A NOTE ON THE CALCULATION OF THE AFTER-TAX COST OF DEBT

Transcription

1 INTERNATIONAL JOURNAL OF BUSINESS, 1(1), 1996 ISSN: A NOTE ON THE CALCULATION OF THE AFTER-TAX COST OF DEBT Wm R McDaiel, Daiel E. McCarty, ad Keeth A. Jessell Whe oe examies stadard fiacial maagemet texts' treatmets of how to calculate the after-tax yield ad cost from the issuig firm's perspective of a particular debt issue, oe fids apparet icosistecies ad, sometimes, outright errors. A geeral model to fid the after-tax cost of debt is developed ad applied to several cases. Ay method that icludes the coversio k d = k d * (1 - T) will be accurate if there are o flotatio costs regardless of the et proceeds to the firm; however, whe both flotatio cost ad a discout (or premium) over face value at issue exist, the coversio is iaccurate. Whether the magitude of the error is sigificat is a questio of the aalyst's subjective judgemet. I. INTRODUCTION Whe oe examies stadard fiacial maagemet texts' treatmets of how to calculate the after-tax yield ad cost from the issuig firm's perspective of a particular debt issue, 1 oe fids apparet icosistecies ad, sometimes, outright errors. Three specific coditios seem to be at the root of the problem: (1) the case where debt, at issue, is sold at a premium or a discout; (2) the case where a flotatio cost makes the proceeds to issuer less tha the face value of the debt; ad (3) the case where both flotatio costs ad premia or discouts chage the proceeds to the issuer. This ote discusses each of these cases, comparig the correct techique to methods foud i fiacial maagemet texts. This ote is i cotrast to the recet paper by Bey ad Collis [1], which examies before- ad after-tax yields from the ivestor's viewpoit. To accomplish this objective, Sectio II develops the basic model used for the aalysis ad presets the problem. I Sectio III, the basic Wm R McDaiel, Daiel E. McCarty ad Keeth A. Jessell, College of Busiess ad Public Admiistratio, Florida Atlatic Uiversity, Boca Rato, FL Copyright 1996 by SMC Premier Holdigs, Ic. All rights of reproductio i ay form reserved.

2 80 McDaiel, McCarty, ad Jessell model is used to prove that a two-equatio approach to estimate the after-tax cost of debt provides the correct result, eve though it may be less appealig tha a sigle-equatio method, regardless of the amout of the et proceeds to the firm. Sectio IV examies the relatioship betwee the two- ad sigle-equatio methods whe flotatio costs are preset. The fial sectio summarizes the research effort. II. THE BASIC MODEL AND PROBLEM Two methods for estimatig a firm's after-tax cost of debt are commoly used. The first is give i Eq(1a) ad Eq(1b). The secod method is give i Eq(2). For example, Gitma [4], ad Gitma, Joehk ad Piches [5] show that the cost of debt is foud by first solvig for k d * i the debt valuatio equatio: B 0 = Σ (I t + P t )/(1 + k * d ) t (1a) where, B 0 = Proceeds from ivestors at issue (may be at par, premium, or discout); I t = iterest i period t; P t = periodic pricipal repaymet; = bod maturity; ad k d * = before-tax cost of debt. I the most geeral applicatio, oe may thik of the sum of each period's iterest ad pricipal repaymet as the periodic cash flow servicig the creditors. The after-tax cost, k d, is foud by: k d = k d * (1 - T), (1b) where T = firm's margial tax rate. Nevertheless, oe might questio whether this method is equivalet to a straightforward applicatio of the valuatio otio. Geerally, a valuatio is a value calculatio based o a stream of cash flows discouted at the appropriate rate; oe ca approach the problem with either the value or the discout rate as the ukow. Specifically, the after-tax cost of debt applicatio is: B 0 = Σ (I t + P t - TI t ' )/(1 + k d ) t (2)

