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Beatrice Presentation February 22, 2005 Robert M. Hull, Beatrice Professor Summer 2005 School of Business, Washburn University, 1700 SW College Avenue, Topeka, KS 66621 Phone : + 1-785‑231‑1010; FAX: + 785‑231‑1063. E‑mail address: [email protected] (R. Hull)

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Beatrice Presentation February 22, 2005 Robert M. Hull, Beatrice Professor Summer 2005

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Beatrice Presentation

February 22, 2005

Robert M. Hull, Beatrice Professor Summer 2005

School of Business, Washburn University,

1700 SW College Avenue, Topeka, KS 66621

Phone: + 1-785‑231‑1010; FAX: + 785‑231‑1063.

E‑mail address: [email protected] (R. Hull)

Earlier Version Published in Proceedings of:

Southwestern Finance Association, Dallas, TX, March 2005

Final Version Accepted for Publication February 4, 2005:

Regional Business Review (Forthcoming 2005)

Firm Value and

The Debt-Equity Choice


Building on the no growth perpetuity framework first developed by Modigliani and Miller (1963), this paper attempts to offer gain to leverage (GL) formulations useable by managers in making debt-equity choices. These formulations focus on how changes in equity and debt discount rates influence firm value. A real world application (using data suggested by independent analysts) seeks to determine the gain to leverage for nine different debt-equity choices. Using our formulation with constant growth, we offer results that can support the suggested target debt-equity choice as the choice that maximizes firm value.

JEL Classification:G32

Key Words:Gain to Leverage; Debt-to-Equity Choice;

Costs of Capital


  • 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 equityor the return required by investors in equity (for most firms equity is just common equity).

  • Debt discount rate is the firm’s cost of borrowing for debt or the return required by investors in debt (for most firms debt is long-term debt such as bonds).

  • Target debt-equity choice 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).

  • 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.

According to Compustat, since the beginning of the century there have been about 1,650 firms per year that on average have reported no long-term debt (which includes capitalized lease obligations).

Theoretical 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 as it implies that financial executives issue unrestricted amounts of debt.

Empirical research (Altman, 1984; Cutler and Summers, 1988; Hull, 1999; Graham and Harvey, 2001) contend that debt affects firm value. Hull (1999) offers event study evidence supporting an optimal debt-equity choice.

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.

Introduction and Background

Prior Research

Prior Research


  • MM (1958): Gain to Leverage (GL) = 0. Value determined solely by operating assets.

  • MM (1963): Gain to Leverage (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): Gain to Leverage (GL) = (1α)D where α = (1TPE)(1TC)/(1TPD) with TPE and TPD the personal tax rates applicable to income from equity and debt and D now equals (1TPD)I/RD.

Prior Research

Prior Research

Prior Research

Motivation for Research

  • Ensuing GL extensions of MM (DeAngelo and Masulis, 1980; Kim, 1982; Modigliani, 1982; Ross, 1985) consider a variety of leverage-related costs and show that an optimal debt level exists even when personal taxes are recognized.

  • Leland and Toft (1996) extend the closed-form results of Leland (1994) to a much richer class of possible debt structures permitting the study of the optimal amount of maturity of debt. Leland (1998) attempts to provide quantitative guidance on the amount and maturity of debt, the financial restructuring, and the optimal risk strategy.

  • The GL extensions are characterized by the inability to make explicit how changes in equity and debt discount rates impact firm value within a model that financial managers might find useable.

  • This paper is motivated to fill in the gap missing in GL research by incorporating discount rates.



We seek to offer GL formulations for an unleveraged firm situation and show the applicability of the GL formulations using empirical data.


GL Formulation withoutPersonal Taxes

  • Starting Point:GL = VL VUwhere VL is leveraged firm value and VU is unleveraged firm value.

  • VU = (1TC)C / RU where C is the unleveraged equity before-tax cash flow with RU > RD.

  • VL = EL + D where EL = (1TC)( C-I) / RL and D = I/RD.

GL Formulation without Personal Taxes

Result of GL Derivation WithoutPersonal Taxes (from Appendix A)

  • In Appendix A we show that:

    GL= [1-(aRD/RL)]D + [(RU/RL)-1]EU (8)

  • Note that [1-(aRD/RL)]D is always positive because aRD/RL < 1 is expected to always hold not only because RD/RL < 1 but a < 1. With no personal taxes, we have: α = (1TC). If RD = RL, then [1-(aRD/RL)]D = [1-(a)]D = [1-(1-TC)]D = TCD which is MM (1963). We see how incorporating discount rates changes the GL formulation.

  • Note that [(RU/RL)-1]EU is always negativebecause RU/RL < 1 is always true. Note that if RU = RL, then [(RU/RL)-1]EU = 0.

