pricing of corporate debt, merton

8/3/2019 Pricing of Corporate Debt, Merton

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American Finance Association

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450 The Journal of Financetheorem [7] is proven, and the required return on debt as a function of thedebt-to-equity ratio is deduced. In Section VI, the analysis is extended toinclude coupon and callable bonds.

II. ON THE PRICING OF CORPORATE LIABILITIESTo develop the Black-Scholes-type pricing model, we make the followingassumptions:A.1 there are no transactions costs, taxes, or problems with indivisibilitiesof assets.A.2 there are a sufficient number of investors with comparablewealth levelsso that each investor believes that he can buy and sell as much of anasset as he wants at the market price.A.3 there exists an exchangemarket for borrowingand lending at the same

rate of interest.A.4 short-sales of all assets, with full use of the proceeds, is allowed.A.5 trading in assets takes place continuously in time.A.6 the Modigliani-Miller theorem that the value of the firm is invariantto its capital structure obtains.A.7 the Term-Structure s "flat" and known with certainty. I.e., the priceof a riskless discount bond which promisesa payment of one dollar attime T n the future is P(T) = exp[-rt] where r is the (instantaneous)riskless rate of interest, the same for all time.A.8 The dynamics for the value of the firm, V, through time can be de-scribed by a diffusion-typestochastic process with stochastic differentialequation

dV= (aV- C) dt+oVdzwherea is the instantaneous expected rate of return on the firm per unittime, C is the total dollar payouts by the firm per unit time to eitherits shareholders or liabilities-holders (e.g., dividends or interest pay-ments) if positive, and it is the net dollars received by the firm fromnew financing if negative; o2 is the instantaneous variance of thereturn on the firm per unit time; dz is a standard Gauss-Wienerprocess.

Many of these assumptionsare not necessary for the model to obtain but arechosen for expositional convenience. In particular, the "perfect market"assumptions (A.1-A.4) can be substantially weakened. A.6 is actually provedas part of the analysis and A.7 is chosen so as to clearly distinguish riskstructure from term structure effects on pricing. A.5 and A.8 are the criticalassumptions. Basically, A.5 requires that the market for these securities isopen for trading most of time. A.8 requires that price movements are con-tinuous and that the (unanticipated) returns on the securities be seriallyindependent which is consistent with the "efficient markets hypothesis" ofFama [3] and Samuelson [9].1

1. Of course, this assumption does not rule out serial dependence in the earnings of the firm.See Samuelson [10] for a discussion.


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On the Pricing of CorporateDebt 451Suppose there exists a security whose market value, Y, at any point intime can be written as a function of the value of the firm and time, i.e.,Y - F(V,t). We can formally write the dynamics of this security's valuein stochastic differentialequation form as

dY = [ayY- Cy]dt + ayYdzy (1)whereay is the instantaneous expected rate of returnper unit time on this security;CY s the dollar payout per unit time to this security; a'y is the instantaneousvariance of the return per unit time; dzy is a standard Gauss-Wienerprocess.However, given that Y F(V, t), there is an explicit functional relationshipbetweenthe ay, oy,and dzyin (1) and the correspondingvariables a, a and dzdefinedin A.8. In particular, by Ito's Lemma,2we can write the dynamics forY asdY=FvdV+ - Fvv(dV)2+Ft (2)21= 4-_ o2 2F + (aV-C)Fv + Ft dt + oVFvdz,romA.8,

where subscripts denote partial derivatives. Comparingterms in (2) and (1),we have thatayY= ayF=-o2V2Fvv + (aV - C)FFv Ft + Cy (3.a)2 VoGYY oYF=oVFv (3.b)dzy dz (3.c)

Note: from (3.c) the instantaneous returns on Y and V are perfectly corre-lated.Following the Merton derivation of the Black-Scholes model presented in[5, p. 164], consider forming a three-security "portfolio"containing the firm,the particularsecurity, and riskless debt such that the aggregate investmentin the portfolio is zero. This is achieved by using the proceeds of short-salesand borrowings to finance the long positions. Let W1 be the (instantaneous)number of dollars of the portfolio invested in the firm, W2 the number ofdollars invested in the security, and W3 (=--[Wl + W2]) be the numberof dollars invested in riskless debt. If dx is the instantaneous dollar returnto the portfolio, then(dV + Cdt) (dY+ Cydt) +W3rdt (4)

[rWIa - r) + W2(al - r)1dt + Wiodz + W2osdz,=[W1 (a - r) + W2(a, - r) dt + [W1 + W22oyIz, from (3.c).Suppose the portfolio strategy Wj- Wj*, is chosen such that the coefficientof dz is always zero. Then, the dollar return on the portfolio, dx*, would benonstochastic. Since the portfolio requires zero net investment, it must be

2. For a rigorous discussion of Ito's Lemma, see McKean [4]. For references to its applicationin portfolio theory, see Merton [5].


