Lectura Merton

8/22/2019 Lectura Merton

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ON THE P R I C I N G O F C O R P O R A T E D E B T : THE R I S K S T R U C T U R EOF IN T E R E ST R A T E S*ROBERTC. MERTON*I. INTRODUCTION

THEVALUE OF a particular issue of corporate debt depends essentially onthree items: (1) the required rate of return on riskless (in terms of default)debt (e.g., government bonds or very high grade corporate bonds); ( 2 ) thevarious provisions and restrictions contained in the indenture (e.g., maturitydate , coupon rate, call terms, seniority in the ev ent of default, sinking f un d,etc .) ; ( 3 ) the probability t ha t the firm will be unable to sat isf y some or allof th e ind entu re requirements (i.e., the probability of d efau lt).While a number of theories and empirical studies has been published onthe term structure of interest rates (item l ) , there has been no systematicdevelopment of a theory for pricing bonds when there is a significant prob-ability of d efault. T h e purpose of this paper is to present such a theory whichmight be called a theory of the risk struc ture of interest rates. T h e use of theterm risk is restricted to the possible gains or losses to bondholders as aresult of (un antic ipate d) changes in the proba bility of d efault and does notinclude the gains or losses inherent to all bonds caused by (unanticipated)changes in interest rates in general. Throughout most of the analysis, a giventerm structure is assumed and hence, the price differentials among bonds willbe solely caused b y differences in the pro bability of default.I n a seminal paper, Black and Scholes [ l ] present a complete generalequilibrium theory of option pricing which is particularly attractive becausethe final formula is a function of observable variables. Therefore, the modelis subject to direct empirical tests which they [2 ] performed with somesuccess. Merton [ 5 1 clarified and extended the Black-Scholes model. Whileoptions are highly specialized and relatively un imp ortant financial instrum ents,both Black and Scholes [ I ] and M erton [ S , 61 recognized that the same basicapproach could be applied in developing a pricing theory for corporate lia-bilities in general.I n Section I1 of t he paper , the basic equation for the pricing of financialinstruments is developed along Black-Scholes lines. In Section 111, the modelis applied to the simplest form of corporate debt, the d iscount bond w hereno coupon payments a re made, and a formula for computing the risk structureof interest rates is presented. In Section IV, comparative statics are used todevelop graphs of the risk structure, and the question of whether the termpremium is a n adeq uate measure of the risk of a bond is answered. I n SectionV, th e validity in the presence of ba nk rup tcy of the famous Modigliani-Miller

* Associate Professor of Finance, Massachusetts Institute of Technology. I thank J. Ingersollfor doing the computer simulations and for general scientific assistance. Aid from the NationalScience Foundation is gratefully acknowledged.449


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450 Th e 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.

11. ON THE PRICING F CORPORATEIABILITIEST o develop the Black-Scholes-type pricing model, we m ake the followingA .l there are no transactions costs, taxes, or problems with indivisibilitiesof assets.A.2 there a re a sufficient number of investors with com parable we alth 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 exchange market for borrowing and lending a t the sam erate 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 th e Modigliani-Miller theorem th at the value of the firm is invar ian tto its c apital structure obtains.A.7 the Term-Structure is flat and known with certainty. I.e., the priceof a riskless discount bond which promises a payment of one dollar attime t n the future is P ( z ) = exp[-rt] where r is the (instantaneous)riskless rate of interest, the same for all time.A.8 T h e dynamics for the value of the firm, V, through time can be de-scribed b y a diffusion-type stochastic process with stochastic differentialequation

assumptions :

dV = (aV - C) dt + oVdzwherea is the instantaneous expected ra te of retu rn on the firm per un ittime, C is the total dollar payouts by the firm per unit time to eitherits shareholder s 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; 19 is the instantan eous variance of thereturn on the firm per unit time; dz is a standard Gauss-Wienerprocess.

Many of these assumptions are not necessary for the model to obtain but arechosen for expositional convenience. In particular, the perfect marketassumptions (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 S and A.8 are the criticalassumptions. Basically, A S 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 ofF a m a [3] and Samuelson [9].1. Of course, this assumption does not rule out serial dependence in the earnings of the firm.See Samuelson [lo] for a discussion.


