Learning and Leverage Dynamics in General Equilibrium Christopher A. Hennessy (LBS, CEPR, and ECGI)
Boris Radnaev (Cornerstone)
Previous Version: November 2014 This Version: November 2015
Abstract This paper develops and empirically tests a tractable general equilibrium model of corporate …nancing and investment dynamics in a trade-o¤ economy where heterogeneous …rms face unobservable disaster risk and engage in rational Bayesian updating. The model sheds light on leverage cycles. During periods absent disasters: equity premia decrease; credit spreads decrease; expected loss given default increases; and leverage ratios increase. Time since prior disaster is the key model conditioning variable. In response to a disaster, risk premia increase while …rms sharply reduce labor, capital and leverage, with response size increasing in time since prior disasters. Firms with high bankruptcy costs are most responsive to the time-since-disaster variable. Disaster responses are more pronounced than in an otherwise equivalent economy featuring observed disaster risk. Empirical tests of novel corporate …nance predictions are conducted. Consistent with the model and simulated model regressions, in the real-world data leverage and investment are increasing in time-since-prior-recessions, with the e¤ect more pronounced for …rms with low recovery ratios.
JEL Classi…cations: E1, E2, G3 Keywords: Capital Structure, Learning, Credit Risk, Disasters
Email:
[email protected]. We thank Andrea Buraschi and Francesca Cornelli for helpful advice on a prior draft. We also thank seminar participants at the London Business School, Norwegian School Economics, New Economic School, University of Bristol, and Bank of England for comments. We thank John Graham, Mark Leary, and Michael Roberts for providing their aggregate data.
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1
Introduction
In recent years, there has been great progress in developing rich structural models of corporate decisions accounting for both dynamics and random shocks. A weakness of some models, e.g. Goldstein, et al. (2001) is the failure to model real investment decisions, which is an important shortcoming inasmuch as it has been shown by Leary and Roberts (2010) that investment is perhaps the most important predictor of …nancing activity. Other models have made advances by modeling jointly …nancing and investment decisions, e.g. Hennessy and Whited (2005), but taken a step back by adopting the assumption of risk-neutrality. This too is an important weakness inasmuch as it has been shown by Gourio (2013) that accounting for risk-adjustments can have an important impact on the present value of distress costs. What all existing capital structure models share in common is an assumption that …rms and investors know the true shock generating functions, i.e. there is no accounting for learning. This too is an important omission since, as surveyed by Pastor and Veronesi (2009), learning has helped resolve some important asset pricing puzzles. The objective of this paper is to introduce learning into a tractable general equilibrium model of corporate …nancing and investment decisions, with an eye for explaining some important stylized facts that existing models cannot, as well as generating and testing some novel empirical implications that arise from the learning mechanism. One of the most striking economic phenomena over the last hundred years is the secular increase in corporate leverage. For example, in their recent study of long-term leverage trends, Graham, et al. (2014) document that aggregate corporate leverage more than tripled between 1945 and 1970, rising from 11% to 35%. While the literature in corporate …nance has tended to focus on crosssectional factors predicting …rms’debt-equity mix, it is apparent that understanding the drivers of long-term leverage trends merits more attention. The Crisis of 2007/2008 (the Crisis below) sharply punctuated a period of what some have termed “leverage excess.” In fact, the conjunction of high leverage, even by those …rms facing very costly bankruptcy, followed by a severe credit crisis has led to some to question the working assumption of rationality embedded in existing dynamic capital structure models and dynamic stochastic general equilibrium (DSGE) models. As further evidence against the hypothesis of rationality, many have cited low credit spreads and low equity risk-premia during the years just prior to the onset of the 2
Crisis. Although many forms of irrationality have been posited, the variant that has perhaps gained the most traction as an explanation for the Crisis is that agents failed to properly understand the risks they faced, e.g. Stiglitz (2010), or worse still su¤ered under the fallacy of induction and concluded that major recessions were a thing of the past, e.g. Taleb (2007). In favor of the behavioral narrative, it is apparent that many …rms, investors and households made investment and …nancing decisions that they regretted with the bene…t of hindsight as of 2008. However, a valid test of rationality must …lter economic time-series just as agents in the economy do, in real-time, not through the rear-view mirror. To this end, this paper considers a dynamic stochastic general equilibrium setting in which …rms observe the economy over time and decide how much to borrow and how much to invest, taking into account risk-premia demanded by a representative agent with Epstein-Zin preferences. As discussed above, we depart from an extant literature discussed below in assuming that …rms do not know the objective probability of the risk of large negative aggregate shocks to total factor productivity and productive capital. Instead, agents engage in a simple and tractable form of Bayesian updating. In addition to facing uncertainty about the objective probability of large negative shocks (“disasters” below), …rms face …nancial market imperfections in the form of tax bene…ts of debt and privately (not socially) costly bankruptcy. In contrast to a standard tradeo¤-theoretic setting, agents in the economy do not know the objective bankruptcy probability since they do not know the objective risk of a macroeconomic disaster. Moreover, …rms are embedded in a general equilibrium where the pricing kernel is determined endogenously. As argued below, the conjunction of Bayesian learning and violations of the perfect …nancial markets assumptions of Modigliani and Miller are two critical elements of the model’s causal mechanism. This is because learning ampli…es ‡uctuations in the risk adjusted costs of distress, leading to sharp swings in optimal leverage ratios. The particulars of the model are as follows. Firms face idiosyncratic and aggregate shocks. We follow the literature in assuming that aggregate-level disasters take the form of large negative shocks to total factor productivity and to the stock of productive physical capital. The true risk of a disaster is either high or low, following a two-state Markov process. Agents do not observe the true risk of a disaster, but instead engage in Bayesian updating of their beliefs based on observed outcomes, with disaster realizations being observable. The idiosyncratic shocks are i.i.d. across
3
…rms and time. In the interest of transparency and facilitating quantitative comparisons with the literature, the model’s speci…cation of …rm-level technology follows closely the elegant model of Gourio (2013). As we hope to make apparent, an attractive feature of the model of Gourio (2013) is the ease with which it can be extended to incorporate learning, as well as …rm-heterogeneity. The conjunction of tractable learning and heterogeneity are the two key ingredients we add to the setup of Gourio. The causal mechanism central to the model, rational Bayesian updating, provides a plausible qualitative and quantitative explanation of recently-observed phenomena in terms of private sector leverage, credit spreads, and equity risk-premia. For example, consider …rst the model’s predictions regarding the behavior of leverage and asset prices during long periods sans disasters, e.g. the Great Moderation discussed by Bernanke (2004). During such periods, agents in the model economy revise upwards their belief regarding the probability of being in the low disaster risk regime. This lowers risk-premia. At the same time, expected bankruptcy costs fall while the expected returns to capital accumulation rise. The conjunction of these three factors causes …rms to increase their level of investment. Further, leverage ratios increase. Another central element of our modeling strategy is to introduce a simple form of cross-sectional heterogeneity. In particular, it is assumed that the proportion of …rm assets recovered by creditors in the event of bankruptcy is either low or high. With this heterogeneity in creditor recovery parameters in mind, we consider again predicted …rm behavior within the model during a prolonged period without disasters. Naturally, the leverage ratios of both …rm types are increasing in time-sincedisaster. However, it is the …rms endowed with low creditor recovery parameters whose behavior is most sensitive to this variable. After all, such …rms are most concerned about the pricing of disaster states since they face greater prospective asset leakage in the event of a bankruptcy …ling. Like building on a mountainside in a seismic zone, such behavior is a rational response by Bayesian learners to prolonged tranquil periods. Of course, the losses will be worse when/if a shock occurs, which agents fully understand. The preceding discussion leads to an important implication of the model regarding bond pricing. During a prolonged quiet period, leverage increases. However, the average yield spread relative to a AAA credit actually declines, re‡ecting a decline in credit risk at each given level of leverage, as
4
well as a decline in risk-premia. However, expected loss-given-default actually increases during such benign periods. Intuitively, during quiet periods the composition of credit risk shifts, with a higher percentage of corporate debt accounted for by the borrowing of …rms with low intrinsic recovery technologies. This is the time-series analog of the cross-sectional prediction of Glover (2015). In fact, as the time since last disaster increases, the distribution of loss-given-default in the model becomes increasingly bi-modal, with a greater probability mass on high loss credits. Having viewed sustained quiet periods such as the Great Moderation through the prism of the model, it is instructive to consider its qualitative and quantitative predictions regarding responses to disasters. As we argue, the learning mechanism central to the model has the potential to explain many stylized facts that other disaster-based models cannot. For example, it has been argued that existing models fail to match observation in terms of delivering sharp swings in real and …nancial variables in response to negative shocks of plausible magnitude. In contrast, we show the conjunction of learning, …nancial market imperfections and …rm heterogeneity greatly ampli…es response magnitudes under otherwise equivalent calibrations. Intuitively, it should come as no surprise to see large changes in …nancing, investment and asset prices when a large negative shock occurs after a prolonged quiet period. After all, disasters necessarily result in negative belief revisions for Bayesian learners. We take the model to the data, contributing two novel predictions to the empirical corporate …nance literature on …rm leverage ratios and investment rates. We begin …rst with mimicking regressions performed on simulated model data, with these regressions motivated by the causal mechanisms discussed above. In particular, we conjecture that leverage ratios and investment rates should be increasing in years-since-disaster, but decreasing in the interaction between yearssince-disaster and recovery rate parameters. Intuitively, beliefs should become more favorable as time-since-disaster increases, with …rms responding by increasing leverage and investment, with the e¤ect being particularly strong for those …rms with low recovery parameters. These conjectured relationships are manifest in our simulated model regressions. We use these simulated model regressions as the basis for conjecturing that similar patterns will be observed in the real-world data. Indeed, we …nd that …rm leverage is increasing in a time-since-recession variable, but decreasing in an interaction between this variable and industry-level recovery ratios. We also …nd that investment
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rates are increasing in time-since-recession, while the interaction variable is negative but statistically insigni…cant. We turn now to other related literature. Learning has received little attention in the corporate …nance literature, especially in the literature that concerns itself with dynamics. The two most notable exceptions are papers by Alti (2003) and Moyen and Platikanov (2013). We depart from these papers in two important respects. First, we consider a general equilibrium setting, with changes in aggregate risk premia being a central mechanism. Second, we consider that …rms can jointly set optimal leverage ratios and investment rates, while they consider equity …nanced …rms. Another paper in the dynamic corporate …nance literature related to our own is recent work by Glover (2015) who considers the e¤ect of heterogeneity on the behavior of recovery ratios. Learning is absent from his model. Disaster risk has been important in the literature in recent years, with Rietz (1988), Barro (2009), Gabaix (2013) and Nakamura, et al. (2013) making in‡uential contributions. The technological setup of our model is closest to that of Gourio (2013), and in fact we build directly on his tractable general equilibrium framework. As in our paper, Gourio focuses on a time-varying probability of disaster. However, in his model the disaster probability is observed. Moreover, the model presented here analyzes the e¤ect of cross-sectional heterogeneity in creditor recovery parameters. We argue the conjunction of learning and heterogeneity can help in better explaining risk premia ‡uctuations and cross-sectional leverage ratio dynamics. The paper is also related to an extant asset pricing literature analyzing the time-varying disaster risk both in non-learning (e.g. Wachter (2013)) and learning setups. In contrast to the model presented here, the asset pricing models generally abstract from capital structure decisions and real investment. This is an important point of departure inasmuch as it will be shown that …nancial market imperfections and …rm heterogeneity represent important ampli…cation mechanisms. Benzoni, et al. (2011), Koulovatianos and Wieland (2011), and Lu and Siemer (2014) introduce learning about rare disasters to explain equity returns in endowment economies. This paper is organized as follows. The model is introduced in Section 2. Section 3 analyzes variations of the simulated model, focusing on …rm leverage, credit risk, and risk-premia. Empirical tests are contained in Section 4. Section 5 o¤ers conclusions.
