Bond & Stock Valuation

Financial Analysis, Planning and Forecasting

Theory and Application

ByAlice C. Lee

San Francisco State UniversityJohn C. Lee

J.P. Morgan ChaseCheng F. Lee

Rutgers University

Chapter 6

Valuation and Capital Structure: Theory and application

Outline 6.1 Introduction 6.2 Bond valuation 6.3 Common-stock valuation 6.4 Financial leverage and its effect on EPS 6.5 Degree of financial leverage and combined

effect 6.6 Optimal capital structure 6.7 Summary and remarks Appendix 6A. Derivation of Dividend Discount Model Appendix 6B. Derivation of DOL, DFL, and CML Appendix 6C. Convertible security valuation theory

6.1 Introduction

Components of capital structure

Opportunity cost, required rate-of-return, and the cost of capital

6.1 Introduction

(6.1)

where

= Expected rate of return for asset j, = Return on a risk-free asset,

= Market risk premium, or the difference in return on the market as a whole and

the return on a risk-free asset, = Beta coefficient for the regression of an

individual’s security return on the market return; the volatility of the individual security’s return relative

to the market return.

,))(()( jfmfj RRERRE

E R j( )

R f

( ( ) )E R Rm f

j

6.2 Bond valuation

Perpetuity Term bonds Preferred stock

6.2 Bond valuation

(6.2)

where

n = Number of periods to maturity,

CFt = Cash flow (interest and principal) received in period t,

kb = Required rate-of-return for bond.

PVCF

kt

bt

t

n

( ),

11

6.2 Bond valuation

(6.3)

(6.4)

where

It = Coupon payment, coupon rate X face value,

p = Principal amount (face value) of the bond,

n = Number of periods to maturity.

PVCF

kb

.

PVI

k

P

kt

bt

bn

t

n

( ) ( ),

1 11

6.2 Bond valuationTABLE 6.1 Convertible bond: conversion options

AdvantagesPurchase PriceOf Bond Grain

(1) Conversion to stock if price rises above $25.

(2)Interest payment if stock price remains less than $25.

(3)Interest payment versus stock dividend.

$1000

$1000

Sell 40 shares at $30, = $1,200, for a return of 12%.

$100 per year, for a return of 10%

Dividend must rise to $2.50 per share before return on stock = 10%.

6.2 Bond valuation

(6.5)

where

dp = Fixed dividend payment, coupon X par on face value of preferred stock;

kp = Required rate-of-return on the preferred stock.

PVd

kp

p

,

6.3 Common-stock valuation

Valuation

Inflation and common-stock valuation

Growth opportunity and common-stock valuation

6.3 Common-stock valuation (6.6a)

whereP0 = Present value, or price, of the

common stock per share, dt = Dividend payment, k = Required rate of return for the stock,

assumed to be a constant term,Pn = Price of the stock in the period when

sold.

Pd

k

d

k

P

kon

n

1 2

21 1 1( ) ( ) ( ),

6.3 Common-stock valuation

(6.6b)

(6.6c)

Pd

knt

tt n

( ).

11

01

,(1 )

tt

t

dP

k

10 .

( )n

dP

k g

6.3 Common-stock valuation

(6.7)

wheregs = Growth rate of dividends during the super-growth period, n = Number of periods before super-growth declines to normal,

gn = Normal growth rate of dividends after the end of the super-growth phase, r = Internal rate-of-return.

n

tn

n

nt

ts

kgr

d

k

gdP

1

100 ,

)1(

1

)()1(

)1(

6.3 Common-stock valuation

where

dt = Dividend payment per share in period t,

p = Proportion of earnings paid out in dividends (the payout ratio, 0 p 1.0),

EPSt = earnings per share in period t.

EPSt td p

6.3 Common-stock valuation

(6.8)

whereQt = Quantity of product sold in period t,Pt = Price of the product in period t,Vt = Variable costs in period t,F = Depreciation and interest expenses in period t, = Firm tax rate.

N

FVPQpd tttt

t

)1)()((

6.3 Common-stock valuation

(6.8a)

where

0{(inflows) (1 ) (outflows) (1 ) }(1 )

(1 ) (1 )

t tt t i t

t t

d p

k k

.(outflows)

and,(inflows)

,outflowscash in the rateinflation annual dAnticipate

,inflowscash in the rateinflation annual dAnticipate

,riskinflation annual dAnticipate

),1)(1()1(

0

tttt

ttt

i

FVQ

QP

kK

6.3 Common-stock valuation (6.9)

where = Current expected earnings per share, b = Investment (It) as a percentage of total

earnings (Xt), r = Internal rate of returnV0 and k = Current market value of a firm and

the required rate of return, respectively.