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452 The Journalof Financethat to avoid arbitrage profits, the expected (and realized) return on theportfolio with this strategy is zero. I.e.,

W1*6+ W2*oy 0 (no risk) (5.a)W1* (a - r) + W2*(ay - r) = 0 (no arbitrage) (5.b)A nontrivial solution (Wi* # 0) to (5) exists if and only if

(a - ) ay - r) (6)a a~~~y

But, from (3a) and (3b), we substitute for ay and (Yand rewrite (6) asa- r I(io2V2FVv + (aV-C)Fv + Ft + Cy-rF /oVFv (6'),

and by rearrangingterms and simplifying, we can rewrite (6') as0 -= 22V.2F + (rV-C)Fv -rF + Ft + Cy (7)2

Equation (7) is a parabolic partial differential equation for F, which mustbe satisfied by any security whose value can be written as a function of thevalue of the firm and time. Of course, a complete description of the partialdifferentialequation requires n addition to (7), a specificationof two boundaryconditions and an initial condition. It is precisely these boundary conditionspecifications which distinguishone security from another (e.g., the debt of afirm from its equity).In closing this section, it is important to note which variables and para-meters appear in (7) (and hence, affect the value of the security) and whichdo not. In addition to the value of the firmand time, F depends on the interestrate, the volatility of the firm's value (or its business risk) as measured bythe variance, the payout policy of the firm, and the promised payout policyto the holders of the security. However, F does not depend on the expectedrate of return on the firm nor on the risk-preferencesof investors nor on thecharacteristics of other assets available to investors beyond the three men-tioned. Thus, two investors with quite different utility functions and differentexpectations for the company's future but who agree on the volatility of thefirm'svalue will for a given interest rate and current firm value, agree on thevalue of the particular security, F. Also all the parameters and variablesexcept the variance are directly observable and the variance can be reasonablyestimated from time series data.

III. ON PRICINGRISKY"DISCOUNT ONDSAs a specific application of the formulation of the previous section, weexamine the simplest case of corporate debt pricing. Suppose the corporationhas two classes of claims: (1) a single, homogenousclass of debt and (2) theresidual claim, equity. Suppose further that the indenture of the bond issuecontains the following provisions and restrictions: (1) the firm promises topay a total of B dollars to the bondholders on the specified calendar date T;


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On the Pricing of CorporateDebt 453(2) in the event this payment is not met, the bondholdersimmediately takeover the company (and the shareholdersreceive nothing); (3) the firm can-not issue any new senior (or of equivalent rank) claims on the firm nor canit pay cash dividends or do share repurchaseprior to the maturity of the debt.

If F is the value of the debt issue, we can write (7) as2V2Fvv + rVFv -rF F-r (83)2

where Cy 0 because there are no couponpayments; C 0 from restriction(3); T T - t is length of time until maturity so that Ft = -F7. To solve(8) for the value of the debt, two boundary conditions and an initial condi-tion must be specified. These boundary conditions are derived from the pro-visions of the indenture and the limited liability of claims. By definition,V F(V, T) + f(V, t) where f is the value of the equity. Because both Fand f can only take on non-negativevalues, we have thatF(OT) = f(Ot) = 0 (9.a)