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2
The Model
The section describes the real and …nancial technologies available to …rms, the nature of shocks, as well as the decision problems faced by …rms and the representative household. We then discuss the determination of equilibrium. Learning about disaster risk and …rm heterogeneity are the two new ingredients we introduce into the benchmark model of Gourio (2013). The exposition focuses on these extensions, with some commentary on key modeling assumptions.
2.1
Learning about Disaster Risk
We begin with a discussion of the learning mechanism. Of particular interest to us is better understanding how …rms that learn di¤er from “standard” …rms that know the true shock generating process each period. Further, we will be interested in assessing the di¤erential impact learning has in a realistic cross-section of heterogeneous …rms. Each period there is a risk of a macroeconomic disaster. As described in detail below, in the event of a disaster, there is a permanent decline in total factor productivity (TFP), as well as destruction of physical capital. Since the notion of disasters is not without controversy, it is worthwhile discussing them brie‡y. A number of real-world events can be properly classi…ed as disasters, e.g. wars, epidemics, earthquakes, terrorist acts, or large-scale industrial accidents. Further, and more generally, one may think of economic agents as discovering that existing production technologies, e.g. asbestos in construction or originate-to-distribute in lending, have negative unintended side-e¤ects that are only observed and abandoned with a lag. Of course, the Crisis of 2007/8 originated in the …nancial sector, lending support to those arguing for intermediary-based risk premia. However, one can think of our model as capturing an economy in which agents posit a number of potential rare-event disasters, with none of them unique in terms of their e¤ect on TFP and capital. To the extent that such a model performs well, there is less urgency in building models predicated upon intermediary-based risk premia. As pointed out by Gourio (2013), the commonality of asset price movements during the Crisis, even those assets that are non-intermediated, lends some support to the aggregate risk premium channel at work in our model. At this stage it is also worth pointing out that we borrow the “disaster” terminology from the literature. However, this language may be unhelpful in that what the model actually features is 7
negative shocks to TFP and the stock of capital. Whether such shocks qualify as disasters clearly depends on the parameterization of the size of the negative shocks. The model is ‡exible enough to account for small shocks (contraction) and large (disaster) shocks with a simple resetting of the two shock parameters. In the interest of transparency, we following Gourio (2013) and model large shocks, so borrow his terminology. A primary point of departure from Gourio (2013) is that we assume that the probability of disaster is not directly observable. Rather, at the start of each period t, agents observe whether or not a disaster hit the prior generation of …rms, with Bayes’rule then used to form updated beliefs. In particular, let xt = 1 be an indicator for a disaster hitting …rms born at date t output at the very start of date t. And let
t
1 who produce
Pr[xt = 1]: Our objective is to account for learning
about disaster risk in a tractable way, with transparency regarding the learning mechanism. To this end, we assume such that 0 h
l
t
<
is a hidden-state Markov process with the two potential states being h
1: That is, the latent state
l
l
and
h
features low disaster risk and the latent state
features high disaster risk. The symmetric transition probability parameter of the
t
process is
s 2 [0; 1=2): Although our model embeds the case of a constant unknown distribution (s = 0) as a special case, we will opt for parameterizations where the underlying disaster shock probability switches over time, with s > 0: The problem with constant disaster probability models is that they have the unattractive feature that the amount of model uncertainty faced by agents declines in a monotonic way over time, so arbitrary assumptions about the length of shock observation become important. A more realistic economic environment is one in which agents always get fresh injections of uncertainty, and are never able to pin down the true distribution. As discussed below, the dramatic nature of disasters in the model, e.g. aggregate capital destruction, renders the actual realization of a disaster to be common knowledge. Therefore, agents can use the entire data series regarding disaster realizations in forming beliefs about the prospective risk of disasters. The state variable pt denotes agents’ common belief at the start of period t regarding the probability of being in the low disaster risk state ( l ) at that point in time based
8
upon the history of disaster realizations up to and including that point in time. We have:
pt
Pr[
t
=
l j(x
)
=t =1 ]:
(1)
Agents share a common belief each period under our maintained assumption of a common initial period prior p0 : For now, we shall think of agents as entering with a non-dogmatic prior p0 2 (0; 1). Further discussion of the prior will follow below. An attractive feature of the model is the simplicity with which the benchmark model of Gourio (2013) is extended. In particular, Gourio posits that the objective probability of disaster is known to all agents, with this probability entering the model as an exogenous state variable. To account for learning, the objective probability of disaster is dropped as a state variable. Instead, we add beliefs p as a state variable, leaving the dimensionality of the problem unchanged. Consider …rst the nature of belief updating conditional on no disaster shock taking place. From Bayes’rule we have: P r[
t
=
l jxt
= 0] =
P r[xt = 0 \ t = P r[xt = 0]
l]
:
(2)
The implied law of motion for the belief state variable is :
xt = 0 ) p t =
(1
l )[pt
1 (1
(1 s) + (1 pt 1 )s] l )[pt 1 (1 s) + (1 pt 1 )s] + (1 pt 1 )(1 h )[(1
s) + pt
1 s]
:
(3)
Notice, contra Taleb (2007), agents in this economy do not su¤er from the fallacy of induction, e.g. reaching the erroneous conclusion that disasters are impossible based upon arbitrarily long sequences without disaster shocks. To see this, notice that so long as disasters are not a certainty under the bad distribution, speci…cally if one assumes the parameter (3) implies that starting from any prior pt
1
h
is less than 1, then equation
< 1, the absence of a disaster shock still results in a
revised belief admitting the possibility of the bad distribution (pt < 1): Further, such a belief always results in a positive assessed probability of a disaster at the end of the period. In particular, we have: If
h
< 1 and pt
1
< 1; then Pr[xt+1 = 1jxt = 0] > 0:
We conjecture that given a su¢ ciently low prior, beliefs will be revised upwards after a period 9
in which no shock has taken place. Indeed, using the preceding law of motion we have: (1
If xt = 0 and
2s)(
+[2s(1
l)
2 l )pt 1
h
(1
s)(
(1 l )]pt
h
l )s;
then pt
pt
1:
(4)
1
The second inequality in the preceding equation is strict if and only if the prior inequality is strict. Therefore, we conjecture an upper bound on beliefs, call it p, which solves the following quadratic equation: (1
2s)(
2 l )p
h
+ [2s(1
l)
(1
s)(
h
l )]p
= (1
l )s:
(5)
To take a particular example, if s = 0 then p = 1: Consider next the nature of belief updating conditional on a disaster shock taking place. Applying Bayes’rule we have:
P r[
t
=
l jxt
= 1] =
P r[xt = 1 \ t = P r[xt = 1]
l]
:
(6)
The implied law of motion for the belief state variable is:
xt = 1 ) pt =
l [pt 1 (1 l [pt 1 (1
s) + (1 pt 1 )s] pt 1 )s] + h [(1 pt 1 )(1
s) + (1
s) + pt
1 s]
:
(7)
We conjecture that given a su¢ ciently higher prior, beliefs will be revised downwards after a period in which a disaster shock has taken place. In particular, using the preceding law of motion we have: 2
6 If xt = 1 and 4
(1 +[2s
2s)(
l+(
h
3
2 l )pt 1
h l )(1
s)]pt
1
7 5
l s;
then pt
pt
1:
(8)
The second inequality in the preceding equation is strict if and only if the prior inequality is strict. Therefore, we conjecture a lower bound on beliefs, call it p; which solves the following quadratic equation: (1
2s)(
h
2 l )p
+ [2s
l
To take a particular example, if s = 0 then p = 0:
10
+(
h
l )(1
s)]p =
l s:
(9)
To ensure the economy has interesting dynamics, we assume agents enter the model with a common prior p0 2 (p; p). A prior falling into this interval ensures that with probability one agents will form revised beliefs p1 6= p0 based on the …rst observed shock x1 . That is, there will at least be some learning over some time interval if p0 2 (p; p). For example, p0 = 1=2 meets the criterion of p0 2 (p; p). There is a strong logical basis for such an initial prior. In particular, the lower bound p corresponds to agents who posit that an in…nite sequence of consecutive disasters took place before they started observing the xt process at t = 1, while the upper bound p corresponds to agents who posit that an in…nite sequence without disasters took place before they started observing the process. Together, equations (3) and (7) deliver a simple tractable law of motion for beliefs. In the special case where the disaster risk probability never changes, s = 0, the updating rules can be simpli…ed as follows:
xt = 0 ) pt =
(1
xt = 1 ) pt =
(1 l )pt 1 l )pt 1 + (1 h )(1 l pt 1 (1
s) pt l pt 1 + h (1
1)
pt
< pt
(10)
1)
1:
(11)
Consider again the fallacy of induction as discussed by Taleb (2008), in the form of “all swans are white.” In our model, agents will never be certain of zero disaster unless s = 0; l
h
= 1 and
= 0; for only then can they conclude there is no risk of disaster after observing no disaster shock.
