Temporary vs Permanent Shocks:
Explaining Corporate Financial Policies∗
Alexander S. Gorbenko
Graduate School of Business
Stanford University
Ilya A. Strebulaev
Graduate School of Business
Stanford University
First version: September 2006
This version: February 2008
∗
We are grateful to Harjoat Bhamra, Sergei Davydenko, Darrell Duffie, Steven Grenadier, Mike Harrison, Dmitry
Livdan, Toni Whited, Jeff Zwiebel, and the participants of Stanford GSB FRILLS, MIT Sloan, and University of
Toronto seminars, as well as of the Western Finance Association meeting in Big Sky (2007) and the 13th Mitsui
Symposium on Finance, University of Michigan, for thoughtful comments. Address for correspondence: 518 Memorial
Way, Stanford Graduate School of Business, Stanford, CA, 94305. Gorbenko: gorbenko@stanford.edu; Strebulaev:
istrebulaev@stanford.edu.
1
Temporary vs Permanent Shocks:
Explaining Corporate Financial Policies
Abstract
We investigate corporate financial policies in the presence of temporary
shocks to firm’s cash flows. In our framework firms can experience negative cash flows, the changes in cash flows and firm value are not perfectly
correlated, and earnings volatility is larger than asset volatility. These
results are consistent with empirical stylized facts on earnings and are opposite to the implications of existing capital structure models which allow
only for permanent shocks to cash flows. Our results show that temporary
shocks increase the importance of financial flexibility and may provide an
intuitively simple and realistic explanation of low leverage phenomena.
They also, however, may lead to managerial short-termism.
Keywords: Financial policy, leverage, debt financing, capital structure,
financing decisions, temporary and permanent shocks, mean-reversion, finance substitution
JEL Classification Numbers: G12, G32, G33
On September 15 2006 all spinach-based food products were taken off the U.S. market as the result
of an outbreak of spinach-borne E.coli illness. A few days later, quoting Joe Pezzini, chairman of
Grower-Shipper Association of Central California: “Right now, the spinach business is closed. It’s
stopped”.1 This scary, and at the time of writing recent, case illustrates a class of typical shocks
to firm’s fortunes which share the following features: their timing is to large extent unexpected,
their initial magnitude can be substantial, and their effect is felt over a limited time (indeed, about
two weeks later spinach was back in consumers’ baskets). To give another example, arguably of
a much larger magnitude, up to 50% of the Western honey bee colonies died unexpectedly in a
number of U.S. states in late 2006, severely disrupting industries relying on bee-pollinating such
as an almond growing industry. Historical evidence shows that such unexpected and scientifically
poorly understood die-offs have been happening periodically over the last hundred years with bee
colonies population recovery on average within the span of several years.2
This paper claims that shocks of this nature are both economically important in explaining a
number of observed phenomena and at the same time have not been sufficiently studied by financial
economists, especially in the corporate finance area. While unexpected arrival and non-trivial shock
magnitudes have been successfully incorporated in various setups (e.g. if asset value follows a random
process with jumps), it is the third component, a transitory nature of shocks, that may help to understand better such phenomena as financial flexibility and conservatism, low/zero-leverage puzzles,
and cash policy of corporations.
Indeed, survey evidence shows that financial managers consider financial flexibility, earnings and
cash flow volatility, and insufficient internal funds to be the most important factors influencing the
financing decisions of their firms (see e.g. Graham and Harvey (2001)). While intuitively appealing,
these factors puzzle corporate finance theorists who find it hard to pin down the exact mechanism
of their influence. The problem is that we know the nature of economic frictions that lead firms
to more cautious financing behavior but our attempts of quantifying their effects have so far left
a lot to desire. In this paper we propose such an economic mechanism and build a novel dynamic
contingent-claims framework to study its impact.
Our starting observation is that the nature of shocks driving the distribution of corporate cash
flows can be summarized by shocks’ initial magnitude and their persistence over time. Some shocks
change long-term prospects of an enterprize and thus influence cash flows permanently. The impact
of other shocks is temporary and subsides over time. To start with, this simple observation helps
to explain why existing dynamic contingent-claims models of firm behavior exhibit so many features
that are in contradiction with empirically observed stylized facts. In standard models (e.g. EBIT cash
flow models of Goldstein, Ju, and Leland (2001), Morellec (2001), and Strebulaev (2007)) current
1
2
Washington Post, September 20 2006, p. D01.
See e.g. New York Times, July 17 2007.
1
period cash flow is proportional to the total value of assets; in other words, the value of assets can be
thought of as a perpetuity with a payment equal to the current cash flow. The critical, though often
overlooked, implication of these models is that all innovations to the value process have permanent
nature: a unit increase or a unit decrease in the current cash flow increases or decreases the expected
value of each future cash flow by exactly this unit adjusted for growth rate. Each innovation therefore
takes the form of a perpetuity as well.
While this critical feature enables us to solve the models, it has a number of undesirable implications. First, non-negativity of value implies that cash flows can never be negative. However, between
1987 and 2005, from about 17% to 25% of all quarterly cash flows for the full COMPUSTAT sample
are negative.3 One commonly used way to reconcile this stylized fact with standard models is to
introduce operating leverage by subtracting fixed costs of production from gross earnings. To the
extent that these costs are also of permanent nature, this approach does not address other stylized
facts. On a broader level, contingent claims models of capital structure ignore the fact that firms
which are highly profitable today may expect to experience negative cash flows in the near future.
To give an example in the context of the trade-off model, consider the economic marginal tax rate
that Scholes and Wolfson (1992) define to account for the present value of the current and future
taxes on an extra dollar of income earned. Firms operating within the context of existing models
have too high economic marginal tax rate relative to what actual firms expect to have. In essence,
they are erroneously assigned high tax rates even though their true marginal tax rate can be low. In
economic terms, firms may value financial flexibility to sustain times when cash flows are very low
even though the future is bright, the scenario not considered in standard models.
Second, the volatilities of cash flow and asset value growth rates are equal in existing models.4
Since asset volatilities for most firms are relatively low,5 this makes it difficult to explain the importance of earnings volatility for financing decisions. To rationalize existing capital structure and
observed prices of corporate securities one needs substantially higher implied asset volatilities which
are unlikely to be observed or justified in practice. Empirical evidence seems to suggest, however,
that earnings or cash flow volatility can be nontrivially larger than asset volatilities. Finally, in
the existing models changes in current cash flow and asset value are perfectly correlated, a result
inconsistent both with intuition and evidence.
Alas, all the features above are caused by modelling all innovations to the cash flow process
as permanent. To see the importance of the distinction between permanent and transitory shocks,
3
The frequency of negative cash flows depends on the definition of the underlying accounting variable. For EBITDA
(COMPUSTAT item 13) 17.5% observations were negative and for cash flow from operations (COMPUSTAT item 308)
25.4% were negative. The sample is from the full quarterly COMPUSTAT/CRSP merged file with the book value
minimum cut-off at $10 million dollars. See also Givoly and Hayn (2000).
4
For the reasons of technical simplicity, these models usually assume constant interest rate. The presence of stochastic discount rate would lead to a decrease in the correlation between current cash flows and their future expectations
but the effect is likely to be minuscule.
5
For example, Schaefer and Strebulaev (2007) estimate annualized asset volatility for an average firm with existing
investment-grade rating to be about 23%.
2
consider the effect of introducing the temporary component in the cash flow process. We are not
concerned with the exact nature or sources of these shocks since the intuition is generic. For example,
these shocks may affect the demand for a firm’s product or the production cost structure. Since their
impact is of limited duration, transitory shocks affect future cash flows disproportionately over the
time horizon. The consequences are many: the total realized current cash flow can turn negative
(while asset values are still economically large), cash flow volatility is larger than asset volatility, and
innovations in current cash flows are imperfectly (though positively) correlated with innovations in
asset values. In a nutshell, introducing temporary shocks makes model’s behavior patterns consistent
with stylized facts along a number of dimensions.
More importantly, what would be the implications of this mechanism for the financing behavior of
firms? Economic intuition would suggest that for any given level of leverage the expected advantage
to debt is more likely to be of lower value with more volatile and more frequently lower cash flows.
For example, since cash flow is now likely to be less than the level of contractual financial obligations
more frequently, expected tax benefits to debt fall in the absence of full loss tax offset provisions.
In other words, incentives to use debt decrease with the probability that a firm will experience
nontaxable states of the world (Kim (1989), Graham (2003)). Alternatively, since free cash flow is
more volatile, lower debt levels are needed to counterbalance empire-building managerial tendencies.
At a first glance, one may think that in the absence of financing frictions equityholders borrow against future earnings and thus temporary negative shocks are unlikely to exert a material
influence. Interestingly, this intuition is not robust and temporary shocks may lead to a more conservative financial behavior even in the absence of additional financing frictions (apart from costs of
bankruptcy) and in the case of symmetric shocks (i.e., the expected value of the shock is zero). Even
though shocks are ex-ante symmetric, firm holders may abandon the firm if the present value of a
negative shock is larger then the present value of future permanent cash flows. The abandonment
option that residual claimants are entitled to due to limited liability, gives rise to asymmetry between
negative and positive shocks. Equityholders benefit disproportionately from positive shocks, while
debtholders bear the worst negative shocks and demand compensation reducing equityholders’ desire
to rely on debt in the first place.
The presence of financing frictions that asymmetrically affects is likely to further exacerbate
the disadvantage of debt financing. In reality frictions are likely to arise whenever cash flows are
non-trivially lower than those expected by the market for a variety of reasons. Firms may find
it difficult to raise funds to finance a temporary shortfall or finance a new profitable investment
project if current cash flow is low for information, liquidity, and institutional reasons. Financing
shortfalls may lead firms to raise expensive external financing or sell assets at a liquidity discount,
events frequently observed empirically (Asquith, Gertner, and Scharfstein (1994), Pulvino (1998)).
In other words, this mechanism extends the notion of financial distress to those asset value levels for
which, under the assumption of perfect correlation of asset values and cash flows, the firm should
3
not experience any financing difficulties.
To develop this intuition further and quantify its effect, we build a dynamic financing model with
both types of shocks. For comparison purposes, we employ the trade-off model, a standard workhorse
of the contingent claims framework. Specifically, our purpose is to compare the optimal financial
behavior recommended by our model to the one implied by an identical model without transitory
shocks. To summarize our quantitative results, we find that the value of financial flexibility for
firms may indeed increase dramatically when firms face the prospects of adverse temporary shocks
and that their desire for financial flexibility may increase with larger financial frictions. For one
benchmark case, the optimal leverage ratios for the firm with and without the presence of temporary
shocks are 65% and 40%, respectively. Such a remarkably large difference is robust to changing the
economic environment. To emphasize the economic message behind this result, what we show is
that while the cause of these shocks is only temporary and their contribution to the unlevered firm
value is relatively small, their impact on financing decisions and resulting securities valuation can be
substantial.
We also find, however, that there exist another side of the same economic mechanism. The presence of adverse shocks may give rise to short-termism on behalf of managers and equityholders who
may try to maximize their value before the temporary shock arrives and who speculate on the shock
being of large positive value. This effect is similar to the asset substitution effect where equityholders
choose riskier investment projects due to the convex nature of their payoffs and therefore we call it
the financial substitution effect. It is also similar to speculative “position-taking” trading behavior.
In essence, debtholders bear the residual responsibility for the negative shocks of larger duration or
magnitude which increase the convexity of equity’s payoff and makes it more option-like. “Hit-andrun” short-term behavior consists of levering up the firm in a hope that the temporary shock will
be favorable. Importantly, this behavior, however, is not a dominant equilibrium outcome for any of
the reasonable parameter values that we consider.
While the underlying economic intuition behind introducing transitory shock to cash flows is
straightforward, building a framework that fully incorporate them represents formidable technical
challenges. Since permanent and temporary shocks should not be perfectly correlated, the presence
of two types of shocks requires necessarily two sources of dynamic uncertainty, which leads us into
largely unexplored field of modelling and makes all financing decisions intrinsically complicated. To
give an example, consider a firm whose cash lows are driven by two stochastic processes: one is
responsible for the long-term prospects (e.g. a geometric Brownian motion) and another – for the
short-term deviations from long-term cash flow (e.g. any process than mean-reverts to long-term cash
flow). In traditional models (with the long-term component only) default decisions consist of finding
a certain low asset value or cash flow threshold that warrants default when reached for the first time.
Importantly, with transitory shocks, the knowledge of total asset value or total realized cash flow by
itself is insufficient to make a well-informed default decision. The total current realized cash flow can
4
be low because firm’s future is bleak and equityholders are better off defaulting or because the future
prospects are bright conditional on the firm surviving a temporary period of decline in profitability
and equityholders find it in their interest to honor contractual financial obligations in a hope to rip
the benefits of the expected bright future. Thus, only by decomposing the shock into the permanent
and temporary components can equityholders now make the optimal default decision.
This leads to two consequences. Empirically, financial contracts frequently include covenants
of two types – net worth covenants (the firm is in violation if asset value is lower than a certain
threshold) and cash flow covenants (the firm should support certain earnings-related ratios such
as interest coverage ratio). Recent research has investigated optimality and prevalence of various
covenants. While traditional models can not distinguish between the two, these covenants naturally
play distinct roles in our framework which allows to pose the question on which firms should use a
particular covenant type. This economic intuition, however, has a reverse side since technically the
default boundary is a curve in a two dimensional permanent-temporary shock space and mathematical
problems related to the first passage to such a boundary in case of two state variables have not been
solved even for simplest cases. This, of course, explains why existing models constrain shocks to be
permanent.
In this light, our contribution becomes clearer. Ideally, one would like to solve the most general
case, with infinite number of shocks of both types where the permanent shock follows the wellknown stochastic process used in existing models. Unfortunately, this structure does not seem to be
yielding to any solution at present. To deal with this problem we succeed in building a simplified
framework which nevertheless incorporates all the intuitive features discussed above. Specifically,
we assume that the temporary component is driven by Poisson jump shocks arriving in discrete
time which are then fading in expectation over time by mean-reverting to the permanent cash flow.
Transitory shocks are characterized by their longevity (or the speed of mean-reversion), their arrival
intensity, and their magnitude. To compare our results to the permanent-only scenario, we introduce
a standard trade-off model and solve for optimal financing decisions including optimal default and
capital structure. We would like to stress that there is nothing special about the trade-off model and
the economic effects of temporary shocks are robust for most other applications but the trade-off
model at present enjoys a certain reputation of being the only structurally quantified set-up.6 To this
extent, we model the permanent component of the cash flow process as in the standard contingent
claims models (such as Leland (1994)). Since shocks of the same magnitude are expected to have
more impact on smaller firms, we assume the additivity structure: the total firm cash flow consists
of the sum of the two components.
We surely have to make a number of heroic assumptions and an important caveat is that we miss a
6
Indeed, information asymmetry issues give rise to a natural setup with temporary shock. While the immediate
magnitude of shocks is observable to all agents, outside investors may not know the duration of shock’s influence. In
other words, while insiders know whether shocks to cash flows are of permanent or temporary nature, the market does
not.
5
number of important avenues (such as optimal investment and cash holdings decisions). Nevertheless,
stylized facts produced by our model both on earnings and financial decisions are consistent with the
empirically observed ones suggesting the influence of temporary shocks. In particular, we reconcile
the evidence that firms care about earnings volatility and at the same time observed asset volatility
is not high enough to justify financial conservatism. Even though asset volatilities are calibrated to
be similar to the observed ones, the model leads to higher cash flow volatility associated on average
with a larger degree of financial conservatism.
Our paper relates to the burgeoning field that tries to extend the contingent claims models to
a broader set of corporate finance issues. Recent examples include Kurshev and Strebulaev (2007),
Hackbarth and Morellec (2008), and Sundaresan and Wang (2007). All of these papers consider the
case of permanent shocks only. Hackbarth, Miao, and Morellec (2006) model the financial policy of
the firm in the presence of business cycles. In their model, the economy can be in two states (boom
and recession) and the probability of being in a particular state is driven by a Markov chain. This
model is related to ours since the change of the state in their model leads to a jump in the cash flow
process which is expected to be of temporary duration. There are also substantial differences in our
approaches due to different nature of problems. In their framework the direction in which cash flows
change upon the change of the state is always known. In essence, their cash flow structure can be
thought of as a partial case of our framework (e.g. recession represents a temporary shock). They
also naturally do not allow for negative shocks. At the same time, temporary shocks in our model
can be thought as both firm-specific and industry- or macro-wide.
Some recent studies justify using the double exponential jump diffusion process (a variation of
which we use to model temporary shocks) in asset pricing models. In particular, Kou (2002) and
Kou and Wang (2004) investigate option pricing when values are affected by this type of jumps. The
shocks that they consider are, however, permanent.
The rest of the paper is organized as follows. The following section presents motivating examples
and introduces notation and initial intuition. Section II presents the benchmark model with permanent shocks following geometric Brownian motion and a single temporary shock. Section III.C
presents empirical evidence and calibrates the model. Section IV extends the model to account for
multiple temporary shocks, financial frictions. Section V concludes. All proofs are given in Appendix
A and Appendix B. Appendix C gives the details of the numerical integration procedure.
I
Motivating examples
Consider a firm with access to a cash flow-generating investment project. At any time t the realized
cash flow, denoted by πt , is the sum of two components: the cash flow generated by the permanent
component, δt , reflecting the long-term prospects of project returns, and the mean-reverting cash
flow ǫt generated by transitory deviations from the permanent component caused by shocks of shorter
6
duration:
πt = δt + ǫt .
(1)
Note that we model the total realized cash flow as the sum of permanent and temporary shocks. In
particular, this implies that shocks of the same magnitude matter more for smaller firms. Empirical
evidence (see Section III.A) demonstrates that large firms are much less likely to experience negative
cash flows than small firms, an observation supportive of the additivity assumption. One potential
explanation is that large firms are more likely to be involved in a number of unrelated lines of business
so that independent temporary shocks dampen each other’s impact.
To give the flavor and intuition of the results of our benchmark model presented in Section
II the next two subsections consider simple and somewhat naive versions of only-permanent and
only-temporary shocks scenarios, respectively.
I.A
Example A: Only permanent shocks
An example of a model with only permanent shocks (ǫt ≡ 0, so δt ≡ πt for any t) is any traditional
contingent claims model where δt is modelled as a lognormal process,
dδt = µδdt + σδdWt ,
(2)
where µ us the instantaneous growth rate under the risk-neutral measure, σ is the instantaneous
volatility of the growth rate, and dWt is a Brownian motion defined on a filtered probability space
(Ω, F, Q, (Ft )t≥0 ). This modelling framework is the workhorse of the burgeoning field in corporate
finance research. The trade-off model in its basic form that allows only for static capital structure
decisions is that of Leland (1994) and the one that allows for dynamic financing decisions is that of
Fisher, Heinkel, and Zechner (1989) and Goldstein, Ju, and Leland (2001), and has been extended
to incorporate, among other dimensions, investment decisions (e.g. Mauer and Triantis (1994)),
strategic behavior (Fan and Sundaresan (2001)), asset liquidity (Morellec (2001)), empire-building
by managers (Morellec (2003)), imperfect competition (Zhdanov (2004)), managerial risk-aversion
(Ju, Parrino, Poteshman, and Weisbach (2005)), macroeconomic fluctuations (Hackbarth, Miao, and
Morellec (2006)), distressed asset sales (Strebulaev (2007)), and strategic mergers and acquisitions
(Hackbarth and Morellec (2006)).
The results of the basic trade-off model are well-known and we will not replicate them here. This
scenario will constitute our benchmark for comparisons in the next section where it is a partial case.
(We introduce the trade-off between debt benefits and financial distress costs that drive optimal
financial decisions in our next example.) For now, it suffices to make an important observation that
any innovation to the cash flow process (2) is permanent, a feature of the log-normal assumption. As
a consequence, if we define asset value as the present value of all future cash flows then asset value
7
is proportional to the current cash flow:
Vt = Et
Z
∞
e−r(s−t) δs ds =
t
δt
,
r−µ
(3)
where r is the interest rate which is assumed constant for the remainder of the paper. Therefore,
this framework implies that asset values and cash flows are perfectly correlated and their growth
rates volatilities are equal. In addition, since cash flows are non-negative, firms never make losses.
Intuitively, all these implications are inconsistent with reality and, as we shall see in Section III.A,
empirical stylized facts clearly emphasize this inconsistency.
I.B
I.B.1
Example B: Only transitory shocks
Modeling transitory shocks
Assume now that the cash flow generated by long-term prospects is a constant. Since it will serve as
a numeraire in this example we let δt = 1. The only uncertainty comes from a single exogenous and
transitory shock to firm’s fortunes in the form of a jump deviation from the long-term cash flow. The
transitory feature of the shock which intuitively relates to any event that is felt only for a limited
time plays a crucial role in this paper so that it is important to have an appropriate discussion of
how we model the concept. There are several sources of uncertainty about the shock that later will
be used in calibration to reflect stylized empirical facts: the random timing of the shock’s arrival,
the initial magnitude of the shock, and its persistence. For pricing purposes, all shock’s stochastic
determinants are assumed to be under the risk-neutral measure. Specifically, the shock arrives at a
random moment Ts (s for shock’s arrival) and its size ǫTs is also random and is determined at Ts .
Figure 1 shows the evolution of cash flows in the model.
The transitory nature of the shock is reflected in the degree of its persistence: the cash flow mean
reverts from δt + ǫTs back to δt at a random time Tr (r for shock’s reversal). The presence of this
mean-reversion is what distinguishes our model from previous asset pricing models of cash flow/asset
value processes with jumps where all shocks are of permanent nature (e.g. Merton (1976), Kou and
Wang (2004), Dao and Jeanblanc (2006)). This will have, as we show, tremendous consequences for
modelling, economic intuition, and quantitative implications.
Both the timing of the shock arrival Ts and its persistence Tr − Ts follow the Poisson process
with intensity λ with the following probability density function:
φ(w) = λw e−λw w , w = {s, r}.
(4)
The intuition offered by this framework is as follows. Starting with a state when the temporary
shock is not in effect, the probability that it will occur in the next short interval of time dt is λs dt,
and when the temporary shock is in effect, the corresponding probability that it will revert to the
