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Risky debt, jump processes, and safety covenants
Journal
of Financial
Economics
RISKY
9 (1981) 281-307.
North-Holland
DEBT, JUMP PROCESSES, SAFETY COVENANTS* Scott
Publishing
Company
AND
P. MASON
Harvard University, Boston, MA 02163, USA
Sudipto
BHATTACHARYA
Stanford University, Stanford, CA 94305, USA
Received
January
1980, final version
received
March
1981
The usual assumptions in the continuous-time contingent claims pricing of risky debt are (1) the firm is in default only when the value of its remaining assets falls short of the currently due promised payment and (2) the firm value follows continuous diffusion-process dynamics. It is the Jomt relaxation of these two simplifying assumptions that motivate this paper in its study of the valuation of risky debt and safety covenants when the firm value follows (possibly) discontinuous sample paths. Exphclt solutions are derived and compared to the work of Black and Cox (1976)
1. Introduction Black and Scholes (1973) identified the isomorphic relationship between the problems of pricing options and ‘discount’ debt with default risk. As a result, the finance literature has witnessed a great growth of research on the pricing of debt claims with more complex payoff patterns, using the techniques of continuous-time contingent claims analysis pioneered by Black and Scholes (1973) and Merton (1973). The usual assumption in most of these models regarding default, and consequent takeover by bondholders, is that a firm is in default only when the value of its remaining assets falls short of the currently due promised payment on the debt. Much of the debt valuation literature - for example, Merton (1974), Galai and Masulis (1976), Ingersoll (1977) - has also assumed continuous diffusion-process dynamics for firm value. It is the joint relaxation of the above two simplifying assumptions that motivates this paper. It is a common observation that firms enter into bankruptcy or reorganization procedures long before the market value of their remaining *The authors wish to thank the referee, Scott dlscussions and editorial suggestions.
0304-405X/81/000&0000/$02.50
Richard,
and E. Philip
0 North-Holland
Jones for many
helpful
282
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
assets drops below currently due debt obligations. Such phenomena have their basis, at least in part, in the distorted stockholder/manager incentives which result from having debt with ‘very’ high default risk. These issues have been discussed by Jensen and Meckling (1976) and Myers (1977). As a procedural matter, this type of default and bankruptcy procedure often arises from the inclusion of various balance sheet and income statement based ‘financial health’ criteria in debt covenants, which have been examined and analyzed in the work of Smith and Warner (1978). One way to recognize these complications in the pricing of risky debt is to postulate the stylized covenant of a ‘safety barrier’, a floor for firm value such that if the firm value reaches or drops below such a barrier, debtholders take over the firm. Black and Cox (1976) developed this approach with diffusion process dynamics for firm value. Discontinuous changes in firm value, resulting from the arrival of information that affects investors’ anticipation of future growth and profitability, appear to be a pervasive feature of economic life, both from casual empiricism as well as the recent work of Rosenfeld (1979). In this paper, we extend the analysis of risky debt pricing with safety barriers to cases where the firm value follows (possibly) discontinuous sample paths. Our goals and achievements in this paper are twofold. The first contribution of this paper is methodological. Most existing analyses of contingent claims, including this paper, make use of solutions and techniques that are transparently related to discounted expected value calculations with simple stochastic processes, as noted by Cox and Ross (1976b). Our analysis makes use of a class of probabilistic ‘first passage time’ techniques for discontinuous processes with somewhat complicated boundary conditions that have not been widely used by financial economists. The incorporation of these techniques may be of more general importance in the analysis of the valuation of financial contracts under increasingly complex and ‘realistic’ assumptions. In this respect, our analysis extends the results on valuation of options with jump processes, pioneered by Cox and Ross (1976a) and Merton (1976a). Second, we develop estimates of the value of safety covenants and compare our results to those of Black and Cox (1976) who consider the value of a finite lived risky discount bond and a safety covenant under the assumption that the firm value follows continuous diffusionprocess dynamics. In section 2 we formulate a pricing equation for contingent claims written on a firm that follows possibly discontinuous sample paths. The problem of valuing finite lived risky discount debt, in the presence of a safety covenant, is posed and an approach to solving this problem is outlined. Section 3 presents a solution to the valuation problem for a specific set of Poisson dynamics, and section 4 compares this solution to the work of Black and Cox (1976). Section 5 contains our concluding remarks.
S P. Mason
and S. Bhattacharya,
Risky
debt and safety covenants
283
2. The valuation problem It is assumed that movements in the value of the firm can be described by a jump process. A Poisson-driven process is used to model these discontinuous changes. The timing of the jumps is random and independent of the impact the jumps have on the value of the firm. The instantaneous probability of a Poisson event in the time interval dt is J_dt, where 2 is the mean number of jumps per unit time. The instantaneous probability of no Poisson event occurring is 1 - idt, since the probability of more than one jump during the interval dt is of an order less than dt. Given that a Poisson event has occurred, the impact of the jump on the value of the firm is determined by a drawing from a distribution, G(Y), where Y = V(t+dt)/V(t) and V(t +dt) - V(t) is the change in the firm value due solely to the Poisson event. Successive drawings from G(Y) are independent. These firm value dynamics can be formally written as
(1) where CI is the instantaneous expected rate of return on the firm per unit time, dz is the Poisson process, and k =E( Y - 1) where (Y - 1) is the random percentage change in firm value given the occurrence of a Poisson event and E is the expectations operator. Thus, in the absence of a jump, the firm value process has a exponential drift rate of (CI--/Zk). If a jump occurs, the change in firm value is the result of this drift plus a drawing from G(Y). of ‘Frictionless Markets’,’ Merton (1976) Under the assumption demonstrated that if the jump component of the firm’s risk can be diversified away, then there results an equilibrium equation which must be satisfied by any claim whose value can be expressed as a function of the value of the firm and time. Let F(V, t) be an arbitrary claim on a firm whose value dynamics are described by (l), then F(V, t) must satisfy (1.4k)V~,(I/~)+Ft(~/,)-rF(I/,t)+~E(F(VY,t)-F(~!))=0,
(2)
which is an integro-differential equation where the expectation is taken over G(Y) and r is the assumed constant instantaneous riskless rate of interest. It is necessary to specify initial and boundary conditions in order to uniquely represent the value of the claim.* ‘There are no transaction costs or dlfferentlal taxes; tradmg takes place continuously m time; borrowmg and short-selling are allowed wlthout restrIctIon and with full proceeds avadable; and the borrowing and lending rates are equal. ‘Smce the firm’s sole source of risk is diversLiable then the expected rate of return on the firm must be the riskless rate, I.e., a=~. The same argument holds for all claims wrItten on the hrm. The diversifiable risk argument IS used to justify (2) because It allows us to most clearly demonstrate the solution techmque employed. An arbitrage argument can be used to derive eq. JFE-
c
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
284
Consider a firm with two classes of claims, equity and a single homogeneous class of discount debt, where the bondholders are promised a principal payment of B on calendar date T. In the event that the promised payment is not made, the entire firm value passes immediately to the bondholders. It is assumed the bond indenture stipulates that, during the life of the debt, the firm cannot make distributions to the equityholders nor can it issue new senior or equivalent rank claims on the firm. It is further assumed that the bond indenture specifies a safety covenant, K(r), such that K(t)
= C
e
-YtTet),
where C and y are constants. of C and y such that C
(3) We will only be concerned
e-Y(T-r)
for t 5
with those choices
T,
where Be-‘(T-‘) is the value of a riskless bond with a promised principal, B, due at t= T. This is the functional form of a safety covenant studied by Black and Cox (1976). If the value of the firm should fall to or below this barrier, then the entire firm value passes immediately to the bondholders. Thus the condition
constitutes a violation of the safety covenant, where V(t)= V is the value of the firm with T-t time periods remaining in the life of the debt. The problem of evaluating the debt in the presence of this safety covenant can be written as (r-~k)~~,(~,t)+F,(I/,t)-rF(~t)+~E(F(~Y,t)-F(I/,t))=O, F(V,t)=V
for
F(V,/,)=min(V,/,),
V~Ce-Y’T-*‘,
(2) (2a) (2b)
given G(Y). Condition (2a) says that the value of the debt equals the value of the firm if the firm value falls to or below the safety barrier. Condition (2b) says that the debt is worth the minimum of the value of the firm or the
(2). Cox and Ross (1975) show that If there exists a sufficient number of assets wtth the same jump risk, then a hedge can be formed. This results in an equilibrium equation like (2), except I and the probabilities associated with G(Y) are transformed. A demonstration of this result is available from the authors upon request.
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
285
promised principal at maturity given that the firm did not violate the safety covenant during the life of the debt. An approach to solving (2) (2a) and (2b) is suggested by Cox and Ross (1976b) who note that the solution to numerous contingent claims valuation problems is consistent with the discounted expected value of all payments made to the claim. In the case of a risky discount bond, in the presence of a safety covenant, a payment may be received either at the maturity date, T, given that the firm did not violate the safety covenant during the life of the bond or prior to the maturity date, due to a violation of the safety covenant. In order to evaluate the expected payment received at the maturity date, the distribution of firm values at time T must be determined. The distribution of firm values, given that the firm did not violate the safety covenant during the life of the debt, is given by a defective distribution. The density function is termed ‘defective’ since it does not integrate to one over the interval (K(T), x). In other words, the probability that the firm is ‘alive’ at date T is less than one since it may have previously violated the safety barrier. To evaluate the expected payment due to a violation of the safety barrier, it is necessary to determine the first passage time distribution. The first passage time distribution describes the random time at which the firm violated the safety covenant. In order to construct a solution along the lines of Cox and Ross (1976b), it is necessary to determine the first passage time distribution and the defective distribution associated with the firm value dynamics, (l), and the safety barrier, (3). At this juncture, our problem becomes isomorphic to the classical ruin problem of Collective Risk Theory.3 Let V, be the firm value at t = 0. Define q(t) to be the first passage time density function and p(V(t),t) the defective density function where I’(t) is the firm value at time t. In studying the ruin problem Prabhu (1961) demonstrates that the function
o(v,,t)=
ji pU’(t)>t)dW), K(f)
associated
with a Poisson-driven
process,
like (1) and a barrier,
like (3) will
‘The theory of collecttve risk deals wtth the business of an Insurance company. A survey of the theory from the point of vtew of stochastic processes IS gtven by Cramer (1954, 1955). The reserves of an insurance company are modeled as growing at a fixed rate due to the payment of premtums. Claims agamst these reserves occur randomly and are modeled by a Poisson process, An example of a ‘posttive’ claim is the payment made m conJunctron with an ordmary whole-hfe msurance policy m the event of the death of the policyholder. An example of a ‘negative’ clatm would be the death of a holder of an ordmary whole-hfe annuity. The ruin problem is concerned with the probabthty distrtbutton of the first time the msurance company’s reserves become negative.
286
S.P. Mason
and S. Bhattacharya,
Risky
debt and safety covenants
satisfy
(r-~k-y)T/oov,(v,,r)-D,(I/,,t)+~E(D(V,Y,t)-D(V,,t))=O, (44 D&,0)=1
for
V,>K(O),
(4b)
given G(Y). The quantity D(&,t) is the probability that the firm is still ‘alive’ at time t, given that it had an initial value of V, at t =O. In other words, D(V,,t) is the probability that the firm has not violated the safety barrier over the time interval [O,t]. The first passage time density function is given by the identity4
q(t)= -w&t).