Agents in the model do engage in Popperian falsi…cation of optimistic beliefs. To see this, suppose the model were parameterized with s = 0 and
l
= 0: That is, suppose the true distribution is
…xed over time, with the low disaster risk state featuring zero risk of a negative shock. From the preceding equation it follows that in the event of a disaster realization, agents would reject the hypothesis that “the probability of disaster risk is zero,” concluding instead “the true probability of disaster is
h
> 0:”
The following proposition o¤ers summary observations based on the preceding analysis. PROPOSITION 1. If agents enter the economy with the prior belief p0 2 [p; p] with p as de…ned in equation (9) and p as de…ned in equation (5), then they will revise beliefs upward in response to each period sans disaster. If
h
< 1; then agents can never rule out the possibility of the bad
distribution, implying pt < 1 for all t
1: Agents will revise beliefs downward in response to each 11
observed disaster. They will exclude the possibility of the good distribution, concluding pt = 0 in response to a disaster at time t if and only if
l
= 0:
Since beliefs are a central causal mechanism we wish to explore, it is useful to develop a better understanding of their behavior. Consider …rst Figure 1 which illustrates Proposition 1. This …gure assumes the high disaster risk is 5% and the low risk is 2.5% with a switching probability of 0.5%. To sharpen the illustration, it is assumed that over the entire sample period the true disaster risk is high, with the …rst 95 years of each century featuring no shocks, with the last 5 years punctuated by 5 consecutive disaster shocks. As one would then expect, beliefs tend to rise then fall each century. However, the behavior of beliefs is shaped by the initiating date belief. For example, Panel A is based upon an initiating date prior of p0 = 1=2 2 (p; p): Here beliefs rise in response to each year sans shock and fall after each year with a shock, even during the …rst century of observed shocks. Panel B is based upon an “unreasonable” initiating date prior of p0 = :95 > p: Notice that here over the early part of the …rst century beliefs fall despite the fact that there is a sequence of years with no disaster shock. It is hard to rationalize a belief that becomes more negative in response to good news, and so we rule out such an initial prior. Figure 2 plots the state variable p over progressively longer periods without the realization of a disaster, with the initial-date belief set at p0 = 1=2. On the vertical axis is the state variable p and on the horizontal axis is the elapsed time since the previous disaster. A key prediction of our analysis is that the time-since-disaster will be a key conditioning variable in understanding the behavior of investment, leverage, and expected returns. Of course, the behavior of beliefs will depend on the speci…c parameterization of the hidden Markov process. To illustrate the role played by the parameterization, Panel A considers variation in the disaster probability parameters while Panel B considers variation in the switching probability. Consider …rst Panel A. As shown, p is increasing and concave in the time-since-disaster. Importantly, one sees that beliefs are more sensitive to time-since-disaster the greater the wedge between
l
and
h:
On the other hand, as shown in Panel
B, beliefs are less sensitive to time-since-disaster the higher the switching probability. We will be interested in generating predictions regarding how the economy can be expected to respond to the realization of a disaster after a prolonged quiet period. To this end, consider Figure 3. On the vertical axis is the change in the state variable p following a disaster shock. On the horizontal 12
axis is the time-since-prior-disaster. To illustrate the role played by the parameterization, Panel A considers variation in the disaster probability parameters while Panel B considers variation in the switching probability. As shown in both panels, after a disaster, agents revise down their beliefs regarding the probability of being in the low disaster risk regime. Importantly, the magnitude of the downward revision is increasing in time-since-last-disaster. Intuitively, following a prolonged period absent large negative shocks, the realization of a disaster will come as a greater surprise and lead to a large change in beliefs. Consider now Panel A. Here one sees that beliefs are more sensitive to the occurrence of a disaster the greater the wedge between
l
and
h:
On the other hand, as
shown in Panel B, beliefs are less sensitive to the occurrence of a disaster the higher the switching probability.
2.2
Production
There is a continuum of mass one of competitive price-taking …rms each period. For simplicity, assume …rms live for only one period. As discussed below, this assumption is without loss of generality under the assumed technologies. Initially, we abstract from heterogeneity in …rm technologies. Ex ante …rms are identical. Ex post, …rms di¤er as a result of facing idiosyncratic shocks to their stock of productive capital as described below. The fact that …rms are identical ex ante is a great simpli…cation since it implies they will adopt the same policies. Timing is as follows. Consider a …rm i born on date t. It has a belief pt and also knows the lagged value of total factor productivity zt : With this information in-hand, the …rm chooses a debt face w : The choice variable K w value Bit+1 and a wished-for capital stock Kit+1 it+1 then maps stochastically
to an e¤ ective capital stock as follows: w Kit+1 = Kit+1 "it+1 (1
xt+1 bk ):
(12)
The e¤ective capital stock is a¤ected by both aggregate and idiosyncratic shocks. The random variable "it+1 is the only idiosyncratic shock in the model. The idiosyncratic shocks are i.i.d., have mean-one, and are drawn from a continuously di¤erentiable cumulative density H having bounded support on the positive real line. As shown, if a disaster occurs, the e¤ective capital stock of each
13
…rm in the economy is scaled down by the factor bk , with bk being a key model parameter. This speci…cation represents a small departure from Gourio (2013) who instead models bk as a random variable. Firms employ constant returns to scale Cobb-Douglas production functions of the form: Yit+1 = Kit+1 (zt+1 Nit+1 )1
:
(13)
In the preceding equation, zt+1 is total factor productivity, Kit+1 is the e¤ective capital stock, and Nit+1 is the amount of labor input chosen by the …rm after observing all shocks. As well as having an impact on e¤ective capital stocks, macroeconomic disasters also impact total factor productivity. The TFP process evolves as follows:
log zt+1 = log zt +
+ et+1 + log(1
xt+1 btf p ):
(14)
In the preceding equation, et+1 is assumed to be i.i.d. and N (0; 1), and so captures the type of small normally distributed shocks commonly found in real business cycle models. The parameter btf p captures the severity of negative productivity shocks resulting from disasters. It is worth noting that this speci…cation represents a small departure from Gourio who instead models btf p as a random variable. After the periodic shocks are observed, each …rm takes into account the equilibrium wage Wt+1 and chooses labor its labor input to maximize operating pro…ts:
it+1 (Kit+1 ; zt+1 ; Wt+1 )
max fKit+1 (zt+1 Nit+1 )1
Nit+1
Wt+1 Nit+1 g:
(15)
The aggregate e¤ective capital stock, aggregate output, and aggregate labor demand are deter-
14
mined as follows:
Kt+1 Yt+1 Nt+1
Z Z
Z
w Kit+1 di = Kt+1 (1
xt+1 bk )
(16)
Yit+1 di = Kt+1 (zt+1 Nt+1 )1 Nit+1 di:
With these aggregates in mind, …rm-level operating pro…ts can be written as:
it+1
= Yit+1
Wt+1 Nit+1 = Yit+1 =
Kit+1 Kt+1
Yt+1 :
(17)
The total resources of the …rm at the end of the period (V ) is the sum of operating pro…ts and capital net of depreciation. Using the preceding equation we have: Yt+1 + (1 ) Kt+1 Yt+1 = "it+1 Kt+1 + (1 Kt+1
Vit+1 = Kit+1
= "it+1 (1
w xt+1 bk )Kt+1
(18) ) Yt+1 + (1 Kt+1
) :
The total gross return on the …rm’s initial capital invested is de…ned as follows: K Rit+1
Vit+1 = "it+1 (1 w Kit+1
xt+1 bk )
Yt+1 + (1 Kt+1
) :
(19)
Notice, the …rm’s return on investment is determined by its idiosyncratic capital shock, the aggregate disaster shock, and the aggregate TFP shock. The aggregate total return on total capital invested in the economy is de…ned as follows: K Rt+1
R
Vit+1 di = (1 w Kt+1
xt+1 bk )
Yt+1 + (1 Kt+1
) :
(20)
It follows that the total end of period value of …rm resources can be written as the product of its idiosyncratic capital shock, the aggregate return on capital, and the aggregate capital invested.
15
We have: K w Vit+1 = "it+1 Rt+1 Kt+1 :
2.3
(21)
Capital Structure
Firms can issue debt or equity to …nance their investment. Financial frictions take the form of tax bene…ts to debt and bankruptcy costs. Thus, the optimal capital structure will re‡ect an equating of marginal bankruptcy costs and tax bene…ts, as in a standard tradeo¤-theoretic model. Tax bene…ts to debt …nance are captured in reduced-form as a government subsidy paid to the …rm at the time debt is issued. The subsidy is equal to
1 > 0 per dollar of debt funding raised.
Given an equilibrium price q per unit of debt face value, the …rm receives q > q per unit of debt face value. The government …nances this debt subsidy through lump sum taxation. If end of period resources are inadequate to pay the promised face value, the …rm defaults, with bondholders receiving a fraction
2 (0; 1) of end-of-period resources. The residual value is assumed
to accrue to the government, who then rebates this value back to the representative household. That is, while the …rm views bankruptcy as costly, bankruptcy costs are not a deadweight loss to the economy. This feature of the model is worth stressing since it follows that the capital structure of …rms has no direct e¤ect on aggregate resources available to the representative household at the end of the period. Finally, we impose
< 1: If this assumption were not satis…ed, the …rm would
…nd it optimal to …nance entirely with debt since the government debt subsidy would swamp default costs. The …rm defaults if its end-of-period terminal resource value is lower then the face value of debt Bt+1 : K w Vit+1 = "it+1 Rt+1 Kt+1 < Bt+1 :
(22)
From the preceding equation it follows that an individual …rm defaults if its idiosyncratic capital shock is less than an economy-wide default threshold:
"t+1 =
1 K Rt+1
!
