T5.1 Chapter Outline
Chapter 5
Introduction to Valuation: The Time Value of Money
Chapter Organization
 5.1 Future Value and Compounding
 5.2 Present Value and Discounting
 5.3 More on Present and Future Values
 5.4 Summary and Conclusions
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T5.2 Time Value Terminology
 Consider the time line below:
0
1
2
3
t
...
PV
FV
 PV is the Present Value, that is, the value today.
 FV is the Future Value, or the value at a future date.
 The number of time periods between the Present Value and
the Future Value is represented by “t”.
 The rate of interest is called “r”.
 All time value questions involve the four values above: PV,
FV, r, and t. Given three of them, it is always possible to
calculate the fourth.
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T5.3 Future Value for a Lump Sum
 Notice that
 1. $110
= $100  (1 + .10)
 2. $121
= $110  (1 + .10) = $100  1.10  1.10 = $100  1.102
 3. $133.10 = $121  (1 + .10) = $100  1.10  1.10  1.10
= $100  ________
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T5.3 Future Value for a Lump Sum
 Notice that
 1. $110
= $100  (1 + .10)
 2. $121
= $110  (1 + .10) = $100  1.10  1.10 = $100  1.102
 3. $133.10 = $121  (1 + .10) = $100  1.10  1.10  1.10
= $100  (1.10)3
 In general, the future value, FVt, of $1 invested today at r%
for t periods is
FVt = $1  (1 + r)t
 The expression (1 + r)t is the future value interest factor.
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T5.4 Chapter 5 Quick Quiz - Part 1 of 5
 Q.
Deposit $5,000 today in an account paying 12%. How
much will you have in 6 years? How much is simple
interest? How much is compound interest?
 A.
Multiply the $5000 by the future value interest factor:
$5000  (1 + r )t = $5000  ___________
= $5000  1.9738227
= $9869.11
At 12%, the simple interest is .12  $5000 = $_____ per
year. After 6 years, this is 6  $600 = $_____ ; the
difference between compound and simple interest is thus
$_____ - $3600 = $_____
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T5.4 Chapter 5 Quick Quiz - Part 1 of 5
 Q.
Deposit $5,000 today in an account paying 12%. How
much will you have in 6 years? How much is simple
interest? How much is compound interest?
 A.
Multiply the $5000 by the future value interest factor:
$5000  (1 + r )t = $5000  (1.12)6
= $5000  1.9738227
= $9869.11
At 12%, the simple interest is .12  $5000 = $600 per
year. After 6 years, this is 6  $600 = $3600; the
difference between compound and simple interest is thus
$4869.11 - $3600 = $1269.11
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T5.5 Interest on Interest Illustration
Q. You have just won a $1 million jackpot in the state lottery. You can
buy a ten year certificate of deposit which pays 6% compounded
annually. Alternatively, you can give the $1 million to your brother-in-law,
who promises to pay you 6% simple interest annually over the ten year
period. Which alternative will provide you with more money at the
end of ten years?
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T5.5 Interest on Interest Illustration
Q. You have just won a $1 million jackpot in the state lottery. You can
buy a ten year certificate of deposit which pays 6% compounded
annually. Alternatively, you can give the $1 million to your brother-in-law,
who promises to pay you 6% simple interest annually over the ten year
period. Which alternative will provide you with more money at the
end of ten years?
A. The future value of the CD is $1 million x (1.06)10 = $1,790,847.70.
The future value of the investment with your brother-in-law, on the
other hand, is $1 million + $1 million (.06)(10) = $1,600,000.
Compounding (or interest on interest), results in incremental
wealth of nearly $191,000. (Of course we haven’t even begun to address
the risk of handing your brother-in-law $1 million!)
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T5.6 Future Value of $100 at 10 Percent (Table 5.1)
Beginning
Year
Amount
Simple Compound
Interest
Interest
1
$100.00
$10.00
$ 0.00
$10.00
$110.00
2
110.00
10.00
1.00
11.00
121.00
3
121.00
10.00
2.10
12.10
133.10
4
133.10
10.00
3.31
13.31
146.41
5
146.41
Totals
10.00
$50.00
4.64
$ 11.05
14.64
$ 61.05
161.05
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Total
Interest Earned
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Ending
Amount
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T5.7 Chapter 5 Quick Quiz - Part 2 of 5
 Want to be a millionaire? No problem!
Suppose you are currently 21 years old, and
can earn 10 percent on your money. How
much must you invest today in order to
accumulate $1 million by the time you reach
age 65?
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T5.7 Chapter 5 Quick Quiz - Part 2 of 5 (concluded)
 First define the variables:
FV = $1 million
r = 10 percent
t = 65 - 21 = 44 years
PV = ?
 Set this up as a future value equation and solve for
the present value:
$1 million = PV  (1.10)44
PV = $1 million/(1.10) 44 = $15,091.
 Of course, we’ve ignored taxes and other
complications, but stay tuned - right now you need to
figure out where to get $15,000!
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T5.8 Present Value for a Lump Sum
 Q. Suppose you need $20,000 in three years to pay your
college tuition. If you can earn 8% on your money, how much
do you need today?
 A. Here we know the future value is $20,000, the rate (8%),
and the number of periods (3). What is the unknown
present amount (i.e., the present value)? From before:
FVt
= PV  (1 + r )t
$20,000 = PV  __________
Rearranging:
PV
= $20,000/(1.08)3
= $ ________
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T5.8 Present Value for a Lump Sum
 Q. Suppose you need $20,000 in three years to pay your
college tuition. If you can earn 8% on your money, how
much do you need today?
 A. Here we know the future value is $20,000, the rate (8%),
and the number of periods (3). What is the unknown
present amount (i.e., the present value)? From before:
FVt
= PV x (1 + r )t
$20,000 = PV x (1.08)3
Rearranging:
PV
= $20,000/(1.08)3
= $15,876.64
The PV of a $1 to be received in t periods when the rate is r is
PV = $1/(1 + r )t
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T5.9 Present Value of $1 for Different Periods and Rates (Figure 5.3)
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T5.10 Example: Finding the Rate
 Benjamin Franklin died on April 17, 1790. In his will, he gave 1,000
pounds sterling to Massachusetts and the city of Boston. He gave
a like amount to Pennsylvania and the city of Philadelphia. The
money was paid to Franklin when he held political office, but he
believed that politicians should not be paid for their service(!).
Franklin originally specified that the money should be paid out
100 years after his death and used to train young people. Later,
however, after some legal wrangling, it was agreed that the money
would be paid out 200 years after Franklin’s death in 1990. By that
time, the Pennsylvania bequest had grown to about $2 million; the
Massachusetts bequest had grown to $4.5 million. The money was
used to fund the Franklin Institutes in Boston and Philadelphia.
 Assuming that 1,000 pounds sterling was equivalent to 1,000
dollars, what rate did the two states earn? (Note: the dollar didn’t
become the official U.S. currency until 1792.)
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T5.10 Example: Finding the Rate (continued)
 Q. Assuming that 1,000 pounds sterling was equivalent
to 1,000 dollars, what rate did the two states earn?
 A. For Pennsylvania, the future value is $________ and
the present value is $______ . There are 200 years
involved, so we need to solve for r in the following:
________ = _____________/(1 + r )200
(1 + r )200 = ________
Solving for r, the Pennsylvania money grew at about 3.87%
per year. The Massachusetts money did better; check that the
rate of return in this case was 4.3%.
Small
differences can add up!
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T5.10 Example: Finding the Rate (concluded)
 Q. Assuming that 1,000 pounds sterling was equivalent
to 1,000 dollars, what rate did the two states earn?
 A. For Pennsylvania, the future value is $ 2 million and
the present value is $ 1,000. There are 200 years
involved, so we need to solve for r in the following:
$ 1,000 = $ 2 million/(1 + r )200
(1 + r )200 = 2,000.00
Solving for r, the Pennsylvania money grew at about
3.87% per year. The Massachusetts money did better;
check that the rate of return in this case was 4.3%.
Small differences can add up!
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T5.11 The Rule of 72
 The “Rule of 72” is a handy rule of thumb that states the
following:
If you earn r % per year, your money will double in about
72/r % years.
So, for example, if you invest at 6%, your money will
double in 12 years.
 Why do we say “about?” Because at higher-than-normal
rates, the rule breaks down.
What if r = 72%?

