Credit spreads and state-dependent volatility: Theory and empirical evidence

Journal of Banking & Finance 55 (2015) 215–231

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Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf

Credit spreads and state-dependent volatility: Theory and empirical evidence q Stylianos Perrakis a,1, Rui Zhong b,⇑ a b

John Molson School of Business, Concordia University, 1455 De Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada Chinese Academy of Finance and Development, Central University of Finance and Economics, 39 South College Road, Haidian District, Beijing 100081, PR China

a r t i c l e

i n f o

Article history: Received 9 June 2014 Accepted 15 February 2015 Available online 3 March 2015 JEL classification: G12 G13 G32 G33

a b s t r a c t We generalize the asset dynamics assumptions of Leland (1994b) and Leland and Toft (1996) to a state dependent variance with constant elasticity process (CEV) and obtain analytical solutions for corporate debt and equity value. We use the GMM technique to extract the parameters by fitting the empirical data in the equity and credit default swap markets simultaneously. We find that the elasticity parameter is significantly different from zero for most of the firms and that the CEV model performs much better than the model with constant volatility in both in-sample fittings and out-of-sample predictions of CDS spreads. Ó 2015 Elsevier B.V. All rights reserved.

Keywords: Structural model Credit spreads Constant elasticity of variance

1. Introduction A very large number of studies, both theoretical and empirical, on corporate bond pricing and the risk structure of interest rates have appeared in the literature following the pioneering work of Merton (1974) and Black and Cox (1976), which in turn were inspired by the seminal Black and Scholes (1973) model of option pricing. These studies adopted the methodological approach of contingent claims valuation in continuous time, in which the value of a firm’s assets played the role of the claim’s underlying asset and

q We thank Regis Balzy, Jean-Claude Cosset, Redouane Elkamhi, Jan Ericssion, Zhiguo He, Sergey Isaenko, Lawrence Kryzanowski, Howard Qi, Baozhong Yang and Fan Yu for the useful discussions, we are also grateful to the participants at Annual Conference of Financial Management Association at Atlanta, the 19th Annual Conference of MFS at Krakow, 6th annual risk management conference at Singapore, the 62nd Annual Conference of Mid-West Finance Association at Chicago, the Mathematical Finance Days at Montreal and the financial seminar at Concordia University for the comments. We also thank Ken Liu for excellent research assistance. Financial support from the Social Sciences and Humanities Research Council, the RBC Distinguished Professorship in Financial Derivatives and the Montreal Institute of Structured Finance and Derivatives (IFSID) is gratefully acknowledged. ⇑ Corresponding author. Tel.: +86 010 62288200. E-mail addresses: [email protected] (S. Perrakis), ruizhong@ cufe.edu.cn (R. Zhong). 1 Tel.: +1 514 848 2424x2963.

http://dx.doi.org/10.1016/j.jbankfin.2015.02.017 0378-4266/Ó 2015 Elsevier B.V. All rights reserved.

allowed the valuation of the various components of the balance sheet under a variety of assumptions. This approach has been shown to be sufficiently flexible to tackle some of the most important problems in corporate finance, such as capital structure, bond valuation and default risk, under a variety of assumptions about the type of bonds included in the firm’s liabilities. The resulting models came to be known as structural models of bond pricing, as distinct from another class of models known as reduced form models, in which there is no direct link between the bonds of a given risk class and the firm’s capital structure.2 Under continuous coupon payment and first-passage default3 assumptions, Leland (1994a,b) and Leland and Toft (1996) first studied corporate debt valuation and optimal capital structure with endogenous default boundary for infinite and finite maturity debt, respectively.4 2 For the reduced form models see Jarrow and Turnbull (1995) and Duffie and Singleton (1999). These models lie outside the topic of this paper. 3 Under the first-passage default assumption, a firm will claim default when the asset value first crosses the pre-determined default boundary. This default boundary can be determined endogenously (Leland, 1994a,b; Leland and Toft, 1996; Duffie and Lando, 2001) or exogenously (Longstaff and Schwartz, 1995). 4 Leland (L, 1994a,b) and LT use the asset value of the unlevered firm as the basic underlying process for the valuation of the various components of the balance sheet of the levered firm. In a variant of the basic model, presented in Goldstein et al., 2001, the firm value is estimated from the dynamics of the earnings before interest and taxes (EBIT), split between the claimholders and the government; see also Sarkar and Zapatero (2003).

