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McGraw-Hill/IrwinCorporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights

Reserved.

CHAPTER

5How to Value

Bonds and Stocks

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Chapter Outline

5.1 Definition and Example of a Bond5.2 How to Value Bonds5.3 Bond Concepts5.4 The Present Value of Common Stocks5.5 Estimates of Parameters in the Dividend-Discount Model5.6 Growth Opportunities5.7 The Dividend Growth Model and the NPVGO Model

(Advanced)5.8 Price Earnings Ratio5.9 Stock Market Reporting5.10 Summary and Conclusions

5.1 Definition and Example of a Bond5.2 How to Value Bonds5.3 Bond Concepts5.4 The Present Value of Common Stocks5.5 Estimates of Parameters in the Dividend-Discount Model5.6 Growth Opportunities5.7 The Dividend Growth Model and the NPVGO Model

(Advanced)5.8 Price Earnings Ratio5.9 Stock Market Reporting5.10 Summary and Conclusions

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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.

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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5.1 Definition and Exampleof a Bond

A bond is a legally binding agreement between a borrower and a lender:

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)

A bond is a legally binding agreement between a borrower and a lender:

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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5.1 Definition and Exampleof a Bond

Consider a U.S. government bond listed as 6 3/8 of December 2009.

The Par Value of the bond is $1,000.

Coupon payments are made semi-annually (June 30 and December 31 for this particular bond).

Since the coupon rate is 6 3/8 the payment is $31.875.

On January 1, 2005 the size and timing of cash flows are:

Consider a U.S. government bond listed as 6 3/8 of December 2009.

The Par Value of the bond is $1,000.

Coupon payments are made semi-annually (June 30 and December 31 for this particular bond).

Since the coupon rate is 6 3/8 the payment is $31.875.


05/1/1

875.31$

05/30/6

875.31$

05/31/12

875.31$

09/30/6

875.031,1$

09/31/12

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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, the yield to maturity is the discount rate.

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, the yield to maturity is the discount rate.

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Pure Discount BondsInformation needed for valuing pure discount bonds:

Time to maturity (T) = Maturity date - today’s dateFace value (F)Discount rate (r)

Information needed for valuing pure discount bonds:Time to maturity (T) = Maturity date - today’s dateFace 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

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Pure Discount Bonds: Example

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


11.174$)06.1(

000,1$

)1( 30

Tr

FPV

0

0$

1

0$

2

0$

29

000,1$

30

0

0$

1

0$

2

0$

29

000,1$

30

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Pure Discount Bonds: Example



PMT

I/Y

FV

PV

N

PV 174.11

6

1,000

30

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Level-Coupon BondsInformation 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

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

TT r

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

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Level-Coupon Bonds: ExampleFind the present value (as of January 1, 2004), of a 6-3/8 coupon T-

bond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent.


Find the present value (as of January 1, 2004), of a 6-3/8 coupon T-bond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent.


04/1/1

875.31$

04/30/6

875.31$

04/31/12

875.31$

09/30/6

875.031,1$

09/31/12

52.070,1$)025.1(

000,1$

)025.1(

11

205.

875.31$1212

PV

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Level-Coupon Bonds: Example

PMT

I/Y

FV

PV

N

PV

31.875 =

5

1,000

– 1,070.52

12

1,000×0.06375

2

Find the present value (as of January 1, 2004), of a6-3/8 coupon T-bond with semi-annual payments, anda maturity date of December 2009 if the YTM is5-percent.

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5.3 Bond Concepts1. Bond prices and market interest rates move in opposite

directions.

2. When coupon rate = YTM, price = par value.When coupon rate > YTM, price > par value (premium bond)When coupon rate < YTM, price < par value (discount bond)

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 change than a higher coupon bond when YTM changes. All other features are identical.

1. Bond prices and market interest rates move in opposite directions.

2. When coupon rate = YTM, price = par value.When coupon rate > YTM, price > par value (premium bond)When coupon rate < YTM, price < par value (discount bond)

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 change than a higher coupon bond when YTM changes. All other features are identical.

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YTM and Bond Value

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.