8
¼
1
0
Tr
Ts
t
1 + ²Ts
Figure 1: The evolution of cash flow process for a case with only one transitory shock. The figure
shows the time-series of firm’s cash flows for a motivating example with only one stochastic temporary shock.
The permanent cash flow is assumed to be 1. At random time Ts shock ǫ arrives. The cash flow reverts to
permanent cash flow at random time Tr .
permanent cash flow is λr dt. While the presence of Poisson intensity offers a dramatic simplification
to our problem (since it is the only process for which the residual waiting time before the event occurs
is unaffected by the past and has the same distribution as the waiting time itself), the economic
intuition derived here would carry to more complicated scenarios. The Poisson intensity is used, for
example, in economic models of depreciation, where the project can die with small probability over
the next instant of time. The economic intuition is similar in our case: we can think of temporary
shock “depreciating” in the future.
Two apparently similar parameters, λs and λr , play distinctly different economic roles. It is
worth to think of λs as the frequency of “big” transitory shocks hitting the firm, with higher value
of λs implying that the firm is “more often in the news”. On the other hand, we can think of λr
as the speed of shock’s mean-reversion. Indeed, the expected value of the shock size attenuates
exponentially with speed λr after occurrence date Ts (if the shock has not disappeared yet). To see
it clearly, for any date t the expected value of the transitory shock to the cash flow process at any
future date τ > t can be written as
Et [ǫτ ] =
(
0,
ǫTs
t < Ts and t ≥ Tr ,
e−λr (τ −t) ,
Ts ≤ t < Tr .
(5)
Figure 2 shows the shock’s expected value for any time Ts + τ at date Ts . Permanent shocks, of
course, represent a partial case in this example if the mean-reversion speed, λr , is zero. In general,
transitory shocks may have two economically similar interpretations. First, the shock is finitely lived
and is dying away according to the Poisson death rate. Alternatively, the shock might have an infinite
life, but its effect exponentially decays over time attenuating to zero. These two scenarios will yield
the same present value in the absence of financial frictions. While we choose to model only the first
case (which turns out to be the simplest of the two), the economic intuition is more general.
9
ETs (¼t )
1
Ts
t
1 + ²Ts
Figure 2: The expected value of the shock at time Ts for a case with only one transitory shock.
The figure shows the expected value at time Ts of future cash flows for a motivating example with only one
stochastic temporary shock given the realization of the transitory cash flow of ǫTs .
The realized transitory shock to the cash flow process can analogously be written for any date t
as
ǫt =
(
0,
t < Ts and t ≥ Tr ,
ǫTs , Ts ≤ t < Tr .
(6)
The initial size of the jump is distributed according to the symmetric double exponential distribution with the following probability density function:
1
f (ǫTs ) = γe−γ|ǫTs | .
2
(7)
The double exponential distribution has been used in the models of option pricing with (permanent) jumps (see, for example, Kou (2002) and Kou and Wang (2004)). One advantage of modelling
shocks in this way is that the expected value of the shock is zero and its positive and negative realizations are symmetric.7 Parameter γ is driving the volatility of the temporary shock, with smaller
γ leading to shocks of larger magnitude.
To sum up our discussion of the nature of the shock, its time of occurrence is unexpected, its
size is discrete, and the magnitude can potentially be very large, and both the expected value of the
shock after its occurrence and its realized value demonstrate that shock is felt only for a limited time.
This contrasts with the typical model of cash flow in contingent claims models, where shocks arrive
continuously, with every instantaneous shock being very small and yielding a permanent effect. In
addition, by model construction the expected cash flow volatility is positive (in the absence of the
7
The reasons why we use double exponential distribution function (rather than normal distribution that has been
used extensively in corporate finance to model shocks) are purely technical and in anticipation of modeling both
permanent and temporary shocks in Section II. We also solved numerically the case with normal shocks with similar
results which are available upon request.
10
temporary shock it is zero) and the expected time-series correlation between firm value and cash
flows is positive but less than one (it is one in the absence of the temporary shock). Thus, this naive
model satisfies all the empirical criteria for firm’s cash flows outlined in the introduction.
I.B.2
Unlevered asset value
While the symmetric shock distribution guarantees that both negative and positive realizations are
equally possible and the expected value of the future shock is zero, the asset value at any time before
the shock occurrence, Vt , is affected by future shocks in the presence of limited liability. In particular,
firm owners will find it in their interest to abandon the firm if the total remaining present value of
cash flows from asset ownership is negative. Note an important difference between the definition of
abandonment which is introduced here and its counterpart in a number of financing models where
it is equal to abandonment by equityholders who default and transfer their ownership rights to
debtholders. In the presence of lognormal shocks cash flows would never turn negative and therefore
the residual claimants will never find it optimal to abandon.8 In reality projects are scrapped and
firms are dissolved routinely, which can be explained by negative net profits and, subsequently,
negative net value. While abandonment may involve costs, it would not change the intuition of what
follows and therefore we assume that owners may abandon immediately and costlessly.9
The threshold level of abandonment is denoted by ǫA , at which firm owners are indifferent between
abandonment and continuation, and which is found from the indifference condition by equating the
residual firm value to zero:10
ETs
Z
Tr
−r(s−Ts )
(1 + ǫA ) e
−Tr
ds + e
Ts
Z
∞
Tr
−r(s−Ts )
e
ds = 0.
(8)
The abandonment level that satisfies (8) is
ǫA = −
λr + r
,
r
(9)
The transitory nature of the shock dictates that the abandonment level is always lower than the
negative value of the permanent cash flow, −δt = −1. Shorter-lived shocks yield lower abandonment
threshold, since shocks of shorter duration are likely to have a marginal impact on the asset value.
Lower interest rates also lead to less abandonment, for they render the bright future more valuable
after the negative shock is over.
8
Abandonment can be optimal only in the presence of negative free cash flows. In real option models the abandonment decision is introduced by modelling δ as the revenue stream and adding production costs (see e.g. Dixit and
Pindyck (1994)). Thus our modelling of abandonment is close to its counterpart in real option models.
9
Introduction of abandonment costs makes residual claimants of the firm willing to hold on to firm’s assets for longer
time since exit is costly and continuation strategy is more profitable even if cash flows are lower.
10
All results in this subsection can be easily proved by a simple adjustment of proofs given in Appendix A for
Section II and therefore we omit all proofs here. Specifically, the intuitive fact that the value of the unlevered firm is
monotonically increasing in ǫ will be established in Section II for a more general case.
11
The asset value (which we define as the present value of all future cash flows streaming from
asset’s cash flow rights) is the sum of the perpetuity paying permanent cash flow and the present
value of transitory deviations and can be shown to be
Vt =
δt
+ Et
r
Z
∞
e−r(s−t) ǫs ds =
t
1 1 λs
1
+
eγǫA , t < Ts .
r 2 r + λs γ(r + λr )
(10)
The second term is always positive, reflecting the asymmetric option-like nature of limited liability. As could be expected, asset value increases with the variance of shock’s size, intensity of shock’s
occurrence, and its persistence.
I.B.3
Canonic securities
To derive the value of any contingent claim security (such as equity and debt) in an intuitive way,
we first introduce simple securities in this economy paying off upon the shock arrival with the payoff
value depending on the region the temporary shock falls within. Since all other securities are a
combination of these, we call them canonic securities. The canonic security is an Arrow-Debreu
security A(g(ǫT ), λ, b) that pays g(ǫT ) at time T of the shock arrival if the initial size of the shock
ǫT is at least as large as b. Note that, importantly, these securities do not assume limited liability
– their payoffs, and therefore values, can be negative. For our purposes, we need canonic securities
with triplets of parameters (1, λs , −∞), (1, λs , b), (ǫTs , λs , b). The first one pays $1 at time Ts for
any value of the shock. The second and third securities pay correspondingly $1 and ǫTs if the initial
size of the shock is at least as large as b.
The values of these securities at time t, t < Ts , are given below:
h
i
A(1, λs , −∞) = Et e−r(Ts −t) =
h
i
A(1, λs , b) = A(1, λs , −∞)Et 1|ǫTs > b =
h
i
A(ǫT (λs ) , λs , b) = A(1, λs , −∞)Et ǫTs |ǫTs > b =
λs
,
r + λs
λs
1
(1 − eγb ),
r + λs
2
λs 1 γb
e (1 − γb),
r + λs 2γ
(11)
(12)
(13)
While the particular values of the last two securities depend on the nature of the distribution of
the initial shock size, the same securities can be defined similarly for any distribution, for example,
normal.
I.B.4
Corporate debt valuation
At date 0 the firm issues perpetual debt that promises interest payments of c per year. Firm’s
equityholders make two decisions over the life of the firm: at date 0 they choose optimal coupon
rate to maximize the firm value (i.e., the sum of equity and debt values) and at some subsequent
date of their choice they can optimally default. In standard models of debt financing (e.g. Leland
12
(1994)) equityholders optimally choose the default boundary, which is the cash flow or asset value
low threshold, with default occurring when the state variable reaches the threshold for the first time
from above. In our example, it is clear that equityholders may optimally choose to default only at
the time of the jump, since before the shock occurs cash flow is a known constant, and if they decide
not to default at the time of the jump, the effect of the shock is of limited duration and so they will
definitely not default afterwards. The default boundary condition is thus equivalent to specifying
the initial magnitude of the jump so that if the shock is at least of this size or more severe (meaning
more negative) equityholders optimally default. In Section II we shall see how this default condition
is transformed when permanent shocks follow a familiar lognormal process.
The default boundary, to be optimally determined by equityholders, is denoted by ǫB ,
ǫB = πB − 1.
(14)
For all the values of the shock below ǫB equityholders transfer the ownership to debtholders.
With this in mind, the debt value at any time t before the shock occurs can be written as follows:
Dt = Et
Z
Ts
−r(s−t)
(1 − τi )ce
ds
Z ∞ Z ∞
+Et
(1 − τi )ce−r(s−t) dsf (ǫTs )dǫTs
(15)
ǫB
Ts
Z ǫB Z ∞
−r(s−t)
+(1 − α)(1 − τe )Et
1 + ǫTs · I{s<Tr } e
ds f (ǫTs )dǫT , t < Ts .
t
ǫA
Ts
The first term is the after-tax present value of coupon payments prior to the shock, where τi is the
marginal tax rate on interest payments. The second term is the present value of coupon payments
after the shock occurs in case equityholders do not default. Finally, the third term in equation (15) is
the residual value of the firm in default that debtholders receive in lieu of future coupon payments by
defaulted equityholders. Debtholders may immediately and costlessly abandon the firm which they
do if the total remaining asset value is negative which occurs at the values of the shock of ǫA , given
in (9), and lower. The post-default value is also reduced by proportional liquidation/bankruptcy
costs α and corporate taxes incurred at the proportional corporate tax rate τe .11
Using the canonic securities we can re-write the value of debt in (15) as
11
Here and throughout the paper, we assume, following the literature, that debtholders do not lever up the firm in
case of default and are thus liable for corporate tax. This assumption is for technical simplicity and its quantitative
impact for reasonable debt levels is negligible.
13
Dt =
(1 − τi )c
(1 − τi )c
(1 − A(1, λs , −∞)) + A(1, λs , ǫB )
(16)
r
r
1
1 +(1 − α)(1 − τe ) (A(1, λs , ǫA ) − A(1, λs , ǫB )) + (A(ǫTs , λs , ǫA ) − A(ǫTs , λs , ǫB ))
.
r
r + λr
The first term gives the riskless bond value if the shock does not happen. It can be alternatively
thought of as an annuity paid until the shock arrives. The second term shows that in case of no
default at the time of the shock, the bondholders are entitled to perpetuity (since this is a single
shock-economy). The last term shows that in case of bankruptcy, the value stems from the perpetuity
generated by the permanent cash flow (the first component of the third term) and the present value
of deviations from the long-term cash flow. To see clearly the intuition behind this representation,
note that we can think of the complex security A(ǫTs , λs , ǫA ) − A(ǫTs , λs , ǫB ) as the present value of
the size of the shock if the debtholders incur all the cost of the shock (i.e the shock is large to default
but not severe enough to abandon the firm) which debtholders are expected to pay until the shock
mean-reverts which is taken into account by the perpetuity formula
1
r+λr .
Since default and abandonment can happen only at negative shocks (ǫB and ǫA are less than
zero), the value of A(ǫTs , λs , ǫA ) − A(ǫTs , λs , ǫB ) is negative and therefore the debt value is always
lower in the presence of temporary shocks for a given coupon rate and other model parameters. This
is a fundamental observation since, as formula (10) demonstrates, the total value of the firm is larger
in the presence of temporary shocks. This apparent inconsistency is a consequence of a conflict
of interests between equity and debt. Debtholders incur costs in bad states of the world but do
not share in benefits from the upside. It follows naturally that debt value is a decreasing function
of temporary shock’s frequency, its persistence and magnitude. As we shall see, this important
mechanism is enforced in the benchmark model and yields a higher reliance on financial flexibility
and preoccupation with cash flow, as opposed to asset value, volatility.
I.B.5
Equity valuation and optimal default boundary
Analogously to debt value, the equity value can be written as follows:
Et = Et
Z
+Et
Ts
tZ
−r(s−t)
(1 − τe )(1 − c)e
ds
Z
∞
∞
−r(s−t)
(1 − τe ) 1 + ǫTs · I{s<Tr } − c e
dsf (ǫTs )dǫTs .
ǫB
(17)
Ts
The first expectation is the present value of equity cash flows before the shock. The second term
gives the present value of cash flows in case of no default which again can be decomposed into the
permanent and temporary components. The equity value can be intuitively represented in terms of
14
canonic securities introduced above as
Et = (1 − τe )
1−c
1−c
1
(1 − A(1, λs , −∞)) + A(1, λs , ǫB )
+ A(ǫTs , λs , ǫB )
r
r
r + λr
. (18)
The last term is the present value of short-term deviations which is strictly positive given equity’s
limited liability. The expected shock has thus an impact on equity value opposite to that on debt
value: equity value increases in shock’s frequency, its persistence and magnitude.
To determine the default boundary ǫB , the equityholders evaluate the continuation value of their
equity ownership rights at time Ts and decide to forfeit them if the continuation value is lower than
their promised payment to debtholders. It turns out that they are indifferent exactly at the negative
shock size when the value of equity is zero12
ETs = ETs
Z
∞
Ts
−r(s−Ts )
(1 + ǫB · I{s<Tr } − c)e
ds = 0,
(19)
which gives us the optimal default boundary ǫB as a function of the coupon rate:
ǫB = −(1 − c)
λr + r
.
r
(20)
The default boundary is always negative and increases in the coupon rate. In particular, if the
coupon level is equal to the permanent cash flow, c = 1, then default happens at any negative shock.
Conversely, if the firm is unlevered, c = 0, the default condition coincides with the abandonment rule,
ǫB = ǫA . The default boundary increases in shock persistence since adverse shocks of longer duration
imply that the firm continues to be in financial distress (cash flows are lower than coupon payments)
for longer. It is interesting to observe that λr in the value of debt can be thought of as another
type of liquidation costs since lower λr implies that negative shock is more long-lived and more of
the firm’s positive cash flows are going to be eroded. However, unlike traditional liquidation costs
α, these costs do enter the default boundary condition of equityholders since equityholders do not
internalize bankruptcy costs incurred by the residual claimants ex-post but they partially internalize
temporary negative shocks to earnings.
Firm owners decide on the optimal capital structure at date 0.13 To find the optimal level of
debt, equityholders maximize the total firm value at that date with respect to the coupon level given
the optimal default boundary:
12
In Section II we show (for a more general case) that equation (19) has only one negative root and is thus an appropriate bankruptcy condition at the time of a discrete jump, an analogue of smooth-pasting conditions for continuous
processes.
13
We assume static capital structure decision for simplicity of exposition. What is important here is that debt is
issued before the shock occurs.
15
max D0 + E0
c
s.t. eq. (20),
(21)
Although there is no analytic formula for c, due to strict convexity of the maximization program
there exists a single solution for a given set of parameters, and the numerical solution is readily obtainable; moreover, many of its properties can be determined in the absence of an explicit formula for
it. The economic implications are non-trivial since the preferences of debtholders and equityholders
are diverging with respect to the parameters that govern temporary shocks.
To understand the implications of this simple model, note that in the absence of temporary
shocks, the optimal coupon level is equal to the permanent cash flow, c = δ = 1, to maximize on
the tax benefits of debt if τi < τe (an assumption supported empirically e.g. Graham (2000)). What
happens if we add the temporary shock? Figure 3 and Figure 4 show how optimal coupon rate and
optimal leverage generally behave as the functions of three parameters of interest: shock’s frequency,
its magnitude, and its persistence.
Debtholders face the asymmetric treatment by bearing the cost of bad shocks without participating in the upside from good shocks. As the expected time of the future shock arrival becomes
shorter, they require higher compensation which implies low debt levels and lower leverage. While
we delegate all quantitative discussions to Section IV where we calibrate the parameters to replicate
a number of stylized empirical facts in a more general model, it is worth mentioning that optimal
financing is convex in λs even for very low values of λs (when the shock arrives with very low frequency and thus cash flows are unlikely to be negative) and thus the effect of temporary shocks on
the financial policy can be substantial.
Coupon
C
1
0
¸s
Leverage
L
1
0
¸s
Figure 3: Optimal leverage ratio and coupon levels as a function of λs for a case with only one
transitory shock. The figure shows the optimal leverage ratio and optimal coupon level as a function of the
shock arrival intensity, λs .
Figure 4 shows that the effects of shock’s magnitude and persistence on the optimal coupon rate
are not monotonic. When persistence is very high (the value of λr is low and so shocks mean-revert
16
very slowly) or the expected size of the shock is very large (the value of γ is low), equityholders
may be tempted to lever up, maximize their value before the shock hits, and default with very
high chances upon shock’s hitting if the shock is negative. This short-sighted behavior reminds to
some extent the asset substitution phenomena in which equityholders choose high-risk investment
projects if their firm is highly levered. This behavior can be thought of as finance substitution where
equityholders lever up to gain from a substantial optionality induced by transitory shocks. In fact,
not shown in Figure 4, for very low (high) values of γ (λr ), which are not likely to occur frequently
in reality, shareholders may prefer to issue coupon higher than one. In other words, they prefer
optimally to put the firm in financial distress from day one since in the case of the positive shock
(about which they care most) the benefits are asymmetrically large.
However, when persistence is lower (the value of λr is large and so shocks mean-revert quickly)
or the expected size of the shock is relatively small (the value of γ is large), transitory shocks matter
less relative to the permanent cash flow component and thus the optimal coupon level is increasing.
C
Coupon
1
0
¸r ; °
L e v e ra g e
L
1
0
¸r ; °
Figure 4: Optimal leverage ratio and coupon levels as a function of λr and γ for a case with only
one transitory shock. The figure shows the optimal leverage ratio and optimal coupon level as a function
of the shock’s speed of mean reversion, λr , and its dispersion, γ.
The leverage ratio is generally increasing as the temporary shock becomes less important (except
unusual and less interesting cases where optimal coupon is higher than one): in the first effect above,
equity value is growing disproportionately to debt value; and in the second effect, larger coupon
level offsets the lower default probability. It is worth mentioning that the leverage ratio is concave in
these two parameters and even for relatively low persistency scenarios we may expect a non-negligible
decrease in the optimal leverage ratio.
17
II
Benchmark model: Permanent and transitory shocks
II.A
Modeling cash flows
After considering simple models with either temporary or permanent shocks, we build a general
framework that includes shocks of both types. To put the framework in perspective to the existing
literature, we can think of the firm’s cash flow process following the lognormal process as in the
standard model (the first motivating example) but that the firm is now also a subject to short-term
fluctuations in cash flow of the type introduced in the second motivating example. For convenience,
we continue to use all the notation introduced above and, for completeness, we briefly review the
nature of shocks introduced in greater detail in Section I. Firm’s earnings, πt , will now depend on
two sources of uncertainty and can be written as
1
2
πt = δ0 e(µ− 2 σP )t+σP t(Wt −W0 ) + ǫt , δ0 ≥ 0, t ≥ 0,
(22)
where δ0 is the initial value of the permanent component of earnings and ǫt is the current value
of short-term deviations. We denote the volatility of the permanent component cash flow σP and
stress that in the presence of the transitory shocks it is lower than the total asset volatility, σV . At
random time Ts either positive or negative jump ǫTs to the permanent cash flow level can happen
with the Poisson intensity λs . The short term nature of shocks is captured by the assumption that
the realized deviation from the long term cash flow level is removed at time Tr which also follows
the Poisson process with intensity λr . We assume that temporary cash flow variable ǫ is distributed
doubly exponentially with intensity γ under the risk-neutral measure. Thus, at any time, if the
temporary shock is present, firm’s instantaneous cash flow is δt + ǫTs . While we consider the case of
only one transitory shock here for reasons of tractability, we explore the multiple-shock scenario in
Section IV.
II.B
Canonic securities
Intuitively, the valuation in the single-shock economy considered in this section depends in which of
the three periods claims are evaluated: before, at the time, or after the temporary shock’s impact.
In the next several subsections we derive the values of all assets in this economy and investigate the
economic intuition behind optimal decisions but first we introduce two basic and intuitive securities
of which all other securities are linear combinations. Similar to Section I.B we call them canonic
securities.
The first security pays off at default or abandonment if these events occur prior to the jump
caused by the transitory shock. Specifically, denote by X(δt , λ, bt ) (X for “exit”) a security that
pays $1 at the moment τ when δt reaches or crosses bt for the first time and if τ < T , where T is the
the random time of temporary shock arrival (Ts ) or mean-reversion (Tr ). In our setting, bt can be
18
either a default or abandonment boundary. The value of this security can be written as
h
i
X(δt , λ, bt ) = Et e−r(τ −t) Iτ <T .
(23)
The second security, on the other hand, pays off at the moment of a jump conditional on the
firm not exiting before that. We can denote the value of this security as A(δt , g(δT ), λ, bt ) which
shows that it pays the expected value of g(δT ) at time T if τ ≥ T . Function g(·) can represent any
corporate claim. We can write the value of this security at time t as
h
i
A(δt , g(δT ), λ, bt ) = Et e−r(T −t) g(δT )Iτ ≥T .
(24)
Both of these claims are Arrow-Debreu securities conditional on the jump structure of the economy. In particular, they are related via the following relationship:
A(δt , g(δT ), λ, bt ) = A(δt , g(δT ), λ, 0) − X(δt , λ, bt )A(bt , g(δT ), λ, 0).
(25)
The analytic values of these two securities are given in Appendix A.
II.C
The value of unlevered firm
We start with deriving the unlevered firm value, V (δt , ǫt ) and V (δt ) in presence and in absence of
temporary shocks, which will be useful for two purposes, as a benchmark in comparisons with the
levered firm value and as debtholders’ liquidation value. The next proposition gives the unlevered
firm value.
Proposition 1 ( Unlevered value) The unlevered before-tax asset value Vt is:
1. For t ≥ Tr ,
V (δt ) =
δt
.
r−µ
(26)
2. For Ts ≤ t < Tr , for any given realization of ǫ:
V (δt , ǫ) =
ǫ
1 − X(δt , λr , δA,t ) − A(δt , 1, λr , δA,t )
r
δA,t
δt
+
−
X(δt , λr , δA,t )
r−µ r−µ
δTr
−A(δt ,
, λr , δA,t ) + A(δt , V (δTr ), λr , δA,t ),
r−µ
where δA,t is the level of cash flow at which the firm is abandoned.
19
(27)
3. For t < Ts ,
V (δt ) =
δt
δTs
− A(δt ,
, λs , 0)
r−µ
r−µ
Z
+ A(δt , V (δTs , ǫTs ), λs , 0)dF (ǫ).
(28)
ǫ
To see the intuition behind the formulas above, consider for example asset value when shock
is present given in (27). The first line gives the present value of transitory shocks (which effect is
present until either abandonment or shock mean-reversion). The remaining lines show the valuation
of permanent cash flows. In the case of abandonment, firm holders lose the right to all future cash
flows (the last term on the second line shows that the expected value of these rights, δA,t /(r − µ),
is multiplied by X(·) to get the present value). Finally, the last line shows that in case of shock’s
mean-reversion firm holders exchange the expected pre-jump firm value for expected post-jump firm
value.
The after-tax value is obtained simply by multiplying the formulas above by (1 − τe ). In case of
only permanent lognormal shocks and costless production the project will never be abandoned. It
implies that the firm holders would be willing to liquidate the firm only if the transitory shock is
present and sufficiently negative, so that the present value of the permanent positive cash flows is
offset by expected losses from transitory shock. To find the threshold level δA,t we apply a standard
smooth-pasting condition to the unlevered firm value. The next proposition gives the closed-form
solution for it.
Proposition 2 ( Abandonment decision) The abandonment level, δA,t , is
δA,t =