FVIF(72,1) = 1.72, not 2.00
And if r = 36%?

FVIF(36,2) = 1.8496
The lesson? The Rule of 72 is a useful rule of thumb, but it
is only a rule of thumb!
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T5.12 Chapter 5 Quick Quiz - Part 4 of 5
 Suppose you deposit $5000 today in an account paying r
percent per year. If you will get $10,000 in 10 years, what rate
of return are you being offered?
 Set this up as present value equation:
FV = $10,000
PV = $ 5,000
PV
=
FVt/(1 + r )t
$5000
=
$10,000/(1 + r)10
t = 10 years
 Now solve for r:
(1 + r)10 = $10,000/$5,000 = 2.00
r = (2.00)1/10 - 1 = .0718 = 7.18 percent
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T5.13 Example: The (Really) Long-Run Return on Common Stocks
According to Stocks for the Long Run, by Jeremy Siegel, the average annual
compound rate of return on common stocks was 8.4% over the period from
1802-1997. Suppose a distant ancestor of yours had invested $1000 in a
diversified common stock portfolio in 1802. Assuming the portfolio remained
untouched, how large would that portfolio be at the end of 1997? (Hint: if you
owned this portfolio, you would never have to work for the rest of your life!)
Common stock values increased by 28.59% in 1998 (as proxied by the growth of
the S&P 500). How much would the above portfolio be worth at the end of 1998?
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T5.13 Example: The (Really) Long-Run Return on Common Stocks
According to Stocks for the Long Run, by Jeremy Siegel, the average annual
return on common stocks was 8.4% over the period from 1802-1997. Suppose a
distant ancestor of yours had invested $1000 in a diversified common stock
portfolio in 1802. Assuming the portfolio remained untouched, how large would
that portfolio be at the end of 1997? (Hint: if you owned this portfolio, you would
never have to work for the rest of your life!)
t = 195 years, r = 8.4%, and FVIF(8.4,195) = 6,771,892.09695
So the value of the portfolio would be: $6,771,892,096.95!
Common stock values increased by 28.59% in 1998 (as proxied by the growth of
the S&P 500). How much would the above portfolio be worth at the end of 1998?
The 1998 value would be $6,771,892,096.95  (1 + .2859) = $8,707,976,047.47!
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T5.14 Summary of Time Value Calculations (Table 5.4)
I. Symbols:
PV = Present value, what future cash flows are worth today
FVt = Future value, what cash flows are worth in the future
r
= Interest rate, rate of return, or discount rate per period
t
= number of periods
C = cash amount
II. Future value of C invested at r percent per period for t periods:
FVt = C  (1 + r )t
The term (1 + r )t is called the future value factor.
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T5.14 Summary of Time Value Calculations (Table 5.4) (concluded)
III. Present value of C to be received in t periods at r percent
per period:
PV = C/(1 + r )t
The term 1/(1 + r )t is called the present value factor.
IV. The basic present value equation giving the relationship
between present and future value is:
PV = FVt/(1 + r )t
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T5.15 Chapter 5 Quick Quiz - Part 5 of 5
 Now let’s see what you remember!
1. Which of the following statements is/are true?