Further, F(V, T) < V which implies the regularity conditionF(VT)/V


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454 The Journal of Financef(VO) =Max[O,V- B] (11)

and boundary conditions (9.a) and (9.b). Inspection of the Black-Scholesequation [1, p. 643, (7)] or Merton [5, p. 65] equation (34) shows that(10) and (11) are identical to the equations for a European call option ona non-dividend-payingcommon stock where firm value in (10)- (11) corre-sponds to stock price and B corresponds o the exercise price. This isomorphicprice relationship between levered equity of the firm and a call option notonly allows us to write down the solution to (10)-(11) directly, but in addi-tion, allows us to immediately apply the comparativestatics results in thesepapers to the equity case and hence, to the debt. From Black-Scholesequation(13) when 02 is a constant, we have thatf(V,t) V(D x1) - Be rr?T(x2) (12)

where l( ) exp [ 2 ]-00and

xi{1log[V/B+ (r+- a02)T}/O V/and

2 1- orFrom (12) and F = V - f, we can write the value of the debt issue as( 1F[V,t] Be-rake [h2(do2t) + - F[hi(d,o2t)] (13)

whered Be-rr7V

1h (do2T)- - ao2T log(d) /oV1h2(da2t) - - 2t + log (d)

Because it is common in discussions of bond pricing to talk in terms of yieldsrather than prices, we can rewrite (13) as-1 1R(t) - r = -log {( [h2(d,o2T)] + - (D[h(da2T] } (14)

whereexp [-R(T)T] F(VT)/B

and R(T) is the yield-to-maturity on the risky debt provided that the firmdoes not default. It seems reasonable to call R(T) - r a risk premium inwhich case equation (14) defines a risk structure of interest rates.For a given maturity, the risk premiumis a function of only two variables:(1) the variance (or volatility) of the firm's operations,2 and (2) the ratioof the present value (at the riskless rate) of the promised payment to the


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On the Pricing of CorporateDebt 455current value of the firm, d. Because d is the debt-to-firm value ratio wheredebt is valued at the riskless rate, it is a biased upward estimate of theactual (market-value) debt-to-firmvalue ratio.Since Merton [5] has solved the option pricing problem when the termstructure is not "flat" and is stochastic, (by again using the isomorphic cor-respondence between options and levered equity) we could deduce the riskstructure with a stochastic term structure. The formulae (13) and (14)would be the same in this case except that we would replace "exp[-rT]" bythe price of a riskless discount bond which pays one dollar at time T in thefuture and "ao2T"y a generalizedvariance term defined in [5, p. 166].

IV. A COMPARATIVE TATICSANALYSIS OF THE RISK STRUCTUREExamination of equation (13) shows that the value of the debt can be

written, showing its full functional dependence, as F[V, T, B, a2, r]. Becauseof the isomorphic relationship between levered equity and a European calloption, we can use analytical results presented in [5], to show that F is afirst-degree homogeneous, concave function of V and B.3 Further, we havethat4FV-11f ?; FB=fBI> O (15)Fr- fr


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456 The Journal of Financeand

PT- -,V(h1)/(2dV/T'5 < O (18)where ?'(x) exp[-x2/2 ]/V2 is the standard normal density function.We now define another ratio which is of critical importance in analyzingthe risk structure: namely, g - ay/o where a, is the instantaneous standarddeviation of the return on the bond and a is the instantaneous standarddevia-tion of the return on the firm. Because these two returns are instantaneouslyperfectly correlated, g is a measure of the relative riskiness of the bond interms of the riskiness of the firm at a given point in time.- From (3b) and(13), we can deduce the formula for g to be

ay = VFv/F0= [h, (d,T) /(P[d,T]d) (19)

g[d,T].In Section V, the characteristics of g are examined in detail. For the purposesof this section, we simply note that g is a function of d and T only, and thatfrom the "no-arbitrage"condition, (6), we have that

_ = g[dT] ((20)a- rwhere (a, - r) is the expected excess return on the debt and (a - r) is theexpected excess returnon the firm as a whole. We can rewrite (17) and (18)in elasticity form in terms of g to be

dPd/P= -g[dT] (21)and

TPT/P = g[d,T lT\/9( h, )1/(21(h ) ) (22As mentionedin Section III, it is commonto use yield to maturity in excessof the riskless rate as a measure of the risk premium on debt. If we define[R(x) - r] H(d, T, 02), then from (14), we have that

Hd -g [dT] > 0; (23)TdHe2 g [dT] [4'(hj)/(D(h1)] > 0; (24)

HT = (log[P] + 2 g[dT] [0(h1)/'I(h) ])/T2 O (25)2As can be seen in Table I and Figures 1 and 2, the term premiumis an increas-ing function of both d and c2. While from (25), the change in the premium

5. Note, for example,that in the context of the Sharpe-Lintner-Mossin apital Asset PricingModel, g is equal to the ratio of the "beta"of the bond to the "beta"of the firm.


