Review of Finance, Volume 22 (1) – Feb 1, 2018

25 pages

/lp/ou_press/learning-and-leverage-cycles-in-general-equilibrium-theory-and-UMLuVvOwQo

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- Oxford University Press
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- © The Authors 2016. Published by Oxford University Press on behalf of the European Finance Association. All rights reserved. For Permissions, please email: journals.permissions@oup.com
- ISSN
- 1572-3097
- eISSN
- 1573-692X
- D.O.I.
- 10.1093/rof/rfw058
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- See Article on Publisher Site

Abstract This article develops and empirically tests a tractable general equilibrium model of corporate financing and investment dynamics in a trade-off economy where heterogeneous firms face unobservable disaster risk and engage in rational Bayesian learning. 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 firms 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 finance predictions are conducted. Consistent with the model, we find empirically that leverage and investment are increasing in time-since-prior-recessions, with the effect more pronounced for firms with low recovery ratios. 1. Introduction 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, Leary, and Roberts (2014) document that aggregate corporate leverage more than tripled between 1945 and 1970, rising from 11 to 35%. While the literature in corporate finance has tended to focus on cross-sectional factors predicting firms’ 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 firms facing very costly bankruptcy, followed by a severe credit crisis has led 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 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, for example, Stiglitz (2010), or worse still suffered under the fallacy of induction and concluded that major recessions were a thing of the past, for example, Taleb (2007). A valid test of rationality must filter economic time series just as agents in the economy do, in real time, not through the rear-view mirror. To this end, this article considers a DSGE setting in which firms 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. Most importantly, we depart from an extant literature discussed below in assuming that agents do not know the objective probability of “disasters”—large negative aggregate shocks to total factor productivity and productive capital. Instead, they engage in Bayesian updating of beliefs based upon the realized history of disasters. In addition to facing uncertainty about the objective probability of disasters, firms face financial market imperfections in the form of tax benefits of debt and privately (not socially) costly bankruptcy, with bankruptcy costs being heterogeneous across firms. 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. During long periods sans disasters, for example, the Great Moderation discussed by Bernanke (2004), agents in the model economy revise upwards their belief regarding the probability of low disaster risk. Risk premia fall and the expected marginal product of capital rises, causing firms to increase investment. Credit risk premia and default probabilities also fall, causing firms to lever-up, with the most aggressive levering done by firms with high loss-given-default. Despite the increase in leverage, the average yield spread relative to AAA credit declines, reflecting a decline in default risk at each leverage level, as well as the 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 firms with low intrinsic recovery technologies.1 The model also generates interesting and plausible qualitative and quantitative predictions regarding responses to disaster realizations. 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 is often argued that existing disaster risk models fail to match observation in that they are unable to deliver large swings in real and financial variables in response to negative shocks of plausible magnitude. In contrast, we show that the conjunction of learning, financing frictions, and firm heterogeneity greatly amplifies response magnitudes under otherwise equivalent calibrations. Intuitively, it should come as no surprise to see large changes in financing, investment, risk premia, and credit spreads when a large negative shock occurs after a prolonged quiet period. After all, a large negative shock after a prolonged tranquil period will cause Bayesian learners to increase substantially their assessment of future disaster risk. Financial frictions then have an accelerator effect, with the response being most pronounced for firms with high intrinsic loss-given-default. We take the model to the data, contributing two novel predictions to the empirical corporate finance literature on firm leverage ratios and investment rates. We begin first with mimicking regressions performed on simulated model data, finding that leverage ratios and investment rates are increasing in years-since-disaster (YSD), and decreasing in the interaction between YSD and recovery rate parameters. Intuitively, beliefs should become more favorable as time-since-disaster increases, with firms responding by increasing leverage and investment, with the effect being particularly strong for those firms with low recovery parameters. We use these simulated model regressions as the basis for conjecturing that similar patterns will be observed in the real-world data. Indeed, we find that firm leverage is increasing in a time-since-recession variable, but decreasing in an interaction between this variable and industry-level recovery ratios. We also find that investment rates are increasing in time-since-recession, while the interaction variable is negative but statistically insignificant. We turn now to related literature. Learning has received little attention in the corporate finance 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 risk premia being a central mechanism. Second, we consider that firms can jointly set optimal leverage ratios and investment rates, while they consider equity financed firms. Another paper in the dynamic corporate finance literature related to our own is recent work by Glover (2016), who considers the effect 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 influential contributions. The technological setup of our model is closest to that of Gourio (2013), and we build directly on his tractable general equilibrium framework. As in our article, 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 effect of cross-sectional heterogeneity in creditor recovery parameters. We argue that the conjunction of learning and heterogeneity can help in better explaining risk premia fluctuations and cross-sectional leverage ratio dynamics. The article 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 (Benzoni, Collin-Dufresne, and Goldstein, 2011; Koulovatianos and Wieland, 2011, and Lu and Siemer, 2014) 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 financial market imperfections and firm heterogeneity represent important amplification mechanisms. This article is organized as follows. The model is introduced in Section 2. Section 3 analyzes variations of the simulated model, focusing on firm leverage, credit risk, and risk premia. Empirical tests are contained in Section 4. Section 5 discusses policy implications and offers conclusions. 