CHAPTER 11
ALTERNATIVE COST OF CAPITAL ANALYSIS
AND ESTIMATION
1. a) The current earning yield method, as shown by Lintner, is equal to the marginal
cost of capital under uncertainty. It is described as r ≥ ye, where r is the marginal
internal rate of return, ye = yo/po is the current earnings yield and yo and po are the
current EPS and PPS. This equation implies that an investment should not be
accepted unless r is larger than ye. Although this method is suggested for use by
Lintner over the average earnings yield method, it is not very popular in practical
applications.
b) The discounted cash flow method (DCF) determines the cost of capital for a
firm’s retained earnings. Using Gordon’s growth equation, po = D1/(Ke – g), we
can rearrange to solve for the cost of equity capital, Ke :Ke  D p  g , using both
average DPS and PPS. These are readily ascertainable, but the estimation of a
proper growth rate can be considerable problem.
c) In order to estimate the weighted average cost of capital (WACC), we (1)
determine the relative proportions of the various financial instruments in a firm’s
capital structure, and (2) calculate the proper cost of each of the components. Thus
we estimate the cost of debt, cost of retained earnings, cost of preferred stock, and
the cost of new equity. To determine the appropriate weights of each capital
component, we use the market value outstanding of each component/total assets to
determine the weight for each financing instrument. Then, WACC equals the sum
of these weighted components.
WACC is a more popular method in applications, although it is subject to
much criticism for its simplicity, ease of calculation, and lack of completeness.
However, Henderson and others have shown that WACC is equivalent
theoretically and empirically with other methods.
d) CAPM expresses the required rate of return for a risky asset or investment;
therefore it can be used to determine the cost of capital for an investment project
undertaken by a firm. CAPM’s advantages over WACC are that (1) the concept of
systematic risk is directly incorporated into the CAPM framework, and (2) it does
not implicitly assume (as WACC does), that the project is to the same risk class as
the firm. However, the simple CAPM does not explicitly consider the effects of
leverage and taxes on a firm’s cost of capital. To overcome this potential bias the
practitioner should use alternative CAPM’s as derived by Brennan (Chapter 7, 230)
and, Litzenberger and Ramaswamy (Chapter 13, 468-471).
e) Under the M & M cross-sectional method, we assume (1) a perpetual stream of
earnings from real assets, and (2) a constant capitalization rate (p) at which the
market discounts the unlevered equity stream of earnings for some risk classes and
perfect markets: the value of a levered firm is
where
V = sum of market value of all securities issued,
p = cost of unlevered equity capital/risk class,
D = market value of a firm’s debt,
x = expected level of average annual earnings generated by current assets,
and
T = corporate tax rate.
A finite supernormal growth model can be applied to an all-equity firm by
partitioning the value of the firm into
V = (S)stock + (D)debt + (P)preferred stock.
Thus it is possible to obtain V  (1   c ) x   c D  kx (1   c )[
*  
]n ,
 (1   )
where k = investment as a percentage of tax adjusted earnings; ρ* = IRR.
By then performing a cross-sectional regression, we can estimate the market
capitalization rate and thus the cost of capital.
2. The Chase cost-of-capital method is a useful blend of the CAPM method and M &
M’s firm value relationship. This three-step process (1) estimates the equity cost
of capital of a firm, (2) adjusts this cost of capital figure for financial risk to arrive
at an estimate of the unlevered equity cost of capital, and (3) estimates the total
cost of capital using the unlevered cost of capital and an estimate of the future
long-term ratio. The main advantage of this method is to more explicitly take
account of the effects of leverage in estimating the firm’s cost of capital. In
addition, the Chase method is structurally proficient enough to consider both time
series and cross-sectional effects on cost of capital estimation.
3. Three methods of growth rate estimation are:
a) Simple growth rate－the average total change in earning over a period of a
year: gˆ 
x(n)  x(1)
n
b) Arithmetic average－growth estimates for short periods are again averaged to
arrive at one growth estimate for the total period:
gˆ1  gˆ 2  gˆ 3  gˆ 4
