Capital Structure Decision-Making with
Growth: An Instructional Class Exercise
Professor Robert M. Hull
Clarence W. King Endowed Chair in Finance
School of Business
Washburn University
1700 SW College Avenue
Topeka, Kansas 66621
(Phone: 7853935630)
Email: rob.hull@washburn.edu
1
Overview
This paper offers an instructional class exercise of the capital
structure decision-making process including a hands-on application
of how to use four gain to leverage (GL ) equations including a recent
equation that is used when managing growth firms.
The latter equation is given by the recent Hull [2010, IMFI] Capital
Structure Model (CSM) with growth.
Given estimates for the costs of capital, tax rates and growth rates, this
equation illustrates how managers of growth firms go about choosing an
optimal debt level.
The exercise demonstrates the interdependency of the plowbackpayout and debt-equity decisions when maximizing firm value.
 By incorporating growth, this paper extends the non-growth
pedagogical exercise of Hull [2008, JFEd].
This growth extension has proven to be successful in helping advanced
business students understand the impact of the plowback and debt choices
on firm value.
2
• Capital structure perpetuity research begins with Modigliani and
Miller, MM, (1963) who derive a gain to leverage (GL) formulation in the
context of an unleveraged firm issuing risk-free debt to replace risky
equity. For MM, GL is the corporate tax rate multiplied by debt value. The
applicability of MM’s GL formulation is limited.
• Miller (1977) and Warner (1977) are among those who argue that debtrelated effects are weak and have no real impact on firm value.
• Altman (1984), Cutler and Summers (1988), Fischer, Heinkel and
Zechner (1989), and Kayhan and Titman (2006) provide contrary
evidence.
• Graham (2000) estimates that the corporate and personal tax benefit of
debt is as low as 4.3% of firm value. Korteweg (2009) finds that the net
benefit of leverage is typically 5.5% of firm value.
• Given the presence of debt in the capital structure of most firms as well
as the evidence concerning leverage-related wealth effects, there is a need
to offer usable equations that can quantify these effects. This paper aims to
fill this void by offering GL formulations quantifying these effects.
3
Definitions
• Gain to Leverage (GL) formulations are formulations that measure the change
in value caused by changing the amount of debt.
• Equity discount rate is the firm’s cost of borrowing for equity or the return
required by investors in equity (for most firms equity is just common equity).
The rate can be for an unleveraged firm (rU) or a leveraged firm (rL).
• Debt discount rate is the firm’s cost of borrowing for debt or the return required
by investors in debt (rD). For most firms debt is long-term debt such as bonds or
short-term debt that is renewed indefinitely.
• Plowback-payout choice determines the unleveraged growth rate (gU). This
choice along with the leverage choice decides the leveraged growth rate (gL).
• Optimal debt-equity choice (ODE) is the optimal amount of debt relative to the
amount equity at which the firm and its manager strive to obtain so as to
maximize firm value.
• Perpetuity with growth involves a perpetual cash, a discount rate, and a growth
rate. Any series of uneven cash flows can be approximated by a perpetuity with
growth.
• Growth-adjusted discount rate refers to the discount rate minus the growth rate.
For an unleveraged firm the equity growth-adjusted rate is rUg = rU  gU. For a
leveraged firm it is rLg = rL  gL.
4
• MM (1958): Gain to Leverage (GL) = 0.
Value determined solely by operating assets.
• MM (1963): GL = TCD
where TC is the applicable corporate tax rate and D = I / rD where I
is the perpetual interest payment and rD is the cost of debt.
• Miller (1977): GL = (1α)D where
α = (1  TE)(1  TC) / (1  TD) with TE and TD the personal tax rates
applicable to income from equity and debt, D now equals
(1  TD)I / rD, and (1α) < TC is expected to hold.
• Hull (2007): GL = [1  (αrD / rL)]D  [1  (rU / rL)]EU.
CSM non-growth equation extends MM and Miller by
incorporating equity discount rates (e.g., the unleveraged equity
rate of rU and the leveraged equity rate of rL).