0,



for t < Ts , t ≥ Tr and Ts ≤ t < Tr if ǫt ≥ 0;
ϑ+
λr
ǫ
r+λr t
− 1+ϑ+



r + λr
 r+λrλ−µ
σ2 d
λr
1
r−µ
+
1+ϑ
λr
1−
1+ϑ−
λr
!,
for Ts ≤ t < Tr if ǫt < 0,
where dw is
dw =
−
ϑ+
w − ϑw
,
2
(29)
(30)
−
and ϑ+
w , and ϑw are positive and negative roots of the quadratic equation
σ2 2
σ2
ϑ − µ−
ϑw − (r + w) = 0.
2 w
2
(31)
As Figure 5 shows, abandonment boundary is a linear function of transitory shock since there is a
constant trade-off between the cost of an additional shock and the present value of future permanent-
20
related cash flows. Linearity, however, is a consequence of the economy facing no future transitory
shocks as we shall see in Section IV.A.
Figure 5 also clearly shows that firmholders’ limited liability leads to the convexity in the payoff
with respect to transitory shock and thus creates a real option value. This limited liability feature
helps to explain that when volatility of permanent cash flows goes up the abandonment boundary is
lower. On the other hand, as transitory shocks becomes more persistent, the abandonment boundary
uniformly increases, reflecting higher importance of current temporary cash flows. We discuss the
relationship between abandonment and default boundaries in the next section.
δ and δ
B
min
1.6
0
−10
0
ε
10
Figure 5: Optimal default and abandonment boundaries for fixed coupon in the benchmark
model with both permanent and temporary shocks. The figure shows the optimal default boundaries
for t ≤ Ts (dotted and dashed line), Ts ≤ t < Tr (solid line), and t ≥ Tr (dashed line), and optimal
abandonment level (dotted line) for the benchmark model with permanent and temporary shocks.
II.D
Value of equity and optimal default decision
The value of equity also consists of both transitory and permanent components. For example, during
a period of transitory shock’s presence, equity value can be written as follows:
E(δt , ǫt ) = (1 − τe )Et
"Z
t
min(τ,Tr )
#
(δs + ǫt − c)e−r(s−t) ds + e−r(Tr −t) E δTr Tr < τ ,
(32)
where min(τ, Tr ) is the first time either shock-mean reversion of default occurs. The first component of the integral is the expected present value of dividends before the shock’s mean-reversion.
21
The second component is the present value of equity payouts after the shock’s mean-reversion.
Similarly, in period before the transitory shock is felt, the value of equity is the sum of the
present value of payout accruing to shareholders before the shock hits and the properly discounted
continuation value:
E(δt ) = (1 − τe )Et
"Z
min(τ,Ts )
−r(s−t)
δs e
t
−r(Ts −t)
ds + e
#
E δTs , ǫTs Ts < τ .
(33)
The next proposition gives an intuitive way to write generic equity value using canonic security
values.
Proposition 3 ( Equity value) Let δB,t be the default boundary at time t. The value of corporate
equity, Et , can be written as:
1. For t ≥ Tr ,
δB,t
E(δt )
δt
c
=
−
X(δt , 0, δB,t ) −
1 − X(δt , 0, δB,t ) .
1 − τe
r−µ r−µ
r
(34)
2. For Ts ≤ t < Tr , the value of equity for any given realization of ǫ is:
E(δt , ǫ)
1 − τe
=
ǫ
c 1 − X(δt , λr , δB,t ) − A(δt , 1, λr , δB,t )
r r
δB,t
δt
+
−
X(δt , λr , δB,t )
r−µ r−µ
δTr
−A(δt ,
, λr , δB,t ) + A(δt , E(δTr ), λr , δB,t ).
r−µ
−
(35)
3. For t < Ts ,
E(δt )
1 − τe
=
δB,t
δt
−
X(δt , λs , δB,t )
r−µ r−µ
Z
δTs
−A(δt ,
, λs , δB,t ) + A(δt , E(δTs , ǫTs ), λs , δB,t )dF (ǫ)
r−µ
ǫ
c
− 1 − X(δt , λs , δB,t ) − A(δt , 1, λs , δB,t ) .
r
(36)
For any given level of interest payments to debtholders, equityholders choose default boundary
to maximize their value. An important distinction between default decision of equityholders in this
22
and earlier models is that now default boundary is time-varying and depends on temporary shock’s
expectations.14 The next proposition derives the default boundary after the transitory shock arrives.
Proposition 4 ( Default boundary)
1. If either (i) t ≥ Tr or (ii) Ts ≤ t < Tr and ǫt = 0, then
δB,t = c
r − µ ϑ+
0
.
r ϑ+
0 +1
(37)
2. If Ts ≤ t < Tr and ǫt < 0, then δB,t is a solution to the following equation:
−ϑ+
ωB,1 + ωB,2 δB,t + ωB,3 δB,t0 = 0
(38)
where ωB,1 , ωB,2 and ωB,3 are constants. Appendix Appendix A gives their values.
3. If Ts ≤ t < Tr and ǫTs ≥ c, then the firm does not default, δB,t = 0.
4. If Ts ≤ t < Tr and 0 < ǫTs < c, then δB,t is a positive solution to the following equation:
−ϑ−
χB,1 + χB,2 δB,t + χB,3 δB,tλr = 0
(39)
where χB,1 , χB,2 , and χB,3 are constants. Appendix Appendix A gives their values.
For the period before shock’s arrival, the time-varying default boundary can be easily found numerically and a number of its properties can be shown analytically. Consider first optimal default
decision when transitory shock is present. When the realization of temporary shock is zero, equityholders find themselves in a permanent-only cash flow world and thus both the equity value and the
bankruptcy threshold are equal to their permanent-only counterparts after the shock mean reverts.
This is shown on Figure 5 where default boundaries for these periods intersect at the value of the
shock equal to zero. The negative values of temporary shock can be intuitively thought of as an
additional fixed coupon, costs of production, or tax penalty temporarily imposed on equityholders.
A temporary reduction in the dividend payout for the duration of the shock decreases the benefits of
cash flow rights triggering a higher default boundary level. Larger adverse shock realizations imply
higher losses to shareholders and lead to even higher bankruptcy thresholds.
For positive realizations of temporary shocks, equityholders experience a temporary coupon reduction or tax subsidy for the duration of the shock and the desire to default earlier diminishes.
When the temporary cash flow component is high enough, it is never optimal for equityholders to
14
The optimal default boundary in Hackbarth, Miao, and Morellec (2006), Bhamra, Kühn, and Strebulaev (2008),
and Chen (2008) are also state-dependent, reflecting the change in macroeconomic regimes.
23
default for the duration of the shock. Even if permanent cash flow falls to zero, equityholders optimally wait until the shock’s mean reversion before defaulting. This explains why for the value of the
shock larger than the level of coupon the optimal default level is zero in Proposition 4 and Figure 5.
An interesting and, as it turns out, economically important feature of default boundary is its
concavity (for values of the shock lower than the coupon level). This concavity is understood by
considering the trade-off between the present value of current cash flows and future cash flows (where
“future” here denotes all earnings after shock’s mean-reversion) or, in other words, comparing a
current default decision with a future default decision. Specifically, equityholders compare the present
value of cash flows until shock’s mean-reversion with future cash flows conditional on not defaulting
at the moment of mean reversion. This conditioning explains concavity. To see it, observe that
for a positive shock, for example, equityholders try to survive until the shock’s mean-reversion even
though they know that they are very likely then to default. In an extreme case, when the permanent
component is absent, they do not default if ǫ = ǫB ≥ c even though default happens with certainty
at some point in the future. This desire quickly evaporates, however, as shocks are getting smaller
since one unit of permanent cash flow is more important than one unit of a transitory component.
As the level of δB increases, however, the expected distance to default upon shock’s mean-reversion
decreases. In other words, it is now more likely that equityholders survive even after shock meanreverts and a smaller compensation for every additional decrease in ǫ is needed. The same intuition
implies to the negative shock scenario. As temporary shock becomes more negative, the distance
to default is becoming so large that the survival is guaranteed after the shock mean-reverts, and
therefore the trade-off becomes a constant and the default boundary converges to a linear boundary.
Moreover, the slopes of the default and abandonment boundaries are equal in the limiting case of
infinitely negative temporary shock since, conditional on surviving it, neither future abandonment
nor future default can damage future cash flow stream.
Empirical evidence suggests that firms often default “too late”, when most of the value is dissipated and bondholders’ recovery rate is too low. In other words, firms frequently default when net
worth is substantially lower than the promised value of debtholders’ payments (see e.g. Davydenko
(2007)). Standard contingent-claim models have difficulty explaining this stylized fact and also the
wide distribution of bondholders’ recovery rates in the cross-section. In our model, when transitory
shock is positive, this is exactly what may happen upon default. It also easily accommodates cases
of zero recovery rates. Moreover, empirical evidence shows that debt covenants can be both of networth type (i.e., the firm is in violation if its value is lower than a certain threshold) and of cash-flow
type (the firm is in violation if, say, interest coverage ratio is lower than a certain threshold). While
standard models can not distinguish between two types of covenants (in a permanent-only case, these
covenants coincide), our model may allow to disentangle them.
When the transitory component of cash flow is absent, default boundary does not depend on
the value of the shock. It is interesting to observe that equityholders always default at lower values
24
of permanent earnings when temporary shocks are expected some time in the future. The limited
liability option they hold creates a wedge between ex ante and ex post default levels and results in
a wealth transfer from debtholders to equityholders.
One argument that can be put forward in considering the nature of temporary shocks is that their
introduction is likely to lead to more frequent defaults and thus make reconciling empirical evidence
on the low frequency of defaults even more difficult. It is important to observe that when the shock
is present, default boundary can actually be both higher and lower than the default threshold in
the permanent-only scenario, and when the shocks are expected in the future, the expected default
frequency is always lower, for the same level of coupon payments. Thus, it is far from certain
that default frequency may increase; moreover, transitory shocks can actually lead to lower default
probabilities even for the same debt commitment levels. Indeed, our calibration results in Section
III.C confirm this intuition.
The next proposition summarizes properties of default and abandonment thresholds.
Proposition 5 Let δB,t and δA,t be the optimal default and abandonment boundaries, respectively.
Then:
1. δB,t > δB,s for any t ≥ Tr and any s ≤ Ts ;
2. δB,t is a continuous function of ǫt for any Ts ≤ t < Tr ;
3. δA,t < δB,t for any Ts ≤ t < Tr ;
4. δB,t − δA,t converges to a constant as ǫTs goes to −∞ for any Ts ≤ t < Tr ;
5. δB,t is a decreasing and concave function of ǫt for any Ts ≤ t < Tr and ǫTs < c.
Figure 6 shows how default boundary behaves with respect to the speed of mean-reversion or
longevity of shocks, λr , asset volatility, σ, and coupon level, c, fixing other parameters. Higher coupon
payments reduce the equityholders’ residual value and thus increase optimal default boundary. Note
that the graph of the optimal default boundary for the period when temporary shock is present
moves to the right so that the new default boundary reaches zero exactly for the value of the shock
equal to the new coupon payment.
As λr increases, shocks are expected to be of shorter duration, and their adverse effect damages
equityholders’ payoffs less. As a result, equityholders prefer to default at uniformly lower levels of
permanent cash flow process for negative shocks. Conversely, for positive values of temporary shocks
equityholders expect to get lower values of benefits for low λr and thus default boundaries increase.
One of the most important parameters that are known to influence optimal financial decisions
is asset volatility. Figure 6 shows how optimal default boundary changes with the change in the
instantaneous volatility of the permanent cash flow growth, σ, when other parameters are fixed.
When the temporary shock is present, the permanent component of asset volatility can substantially
25
±B;t 2 [Ts ; Tr ]
Low σ
Med σ
High σ
0
²
0
±B;t 2 [Ts ; Tr ]
±B;t 2 [Ts ; Tr ]
Low c
Med c
High c
Low λ
r
Med λ
r
High λr
0
0
²
0
0
²
Figure 6: Comparative statics for the case of both permanent and temporary shocks with
respect to σ, λr , c. The figure shows optimal default boundaries at t ∈ [Ts , Tr ) for the benchmark model
of both permanent and temporary shocks with respect to variance of permanent cash flows, σ, speed of
mean-reversion, λr , and coupon level, c.
influence default boundary. Higher σ increases the optionality of equityholders’ claim and make them
wait longer before defaulting.
II.E
Value of debt and optimal financing
To determine an optimal capital structure decisions of the firm, we first give the value of debt for
any fixed coupon payment c. The next proposition accomplishes this task using canonic securities.
Proposition 6 ( Debt Value) Debt value Dt is given by:
1. For t ≥ Tr ,
D(δt ) =
δB,t
(1 − τi )c 1 − X(δt , 0, δB,t ) + (1 − α)(1 − τe )
X(δt , 0, δB,t ).
r
r−µ
(40)
2. For Ts ≤ t < Tr , the value of debt for any given realization of ǫ is:
D(δt , ǫ) =
(1 − τi )c 1 − X(δt , λr , δB,t ) − A(δt , 1, λr , δB,t )
r
+(1 − α)(1 − τe )V (δB,t , ǫ)X(δt , λr , δB,t ) + A(δt , D(δTr ), λr , δB,t ).
3. For t < Ts ,
26
(41)
(1 − τi )c 1 − X(δt , λs , δB,t ) − A(δt , 1, λs , δB,t )
r
Z
D(δt ) =
+(1 − α)(1 − τe )V (δB,t )X(δt , λs , δB,t ) +
A(δt , D(δTs , ǫTs ), λs , δB,t )dF (ǫ).(42)
ǫ
The conflict of interests between equity- and debtholders is deepened by the different effect of
short-term fluctuations on their respective value. In particular, when shocks are expected to last
longer, equityholders are on average better off ex post shock realization and debtholders are on
average worse off. In deciding on optimal financing, however, equityholders maximize the total value
of the firm and it is easy to see how conflicting interests of equity and debt can lead to non-trivial
implications.
Equityholders choose at time 0 coupon level c to solve the maximization program similar to (21):
max D(δ0 ) + E(δ0 ) s.t. default boundary conditions (37) − (39).
c
(43)
We investigate the impact of transitory shocks on optimal financing and calibrate the model
empirically in the next section.
III
How important are transitory shocks? Empirical evidence and
calibration
III.A
Empirical evidence
In this section, we present and discuss empirical evidence on firm earnings. Our task here is two-fold.
First, the main reason we built a new framework of earnings that incorporate both permanent and
transitory shocks to cash flows is the apparent inconsistency between actually observed earnings and
earnings generated by the permanent-shock only model. Therefore, here we document a number of
empirical stylized facts that support this claim and motivate our theoretical framework. Second,
to investigate whether transitory shocks can substantially influence financial behavior, we use this
empirical evidence to calibrate a number of parameters.
For our purposes of documenting stylized facts, we start with a comprehensive sample of merged
COMPUSTAT and CRSP firm-year observations over the period from 1987 to 2005.15 We exclude
financial companies (SIC 6000–6999), utilities (SIC 4900–4999), non-US companies (entries with
incorporation code variable equal to 99), and non-publicly traded firms and subsidiaries (entries
15
The stylized facts that we document are robust to any changes in the data set, variables definitions, and empirical
methods that we tried. For example, while we start in 1987 since cash flow from operations is reported from this
year, the results are qualitatively unchanged when we investigate the 1962 from 1986 sample (using either EBITDA or
reconstructed cash flow from operations). All robustness results are available upon request.
27
with stock-ownership variable equal to 1 or 2). We also exclude firm-years with total book value
of assets (COMPUSTAT data item 6) of less than $10 million. We also require the observations to
have valid quasi-market leverage and book leverage ratios, and exclude observations with market-tobook ratios above ten. This leaves us with 68,625 firm-year observations from 9,175 unique firms (as
defined by unique NPERMNO, a CRSP identifier), from a minimum of 2,914 observations in 1990
to a maximum of 4,581 in 1997. All nominal values are converted into year 2000 dollar values using
CPI index from the Bureau of Labor Statistics of the United States Department of Labor.
Table I reports descriptive statistics on a number of variables of interest for the final sample and
size subsamples. Since we are interested in the cross-sectional distribution properties of firm financial
characteristics, we choose first to produce a cross-sectional distribution for each characteristic in each
year and then average across years.16 We define Firm size as the book value of assets and the Quasimarket leverage ratio of firm i in year t by
QM Lit =
D9it + D34it
,
D9it + D34it + D199it × D25it
(44)
where Dx is a COMPUSTAT annual data item x. D9 is the amount of long-term debt exceeding
maturity of one year, D34 is debt in current liabilities, including the portion of long-term due within
one year, D199 is the year-end common share price, and D25 is the year-end volume of shares
outstanding.
We take operating cash flow (D308) which is available after 1986 as our measure of firm Cash
flow. A number of alternative measures could be used, for example, EBITDA (D13) which is in
general larger than the benchmark measure but does not change the results substantially.17 Another
possibility is to exclude depreciation and amortization from the estimation of earnings which generally
lowers cash flow values. It is important that our measure includes both interest payments and taxes
since model state variable δt does. We define Profitability as the ratio of operating cash flow to book
assets and Coverage ratio as the ratio of operating cash flow to interest expense (D15) (in producing
statistics for this variable, we exclude all observations for which interest expense is zero).
The average and median values of statistics for the final sample in Panel A of Table I do not
produce any surprising results and are similar to those reported elsewhere in the empirical literature.
The distribution, however, shows that at the 5th percentile, firms experience losses of nineteen million
dollars per year or 21% of their book assets, and almost a quarter of all firm-year observations have
negative operating cash flow. This result is bluntly inconsistent with the standard lognormal process
16
Unreported, we also replicate these results by either constructing a cross-sectional distribution that includes all
panel or first averaging all statistics for each firm and then averaging over all firms. There is no substantial difference
between these three methods; the latter procedure produces higher frequency of negative cash flows and overall stronger
support for the modeling framework along all dimensions. These findings, however, may be biased since the results are
likely to be dominated by small non-surviving firms.
17
See Givoly and Hayn (2000) for a detailed discussion of these two measures and analysis of conservatism in
accounting reporting.
28
used so widely in contingent claims models since it implies positive cash flows. The panel also shows
that a non-trivial number of firms are likely to experience cash flow shortages since their interest
coverage ratio is lower than one.
Importantly, Panel A shows that there is a wide cross-sectional distribution of within-year profitability and related variables. Intuitively, firm size should explain a significant portion of this crosssectional variation (since profitability is persistent and firm size is conditioned on past profitability)
and Panels B, C, and D, which show the same statistics for firms of different size tertiles, support
this intuition. The results unambiguously show that smaller firms are more likely to experience losses
and find themselves in financial distress. In an average year, between a quarter and a half of all firms
in the smallest size tertile report negative cash flows and for a similar fraction of firms operating
cash flow is not sufficient to cover their financial obligations. The relationship between size, on the
one hand, and losses and financial distress, on the other hand, exhibits a monotonic relationship.
For example, firms in the largest tertile experience negative cash flows in less than 10% of cases.
This evidence supports our choice of modeling the total cash flow in the additive structure (the sum
of permanent and transitory components), since multiplicative structure of the kind δR = δ(1 + ǫ)
would lead to shocks proportional to firm size. Additive structure produces shocks which are similar
to fixed costs of production in real option models and are independent of size. A part of the difference
between small and large firms are driven by firms’ survival. If small firms experience losses, they
are more likely to leave the sample. Conversely, larger firms are less likely to experience losses since
size is a consequence of positive cash flows in the past and cash flows are persistent. Importantly,
the same behavior would be observed in our model. When firms are small (as measured by δt or
V (δt , ǫt )), they are more likely to hit the abandonment level for the same shock magnitude and leave
the sample. Large firms experiences positive shocks in the past and are less likely to face negative
cash flows in the future.
We now investigate the empirical values of parameters in the data that further shed light on three
criteria that we identified in the introduction and produce related stylized facts. These variables are
also more closely directed to our calibration exercise below. In particular, we investigate: (1) How
prevalent the experience of negative cash flows and financial distress is in the data; (2) Conditional
on experiencing losses, what is the duration of staying in the red; (3) Whether cash flow volatility is