(5)
Thus, if the proper form of the defective density function can be determined then a solution to (2), (2a), (2b) can be constructed along the line of Cox and Ross (1976b). To implement the suggested solution technique of Cox and Ross (1976b), the discounted value of all expected payments to the claim must be computed. As an example, consider the equity claim, f(Kt), in the presence of a discount bond with a safety covenant. The equity will receive nothing if the firm violates the safety covenant during the life of the bond, thus there are no payments to the equity to be evaluated prior to the maturity date of the bond, t = T. At t = T, the equity claim is worth f(V(T),T)=max[O,V(T)-B]. The value of the equity can be written f(~t)=e-‘(T-‘)
as
$U'(T)-WPU'(T),
Now consider the discount bond, F(V, t). upon a violation of the safety barrier or payments due to violation F, (V, t) and the F2(V,r). Thus F(V, tj=F,(V,/,)+F,(V,t). bond due to a violation. The discounted
T)dW-1. This claim may receive payments at maturity. Call the value of the value of the payments at maturity First consider payments to the expected value of these can be
4D( V,, t) is the probability that the firm IS alive at time t. Now consider the same quantity instant later, D(V,,t+dt). It is clear from the definition of D(V,, t) that D(V,,t+dt)-D(V,,t) 40. A decrease in the quantity D(V&t) over the interval (t,t+dt) must be precisely probability that a violation of the safety barrier occurred in the interval (t, t + dt).
one the
S.P. Muson and S. Bhattacharya,
Ruky debt und safety covenants
287
written F,(V,t)=ie-““-“P(s)q(s)ds, i
where V(s) is the value of the firm upon violation of the safety barrier at time s. The payment at maturity is min [V,B], conditional on not having violated the barrier during the life of the debt. The evaluation of this payment may be broken up into two parts F2(V,f)=e-‘(T-r)
i,
I’(T)P(I’(~-),
T)dI’(T)
+e -r(~-“%Bp(V(T),T)dl/(T).
Summarizing the problem to this point, the object is to evaluate a risky discount bond, with a safety covenant, issued by a firm that follows a jump Using either a diversifiable risk assumption or an arbitrage process. argument, it is easily shown that (2) (2a), (2b) is the formal statement of our valuation problem. Cox and Ross (1976b) outline an approach to solving (2), (2a), (2b) which is based upon the discounted expected value of all payments to the bondholders. In order to implement this solution technique, it is necessary to determine the form of the defective distribution associated with the firm value dynamics (1) and safety barrier (3). The defective distribution must be consistent with (4a), (4b). Given the defective distribution, the first passage time distribution follows from (5) and the solution technique of Cox and Ross (1976b) can be applied. As a final check, the resulting solution must satisfy (2), (2a), (2b). 3. The binomial case In this section it will be assumed that G(Y) is a binomial distribution. The form of the defective distribution associated with these dynamics will be derived. We will proceed by first solving the ‘parallel’ case, where the firm value, in the absence of jumps, has the same exponential drift rate as the barrier. This is termed the ‘parallel’ case since log V(t), in the absence of jumps, will be drifting parallel to logK(t). Explicit solutions for the value of discount debt and a safety covenant are presented. It will then be demonstrated that the solution to this case will serve as an integral part in solving the more general case of arbitrary y. Recall the firm value dynamics dV/I’=(r-M)dt+dn,
(14
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
288
where the expected rate of return on the firm is r, given the assumption that the firm’s jump risk is diversifiable. Given that a jump occurs, the impact on the value of the firm is determined by a-drawing from G(Y). Assume that G(Y) is a binomial distribution and the associated distribution function, g(Y), is represented as g(Y)=+
for
Y=ed,
=+
for
Y=em6,
=0
otherwise,
where 6 is a positive written V(t)=
constant,
(6) The
random
firm value
at time
V,exp((r-H~)t+z(t)6),
where z(t) is the sum of a random number, N, of mutually random variables, x,, i= 1,2,. . ., N, with the common density g(x)=+
for
x=1,
=$
for
x= -1,
=0
otherwise,
t can
be (7)
independent
and z(O)=O. We will refer to z(t) as the random sum of jumps. The random variable z(t), in the absence of a safety barrier, is distributed compound Poisson, h(z(t)), where h(z(t))=
f
e-A;t)N
{g(~)}~*,
N=O
and where N is the Poisson distributed random variable number of jumps in t time periods, N and x are independent and {g(~)}~* is the N-fold convolution of g(x). Feller (1966) shows that this is equivalent to
(8) where
is a modified
Bessel function
of the first kind of order z where z-z(t).
S.P. Mason and S. Bhattacharya, Rsky debt and safety cmenant.7
It is clear from the one-to-one V(t) = V, exp ((r -2k)t
correspondence
between
289
V(t) and z(t),
+ z(t)6),
that V(t) is a discrete random variable since z(t) takes on only integer values. Thus the distribution of V(t), in the absence uf a safety barrier, can be written as (9) given (7), the relationship
between
V(t) and z(t).
3.1. A ‘purallel’ barrier
Consider the barrier, K(t) = C e-y(T-f), where y = r -%k. This case is of interest since its solution will serve as an integral part of the solution to the more general case of arbitrary y. When y=r-Ak, the barrier and the firm value, in the absence of jumps, are growing at the same exponential rate. Now define the integer n, n=[(log(C/V,)-yT)/6]<0,
(10)
where [X] is the largest integer smaller than X. In other words, z(t)>n defines all those states in which V(t)>K(t) and z(t)=n defines that state in which V(t)SK(t). It is possible to solve for the defective density function, p(V(t), t), borrowing from the method of images. The proper representation of the defective density function is
(11) for z2n5 The means by which the derivation of (11) is related of images can be seen by first considering the distribution absence CJJ a safety barrier. It is clear that this distribution symmetric about zero, and ranges from - cx) to SC. However, distribution of z(t) has been characterized as a triangle, A, This will not alter the validity of this explanation. Clearly, all sWe assumed that prob(Y=e6)=prob(Y define prob(Y=@)=q and prob(Y=e-6)=q’, p(V(t),t)=(qlq’)i”e-*’
to the method of z(t) in the is discrete, in fig. 1, the for simplicity. of the sample
=e-“)=f. Tlus was done for ease of exposltlon. If we then a more general form of (11) can be written
(1z(2(44’)*nt)-r,-,,(2(4q’)*~,t)),
The sigmficance of this generahzatlon IS that the solution technique easdy handle the transformations referred to m footnote 2.
outlined
in this paper
can
290
S.P. Mason and S. Bhattacharya, Rsky debt and safety couenants
paths that end up in the right-hand tail of distribution A, to the right of n, traveled through the state n at least once. In the presence of a safety barrier, all of these sample paths would be associated with a violation. However, some sample paths traveled through n but ended up to the left of n at time t. How do we account for these sample paths? The insight is to realize that all of the sample paths that travel through II will end up distributed symmetrically around n at time t. Thus the right-hand tail of distribution A, to the right of n, represents one-half of all the sample paths that violated n during t time periods. The other half of these sample paths is distributed similarly to the left of n. The distribution of those sample paths to the left of n can be represented through the convention of mirror images. Consider an identical process, z’(t), that initiates at 2n, i.e., z’(O)=2n. The process z’(t) is termed the mirror image of z(t), given a mirror placed at n. The distribution of z’(t) will be distribution B in fig. 1. Clearly distribution B is identical to distribution A except it has been displaced by 2n. Now, if distribution B is subtracted from distribution A for z(t)zn, this will precisely account for those sample paths that had violated the barrier but had ended up to the left
z(O)
z(t) I
I
(+I
(+I
F
3 5
Timei” E ” _----8 5 2n -------__-
a
A
----_-_---------
B
(4
1
(4
Fig. 1. The random sum of jumps has an rmtial value of z(O)=O. An up Jump adds + 1 to the sum and a down jump adds - 1 to the sum. When z(s)=n, where n is a negative integer, a violation of the safety barrier has occurred at time s. We are Interested in the distribution of all those sample paths of z(t) which avoid a violation of the safety barrier durmg the time interval [O, t]. This is termed the defective distrlbutlon which can be derived from the method of images Distrlbutlon A is the distribution of z(t) m the absence of a safety barrier. Distribution B 1s the dlstributlon of a process which IS a mxror Image (mirror placed at n) of z(l) and has an imtlal value of 2n. The defective distribution C can be represented as the difference between dlstrlbution A and distribution B for z(t)Ln.
S.P. Mason and S. Bhattacharya,
Risky debt and safety covenants
291
of n. The resulting distribution defined by the difference between A and B is the defective distribution C in fig. 1. Thus in (ll), the quantity e-“‘I,(At) is analogous to A in fig. 1, the quantity e -“ZZ_2n(lZt) is analogous to B in fig. 1, and p(V(t),t) is analogous to C in fig. 1. D(V&t), as defined by eq. (4), is the probability that the firm has not violated the safety barrier in the first t time periods. For the case of the binomial jump dynamics considered here, this quantity can be formed by summing over the defective density for all zzn,
D(V,,t)=
:
e-“’
(12)
(WCL,,(W).
I=”
D(&, t) must satisfy (4a), (4b) if our derivation of (11) is correct. The fact that (12) satisfies (4a), (4b) is demonstrated in appendix A. The value of the debt, F(V,r), in the presence of a safety covenant can be written as m-l F(V;t)=
e-“‘T-“(1Z(~(T-t)))Vexp(-Ik(T-t)+z6)
1
+ f e- “‘T-‘)(ZZ(A(T.?=m
t)))Bexp(
- r(T-t))
(13)
co
+
,_z.(l(T-t))(Vexp(-Ik(T-t)+z~) c e-i(T-*)z S?=lll
-Bexp(-r(T-t))), for V>K(t),
where m=[(log(B/V)-(r-Ik)(T-t))/6]+1,
(134
and
The expression for m defines that integer such that (z(T) 2 m)o( V(T) 2 B). In other words, z(T)Zm, defines all those states, at time T, in which the firm is able to pay the debtholders their promised principal, B. The expression for n is similar to (10) in that it defines that negative integer such that (z(t) > n)o( V(t) >K(t)) and (z(t) = n)*( V(t) SK(t)). The only difference in these two expressions for n is notation. Implicit in (10) is the assumption that current time is t=O. This was done for exposition purposes. The definition
292
S.P. Mason and S. Bhattacharya,
Risky debt and safety covenants
of n used in (13b) assumes that current time is t. The first two terms in (13) correspond to the value of the debt in the absence of a safety covenant and the third represents the value of the safety covenant. The fact that (13) satisfies (2), (2a), (2b) is demonstrated in appendix B. The easiest way to understand why (13) is the value of the discount debt is to consider the value of the equity, f(V, r), where f(v,t)+F(V¶r)=V. From the definition of m, it is clear that f(v,r)=
f e-“(T-f)(Zn(A(T-t)) ‘Z=m
-ZZ_zn(A(T--t)))
X (Vexp(-Ak(T-t)+z6)-Be-“T-‘)). This is simply the expected value of the payment to equity, (V(T)-B), over the defective distribution and discounted by the riskless rate r. It is possible to write
taken
V= e-r(T-‘)E[ V( T)] , where the expectation is taken over p’(V(T)), (9),
Grouping the terms as we have done in (13), it is straightforward to isolate the value of the covenant. Remembering the definition of m, and that p’(V(T)), (9), will describe the distribution of V(T) in the absence of a safety covenant, then clearly the first two terms represent the value of debt without a safety covenant. Therefore, the last term in (13) is the value of the safety covenant. 3.2. A ‘non-parallel’ barrier It was earlier asserted that (ll), the defective density function when y= would serve an integral part in determining the defective density function for arbitrary y. To see this, consider a y <(r -Ak). This would describe a case where the firm, in the absence of jumps, is drifting away from the barrier. Fig. 2 depicts this situation as it relates to the random walk of z(t). When y is less than r-ilk, the barrier and the line z(t)=0 are diverging at the rate of r-lk-y. In the previous section, which dealt with a ‘parallel r -Ik,
S.P. Mason and S. Bhattacharya,
z(O) z(t,)
Risky debt and safety covenants
z(t*)
z(tg)
I
I
I
+3 .
--
+2 ‘.
--
8 ,. E +’ 4 B
__
-----
- ----.-
_t ,t3
\
“3,
Fig. 2. When the barrier is non-parallel the critical values for the random sum of jumps, z(t), that describe a violation will be time dependent. For example, given the above non-parallel barrier, if z(t)=n,, where n, is a negative integer and tf t,, then the barrier is violated. However, if z(t)=nl for t > t,, then the barrier is not violated.
barrier, the barrier is parallel to z(t)=O,Vt. See fig. 1. With a ‘parallel barrier, z(r)= n describes a violation Vt. However, when the barrier is nonparallel, the critical values for z(t) that describe a violation will be time dependent. Assume that V, is the firm value at t =O. It is then true that z(t)=n,, where n, rn and n is given by (lo), would describe a violation of the safety barrier for t 5 t, where (log(C/&)-y(T-t,)-(r-Ak)t,)/6=n,
defines t,. For t E (cl, tJ, z(t)=n, - 1 =n2 would describe a violation of the safety barrier conditional on z(t) > n,, Vt E [0, cl]. The expression (log(C/T/,)-y(T-t,)-(r-lk)t,)/6=n,
defines t,. Continuing integers ni+t
=ni-1
in this manner, we are able to define a sequence of
for
i=1,2,...,
294
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
where n, is given by (10) and a sequence of points in time
It is clear that the barrier in fig. 3 is equivalent to the barrier in fig. 2. That is to say that any sample path for z(t) which violates the barrier in fig. 2 will also violate the barrier in fig. 3 at the same time, and the value of the payment received by the bondholders upon violation will be the same in both cases. This is the first hint that the solution to the ‘parallel’ case will serve an important part in solving the arbitrary y case.
