“ Capital Structure Model ( CSM ) : correction , constraints , and applications

This paper extends the Capital Structure Model (CSM) research by performing the following tasks. First, a correction is offered on the corporate tax rate adjustment found in the break-through concept of the levered equity growth rate (g L ) given by Hull (2010). This correction is important because g L links the plowback-payout and debt-equity choices and so its accuracy is paramount. Second, this paper introduces a retained earnings (RE) constraint missing from the CSM growth research when a firm finances with internal equity. The RE constraint governs the plowback-payout and debt-equity choices through the interdependent relation between RE and interest payments (I). Third, a by-product of the RE constraint is a second constraint that governs a no-growth situation so that I does not exceed the available cash flows. Fourth, with the g L correction and two constraints in place, updated applications of prior research and new applications are provided. These applications reveal lower gain to leverage (G L ) values than previously reported with more symmetry around the optimal debt-toequity ratio (ODE) while minimizing steep drop-offs in firm value. For larger plowback ratios, the optimal debt level choice can change. The new constraints serve to point out the need for further research to incorporate external financing within the CSM framework. Robert M. Hull (USA) BUSINESS PERSPECTIVES LLC “СPС “Business Perspectives” Hryhorii Skovoroda lane, 10, Sumy, 40022, Ukraine www.businessperspectives.org Capital Structure Model (CSM): Correction, Constraints, and Applications Received on: 7th of December, 2017 Accepted on: 22 of January, 2018


INTRODUCTION
Since the publication of Capital Structure Model (CSM) with growth by Hull (2010), five papers on the CSM have been published.They include two theoretical extensions, two instructional papers, and one applied paper.The two theoretical extensions cover wealth transfers (Hull, 2012) and changes in tax rates (Hull, 2014b).The two instructional papers provide pedagogical exercises on growth (Hull, 2011) and wealth transfers (Hull, 2014a).The applied paper (Hull & Price, 2015) concerns pass-through enterprises where corporate tax rates are nonexistent.
At the root of Hull (2010) CSM growth research is the concept of the levered equity growth rate ( ).

L g
Prior to the development of the CSM, the finance world had no concept of L g and thus no variable to di- rectly link the plowback-payout and debt-equity choices as interdependent selections when applied to a perpetuity gain to leverage ( ) L G equation resulting from a debt-equity exchange.The concept of L g remains absent in the dividend valuation model (DVM) with growth.Unlike the Hull (2010) CSM, the DVM with growth offers a growth rate that does not distinguish between a firm having debt and not having debt.
In this paper, the primary goals are to correct the L g equation given by Hull (2010) where ( ) was misplaced, provide two missing constraints not found in the extant CSM research, and offer updated and new applications using the new L g and constraints.One constraint governs the use of retained earnings (RE) and tells us when RE cannot be maintained because of too much interest ( ) The RE constraint also embodies a constraint for a no growth situation that was missing in Hull (2007)  1 .Both constraints involve monitoring large debt issues that can lead to increasing I values that cause the firm to exhaust cash flows needed to satisfy the plowback-payout decision.When a constraint is violated before reaching an optimal debt-equity ratio ( ), ODE then it signals the firm needs external financing to attain its ODE.
Our applications highlight the differences between the old L g without constraints and our new L g with constraints.With the L g correction in place, we find lower L G values than previously reported and more symmetry around the ODE while minimizing steep drop-offs in firm value.Except for lower plowback ratios ( ), PBRs managerial decision-making in terms of choosing an optimal debt choice can be affected by the L g correction.Finally, it appears that maximum L G values are often achieved before a constraint sets in with an exception being when larger PBRs are used with larger debt levels.
The remainder of our paper is as follows.Section 1 provides a literature review of capital structure research.Section 2 corrects three equations related to L g and introduce the two new constraints.Section 3 presents an overview of major CSM equations used in our applications.Section 4 reports results from updated and new applications using the new L g and the two constraints.Section 5 provides a discus- sion of results.Final section gives conclusions and future research possibilities.
1 While on the subject of corrections, we would note an error in a footnote, page 7, in Hull (2007) where it was stated that an ODE could, under simplifying assumptions, be approximated by .UD ar r Even withstanding the advances introduced since that article, tests indicate this approximation overestimates ODE and so the derivational process behind this claim needs to be revisited.

