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Instructor's Manual Sample
Chapter 4: Introduction to Valuation:
The Time Value of Money
TRANSPARENCIES
T4.1: Chapter Outline
T4.2: Future Value for a Lump Sum (2 pages)
T4.3: Chapter 4 Quick Quiz – Part 1 of 4
T4.4: Future Value of $100 at 10 Percent (Table 4.1)
T4.5: Interest on Interest Illustration
T4.6: Chapter 4 Quick Quiz – Part 2 of 4 (2 pages)
T4.7: Present Value for a Lump Sum
T4.8: Present Value of $1 for Different Periods and Rates (Fig. 4.3)
T4.9: Example: A Penny Saved (2 pages)
T4.10: Chapter 4 Quick Quiz – Part 3 of 4
T4.11: Summary of Time Value Calculations (Table 4.4)
T4.12: Chapter 4 Quick Quiz – Part 4 of 4 (2 pages)
T4.13: Solution to Problem 4.6 (2 pages)
T4.14: Solution to Problem 4.10
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CHAPTER ORGANIZATION
4.1 FUTURE VALUE AND COMPOUNDING
Investing for a Single Period
Investing for More than One Period
4.2 PRESENT VALUE AND DISCOUNTING
The Single-Period Case
Present Values for Multiple Periods
4.3 MORE ON PRESENT AND FUTURE VALUES
Present versus Future Value
Determining the Discount Rate
Finding the Number of Periods
ANNOTATED CHAPTER OUTLINE
4.1 FUTURE VALUE AND COMPOUNDING
Lecture Tip, page 84: Many students find
the phrases "time value of money" and "a dollar today is worth more than
a dollar later" to be somewhat cryptic. In some ways, it might be better
to say "the money value of time" and to state that a dollar today doesn't
(and indeed, cannot) trade for less than a dollar later.
-----Indeed, many of the phrases
and much of the terminology surrounding exchanges of money now for money
later are 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 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 (or leftmost and rightmost amounts
on the time line).
-----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
and vice-versa, present value 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.
A. Investing for a Single Period
T4.2: Future Value for
a Lump Sum (2 pages)
T4.3: Chapter 4 Quick Quiz – Part 1 of 43 |
Given r, the interest rate, every $1 today will produce (1 + r) of future value (FV). So,
FV = $X(1 + r), where $X is principal.
Example:
$100 at 10% interest gives $100(1.10) = $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: $100(1.10)(1.10) = $100(1.10)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: $100(1.10)10= $259.37
FVIF(r,t): Factor can be obtained in various ways
-factor tables such as A.1 of Appendix in text
-scientific calculator with yx key
-financial calculator
Lecture Tip, page 84: It may be helpful
to emphasize this compounding example on the chalkboard. Demonstrate the
compounding of $100 at 10 percent by showing the future value at the end
of year one. Then separate the $110 into $100 principal and $10 interest.
Now demonstrate that the $100 principal will then earn another $10 over
the second year and the $10 interest earned at the end of the first year
will earn $1 interest over the second year, resulting in a $121 end-of-year-two
value
Example:
Present----- End
Yr. 1 Value ----- End Yr. 2 Value
$100 ------> $100----------Principal------>
$110
---------------$
10----------Interest
-------> $ 11
$100 --------
$110 ---------------------------$121
By stressing this paragraph's example and the initial example
in the text, the students' intuition of compounding or interest-on-interest
may be enhanced. This example is extended over five periods in Table 4.1
in the text. A failure to understand this compounding impact will create
trouble for some students throughout the course.
T4.4: Future Value of $100
at 10 Percent (Table 4.1)
T4.5: Interest on Interest Illustration4 |
Lecture Tip, page 89: Students are often
helped by concrete examples tied to real life. For example, one might 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 that the rate of inflation stays at around 4% annually for the next
40 years, and you receive annual cost-of-living increases tied to the inflation
rate. What will your ending salary be?"
-----Most students are happy to hear that their
final annual salary will be $120,025. (=$25,000 ´
(1.04)40) They are often less happy, however, when they find
that today's $15,000 automobile will cost $72,015 under the same assumptions.
-----This example can be extended in many directions.
For example, you might next ask how much their final salary will be 40 years
hence, should they receive better-than-average raises of, say, 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! (Admittedly, the difference is smaller than it appears when
one realizes that it is quoted in future dollars and wouldn't be enough
to buy us that $15,000 car 40 years hence.)
| T4.6: Chapter
4 Quick Quiz – Part 2 of 4 |
4.2 PRESENT VALUE AND DISCOUNTING
A. The Single Period Case
Given r, what amount today (Present Value or PV)
will produce a given future amount?
Since future amount = $X(1 + r), PV = future amount/(1 + r).
Example:
$110 in 1 period at 10% has a PV of $110/(1.10) = $100.