2’
I
z(t1)
z
:)
z(tg)
+3
+2
K +1
4 %
___---__ _____.:__
co E
t az “1 “2
“3
Fig. 3. This barrier is equivalent to the barrier depicted in fig. 2. Any sample path for the random sum of jumps, r(t), which violates the barrier in fig. 2 will violate this barrier at the same time.
For t I t,, the defective density can be written as (14) for zln,, and where V(t)= Voexp ((I- Ak)t + ~6). However what is the defective density function for t E (Cl, t2]? As a specific example, consider the defective density at tZ,Pz(V(t,), tz). Because the process for z(t) is Markov, the Chapman-Kolmogorov condition can be applied to the defective density function. To understand the Chapman-Kolmogorov condition, consider a
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
295
specific state, z*(t,), at t,. The initial state for z(t) is z(O)=O. The ChapmanKolmogorov condition says that the probability of going from z(O)=0 to z(t2)=z*(t2) can be written as prob(z(O)+z*(tz))=
2 prob(z(O)+z(tr)) (+r)Ln,l x prob(z(t,)+z*(t,)).
(15)
That is to say that the probability of going from z(0) to z*(t2) is the product of the probability of going from z(0) to z(tl), times the probability of going from z(tl) to z*(t2), summed over all admissible z(tl). Clearly the probability of going from z(0) to a specific z(tl)=j’ is given by p1 (V(t,), tl), (14), for z =j’. Now it is left to determine the probability of going from z(t, ) =j’ to z(t*) = z*(tz).
The insight here is to realize that this is simply the ‘parallel’ solution, (ll), applied over a time span of (tz -t, ), where the initial state is j’, not 0, and the distance to the barrier is n2 -j’, not n,. In other words,
where n2 =nl - 1 and it is understood that z* = z*(t,). It then follows that
Expression (16) is simply a restatement for z&n,. Chapman-Kolmogorov condition. This allows us to define the recursive system
Pj+l(V(ti+l),ti+l)=
f
of expression
(15), the
Pi(~(ti),ti)e-““t+‘-‘3
j=n,
(17)
296
S.P.
where i=l,2,...,
Mason and
S. Bhattacharya,
pi(V(r,),t,)
Risky
debt and safety covenants
is given by (14) and
v(ti)= Voexp((r-;lk)ti+j6),
for
Z>,lli+l.
The defective distribution of V(t) for t E (ri, ri+ 1] is p(V(t),t)=
2 pi(V(ti),ti)
e-‘(‘-‘J
j=n,
(18) where i = 1,2,. . ., and p,(V(t,), ti) is given by (17) and V(r) = V, exp ((I - Ak)r+ z6), for zlr~+~.
A similar analysis would hold for the case where y > r - Ak.
Thus, it is possible to construct a solution to (2), (2a), (2b) for arbitrary y, along the lines of Cox and Ross (1976b), by utilizing (18), the defective density function. For instance, the value of equity,f(V, t), in the presence of a safety covenant can be written f(Kr)=
f
p(V(T),T)(Vexp(-Ik(T-r)+z~5)-Be-’(~-’)),
z=Wl
(19)
where p(V(T),T) is given by (18) for (T-r)E(r,_l,ri) and m is given by (13a). The value of the debt, F(I/,r), follows from the identity, F(V, r)=
v-f(K r). 4. Comparison with Black and Cox solution Black and Cox (1976) provide solutions for the value of a finite lived risky discount bond and a safety barrier, under the assumption that the firm value follows continuous diffusion-process dynamics. In their conclusion they state: It should be noted that if the value of the firm follows a jump process, the value of a safety covenant may be drastically altered since the value of the firm could then reach points below the bankruptcy level without first passing through it.
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
291
This section briefly examines this conjecture for the binomial jump dynamics assumed in the previous section.6 Consider the Black and Cox (1976) problem of valuing a discount bond in the presence of a safety covenant when it is believed that the firm value dynamics are dV/V=udt+odz,
(20)
where a2 is the variance rate per unit time and dz is a standard GaussWiener process. These dynamics will lead to a continuous sample path for the value of the firm. Assume that the firm value dynamics are in truth described by a jump process (l), with G(Y) being the binomial distribution. What will be the nature of the valuation errors induced by using the Black and Cox (1976) valuation for the value of the discount bond with a safety covenant? In order to utilize the Black and Cox (1976) solution (BC solution), the variance rate must be estimated. Assume that the estimate is the truth variance per unit time,
The BC solution, using 6’ and T= T-t,
can be written
logV-logB+(r-1/2&2)t BJ;
F'(y7)=Be-'W
+V@
log B - log V- (r + 1/2e2)2 fl^X!fz q+l
210gC-logB-logV+(r-2Y+1/2~2)~
X@
&/G
>
xQi 2logC-logB-logV+(r-2y-1/2&‘)r ( 6Black and Cox (1976) did not assume their solution will hold in such a world, results to the results of this paper.
&fi
(21)
that the firm’s diffusion risk was diversifiable. Since it is permissible to compare Black and Cox (1976)
S.P. Mason and S. Bhattacharya,
Risky debt and sajety covenants
298
where q=2((r-y)/G2) and Qi is the unit normal distribution function. The first two terms correspond to the risky discount bond valuation of Merton (1974) and the last two terms represent the value of the safety covenant. One insight into the nature of the valuation errors is provided by the realization that the binomial jump dynamics have a fundamental relationship to the Gauss-Wiener dynamics assumed by Black and Cox (1976). It is easily demonstrated that the binomial jump dynamics will converge to the GaussWiener dynamics assumed by Black and Cox (1976), if 2+co while 6+0 such that U2 = c2. In other words, if the mean number of jumps per unit time is made progressively larger, J.+co, as the amplitude of the jump is made progressively smaller, 6+0, in such a way as to keep the quantity U2 equal to a constant, c?, then these binomial jump dynamics will converge to the Gauss-Wiener dynamics used by Black and Cox (1976). Therefore, for a given rr2, independent of the nature of the error, i.e., overestimate or underestimate, we would expect the magnitude of the error to approach zero as A--r00 and 6-+0 such that Xi2=ts2. As a more direct assault on the valuation question, a computer program was written around (18), the defective density function. The procedure was to evaluate the debt, F( V,z), via a direct evaluation of the equity, f( I’, T), and to use the fact that f(V, 7) +F( V, 7) = V. Eq. (19) was used to evaluate f(V, T). Comparisons of F(K r) and F’( K r) were made over a wide range of the variables, V and 7, and the parameters, ;1, 6, C, y, r and B. The valuation of F(V, 7) was broken down into the valuation of a risky discount bond without a safety covenant, G(V,T), and the valuation of the safety covenant, g( V,7). Therefore F( V,z) = G( V,T) + g( I’, z). With regards to the BC solution, F’(V, z), the discount bond portion was labeled G’(V, 7) and the safety covenant portion g’( I’, 7). Therefore F’( V,r) = G’( V,z) + g’( I’, t). The nature of the valuation errors associated with G’(V, T) are as follows. For V$=>Band VeB it was found that G’(V,‘,r)>G(V,7). That is to say, if the value of the firm, V, is significantly greater than the promised payment, B, then the fact that the firm value can jump is detrimental to the bondholders and therefore G’(V, T) > G(V, 7). If the value of the firm is significantly less than the promised payment, then the fact that the firm value can jump will clearly benefit the equityholders, thus hurt the bondholders, and therefore G’(V,2)> G(V,T). In addition, it was found that if V=.Be-” then G’(V,T) < G(I’, 7). That is to say, if the value of the firm was approximately equal to the present value of the promised payment then the discount bond portion of the BC solution is an underestimate of the true value.7 It was rare to observe ‘These results are similar to those of Merton (1974, 1976a) in a study of misspecification error in the pricing of call options. Merton (1976) presents a valuation formula for a call option, where the underlying stock price, S, dynamics contain both a diffusion and a jump component. For the case where the jump magmtude, Y, is drawn from a lognormal distribution, Merton (1976a) compares his valuation formula to the Black and Scholes (1973) call option valuation, which assumes that the stock price dynamics contain only a diffusion component. Merton
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
299
valuation errors, in relative terms, of greater than 5 %. Finally, just as would be expected from the relationship between the binomial jump dynamics and the Gauss-Wiener dynamics, G(V,r)+G’(V,r) as 2-03, and 6-O such that As = a*. The nature of the valuation errors associated with g(V,r), the value of the safety covenant, is different. For all I/ and r, it was found that g’(V, r) > g(V, r). The safety covenant portion of the BC solution, g’(V, r), is an overestimate of the true safety covenant value, g( V, z), for all firm values and time to maturity. Furthermore, it was not rare to encounter valuation errors, in relative terms, of more than 25 7’. This confirms, for the binomial jump dynamics considered here, the Black and Cox (1976) conjecture that the value of a safety covenant could be materially affected if the firm followed a jump process. Lastly, it is true that g(V,r)-+g’(V,r) as A-+ cc and 6+0 such that 116*= a2.8
5. Conclusions This paper considered the valuation of risky debt and safety covenants when the firm value follows (possibly) discontinuous sample paths. The analysis made use of a class of probabilistic first passage time techniques, developed within Collective Risk Theory, for Poisson processes. For the Poisson dynamics considered, it was possible to derive the form of the defective distribution of firm value, and to construct solutions to the valuation problem along the lines of Cox and Ross (1976b). The results of this paper were compared to the work of Black and Cox (1976) which
(1976b) shows that if the stock prrce dynamics do contain both a diffusion and a jump component then the Black and Scholes (1973) call option valuation will systematically underestimate the true value of the call option for almost all regions of S and r. The exception is SzE’e-“, where E’ is the exercise price of the call option. In this region the Black and Scholes (1973) valuation is an overestimate. Since Merton (1974) demonstrated that equity, in the presence of a risky discount bond, can be viewed as a call option, we are not surprised to find the nature of our errors to be the opposite of Merton’s since F( V, T) +/( V, 7) = V. 81t is easy to extend our comparison of safety-covenant solutions, with and without jumps, to perpetual consol bonds. Consider lirm value dynamics that contam both a diffusion and a jump component. Assume diversifiability of jump risk, and payout rates of P(V) to all securityholders and C to the bondholders. Further assume that a safety covenant exists such that VJK constitutes a violation. The valuatton equation, $~*v~F,,+[(r-lk)V-P(V)]F,-rF+C+IE[F(YV)-F(V)]=O, with the appropriate boundary conditions, including F(V)= V for VJK. If Y IS deterministic then straightforward solutions are obtained for Y> 1 (jumping through the safety barrier is impossible) or Y=O (the Samuelson ruin jump), especially if P(V) 1s proportional to V. These solutions are simple transformations of the corresponding diffusion-only solution, e.g., r is simply replaced by (r+n) m the Y=O case. When Y< 1 and deterministic, the solution has a tractable recursive structure over stages [K s VsK/Y]. . [K/Y” s VsK/Y”+‘], etc., for n=0,1,2 ,.... Details of these solutions are available from the authors upon request.
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
300
addresses the same valuation problem under the assumption of continuous firm value dynamics. It was shown that discontinuous, versus continuous, firm value dynamics can have a significant differential impact on the value of a safety covenant, as conjectured by Black and Cox (1976).
Appendix A Below it is verified that eq. (12),
wG’,,t)= f e-“‘(Z,(nt)-z,_,,(nt)), Z=”
satisfies (4a) and (4b), (r-ik-y)K&)(Kj,t)D&O)=1
for
D,(v&t)+wD(I/,
r,t)--D(vmt))=O,
V,>K(O).