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O n the Pricing of Corporate Deb t 45 1Suppose 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). W e can form ally write the dynam ics of this securitys valuein stochastic differential equation form as

dY = [a,Y - ,] dt + o,Ydz, ( 1 )wherea, is the instantaneou s expected rate of retu rn per un it time on this security;C, is th e dollar payout per unit time to this security; o, is the instantaneousvariance of the return per unit time; dz, is a standard Gauss-Wiener process.However, given that Y = F(V , t ) , there is an explicit functional relationshipbetween the a,, oy, an d dz, in ( 1 ) an d th e corresponding variables a , o and dzdefined in A.8. In particular, by It6s Lemma,2 we can write the dynamics forY a S

(2)1dY = FVdV +- ,,, (dV) +Ft21= [ 02V2F,, + aV - )F , + Ft ] t+ oVF,dz, from A.8,where subscripts denote partial derivatives. Comparing terms in ( 2 ) and ( 1 ) ,we have that

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a,Y = a,F =- 2V2F,, + (aV - C)F, + Ft + C ,o,Y = o,F = oVF,dz, = z

( 3 . 4

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. 1641 ) consider forming a three-security portfolio containing the firm,the particular security, 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 W, be the (instantaneous)number of dollars of the portfolio invested in the firm, W, the number ofdollars invested in the security, and W3 (=- [W , + W,] ) be the numberof dollars invested in riskless debt. If d x is the instantaneous dollar returnto the portfolio, then

dX = W 1 +w2 (dY + C d t ) + WardtdV +Cdt)V (4 )= [ W , ( a - ) + WZ(a, - ) ] dt + Wlodz+W 2 9 . d ~ ~= [ W l ( a - ) + W2(a, - ) ] dt + [Wlo + W2oY]dz, 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 ItBs Lemma, see McKean [41. For references to its app licationin portfolio theory, see Merton 151.


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

Wl*O + W2*ay = 0 (no risk) (5.4W1* ( a- ) + W2*(a, - ) = 0 (5.b)A nontrivial solution (Wf # 0) to ( 5 ) exists if and only if

(no arbitrage)

(+)=(y)But, from (3a) and (3b), we substitute for ay and oy and rewrite ( 6 ) as

a - r 1d 2-- - (- 2v2Fv,+ ( aV - C)F, + Ft + C, - F)/oVF, (61,and by rearranging terms and simplifying, we can rewrite (6) as

(7)=- 2V2Fvv+ (rV- )F , - F + Ft + C,Equation (7) is a parabolic partial differential equation for F, which mustbe satisfied by a n y security whose value can be written as a function of thevalue of the firm and time. Of course, a complete description of the partialdifferential equation requires in a ddition to (7), a specification of tw o boundaryconditions and an initial condition. It is precisely these boundary conditionspecifications which distinguish one security from another (e.g., the debt of afirm from its equi ty ) .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 firm an d time, F depends on the interestrate, the volatility of the firms value (or its business risk) as measured byth e variance, the pa yout policy of the firm, an d th e promised payo ut policyto the holders of the security. However, F does not depend on the expectedra te of retu rn on the firm nor on the risk-preferences of investors nor on thecha racteristics of other assets available to investors beyond the th ree m en-tioned. Thus, two investors with quite different utility functions and differentexpectations for the companys futu re bu t who agree on the volatility of thefirms value 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 a re directly observable and the variance can be reasonablyestimated from time series data.

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111. ON PRICING RISKYISCOUNTONDSAs 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, homogenous class 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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O n th e Pricing of Corporate Debt 453( 2 ) in the event this payment is not met, the bondholders immediately takeover the company (and the shareholders receive nothing); ( 3 ) the firm can-not issue an y new senior (or of equ ivalent ran k ) claims on the firm nor canit pay cash dividends or do share repurchase prior to the m aturity of the debt.If F is the value of th e debt issue, we can write ( 7 ) as

1- 2V2Fvv+ VF, - F - F, = 02where C, = 0 because there are no coupon payments; C = 0 from restriction( 3 ) ; t=T - is length of time until maturity so tha t F, = -F,. T o 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 th e indenture and the limited liability of claims. B y 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-negative values, we have that

