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Capital Structure Decision-Making with Growth: An Instructional Class Exercise

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### Capital Structure Decision-Making with Growth: An Instructional Class Exercise

Professor Robert M. Hull

Clarence W. King Endowed Chair in Finance

School of Business

Washburn University

1700 SW College Avenue

Topeka, Kansas 66621

(Phone: 7853935630)

Email: [email protected]

Overview

- This paper offers an instructional class exercise of the capital structure decision-making process including a hands-on application of how to use four gain to leverage (GL) equations including a recent equation that is used when managing growth firms.
- The latter equation is given by the recent Hull [2010, IMFI] Capital Structure Model (CSM) with growth.
- Given estimates for the costs of capital, tax rates and growth rates, this equation illustrates how managers of growth firms go about choosing an optimal debt level.
- The exercise demonstrates the interdependency of the plowback-payout and debt-equity decisions when maximizing firm value.
- By incorporating growth, this paper extends the non-growth pedagogical exercise of Hull [2008, JFEd].
- This growth extension has proven to be successful in helping advanced business students understand the impact of the plowback and debt choices on firm value.

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, GLis the corporate tax rate multiplied by debt value. The applicability of MM’s GLformulation 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 GLformulations quantifying these effects.

Background

Definitions

- Gain to Leverage (GL) formulations are formulations that measure the change in value caused by changing the amount of debt.
- Equity discount rateis the firm’s cost of borrowing for equityor the return required by investors in equity (for most firms equity is just common equity). The rate can be for an unleveraged firm (rU) or a leveraged firm (rL).
- Debt discount rate is the firm’s cost of borrowing for debt or the return required by investors in debt (rD). For most firms debt is long-term debt such as bonds or short-term debt that is renewed indefinitely.
- Plowback-payout choice determines the unleveraged growth rate (gU). This choice along with the leverage choice decides the leveraged growth rate (gL).
- Optimal debt-equity choice(ODE) is the optimal amount of debt relative to the amount equity at which the firm and its manager strive to obtain so as to maximize firm value.
- Perpetuity with growth involves a perpetual cash, a discount rate, and a growth rate. Any series of uneven cash flows can be approximated by a perpetuity with growth.
- Growth-adjusted discount rate refers to the discount rate minus the growth rate. For an unleveraged firm the equity growth-adjusted rate is rUg = rUgU. For a leveraged firm it is rLg = rLgL.

Prior GL Formulations

- MM (1958): Gain to Leverage (GL) = 0.

Value determined solely by operating assets.

- MM (1963): GL= TCD

where TCis the applicable corporate tax rate and D = I/rDwhere I is the perpetual interest payment and rDis the cost of debt.

- Miller (1977): GL= (1α)D where

α= (1TE)(1TC) / (1TD) with TEand TDthe personal tax rates applicable to income from equity and debt, Dnow equals (1TD)I/rD, and (1α) < TC is expected to hold.

- Hull (2007): GL = [1(αrD/rL)]D[1(rU/rL)]EU.

CSM non-growth equation extends MM and Miller by incorporating equity discount rates (e.g., the unleveraged equity rate of rU and the leveraged equity rate of rL).

Two Current CSM GLFormulations

- Hull (2007): GL = [1(αrD/rL)]D[1(rU/rL)]EU.

CSM non-growth equation extends MM and Miller by incorporating equity discount rates (e.g., the unleveraged equity rate of rU and the leveraged equity rate of rL).

- Hull (2010): GL = [1(αrD/rLg)]D[1(rUg/rLg)]EU.

CSM growth equation extends CSM non-growth equation in incorporate growth-adjusted discount rates: rUg = rU – gU and rLg = rL – gL.

Features of Growth Model Used In Exercise

The Capital Structure Model (CSM) with growth (2010, IMFI) enables this paper to broaden the non-growth pedagogical application of Hull (2008, JFEd).

The model recognizes (in its application) the notion that the plowback ratio decides the minimum unleveraged growth rate (gU) and that this leads to choosing a minimum plowback ratio (PBR).

The model uses a break-through concept: the leveraged growth rate for equity (gL), which depends on both the plowback-payout decision and the debt-equity decision, thus showing how these decisions are intertwined.

The model can illustrate why firms with moderate growth can have larger optimal debt-equity (ODE) ratios, while firms with more pronounced growth must have lower ODEs.