larger than asset volatility; (4) Whether the correlation between cash flow and asset values (both in
levels and changes) is lower than one. For this exercise, we need to use time-series data on individual
firms and therefore we choose an individual firm to be the unit of observation in Table II. We report
our findings, in addition to the final sample, again for firms in different book size tertiles. For this
exercise, we define firm size as the average inflation-adjusted book value of assets for all years the
firm is present in the final sample.
Table II shows that an average firm experiences negative operating cash flows every third year.
Given that our chosen measure of profitability is potentially less conservative than other measures, we
29
also re-check this result using EBITDA (COMPUSTAT data item 13) as a more conservative proxy
for cash flow. With this measure, the result is similar (31% vs 34%) for the same period. Operating
cash flows are on average insufficient to cover contractual financial obligations more than 40% of the
time. Both of these numbers are telling. These mean statistics, however, hide substantial variation
in the data. Indeed, a median firm reports negative cash flows only every fifth year and interest
coverage ratio lower than one every third year. At the same time, a quarter firms report neither of
these events in any year and almost a quarter of firms experiences low interest coverage ratio almost
every year. The difference between small and large firms is startling: while small firms spend 52%
of time on average making losses and are unable to cover interest expense by their earnings 58% of
time, large firms find themselves in the same situations only 15% and 27%, respectively.
Once an average firm makes losses, it is also likely to be unprofitable next year, and to return
to positive earnings in the third year (conditional on survival). The duration of loss-making period
decreases with firm size. Since small firms are less likely to survive losses, the higher duration
reported for them is even more significant. About a quarter of small firms experience on average
three consecutive years of losses. Once in financial distress, firms are likely to stay in the red for two
more years. This duration, however, is lower for small firms, suggesting that small firms are more
likely to exit the data (via liquidation or acquisition) once they are unable to cover their interest
expense.
The task of estimating cash flow and asset volatilities is not an easy one: we need measures
that would be consistent with each other and also produce the expected results when applied to the
benchmark model without transitory shocks. Importantly, note that within the realm of a standard
contingent claims model, we would of course take the ratio of cash flow (asset value) changes to
the cash flow (asset) value in the previous year to estimate cash flow (asset) growth volatility.
However, such a definition requires a proportionality assumption which is standard for asset prices
but empirically unjustified for cash flows. As we show above, cash flows can be negative. They also
can have the value of zero or being very close to zero making the standard estimation unreasonable.
The problem then, however, in comparing asset and cash flow volatilities since the ratio of cash flow
changes to asset values is expected to be lower than the ratio of asset value changes to asset value.
To estimate cash flow and asset volatilities, we therefore proceed as follows. For each firm we
estimate the ratio of the standard deviation of annual changes in cash flows to the average of absolute
values of operating cash flows. For asset volatility, we estimate for each firm the standard deviation
of the ratio of annual changes in their quasi-market asset value (the sum of market equity and
book debt) to the quasi-market value in the previous year. While measures based on more frequent
equity prices undoubtedly produce much better estimates of asset volatility (see e.g. Schaefer and
Strebulaev (2007)), they would hardly be compared with cash flow volatility which is our main goal.
We therefore use data of the same frequency to estimate asset and cash flow volatilities. Also note
that both of our measures are equivalent (aside from the frequency issue) to the instantaneous growth
30
rate of asset value in standard contingent claims models.
As Table II shows, cash flow volatility is much larger for an average and a median firm both for
the final sample and all subsamples.18 This result is consistent with our model of temporary shocks
and intuition, but inconsistent with any model where only permanent shocks are allowed for the
earnings process. The difference is also economically non-trivial. Finally, we estimate the correlation
between cash flow and asset values both in levels and changes. The permanent-only models produce
a perfect correlation both for levels and changes. The data shows that correlation is about 0.07 for
changes and about 0.22 for levels. Since transitory shocks matter less for large firms, correlation
measures increase with size. On a qualitative level, the result of positive imperfect correlation is
supportive of our framework. We now turn to the calibration of the model.
III.B
Model calibration
This section describes how firm parameters are calibrated for our benchmark case. An important
caveat is that for most parameters of interest, there is little empirical evidence permitting precise
estimation of averages, sampling distributions or even their ranges. In addition, the model requires
that all parameters be estimated as time-invariant. Overall then, while the empirical evidence in the
previous section is suggestive, the parameters used in the benchmark case must inevitably be regarded
as ad hoc and approximate. There are two ways we deal with this problem. First, whenever possible
(for example, for tax rates), we use established empirical estimates. Second, and more importantly,
we perform numerous robustness checks. These robustness tests show that the economic implications
of the model are not qualitatively affected by changing the parameters within a feasible range.
Finally, while we calibrate and present in this section the benchmark case, there is a substantial
cross-sectional heterogeneity in the data, most importantly, on the importance of transitory shocks.
In Section III.C we therefore carefully analyze various comparative statics. Table summarizes the
descriptive information for the parameters described below.
The average value of firm asset volatility, σV , is taken to be the average asset volatility from
the distribution of the standard deviation of rates of return on firm assets reported by Schaefer and
Strebulaev (2007). While the sample used in that paper is confined to firms that issue public debt,
we would like to be able to compare our results with similar studies by, for example, Leland (1994)
and Goldstein, Ju, and Leland (2001), which also use asset volatility implied from firms issuing
public debt. Since Faulkender and Petersen (2006) show that for firms without access to public debt
markets implied volatility is much higher and our empirical evidence corroborates this evidence for
the total COMPUSTAT sample, we will pay a particular attention to asset volatility in analyzing
comparative statics.
For the proportional cost of restructuring in default, α, we assume the value of 0.10. Most of
18
We also check the robustness of this result by excluding firms with short history to reduce the noisiness of the
results. The estimates are unchanged.
31
the empirical values reported in early literature, for example, Weiss (1990) and Altman (1984), are
somewhat lower. Recent evidence by Andrade and Kaplan (1998) suggests somewhat higher values.
Leland (1994) uses a similarly defined cost of 0.5, Leland (1998) uses 0.25, and Goldstein, Ju, and
Leland (2001) use 0.05.
To choose the effective tax rate on equityholders (which includes both corporate tax rate and
marginal person tax rate on dividends) and marginal personal tax rates on interest income, we
follow recent evidence (for example, Graham (2000)). We assume that the effective tax rate on
equityholders, τe , is 40% and the marginal tax rate on interest income, τi ,to be 30%.
The after-tax risk-free interest rate is assumed 0.055. We also choose the risk-neutral growth rate
of cash flows, µ, to be 0.01, which is consistent with the average payout rate used in earlier studies.
The rest of benchmark structural parameters, responsible directly for the transitory nature of
shocks (shock frequency λs , persistence λr , magnitude γ and the volatility of permanent cash flows
σP ), are chosen as follows based on the empirical evidence in the previous section. Persistence
parameter λr is taken to be 0.5 (which means that on average the shock mean reverts in two years)
to reflect the time that firms of different size spend on average in the area of negative cash flows
once they entered it (1.56–2.16 years according to Table II). Shock frequency parameter λs is taken
to be 1 (the new transitory shock arrives on average in one year) to reflect the frequency with which
firms experience negative cash flows again after exiting the negative cash flow period. In choosing
this number we assume for simplification that each new transitory shock makes cash flows negative
with probability 1/2 and then fit the simulated frequency of negative cash flows to the empirical
frequency. Since in reality realized cash flows do not become negative for each negative realization
of transitory shock, our calibrated shock frequency may slightly be biased downwards. We show in
comparative statics that by taking λs lower than one year to account for the bias, our results and
intuition of temporary shock importance are only further enforced.
We calibrate the permanent component of asset volatility σP and magnitude parameter γ from
the following two moments. First, the realized volatility of firm’s assets σA should be equal to 0.25 as
discussed above. Higher σP and lower γ increase asset volatility σV . Second, we choose the influence
of transitory shocks on unlevered asset value by estimating the premium that the real option of
abandonment in presence of transitory shocks adds to the permanent shock only asset value. We
take the expected value of this premium, which we denote φ, to be 2% of the permanent-only asset
value. Parameter φ is high when either γ is low or σP is high, reflecting the call option nature of
assets in presence of transitory shocks. Parameters σP and γ are then chosen to match σA and φ and
are, correspondingly, 0.20 and 0.098. This calibration also shows that 80% of variability of unlevered
asset values comes from the permanent shock component.
32
III.C
Results
Can the existence of transitory shocks explain well-known stylized facts about corporate financial
policies? The first set of predictions is presented in Table IV. The first row of the table shows the
benchmark case of the standard contingent claims model with permanent shocks only (for example,
the model of Leland (1994)). For the asset volatility value of 25% the optimal leverage ratio is about
66%. We compare this optimal leverage decision with the benchmark case of our model. Importantly,
for this comparison to be viable, we fix total asset volatility of unlevered firm value at the same level
of 25%. In other words, even though the source of cash flow variability differs between these firms,
we would like for an empiricist to estimate the same asset volatility from the data on a firm with
permanent shocks only and on a firm with both types of shocks. Otherwise, higher volatility in our
benchmark case can lower optimal leverage ratio and lead us to believe that financial policy is more
conservative even if it is not. We also choose a conservative assumption about the importance of
transitory shocks on the present value of total cash flows by assuming that the real option implied
by their presence constitutes only φ = 2% of the unlevered asset value. While empirical evidence is
not easy to obtain, it is likely that for most firms this effect will be at least of this magnitude (for
growth firms, obviously, we expect to have a much higher value of φ).
As the second row of Table IV demonstrates for our benchmark calibration scenario, when transitory shocks are present, firms’ optimal financial response is to behave substantially more conservatively. The optimal leverage ratio is now about 42% or almost 24% smaller than in the permanentonly case. This is a remarkable result and it is worth understanding its implications better. Empirical
evidence in Section III.A and elsewhere in the literature widely reports low leverage ratios of the
order of 25% for an average COMPUSTAT firm, with a number of firms exhibiting much smaller
leverage ratios. This is often regarded as the “low leverage puzzle” since standard trade-off models
have difficulty explaining such low reliability on debt financing. While for reasons of tractability
we can not obtain the solution for a dynamic financing case in our model (where firms refinance
infrequently as in models of Goldstein, Ju, and Leland (2001) and Strebulaev (2007)), it is a commonplace intuition, implied by real option economic considerations, that in the presence of future
refinancing options firms prefer to reduce their initial leverage (since they will be able to increase
it later conditional on experiencing good cash flows). For example, the introduction of dynamic
refinancing in Goldstein, Ju and Leland (2001) reduces optimal leverage from about 50% (for their
benchmark case; differs from ours due to different tax assumptions) to about 37%. It seems intuitive
then that the presence of transitory shocks coupled with dynamic refinancing is able to explain the
low leverage puzzle.
An important concern that one could raise about this result is that the presence of a real option
driven by the expectation of transitory shocks increases levered equity value and may mechanically
lead to a reduction in quasi-market leverage measure even if the reliance on debt has not actually
changed. We address this concern by comparing optimal coupon levels in both models. As the second
33
column shows, it is truly the reduction of coupon level which is responsible for low leverage and not
the mechanic relationship between market value of equity and leverage. Thus, book leverage ratios
should follow the same pattern.
The remaining rows of Table IV consider the comparative statics and robustness of the basic
result when we vary major parameters of influence. Importantly, we proceed two-fold to continue
comparing the case with transitory shocks to the case without transitory shocks. First, for each
case we require asset volatility to be at the same 25% level and the real optionality implied by
temporary shocks as a fraction of the unlevered asset value to be at the same 2% (apart from the
cases when we perform comparative statics with respect to these two variables). Second, for each
case we re-estimate optimal financing decisions for the permanent-only case for every variation in
parameters (not reported). The fourth and fifth columns show the difference in optimal leverage
(∆L) and coupon levels (∆c) between the adjusted permanent-only case and the benchmark model.
The bottom line of this table is that for a wide variation in parameters the presence of transitory
shocks leads to a substantial adjustment in financing behavior, from a minimum decrease of 20%
(when asset volatility is large) to a maximum reduction of 40% (when asset volatility is small), and
that almost all of its variation is caused by changes in book debt.
IV
The effect of financing constraints and multiple shocks
Two stylized facts that the benchmark model of the previous section does not consider for the reasons
of tractability are the multiplicity of temporary shocks and asymmetric costs of financing. The first
stylized fact is that shocks occur in reality multiple times, at different scale and with different waiting
periods. Over the sufficiently long horizon firms always expect to be hit by future shocks, both of
positive and negative nature. Section IV.A considers the model with multiple shocks and compares
its implications with those of the benchmark model.
The second stylized fact refers to the asymmetry that is usually observed in firms’ financing.
When firms pay interest out of current cash flow, there are unlikely to be any additional penalties
imposed. However, when cash flow is insufficient to pay interest and equityholders, in an attempt to
stave off default, raise additional financing by other means, their action is usually costly, whether by
paying higher interest rate on new financing, incurring fixed transaction costs on issuing new external
equity or debt, selling assets at fire-sale prices, or reducing financial slack.19 Section ?? extends the
benchmark model to explore to what extent financing constraints effect the conservative nature of
firm’s financial behavior.
19
Contingent claims models of capital structure usually assume this cost by allowing equityholders to restructure in
good times but not allowing to reduce debt in bad times (see, for example, Goldstein, Ju, and Leland (2001)). These
models, however, allow equityholders issue new equity costlessly in financial distress. Strebulaev (2007) models directly
the higher costs of financing in financial distress in a way that is similar to what we do here (he also assumes that
assets could be sold at fire-sale prices).
34
IV.A
A model with multiple shocks
We develop two models in this section. First, we extend a motivating example of Section I.B to
include infinite number of shocks. For this scenario, we are able to find the closed-form solution
which lends itself easier to an investigation of economic properties of equilibrium. We then extend
the benchmark model of both permanent and transitory shocks to include infinite number of shocks.
While that model does not yield a closed-form solution, we are also able to characterize equilibrium
properties both qualitatively and quantitatively.
IV.A.1
A model with multiple transitory shocks
In a motivating example of Section I.B, only one transitory shock could hit firm’s cash flow. Assume now that the economy is subject to infinite number of shocks in expectation, even though for
simplicity we continue to assume that at any point in time only one temporary shock may effect
earnings. Specifically, we assume that the state of the economy follows a 2-state continuous-time
Markov chain. In the first state the permanent cash flow is the only component of the cash flow,
while in the second state both shock types are present. The probability per unit of time of switching
from permanent-only state to both-shock state is λs and the probability of switching from both-shock
state to permanent-only state is λr . Thus, on average, a fraction λr /(λs + λr ) of time firm’s cash
flow is affected by transitory shocks.
As before, Ts denotes any date when the temporary shock hits and Tr – any date when the shock
mean reverts. At any Ts , the magnitude of the shock is drawn randomly from the same double
exponential distribution function (7). In other words, the expected magnitude of the new shock
is the same but its actual realization upon arrival is generally different. The firm expects infinite
number of shocks to arrive and get reversed in the future.20 .
To value any security, note that the economy at any time is in one of two states depending
on whether current earnings are affected by the transitory shock. For the before-tax value of the
unlevered firm, V , denote V (δ) to be the value in the no-shock state and V (δ, ǫ) to be the value when
shock of size ǫ is present. Note that V (δ, 0) and V (δ) represent values in two different states of the
world and generally are not equal to each other.21
In both states these values can be written intuitively, using recursive formulation, as
20
This is one of a number of potential “shock structures” that we could consider. For example, shocks can be also
reversed by the arrival of the new shock. Alternatively, it can be thought that shocks can be reversed only by new
shocks. While all these models would lead to similar implications, they do not in general allow disentangling two
quantities with vastly different intuition: waiting times of shocks and their mean-reversion speed. In our model λs is
responsible for the first effect and λr reflects the speed of mean-reversion. This distinction is helpful in understanding
economic intuition.
21
Since shock of size zero happens on a probability measure of zero, we do not consider informational aspects of the
shock existence, even though managers most probably would not know for certain whether the firm is in the shock or
no-shock state, when the realization of the shock is zero.
35
(
V (δ, ǫ) =
V (δ) = E
E
hR
Tr
0
0,
h Z Ts
0
i
(δ + ǫ)e−rt dt + V (δ)e−rTr , ǫ > ǫA ,
ǫ ≤ ǫA ,
i
δe−rt dt + V (δ, ǫ)e−rTs .
(45)
As before, ǫA is the abandonment level of the shock, below which firmholders find it in their
interest to liquidate the project to mitigate losses. In the formula for V (δ, ǫ) above, the first term
is the present value of cash flows until the shock reverts to the long-run mean and the second term
is the firm value at the moment of shock’s mean reversion discounted to take into account that
mean-reversion occurs in the future.
To simplify the exposition, we use the same canonic security A(δ, g(ǫT ), λ, b) that we introduced
in the motivating example. The only important distinction is that its value now depends on whether
transitory shock is present. For example, we can re-write the unlevered value in the absence of the
transitory shock as
V (δ) = E
hZ
Ts
0
i
δe−rt dt + e−rTs A(δ, V (δ, ǫ), λs , ǫA ) .
(46)
Since this value is larger than the value of perpetuity paying the permanent cash flow, once again
firmholders possess a real option in the presence of transitory shocks.
Appendix B derives the closed form expressions for the values of this and all other securities
introduced below. To find ǫA , we solve the following abandonment condition:
V (δ, ǫA ) = 0.
(47)
Turning now to the value of equity, and assuming for the moment that the coupon level is given,
the value of equity in two states can be written as follows:
E(δ, ǫ) =

