Chapter 4
DISCOUNTED CASH FLOW VALUATION
SLIDES
4.41 Int
ere
st-
Only Loans
4.1 Key Concepts and Skills
4.2 Chapter Outline
4.3 The One-Period Case
4.4 Future Value
4.5 Present Value
4.6 Present Value
4.7 Net Present Value
4.8 Net Present Value
4.9 Net Present Value
4.10 The Multiperiod Case
4.11 Future Value
4.12 Future Value and Compounding
4.13 Future Value and Compounding
4.14 Present Value and Discounting
4.15 Finding the Number of Periods
4.16 What Rate is Enough?
4.17 Calculator Keys
4.18 Multiple Cash Flows
4.19 Multiple Cash Flows
4.20 Valuing “Lumpy” Cash Flows
4.21 Compounding Periods
4.22 Compounding Periods
4.23 Effective Annual Rates of Interest
4.24 Effective Annual Rates of Interest
4.25 Effective Annual Rates of Interest
4.26 EAR on a Financial Calculator
4.27 Continuous Compounding
4.28 Simplifications
4.29 Perpetuity
4.30 Perpetuity: Example
4.31 Growing Perpetuity
4.32 Growing Perpetuity: Example
4.33 Annuity
4.34 Annuity: Example
4.35 Continued
4.36 Growing Annuity
4.37 Growing Annuity: Example
4.38 Growing Annuity: Example
4.39 Loan Amortization
4.40 Pure Discount Loans
4.42 Amortized Loan with Fixed Principal Payment
4.43 Amortized Loan with Fixed Payment
4.44 What Is a Firm Worth?
4.45 Quick Quiz
CHAPTER WEB SITES
Section Web Address
4.2 www.studyfinance.com
CHAPTER ORGANIZATION
4.1 Valuation: The One-Period Case
4.2 The Multiperiod Case
Future Value and Compounding
The Power of Compounding: A Digression
Present Value and Discounting
Finding the Number of Periods
The Algebraic Formula
4.3 Compounding Periods
Distinction between Annual Percentage Rate and Effective Annual Rate
Compounding over Many Years
Continuous Compounding
4.4 Simplifications
Perpetuity
Growing Perpetuity
Annuity
Growing Annuity
4.5 Loan Amortization
4.6 What is a Firm Worth?
ANNOTATED CHAPTER OUTLINE
Slide 4.0 Chapter 4 Title Slide
Slide 4.1 Key Concepts and Skills
Slide 4.2 Chapter Outline
This chapter explains the algebra of the time value of money and net present
value. NPV depends upon the size, timing, and riskiness of expected cash flows,
which is consistent with the maximization of shareholder wealth discussed in
Chapter 1.
There are three ways to compute time-value-of-money problems: with a financial
calculator (or spreadsheet), with formulas, and with time value factor tables. A
good understanding of the formulas is necessary to value more complex cash flow
streams in later chapters; however, the understanding of financial calculators and
spreadsheets is just as important.
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 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 are not the same thing. And just as U.S.
dollars rarely trade 1 to 1 for Canadian dollars, neither do present dollars trade
1 to 1 for future dollars (except if r=0). 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. Valuation: The One-Period Case
Slide 4.3 The One-Period Case
Slide 4.4 Future Value
If you invest $C today at an interest rate of r, you will have $C +
$C(r) = $C(1 + r) in one period.
The general form is: FV = C0×(1 + r)
where
r is the interest rate per period (or opportunity cost)
C0 (also called PV) is the value at period 0
FV (also called Ct+T) is the value at period t+T
Compounding solves for the value at the end of the investment duration
(FV), and discounting solves for the value at the beginning of the
investment duration (PV).
Example: $10,000 at 5% interest gives $10,000(1.05) = $10,500
Slide 4.5 –
Slide 4.6 Present Value
Given r, what amount today (Present Value or PV) will produce a given
future amount? Remember that FV = $C0 (1 + r). Rearrange and solve for
$C0, which is the present value. Therefore,
PV = FV / (1 + r) = C1 / (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: 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 cost. 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 cannot (or should not) 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?” When the possibility of multiple cash flows is added, it
becomes five variables, with four being known.
Slide 4.7 –
Slide 4.9 Net Present Value
Some tips for computing NPV:
Only add (subtract) cash ows from the same time period
Use the Time Line
Specify a cash ow for each time period (even when it is
$0)
Use an appropriate discount rate
The general form for calculating NPV:
NPV =−C0+C1
(1+r)1+C2
(1+r)2+⋯+ CT
(1+r)T
The initial cash flow is typically an investment and is subtracted to
compute the NPV. Note that the formula from the text assumes that the
discount rate (r) is the same each period. Even though this is often the
case, you may wish to point out this assumption to students and go over an
example that uses two discount rates at different time periods.
An Example: You have an opportunity to invest in a business that will pay
$200,000 in one year, $400,000 in two years, $600,000 in three years and
$800,000 in four years. You can earn 12% per year compounded annually
on a mutual fund that has similar risk. If it costs $1.2 million to start this
business, should you invest?
0 1 2 3 4 years
| | | | |
|||||
CF –$1.2 mil $200,000 $400,000 $600,000 $800,000
Discount rate = 12%
NPV =−C0+C1
(1+r)1+C2
(1+r)2+⋯+CT
(1+r)T
NPV =−1, 200 ,000+200 ,000
(1 .12 )1+400 ,000
(1 .12 )2+600 ,000
(1. 12)3+800 ,000
(1 .12 )4
= $232,932
Lecture Tip: Net present value is essentially a marginal benefit versus
marginal cost comparison. The marginal benefit is the present value of the
future cash flows generated by the project (or investment), and the
marginal cost is the outflow required at time zero (i.e., today). If the
marginal benefit is greater than the marginal cost (i.e., NPV is positive),
the project should be accepted. Given that students generally have had an
economics class, this comparison seems to help with understanding.
2. The Multiperiod Case
A. Future Value and Compounding
Slide 4.10 The Multiperiod Case
Slide 4.11 Future Value
Slide 4.12 –
Slide 4.13 Future Value and Compounding
Reinvesting interest, we earn interest on interest, i.e., compounding
FV = $C0 (1 + r)(1 + r) = $C0 (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 = $C0 (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
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.
Lecture Tip: Slides 4.12 and 4.13 distinguish between simple interest and
compound interest and can be used to emphasize the effects of earning
interest on interest. 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.
The Power of Compounding: A Digression
Lecture 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
ending salary be?”
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 $15,000 automobile will cost $72,015
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!
B. Present Value and Discounting
Slide 4.14 Present Value and Discounting
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
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
Lecture Tip: The present value decreases as interest rates increase.
Since there is a reciprocal relationship between PVIF’s and
FVIF’s, 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.
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
C. Finding the Number of Periods
Slide 4.15 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.
If you compute a negative N (or receive an error), you have likely
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.
Lecture Tip: It may be beneficial to illustrate to students that
solving for the interest rate in multiple cash flow problems is
essentially an iterative process. Thus, financial calculators are
extremely useful.
Slide 4.16 What Rate is Enough?
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.
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 =
2.5937. 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.