Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 11
Alternative Cost of Capital Analysis and Estimation
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
Outline










11.1 Introduction
11.2 Overview of Cost of Capital
11.3 Average earnings yield vs. current earnings yield
method
11.4 Discounting cash-flow method
11.5 Weighted average cost of capital
11.6 The CAPM method
11.7 M&M’s cross-sectional method
11.8 Chase cost of capital
11.9 Summary and conclusion remarks
Appendix 11A. Derivative of the basic equilibrium market price
of stock and its implications
11.3 Average earnings yield vs. current
earnings yield method
The market value of the firm, V, can be defined as:
x
V=
k
(11.1)
where x is the total expected future earnings and k is the cost of capital,
then cost of capital can be estimated from
x
k=
V
(11.2)
Lintner (1963) derived the rule for the marginal cost-of-capital decision as:
r  Ye
(11.3)
where r = the marginal internal rate of return, Ye = Y0/P0 is the current
earnings yield,Y0 is current earnings per share, and P0 is current price per
share..
11.4 Discounting cash-flow method
P0 =
d1
d
, then K e = 1 + g
p0
Ke - g
(11.4), (11.5)
Percentage change method
gˆ 1 + gˆ 2 + gˆ 3 + gˆ 4
ˆ
= g,
4
= earnings in year t, year (t -1), respectively, and
X
t - X t -1
,
gˆ t =
X t -1
where Xt, Xt-1
gˆ t is the estimate of the growth rate in period t.
Regression method
X
t
X 0 (1  g )  X t , then ( 1+ g ) =
X0
t
t
11.4 Discounting cash-flow method
loge EPSt  a0  a1T  1t
loge DPSt  b0  bT
1   2t
log EPSt = 0.910 + 0.015T,
(0.026*) (0.020)
log DPSt = -0.137 + 0.022T
(0.145) (0.020)
Where t-statistics are in parentheses and * indicated statistical
significance under 5% significant level.
11.4 Discounting cash-flow method
TABLE 11.1 EPS and DPS of Johnson & Johnson (1995-2006)
Year
EPSt
DPSt
LogEPSt
LogDPSt
T
1995
3.72
1.28
1.314
0.247
1
1996
2.17
0.74
0.775
-0.308
2
1997
2.47
0.85
0.904
-0.163
3
1998
2.27
0.97
0.820
-0.030
4
1999
3.00
1.09
1.099
0.086
5
2000
3.45
1.24
1.238
0.215
6
2001
1.87
0.70
0.626
-0.357
7
2002
2.20
0.80
0.788
-0.229
8
2003
2.42
0.93
0.884
-0.078
9
2004
2.87
1.10
1.054
0.091
10
2005
3.50
1.28
1.253
0.243
11
2006
3.76
1.46
1.324
0.375
12
* Standard errors are in parentheses.
11.5 Weighted average cost of capital
n
P
I
t
M =
+
t
n
(1 + k d )
t=1 (1 + k d )
Interest payment
kd 
Principle borrowed
(11.6)
(11.6′)
(0.075)($1,000,000)
$75,000
=
= 7.5%.
kd =
$1,000,000
$1,000,000
Cost of debt
K d  (1   c )kd

kd  (0.075)(1  0.50)  3.75%.
(11.7)
11.5 Weighted average cost of capital
'
Ct  ( M  M ) / n
'
(11.6′′)
kd 
'
(
M

M
)/2
where
M = Price at which the bond is sold in the market:
M′ = Issue price of the bond (the price actually
received by the issuing company);
(M - M′) = Flotation cost;
n = Life of the bond;
Ct = Interest expense per period on one bond.
[75  (1000  955) / 30] 75  (45 / 30)
k 