2. The Model This section describes the real and financial technologies available to firms, the nature of shocks, as well as the decision problems faced by firms and the representative household. We then discuss the determination of equilibrium. 2.1 Beliefs Each period there is a risk of a macroeconomic “disaster” taking the form of a negative shock to total factor productivity and the capital stock. Departing from Gourio (2013), the probability of disaster is not directly observable to any agent. Rather, at the start of each period t, all agents observe whether or not a disaster hit the prior generation of firms, with Bayes’ rule then used to form updated beliefs. The random variable xt is equal to one if a disaster hits firms born at date t − 1 and producing output at the start of date t, and is equal to zero if no disaster hits these firms. The objective probability of a disaster hitting these firms is λt≡Pr[xt=1]. The process λt is a hidden-state Markov process with the two potential states being λl and λh such that 0≤λl<λh≤1. The symmetric transition probability parameter of the λt process is s∈[0,1/2). Agents are rational and use the entire history 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 (λl) at that point in time based upon the history of disasters up to and including that point in time. We have: pt≡Pr[λt=λl|(xτ)τ=1τ=t]. (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∈(0,1). Further discussion of the prior follows below. Consider first the nature of belief updating conditional on no disaster shock taking place. From Bayes’ rule, we have: Pr[λt=λl|xt=0]=Pr[xt=0∩λt=λl]Pr[xt=0]. (2) The implied law of motion for the belief state variable is: xt=0⇒pt=(1−λl)[pt−1(1−s)+(1−pt−1)s](1−λl)[pt−1(1−s)+(1−pt−1)s]+(1−λh)[(1−pt−1)(1−s)+pt−1s]. (3) Provided that the prior period belief is not too high, the belief will be revised upwards after a period with no disaster taking place. Using the preceding law of motion, we have: If xt=0 and (1−2s)(λh−λl)pt−12+[2s(1−λl)−(1−s)(λh−λl)]pt−1≤(1−λl)s, then pt≥pt−1. (4) The second inequality in the preceding equation is strict if and only if the inequality preceding it is strict. Therefore, we conjecture an upper bound on beliefs, call it p¯, which solves the following quadratic equation: (1−2s)(λh−λl)p¯2+[2s(1−λl)−(1−s)(λh−λl)]p¯=(1−λl)s. (5) To take a particular example, if s = 0, then p¯=1. Consider next the nature of belief updating conditional on a disaster taking place. Applying Bayes’ rule, we have: Pr[λt=λl|xt=1]=Pr[xt=1∩λt=λl]Pr[xt=1]. (6) The implied law of motion for the belief state variable is: xt=1⇒pt=λl[pt−1(1−s)+(1−pt−1)s]λl[pt−1(1−s)+(1−pt−1)s]+λh[(1−pt−1)(1−s)+pt−1s]. (7) Given a sufficiently higher prior, beliefs will be revised downwards after a period in which a disaster shock has taken place. Using the preceding law of motion, we have: If xt=1 and [−(1−2s)(λh−λl)pt−12+[2sλl+(λh−λl)(1−s)]pt−1]≥λls, then pt≤pt−1. (8) The second inequality in the preceding equation is strict if and only if the inequality preceding it is strict. Therefore, we conjecture a lower bound on beliefs, call it p¯, which solves the following quadratic equation: −(1−2s)(λh−λl)p¯2+[2sλl+(λh−λl)(1−s)]p¯=λls. (9) To take a particular example, if s = 0, then p¯=0. The following remark offers summary observations based on the preceding analysis. Remark If agents enter the economy with the belief p0∈[p¯,p¯]with p¯as defined in Equation (9) and p¯as defined in Equation (5), then they will revise beliefs upward in response to each period sans disaster and downward in response to each observed disaster. To ensure the economy has interesting dynamics, we assume all agents enter the model holding a belief p0∈(p¯,p¯). A belief falling into this interval ensures that with probability one agents will form revised beliefs p1≠p0 based on the first observed shock x1. That is, there will at least be some learning over some time interval if p0∈(p¯,p¯). For example, p0=1/2 meets the criterion of p0∈(p¯,p¯). Since beliefs are a central causal mechanism we wish to explore, it is useful to develop a better understanding of their behavior. On the vertical axis of Figure 1 is the state variable p and on the horizontal axis is the elapsed time since the previous disaster, starting at an initial belief p0=1/2. To illustrate the role played by the parameterization of the disaster risk process, the top panel considers variation in the disaster probability parameters while the bottom panel considers variation in the switching probability. As shown in the top panel, p is increasing and concave in the time-since-disaster. One sees that beliefs are more responsive to time-since-disaster the greater the wedge between λl and λh. On the other hand, as shown in the bottom panel, beliefs are less sensitive to time-since-disaster the higher the switching probability. Figure 1 View largeDownload slide The dynamics of the state variable p. Note: This figure plots the probability of being in the λl regime p after consecutive series of no disasters. Prior belief is set to 0.5. Figure 1 View largeDownload slide The dynamics of the state variable p. Note: This figure plots the probability of being in the λl regime p after consecutive series of no disasters. Prior belief is set to 0.5. We will be interested in understanding how the economy can be expected to respond to the realization of a disaster after a prolonged quiet period. To this end, consider Figure 2. On the vertical axis is the change in the state variable p following a disaster shock. On the horizontal axis is the time-since-prior-disaster. To illustrate the role played by the parameterization of the disaster risk process, the top panel considers variation in the disaster probability parameters while the bottom panel 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 comes as a greater surprise and leads to a larger change in beliefs. In the top panel, one sees that beliefs are more sensitive to the occurrence of a disaster the greater the wedge between λl and λh. The bottom panel shows that beliefs are less sensitive to the occurrence of a disaster the higher the switching probability. Figure 2 View largeDownload slide The change in the state variable p after a disaster. Note: This figure plots the change in probability of being in the λl regime p after a disaster as a function of a time-since-prior-disaster. Prior belief is set to 0.5. Figure 2 View largeDownload slide The change in the state variable p after a disaster. Note: This figure plots the change in probability of being in the λl regime p after a disaster as a function of a time-since-prior-disaster. Prior belief is set to 0.5. 