  • Thus, we have GL = Positive Components + Negative Component. Which ever component dominates will determine whether GL is positive or negative.

GL Derivation Without Personal Taxes

Illustration:No Personal Taxes Versus Personal Taxes

  • TC = 0.3 (30%); RU = 0.1 (10%); C = $10

  • VU = (1-TC)C / RU = (1-0.3)$10 / 0.1 = $70

  • TPE = 0.05 (5%);TC = 0.3 (30%); RU = 0.1 (10%)

  • VU = (1-TPE)(1-TC)C / RU = (1-0.05)(1-0.3)$10 / 0.1 = $66.50

  • Difference is $70.00 - $66.50 = $3.50

  • For D, where TPD = 0.20 differences would be four times greater per $100.

  • For example, if D = (I / RD) = $100, we have: (1-TPD)D = (1−0.2)($100) = $80. Here the difference is $100 - $80 = $20 $20 /100= 0.2 on the dollar compared to $3.50 / $70 = 0.05because TPD / TPE = 0.2 / 0.05 = 4 (four times greater).

Result of GL Derivation With PersonalTaxes and Constant Growth (Appendix B)

  • In Appendix B, we derive GL with personal taxes and constant growth and show that GL can still be expressed as two components.

  • For the first component we now have:

    {1 – [ αRD / (RL – γL) ] }D

  • For the second component we now have:

    {1 – [ (RU – γL) / (RL – γL) ] } EU

  • Putting these together, we have:

    GL= {1 – [ αRD / (RL – γL) ] }D + {1 – [ (RU – γL) / (RL – γL) ] } EU (12)

  • Note that the first component is no longer always positive and the second component is no longer always negative.

GL with Personal Taxes and Constant Growth

Growth Makes A Difference

Consider the following perpetuity:

Cash Flow = $150

Discount Rate = 0.15

Value of Perpetuity = $150 / 0.15 = $1,000

If Cash Flow growing by 0.05 (5%), we have:

$150 / (0.15 – 0.05) = $150 / 0.10 = $1,500

Difference is $500 of $500 / $1,000 = 0.5 or 50%.

Application UsingCompany Data

  • In our application, we consider Australian Gas Light Company (AGL Co.) which is a major retailer of gas and electricity with about three million customers.

  • We attempt to determine GL if the suggested target debt-equity choice is reached and simultaneously try to show that this can be the optimal.

  • To achieve this aim we gather market data and company data from sources including independent analysis from a firm offering audit, tax, and advisory services (KPMG International) and a brokerage firm (State One Stockbroking Ltd.).

  • To compute GL, we will use equation (12) which incorporates personal taxes and constant growth

  • All monetary values will be given in Australian dollars (A$).

Application Using Company Data

Table 1. Market and Tax Rate Data

RR= Real Rate = 3.42% andRINF = Inflation Rate = 2.17%

RF= Risk-Free Rate = RR + RINF + (RR)( RINF) = 5.664%

MPREM = Market Premium = (RM – RF) = 6.00%

TC = Corporate Tax Rate = 30.00%

λ = Imputation Tax Credit = 40.00%

TE= Effective Tax Rate = TC (1–λ) = 18.00%

TE = Effective Tax Rate = {1 – TC)} / {[1 – [TC(1–λ)]} = 14.634%

Average TE = (18.00% + 14.634%) = 16.317%≈ 16.32%

Average (1 – TE) = 1 – 0.16317 = 0.83683 ≈ 83.68%

TPE= 4.77% and TPD = 20.34%

α = (1TPE)(1TC)/(1TPD) ≈ 83.68% ≈ Average (1 – TE)

(1 – α) = 0.163181 ≈ 16.32% ≈ Average TE

Table 1. Market & Tax Rate Data

Table 2. Cost of Capital Data


Table 2

Cost of Capital

Cost of Debt for

Choices from 0.10 to 0.90

Debt Betas &

Table 3. Data Related to Unleveraged Situation And Target Market Debt-Equity Choice

Table 4. Company Data Related to Market Debt-Equity Target(Values for Market Debt-Equity Choice ≈ 1.5, which is Book Debt-Firm Value Choice = 0.7)

Results for All Nine Debt Level Choices for the Application

Table 5Application of Gain to Leverage Formulation for a Real World Firm Assuming Risky Debt, Personal Taxes, and Constant Growth Rate

Table 5.Application of Gain to Leverage Formulation for a Real World Firm Assuming Risky Debt, Personal Taxes, and Constant Dollar Grow

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).

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 workablegiven 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 interpolatefrom 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 (δg) based upon a chosen growth rate at the target debt-equity choice (which was γL= 5.4%). Using a software program, like Excel, we were able to solve for δg by first computing the interest paid.

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.

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