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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.edurob.hull@washburn.edu 1

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1) 1)Broadens perpetuity gain to leverage (G L ) research by incorporating and analyzing growth within the capital structure model (CSM) formalized by Hull (2007).2007 2) 2)Argues that, contrary to pecking order theory (POT), internal equity is more expensive than external equity. 3) 3)Discusses how the plowback ratio decides the minimum unleveraged growth rate (g U ) and how leads to choosing a target leverage. 4) 4)Develops a break-through concept: the leveraged growth rate for equity (g L ). 5) 5)Establishes that g L depends on both the plowback-payout decision and the target leverage decision, thus showing how these decisions are intertwined. 6) 6)Derives a growth-adjusted CSM equation and analyzes the role of the growth- adjusted discount rate within this equation. 7) 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 g L. 8) 8)Illustrates why firms with growth often have lower optimal debt-equity (ODE) ratios. 9) 9)Suggests why paying a high coupon rate diminishes a positive agency shield effect. 10) 10)Yields a stunning (yet simple) finding: the starting point for an ODE is the cost of leveraged equity adjusted for growth (r L g ) divided the cost of debt (r D ). 11) 11)Brings us closer to developing a full CSM model by setting the stage for further enhancements within the CSM framework. Overview of Paper 2

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Definitions Gain to Leverage (G L ) 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 (r U ) or a leveraged firm (r L ). Debt discount rate is the firm’s cost of borrowing for debt or the return required by investors in debt (r D ). 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 (g U ). This choice along with the leverage choice decides the leveraged growth rate (g L ). 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 g U and for a leveraged firm as g L. Growth-adjusted discount rate refers to the discount rate minus the growth rate. For an unleveraged firm the equity growth-adjusted rate is r U g = r U g U. For a leveraged firm it is r L g = r L g L. 3

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4 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 (G L ) formulation in the context of an unleveraged firm issuing risk-free debt to replace risky equity. For MM, G L is the corporate tax rate multiplied by debt value. The applicability of MM’s G L 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 G L 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 plowback- payout and leverage decisions within a perpetuity G L equation with growth.

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MM (1958): Gain to Leverage (G L ) = 0. Value determined solely by operating assets. MM (1963): G L = T C D where T C is the applicable corporate tax rate and D = I/r D where I is the perpetual interest payment and r D is the cost of debt. Miller (1977): G L = (1 α)D where α = (1 T E )(1 T C ) / (1 T D ) with T E and T D the personal tax rates applicable to income from equity and debt, D now equals (1 T D )I/r D, and (1 α) < T C is expected to hold. Hull (2007): G L = [1 (αr D /r L )]D [1 (r U /r L )]E U. Hull’s CSM equation is the only equation with equity discount rates (e.g., the unleveraged equity rate of r U and the leveraged equity rate of r L ). 5

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6 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 T C )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 T C )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 (T C ) 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 T C )RE = (1–0.2)$0.3B = $0.24B. Thus, the adjusted PBR = $0.24B/$1.0B = 0.24 or simply (1–T C )PBR = (1–0.2)3 = 0.8(3) = 0.24. (We could further adjust the denominator and get an adjusted PBR of 0.256.)

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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) = (–T C )(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 T C > F should hold since T C should be close to five times greater than F. This is based on reported estimates of 5.5% for F and 26% for T C. 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.Tax NotesTreasury Department 7 II. Why Internal Equity Is More Costly than External Equity: The Pecking Order Theory Debunked

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Minimum Unleveraged Growth Rate What is the minimum unleveraged growth rate (g U ) that an unleveraged firm must attain so that unleveraged equity value (E U ) 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 (E U ): E U (no growth ) = (1 T E )(1 T C )C / r U where T E is the effective personal tax rate paid by equity owners, T C 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–T E )(1–T C )C is replaced by (1–T E )(1–T C )C(1–PBR). This means that the discount rate of r U must be lowered by at least (1–PBR) if E U is not to decrease when it chooses growth. For this lowered discount rate, we have: (1–PBR)(r U ) = r U –(PBR)r U where the minimum unleveraged growth rate (g U ) must equal (PBR)r U making the making the growth-adjusted denominator equal to r U –g U. With g U = (PBR)r U, the two E U values are equal: E U (growth) = E U (no growth) or (1–T E )(1–T C )C(1–PBR) / r U –g U = (1–T E )(1–T C )C/r U. 8