LITERATURE REVIEW
In this section, we overview the MM and Miller tax models, trade-off theory (TOT) and pecking order theory (POT).

MM and Miller tax models
The perpetuity gain to leverage ( ) L G research originates with Modigliani and Miller (1963), referred to as MM.Given their simplifying assumptions of an unlevered situation, no growth, no personal taxes and riskless debt, MM contend that: where C T is the effective corporate tax rate and D is perpetual riskless debt.With no personal taxes, D is: where I is the perpetual interest payment and F r is the riskless cost of debt.Miller (1977) extends (1) by including personal taxes so that: ( ) where ( ) ( ) T and D T are the effective personal tax rates on equity and debt, respectively, and D now includes personal taxes and risky debt ( ) D r such that: ( )

TOT versus POT
The MM (1963) research stresses the benefits of debt.MM extensions focus on debt-related costs consisting of bankruptcy costs (Baxter, 1967;Kraus & Litzenberger, 1973) and agency costs (Jensen & Meckling, 1976;Jensen, 1986).This line of research, referred to as trade-off theory (TOT), emphasizes both the benefits and costs of leverage and argues for the existence of an optimal debt-equity ratio ( ) In contrast to TOT research, Miller (1977) and Warner (1977) argue that debt-related effects can be inconsequential such that no unique ODE occurs.However, sub- sequent researchers (Altman, 1984;Fischer et al., 1989;Kayhan & Titman, 2007) provide evidence that debt-related effects are important such that ODE exists.
The Capital Structure Model (CSM) research, overviewed in section 3, is consistent with TOT as its equations allow for both positive and negative effects from debt that lead to an ODE.As seen in Section 4, the CSM produces values consistent with TOT researchers (Graham, 2000;Korteweg, 2010;Van Binsbergen et al., 2010) who find maximum L G values that increase firm val- ue from 4% to 10%.
Pecking order theory (POT) provides the main challenge to TOT.Donaldson (1962), an early POT proponent, offers a pecking order where managers prefer internal equity financing for growth.If internal equity financing is lacking, he recommends asset conversion followed by debt issuance with external equity issuance the last resort.Myers and Majluf (1984) extend Donaldson emphasizing asymmetric information between managers and investors.Since investors lack information on the firm's prospects, they fear that managers will issue equity when overvalued and so will bid the price down if a new issue takes place.Consequently, prohibitive asymmetric information costs can result when using external equity.Besides asymmetric information, the financing resource ordering can stem from agency conflicts and taxes.

EXTENDING THE CSM WITH A L g CORRECTION AND NEW CONSTRAINTS
In this section, we provide background for the CSM unlevered equity growth rate ( ) U g and levered equity growth rate ( ).

L g
All L g related- formulations include our correction on how cash flows are adjusted for corporate taxes.We also introduce the no growth and RE constraints.Hull (2010) provides two definitions for the beforetax perpetual unlevered cash flow that results from growth (R U ).For the first definition, Hull has:

Double taxation on retained earnings
Hull notes that the cost to produce U R is the cor- porate taxes paid on RE before it can produce its own taxable income subject to corporate taxes.This double corporate tax when using internal financing is a fact researchers overlook when accounting for the cost of using internal equity.In response to this fact, Hull offers a second definition for U R given as ( ) ( ) The flotation expenses ( ) EF are a cheaper form of financing for Hull.For this form, ( ) ( ) where L r is the levered equity rate.For external financing, we have ( ) With debt, there are added funds available to equity, namely, the tax shield from the interest payment.If retained earnings gets the same percent given by the firm's PBR, then an incremental retained earnings from the tax shield (∆RE TS ) of PBR(T C )I could be added to (5b) to prevent underestimating R L .However, it can be argued that the cash flow effect from this tax shield is already captured by the gain to leverage and, in that sense, it is already accounted for in firm value as a residual cash flow for equity.As seen later, this cash flow is in the variable G when used to formulate the levered equity growth rate (g L ) equations given in (7) and (7a) where we imply that G contains the tax shield being used for RE.Thus, at least for now, we will ignore this possibility that R L is underestimated and reserve it for future exploration when a PBR, other than a before-tax PBR, is the focus.