00

( )1 ,

X b r kV

k k br

0X

6.3 Common-stock valuation

(6.9a)

(6.9b)

0 10

(1 ),

X b DV

k br k g

Pd

k g01

.

6.4 Financial leverage and its effect on EPS

6.4.1 Measurement

6.4.2 Effect

6.4 Financial leverage and its effect on EPS

(6.10)

where

ke = Return on equity,

r = Return on total assets (return on equity without leverage)

i = Interest rate on outstanding debt,

D = Outstanding debt,

E = Book value of equity.

E

Dirrke )(

6.4 Financial leverage and its effect on EPS

(6.11)

(6.10a)

(6.12a)

krA iD

Ee

,

( ) ,e

Dk r r i

E

Mean of ( ) ( ) ,e eD

k k r r iE

6.4 Financial leverage and its effect on EPS

(6.12b)

(6.10b)

(6.13)

2

Variance of ( ) 1 Var( ) .eD

k rE

[( ) ( )],e

rA iD rA iDk

E

( ) (1 ).eD

k r r iE

6.4 Financial leverage and its effect on EPS

(6.14)

(6.15a)

(6.15b)

).1()~(~~

E

Dirrke

)1()()~

( Mean

E

Dirrkk e

)~(Var 1)1()~

(Var 2

2 rE

Dke

6.4 Financial leverage and its effect on EPS

(6.16)

(6.17)

(6.18a)

EPS ( ( ))

,rA iD rA iD

N

hE

N ,

hE

Dirr )1()( EPS

6.4 Financial leverage and its effect on EPS

(6.18b)

(6.18c)

(6.18d)

).~(Var1)1(Var(EPS)2

22 rE

Dh

EPS r h( )1

).~(Var1)1( Var(EPS)2

22 rE

Dh

6.4 Financial leverage and its effect on EPS

Figure 6.1

6.4 Financial leverage and its effect on EPS

(6.19)

(6.20)

CVStandard Deviation of EPS

Mean(EPS)EPS

r D E

r r i D E

[ ( / )]

( )( / )

1

ir

r

E

DH

ir

r

E

DH

1

1if ,1

1

1if ,1

k

k

( % ( % %)( . ))( . ) . %,

( . )( . )( %) . %.

18 18 15 0 6 0 5 9 9

1 0 5 1 0 6 2 1 6

6.5 Degree of financial leverage and combined effect

(6.21)

(6.22)

(6.23)

EPS / EPS

EBIT / EBIT

EBIT / (EBIT

EBIT / EBIT

EBIT

EBIT

iD

iD

),

( )DFL

( )

Q P V F

Q P V F iD

CLE DFL DOL,

( )Combined Leverage Effect (CLE) ,

( )

Q P V

Q P V F iD

6.5 Degree of financial leverage and combined effect

( )

DOL ,( )

Q P V

Q P V F

( )

DFL ,( )

Q P V F

Q P V F iD

( )

CLE .( )

Q P V

Q P V F iD

6.6 Optimal capital structure

Overall discussion

Arbitrage process and the proof of M&M Proposition I

6.6.1 Overall Discussion

ij

jt

it jt

it jt

it jt

X

X X

CX X

C X X

Cov(X Cov(it, )

( ) ( )

, )

( ) ( ),1

,

,

RX X

Xitit it

i t

1

1

RCX CX

CXRjt

jt jt

jtit

1

1

6.6 Optimal capital structure

(6.24)

(6.25)

(6.26)

V S DX

j j jj ( ) ,

VX I

rV Dj

L j j j jjU

j j

( )

.1

kr D

Sjj

j

( )

,

6.6 Optimal capital structure

(6.27)

(6.28)

(6.29)

kS

r D

Sjj

jj

j

( )[ ]

,1

Ys

D SX rD X rD2

2

2 22 2 2 2

( ) ( ),

Ys

SX X1

1

11 1 ,

6.6 Optimal capital structure

(6.30)

(6.31)

YS D

SX r D

V

VX r D1

1 2 2

11 2 1

2

12

( ).