 gˆ t
4
c) Ordinary least squares regression－using EPS and/or DPS information,
Loge EPSt = ao + a1T1 + ε1t
and/or
Loge DPSt = bo + b1T1 + ε2t.
The regression method takes more information into account. Also, a null
hypothesis test can be made to determine if the growth rate obtained is statistically
significant from zero.
4. Yes, WACC can be theoretically justified. As Henderson derived in his paper, the
WACC can be transformed into a generalized form of WACC:
Ko  K (s I )  r (1  )(D I )  (K I )S  (K I )(1   ) D
which proves to be theoretically and conceptually consistent with the equity cost
of capital expressions that emerge from the M & M theory as well as from CAPM.
5. The information needed to use CAPM in cost-of-equity capital estimation include:
a) the risk-free rate, which can be calculated or estimated using 90-day T-bills;
b) the risk premium (Rm – Rf), which can be approximated by the long-term
market risk premium of a market index such as the Dow Jones 30 or S & P 500
index;
c) each asset’s or investment’s beta, which represents the systematic risk
associated with undertaking the project or buying the asset.
6.
The discount rate for the project is equal to the expected return for the security, RS,
since the project has the same risk as the firm as a whole. Apply the CAPM to
express the firm’s required return, RS, in terms of the firm’s beta, , the risk-free rate,
RF, and the expected market return, R M.
RS
= RF +   ( R M – RF)
= 0.06 + 0.8 (0.10)
= 0.14
Subtract the initial investment at year 0. To calculate the PV of the cash inflows,
apply the annuity formula, discounted at 0.14.
NPV
= C0 + C1 ATr
= -$1,000,000 + $340,000 A50.14
= -$201440
Do not undertake the project since the NPV is negative.
7.
Calculate the square root of the stock’s variance and the market’s variance to find the
standard deviation, , of each.
C = (2C)1/2
= (0.003025)1/2
= 0.055
M = (2M)1/2
= (0.001764)1/2
= 0.042
Use the formula for beta:
C = [Corr (RC, RM)  C] / M
= [(0.75) (0.055)] / (0.042)
= 0.982
The beta of Company BBB is 0.982.
8.
In order to calculate the weighted average cost of capital, one has to determine the
proper weights expressing the firm’s proportion of debt in terms of the firm’s
proportion of equity. Solve for the equity to value ratio [S / (S+B)] and the debt to
value ratio [B / (S+B)].
B/S
= 0.45
B = 0.45 S
[S / (S+B)]
= [S / (S + 0.45S)]
= [S / (1.45  S)]
= 1 / 1.45
= 0.69
[B / (S+B]
= [0.45  S / (S + 0.45  S)]
= [(0.45  S) / (1.45  S)]
=1/5
= 0.31
Interest on debt is tax deductible at the corporate level, which leads to
tax benefits for the firm. Multiply the cost of debt, rB, by (1 - TC), to
include the tax benefit created by the interest tax shield. The tax
benefit of debt will lower the firm’s overall cost of capital.
rWACC
= [S / (S+B)]  rS + [B / (S+B)]  rB  (1 – TC)
= (0.69) (0.15) + (0.31) (0.08) (0.70)
= 0.1209
The weighted average cost of capital is 12.09%.
9.
Apply the CAPM to estimate DDD’s cost of equity, RS. The following equation
expresses the firm’s required return, RS, in terms of the firm’s beta, S, the risk-free
rate, RF, and the market return, R M.
RS
= RF +   ( R M – RF)
= 0.06+ 1.2 (0.15 – 0.06)
= 0.168
DDD’s cost of equity is 16.8%.
Use the debt-to-equity ratio to determine the weighted average cost of capital.
Calculate the equity to value ratio [S / (S+B)] and the debt to value ratio [B / (S+B)]
by substitution.
B/S
= 1.2
B = S
[S / (S+B)]
= [S / (1.2S + S)]
= [S / (2.2  S)]
= 1 / 2.2
= 0.4545
[B / (S+B]
= [1.2  S / (S + 1.2  S)]
= [(1.2S) / (2.2  S)]
= 1.2 / 2.2
= 0.5455
Remember that interest on debt is tax deductible at the corporate level,
which will lower the firm’s overall cost of capital. Multiply the cost
of debt, rB, by (1 - TC) to include the interest tax shield in the weighted
average cost of capital.
rWACC
= [S / (S+B)]  rS + [B / (S+B)]  rB  (1 – TC)
= (0.4545) (0.168) + (0.5455) (0.09) (1 – 0.35)
= 0.09645
The weighted average cost of capital is 10.83%.
10.
Use the market value of debt to compute the weighted average cost of capital.
To
calculate the market value of the debt, multiply the book value of EEE’s debt by 120
percent.
B
= $50,000,000  1.3
= $65,000,000
The market value of EEE’s stock is equal to the number of shares outstanding
multiplied by the market price per share.
S
= 6,000,000  $25
= $150,000,000
The value of the firm is the market value of EEE’s debt plus the market value of
EEE’s stock.
V
=B+S
= $65,000,000 + $150,000,000
= $215,000,000
