ch5 HR Management

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McGraw-Hill Ryerson 2003 McGrawHi ll Ryerson Limited

5-0

Corporate FinanceRoss Westerfield Jaffe

Sixth Edition

5

Chapter Five

How to Value Bonds

and Stocks

Prepared by

Gady Jacoby

University of Manitoba

and

Sebouh Aintablian

American University of

Beirut


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5-1

Chapter Outline

5.1 Definition and Example of a Bond

5.2 How to Value Bonds

5.3 Bond Concepts

5.4 The Present Value of Common Stocks

5.5 Estimates of Parameters in the Dividend-Discount Model

5.6 Growth Opportunities

5.7 The Dividend Growth Model and the NPVGO

Model (Advanced)5.8 Price Earnings Ratio

5.9 Stock Market Reporting

5.10 Summary and Conclusions


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5-2

Valuation of Bonds and Stock

First Principles:Value of financial securities = PV of expected

future cash flows

To value bonds and stocks we need to:Estimate future cash flows:

Size (how much) and

Timing (when)

Discount future cash flows at an appropriate rate:

The rate should be appropriate to the risk presented by

the security.


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5-3


A bond is a legally binding agreement between aborrower (bond issuer) and a lender (bondholder):

Specifies the principal amount of the loan.

Specifies the size and timing of the cash flows: In dollar terms (fixed-rate borrowing)

As a formula (adjustable-rate borrowing)


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5-4


Consider a Government of Canada bond listed as

6.375 of December 2009.

ThePar Valueof the bond is $1,000.

Coupon paymentsare made semi-annually (June 30 and

December 31 for this particular bond). Since the coupon rateis 6.375 the payment is $31.875.

On January 1, 2002 the size and timing of cash flows are:

02/1/1

875.31$

02/30/6

875.31$

02/31/12

875.31$

09/30/6

875.031,1$

09/31/12


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5-5

5.2 How to Value Bonds

Identify the size and timing of cash flows.

Discount at the correct discount rate.

If you know the price of a bond and the size and

timing of cash flows, theyield to maturityis thediscount rate.


7/49McGraw-Hill Ryerson 2003 McGrawHi ll Ryerson Limited

5-6

Pure Discount Bonds

Information needed for valuing pure discount bonds:

Time to maturity (T) = Maturity date - todays date

Face value (F)

Discount rate (r)

Tr

FPV

)1(

Present value of a pure discount bond at time 0:

0

0$

1

0$

2

0$

1T

F$

T



5-7

Pure Discount Bonds: Example

Find the value of a 30-year zero-coupon bondwith a $1,000 par value and a YTM of 6%.

11.174$)06.1(

000,1$

)1( 30

Tr

FPV

0$0$0$

29

0001$

0

0$

1

0$

2

0$

29

000,1$

30



5-8

Level-Coupon Bonds

Information needed to value level-coupon bonds:

Coupon payment dates and time to maturity (T)

Coupon payment (C) per period and Face value (F)

Discount rate

TTr

F

rr

CPV

)1()1(

11

Value of a Level-coupon bond= PV of coupon payment annuity + PV of face value

0

C$

1

C$

2

C$

1T

FC $$

T



5-9

Level-Coupon Bonds: Example

Find the present value (as of January 1, 2002), of a 6.375

coupon Government of Canada bond with semi-annual

payments, and a maturity date of December 31, 2009 if the

YTM is 5-percent.

On January 1, 2002 the size and timing of cash flows are:

02/1/1

875.31$

02/30/6

875.31$

02/31/12

875.31$

09/30/6

875.031,1$

09/31/12

75.089,1$)025.1(

000,1$

)025.1(

11

205.

875.31$1616

PV



5-10

Bond Market Reporting

CANADA

Coupon Mat. Date Bid $ Yld%Canada 6.375 Dec 31/09 108.98 5.00

The Governmentof Canada issued

this bond

The bond paysan annual

coupon rate of

6.375%

The bondmatures on

December 31,

2009

The bond is selling

at 108.98% of the

face value of

$1,000

The bondsquoted annual

yield to

maturity is 5%



5-11

5.3 Bond Concepts

1. Bond prices and market interest rates move in oppositedirections.

2. When coupon rate = YTM, price = par value.When coupon rate > YTM, price > par value (premium

bond)When coupon rate < YTM, price < par value (discountbond)

3. A bond with longer maturity has higher relative (%) price

change than one with shorter maturity when interest rate(YTM) changes. All other features are identical.