Given r and t greater than zero, future value interest factors
FVIF(r,t ) are always greater than 1.00.

Given r and t greater than zero, present value interest factors
PVIF(r,t ) are always less than 1.00.
2. True or False: For given levels of r and t, PVIF(r,t ) is the
reciprocal of FVIF(r,t ).
3. All else equal, the higher the discount rate, the
(lower/higher) the present value of a set of cash flows.
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T5.15 Chapter 5 Quick Quiz - Part 5 of 5 (concluded)
1. Both statements are true. If you use time value tables, use
this information to be sure that you are looking at the
correct table.
2. This statement is also true. PVIF(r,t ) = 1/FVIF(r,t ).
3. The answer is lower - discounting cash flows at higher
rates results in lower present values. And compounding
cash flows at higher rates results in higher future values.
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T5.16 Solution to Problem 5.6
 Assume the total cost of a college education will be
$200,000 when your child enters college in 18 years. You
have $15,000 to invest. What annual rate of interest must
you earn on your investment to cover the cost of your
child’s college education?
Present value = $15,000
Future value
t = 18
= $200,000
r=?
 Solution: Set this up as a future value problem.
$200,000
= $15,000  FVIF(r,18)
FVIF(r,18)
= $200,000 / $15,000 = 13.333 . . .
Solving for r gives 15.48%.
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T5.17 Solution to Problem 5.10
 Imprudential, Inc. has an unfunded pension liability of $425
million that must be paid in 20 years. To assess the value
of the firm’s stock, financial analysts want to discount this
liability back to the present. If the relevant discount rate is
8 percent, what is the present value of this liability?
Future value = FV = $425 million
t = 20
r = 8 percent
Present value = ?
 Solution: Set this up as a present value problem.
PV = $425 million  PVIF(8,20)
PV = $91,182,988.15 or about $91.18 million
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T6.1 Chapter Outline
Chapter 6
Discounted Cash Flow Valuation
Chapter Organization
 6.1 Future and Present Values of Multiple Cash Flows
 6.2 Valuing Level Cash Flows: Annuities and Perpetuities
 6.3 Comparing Rates: The Effect of Compounding
 6.4 Loan Types and Loan Amortization
 6.5 Summary and Conclusions
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T6.2 Future Value Calculated (Fig. 6.3-6.4)
Future value calculated by compounding forward one period at a time
Future value calculated by compounding each cash flow separately
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T6.3 Present Value Calculated (Fig 6.5-6.6)
Present value
calculated by
discounting each
cash flow separately
Present value
calculated by
discounting back one
period at a time
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T6.4 Chapter 6 Quick Quiz: Part 1 of 4
Example: Finding C
 Q. You want to buy a Mazda Miata to go cruising. It costs $25,000.
With a 10% down payment, the bank will loan you the rest at 12% per
year (1% per month) for 60 months. What will your monthly payment
be?
 A. You will borrow ___  $25,000 = $______ . This is the amount today, so
it’s the ___________ . The rate is ___ , and there are __ periods:
$ ______ = C  { ____________}/.01
= C  {1 - .55045}/.01
= C  44.955
C
C
= $22,500/44.955
= $________
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T6.4 Chapter 6 Quick Quiz: Part 1 of 4 (concluded)
Example: Finding C
 Q. You want to buy a Mazda Miata to go cruising. It costs $25,000.
With a 10% down payment, the bank will loan you the rest at 12% per
year (1% per month) for 60 months. What will your monthly payment
be?
 A. You will borrow .90  $25,000 = $22,500 . This is the amount today, so
it’s the present value. The rate is 1%, and there are 60 periods:
$ 22,500
C
C
= C  {1 - (1/(1.01)60}/.01
= C  {1 - .55045}/.01
= C  44.955
= $22,500/44.955
= $500.50 per month
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T6.5 Annuities and Perpetuities -- Basic Formulas
 Annuity Present Value
PV = C  {1 - [1/(1 + r )t]}/r