Remembering that (12) was derived under the assumption rewrite (4) as
of y= r-lk,
D,(I/,,t)+~D(I/,,t)=~E(D(~o’,Y,)).
we
(A.11
A rule for the differentiation of Bessel functions, given in Olver (1972), is
Applying this rule, it is easily verified that D&t)=
-1D(V,,t)+(L/2)
2 e-“’ z=fl
Now consider the quantity E(D(F$Y,t)) where the expectation is taken over g(Y), (6), the binomial density function. The quantity Y can be either an upjump, Y+ =ed, or a down-jump, Y- = e-* with equal probability. Thus
D(V,,Y’,t)
is
simply the solution, D(V,,t), where the initial firm value, V,,
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
301
has been increased by one jump. D(VOY-, t) is the solution, D(VO,t), where the initial firm value has been decreased by one jump.
D(V,Y’,t)=
f I=“-
e-“‘(Zl(~t)-Zz-z(n-l)(~T)). 1
The quantity (n - 1) appears in the place of n because the distance between the initial firm value, V,Y’, and the barrier has increased by one jump. D(V,Y-,t)=
F
e-*‘(ZI(IZt)-Zl-2(n+1)(IZt)).
r=n+1
The quantity (n+ 1) appears in the place of n because the distance between the initial firm value, V,Y-, and the barrier has decreased by one jump. Thus
J-w(b’,~t))=(w2)
e-“‘(Z.(~t)-zn-z(n-l)(~t))
f 2=fl-1
+(a)
f
e-“‘(z,(~t)-z,-,,,-,,(lt)).
z=n+l
In the first summation in the above expression, change the summation index to z’=z+ 1. In the second summation, change the index to z”=z- 1. AE(D(VoY,t))=(A/2)
jJ e-“‘(ZI,_l(~t)-Z,,_2n+l(lZt)) I’=#8 +(A/21
f e-“‘(Z,..+,(lr)-Z,.._,,_,(lt)), Z”E”
which is equal to D,(I/,, t) + lD(I/,, r). Thus D(I/,, t), (12), satisfies (4). Consider the initial condition, (4a). The following is true of modified Bessel functions: Z,(O)=O,
Vz,
except
2 =0,
where Z,(O)= 1.
Thus D( V,, 0) = 1 for n < 0, which is to say, for V, >K(O).
302
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
Appendix B Below it is verified that eq. (13), m-l FK’,t)=
c
z=-* +
e-“‘r-“I,(~(T-t))Vexp(-~k(T-t)+zs)
F e-"(T-')Z,(A(T-
z=m + f
t))B exp (- r(T-
c))
e-"(r-r)Zr_zn(lZ(T-t))
z=lll
x (Vexp(-Lk(T-t)+z6)
-Bexp(-r(T-r))),
satisfies (2), (2a), (2b), (r-A.k)VF,(V,t)+F,(V,t)-rF(V,t)+AE(F(VY,t)-A
F(V,t)=V
for
,/,t))=O,
V~Ce-y(r-r),
F(V, T)=min(V,‘,).
Clearly we can rewrite (2) as (r+l)F(V,t)-(r-Ak)W,(V,t)-F,(V,t)=AE(F(VY,t)).
(B.1)
First evaluate the RHS of (B.l). We will use the same argument used to evaluate E(D(VoY, t)) in appendix A. The expectation is taken over the binomial density, g(Y), (6), where there is equal probability of an upward jump, Y+ =e’, and a downward jump, Y- =e-*.
The quantity F(VY+,t) is simply the value of the debt assuming that the current firm value is one jump further away from the barrier. The quantity F(VY-,t) is the value of the debt assuming the current firm value is one jump closer to the barrier.
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
303
m-2
c
F(vY+,t)=
e-“‘T-“ZZ(A(T-t))Vexp(-A.k(T-t)+(z+1)6)
z=-m
jJ
+
e-’ ““-“Z,(A(T-t))(Bexp(-r(T-t))
z=m-1
x (Vexp(-lk(T-t)
F(VY_,t)=
f
+(z+l)S)-Bexp(-r(T-t))),
e-“‘T-“Z,(~(T-t))Vexp(-A.k(T-t)+(z-l)b)
.7=-m
+
1
e-“‘*-‘)Z,(A(T-t))Bexp(-r(T-t))
z=m+l
m
i(=-f)zz_2(n+1)(A(T-t)) +z=;+lex(Vexp(-Ak(T-t)
+(z-1)6)-Bexp(-r(T-t))).
In the above expression for F( VY+, t), change the summation z+ 1. In the expression for F(VY-, t) change the summation z-l. We are now able to write m-l
/ZE(F(VY,t))=(A/2)
C
e-l’T-z)ZI,_l(~(T-t))
x Vexp(-Ak(T-t)+z’d)
+(A/2)
2
e-“‘T-“Z,._l(~(T-t))
z’=m
xBexp(-r(T-t))
+ (A/2) C e-“(T-‘)Z,._2n+l(~(Tz’=m
x (Vexp(-Ak(T-t)+z’S)-Bexp(-r(T-t)))
t))
index to z’ = index to z”=
304
Risky debt and safety covenants
S.P. Mason and S. Bhattacharya, m-l +
(A/2)
C
e-A(T-l)Z.SS+1(A(T- t))
z”Z - 00
x Vexp(-Ak(T-t)+z”S)
+ (A/2) f
L”C#j
e-A(T-r)Zzsf+ 1(l(T-
t))
xBexp(-r(T-t))
2 e-“(T-‘)Z,,,-2,-,(I(T-t))
+W)
n”=m
x (Vexp(-Ik(T-t)+z”6)-Bexp(-r(T-t))). Now consider the LHS of (B.l). The first term is clearly m-l (r+L)F(V;
t)= (r+l)
C e-“(T-r)Zz(l(Tz=-03
t))
x Vexp(-Ak(T-t)+&) e-“(T-‘)Z,(A(T-t)) t=lPl
+@+A) f
xBexp(-r(T-t)) + (r+l)
f e-“(T-r)Z,_2n(~(T-t)) 1=llI
x (Yexp(-Ak(T-t)+zd)-Bexp(-r(T-t))). The second term is equally clear m-l -(r-Ak)W,(I/,t)=
-(r-lk)
c
e-A(T-f)1z(A(7’-t))
z=-m
x Vexp(-Ak(T-t)+zS) -(r-ilk)
f e-A(r-z)Z,_,,(I(T*=lVl
x Vexp(-Ak(T-t)+za).
t))
S.P. Mason and S. Bhattacharya,
Risky debt and safety covenants
In order to evaluate the third term, -F,(V,t), differentiation of a Bessel function.
305
again recall the rule for the
m-l
-F,(K:t)= - (1 +k)A
C e-“‘T-r)Z~(A(T-t)) I=-CO
x Vexp(-Ik(T-t)+zd) -(r+jl)
f
e-“(T-“Z,(I(T-
t))
xBexp(-r(T-t)) -(l+k)l
5 e-“‘T-‘)Z,_,,(~(T-t)) z=fPl
x (Vexp(-Ak(T-t)-tzh))
+(r+l)
f
e-“(T-“Z,_2n(lZ(T-_))
z=lll
xBexp(-r(T-t)) m-1
+(W
Cme-“(T-r)(ll+l(~(T-t))
+Z,-l(A(T-t)))
*=-co
x Vexp(-Ak(T-t)+z6) + (A/2) $ e-“‘T-ll(Zt+l(~(T-t)) 1=llJ X
+Z,-l(I(T-t)))
Bexp(-r(T-t))
+(1/2)
5 eyACTVt) Z=m
x(L-,,+1(4~--~)) +I,-2,-l(R(T--t))) x (Vexp(-Ik(T-t)+zd)
-Bexp(-r(T-t))).
Upon careful inspection, it becomes clear that the sum of the expressions for
S.P. Mason
306
Risky debt and safety dovenants
und S. Bhuttacharya,
(I+ i)F(V, t), - (r - ik) VF,(V, t) and the first four terms in the expression for -F,(V,t) sum to zero. It is then seen that the last three terms in the equals the expression for Z(F(VX:t)) thus (13) expression for -F,(V,r) satisfies eq. (2). Now consider the boundary condition, F(V,t)=V From
the definition
for
VSCe-Y(T-l).
of n.
VgCe-y(T-‘)
for
n=O,
m-l F(I/,t)=
e-“‘T-“I,(~(T-t))‘Vexp(-~k(T-t)+z6)
c
2=-03 + f e-“‘T-‘)ZZ(~(T-t))Bexp(-r(T-t)) Z=m + f
e-“(T-‘)Z,(A(T-t))
z=lll
x (Vexp(-Ak(T-t)+za)
=
f
e-A(T-f)Zz(~(T-
-Bexp(-r(T-t)))
t))Vexp(-Ak(T-t)+zd)
z=-03
=e-r(T-r)E(V(T))= Finally,
consider
condition
I/.
(2b),
F(I/,T)=min[1!B], m-l
C
F(VT)=
L=--m +
Z,(O)Vexp(zd)+
2 Z,(O)B z=lll
5
Z,_,,(O)(Vexp(zG)-B). *=Wl
Given the nature of the safety barrier, B>C, and from the definitions of n and m, m > n. Thus if I/ 2 B, then that implies 0 2 m > n. The first and third terms equal zero since I,(O) =0, Vz except z =O. The second term equals B since ZO(0)=l. Therefore F(V;T)=B if VZB. If CO >n. The second and third term equal zero and the first term equals V. Thus F(V, T)= V if c< V
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
307
References Black, F. and J.C. Cox, 1976, Valuing corporate securittes: Some effects of bond indenture provision, Journal of Finance 31, 351-367. Black, F. and M. &holes, 1973, The pricing of options and corporate liabtlities, Journal of Political Economy 81, 637459. COX, J.C. and S.A. Ross, 1975, The general structure of contingent claims pricmg, Mimeo. (University of Pennsylvania, Philadelphia, PA). Cox, J.C. and S.A. Ross, 1976a, The valuation of options for alternative stochastic processes, Journal of Financial Economtcs 3, 145-166. Cox, J.C. and S.A. Ross, 1976b, A survey of some new results in financial option pricing theory, Journal of Fmance 31, 383402. Cramer, Harold, 1954, On some questions connected with mathemattcal risk, University of Cahforma Publications m Statistics 2. 99-124. Cramer, Harold, 1955, Collective risk theory, Jubilee volume of the Skandia Insurance Co. (Ab Nordiska Bokhandeln, Stockholm) S-92. Feller, W., 1966, Inlinitely divisible distributions and Bessel functions associated with random walks, S.I.A.M. Journal of Apphed Mathematics. Galai, D. and R.W. Masulis, 1976, The optton pricing model and the risk factor of stock, Journal of Fmancial Economics 3, 53381. Ingersoll, J.E., 1977, A contingent claims valuation of convertible securities, Journal of Financial Economics 4, 289-321. Jensen, M.C and W.H. Meckling, 1976, Theory of the firm. Managerial behavior, agency costs and ownership structure, Journal of Financtal Economics 3, 305-360. Jones, E.P., 1978, Arbitrage pricmg of opttons for mixed dtffusion - Jump processes, Working paper (M.I.T., Cambridge, MA). Merton, R.C., 1973, A rational theory of option pricmg, Bell Journal ot Economics and Management Science 4, 141-183. Merton, R.C., 1974, On the pricmg of corporate debt: The risk structure of interest rates, Jouri al of Finance 29, 449470. Merton, R.C., 1976a, Option pricing when underlying stock returns are discontmuous, Journal of Financial Economics 3, 125-144. Merton, R.C., 1976b. The impact on option pricing of specification error m the underlying stock price returns, Journal of Finance 31, 333-350. Myers, S.C., 1977, Determinants of corporate borrowing, Journal of Financial Economics 5, 147175. Olver, F.W.J, 1966, Bessel functions of integer order, Ch. 9 m: Handbook of mathematical functions, National Bureau of Standards Applied Math Series 55 (U.S. Government Printmg Office, Washmgton, DC). Prabhu, N.U., 1961, On the ruin problem of collective risk theory, Annals of Mathematics and Statistics. Rosenfeld, E., 1979, Stochastic processes of common stock returns: An empirical examination, Ph.D. dissertation (M.I.T., Cambridge, MA). Smith, C.W. and J.B. Warner, On financial contracting: An analysis of bond covenants, Journal of Financial Economics 7. 117-161.
of Financial
Economics
RISKY
9 (1981) 281-307.