2.2 Production There is a continuum of mass one of competitive price-taking firms born each period. For simplicity, assume firms 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 firm technologies. Ex ante firms are identical. Ex post, firms differ as a result of facing idiosyncratic shocks to their stock of productive capital as described below. Since firms are identical ex ante, they will adopt the same policies. Timing is as follows. Consider a firm 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 firm chooses a debt face value Bit+1 and a wished-for capital stock Kit+1w. The choice variable Kit+1w then maps stochastically to an effective capital stock as follows: Kit+1=Kit+1wεit+1(1−xt+1bk). (10) The effective capital stock is determined by both aggregate and idiosyncratic shocks. The random variable εit+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 differentiable cumulative density H having bounded support on the positive real line. Notice, if a disaster occurs, the effective capital stock of each firm is scaled down by the factor bk.2 Firms employ constant returns to scale Cobb–Douglas production functions with output: Yit+1=Kit+1α(zt+1Nit+1)1−α. (11) In the preceding equation, zt+1 is total factor productivity, Kit+1 is the effective capital stock, and Nit+1 is the labor input chosen after observing all shocks. Disasters also impact total factor productivity. The total factor productivity (TFP) process evolves as follows: log zt+1= log zt+μ+σet+1+ log (1−xt+1btfp). (12) 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 btfp captures the severity of negative productivity shocks resulting from disasters.3 After the periodic shocks are observed, each firm takes into account the equilibrium wage Wt+1 and chooses its labor input to maximize operating profits: πit+1(Kit+1,zt+1;Wt+1)≡maxNit+1{Kit+1α(zt+1Nit+1)1−α−Wt+1Nit+1}. (13) The aggregate effective capital stock, aggregate output, and aggregate labor demand are determined as follows: Kt+1≡∫Kit+1di=Kt+1w(1−xt+1bk)Yt+1≡∫Yit+1di=Kt+1α(zt+1Nt+1)1−αNt+1≡∫Nit+1di. (14) With these aggregates in mind, firm-level operating profits can be written as: πit+1=Yit+1−Wt+1Nit+1=αYit+1=α(Kit+1Kt+1)Yt+1. (15) The total resources of the firm at the end of the period (V) are the sum of operating profits and effective capital net of depreciation. Using the preceding equation, we have: Vit+1=Kit+1(αYt+1Kt+1+(1−δ))=εit+1Kt+1(αYt+1Kt+1+(1−δ))=εit+1(1−xt+1bk)Kt+1w(αYt+1Kt+1+(1−δ)). (16) The total gross return on the firm’s initial capital invested is defined as follows: Rit+1K≡Vit+1Kit+1w=εit+1(1−xt+1bk)(αYt+1Kt+1+(1−δ)). (17) Notice, the firm’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 defined as follows: Rt+1K≡∫Vit+1diKt+1w=(1−xt+1bk)(αYt+1Kt+1+(1−δ)). (18) It follows that the total end of period value of firm resources can be written as the product of its idiosyncratic capital shock, the aggregate return on capital, and the aggregate capital invested. We have: Vit+1=εit+1Rt+1KKt+1w. (19) 2.3 Capital Structure Firms can issue debt or equity to finance their investment. Financial frictions take the form of tax benefits to debt and bankruptcy costs. Tax benefits to debt are captured in reduced form as a government subsidy paid to the firm at the time debt is issued. The subsidy is equal to χ−1>0 per dollar of debt funding raised. Given an equilibrium price q per unit of debt face value, the firm receives χq per unit of debt face value. The government finances the debt subsidy through lump sum taxation. Here it is worth noting that the deadweight loss attributable to the debt tax subsidy would be higher if lump sum taxation were infeasible. If end of period resources are inadequate to pay the promised face value, the firm defaults, with bondholders receiving a fraction θ∈(0,1) of end-of-period resources. All remaining resources accrue to the government, which then rebates this value back to the representative household. Since bankruptcy is privately costly, not socially costly, the capital structure chosen by firms has no direct effect on aggregate resources available to the representative household at the end of the period. We impose χθ<1. If this assumption were not satisfied, the firm would find it optimal to finance entirely with debt since the government debt subsidy would more than cover default costs. The only social inefficiency arising from the posited financial market imperfections is that the tax subsidy to debt distorts real investment by artificially lowering the cost of debt capital. This distortion lowers the utility of the representative household. The socially optimal policy in the present economy would be for the government to eliminate the tax subsidy to debt (χ = 1) inducing firms to finance exclusively with equity, with zero investment distortions. The firm defaults if its end-of-period terminal resource value is less than the face value of debt Bt+1, or: Vit+1=εit+1Rt+1KKt+1w<Bt+1. (20) From the preceding equation, it follows that an individual firm defaults if its idiosyncratic capital shock is less than the following economy-wide default threshold: εt+1∗=(1Rt+1K)(Bt+1Kt+1w). (21) Notice that the common default threshold is decreasing in the aggregate return on capital and increasing in the leverage ratio. It follows that defaults will be correlated since a low aggregate return on capital implies each firm has higher default threshold, implying a greater mass of defaulting firms. Moreover, there will be particularly pronounced clustering of defaults in the event of a disaster shock. Letting M denote the endogenous pricing kernel, the market value of debt is: qtBt+1=Et[Mt+1(Bt+1[1−H(εt+1∗)]+θRt+1KKt+1w∫0εt+1∗εh(ε)dε)]. (22) At this stage, it is worth commenting on the assumption that firms live for only one period. To see that this assumption is without loss of generality, notice that each firm has a going-concern value equal to zero under the stated assumptions of constant returns, free entry and price taking. Thus, even if firms were to live for multiple periods, their decision rules would remain the same as that discussed above. Departing from Gourio (2013), we introduce heterogeneous firms as follows. Again there is free entry. But now assume there is a start-up cost, for example, a license fee, that must be paid to the government at the start of each period, with this cost varying with the technology employed. License fees collected by the government will be returned to households via lump sum transfer. Entrants can acquire a license to operate one of two alternative technologies. There is an expensive technology featuring a high recovery parameter θH and an inexpensive technology featuring a low recovery parameter θL. 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. 