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Because of the computational complexity of the valuation expressions a major emphasis in the structural models was placed on the derivation of closed form expressions, rather than numerical results based on approximations5 or simulations.6 Such a focus allowed relatively easy estimations of numerical values given the parameters of the model, but at the cost of maintaining simple formulations of the mathematical structure of the asset value dynamics, in which a univariate diffusion process still follows the original Black and Scholes (1973) and Merton (1974) assumption of a lognormal diffusion with constant volatility.7 This is all the more surprising, in view of the fact that the option pricing literature has long recognized that such an assumption is no longer adequate to represent underlying assets in option markets, and has introduced factors such as rare events, stochastic volatility and transaction costs. Choi and Richardson (2009) studied the conditional volatility of the firm’s asset by a weighted average of equity, bond and loan prices and found that asset volatility is time varying. Similarly, Huang and Zhou (2008), in their study of the term structure of credit default swaps (CDS), note that time varying asset volatility should potentially play a role in structural models in order to fit into the empirical credit default spread. In this paper we generalize the dynamics of the asset value by assuming that the diffusion volatility is state-dependent, varying with the asset value. We use a particular form of volatility state dependence, known as the Constant Elasticity of Variance (CEV) model, originally formulated by Cox (1975) in the context of option pricing.8 Compared to constant volatility diffusion, the CEV model has only one extra parameter, the elasticity of variance, and includes constant volatility as a special case. Although this extension introduces significant additional computational complexity, we manage to derive closed form expressions for almost all the variables of interest, including corporate debt value, total levered firm value, optimal leverage and equity value. Both Leland (1994b) and LT are special cases of the CEV model with zero elasticity of variance. The results presented in the body of the paper refer to the L model, while the equivalent results for LT are presented in the appendix. Our numerical simulations show that the elasticity parameter plays a major role in the determination of the endogenous default boundary of the L and LT models, with a negative (positive) parameter decreasing (increasing) the boundary compared to the constant volatility case. These relative sizes of the default boundary hold for all maturities and all leverage ratios. The elasticity parameter also has a strong impact on credit spreads, with negative (positive) values widening (narrowing) the spread for all maturities in comparison with the constant volatility case, especially for exogenously set default boundary; this effect still persists but in a weaker form when the boundary is endogenous. Similar elasticity parameter effects are present in the determination of the optimal leverage and the volatilities of the equity and debt. As in Huang and Huang (HH, 2012), we also estimate the empirical default probability, the equity premium and the leverage ratio and find that the empirically

documented negative elasticity parameter boosts the percentage of yield spread due to default significantly, especially for debt with longer maturity and lower ratings. The main empirical results of this paper, however, consist of a comparison of the performance of the two alternative volatility assumptions (constant and CEV) within the context of the debt assumptions of the Leland (1994b) model. Using time series data from equity and CDS market for a sample of firms, we estimate the parameters using the Generalized Method of Moments (GMM) method by fitting the two competing models to all the available historical data, including both the implied equity volatilities and the credit default swap (CDS) data. We document that the CEV structural model with an elasticity parameter of around 0.54 on average exhibits a superior fitting in the CDS spreads across all maturities. The relationship between the sign and value of the elasticity parameter and the firm specific measures of default risk, such as leverage ratios, CDS spreads and current ratios indicates that there is a tendency for b to increase as the risk of the firm decreases, but that the tendency is weak and fluctuates. We confirm the superiority of the CEV model by systematic comparisons of the goodness of fit of the CDS data for each firm and each maturity, both in- and out-of-sample. We find that the constant volatility model systematically under-predicts the size of the spreads for all maturities but especially for medium term debt compared to CEV. In the latter model the CDS predictions are quite good, especially for the junk rated CDS contracts with intermediate and long-term maturities, for which the errors are less than half than those of constant volatility. The superiority persists also in out-of-sample tests, in which the model-predicted CDS spread is compared to the actual observed spread. The importance of this predictor stems from the fact that the CDS market is much more liquid than the bond market and estimates of credit spreads extracted from it are correspondingly less contaminated by liquidity factors. In what follows we complete the review of the earlier empirical work involving structural models and default risk, which is extended by this paper. A common result of these earlier papers that are based on constant volatility diffusion is the underestimation of the corporate bond yield spreads,9 the so-called ‘‘credit spread puzzle’’.10 One possible solution to the puzzle is an alternative stochastic process of asset (or cash flow) dynamics.11 Motivated by recent empirical evidence that asset volatility is time-varying12 and that the presence of stochastic volatility and jumps could improve the fitting of credit spreads even in the context of a Merton-type structural model,13,14 Elkamhi et al. (2011) try to explain this puzzle in the context of stochastic volatility asset dynamics. They use the two-dimensional Fortet equation approximation to calculate numerically the first passage default probability and then estimate the equity value by assuming that the maturity of the debt is infinite. Although their work is more general in its asset dynamics assumptions, our CEV approach yields analytical solutions for the firstpassage default probability and the equity and debt values. From these, we can derive the endogenous default boundary and the optimal capital structure, which are not available in their