hR
i
Tr
−rt dt + E(δ)e−rTr

(1
−
τ
)E
(δ
+
ǫ
−
c)e

e
0



 0,
E(δ) = (1 − τe )E
hZ
Ts
i
(δ − c)e−rt dt + E(δ, ǫ)e−rTs .
0
ǫ > ǫA ,
ǫ ≤ ǫA ,
(48)
In both expressions above, the first term is the equity payout in the current state. The equityholders will choose the default threshold level of ǫB by solving for the negative root of the following
36
value-matching condition:
E(δ, ǫB ) = 0.
(49)
The value of debt consists of the sum of two components, the present value of coupon payments
until default and the liquidating payment in default. The first component, which we denote in two
states by C(δ, ǫ) and C(δ), respectively, can be written as:
C(δ, ǫ) =
(
C(δ) = E
E
hR
Tr
0 (1
0,
h Z Ts
− τi )ce−rt dt + C(δ)e−rTr
i
ǫ > ǫB ,
ǫ ≤ ǫB ,
i
(1 − τi )ce−rt dt + C(δ, ǫ)e−rTs .
0
(50)
It is easy to see that the value of debt before bankruptcy C(δ, ǫ) does not depend on the realization
of the temporary shock, as long as the shock does not lead to default.
The expected continuation value of liquidating payments, which we denote L(δ, ǫ) and L(δ),
respectively, can analogously be written as:
L(δ, ǫ) =
L(δ) =
 h
i
 E L(δ)e−rTr
 (1 − τe )(1 − α)E
 h
i
 E L(δ, ǫ)e−rTs
 (1 − τe )(1 − α)E
ǫ > ǫB ,
hR
δe−rt dt + L(δ, ǫ)e−rTr ], ǫ ≤ ǫB ,
hR
δe−rt dt + L(δ)e−rTs ], ǫ ≤ ǫB .
Tr
0
Ts
0
ǫ > ǫB ,
(51)
To gauge the intuition behind the expression above, note that when the realized shock does not
lead to default, there are no liquidating payments until the next shock.
The optimal coupon choice follows from the optimization problem
max D(δ) + E(δ),
c
(52)
given expressions for ǫB (c) and ǫA (c) obtained above. In the above expression, we explicitly assume
that at the time of optimal coupon choice there are no transitory shocks, although the results are
without loss of generality.
To get the intuition on how the model works, Figure 7 shows the optimal default threshold, optimal coupon, optimal leverage ratio, and firm value for different values of the shock arrival intensity
λs . As shocks arrive more frequently (λs is larger), the firm starts valuing financial flexibility more
by reducing optimal coupon. Since the optimal leverage ratio is a convex function of shock’s arrival
intensity, leverage is small even for relatively infrequent shocks.
Figure 8 shows optimal decisions as a function of the dispersion of the shock. Smaller values of
37
Threshold
Coupon
0.8
−3
0.7
−4
0.6
C
πB
−2
−5
0.5
−6
0.4
−7
0
1
2
3
λ
4
5
6
0
1
2
s
3
λ
4
5
6
5
6
s
Leverage
Unlevered Firm Value
0.7
13.4
0.65
13.2
0.55
13
0.5
V
Leverage
0.6
12.8
0.45
0.4
12.6
0.35
0
1
2
3
λ
4
5
12.4
6
0
1
2
s
3
λ
4
s
Figure 7: Optimal default boundary, leverage ratio, coupon and firm value as a function of λs
for a case with multiple transitory shocks. The figure shows the optimal default boundary, optimal
leverage ratio, optimal coupon level, and optimal firm value as a function of the shock’s speed of arrival, λs .
γ lead to shocks of larger magnitude. When γ goes to infinity, the variance of jumps is negligible
- equityholders behave as if there were no jumps at all and prefer to choose optimal coupon equal
to δ to realize full benefits of debt taxation. On the other hand, when γ goes to zero, the variance
of jumps is so high that highly negative shocks arrive with probability close to 0.5; but the same
is true for highly positive shocks which increase the fortunes of shareholders. To reap full gains of
these positive shocks, it is optimal to set the coupon level at the highest possible value, sometimes
even at expense of being in financial distress initially, get full benefits of debt taxation and default
only for highly adverse realizations of the shock. In between the two extremes, the two effects move
in different directions and the optimal coupon level falls below δ.
IV.A.2
A model with permanent and multiple transitory shocks
We now incorporate multiple temporary shocks into a model with permanent Brownian shocks.
Following the discussion of Sections II and IV.A.1, the values of all securities can be found from a
system of recursive equations. For example, the unlevered firm value can be written as
38
Coupon
Threshold
2
1
0
0.8
−2
C
πB
−4
0.6
−6
−8
0.4
−10
−12
0
1
2
3
γ
4
5
0.2
6
0
1
Leverage
2
3
γ
4
5
6
5
6
Unlevered Firm Value
1
16
15.5
0.8
14.5
V
Leverage
15
0.6
14
0.4
13.5
0.2
13
0
0
1
2
3
γ
4
5
12.5
6
0
1
2
3
γ
4
Figure 8: Optimal default boundary, leverage ratio, coupon and firm value as a function of γ for
a case with multiple transitory shocks. The figure shows the optimal default boundary, optimal leverage
ratio, optimal coupon level, and optimal firm value as a function of the shock’s volatility, γ.
V (δt , ǫt ) =
(
V (δt ) = E
E
0,
Z
hR
min(Tr ,τ (δA (ǫt )))
(δs
t
Ts
i
+ ǫs )e−r(s−t) ds + e−rTr ITr <τ (δA (ǫt )) V (δTr ) , δt > δA (ǫt ),
δt ≤ δA (ǫt ),
δs e−r(s−t) ds + e−rTs V (δTs , ǫTs ),
t
(53)
where δA is found from smooth-pasting condition that the unlevered firm is abandoned given low
enough realization of temporary shock:
∂V (δt , ǫt ) = 0.
∂δt
δt =δA (ǫt )
(54)
Analogous expressions can be written for equity and debt values. Equityholders default optimally
according to the following smooth-pasting conditions
∂E(δt , ǫt ) = 0,
∂δt
δt =δB (ǫt )
∂E(δt ) = 0.
∂δt δt =δB
39
(55)
While the above equations do not have a closed form solution for corporate securities, many
intuitive properties of solutions can be established. Figure 9 compares optimal abandonment and
default boundaries for this and the benchmark models, and Appendix Appendix C provides details
of the numerical procedure used to obtain the values of optimal financial decisions. Even though it
is clear that both models are driven by the same forces, Figure 9 shows several interesting differences
between the two models. First, in the model with single temporary shock Proposition 2 establishes
that it is optimal to abandon the firm for any negative realization of temporary shock if the permanent
component is sufficiently low. This is not the case in the model with multiple shocks since even if
δ = 0 firm holders hold a call option on future ex-ante symmetric realizations of shocks. For the
same reason, equityholders now do not default even for realizations of shock lower than coupon level
(if δ = 0). In other words, multiple shocks lead to further dissipation of asset value prior to default
and further corroborate the need for net worth covenants.
3.5
0
−10
0
10
Figure 9: Optimal default boundaries. The figure shows the permanent shock-only optimal default
boundary (solid line), the optimal default boundary before and during the shock as well as abandonment
boundary during the shock in the model with single temporary shock (dashed line) and in the model with
multiple temporary shocks (dashed and dotted line).
Second, the abandonment threshold is now intuitively concave since the abandonment decision
accounts for future abandonment thresholds (similar to default boundary in the single shock case).
Finally, the slope of abandonment boundary when ǫ → −∞ is smaller in absolute value than that in
the model with single shock. This is so because in addition to permanent cash flows, future stream
of cash flows also includes future realizations of temporary shocks subject to limited liability on the
part of firm owner and therefore weighs heavier relative to present temporary stream in owner’s
40
decision to abandon.
Finally, the increased optionality on the part of the equityholders implies that the default boundary is now uniformly below the default boundary in the single shock case and its slope for ǫ → −∞
is smaller in absolute value than that in the model with single shock exactly for the same reasons as
the abandonment threshold.
The calibration of the model with multiple jumps proceeds in the same fashion as in Section III.B.
Importantly, for the same realized volatility of assets σV and the same temporary shock premium φ
as in the model with single jump, the calibrated multiple jump model parameters σδ and γ should fall
to compensate for multiple jumps and their optionality in presence of firm owners’ limited liability.
V
Concluding Remarks
Firms’ earnings are affected by shocks of various duration. While some shocks are long-term and
can be thought of as permanent, most other shocks, however strong they are in the short run, are
of limited duration. In this paper we implement this simple intuition to investigate the impact of
temporary shocks on corporate financial policies.
Our contributions at a theory level can be summarized as follows. By incorporating temporary
deviations from permanent cash flows our model can match the empirical evidence along three
dimensions. First, firms in the model may experience negative cash flows. Empirical evidence
suggests that for a broad set of COMPUSTAT firms up to a third of firm-year observations exhibit
negative earnings. Second, the correlation between changes in current cash flows and asset values
is less than one. Third, earnings volatility in the model plays a separate role and larger than asset
volatility. These model implications are important since extant contingent claims models of financing
decisions utilize only permanent shock structure and in these models cash flow is proportional to
asset values. As a result, cash flows are always positive, changes in cash flows and asset values are
perfectly correlated, and earnings volatility is equal to asset volatility, inconsistent with stylized facts
on earnings.
We show that firms in our model are much more likely to be financially conservative than in extant
models. For the benchmark calibration scenario, the optimal leverage ratio drops by about 25%. The
main results is that transitory shocks provide a natural explanation of conservative financial behavior.
The convex relationship between the arrival intensity of transitory shocks and optimal coupon levels
suggests that this result is robust.
The results of this paper call for rethinking of both theoretical and empirical research on capital
structure. Too much attention has been devoted to long-term changes in firm’s fortunes at the expense of investigating the role of short-term shocks. Modeling optimal cash and short-term financing
policies is the next natural step in understanding financial behavior of corporations.
41
Appendix A
I.A
Proofs: Benchmark model
Derivation of values for X(δt , λ, bt ) and A(δt , g(δT ), λ, bt )
−
We define ϑ+
w and ϑw - correspondingly positive and negative root of the quadratic equation
−
ϑ+
w −ϑw
.
2
σ2 2
2 ϑw −
µ−
σ2
2
ϑw −
(r + w) = 0. We also define dw =
First, using properties of first-passage times for geometric Brownian motion, we price the canonic security
X(δt , λ, bt ):
" + #−
−ϑλ
h
i
δt
−r(τ −t)
X(δt , λ, bt ) = Et e
Iτ <T = 1,
(A1)
bt
−ϑ+
λ
We have to take the minimum of 1 and bδtt
because bt changes with jumps over time and can become
lower than δt , even though δ is continuous process.
Next, since Poisson process of shock arrival and reversal exhibits memoryless property, for every claim
g(δT ) that pays off at time T of shock arrival (Ts ) or reversal (Tr ) conditional on not crossing the lower
boundary bt before T ,
h
i
Et e−r(T −t) g(δT )Iτ ≥T
h
i
h
i
= Et e−r(T −t) g(δT ) − Et e−r(T −t) g(δT )Iτ <T
h
i h
i
h
i
= Et e−r(T −t) g(δT ) − Et e−r(τ −t)Iτ <T Et e−r(T −t) g(δT ) | δt = bt (A2)
where bt is the boundary that is crossed at time τ . It is easy to see that the above equation is exactly (25)
for canonic securities A(δt , g(δT ), λ, bt ), in which the argument λ stands for the frequency of shock arrival at
time T .
We are interested in pricing of A(δt , g(δT ), λ, bt ) for any function g(δT ) that is encountered in our problem.
As we will see later, these take the form of piecewise quasipolynomials of the form
g(x) =
N
X
Izi−1 ≤x<zi
i=1
X
αi,j xj ,
j∈Ji
where quasipolynomial pieces j∈Ji αi,j xj are smoothly connected at points {zi }N
i=0 such that z0 = 0,
zN = ∞, and Ji is a finite set of real numbers for each i. The expected value of security that promises to pay
out g(δT ) of such form at the time T of shock arrival or reversal and is not contingent on τ is
P
N X
X
αi,j azzii−1 (δt , δTj , λ, 0).
(A3)
h
i
azzii−1 (δt , δTj , λ, 0) = Et e−r(T −t) δTj Izi−1 ≤δT <zi
(A4)
A(δt , g(δT ), λ, 0) =
i=1 j∈Ji
where
and A(δt , g(δT ), λ, bt ) is defined as above.
Using properties of geometric Brownian motion, we obtain
42
h
i
azzii−1 (δt , 1, λ, 0) = Et e−r(T −t) Izi−1 <δT ≤zi =







−
λ
r+λ






−
λ
σ 2 dλ ϑ−
λ
λ
σ 2 dλ ϑ+
λ
λ λ
h
i
azzii−1 (δt , δTk , λ, 0) = E e−r(T −t) δTk Izi−1 <δT ≤zi =
λ
zi−1
− σ 2 dλ ϑ+
−ϑ−
δt
δt
zi−1
δt
zi−1
λ
σ 2 dλ ϑ−
λ
+
−ϑ+
λ
+
−ϑ+
λ
λ
σ 2 dλ ϑ−
λ
+
δt
zi
λ
σ 2 dλ ϑ+
λ
δt
zi
−ϑ−
λ
, δt < zi−1 ,
−ϑ−
λ
, δt ∈ [zi−1 , zi ),(A5)
−ϑ+
λ
δt
, δt ≥ zi
zi

−
−
−
−
ϑ
ϑ +k
ϑ
ϑ +k
λ

λδt λ zi−1
λδt λ zi λ


−
+
, δt < zi−1 ,

−
−

σ 2 dλ (ϑλ +k) σ 2 dλ (ϑλ +k) 


k

λδt

1

− −1

σ 2 dλ
ϑ+ +k
ϑ +k
+
λ
−
+
λ
−
ϑ
ϑ +k
ϑ
ϑ +k
λ

λδt λ zi−1
λδ λ z λ


−
+ σ 2td (ϑi− +k) , δt ∈ [zi−1 , zi ),

2 d (ϑ+ +k)

σ

λ
λ
λ
λ

+
+
+
+

ϑ
ϑ +k
ϑ
ϑ +k

λ

λδt λ zi−1
λδt λ zi λ


+
,δ ≥z
−
σ 2 dλ (ϑ+
+k)
λ
t
σ 2 dλ (ϑ+
+k)
λ
(A6)
i
Note that, being a particular case of A(δt , g(δT ), λ, bt ), azzii−1 (δt , δTj , λ, bt ) is also defined by a version of
(25) which takes into account that at time T the low boundary zi−1 cannot be below the exit threshold bt :
i
i
azzii−1 (δt , δTj , λ, bt ) = azmax(b
(δt , δTj , λ, 0) − X(δt , λ, bt )azmax(b
(bt , g(δT ), λ, 0).
t ,zi−1 )
t ,zi−1 )
(A7)
As an important particular case of (A5)-(A6) which we will use extensively,
azzN
(δt , 1, λ, 0) = a∞
zN −1 (δt , 1, λ, 0) =
N −1



λ
r+λ
k
azzN
(δ, δTk , λ, 0) = a∞
zN −1 (δ, δT , λ, 0) =
N −1
I.B
Proofs of Propositions 1–6
λδtk
σ 2 dλ



λ
σ2 d
+
λ ϑλ
− σ2 dλ ϑ−






−
λ
1
ϑ+
λ +k
λ
−
−
−ϑ+
λ
δt
zN −1
−ϑ−
λ
δt
zN −1
1
ϑ−
λ +k
, δt ≥ zN −1 ,
(A8)
, δt < zN −1
+
+
ϑ +k
λz λ
N −1
σ2 dλ (ϑ+
λ +k)
−ϑ
−
λδt
−
−
−ϑ
ϑ +k
λδt λ zNλ−1
σ2 dλ (ϑ−
λ +k)
, δt ≥ zN −1 ,
(A9)
, δt < zN −1
Proposition 1. Proof. See proof of Proposition 3 since unlevered firm value is a particular case of equity
value with zero debt.
Proposition 2. Proof. If ǫTs ≥ 0, it is never optimal to abandon the firm, since cash flows can never be
negative and their present value is always higher than zero.
The equation is obtained by explicitly writing down the derivative for V (δt , ǫTs ) for ǫTs < 0 and applying
∂V (δt ,ǫTs )
the smooth-pasting condition
|δt =δA,t (ǫTs ) = 0. ǫTs enters the condition linearly and the closed-form
∂δt
solution is possible.
Proposition 3. Proof. The value of equity for t ≥ Tr is the standard result of any Leland-type model.
43
The value of equity in time interval [Ts , Tr ) is
E(δt , ǫt )
=E
1 − τe
"Z
min(τ,Tr )
(δs + ǫt − c)e
−r(s−t)
dt + e
−r(Tr −t)
#
Iτ ≥T E(δTr )
t
(A10)
The first part of the expectation can be thought of as a perpetuity that pays δs + ǫt − c for every s ≥ t
starting at δt that is stopped either by the endogenous default time τ or by jump reversal Tr . If τ < Tr , then
the value of the above perpetuity is decreased by the value of similar perpetuity starting at δB,t multiplied
by the price of a security that
moment τ when δt reaches or crosses δB,t for the first time
pays $1 at the
δB,t
ǫt
c
and if τ < T . This is exactly r−µ + r − r X(δt , λr , δB,t ). If τ < Tr , the value is further decreased by the
perpetuity starting at random δTr such that neither of paths of δ starting
at δt and ending at δT ever crosses
δTr
low barrier δB,t . The amount of decrease is thus A(δt , r−µ
+ ǫrt − rc , λ, δB,t ).
The second part of the expectation is A(δt , E(δTr ), λ, δB,t ) by definition of canonic security A(·). Recombining the above terms, we get (35). In the similar fashion, (36) is true, the only difference is that we have to
integrate over potential realizations of ǫTr in the second part of the expectation in (33). Note that (34)-(35)
are obtained in closed form, and also for every future realization of shock size ǫTs (36) is obtained in closed
form (the only step left is numerically integrate over the distribution of shock size).
The exact (up to the set of securities azzii−1 the closed form expressions for which are given above) values
of equity in each time period are:

















1
r−µ
δt − δB,t
δt
δB,t
−ϑ+
0
−ϑ+
λ
−
c
r
−ϑ+
0
δt
1 − δB,t
,
for t ≥ Tr ,
r
δt
1
∞
δ
−
δ
−
a
(δ
,
δ
,
λ
,
δ
)
+
t
B,t
t
T
r
B,t
r
δB,Tr
r−µ
δB,t
+
−ϑλ
r
δt
ǫ
c
1 − δB,t
− a∞
δB,Tr (δt , 1, λr , δB,t ) +
r − r
E(δt , ǫt )
=

1 − τe


1
∞



r−µ aδB,Tr (δt , δTr , λr , δB,t )−

+
+


−ϑ+
(δB,Tr )1+ϑ0
c(δB,Tr )ϑ0

0

−
a∞

δB,Tr (δt , δTr , λr , δB,t )−
r−µ
r



 c ∞
r aδB,Tr (δt , 1, λr , δB,t ),
44
for Ts ≤ t < Tr ,
For any t < Ts and given ǫTs the value of equity is

−ϑ+
λs

δt
1
∞

− aδB,Ts (δt , δTs , λs , δB,t ) −

r−µ δt − δB,t δB,t



+


−ϑ
λs

δt
c
∞


− aδB,Ts (δt , 1, λs , δB,t ) +

r 1 − δB,t




∞

+βE,1 a∞

δB,Tr (δt , 1, λs , δB,t ) + βE,2 aδB,Tr (δt , δTs , λs , δB,t )+


+

−ϑ+
−ϑ0

λr
∞

βE,3 a∞
, λs , δB,t )+

δB,Tr (δt , δTs , λs , δB,t ) + βE,4 aδB,Tr (δt , δTs



 γE,1 aδB,Tr (δt , 1, λs , δB,t ) + γE,2 aδB,Tr (δt , δTs , λs , δB,t )+
δB,Ts
δB,Ts
E(δt , ǫt )
=
−ϑ−
−ϑ+
δ
δB,Tr
λr
λr
r

1 − τe
γE,3 aδB,T
(δ
,
δ
,
λ
,
δ
)
+
γ
a
(δ
,
δ
, λs , δB,t ),

t
s
B,t
E,4
t
T
T
δ

s
s
B,Ts

B,Ts
+

−ϑλs
 1

δt

− a∞

δB,Ts (δt , δTs , λs , δB,t ) −
 r−µ δt − δB,t δB,t



+

−ϑλs

δ
c

∞
t

− aδB,Ts (δt , 1, λs , δB,t ) +

r 1 − δB,t




∞

+ζE,1 a∞

δB,Tr (δt , 1, λs , δB,t ) + ζE,2 aδB,Tr (δt , δTs , λs , δB,t )+


+

−ϑ+

−ϑ0
λr
∞
ζE,3 a∞
, λs , δB,t ),
δB,Tr (δt , δTs , λs , δB,t ) + ζE,4 aδB,Tr (δt , δTs
for t < Ts and ǫTs ≥ 0
for t < Ts and ǫTs < 0
−ϑ+
0
where coefficients βE,j , γE,j and ζE,j j = 1, 2, 3, 4, are such that E(δt , ǫt ) = βE,1 + βE2 δt + βE,3 δt
−ϑ+
βE,4 δt λr
for Ts ≤ t < Tr , ǫt ≥ 0 and δt ≥ δB,Tr , E(δt , ǫt ) =
Ts ≤ t < Tr , ǫt ≥ 0 and δt < δB,Tr , E(δt , ǫt ) = ζE,1 + ζE,2 δt +
ǫt < 0. These coefficients are given by
=
βE,1
=
βE,3
−
c − ǫTs
r + λr
−
c
λr
r r + λr

+

; βE,2 = 
1
r + λr − µ
=

+
+1
λr
δB,T
s

 
 
−
− 
  r + λr − µ

ϑ
+
ϑ
+1
+

0
0
cδB,T
 δB,Tr

λr
r 
+ 
 r−µ −
 σ2 d
r
λr
γE,1
γE,3
γE,4
=
=
=
−

c − ǫTs
r + λr

ϑ
+
; γE,2 =
1
r + λr − µ
ϑ
+1
r − µ σ 2 dλ r

ϑ+
+1
λ
r
−
1
ϑ−
+1
λ
r

ϑ
+
+1
ϑ
−
+1 ϑ
+
−ϑ
λr
λr
 δ λr
δB,T
δB,T
 B,Tr
r
s
−
 −

−

r − µ σ 2 dλr  ϑ+ + 1
ϑ
+
1
λr
λr
ϑ
+
−ϑ
1
λr
+
ϑ
−
−ϑ
+
ϑ
+
−ϑ
0
0
λr
λr
 δ λr
δB,T
δB,T
 B,Tr
r
s
−
 +
+
−
+
ϑ
−
ϑ
ϑ
−
ϑ
0
0
λ
λ
r
+
r

−
λr 
c λr

 +

r σ 2 dλ r

−
+
−
+
+
ϑ
+1 ϑ
−ϑ
ϑ +1
ϑ
λr
λr
λr
0
0

λr δB,T
δB,T
cδB,T
 δB,T
1

r
s
r
r

−
−
 +
 r−µ
−
2

r−µ
r
σ dλr (ϑλ + 1)