North-Holland
DEBT, JUMP PROCESSES, SAFETY COVENANTS* Scott
Publishing
Company
AND
P. MASON
Harvard University, Boston, MA 02163, USA
Sudipto
BHATTACHARYA
Stanford University, Stanford, CA 94305, USA
Received
January
1980, final version
received
March
1981
The usual assumptions in the continuous-time contingent claims pricing of risky debt are (1) the firm is in default only when the value of its remaining assets falls short of the currently due promised payment and (2) the firm value follows continuous diffusion-process dynamics. It is the Jomt relaxation of these two simplifying assumptions that motivate this paper in its study of the valuation of risky debt and safety covenants when the firm value follows (possibly) discontinuous sample paths. Exphclt solutions are derived and compared to the work of Black and Cox (1976)
1. Introduction Black and Scholes (1973) identified the isomorphic relationship between the problems of pricing options and ‘discount’ debt with default risk. As a result, the finance literature has witnessed a great growth of research on the pricing of debt claims with more complex payoff patterns, using the techniques of continuous-time contingent claims analysis pioneered by Black and Scholes (1973) and Merton (1973). The usual assumption in most of these models regarding default, and consequent takeover by bondholders, is that a firm is in default only when the value of its remaining assets falls short of the currently due promised payment on the debt. Much of the debt valuation literature - for example, Merton (1974), Galai and Masulis (1976), Ingersoll (1977) - has also assumed continuous diffusion-process dynamics for firm value. It is the joint relaxation of the above two simplifying assumptions that motivates this paper. It is a common observation that firms enter into bankruptcy or reorganization procedures long before the market value of their remaining *The authors wish to thank the referee, Scott dlscussions and editorial suggestions.
0304-405X/81/000&0000/$02.50
Richard,
and E. Philip
0 North-Holland
Jones for many
helpful
282
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
assets drops below currently due debt obligations. Such phenomena have their basis, at least in part, in the distorted stockholder/manager incentives which result from having debt with ‘very’ high default risk. These issues have been discussed by Jensen and Meckling (1976) and Myers (1977). As a procedural matter, this type of default and bankruptcy procedure often arises from the inclusion of various balance sheet and income statement based ‘financial health’ criteria in debt covenants, which have been examined and analyzed in the work of Smith and Warner (1978). One way to recognize these complications in the pricing of risky debt is to postulate the stylized covenant of a ‘safety barrier’, a floor for firm value such that if the firm value reaches or drops below such a barrier, debtholders take over the firm. Black and Cox (1976) developed this approach with diffusion process dynamics for firm value. Discontinuous changes in firm value, resulting from the arrival of information that affects investors’ anticipation of future growth and profitability, appear to be a pervasive feature of economic life, both from casual empiricism as well as the recent work of Rosenfeld (1979). In this paper, we extend the analysis of risky debt pricing with safety barriers to cases where the firm value follows (possibly) discontinuous sample paths. Our goals and achievements in this paper are twofold. The first contribution of this paper is methodological. Most existing analyses of contingent claims, including this paper, make use of solutions and techniques that are transparently related to discounted expected value calculations with simple stochastic processes, as noted by Cox and Ross (1976b). Our analysis makes use of a class of probabilistic ‘first passage time’ techniques for discontinuous processes with somewhat complicated boundary conditions that have not been widely used by financial economists. The incorporation of these techniques may be of more general importance in the analysis of the valuation of financial contracts under increasingly complex and ‘realistic’ assumptions. In this respect, our analysis extends the results on valuation of options with jump processes, pioneered by Cox and Ross (1976a) and Merton (1976a). Second, we develop estimates of the value of safety covenants and compare our results to those of Black and Cox (1976) who consider the value of a finite lived risky discount bond and a safety covenant under the assumption that the firm value follows continuous diffusionprocess dynamics. In section 2 we formulate a pricing equation for contingent claims written on a firm that follows possibly discontinuous sample paths. The problem of valuing finite lived risky discount debt, in the presence of a safety covenant, is posed and an approach to solving this problem is outlined. Section 3 presents a solution to the valuation problem for a specific set of Poisson dynamics, and section 4 compares this solution to the work of Black and Cox (1976). Section 5 contains our concluding remarks.
S P. Mason
and S. Bhattacharya,
Risky
debt and safety covenants
283
2. The valuation problem It is assumed that movements in the value of the firm can be described by a jump process. A Poisson-driven process is used to model these discontinuous changes. The timing of the jumps is random and independent of the impact the jumps have on the value of the firm. The instantaneous probability of a Poisson event in the time interval dt is J_dt, where 2 is the mean number of jumps per unit time. The instantaneous probability of no Poisson event occurring is 1 - idt, since the probability of more than one jump during the interval dt is of an order less than dt. Given that a Poisson event has occurred, the impact of the jump on the value of the firm is determined by a drawing from a distribution, G(Y), where Y = V(t+dt)/V(t) and V(t +dt) - V(t) is the change in the firm value due solely to the Poisson event. Successive drawings from G(Y) are independent. These firm value dynamics can be formally written as
(1) where CI is the instantaneous expected rate of return on the firm per unit time, dz is the Poisson process, and k =E( Y - 1) where (Y - 1) is the random percentage change in firm value given the occurrence of a Poisson event and E is the expectations operator. Thus, in the absence of a jump, the firm value process has a exponential drift rate of (CI--/Zk). If a jump occurs, the change in firm value is the result of this drift plus a drawing from G(Y). of ‘Frictionless Markets’,’ Merton (1976) Under the assumption demonstrated that if the jump component of the firm’s risk can be diversified away, then there results an equilibrium equation which must be satisfied by any claim whose value can be expressed as a function of the value of the firm and time. Let F(V, t) be an arbitrary claim on a firm whose value dynamics are described by (l), then F(V, t) must satisfy (1.4k)V~,(I/~)+Ft(~/,)-rF(I/,t)+~E(F(VY,t)-F(~!))=0,
(2)
which is an integro-differential equation where the expectation is taken over G(Y) and r is the assumed constant instantaneous riskless rate of interest. It is necessary to specify initial and boundary conditions in order to uniquely represent the value of the claim.* ‘There are no transaction costs or dlfferentlal taxes; tradmg takes place continuously m time; borrowmg and short-selling are allowed wlthout restrIctIon and with full proceeds avadable; and the borrowing and lending rates are equal. ‘Smce the firm’s sole source of risk is diversLiable then the expected rate of return on the firm must be the riskless rate, I.e., a=~. The same argument holds for all claims wrItten on the hrm. The diversifiable risk argument IS used to justify (2) because It allows us to most clearly demonstrate the solution techmque employed. An arbitrage argument can be used to derive eq. JFE-
c
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
284
Consider a firm with two classes of claims, equity and a single homogeneous class of discount debt, where the bondholders are promised a principal payment of B on calendar date T. In the event that the promised payment is not made, the entire firm value passes immediately to the bondholders. It is assumed the bond indenture stipulates that, during the life of the debt, the firm cannot make distributions to the equityholders nor can it issue new senior or equivalent rank claims on the firm. It is further assumed that the bond indenture specifies a safety covenant, K(r), such that K(t)
= C
e
-YtTet),
where C and y are constants. of C and y such that C
(3) We will only be concerned
e-Y(T-r)
for t 5
with those choices
T,
where Be-‘(T-‘) is the value of a riskless bond with a promised principal, B, due at t= T. This is the functional form of a safety covenant studied by Black and Cox (1976). If the value of the firm should fall to or below this barrier, then the entire firm value passes immediately to the bondholders. Thus the condition
constitutes a violation of the safety covenant, where V(t)= V is the value of the firm with T-t time periods remaining in the life of the debt. The problem of evaluating the debt in the presence of this safety covenant can be written as (r-~k)~~,(~,t)+F,(I/,t)-rF(~t)+~E(F(~Y,t)-F(I/,t))=O, F(V,t)=V
for
F(V,/,)=min(V,/,),
V~Ce-Y’T-*‘,
(2) (2a) (2b)
given G(Y). Condition (2a) says that the value of the debt equals the value of the firm if the firm value falls to or below the safety barrier. Condition (2b) says that the debt is worth the minimum of the value of the firm or the
(2). Cox and Ross (1975) show that If there exists a sufficient number of assets wtth the same jump risk, then a hedge can be formed. This results in an equilibrium equation like (2), except I and the probabilities associated with G(Y) are transformed. A demonstration of this result is available from the authors upon request.
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
285
promised principal at maturity given that the firm did not violate the safety covenant during the life of the debt. An approach to solving (2) (2a) and (2b) is suggested by Cox and Ross (1976b) who note that the solution to numerous contingent claims valuation problems is consistent with the discounted expected value of all payments made to the claim. In the case of a risky discount bond, in the presence of a safety covenant, a payment may be received either at the maturity date, T, given that the firm did not violate the safety covenant during the life of the bond or prior to the maturity date, due to a violation of the safety covenant. In order to evaluate the expected payment received at the maturity date, the distribution of firm values at time T must be determined. The distribution of firm values, given that the firm did not violate the safety covenant during the life of the debt, is given by a defective distribution. The density function is termed ‘defective’ since it does not integrate to one over the interval (K(T), x). In other words, the probability that the firm is ‘alive’ at date T is less than one since it may have previously violated the safety barrier. To evaluate the expected payment due to a violation of the safety barrier, it is necessary to determine the first passage time distribution. The first passage time distribution describes the random time at which the firm violated the safety covenant. In order to construct a solution along the lines of Cox and Ross (1976b), it is necessary to determine the first passage time distribution and the defective distribution associated with the firm value dynamics, (l), and the safety barrier, (3). At this juncture, our problem becomes isomorphic to the classical ruin problem of Collective Risk Theory.3 Let V, be the firm value at t = 0. Define q(t) to be the first passage time density function and p(V(t),t) the defective density function where I’(t) is the firm value at time t. In studying the ruin problem Prabhu (1961) demonstrates that the function
o(v,,t)=
ji pU’(t)>t)dW), K(f)
associated
with a Poisson-driven
process,
like (1) and a barrier,
like (3) will
‘The theory of collecttve risk deals wtth the business of an Insurance company. A survey of the theory from the point of vtew of stochastic processes IS gtven by Cramer (1954, 1955). The reserves of an insurance company are modeled as growing at a fixed rate due to the payment of premtums. Claims agamst these reserves occur randomly and are modeled by a Poisson process, An example of a ‘posttive’ claim is the payment made m conJunctron with an ordmary whole-hfe msurance policy m the event of the death of the policyholder. An example of a ‘negative’ clatm would be the death of a holder of an ordmary whole-hfe annuity. The ruin problem is concerned with the probabthty distrtbutton of the first time the msurance company’s reserves become negative.
286
S.P. Mason
and S. Bhattacharya,
Risky
debt and safety covenants
satisfy
(r-~k-y)T/oov,(v,,r)-D,(I/,,t)+~E(D(V,Y,t)-D(V,,t))=O, (44 D&,0)=1
for
V,>K(O),
(4b)
given G(Y). The quantity D(&,t) is the probability that the firm is still ‘alive’ at time t, given that it had an initial value of V, at t =O. In other words, D(V,,t) is the probability that the firm has not violated the safety barrier over the time interval [O,t]. The first passage time density function is given by the identity4
q(t)= -w&t).