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Unleveraged Growth Rate We can express unleveraged growth rate (g U ) as: g U = R U /C where R U is the change in C (or ΔC) with R U = r U (1 T C )RE. Inserting this value for R U into the above g U equation, we get: g U = r U (1 T C )RE/C where RE is the retained earnings determined by the plowback ratio (PBR). Recalling that the minimum unleveraged growth rate (g U ) must equal (PBR)r U and rearranging g U = R U /C so that R U = g U (C), we can insert g U = (PBR)r U into R U = g U (C) to get R U = r U (PBR)C and use this equation along with R U = r U (1 T C )RE to show that T C = PBR. (Proof in paper.) Thus, if R U = r U (PBR)C and R U = r U (1 T C )RE gives the same value for R U, then T C equals PBR. Interpretation: T C determines the ideal starting point for setting PBR to achieve minimum g U. [NOTE: We have found that G L is maximized at some threshold point where PBR > T C holds; beyond this threshold G L can start falling dramatically. Even if the firm can not achieve a PBR as high as T C or higher, it can still achieve a higher G L to the extent that it can increase its PBR. If T C is low, then PBR must be greater than T C for firm maximization.] Unlike the equilibrating g L described next, the value for the g U using g U = r U (1 T C )RE/C is the same as that using g U = R U /C. Both g U values also change but remain equal when we change a value for either T C or PBR. 9

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I. Leveraged Growth Rate An equilibrating leveraged growth rate (g L ) is derived based on two definitions for R L where R L refers to the change in cash flow paid to leveraged equity owners (C). The two definitions are: R L = g L [C+G I/(1 T C ) ] and R L = r L (1 T C )RE. From these two definitions, we get: equilibrating g L = r L (1 T C )RE/[C+G I/(1 T C )]. In comparing to the equilibrating g L to g U from the equation of g U = r U (1 T C )RE/C, we can see that equilibrating g L > g U should hold. This because the equilibrating g L equation has a larger numerator (e.g., r L > r U ) and also has a smaller denominator for most situations (e.g., I/(1 T C ) > G should hold). 10

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II. Leveraged Growth Rate The equilibrating g L equation of r L (1 T C )RE/[C+G I/(1 T C )] 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 T C ) 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 g L at a large and unsustainable rate for a long-term horizon (and possibly make the denominator negative). We would expect that g L will become negative if I dominates (C+G) for more extreme higher levels of debt. At some point, a negative g L will cause a large positive r Lg (even though in practice that point would never be desired). If G L becomes negative then G also becomes negative making g L increase rapidly as I increases before things totally break down and g L becomes negative. 11

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Assume G L > 0 and that it can be represented as a positive perpetual cash flow of G discounted by the same rate as E L = (1 T E )(1 T C )(C+I) / r L g. If so, we have: G L = (1 T E )(1 T C )G/r L g. Rearranging to solve for G, we have: G = r L g (G L )/(1 T E )(1 T C ). As found in the paper, we have two definitions for leveraged equity (E L ) that are equal but which have different cash flows and discount rates. We have: (1) E L = (1 T E )(1 T C )(C+I) / r L g, and (2) E L = (1 T E )(1 T C )(C+I+G) / r L g’. By adding the perpetual cash flow of G in the leveraged equity’s cash flows in (2), the discount rate of r L g must increase to r L g’ for E L to remain unchanged. From this we can generate a G value equal to G but with a definition that includes some different variables: G = [r L g’ (E L )/(1 T E )(1 T C )] C + I where G = G and r L g’ > r L g. When breaking down G L so that G = G, we find two r L g values (r L g and r L g’ ) result to give the same G L value; thus, an r L g value depends on how we define E L. The understanding of G is crucial because to compute g L we must first compute G. When working in Excel, one must use the iterative command to achieve equality for G and G. 12