5

Zhou (2001) and Collin-Dufresne and Goldstein (2001). Brennan and Schwartz (1978), and more recently Titman and Tsyplakov (2007) are examples of studies that rely on numerical simulations. 7 Most structural models are univariate and assume a constant riskless rate of interest. Longstaff and Schwartz (1995), Briys and de Varenne (1997), and CollinDufresne and Goldstein (2001) use bivariate diffusion models, in which the term structure of interest rates follows the Vacisek (1977) model and the asset value is a constant volatility diffusion. As the empirical work in Chan et al. (1992) shows, the Vacisek model does not fit actual term structure data. Further, Leland and Toft (1996) note that this bivariate diffusion refinement plays a very small role in the yield spreads of corporate bonds. 8 See also Cox and Ross (1976), Emmanuel and MacBeth (1982), Cox and Rubinstein (1985), and Schroder (1989). 6

9

See Elton et al. (2001), Huang and Huang (2012) and Eom et al. (2004) See Chen et al. (2009). 11 See Collin-Dufresne and Goldstein (2001) for the mean reverting leverage ratio, Sarkar and Zapatero (2003) for mean reverting cash flow, Leland (1998) for risk hedging with two risk levels, Huang and Huang (2012) for the double exponential jumps and Elkamhi et al. (2011) for the stochastic volatility. 12 See Choi and Richardson (2009). 13 See Zhang et al. (2009). 14 The Merton-type structural model assumes that the default event only occurs at the maturity of the zero-coupon debt. 10

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formulation. Second, our model is more general in allowing for stationary debt structures with both infinite and finite maturities, and it is well known that the maturity of the debt is an important factor driving the variables of interest. Our work also generalizes some aspects of the results of Bhamra et al. (2010) and Chen (2010), who show the close relationship between credit spreads and macroeconomics conditions. They use regime switching models under which there are only two possible states. In this paper we incorporate a more general framework that includes continuous states in which the volatilities are different. Our extensions also have implications for several strands of literature that have dealt with different problems in corporate finance. One of the main issues in corporate finance is the optimal capital structure choice, which originates in the classic Modigliani and Miller (1963) analysis of the levered firm in the presence of taxes, according to which capital structure is chosen as a tradeoff between the tax advantage of debt and the costs of possible bankruptcy. The pioneering work in this area that comes closest to our own approach is that of Leland (1994a,b, 1998), and LT.15 Since several authors have raised doubts on whether the trade-off approach can really be invoked to justify the observed leverage ratios, several studies have focused on agency problems between stockholders and debtholders or stockholders and managers.16 These and other related studies show clearly the importance of structural models in linking the default probabilities and yield spreads to the capital structure decision, a linkage that is missing from the reduced form models. The rest of paper is organized as follows. Section 2 presents the CEV model and summarizes some pertinent results from the option pricing literature. Section 3 contains the main theoretical results of the structural model, including the estimating equations for equity and debt, the endogenous default boundary and the optimal leverage. Section 4 presents simulated results comparing CEV to the equivalent constant volatility case under controlled conditions. Sections 5 and 6 contain the empirical work, with Section 5 presenting the firm-level results and the comparisons of the CDS estimates between CEV and constant volatility. Section 7 concludes. 2. Economic setup 2.1. Unlevered asset dynamics Following L (1994a, b), we consider a firm whose assets are financed by equity and finite maturity debt with tax-deductible coupons. As in all previous related literature, the values of the components on the firm’s balance sheet are estimated as contingent claims of the state variable V, the value of the unlevered firm’s assets representing its economics activities, which follows a mixture of a continuous diffusion process V with state-dependent volatility rðVÞ under the physical measure (or P- distribution):