Why Internal Equity Is More Costly than External Equity

Double Taxation: Because corporate taxes are paid before internal equity or retained earnings (RE) can be used for growth purposes, a firm actually has (1TC)REavailable to reinvest for these purposes. This gives a double taxation situation because the cash flow generated from retained earnings is taxed again at the corporate level before being paid out to owners.

How does this double taxation affect the before-tax plowback ratio (PBR)? In terms of cash earnings available before taxes for distribution or plowback, we have: (1) cash that is retained (RE) and (2) cash that is paid out (C). We have (1TC)RE useable for reinvestment after we adjust for double taxation.

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) = (1TE)(1TC)(1–PBR)CFBT / rU = (1TE)(1TC)CFBT / rU

where TE is the effective personal tax rate paid by equity owners, TC is effective corporate tax rate, PBR is the before-tax plowback ratio, CFBT the before-tax cash flow paid out to equity owners, and rU is unleveraged equity cost of borrowing.

- For non-growth PBR = 0 but with growth PBR > 0. Thus the numerator becomes multiplies by (1–PBR). This means that the discount rate of rU must be lowered by at least (1–PBR) if EU is not to decrease when it chooses growth. For this lowered discount rate, we have: (1–PBR)(rU) = rU–(PBR)rUwhere the minimum unleveraged growth rate (gU) must equal (PBR)rUmaking the making the growth-adjusted denominator equal to rU–gU .With gU = (PBR)rU, the two EU values are equal. For example, = EU(no growth) =EU(growth) implies

(1–TE)(1–TC) CFBT /rU =(1–TE)(1–TC) (1–PBR)CFBT / rU–gU.

Equilibrating Growth Rates

- Equilibrating gU=

rU(1TC)RE/C

- Equilibrating gL=

rL(1TC)RE/[C+GI/(1TC)]

- In comparing to equilibrating gL and gU formulas, we see that the equilibrating gL > equilibrating gU .

Question 1: Computing MM and Miller Values

- Unlevgrowth Inc. (UGI) is an unlevered growth firm. UGI’s managers believe it can increase its equity value by retiring a proportion of its outstanding equity through a new debt issue. UGI will treat its new debt as perpetual since it plans to continuously roll it over whenever it reaches maturity. Besides increasing its value through the use of debt, UGI is also considering expansion through its technological innovation. The expansion will involve a new line of marketable products for which future patents will assure constant long-term growth in cash payable to equity owners. UGI will make no choice about growth until it makes its leverage decision.
- After considerable discussion, UGI’s managers decide to issue debt and retire one-half of its outstanding equity shares. UGI’s managers estimate values for key variables needed when using the MM and Miller GL equations. These values are given in Table 1… Fill in the blank cells in Exhibit 1.

[Note. When different, the MM and Miller values are denoted in subscripts.]

TEMM = personal tax rate on equity income = 0%TEMiller= 5.00%

TDMM = personal tax rate on debt income = 0%TDMiller = 15.00%

TC = corporate tax rate = 30.00% αMiller = α = (1TE)(1TC) / (1TD)

GLMM = TC(DMM) GLMiller= [1−αMiller]DMiller

rU = cost of capital for unlevered equity = 11.00%

rF = risk-free rate = 5.00%

I = Interest = rD(D) where I = 0 for an unlevered firm because D = 0

CFBT = perpetual before-tax cash flow generated by operating assets = $1,654,135,338.34

PBR = plowbackratio used on CFBT (PBR = 0 with no growth)

POR = payoutratio = 1 – PBR

RE = before-tax retained earnings = PBR(CFBT) with RE = $0 for no growth because PBR = 0

C = before-tax cash to equity = POR(CFBT) with C = $1,654,135,338.34 for no growth because POR = 1

Question 2. Computing CSM Values without Growth

- UGI is not satisfied with the results from MM and Miller models and so it decides to turn to Capital Structure Model (CSM) without growth. This CSM equation is: GL = ; Before using the CSM, UGI estimates the costs of capital (rD and rL) for each debt level choice. The values for rD and rL are in Exhibit 2. The CSM non-growth value for VU and D in Exhibit 2 are the same as Miller’s VU and debt values because both consider personal and corporate taxes while assuming no growth.
- Fill in the blank cells in Exhibit 2. After you fill in all cells, identify and comment on the debt choice for UGI’s maximum GL, the largest increase in its firm value (as given by the “%ΔV” row), and the optimal D/VL. Finally, explain the significance of the “Incremental ΔGL” and “Incremental %ΔV” rows and what their first negative values indicate.