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

Bon

d V

alu

e

6 3/8

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Maturity and Bond Price Volatility

Consider two otherwise identical bonds.

The long-maturity bond will have much more volatility with respect to

changes in the discount rate

C Discount Rate

Bon

d V

alu

e

Par

Short Maturity Bond

Long Maturity Bond

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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

Bon

d V

alu

e

High Coupon Bond

Low Coupon Bond

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5.4 The Present Valueof Common Stocks

Dividends versus Capital Gains

Valuation of Different Types of StocksZero Growth

Constant Growth

Differential Growth

Dividends versus Capital Gains

Valuation of Different Types of StocksZero Growth

Constant Growth

Differential Growth

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Case 1: Zero GrowthAssume that dividends will remain at the same level foreverAssume that dividends will remain at the same level forever

rP

rrrP

Div

)1(

Div

)1(

Div

)1(

Div

0

33

22

11

0

321 DivDivDiv

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

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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 present value of a growing perpetuity:

grP

1

0

Div

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

2012 )1(Div)1(DivDiv gg

..

3023 )1(Div)1(DivDiv gg

.

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Case 3: Differential Growth

Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter.To value a Differential Growth Stock, we need to:

Estimate future dividends in the foreseeable future.Estimate the future stock price when the stock becomes a Constant Growth Stock (case 2).Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate.

Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter.To value a Differential Growth Stock, we need to:

Estimate future dividends in the foreseeable future.Estimate the future stock price when the stock becomes a Constant Growth Stock (case 2).Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate.

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)(1DivDiv 101 g

Assume that dividends will grow at rate g1 for N years and grow at rate g2 thereafter

210112 )(1Div)(1DivDiv gg

NNN gg )(1Div)(1DivDiv 1011

)(1)(1Div)(1DivDiv 21021 ggg NNN

...

...

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Case 3: Differential Growth Dividends will grow at rate g1 for N years

and grow at rate g2 thereafter

)(1Div 10 g 210 )(1Div g

…0 1 2

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

)(1Div

210

2

gg

gN

N

…N N+1

…

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Case 3: Differential GrowthWe can value this as the sum of:

an N-year annuity growing at rate g1

T

T

Ar

g

gr

CP

)1(

)1(1 1

1

plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1

NBr

grP

)1(

Div

2

1N

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To value a Differential Growth Stock, we can use

NT

T

r

gr

r

g

gr

CP

)1(

Div

)1(

)1(1 2

1N

1

1

Or we can cash flow it out.

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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.

What is the stock worth? The discount rate is 12%.

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.

What is the stock worth? The discount rate is 12%.

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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(

108.12.

)08.1(2$

P

3)12.1(

75.32$8966.154$ P

31.23$58.5$ P 89.28$P

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A Differential Growth Example (continued)

08).2(1$ 208).2(1$…

0 1 2 3 4

308).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 phase

beginning in year 4 can be valued as a

growing perpetuity at time 3.

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A Differential Growth Example (continued)

PMT

I/Y

FV

PV

N

$2 =

3.70 =

0

– 5.58

3

PV

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. What is the stock worth?

$28.89 = 5.58 + 23.31First find the PV of the supernormal dividend stream then find

the PV of the steady-state dividend stream.

2×1.08

1.08

1.12

1.08–1 ×100

PMT

I/Y

FV

PV

N

0

12

32.75 =

3

PV – 23.31

2×(1.08)3 ×(1.04).08

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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 does g come from?

Where does r come from?

The value of a firm depends upon its growth rate, g, and its discount rate, r.



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g = Retention ratio × Return on retained earningsg = Retention ratio × Return on retained earnings

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The discount rate can be broken into two parts. The dividend yield

The growth rate (in dividends)

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

The discount rate can be broken into two parts. The dividend yield

The growth rate (in dividends)

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

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5.6 Growth Opportunities

Growth opportunities are opportunities to invest in positive NPV projects.

The value of a firm can be conceptualized as the sum of the value of a firm that pays out 100-percent of its earnings as dividends and the net present value of the growth opportunities.

Growth opportunities are opportunities to invest in positive NPV projects.