dV ¼ ðl  qÞdt þ rðVÞdW V

ð2:1Þ

where l is the instantaneous expected rate of return of asset; q is the payout rate to the asset holders, including coupon payments to the debt holders and dividends to the equity holders; rðVÞ is a state dependent volatility; and W is a standard Brownian motion. The constant risk free rate is denoted by r. Under the risk neutral measure (Q-distribution), Eq. (2.1) becomes

15 See also, Sarkar and Zapatero (2003), Ju et al. (2005) and Titman and Tsyplakov (2007), 16 See Mella-Barral and Perraudin (1997), Leland (1998), Morellec (2004), and Ju et al. (2005).

dV Q ¼ ðr  qÞdt þ rðVÞdW V

ð2:2Þ

This mixed process continues until the asset value hits or falls below a threshold value (the default boundary), denoted by K, for the first time. In such a case, a default event will be triggered and liquidation comes in immediately. Assuming the absolute priority rule is respected, the bond holders receive ð1  aÞK, while the equity holders receive nothing. The remaining asset value that equals aK is considered a bankruptcy cost. Denoting the bond maturity by T and the first passage time when the asset value hits the default boundary by s, the asset value dynamics then become

(

dV t Vt

Q

¼ ðr  qÞdt þ rðVÞdW ; if 0 < t < s < T

V t ¼ minfV s ; Kg if 0 < s 6 t < T

ð2:3Þ

Under state-dependent volatility, we set rðV t Þ ¼ hV bt in which case the diffusion process in (2.3) becomes a CEV (constant elasticity variance or volatility) diffusion process. The parameter b, the elasticity of the local volatility, is a key feature of the CEV model. When b ¼ 0 the model becomes a geometric Brownian motion with constant volatility. b > 0 (b < 0) indicates that the state-dependent asset volatility is positively (negatively) correlated with the asset price.17 In equity markets, the well-known ‘‘leverage effect’’ shows generally a negative relationship between equity volatility and price. There are also some suggestions that the economically appropriate range is 0 > b > 1,18 even though empirical evidence in the case of the implied risk neutral distribution of index options finds negative values significantly below this range. Jackwerth and Rubinstein (2012) find that the unrestricted CEV model when applied to the risk neutral distribution extracted from S&P 500 index options is able to generate as good out-of-sample option prices as the better known stochastic volatility model of Heston (1993). Hereafter we shall adopt b 6 0 without any further restrictions as our base case. The CEV model yields a distribution of the asset value V T conditional on the initial value V 0 that has the form of a non-central chisquare v2 ðz; u; v Þ, denoting the probability that a chi-square-distributed variable with u degrees of freedom and non-centrality parameter v would be less than z. The shape of this distribution is given analytically most often in terms of its complementary form 1  v2 ðz; u; v Þ, denoting in our case the probability V T P v T . For b < 0 this probability is given analytically by19

ProbðV T P v T Þ ¼ 1  v2 ðc; b; aÞ ¼ v2 ða; 2  b; cÞ;

ð2:4Þ

Where

a ¼ tv 2b T ;

c ¼ tðV t eðrqÞT Þ

2ðrqÞ

t ¼  h2 b½e2ðrqÞbT 1

2b

;

b ¼ b1 ;