−
+
+
ϑ
+1
ϑ +1
ϑ
λr
0
0

λr δB,T
cδB,T
 δB,Tr
1

r
r

+
−
−
 r−µ
 r − µ σ 2 d (ϑ− + 1)
r
λr
λ
r
1



−
λr 




ϑ
+
ϑ
−
ϑ
+
−ϑ
λr
λ
 δ λr
δB,T
δ r
 B,Tr
r B,Ts
−

−
 ϑ+
ϑ
λ
λ
r
r

−
λr  



λr
  δ λr
(c − ǫTs )δB,T
B,Ts
 
s
−
− 
  r + λr − µ
r + λr


r + λr

r
+
λr 
(c − ǫTs )δB,T
s 
ϑ
ϑ

λr
+


ϑ +1
ϑ
0
0
cδB,T
 δB,T

λr
1
1
r
r 


−
−
−
 r−µ
 σ2d
r
ϑ+
− ϑ+
ϑ−
− ϑ+
λr
0
0
λ
λ

r
βE,4
1
+
+
−ϑ+
−ϑ−
γE,1 + γE,2 δt + γE,3 δt λr + γE,4 δt λr for
−ϑ+
−ϑ+
ζE,3 δt 0 + ζE,4 δt λr for Ts ≤ t < Tr and
r




−
+
−
−ϑ0
ϑ
λ
λr
c λr δB,Tr 

r
+

−
+
2
2
r σ dλ ϑ− 
σ dλr (ϑλ − ϑ0 )
r
λ
r
r
ϑ
r
λr δB,T
45


ϑ
−
−ϑ
+
ϑ
+
−ϑ
−
ϑ
−
ϑ
+
−ϑ
0
λr
λr
λr
λr
λr
δB,T
 λr δB,Tr
c λr δB,Tr δB,Ts
s

−

−
+
2
2
r
σ dλr (ϑλ − ϑ0 )
σ dλr ϑ−
λ
r
r

−
λr 



ζE,1
ζE,3
=
=
−
c − ǫTs
r + λr
−
c
λr
r r + λr

+

; ζE,2 = 
1
r + λr − µ
=
1
λr
r − µ σ 2 dλ r

+


ϑ
ϑ +1
0
0
cδB,T

 δB,Tr
λr
1
1
r 


−
−
−
 σ2 d
 r−µ
r
ϑ+
− ϑ+
ϑ−
− ϑ+
λr
0
0
λ
λ
r
ζE,4
+

+
+1
λr
δB,T
s

 
 