F(0,t) = (0,t) = 0 (9.4F ( V , t ) / V < 1 (9.b)

which substitutes for the other boundary condition in a semi-infinite boundaryproblem where 0 < V < 03. The initial condition follows from indentureconditions (1) an d ( 2 ) and the fact that management is elected by the equityowners and hence, must act in their best interests. On the maturity date T(i.e., t = 0), th e firm must either pay the promised paym ent of B to thedebtholders or else the current equity will be valueless. Clearly, if at timeT, V ( T ) > B , the firm should pay th e bondholders because th e value of equitywill be V (T ) - B > 0 whereas if they do not, the value of equity would bezero. If V ( T ) < B, then the firm will not make the payment and defaultthe firm to the bondholders because otherwise the equity holders would haveto pay in additional money and the (form al) value of equity prior to suchpayments would be (V(T) - B ) < 0. Thu s, the initial condition for the d ebtat t = 0 is

Fur the r, F (V , t) < V which implies the regularity condition

F(V,O) = min[V,B] (9.c)Armed with boundary conditions ( 9 ) , one could solve (8) directly for thevalue of the deb t b y the standa rd m ethods of Fourier transforms or separationof variables. However, we avoid these calculations by looking at a relatedproblem and showing its correspondence to a problem already solved in theliterature.T o determine the value of equity, f (V , t), we note that f (V , z) = V -F ( V , t), and substitute for F in (8) and ( 9 ) , to deduce the partial differentialequation for f . Namely,

1- 2V2 ,, + Vf, - f - , = 0 (10)2Subject to:


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

and boundary conditions (9.a) and (9.b). Inspection of the Black-Scholesequation [ l , p. 643, (7)] or Merton [ S , p. 651 equation (34) shows that(10) and (11) are identical to the equations for a European call option ona non-dividend-paying common stock where firm value in (10)- (11) corre-sponds to stock price a nd B corresponds to 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 comparative statics results in thesepapers to the equity case and hence, to the debt. From Black-Scholes equation(13) when o2 is a constant, we have tha tf(S,t)= V 0 xl)- e-rT0 X Z )where

--mandxl={ log [V/Bl + (+T 0 2 ) t } / o f i

F[V,t] = Be-rT{ CP [h2(d,02t)]+d @ [ h 1 ( d , o 2 t ) 1}and

x2= 1 - f iFrom (12 ) and F = V - , we can write th e value of the debt issue as1 ( 1 3 )

where

02 t- og(d) / o f i1Because it is common in discussions of bond pricing to talk in terms of yieldsrather than prices, we can rewrite (13) as

-1 log { 0[h2(d,02t)] + i - @ [ h ~ ( d , ~ ~ ~ ) l} (14)R(t) - =-twhere

exp [- R(t ) t ] 5 F(V, t ) /Ba n d R(t) is the yield-to-maturity on the risky debt provided that the firmdoes not default. It seems reasonable to call R ( t ) - a risk premium inwhich case equation (14) defines a risk structure of interest rates.For a given maturity, the risk premium is a function of only two variables:(1) the variance (or volatility) of the firms operations, o2 and ( 2 ) the ratioof the present value (at the riskless rate) of the promised payment to the


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On th e Pricing of Corporate Deb t 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-firm value ratio.Since Merton [S] 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) an d ( 1 4 )would be the same in this case except th at we would replace exp[-rt] b ythe price of a riskless discount bond which pays one dollar at time t in thefuture and oat by a generalized variance term defined in [5, p. 1663.

IV. A COMPARATIVETATICS NALYSIS F THE RISK TRUCTUREExamination of equation ( 1 3 ) shows th at the value of the deb t can bewritten, showing its full functional dependence, as F[V, z, B, o, 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 havetha t4

(15)v = 1 - V 2 ; Fg = fB > 0F,=--f,


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456 The Journat of Financean d

PT= @(hi)/(Zdfi < 0 (18)where @(x) = exp[-x2/2]/- is the stand ard norm al den sity function .W e now define anothe r ratio which is of critical im portanc e in analyzingthe risk structure: namely, g = o,/o where T is the instantaneous standarddeviation of the return on the bond and o is the instantaneous standard devia-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! Fro m ( 3 b ) and( 1 3 ) , we can deduce the formula for g to be

TY-= VFy/FT = @[hi(d ,T) l / (P[d ,Tld) (19)5 [d,T].