Table 2. CSM (without growth) Values

PBR = plowbackratio used on CFBT = 0.35 POR = payoutratio = 1 – 0.35 = 0.65

TC = corporate tax rate = 30.00% TD = personal tax rate on debt income = 15.00%

α = 0.78235294118 rU = 11.00%%

CFBT = perpetual before-tax cash flow generated by operating assets = $1,654,135,338.34

RE = before-tax retained earnings = PBR(CFBT) = 0.35($1,654,135,338.34) = $578,947,368.42

C = before-tax cash to equity = POR(CFBT) = 0.65($1,654,135,338.34) = $1,075,187,969.92

I = Interest = where D is DMiller and I must be computed for each D choice.

[Note. The CSM, like Miller, assumes personal taxes, and so we divide D by (1–TD) because the below gL equation uses an I value before personal taxes are considered.]

G = perpetuity cash flow beyond I created with debt when GL ≠ 0. (Supplied for each D choice.)

gU = unlevered equity growth rate = gU= rU(1TC)RE/C = 4.146153846%

gL = levered equity growth rate = gL= rL(1TC)RE/[C+GI/(1TC)]

(Computed for each D choice.)

rUg = growth-adjusted unlevered equity rate of return = rU gU = 11% 4.146154% = 6.853846%

rLg = growth-adjusted levered equity rate of return = rL gL (Computed for each D choice.)

VU(no growth) = (1–TE)(1–TC) CFBT /rU =$10,000,000,000

VU (growth) = (1–TE)(1–TC) (1–PBR)CFBT / rUg = $10,432,098,765.43.

Computing Growth-Adjusted Costs of Borrowing

While UGI’s managers are satisfied with the valuation results using the CSM model with no growth, they still want to know if UGI can improve its value through a new line of marketable products for which future patents can assure constant long-term growth in cash payable to equity. To determine if growth can add to its current value, UGI will use the GL equation given by the CSM with growth. But first it must make some needed computations concerning the value of growth as well as its levered growth rates for various debt choices. Using the values and equations in Table 2, supply answers to the below questions…Fill in the blank cells in Exhibit 3.

Computing CSM Values Using the CSM with Growth

Fill in the blank cells in Exhibit 4. For the rLg row, copy in the values computed previously. After you fill in all cells, identify and comment on the debt choice for UGI’s maximum GL, the largest increase in its firm value (as given by the “%ΔV” row), and the optimal D/VL. Finally, explain the significance of the “Incremental ΔGL” and “Incremental %ΔV” rows and what their first negative values indicate. Comment on how values for these two rows differ from Exhibit 2 when the CSM GL equation was used without growth. Can you offer an explanation or two to explain the difference?

Question 5. Computing and Comparing GL Values

In Questions 1, 2, and 4, you computed the GL values using the MM, Miller and CSM equations for the nine after-tax perpetual debt choices. From your answers for these three questions, fill in Exhibit 5 expressing all values in billions of dollars to four decimals.

After filling in Exhibit 5 and studying how the GL values change as the proportion changes, answer the below questions.

(a) The GL answers in Exhibit 5 use the MM, Miller and CSM equations. In comparing these answers, which (if any) of the three equations is consistent with trade-off theory? Explain.

…..

(e) Which equation would you feel more comfortable with if you were a UGI manager charged with the capital structure decision? Explain.

Relation between Optimal Payback and Leverage Choices: Fix TC = 0.10; gU = 4.45%; gL = 7.54%

Relation between Optimal Payback and Leverage Choices: Fix TC = 0.30; gU= 4.92%; gL= 8.35

Final Remarks

- Student feedback when using the CSM exercises has been positive over the years from both upper level undergraduate finance students and graduate students. The below quote typifies students’ feelings on the exercise:
- “The CSM with Growth model is most complete because it provisions for more scenarios which is keeping in line with diversity of situations faced by managers in trying to determine the optimum leverage. CSM with growth gives due consideration to the growth that can be brought about if the company keeps aside some of its earning to fuel expansion.”
- The approval from students concerning the CSM applications has been received not only for courses taught in the classroom but also online.

Celebrate Relax

It’s Over!

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