Chapter 3
Stock and Bond
Valuation:
Annuities and
Perpetuities
Copyright © 2009 Pearson Prentice Hall. All rights reserved.
Chapter 3 Outline
3.1 Perpetuities
3.2 Annuities
3.3 The Four Formulas Summarized
Appendix: Advanced Material
3.4 Projects With Different Lives and Rental
Equivalents
3.5 Perpetuity and Annuity Derivations
Copyright © 2009 Pearson Prentice Hall. All rights reserved.
3-2
Stock and Bond Valuation
Annuities and Perpetuities
• The present value formula is the primary way to find value.
• Formulas can provide a shortcut to having to write out all the
cash flows to infinity. Whew!
• The perpetuity formula values a series of infinite payments.
• The annuity formula values payments that span a time
period.
• Both formulas need the payments or cash flows to be equal
dollar amounts every period.
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3-3
Stock and Bond Valuation:
Example: $2 Perpetuity at a 10% Interest Rate
•
What would you have to invest today to receive the same value as $2 of
interest paid to you each year forever, starting next year, if the constant
interest rate is 10%?
•
You could add $2 X 1/(1+r)t for every t until infinity on a spreadsheet but
that might take a while….probably more time than you have.
•
Using the perpetuity formula,
we can calculate the answer:
•
You find the answer is $20.
•
And you don’t miss meals……just keep r and C constant; the formula
works.
Copyright © 2009 Pearson Prentice Hall. All rights reserved.
C1
r
$2
PV 
.10
PV  $20
PV 
3-4
Stock and Bond Valuation
Perpetuities
•
A perpetuity is a project with a set of constant cash flows that repeats
forever.
•
If the cost of capital (r) and the cash flow per period is the same from today
until infinity, then you need the perpetuity formula.
•
Perpetuity Formula:
•
Since the first cash flow occurs next year, C1 is the proper cash flow.
•
The formula is the mathematical upper limit of a infinite series.
C1
PV 
r
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3-5
Stock and Bond Valuation
Simple Perpetuity Table for $2 at a rate of 10%
TABLE 3.1 Perpetuity Stream of $2 with Interest Rate r = 10%
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3-6
Stock and Bond Valuation
Growing Perpetuity Formula
•
What if the cash flow grows by the same percentage every year?
•
Now we need a growing perpetuity formula (the Gordon Growth Model).
• Growing perpetuity or Gordon Growth Model:
PV 
C1
rg
•
We are investing to receive the firm’s future cash flows, but if the cash
flows grow at a constant rate, we reduce the discount rate in the
denominator by the growth rate.
•
Growth is a benefit to us.
•
This formula is very useful when valuing stocks since corporations have
infinite lives and many pay steadily increasing dividends.
•
By reducing the denominator, ‘g’ will increase the PV of the cash flows.
• …..the average investor hopes dividends increase!
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3-7
Stock and Bond Valuation
Growing Perpetuity Example: $2 growing at 5%, r = 10%
•
What if next year’s $2 payment grew at 5% every year forever when
investors require a return of 10%?
•
We could grow cash flow from $2 to $2.10 to $2.205 to forever, and then
discount…..but we might miss a meal, so use the formula.
•
The growing perpetuity formula:
•
We find the answer is $40, double the “no growth” answer.
•
We assume r and g are constant to infinity. If g isn’t greater or equal to
the discount rate, we can use this formula on stocks with dividends.
C1
rg
$2
PV 
.10  .05
PV  $40
PV 
• Note: growth is very important to valuation!
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3-8
Stock and Bond Valuation
Growing Perpetuity Example: $2 growing at 5%, r = 10%
TABLE 3.2 Perpetuity Stream with C1 = $2, Growth Rate g = 5%, and
Interest Rate r = 10%
Copyright © 2009 Pearson Prentice Hall. All rights reserved.
3-9
Stock and Bond Valuation
Business Valuation with Gordon Growth Model
•
What is today’s value of a stable business that grows with the inflation rate
of 2% and is expected to have $1,000,000 in profit next year when the
firm’s cost of capital (required return) is 8%?
•
Use the growing perpetuity formula as follows:
C1
rg
$1, 000, 000
PV 
 $1, 666, 667
8%  2%
BusinessValue  $1, 666, 667
PV 
•
We find today’s value of the business to be $16,666,667.
•
Since many good and bad things can happen to the business, consider this
one estimate of its value. Its true value could be higher or lower if more
factors were considered.
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3-10
Stock and Bond Valuation
Stock Valuation Example with a Growing Perpetuity
•
Applying the Gordon growth model, we can find today’s value of a stock
with a $10 dividend next year that’s growing by 5% (forever) when we
expect a return equal to the S&P500 long-run return of 10%.
•
Using the growing perpetuity formula, we can calculate the stock’s value:
C1
rg
$10
PV 
 $200
10%  5%
Stock Value  $200
PV 
•
•
We find the value to be $200 per share.
Rearranging the Gordon growth formula, we can observe that:
r
•