Bt+1 w Kt+1
:
(23)
Notice the common default threshold is decreasing in the aggregate return on capital and increasing
16
in the leverage ratio. It follows that defaults will be correlated since a low aggregate return on capital implies each …rm has higher default threshold, implying a greater mass of defaulting …rms. Moreover, there will be particularly pronounced clustering of defaults in the event of a disaster shock. But again, since defaults have no e¤ect on aggregate resources, clustered defaults are here properly viewed as a symptom of underlying economic weakness, not a cause. Letting M denote the endogenous pricing kernel, the market value of debt is: "
qt Bt+1 = Et Mt+1
Bt+1 [1
H("t+1 )] +
K w Rt+1 Kt+1
Z
0
"t+1
!#
"h(")d"
:
(24)
At this stage it is worth commenting on the assumption that …rms live for only one period. To see that this assumption is without loss of generality, notice that each …rm has a going-concern value equal to zero under the stated assumptions of constant returns, free-entry and price-taking. Thus, even if …rms were to live for multiple periods, their decision rules would remain the same as that discussed above. A …rm can simply shut down at the end of each period and restart the next with no cost. A key objective is for us to generate a more realistic cross-section of …rms. In particular, in the model of Gourio (2013), …rms are identical ex ante and so adopt identical policies. In reality, there is cross-sectional dispersion in …rm policies, e.g. wide variation in leverage ratios. In addition to being more realistic, we will be interested in generating predictions regarding how …rms in the cross-section will react to changes in economy-wide beliefs. We introduce heterogeneous …rms into the model as follows. Suppose again there is free-entry. But now assume there is a start-up cost that must be paid at the start of each period, with this cost varying with the technology employed. To …x ideas, we shall think of the start-up cost as being a per-period license fee charged by the government. The license fees collected by the government will be returned to households via lump sum transfers. With this in mind, assume entrants can acquire a license to operate one of two alternative technologies. There is an expensive technology featuring a high recovery parameter L.
H
and an inexpensive technology featuring a low recovery parameter
The measure of each class of licenses is equal to one-half. With free-entry, the net present value
to operating either technology, net of the respective license fee, must be equal to zero. Once the
17
license fee is paid then …rms will set optimal policies as above.
2.4
Households
As is standard in the asset pricing literature, we consider an in…nitely-lived representative household with recursive preferences, as in Epstein and Zin (1989). Letting C denote consumption and N hours worked, the representative household has the following recursive utility:
Ut =
(1
)(Ctv (1
Nt )
1 v 1
)
+ Et
1 Ut+1
1
1 1
1
:
(25)
The household budget constraint demands that uses of funds is no greater than sources of funds. In particular, we demand:
Ct + Pt + qt Bt
Wt Nt +
t Bt 1
+ Dt + Tt :
(26)
In the preceding equation, P denotes the market price of a 100% equity interest in all the …rms in the economy. The variable B denotes the face value of debt and q the market price per unit of face value, with the natural generalization if there are heterogeneous …rms. The variable W denotes the wage rate. The variable
denotes the realized value received by bondholders for each unit of
debt face value purchased in the prior period. The variable D is the realized dividend payo¤ to shareholders at the start of the current period. Finally, the term T captures lump sum transfers from the government, a portion of which consists of the rebating of bankruptcy costs and licensing fees. From the representative household’s …rst-order condition, the equilibrium wage is determined as follows: Wt =
1
v v
Ct : 1 Nt
(27)
The endogenous pricing kernel is:
Mt+1 =
Ct+1 Ct
v(1
) 1
1 Nt+1 1 Nt
18
(1 v)(1
)
Ut+1 Et
1 Ut+1
: 1
(28)
2.5
Equilibrium
In equilibrium, the labor market must clear. From the household’s optimality condition we demand:
(1
Yt 1 v Ct = Wt = : Nt v 1 Nt
)
(29)
Goods market clearing requires that the sum of consumption and investment is equal to output: Ct + [Ktw |
(1 {z It
)Kt ] = Yt : }
(30)
Securities market equilibrium is pinned down by equations (25)and (28). Finally, …rms must optimize over their capital and leverage. The …rst-order condition pinning down the wished-for capital stock is:
Et
(
K Mt+1 Rt+1
"
1
(1
)
Z
#)
"t+1
"h(")d" + (
0
1)"t+1 (1
H("t+1 ))
= 1:
(31)
At an optimum, …rms equate the marginal product of capital with the unit price of capital. The …rst-order condition for pinning down the optimal face value (Bt+1 ) is:
(1
)Et [Mt+1 "t+1 h("t+1 )] =
1
1
Et [Mt+1 (1
H("t+1 ))]:
(32)
At an optimum, …rms equate marginal expected bankruptcy costs (left-side) with the tax bene…ts of debt (right-side). To solve the model we use the projection methods of Aruoba, et al. (2006) and Caldara, et al. (2012). To this end, one can begin by de…ning the following rescaled variables:
y = Y =z; c = C=z; i = I=z; g = U=z:
(33)
We approximate policy functions c(k; p); L(k; p); N (k; p); g(k; p) with Chebyshev polynomials. We use a grid for p with Np values. For each discrete value of p, we approximate the policy function by a one-dimensional Chebyshev polynomial. We evaluate …rst-order conditions at Nc Chebyshev
19
nodes to …nd coe¢ cients that minimize the residual function and Euler equation errors. Due to slow convergence, we …rst solve the model employing low values for Nc , Np and risk aversion. Using the previous solution, we solve the model for consecutively increasing parameter values. The policy functions can be used to calculate asset prices.
3
Model Simulation
This section begins with a discussion of the parameters chosen and model variations considered. We then move on to a presentation of simulation results.
3.1
Parameterization and Simulation Approach
Table 1 lists the parameter values. The values for the parameters ( ; ; v; ; ; ) are standard in the literature, e.g. Cooley and Prescott (1995). For the other parameter values we follow closely Gourio (2013). For example, we set the intertemporal elasticity of substitution of consumption (IES) equal to 2, an assumption consistent with the more recent asset pricing literature. Parameters of the disaster shock process are chosen so that the average probability of a disaster is approximately 2%. In particular, we set disaster risk is (
l
+
l
= 0:07% and
h )=2
h
= 3:9% so that the unconditional asymptotic expected
= 1:985%. The regime switching probability is set to s = 0:1, implying
an expected regime life of ten years. The capital and TFP disaster size parameters are both set to 15%, matching the averages in Gourio (2013). The cumulative distribution for the idiosyncratic depreciation shocks H is based upon a lognormal distribution with mean unity. In the model with homogeneous …rms we set the creditor recovery parameter to 0:7 as in Gourio (2013). In the model with heterogeneous …rms, half the …rms have recovery parameters equal to 0:6 and the other half have recovery parameters equal to 0:8. The tax shield
and volatility
parameters are chosen to roughly match the average default rate for BAA …rms (0.5%) and an average book leverage ratio of 55%. The average …rm in the model thus has characteristics similar to a …rm rated BAA. The resulting calibrated tax subsidy to debt funding is 3:3 cents per dollar of debt capital raised. This is not a bad approximation of reality inasmuch as a corporation with a debt yield of 8% facing a combined federal and state corporate income tax rate of 40% captures a
20
tax savings equal 3:2 cents per dollar of debt capital raised. To compute credit spreads we add a measure-zero fringe of AAA …rms which have no in‡uence on the economy. We choose the tax shield value for this fringe of …rms (
aaa )
to replicate the
average unconditional default rate for AAA …rms, 4 bps, and average leverage of 45%. We run 1000 simulated histories, with each simulated history lasting 1000 periods. We discard the …rst 100 periods in each simulated history in order to reduce the importance of date-zero beliefs. The initial prior belief is set at one-half. One of the goals in the numerical analysis is to get a sense of the role played by the various causal mechanisms at work in the model. This is important in as much as many prior disaster risk models struggle to match real-world moments under plausible parameterizations. To understand the role played by each model element, we consider alternative variants of the model, recalling that the full model features heterogeneous levered …rms who learn about disaster risk. Thus, the causal channels in the full model are learning, …rm heterogeneity, and optimal responses to …nancial market imperfections. To get a sense of the role played by the learning mechanism, we consider …rst the behavior of …rms in a so-called Non-Learning economy in which the true probability of a disaster is an observable twostate Markov process with identical parameters to the hidden Markov process described above. This model variation is most similar to that of Gourio (2013), since he assumes the objective probability of a disaster is common knowledge. In order to gauge the e¤ect of the learning mechanism, the behavior of the Non-Learning …rms can then be compared with that of otherwise identical …rms in the Learning economy. Within the two broad categories of Non-Learning and Learning …rms, three variations are considered. First, to get a sense of the role played by violations of the Modigliani-Miller assumptions, we …rst consider …rms embedded in the respective economies who do not face any …nancing frictions and so choose to …nance with equity. Second, …nancial market imperfections are introduced into the Non-Learning and Learning economies, but …rms are assumed to be homogeneous in terms of their lender recovery parameters. Finally, heterogeneous recovery parameters are introduced.
