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
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 rateof-return, and the cost of capital
3
6.1 Introduction
E ( R j )  R f  ( E ( Rm )  R f )  j ,
(6.1)
where
E ( Rj )
= Expected rate of return for asset j,
R f = Return on a risk-free asset,
( E ( Rm )  R f ) = Market risk premium, or the difference
in return on the market as a whole and
the return on a risk-free asset,
 j = 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.
4
6.2 Bond valuation



Perpetuity
Term bonds
Preferred stock
5
6.2 Bond valuation
n
CFt
PV  
,
t
t 1 (1  kb )
(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.
6
6.2 Bond valuation
CF
PV 
.
kb
(6.3)
n
It
P
PV  

, (6.4)
t
n
(1  kb )
t 1 (1  kb )
where
It = Coupon payment, coupon rate X face value,
p = Principal amount (face value) of the bond,
n = Number of periods to maturity.
7
6.2 Bond valuation
TABLE 6.1 Convertible bond: conversion options
Advantages
Purchase Price
Of Bond
Gain
(1) Conversion to stock if
price rises above $25.
$1000
Sell 40 shares at $30, =
$1,200, for a return of 12%.
(2)Interest payment if
stock price remains
less than $25.
$1000
$100 per year, for a return of
10%
(3)Interest payment versus
stock dividend.
Dividend must rise to $2.50
per share before return on
stock = 10%.
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.
8
6.2 Bond valuation
PV 
dp
kp
,
(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.
9
6.3 Common-stock valuation

Valuation

Inflation and common-stock
valuation

Growth opportunity and
common-stock valuation
10
6.3 Common-stock valuation
d1
d2
Pn
Po 


,
2
n
(1  k ) (1  k )
(1  k )
(6.6a)
where
P0 = 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
11
sold.
6.3 Common-stock valuation

dt
Pn  
.
t
t  n 1 (1  k )

P0  
dt
t 1 (1  k )
t
,
d1
P0 
.
(k  g n )
(6.6b)
(6.6c)
12
6.3 Common-stock valuation
d 0 (1  g s ) t
d n 1  1 
P0  


, (6.7)
t
n 
(1  k )
( r  g n )  (1  k ) 
t 1
n
where
gs = 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,
13
r = Internal rate-of-return.
6.3 Common-stock valuation
dt  pEPSt
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.
14
6.3 Common-stock valuation
p(Qt ( Pt  Vt )  Ft)(1   )
dt 
N
(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.
15
6.3 Common-stock valuation
dt
p{(inflows)t (1  i )t  (outflows)t (1  0 )t }(1   )

t
(1  k )
(1  k )t
(6.8a)
where
(1  K )  (1  k )(1   ),
  Anticipate d annual inflation risk ,
 i  Anticipate d annual inflation rate in the cash inflows ,
 0  Anticipate d annual inflation rate in the cash outflows ,
(inflows) t  Pt Qt , and
(outflows) t  QtVt  Ft .
The equation (6.8) is related to operating-income hypothesis which
has been discussed in chapter 5 on pages 158-160.
16
6.3 Common-stock valuation
X0
V0 
k
 b( r  k ) 
1

,

k  br 
(6.9)
where
X 0 = Current expected earnings per share,
b = Investment (It) as a percentage of total
earnings (Xt),
r = Internal rate of return
V0 and k = Current market value of a firm and
the required rate of return, respectively.
17
6.3 Common-stock valuation
X 0 (1  b)
D1
V0 

,
k  br
kg
d1
P0 
.
kg
(6.9a)
(6.9b)
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
D
k e  r  (r  i ) 
E
(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.
20
6.4 Financial leverage and its effect on EPS
rA  iD
ke 
,
E
(6.11)
D
ke  r  ( r  i )   ,
E
(6.10a)
D
Mean of ( ke )  ke  r  (r  i )   ,
E
(6.12a)
21
6.4 Financial leverage and its effect on EPS
2
 D
Variance of (ke )  1    Var(r ) .
E

[(rA  iD)   (rA  iD)]
ke 
,
E

 D 
ke   r  (r  i )    (1   ).
 E 

(6.12b)
(6.10b)
(6.13)
22
6.4 Financial leverage and its effect on EPS
~ ~ ~
 D 
k e   r  (r  i )  (1   ).
 E 

~

 D 
Mean (k )  k e  r  ( r  i ) (1  )
 E 

(6.14)
(6.15a)
2
~
D
2
Var (k e )  (1   ) 1   Var (~
r)
E

(6.15b)
23
6.4 Financial leverage and its effect on EPS
(rA  iD  ( rA  iD ))
EPS 
,
N
E
h ,
N

 D 
EPS  r  ( r  i ) (1   )h
 E 

(6.16)
(6.17)
(6.18a)
24
6.4 Financial leverage and its effect on EPS
2
D

Var(EPS)  h (1   ) 1   Var ( ~
r ).
E

2
2
EPS  r (1  )h
(6.18b)
(6.18c)
2
D

Var(EPS)  (1   ) h 1   Var (~
r ).
E

2
2
(6.18d)
25
6.4 Financial leverage and its effect on EPS
Figure 6.1
26
6.4 Financial leverage and its effect on EPS
Standard Deviation of EPS
 r [1  ( D / E )]
CVEPS 