(5)
Thus, if the proper form of the defective density function can be determined then a solution to (2), (2a), (2b) can be constructed along the line of Cox and Ross (1976b). To implement the suggested solution technique of Cox and Ross (1976b), the discounted value of all expected payments to the claim must be computed. As an example, consider the equity claim, f(Kt), in the presence of a discount bond with a safety covenant. The equity will receive nothing if the firm violates the safety covenant during the life of the bond, thus there are no payments to the equity to be evaluated prior to the maturity date of the bond, t = T. At t = T, the equity claim is worth f(V(T),T)=max[O,V(T)-B]. The value of the equity can be written f(~t)=e-‘(T-‘)
as
$U'(T)-WPU'(T),
Now consider the discount bond, F(V, t). upon a violation of the safety barrier or payments due to violation F, (V, t) and the F2(V,r). Thus F(V, tj=F,(V,/,)+F,(V,t). bond due to a violation. The discounted
T)dW-1. This claim may receive payments at maturity. Call the value of the value of the payments at maturity First consider payments to the expected value of these can be
4D( V,, t) is the probability that the firm IS alive at time t. Now consider the same quantity instant later, D(V,,t+dt). It is clear from the definition of D(V,, t) that D(V,,t+dt)-D(V,,t) 40. A decrease in the quantity D(V&t) over the interval (t,t+dt) must be precisely probability that a violation of the safety barrier occurred in the interval (t, t + dt).
one the
S.P. Muson and S. Bhattacharya,
Ruky debt und safety covenants
287
written F,(V,t)=ie-““-“P(s)q(s)ds, i
where V(s) is the value of the firm upon violation of the safety barrier at time s. The payment at maturity is min [V,B], conditional on not having violated the barrier during the life of the debt. The evaluation of this payment may be broken up into two parts F2(V,f)=e-‘(T-r)
i,
I’(T)P(I’(~-),
T)dI’(T)
+e -r(~-“%Bp(V(T),T)dl/(T).
Summarizing the problem to this point, the object is to evaluate a risky discount bond, with a safety covenant, issued by a firm that follows a jump Using either a diversifiable risk assumption or an arbitrage process. argument, it is easily shown that (2) (2a), (2b) is the formal statement of our valuation problem. Cox and Ross (1976b) outline an approach to solving (2), (2a), (2b) which is based upon the discounted expected value of all payments to the bondholders. In order to implement this solution technique, it is necessary to determine the form of the defective distribution associated with the firm value dynamics (1) and safety barrier (3). The defective distribution must be consistent with (4a), (4b). Given the defective distribution, the first passage time distribution follows from (5) and the solution technique of Cox and Ross (1976b) can be applied. As a final check, the resulting solution must satisfy (2), (2a), (2b). 3. The binomial case In this section it will be assumed that G(Y) is a binomial distribution. The form of the defective distribution associated with these dynamics will be derived. We will proceed by first solving the ‘parallel’ case, where the firm value, in the absence of jumps, has the same exponential drift rate as the barrier. This is termed the ‘parallel’ case since log V(t), in the absence of jumps, will be drifting parallel to logK(t). Explicit solutions for the value of discount debt and a safety covenant are presented. It will then be demonstrated that the solution to this case will serve as an integral part in solving the more general case of arbitrary y. Recall the firm value dynamics dV/I’=(r-M)dt+dn,
(14
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
288
where the expected rate of return on the firm is r, given the assumption that the firm’s jump risk is diversifiable. Given that a jump occurs, the impact on the value of the firm is determined by a-drawing from G(Y). Assume that G(Y) is a binomial distribution and the associated distribution function, g(Y), is represented as g(Y)=+
for
Y=ed,
=+
for
Y=em6,
=0
otherwise,
where 6 is a positive written V(t)=
constant,
(6) The
random
firm value
at time
V,exp((r-H~)t+z(t)6),
where z(t) is the sum of a random number, N, of mutually random variables, x,, i= 1,2,. . ., N, with the common density g(x)=+
for
x=1,
=$
for
x= -1,
=0
otherwise,
t can
be (7)
independent
and z(O)=O. We will refer to z(t) as the random sum of jumps. The random variable z(t), in the absence of a safety barrier, is distributed compound Poisson, h(z(t)), where h(z(t))=
f
e-A;t)N
{g(~)}~*,
N=O
and where N is the Poisson distributed random variable number of jumps in t time periods, N and x are independent and {g(~)}~* is the N-fold convolution of g(x). Feller (1966) shows that this is equivalent to
(8) where
is a modified
Bessel function
of the first kind of order z where z-z(t).
S.P. Mason and S. Bhattacharya, Rsky debt and safety cmenant.7
It is clear from the one-to-one V(t) = V, exp ((r -2k)t
correspondence
between
289
V(t) and z(t),
+ z(t)6),
that V(t) is a discrete random variable since z(t) takes on only integer values. Thus the distribution of V(t), in the absence uf a safety barrier, can be written as (9) given (7), the relationship
between
V(t) and z(t).
3.1. A ‘purallel’ barrier
Consider the barrier, K(t) = C e-y(T-f), where y = r -%k. This case is of interest since its solution will serve as an integral part of the solution to the more general case of arbitrary y. When y=r-Ak, the barrier and the firm value, in the absence of jumps, are growing at the same exponential rate. Now define the integer n, n=[(log(C/V,)-yT)/6]<0,
(10)
where [X] is the largest integer smaller than X. In other words, z(t)>n defines all those states in which V(t)>K(t) and z(t)=n defines that state in which V(t)SK(t). It is possible to solve for the defective density function, p(V(t), t), borrowing from the method of images. The proper representation of the defective density function is
(11) for z2n5 The means by which the derivation of (11) is related of images can be seen by first considering the distribution absence CJJ a safety barrier. It is clear that this distribution symmetric about zero, and ranges from - cx) to SC. However, distribution of z(t) has been characterized as a triangle, A, This will not alter the validity of this explanation. Clearly, all sWe assumed that prob(Y=e6)=prob(Y define prob(Y=@)=q and prob(Y=e-6)=q’, p(V(t),t)=(qlq’)i”e-*’
to the method of z(t) in the is discrete, in fig. 1, the for simplicity. of the sample
=e-“)=f. Tlus was done for ease of exposltlon. If we then a more general form of (11) can be written
(1z(2(44’)*nt)-r,-,,(2(4q’)*~,t)),
The sigmficance of this generahzatlon IS that the solution technique easdy handle the transformations referred to m footnote 2.
outlined
in this paper
can
290
S.P. Mason and S. Bhattacharya, Rsky debt and safety couenants
paths that end up in the right-hand tail of distribution A, to the right of n, traveled through the state n at least once. In the presence of a safety barrier, all of these sample paths would be associated with a violation. However, some sample paths traveled through n but ended up to the left of n at time t. How do we account for these sample paths? The insight is to realize that all of the sample paths that travel through II will end up distributed symmetrically around n at time t. Thus the right-hand tail of distribution A, to the right of n, represents one-half of all the sample paths that violated n during t time periods. The other half of these sample paths is distributed similarly to the left of n. The distribution of those sample paths to the left of n can be represented through the convention of mirror images. Consider an identical process, z’(t), that initiates at 2n, i.e., z’(O)=2n. The process z’(t) is termed the mirror image of z(t), given a mirror placed at n. The distribution of z’(t) will be distribution B in fig. 1. Clearly distribution B is identical to distribution A except it has been displaced by 2n. Now, if distribution B is subtracted from distribution A for z(t)zn, this will precisely account for those sample paths that had violated the barrier but had ended up to the left
z(O)
z(t) I
I
(+I
(+I
F
3 5
Timei” E ” _----8 5 2n -------__-
a
A
----_-_---------
B
(4
1
(4
Fig. 1. The random sum of jumps has an rmtial value of z(O)=O. An up Jump adds + 1 to the sum and a down jump adds - 1 to the sum. When z(s)=n, where n is a negative integer, a violation of the safety barrier has occurred at time s. We are Interested in the distribution of all those sample paths of z(t) which avoid a violation of the safety barrier durmg the time interval [O, t]. This is termed the defective distrlbutlon which can be derived from the method of images Distrlbutlon A is the distribution of z(t) m the absence of a safety barrier. Distribution B 1s the dlstributlon of a process which IS a mxror Image (mirror placed at n) of z(l) and has an imtlal value of 2n. The defective distribution C can be represented as the difference between dlstrlbution A and distribution B for z(t)Ln.
S.P. Mason and S. Bhattacharya,
Risky debt and safety covenants
291
of n. The resulting distribution defined by the difference between A and B is the defective distribution C in fig. 1. Thus in (ll), the quantity e-“‘I,(At) is analogous to A in fig. 1, the quantity e -“ZZ_2n(lZt) is analogous to B in fig. 1, and p(V(t),t) is analogous to C in fig. 1. D(V&t), as defined by eq. (4), is the probability that the firm has not violated the safety barrier in the first t time periods. For the case of the binomial jump dynamics considered here, this quantity can be formed by summing over the defective density for all zzn,
D(V,,t)=
:
e-“’
(12)
(WCL,,(W).
I=”
D(&, t) must satisfy (4a), (4b) if our derivation of (11) is correct. The fact that (12) satisfies (4a), (4b) is demonstrated in appendix A. The value of the debt, F(V,r), in the presence of a safety covenant can be written as m-l F(V;t)=
e-“‘T-“(1Z(~(T-t)))Vexp(-Ik(T-t)+z6)
1
+ f e- “‘T-‘)(ZZ(A(T.?=m
t)))Bexp(
- r(T-t))
(13)
co
+
,_z.(l(T-t))(Vexp(-Ik(T-t)+z~) c e-i(T-*)z S?=lll
-Bexp(-r(T-t))), for V>K(t),
where m=[(log(B/V)-(r-Ik)(T-t))/6]+1,
(134
and
The expression for m defines that integer such that (z(T) 2 m)o( V(T) 2 B). In other words, z(T)Zm, defines all those states, at time T, in which the firm is able to pay the debtholders their promised principal, B. The expression for n is similar to (10) in that it defines that negative integer such that (z(t) > n)o( V(t) >K(t)) and (z(t) = n)*( V(t) SK(t)). The only difference in these two expressions for n is notation. Implicit in (10) is the assumption that current time is t=O. This was done for exposition purposes. The definition
292
S.P. Mason and S. Bhattacharya,
Risky debt and safety covenants
of n used in (13b) assumes that current time is t. The first two terms in (13) correspond to the value of the debt in the absence of a safety covenant and the third represents the value of the safety covenant. The fact that (13) satisfies (2), (2a), (2b) is demonstrated in appendix B. The easiest way to understand why (13) is the value of the discount debt is to consider the value of the equity, f(V, r), where f(v,t)+F(V¶r)=V. From the definition of m, it is clear that f(v,r)=
f e-“(T-f)(Zn(A(T-t)) ‘Z=m
-ZZ_zn(A(T--t)))
X (Vexp(-Ak(T-t)+z6)-Be-“T-‘)). This is simply the expected value of the payment to equity, (V(T)-B), over the defective distribution and discounted by the riskless rate r. It is possible to write
taken
V= e-r(T-‘)E[ V( T)] , where the expectation is taken over p’(V(T)), (9),
Grouping the terms as we have done in (13), it is straightforward to isolate the value of the covenant. Remembering the definition of m, and that p’(V(T)), (9), will describe the distribution of V(T) in the absence of a safety covenant, then clearly the first two terms represent the value of debt without a safety covenant. Therefore, the last term in (13) is the value of the safety covenant. 3.2. A ‘non-parallel’ barrier It was earlier asserted that (ll), the defective density function when y= would serve an integral part in determining the defective density function for arbitrary y. To see this, consider a y <(r -Ak). This would describe a case where the firm, in the absence of jumps, is drifting away from the barrier. Fig. 2 depicts this situation as it relates to the random walk of z(t). When y is less than r-ilk, the barrier and the line z(t)=0 are diverging at the rate of r-lk-y. In the previous section, which dealt with a ‘parallel r -Ik,
S.P. Mason and S. Bhattacharya,
z(O) z(t,)
Risky debt and safety covenants
z(t*)
z(tg)
I
I
I
+3 .
--
+2 ‘.
--
8 ,. E +’ 4 B
__
-----
- ----.-
_t ,t3
\
“3,
Fig. 2. When the barrier is non-parallel the critical values for the random sum of jumps, z(t), that describe a violation will be time dependent. For example, given the above non-parallel barrier, if z(t)=n,, where n, is a negative integer and tf t,, then the barrier is violated. However, if z(t)=nl for t > t,, then the barrier is not violated.
barrier, the barrier is parallel to z(t)=O,Vt. See fig. 1. With a ‘parallel barrier, z(r)= n describes a violation Vt. However, when the barrier is nonparallel, the critical values for z(t) that describe a violation will be time dependent. Assume that V, is the firm value at t =O. It is then true that z(t)=n,, where n, rn and n is given by (lo), would describe a violation of the safety barrier for t 5 t, where (log(C/&)-y(T-t,)-(r-Ak)t,)/6=n,
defines t,. For t E (cl, tJ, z(t)=n, - 1 =n2 would describe a violation of the safety barrier conditional on z(t) > n,, Vt E [0, cl]. The expression (log(C/T/,)-y(T-t,)-(r-lk)t,)/6=n,
defines t,. Continuing integers ni+t
=ni-1
in this manner, we are able to define a sequence of
for
i=1,2,...,
294
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
where n, is given by (10) and a sequence of points in time
It is clear that the barrier in fig. 3 is equivalent to the barrier in fig. 2. That is to say that any sample path for z(t) which violates the barrier in fig. 2 will also violate the barrier in fig. 3 at the same time, and the value of the payment received by the bondholders upon violation will be the same in both cases. This is the first hint that the solution to the ‘parallel’ case will serve an important part in solving the arbitrary y case.