−
− 
  r + λr − µ

ϑ
+
ϑ
+1

ϑ
+

ϑ
ϑ
1
 −

r−µ
r + λr
+
+ 
−ϑ
0
λ
0
r
0
cδB,T
 λr δB,Ts
 δB,Tr
r 
+ 
 r−µ −

r
σ 2 dλ r

1
ϑ+
+1
λ
1
−
ϑ−
+1
λ
r
r


r
+
λr 
(c − ǫTs )δB,T
s 
ϑ


+
+1 
λ
s

r
λr δB,T
σ 2 dλ r
1
ϑ+
− ϑ+
0
λr
−
1
ϑ+
+1
λ
−
r
1
ϑ−
− ϑ+
0
λr

1
ϑ−
+1
λ
r
ϑ
 +
+
λ
r
c λr δB,Ts
r
σ 2 dλ r



1

ϑ+
λr
−
1
ϑ−
λr






Proposition 4. Proof.
1. When t ≥ Tr , we obtain the same result as in Leland-type model. If t ∈ [Ts , Tr ) and ǫTs = 0, then
λTr -driven Poisson process does not in fact kill any temporary cash flows; therefore, the equity value
can as well be rewritten as E(δt , 0) = E(δTr ). The solution of the smooth-pasting condition is exactly
equivalent to that of E(δB,Tr ) and gives the same result (37).
2. If ǫTs < 0, temporary cash flows act as a temporary increase in fixed coupon payments. Intuitively, a
temporary increase in coupon payments puts larger burden on shareholders and depletes their value; they
optimally default at higher levels of permanent cash flows. This means that δB,t > δB,Tr for t ∈ [Ts , Tr ).
Differentiating the value of equity for t ∈ [Ts , Tr ) obtained in Proposition 3 with respect to δt at point
ϑ+ (c−ǫt )
−ϑ+
r
δt = δB,t (ǫt ) yields the equation for δB,t : ωB,1 + ωB,2 δB,t + ωB,3 δB,t0 = 0 where ωB,1 = − λr+λ
−
r
!
+
+
ϑ
1+ϑ
+
0
1+ϑ+
ϑ+ +1
ϑ+ −ϑ+
δB,Tr0
cδB,T
λr
λr
c λr ϑλr
1
r
1 − ϑλ−r +1 r−µ
, and ωB,3 = − σ2λdrλ 1 − ϑλ−r −ϑ0+
.
r r+λr , ωB,2 = r+λr −µ + σ2 dλ
r−µ −
r
r
r
λr
λr
0
3. If ǫTs > 0, we essentially have a temporary reduction in cash flows. If ǫTs ≥ c, then equityholders get
positive cash flows unconditional on the realization of permanent shock δt until the time Tr of shock
reversal - therefore it is never optimal to go bankrupt before Tr .
4. If ǫTs ∈ (0, c), temporary cash flows act as a temporary decrease in fixed coupon payments. Intuitively, lower coupons make shareholders delay their default decision so that δB,t < δB,Tr for t ∈
[Ts , Tr ). Differentiating the value of equity for t ∈ [Ts , Tr ) obtained in Proposition 3 with respect
−ϑ−
to δt at point δt = δB,t (ǫt ) yields the equation for δB,t : χB,1 + χB,2 δB,t + χB,3 δB,tλr = 0 where
χB,1 =
ϑ+r (c−ǫt )
− λr+λ
,
r
χB,2 =
−
+
−ϑ
0
λr
+
−
λr (ϑλr −ϑλr )δB,Tr
−
+
σ 2 d λr
ϑλr −ϑ0
ϑ
1+ϑ+
λr
r+λr −µ ,
+
1+ϑ
δB,Tr0
r−µ
−
ϑ
and χB,3 =
!
+
ϑ
0
cδB,T
r
r
−
+1
δ λrr
− B,T
r−µ
+
−
λr ϑλr −ϑλr
σ2 dλr ϑ− +1
λr
ϑ
+
−
ϑ
−
λr
λr
+
−
cδB,T
λr (ϑλr −ϑλr )δB,Tr
r
−
r
σ 2 d λr
ϑ
+
λr
.
Proposition 5. Proof.
1. In comparison to the case t ≥ Tr , equityholders at time t < Ts expect an ex-ante symmetric temporary
shock subject to limited liability. Thus, a temporary shock acts as the additional source of future
uncertainty and increases the value of equity (a call option on firm’s cash flows) for any realization of
permanent cash flow component δt . Hence, optimally at time t < Ts for equityholders δB,t < δB,Tr .
46
2. This directly follows from the implicit function theorem applied to (38) and (39) for intervals ǫTs < 0
ϑ+
0
and ǫTs > 0. At point ǫTs = 0, the only solution δB,t = c r−µ
satisfies both (38) and (39). Hence,
r ϑ+
0 +1
δB,t is a continuous function of ǫTs . In addition, implicit function theorem establishes continuous
differentiability of δB,t at any ǫTs .
ϑ
3. First, ωB,2 > 0, ωB,3 > 0, χB,2 > 0 and χB,3 < 0. Since
ϑ
dδB,t
dǫ
=
dδB,t
dǫ
=
+
λr
− r+λ
r
−ϑ
+
0
ωB,2 −ϑ+
0 ωB,3 δB,t
−1
if ǫTs < 0 and
+
λr
− r+λ
r
+
−ϑ
−1
λr
χB,2 −ϑ−
λr χB,3 δB,t
if ǫTs > 0, δB,t is a decreasing function of ǫ. Second, for any ǫ,
dδB,t
dǫ
is
+
ϑ
λr
ǫ
r+λr t
ωB,2
dδA,t
dǫ
below the constant slope of abandonment boundary
= −
, so default boundary decreases
faster. Finally, since δB,t > δAt = 0 for ǫTs = 0, δB,t > δA,t for every ǫTs .
4. As ǫTs goes to −∞, the limit of
dδB,t
dǫ
is exactly the slope of the abandonment boundary.
5. That δB,t is decreasing in ǫTs is proven in 3. Differentiating continuously differentiable
to δB,t at any ǫTs yields concavity.
dδB,t
dǫ
with respect
Proposition 6. Proof. The proof follows the lines of Proposition 3 for the value of equity.
Appendix B
Proofs: Model with multiple transitory shocks
Proposition 7 The unlevered before-tax value of the firm in periods where transitory shock is not present,
V (δ), and in periods where transitory shocks are present, V (δ, ǫ), is:
δ+
V (δ) =
r+λs
r+λr
(δA(1, λs , ǫA ) + A(ǫTs , λs , ǫA ))
h
i
,
λr
(r + λs ) 1 − r+λ
A(1, λs , ǫA )
r
λr
δ+ǫ
+
V (δ).
r + λr
r + λr
V (δ, ǫ | ǫ ≥ ǫA ) =
(B1)
Proof. From
V (δ, ǫ | ǫ ≥ ǫA ) = E
V (δ) = E
"Z
"Z
Tr
(δ + ǫ)e
−rt
dt + V (δ)e
−rTr
0
Ts
δe−rt dt + V (δ, ǫ)e−rTs
0
#
#
we obtain the recursive formulation
δ+ǫ
λr
+
V (δ)
r + λr
r + λr
Z ∞
δ
λs
V (δ) =
+
V (δ, ǫ)dF (ǫ)
r + λs
r + λs ǫA
V (δ, ǫ | ǫ ≥ ǫA ) =
(B2)
where ǫA is negative temporary shock realization at which the present value of unlevered firm’s cash flows
47
becomes zero. To exclude the integral term from the second equation above, we treat ǫ as a random variable
in the first equation and take expectations of both parts of the equation with respect to ǫ.
Z
∞
ǫA
r + λs h
A(1, λs , ǫA )
V (δ, ǫ)dF (ǫ) =
λs
δ
r
+ V (δ)
r + λr
r + λr
+
A(ǫTs , λs , ǫA ) i
r + λr
(B3)
Now, inserting this into the second equation of (B2), we obtain
V (δ) =
δ+
r+λs
r+λr (δA(1, λs , ǫA )
(r + λs ) 1 −
+ A(ǫTs , λs , ǫA ))
λr
r+λr A(1, λs , ǫA )
(B4)
The second formula of the statement follows directly from (B2) and (B4).
Proposition 8 The value of equity in periods when transitory shock is absent is given by
E(δ) = (1 − τe )
r+λs
r+λr
(δ − c) +
(A(1, λs , ǫB )(δ − c) + A(ǫTs , λs , ǫB ))
.
λr
(r + λs ) 1 − r+λ
A(1,
λ
,
ǫ
)
s
B
r
(B5)
And the value of the equity in the presence of shock can be then written as:
E(δ, ǫ | ǫ ≥ ǫB ) =
(1 − τe )
δ+ǫ−c
λr
+
E(δ)
r + λr
r + λr
Proof. From
E(δ, ǫ | ǫ ≥ ǫB ) =
hZ
E
Tr
0
E(δ)
=
hZ
E
0
Ts
i
(1 − τe )(δ + ǫ − c)e−rt dt + E(δ)e−rTr ,
i
(1 − τe )(δ − c)e−rt dt + E(δ, ǫ)e−rTs .
(B6)
we obtain the recursive formulation
δ+ǫ−c
λr
+
E(δ)
r + λr
r + λr
Z ∞
δ−c
λs
E(δ) = (1 − τe )
+
E(δ, ǫ)dF (ǫ)
r + λs
r + λs ǫB
E(δ, ǫ | ǫ ≥ ǫB ) = (1 − τe )
(B7)
Using the same technique as in Proposition 7, we treat ǫ as a random variable in the first equation and take
expectations of both parts of the equation with respect to ǫ.
Z
∞
r + λs h
δ−c
λr
(1 − τe )
A(1, λs , ǫB ) +
E(δ)A(1, λs , ǫB ) +
λs
r + λr
r + λr
i
A(ǫTs , λs , ǫB )
E(δ, ǫ)dF (ǫ) =
ǫB
+(1 − τe )
1
r + λr
48
(B8)
Inserting this into the second equation of (B7), we obtain
E(δ) = (1 − τe )
(δ − c) +
r+λs
r+λr
(A(1, λs , ǫB )(δ − c) + A(ǫTs , λs , ǫB ))
λr
(r + λs ) 1 − r+λ
A(1,
λ
,
ǫ
)
s
B
r
(B9)
The value of the equity in presence of the temporary shock is then the combination of (B7) and (B9).
Proposition 9 The value of debt in two states (with and without temporary shock), D(δ, ǫ) and D(δ), respectively, is given by
D(δ) = (1 − τi )
r+λs
c + A(1, λs , ǫB )c r+λ
r
(B10)
r+λs
r + λs − A(1, λs , ǫB ) r+λ
r
λs
r+λs
δ
1
A(1,
λ
,
ǫ
)
−
A(1,
λ
,
ǫ
)(
+
V
(δ))
+
(A(ǫ
,
λ
,
ǫ
)
−
A(ǫ
,
λ
,
ǫ
))
s
A
s
A
T
s
A
T
s
B
s
s
r+λr
r+λr
r+λr
r+λr
+(1 − τe )(1 − α)
,
r+λs
r + λs − A(1, λs , ǫB ) r+λ
r
D(δ, ǫ | ǫ ≥ ǫB ) = (1 − τi )
c
λr
+
D(δ).
r + λr
r + λr
(B11)
Proof. To calculate the value of debt, we will first separately find expressions for continuation values of debt
before and after bankruptcy.
Continuation values of debt before bankruptcy
C(δ, ǫ | ǫ ≥ ǫB ) =
hZ
E
Tr
0
C(δ)
=
hZ
E
0
Ts
i
(1 − τi )ce−rt dt + C(δ)e−rTr ,
i
(1 − τi )ce−rt dt + C(δ, ǫ)e−rTs .
(B12)
give us the system of recursive equations
c
λr
+
C(δ)
r + λr
r + λr
Z ∞
c
λs
C(δ) = (1 − τi )
+
C(δ, ǫ)dF (ǫ)
r + λs
r + λs ǫB
C(δ, ǫ | ǫ ≥ ǫB ) = (1 − τi )
(B13)
From the first equation,
Z
∞
ǫB
C(δ, ǫ)dF (ǫ) =
r + λs
c
λr
A(1, λs , ǫB ) (1 − τi )
+
C(δ)
λs
r + λr
r + λr
(B14)
and, consequently,
C(δ) = (1 − τi )
r+λs
c + r+λ
cA(1, λs , ǫB )
r
λr
(r + λs ) 1 − r+λ
A(1,
λ
,
ǫ
)
s
B
r
49
(B15)
Continuation values of debt after bankruptcy
L(δ, ǫ | ǫ ≥ ǫB ) =
L(δ) =
h
i
E L(δ)e−rTr ,
h
i
E L(δ, ǫ)e−rTs + (1 − τe )(1 − α)V (δ, ǫ)e−rTs .
(B16)
gives the system of equations
λr
L(δ)
r + λr
Z ǫB
Z ∞
λs
λs
L(δ) = (1 − τe )(1 − α)
V (δ, ǫ)dF (ǫ) +
L(δ, ǫ)dF (ǫ)
r + λs ǫA
r + λs ǫB
L(δ, ǫ | ǫ ≥ ǫB ) =
(B17)
The transformation of equations gives
Rǫ
(1 − τe )(1 − α)λs ǫAB V (δ, ǫ)dF (ǫ)
L(δ) =
λr
(r + λs ) 1 − r+λ
A(1,
λ
,
ǫ
)
s
B
r
(B18)
and the integral above can be calculated explicitly in terms of securities A(·), from expressions (B2) and (B4).
Finally,
Rǫ
r+λs
(1 − τe )(1 − α)λs ǫAB V (δ, ǫ)dF (ǫ)
cA(1, λs , ǫB )
c + r+λ
r
+
D(δ) = (1 − τi )
λr
λr
A(1,
λ
,
ǫ
)
A(1,
λ
,
ǫ
)
(r + λs ) 1 − r+λ
(r
+
λ
)
1
−
s
B
s
B
s
r+λr
r
(B19)
and summing first equations of (B13) and (B17),
D(δ, ǫ) = (1 − τi )
Appendix C
c
λr
+
D(δ).
r + λr
r + λr
(B20)
Details of numerical procedures
The model in Section IV.A.2 with permanent and multiple transitory shocks allows for finding a solution to
unlevered firm value, securities value (equity and debt), and optimal default and abandonment boundaries
in terms of non-linear equations but does not allow for a closed form analytical solution. In particular, the
default boundary, δB (ǫ), can be written out, using the the expression of equity value obtained by solving the
Hamilton-Jacobi-Bellman (HJB) equation (54)–(55) and applying standard smooth-pasting condition, in the
following functional form:
∞ X
−ϑ+
r
ã+
i δB
−ϑ+
r
logi δB
−ϑ−
r
+ ã−
i δB
−ϑ−
r
logi δB
−ϑ+
s
+ b̃+
i δB
i=0
if the transitory shock is present and
50
−ϑ+
s
logi δB
−ϑ−
s
+ b̃−
i δB
−ϑ−
s
logi δB
+ Ã + B̃ǫt = 0
(C1)
∞ X
+
−
−
+
+
−
−
−ϑ+
i −ϑr
i −ϑr
i −ϑs
i −ϑs
− −ϑr
+ −ϑs
− −ϑs
r
a+
δ
log
δ
+
a
δ
log
δ
+
b
δ
log
δ
+
b
δ
log
δ
+A=0
i B
i B
i B
i B
B
B
B
B
(C2)
i=0
if the transitory shock is absent. A similar functional form can be written for the abandonment boundary
by solving the HJB equation (53)–(54).
Since solving for all the unknown coefficients {ai , ãi , bi , b̃i }∞
i=0 in the above equations is not possible in
closed form, we use the following numerical procedure to obtain a close approximation of both abandonment
and default boundaries and then make sure that this procedure converges to the true boundaries.
To find the default boundary (the abandonment boundary is found in a similar way and commented on
below), we consider the following functional form:
f− (ǫ, δB ) =
+
f (ǫ, δB ) =
f(δB ) =
−ϑ+
r
ǫ + Ã+ + ã+
0 δB
−
ǫ + Ã +
−ϑ−
r
ã−
0 δB
−ϑ+
s
+ b̃+
0 δB
+
−ϑ−
s
b̃−
0 δB
= 0, ǫ < ǫ̃
= 0, ǫ ≥ ǫ̃
δB = A,
(C3)
where first two equations approximate the default boundary in the presence of temporary shock (and thus
depend both on the temporary and permanent cash flow realization), and the last equation approximates the
default boundary in the absence of temporary shock (and thus depends only on the level of permanent cash
+
− − +
flow). Thus, we need to find the optimal set of parameters ΩB = {Ã+ , ã+
0 , b̃0 , Ã , ã0 , b̃0 , A} that maximizes
the value of levered equity (the value of ǫ̃ is discussed below). This approximation works well since the default
boundary in the multiple temporary shock case is functionally similar to the one in the single temporary shock
case and (C3) is very similar to the default boundary in the single temporary shock case. For the abandonment
boundary, the approximation is similar and results in finding the set of parameters ΩA with two changes: first,
since since its curvature is much less pronounced due to a smaller effect of future abandonment opportunities,
the negative part of the abandonment threshold in presence of temporary shocks is assumed to be linear (term
−ϑ+
s
b̃+
is not included); second, there is no abandonment when temporary jump is absent. Therefore, ΩA
0 δA
includes five parameters.
Next, using our knowledge about the model solution, we introduce additional constraints that ΩB and ΩA
must satisfy:
f+ (ǫB , 0) =
+
f (ǫ̃, δB ) =
∂f (ǫ, δB ) =
∂δB
ǫ=ǫ̃
dδB =
dǫ +
ǫ→−∞
0;
(C4)
−
(C5)
f (ǫ̃, δB ) = 0;
∂f− (ǫ, δB ) ;
∂δB
ǫ=ǫ̃
∂f− /∂ǫ − −
∂f /∂δB ǫ→−∞
(C6)
=K=
dδA .
dǫ ǫ→−∞
(C7)
In (C4), ǫB = minǫ∈R [δB (ǫ) = 0]. For all values of transitory shock ǫ > ǫB the firm does not default for
any permanent cash flow and since obviously ǫ̃ < ǫB , the boundary condition applies to the f + function. This
51
condition directly corresponds to (49) with δ = 0: in other words, ǫB is the largest realization of temporary
shock that makes shareholders default in case δ = 0 when only temporary shocks are left. For the single
transitory shock case, ǫB = c, and so in this case, ǫB ≤ c.
Conditions (C5) and (C6) hold since the default boundary is smooth and continuous. The value of ǫ̃
should be such that ǫB − ǫ̃ ≥ c since the f + part of the default boundary, which is mostly responsible for
its curvature, should be at least as large as in the single temporary shock case since the likelihood of future
default in the multiple case for the same coupon level is higher.
All three above conditions also hold for the abandonment threshold, except that the first condition corresponds to (47) with δ = 0, with ǫA being the largest realization of temporary shock that leads to abandonment
when only temporary shocks are left (in comparison to the single temporary shock case, ǫA ≤ 0).
Finally, condition (C7) imposes the condition that the limit K of the slope of default threshold is equal
to the slope of the abandonment threshold when ǫ → −∞ and the slope of the corresponding approximation
functions. The slopes of abandonment and default thresholds are equal when ǫ → −∞ since, as in the single
temporary shock case, the possibility of future abandonment/default on the tradeoff between temporary and
permanent shocks which yields the present abandonment/default decision is insignificant.