 7.8%
(100  955) / 2
977.5
'
d
Ct  [( M  P) / n]
k 
( M  P) / 2
''
d
(11.6”’)
11.5 Weighted average cost of capital
N
V
C
t
M =
+
t
N
(1 + k dc )
t=1 (1 + k dc )
where
(11.8)
MC = Market price of the convertible bond;
Ct = Interest payment on the convertible bond in
period t;
N = Time to conversion;
V = Forecast value of the bond on termination.
( 1/n)(V - M c )
.
k dc = C t + 1
2 (V + M c )
11.5 Weighted average cost of capital
PN = P0(1 - Cf)
(11.9)
where
PN = Net price of the stock,
P0 = Market price of the new stock,
Cf = Percentage flotation cost.
PN = 22(1 - 0.05) = 20.90.
Use PN into (11.5)
1.10
d1
+ 0.05 = 10.3.
+ g , ke=
Ke=
20.90
PN
1.00
D
P
(11.10)
= 10.127%.
,
k p=
K P=
9.875
Pp
11.5 Weighted average cost of capital
TABLE 11.2 XYZ financing
Component
Calculation
Cost of Component
Debt(no flotation cost)
Total interest
$75, 000

Principal borrowed $1, 000, 000
7.5%
After tax
(1-0.50)(7.5%)
3.75%
(100  95)
30
100  95
2
7.5 
Debt (with flotation cost)
After tax
7.87%
(1-0.50)(7.8%)
3.9%
Retained earnings
1.10
 0.05
22.00
10.0%
New preferred stock
1.00
9.875
10.125%
New equity
1.10
 0.05
20.90
10.3%
11.5 Weighted average cost of capital
Market value outstanding of component
Weight=
Total assets
(11.11)
(Book value of retained earnings)  (Market value of common stock)
(Book value of total assets)
(11.11a)
n
WACC =  W i K i
i
(11.12)
11.5.1 Theoretical Justification of the WACC

 D  

 ΔD  

,
=

1
WACC =  1 -  c 
  WACC
 τ c  ΔV  

A







 D 
K e =  + (1   c )(  k d ) 
,
 S 
(11.13), (11.14)
(11.15)
D / S is changes in market value ratio of debt to equity
D
(11.15’)
K e =  + 1   c    kd   
S
 D 
 S 

WACC = (1 - τ c) K d 
+ Ke
 (11.16)
D

S
D

S




11.5.1 Theoretical Justification of the WACC
D S 
 D  
WACC  (1   c )kd 



(1


)(


k
)
c
d



S   D  S 
 D  S  
 D 
 S 
 D  S 
 D  S 
 (1   c )kd 



(1


)


(1


)
k
c
c
d 



 



 DS 
 DS 
 S  D  S 
 S  D  S 
D 
 D 
 S
 D 
 D 
 (1   c )kd 






(1


)
k
c
c
d 



 DS 
 D  S D  S 
 DS 
 DS 

 D 
  1   c 
 ,
 D  S 

11.5.1 Theoretical Justification of the WACC

(1 - τ c)(1 - τ ps) 
V =V +  1D
(1 - τ pD) 

L
U
(6.32)
where
VU = Market value of unlevered firm,
 c = Corporate tax rate,
 ps = Capital gains tax rate,
 pD = Tax rate on ordinary income,
D = Market value of debt.
(1   pD )  (1   ps )(1   c )
11.6 The CAPM method
Fig. 11.1 Application of the asset-expansion criterion.
11.6 The CAPM method