5
• Hull (2007): GL = [1(αrD / rL)]D  [1  (rU / rL)]EU.
CSM non-growth equation extends MM and Miller by
incorporating equity discount rates (e.g., the unleveraged
equity rate of rU and the leveraged equity rate of rL).
• Hull (2010): GL = [1  (αrD / rLg)]D  [1  (rUg / rLg)]EU.
CSM growth equation extends CSM non-growth
equation in incorporate growth-adjusted discount rates:
rUg = rU – gU and rLg = rL – gL.
6
Features of Growth Model Used In Exercise
1) The Capital Structure Model (CSM) with growth (2010,
IMFI) enables this paper to broaden the non-growth
pedagogical application of Hull (2008, JFEd).
2) The model recognizes (in its application) the notion that
the plowback ratio decides the minimum unleveraged
growth rate (gU) and that this leads to choosing a
minimum plowback ratio (PBR).
3) The model uses a break-through concept: the leveraged
growth rate for equity (gL), which depends on both the
plowback-payout decision and the debt-equity decision,
thus showing how these decisions are intertwined.
4) The model can illustrate why firms with moderate
growth can have larger optimal debt-equity (ODE)
ratios, while firms with more pronounced growth must
have lower ODEs.
7
Why Internal Equity Is More Costly than
External Equity
Double Taxation: Because corporate taxes are paid before
internal equity or retained earnings (RE) can be used for
growth purposes, a firm actually has (1  TC)RE available
to reinvest for these purposes. This gives a double taxation
situation because the cash flow generated from retained
earnings is taxed again at the corporate level before being
paid out to owners.
How does this double taxation affect the before-tax
plowback ratio (PBR)? In terms of cash earnings
available before taxes for distribution or plowback, we
have: (1) cash that is retained (RE) and (2) cash that is
paid out (C). We have (1  TC)RE useable for reinvestment
after we adjust for double taxation.
8
Minimum Unleveraged Growth Rate
What is the minimum unleveraged growth rate (gU ) that an unleveraged firm must
attain so that unleveraged equity value (EU ) will not fall when the firm chooses
to reinvest its retained earnings?
 This can be shown to depend on the plowback ratio (PBR). For example,
consider the value of an unleveraged firm with no growth (EU):
EU (no growth) = (1TE)(1TC)(1 – PBR)CFBT / rU = (1TE)(1TC)CFBT / rU
where TE is the effective personal tax rate paid by equity owners, TC is effective
corporate tax rate, PBR is the before-tax plowback ratio, CFBT the before-tax
cash flow paid out to equity owners, and rU is unleveraged equity cost of
borrowing.
 For non-growth PBR = 0 but with growth PBR > 0. Thus the numerator becomes
multiplies by ( 1 – PBR). This means that the discount rate of rU must be lowered
by at least (1 – PBR) if EU is not to decrease when it chooses growth. For this
lowered discount rate, we have: (1 – PBR)(rU) = rU – (PBR)rU where the
minimum unleveraged growth rate (gU) must equal (PBR)r U making the making
the growth-adjusted denominator equal to rU – gU . With gU = (PBR)rU , the two
EU values are equal. For example, = EU (no growth) = EU (growth) implies
(1 – TE)(1 – TC) CFBT / rU = (1 – TE)(1 – TC) (1 – PBR)CFBT / rU – gU.
9
Equilibrating Growth Rates
Equilibrating gU =
rU(1TC)RE/C
Equilibrating gL =
rL(1TC)RE / [C+G  I/(1TC)]
In comparing to equilibrating gL and
gU formulas, we see that the
equilibrating gL > equilibrating gU .
10
Question 1: Computing MM and Miller Values
• Unlevgrowth Inc. (UGI) is an unlevered growth firm. UGI’s
managers believe it can increase its equity value by retiring a
proportion of its outstanding equity through a new debt issue.
UGI will treat its new debt as perpetual since it plans to
continuously roll it over whenever it reaches maturity. Besides
increasing its value through the use of debt, UGI is also
considering expansion through its technological innovation.