These conditions reduce the number of parameters in ΩA and ΩB . We also include parameters K and ǫ̃
into the modified set of parameters for abandonment boundary, so that Ω∗A = {b̃−
0 , ǫ̃, K}. Note that since K
is the same for abandonment and default boundaries, the modified set of parameters for default boundary is
Ω∗B = {A, b̃−
0 , δB (ǫ̃)}.
We find both Ω∗B and Ω∗A by maximizing the simulated value of levered equity (for given coupon level c)
and unlevered firm value, correspondingly. Below we give the description of the sequence of this procedure.
1. We set the value of all exogenous model parameters (δ0 , µ, σP , r, λs , λr , γ, τe , τc , α).
2. We simulate 1 million paths of the geometric Brownian motion for 120 years with a quarter frequency.
3. The initial ten values of each parameter are chosen for both Ω∗ A and Ω∗ B, so that for each of them the
initial search iss over 10×10×10 in a three-dimensional parameter space. For each triplet, we first check
whether it satisfies the following necessary condition: the multiple transitory shock threshold is uniformly
lower than the single transitory shock threshold (which is known in closed form; see Section IV.A). For
all triplets that satisfy this condition, we simulate corresponding (equity or unlevered firm) values and
V ∗ −V (ω)
find all parameter triplets Ω∗1 that satisfy the following condition: Ω̃∗1 = ω ∈ Ω∗ : k 1 V ∗ k ≤ m%,
1
where V (ω) is the simulated (unlevered firm or levered) value for a particular triplet and V1∗ is the
maximum attained value for all triplets. In other words, at the first stage we choose all parameter sets
for which the simulated values are close to the maximum value. In the benchmark procedure, we choose
the cutoff level m = 0.1%.
4. Minimal and maximal values of the three parameters of points that constitute Ω∗1 are used as boundaries
on the search region at the second stage. Step 3 is repeated using newly obtained boundaries on the
search region in addition to original parameter restrictions to form a finer grid. The cutoff level of m at
this stage is chosen to be m = 0.01%. As a result, V2∗ and Ω∗2 are found. In the same fashion, this step
is repeated again to find V3∗ and the corresponding optimal parameter set. For abandonment boundary
the numerical procedure is completed.
5. To find optimal coupon, we consider initially 20 points in the set [c, c], where For each level c, we
simulate the optimal default boundary, equity and debt values using the optimal parameter set from
52
above. Then find the optimal coupon value c∗1 that maximized the levered firm value (E1∗ + D1∗ ). Strict
convexity of levered firm value with respect to coupon implies that the optimal coupon c∗ lies between
the grid points directly below and above c∗1 . On the second step, we repeat the above step over the
grid of new 20 points in the interval [c∗1 − step1 , c∗1 + step1 ] where step1 is initial grid step size and find
(E2∗ + D2∗ ) and c∗2 , which are chosen to be the optimal coupon and firm values.
In order to check how good our approximation is, we add further terms from the theoretical functional
− + −
form of the boundary (with coefficients ã+
1 , ã1 , b̃1 , b̃1 , one at a time) to the reduced functional form (C3),
solve for the extended optimal set of parameters, and compare the newly approximated optimal value of equity
or unlevered firm to the original approximation. While the extension of the parameter set significantly slows
down the search for optimal boundary parametrization, the difference between approximated optimal values
obtained under the extended parametrization and under the original parametrization is found to be always
smaller than 0.1% of the original approximated optimal value, a negligible difference for our purposes. We
also check the measurement error at the simulation stage and find that the two step procedure that we employ
allows us to find the approximated optimal coupon c∗ and optimal leverage with the error of at most 1%.
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54
Table I. Descriptive statistics
This table reports the descriptive statistics for firm size, leverage ratio, cash flow, profitability,
and interest coverage ratio for the COMPUSTAT data set over the period from 1987 to 2005.
The final sample excludes financial companies, utilities, non-US companies, non-publicly traded
firms and subsidiaries, firm-years with total book value of assets of less than $10 million, nonexistent quasi-market leverage and book leverage ratios, and market-to-book ratios above ten. All
nominal values are converted into year 2000 dollar values using CPI index from the Bureau of
Labor Statistics of the United States Department of Labor. Firm size is the book value of asD9+D34
sets (COMPUSTAT data item D6); Quasi-market leverage ratio is QM L = D9+D34+D199×D25
,
where D9 is the amount of long-term debt exceeding maturity of one year, D34 is debt in current
liabilities, D199 is the year-end common share price, and D25 is the year-end volume of shares
outstanding; Cash flow is cash flow from operations (D308); Profitability is the ratio of cash
flow of operations to book assets; Coverage ratio is the ratio of cash flow from operations to
interest expense (D15). For each variable all statistics are first computed for each year and then
averaged across years. N is the total number of firm-years. Panels B, C, and D report the same
statistics for small, medium-size, and large firms. For each year, observations are classified into
corresponding size bins and then all statistics are averaged across years.
Panel A:
Firm Size
Quasi-market leverage
Cash Flow
Profitability
Coverage Ratio
Panel B
Firm Size
Quasi-market leverage
Cash Flow
Profitability
Coverage Ratio
Panel C
Firm Size
Quasi-market leverage
Cash Flow
Profitability
Coverage Ratio
Panel D
Firm Size
Quasi-market leverage
Cash Flow
Profitability
Coverage Ratio
ratio
ratio
ratio
ratio
Mean
5%
25%
Final sample
1,704.98
17.70
52.94
26.07
0.00
4.05
155.87
-19.05
0.18
5.96
-21.32
0.30
28.59
-27.72
0.22
Small firms
38.16
13.58
21.78
22.59
0.00
1.60
0.20
-13.17
-2.50
0.35
-35.88
-7.87
-5.08
-109.82
-3.81
Medium-size firms
191.06
83.67
113.62
25.51
0.00
3.15
12.83
-23.26
1.17
7.15
-15.62
0.87
61.14
-22.71
0.30
Large firms
4,876.15 436.07 657.49
30.10
0.55
10.99
449.02
-31.24
42.64
10.26
-3.57
4.95
27.54
-1.18
1.52
55
Median
75%
Std. Dev.
N
171.00
19.36
9.73
7.51
3.19
659.39
41.90
54.85
14.05
9.79
10,527.03
24.92
886.70
16.49
644.44
68,625
68,623
65,761
65,761
60,584
34.72
13.33
0.92
3.14
1.31
52.86
37.67
3.93
11.60
8.00
18.73
24.73
7.72
20.63
648.26
22,850
22,848
21,816
21,816
18,739
170.68
18.03
11.32
7.67
3.47
256.13
41.77
23.96
14.59
12.73
89.03
25.48
24.83
15.19
747.94
22,854
22,854
21,889
21,889
19,924
1,265.18
24.97
107.67
9.83
4.16
3,196.69
44.79
301.12
14.99
9.15
17,796.43
23.85
1485.33
10.06
331.68
22,921
22,921
22,056
22,056
21,921
Table II. Descriptive statistics on parameters of interest
This table reports the descriptive statistics for coverage ratio, the frequency of loss-making, duration of
loss-making and financial distress, cash flow and asset volatility, and correlation between asset values and
cash flows estimated for the COMPUSTAT data set over the period from 1987 to 2005. The final sample
excludes financial companies, utilities, non-US companies, non-publicly traded firms and subsidiaries, firmyears with asset book values less than $10 million, nonexistent quasi-market and book leverage ratios, and
market-to-book ratios above ten. All statistics are first estimated for an individual firm (as identified
by NPERMNO) and then averaged over all firms. Coverage ratio < 1 is the fraction of years with the
coverage ratio lower than one. Negative cash flows is the fraction of years with negative operating cash
flow. Duration of loss-making is the average number of consecutive years of negative operating cash flows
(conditional on having at least one such year). Duration of financial distress is the average number of
consecutive years of the coverage ratio lower than one (conditional on having at least one such year).
Cash flow volatility is the standard deviation of the annual changes in operating cash flow to the mean
of absolute operating cash flows. Asset volatility is the standard deviation of the ratio of annual changes
in asset value to asset value. Correlation in changes is the correlation between the changes in operating
cash flows and asset values. Correlation in levels is the correlation between operating cash flows and asset
values. Duration measures are in years. Frequency measures are in per cent. Volatility and correlation
measures are in per cent divided by 100. N is the total number of firms. Panels B, C, and D report the
same statistics for small, medium-size, and large firms. For each firm size is defined as average of CPI
adjusted book values across all years the firm is recorded in the data set.
Panel A:
Coverage ratio < 1
Negative cash flows
Duration of loss-making
Duration of financial distress
Cash flow volatility
Asset volatility
Correlation of changes
Correlation of levels
Panel B:
Coverage ratio < 1
Negative cash flows
Duration of loss-making
Duration of financial distress
Cash flow volatility
Asset volatility
Correlation of changes
Correlation of levels
Panel B:
Coverage ratio < 1
Negative cash flows
Duration of loss-making
Duration of financial distress
Cash flow volatility
Asset volatility
Correlation of changes
Correlation of levels
Panel D:
Coverage ratio < 1
Negative cash flows
Duration of loss-making
Duration of financial distress
Cash flow volatility
Asset volatility
Correlation of changes
Correlation of levels
Mean
5%
25%
Final sample
41.97 0.00
0.00
33.71 0.00
0.00
1.94
1.00
1.00
2.46
1.00
1.00
1.04
0.17
0.47
0.70
0.11
0.26
0.07
-1.00 -0.37
0.22
-0.80 -0.21
Small firms
58.15 0.00 25.00
52.52 0.00 14.29
2.16
1.00
1.00
2.47
1.00
1.00
1.20
0.19
0.60
0.84
0.11
0.31
0.03
-1.00 -0.50
0.11
-0.85 -0.34
Medium-size firms
41.20 0.00
0.00
33.31 0.00
0.00
2.01
1.00
1.00
2.54
1.00
1.00
1.09
0.19
0.53
0.75
0.11
0.28
0.07
-1.00 -0.35
0.20
-0.81 -0.23
Large firms
27.15 0.00
0.00
15.75 0.00
0.00
1.56
1.00
1.00
2.35
1.00
1.00
0.84
0.16
0.36
0.55
0.12
0.23
0.11
-1.00 -0.27
0.34
-0.71 -0.01
Median
75%
Std. Dev.
N
33.33
18.18
1.25
2.00
0.90
0.43
0.11
0.31
80.00
60.00
2.00
3.00
1.49
0.76
0.54
0.71
38.96
37.82
1.59
1.98
0.70
1.31
0.60
0.56
8,891
8,897
6,409
6,711
4,834
6860
4,834
4,834
60.00
50.00
1.50
2.00
1.13
0.52
0.05
0.15
100.00
100.00
2.50
3.00
1.69
0.93
0.58
0.60
38.42
39.42
1.77
2.00
0.73
1.62
0.65
0.56
2,921
2,929
2,467
2,551
1,504
1,986
1,504
1,504
33.33
20.00
1.40
2.00
0.98
0.49
0.10
0.28
75.00
60.00
2.00
3.00
1.54
0.83
0.53
0.68
37.74
36.55
1.68
2.03
0.69
1.05
0.59
0.56
2,935
2,967
2,147
2,294
1,685
2,350
1,685
1,685
10.53
0.00
1.00
2.00
0.65
0.35
0.17
0.48
47.37
21.05
2.00
3.00
1.14
0.57
0.54
0.80
34.30
27.13
1.09
1.88
0.63
1.23
0.57
0.53
3,035
3,001
1,795
1,866
1,645
2524
1,645
1,645
Table III. Parameters for the benchmark model case
This table reports parameter values for the benchmark model
with permanent and one transitory shock. Details on calibration are given in Section III.B.
Variable
λs
λr
γ
σV
σP
τe
τi
α
r
µ
Description
Shock frequency
Shock persistence
Shock magnitude
Asset volatility
Volatility of permanent component
Effective tax rate on dividends
Marginal income tax rate
Bankruptcy costs
Risk-free rate
Risk-neutral growth rate
57
Value
1
0.5
0.098
0.01
0.20
0.4
0.3
0.1
0.055
0.01
Table IV. Financial decisions and transitory shocks
This table reports a number of financial decision variables predicted by the benchmark model’s with permanent and one transitory shocks. “Permanent only” case
refers to the model without transitory shocks. “Benchmark case” refers to the
benchmark calibration described in Section III.B with all parameters given in III.
All other rows report results for the benchmark calibration with the change in one of
the parameters: σV is the total asset volatility, λs is the shock frequency parameter,
λr is the shock persistence parameter, τe is the effective tax on dividends, and δ0 is
the initial value of the permanent cash flow component. L is the optimal leverage
ratio, c is the optimal level of coupon, s is the credit spread (in basis points), ∆L
(∆c) is the difference between the permanent-only leverage (coupon) for the same
value of σV , σP is the volatility of the permanent cash flow component, γ is the
shock magnitude parameter, and φ is the fraction of the unlevered firm value due to
the present value of transitory shocks. For the benchmark calibration, parameters
γ and σP are chosen to match σV = 0.25 and φ = 0.02.
Permanent only
Benchmark case
σV = 0.1
σV = 0.5
λs = 0.5
λs = 2
λr = 0.25
λr = 1
τe = 0.35
τe = 0.5
δ0 = 0.2
δ0 = 4
φ = 0.01
φ = 0.05
L
c
s
∆L
∆c
σP
σV
γ
φ
0.646
0.419
0.434
0.408
0.423
0.414
0.432
0.398
0.319
0.537
0.429
0.427
0.440
0.381
1.872
1.155
1.123
1.509
1.164
1.151
1.230
1.122
0.877
1.462
0.113
2.252
1.206
1.104
235
187
123
529
182
193
216
212
132
278
137
140
187
208
0.000
0.227
0.393
0.196
0.250
0.253
0.230
0.259
0.247
0.218
0.286
0.286
0.220
0.374
0.000
0.717
0.864
0.754
0.639
0.661
0.593
0.715
0.623
0.567
0.067
1.338
0.620
0.925
0.25
0.20
0.06
0.4
0.195
0.205
0.216
0.227
0.200
0.200
0.138
0.140
0.218
0.170
0.25
0.25
0.1
0.5
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
∞
0.098
0.092
0.114
0.097
0.097
0.180
0.052
0.098
0.098
0.950
0.047
0.119
0.070
0.00
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.01
0.05
58