2.4 Households We consider an infinitely 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−β)(Ctv(1−Nt)1−v)1−ψ+βEt(Ut+11−γ)1−ψ1−γ)11−ψ. (23) The household budget constraint demands that uses of funds are no greater than sources of funds. We demand: Ct+Pt+qtBt≤WtNt+ρtBt−1+Dt+Tt. (24) In the preceding equation, P denotes the market price of a 100% equity interest in all the firms 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 firms. The variable W denotes the wage rate. The variable ρ 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 payoff to shareholders at the start of the current period. Finally, the term T captures lump sum transfers from the government. This transfer is equal to tax subsidies to debt less bankruptcy costs and less licensing fees. 2.5 Equilibrium In addition to goods market clearing, we have the following definition of Equilibrium: Household optimization: Wt=1−vvCt1−Nt. (25) Mt+1=β(Ct+1Ct)v(1−ψ)−1(1−Nt+11−Nt)(1−v)(1−ψ)Ut+1ψ−γEt(Ut+11−γ)ψ−γ1−γ. (26) Labor market clearing: (1−α)YtNt=Wt=1−vvCt1−Nt. (27) Optimal firm investment: Et{Mt+1Rt+1K[1−(1−χθ)∫0εt+1∗εh(ε)dε+(χ−1)εt+1∗(1−H(εt+1∗))]}=1. (28) Optimal firm capital structure: (1−θ)Et[Mt+1εt+1∗h(εt+1∗)]=(χ−1χ)Et[Mt+1(1−H(εt+1∗))]. (29) The last two requirements for equilibrium merit brief discussion. First, at an optimum, firms equate the marginal product of capital with the unit price of capital, with the debt tax subsidy distorting investment decisions at the margin. Second, firms equate marginal expected bankruptcy costs with the tax benefits of debt. To solve the model we use the projection methods of Aruoba, Fernandez-Villaverde, and Rubio-Ramirez (2006) and Caldara et al. (2012). To this end, one can begin by defining the following rescaled variables: y=Y/z,c=C/z,i=I/z,g=U/z. (30) We approximate policy functions c(k,p), L(k,p), N(k,p), and 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 1D Chebyshev polynomial. We evaluate first-order conditions at Nc Chebyshev nodes to find coefficients that minimize the residual function and Euler equation errors. Due to slow convergence, we first solve the model employing low values for Nc, Np, and risk aversion. Using the previous solution, we solve the model by consecutively increasing these parameter values. The policy functions are then 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 Parametrization and Simulation Approach Table I lists the parameter values. The values for the parameters ( α,δ,v,β,μ,σ) are standard in the literature, for example, Cooley and Prescott (1995). For the other parameters, we follow closely Gourio (2013) to facilitate comparison. The intertemporal elasticity of substitution of consumption (IES) is set at 2, consistent with recent asset pricing literature. Parameters of the disaster shock are chosen so that the average disaster probability is approximately 2%, with λl=0.07% and λh=3.9%. The regime switching probability is set to s = 0.1, implying an expected regime life of 10 years. The capital and TFP disaster size parameters are both set to 15%, so as to match the averages in Gourio (2013). The cumulative distribution for the idiosyncratic capital shocks H is the lognormal distribution with mean one. Table I. Parameter values used in model numerical solution and simulations Description Parameter Value Recovery rate for average firm θ 0.7 Recovery rate for bad firm θL 0.6 Recovery rate for good firm θH 0.8 Average probability of disaster λa 0.02 Probability of disaster in the good regime λl 0.007 Probability of disaster in the bad regime λh 0.039 Average disaster size b 15% Switching probability s 0.1 Tax subsidy χ−1 0.033 Tax subsidy for a fringe of AAA firms χaaa−1 0.0163 Elasticity of capital α 0.3 Depreciation rate δ 0.08 Share of consumption in utility v 0.3 Discount factor β 0.987 Trend growth of aggregate shock μz 0.01 Standard deviation of aggregate shock σe 0.015 Trend growth of idios. shock μx 0.01 Persistence of idios. shock ρx 0.767 Standard deviation of idios. shock σε 0.015 IES 1/ψ 2 Risk aversion γ 6 Description Parameter Value Recovery rate for average firm θ 0.7 Recovery rate for bad firm θL 0.6 Recovery rate for good firm θH 0.8 Average probability of disaster λa 0.02 Probability of disaster in the good regime λl 0.007 Probability of disaster in the bad regime λh 0.039 Average disaster size b 15% Switching probability s 0.1 Tax subsidy χ−1 0.033 Tax subsidy for a fringe of AAA firms χaaa−1 0.0163 Elasticity of capital α 0.3 Depreciation rate δ 0.08 Share of consumption in utility v 0.3 Discount factor β 0.987 Trend growth of aggregate shock μz 0.01 Standard deviation of aggregate shock σe 0.015 Trend growth of idios. shock μx 0.01 Persistence of idios. shock ρx 0.767 Standard deviation of idios. shock σε 0.015 IES 1/ψ 2 Risk aversion γ 6 In the model with homogeneous firms, we set the creditor recovery parameter θ to 0.7 as in Gourio (2013). In the model with heterogeneous firms, half the firms have recovery parameters equal to 0.6 and the other half have recovery parameters equal to 0.8. The tax shield χ and volatility σϵ parameters are chosen to roughly match the average default rate for BAA firms (0.5%) and an average book leverage ratio of 55%. The resulting calibrated tax subsidy to debt funding is 3.3 cents per dollar of debt capital raised. To compute credit spreads we add a measure-zero fringe of AAA firms having no influence on the economy. We choose the tax shield value for this fringe of firms (χaaa) to replicate the average unconditional default rate for AAA firms, 4 bps, and average leverage of 45%. We run 1,000 simulated histories, with each simulated history lasting 1,000 periods. We discard the first 100 periods in each simulated history in order to reduce the importance of date-zero beliefs. The date zero prior belief is set at one-half. A number of model variations are explored. To shed light on the role of learning, we consider the behavior of firms in a so-called Non-Learning economy in which the true probability of a disaster is an observable two-state Markov process with identical parameters to the hidden Markov process described above. Within the two broad categories of Non-Learning and Learning firms, three variations are considered. First, to get a sense of the role played by violations of the Modigliani–Miller assumptions, we first consider firms embedded in the respective economies who do not face any financing frictions and so choose to finance with equity. Second, financial market imperfections are introduced into the Non-Learning and Learning economies, but firms are assumed to be homogeneous. Finally, heterogeneous recovery parameters are introduced. 3.2 Financing and Investment Table II reports business cycle statistics from real-world data (first 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). Table II. 