ð2:5Þ

This distribution is the equivalent of the lognormal when the volatility is constant. It has been tabulated and is easily available numerically. Several additional results hold about the v2 ðz; u; v Þ distribution when the parameter v is an even integer that can simplify the computations. 17 As Emmanuel and Macbeth (1982, p. 536) were the first to point out, for b > 0 the local volatility becomes unbounded for very large values of V, and there are technical issues concerning the mean of the process under both the physical and the risk neutral distributions. This problem is solved by assuming that the volatility is bounded and becomes constant for V exceeding an upper bound; see Davydov and Linetsky (2001, p. 963), A similar lower bound when b is <0 prevents the formation of an absorbing state at 0. 18 See Cox (1975), and also Jackwerth and Rubinstein (2012), who term this model the restricted CEV. The arguments in favor of the restricted CEV model are mostly applicable to index options and will not affect our formulation. 19 See Schroder (1989, pp. 213–214).

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2.2. Stationary debt structure

Yðg; V; KÞ ¼

We consider an exponential stationary debt structure, proposed by Leland in 1994, under which a firm keep issuing and retiring pieces of corporate debt of infinite maturity at a proportional rate g. Denote the face value by p and the continuous coupon payment by c per unit time for a piece of debt. Since all the pieces of debts outstanding are identical, the total outstanding principal and coupon payments are P ¼ p=g and C ¼ c=g, respectively. As time goes by, the remaining principal value of this debt at time t is egt P and the debt holders receive a cash flow egt ðC þ gPÞ at time t, provided the firm remains solvent. Hence, the average maturity of this debt will be, given that no default occurs,

Ta ¼

Z

1

gtegt dt ¼ g 1

ð2:6Þ

0

Thus, the average maturity under the L model is the reciprocal of the proportional retirement rate. In order to get a stationary debt structure we assume that the firm replaces the retired debt with newly issued debt having the same principal and coupon so as to keep the total principal and total coupon payments independent of time. Denoting the value of a piece of debt and total outstanding debts by dðg; V; KÞ and Dðg; V; KÞ, respectively, we have

Dðg; V; KÞ ¼

Z

1

0

dðg; V; KÞ dðg; V; KÞdt ¼ g

ð2:7Þ

When the volatility of the unlevered asset value is constant, Leland derived the debt value analytically. If the volatility is state dependent, the debt value depends on the structure of statedependent volatility. In the following section we derive an analytical solution for a particular case of state dependent volatility when

rðVÞ ¼ hV b which is the CEV form of the volatility.

3. Corporate debt and equity valuation and endogenous default boundary

Consider a piece of outstanding debt denoted by dðg; V; KÞ with proportional retirement rate g and first passage default time s. If there is no default event before the maturity (0 < t < s),20 it pays a nonnegative coupon egt c and also retires the principal gegt p at time t. If the asset value hits the default boundary K before maturity for the first time the debt defaults and the debt holders recover gð1  aÞK. Thus, under the risk neutral measure, given the risk free rate r, the value of this debt can be expressed as

Z s

ers ðegs c þ gegs pÞds þ eðrþgÞs gð1  aÞK

 ð3:1Þ

0

Define the first passage probability density function as f ðs; V; K jg Þ and the value of the digital option, the present value of one dollar payment upon default as Yðg; V; KÞ. We have,

Yðg; V; KÞ ¼

Z

1

ers f ðs; V; KÞds

ð3:2Þ

0

Lemma 1. When the state dependent volatility is given by the CEV process rðVÞ ¼ hV b , the present value of one dollar payment upon default is

20 The firm is solvent only when the asset value is above the threshold value for bankruptcy all the time and never below it, starting from the issue date of this bond.

ð3:3Þ

where,

( /r ðVÞ ¼

1



V bþ2 e2x W k;m ðxÞ; b < 0; V

bþ12 x

e2 M k;m ðxÞ; b > 0;

x ¼ jrqj V 2b ; h2 jbj

 ¼ signððr  qÞbÞ; 

k¼

1 2



r–0 r–0

ð3:4Þ

1 m ¼ 4jbj

1 r  2jðrqÞbj þ 4b

W k;m ðxÞ and Mk;m ðxÞ are the Whittaker functions.21 Proof. See appendix. h Observe also from (3.2) that (3.3) and (3.4) allow the quasianalytical derivation of the CEV first passage to default probability density f ðs; V; KjgÞ, by inverting its Laplace transform h i f ðs; V; KjgÞ ¼ L1 //kk ðVÞ . From the properties of the Laplace transðKÞ form, the cumulative first passage to default probability (CDP), which is studied in detail in Leland (2004) and Huang and Huang (2012), is then given by the inversion Fðs; V; K jg Þ ¼ L1 ½1k