The expansion will involve a new line of marketable products
for which future patents will assure constant long-term growth
in cash payable to equity owners. UGI will make no choice
about growth until it makes its leverage decision.
• After considerable discussion, UGI’s managers decide to issue
debt and retire one-half of its outstanding equity shares. UGI’s
managers estimate values for key variables needed when using
the MM and Miller GL equations. These values are given in
Table 1… Fill in the blank cells in Exhibit 1.
11
Table 1. MM and Miller Values
[Note. When different, the MM and Miller values are denoted in subscripts.]
TEMM = personal tax rate on equity income = 0%
TEMiller = 5.00%
TDMM = personal tax rate on debt income = 0%
TDMiller = 15.00%
TC = corporate tax rate = 30.00%
αMiller = α = (1  TE)(1  TC) / (1  TD)
GLMM = TC(DMM)
GLMiller = [1−αMiller]DMiller
rU = cost of capital for unlevered equity = 11.00%
rF = risk-free rate = 5.00%
I = Interest = rD(D) where I = 0 for an unlevered firm because D = 0
CFBT = perpetual before-tax cash flow generated by operating assets =
$1,654,135,338.34
PBR = plowback ratio used on CFBT (PBR = 0 with no growth)
POR = payout ratio = 1 – PBR
RE = before-tax retained earnings = PBR(CFBT) with RE = $0 for no growth because
PBR = 0
C = before-tax cash to equity = POR(CFBT) with C = $1,654,135,338.34 for no growth
because POR = 1
12
Question 2. Computing CSM Values without Growth
•
UGI is not satisfied with the results from MM and
Miller models and so it decides to turn to Capital
Structure Model (CSM) without growth. This CSM
equation is: GL = ; Before using the CSM, UGI
estimates the costs of capital (rD and rL) for each debt
level choice. The values for rD and rL are in Exhibit 2.
The CSM non-growth value for VU and D in Exhibit 2
are the same as Miller’s VU and debt values because
both consider personal and corporate taxes while
assuming no growth.
•
Fill in the blank cells in Exhibit 2. After you fill in
all cells, identify and comment on the debt choice for
UGI’s maximum GL, the largest increase in its firm
value (as given by the “%ΔV” row), and the optimal
D/VL. Finally, explain the significance of the
“Incremental ΔGL” and “Incremental %ΔV” rows and
what their first negative values indicate.
13
Table 2. CSM (without growth) Values
PBR = plowback ratio used on CFBT = 0.35
POR = payout ratio = 1 – 0.35 = 0.65
TC = corporate tax rate = 30.00%
TD = personal tax rate on debt income = 15.00%
α = 0.78235294118
rU = 11.00%%
CFBT = perpetual before-tax cash flow generated by operating assets = $1,654,135,338.34
RE = before-tax retained earnings = PBR(CFBT) = 0.35($1,654,135,338.34) =
$578,947,368.42
C = before-tax cash to equity = POR(CFBT) = 0.65($1,654,135,338.34) = $1,075,187,969.92
I = Interest = where D is DMiller and I must be computed for each D choice.
[Note. The CSM, like Miller, assumes personal taxes, and so we divide D by (1–TD) because
the below gL equation uses an I value before personal taxes are considered.]
G = perpetuity cash flow beyond I created with debt when GL ≠ 0. (Supplied for each D
choice.)
gU = unlevered equity growth rate = gU = rU(1TC)RE / C = 4.146153846%
gL = levered equity growth rate = gL = rL(1TC)RE / [ C + G  I / ( 1TC)]
(Computed for each D choice.)
rUg = growth-adjusted unlevered equity rate of return = rU  gU = 11%  4.146154% =
6.853846%
rLg = growth-adjusted levered equity rate of return = rL  gL (Computed for each D choice.)
VU (no growth) = (1 – TE)(1 – TC) CFBT / rU = $10,000,000,000
VU (growth) = (1 – TE)(1 – TC) (1 – PBR)CFBT / rUg = $10,432,098,765.43.
14
Question 3.