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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 R U and g U. Thus, C, RE, R U and g U are determined endogenously (subject to finite operating cash flows) when an unleveraged firm chooses its plowback- payout ratios. The plowback decision or payout decision (or both because one implies the other) can drive g U. But this was for an unleveraged situation. What if managers of an unleveraged firm decide it can maximize its value by becoming leveraged because G L > 0 holds for at least one debt level choice? The answer is that we must now consider how the change from g U to g L 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

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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 G L 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 T C, the process could be considerably reduced because (as suggested previously) T C gives the starting point for the plowback-payout choice. 14

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Starting Point: G L = V L V U where V L is leveraged firm value and V U is unleveraged firm value. V U = (1 T E )(1 T C )C/r Ug where C is the unleveraged equity before-tax cash flow with r U g >r D and r U g = r U g U. V L = E L +D where E L = (1 T E )(1 T C )(C I)/r L g with r L g = r L g L and D = I/r D. V L ’ = E L ’ + D where E L ’ = (1 T E )(1 T C )(C I+G)/r L g ’ with r L g ’ > r L g. 15

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In Appendix 1 we show that: G L = [1 (ar D /r Lg )]D + [1 (r Ug /r Lg )]E U. (13)13 Equation (13) is a more general equation reducing to:13 (i) Hull’s non-growth G L equation when growth is zero; (ii) Miller’s G L equation when growth is zero and discount rates are equal; (iii) MM’s G L equation when growth is zero, discount rates are equal, personal taxes are zero, and debt is risk-free. Note that the 1 st component of (13) typically remains positive and the 2 nd component typically remains negative until things break down with large positive (and unsustainable) leveraged growth rates (g L ) that then becomes negative due to a large I values. For example, a negative G L causes G to become negative, which in turn causes g L to become negative. The latter can be seen from the equation of equilibrating g L = r L (1 T C )RE/[C+G I/(1 T C )].13 16

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Assume two unleveraged firms, A and B, where B is younger but more risky with greater growth opportunities. A and B have respective r U values of 0.08 and 0.10, and g U values of 0.01 and 0.05. Thus, the growth-adjusted unleveraged costs of equity for A is r Ug = r U g U = 0.08 0.01 = 0.07 and for B is r Ug = r U g U = 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 r Lg = r L g L = 0.10 – 0.03 = 0.07 and for B is r Lg = r L g L = 0.13 – 0.08 = 0.05. Suppose that A and B have respective costs of debt (r D ) 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 ar D = 0.8(0.05) = 0.04 and, for B, we have: ar D = 0.8(0.07) = 0.056. We see that the 2 nd component of (13), G L = [1 (ar D /r Lg )]D + [1 (r Ug /r Lg )]E U, will be zero since r Ug = r Lg for both A and B, while the 1 st component will be positive for A:13 [1 (ar D /r Lg )]D = [1 (0.04 /0.056)]D = [1 – 0.571429]D = +0.428571D; but, it will negative for B: [1 (ar D /r Lg )]D = [1 (0.056 /0.500)]D = [1 – 1.12000]D = –0.120000D. Thus, the 1 st 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.13 17

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Consider G L = n 1 D + n 2 E U where n 1 = [1 (ar D /r Lg )] and n 2 = [1 (r Ug /r Lg )]. Together, n 1 and n 2 emphasize that G L 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 G L = n 1 D + n 2 E U suggests that│n 1 │must be sizeable compared to│n 2 │; otherwise, G L may have little chance of being positive unless it increases its amount of debt issued. 18

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The paper offers a qualitative argument that the gain to leverage (G L ) 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 G L = 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 G L equation. We show that ODE = r U g /ar D. When using ODE = r U g /ar D 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 r D that could reverse the positive agency effect in the 1 st 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 r U g /ar D. The paper offers an illustration of ODE = r U g /ar D 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

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In Appendix 2, we derive G L for a leveraged firm undergoing an equity-for-debt transaction with personal taxes and constant growth and show that G L can still be expressed as two components with components reversed from debt-for-equity transaction. For the first component we now have: [1 (r Ug /r Lg )]E U. For the second component we now have: [1 (ar D /r Lg )]D. Putting these together, we have: G L = [1 (r Ug /r Lg )]E U [1 (ar D /r Lg )]D. 20

