A Capital Structure
Model with Growth
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 of Paper
1) Broadens perpetuity gain to leverage (GL) research by incorporating and
analyzing growth within the capital structure model (CSM) formalized by
Hull (2007).
2) Argues that, contrary to pecking order theory (POT), internal equity is more
expensive than external equity.
3) Discusses how the plowback ratio decides the minimum unleveraged growth
rate (gU) and how leads to choosing a target leverage.
4) Develops a break-through concept: the leveraged growth rate for equity (gL).
5) Establishes that gL depends on both the plowback-payout decision and the
target leverage decision, thus showing how these decisions are intertwined.
6) Derives a growth-adjusted CSM equation and analyzes the role of the growthadjusted discount rate within this equation.
7) Examines the growth-adjusted CSM equation to show how a change in the
debt-to-equity mix influences firm value through its impact on gL.
8) Illustrates why firms with growth often have lower optimal debt-equity
(ODE) ratios.
9) Suggests why paying a high coupon rate diminishes a positive agency shield
effect.
10) Yields a stunning (yet simple) finding: the starting point for an ODE is the
cost of leveraged equity adjusted for growth (rLg) divided the cost of debt (rD).
11) Brings us closer to developing a full CSM model by setting the stage for
further enhancements within the CSM framework.
2
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).
• Target debt-equity choice (Target DE) is the amount of debt relative to the
amount equity at which the firm and its manager strive to obtain (the target
can be viewed as involving the amount of debt that maximizes firm value).
Optimal debt-equity ratio referred to as ODE.
• 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. We refer to the growth rate for an unleveraged firm as gU and for
a leveraged firm as gL.
• 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.
3
• There are thousands of firms per year that on average report no long-term debt
(including capitalized lease obligations). The starting point of this paper is an
unleveraged firm seeking to increase its value by issuing debt.
• 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 debt-related
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.
• This paper extends the Capital Structure Model (CSM) of Hull (2007) by
incorporating growth and in the process shows the interrelationship of the plowbackpayout and leverage decisions within a perpetuity GL equation with growth.
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.
Hull’s CSM equation is the only equation with equity discount
rates (e.g., the unleveraged equity rate of rU and the leveraged
equity rate of rL).
5
I. Why Internal Equity Is More Costly than External Equity:
The Pecking Order Theory Debunked
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 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. This means that the real PBR is below
what is conventionally stated and believed to be the PBR. We call this real PBR by the
name of the adjusted PBR.
Let us illustrate the adjusted PBR.
Suppose the following: (i) cash earnings available before taxes for distribution or
plowback (C + RE) is $1B (B = billions); (ii) retained earnings before double taxation
consideration (RE) is $0.3B; and, (iii) the effective corporate tax rate (TC) is 0.2. What
are the PBR and the adjusted PBR?
We have: PBR = RE / (C + RE) = $0.3B / $1.0B= 0.30.
Adjusting for double taxation, we have: (1  TC)RE = (1 – 0.2)$0.3B = $0.24B. Thus,
the adjusted PBR = $0.24B / $1.0B = 0.24 or simply (1 – TC)PBR = (1 – 0.2)3 = 0.8(3) =
0.24. (We could further adjust the denominator and get an adjusted PBR of 0.256.)
6
II. Why Internal Equity Is More Costly than External Equity:
The Pecking Order Theory Debunked
Considering taxes, flotation fees, and momentarily ignoring asymmetric
information effects associated with external equity, the cost to equity owners to
raise funds for growth can be represented as a negative cash outflow in one of two
ways:
• cost from using internal equity (e.g., retained earnings) = (–TC )(Gross Funds Raised)
• cost from using external equity = (–F)(Gross Funds Raised)
where F is the flotation fees as a portion of Gross Funds Raised with the latter
referring to funds raised before taxes and flotation fees are considered. The
expression representing the cost from using internal equity would be much more
expensive than the cost from using external equity because TC > F should hold
since TC should be close to five times greater than F. This is based on reported
estimates of 5.5% for F and 26% for TC . In light of double taxation for internal
equity, POT’s prediction that internal equity is cheaper does not hold.