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. σ(Δlog(Y)) σ( Δlog(C)) σ(Δlog(Inv)) σ(Δlog(N)) Data 2.78 1.81 7.01 2.67 Non-Learning All-Equity 3.39 3.84 5.74 0.85 Homogeneous 5.44 5.38 8.11 1.41 Heterogeneous 5.60 5.57 8.46 1.58 Learning All-Equity 3.24 3.78 5.63 0.46 Homogeneous 5.47 5.21 6.43 0.51 Heterogeneous 5.61 5.43 6.64 0.72 σ(Δlog(Y)) σ( Δlog(C)) σ(Δlog(Inv)) σ(Δlog(N)) Data 2.78 1.81 7.01 2.67 Non-Learning All-Equity 3.39 3.84 5.74 0.85 Homogeneous 5.44 5.38 8.11 1.41 Heterogeneous 5.60 5.57 8.46 1.58 Learning All-Equity 3.24 3.78 5.63 0.46 Homogeneous 5.47 5.21 6.43 0.51 Heterogeneous 5.61 5.43 6.64 0.72 If one first compares the first row in the table (Non-Learning All-Equity) with the final row (Learning, Levered, Heterogeneous), it is apparent that the three mechanisms at work in the full model, learning, financing frictions, and heterogeneity, lead to a significant amplification of volatility in consumption and income, as well as a non-trivial increase in the volatility of investment. To understand the contribution of each causal mechanism, it is interesting to note that the learning mechanism by itself actually tends to reduce unconditional macroeconomic volatility. To see this, note that each of the four macroeconomic series is a bit less volatile if one compares All-Equity firms in the Non-Learning Economy with All-Equity firms in the Learning Economy. Intuitively, agents in the Non-Learning economy have more extreme belief fluctuations since the state variable p alternates between 0 and 1 for them. Agents in the Learning economy form less extreme beliefs, with p falling in the interval [p¯,p¯]. As shown in Table II, within each class of model, Non-Learning and Learning, financial frictions amplify macroeconomic volatility. It is also apparent from Table II that introducing heterogeneity into firm technologies further amplifies financial accelerator effects. Table III reports unconditional moments for expected returns, default rates, loss-given-default and leverage. The real-world leverage and default probability data are taken from Chen, Collin-Dufresne, and Goldstein (2009). Credit spreads are from FRED. As shown, the model tends to overshoot observed leverage ratios.4 The model also has a slight tendency to overshoot the unconditional default probability and to overshoot expected loss-given-default. As shown, and in the spirit of Gourio (2013), disaster risk is sufficient to generate large corporate bond risk premia and yield spreads. 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, creditor losses are countercyclical, resulting in high risk premia. In fact, under the stated parameterization the model overshoots credit spreads. This implies the model can replicate observed spreads with less severe disaster shocks. Table III. 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. Data Heterogeneous Homogeneous E(book leverage), % 45 63.98 62.49 E(Default Prob.), bp 50 59.42 57.93 E(Loss given default), % 45 34.97 32.83 E(Rf), % 1.6 1.43 1.41 E( Rc−Rf), % 0.8 2.02 1.98 E( Re−Rf), % 5.6 6.48 6.37 E( y−yAAA), % 0.94 1.94 1.90 Data Heterogeneous Homogeneous E(book leverage), % 45 63.98 62.49 E(Default Prob.), bp 50 59.42 57.93 E(Loss given default), % 45 34.97 32.83 E(Rf), % 1.6 1.43 1.41 E( Rc−Rf), % 0.8 2.02 1.98 E( Re−Rf), % 5.6 6.48 6.37 E( y−yAAA), % 0.94 1.94 1.90 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 IV considers the Learning Economy, exhibiting simulated model moments for leverage, capital, labor, and consumption, conditional on years since the last disaster. Table IV shows that successively longer quiet periods induce the simulated firms to increase their capital stocks. Intuitively, as shown in Figure 2, the longer the elapsed time since the last disaster, the more favorable are agent beliefs. The capital accumulation of firms with low recovery parameters (θL) is particularly responsive to the time-since-disaster variable. Moving from the first to the last column in Table IV, the capital stock of the high recovery firms only increases by 6.3% while that of the low recovery firms increases by 8.9%. Intuitively, low recovery rate firms are especially concerned about bankruptcy states, and the pricing kernel in such states. Table IV. 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 the Homogeneous and Heterogeneous Learning models and Heterogeneous Non-Learning model. The book leverage moment is shown for the Learning Heterogeneous model with a breakdown for θL and θH firms. Time-since-disaster 1 5 10 15 20 E(book leverage), % Homog. 51.75 60.89 62.98 63.63 63.69 Heterog. 46.01 53.02 57.06 60.03 62.02 θL 31.79 37.93 47.51 50.37 54.86 θH 50.11 61.33 63.91 64.23 65.44 Capital Homog. 16.35 16.60 17.01 17.43 17.59 Heterog. 16.35 16.60 17.01 17.43 17.59 θL 16.02 16.34 16.82 17.26 17.45 θH 16.68 16.86 17.2 17.6 17.73 Labor Homog. 0.28 0.29 0.30 0.30 0.30 Heterog. 0.29 0.30 0.31 0.32 0.32 C/Y Homog. 0.68 0.70 0.71 0.72 0.73 Heterog. 0.69 0.70 0.71 0.72 0.73 Time-since-disaster 1 5 10 15 20 E(book leverage), % Homog. 51.75 60.89 62.98 63.63 63.69 Heterog. 46.01 53.02 57.06 60.03 62.02 θL 31.79 37.93 47.51 50.37 54.86 θH 50.11 61.33 63.91 64.23 65.44 Capital Homog. 16.35 16.60 17.01 17.43 17.59 Heterog. 16.35 16.60 17.01 17.43 17.59 θL 16.02 16.34 16.82 17.26 17.45 θH 16.68 16.86 17.2 17.6 17.73 Labor Homog. 0.28 0.29 0.30 0.30 0.30 Heterog. 0.29 0.30 0.31 0.32 0.32 C/Y Homog. 0.68 0.70 0.71 0.72 0.73 Heterog. 0.69 0.70 0.71 0.72 0.73 Consider next the behavior of leverage ratios as shown in Table IV. The book leverage ratio of the simulated firms is particularly responsive to the time-since-disaster variable. And this pronounced leverage increase occurs despite the large increase in the capital stock denominator. Apparently, debt issuance is more sensitive to the time-since-disaster variable than real investment. With homogeneous firms, there is a 23% increase in leverage as one moves from the first to last column. In contrast, with heterogeneous firms, the leverage increase is 35%. Thus, cross-sectional heterogeneity may be understood as playing an important part in amplifying leverage cyclicality. Further inspection of the simulated heterogeneous firms reveals that this pattern is due to firms with low recovery parameters. In particular, firms with low recovery parameters exhibit leverage ratios that are particularly responsive to time-since-disaster, increasing by 73% moving from the first to last columns. Again, this subset of firms is particularly concerned about bankruptcy states, and the pricing kernel in such states. Thus, the model predicts that during a period such as the Great Moderation, firms will rationally increase their capital stocks and book leverage ratios, with the most aggressive reactions by those firms with higher bankruptcy costs. Having considered firm behavior during tranquil periods, Table V considers responses to the realization of a disaster shock, evaluated as a function of time-since-disaster. The table reveals that time-since-disaster can be understood as a key conditioning variable in terms of predicting how firms will respond to a disaster realization—especially in the Learning Economy. In the Learning Economy, a disaster realization leads to a revision of beliefs, and the revision is especially large if the time-since-disaster is long. No such belief-revision mechanism is operative 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 (consisting of learning agents) after a long tranquil period. Rather than being an indication of irrational panic or clogged debt markets, large declines in investment and leverage can be understood as natural byproducts of rational Bayesian updating. Finally, Table V also reveals that the conjunction of learning and heterogeneity is particularly powerful as a disaster amplification mechanism. Table V. 