Mean(EPS)
r  ( r  i )( D / E )
H  1,
H  1,
D
1 r
if

E 1 r  i
D
1 r
if

E 1 r  i
(6.19)
(6.20)
k  (18%  (18%  15%)( 0. 6))( 0.5)  9. 9%,
 k  (1  0.5)(1  0. 6)(2%)  1. 6%.
27
6.5 Degree of financial leverage and combined effect
EPS/EPS
EBIT

,
EBIT/EBIT EBIT  iD
(6.21)
Q( P  V )  F
DFL 
Q( P  V )  F  iD
(6.22)
CLE  DFL  DOL,
Q( P  V )
Combined Leverage Effect (CLE) 
,
Q( P  V )  F  iD
(6.23)
28
6.5 Degree of financial leverage and combined effect
Q( P  V )
DOL 
,
Q( P  V )  F
Q( P  V )  F
DFL 
,
Q( P  V )  F  iD
Q( P  V )
CLE 
.
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 
 it, X jt )
Cov(X
 ( X it )  ( X jt )

Cov( CX it , X jt )
C ( X it )  ( X jt )
 1,
 it  X it 1
X
Rit 
,
X i ,t 1
CX jt  CX jt 1
R jt 
 Rit
CX jt 1
31
6.6 Optimal capital structure
V j  ( S j  Dj ) 
V 
L
j
(1   j ) X j

kj  

Xj

jIj
Sj
 V jU   j D j .
r
(  r ) D j
,
,
(6.24)
(6.25)
(6.26)
32
6.6 Optimal capital structure
kj 


Sj

   (1   j )
[  r ] D j
Sj
,
(6.27)
s2
Y2 
( X 2  rD2 )   2 ( X  rD2 ), (6.28)
D2  S2
s1
Y1  X 1  1 X ,
S1
(6.29)
33
6.6 Optimal capital structure
Y1
1 ( S2  D2 )
S1
V2
X  r1 D2  1 X  rD2 . (6.30)
V1
s2
Y2  ( X  rD2 )  rd
S2
s1
D2
 ( X  rD2 )  r
V2
V2 s1
(6.31)
s1
s1

X  1 X ,
V2
V2
34
6.6 Optimal capital structure
TABLE 6.2 Valuation of two companies in accordance with Modigliani
and Miller’s Proposition 1
Initial Disequilibrium
Total Market Value ( V j )
Debt ( D j )
Equity ( S j )
Expected Net Operating
Income ( X )
Interest (rj D j )
Net Income ( X  r D )
j j
Cost of Common
Equity ( k j )
W1  D j V j
W2  S j V j
Average Cost of
Capital ( p j )
Final Equilibrium
Company
1
Company
2
Company
1
Company
2
$500
0
500
$600
300
300
$550
0
550
$550
300
250
50
0
50
50
21
29
50
0
50
50
21
29
10.00%
0
1
9.67%
11.6%
1
2
9.09%
0
1
10.00%
8.34%
9.09%
9.09%
1
2
6
11
5
11
35
6.6 Optimal capital structure
C
PS



(
1


)(
1


j
j )
L
U
V j  V j  1  
 D j ,
PD
1 j

 
(6.32)
 (1   Cj )(1   PS
j 
C
1





j
PD
(1   j ) 


1   Cj ) 
Dj.
1 
PD 
 (1   j ) 
r0
r0
rs 
 rd 
,
C
PD
1  j
1  j
(6.33)
(6.34)
36
6.6 Optimal capital structure
Fig. 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
X   (1   C )( X  R )  R  (1   C ) X   C R  (1   C ) X Z   C R ,
(6.35)
Var ( X  )  Var [( X    C R ) Z   C R ]
    CR 

C
 Var  X 1    Z   R 
X 
 

(6.36)
2

C R 
  ( X ) 1  
.
 
X 

2
z

38
6.6 Optimal capital structure
Y
U
 m
 U
V

C
(1   ) XZ ,

(6.37)


m
C
Y  L
[(1   ) XZ ].
C
L 
(6.38)
 ( S  (1   ) D ) 
L
S  (1   ) D  S  D   D  V   D  V .
L
C
L
L
L
C
L
L
C
L
U
39
6.6 Possible Reason for Optimal Capital Structure
•
•
•
The traditional Approach of Optimal Capital
Structure
Bankruptcy Cost
Agency Cost
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
j m
Pt  
t 1
rF (1   )  [( P0  F )( n  m)]
F

(1  ki ) t
(1  ki ) j  m
(6.A.1)
where
P = 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.
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
C  max(Cs , Cb ),
Cs  ps  
B / ps
0
Cb  B  