NOTE. For seasoned offerings, Hull and Kerchner (1996) reported average cash costs of about
5.5% with smaller firms having greater costs (near the 7.0% average cash costs commonly found
for IPOs). The effective corporate tax rate was given as 25% for 2002-2006 according to Tax
Notes, January 22, 2007. It is given as 27% by the Treasury Department, July 23, 2007. The
average corporate tax rate of 26% suggested by these two sources is less than the 39% combined
statutory federal tax rate and average state tax rate. Reasons as to why the effective rate is below
the statutory rate include accelerated depreciation, tax deduction from employee stock option
profits, tax credits, and offshore tax sheltering.
7
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)C / rU
where TE is the effective personal tax rate paid by equity owners, TC is effective
corporate tax rate, and C the before-tax cash flow paid out to equity owners.
 One minus the plowback ratio, (1 – PBR), which is the payout ratio (POR),
determines the after-tax cash paid out over time with growth such that the
numerator of (1 – TE)(1 – TC)C is replaced by (1 – TE)(1 – TC)C(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:
EU (growth) = EU (no growth) or
(1 – TE)(1 – TC)C(1 – PBR) / rU – gU = (1 – TE)(1 – TC)C / rU .
8
Unleveraged Growth Rate
We can express unleveraged growth rate (gU) as:
gU = RU / C
where RU is the change in C (or ΔC) with RU = rU (1  TC)RE. Inserting this value for RU
into the above gU equation, we get:
gU = rU (1  TC)RE / C
where RE is the retained earnings determined by the plowback ratio (PBR).
Recalling that the minimum unleveraged growth rate (gU) must equal (PBR)rU and
rearranging gU = RU / C so that RU = gU (C), we can insert gU = (PBR)rU into RU =
gU (C) to get RU = rU (PBR)C and use this equation along with RU = rU (1  TC)RE to
show that TC = PBR. (Proof in paper.)
Thus, if RU = rU(PBR)C and RU = rU(1  TC )RE gives the same value for RU , then TC
equals PBR. Interpretation: TC determines the ideal starting point for setting PBR to
achieve minimum gU . [NOTE: We have found that GL is maximized at some threshold point
where PBR > TC holds; beyond this threshold GL can start falling dramatically. Even if the firm
can not achieve a PBR as high as TC or higher, it can still achieve a higher GL to the extent that it
can increase its PBR. If TC is low, then PBR must be greater than TC for firm maximization.]
Unlike the equilibrating gL described next, the value for the gU using gU =
rU (1  TC)RE / C is the same as that using gU = RU / C. Both gU values also change but
remain equal when we change a value for either TC or PBR.
9
I. Leveraged Growth Rate
 An equilibrating leveraged growth rate (gL) is derived based on
two definitions for RL where RL refers to the change in cash flow
paid to leveraged equity owners (C).
 The two definitions are: RL = gL[ C + G  I / ( 1TC) ] and RL =
rL(1TC)RE.
 From these two definitions, we get:
equilibrating gL = rL(1TC)RE / [ C + G  I / ( 1TC)].
 In comparing to the equilibrating gL to gU from the equation of
gU = rU(1TC)RE / C, we can see that equilibrating gL > gU
should hold.
 This because the equilibrating gL equation has a larger
numerator (e.g., rL > rU) and also has a smaller denominator for
most situations (e.g., I / ( 1TC ) > G should hold).
10
II. Leveraged Growth Rate
 The equilibrating gL equation of rL(1TC)RE / [ C + G  I / ( 1TC)] shows
the role of the plowback choice (RE) and the leverage choice (I) . Thus, the
concept of the leveraged growth rate ties in these two choices.
 G represents the perpetual cash flow from the gain to leverage. (See next
overhead for more information on G.)
 We divide I by (1TC) because, unlike C or G, I is not subject to corporate
taxes.
 The amount of debt issued must be reasonable such that the interest of I
would not set the equilibrating leveraged growth rate of gL at a large and
unsustainable rate for a long-term horizon (and possibly make the
denominator negative).
 We would expect that gL will become negative if I dominates (C + G) for
more extreme higher levels of debt. At some point, a negative gL will cause a
large positive rLg (even though in practice that point would never be desired).
 If GL becomes negative then G also becomes negative making gL increase
rapidly as I increases before things totally break down and gL becomes
negative.
