Chapter 05 – Introduction to Valuation: The Time Value of Money
Chapter 5
INTRODUCTION TO VALUATION: THE TIME VALUE OF
MONEY
CHAPTER WEB SITES
Section Web Address
5.1 www.financeprofessor.com
www.teachmefinance.com
5.3 www.calculator.org
www.moneychimp.com
www.about.com/
www.studyfinance.com
www.investopedia.com
CHAPTER ORGANIZATION
5.1 Future Value and Compounding
Investing for a Single Period
Investing for More Than One Period
A Note about Compound Growth
5.2 Present Value and Discounting
The Single-Period Case
Present Values for Multiple Periods
5.3 More about Present and Future Values
Present versus Future Value
Determining the Discount Rate
Finding the Number of Periods
5.4 Summary and Conclusions
ANNOTATED CHAPTER OUTLINE
Lecture Tip: Many students find the phrases “time value of
money” and “a dollar today is worth more than a dollar later” a
bit confusing. In some ways it might be better to say the “money
value of time.”
Indeed, much of the terminology surrounding exchanges of
money now for money later is confusing to students. For example,
present value as the name for money paid or received earlier in
time and future value as the name for money paid or received later
in time are a constant source of confusion. How, students ask, can
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Chapter 05 – Introduction to Valuation: The Time Value of Money
money to be paid next year be a “present” value; how can money
received today be a “future” value? They must be made aware that
we mean earlier money and later money. Many students never fully
comprehend that present value, future value, interest rates, and
interest rate factors are simply a convenient means for
communicating the terms of exchange for what are essentially
different kinds of money.
One way to emphasize both the exchange aspect of the time value
of money and that present dollars and future dollars are different
kinds of money is to compare them to U.S. dollars and Canadian
dollars. Both are called dollars, but they’re not the same thing.
And just as U.S. dollars rarely trade 1 for 1 for Canadian dollars,
neither do present dollars trade 1 for 1 for future dollars. Just as
there are exchange rates for U.S. dollars into Canadian dollars
and vice-versa, so present value factors and future value factors
represent exchange rates between earlier money and later money.
Also, the same reciprocity that exists between the foreign exchange
rates exists between future value and present value interest factors.
1. Future Value and Compounding
A. Investing for a single period
If you invest $X today at an interest rate of r, you will have $X +
$X(r) = $X(1 + r) in one period.
Example: $100 at 10% interest gives $100(1.1) = $110
B. Investing for more than one period
Reinvesting the interest, we earn interest on interest, i.e.,
compounding
FV = $X(1 + r)(1 + r) = $X(1 + r)2
Example: $100 at 10% for 2 periods gives $100(1.1)(1.1) =
$100(1.1)2 = $121
In general, for t periods, FV = $X(1 + r)t, where (1 + r)t is the
future value interest factor, FVIF(r,t)
Example: $100 at 10% for 10 periods gives $100(1.1)10 = $259.37
Lecture Tip: It is important that students understand the impact of
compounding now, or they will have more difficulty distinguishing
when it is appropriate to use the APR and when it is appropriate to
use the effective annual rate.
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Chapter 05 – Introduction to Valuation: The Time Value of Money
Real-World Tip: Students are often helped by concrete examples
tied to real life. For example, you can illustrate the effect of
compound growth by asking the following question in class:
“Assume you just started a new job and your current annual
salary is $25,000. Suppose the rate of inflation is about 4%
annually for
the next 40 years, and you receive annual cost-of-living increases
tied to the inflation rate. What will your salary be in 40 years?
Most students are happy to hear that their final annual salary
will be 25,000(1.04)40 = $120,025. They are often less happy,
however, when they find that today’s $20,000 automobile will cost
$96,020 under the same assumptions.
This example can be extended in many directions. For example,
you might ask how much their final salary will be should they
receive average raises of 5% annually. The difference is striking:
25,000(1.05)40 = $176,000; or approximately $56,000 in
additional purchasing power in that year alone!
C. A Note about Compound Growth
The interest rate is really just the “growth” rate of money, and the
future value formula can be used more generally to find the future
amount of anything that is expected to grow at a constant rate over
a set number of periods. The book illustrates this with employees
and sales.
Lecture Tip: You may wish to take this opportunity to remind
students that, since compound growth rates are found using only
the beginning and ending values of a series, they convey nothing
about the values in between. For example, a firm may state that
“EPS has grown at a 10% annually compounded rate over the last
decade” in an attempt to impress investors of the quality of
earnings. However, this just depends on EPS in year 1 and year 11.
For example, if EPS in year 1 = $1, then a “10% annually
compounded rate” implies that EPS in year 11 is (1.10)10 = 2.59.
So, the firm could have earned $1 per share 10 years ago, suffered
a string of losses, and then earned $2.59 per share this year.
Clearly, this is not what is implied by management’s statement
above.
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Chapter 05 – Introduction to Valuation: The Time Value of Money
2. Present Value and Discounting
A. The Single-Period Case
Given r, what amount today (Present Value or PV) will produce a
given future amount? Remember that FV = $X(1 + r). Rearrange
and solve for $X, which is the present value. Therefore,
PV = FV / (1 + r).
Example: $110 in 1 period with an interest rate of 10% has a PV =
110 / (1.1) = $100
Discounting – the process of finding the present value.