C1
g
PV

r

Div1
g
P
We can find the value of the firm’s cost of capital r if we know a stock’s
dividend, its stock’s price, and its growth rate.
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3-11
Stock and Bond Valuation
Finding Cost of Capital with the Gordon Growth Model
•
Looking at GE’s stock data for October of 2004, we observe:
•
Dividend yield ( Div1 / Stock price )

2.4%
•
Growth of dividends

9.6%
•
Using the Gordon Growth Formula,
• Cost of capital, or r
•
r

r

r
r


C1
g
PV
Div1
g
P
2.4%  9.6%
12%

12%
We find that investors in GE must be using a cost of capital of 12%.
Copyright © 2009 Pearson Prentice Hall. All rights reserved.
3-12
Stock and Bond Valuation
The Gordon Growth Model and Forward Earnings Yield, 1/P/E
•
For the GE data, if we assume the earnings yield equals the dividend yield, we find:
•
•
•
•
Forward P/E
= 18.5 (forward looks at next year’s EPS)
Growth in earnings
= 6.3%
1 / Forward P/E = E1 / P and we’ll substitute it for Div1 / Stock Price
Next, we rearrange the Gordon Growth Model to find cost of capital, r:
r

r

r

r

C1
g
PV
E1
g
P
1
 6.3%  5.4%  6.3%
18.5
11.7%
•
Thus, investors in GE who use earnings as the source of their valuation (and dividends), find a cost
of capital of 11.7%, which is close to the 12% we found before .
•
Warning! Use of these simple relationships will do your portfolio harm! The world of investing is
not so simple. Industries and markets change too much to allow investors to assume the
relationships stay fixed until infinity, or even next month.
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3-13
Stock and Bond Valuation
Annuities
•
Annuities value a series of equal dollar cash flows per period over time.
•
Rather than separately calculating the present value of $5 in Year 1, Year 2, and
Year 3 at 10%, we can do it all in one formula.
•
The Annuity Formula:
PV

C1 
1 
1

r 
(1  r)T 
PV


$5 
1
1

3
.10 
(1  .10) 
Solving,
PV
•

$12.43
This formula works so you don’t miss any meals when someone asks how much
that mortgage payment for 360 months is worth.
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3-14
Stock and Bond Valuation
Annuity Application:
•
•
30-Year, Fixed-Rate Mortgage
Most mortgages on homes are repaid over 360 months (30 years) in equal payments
and have a fixed interest rate (think annuity formula). Take the annual loan rate and
divide it by 12 to find the monthly rate used over the 360 months.
•
What is your monthly payment if you borrow $500,000 and the annual rate is 7.5%?
•
We are solving for the payment C1 for a 360-month loan (use PV annuity formula).
•
Find monthly r first, which is .075/12 = .00625.
PV