21
3.2
Financing and Investment
Table 2 reports business cycle statistics from real-world data (…rst row) in relation to the corresponding statistics generated under the alternative model variations. Data on GDP, consumption, employment, and real investment are sourced from the Federal Reserve (FRED). If one …rst compares the second row in the table (Non-Learning All-Equity) with the …nal row (Learning, Levered, Heterogeneous), it is apparent that the three mechanisms at work in the full model, learning, …nancing frictions, and heterogeneity, lead to a signi…cant ampli…cation of volatility in consumption and income, as well as a non-trivial increase in the volatility of investment. However, the contribution of each mechanism is distinct, and so it will be useful to unpack them. To understand the contribution of each causal mechanism, it is interesting to note that the learning mechanism by itself actually tends to reduce macroeconomic volatility unconditionally. To see this, note that each of the four macroeconomic series is a bit less volatile if one compares All-Equity …rms in the Non-Learning Economy with All-Equity …rms in the Learning Economy. Intuitively, agents in the Non-Learning economy can be viewed as having more extreme belief ‡uctuations in that the state variable p alternates between 0 and 1 for them. After all, in the Non-Learning Economy the true disaster risk regime is observed each period. In contrast, agents in the Learning economy form less extreme beliefs, with p falling in the interval [p; p]: Consider next the role of …nancial market imperfections illustrated by Table 2. Notice, within each class of model, Non-Learning and Learning, …nancial frictions amplify macroeconomic volatility, a variant of the …nancial accelerator. In the present setting, the presence of the tax subsidy to debt causes corporations to overinvest relative to a perfect …nancial market. On the other hand, the cost of external capital is volatile since bankruptcy costs are especially sensitive to systematic risk factors. Finally, it is apparent from Table 2 that introducing heterogeneity into …rm technologies further ampli…es …nancial accelerator e¤ects. Worthy of particular note is the fact that the introduction of heterogeneity increases employment volatility by 42% in the Learning Economy. Table 3 reports unconditional moments for expected asset returns, default rates, loss-givendefault and leverage. The real-world leverage and default probability data are taken from Chen, et al. (2009). Credit spreads are from FRED. As shown, the model tends to overshoot observed leverage ratios. This is typical of many structural models featuring a strictly linear tax bene…t of 22
debt. Real-world frictions such as loss limitations in the tax code could be introduced to bring down model-implied leverage, but at the cost of considerable complexity. Further, the only assets booked by the simulated …rm are physical capital, whereas real-world …rms have other items entering the book value of capital, such as intangible assets, which results in lower book leverage. As shown in Table 3, the model has a slight tendency to overshoot the unconditional default probability and to undershoot the unconditional expected loss-given-default. Consistent with a main argument made by Gourio (2013), it is apparent that the introduction of systematic disaster risk is su¢ cient to generate extremely large corporate bond risk premia and large corporate bond yield spreads in relation to expected losses in the event of default. For example, multiplying the probability of default by the loss-given-default one arrives at an expected loss of approximately 20 bps in the two models. However, yield spreads amount to approximately 200 bps. Intuitively, in the model lender losses are concentrated during bad macroeconomic periods, resulting in high riskpremia. In fact, it is apparent that under the stated parameterization the model actually overshoots credit spreads. However, this simply means that the model is capable of replicating observed spreads under parameterizations featuring much less severe disaster shocks, with the severity of disaster shocks being a prime point of criticism of existing disaster risk models. A novel feature of the model relative to Gourio (2013) is the fact that it generates secular trends in leverage ratios resembling those observed during the post-war period. Table 4 considers the Learning Economy, exhibiting simulated model moments for leverage, capital, labor, and consumption, conditional on years since the last disaster. Table 4 shows that successively longer quiet periods induce the simulated …rms to increase their capital stocks. Intuitively, as shown in Figure 2, the longer the elapsed interval since the last disaster, the more favorable are agent beliefs. More favorable beliefs imply higher expected marginal product of capital. Of particular interest in Table 4 is the fact that the capital accumulation of …rms with low recovery parameters (
L)
is particularly responsive to the time-since-disaster variable. Intuitively,
this subset of …rms is especially concerned about bankruptcy states, and the pricing kernel in such states, due to their low recovery parameters. It follows that lower assessed probabilities of disasters have a greater quantitative e¤ect on the marginal product of capital for such …rms. In fact, moving from the …rst to the last column in Table 4, the capital stock increases by 7.6% for the typical …rm.
23
However, the capital stock of the high recovery …rms only increases by 6.3% while that of the low recovery …rms increases by 8.9%. That is, during a prolonged quiet period, the model implies that one should expect to see a shift in aggregate investment towards “riskier” sectors, with no lack of prudence implied. Consider next the behavior of leverage ratios as shown in Table 4. The book leverage ratio of the simulated …rms is particularly responsive to the time-since-disaster variable. And this leverage uptick occurs despite the large increase in the capital stock denominator. Apparently, …nancing policy is more sensitive to the time-since-disaster variable than real investment. It is particularly instructive to contrast the behavior of leverage ratios for the simulated samples of homogeneous versus heterogeneous …rms. It is apparent that the book leverage ratios of the heterogeneous …rms have a higher slope with respect to time-since-disaster. In particular, with homogeneous …rms, there is a 23% increase in leverage as one moves from the …rst to last column. In contrast, with heterogeneous …rms, the increase amounts to 35%. Thus, cross-sectional heterogeneity may be understood as playing an important part in amplifying leverage trends. Further inspection of the simulated heterogeneous …rms reveals that this pattern is due to the behavior of …rms with low recovery parameters. In particular, …rms with low recovery parameters exhibit leverage ratios that are particularly responsive to time-since-disaster, increasing by 73% moving from the …rst to last columns. Intuitively, in making their decisions, this subset of …rms is particularly concerned about bankruptcy states, and the pricing kernel in such states, due to their low recovery parameters. It follows that lower assessed probabilities of disasters have a greater quantitative e¤ect on marginal expected bankruptcy costs for such …rms, implying higher leverage slopes. Thus, the model predicts that during a period like the Great Moderation, …rms will rationally increase their real capital stocks and book leverage ratios, with the most aggressive reactions exhibited by those …rms with higher bankruptcy costs. Having considered …rm behavior during tranquil periods, we turn next to their responses to the realization of disaster shocks. In the immediate aftermath of the Crisis of 2007/8, policymakers expressed surprise with the sheer magnitude of the decline in real and …nancial activity. To explain the magnitude of the downturn, many policymakers made appeals to “a clogged …nancial system” and “bank reluctance to lend” (see e.g. Geithner (2014)). Related to this story, some academics
24
have argued that introducing time-varying intermediation spreads, resulting from bank balance sheet e¤ects, might be useful in explaining the observed large negative response. While there is surely some truth to these narratives, it is worthwhile to use the model in order to evaluate how large a response one would expect in the absence of such channels. Notice, there is no “supply of intermediated credit” mechanism operative in our model, and yet, as will be shown, it delivers the type of large negative responses that many attributed to a collapse in intermediation. In our model, e¤ects operate through changes in aggregate risk-premia as opposed to intermediary-demandedpremia. As shown in Figure 3, negative belief revisions should be especially large if a long time interval has elapsed since the preceding disaster shock. Thus, we would expect that a rational response to the end of a prolonged tranquil period will be sharp changes in real and …nancial policies. To see the argument at a quantitative level, consider Tables 5 and 6. Both tables evaluate …rm behavior in response to a disaster shock, as a function of time-since-disaster, with the former focusing on levels and the latter focusing on percentage changes. Before analyzing the …ndings, we describe the sample construction. Within the simulated sample time-series there will be some time windows with exactly N years, say N=10, between two consecutive disasters. For each such window, we compute analytically the di¤erence in average …rm-level investment rates and leverage ratios for the year just prior to the disaster versus the year just after the disaster. We then average over all such sample windows. The tables reveal that time-since-disaster can be understood as a key conditioning variable in terms of predicting how …rms will respond to a disaster– but only in the Learning Economy. This is because a disaster realization leads to a large revision of beliefs in the Learning Economy, with concomitant implications for the marginal product of capital and the costs of debt capital. There is no such belief-revision mechanism operative in the Non-Learning Economy. In particular, a disaster realization by itself contains no new information in the Non-Learning Economy. Critically, one sees that the magnitudes of disinvestment and delevering in the Learning Economy increase dramatically with the length of the quiet period preceding the disaster shock. That is, a large decline in corporate sector demand for real capital and debt capital is to be expected if a negative shock hits an economy after a long tranquil period. Rather than being an indication of irrational panic or clogged debt
25
markets, large declines in investment and leverage can be understood as natural byproducts of rational Bayesian updating. To see that learning is the underlying mechanism at work here, note that the time-since-disaster variable plays an insigni…cant role in predicting the shock response in the Non-Learning economy. Finally, Tables 5 and 6 also reveal that the conjunction of learning and heterogeneity is particularly powerful as a disaster ampli…cation mechanism.