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Relation between Optimal Payback and Leverage Choices: Fix T C = 0.10; g U = 4.45%; g L = 7.54% PBR: DE 0.100.200.300.400.500.600.70 0.10: 0.402 $406,133,726$584,060,566$595,066,819$490,277,818$322,277,754-$75,930,001-$516,358,312 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 21

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Relation between Optimal Payback and Leverage Choices: Fix T C = 0.30; g U = 4.92%; g L = 8.35 PBR: DE 0.100.200.300.400.500.600.70 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

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Results for All Nine Debt Level Choices for the Application Table 1 gives gain to leverage (G L ) 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 G L using (12). (a) debt is risky with r D > r F = 5.6642% and r D positively related to debt (b) tax rates are relevant with T E = 4.77%, T D = 20.34%, and T C = 30% (c) perpetual before-tax cash flows: C = $905,200,000 (d) constant growth rate when target market approximated: g L = 5.4% with dollar growth = R L = $14,834,558 (e) an unleveraged firm with risky equity faces a finite set of perpetual debt-for-equity choices with r L > r U = 10.0907%. 25

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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 Book 1 st 2 nd Market Book 1 st 2 nd Market D/V r D r L ComponentComponent G L D E L V L D/E D/V r D r L ComponentComponent G L D E L V L D/E 0.1000.0600.105 0.238-0.232 0.0060.6617.2507.9120.091 0.2000.0640.108 0.448-0.417 0.0321.3236.6157.9380.200 0.3000.0670.112 0.625-0.547 0.0781.9846.0007.9840.331 0.4000.0710.115 0.755-0.611 0.1442.6455.4058.0500.489 0.5000.0740.120 0.817-0.583 0.2343.3074.8348.1400.684 0.6000.0800.123 0.656-0.334 0.3223.9684.2608.2280.932 0.7000.0850.127 0.135 0.344 0.4804.6293.7568.3861.232 0.8000.0900.142-0.485 0.821 0.3365.2912.9518.2461.793 0.9000.0960.177-1.945 2.106 0.1615.9522.1158.0672.814 26

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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 Book 1 st 2 nd Market Book 1 st 2 nd Market D/V r D r L Component Component G L D E L V L D/E D/V r D r L Component Component G L D E L V L D/E 0.1000.0600.105 0.386-0.244 0.1420.8307.6148.444 0.109 0.2000.0640.108 0.742-0.437 0.3051.6606.9478.6070.239 0.3000.0670.112 1.063-0.574 0.4892.4916.3008.7910.395 0.4000.0710.116 1.335-0.641 0.6943.3215.6768.9960.585 0.5000.0740.120 1.536-0.612 0.9254.1515.0769.2270.818 0.6000.0800.123 1.503-0.351 1.1524.9814.4739.4541.114 0.7000.0850.127 1.092 0.362 1.4545.8813.9449.7561.473 0.8000.0900.142 0.577 0.862 1.4396.6423.0999.7412.143 0.9000.0960.177-0.821 2.211 1.3917.4722.2219.6933.364 27

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28 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 G L 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 G L = $1.4537 billion on a before personal tax basis (which is what the market sees). For this row, dividing E L by the number of outstanding shares (N L ), we get a share price ≈ $14.42. For example, with D/V = 0.7 (or E/V = 0.3), we have N L = (E/V)(NU) = 0.3(912,000,000) = 273,600,000 shares giving the share price as: P Before Personal Tax = E L / N L = $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 N L = 456,000,000 shares. We have: P Before Personal Tax = E L / N L = $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).

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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 β D ’s for each debt level choice. From there we could proceed to get β U and then obtain β L ’s for the nine debt level choices by using a standard formula. However, unless adjusted upward, those β L 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 (R L ) based upon a chosen growth rate at the target debt-equity choice (which was g L = 5.4%). Using the iterative command in Excel, we were able to solve for R L by first computing the interest paid. 29 Shortcomings of Application

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Summary & Conclusion This research derives G L 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 G L values to eventually decrease with increasing debt levels. Three G L 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 G L 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 G L did conform to the suggested market target D/E of 1.5. Prior research offers G L 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 G L 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 G L formulations found in this paper reaffirm, synthesize, and extend prior G L 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

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Bond Valuation and Risk

Bond Valuation and Risk

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