I n 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

where (a,- ) is the expected excess return on the debt and (a- ) is theexpected excess return on the firm as a whole. We can rewrite (17) and (IS)in elasticity form in term s of g to bedPd/P = g[d,T] (21)

(22)an d

TPT/P = g [d,T] m ( h i /( 2@ (h i 1As mentioned in Section 111,it is common to use yield to maturity in excessof the riskless rate as a measure of the risk premium on debt. If we define[ R ( t ) - ] =H(d, z, 0 2 ) , then from (14) , we have that

( 2 3 )

(24)

1zdd = g[d,Tl > 0 ;2 f l

1H n 2 = - g[d,Tl [@(hl)/@(hl) > 0 ;

As can be seen in Tab le I and Figures 1 and 2, the te rm premium is an increas-ing function of both d and 02. While from (25) , the change in the premium5. Note, for example, that in the context of the Sharpe-Lintner-Mossin Capital 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 Corporate DebtTABLE 1

REPRESENTATIVE VALUES F THE TERM REMIUM,-457

Time Until Maturity = 2 Time Until Maturity =6 d R - (% ) 02 d R - (%)

0.030.030.030.030.030.100.100.100.100.100.200.200.200.200.20

0.20.51o1.53.00.20.51o1.53.00.20.51o1.53.O

0.000.025.13

20.5854.940.010.829.7423.03

55.020.123.09

14.2726.6055.82

0.030.030.030.030.030.100.100.100.100.100.200.200.200.200.20

0.20.51o1.53.O0.20.51o1.53.O0.20.51o1.53.0

0.010.163.348.84

21.990.121.746.47

11.3122.590.954.239.66

14.2424.30

Time Until Maturity = 10 Time Until Maturity= 50 2 d R - (%) 02 d R - (% )

0.030.030.030.030.030.1.00.100.100.100.100.200.200.200.200.20

0.20.51.01.53.00.20.51o1.53.O0.20.51o1.53.0

0.010.382.444.98

11.070.482.124.837.12

12.151.884.387.369.5514.08

0.030.030.030.030.030.100.100.100.100.100.200.200.200.200.20

0.20.51 o1.53.O0.20.51o1.53.00.20.51o1.53.O

0.090.601.642.574.681.072.173.394.266.012.694.065.346.197.81

with respect to a change in maturity can be either sign, Figure 3 shows thatfor d 2 1, it will be negative. To complete the analysis of the risk structureas measured b y the term premium, we show that the premium is a decreasingfunction of the riskless rate of interest. I.e.,dH _ _ ad

= -g[d,T] < 0. (26)It still remains to be determined whether R - is a valid measure of theriskiness of the bond. I.e., can one assert that if R - is larger for one bondthan for another, then the former is riskier than the latter? To answer thisques tion, one mu st first establish an app rop riate definition of riskier. Since


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C

Th e Journal of Finance

"QUASI" DEBT FIRM VALUE RATIOFIGURE

V A R I A N C E O F T H E F I R MFIGURE


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459n the Pricing of Corporate Debt

T I M E UNTIL MATURITYFIGUREthe risk structure like the corresponding term structure is a snap shot atone point in time, it seems natura l to define the riskiness in term s of th e 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, oy = og[d, TI = G (d , o, t). Inaddition, for the type of dynamics postulated, I have shown elsewheree thatthe stan dar d deviation is a sufficient statistic for comparing the relative riski-ness of securities in the Rothschild-Stiglitz [S] 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 sense 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 knowledge of such correlations, it can not reflect such differencesexcept indirectly through t he ma rke t value of the firm. Th us, as, at least, anecessary condition for R - to be a valid measure of risk, it should move inthe same direction as G does in response to changes in th e underlying variables.From the definition of G and (19), we have that

6. See Merton [S, Appendix 21.7. 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 t h e beta on the bond with the smaller standarddeviation.