 D  


=
+

E

1





WACC  R f
Rm
Rf  C


A



τ
(11.17)
E( R j ) = R f + [E( R m ) - R f ]  j + E( R z )(1 -  j ) + E( R h )( d i -  j d m )
(11.18)
where Rj, Rm, and ßj are defined in Chapter 6,
E( Rz)= The risk premium on a portfolio having a
zero beta and zero dividend yield,
E( Rh) = Expected rate of return on a hedge
portfolio having zero beta and dividend
yield of unity,
di = Dividend yield on stock i, and
dm = Dividend yield on the market portfolio.
11.6 The CAPM method
TABLE 11.3
Means and standard
deviations for three
estimates of the cost of
equity for the electric
utility industry (standard
deviations in parentheses)
Year
E/P
Ke
Rj
1967
.0558
(.0077)
.1033
(.0169)
.1054
(.0140)
1968
.0589
(.0077)
.1119
(.0186)
.1063
(.0149)
1969
.0663
(.0088)
.1340
(.0225)
.1209
(.0140)
1970
.0713
(.0092)
.1451
(.0283)
.1252
(.0143)
1971
.0752
(.0090)
.1576
(.0330)
.1010
(.0133)
1972
.0788
(.0091)
.1657
(.0354)
.1034
(.0152)
1973
.0880
(.0088)
.1891
(.0395)
.1285
(.0157)
1974
.1031
(.0119)
.2381
(.0566)
.1313
(.0156)
1975
.1115
(.0146)
.2009
(.0388)
.1258
(.0163)
1976
.1167
(.0166)
.1905
(.0350)
.1202
(.0159)
11.7 M&M’s cross-sectional method
 The cost of capital
 Regression formulation and empirical results
11.7 M&M’s cross-sectional method
11.7.1 The cost of capital
V=
X (1  c )
k
+  cD
(11.19)
where
V = Sum of the market value of all securities
issued by the firm,
X = Expected level of average annual earnings
generated by current assets,
 c = Corporate tax rate,
 k = Cost of unlevered equity capital in a certain
designated risk class,
D = Market value of a firm’s debt.
11.7 M&M’s cross-sectional method
V = S + D + P,
(11.20)
Where S = Common equity, D = Debt, P = Preferred equity
V  S 0  S n P D  S 0
=
+
+
+
=
+1
A A A A A A
(11.21)
where V = Change of market value of a firm,
A =  Sn + P +  D = New investment in real asset,
S0 = Change in the market value of the shares
held by the current owners of the firm,
 Sn = Value of any new common stock issued,
 P = Value of any new preferred stock issued,
 D = Value of any new debt issued.
11.7 M&M’s cross-sectional method
V  S
 X (1 -  c )  cD
=
+ 1=
+
A A
A k
A
(11.22)
X
ΔD 

(1 - τ c)
>  k 1  τ c (
) = C
A
ΔA 

(11.23)
0
 * - C
V = 1 -  c  X +  c D + K X 1 -  c  
 C(1 + C)

n

(11.24)
11.7 M&M’s cross-sectional method
11.7.2 Regression formulation and empirical results
(V  c D)  a0  a1 X (1   c )  a2 (growth potential)   (11.25)
A
t - At-5
 A= (
)
At
1
5
At
(V - τ cD) a 0
(1 - τ c)
A
= + a1 X
+ a2
+u
A
A
A
A
Vi *   X i*    j Z ij  ui
j
(11.26)
(11.27)
(11.28a)
X i  X  vi , X    j Z ij  wi (11.28b), (11.28c)
*
i
*
i
j
11.7 M&M’s cross-sectional method
Vi   X i    j Zij  U , Xˆ i*   ˆ j Zij
*
(11.29), (11.30)
(r - k)  I 0 (g -k)t
dt
e

k
0
T
The integral yields
(11.31)
 1  (g-k)T
- 1]

 [e
g-k
 1 
(g-k)T

 =

1
1





e


g  k
 1 

   1 k  g  T  = T
g - k
(11.32)
(r  k )  I 0T S = Y 0 + (r - k) I 0T , k = Y 0 + r I 0T
(11.33), (11.34)
, 0 k
k
S 0 +  I 0T , and (11.35)
k
11.7 M&M’s cross-sectional method
(r  k )  I 0 Y 0k   I 0k + r I 0  gY 0
Y
0