Discounting- the
process of finding PV
| T4.7: Present
Value for a Lump Sum |
Lecture Tip, page 92: 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: PV ´
(1 + r)t = FV; therefore, PV = FV ´
[1 / (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 percent. Now demonstrate
the mechanics:
$100 = $110 ´ [1 / (1
+ .10)1] = $110 ´ .0909
$100 = $121 ´ [1 / (1 + .10)2]
= $121 ´ .8264
The students will recognize that it was an initial investment
of $100 and an interest rate of 10 percent that created these two future
values.
B. Present Values for Multiple Periods
PV of future amount in t periods at r is:
PV = future amount ´
[1/(1 + r)t], where [1/(1 + r)t] is the discount
factor or Present Value Interest Factor, PVIF(r,t).
Example:
$259.37 10 periods from now has a PV at 10% of $259.37 ´
[1/(1.10)10] = $100 (the PVIF is .3855).
DCF (Discounted Cash Flow)-
the process of valuation by finding the present value.
Lecture Tip, page 93: The following example
can be used to dramatize the effect of discounting over long periods.
Vincent Van Gogh's 'Sunflowers' was sold at auction in
1987 for approximately $36 million. It had sold in 1889 for $125. At what
discount rate is $125 the present value of $36 million, given a 98-year
timespan?
$125/$36 million = PVIF(r,98) = .0000034722 = (1 + r)1/98.
Solving for r, we find that the implied discount rate is approximately 13.685%.
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.
| T4.8: Present
Value of $1 for Different Periods and Rates (Fig. 4.3)7
|
4.3 MORE ON 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) = (1.10)4 = 1.464 and PVIF(10%,4)
= 1/(1.10)4 = .683
Basic present value equation: PV = FV ´
[1/(1 + r)t]
Lecture Tip, page 96: 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, $11 to be received
in one year is equivalent to having $10 today. Or $9.09 one year
ago. Or $8.26 two years ago, etc. At its most fundamental level, compounding
and discounting is 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 $10 today and $11 in one year, given
a 10% opportunity rate, because s/he could (a) take the $10 today and
invest it to have $11 in one year, or (b) sell the right to receive
$11 in one year for $10 today.
-----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 stated in equivalent
terms, i.e., at a single point in time.
B. Determining the Discount Rate
Finding r in the basic present value equation
PV = FV ´ [1/(1 + r)t]
or FV = PV(1 + r)t
-financial calculator (supply PV, FV, and t; compute r)
-[tth root of FV/PV] - 1
-look up either PVIF (PV/FV) or FVIF (FV/PV) in appropriate table
Example:
What interest rate makes a PV of $100 become a FV of $150 in
6 periods?
-FV/PV = $150/100 =1.5, the 6th root of 1.5
is 1.0699, making r = 7%
-FV/PV gives an FVIF of 1.5, look across 6 periods in Table A.1, r = 7%
-PV/FV gives a PVIF of .6666, look across 6 periods in Table A.2, r = 7%
Lecture Tip, page 96: This paragraph in
the text should be strongly emphasized. With the examples used thus far,
there have always been four parts to the equation, one of which is
unknown. In finding the future value, we must know the present value,
the interest rate and the time of the investment. In finding the present
value, we must know the future value, interest rate, and time of the
investment. Although it may seem obvious, ask the class "what must be known
if we are attempting to determine the discount rate of an investment?" (PV,
FV and t). Then use the previous example and ask the class to determine
the discount rate or return on an investment if they invested $100 today
and received $121 in two years. The students should remember that 10 percent
generated the identity. Finally, show the solution process using Table A.1
or A.2.
T4.9: Example: A Penny
Saved (2 pages)
T4.10: Chapter 4 Quick Quiz – Part 3 of 4 |
C. Finding the Number of Periods
Finding t in the basic present value equation
PV = FV ´
[1/(1 + r)t] (or FV = PV(1 + r)t)
-financial calculator (supply PV, FV, and r; compute t)
-using FV = PV(1 + r)t, make (FV/PV) = (1 + r)t, ln(FV/PV) = t[ln(1 + r)], then t = [ln(FV/PV)/ln(1 + r)]
-using PVIF ( = PV/FV) or FVIF ( = FV/PV), look under r in appropriate table.
Example: How many periods before $100 today grows to be $150 at 7%?
-FV/PV = 150/100 = 1.5; 1 + r = 1.07;
-ln(1.5) = .405465 and ln(1.07) = .067659,
so ln(FV/PV)/ln(1 + r) = .405465/.067659 = 6 periods.
-FV/PV gives FVIF of 1.5, look under 7% in Appendix A, Table A.1, t = 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.
Example: To double your money in 6 years takes approximately (72/6) = 12%.
T4.11: Summary of Time
Value Calculations (Table 4.4)
T4.12: Chapter 4 Quick Quiz – Part 4 of 4 (2 pages)
T4.13: Solution to Problem 4.6 (2 pages)
T4.14: Solution to Problem 4.10 |
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