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On the Pricing of CorporateDebt 457TABLE 1

REPRESENTATIVE VALUES OF THE TERM PREMIUM, R - rTime UntilMaturity= 2 Time UntilMaturity= 5


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458 The Journalof Finance

w2/a-wcc

O _Cd0 QUASI DEBT FIRM VALUE RATIOFIGURE 1

w0Dw

H ~ ~ ~

O cr2Cr

VARIANCE OF THE FIRMFIGURE


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On the Pricing of CorporateDebt 459

0~

to 0 TIME UNTIL MATURITYFIGURE 3

the risk structure like the corresponding term structure is a "snap shot" atone point in time, it seems natural to define the riskiness in terms of the un-certainty of the rate of return over the next trading interval. In this sense ofriskier, the natural choice as a measure of risk is the (instantaneous) stan-dard deviation of the return on the bond, y ag[d, T] = G(d, a, T). Inaddition, for the type of dynamics postulated, I have shown elsewhere' thatthe standarddeviation is a sufficient statistic for comparingthe relative riski-ness of securities in the Rothschild-Stiglitz [8] sense. However, it should bepointed out that the standard deviation is not sufficient for comparing theriskiness of the debt of different companies in a portfolio sense7 because thecorrelations of the returns of the two firms with other assets in the economymay be different. However, since R - r can be computed for each bondwithout the knowledgeof such correlations,it can not reflect such differencesexcept indirectly through the market value of the firm. Thus, as, at least, anecessary condition for R - r to be a valid measureof risk, it should move inthe same directionas G does in responseto changes in the underlyingvariables.From the definitionof G and (19), we have that

6. See Merton [5, Appendix 2].7. For example, in the context of the Capital Asset Pricing Model, the correlations of the twofirms with the market portfolio could be sufficiently different so as to make the beta of the bondwith the larger standard deviation smaller than the beta on the bond with the smaller standarddeviation.


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460 The Journalof Financeog2 D(h2) (DP(h2) V(hi) 1 (27

Gd ';' 4(h1) - 0(h2) + 0(h1) + + | (27)> Q;8

G =g ( 0 for - oc < x c o.9. While inspectionof (25) shows that HT 1, HT can be eithersignedfor d < 1 which does not agreewith the positivesign on GT.


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On the Pricing of CorporateDebt 461TABLE 2

REPRESENTATIVE VALUES OF THE STANDARD DEVIATION OF THE DEBT, G AND THE RATIO OFTHE STANDARD DEVIATION OF THE DEBT TO THE FrIM, g

Time Until Maturity =2 Time Until Maturity = 5a2 d g G a2 d g G

0.03 0.2 0.000 0.000 0.03 0.2 0.000 0.0000.03 0.5 0.003 0.001 0.03 0.5 0.048 0.0080.03 1.0 0.500 0.087 0.03 1.0 0.500 0.0870.03 1.5 0.943 0.163 0.03 1.5 0.833 0.1440.03 3.0 1.000 0.173 0.03 3.0 0.996 0.1730.10 0.2 0.000 0.000 0.10 0.2 0.021 0.0070.10 0.5 0.077 0.024 0.10 0.5 0.199 0.0630.10 1.0 0.500 0.158 0.10 1.0 0.500 0.1580.10 1.5 0.795 0.251 0.10 1.5 0.689 0.2180.10 3.0 0.989 0.313 0.10 3.0 0.913 0.2890.20 0.2 0.011 0.005 0.20 0.2 0.092 0.0410.20 0.5 0.168 0.075 0.20 0.5 0.288 0.1290.20 1.0 0.500 0.224 0.20 1.0 0.500 0.2240.20 1.5 0.712 0.318 0.20 1.5 0.628 0.2810.20 3.0 0.939 0.420 0.20 3.0 0.815 0.364