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460 The Journal ofFinance

Table I1 and Figures 4-6 plot the standard deviation for typical values ofd, 0, and z. Comparing ( 2 7 ) - ( 2 9 ) with ( 2 3 ) - ( 2 5 ) , we see that the termpremium and the standard deviation change in the same direction in responseto a change in the quasi debt-to-firm value ratio or the business risk of thefirm. However, they need not change in the same direction with a change inmaturity as a comparison of Figures 3 and 6 readily demonstrate. Hence,while comparing the term premium s on bonds of the same m atur ity does pro-vide a valid comparison of the riskiness of such bonds, one cannot concludetha t a higher term premium on bonds of different maturities implies a higherstandard deviation?T o complete the comparison between R - and G, the standard deviationis a decreasing function of the riskless rate of interest as was the case for theterm premium in ( 2 6 ) . Namely, we have th at

ad--- d -Gdr ar- -td G,, 0.V. ON THE MODIGLIANI-MILLERHEOREMITH BANKRUPTCY

I n the derivation of the fundam ental equation for pricing of corpo rateliabilities, ( 7 ) , it was assumed that the Modigliani-Miller theorem held sothat the value of the firm could be treated as exogeneous to the analysis. If ,for example, due to bankruptcy costs or corporate taxes, the M-M theoremdoes not obtain and the value of the firm does depend on the debt-equityratio, then the formal analysis of the paper is still valid. However, the linearproperty of ( 7 ) would be lost, and instead, a non-linear, simultaneous solution,F = F [ V ( F ) , z], would be required.Fortunately, in the absence of these imperfections, the formal hedginganalysis used in Section I1 to deduce ( 7 ) , simultaneously, stands as a proof

8. It is well known that W( x ) +x @(x ) >0 or- w


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On the Pricing of Corporate Deb tTABLE 2

REPRESENTATIVE VALUES OF THE STANDARDEVIATIONF THE DEBT,G AND THE RATIOOF461

THE STANDARD DEVIATIONF THE DEBT O THE FIRM,Time Until Maturity= Time Until Maturity= 5

6 2 d g G 0 2 d E G0.03 0.2 0.000 0.000 0.03 0.2 0.000 .o.ooo0.03 0.5 0.003 0.001 0.03 0.5 0.048 0.0080.03 1o 0.500 0.087 0.03 1o 0.500 0.0870.03 1.5 0.943 0.163 0.03 1.5 0.833 0.1440.03 3.O 1 ooo 0.173 0.03 3.O 0.996 0.1730.10 0.2 0.000 0.000 0.10 0.2 0.02 1 0.0070.10 0.5 0.077 0.024 0.10 0.5 0.199 0.0630.10 1o 0.500 0.158 0.10 1o 0.500 0.1580.10 1.5 0.795 0.251 0.10 1.5 0.689 0.2180.10 3.O 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 1o 0.500 0.224 0.20 1o 0.500 0.2240.20 1.5 0.712 0.318 0.20 1.5 0.628 0.2810.20 3.O 0.939 0.420 0.20 3.0 0.815 , 0.364

Time Until Maturity = 10 Time Until Maturity = 256 2 d I7 G 6 2 d g G0.03 0.2 0.003 0.001 0.03 0.2 0.056 0.0100.03 0.5 0.128 0.02 2 0.03 0.5 0.253 0.0440.03 1o 0.500 0.087 0.03 1o 0,500 0.0870.03 1.5 0.745 0.129 0.03 1.5 0.651 0.1130.03 3.O 0.966 0.167 0.03 3.O 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 0.3 7 7 0.1190.10 1o 0.500 0.158 0.10 1.o 0.500 0.1580.10 1.5 0.628 0.199 0.10 1.5 0.573 0.1810.10 3.O 0.8 15 0.258 0.10 3.O 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 1o 0.500 0.224 0.20 1o 0.500 0.2240.20 1.5 0.584 0.261 0.20 1.5 0.545 0.2440.20 3.O 0.719 0.32 1 0.20 3.O 0.622 0.278of the M-M theorem even 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 (FlrV) dollars of theequity an d (F - F,V) dollars of riskless bonds where V is th e 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. I n a similar fashion, one couldcreate levered equity by a portfolio strategy of holding (fvV) dollars of theunlevered equity and ( f - vV ) dollars of borrowing on margin which would