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 Heterogeneous Non-Learning model. The book leverage moment is shown for Learning Heterogeneous model with a breakdown for θL and θH firms. Time-since-disaster Change after disaster: 1 5 10 15 20 E(book leverage), % Homog. −3% −11% −13% −14% −14% Heterog. −7% −14% −16% −17% −17% θL −8% −18% −16% −15% −15% θH −7% −13% −17% −19% −21% Non-Learning −0.02% −0.06% −0.06% −0.08% −0.08% Capital Homog. −4.7% −4.7% −4.7% −4.7% −4.7% Heterog. −5.0% −5.0% −5.0% −5.1% −5.1% θL −5.3% −5.3% −5.3% −5.3% −5.4% θH −4.7% −4.7% −4.7% −4.7% −4.7% Non-Learning −4.5% −4.5% −4.6% −4.7% −4.7% Labor Homog. −0.4% −0.7% −0.7% −1.0% −1.0% Heterog. −0.3% −0.7% −1.0% −1.3% −1.6% Non-Learning −0.2% −0.6% −0.8% −0.9% −1.3% Time-since-disaster Change after disaster: 1 5 10 15 20 E(book leverage), % Homog. −3% −11% −13% −14% −14% Heterog. −7% −14% −16% −17% −17% θL −8% −18% −16% −15% −15% θH −7% −13% −17% −19% −21% Non-Learning −0.02% −0.06% −0.06% −0.08% −0.08% Capital Homog. −4.7% −4.7% −4.7% −4.7% −4.7% Heterog. −5.0% −5.0% −5.0% −5.1% −5.1% θL −5.3% −5.3% −5.3% −5.3% −5.4% θH −4.7% −4.7% −4.7% −4.7% −4.7% Non-Learning −4.5% −4.5% −4.6% −4.7% −4.7% Labor Homog. −0.4% −0.7% −0.7% −1.0% −1.0% Heterog. −0.3% −0.7% −1.0% −1.3% −1.6% Non-Learning −0.2% −0.6% −0.8% −0.9% −1.3% 3.3 Asset Returns This subsection considers the model’s asset pricing implications, beginning with predictions regarding pricing during prolonged quiet periods. Table VI reports expected returns, expected default rates, expected loss-given-default, credit spreads and expected yield spreads, with data grouped by time-since-prior-disaster. Table VI. Asset pricing moments: quiet periods The table reports moments for asset pricing variables and book leverage conditional on time-since-disaster. All moments are shown for both Homogeneous and Heterogeneous Learning models. Time-since-disaster 1 5 10 15 20 E(Rf), % Homog. 0.02 0.98 1.46 1.63 1.63 Heterog. 0.01 0.84 1.34 1.44 1.45 E(D/E) Homog. 35.57 39.75 41.48 43.29 45.88 Heterog. 30.45 35.95 43.51 49.72 53.87 E(book leverage), % Homog. 51.75 60.89 62.98 63.63 63.69 Heterog. 50.11 61.33 63.91 64.23 65.44 E(Default Prob.), bp Homog. 0.01 2.83 9.18 12.13 12.74 Heterog. 1.82 3.46 13.38 22.87 29.85 E(Loss given default), % Homog. 32.01 32.57 32.78 32.84 32.85 Heterog. 31.44 32.4 33.12 33.76 34.03 E( Rc−Rf), % Homog. 1.47 2.08 2.09 2.11 2.14 Heterog. 1.48 2.09 2.11 2.12 2.15 E(yBAA-Rf), % Homog. 1.94 1.79 1.59 1.36 1.32 Heterog. 1.98 1.82 1.61 1.38 1.33 E(yAAA-Rf), % Homog. 0.22 0.21 0.20 0.19 0.18 Heterog. 0.23 0.21 0.20 0.19 0.17 E(yBAA-yAAA), % Homog. 1.72 1.58 1.39 1.17 1.14 Heterog. 1.75 1.61 1.41 1.19 1.16 E( Re−Rf), % Homog. 7.01 6.4 6.32 6.27 6.23 Heterog. 9.45 7.01 5.03 4.31 4.02 Time-since-disaster 1 5 10 15 20 E(Rf), % Homog. 0.02 0.98 1.46 1.63 1.63 Heterog. 0.01 0.84 1.34 1.44 1.45 E(D/E) Homog. 35.57 39.75 41.48 43.29 45.88 Heterog. 30.45 35.95 43.51 49.72 53.87 E(book leverage), % Homog. 51.75 60.89 62.98 63.63 63.69 Heterog. 50.11 61.33 63.91 64.23 65.44 E(Default Prob.), bp Homog. 0.01 2.83 9.18 12.13 12.74 Heterog. 1.82 3.46 13.38 22.87 29.85 E(Loss given default), % Homog. 32.01 32.57 32.78 32.84 32.85 Heterog. 31.44 32.4 33.12 33.76 34.03 E( Rc−Rf), % Homog. 1.47 2.08 2.09 2.11 2.14 Heterog. 1.48 2.09 2.11 2.12 2.15 E(yBAA-Rf), % Homog. 1.94 1.79 1.59 1.36 1.32 Heterog. 1.98 1.82 1.61 1.38 1.33 E(yAAA-Rf), % Homog. 0.22 0.21 0.20 0.19 0.18 Heterog. 0.23 0.21 0.20 0.19 0.17 E(yBAA-yAAA), % Homog. 1.72 1.58 1.39 1.17 1.14 Heterog. 1.75 1.61 1.41 1.19 1.16 E( Re−Rf), % Homog. 7.01 6.4 6.32 6.27 6.23 Heterog. 9.45 7.01 5.03 4.31 4.02 A number of predictions stand out. First, we see a large decline in the equity risk premium as the quiet period lengthens. For example, in the Learning Economy with homogeneous firms, the typical equity risk premium declines by seventy-eight basis points as the time-since-disaster lengthens from 1 to 20 years. Further, in the Learning Economy with heterogeneous firms, the equity risk premium declines by 543 basis points as the time-since-disaster lengthens from 1 to 20 years. Thus, firm heterogeneity can have an important effect on the dynamics of equity premia. It is also worth noting that this effect is present despite the fact that the simulated firms increase their debt-to-equity ratios during such tranquil periods. 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 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-default as the quiet period lengthens. Recall, it is the firms with low creditor recovery parameters who lever up most aggressively in response to lengthy periods without a disaster. This leads to a composition effect in which average loss-given-default is more heavily influenced by firms with low intrinsic creditor recovery parameters. Having analyzed risk premia and credit spreads during tranquil periods, we turn next to the effects 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 demand much higher expected returns after the occurrence of a disaster. Table VII examines how disaster realizations affect 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 effect is further amplified if one considers the realistic possibility of firm heterogeneity. Table VII. 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 Change after disaster in: 1 5 10 15 20 E( Rc−Rf), % Learning Homog. 3% 2% 3% 3% 4% Heterog. 4% 3% 4% 4% 5% Non-Learning Homog. 1% 0% 0% 0% 0% Heterog. 1% 0% 0% 0% 0% E( Re−Rf), % Learning Homog. 2% 29% 53% 62% 62% Heterog. 2% 36% 88% 119% 129% Non-Learning Homog. 2% 2% 3% 3% 3% Heterog. 2% 3% 5% 6% 7% E( yBAA−yAAA), % Learning Homog. 8% 25% 42% 57% 59% Heterog. 9% 28% 47% 63% 66% Non-Learning Homog. 1% 1% 1% 2% 2% Heterog. 1% 1% 1% 2% 2% Time-since-disaster Change after disaster in: 1 5 10 15 20 E( Rc−Rf), % Learning Homog. 3% 2% 3% 3% 4% Heterog. 4% 3% 4% 4% 5% Non-Learning Homog. 1% 0% 0% 0% 0% Heterog. 1% 0% 0% 0% 0% E( Re−Rf), % Learning Homog. 2% 29% 53% 62% 62% Heterog. 2% 36% 88% 119% 129% Non-Learning Homog. 2% 2% 3% 3% 3% Heterog. 2% 3% 5% 6% 7% E( yBAA−yAAA), % Learning Homog. 8% 25% 42% 57% 59% Heterog. 9% 28% 47% 63% 66% Non-Learning Homog. 1% 1% 1% 2% 2% Heterog. 1% 1% 1% 2% 2% Finally, Figure 3 depicts the simulated statistical distribution of loss-given-default within the heterogeneous firm model-generated time series. The three panels of the figure illustrate the effect 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 10 years having elapsed between the prior disaster and the current year, where the current year may or may not contain a disaster. A number of interesting observations emerge. First, the mean loss-given-default actually increases with the time since prior disaster. Second, as the time since prior disaster increases, the distribution of loss-given-default becomes increasingly bi-modal. Both features of the simulated data arise from 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 firms with low intrinsic recovery technologies. These patterns are reminiscent of the selection effect discussed in Glover (2016). However, here the selection effect is more subtle in that it is time-varying, due to the evolution of beliefs leading to changes in the relative importance of different credit types. Figure 3 View largeDownload slide Conditional loss-given-default distributions. Note: This figure plots loss-given-default distributions conditional on time-since-disaster for the Learning Heterogeneous model. Figure 3 View largeDownload slide Conditional loss-given-default distributions. Note: This figure plots loss-given-default distributions conditional on time-since-disaster for the Learning Heterogeneous model. 