/k ðVÞ . /k ðKÞ

The next result is also proven in the appendix. Lemma 2. The value of corporate debt with proportional retirement rate g is equal to,

dðg; V; KÞ ¼

c þ gp ½1  Y rþg  þ gð1  aÞKY rþg ; rþg

ð3:5Þ

where Y rþg is given by (3.3). The value of corporate debt in (3.5) is equivalent to Eq. (10) in Leland (1994b) but with different probability density functions. Equation (2.7) allows the derivation of the analytical solution for the value of total outstanding debt, which by (2.7) is equal to,

Dðg; V; KÞ ¼

3.1. Corporate debt valuation

dðg; V; KÞ ¼ E

/rþg ðVÞ  Y rþg /rþg ðKÞ

C þ gP ½1  Y rþg  þ ð1  aÞKY rþg rþg

ð3:6Þ

3.2. Equity valuation The equity holders have the right to claim the residual cash flows in the form of dividends, after coupon payments and the costs of retiring debt continuously. Under the stationary debt structure a firm continuously issues and retires the bonds to keep the value of total outstanding debt time invariant. Let EðV; gÞ denote the value of the equity and w denote the corporate tax rate. Similar to Eq. (11) of He and Xiong (2012), the equity value for general state-dependent volatility rðVÞ satisfies the following equation,

1 rEðV; gÞ ¼ ðr  qÞVEV þ ðrðVÞÞ2 V 2 EVV þ qV  ð1  wÞC 2 þ gDðV; gÞ  gP

ð3:7Þ

The first two terms in equation (3.7) reflect the change in equity value because of the dynamic change in the unlevered firm’s assets, the third and fourth terms are cash inflows from dividends and after tax cost of debt coupon, while the last two terms represent the change in equity value by the debt issuance cost absorption, with debt retired at face value but refinanced at market value. For instance, if the new debts are issued at a discount the difference between the last two terms becomes negative and imposes costs 21 See Whittaker and Watson (1990, pp. 339–351). The Whittaker functions are the fundamental solutions for the Whittaker equation.

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to equity holders through the debt rollover channel. On the other hand, the coupon payments are relatively lower given the new debts are issued at discount. Presumably, these two effects will cancel each other provided the new debts are fairly priced. To derive the analytical solution for the equity value we need to solve equation (3.7) with two boundary conditions. First, when the asset value hits the default boundary the equity value should be zero. Second, when the asset value approaches infinity the equity value should increase approximately linearly with respect to the unlevered asset value. Lemma 3. The analytical solution for equation (3.7) given the two boundary conditions above is,

wC EðV; gÞ ¼ V þ ½1  Y r ðV; KÞ  aKY r ðV; KÞ r     C þ gP C þ gP Y rþg ðV; KÞ  þ ð1  aÞK  rþg rþg Proof. See appendix. h

L (1994a, b) and LT derived the equity value by the trade-off theory. Specifically, since a firm’s asset base consists of debt and equity only, the value of equity could also be evaluated as a residual from the value of total debts, the tax benefits created by the tax deductible coupon payments and the bankruptcy costs. The tax benefit available to the firm equals the total tax benefit for a risk-free bond minus the tax benefit losses due to the default event, which yields the second term in (3.8). The third term is the present value of the bankruptcy loss due to the default event and the last term is the present value of all the outstanding debts. In the previous sections we assumed that default happens when the state variable V drops below a default boundary, K. This default trigger value can be determined exogenously or endogenously. If a firm cannot choose its default boundary value then this boundary can be determined by a zero-net worth trigger22 or by a zero cash flow trigger.23 Under the zero-net worth trigger assumption, the default occurs when the net worth of the firm becomes negative for the first time, which implies that the default trigger value equals the total face value of the outstanding debt, namely K ¼ P. However, we often observe that firms are still alive even though their net worth is negative in the financial markets. Thus, in order to improve the simple zero net worth trigger, Moody’s KMV defines the trigger value K ¼ P Short þ 0:5 PLong . Under zero cash-flow trigger, a firm claims default when the current net cash flow to the security holders cannot meet the current coupon payments, which implies K ¼ C=d, where d is the net cash flow to the security holders. The problem for this trigger value is that sometimes the equity value is still positive even though the current net cash flow is zero. In this case a firm will prefer to issue more equity so as to meet the current coupon payment, instead of announcing default. On the other hand, if a firm is capable to choose its default boundary value, this default boundary value will be set endogenously by maximizing total firm value. Following Leland (1994b) and LT, we may find the optimal endogenous default boundary by the smooth-pasting condition,24