Ys

SX rD rd

s

VX rD r

D

V s

s

VX

s

VX

22

22

1

22

2

2 1

1

21

1

2

( )

( )

,

6.6 Optimal capital structureTABLE 6.2 Valuation of two companies in accordance with Modigliani

and Miller’s Proposition 1

jVjD

jS

Xj jr D

j jX r D

jk

1 j jW D V

2 j jW S V

jp

1

2

1

2

Initial Disequilibrium Final Equilibrium

Company 1

Company2

Company 1

Company2

Total Market Value ( )Debt ( )Equity ( )Expected Net Operating Income ( )Interest ( )Net Income ( )Cost of Common Equity ( )

Average Cost of Capital ( )

$5000

500

500

50

10.00%01

10.00%

$600300300

502129

9.67%

8.34%

$5500

550

500

50

9.09%01

9.09%

$550300250

502129

11.6%

9.09%

6

115

11

6.6 Optimal capital structure

(6.32)

(6.33)

(6.34)

,1

)1)(1(1 jPD

j

PSj

CjU

jLj DVV

CjPD

j

PSj

Cj

)1(

1)(1(1

.)1(

)11 jPD

j

Cj D

rr

rr

sjC d

jPD

0 0

1 1 ,

6.6 Optimal capital structureFig. 6.2 Aggregated supply and demand for corporate bonds (before tax rates). From

Miller, M., “Debt and Taxes,” The Journal of Finance 29 (1977): 261-275. Reprinted by permission.

6.6 Optimal capital structure

(6.35)

(6.36)

X X R R X R X Z RC C C C C ( )( ) ( ) ( ) ,1 1 1

.1)(

1

])[(Var)(Var

22

X

RX

RZX

RXVar

RZRXX

Cz

CC

CC

6.6 Optimal capital structure

(6.37)

(6.38)

,)1( ZXV

mY C

UU

].)1[())1((

ZXDS

mY C

LCLL

S D S D D V D VL C L L L C L L C L U ( ) .1

6.7 Summary and remarks In this chapter the basic concepts of valuation and capital structure are

discussed in detail. First, the bond-valuation procedure is carefully discussed. Secondly, common-stock valuation is discussed in terms of (i) dividend-stream valuation and (ii) investment-opportunity valuation. It is shown that the first approach can be used to determine the value of a firm and estimate the cost of capital. The second method has decomposed the market value of a firm into two components, i.e., perpetual value and the value associated with growth opportunity. The criteria for undertaking the growth opportunity are also developed.

An overall view on the optimal capital structure has been discussed in accordance with classical, new classical, and some modern finance theories. Modigliani and Miller’s Proposition I with and without tax has been reviewed in detail. It is argued that Proposition I indicates that a firm should use either no debt or 100 percent debt. In other words, there exists no optimal capital structure for a firm. However, both classical and some of the modern theories demonstrate that there exists an optimal capital structure for a firm. In summary, the results of valuation and optimal capital structure will be useful for financial planning and forecasting.

Appendix 6A. Convertible security valuation theory

(6.A.1)

whereP = Market value of the convertible bond,r = Coupon rate on the bond,F = Face value of the bond,

P0 = Initial market value, ki = Effective rate of interest on the bond at the end of the period m (now),

n = Original maturity of the bond,m = Number of periods since the bond was issued,j = Number of periods from the time the bond was issued

till the time of conversion, F’= Value of the stock on date of conversion,

t = Marginal corporate tax rate.

PrF P F n m

k

F

kti

ti

j mt

j m

( ) [( )( )]

( ) ( )

1

1 10

1

Appendix 6A. Convertible security valuation theory Fig. 6.A.1 Hypothetical model of a convertible years’ bond. (From Brigham,

E. F. “An analysis of convertible debentures: theory and some empirical evidence,” Journal of Finance 21 (1966), p. 37) Reprinted by permission.

Appendix 6A. Convertible security valuation theory

(6.A.2)

(6.A.2a)

(6.A.2b)

(6.A.3)

C C Cs bmax( , ),

psB

s tdipstiBtifpsC/

0 0 ),(])()[,(

psBb tdiBpstitifBC

/ 0 ),(])()[,(

psB

tdipstiBtiftditipsftiC/

0 00 0 ),(])()[,()(),()(

Appendix 6A. Convertible security valuation theory (6.A.4)

(6.A.4′)

(6.A.5)

(6.A.6)

, )( )(00

dyygdxy

xxhdx

y

xhyPE

y

y

0 0( ) ( , ) ,

y

y

xE P y h dx xh x y dx dy

y

,),()(),()(

guarantee floor of Value

0

alue v

stock d Expecte

0 0dydxyxhxydxyxxhPE

y

,),()()()(

option conversion the

of valueExpected

0

valuedebt straight

Expected

0

y

dydxyxhyxdyyygPE

Appendix 6A. Convertible security valuation theory

(6.A.6′)

(6.A.7)

(6.A.8)

ydxxfyxdyyygPE ,)()()()(

0

y

ydxxxbdxxyhCBE

0,)()()(

,1

x

a

m

xBeta

Appendix 6A. Convertible security valuation theory

(6.A.9)