4. A lower coupon bond has a higher relative price changethan a higher coupon bond when YTM changes. All other

features are identical.



5-12

YTM and Bond Value

800

1000

1100

1200

1300

$1400

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Discount Rate

BondValue

6 3/8

When the YTM < coupon, the bond

trades at a premium.

When the YTM = coupon, the

bond trades at par.

When the YTM > coupon, the bond trades at a discount.



5-13

Maturity and Bond Price Volatility

C

Consider two otherwise identical bonds.

The long-maturity bond will have much more

volatility with respect to changes in the

discount rate

Discount Rate

BondValue

Par

Short Maturity Bond

Long Maturity

Bond



5-14

Coupon Rate and Bond Price Volatility

Consider two otherwise identical bonds.

The low-coupon bond will have much more

volatility with respect to changes in the

discount rate

Discount Rate

BondValue

High Coupon Bond

Low Coupon Bond



5-15

Holding Period Return

Suppose that on January 1, 2002, you bought the above6.375 coupon Government of Canada bond with semi-annual

payments, and a maturity date of December 31, 2009.

At that time the YTM was 5-percent, and you paid $1,089.75

(the PV of the bond). Six months later (July 1, 2002), You sold the bond when the

YTM was 4-percent. The size and timing of cash flows (as of

July 1, 2002 ) were:

02/1/7

875.31$

02/31/12

875.31$

03/30/6

875.31$

09/30/6

875.031,1$

09/31/12



5-16

Your holding period returnwas:

This annualizes to an effective rate of:

Given that the YTM at that time was 4-percent, you sold thebond for:

Holding Period Return (continued)

59.152,1$

)02.1(

000,1$

)02.1(

11

204.

875.31$1515

PV

%77.5$1,089.75

1,089.75$59.152,1$

%87.111)0577.1( 2



5-17

5.4 The Present Value of Common Stocks

Dividends versus Capital Gains Valuation of Different Types of Stocks

Zero Growth

Constant Growth

Differential Growth



5-18

Case 1: Zero Growth

Assume that dividends will remain at the same levelforever

rP

rrrP

Div

)1(

Div

)1(

Div

)1(

Div

0

3

3

2

2

1

1

0

321 DivDivDiv

Since future cash flows are constant, the value of a zerogrowth stock is the present value of a perpetuity:



5-19

P0= Div1/ r

= 0.75/0.12 = $6.25

ABC Corp. is expected to pay $0.75 dividend per annum,starting a year from now, in perpetuity. If stocks of similar

risk earn 12% annual return, what is the price of a share of

ABC stock?

The stock price is given by the present value of theperpetual stream of dividends:

A Zero Growth Example

0 1 2 3 4

$0.75 $0.75 $0.75 $0.75


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5-20

Case 2: Constant Growth

)1(DivDiv 01 g

Since future cash flows grow at a constant rate forever,

the value of a constant growth stock is the presentvalue of a growing perpetuity:

gr

P

10Div

Assume that dividends will grow at a constant rate,g,forever. i.e.

2

012 )1(Div)1(DivDiv gg

3

023 )1(Div)1(DivDiv gg ...


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5-21

A Constant Growth Example

XYZ Corp. has a common stock that paid itsannual dividend this morning. It is expected to

pay a $3.60 dividend one year from now, and

following dividends are expected to grow at arate of 4% per year forever.

If stocks of similar risk earn 16% effectiveannual return, what is the price of a share of

XYZ stock?


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5-22

A Constant Growth Example (continued)

The stock price is given by the the present valueof the perpetual stream of growing dividends:

$3.60 $3.601.04 $3.601.042 $3.601.043

P0= Div1/ (r-g)

= 3.60/(0.16-0.04)

= $30.00

0 1 2 3 4


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5-23

Case 3: Differential Growth

Assume that dividends will grow at differentrates in the foreseeable future and then willgrow at a constant rate thereafter.

To value a Differential Growth Stock, we needto:

Estimate future dividends in the foreseeablefuture.

Estimate the future stock price when the stockbecomes a Constant Growth Stock (case 2).

Compute the total present value of the estimatedfuture dividends and future stock price at the

appropriate discount rate.


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5-24


)(1DivDiv 101 g

Assume that dividends will grow at rateg1forNyears and grow at rateg2thereafter

2

10112 )(1Div)(1DivDiv gg

N

NN gg )(1Div)(1DivDiv 1011

)(1)(1Div)(1DivDiv 21021 ggg N

NN

.