3 INTERNATIONAL JOURNAL OF BUSINESS, 1(1), The quatity, I t ', is the portio of the aual cash flow that is tax deductible by IRS regulatio. 2 The calculatio of k d through the two-equatio method (Eq(1a) ad Eq(1b)) is ot the ituitively obvious equivalet to k d foud through the sigle equatio method (Eq(2)). Bierma ad Smidt [2: 359] demostrate that both methods yield the same results i the limited case where either discouts (or premia) or flotatio costs exist. The broader questio here is whether the results differ i the presece of discouts (or premia). III. A GENERAL PROOF To aswer the questio, cosider a special case (for referece purposes, this is called Case Q) of Equatio 1a, where the iterest paymets are i each period equal to k * d times the remaiig balace. Whe the iterest paymets are subtracted from their respective periodic debt service cash flows, the result is the true stream of paymets applied to the pricipal. I some applicatios, the true pricipal paymet will be o-positive.with the assumptio that the etire pricipal will have bee repaid at period, the sum of the pricipal paymets will equal the amout B 0. This is true whether the cash flow stream is equal paymets, equal pricipal paymets, egative amortizatio, a sikig fud with optioal double paymets, or ay other imagiable redemptio scheme. The IRS Code [12] states that the deductible iterest o discout (premium) at issue debt is foud by first solvig Equatio 1a for k * d --this rate is the true iterest rate. This rate is the used i a amortizatio table to determie the schedule of deductible iterest paymets over the debt's life. I other words, the deductible iterest is k * d times the true remaiig balace. This is the "effective iterest rate" method, as demostrated i Daos ad Imhoff [3]. Now cosider the periodic cash flow to service the debt, CF t = P t + I t, i.e., the sum of the omial iterest ad pricipal repaymets. I each period, oe may restate CF to be the true iterest paymet, I ' t, plus the true pricipal repaymet, P j = CF t - I ' ' t. Agai, the sum of P t = B 0. Equatio 2, by substitutio of I t + P t = I ' t + P ' t becomes: ' ' B 0 = Σ (I t + P t - TI ' t )/(1 + k d ) t. (3) Rearragemet of terms yields:

4 82 McDaiel, McCarty, ad Jessell ' B 0 = Σ (I t (1-T) + P ' t )/(1 + k d ) t. (4) Notice that Equatio 4 ca be thought of as modellig a debt issue whose rate, as applied to the upaid balace, is k d * (1 - T). Also, this debt issue's B 0 equals the sum of its pricipal paymets. Thus, the factors fit precisely the defiitios for special Case Q of Equatio 1a, if ad oly if the discout rate is k d * (1 - T). Therefore, regardless of whether debt is issued at par, a discout, or a premium (i cosideratio of its face value ad coupo), its after-tax cost is equal to the before-tax cost times (1 - T); i.e., k d = k d * (1 - T). As demostrated by Equatio 4, as log as the iterest expese is the effective iterest as required by the IRS, the equality betwee the two methods exists. To illustrate the equivalece of the two methods i the presece of taxes ad discouts, assume a bod with a par value of $1,000, a 5 percet coupo, a maturity of 10 years, a margial tax rate of 34 percet, ad a discout of $ The yield to maturity o this bod is 10 percet. The umerical calculatios are show i Table 1. Table 1: Illustratio of the Basic Model Year Startig Balace Coupo Iterest Pricipal Reductio Edig Balace Iterest Tax Shield Cash Flow 1 $ $50 $69.28 $ $ $23.56 $ , Solvig for the after-tax cost of debt, usig the after-tax cash flows, yields a after-tax cost of debt of 6.6 percet. The same result is foud by multiplyig the before-tax cost of debt by (1-T),.10(1-.34)=.066 or 6.6 percet. IV. THE ANALYSIS WITH FLOTATION COSTS

5 INTERNATIONAL JOURNAL OF BUSINESS, 1(1), The chief cause of questioable discussios of the cost of debt with regard to flotatio cost is the omissio of tax cosequeces. The IRS Reveue Rulig [11] states that direct flotatio costs for debt may be writte off i straightlie fashio over the life of the debt. Oe plausible explaatio for the omissio of tax cosequeces may be that the typical flotatio cost of debt is quite small, so that its tax beefits are egligible. Va Hore [13] gives a more sophisticated reaso, statig that attachig the flotatio cost to each project to be fiaced is a superior method because attachig it to the cost of capital causes a "excessive pealty." (See Hubbard [7] for a complete discussio ad bibliography o this otio.) This argumet has merit but rus ito some practical difficulties: (1) whe several projects are fuded at the same time by several differet sources, how does oe determie which source's flotatio cost goes with which project?; (2) the method deals with flotatio costs, excludig their tax effect; ad (3) as evideced by may cites i the referece list [2, 4, 5, 6, 8, 9, ad 10] ad may other fiacial maagemet texts, academicias ad practitioers seem to use the method attachig the flotatio costs to the cost of capital. Additio of the write-off to the cash flow stream to the valuatio otio yields a modified form of Eq(2), where F is the direct flotatio cost i dollars: ' B 0 - F = Σ (I t + P ' ' t - TI t - TF/)/(1 + k d ) t. (5) Note that the simple coversio, k d = k d * (1 - T), is iaccurate for this case (Ufortuately, several authors icorrectly advocate this method, e.g., Moyer, McGuiga ad Kretlow [9]; Petty, Scott, Keow, ad Marti [8]; Gup [6]; ad perhaps Piches [10].) This "popular" method appears as Equatio 6: B 0 - F = Σ (I t + P t )/(1 + k d ) t (6) For example, cosider a bod with the followig traits: B 0 = $692.77; =10 years; I 1 to10 =$50; P 10 = $1,000; F =34.64 or 5 percet of B 0 ;T =.34 If oe applies the method iitiated by Equatio 1a, k d * is 10 percet, igorig flotatio costs, ad Equatio 1b will produce k d = 7.09 percet. The accurate method foud through Equatio 5 fids k d to be 7.08 percet from the cash flows i Table 2. Thus, a small error comes from igorig the