B / ps
(6.A.2)
f (i, t0 )[ B  i (t ) ps ]di (t ), (6.A.2a)
f (i, t0 )[i (t ) ps  B ]di (t ), (6.A.2b)

B / ps
0
0
C   i (t ) psf (i, t0 )di (t )  
f (i, t0 )[ B  i(t ) ps ]di (t ),
(6.A.3)
45
Appendix 6A. Convertible security valuation theory
E ( P)  

0
E ( P)  

0
  x
 y x

 y 0 h dx  yxh dx  g ( y ) dy,
 y
 y 


 y x

 y 0 h   dx  y xh( x, y)dx  dy,
 y


(6.A.4)
(6.A.4′)

y

E ( P )    xh( x, y )dx   ( y  x )h( x, y )dx  dy ,

0 
0
0

 
(6.A.5)

Expecte d stock
v alue

Value of floor
guarantee


E ( P )   yg ( y )dy    ( x  y )h( x, y )dxdy,
0
0 y

 


Expected
straightdebt
value
Expected value of
the conversion
option
(6.A.6)
46
Appendix 6A. Convertible security valuation theory


0
y
E ( P )   yg ( y )dy   ( x  y ) f ( x )dx,
y

E (CB)   yh ( x )dx   xb( x )dx,
0
(6.A.6′)
y
x 
 a 
Beta  
1     ,

m 
 x 
(6.A.7)
(6.A.8)
47
Appendix 6A. Convertible security valuation theory
E en ( i ( t )) CB  a

a

1
2 x
e
(1 / 2 )[( x   x ) /  x ]2
dx
(6.A.9)

x
a
1
2 x
e
(1 / 2 )[( x   x ) /  x ]2
dx.
G (V , ; B , c,  )  F (V , ; B , c,)  W (  V , ; B ),
(6.A.10)
H (V ,  )  F (V , ; B , c)  W (  V , ; B )  Z
 F (  V  , ; B  /  , c)],
2 ( r  p )/  2
[ F (  V , ; B  , c)
(6.A.11)
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
I. DOL
Let Sales = P×Q′
EBIT = Q (P – V) – F
Q′ = new quantities sold
The definition of DOL can be defined as:
DOL (Degree of operating leverage) 
Percentage Change in Profits
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
50
Appendix 6B. Derivation of DOL, DFL, and CML
I. DOL

(Q  Q) ( P  V )
Q (P V )  F

P Q
P (Q  Q)

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
 1
F
Q (P V )  F
 1
Fixed Costs
Profits
51
Appendix 6B. Derivation of DOL, DFL, and CML
II. DFL
Let i = interest rate on outstanding debt (or iD = interest
payment on debt )
D = outstanding debt
N = the total number of shares outstanding
τ = corporate tax rate
EAIT = [Q(P – V)– F– iD] (1–τ)
The definition of DFL can be defined as:
52
Appendix 6B. Derivation of DOL, DFL, and CML
II. DFL
DFL (Degree of financial leverage)
EPS EPS
(EAIT N ) (EAIT N )


EBIT EBIT
EBIT EBIT

EAIT EAIT
EBIT EBIT
[Q( P  V )  F  iD](1   )  [Q( P  V )  F  iD](1   )
[Q( P  V )  F  iD](1   )

[Q( P  V )  F ]  [Q( P  V )  F ]
[Q( P  V )  F ]
53
Appendix 6B. Derivation of DOL, DFL, and CML
II. DFL
[Q( P  V )](1   )  [Q( P  V )](1   )
[Q( P  V )  F  iD ](1   )

[Q( P  V )]  [Q( P  V )]
[Q( P  V )  F ]
[ (Q  Q)( P  V ) ] (1   )
Q( P  V )  F


[Q( P  V )  F  iD ] (1   ) (Q  Q)( P  V )
Q( P  V )  F

Q( P  V )  F  iD
EBIT 



 EBIT  iD 
54
Appendix 6B. Derivation of DOL, DFL, and CML
III. DCL (degree of combined leverage)
= DOL × DFL
Q( P  V )  F
Q( P  V )
Q( P  V )


Q( P  V )  F 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
A(1  R )
S
1 R
n
(6.C.3)
S∞ = A + AR + AR2 +…+ ARn −1 +…+ AR∞,
(6.C.4)
A
S 
1 R
(6.C.5)
58
Appendix 6C. Derivation of Dividend Discount Model
D3
D1
D2
P0 



(6.C.6)
2
3
1  k 1  k  1  k 
D1
D1 (1  g ) D1 (1  g )2
P0 



2
3
1  k 1  k 
1  k 
or
D1
D1 (1  g ) D1 (1  g )
P0 





2
1  k 1  k 1  k  1  k 1  k 
2
(6.C.7)
D1 (1  k )
D1 (1  k )
P0 

1  [(1  g ) (1  k )] [1  k  (1  g ) (1  k )]
D0 (1  g )
D1 (1  k )
D1



(k  g ) (1  k ) (k  g )
kg
59