@EðV; KÞ jV¼K ¼ 0 @V

22

ð3:9Þ

See Brennan and Schwartz (1978), and Longstaff and Schwartz (1995). See Kim et al., 1993. 24 LT verified numerically that the smooth-pasting condition has the property of maximizing both the value of equity and the firm simultaneously subject to the limited liability of equity. Chen and Kou (2009) prove this result analytically using local convexity at the default boundary. We verify the local convexity at the default boundary numerically for all the cases discussed in this paper. 23

According to (3.9), the endogenous default boundary under the CEV diffusion process, denoted by K e , can be obtained by solving the equation for given assumptions about the volatility dynamics. The following result is also proven in the appendix. Proposition 1. According to the smooth pasting condition (3.9), the endogenous default value under the CEV diffusion process, denoted by K e , can be obtained by solving the following equation for given parameter values,

  wC 1 @/r ðK e Þ þ aK e r /r ðK e Þ @K e   @/rþg ðK e Þ C þ gP 1 þ  ð1  aÞK e rþg /rþg ðK e Þ @K e

1



¼0 ð3:8Þ

219

ð3:10Þ

where

h i 8 W ðxÞ bþ0:5 > þ 0:5e þ 0:5  kx  Wkþ1;mðxÞx x0 ; if b < 0 > K k;m < 1 @/r ðKÞ h i Mkþ1;m ðxÞðkþmþ0:5Þ ¼ bþ0:5  kx þ 12 x0 ; if b > 0 > /r ðKÞ @K Mk;m ðxÞx > K þ 0:5e þ : ð3:11Þ Although there is no explicit solution for the endogenous default values, it is straightforward to find it from (3.10) by using a root finding algorithm, provided a positive root exists. We examine the properties of the solution as well as the other variables of interest of the model in numerical examples in the following sections. 4. Calibrations In the calibration exercises of this section we consider a base case with the following parameters: current asset value V ¼ 100, risk free rate r ¼ 0:08, firm’s payout rate q ¼ 0:06, tax rate w ¼ 0:35, proportional bankruptcy cost a ¼ 0:5, and initial volatility of asset r0 ¼ 20%. Although some of these parameters may not reflect current conditions, they were chosen based on previous studies closely related to this paper, such as LT and Leland (2004), with which the results of this study need to be compared in order to assess the impact of its more general formulation. The remaining parameters will assume various values according to the studied topic. The first three subsections study the effects of the parameter b on the endogenous variables of interest, while Section 4.4 focuses on important economic variables implied by historically observed endogenous quantities under the CEV model with b ¼ 0:5, the average elasticity estimated from our sample in Section 5. 4.1. Endogenous default triggers Fig. 1 shows the endogenous default boundaries for the L model, which is a special case of the CEV structural model when b equals zero, and four CEV structural models with b ¼ 1; 0:5; 0:5; 1. As expected from the L and LT models’ results, the endogenous default trigger declines as the average maturity of the bond increases in all cases. After incorporating the CEV process, we find that the smaller b is, the faster the endogenous default boundary declines. If b is negative and large in absolute value, and the average maturity of the debt is long enough, the corresponding endogenous default boundary could be close to zero, which implies that this firm would never choose to go bankrupt endogenously even though the net worth of the firm may be negative. On the other hand, if b is positive and large in absolute value, the endogenous default trigger of the CEV model decreases slowly and is greater than that of the L model.