3.3
Asset Returns
Having considered the model’s implications for corporate …nancing and investment, we turn now to a discussion of asset pricing implications. We begin …rst with an analysis of model predictions regarding the behavior of asset pricing during prolonged quiet periods. Table 7 reports expected returns, expected default rates, expected loss-given-default, credit spreads and expected yield spreads. The data are grouped by time-since-prior-disaster. For example, the middle column (“10”) reports moments for those simulated sample years where exactly ten years have passed since a disaster. Note, we report expected returns demanded by the representative household, computed analytically, rather than realized returns. A number of predictions stand out. First, we see a striking decline in the equity risk premium as the quiet period lengthens. For example, in the Learning Economy with homogeneous …rms, the typical equity risk premium declines by 78 basis points as the time-since-disaster lengthens from 1 to 20 years. By contrast, in the Learning Economy with heterogeneous …rms, the equity risk premium declines by 543 basis points as the time-since-disaster lengthens from 1 to 20 years. Thus, …rm heterogeneity can have an important e¤ect on the dynamics of equity premia. It is also worth noting that this e¤ect is present despite the fact that the simulated …rms increase their debt-to-equity ratios during such tranquil periods. In particular, the heterogenous …rms increase market debt-to-equity ratios from 30.45% to 53.87% on average which would otherwise drive up equity risk premia. Thus, the decline in equity premia must be understood as arising from a sharp decline in the unlevered cost of capital. Similarly, sharp declines in risk premia demanded by the representative household must explain the decline in credit spreads despite the increase in default probabilities and expected loss-given-default apparent in the table. Another interesting, yet more subtle, quantitative prediction is the rise in expected loss-given-
26
default as the quiet period lengthens. Here we recall that it is the …rms with low creditor recovery parameters who lever up most aggressively in response to lengthy periods without a disaster. This leads to a composition e¤ect in which average loss-given-default is more heavily in‡uenced by …rms with low intrinsic creditor recovery parameters. Having analyzed risk premia and credit spreads during tranquil periods, we turn next to the e¤ects of disaster realizations. An obvious question post-Crisis is how expected returns demanded by rational investors should change after such events. In our model, the realization of a disaster implies a negative revision of investor beliefs regarding the probability of disasters going forward. In a risk-neutral economy this would not lead to any change in expected returns. However, the fact that disasters are systematic events implies that risk-averse investors should also demand higher expected returns after the occurrence of a disaster. Tables 8 and 9 examine this causal mechanism in greater detail, showing how disaster realizations a¤ect expected returns on corporate equity and debt. The table reveals that time-since-prior-disaster is a key conditioning variable in terms of understanding the expected returns that will be demanded by rational investors–but only in the Learning Economy. And the learning e¤ect is further ampli…ed if one considers the realistic possibility of …rm-level heterogeneity. To see these arguments, consider …rst the levered equity risk premium. In the Non-Learning Economy, if a disaster occurs after a 20 year quiet period, the equity risk premium increases by only 20 (homogeneous) to 28 (heterogeneous) basis points. In contrast, in the Learning Economy, the equity risk premium increases by 389 basis points if …rms are homogeneous and by 518 basis points if they are heterogeneous. This increase in the equity risk premium is even more striking given that …rms de-lever after the realization of a disaster. A similar pattern is observed if one instead inspects expected excess returns on corporate bonds over the risk-free rate and the yield spread between BAA and AAA corporate bonds. Thus, the conjunction of learning, frictions and heterogeneity can be understood as critical to understanding sharp ‡uctuations in investment, …nancing, spreads, and risk-premia. Finally, Figure 4 demonstrates the simulated statistical distribution of loss-given-default within the heterogeneous …rm model-generated time-series. The three panels of the …gure illustrate the e¤ect of time-since-disaster on the distribution of loss-given-default. For example, the middle panel consists of those sample time windows meeting the criteria of exactly ten years having elapsed
27
between the prior disaster and the current year. It should be emphasized that there is not necessarily a disaster in the current year. Indeed, for the majority of observations in the simulated sample there will not be a disaster. A number of interesting observations emerge. First, we see that the mean loss-given-default actually increases with the time since prior disaster. Again, this is due to the fact that all …rms, but especially those with low recovery parameters, take on higher debt face values in response to quiet periods, implying lower recoveries in the event of default. Second, we see that as the time since prior disaster increases, the distribution of loss-given-default becomes increasingly bi-modal, with fatter right tails corresponding to greater probability mass on high loss credits. This is due to the fact that during quiet periods the composition of credit risk shifts, with a higher percentage of corporate debt accounted for by the borrowing of …rms with low intrinsic recovery technologies. In other words, the change in the statistical distribution of loss-given-default is driven by aggressive leveraging by
L
…rms as the state variable p increases. This pattern was
shown earlier in the conditional moments for book leverage. These patterns are reminiscent of the selection e¤ect discussed in Glover (2015). However, here the selection e¤ect is more subtle in that it is time-varying, due to the evolution of beliefs leading to changes in the relative importance of various credit types, here determined by deep recovery parameters.
4
Empirical Tests
As the preceding discussion indicates, the model generates a wide range of empirical predictions. Rather than conduct tests of predictions that are likely to be shared with other models, we instead focus on tests of particularly novel predictions generated by the model. A key advantage of having a fully dynamic model with stock and ‡ow variables resembling real-world data is that we can replicate in the simulated model the regressions that will be conducted using the real-world data. This is a more reliable way of generating null hypotheses than simple verbal plausibility arguments, especially in complex environments in which ratios are used as dependent variables, with multiple regressors employed. A particularly salient and novel feature of the model is the role played by the amount of time that has elapsed since a large negative shock. To the best of our knowledge, this variable has not
28
yet been employed in an extant corporate …nance literature that seeks to explain corporate leverage and investment decisions. Here we recall that in the model, on an unconditional basis, leverage is increasing in time-since-disaster, with the e¤ect being less pronounced for …rms with high intrinsic creditor recovery parameters. Intuition suggests that these variables may well play a similar role in explaining investment rates. Of course, it will be important to …rst ensure that within the simulated data these predictions hold conditional upon the inclusion of standard regressors since these same regressors will be employed in the real-world regressions.
4.1
Data
We collect …nancial statement data from the Compustat-CRSP Merged Database. Following the literature, we remove all regulated (SIC 4900-4999) and …nancial …rms (6000-6999). Observations with missing entries for SIC codes, total assets, gross capital stock, market value, long-term debt, debt in current liabilities, and cash and short-term investments are excluded. We also require that …rms have at least two consecutive years of data because we need to lag some of the variables. We focus on the time period between 1950 and 2013 at the annual frequency. Data de…nitions are standard, and we follow Rajan and Zingales (1995). We treat NBER dated recessions as negative macroeconomic shocks. The variable Years-SinceRecession (YSR) is set to 0 if a year belongs to an NBER recession period, with the variable increasing by 1 each year after a recession year. Each recession resets the YSR variable to 0. While alternative proxies for negative events exist, NBER recessions represent a relatively standard proxy for negative macroeconomic shocks. The only other non-standard data item employed is recovery rates on defaulted debt. Here we utilize recovery rates on defaulted debt as the empirical corollary of the
4.2
parameter. Recovery rates are from Altman (1996) at the two-digit SIC industry level.
Findings
In the …rst set of regressions, we focus on novel predictions of the model in terms of explaining book leverage. To begin, we run mimicking regressions using the simulated model data in Table 10. The dependent variable is book leverage. The …rst reported simulated regression features a standard regression of leverage on Tobin’s q and cash ‡ow normalized by assets. The next reported regression
29
adds a years-since-disaster (YSD) variable to the standard leverage regressors. This variable enters with a positive coe¢ cient, consistent with the unconditional correlation between leverage and YSD discussed in Section 3. The …nal column adds an interaction between the YSD variable and the …rm-level creditor recovery parameter. Consistent with the unconditional correlation discussed in Section 3, this interaction term enters with a signi…cant negative coe¢ cient. Thus, in the simulated data one sees that even conditionally, leverage increases in years-since-disaster, with the e¤ect being less pronounced for …rms with high creditor recovery parameters. We now test these model-implied predictions in the real-world data. Table 11 reports the results when we add a years-since-recession variable and the YSRxRecovery interaction variable to standard leverage regressions. The …rst columns feature standard conditioning variables designed to capture real-world heterogeneity of the sort absent from the simulated model data. The last two columns feature tests of the model’s novel predictions. Consistent with the simulated model regressions, the penultimate column shows that years-since-recession enters with a signi…cant positive coe¢ cient. That is, …rms lever-up as the time-since-recession increases. In the …nal column, we see that the coe¢ cient on years-since-recession remains positive and signi…cant, while the interaction term enters with a signi…cant negative coe¢ cient. Apparently, …rms with low creditor recovery technologies respond most aggressively to time-since-recession. In the next set of regressions, we focus on novel predictions of the model in terms of explaining investment rates. To begin, we run mimicking regressions using the simulated model data in Table 12. The dependent variable is the investment rate. The …rst reported simulated regression results correspond to standard regressions of the investment rate on Tobin’s q and cash ‡ow normalized by assets. Both variables enter with positive coe¢ cients, with the coe¢ cient on q being particularly high in the simulated data, perhaps due to the fact that the simulated data is not contaminated by measurement error. The second reported regression using the simulated data adds a yearssince-disaster variable to the …rst two standard regressors. This variable enters with a positive coe¢ cient, consistent with the unconditional correlations discussed in Section 3. The …nal column adds an interaction between the years-since-disaster variable and the …rm-level creditor recovery parameter. Consistent with the unconditional model correlation, this interaction term enters with a negative coe¢ cient. Thus, in the simulated data one sees that even conditionally, the investment
30
rate increases in years-since-disaster, with the e¤ect being less pronounced for …rms with high creditor recovery parameters. We now test these model-implied predictions regarding investment rates in the real-world data. Table 13 reports the results when we add a years-since-recession variable and an interaction term to standard investment regressions. As shown in the …rst column, investment is increasing in Tobin’s q and normalized cash ‡ow, standard …ndings. The next columns add standard conditioning variables designed to capture real-world heterogeneity in investment drivers of the sort absent from the simulated model data. The …nal two columns in the table feature tests of the model’s novel predictions regarding investment. Consistent with the model, the penultimate column shows that years-since-recession enters with a signi…cant positive coe¢ cient. Apparently, investment rates increase as the amount of time since the prior recession increases. In the …nal column, we see that the coe¢ cient on years-since-recession remains positive and signi…cant, while the interaction term enters with the predicted negative sign, but is insigni…cant.
5
Conclusion
This paper develops a dynamic stochastic general equilibrium model featuring disaster risk and …nancial market frictions. We depart from the prior literature in assuming that the objective probability of disaster cannot be directly observed, forcing agents to instead form rational inferences about the latent risk state based upon the historical sequence of events. A contribution of the model is to show how learning and …nancial frictions can be readily integrated in a transparent and tractable way into canonical DSGE models used in the asset pricing literature. The model sheds light on recent leverage cycles. During periods absent disasters, equity premia decrease, credit spreads decrease, and leverage ratios increase, especially amongst …rms with high bankruptcy costs. Time since prior disasters is the key model conditioning variable. In response to a disaster, risk premia increase sharply while …rms shed labor, capital and leverage, with response size increasing in time since prior disasters. Disaster responses are more pronounced than in an otherwise equivalent economy featuring observed disaster risk. Further, business cycles are more pronounced than in an otherwise equivalent economy with frictionless …nancing. Firms with low
31
recovery parameters are the most sensitive to time-since-disaster. Using the simulated model as a laboratory we …rst run mimicking regressions in order to generate novel empirical predictions. In the simulated data, leverage ratios and investment rates vary positively with time-since-disasters, with the e¤ect attenuated for …rms with high creditor recovery parameters. Empirical tests o¤er support for these novel predictions.