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462 The Journal of Finance

20 UI-> m-

2u)c3 0

0

t 7 -= TIe7 -= 7-2r, < 7-2------

IIII

~~ Ih[/ I ad0 I

Q U A S I D E B T I F I R M VALUE R A T I OFIGURE

0 S T A N D A R D D E V I A T I O N O F THE F I R MFIGURE


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On the Pricing of Corporate Debt 463

0T I M E U N T I L M A T U R I T Y

FIGURE

have a payoff structure identical to the equity issued by the levering firm.Hence, the value of the levered firm's equity can never sell for more than f.But, b y construction, f + F = V, the value of the unlevered firm. Therefore,th e value of th e levered firm can be no larger than th e unlevered firm, and itcannot be less.Note, unlike in the analysis b y S tiglitz [111, we did not require a specializedtheory of capital market equilibrium (e.g., the Arrow-Debreu model or thecapital asset pricing model) to prove the theorem when bankruptcy is possible.I n the previous section, a cross-section of bonds across firms at a point intime were analyzed to describe a risk structure of interest rates. We nowexamine a debt issue for a single firm. In this context, we are interested inmeasuring the risk of the debt relative to the risk of the firm. As discussed inSection IV, the correct measure of this relative riskiness is OJG = g [ d , T ]defined in (19):From (16) and (19), we have that

From ( 3 1 ) , we have 0 < g < 1. I.e., the debt of the firm can never be morerisky than the firm as a whole, and as a corollary, the equity of a levered firmmust always be at least as risky as the firm. In particular, from (13 ) a n d (3 1 ) ,the limit as d -j of F[V, z] = V and of g[d, TI = 1. Thus, as the ratio ofth e present valu e of the promised pay me nt to the c urre nt value of the firmbecomes large an d therefo re the prob ability of eventu al default becomes large,the market value of the debt approaches that of the firm and the risk charac-


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464 Th e Journal of Financeteristics of th e deb t approaches tha t of (unlevere d) equity. As d -+ 0, theprobability of default approaches zero, and F[V, t] + B exp[-rt], the valueof a riskless bond, and g += 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 combination of riskless d ebt a nd equ ity, and will change ina continuous fashion. To see this, note that in the portfolio used to replicateth e risky de bt by com bining the equity of an unlevered firm with risklessbonds, g is the fraction of that portfolio invested in the equity and (1 - )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 ( 3 1 ) , we have thati.e., the relative riskiness of th e debt is an increasing func tion of d , and

-&'(hi)

Further, we have that

and1limit g[d,T] = -, 0 < d < 00P" 2 ( 3 5 )

Thus , for d = 1, independent of the business risk of the firm or the length of timeuntil maturity, the standa rd deviation of the retu rn on the de bt equals half th estan dard deviation of th e return on the whole firm. Fro m ( 3 5 ) ,as the businessrisk of the firm or the time to maturity get large, (T + 0/2, or all d. Figures7 and 8 plot g as a function of d and T.Co ntra ry to w hat m any m ight believe, the relative ,riskiness of th e debt candecline as either the business risk of the firm or the time until maturity in-creases. Inspection of ( 3 3 ) 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., o2 or t small), the chances that the debt will become equitythrough default are large, and this will be reflected in the risk characteristicsof the debt through a large g. B y increasing T (through an increase in 0 ort), the chances are better that the firm value will increase enough to meet thepromised payment. It is also true that the chances that the firm value will belower are increased. However, remember that 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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gRATIOOFSTANDARD