.

.

.

.

.

2


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5-25


)(1Div 10 g

Dividends will grow at rateg1forNyears andgrow at rateg2thereafter

2

10 )(1Div g

Ng)(1Div 10 )(1)(1Div

)(1Div

210

2

gg

g

NN

0 1 2

N N+1

5 26


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5-26


We can value this as the sum of:

anN-year annuity growing at rateg1

T

T

A

r

g

gr

CP

)1(

)1(1 1

1

plus the discounted value of a perpetuity growing at rateg2that starts in yearN+1

NB r

grP

)1(

Div

2

1N

5 27


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5-27


To value a Differential Growth Stock, we can use

NT

T

rgr

rg

grCP

)1(

Div

)1()1(1 2

1N

1

1

Or we can cash flow it out.

5 28


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5-28

A Differential Growth Example

A common stock just paid a dividend of $2.The dividend is expected to grow at 8% for 3

years, then it will grow at 4% in perpetuity.

If stocks of similar risk earn 12% effective

annual return, what is the stock worth?

5 29


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5-29

With the Formula

NT

T

r

gr

r

g

gr

CP

)1(

Div

)1(

)1(1 2

1N

1

1

3

3

3

3

)12.1(

04.12.

)04.1()08.1(2$

)12.1(

)08.1(1

08.12.

)08.1(2$

P

3)12.1(75.32$8966.154$ P

31.23$58.5$ P 89.28$P

5 30


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5-30

A Differential Growth Example (continued)

08).2(1$ 2

08).2(1$

0 1 2 3 4

3

08).2(1$ )04.1(08).2(1$ 3

16.2$ 33.2$

0 1 2 3

08.

62.2$52.2$

89.28$

)12.1(

75.32$52.2$

)12.1(

33.2$

12.1

16.2$320

P

75.32$08.

62.2$3 P

The constant

growth phasebeginning in year 4

can be valued as a

growing perpetuity

at time 3.

5 31 i f i h


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5-31 5.5 Estimates of Parameters in the

Dividend-Discount Model

The value of a firm depends upon its growth

rate,g, and its discount rate, r.

Where doesg come from?Where does rcome from?

5 32


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5-32

Where doesgcome from?

Formula for Firms Growth Rate (g):

The firm will experience earnings growth if its net

investment (total investment-depreciation) is

positive.

To grow, the firm must retain some of its earnings.

This leads to:

Earnings

next Year

Earnings

this Year

Retained

earnings

this Year

Return on

retained

earnings= +

5 33


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5-33

This leads to the formula for the firms growth rate:

g= Retention ratio Return on retained earnings

The return on retained earnings can be estimatedusing the firms historical return on equity (ROE)

Where doesgcome from?

Dividing both sides by this years earnings, we get:

Earnings this Year

Earnings next Year1

Retained earnings

this YearReturn on

retained

earnings= +

Earnings this Year

1 +g Retention ratio

5 34


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5-34

Where doesgcome from? An Example

Ontario Book Publishers (OBP) just reportedearnings of $1.6 million, and it plans to retain 28-

percent of its earnings.

If OBPs historical ROE was 12-percent, what is the

expected growth rate for OBPs earnings?

With the above formula:

g= 0.28 0.12 = 0.0336 = 3.36%

Or:

Total earnings

Change in earnings=

$1.6 million

(0.28$1.6 million)0.12= 0.0336

5 35


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5-35

In practice, there is a great deal of estimationerror involved in estimating r.

The discount rate can be broken into twoparts.

The dividend yield

The growth rate (in dividends)

From the constant growth cas, we can write:

Where does rcome from?

gP

r 0

1Div

5-36


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5-36

Where does rcome from? An Example

Manitoba Shipping Co. (MSC) is expected to pay adividend next year of $8.06 per share. Future

Dividends for MSC are expected to grow at a rate of

2% per year indefinitely.

If an investor is currently willing to pay $62.00 per

one MSC share, what is her required return for this

investment?

With the above formula:

r = (8.06/62) + 0.02 = 0.15 = 15%

5-37


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5-37

5.6 Growth Opportunities

Growth opportunities are opportunities toinvest in positive NPV projects.

The value of a firm can be conceptualized as

the sum of the value of a firm that pays out100-percent of its earnings as dividends and

the net present value of the growth

opportunities.