32
References Alti, Aydogan, 2003, How Sensitive Is Investment to Cash Flow When Financing Is Frictionless?, The Journal of Finance 58, 707–722. Altman, Edward I., and Vellore M. Kishore, 1996, Almost everything you wanted to know about recoveries on defaulted bonds, Financial Analysts Journal 52, 57–64. Aruoba, S. Boragan, Jesus Fernandez-Villaverde, and Juan F. Rubio-Ramirez, 2006, Comparing solution methods for dynamic equilibrium economies, Journal of Economic Dynamics and Control 30, 2477–2508. Barro, Robert, 2009, Rare Disasters, Asset Prices, and Welfare Costs, American Economic Review 99, 243–64. , and Tao Jin, 2011, On the size distribution of macroeconomic disasters, Econometrica 79, 1567–89. Barro, Robert, and Jose Ursua, 2008, Macroeconomic crisis since 1870, Brookings Papers on Economic Activity. Benzoni, Luca, Pierre Collin-Dufresne, and Robert S. Goldstein, 2011, Explaining asset pricing puzzles associated with the 1987 market crash, Journal of Financial Economics 101, 552–573. Bernanke, Ben, 2004, The great moderation, Speech at the meetings of the Eastern Economic Association, Washington, DC, Federal Reserve Board. Caldara, Dario, Jess Fernandez-Villaverde, Juan F. Rubio-Ramrez, and Wen Yao, 2012, Computing dsge models with recursive preferences and stochastic volatility, Review of Economic Dynamics 15, 188–206. Chen, Long, Pierre Collin-Dufresne, and Robert S. Goldstein, 2009, On the relation between the credit spread puzzle and the equity premium puzzle, Review of Financial Studies 22, 3367–3409. Cooley, Thomas, and Edward Prescott, 1995, Frontiers of Business Cycle Research (Princeton University Press). 33
Epstein, Larry G., and Stanley E. Zin, 1989, Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework, Econometrica 57, 937–969. Gabaix, Xavier, 2013, Variable rare disasters: An exactly solved framework for ten puzzles in macro-finance, Quarterly Journal of Economics 127, 645–700. Geithner, Timothy, 2014, Stress Test: Reflections on Financial Crises (Random House). Glover, Brent, Forthcoming, The expected cost of default, Journal of Financial Economics. Goldstein, Robert, Nengjiu Ju, and Hayne Leland, 2001, An ebit-based model of dynamic capital structure, The Journal of Business 74, 483–512. Gourio, Francois, 2013, Credit risk and disaster risk, American Economic Journal: Macroeconomics 5, 1–34. Graham, John R., Mark T. Leary, and Michael R. Roberts, 2014, A century of capital structure: The leveraging of corporate america, Journal of Financial Economics. Hennessy, Christopher A, and Toni M Whited, 2005, Debt Dynamics, The Journal of Finance 60, 1129–1165. Koulovatianos, Christos, and Volker Wieland, 2011, Asset pricing under rational learning about rare disasters, Working paper. Leary, Mark T., and Michael R. Roberts, 2010, The pecking order, debt capacity, and information asymmetry, Journal of Financial Economics 95, 332 – 355. Lu, Yang K., and Michael Siemer, 2014, Learning, rare disasters, and asset prices, Working paper. Moyen, Nathalie, and Stefan Platikanov, 2012, Corporate investments and learning, Review of Finance. Nakamura, Emi, Jon Steinsson, Robert Barro, and Jose Ursua, 2013, Crises and recoveries in an empirical model of consumption disasters, American Economic Journal: Macroeconomics 5, 35– 74.
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P´astor, Lubos, and Pietro Veronesi, 2009, Learning in Financial Markets, Annual Review of Financial Economics 1, 361–381. Rajan, Raghuram G., and Luigi Zingales, 1995, What do we know about capital structure? some evidence from international data, Journal of Finance 50, 1421–1460. Rietz, Thomas, 1988, The equity risk premium a solution, Journal of Monetary Economics 22, 117–131. Stiglitz, Joseph, 2010, Freefall: America, Free Markets, and the Sinking of the World Economy (W. W. Norton and Company). Wachter, Jessica A., 2013, Can time-varying risk of rare disasters explain aggregate stock market volatility?, Journal of Finance 68, 3987–1035.
35
Figure 1: The Evolution of State Variable p The figure presents the evolution of state variable, belief p.
Panel A 0.9 0.8 0.7 Belief [p]
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
200
400
600 Time
800
1000
1200
800
1000
1200
Panel B 1.0 0.9 0.8
Belief [p]
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
200
400
600 Time
36
Figure 2: The Dynamics of the State Variable p The figure plots a probability of being in λl regime p after consecutive series of no disasters. Prior belief is set to 0.5 Panel A
0.58
0.57
Probability of 6 l regime, p
0.56
0.55
0.54
0.53
6 h =0.066, 6 l =0.01, s=0.1 6 h =0.056, 6 l =0.02, s=0.1
0.52
0.51
0.5
2
4
6
8
10
12
14
16
18
20
Number of periods with consecutive no disasters
Panel B
0.8
s=0, 6 h =0.066, 6 l =0.01 s=0.2, 6 h =0.066, 6 l =0.01 s=0.4, 6 h =0.066, 6 l =0.01
Probability of 6 l regime, p
0.75
0.7
0.65
0.6
0.55
0.5
2
4
6
8
10
12
14
16
Number of periods with consecutive no disasters
37
18
20
Figure 3: The Change in the State Variable p after a Disaster The figure plots a change in probability of being in λl regime p after a disaster as a function of a time-since-prior-disaster. Prior belief is set to 0.5 Panel A
-0.22
-0.24
-0.26
Change in Prob (p)
-0.28
6 h =0.066, 6 l =0.01, s=0.1 6 h =0.056, 6 l =0.02, s=0.1
-0.3
-0.32
-0.34
-0.36
-0.38
-0.4
-0.42
2
4
6
8
10
12
14
16
18
20
Number of periods with consecutive no disasters
Panel B
-0.35
-0.4
s=0, 6 h =0.066, 6 l =0.01 s=0.2, 6 h =0.066, 6 l =0.01 s=0.4, 6 h =0.066, 6 l =0.01
Change in Prob (p)
-0.45
-0.5
-0.55
-0.6
-0.65
-0.7
-0.75
-0.8
2
4
6
8
10
12
14
16
Number of periods with consecutive no disasters
38
18
20
Figure 4: Conditional Loss-Given-Default Distributions The figure plots loss-given-default distributions conditional on time-since-disaster for Learning Heterogeneous model
% of observations
Time−since−disaster=1 10
5
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
% of observations
Time−since−disaster=10 8 6 4 2 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
% of observations
Time−since−disaster=20 6 4 2 0 0.1
0.2
0.3
0.4
0.5
39
0.6
0.7
0.8
Table 1: Parameter Values The table reports parameter values used in model numerical solution and simulations Description Recovery rate for average firm Recovery rate for bad firm Recovery rate for good firm Average probability of disaster Probability of disaster in the good regime Probability of disaster in the bad regime Average disaster size Switching probability Tax subsidy Tax subsidy for a fringe of AAA firms Elasticity of capital Depreciation rate Share of consumption in utility Discount factor Trend growth of aggregate shock Standard deviation of aggregate shock Trend growth of idios. shock Persistence of idios. shock Standard deviation of idios. shock IES Risk aversion
40
Parameter
Value
θ
0.7 0.6 0.8 0.02 0.007 0.039 15% 0.1 0.033 0.063 0.3 0.08 0.3 0.987 0.01 0.015 0.01 0.767 0.015 2 6
θL θH λa λl λh b s χ-1 χaaa -1 α δ v β µz σe µx ρx σε 1/ψ γ
Table 2: Business Cycle Statistics The table reports annual volatility of the growth rates of investment, consumption, hours and output measured in the data, Learning and Non-Learning models. All-Equity models are absent financial frictions
Non-Learning
Learning
σ(∆log(Y))
σ(∆log(C))
σ(∆log(Inv))
σ(∆log(N))
Data
2.78
1.81
7.01
2.67
All-equity Homogeneous Heterogeneous All-Equity Homogeneous Heterogeneous
3.39 5.44 5.60 3.24 5.47 5.61
3.84 5.38 5.57 3.78 5.21 5.43
5.74 8.11 8.46 5.63 6.43 6.64
0.85 1.41 1.58 0.46 0.51 0.72
Table 3: Asset Returns and Leverage Statistics The table reports unconditional moments for asset returns and leverage statistics. The simulated moments are presented for both Heterogeneous and Homogeneous Learning models
E(Book Leverage), % E(Default Prob), bp E(LGD), % E(Rf ), % E(Rc − Rf ), % E(Re − Rf ), % E(y − yAAA ), %
Data
Heterogeneous
Homogeneous
45 50 45 1.6 0.8 5.6 0.94
63.98 59.42 34.97 1.43 2.02 6.48 1.94
62.49 57.93 32.83 1.41 1.98 6.37 1.90
41
Table 4: Book Leverage: Quiet Periods - Learning The table reports moments for book leverage, capital and labor conditional on time-since-disaster. All moments are shown for both Homogeneous and Heterogeneous Learning models. The book leverage moment is shown for Learning Heterogeneous model with a breakdown for θL and θH firms.
1
Time-since-disaster 5 10 15
20
E(Book Leverage), %
Homog Heterog θL θH
51.75 46.01 31.79 50.11
60.89 53.02 37.93 61.33
62.98 57.06 47.51 63.91
63.63 60.03 50.37 64.23
63.69 62.02 54.86 65.44
Capital
Homog Heterog θL θH
16.35 16.35 16.02 16.68
16.60 16.60 16.34 16.86
17.01 17.01 16.82 17.2
17.43 17.43 17.26 17.6
17.59 17.59 17.45 17.73
Labor
Homog Heterog
0.28 0.29
0.29 0.30
0.30 0.31
0.30 0.32
0.30 0.32
C/Y
Homog Heterog
0.68 0.69
0.70 0.70
0.71 0.71
0.72 0.72
0.73 0.73
42
Table 5: Book Leverage: Response to a Disaster - Change in Levels The table reports moments for book leverage, capital and labor conditional on time-since-disaster. All moments are shown for both Homogeneous and Heterogeneous Learning models and Homogeneous Non-Learning model. The book leverage moment is shown for Learning Heterogeneous model with a breakdown for θL and θH firms.