Equity growth rates and critical points
Rearranging (5) gives the unlevered equity growth rate as: where C grows at U g for far-reaching periods.Since C increases each period, it stands to reason that the financing to support that increase would also be increasing in a similar manner such that U g could also be expressed as RE RE

∆
It should be noted that these increases are on a per share basis.Hull (2010)  as the sustainable growth rate, which can imply a maximum growth rate based on the company's .RE For example, Investopedia states that a company's sustainable growth rate is the product of its return on equity and the fraction of its profits that is plowed back into the firm (i.e., L r PBR ) 2 .It adds that this means a firm can safely grow at this rate using its own revenues to remain self-sustaining and can seek outside funding if it wants to accelerate its growth at higher rate.However, nothing from standard sources appears to mention that this sustainable growth rate must be greater than the cost of financing if firm value is to increase.The standard discussions also do not distinguish between unlevered and levered growth rates.This is akin to the Dividend Valuation Model with growth that also does not differentiate between unlevered and levered growth rates.

Correction on the break-through concept of the levered equity growth rate
We will now correct the levered equity growth rate ( ) L g given by Hull (2010).Equations (5c), ( 7) and (7a) are the equations affected by the correction 3 .
The simplest way to create L g is to adjust (6) where for a levered situation.In doing this, we begin by replacing U R with L R where the lat- ter was given in (5b).We next replace C with all levered cash flows not earmarked for .RE Besides , C we have the perpetual cash flow from the gain to leverage ( ) G that is defined on a before-tax basis as given later in (14).We not only know that C will be reduced by I but we also know that I has a corporate tax benefit that occurs before C is taxed.Thus, we multiply I by ( ) Given the above adjustments, we define the levered equity growth rate as: ( ) where the amount of debt issued must not be too high so that I in (7) causes L g to become large and unsustainable.In fact, if ( ) ⋅> + holds, then RE would have to fall, since debt own- ers have to be paid first.Thus, the break-through concept of L g indicates that a growth firm is lim- ited in its debt-equity choices and managers must exercise prudence and caution in choosing reasonable plowback and leverage choices if they are to avoid financial loss.
While perhaps not the last word, we think for now the new L g in ( 7) is a better representative than given by Hull (2010) where ( ) was misplaced in the denominator.Two points favor the new L g as given in (7).First, the tax deduction given by I occurs prior to the taxing of C as part of net income (similar for , G which is computed on a before-tax basis).Thus, it should arguably be accounted for in the manner now found in (7).Second, as will be shown later in our applications when using the new , L g we find results that are more symmetrical about ODE while minimizing steep drop-offs in value.We believe these results better represent what actually occurs.

Equilibrating unlevered and levered growth rates
Hull (2010) uses equations ( 5) and (5a) for U R to get what he calls an equilibrating unlevered equity growth rate ( ) , U equilibrating g which is the rate that balances the two formulations for .
U R Equating these two equations and solving for U g gives: ( ) where (6a) provides a U g value such that ( 5) and (5a) will have the same U R value.Similarly, Hull (2010) notes there are two equations involving L R that can be used to get what he calls an equilibrating levered equity growth rate

L equilibrating g
First, we have equation (5b) where ( ) ⋅ Second, we have the below equation that results from rearranging (7): ( ) Equating these equations and solving for our equilibrating , L g we get: where (7a) gives a L g value such that (5b) and (5c) give the same .L R

Constraints and maximum equilibrating L g
Assume an unlevered firm with PBR set to achieve its equilibrating L g at it ODE where L G is maxi- mized.A key question is as follows.With internal financing, is it possible that a firm cannot issue enough debt to achieve its ODE before the firm pays out so much I that RE cannot be main- tained?We answer this question by investigating (7a).With RE fixed by the chosen PBR to achieve the firm's desired growth, the denominator of ( ) If not, then RE would have to relinquish some of its funds to service debt.From (7a), we see that the following RE constraint must hold: ( ) If this constraint does not hold, we no longer have enough RE to maintain the chosen growth.To il- lustrate, assume an unlevered firm investing $30 of every $100 of its BT CF in .RE This leaves $70 for C. Suppose the firm is aggressive in its lever- age choice and chooses to retire 60% of its unlevered firm value ( ).