Computing Growth-Adjusted Costs of Borrowing
While UGI’s managers are satisfied with the valuation results
using the CSM model with no growth, they still want to
know if UGI can improve its value through a new line of
marketable products for which future patents can assure
constant long-term growth in cash payable to equity. To
determine if growth can add to its current value, UGI will
use the GL equation given by the CSM with growth. But
first it must make some needed computations concerning
the value of growth as well as its levered growth rates for
various debt choices. Using the values and equations in
Table 2, supply answers to the below questions…Fill in the
blank cells in Exhibit 3.
15
Question 4.
Computing CSM Values Using the CSM with Growth
Fill in the blank cells in Exhibit 4. For the rLg row, copy in
the values computed previously. After you fill in all
cells, identify and comment on the debt choice for
UGI’s maximum GL, the largest increase in its firm
value (as given by the “%ΔV” row), and the optimal
D/VL. Finally, explain the significance of the
“Incremental ΔGL” and “Incremental %ΔV” rows and
what their first negative values indicate. Comment on
how values for these two rows differ from Exhibit 2
when the CSM GL equation was used without growth.
Can you offer an explanation or two to explain the
difference?
16
Question 5. Computing and Comparing GL Values
In Questions 1, 2, and 4, you computed the GL values using the
MM, Miller and CSM equations for the nine after-tax
perpetual debt choices. From your answers for these three
questions, fill in Exhibit 5 expressing all values in billions of
dollars to four decimals.
After filling in Exhibit 5 and studying how the GL values
change as the proportion changes, answer the below questions.
(a) The GL answers in Exhibit 5 use the MM, Miller and CSM
equations. In comparing these answers, which (if any) of the
three equations is consistent with trade-off theory? Explain.
…..
(e) Which equation would you feel more comfortable with if
you were a UGI manager charged with the capital structure
decision? Explain.
17
Exhibit 5. Comparison of Values Given By Four GL Equations
[Note. All values are expressed in billions of dollars to four decimals.]
P = Debt Choice (proportion of unlevered equity retired by debt)
GL Model
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
MM's GL
0.3158
0.6316
0.9474
1.2632
1.5789
1.8947
2.2105
2.5263
2.8421
Miller's GL
0.2176
0.4353
0.6529
0.8706
1.0882
1.3059
1.5235
1.7412
1.9588
CSM's GL
0.5361
0.9531
1.1804
1.2929
1.3331
1.2829
1.2066
1.1276
1.0400
CSM's GL
0.5326
1.0114
1.4110
1.8429
2.5356 2.6564 2.1150 1.6176 1.1985
18
0.20
0.15
0.10
0.05
0.00
0.00
← DC
0.10
0.20
0.30
0.40
0.50
0.60
0.70
-0.05
-0.10
-0.15
-0.20
19
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0.20
← PBR
0.26
0.30
0.33
0.36
0.39
0.40
0.42
20
Relation between Optimal Payback and Leverage Choices: Fix TC = 0.10; gU = 4.45%; gL = 7.54%
PBR: DE
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.10: 0.402
$406,133,726
$584,060,566
$595,066,819
$490,277,818
$322,277,754
-$75,930,001
0.20: 0.401
$406,204,799
$593,671,035
$644,796,555
$608,180,772
$545,212,133
$383,617,482 -$2,981,936,944
0.22: 0.615
$414,538,198
$612,697,475
$683,739,702
$678,297,490
$666,331,387
$630,148,421 -$3,165,641,262
0.23: 1.307
$420,444,325
$625,885,851
$709,609,103
$724,029,509
$745,686,775
$798,467,867 -$3,267,207,714