Time Until Maturity =10 Time Until Maturity = 25ar2 d g G ar2 d g G

0.03 0.2 0.003 0.001 0.03 0.2 0.056 0.0100.03 0.5 0.128 0.022 0.03 0.5 0.253 0.0440.03 1.0 0.500 0.087 0.03 1.0 0.500 0.0870.03 1.5 0.745 0.129 0.03 1.5 0.651 0.1130.03 3.0 0.966 0.167 0.03 3.0 0.857 0.1480.10 0.2 0.092 0.029 0.10 0.2 0.230 0.0730.10 0.5 0.288 0.091 0.10 0.5 0.377 0.1190.10 1.0 0.500 0.158 0.10 1.0 0.500 0.1580.10 1.5 0.628 0.199 0.10 1.5 0.573 0.1810.10 3.0 0.815 0.258 0.10 3.0 0.691 0.2190.20 0.2 0.196 0.088 0.20 0.2 0.324 0.1450.20 0.5 0.358 0.160 0.20 0.5 0.422 0.1890.20 1.0 0.500 0.224 0.20 1.0 0.500 0.2240.20 1.5 0.584 0.261 0.20 1.5 0.545 0.2440.20 3.0 0.719 0.321 0.20 3.0 0.622 0.278

of the M-M theoremeven in the presence of bankruptcy.To see this, imaginethat there are two firms identical with respect to their investment decisions,but one firm issues debt and the other does not. The investor can "create" asecurity with a payoff structure identical to the risky bond by following aportfolio strategy of mixing the equity of the unlevered firm with holdings ofriskless debt. The correct portfolio strategy is to hold (F,-V) dollars of theequity and (F - FIV) dollars of riskless bonds where V is the value of theunlevered firm, and F and F, are determined by the solution of (7). Sincethe value of the "manufactured"risky debt is always F, the debt issued bythe other firm can never sell for more than F. In a similar fashion, one couldcreate levered equity by a portfolio strategy of holding (fvV) dollars of theunleveredequity and (f - fV) dollars of borrowingon margin which would


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462 The Journal of Finance

zaWLLI ~~~~~~~~~~~~~~T=

Uj ~~~~~~~~~T~T2QLAZ 0

00 d0I QUASI DEBT/FIRM VALUE RATIO

FIGURFE

z0

> mT=_ //

O // I~~~~

WW /FIGuJRE 4

/

QLLLzoI- TNADDVATO FTEFRCO~~~~~~~~~FGR


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z0a d>Iw0L 2zO d=I

d


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464 The Journalof Financeteristics of the debt approaches that of (unlevered) equity. As d -O 0, theprobabilityof default approacheszero, and F[V, T] B exp[-rt], the valueof a riskless bond, and g -O 0. So, in this case, the risk characteristics of thedebt become the same as riskless debt. Between these two extremes, the debtwill behave like a combinationof riskless debt and equity, and will change ina continuous fashion. To see this, note that in the portfolio used to replicatethe risky debt by combining the equity of an unlevered firm with risklessbonds, g is the fraction of that portfolio invested in the equity and (1 - g)is the fraction invested in riskless bonds. Thus, as g increases, the portfoliowill contain a larger fraction of equity until in the limit as g -* 1, it is allequity.From (19) and (31), we have that

gd =- -g) + j. oD(hj)] > 0 (32)i.e., the relative riskinessof the debt is an increasingfunction of d, and

-0gD'(h1) 1 log d2V (h1) 2 (1 2g) + T (33)

-Oasdd 1.Further, we have that

g[1,T] -I T> 0 (34)2andlimitg[d,T] -, < d < oo (35)2

Thus, for d -1, independentof the businessrisk of the firmor the length of timeuntil maturity,the standarddeviation of the returnon the debt equals half thestandarddeviationof the returnon the whole firm. From (35), as the businessrisk of the firm or the time to maturityget large, a',- o/2, for all d. Figures7 and 8 plot g as a function of d and T.Contraryto what many might believe, the relative riskiness of the debt candecline as either the business risk of the firm or the time until maturity in-creases. Inspection of (33) shows that this is the case if d > 1 (i.e., thepresent value of the promised payment is less than the current value of thefirm). To see why this result is not unreasonable, consider the following:for small T (i.e., y2 or T small), the chances that the debt will become equitythroughdefault are large, and this will be reflected in the risk characteristicsof the debt through a large g. By increasing T (through an increase in a2 ort), the chances are better that the firmvalue will increase enough to meet thepromisedpayment. It is also true that the chances that the firm value will belower are increased. However, rememberthat g is a measure of how muchthe risky debt behaves like equity versus debt. Since for g large, the debt is