C1 
1 
1

r 
(1  r)T 
$500,000



C1
1
1
360 

.0625 
(1  .0625) 
Solving,
C1

$3, 496.07
Your monthly payment equals $3,496.07 and that includes the loan’s principal and
interest. Most payments have some real estate taxes and home insurance, too.
Some even have you pay insurance to protect the lender; that’s PMI, private
mortgage insurance. If you only borrow 80% of the value, then you don’t pay PMI.
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3-15
Stock and Bond Valuation
Bonds and Annuities Example: Bond Valuation
•
A typical American coupon bond pays semiannual interest (also called coupon
payments) until its maturity date when the bond’s principal or face value is repaid.
•
A typical 2-year semiannual-payment bond’s cash flows are:
6 months
coupon
•
1 year
coupon
1.5 years
coupon
2 years
coupon + face value
The coupon yield describes the annual interest payments. It is calculated on the value
of the principal (face value). A bond with 3% coupon yield (cy) and a principal value
of $100,000 has coupon payments every year of $3,000.
•
To find the semiannual payments, divide by 2 to find payments of $1,500 every 6 months.
•
Bond coupon payment = ½ X Cy X face = ½ X 3% X $100,000
•
Bond coupon payment = $1,500
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3-16
Stock and Bond Valuation
Bonds Example: Bond Valuation -- continued
•
The 2-year ( 4-period ) bond cash flows look like this on a timeline:
Period 1
6 months
$1500
•
Period 2
1 year
$1500
Period 3
1.5 years
$1500
Period 4
2 years
$1500 + $100,000
To value these cash flows you would need to know the bond’s discount
rate, called a yield-to-maturity, and convert it to a semiannual rate.
•
Assume the Yield to Maturity is 5.0%; this is the bond’s discount rate.
•
We’ll find our semiannual discount rate r using
Semi-annual Rate  1  YTM  1
1  YTM  1  r
r
Semi-annual Rate  1.05  1  .0247  r
•
The semiannual rate is 2.47%.
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3-17
Stock and Bond Valuation:
Bonds Example: Bond Valuation -- continued 2
•
We have a 2-year (4-period) bond with the following cash flows:
Period
1
2
3
4
•
Cash flow
1,500
1,500
1,500
101,500
x
Discount rate
1/1.02471
1/1.02472
1/1.02473
1/1.02474
= PVs
Find the PV of the cash flows, sum, and you’ll find the value of the bond:
•
Value equals $96,348.25.
•
Since the bond only promises to pay 3% and the discount rate is 5%
annually, the bond sells at a discount to its principal or face value.
•
Remember that the coupon rate (cy) is the company’s promised rate, and
the yield to maturity (YTM) is the market’s desired or discount rate.
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3-18
Stock and Bond Valuation
Level-Coupon Bond, 5 Years to Maturity, 3% Coupon, 5% YTM
Step 1: Write
down the bond’s
payment stream.
Step 2: Find the
appropriate cost
of capital for each
payment.
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3-19
Stock and Bond Valuation
Level-Coupon Bond, 5 Years to Maturity, 3% Coupon, 5% YTM
Step 3: Compute
the discount
factor—it is
1/(1+r1).
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3-20
Stock and Bond Valuation
Formula Method: Bond 5-Year Maturity, 3% Coupon, 5% YTM
•
What if the 3% level-coupon bond matures in 5 years and has a YTM of 5.0%?
We could do a table or we could use the annuity formula as a shortcut.
•
Our bond is a ten-payment annuity ($1500 every 6 months), and the $100,000 face value is
paid with the final coupon. The semiannual discount rate is 2.47%.
•
Value of bond = PV of 10-period coupon annuity + PV of face value from 10 periods

Coupon1 
1 
Face
1

T 



r
(1  r)
(1  r)T

1
cy Face 
1 
Face
2
1

T 


r
(1  r)  (1  r)T
Value of bond

1
3% $1000 

1
$1000
2
1

10 


.0247
(1  .0247)  (1  .0247)10
Value of bond


$1500 
1
$1000
1

10 

.0247 
(1  .0247)  (1  .0247)10
Value of bond

$91, 497.32
Value of bond
Value of bond
The value of the bond equals $91,497.32.
•
The bond offers less return than desired by investors, so they pay less than face value.
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3-21
Stock and Bond Valuation
An Aside on Discount, Premium, and Par Bonds
•
A discount bond has a coupon rate that is less than its yield-to-maturity. If
a bond offers less than what investors want, they will pay less for it.
•
•
A premium bond has a coupon rate greater than its YTM. Some bonds pay
more than the market requires. Those bonds sell at a premium to face.
•
•
Discount Bond (Coupon rate < YTM)
$91,000 valuation and $100,000 face amount
Premium Bond (Coupon rate > YTM)
$105,000 valuation and $100,000 face amount
A par bond has a coupon rate equal to its yield-to-maturity. The bond sells
for its face (or principal) value.
•
Par Bond (Coupon rate = YTM)
$100,000 valuation and $100,000 face amount
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3-22
Stock and Bond Valuation
The Four Formulas (three we use regularly)
•
The Perpetuity Formula:
PV 
•
The Gordon Growth Model or Growing Perpetuity Formula:
PV 
•
C1
r
C1
(r  g)
The Simple Annuity Formula:
C1 
PV
1 1
r  (1 r)


T

•
The fourth formula (from pension finance) is the Growing Annuity Formula:
PV 
•
C1  (1 g)
1
(rg)  (1 r)
T
T



These four formulas are useful for many different types of corporate decisions.
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3-23
Stock and Bond Valuation:
The Four Payoff Streams and Their Present Values
FIGURE 3.1
The Four Payoff
Streams and Their
Present Values
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3-24