Annuity Future Value
FVt = C  {[(1 + r )t - 1]/r}
 Perpetuity Present Value
PV = C/r
 The formulas above are the basis of many of the calculations in
Corporate Finance. It will be worthwhile to keep them in mind!
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T6.6 Examples: Annuity Present Value
Annuity Present Value
 Suppose you need $20,000 each year for the next three
years to make your tuition payments.
Assume you need the first $20,000 in exactly one year.
Suppose you can place your money in a savings
account yielding 8% compounded annually. How much
do you need to have in the account today?
(Note: Ignore taxes, and keep in mind that you don’t
want any funds to be left in the account after the third
withdrawal, nor do you want to run short of money.)
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T6.6 Examples: Annuity Present Value (continued)
Annuity Present Value - Solution
Here we know the periodic cash flows are $20,000
each. Using the most basic approach:
PV = $20,000/1.08 + $20,000/1.082 + $20,000/1.083
= $18,518.52 + $_______ + $15,876.65
= $51,541.94
Here’s a shortcut method for solving the problem using the
annuity present value factor:
PV = $20,000 [____________]/__________
= $20,000 2.577097
= $________________
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T6.6 Examples: Annuity Present Value (continued)
Annuity Present Value - Solution
Here we know the periodic cash flows are $20,000
each. Using the most basic approach:
PV = $20,000/1.08 + $20,000/1.082 + $20,000/1.083
= $18,518.52 + $17,146.77 + $15,876.65
= $51,541.94
Here’s a shortcut method for solving the problem using the
annuity present value factor:
PV = $20,000  [1 - 1/(1.08)3]/.08
= $20,000  2.577097
= $51,541.94
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T6.6 Examples: Annuity Present Value (continued)
Annuity Present Value
 Let’s continue our tuition problem.
Assume the same facts apply, but that you can only
earn 4% compounded annually. Now how much do you
need to have in the account today?
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T6.6 Examples: Annuity Present Value (concluded)
Annuity Present Value - Solution
Again we know the periodic cash flows are $20,000
each. Using the basic approach:
PV = $20,000/1.04 + $20,000/1.042 + $20,000/1.043
= $19,230.77 + $18,491.12 + $17,779.93
= $55,501.82
Here’s a shortcut method for solving the problem using the
annuity present value factor:
PV = $20,000  [1 - 1/(1.04)3]/.04
= $20,000  2.775091
= $55,501.82
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T6.7 Chapter 6 Quick Quiz -- Part 2 of 4
Example 1: Finding t
 Q.
Suppose you owe $2000 on a Visa card, and the
interest rate is 2% per month. If you make the
minimum monthly payments of $50, how long will it
take you to pay off the debt? (Assume you quit
charging stuff immediately!)
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T6.7 Chapter 6 Quick Quiz -- Part 2 of 4
Example 1: Finding t
 Q.
Suppose you owe $2000 on a Visa card, and the
interest rate is 2% per month. If you make the
minimum monthly payments of $50, how long will it
take you to pay off the debt? (Assume you quit
charging stuff immediately!)
A.
A long time:
$2000
.80 t =
1.02 =
t
=
years!
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=
$50
 {1 - 1/(1.02)t}/.02
t
1 - 1/1.02
5.0
81.3 months, or about 6.78
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T6.7 Chapter 6 Quick Quiz -- Part 2 of 4
Example 2: Finding C
 Previously we determined that a 21-year old could accumulate $1
million by age 65 by investing $15,091 today and letting it earn
interest (at 10%compounded annually) for 44 years.
Now, rather than plunking down $15,091 in one chunk, suppose
she would rather invest smaller amounts annually to accumulate
the million. If the first deposit is made in one year, and deposits
will continue through age 65, how large must they be?
 Set this up as a FV problem:
$1,000,000 = C  [(1.10)44 - 1]/.10
C = $1,000,000/652.6408 = $1,532.24
Becoming a millionaire just got easier!
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T6.8 Example: Annuity Future Value
 Previously we found that, if one begins saving at age 21,
accumulating $1 million by age 65 requires saving only
$1,532.24 per year.
Unfortunately, most people don’t start saving for retirement
that early in life. (Many don’t start at all!)
Suppose Bill just turned 40 and has decided it’s time to get
serious about saving. Assuming that he wishes to accumulate
$1 million by age 65, he can earn 10% compounded annually,
and will begin making equal annual deposits in one year and
make the last one at age 65, how much must each deposit be?

Setup:
$1 million = C  [(1.10)25 - 1]/.10
Solve for C: C = $1 million/98.34706 = $10,168.07
By waiting, Bill has to set aside over six times as much money
each year!
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T6.9 Chapter 6 Quick Quiz -- Part 3 of 4
 Consider Bill’s retirement plans one more time.
Again assume he just turned 40, but, recognizing that he has
a lot of time to make up for, he decides to invest in some highrisk ventures that may yield 20% annually. (Or he may lose his
money completely!) Anyway, assuming that Bill still wishes to
accumulate $1 million by age 65, and will begin making equal
annual deposits in one year and make the last one at age 65,
now how much must each deposit be?