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 novel predictions. Recall, in the model, on an unconditional basis, leverage, and capital investment were shown to be increasing in time-since-disaster, with the effect being less pronounced for firms with high intrinsic creditor recovery parameters. In this section, we first use the simulated model data to verify that these predictions remain valid conditional upon the inclusion of standard regressors. We then test these predictions in the real-world data. 4.1 Data We collect annual financial statement data from the Compustat-CRSP Merged Database for the time period 1950–2013. Following the literature, we remove all regulated (SIC 4900–4999) and financial firms (6000–6999). We exclude observations 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. We require firms to have at least 2 consecutive years of data since we need to lag some variables. We treat National Bureau of Economic Research (NBER)-dated recessions as negative macroeconomic shocks. The variable Years-Since-Recession (YSR) is set to 0 if a year belongs to an NBER recession, with the variable increasing by 1 each year after the last recession year. We utilize the recovery rates on defaulted debt from Altman and Kishore (1996), at the two-digit SIC level, as the empirical analog of the θ parameter. The remaining data definitions are standard following Rajan and Zingales (1995). 4.2 Findings In the first 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 VIII. The dependent variable is book leverage. The first reported simulated regression features a standard regression of leverage on Tobin’s q and cash flow normalized by assets. The next reported regression adds a YSD variable. This variable enters with a positive coefficient, consistent with the model-implied unconditional correlation between leverage and YSD discussed in Section 3. The final column adds an interaction between the YSD variable and the firm-level creditor recovery parameter. Consistent with the unconditional correlation discussed in Section 3, this interaction term enters with a significant negative coefficient. Thus, in the simulated data one sees that even conditionally, leverage increases in YSD, with the effect being less pronounced for firms with high creditor recovery parameters. Table VIII. Book leverage regressions: model The table reports book leverage regressions output for Learning Heterogeneous model. YSD denotes YSD. Standard errors are in parenthesis. ***, **, and * indicate significance at the 1, 5, and 10% levels, respectively. (1) (2) (3) Tobin’s q −89.14*** −72.75*** −92.66*** (17.36) (16.07) (15.00) CashFlow/Assets 30.11*** 24.76*** 31.92*** (5.809) (5.379) (5.018) YSD 0.000307*** 1.375*** (2.84e−06) (0.0134) YSD × Recovery −2.047*** (0.0200) Observations 70,329 70,329 70,329 R-squared 0.004 0.146 0.257 (1) (2) (3) Tobin’s q −89.14*** −72.75*** −92.66*** (17.36) (16.07) (15.00) CashFlow/Assets 30.11*** 24.76*** 31.92*** (5.809) (5.379) (5.018) YSD 0.000307*** 1.375*** (2.84e−06) (0.0134) YSD × Recovery −2.047*** (0.0200) Observations 70,329 70,329 70,329 R-squared 0.004 0.146 0.257 We next test these model-implied predictions in the real-world data. Table IX reports the results when we add the YSR variable and the YSR × Recovery interaction variable to standard leverage regressions. The first 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 YSR enters with a significant positive coefficient. In the final column, we see that the coefficient on YSR remains positive and significant, while the interaction term enters with a significant negative coefficient, consistent with model predictions. Table IX. Book leverage regressions: data This table reports the effect of a YSR variable and an interaction between the YSR and recovery in a standard leverage regressions. Variable definitions are provided in Appendix A. Standard errors are in parenthesis. ***, **, and * indicate significance at the 1, 5, and 10% levels, respectively. (1) (2) (3) (4) (5) (6) Tobin’s q −0.0240*** −0.0211*** −0.0205*** −0.0205*** −0.0214*** −0.0214*** (0.000646) (0.000646) (0.000643) (0.000643) (0.000646) (0.000646) CashFlow/Assets −0.0202*** −0.0208*** −0.0200*** −0.0200*** −0.0200*** −0.0200*** (0.000449) (0.000446) (0.000445) (0.000445) (0.000445) (0.000445) Size 0.0227*** 0.0243*** 0.0245*** 0.0237*** 0.0237*** (0.000593) (0.000593) (0.000617) (0.000620) (0.000620) Tangibility 0.169*** 0.169*** 0.170*** 0.170*** (0.00581) (0.00581) (0.00581) (0.00581) Growth −0.000770 −0.000737 −0.000732 (0.000704) (0.000703) (0.000703) YSR 0.00262*** 0.00429*** (0.000196) (0.000948) YSR × Recovery −0.00379* (0.00210) Firm-FE Y Y Y Y Y Y Observations 110,401 110,401 110,384 110,378 110,378 110,378 R-squared 0.603 0.608 0.612 0.612 0.612 0.612 (1) (2) (3) (4) (5) (6) Tobin’s q −0.0240*** −0.0211*** −0.0205*** −0.0205*** −0.0214*** −0.0214*** (0.000646) (0.000646) (0.000643) (0.000643) (0.000646) (0.000646) CashFlow/Assets −0.0202*** −0.0208*** −0.0200*** −0.0200*** −0.0200*** −0.0200*** (0.000449) (0.000446) (0.000445) (0.000445) (0.000445) (0.000445) Size 0.0227*** 0.0243*** 0.0245*** 0.0237*** 0.0237*** (0.000593) (0.000593) (0.000617) (0.000620) (0.000620) Tangibility 0.169*** 0.169*** 0.170*** 0.170*** (0.00581) (0.00581) (0.00581) (0.00581) Growth −0.000770 −0.000737 −0.000732 (0.000704) (0.000703) (0.000703) YSR 0.00262*** 0.00429*** (0.000196) (0.000948) YSR × Recovery −0.00379* (0.00210) Firm-FE Y Y Y Y Y Y Observations 110,401 110,401 110,384 110,378 110,378 110,378 R-squared 0.603 0.608 0.612 0.612 0.612 0.612 The next set of regressions focuses on investment rates. To begin, we run mimicking regressions using the simulated model data in Table X. The dependent variable is the investment rate. The first reported simulated regression results correspond to standard regressions of the investment rate on Tobin’s q and cash flow normalized by assets. Both variables enter with positive coefficients, with the coefficient 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 adds a YSD variable to the first two standard regressors. This variable enters with a positive coefficient, consistent with the unconditional correlations discussed in Section 3. The final column adds an interaction between the YSD variable and the firm-level creditor recovery parameter. Consistent with the unconditional model correlation, this interaction term enters with a negative coefficient. Table X. 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. (1) (2) (3) Tobin’s q 1.766*** 2.213*** 1.587*** (0.409) (0.368) (0.320) CashFlow / Assets 0.102 −0.0435 0.182* (0.137) (0.123) (0.107) YSD 8.36e−06*** 0.0433*** (6.49e−08) (0.000286) YSD × Recovery −0.0644*** (0.000426) Observations 70,329 70,329 70,329 R-squared 0.973 0.978 0.984 (1) (2) (3) Tobin’s q 1.766*** 2.213*** 1.587*** (0.409) (0.368) (0.320) CashFlow / Assets 0.102 −0.0435 0.182* (0.137) (0.123) (0.107) YSD 8.36e−06*** 0.0433*** (6.49e−08) (0.000286) YSD × Recovery −0.0644*** (0.000426) Observations 70,329 70,329 70,329 R-squared 0.973 0.978 0.984 Table XI tests these model-implied predictions regarding investment rates in the real-world data. As shown in the first column, investment is increasing in Tobin’s q and normalized cash flow, standard findings. 