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a I 2s m ~~T=T;

? To< T

wO dO I

QUASI' DEBT/FIRM VALUE RATIOFIGURE 7

)ixaILL(100

0 0 2 n

ma a 1,g > i and the "replicating" portfolio will contain more than half equity.)Thus, the increased probability of meeting the promised payment dominates,and g declines. For d < 1, g will be less than a half, and the argument goesjust the opposite way. In the "watershed"case when d 1, g equals a half;the "replicating"portfolio is exactly half equity and half riskless debt, andthe two effects cancel leaving g unchanged.In closing this section, we examine a classical problemin corporatefinance:given a fixed investment decision, how does the required return on debt andequity change,as alternativedebt-equitymixes are chosen? Because the invest-ment decision is assumed fixed, and the Modigliani-Millertheoremobtains, V,

2, and a (the requiredexpected returnon the firm) are fixed. For simplicity,suppose that the maturity of the debt, T, is fixed, and the promisedpayment atmaturity per bond is $1. Then, the debt-equity mix is determinedby choosingthe number of bonds to be issued. Since in our previous analysis, F is thevalue of the wholedebt issue and B is the total promisedpayment for the wholeissue, B will be the number of bonds (promising$1 at maturity) in the currentanalysis, and F/B will be the price of one bond.Define the market debt-to-equity ratio to be X which is equal to (F/f) -F/(V-F). From (20), the required expected rate of return on the debt, ay,will equal r + (a - r)g. Thus, for a fixed investmentpolicy,

day dg /dXdX dB dB (36)

providedthat dX/dB #&0. Fromthe definition of X and (13), we have thatdX X(1 +X)(1 -g)dB B ~~~~~> (37)B B

Since dg/dB -gdd/B, we have from (32), (36), and (37) thatday d(a -r)gd > 0 (38)dX X(1+X)( -g)(a -r) F 1 F'(h2)1

X ( 1 + X) - N/VT F (h2) -Further analysis of (38) shows that aVstarts out as a convex function of X;passes through an inflection point where it becomes concave and approachesa asymptotically as X tends to infinity.To determine the path of the required return on equity, ae, as X movesbetween zero and infinity, we use the well known identity that the equityreturnis a weighted average of the returnon debt and the return on the firm.I.e.,

a, a+X(a- ay) (39)h saf+ (Irt g) X(ac r)f

aehasa slope of (a -r) at X - 0 and is a concave function bounded from


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Z/zD aHI-X- a

r X0 MARKET DEBT/EQUITY RATIO

FIGURE 9

above by the line a + (a - r)X. Figure 9 displays both ay and ae. WhileFigure 9 was not produced from computersimulation, it should be emphasizedthat because both (a, - r)/(a - r) and (ae - r)/(a - r) do not depend on a,such curves can be computed up to the scale factor (a - r) without knowledgeof a.VI. ON THEPRICINGOF RISKY COUPONBONDS

In the usual analysis of (default-free) bonds in term structure studies,the derivation of a pricing relationship for pure discount bonds for everymaturity would be sufficient because the value of a default-free coupon bondcan be written as the sum of discount bonds' values weighted by the size ofthe coupon payment at each maturity. Unfortunately, no such simple formulaexists for risky coupon bonds. The reason for this is that if the firm defaultson a coupon payment, then all subsequent coupon payments (and payments ofprincipal) are also defaulted on. Thus, the default on one of the "mini"bonds associated with a given maturity is not independent of the event ofdefault on the "mini" bond associated with a later maturity. However, theapparatus developed in the previous sections is sufficient to solve the couponproblem.Assume the same simple capital structure and indenture conditions as inSection III except modify the indenture condition to require (continuous)


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468 The Journalof Financepayments at a coupon rate per unit time, C. From indenture restriction (3),we have that in equation (7), C -C = C and hence, the coupon bond valuewill satisfy the partial differential equation

120?---2V2Fvv+ (rV-C) Fv-rF-F,+C=O (40)2subject to the same boundaryconditions (9). The correspondingequation forequity, f, will be