2’
I
z(t1)
z
:)
z(tg)
+3
+2
K +1
4 %
___---__ _____.:__
co E
t az “1 “2
“3
Fig. 3. This barrier is equivalent to the barrier depicted in fig. 2. Any sample path for the random sum of jumps, r(t), which violates the barrier in fig. 2 will violate this barrier at the same time.
For t I t,, the defective density can be written as (14) for zln,, and where V(t)= Voexp ((I- Ak)t + ~6). However what is the defective density function for t E (Cl, t2]? As a specific example, consider the defective density at tZ,Pz(V(t,), tz). Because the process for z(t) is Markov, the Chapman-Kolmogorov condition can be applied to the defective density function. To understand the Chapman-Kolmogorov condition, consider a
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
295
specific state, z*(t,), at t,. The initial state for z(t) is z(O)=O. The ChapmanKolmogorov condition says that the probability of going from z(O)=0 to z(t2)=z*(t2) can be written as prob(z(O)+z*(tz))=
2 prob(z(O)+z(tr)) (+r)Ln,l x prob(z(t,)+z*(t,)).
(15)
That is to say that the probability of going from z(0) to z*(t2) is the product of the probability of going from z(0) to z(tl), times the probability of going from z(tl) to z*(t2), summed over all admissible z(tl). Clearly the probability of going from z(0) to a specific z(tl)=j’ is given by p1 (V(t,), tl), (14), for z =j’. Now it is left to determine the probability of going from z(t, ) =j’ to z(t*) = z*(tz).
The insight here is to realize that this is simply the ‘parallel’ solution, (ll), applied over a time span of (tz -t, ), where the initial state is j’, not 0, and the distance to the barrier is n2 -j’, not n,. In other words,
where n2 =nl - 1 and it is understood that z* = z*(t,). It then follows that
Expression (16) is simply a restatement for z&n,. Chapman-Kolmogorov condition. This allows us to define the recursive system
Pj+l(V(ti+l),ti+l)=
f
of expression
(15), the
Pi(~(ti),ti)e-““t+‘-‘3
j=n,
(17)
296
S.P.
where i=l,2,...,
Mason and
S. Bhattacharya,
pi(V(r,),t,)
Risky
debt and safety covenants
is given by (14) and
v(ti)= Voexp((r-;lk)ti+j6),
for
Z>,lli+l.
The defective distribution of V(t) for t E (ri, ri+ 1] is p(V(t),t)=
2 pi(V(ti),ti)
e-‘(‘-‘J
j=n,
(18) where i = 1,2,. . ., and p,(V(t,), ti) is given by (17) and V(r) = V, exp ((I - Ak)r+ z6), for zlr~+~.
A similar analysis would hold for the case where y > r - Ak.
Thus, it is possible to construct a solution to (2), (2a), (2b) for arbitrary y, along the lines of Cox and Ross (1976b), by utilizing (18), the defective density function. For instance, the value of equity,f(V, t), in the presence of a safety covenant can be written f(Kr)=
f
p(V(T),T)(Vexp(-Ik(T-r)+z~5)-Be-’(~-’)),
z=Wl
(19)
where p(V(T),T) is given by (18) for (T-r)E(r,_l,ri) and m is given by (13a). The value of the debt, F(I/,r), follows from the identity, F(V, r)=
v-f(K r). 4. Comparison with Black and Cox solution Black and Cox (1976) provide solutions for the value of a finite lived risky discount bond and a safety barrier, under the assumption that the firm value follows continuous diffusion-process dynamics. In their conclusion they state: It should be noted that if the value of the firm follows a jump process, the value of a safety covenant may be drastically altered since the value of the firm could then reach points below the bankruptcy level without first passing through it.
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
291
This section briefly examines this conjecture for the binomial jump dynamics assumed in the previous section.6 Consider the Black and Cox (1976) problem of valuing a discount bond in the presence of a safety covenant when it is believed that the firm value dynamics are dV/V=udt+odz,
(20)
where a2 is the variance rate per unit time and dz is a standard GaussWiener process. These dynamics will lead to a continuous sample path for the value of the firm. Assume that the firm value dynamics are in truth described by a jump process (l), with G(Y) being the binomial distribution. What will be the nature of the valuation errors induced by using the Black and Cox (1976) valuation for the value of the discount bond with a safety covenant? In order to utilize the Black and Cox (1976) solution (BC solution), the variance rate must be estimated. Assume that the estimate is the truth variance per unit time,
The BC solution, using 6’ and T= T-t,
can be written
logV-logB+(r-1/2&2)t BJ;
F'(y7)=Be-'W
+V@
log B - log V- (r + 1/2e2)2 fl^X!fz q+l
210gC-logB-logV+(r-2Y+1/2~2)~
X@
&/G
>
xQi 2logC-logB-logV+(r-2y-1/2&‘)r ( 6Black and Cox (1976) did not assume their solution will hold in such a world, results to the results of this paper.
&fi
(21)
that the firm’s diffusion risk was diversifiable. Since it is permissible to compare Black and Cox (1976)
S.P. Mason and S. Bhattacharya,
Risky debt and sajety covenants
298
where q=2((r-y)/G2) and Qi is the unit normal distribution function. The first two terms correspond to the risky discount bond valuation of Merton (1974) and the last two terms represent the value of the safety covenant. One insight into the nature of the valuation errors is provided by the realization that the binomial jump dynamics have a fundamental relationship to the Gauss-Wiener dynamics assumed by Black and Cox (1976). It is easily demonstrated that the binomial jump dynamics will converge to the GaussWiener dynamics assumed by Black and Cox (1976), if 2+co while 6+0 such that U2 = c2. In other words, if the mean number of jumps per unit time is made progressively larger, J.+co, as the amplitude of the jump is made progressively smaller, 6+0, in such a way as to keep the quantity U2 equal to a constant, c?, then these binomial jump dynamics will converge to the Gauss-Wiener dynamics used by Black and Cox (1976). Therefore, for a given rr2, independent of the nature of the error, i.e., overestimate or underestimate, we would expect the magnitude of the error to approach zero as A--r00 and 6-+0 such that Xi2=ts2. As a more direct assault on the valuation question, a computer program was written around (18), the defective density function. The procedure was to evaluate the debt, F( V,z), via a direct evaluation of the equity, f( I’, T), and to use the fact that f(V, 7) +F( V, 7) = V. Eq. (19) was used to evaluate f(V, T). Comparisons of F(K r) and F’( K r) were made over a wide range of the variables, V and 7, and the parameters, ;1, 6, C, y, r and B. The valuation of F(V, 7) was broken down into the valuation of a risky discount bond without a safety covenant, G(V,T), and the valuation of the safety covenant, g( V,7). Therefore F( V,z) = G( V,T) + g( I’, z). With regards to the BC solution, F’(V, z), the discount bond portion was labeled G’(V, 7) and the safety covenant portion g’( I’, 7). Therefore F’( V,r) = G’( V,z) + g’( I’, t). The nature of the valuation errors associated with G’(V, T) are as follows. For V$=>Band VeB it was found that G’(V,‘,r)>G(V,7). That is to say, if the value of the firm, V, is significantly greater than the promised payment, B, then the fact that the firm value can jump is detrimental to the bondholders and therefore G’(V, T) > G(V, 7). If the value of the firm is significantly less than the promised payment, then the fact that the firm value can jump will clearly benefit the equityholders, thus hurt the bondholders, and therefore G’(V,2)> G(V,T). In addition, it was found that if V=.Be-” then G’(V,T) < G(I’, 7). That is to say, if the value of the firm was approximately equal to the present value of the promised payment then the discount bond portion of the BC solution is an underestimate of the true value.7 It was rare to observe ‘These results are similar to those of Merton (1974, 1976a) in a study of misspecification error in the pricing of call options. Merton (1976) presents a valuation formula for a call option, where the underlying stock price, S, dynamics contain both a diffusion and a jump component. For the case where the jump magmtude, Y, is drawn from a lognormal distribution, Merton (1976a) compares his valuation formula to the Black and Scholes (1973) call option valuation, which assumes that the stock price dynamics contain only a diffusion component. Merton
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
299
valuation errors, in relative terms, of greater than 5 %. Finally, just as would be expected from the relationship between the binomial jump dynamics and the Gauss-Wiener dynamics, G(V,r)+G’(V,r) as 2-03, and 6-O such that As = a*. The nature of the valuation errors associated with g(V,r), the value of the safety covenant, is different. For all I/ and r, it was found that g’(V, r) > g(V, r). The safety covenant portion of the BC solution, g’(V, r), is an overestimate of the true safety covenant value, g( V, z), for all firm values and time to maturity. Furthermore, it was not rare to encounter valuation errors, in relative terms, of more than 25 7’. This confirms, for the binomial jump dynamics considered here, the Black and Cox (1976) conjecture that the value of a safety covenant could be materially affected if the firm followed a jump process. Lastly, it is true that g(V,r)-+g’(V,r) as A-+ cc and 6+0 such that 116*= a2.8
5. Conclusions This paper considered the valuation of risky debt and safety covenants when the firm value follows (possibly) discontinuous sample paths. The analysis made use of a class of probabilistic first passage time techniques, developed within Collective Risk Theory, for Poisson processes. For the Poisson dynamics considered, it was possible to derive the form of the defective distribution of firm value, and to construct solutions to the valuation problem along the lines of Cox and Ross (1976b). The results of this paper were compared to the work of Black and Cox (1976) which
(1976b) shows that if the stock prrce dynamics do contain both a diffusion and a jump component then the Black and Scholes (1973) call option valuation will systematically underestimate the true value of the call option for almost all regions of S and r. The exception is SzE’e-“, where E’ is the exercise price of the call option. In this region the Black and Scholes (1973) valuation is an overestimate. Since Merton (1974) demonstrated that equity, in the presence of a risky discount bond, can be viewed as a call option, we are not surprised to find the nature of our errors to be the opposite of Merton’s since F( V, T) +/( V, 7) = V. 81t is easy to extend our comparison of safety-covenant solutions, with and without jumps, to perpetual consol bonds. Consider lirm value dynamics that contam both a diffusion and a jump component. Assume diversifiability of jump risk, and payout rates of P(V) to all securityholders and C to the bondholders. Further assume that a safety covenant exists such that VJK constitutes a violation. The valuatton equation, $~*v~F,,+[(r-lk)V-P(V)]F,-rF+C+IE[F(YV)-F(V)]=O, with the appropriate boundary conditions, including F(V)= V for VJK. If Y IS deterministic then straightforward solutions are obtained for Y> 1 (jumping through the safety barrier is impossible) or Y=O (the Samuelson ruin jump), especially if P(V) 1s proportional to V. These solutions are simple transformations of the corresponding diffusion-only solution, e.g., r is simply replaced by (r+n) m the Y=O case. When Y< 1 and deterministic, the solution has a tractable recursive structure over stages [K s VsK/Y]. . [K/Y” s VsK/Y”+‘], etc., for n=0,1,2 ,.... Details of these solutions are available from the authors upon request.
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
300
addresses the same valuation problem under the assumption of continuous firm value dynamics. It was shown that discontinuous, versus continuous, firm value dynamics can have a significant differential impact on the value of a safety covenant, as conjectured by Black and Cox (1976).
Appendix A Below it is verified that eq. (12),
wG’,,t)= f e-“‘(Z,(nt)-z,_,,(nt)), Z=”
satisfies (4a) and (4b), (r-ik-y)K&)(Kj,t)D&O)=1
for
D,(v&t)+wD(I/,
r,t)--D(vmt))=O,
V,>K(O).