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 final two columns in the table feature tests of the model’s novel predictions regarding investment. Consistent with the model, the penultimate column shows that YSR enters with a significant positive coefficient. In the final column, we see that the coefficient on YSR remains positive and significant, while the interaction term enters with the predicted negative sign, but is insignificant. Table XI. Investments regressions: data This table reports the effect of a YSR variable and an interaction between the YSR and recovery in a standard investment regressions. Variable definitions are provided in Appendix A. Standard errors are in parenthesis. ***, **, and * indicate significance at the 1, 5, and 10% levels, respectively. (1) (2) (3) (4) (5) (6) Tobin’s q 0.0196*** 0.0191*** 0.0200*** 0.0200*** 0.0199*** 0.0199*** (0.000534) (0.000536) (0.000526) (0.000526) (0.000528) (0.000528) CashFlow / Assets 0.00324*** 0.00343*** 0.00447*** 0.00447*** 0.00447*** 0.00446*** (0.000354) (0.000354) (0.000349) (0.000349) (0.000348) (0.000348) Size −0.00583*** −0.00315*** −0.00337*** −0.00363*** −0.00364*** (0.000545) (0.000538) (0.000557) (0.000562) (0.000562) Tangibility 0.254*** 0.254*** 0.254*** 0.254*** (0.00497) (0.00497) (0.00497) (0.00497) Growth 0.000846 0.000851 0.000856 (0.000589) (0.000589) (0.000589) YSR 0.000602*** 0.00193** (0.000168) (0.000827) YSR × Recovery −0.00299 (0.00183) Firm-FE Y Y Y Y Y Y Observations 80,655 80,655 80,648 80,646 80,646 80,646 R-squared 0.444 0.445 0.465 0.465 0.465 0.465 (1) (2) (3) (4) (5) (6) Tobin’s q 0.0196*** 0.0191*** 0.0200*** 0.0200*** 0.0199*** 0.0199*** (0.000534) (0.000536) (0.000526) (0.000526) (0.000528) (0.000528) CashFlow / Assets 0.00324*** 0.00343*** 0.00447*** 0.00447*** 0.00447*** 0.00446*** (0.000354) (0.000354) (0.000349) (0.000349) (0.000348) (0.000348) Size −0.00583*** −0.00315*** −0.00337*** −0.00363*** −0.00364*** (0.000545) (0.000538) (0.000557) (0.000562) (0.000562) Tangibility 0.254*** 0.254*** 0.254*** 0.254*** (0.00497) (0.00497) (0.00497) (0.00497) Growth 0.000846 0.000851 0.000856 (0.000589) (0.000589) (0.000589) YSR 0.000602*** 0.00193** (0.000168) (0.000827) YSR × Recovery −0.00299 (0.00183) Firm-FE Y Y Y Y Y Y Observations 80,655 80,655 80,648 80,646 80,646 80,646 R-squared 0.444 0.445 0.465 0.465 0.465 0.465 5. Conclusion This article develops a DSGE model featuring disaster risk and financial 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 financial 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 firms with high bankruptcy costs. Time since prior disasters is the key model conditioning variable. In response to a disaster, risk premia increase sharply while firms 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 observable disaster risk. Further, business cycles are more pronounced than in an otherwise equivalent economy with frictionless financing. Firms with low recovery parameters are the most sensitive to time-since-disaster. Using the simulated model as a laboratory, we first 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 effect attenuated for firms with high creditor recovery parameters. Empirical tests offer support for these novel predictions. In terms of policy, the model shows that large fluctuations in real investment, leverage, and risk premia are not in themselves indicative of irrationality or of some inherent need for government intervention. Rather, an economy populated by rational learning agents can be expected to exhibit relatively large fluctuations. In the economy considered, it is government intervention that causes lower welfare. Specifically, the tax subsidy artificially lowers the cost of debt capital. This distorts real investment and increases volatility, thus lowering welfare. Taking a step away from the model, it is apparent that mid-crisis and post-crisis governmental interventions in the debt market served to increase the perceived subsidy to debt financing. The present model shows that the anticipation of ex ante (tax) and ex post (bailouts) debt subsidies, although well intended, has the potential to exacerbate cyclicality and lower welfare. Footnotes 1 This is the time-series analog of the cross-sectional prediction of Glover (2016). 2 Gourio (2013) models bk as a random variable. 3 Gourio (2013) instead models btfp as a random variable. 4 Tax loss limitations would bring down model leverage. 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Google Scholar CrossRef Search ADS Pastor L., Veronesi P. ( 2009) Learning in financial markets, Annual Review of Financial Economics 1, 361– 381. Google Scholar CrossRef Search ADS Rajan R., Zingales L. ( 1995) What do we know about capital structure? Some evidence from international data, Journal of Finance 50, 1421– 1460. Google Scholar CrossRef Search ADS Rietz T. ( 1988) The equity risk premium a solution, Journal of Monetary Economics 22, 117– 131. Google Scholar CrossRef Search ADS Stiglitz J. ( 2010) Freefall: America, Free Markets, and the Sinking of the World Economy , W. W. Norton and Company, New York. Wachter J. ( 2013) Can time-varying risk of rare disasters explain aggregate stock market volatility?, Journal of Finance 68, 3987–1035. Appendix A: List of Variables We use conventional variable definitions as in, for example, Rajan and Zingales (1995) and Graham, Leary, and Roberts (2014). Variable name Definition Total debt Total debt in current liabilities + Total long-term debt Book leverage Total debt / (Total debt + Book value of equity) Investment rate (Capital expenditures − Sale of property) / Total book assets CashFlow / Assets Operating income before depreciation / Total book assets Size Natural log of total book assets Tangibility Net property, plant, and equipment / Total book assets Growth Annual growth rate in Total book assets Market-to-book (Tobin’s q) (Market equity + Total debt) / Total book assets YSR 0 in NBER recession year and increases by 1 each Non-recession year (note: recessions reset YSR to 0). Variable name Definition Total debt Total debt in current liabilities + Total long-term debt Book leverage Total debt / (Total debt + Book value of equity) Investment rate (Capital expenditures − Sale of property) / Total book assets CashFlow / Assets Operating income before depreciation / Total book assets Size Natural log of total book assets Tangibility Net property, plant, and equipment / Total book assets Growth Annual growth rate in Total book assets Market-to-book (Tobin’s q) (Market equity + Total debt) / Total book assets YSR 0 in NBER recession year and increases by 1 each Non-recession year (note: recessions reset YSR to 0). © The Authors 2016. Published by Oxford University Press on behalf of the European Finance Association. All rights reserved. For Permissions, please email: journals.permissions@oup.com

Review of Finance – Oxford University Press

**Published: ** Feb 1, 2018

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