Setup:
$1 million = C  [(1.20)25 - 1]/.20
Solve for C: C = $1 million/471.98108 = $2,118.73
So Bill can catch up, but only if he can earn a much higher
return (which will probably require taking a lot more risk!).
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2000
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T6.10 Summary of Annuity and Perpetuity Calculations (Table 6.2)
I. Symbols
PV
FVt
r
t
C
=
=
=
=
=
Present value, what future cash flows bring today
Future value, what cash flows are worth in the future
Interest rate, rate of return, or discount rate per period
Number of time periods
Cash amount
II. FV of C per period for t periods at r percent per period:
FVt = C  {[(1 + r )t - 1]/r}
III. PV of C per period for t periods at r percent per period:
PV = C  {1 - [1/(1 + r )t]}/r
IV. PV of a perpetuity of C per period:
PV = C/r
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2000
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T6.11 Example: Perpetuity Calculations
 Suppose we expect to receive $1000 at the end of each of the
next 5 years. Our opportunity rate is 6%. What is the value
today of this set of cash flows?
PV = $1000  {1 - 1/(1.06)5}/.06
= $1000  {1 - .74726}/.06
= $1000  4.212364
= $4212.36
 Now suppose the cash flow is $1000 per year forever. This is
called a perpetuity. And the PV is easy to calculate:
PV = C/r = $1000/.06 = $16,666.66…
 So, payments in years 6 thru  have a total PV of $12,454.30!
Irwin/McGraw-Hill
2000
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T6.12 Chapter 6 Quick Quiz -- Part 4 of 4
Consider the following questions.
 The present value of a perpetual cash flow stream has a finite value
(as long as the discount rate, r, is greater than 0). Here’s a question
for you: How can an infinite number of cash payments have a finite
value?
 Here’s an example related to the question above. Suppose you are
considering the purchase of a perpetual bond. The issuer of the
bond promises to pay the holder $100 per year forever. If your
opportunity rate is 10%, what is the most you would pay for the bond
today?
 One more question: Assume you are offered a bond identical to the
one described above, but with a life of 50 years. What is the
difference in value between the 50-year bond and the perpetual
bond?
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2000
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T6.12 Solution to Chapter 6 Quick Quiz -- Part 4 of 4
 An infinite number of cash payments has a finite present
value is because the present values of the cash flows in the
distant future become infinitesimally small.
 The value today of the perpetual bond = $100/.10 = $1,000.
 Using Table A.3, the value of the 50-year bond equals
$100  9.9148 = $991.48
So what is the present value of payments 51 through infinity
(also an infinite stream)?
Since the perpetual bond has a PV of $1,000 and the
otherwise identical 50-year bond has a PV of $991.48, the
value today of payments 51 through infinity must be
$1,000 - 991.48 = $8.52 (!)
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2000
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T6.13 Compounding Periods, EARs, and APRs
Compounding
period
Effective
compounded
annual rate
 Year
1
10.00000%
 Quarter
4
10.38129
 Month
12
10.47131
 Week
52
10.50648
365
10.51558
8,760
10.51703
525,600
10.51709
 Day
 Hour
 Minute
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Number of times
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T6.13 Compounding Periods, EARs, and APRs (continued)
 EARs and APRs
 Q. If a rate is quoted at 16%, compounded semiannually,
then the actual rate is 8% per six months. Is 8% per six
months the same as 16% per year?
 A. If you invest $1000 for one year at 16%, then you’ll
have $1160 at the end of the year. If you invest at 8%
per period for two periods, you’ll have

FV =
$1000  (1.08)2
=
$1000  1.1664
=
$1166.40,
or $6.40 more. Why? What rate per year is the
same as 8% per six months?
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2000
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T6.13 Compounding Periods, EARs, and APRs (concluded)
 The Effective Annual Rate (EAR) is _____%. The “16%
compounded semiannually” is the quoted or stated rate,
not the effective rate.
 By law, in consumer lending, the rate that must be quoted
on a loan agreement is equal to the rate per period
multiplied by the number of periods. This rate is called the
_________________ (____).
 Q. A bank charges 1% per month on car loans. What is the
APR? What is the EAR?
 A. The APR is __  __ = ___%. The EAR is:
EAR = _________ - 1 = 1.126825 - 1 = 12.6825%
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2000
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T6.13 Compounding Periods, EARs, and APRs (concluded)
 The Effective Annual Rate (EAR) is 16.64%. The “16%
compounded semiannually” is the quoted or stated rate,
not the effective rate.
 By law, in consumer lending, the rate that must be quoted
on a loan agreement is equal to the rate per period
multiplied by the number of periods. This rate is called the
Annual Percentage Rate (APR).
 Q. A bank charges 1% per month on car loans. What is the
APR? What is the EAR?
 A. The APR is 1%  12 = 12%. The EAR is:
EAR = (1.01)12 - 1 = 1.126825 - 1 = 12.6825%
The APR is thus a quoted rate, not an effective rate!
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2000
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McGraw-Hill Companies, Inc.
T6.14 Example: Amortization Schedule - Fixed Principal
Beginning
Balance
Total
Payment
Interest
Paid
Principal
Paid
Ending
Balance
1
$5,000
$1,450
$450
$1,000
$4,000
2
4,000
1,360
360
1,000
3,000
3
3,000
1,270
270
1,000
2,000
4
2,000
1,180
180
1,000
1,000
5
1,000
1,090
90
1,000
0
$6,350
$1,350
$5,000
Year
Totals
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2000
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T6.15 Example: Amortization Schedule - Fixed Payments
Year
Beginning
Balance
Total
Payment
Interest
Paid
Principal
Paid
Ending
Balance
1
$5,000.00
$1,285.46
$ 450.00
$ 835.46
$4,164.54
2
4,164.54
1,285.46
374.81
910.65
3,253.88
3
3,253.88
1,285.46
292.85
992.61
2,261.27
4
2,261.27
1,285.46
203.51
1,081.95
1,179.32
5
1,179.32
1,285.46
106.14
1,179.32
0.00
$6,427.30
$1,427.31
$5,000.00
Totals
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2000
©The
McGraw-Hill Companies, Inc.
T6.16 Chapter 6 Quick Quiz -- Part 4 of 4
How to lie, cheat, and steal with interest rates:
RIPOV RETAILING
Going out for business sale!
$1,000 instant credit!
12% simple interest!
Three years to pay!
Low, low monthly payments!
Assume you buy $1,000 worth of furniture
from this store and agree to the above credit
terms. What is the APR of this loan? The
EAR?
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2000
©The
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T6.16 Solution to Chapter 6 Quick Quiz -- Part 4 of 4 (concluded)
 Your payment is calculated as:

1. Borrow $1,000 today at 12% per year for three years, you
will owe $1,000 + $1000(.12)(3) = $1,360.

2. To make it easy on you, make 36 low, low payments of
$1,360/36 = $37.78.

3. Is this a 12% loan?
$1,000
Irwin/McGraw-Hill
2000
= $37.78 x (1 - 1/(1 + r )36)/r
r
= 1.767% per month
APR
EAR
= 12(1.767%) = 21.204%
= 1.0176712 - 1 = 23.39% (!)
©The
McGraw-Hill Companies, Inc.
T6.17 Solution to Problem 6.10
 Seinfeld’s Life Insurance Co. is trying to sell you an
investment policy that will pay you and your heirs $1,000
per year forever. If the required return on this investment is
12 percent, how much will you pay for the policy?
 The present value of a perpetuity equals C/r. So, the most a
rational buyer would pay for the promised cash flows is
C/r = $1,000/.12 = $8,333.33
 Notice: $8,333.33 is the amount which, invested at 12%,
would throw off cash flows of $1,000 per year forever. (That
is, $8,333.33  .12 = $1,000.)
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2000
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T6.18 Solution to Problem 6.11
 In the previous problem, Seinfeld’s Life Insurance Co. is trying to
sell you an investment policy that will pay you and your heirs
$1,000 per year forever. Seinfeld told you the policy costs $10,000.
At what interest rate would this be a fair deal?
 Again, the present value of a perpetuity equals C/r. Now solve the
following equation:
$10,000 = C/r = $1,000/r
r = .10 = 10.00%
 Notice: If your opportunity rate is less than 10.00%, this is a good
deal for you; but if you can earn more than 10.00%, you can do
better by investing the $10,000 yourself!
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2000
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T6.18 Solution to Problem 6.11
 Congratulations! You’ve just won the $20 million first prize in
the Subscriptions R Us Sweepstakes. Unfortunately, the
sweepstakes will actually give you the $20 million in $500,000
annual installments over the next 40 years, beginning next
year. If your appropriate discount rate is 12 percent per year,
how much money did you really win?
 “How much money did you really win?” translates to, “What is
the value today of your winnings?” So, this is a present value
problem.
PV = $ 500,000  [1 - 1/(1.12)40]/.12
= $ 500,000  [1 - .0107468]/.12
= $ 500,000  8.243776
= $4,121,888.34
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2000
(Not quite $20 million, eh?)
©The
McGraw-Hill Companies, Inc.
T9.1 Chapter Outline
Chapter 9
Net Present Value and Other Investment Criteria
Chapter Organization
 9.1 Net Present Value
 9.2 The Payback Rule
 9.3 The Discounted Payback
 9.4 The Average Accounting Return
 9.5 The Internal Rate of Return
 9.6 The Profitability Index
 9.7 The Practice of Capital Budgeting
 9.8 Summary and Conclusions
CLICK MOUSE OR HIT
SPACEBAR TO ADVANCE
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T9.2 NPV Illustrated

Assume you have the following information on Project X:
Initial outlay -$1,100
Required return = 10%
Annual cash revenues and expenses are as follows:

Year
Revenues
Expenses
1
2
$1,000
2,000
$500
1,000
Draw a time line and compute the NPV of project X.
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2000
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T9.2 NPV Illustrated (concluded)
0
Initial outlay
($1,100)
1
Revenues
Expenses
$1,000
500
Cash flow
$500
– $1,100.00
$500 x
+454.55
2
Revenues
Expenses
$2,000
1,000
Cash flow $1,000
1
1.10
$1,000 x
1
1.10 2
+826.45
+$181.00 NPV
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2000
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T9.3 Underpinnings of the NPV Rule
 Why does the NPV rule work? And what does “work” mean?
Look at it this way:
A “firm” is created when securityholders supply the funds to acquire
assets that will be used to produce and sell a good or a service;
The market value of the firm is based on the present value of the
cash flows it is expected to generate;
Additional investments are “good” if the present value of the
incremental expected cash flows exceeds their cost;
Thus, “good” projects are those which increase firm value - or, put
another way, good projects are those projects that have positive
NPVs!
Moral of the story: Invest only in projects with positive NPVs.
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2000
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T9.4 Payback Rule Illustrated
Initial outlay -$1,000
Year
1
2
3
Year
1
2
3
Cash flow
$200
400
600
Accumulated
Cash flow
$200
600
1,200
Payback period = 2 2/3 years
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2000
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T9.5 Discounted Payback Illustrated
Year
1
2
3
4
Year
1
2
3
4
Initial outlay -$1,000
R = 10%
PV of
Cash flow
Cash flow
$ 200
400
700
300
$ 182
331
526
205
Accumulated
discounted cash flow
$ 182
513
1,039
1,244
Discounted payback period is just under 3 years
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2000
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T9.6 Ordinary and Discounted Payback (Table 9.3)
Cash Flow
Year
Undiscounted Discounted
Undiscounted
Discounted
1
$100
$89
$100
$89
2
100
79
200
168
3
100
70
300
238
4
100
62
400
300
5
100
55
500
355
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2000
Accumulated Cash Flow
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T9.7 Average Accounting Return Illustrated
 Average net income:
Year
1
2
3
Sales
$440
$240
$160
Costs
220
120
80
Gross profit
220
120
80
Depreciation
80
80
80
140
40
0
35
10
0
$105
$30
$0
Earnings before taxes
Taxes (25%)
Net income
Average net income = ($105 + 30 + 0)/3 = $45
Irwin/McGraw-Hill
2000
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T9.7 Average Accounting Return Illustrated (concluded)
 Average book value:
Initial investment = $240
Average investment = ($240 + 0)/2 = $120
 Average accounting return (AAR):
Average net income
AAR =
Irwin/McGraw-Hill
2000
Average book value
$45
=
$120
©The
= 37.5%
McGraw-Hill Companies, Inc.
T9.8 Internal Rate of Return Illustrated
Initial outlay = -$200
Year
Cash flow
1
2
3
$ 50
100
150
 Find the IRR such that NPV = 0
50
0 = -200 +
100
(1+IRR)1
50
200 =
Irwin/McGraw-Hill
2000
(1+IRR)1
+
(1+IRR)2
100
+
150
(1+IRR)2
+
(1+IRR)3
150
+
(1+IRR)3
©The
McGraw-Hill Companies, Inc.
T9.8 Internal Rate of Return Illustrated (concluded)
 Trial and Error
Discount rates
NPV
0%
$100
5%
68
10%
41
15%
18
20%
-2
IRR is just under 20% -- about 19.44%
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2000
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T9.9 Net Present Value Profile
Net present value
120
100
80
Year
Cash flow
0
1
2
3
4
– $275
100
100
100
100
60
40
20
0
– 20
– 40
Discount rate
2%
6%
10%
14%
18%
22%
IRR
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2000
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T9.10 Multiple Rates of Return
 Assume you are considering a project for
which the cash flows are as follows:
Year
Irwin/McGraw-Hill
2000
Cash flows
0
-$252
1
1,431
2
-3,035
3
2,850
4
-1,000
©The
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T9.10 Multiple Rates of Return (continued)
 What’s the IRR? Find the rate at which
the computed NPV = 0:
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2000
at 25.00%:
NPV = _______
at 33.33%:
NPV = _______
at 42.86%:
NPV = _______
at 66.67%:
NPV = _______
©The
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T9.10 Multiple Rates of Return (continued)
 What’s the IRR? Find the rate at which
the computed NPV = 0:
at 25.00%:
NPV =
0
at 33.33%:
NPV =
0
at 42.86%:
NPV =
0
at 66.67%:
NPV =
0
 Two questions:


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1. What’s going on here?
2. How many IRRs can there be?
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T9.10 Multiple Rates of Return (concluded)
NPV
$0.06
$0.04
IRR = 1/4
$0.02
$0.00
($0.02)
IRR = 1/3
IRR = 2/3
IRR = 3/7
($0.04)
($0.06)
($0.08)
0.2
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2000
0.28
0.36
0.44
0.52
Discount rate
0.6
©The
0.68
McGraw-Hill Companies, Inc.
T9.11 IRR, NPV, and Mutually Exclusive Projects
Net present value
Year
0
160
140
120
100
80
60
40
20
0
1
2
3
4
Project A:
– $350
50
100
150
200
Project B:
– $250
125
100
75
50
Crossover Point
– 20
– 40
– 60
– 80
– 100
Discount rate
0
2%
6%
10%
14%
IRR A
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2000
18%
22%
26%
IRR B
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T9.12 Profitability Index Illustrated

Now let’s go back to the initial example - we assumed the
following information on Project X:
Initial outlay -$1,100Required return = 10%
Annual cash benefits:
Year
1
2