0= -2V VV (rV-C) fv -rf - fT (41)2subject to boundary conditions (9a), (9b), and (11). Again, equation (41)has an isomorphiccorrespondencewith an option pricing problem previouslystudied. Equation (41) is identical to equation (44) in Merton [5, p. 170]which is the equation for the European option value on a stock which paysdividends at a constant rate per unit time of C. While a closed-form solutionto (41) for finite Thas not yet been found, one has been found for the limitingcase of a perpetuity (T - oo), and is presentedin Merton [5, p. 172, equation(46)]. Using the identity F = V - f, we can write the solution for theperpetualrisky couponbond as

(2C u2_C t a2r (-2r 2r -2C } 42F(voo) =- I 1- F M2+) 2 +' 2 o


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On the Pricing of CorporateDebt 469for the firm, call it V(l), such that for all V(T) > V(T), it would be advan-tageous for the firm to redeemthe bonds. Hence, the new boundaryconditionwill be

F [V(T), I] =K ( T) (43)Equation (40), (9.a), (9.c), and (43) provide a well-posed problem to solvefor F provided that the V(l) function were known. But, of course, it is not.Fortunately, economic theory is rich enough to provide us with an answer.First, imagine that we solved the problem as if we knew V(T) to get F[V, T;V(T)] as a function of V(T). Second, recognize that it is at management'soption to redeem the bonds and that managementoperates in the best interestsof the equity holders. Hence, as a bondholder,one must presumethat manage-ment will select the V(T) function so as to maximize the value of equity, f.But, from the identity F V - f, this implies that the V(T) function chosenwill be the one which minimizes F [V, T; V(T)]. Therefore, the additionalcondition is that

F[Vr] =min F[V,T; V(T)] (44){V(T)}

To put this in appropriate boundary condition form for solution, we againrely on the isomorphic correspondencewith options and refer the reader tothe discussion in Merton [5] where it is shown that condition (44) is equiva-lent to the conditionFv[V(T),T] = 0 (45)

Hence, appending (45) to (40), (9.a), (9.c) and (43), we solve the problemfor the F[V, T] and V(T) functions simultaneously.VII. CONCLUSION

We have developed a method for pricing corporate liabilities which isgrounded in solid economic analysis, requires inputs which are on the wholeobservable; can be used to price almost any type of financial instrument.Themethod was applied to risky discount bonds to deduce a risk structure ofinterest rates. The Modigliani-Miller theorem was shown to obtain in thepresence of bankruptcy provided that there are no differential tax benefits tocorporations or transactions costs. The analysis was extended to includecallable, coupon bonds.REFERENCES

1. F. Black and M. Scholes. "The Pricing of Options and Corporate Liabilities," Journal of Po-litical Economy (May-June 1973).2. . "The Valuation of Option Contracts and a Test of Market Efficiency," Journalof Finance (May 1972).3. E. F. Fama. "Efficient Capital Markets: A Review of Theory and Empirical Work," Journalof Finance (May 1970).4. H. P. McKean, Jr., Stochastic Integrals, New York, Academic Press, 1969.5. R. C. Merton. "A Rational Theory of Option Pricing," Bell Journal of Economics and Man-agement Science (Spring 1973).6. . "Dynamic General Equilibrium Model of the Asset Market and Its Applicationto the Pricing of the Capital Structure of the Firm," SSM W.P. #497-70, M.I.T. (December1970).


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470 The Journalof Finance7. M. Miller and F. Modigliani. "The Cost of Capital, Corporation Finance, and the Theory ofInvestment," American Economic Review (June 1958).8. M. Rothschild and J. E. Stiglitz. "Increasing Risk: I. A. Definition," Journal of EconomicTheory, Vol. 2, No. 3 (September 1970).9. P. A. Samuelson. "Proof that Properly Anticipated Prices Fluctuate Randomly," IndustrialManagement Review (Spring 1965).10. . "Proof that Properly Discounted Present Values of Assets Vibrate Randomly,"Bell Journal of Economics and Management Science, Vol. 4, No. 2 (Autumn 1973).11. J. E. Stiglitz. "A Re-Examination of the Modigliani-Miller Theorem," American EconomicReview, Vol. 59, No. 5 (December 1969).


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pricing of corporate debt, merton

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