11
 Assume GL > 0 and that it can be represented as a positive perpetual cash flow of G
discounted by the same rate as EL = (1TE)(1TC)(C + I) / rLg. If so, we have:
GL = (1TE)(1TC)G / rLg.
 Rearranging to solve for G , we have: G = rLg(GL) / (1TE)(1TC).
 As found in the paper, we have two definitions for leveraged equity (EL) that are equal
but which have different cash flows and discount rates. We have:
(1) EL = (1TE)(1TC)(C + I) / rLg, and
(2) EL = (1TE)(1TC)(C + I + G ) / rLg’.
 By adding the perpetual cash flow of G in the leveraged equity’s cash flows in (2), the
discount rate of rLg must increase to rLg’ for EL to remain unchanged. From this we can
generate a G value equal to G but with a definition that includes some different
variables:
G = [rLg’ (EL) / (1TE)(1TC)]  C + I
where G = G and rLg’ > rLg.
 When breaking down GL so that G = G , we find two rLg values (rLg and rLg’ ) result to
give the same GL value; thus, an rLg value depends on how we define EL. The
understanding of G is crucial because to compute gL we must first compute G. When
working in Excel, one must use the iterative command to achieve equality for G and G .
12
I. Interdependence of Plowback-Payout
Decision and Target Leverage Decision
• For an unleveraged firm financing strictly with internal funds to
achieve a specified level of expansion, the plowback-payout decision is
inseparable from determining the amounts of RE and C where these
two amounts in turn establish RU and gU.
• Thus, C, RE, RU and gU are determined endogenously (subject to finite
operating cash flows) when an unleveraged firm chooses its plowbackpayout ratios.
• The plowback decision or payout decision (or both because one implies
the other) can drive gU . But this was for an unleveraged situation. What
if managers of an unleveraged firm decide it can maximize its value by
becoming leveraged because GL > 0 holds for at least one debt level
choice?
• The answer is that we must now consider how the change from gU to gL
intrinsically leads to maximizing firm value in such a way that the
plowback-payout decision would be determined based on
consideration of leverage choices.
13
II. Interdependence of Plowback-Payout
Decision and Target Leverage Decision
•
•
•
•
•
A firm’s plowback-payout and leverage decisions are both intertwined, and even
inseparable, and any GL model dependent on the usage of a target leveraged growth
rate reveals that firm maximization depends on recognizing both decisions in tandem.
When undergoing a debt-for-equity exchange, how would a manager of an unleveraged
firm go about determining an optimal firm value if the plowback payout choice is
indeed inseparable from on optimal leverage choice?
A manager would begin by considering a spectrum of possible sets of choices for C
and RE. Each C and RE set would then be combined with a range of feasible interest
payment (I) choices to compute firm values.
Suppose a manager finds there is one set of C and RE values that combines with one I
value that renders a maximum firm value. If this be case, we can say that C, RE, and I
all simultaneously determine the maximum firm value and the plowback-payout and
leverage decisions are inseparable in this maximization process.
If a firm has ten debt choices and ten plowback-payout choices, then 100 different
combinations would have to be analyzed. With an Excel spreadsheet this task is not
hard once the proper variables are inputted. Furthermore, through the analysis of the
100 different combinations, one could see which combinations give the greatest firm
value. One could then further refine the choices so that the correct leverage and
plowback-payout choices are made to achieve any desired precision. Given TC , the
process could be considerably reduced because (as suggested previously) TC gives the
starting point for the plowback-payout choice.
14
 Starting Point: GL = VL  VU where
VL is leveraged firm value and VU is unleveraged
firm value.
 VU = (1TE)(1TC)C / rUg where C is the unleveraged
equity before-tax cash flow with rUg > rD and rUg =
rU  g U.
 VL = EL + D where EL = (1TE)(1TC)(CI) / rLg with
rLg = rL  gL and D = I / rD.
 VL’ = EL’ + D where EL’ = (1TE)(1TC)(CI+G) / rLg’
with rLg’ > rLg.
15
 In Appendix 1 we show that:
GL = [1  (arD / rLg )]D + [1  (rUg / rLg )]EU. (13)
 Equation (13) is a more general equation reducing to:
(i) Hull’s non-growth GL equation when growth is zero;
(ii) Miller’s GL equation when growth is zero and discount
rates are equal;
(iii) MM’s GL equation when growth is zero, discount rates
are equal, personal taxes are zero, and debt is risk-free.