Lecture Tip: It may be helpful to utilize the example of $100
compounded at 10 percent to emphasize the present value concept.
Start with the basic formula: FV = PV(1 + r)t and rearrange to
find PV = FV / (1 + r)t. Students should recognize that the
discount factor is the inverse of the compounding factor. Ask the
class to determine the present value of $110 and $121 if the
amounts are received in one year and two years, respectively, and
the interest rate is 10%. Then demonstrate the mechanics:
$100 = $110 (1 / 1.1) = 110 (.9091)
$100 = $121 (1 / 1.12) = 121(.8264)
The students should recognize that it was an initial investment of
$100 invested at 10% that created these two future values.
B. Present Values for Multiple Periods
PV of future amount in t periods at r is:
PV = FV [1 / (1 + r)t], where [1 / (1 + r)t] is the discount factor,
or the present value interest factor, PVIF(r,t)
Example: If you have $259.37 in 10 periods and the interest rate
was 10%, how much did you deposit initially?
PV = 259.37 [1/(1.1)10] = 259.37(.3855) = $100
Discounted Cash Flow (DCF) – the process of valuation by finding
the present value
Lecture Tip: The present value decreases as interest rates
increase. Since there is a reciprocal relationship between PVIFs
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Chapter 05 – Introduction to Valuation: The Time Value of Money
and FVIFs, you should also point out that future values increase as
the interest rate increases. You can illustrate this by
starting with a present value of $100 and computing the future
value under different interest rate scenarios.
Example: Future Value of $100 at 10% for 5 years = 100(1.1)5 =
$161.05
Future Value of $100 at 12% for 5 years = 100(1.12)5 = 176.23
Future Value of $100 at 14% for 5 years = 100(1.14)5 = 192.54
3. More about Present and Future Values
A. Present versus Future Value
Present Value factors are reciprocals of Future Value factors:
PVIF(r,t) = 1 / (1 + r)t and FVIF(r,t) = (1 + r)t
Example: FVIF(10%,4) = 1.14 = 1.464
PVIF(10%,4) = 1 / 1.14 = .683
Basic present value equation: PV = FV [1 / (1 + r)t]
Lecture Tip: Students who fail to grasp the concept of time value
often do so because it is never really clear to them that given a
10% opportunity rate, $110 to be received in one year is
equivalent to having $100 today (or $90.90 one year ago, or
$82.64 two years ago, etc.). At its most fundamental level,
compounding and discounting are nothing more than using a set of
formulas to find equivalent values at any two points in time. In
economic terms, one might stress that equivalence just means that
a rational person will be indifferent between $100 today and $110
in one year, given a 10% opportunity. This is true because she
could (a) take the $100 today and invest it to have $110 in one
year or (b) she could borrow $100 today and repay the loan with
$110 in one year. A corollary to this concept is that one can’t (or
shouldn’t) add, subtract, multiply or divide money values in
different time periods unless those values are expressed in
equivalent terms, i.e., at a single point in time.
Lecture Tip: It is important to emphasize that there are four
variables in the basic time value equation. If we know three of the
four, we can always solve for the fourth. You can reinforce this
concept by asking the class “what must be known if we are
attempting to determine the discount rate of an investment?”
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Chapter 05 – Introduction to Valuation: The Time Value of Money
B. Determining the Discount Rate
Start with the basic time value of money equation and rearrange to
solve for r:
FV = PV(1 + r)t
r = (FV / PV)1/t – 1
Or, you can use a financial calculator to solve for r (I/Y on the
calculator). It is important to remember the sign convention on
most calculators and enter either the PV or the FV as negative.
Example: What interest rate makes a PV of $100 become a FV of
$150 in 6 periods?
r = (150 / 100)1/6 – 1 = 7%
or PV = -100; FV = 150; N = 6; CPT I/Y = 7%
Lecture Tip: The following example can be used to demonstrate
the effects of compounding over long periods.
Vincent Van Gogh’s “Sunflowers” was sold at auction in 1987
for approximately $36 million. It had been sold in 1889 for $125.
At what discount rate is $125 the present value of $36 million,
given a 98-year time span.
125 = 36,000,000 [1 / (1 + r)98]
(36,000,000 / 125)1/98 – 1 = r = .13686 = 13.686%
or use a financial calculator N = 98; PV = -125; FV =
36,000,000; CPT I/Y = 13.686%.
Of course, the example can be turned around. “If your great-
grandfather had purchased the painting in 1889 and your family
sold it for $36 million, the average annually compounded rate of
return on the $125 investment was ____?” Stating the problem this
way and working it as a compounding problem helps students to
see the relationship between discounting and compounding.
C. Finding the Number of Periods
FV = PV(1 + r)t – rearrange and solve for t. Remember your logs!
t = ln(FV / PV) / ln(1 + r)
Or use the financial calculator, just remember the sign convention.
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Chapter 05 – Introduction to Valuation: The Time Value of Money
If you compute a negative N, you have forgotten the sign
convention!
Example: How many periods before $100 today grows to $150 at
7%?
t = ln(150 / 100) / ln(1.07) = 6 periods
Rule of 72 – the time to double your money, (FV / PV) = 2.00, is
approximately (72 / r%) periods. The rate needed to double your
money is approximately (72 / t)%.
Example: To double your money at 10% takes approximately
(72/10) = 7.2 periods.
4. Summary and Conclusions
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