U E
Further assume 0 25, Inserting the given values into (8) and solving, we have $29 5 $30 .. ≥ Since $29 5 . is not greater than or equal to $30, our constraint does not hold and the firm has issued too much debt and cannot cover its RE require- ment.Such a firm would have to turn to external equity to achieve its desired growth.
If 0 RE = (no growth situation where 0 PBR = ), then (8) implies that the following no-growth constraint must hold: ( ) If this constraint does not hold, we no longer have enough cash flows to cover debt payments.The constraint given by (8a) would likely only be violated for very high debt levels where 0 G < occurs due to a negative L G.

CSM EQUATIONS NEEDED FOR APPLICATIONS
Before we offer updated and new applications using the new , L g it is necessary to overview the major CSM equations used in these applications.Thus, in this section, we briefly present those equations covering situations for no growth, growth, wealth transfers, and changes in tax rates.

No growth CSM
Keeping the MM and Miller unlevered and nogrowth conditions, Hull (2007) derives a CSM equation incorporating discount rates dependent on the leverage change.This equation is: where DE → indicates debt-for-equity exchange, the first component captures a positive tax-agency effect, and the second component represents financial distress costs (captured by increasing L r values as debt increases) such that this compo- nent's negativity can offset the positive first component as debt increases.The reverse of equation ( 9) can be derived if a levered firm becomes unlevered.Equity-for-debt equations can also be derived for the other CSM extensions involving growth, wealth transfers, and changes in tax rates.

Growth CSM
Hull (2010) extends ( 9) by incorporating growth.His growth CSM equation is: where Ug r and Lg r are the growth-adjusted discount rates on unlevered and levered equity, Ug U U r rg = − with U r and U g the borrow- ing and growth rates for unlevered equity, and with L r and L g the borrowing and growth rates for levered equity.

CSM for levered situation with a wealth transfer
Hull (2012) incorporates a levered situation within the CSM framework and derives L G equations showing how a wealth transfer (linked to a shift in risk between debt and equity) impacts firm value for incremental leverage changes.For incremental changes over time, we have to distinguish between values before and after the increment.Hull has "1" denotes less levered values and "2" signifying more levered value and is also used to refer to the new debt for debt-for-equity increments and the retired debt for equity-for-debt increments.
Debt-equity CSM equations for a levered situation focus on how the less levered cost of debt ( ) 1, where the "2" in where the last component of ( ) decreases with debt.

The G variable in L g equations
As residual owners, Hull (2010) argues that the perpetual before-tax cash flow from L G falls with- in the domain of the equity owners and thus is discounted at the same rate as .
BT CF This perpetuity cash flow is called G.
In terms of (10), L G can be expressed as: ( ) ( ) Solving for G in (13), we get: where G can be positive or negative depending on the value for .
L G G influences the cash flows available for payout as G belongs to the residual equity owners.As seen earlier, it influences .L g

Coefficients in CSM equations
Hull (2010) represents the CSM equations for L G in terms of positive and negative coefficients that multiply security factors.For example, Hull represents equation (10) as:

RESULTS AND DISCUSSION FROM THE CSMGL APPLICATIONS
This section gives updated and new applications.
Based on these applications, we report findings comparing the old L g versus the new L g and the RE constraint.We also report results using the no growth constraint.

CSM GL application with growth, no wealth transfer, and no tax change
Appendix A and Appendix B present applications that repeat those given by Hull (2010) using (10) except we use the new L g given in (7a) and the two new constraints.The four gray-shaded cells (with bold print) in the rows above the chart in Appendix A correspond to the maximum (max) L G as a fraction of unlevered equity ( ) nn − Fifth, compared to their peak , ODEs the old L g and new L g show lower ODEs for either lower PBRs or higher .PBRs Finally, the number of times the RE constraint is violated is given in the last row.As expected, for higher , PBRs there are more violations of the RE con- straint.For a PBR of 0.42, there are six violations among the nine DC choices.This compares to zero violations for a PBR of 0.05.
Appendix C provides three charts that updates the instructional CSM growth paper of Hull (2011) using (10) = for the old , L g as the old L g be- comes negative after this point.After 0 7, DC .= the free fall would begin for the new L g except for the fact that the RE constraint is violated stop- ping the trajectory.From the third chart, we see the debt-to-firm value ratio DV increases for the old L g and new L g.For the old , L g there is a ma- jor rise of about 100% from 0.4 to 0.81 when the DC goes from 0.5 to 0.6.ODEs using the old L g and new L g are similar at 0.4 and 0.46, respective- ly, reflecting the fact both have the same optimal DC of 0.5.