 Note that the 1st component of (13) typically remains positive
and the 2nd component typically remains negative until things
break down with large positive (and unsustainable) leveraged
growth rates (gL) that then becomes negative due to a large I
values. For example, a negative GL causes G to become
negative, which in turn causes gL to become negative. The
latter can be seen from the equation of equilibrating gL =
rL(1TC)RE / [ C + G  I / ( 1TC)].
16
• Assume two unleveraged firms, A and B, where B is younger but more risky with
greater growth opportunities. A and B have respective rU values of 0.08 and 0.10, and gU
values of 0.01 and 0.05. Thus, the growth-adjusted unleveraged costs of equity for A is rUg
= rU  gU = 0.08  0.01 = 0.07 and for B is rUg = rU  gU = 0.10  0.05 = 0.05.
• Now assume both achieve identical debt-for-equity exchanges retiring 30% of their
outstanding equity shares. Suppose A and B estimate their respective leveraged equity
discount rates to be 0.10 and 0.13 and their leveraged growth rates to be 0.03 and 0.08
after their debt-for-equity exchanges. Thus, the growth-adjusted discount rate on
leveraged equity for A is rLg = rL  gL = 0.10 – 0.03 = 0.07 and for B is rLg = rL  gL =
0.13 – 0.08 = 0.05.
• Suppose that A and B have respective costs of debt (rD) to be 0.05 and 0.07 when
issuing enough debt to retire 30% of their equity shares and a is 0.8 for both. Thus, for A,
we have arD = 0.8(0.05) = 0.04 and, for B, we have: arD = 0.8(0.07) = 0.056.
• We see that the 2nd component of (13), GL = [1  (arD / rLg )]D + [1  (rUg / rLg )]EU, will
be zero since rUg = rLg for both A and B, while the 1st component will be positive for A:
[1  (arD / rLg )]D = [1  (0.04 / 0.056)]D = [1 – 0.571429]D = +0.428571D;
but, it will negative for B:
[1  (arD / rLg )]D = [1  (0.056 / 0.500)]D = [1 – 1.12000]D = –0.120000D.
• Thus, the 1st component gives a 0.428571D – (–0.120000D) = 0.548571D advantage to
A if 30% of equity is retired.
• Using the equation (13) the above numbers indicate that A could increase its value by
exchanging 30% debt for equity, while B would lower its value if it did the same.
17
• Consider GL = n1D + n2EU where n1 = [1  (arD / rLg )] and n2 =
[1  (rUg / rLg )].
• Together, n1 and n2 emphasize that GL is a function of tax rates, growth rates,
and discount rates for debt and equity.
• Such factors might include investor clientele tax rates, current tax legislation,
nondebt tax shields, employee stock options, alternative minimum tax, tax
credits, industrial factors, expected growth in GNP, riskless rate, betas, expected
market return, outstanding debt, need for financial slack, financial risk, business
risk, free cash flows, managerial autonomy and inside ownership level.
• The dollar sizes of the values for equity and debt are more a function of a
firm’s age and industry. We can describe two divergent scenarios for these
security size factors.
1) For a younger and growing firm (say in the computer software industry), we
expect equity value to be greater than debt value, while for an older and mature
firm (say in the electric utility industry), we expect debt and equity values to be
more similar.
2) For an unleveraged firm issuing debt that is small relative to E U , the equation of
GL = n1D + n2EU suggests that│n1│must be sizeable compared to│n2│;
otherwise, GL may have little chance of being positive unless it increases its
amount of debt issued.
18
• The paper offers a qualitative argument that the gain to leverage (GL) is not a
minimum and the chosen debt level is not an endpoint. The argument is based on
(i) Rolle's Theorem where we have two endpoints where GL = 0 holds and (ii) the
consideration that firms that issue more debt would not do so unless there is an
expected gain. For a leveraged firm that chooses not to issue more debt, one
might assume the firm already believes that its attained debt level is its optimal
debt-equity level (ODE).
• The paper offers a test for downward concavity and the existence of a
maximum ODE for its CSM GL equation. We show that ODE = rUg / arD .
