
▪ The more often interest compounds, the larger the future
amount that will be accumulated, and the higher the effective,
or true, annual rate (EAR)
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Review of Learning Goals (9 of 11)
• LG 5 (Cont.)
– Understand the effect that compounding interest more
frequently than annually has on future value and on the
effective annual rate of interest.
▪ The annual percentage rate (APR)—a nominal annual
rate—is quoted on credit cards and loans
▪ The annual percentage yield (APY)—an effective annual
rate—is quoted on savings products
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Review of Learning Goals (10 of 11)
• 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.
▪ (1) The periodic deposit to accumulate a given future sum can
be found by solving the equation for the future value of an
annuity for the annual payment
▪ (2) A loan can be amortized into equal periodic payments by
solving the equation for the present value of an annuity for the
periodic payment
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Review of Learning Goals (11 of 11)
• LG 6 (Cont.)
– 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.
▪ (3) Interest or growth rates can be estimated by finding the
unknown interest rate in the equation for the present value of a
single amount or an annuity
▪ (4) The number of periods can be estimated by finding the
unknown number of periods in the equation for the present
value of a single amount or an annuity
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