(6.A.10)

(6.A.11)

.2

1

2

1

2

2

]/))[(2/1(

]/))[(2/1())((

dxex

dxeaE

xx

xx

x

xa

a x

x

CBtien

G V B c F V B c W V B( , ; , , ) ( , ; , , ) ( , ; ),

H V F V B c W V B Z F V B c

F V B c

r p( , ) ( , ; , ) ( , ; ) [ ( , ; , )

( , ; / , )],

( )/

2 2

Appendix 6B. Derivation of DOL, DFL, and CML

I. DOL II. DFL III. DCL (degree of combined leverage)

Appendix 6B. Derivation of DOL, DFL, and CML Let Sales = P×Q′

EBIT = Q (P – V) – F

Q′ = new quantities sold

The definition of DOL can be defined as:Percentage Change in Profits

DOL (Degree of operating leverage) Percentage Change in Sales

EBIT EBIT

Sales Sales

{[ ( ) ] [ ( ) ]} ( )

( ) ( )

( ) (

Q P V F Q P V F Q P V F

P Q P Q P Q

Q P V Q

) ( )

( )

( )( ) [ ( ) ]

( )

P V Q P V F

P Q Q P Q

Q Q P V Q P V F

P Q Q P Q

I. DOL

Appendix 6B. Derivation of DOL, DFL, and CML

( )

Q Q

( )

( ) ( )

P V P Q

Q P V F P Q Q

( )

( )

( ) ( )

( ) ( ) ( )

1 ( )

Fixed Costs 1

Profits

Q P V

Q P V F

Q P V F F Q P V F F

Q P V F Q P V F Q P V F

F

Q P V F

I. DOL

Appendix 6B. Derivation of DOL, DFL, and CML II. DFL

Let i = interest rate on outstanding debt

D = outstanding debt

N = the total number of shares outstanding τ = corporate tax rateEAIT = [Q(P – V)– F– iD] (1–τ)

The definition of DFL can be defined as:

(or iD = interest payment on debt )

Appendix 6B. Derivation of DOL, DFL, and CML II. DFL

DFL (Degree of financial leverage)

EPS EPS ( EAIT ) (EAIT )

EBIT EBIT EBIT EBIT

EAIT EAIT

EBIT EBIT

[ ( ) ](

1 ) [ ( ) ](1 )[ ( ) ](1 )[ ( ) ] [

(

)

N N

Q P V F iD Q P V F iDQ P V F iD

Q P V F Q P V F

][ ( ) ]Q P V F

Appendix 6B. Derivation of DOL, DFL, and CML [ ( )](1 ) [ ( )](1 )

[ ( ) ] (1 )[ ( )

] [ ( )][ ( ) ]

[ (

)( )

Q P V Q P VQ P V F iD

Q P V Q P VQ P V F

Q Q P V

] (1 )[ ( ) ] (1 )Q P V F iD

( )

( )( )

Q P V F

Q Q P V

( ) EBIT

( ) EB

IT

Q P V F

Q P V F iD iD

II. DFL

Appendix 6B. Derivation of DOL, DFL, and CML

III. DCL (degree of combined leverage) = DOL × DFL

( )

( )

Q P V

Q P V F

( )Q P V F

( )

( ) ( )

Q P V

Q P V F iD Q P V F iD

Appendix 6C. Derivation of Dividend Discount Model

I. Summation of infinite geometric series

II. Dividend Discount Model

Appendix 6C. Derivation of Dividend Discount Model

S = A + AR + AR2 + … + ARn −1 (6.C.1)

RS = AR + AR2 + … + ARn −1 + ARn (6.C.2)

S − RS = A − ARn

Appendix 6C. Derivation of Dividend Discount Model

(6.C.3)

S∞ = A + AR + AR2 +…+ ARn −1 +…+ AR∞, (6.C.4)

(6.C.5)

(1 )

1

nA RS

R

1

AS

R

Appendix 6C. Derivation of Dividend Discount Model

(6.C.6)

or

(6.C.7)

31 2

0 2 31 1 1

DD DP

k k k

21 1 1

0 2 3

(1 ) (1 )

1 1 1

D D g D gP

k k k

21 1 1

0 2

(1 ) (1 )

1 1 1 1 1

D D Dg gP

k k k k k

1 10

01 1

(1 ) (1 )

1 [(1 ) (1 )] [1 (1 ) (1 )]

(1 )(1 )

( ) (1 ) ( )

D k D kP

g k k g k

D gD k D

k g k k g k g