The value of a firm can be conceptualized as the sum of the value of a firm that pays out 100-percent of its earnings as dividends and the net present value of the growth opportunities.

NPVGOr

EPSP

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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.

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.

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The Dividend Growth Model and the NPVGO Model

Consider a firm that has EPS of $5 at the end of the first year, a dividend-payout ratio of 30%, 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:

Consider a firm that has EPS of $5 at the end of the first year, a dividend-payout ratio of 30%, 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$Div10

grP

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The NPVGO Model First, we must calculate the value of the firm

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

as a cash cow.

25.31$16.

5$Div10

rP

Second, we must calculate the value of the growth opportunities.

75.43$14.16.

875$.16.20.50.3

50.3

0

gr

P

Finally, 75$75.4325.310 P

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5.8 Price Earnings RatioMany analysts frequently relate earnings per share to price.

The price earnings ratio is a.k.a. the multipleCalculated as current stock price divided by annual EPS

The Wall Street Journal uses last 4 quarter’s earnings

Many analysts frequently relate earnings per share to price.

The price earnings ratio is a.k.a. the multipleCalculated as current stock price divided by annual EPS

The Wall Street Journal uses last 4 quarter’s earnings

EPS

shareper Priceratio P/E

Firms whose shares are “in fashion” sell at high multiples. Growth stocks for example.

Firms whose shares are out of favor sell at low multiples. Value stocks for example.

Firms whose shares are “in fashion” sell at high multiples. Growth stocks for example.

Firms whose shares are out of favor sell at low multiples. Value stocks for example.

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Other Price Ratio AnalysisMany analysts frequently relate earnings per share to variables other than price, e.g.:

Price/Cash Flow Ratiocash flow = net income + depreciation = cash flow from operations or operating cash flow

Price/Salescurrent 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

Many analysts frequently relate earnings per share to variables other than price, e.g.:

Price/Cash Flow Ratiocash flow = net income + depreciation = cash flow from operations or operating cash flow

Price/Salescurrent 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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5.9 Stock Market Reporting52WEEKS YLD VOL NET

HI LO STOCKSYMDIV % PE 100s HI LOCLOSE CHG52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75

Gap has been as high as $52.75 in the last year.

Gap has been as low as $19.06 in the last year.

Gap pays a dividend of 9 cents/share

Given the current price, the dividend yield is ½ %

Given the current price, the PE ratio is 15 times earnings

6,517,200 shares traded hands in the last day’s trading

Gap ended trading at $19.25, down $1.75 from yesterday’s close

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5.9 Stock Market Reporting52WEEKS YLD VOL NET

HI LO STOCKSYMDIV % PE 100s HI LOCLOSE CHG52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75

Gap Incorporated is having a tough year, trading near their 52-week low. Imagine how you would feel if within the past year you had paid $52.75 for a share of Gap and now had a share worth $19.25! That 9-cent dividend wouldn’t go very far in making amends.

Yesterday, Gap had another rough day in a rough year. Gap “opened the day down” beginning trading at $20.50, which was down from the previous close of $21.00 = $19.25 + $1.75

Looks like cargo pants aren’t the only things on sale at Gap.

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5.10 Summary and Conclusions

In this chapter, we used the time value of money 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

In this chapter, we used the time value of money 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

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Reserved.

5.10 Summary and Conclusions (continued)

3. The value of a coupon bond is the sum of the PV of the annuity of coupon payments plus the PV of the par value at maturity.

3. The value of a coupon bond is the sum of the PV of the annuity of coupon payments plus the PV of the par value at maturity.

TT r

F

rr

CPV

)1()1(

11

The yield to maturity (YTM) of a bond is that single rate that discounts the payments on the bond to the purchase price.

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5. A stock can be valued by discounting its dividends. There are three cases:

5. A stock can be valued by discounting its dividends. There are three cases:

NT

T

r

gr

rg

grC

P)1(

Div

)1()1(

1 2

1N

1

1

rP

Div0Zero growth in dividends

grP

1

0

DivConstant growth in dividends

Differential growth in dividends

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Reserved.


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.

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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