Remembering that (12) was derived under the assumption rewrite (4) as
of y= r-lk,
D,(I/,,t)+~D(I/,,t)=~E(D(~o’,Y,)).
we
(A.11
A rule for the differentiation of Bessel functions, given in Olver (1972), is
Applying this rule, it is easily verified that D&t)=
-1D(V,,t)+(L/2)
2 e-“’ z=fl
Now consider the quantity E(D(F$Y,t)) where the expectation is taken over g(Y), (6), the binomial density function. The quantity Y can be either an upjump, Y+ =ed, or a down-jump, Y- = e-* with equal probability. Thus
D(V,,Y’,t)
is
simply the solution, D(V,,t), where the initial firm value, V,,
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
301
has been increased by one jump. D(VOY-, t) is the solution, D(VO,t), where the initial firm value has been decreased by one jump.
D(V,Y’,t)=
f I=“-
e-“‘(Zl(~t)-Zz-z(n-l)(~T)). 1
The quantity (n - 1) appears in the place of n because the distance between the initial firm value, V,Y’, and the barrier has increased by one jump. D(V,Y-,t)=
F
e-*‘(ZI(IZt)-Zl-2(n+1)(IZt)).
r=n+1
The quantity (n+ 1) appears in the place of n because the distance between the initial firm value, V,Y-, and the barrier has decreased by one jump. Thus
J-w(b’,~t))=(w2)
e-“‘(Z.(~t)-zn-z(n-l)(~t))
f 2=fl-1
+(a)
f
e-“‘(z,(~t)-z,-,,,-,,(lt)).
z=n+l
In the first summation in the above expression, change the summation index to z’=z+ 1. In the second summation, change the index to z”=z- 1. AE(D(VoY,t))=(A/2)
jJ e-“‘(ZI,_l(~t)-Z,,_2n+l(lZt)) I’=#8 +(A/21
f e-“‘(Z,..+,(lr)-Z,.._,,_,(lt)), Z”E”
which is equal to D,(I/,, t) + lD(I/,, r). Thus D(I/,, t), (12), satisfies (4). Consider the initial condition, (4a). The following is true of modified Bessel functions: Z,(O)=O,
Vz,
except
2 =0,
where Z,(O)= 1.
Thus D( V,, 0) = 1 for n < 0, which is to say, for V, >K(O).
302
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
Appendix B Below it is verified that eq. (13), m-l FK’,t)=
c
z=-* +
e-“‘r-“I,(~(T-t))Vexp(-~k(T-t)+zs)
F e-"(T-')Z,(A(T-
z=m + f
t))B exp (- r(T-
c))
e-"(r-r)Zr_zn(lZ(T-t))
z=lll
x (Vexp(-Lk(T-t)+z6)
-Bexp(-r(T-r))),
satisfies (2), (2a), (2b), (r-A.k)VF,(V,t)+F,(V,t)-rF(V,t)+AE(F(VY,t)-A
F(V,t)=V
for
,/,t))=O,
V~Ce-y(r-r),
F(V, T)=min(V,‘,).
Clearly we can rewrite (2) as (r+l)F(V,t)-(r-Ak)W,(V,t)-F,(V,t)=AE(F(VY,t)).
(B.1)
First evaluate the RHS of (B.l). We will use the same argument used to evaluate E(D(VoY, t)) in appendix A. The expectation is taken over the binomial density, g(Y), (6), where there is equal probability of an upward jump, Y+ =e’, and a downward jump, Y- =e-*.
The quantity F(VY+,t) is simply the value of the debt assuming that the current firm value is one jump further away from the barrier. The quantity F(VY-,t) is the value of the debt assuming the current firm value is one jump closer to the barrier.
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
303
m-2
c
F(vY+,t)=
e-“‘T-“ZZ(A(T-t))Vexp(-A.k(T-t)+(z+1)6)
z=-m
jJ
+
e-’ ““-“Z,(A(T-t))(Bexp(-r(T-t))
z=m-1
x (Vexp(-lk(T-t)
F(VY_,t)=
f
+(z+l)S)-Bexp(-r(T-t))),
e-“‘T-“Z,(~(T-t))Vexp(-A.k(T-t)+(z-l)b)
.7=-m
+
1
e-“‘*-‘)Z,(A(T-t))Bexp(-r(T-t))
z=m+l
m
i(=-f)zz_2(n+1)(A(T-t)) +z=;+lex(Vexp(-Ak(T-t)
+(z-1)6)-Bexp(-r(T-t))).
In the above expression for F( VY+, t), change the summation z+ 1. In the expression for F(VY-, t) change the summation z-l. We are now able to write m-l
/ZE(F(VY,t))=(A/2)
C
e-l’T-z)ZI,_l(~(T-t))
x Vexp(-Ak(T-t)+z’d)
+(A/2)
2
e-“‘T-“Z,._l(~(T-t))
z’=m
xBexp(-r(T-t))
+ (A/2) C e-“(T-‘)Z,._2n+l(~(Tz’=m
x (Vexp(-Ak(T-t)+z’S)-Bexp(-r(T-t)))
t))
index to z’ = index to z”=
304
Risky debt and safety covenants
S.P. Mason and S. Bhattacharya, m-l +
(A/2)
C
e-A(T-l)Z.SS+1(A(T- t))
z”Z - 00
x Vexp(-Ak(T-t)+z”S)
+ (A/2) f
L”C#j
e-A(T-r)Zzsf+ 1(l(T-
t))
xBexp(-r(T-t))
2 e-“(T-‘)Z,,,-2,-,(I(T-t))
+W)
n”=m
x (Vexp(-Ik(T-t)+z”6)-Bexp(-r(T-t))). Now consider the LHS of (B.l). The first term is clearly m-l (r+L)F(V;
t)= (r+l)
C e-“(T-r)Zz(l(Tz=-03
t))
x Vexp(-Ak(T-t)+&) e-“(T-‘)Z,(A(T-t)) t=lPl
+@+A) f
xBexp(-r(T-t)) + (r+l)
f e-“(T-r)Z,_2n(~(T-t)) 1=llI
x (Yexp(-Ak(T-t)+zd)-Bexp(-r(T-t))). The second term is equally clear m-l -(r-Ak)W,(I/,t)=
-(r-lk)
c
e-A(T-f)1z(A(7’-t))
z=-m
x Vexp(-Ak(T-t)+zS) -(r-ilk)
f e-A(r-z)Z,_,,(I(T*=lVl
x Vexp(-Ak(T-t)+za).
t))
S.P. Mason and S. Bhattacharya,
Risky debt and safety covenants
In order to evaluate the third term, -F,(V,t), differentiation of a Bessel function.
305
again recall the rule for the
m-l
-F,(K:t)= - (1 +k)A
C e-“‘T-r)Z~(A(T-t)) I=-CO
x Vexp(-Ik(T-t)+zd) -(r+jl)
f
e-“(T-“Z,(I(T-
t))
xBexp(-r(T-t)) -(l+k)l
5 e-“‘T-‘)Z,_,,(~(T-t)) z=fPl
x (Vexp(-Ak(T-t)-tzh))
+(r+l)
f
e-“(T-“Z,_2n(lZ(T-_))
z=lll
xBexp(-r(T-t)) m-1
+(W
Cme-“(T-r)(ll+l(~(T-t))
+Z,-l(A(T-t)))
*=-co
x Vexp(-Ak(T-t)+z6) + (A/2) $ e-“‘T-ll(Zt+l(~(T-t)) 1=llJ X
+Z,-l(I(T-t)))
Bexp(-r(T-t))
+(1/2)
5 eyACTVt) Z=m
x(L-,,+1(4~--~)) +I,-2,-l(R(T--t))) x (Vexp(-Ik(T-t)+zd)
-Bexp(-r(T-t))).
Upon careful inspection, it becomes clear that the sum of the expressions for
S.P. Mason
306
Risky debt and safety dovenants
und S. Bhuttacharya,
(I+ i)F(V, t), - (r - ik) VF,(V, t) and the first four terms in the expression for -F,(V,t) sum to zero. It is then seen that the last three terms in the equals the expression for Z(F(VX:t)) thus (13) expression for -F,(V,r) satisfies eq. (2). Now consider the boundary condition, F(V,t)=V From
the definition
for
VSCe-Y(T-l).
of n.
VgCe-y(T-‘)
for
n=O,
m-l F(I/,t)=
e-“‘T-“I,(~(T-t))‘Vexp(-~k(T-t)+z6)
c
2=-03 + f e-“‘T-‘)ZZ(~(T-t))Bexp(-r(T-t)) Z=m + f
e-“(T-‘)Z,(A(T-t))
z=lll
x (Vexp(-Ak(T-t)+za)
=
f
e-A(T-f)Zz(~(T-
-Bexp(-r(T-t)))
t))Vexp(-Ak(T-t)+zd)
z=-03
=e-r(T-r)E(V(T))= Finally,
consider
condition
I/.
(2b),
F(I/,T)=min[1!B], m-l
C
F(VT)=
L=--m +
Z,(O)Vexp(zd)+
2 Z,(O)B z=lll
5
Z,_,,(O)(Vexp(zG)-B). *=Wl
Given the nature of the safety barrier, B>C, and from the definitions of n and m, m > n. Thus if I/ 2 B, then that implies 0 2 m > n. The first and third terms equal zero since I,(O) =0, Vz except z =O. The second term equals B since ZO(0)=l. Therefore F(V;T)=B if VZB. If C
S.P. Mason and S. Bhattacharya, Risky debt and safety covenants
307
References Black, F. and J.C. Cox, 1976, Valuing corporate securittes: Some effects of bond indenture provision, Journal of Finance 31, 351-367. Black, F. and M. &holes, 1973, The pricing of options and corporate liabtlities, Journal of Political Economy 81, 637459. COX, J.C. and S.A. Ross, 1975, The general structure of contingent claims pricmg, Mimeo. (University of Pennsylvania, Philadelphia, PA). Cox, J.C. and S.A. Ross, 1976a, The valuation of options for alternative stochastic processes, Journal of Financial Economtcs 3, 145-166. Cox, J.C. and S.A. Ross, 1976b, A survey of some new results in financial option pricing theory, Journal of Fmance 31, 383402. Cramer, Harold, 1954, On some questions connected with mathemattcal risk, University of Cahforma Publications m Statistics 2. 99-124. Cramer, Harold, 1955, Collective risk theory, Jubilee volume of the Skandia Insurance Co. (Ab Nordiska Bokhandeln, Stockholm) S-92. Feller, W., 1966, Inlinitely divisible distributions and Bessel functions associated with random walks, S.I.A.M. Journal of Apphed Mathematics. Galai, D. and R.W. Masulis, 1976, The optton pricing model and the risk factor of stock, Journal of Fmancial Economics 3, 53381. Ingersoll, J.E., 1977, A contingent claims valuation of convertible securities, Journal of Financial Economics 4, 289-321. Jensen, M.C and W.H. Meckling, 1976, Theory of the firm. Managerial behavior, agency costs and ownership structure, Journal of Financtal Economics 3, 305-360. Jones, E.P., 1978, Arbitrage pricmg of opttons for mixed dtffusion - Jump processes, Working paper (M.I.T., Cambridge, MA). Merton, R.C., 1973, A rational theory of option pricmg, Bell Journal ot Economics and Management Science 4, 141-183. Merton, R.C., 1974, On the pricmg of corporate debt: The risk structure of interest rates, Jouri al of Finance 29, 449470. Merton, R.C., 1976a, Option pricing when underlying stock returns are discontmuous, Journal of Financial Economics 3, 125-144. Merton, R.C., 1976b. The impact on option pricing of specification error m the underlying stock price returns, Journal of Finance 31, 333-350. Myers, S.C., 1977, Determinants of corporate borrowing, Journal of Financial Economics 5, 147175. Olver, F.W.J, 1966, Bessel functions of integer order, Ch. 9 m: Handbook of mathematical functions, National Bureau of Standards Applied Math Series 55 (U.S. Government Printmg Office, Washmgton, DC). Prabhu, N.U., 1961, On the ruin problem of collective risk theory, Annals of Mathematics and Statistics. Rosenfeld, E., 1979, Stochastic processes of common stock returns: An empirical examination, Ph.D. dissertation (M.I.T., Cambridge, MA). Smith, C.W. and J.B. Warner, On financial contracting: An analysis of bond covenants, Journal of Financial Economics 7. 117-161.