S0  +
k
k (k  g )
k (k - g )
gY0  rbY0 , r I 0  rbY0 ,
(11.36)
Y
Y
0k -  I 0k
0 - I0
=
S0=
k(k - g)
k-g
(11.37), (11.38)
D0 ,
=
S0
k-g
S  a2Y  a3r I  a4 I  
S
1
Y
r I
I
= a0 + a1 + a 2 + a 3
+ a4
+e
A
A
A
A
A
(11.39)
11.7 M&M’s cross-sectional method
The direct least squares estimates
ke with Constant
ke without Constant
1957
0.0637
0.0625
1956
0.0641
0.0602
1954
0.0730
0.0521
The two-stage estimates
ke with Constant
1957
1956
1951
0.0617
0.0641
0.0552
Direct
Two-Stage
1957
0.164
0.004
1956
0.057
0.054
1954
0.274
0.072
ke without Constant
0.0621
0.0599
0.0508
11.8 Chase cost of capital
NOPAT
V=
+  cD,
C
V=E+D
NOPAT
E + D=
+  cD
C
NOPAT = C(E + D - cD)
(11.40), (11.41)
(11.42)
(11.43)
NE NOPAT - bD(1 -  c )
Y=
=
E
E
(11.44)
C(E + D -  cD) - bD(1 -  c )
Y=
E
(11.45)
11.8 Chase cost of capital
Y + (1 -  c )(D/E)b
C=
1 + (1 -  c )(D/E)

 D 
C = C 1 - ( τ c) 

 CE  

(11.46)
*
Y = R + ß(P)
= 8% + 0.95(5%) = 12.75%.
(11.47)
11.8 Chase cost of capital
D 
Y = C + ( 1   c )(C - b)  
E 
(11.48)
where
 c = Marginal tax rate over the past five years;
b = Interest rate on all debt over the past five
years;
D/E = Average total debt to total equity over the
past five years. (Debt included capitalized
leases, and equity includes deferred items
and minority interest.)
11.8 Chase cost of capital
Y + (1 -  c )(D/E)(b)
C=
.
1 + (1 -  c )(D/E)
12.75% + (1  0.5)(0.60)(7%)
C
1 + (1  0.5)(0.60)

12.75%+2.10% 14.85%

 11.42%
1+0.30
1.30

 D 
C*  C 1  ( c ) 
  = 11.42%[1  0.5(0.40)]
 CE  

 11.42%(.80)  9.14% (all figures are hypothetical)
11.9 Summary and conclusion remarks
Based upon the valuation models and capital-structure
theories presented earlier, six alternative cost-of-capital
determination and estimation methods are discussed in
detail. These methods are (i) average earnings yield
method, (ii) DCF method, (iii) WACC method, (iv) CAPM
method, (v) M&M’s cross-section method, and (vi) Chase’s
method.
The interrelationship among different cost-of-capital
estimation methods were explored in some detail. The
relative advantages between different estimation methods
were also indirectly explored. The six cost-of-capital
estimation methods that were discussed in this chapter
give managers enough background to choose the
appropriate cost-of-capital estimation method for utilityregulation determination, capital-budgeting decisions, and
financial planning and forecasting.
Appendix 11A. Derivative of the basic equilibrium market
price of stock and its implications
yd + d ln Pt/dt = kt
(11.A.1)
where
yd  Dt / Pt  dividend yield,
1 dP
d ln pt / dt 
 Capital gain yield,
P dt
kt  Required rates-of-return.
Appendix 11A. Derivative of the basic equilibrium market
price of stock and its implications
dPt
 kPt   Dt
dt
(11.A.2)
kdt

Pt  e [  D0 e gt e  kt  C ]
gt  kt
e
 e kt [ D0 
d ( gt  kt )  C ]
g k
 kt  D0 gt  kt

e
 C  (general solution),
e 
 k  g


gt
e
D
(special solution).
 0 k  g
(11.A.3a)
(11.A.3b)
Appendix 11A. Derivative of the basic equilibrium market
price of stock and its implications
XY0
XY0
D
0
=
=
(k > g)
P0 =
*
k - g k - g k - ( g - n)
 1 dg / db 
dP0
 Y0   
0

db
 b kg 
(11.A.4)
dP0
k  g Y0
r

 ye
db
b
P0
(11.A.5)
V0
r  k 1  ( ) [k 1 - k 0]
I
(11.A.6)