6 84 McDaiel, McCarty, ad Jessell tax effect of the flotatio costs. Flotatio cost always icreases the cost of ew debt. The popular method overstates the amout of the icrease i the after-tax cost of debt. Examiatio of Equatio 5 shows that the best estimate of the after-tax cost of debt lies betwee the rates yielded by the k d = k d * (1 - T) method ad the popular method. Table 3 shows the magitude of the differece for various debt characteristics ad flotatio costs. Table 2: Illustratio of Equatio 5 After-Tax Flotatio Cost Cash Year Cash Flow* Tax Shield Flow 1 $ $-1.18 $ , , V. CONCLUSION Eq(5) applies geerally to all cases whether B 0 is at a premium or a discout. Ay method that icludes the coversio k d = k d * (1 - T) will be accurate if there are o flotatio costs regardless of the et proceeds to the firm; however, whe both flotatio cost ad a discout (or premium) over face value at issue exist, the coversio is iaccurate.whether the magitude of the error is sigificat is a questio of the aalyst's subjective judgmet. Table 3: After-Tax Cost of Debt Computatios Usig Equatio 5 Market rate=10.0 %; Face value=$1,000; Maturity=10 years; Tax rate=34% After-Tax Cost of Debt with Flotatio Cost at: Rate Issue O% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

7 INTERNATIONAL JOURNAL OF BUSINESS, 1(1), % $1, NOTES 1. Oe should ot cofuse the otio of the cost of debt for a particular issue with the margial cost of debt capital. The latter is a fuctio of may factors, icludig the firm s operatioal ad fiacial risk, iterest rates i the market, ad portfolio effects with other securities. The former may differ because of market imperfectios, callability, sikig fud provisios, ad other ideture items. 2. Yawitz ad Maloey [14] i their study of origial issue discout bods discuss the tax effects to both issuer ad ivestor, whe the effective iterest rate method is used. REFERENCES [1] Bey, Roger P., & Collis, J. Markham. (1988). The Relatioship Betwee Before- Ad After-Tax Yields o Fiacial Assets, The Fiacial Review. August, [2] Bierma, Harold Jr., & Smidt, Seymour. (1986). The Capital Budgetig Decisio. MacMilla, 359. [3] Daos, Paul, & Imhoff, Eugee A. (1986). Itermediate Accoutig. 2d Editio, Pretice-Hall, [4] Gitma, Lawrece J. (1994). Priciples of Maagerial Fiace. 7th ed., Harper ad Row,

8 86 McDaiel, McCarty, ad Jessell [5] Gitma, Lawrece J., Joehk, Michael D., & Piches, George E. (1985). Maagerial Fiace. Harper ad Row, 702, 732. [6] Gup, Beto E. (1987). Priciples of Fiacial Maagemet. Wiley, [7] Hubbard, Carl M. (1984). Floatatio Costs i Capital Budgetig: A Note o the Tax Effect, Fiacial Maagemet. (Summer), [8] Petty, J. William, Scott Jr., David F., Keow, Arthur J., & Marti, Joh D. (1993). Basic Fiacial Maagemet. 6th ed., Harper ad Row, [9] Moyer, R. Charlers, McGuiga, James R., & Kretlow, Wm. J. (1992). Cotemporary Fiacial Maagemet. 5th ed., West Publishig Co., [10] Piches, George E. (1994). Essetials of Fiacial Maagemet. Harper ad Row, [11] Reveue Rulig , Helverig v. Uio Pacific Railroad Compay, 293 U.S. 282 (1934) CB56. [12] Sectios 163(e) ad 1272(a), Iteral Reveue Code. [13] Va Hore, James C. (1992). Fiacial Maagemet ad Policy. 9th ed., Pretice-Hall, [14] Yawitz, Jess B., & Maloey, Kevi J. (1983). Evaluatig the Decisio to Issue Discout Bods: Term Structure ad Tax Effects, Fiacial Maagemet. (Witer),