CSM L G application with growth, a wealth transfer, and no tax change
We now repeat the applications found in Hull (2012) using the CSM with wealth transfers and the new L g equation.

Asset substitution problem
The application for the asset substitution problem involves the claim by Leland (1998) that a tax shield effect from debt is greater than an agency costs of debt related to asset substitution.To examine this claim, we compare the tax shield component of (11a), Due to space constraints, we do not report all details with numbers, but only the most important outcomes.When going from the 20% to 40% debt levels like Hull, we find that the Leland claim holds.This is true even if we adjust for a wealth transfer due to risk shift from debt to equity.Using the Hull optimal debt level of 50% and extrapolating to get 25% debt level values, we find that the Leland claim still holds.This latter application using the new L g differs from Hull (2012) where the Leland claim did not hold when using the old L g. Disregarding the fact 21 DD > (which can heavily favor rejection of Leland), if we go from the 10% to 45% debt level and adjust for risk, we discover the Leland claim does not hold.Thus, even ignoring the fact 21 , DD > we see the possibility of still rejecting the Leland notion as a firm attempt to reach its .ODE In conclusion, while the results us- ing the new L g is more likely to favor the Leland claim, we can see there are still scenarios where the Leland claim would not hold.

Underinvestment problem
In regards to the underinvestment notion, Myers (1977) suggests that equity would not want to plow RE into lower risk projects that favor debt.Similarly, equity would not want to approve an equity-for-debt transaction if the new equity favors remaining debt owners by making their cash flows safer at equity's expense.For both cases, the decision to increase equity would not be desired by equity owners if debt profited at their expense.
To examine the underinvestment problem, we use the equity-for-debt equation of Hull (2012) given as: In revisiting the underinvestment problem of Hull (2012) using his numbers but with the new , L g we find that an optimal debt-to-firm value ( ) ODV of 0.46 is achieved with an optimal DC of 0.5.For the old , L g an ODV of 0.40 is attained with same optimal DC of 0.5.Assuming the fall in debt's dis- count rate is from a debt level of 60% to 50%, we get ( The negative value of $0 251 .B − in the third component indicates that debt experiences a loss in this transaction with the overall L G value being negative indicating ODV is not attained by retir- ing between 50% and 60% debt.Thus, unlike the Hull (2012) finding of a positive value when going from 60% to 50%, we find a negative value when using the new .

L g
Repeating Hull (2012), we go from a 50% debt level to a 40% debt level.For this example, we would not expect to get a positive L G because we are mov- ing away from the ODV of 0.46.This expectation holds as we get ( ) The absolute magnitude of 9 $0 65 .B − (when going from 60% to 50%) is greater than that of 3 $0 15 .B − (when going from 50% to 40%) with at least some of this due to asymmetry about the optimal leverage ratio where overshooting the optimal is more costly than undershooting.While this overshooting result is consistent with the empirical results of Hull (1999), the difference is less than Hull (2012) for the old .L g Regardless, we see the possibility of increasing value when equity is lowered, which is central to the underinvestment claim.

Examination of the notion that leverage increases when wealth is transferred
Leland (1998) suggests that ODV may increase with asset substitution.In examining this notion by considering the wealth transfer aspect of asset substitution, we find disagreement with the Leland assertion that an asset substitution increases the optimal leverage ratio.Like Hull (2012), we find just the opposite of the Leland assertion when using (11a) and assuming that the wealth transfer component captures an effect similar to an asset substitution effect.For example, we find that 0 849 ODE .= before adjusting for a wealth transfer and 0 786 ODE .= after adjusting.These numbers that indicate leverage fall when wealth is transferred are qualitatively similar to the corresponding numbers of 0.673 and 0.627 found by Hull (2012) when using the old .
L g Finally, like Hull, we can confirm the using (11b) would produce the desired Leland results.However, as suggested by Hull, the use (11b) for this situation would be unlikely.