DEVIATONSDEBTIFIRM

0

ml-

gRATIOOFSTANDARD

DEVIATONSDEBTIF1RM

-

0

I-kII II II II II IIR----- I Iu'a


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466 The Journal of Financealready more aptly described by equity than riskless debt. (E.g., for d > 1,g > 4 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 portfoIio is exactly half equity and half riskless debt, andthe two effects cancel leaving g unchanged.I n closing this section, we examine a classical problem in corporate finance:given a fixed investment decision, how does the required return on debt andequity change, as alternative debt-equity m ixes ar e chosen? Because the invest-me nt decision is assumed fixed, and the Modigliani-Miller theorem obtains, V,02, and a (the required expected return on the firm) are fixed. For simplicity,suppose th at the ma turity of the debt, t, is fixed, and the promised payment atmaturity p er bond is $1. Then, the debt-equity mix is determined by choosingthe numb er of bonds to be issued. Since in our p revious analysis, F is thevalue of the whole debt issue and B is the total promised payment for the wholeissue, B will be the num ber of bonds (promising $1 a t maturi ty) 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 (ZO), the required expected rate of retu rn on the debt, ay,will equal r + ( a - )g. Thus, for a fixed investment policy,

dXprovided tha t dX/d B # 0. From the definition of X and (13), we have that

Since dg/dB = gdd/B, we have from (32), ( 3 6 ) , and (37) that

Fu rth er analysis of (3 8) shows that ay s tarts 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, a,, as X movesbetween zero and infinity, we use the well known identity that the equityreturn is a weighted average of the return on debt and the return on the firm.I.e.,

(39)e= a +X(a- %I= a + ( l - g ) X ( a - r ) .

a, has a slope of ( a - ) at X = 0 and is a concave function bounded from


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O n th e Pricing of Corporate D ebt 467

zII:3I-WU

n~

aI-XwXW

I0 M A R K E T D E B T I E Q U I T Y RATIO

FIGUREabove by the l ine a + ( a - )X. F igure 9 displays both ay and a,. WhileFigure 9 was no t produced from com puter simulation, it should be emphasizedth at because both (a , - ) / ( a - ) a n d (a ,- ) / ( a - ) do not depend on a ,such curves can be computed u p to th e scale factor (a- ) without knowledgeof a.

VI. ON HE PRICINGF RISKYCOUPON ONDSI n the usual analysis of (default-free) bonds in term struc ture 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 th e sum of d iscount bonds values weighted by t he size ofthe coupon paym ent at each maturity. Unfortunately, no such simple formulaexists for risky coupon bonds. T h e reason for this is th a t if th e firm defaultson a coupon paym ent, then all subsequent coupon paymen ts (a nd paym ents ofprincipal) are also defaulted on. Thus, the default on one of the minibonds associated with a given maturity is not independent of the event ofdefault on the mini bond associated with a later maturity. However, theap pa ra tus developed in the previous sections is sufficient to solve the couponproblem.Assume the same simple capital structure and indenture conditions as inSection 111 except modify the indenture condition to require (continuous)


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468 Th e Journal of Financepayments at a coupon ra te per un it time,-c. Fro m inden ture restriction ( 3 ) ,we have tha t in equation (7 ), C = C, = C and hence, the coupon bond valuewill satisfy the partial differential equation1

2O = - G ~ V ~ F ~ , +r V - c ) F , - r F - F F , + c = Osubject to the same boundary conditions (9). The corresponding equation forequity, f , will be

12O =- 2V2 ,, + rV - C) , - f - ,

subject to boundary conditions (9a), (9b), and (11). Again, equation (41)has an isomorphic correspondence with an option pricing problem previouslystudied. Equation (41) is identical to equation (44) in Merton [ S , p. 1701which is the equation for the European op tio z value on a stock which paysdividends at a constant rate per unit time of C. While a closed-form solutionto (41) for finite t has not yet been found, one has been found for the limitingcase of a perpetuity (z = a), nd is presented in M erton [S , p. 172 , equation(46)]. Using the identity F = V - , we can write the solution for theperpetual risky coupon bond asr 2r

1where ( ) is the gamma function and M ( ) is the confluent hypergeo-metric function. While perpetual, non-callable bonds are non-existent in theUnited States, there are preferred stocks with no maturity date and (42)would be the correct pricing function for them.