Change after disaster:
1
Time-since-disaster 5 10 15
20
E(Book Leverage), %
Homog Heterog θL θH Non-Learning
-1.40 -3.00 -2.60 -3.40 -0.01
-7.00 -7.49 -6.83 -8.15 -0.04
-8.24 -9.24 -7.83 -10.65 -0.04
-8.94 -9.94 -7.75 -12.13 -0.05
-9.01 -10.78 -8.01 -13.55 -0.05
Capital
Homog Heterog θL θH Non-Learning
-0.77 -0.82 -0.85 -0.79 -0.75
-0.78 -0.82 -0.87 -0.79 -0.75
-0.80 -0.85 -0.89 -0.81 -0.77
-0.82 -0.87 -0.92 -0.82 -0.79
-0.83 -0.89 -0.95 -0.83 -0.81
Labor
Homog Heterog Non-Learning
-0.001 -0.001 -0.001
-0.002 -0.002 -0.001
-0.002 -0.003 -0.002
-0.003 -0.004 -0.003
-0.003 -0.005 -0.004
43
Table 6: Book Leverage: Response to a Disaster - Percentage Change The table reports moments for book leverage, capital and labor conditional on time-since-disaster. All moments are shown for both Homogeneous and Heterogeneous Learning models and Homogeneous Non-Learning model. The book leverage moment is shown for Learning Heterogeneous model with a breakdown for θL and θH firms.
Change after disaster:
1
Time-since-disaster 5 10 15
20
E(Book Leverage), %
Homog Heterog θL θH Non-Learning
-3% -7% -8% -7% -0.02%
-11% -14% -18% -13% -0.06%
-13% -16% -16% -17% -0.06%
-14% -17% -15% -19% -0.08%
-14% -17% -15% -21% -0.08%
Capital
Homog Heterog θL θH Non-Learning
-4.7% -5.0% -5.3% -4.7% -4.5%
-4.7% -5.0% -5.3% -4.7% -4.5%
-4.7% -5.0% -5.3% -4.7% -4.6%
-4.7% -5.1% -5.3% -4.7% -4.7%
-4.7% -5.1% -5.4% -4.7% -4.7%
Labor
Homog Heterog Non-Learning
-0.4% -0.3% -0.2%
-0.7% -0.7% -0.6%
-0.7% -1.0% -0.8%
-1.0% -1.3% -0.9%
-1.0% -1.6% -1.3%
44
Table 7: Asset Pricing Moments: Quiet Periods The table reports moments for asset pricing variables and book leverage conditional on time-sincedisaster. All moments are shown for both Homogeneous and Heterogeneous Learning models
1 E(Rf), % E(D/E) E(Book Leverage), % E(Default Prob), bp E(LGD), % E(Rc − Rf ), % E(yBAA -Rf ), % E(yAAA -Rf ), % E(yBAA -yAAA ), % E(Re − Rf ), %
Homog Heterog Homog Heterog Homog Heterog Homog Heterog Homog Heterog Homog Heterog Homog Heterog Homog Heterog Homog Heterog Homog Heterog
0.02 0.01 35.57 30.45 51.75 50.11 0.01 1.82 32.01 31.44 1.47 1.48 1.94 1.98 0.22 0.23 1.72 1.75 7.01 9.45
45
Time-since-disaster 5 10 15 0.98 0.84 39.75 35.95 60.89 61.33 2.83 3.46 32.57 32.4 2.08 2.09 1.79 1.82 0.21 0.21 1.58 1.61 6.4 7.01
1.46 1.34 41.48 43.51 62.98 63.91 9.18 13.38 32.78 33.12 2.09 2.11 1.59 1.61 0.20 0.20 1.39 1.41 6.32 5.03
1.63 1.44 43.29 49.72 63.63 64.23 12.13 22.87 32.84 33.76 2.11 2.12 1.36 1.38 0.19 0.19 1.17 1.19 6.27 4.31
20 1.63 1.45 45.88 53.87 63.69 65.44 12.74 29.85 32.85 34.03 2.14 2.15 1.32 1.33 0.18 0.17 1.14 1.16 6.23 4.02
Table 8: Asset Pricing Moments: Response to a Disaster - Change in Levels The table reports changes in moments for expected excess bond and equity returns, spreads between an average firm and hypothetical AAA firm. All moments are shown for Non-Learning and Learning models Time-since-disaster 1 5 10
Change after disaster in: E(Rc − Rf ), %
Learning Non-Learning
E(Re − Rf ), %
Learning Non-Learning
E(yBAA − yAAA ), %
Learning Non-Learning
Homog Heterog Homog Heterog Homog Heterog Homog Heterog Homog Heterog Homog Heterog
46
0.04 0.06 0.01 0.01 0.17 0.22 0.14 0.20 0.13 0.17 0.01 0.01
0.05 0.07 0.01 0.01 1.88 2.51 0.15 0.21 0.39 0.51 0.02 0.02
0.06 0.08 0.01 0.01 3.34 4.45 0.18 0.25 0.59 0.76 0.02 0.02
15
20
0.07 0.09 0.01 0.01 3.86 5.15 0.18 0.25 0.67 0.87 0.02 0.02
0.08 0.10 0.01 0.01 3.89 5.18 0.20 0.28 0.68 0.88 0.02 0.02
Table 9: Asset Pricing Moments: Response to a Disaster - Percentage Change The table reports changes in moments for expected excess bond and equity returns, spreads between an average firm and hypothetical AAA firm. All moments are shown for Non-Learning and Learning models Time-since-disaster 1 5 10
Change after disaster in: E(Rc − Rf ), %
Learning Non-Learning
E(Re − Rf ), %
Learning Non-Learning
E(yBAA − yAAA ), %
Learning Non-Learning
Homog Heterog Homog Heterog Homog Heterog Homog Heterog Homog Heterog Homog Heterog
47
3% 4% 1% 1% 2% 2% 2% 2% 8% 9% 1% 1%
2% 3% 0% 0% 29% 36% 2% 3% 25% 28% 1% 1%
3% 4% 0% 0% 53% 88% 3% 5% 42% 47% 1% 1%
15
20
3% 4% 0% 0% 62% 119% 3% 6% 57% 63% 2% 2%
4% 5% 0% 0% 62% 129% 3% 7% 59% 66% 2% 2%
Table 10: Book Leverage Regressions: Model The table reports book leverage regressions output for Learning Heterogeneous model. YSD denotes years-since-disaster. Standard errors are in parenthesis. ***, **, and * indicate significance at the 1, 5, and 10% levels, respectively
Tobin’s q CashFlow/Assets
(1)
(2)
(3)
-89.14*** (17.36) 30.11*** (5.809)
-72.75*** (16.07) 24.76*** (5.379) 0.000307*** (2.84e-06)
-92.66*** (15.00) 31.92*** (5.018) 1.375*** (0.0134) -2.047*** (0.0200)
70,329 0.004
70,329 0.146
70,329 0.257
YSD YSD x Recovery
Observations R-squared
48
Table 11: Book Leverage Regressions: Data This table reports the effect of a years-since-recession variable and an interaction between the yearssince-recession and recovery in a standard leverage regressions. Variable definitions are provided in Appendix A.1. Standard errors are in parenthesis. ***, **, and * indicate significance at the 1, 5, and 10% levels, respectively
Tobin’s q CashFlow/Assets
(1)
(2)
(3)
(4)
(5)
(6)
-0.0240*** (0.000646) -0.0202*** (0.000449)
-0.0211*** (0.000646) -0.0208*** (0.000446) 0.0227*** (0.000593)
-0.0205*** (0.000643) -0.0200*** (0.000445) 0.0243*** (0.000593) 0.169*** (0.00581)
-0.0205*** (0.000643) -0.0200*** (0.000445) 0.0245*** (0.000617) 0.169*** (0.00581) -0.000770 (0.000704)
-0.0214*** (0.000646) -0.0200*** (0.000445) 0.0237*** (0.000620) 0.170*** (0.00581) -0.000737 (0.000703) 0.00262*** (0.000196)
-0.0214*** (0.000646) -0.0200*** (0.000445) 0.0237*** (0.000620) 0.170*** (0.00581) -0.000732 (0.000703) 0.00429*** (0.000948) -0.00379* (0.00210)
Y 110,401 0.603
Y 110,401 0.608
Y 110,384 0.612
Y 110,378 0.612
Y 110,378 0.612
Y 110,378 0.612
Size Tangibility Growth YSR YSR x Recovery Firm-FE Observations R-squared
49
Table 12: Investments Regressions: Model The table reports investments regressions output for Learning Heterogeneous model. YSD denotes years-since-disaster. Standard errors are in parenthesis. ***, **, and * indicate significance at the 1, 5, and 10% levels, respectively
Tobin’s q CashFlow/Assets
(1)
(2)
(3)
1.766*** (0.409) 0.102 (0.137)
2.213*** (0.368) -0.0435 (0.123) 8.36e-06*** (6.49e-08)
1.587*** (0.320) 0.182* (0.107) 0.0433*** (0.000286) -0.0644*** (0.000426)
70,329 0.973
70,329 0.978
70,329 0.984
YSD YSD x Recovery
Observations R-squared
50
Table 13: Investments Regressions: Data This table reports the effect of a years-since-recession variable and an interaction between the yearssince-recession and recovery in a standard investment regressions. Variable definitions are provided in Appendix A.1. Standard errors are in parenthesis. ***, **, and * indicate significance at the 1, 5, and 10% levels, respectively
Tobin’s q CashFlow/Assets
(1)
(2)
(3)
(4)
(5)
(6)
0.0196*** (0.000534) 0.00324*** (0.000354)
0.0191*** (0.000536) 0.00343*** (0.000354) -0.00583*** (0.000545)
0.0200*** (0.000526) 0.00447*** (0.000349) -0.00315*** (0.000538) 0.254*** (0.00497)
0.0200*** (0.000526) 0.00447*** (0.000349) -0.00337*** (0.000557) 0.254*** (0.00497) 0.000846 (0.000589)
0.0199*** (0.000528) 0.00447*** (0.000348) -0.00363*** (0.000562) 0.254*** (0.00497) 0.000851 (0.000589) 0.000602*** (0.000168)
0.0199*** (0.000528) 0.00446*** (0.000348) -0.00364*** (0.000562) 0.254*** (0.00497) 0.000856 (0.000589) 0.00193** (0.000827) -0.00299 (0.00183)
Y 80,655 0.444
Y 80,655 0.445
Y 80,648 0.465
Y 80,646 0.465
Y 80,646 0.465
Y 80,646 0.465
Size Tangibility Growth YSR YSR x Recovery Firm-FE Observations R-squared
51