Calculate the debt-to-value and equity-to-value ratios. These ratios will be used to
compute the weighted average cost of capital.
B / (S + B)
= $65,000,000 / $215,000,000
= 0.3023
S / (S + B)
= $150,000,000 / $215,000,000
= 0.6977
Use the cost of debt and cost of equity, 10 percent and 16 percent, respectively, and
the weights calculated above, to determine the weighted average cost of capital.
Remember that interest on debt is tax deductible at the corporate level, which will
lower the firm’s overall cost of capital. Multiply the cost of debt, rB, by (1 - TC) to
include the interest tax shield in the weighted average cost of capital.
rWACC
= [S / (S+B)]  rS + [B / (S+B)]  rB  (1 – TC)
= (0.6977) (0.16) + (0.3023) (0.10) (0.65)
= .1313
The weighted average cost of capital for Company EEE is 13.13%.
11.
a. The average earnings yield is the most basic method of estimating the cost of
capital and defines the yield as average expected future earnings ( X ) divided
by the market value of the firm (V). The current yield is defined as EPS
divided by price per share. It is considered superior to the average earnings
yield in that the current yield accounts for expected future growth.
b. The cost of retained earnings. KRE, is the rate of return stockholders require
on the firm's common stock. It is defined as the dividend yield,
D1
, plus the
Po
expected growth rate in dividends, g. The cost of new common stock is similar
to the cost of retained earnings in that it reflects of rate of return stockholders
require; however, flotation costs are incurred when selling new common stock.
Thus, the cost of new common stock, K≠, reflects these costs by dividing the
cost of retained earnings, KRE by (1 – f) where f represents the percentage
flotation cost.
c. The weighted cost of capital (WACC) or therefoe the marginal cost of capital
(MCC) is defined as the cost of an additional unit of financing. It is measured
as the weighted average of the component sources of financing where the
weights are determined by the percentage obtained from each source. The
WACC should not be confused with the ACC which the authors refer to as the
weighted average of the marginal costs of each interval. (See pp.239-240 for a
discussion of WACC, MCC, and ACC.)
d. The marginal cost of capital (MCC) is defined as the cost of an additional unit
of financing. See the previous discussion for a comparison to ACC and
WACC.
e. The unleveraged cost of capital is that which would exist if the firm were
unlevered. A common method of estimating the unlevered cost of capital is to
use an unlevered beta in a CAPM formulation. An unlevered beta is the beta
for the company's equity if there were no debt financing. A discussion and
example of levered and unlevered betas is given on pp. 716 .
12.
The methods of estimating the growth rate are given as follows:
a. The simple growth rate between two successive periods is estimated as
follows:
g
EPSt  EPSt 1
EPSt 1
where EPSt = earnings per share at time t.
b. The arithmetic growth rate estimate is just the arithmetic average of
individual growth rates calculated for short periods according to the simple
growth rate formulation defined above. Specifically,
g
g1  g 2 
n
 gn
c. The compound growth rate is estimated as g in the following equation:
(1  g )n 
EPSn
EPS0
d. The fourth growth rate estimation procedure is obtained using ordinary least
squares.
loge(EPSt) = a0 + a1t + εt,
The time indicator is t. Instead of EPS, DPS can be used in the equation.
13.
The Chase Manhattan Model defines the cost of capital without leverage as
follows:
C
Y  (1   c )( B S ) i
1  (1   c )( B S )
where C represents the cost of equity estimated using the company's equity beta
which has the use of financial leverage impounded in it. B/S is the company's debt
to equity ratio and TC is the company's corporate tax rate. See answer to problem
11-6 or the text for an application of the Chase Manhattan Model.
14.
K
D1
10
g
 .05  .15 or 15%
P
100
15.
a. after tax cost of debt = (1 – .0)(10%) = 10%
b. after tax cost of debt = (1 –.5)(10%) = 5%
c. after tax cost of debt = (1 – 1.0)(10%) = 0
16.
a. Chase's cost of capital:
Y can be estimated as follows:
Y = Rf + β[E(Rm) – Rf]
= 8.5 + 0.46[10.55 – 8.5] = 9.443%
b. Leveraged Cost of Equity = Y =
C
NOPAT  bB (1   c )
S
9 . 4 43 ( 1 . 4 ) ( 0 . 3 6 ) ( 1 0 )
1  ( 1 . 4 ) ( 0 . 3 6 )
 9.54%
Give that the target D/E = 0.4,
Target CE = 0.4/(1 + 0.4) = 0.2857
C *  C[1  (40%)(
0.4
)]  0.0954(88.57)  8.45%
1  0.4
βL = .46
U ( 1
U 
L
B
(1  )
S
B
)  L
S

