financial analysis, planning and forecasting theory and application
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Financial Analysis, Planning and Forecasting Theory and Application. Chapter 6. Valuation and Capital Structure: Theory and application. By Alice C. Lee San Francisco State University John C. Lee J.P. Morgan Chase Cheng F. Lee Rutgers University. Outline. 6.1Introduction - PowerPoint PPT PresentationTRANSCRIPT
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
1
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
2
6.1 Introduction
Components of capital structure
Opportunity cost, required rate-of-return, and the cost of capital
3
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
4
6.2 Bond valuation
Perpetuity Term bonds Preferred stock
5
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
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
7
6.2 Bond valuationTABLE 6.1 Convertible bond: conversion options
AdvantagesPurchase PriceOf Bond Gain
(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%.
8
The results in this table are based on a $1000 face-value bond with 10% coupon rate, convertible to 40 shares of stock at $25 each.
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
,
9
6.3 Common-stock valuation
Valuation
Inflation and common-stock valuation
Growth opportunity and common-stock valuation
10
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( ) ( ) ( ),
11
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
12
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(
13
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
14
6.3 Common-stock valuation
(6.8)
Where Qt = 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)()((
15
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
16
The equation (6.8) is related to operating-income hypothesis which has been discussed in chapter 5 on pages 158-160.
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
17
6.3 Common-stock valuation
(6.9a)
(6.9b)
0 10
(1 ),
X b DV
k br k g
Pd
k g01
.
18
6.4 Financial leverage and its effect on EPS
6.4.1 Measurement
6.4.2 Effect
19
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 )(
20
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
21
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
22
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
23
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
24
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
25
6.4 Financial leverage and its effect on EPS
Figure 6.1
26
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 27
6.5 Degree of financial leverage and combined effect
(6.21)
(6.22)
(6.23)
EPS/EPS EBIT,
EBIT/EBIT EBIT 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
28
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
29
6.6 Optimal capital structure
Overall discussion
Arbitrage process and the proof of M&M Proposition I
30
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
31
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
( )
,
32
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 ,
33
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
( )
( )
,
34
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
35
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 ,
36
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.
37
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
38
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
39
• The traditional Approach of Optimal Capital Structure
• Bankruptcy Cost• Agency Cost
6.6 Possible Reason for Optimal Capital Structure
40
Possibility of Optimal Debt Ratio when Bankruptcy Allowed
41
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.
42
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
43
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.
44
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 ),(])()[,()(),()(
45
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
46
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
47
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
48
Appendix 6B. Derivation of DOL, DFL, and CML
I. DOL II. DFL III. DCL (degree of combined leverage)
49
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
50
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
51
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 )
52
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
53
Appendix 6B. Derivation of DOL, DFL, and CML [ ( )](1 ) [ ( )](1 )
[ ( ) ](1 )
[ ( )] [ ( )][ ( ) ]
[ ( )( )
Q P V Q P VQ P V F iDQ P V Q P V
Q P V F
Q Q P V
] (1 )[ ( ) ] (1 )Q P V F iD
( )
( )( )
Q P V F
Q Q P V
( ) EBIT
( ) EBIT
Q P V F
Q P V F iD iD
II. DFL
54
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
55
Appendix 6C. Derivation of Dividend Discount Model
I. Summation of infinite geometric series
II. Dividend Discount Model
56
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
57
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
58
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
59