0.25: 0.878
$436,681,828
$661,749,409
$778,133,891
$844,232,463
$957,565,742 -$3,877,540,403 -$3,493,225,315
0.30: 0.573
$517,571,211
$840,502,919 $1,113,353,404 $1,444,840,597 -$5,179,813,748 -$4,603,741,882 -$4,238,732,092
0.31: 0.562
$544,614,239
$901,053,893 $1,228,041,530 $1,659,219,239 -$5,372,799,978 -$4,794,845,068 -$4,430,248,371
0.32: 0.380
$577,145,549
$974,526,842 $1,368,699,081 -$6,309,293,696 -$5,587,401,266 -$5,006,406,071 -$4,640,736,191
0.35: 0.106
$721,927,950 -$8,830,311,051 -$7,969,604,746 -$7,158,211,444 -$6,398,231,232 -$5,798,700,437 -$5,418,646,822
0.355: 0.00
-$1,533,815,988 -$9,024,397,854 -$8,152,700,907 -$7,331,968,915 -$6,564,211,949 -$5,959,794,343 -$5,575,324,257
-$516,358,312
21
4000000000
2000000000
0
PBR: DE 0.10: 0.402 0.20: 0.401 0.22: 0.615 0.23: 1.307 0.25: 0.878 0.30: 0.573 0.31: 0.562 0.32: 0.380 0.35: 0.106 0.355: 0.00
Series1
-2000000000
Series2
Series3
Series4
Series5
-4000000000
Series6
Series7
-6000000000
-8000000000
-10000000000
22
Relation between Optimal Payback and Leverage Choices: Fix TC = 0.30; gU = 4.92%; gL = 8.35
PBR: DE
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.000: 0.3873
$434,632,560
$669,654,615
$745,744,788
$710,471,209
$609,654,045
$288,670,869
-$173,109,185
0.200: 0.5877
$429,042,179
$669,610,812
$775,645,491
$781,888,862
$725,918,778
$488,193,881
$223,564,278
0.300: 1.1343
$483,672,218
$783,812,446
$973,627,922 $1,090,987,281 $1,193,367,665 $1,289,652,011 -$2,331,871,756
0.350: 0.7283
$557,393,968
$939,614,274 $1,243,709,710 $1,531,937,010 $1,945,446,020 -$3,093,139,569 -$2,707,461,618
0.380: 0.5061
$635,524,783 $1,109,375,208 $1,548,349,336 $2,066,314,473 -$3,913,930,913 -$3,416,091,195 -$3,031,359,376
0.390: 0.4951
$670,907,102 $1,187,833,426 $1,693,225,511 $2,296,612,998 -$4,052,022,144 -$3,549,308,769 -$3,163,204,261
0.400: 0.3463
$712,724,243 $1,281,807,758 $1,870,256,428 -$4,791,952,920 -$4,207,378,575 -$3,698,760,134 -$3,310,175,742
0.410: 0.2170
$762,518,798 $1,395,353,990 -$5,501,499,522 -$4,979,330,361 -$4,382,991,184 -$3,867,218,922 -$3,474,837,977
0.4185: 0.1068
$425,397,181 -$6,213,307,287 -$5,787,663,077 -$5,158,441,671 -$4,550,929,429 -$4,027,899,478 -$3,631,064,808
0.419: 0.0000
-$41,516,035 -$6,357,884,696 -$5,799,645,666 -$5,169,615,196 -$4,561,403,704 -$4,037,908,769 -$3,640,772,740
23
3000000000
2000000000
1000000000
0
PBR: DE
-1000000000
0.000:
0.3873
0.200:
0.5877
0.300:
1.1343
0.350:
0.7283
0.380:
0.5061
0.390:
0.4951
0.400:
0.3463
0.410:
0.2170
0.4185:
0.1068
0.419:
0.0000
Series1
Series2
Series3
-2000000000
Series4
Series5
Series6
-3000000000
Series7
-4000000000
-5000000000
-6000000000
-7000000000
24
Final Remarks
•
Student feedback when using the CSM exercises has been
positive over the years from both upper level undergraduate
finance students and graduate students. The below quote
typifies students’ feelings on the exercise:
• “The CSM with Growth model is most complete because it
provisions for more scenarios which is keeping in line with
diversity of situations faced by managers in trying to determine
the optimum leverage. CSM with growth gives due
consideration to the growth that can be brought about if the
company keeps aside some of its earning to fuel expansion.”
• The approval from students concerning the CSM applications
has been received not only for courses taught in the classroom
but also online.
25
Celebrate
It’s Over!
Relax
26