Moreover, even for those cases where closed-form solutions cannot befound, powerful numerical integration techniques have been developed forsolving equations like (7) or (41). Hence, computation and empirical testingof these pricing theories is entirely feasible.Note that in deducing (40), t was assumed that coupon payments weremade uniformly and continuously. In fact, coupon payments are usually onlym ade semi-annually or annually in discrete lu_mps. However, i t j s a simplematter to take this into account by replacing C n (40) by X , C P ( z- i)where 6 ( ) is the dirac delta function and ti is the length of time untilmaturity when the ithcoupon payment of c, ollars is made.As a final illustration, we consider the case of callable bonds. Again, assumethe same capital structure but modify the indenture to sta te tha t the firm canredeem the bonds at its option for a stated price of K ( t ) dollars where Kmay depend on the length of time until maturity. Formally, equation (40)and boundary conditions (9.a) and (9.c) are still valid. However, instead ofthe boundary condition (9.b) we have th at for each t, there will be some value


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On he Pricing of Corporate Deb t 469for the firm, call it V(t) , such that for a l l V(t) 3 V(t), it would be advan-tageous for the firm to redeem the bonds. Hence, the new boundary conditionwill b e F [ V ( t ) , t ] = K ( t ) (431Equation (40), (9.a), (9.c), and (43) provide a well-posed problem to solvefo r F provided that the v ( z ) function were known. But, of course, it is not.Fortunately, economic theory is rich enough to provide us with an answer.-irst, imagine that we solved the problem as if we knew v(t) o get F[V, t;V ( t ) ] a s a function of v(t) . econd, recognize that it is at managementsoption to redeem th e bonds an d tha t management operates in the best interestsof th e equ ity holders. Hence, as a bondholder, one must presume th at m anage-ment will select the v(t) function so as to maximize the value of equity, f .But, from the identity F = V - , this implies th at the V ( t ) function chosenwill be the one which minimizes F [ V , t ; v ( t ) ] . Therefore, the additionalcondition is tha t

F[V,t ] = min F [ V , t ; V ( z ) ]W ( t ) 1 (44)

T o put this in appropriate b ound ary condition form for solution, we againrely on the isomorphic correspondence with options and refer the reader tothe discussion in Merton [5 1 where it is shown th at condition (4 4) is equiva-lent to the condition

F v [ 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 . CONCLUSIONWe have developed a method for pricing corporate liabilities which isgrounded in solid economic analysis, requires inputs which are on the wholeobserva ble; can be used to price almost an y typ e of financial instrument. T h emethod 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.

REFERENCES1. F . Black and M. S choles. The Pricing of Options and Corporate Liabilities, Journal of Po-2 . . The Valuation of Option Contracts and a Test of Market Efficiency, Journal3. E. F . Fam a. Efficient Capital Markets: A Review of Theory and Empirical Work, Journal4. 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).

litical Economy (May-June 1973).of Finance (May 1972).o f Finance (May 1970).


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47 0 The Journal of FinanceInvestment, American Economic Review (June 1958).Theory,Vol. 2, No. 3 (September 1970).Management Review (Spring 1965).Bell Journal of Economics and Management Science,Vol. 4, No. 2 (Autumn 1973).Review, Vol. 59, No. 5 (December 1969).

7. M. Miller and F. Modigliani. The Cost of Capital, Corporation Finance, and th e Theory of8. M. Rothschild and J. E. Stiglitz. Increasing Risk: I. A. Definition, Journal of Economic9. P. A. Samuelson. Proof that Properly Anticipated Prices Fluctuate Randomly, Zndustrial

10. . Proof that Properly Discounted Present Values of Assets Vibrate Randomly,11. J. E. Stiglitz. A Re-Examination of the Modigliani-Miller Theorem, American Economic
http://dx.doi.org/10.1016/0022-0531(70)90038-4http://dx.doi.org/10.2307/3003046http://dx.doi.org/10.2307/3003046http://dx.doi.org/10.2307/3003046http://dx.doi.org/10.2307/3003046http://dx.doi.org/10.1016/0022-0531(70)90038-4http://dx.doi.org/10.1016/0022-0531(70)90038-4http://dx.doi.org/10.1016/0022-0531(70)90038-4http://dx.doi.org/10.2307/3003046http://dx.doi.org/10.1016/0022-0531(70)90038-4http://dx.doi.org/10.1016/0022-0531(70)90038-4

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Lectura Merton

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