.46
 0.338
(1  .36)
Since the target B/S has changed, the new  L* can be estimated as follows:
 L* = 0.338(1 + .4) = 0.473
then,
E(Ri) = .085 + 0.473(.1055 – .085) = 0.0947
Therefore, the new cost of equity capital with leverage is 9.47%.
17.
WACC = (.6)(12%) + (.02)(14%) + (1 – .4)(.38)(10%) = 9.76%
18.
a. cost of equity = k =
5
 .10  .15 or 15%
100
Cost of debt = .08(1 – .4) = 0.048 or 4.8%
b. The marginal cost of capital is given as follows:
WACC  (15%)
c. WACC 
19.
300, 000
200, 000
 (4.8%)
 10.92%
500, 000
500, 000
15% 300, 000
200, 000
(
)  (4.8%)
 11.92%
1  .10 500, 000
500, 000
P0 = 40; D0 = 150; g = 10%; f = 5%
a.
D0 (1  g )
g
P0
K RE 

1.50(1  .10)
 .10
40
 14.13%
b.
K NE 5 

D1
g
P0 (1  f )
1.65
 .10
38
 14.34%
20.
It
principal

t
t 1 (1  k )
(1  kB )n
B
n
P
80
1000

t
t 1 (1  k )
(1  kB )10
B
10
975  
kb = 8.38% (approximately)
After tax kB = 8.38% (1 – .40) = 5.03%
21.
kp 
Dp
P0

8
 8.89%
90
The cost of new preferred stock =
22.
a. KNE = Rf + (Rm – Rf)β
kp
1 f

8.89
 9.36
(1  .05)100
= 6% + (13% – 6%)(1.4) = 15.8 %
b. K NE 
kRE
15.8

 16.63%
1  f 1  .05
23.
a. Weights for Components:
600, 000
 30%
2, 000, 000
200, 000
Preferred Stock 
 10%
2, 000, 000
1, 200, 000
Common Equity 
 60%
2, 000, 000
Debt 
b. Break Points:
First Break: 15,000/.60 = $25,000 (attributable to equity)
Second Break: 45,000/.60 = $75,000 (attributable to equity)
c. Component Costs:
1) After tax cost of debt = 10%(1 – .46) = 5.40%
2) Preferred Stock =
Dp
P0 (1  f )
3) Retained Earnings =