• When using ODE = rUg / arD as an estimate of a firm’s optimal leverage ratio,
one should note that this equation may not properly allow for (i) the impact of an
increasing rD that could reverse the positive agency effect in the 1st component or
(ii) a wealth transfer effect among security holders. Both of these agency effects
could severely limit the ODE from achieving a value as great as rUg / arD.
• The paper offers an illustration of ODE = rUg / arD using a practical example.
The expression for ODE works well for this example using real world data but it
is only a sample of one.
19
 In Appendix 2, we derive GL for a leveraged firm
undergoing an equity-for-debt transaction with personal
taxes and constant growth and show that GL can still be
expressed as two components with components reversed
from debt-for-equity transaction.
 For the first component we now have:
[1 (rUg / rLg )]EU.
 For the second component we now have:
[1  (arD / rLg )]D.
 Putting these together, we have:
GL = [1 (rUg / rLg )]EU  [1  (arD / rLg )]D.
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
Results for All Nine Debt Level Choices
for the Application
Table 1 gives gain to leverage (GL ) results for the unleveraged application
for AGL Co. for all nine debt level choice. The application assumes the
previously mentioned data including the betas needed to compute the
costs of capital. The below conditions are formally stated so as to include
values for key variables. From these values, we can determine values for
other variables all of which are needed to compute GL using (12).
(a) debt is risky with rD > rF = 5.6642% and rD positively related to debt
(b) tax rates are relevant with TE = 4.77%, TD = 20.34%, and TC = 30%
(c) perpetual before-tax cash flows: C = $905,200,000
(d) constant growth rate when target market approximated: gL = 5.4%
with dollar growth = RL = $14,834,558
(e) an unleveraged firm with risky equity faces a finite set of perpetual
debt-for-equity choices with rL > rU = 10.0907%.
25
Table 1
Application of Gain to Leverage Formulation for a Real World Firm
Assuming Risky Debt, Personal Taxes, and Constant Growth Rate
Panel A. On After Personal Tax Basis
with Currency in Billions of Australian Dollars
1st
Book
D/V
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
rD
0.060
0.064
0.067
0.071
0.074
0.080
0.085
0.090
0.096
rL
0.105
0.108
0.112
0.115
0.120
0.123
0.127
0.142
0.177
2nd
Component Component
0.238
0.448
0.625
0.755
0.817
0.656
0.135
-0.485
-1.945
-0.232
-0.417
-0.547
-0.611
-0.583
-0.334
0.344
0.821
2.106
Market
GL
0.006
0.032
0.078
0.144
0.234
0.322
0.480
0.336
0.161
D
0.661
1.323
1.984
2.645
3.307
3.968
4.629
5.291
5.952
EL
7.250
6.615
6.000
5.405
4.834
4.260
3.756
2.951
2.115
VL
7.912
7.938
7.984
8.050
8.140
8.228
8.386
8.246
8.067
D/E
0.091
0.200
0.331
0.489
0.684
0.932
1.232
1.793
26
2.814
Table 1.
Application of Gain to Leverage Formulation for a Real World Firm
Assuming Risky Debt, Personal Taxes, and Constant Dollar Grow
Panel B. On Before Personal Tax Basis
with Currency in Billions of Australian Dollars
1st
Book
D/V
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
rD
0.060
0.064
0.067
0.071
0.074
0.080
0.085
0.090
0.096
rL
0.105
0.108
0.112
0.116
0.120
0.123
0.127
0.142
0.177
2nd
Component Component
0.386
0.742
1.063
1.335
1.536
1.503
1.092
0.577
-0.821
-0.244
-0.437
-0.574
-0.641
-0.612
-0.351
0.362
0.862
2.211
Market
GL
0.142
0.305
0.489
0.694
0.925
1.152
1.454
1.439
1.391
D
0.830
1.660
2.491
3.321
4.151
4.981
5.881
6.642
7.472
EL
7.614
6.947
6.300
5.676
5.076
4.473
3.944
3.099
2.221
VL
8.444
8.607
8.791
8.996
9.227
9.454
9.756
9.741
9.693
D/E
0.109
0.239
0.395
0.585
0.818
1.114
1.473
2.143
3.364
27
•
•
•
•
•
Each panel has two bold-faced rows. The 1st bold-faced row is for the current
situation where book D/V = 0.5, while the 2nd bold-faced row is for book D/V
= 0.7, which is where GL is maximized for both panels. As seen in the last
column of Panel B, it is also the row which is nearest the market target D/E of
1.5. For this row, we get GL = $1.4537 billion on a before personal tax basis
(which is what the market sees). For this row, dividing EL by the number of
outstanding shares (NL), we get a share price ≈ $14.42. For example, with D/V
= 0.7 (or E/V = 0.3), we have NL = (E/V)(NU) = 0.3(912,000,000) =
273,600,000 shares giving the share price as:
PBefore Personal Tax = EL / NL = $3,944,340,023 / 273,600,000 shares = $14.4164
per share ≈ $14.42.