Debt-equity decision-making with wealth transfers
In Appendix D, we revisit the Hull (2014a) inlstructional exercise.The numbers used in this exercise for tax rates, costs of capital, and PBR are like those described in Appendix C. Appendix D illustrates L V for no growth applications (with and without wealth transfer) and growth applications (with and without wealth transfers).The first chart in this appendix plots the relation between L V and the debt choice ( ) DC using the old L g for the four applications.The second chart repeats the first chart but uses the new L g in conjunction with the constraints given in ( 8) and (8a).
In examining the two charts, we discover several points of interest.First, as before, we find lower maximum firm ( ) = for all eight trajectories.Third, unlike the first chart where there is no RE constraint, the second chart uses the RE constraint and so this constraint pre- vents a steep drop off for the two growth trajectories.NOTE: Appendix D was updated because Growth ( ) WT was Growth ( ) Fourth, the second chart reveals that a greater max L V occurs when a WT is present for a le- vered growth situation, but a lower max L V for a non growth levered situation with a WT .While these two relations also hold for the first chart, the max L V are much more similar in the first chart.Fifth, the constraint given in (8a) is for the situation of non growth.As seen in the second chart, when used with a with no constraint.The second trajectory is the same as the first but with the constraint given in (8a).The third trajectory is the like the first trajectory but is for a levered growth situation using the old L g with no con- straint.The fourth trajectory is like the third trajectory but uses the new L g with the RE con- straint given in (8) = where the trajectory ends.
From the second figure, we find L V results mirror those found in the first figure for L G. Thus, we can offer the same general conclusions for both figures.First, TR ∆ results are like prior results in that the old L g produces greater max L G and max L V val- ues.Second, the constraints do not necessarily affect decision-making in terms of choosing an optimal DC.For example, the non growth constraint has no real affect on decision-making as max $2 25 L G .B = occurs before the constraint kicks in.For growth, it is more difficult to ascertain because the RE constraint sets in when L G is still rising.

DISCUSSION OF RESULTS
In this section, we will offer a brief interpretation of the salient results documented in Section 4. We also call attention to our findings compared to prior research that used the old L g without constraints.
We interpret the results in Appendix A as follows.Using the new L g has practical ramification for managers as a lower max LU GE results.This means there is a lower maximum firm value from leverage than suggested by prior CSM growth research.Compared to the results of Hull (2010), we find more symmetry around the optimal leverage ratio consistent with the fact that the decline in trajectories are not as steep when using the new L g. From Appendix B, we see that the number of violations of the RE constraint increase as the PBR increases.We interpret this as indicating to managers that ex-ternal financing is needed if they want to maintain larger PBRs with larger .DCs Appendix C directly compares our results with prior research.From the comparisons in the three charts, we can visualize how the L g correction explains our findings relat- ed to lower L G values with greater symmetry about ODEs and less dramatic falls in firm value.
From revisiting the asset substitution problem, we find that (compared to prior research) it is more difficult to reject the Leland claim that a tax shield effect from debt is greater than an agency costs of debt related to asset substitution.We interpret this as meaning that managers should not underestimate the effect of a tax shield effect compared to an agency effect.From investigating the underinvestment problem, we discover results consistent with the notion that equity can profit by underinvesting.Practically speaking, this means that equity-for-debt transactions can be valuable undertakings by managers.This latter result with the new L g is like that using the old L g.When examining Leland' claim that leverage increases when wealth is transferred, we cannot confirm this claim.Our finding using the new L g is qualitatively similar to prior CSM research.Managers can take notice that a wealth transfer should typically decrease leverage.
From the applications in Appendix D that incorporate a wealth transfer and Appendix E that give new applications with changes in tax rates, we interpret results from these applications as consistent with our prior applications.In summary, once again, we find that using the new L g ren- ders lower firm values.Regardless, managers can take notice that general conclusions about the optimal DC are similar when using either the old L g or new L g.We interpret this as meaning that optimal leverage choices can still be made even if miscalculations about growth rates occur.