$ 500
1,000
What’s the Profitability Index (PI)?
Irwin/McGraw-Hill
2000
Cash flows
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McGraw-Hill Companies, Inc.
T9.12 Profitability Index Illustrated (concluded)
 Previously we found that the NPV of Project X is equal to:
($454.55 + 826.45) - 1,100 = $1,281.00 - 1,100 = $181.00.
 The PI = PV inflows/PV outlay = $1,281.00/1,100 = 1.1645.
 This is a good project according to the PI rule. Can you explain
why?
It’s a good project because the present value of the inflows
exceeds the outlay.
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2000
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T9.13 Summary of Investment Criteria
 I. Discounted cash flow criteria
A. Net present value (NPV). The NPV of an investment is the
difference between its market value and its cost. The NPV
rule is to take a project if its NPV is positive. NPV has no
serious flaws; it is the preferred decision criterion.
B. Internal rate of return (IRR). The IRR is the discount rate that
makes the estimated NPV of an investment equal to zero. The IRR
rule is to take a project when its IRR exceeds the required return. When
project cash flows are not conventional, there may be no IRR or there
may be more than one.
C. Profitability index (PI). The PI, also called the benefit-cost ratio, is
the ratio of present value to cost. The profitability index rule is to
take an investment if the index exceeds 1.0. The PI
measures the present value per dollar invested.
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2000
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T9.13 Summary of Investment Criteria (concluded)
 II. Payback criteria
A. Payback period. The payback period is the length of time until the
sum of an investment’s cash flows equals its cost. The payback period
rule is to take a project if its payback period is less than some
prespecified cutoff.
B. Discounted payback period. The discounted payback period is the
length of time until the sum of an investment’s discounted cash flows
equals its cost. The discounted payback period rule is to take an
investment if the discounted payback is less than some prespecified
cutoff.
 III. Accounting criterion
A. Average accounting return (AAR). The AAR is a measure of
accounting profit relative to book value. The AAR rule is to
take an investment if its AAR exceeds a benchmark.
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2000
©The
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T9.14 Chapter 9 Quick Quiz
1. Which of the capital budgeting techniques do account for both the time
value of money and risk?
2. The change in firm value associated with investment in a project is
measured by the project’s _____________ .
a. Payback period
b. Discounted payback period
c. Net present value
d. Internal rate of return
3. Why might one use several evaluation techniques to assess a given
project?
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2000
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McGraw-Hill Companies, Inc.
T9.14 Chapter 9 Quick Quiz
1. Which of the capital budgeting techniques do account for both the time
value of money and risk?
Discounted payback period, NPV, IRR, and PI
2. The change in firm value associated with investment in a project is
measured by the project’s Net present value.
3. Why might one use several evaluation techniques to assess a given
project?
To measure different aspects of the project; e.g., the payback period
measures liquidity, the NPV measures the change in firm value, and the
IRR measures the rate of return on the initial outlay.
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2000
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T9.15 Solution to Problem 9.3
 Offshore Drilling Products, Inc. imposes a payback cutoff of 3
years for its international investment projects. If the company
has the following two projects available, should they accept
either of them?
Irwin/McGraw-Hill
2000
Year
Cash Flows A
Cash Flows B
0
-$30,000
-$45,000
1
15,000
5,000
2
10,000
10,000
3
10,000
20,000
4
5,000
250,000
©The
McGraw-Hill Companies, Inc.
T9.15 Solution to Problem 9.3 (concluded)
 Project A:
Payback period
= 1 + 1 + ($30,000 - 25,000)/10,000
= 2.50 years
 Project B:
Payback period
= 1 + 1 + 1 + ($45,000 - 35,000)/$250,000
= 3.04 years
 Project A’s payback period is 2.50 years and project B’s
payback period is 3.04 years. Since the maximum acceptable
payback period is 3 years, the firm should accept project A and
reject project B.
Irwin/McGraw-Hill
2000
©The
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T9.16 Solution to Problem 9.7
 A firm evaluates all of its projects by applying the IRR
rule. If the required return is 18 percent, should the firm
accept the following project?
Irwin/McGraw-Hill
2000
Year
Cash Flow
0
-$30,000
1
25,000
2
0
3
15,000
©The
McGraw-Hill Companies, Inc.
T9.16 Solution of Problem 9.7 (concluded)
 To find the IRR, set the NPV equal to 0 and solve for the
discount rate:
NPV = 0 = -$30,000 + $25,000/(1 + IRR)1 + $0/(1 + IRR) 2
+$15,000/(1 + IRR)3
 At 18 percent, the computed NPV is ____.
 So the IRR must be (greater/less) than 18 percent. How did
you know?
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2000
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T9.16 Solution of Problem 9.7 (concluded)
 To find the IRR, set the NPV equal to 0 and solve for the
discount rate:
NPV = 0 = -$30,000 + $25,000/(1 + IRR)1 + $0/(1 + IRR)2
+$15,000/(1 + IRR)3
 At 18 percent, the computed NPV is $316.
 So the IRR must be greater than 18 percent. We know this
because the computed NPV is positive.
 By trial-and-error, we find that the IRR is 18.78 percent.
Irwin/McGraw-Hill
2000
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McGraw-Hill Companies, Inc.