10
 10.53%
100(1  .05)
D1
1
g
 .08  .13 or 13%
P0
20
4) New Common Stock
Up to $30,00 new common stock:
K NE 
D1
1
 g   .08  13.56%
P0 (1  f )
18
Over $30,000 new common stock:
K NE 
1
 .08  .1388 or 13.88%
17
d.
1) Common Equity Component = 30,000 (.60) = $18,000
New Common Stock = $18,000 – 15,000 = $3,000
2) Common Equity = 50,000 (.60) = $15,000
New Stock = $30,000 – 15,000 = $15,000
3) Common Equity = 100,000 (.60) = $60,000
New Stock = 60,000 – 15,000 = $45,000
e. Weighted Marginal Cost of Capital Schedule
Total Financing
0 – $25,000
Component
Weight
Cost
Debt
0.30
5.40%
1.62%
Preferred
Common
Total
0.10
0.60
1.00
10.53%
13.00%
1.05%
7.80%
10.47%
Debt
Preferred
Common
Total
0.30
0.10
0.60
5.40%
10.53%
13.56%
1.62%
1.05%
8.14%
10.81%
Debt
0.30
5.40%
1.62%
Preferred
Common
Total
0.10
0.60
10.53%
13.88%
1.05%
8.33%
11.00%
$25,001 – 75,000:
Over $75,000
f. Weighted Average Cost of Capital
25,000
50,000
25,000
$100,000
.25
.50
.25
1.00
24.
a. Cost of Debt:
Up to $200,000:
I
principal

t
t 1 (1  k )
(1  kB )n
B
n
P
10.47%
10.81%
11.00
2.62
5.41
275
10.78%
100
1000

t
t 1 (1  k )
(1  kB )10
B
10
960  
Before tax cost of debt = approximately 10.67%
After tax cost of debt = (10.67)(1 – .50) = 5.34%
Over $200,000:
100
1000

t
t 1 (1  k )
(1  kB )10
B
10
940  
Before tax cost of debt = approximately 11.02%
After tax cost of debt = 11.02% (1 – .50) = 5.51%
b. k p 
Dp
P0

7
 9.33%
75
c. kRE 
D1
2
g
 .07  .1609 or 16.09%
P0
22
d. kNC 
D1
2
g
 .07  .17 or 17%
P0 (1  f )
20
e. Weights of capital components:
Debt 
1, 200, 000
 .30 or 30%
4, 000, 000
Preferred 
Equity 
800, 000
 .20 or 20%
4, 000, 000
200, 000
 .50 or 50%
4, 000, 000
f. Break Points
First Break (attributable to equity) =
200, 000
 $400, 000
.50
Second Break (attributable to debt) =
200, 000
 $666, 667
.30
g. Common equity Component = 900,000 × .50 = $450,000
New Common Stock = $450,000 – 200,000 = $250,000
h. WMCC Schedule:
Up to $400,000
Component
Debt
Pref
Common
Total
$400,001 to 666,667
Debt
Pref
Common
Total
over $667,667
Debt
Pref
Equity
Total
Weight
(1)
0.30
Cost
(2)
5.34%
0.20
0.50
1.00
9.33%
16.09%
1.87%
8.05%
11.52%
0.30
0.20
0.50
1.00
5.34%
9.33
17.00
1.60%
1.87%
8 50%
11.97%
0.30
5.51%
1.65%
0.20
0.50
1.00
9.33%
17.00
1.87%
8.50%
12.02%
0.44
0.30
0.26
1.00
11.52%
11.97%
12.02%
5.07%
3.59%
3.13%
(1)×(2)
1.60%
i. WACC:
$400,000
266,667
233,333
$900,000
j.
11.79%
k. Projects A,B,C,D, and E should be accepted.