This is less than the average market price at the time of this writing, which
has averaged $13.83 for January 2005. Thus, $14.42 can be considered a
prediction of the future price (absent effects beyond those stemming from the
increased debt) if the market target is achieved. The prediction for the stock
price at the time we begin estimating values for our variables (February 2004)
can be computed for the 1st bold-faced row where NL = 456,000,000 shares.
We have:
PBefore Personal Tax = EL / NL = $5,075,687,792 / 456,000,000 shares = $11.1309
per share ≈ $11.13.
This price is consistent with both the average price of $11.06 for AGL Co. for
February 2004 and also for the average price of $11.29 for the year of the
2003 annual report (7/1/03 to 6/30/04).
28
Shortcomings of Application
We can point out four shortcomings of our application, which in general are
found in all models that rely on accurate estimates of values for given
variables. First, personal tax rates were not directly known. This problem was
ameliorated through use of an effective tax rate and analysis of before personal
tax values. Second, we had to unleverage our firm in an attempt to estimate the
number of shares outstanding if it had no debt. The estimate appears to be
workable given that our share estimates gave predictions for stock prices that
were quite consistent with given market prices. Third, we encountered
problems when approximating betas. For example, we had to interpolate from
endpoints and a midpoint to get reasonable βD’s for each debt level choice.
From there we could proceed to get βU and then obtain βL’s for the nine debt
level choices by using a standard formula. However, unless adjusted upward,
those βL computations for higher debt levels would suggest that firms aim for
extremely high leverage targets that we do not find in the real world. This
caused us to make intuitive assignments for several leveraged equity betas.
Future research needs to explore other ways of estimating betas and costs of
capital such as suggested by Fama and French (1997) and Lally (2004). Fourth,
the application had to estimate a constant dollar level of growth (RL) based
upon a chosen growth rate at the target debt-equity choice (which was gL=
5.4%). Using the iterative command in Excel, we were able to solve for RL by
first computing the interest paid.
29
Summary & Conclusion
•
•
•
•
•
This research derives GL formulations based on definitions for unleveraged and
leveraged firm values. Such formulations include discount rates for unleveraged
equity, leveraged equity, and debt. The inclusion of these rates makes it possible for
GL values to eventually decrease with increasing debt levels. Three GL formulations
for an unleveraged situation are offered to aid managers (when making the debt-equity
choice) and educators (when explaining the ramifications of this choice).
The application using analysts’ data for AGL Co. showed how managers can use the
GL formulation with growth to estimate how issuing debt changes firm value. While
this paper’s model (like any model) relies on accurate estimates of values for
variables, its optimal GL did conform to the suggested market target D/E of 1.5.
Prior research offers GL formulations difficult for practitioners. They tend to include
variables virtually immeasurable in themselves (e.g., bankruptcy and agency costs).
As such, financial managers are hard pressed to find utility in their application. To the
extent changes in discount rates are easier to estimate, this paper's GL formulations
offer more practical potential.
This paper’s practical application suggests a wealth maximizing D/E choice. The
choice depends not only on changes in discount rates but also tax and growth rates.
The application’s results are consistent with prior empirical and theoretical research in
regard to the belief that taxes, bankruptcy costs, and agency effects can determine a
firm’s optimal debt-equity choice.
The GL formulations found in this paper reaffirm, synthesize, and extend prior GL
formulations, while opening up a fresh vista from which to view the D/E choice faced
by managers. This vista offers a practical vantage point in that capital structure
decision-making can be based on variables heretofore not fully utilized.
30
Celebrate
It’s Over!
Relax
31