CONCLUSION AND FUTURE RESEARCH
In this paper, we further develop the CSM research through the following achievements.First, we offer an important modification to the L g equation given by Hull (2010).The modification of the L g equa- tion concerns a correction on the corporate tax adjustment for the variables used in the L g equation.Second, we introduce a constraint previously missing when a firm grows strictly by internal equity or retained earnings ( ) RE .This RE constraint governs the plowback-payout and debt-equity choices and, in particular, the relation between RE and interest payments ( ) I. When the constraint is violated before the optimal debt-equity choice is achieved, it signals that external financing is needed.Third, a by-product of the RE constraint is a second constraint that governs a no growth situation so that inter- est payments do not exceed the maximum payout and any gains from leverage that enhance the payout.Fourth, with the L g correction and two constraints in place, we provide updated applications of prior research along with new applications.
From our applications, we obtain the following results that have practical ramifications for managers.We find lower L G and L V values with more symmetry around ODE and less steepness in the fall in firm value.Managers can note that growth is less risky than indicated by prior CSM research.We also show that general managerial decision-making in terms of choosing an optimal debt choice is not materially affected by the L g correction.Except for larger PBRs and larger , DCs we discover that maximum L G values are achieved before the RE constraint set in.Thus, managers can often count on internal financing fulfilling their growth needs if that is desired.
The new constraints developed in this paper serve to point out the need for further research to incorporate external financing within the CSM framework.This incorporation should be valuable because it is cheaper and thus has a lower critical point making growth more profitable.We can point out that, while both constraints are not affected on a per share basis, we still cannot rule out the possibility that a different optimal DC might be chosen if we maximize equity on a per share basis.Thus, considering per share values is another subject that future research can explore.This appendix updates the Hull (2011) growth exercise by comparing its "old" results (dotted line) with the "new" results using the new levered equity growth rate ( ) L g given in (7a) and used with the RE constraint given in (8).Key values used by Hull (2011)

R
less external funds are needed to generate the same .UFor a levered firm, the perpetuity before-tax cash flow from growth ( ) L R is: change.The three outcomes for 1 D r are no change, an increase, and a decrease.First, for no change in 1 ,

G 2 D r is the cost of 2 , D 2 Lgr 2 Lg 2 LE 1 D 1 Dr ↑ is 1 Dr
→ indicates at least one prior leverage change; 2 D is the new debt; is the growth-adjusted levered equity discount rate after the debt-for-equity increment with are equity's discount and growth rates; 1 Lg r is the growthadjusted levered equity discount rate before the increment with less levered equity value that occurs before the increment.Equation(11) represents the situation with no wealth transfer from 1 D (older debt) to (remaining equity).This is not the case for the next two derivations where 1 D is affected through the change in 1 .are diluted by the new debt ( ) after its risk shifts upward from issuing 2 D. The last component is negative and identical to the fall in 1 D caused when its dis- count rate increases from

4 .
CSM with change in tax ratesHull (2014b) extends prior CSM equations by allowing tax rates to be dependent on leverage.In the process, he discovers a new α variable found in the second component of CSM equations that he labels 2 .αThis adds to the prior α vari- able in the first component that he now calls 1 .αHull shows that managers should not ignore 2 α due to its potential strong effects on the debt-forequity choice.To derive his new CSM equation with tax rate changes, Hull labels the tax rates prior to the debt-equity increment as 1 , With wealth transfers included along with tax rate changes, Hull shows:

Following 1 D 2 D
with the asset substitution or wealth transfer component of (11a), Hull (2012), we set the outstanding debt ( ) equal to the new debt ( ) so such as when a positive first component dominates negative second and third components.

9 V
Tax rates do not change.The sequence for costs of debt and equity can differ compared to the prior two appendices with one reason being the change in F r.This appendix plots , DCs for the old L g and the new L g.Each DC is the debt choice that represents the fraction of unlevered equity ( ) U E tat is retired by debt.L versus Debt Choice (DC) Using New g L with Constraints DC→ V L ↓ (in billions)APPENDIX D.Update of Hull (2014a) application

discusses the
PBRs are chosen.These two DCs agree with Hull (2010) for his PBRs of 0 and 0.15 that use the old .
. As seen in the first chart, old L .

Gain to Leverage (G L ) Comparison: Old g L versus New g L G
L ↓ (in billions)