NPVGOr

EPSP

5-38 5 7 Th Di id d G th M d l d th


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5 38 5.7 The Dividend Growth Model and the

NPVGO Model (Advanced)

We have two ways to value a stock:

The dividend discount model.

The price of a share of stock can be calculated as

the sum of its price as a cash cow plus the per-share value of its growth opportunities.

5-39 Th Di id d G th M d l d th


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5 39 The Dividend Growth Model and the

NPVGO Model

Consider a firm that has EPS of $5 at the end of thefirst year, a dividend-payout ratio of 30-percent, a

discount rate of 16-percent, and a return on retained

earnings of 20-percent.

The dividend at year one will be $5 .30 = $1.50 per share.

The retention ratio is .70 ( = 1 -.30) implying a growth rate

in dividends of 14% = .70 20%

From the dividend growth model, the price of a share is:

75$14.16.

50.1$Div 10

grP

5-40


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5 40

The NPVGO Model

First, we must calculate the value of the firm as acash cow.

25.31$16.

5$EPS

r

Second, we must calculate the value of the growthopportunities.

75.43$14.16.

875$.16.

20.50.350.3

grNPVGO

Finally, 75$75.4325.310 P

5-41


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5 41

5.8 Price Earnings Ratio

Many analysts frequently relate earnings per share to

price.

The price earnings ratio is a.k.a the multiple

Calculated as current stock price divided by annual EPS

The National Postuses last 4 quarters earnings

Firms whose shares are in fashion sell at highmultiples. Growth stocksfor example.

Firms whose shares are out of favour sell at low

multiples. Value stocksfor example.

EPS

shareperPriceratioP/E

5-42


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5 42

Other Price Ratio Analysis

Many analysts frequently relate earnings per

share to variables other than price, e.g.:

Price/Cash Flow Ratio

cash flow = net income + depreciation = cash flow

from operations or operating cash flow

Price/Sales

current stock price divided by annual sales per share

Price/Book (a.k.a. Market to Book Ratio) price divided by book value of equity, which is

measured as assets - liabilities

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52W 52W Yield Vol Net

high low Stock Ticker Div % P/E 00s High Low Close chg42.10 32.25 BCE Inc BCE 1.20 3.5 11.8 19210 34.59 33.80 34.50 -0.47


BCE has

been as

high as

$42.10 in

the last

year.BCE has

been as low

as $32.25 in

the last year.

Given the

current price,

the dividend

yield is 3 %Given the

current price, the

P/E ratio is 11.8

times earnings

1,921,000 shares

traded hands in the

last days trading

BCE ended

trading at $34.50,down $0.47 from

yesterdays close

BCE pays a

dividend of 1.2

dollars/share

5-44


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BCE Incorporated is having a tough year, trading near their 52-

week low. Imagine how you would feel if within the past yearyou had paid $42.10 for a share of BCE and now had a share

worth $34.50! That $1.20 dividend wouldnt go very far in

making amends.

Yesterday, BCE had another rough day in a rough year. BCE

opened the day down beginning trading at $34.59, which was

down from the previous close of $34.97 = $34.50 + $0.47

52W 52W Yield Vol Net

high low Stock Ticker Div % P/E 00s High Low Close chg

42.10 32.25 BCE Inc BCE 1.20 3.5 11.8 19210 34.59 33.80 34.50 -0.47

5-45


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5.10 Summary and Conclusions

In this chapter, we used the time value ofmoney formulae from previous chapters to

value bonds and stocks.

1. The value of a zero-coupon bond is

2. The value of a perpetuity is

Tr

FPV

)1(

r

CPV

5-46


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5.10 Summary and Conclusions (continued)

3. The value of a coupon bond is the sum ofthe PV of the annuity of coupon payments

plus the PV of the par value at maturity.

4. The yield to maturity (YTM) of a bond isthat single rate that discounts the payments

on the bond to the purchase price.

TTr

Frr

CPV)1()1(

11

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5. A stock can be valued by discounting its

dividends. There are three cases:

1. Zero growth in dividends

2. Constant growth in dividends

3. Differential growth in dividends

r

P Div

0

grP

10Div

NT

T

r

gr

r

g

gr

CP

)1(

Div

)1(

)1(1 2

1N

1

1

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6. The growth rate can be estimated as:

g= Retention ratio Return on retained earnings

7. An alternative method of valuing a stock

was presented. The NPVGO values a stock

as the sum of its cash cow value plus the

present value of growth opportunities.

NPVGOr

EPSP

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