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alive. Thus, the equity holders will choose to issue equity in order to fund the coupon payment for the debt holders until the equity value goes to zero. This behavior leads to a lower recovery value for the bond holders, which increases the risk of corporate bonds.25

70

Endogenous Default Trigger

60

50

4.2. Credit spreads 40

30

20

10

0 0

2

4

6

8

10

12

14

16

18

20

Average Maturity (T) Fig. 1. Endogenous default boundaries. This figure depicts the values of endogenous default triggers for the Leland (1994b) model (bold solid line) and CEV structural models with b ¼ 1 (dashed line), b ¼ 0:5 (dotted line), b ¼ 0:5 (bold dotted line) and b ¼ 1 (bold dashed line). It is assumed that current asset value V ¼ 100, current debt value D ¼ 50, risk free rate r ¼ 0:08, the firm’s payout rate q ¼ 0:06, tax rate w ¼ 0:35, and proportional bankruptcy cost a ¼ 0:5. The coupon rate is determined by making the debt issued at par value under the endogenous default boundary. The initial volatilities are the same for all these scenarios, and r ¼ 20%. For each given b under the CEV diffusion process, h ¼ r0 =V b .

These changes of endogenous default triggers under the CEV structural model can be understood economically from the point of view of the relationship between anticipated equity value and volatility. When b is negative (positive), the volatility of the asset increases (decreases) when the asset value decreases. Recall that it is well known since Merton (1974) that the equity in a levered firm can be interpreted as a call option on the value of the assets. Similarly, Merton (1973) showed that in many cases of underlying asset dynamics, including those used in this paper, the value of the option is an increasing function of the volatility. It follows that ceteris paribus the anticipated increase in the value of the equity will be inversely proportional to the value of b, while the default boundary will also vary inversely with this anticipated equity appreciation. In other words, at low values of V where the probability of default is high, the increase in volatility when b is negative will counteract the fall in the value of equity because of the fall in V. Next we examine the effect of the leverage ratio, DðV; K; gÞ=v ðV; K; gÞ, on the default boundary for two different values of maturity, or of its inverse g. The debts are always issued at par, implying that the RHS of equations (3.6) is set equal to P. The L model shows a strictly increasing function of this endogenous default boundary with respect to the leverage ratio, depicted by the solid lines in Fig. 2. As expected, the monotone increasing property of the default boundary as a function of the leverage ratio is preserved, but the speed of increase depends on the value of b. Note also that for firms with a low leverage ratio and a negative b there is a non-endogenous default zone in which a firm would never choose to go bankrupt endogenously, especially for the lowest value of b ¼ 1. By comparing the endogenous default triggers for CEV structural models with b ¼ 1 and b ¼ 0:5 we see the non-endogenous default zone becomes wider when b decreases. Again, this lowering of the default boundary to about zero is consistent with the volatility effect that causes ceteris paribus an appreciation of the equity treated as an option whenever the underlying value V decreases. In the non-endogenous default zone, the anticipated equity value is high enough to dominate the required cash outflow for debt required to keep the firm

Following Leland (1994b), we calculate the credit spreads of newly issued debt by (C=D  r) for various scenarios under both exogenous and endogenous default boundaries, and report the results in Table 1. Under the exogenous default boundaries in Panel A, we let the coupon payments be the same across all three scenarios and equal to the coupon payment that make the debt issued at par with the endogenous default boundary when the asset volatility is constant. We also use this endogenous default boundary as the exogenous default boundary for the cases under which the elasticity parameters equal 1, 0.5 and 1, respectively.26 We find very consistent results: as the value of the elasticity parameter decreases, the credit spread increases, indicating that a decrease of the elasticity parameter implies an increase of the expected default probability. Panel B exhibits the results in the cases of the endogenous default boundaries that are determined by a firm itself in order to maximize the total firm value. As discussed in the previous section, the endogenous default boundary decreases as the value of the elasticity parameter decreases because the negative (positive) elasticity parameter increases (decreases) the anticipated equity appreciation, which motivates the equity holders to lower (increase) the default boundary to keep the firm solvent. Intuitively, the expected default probability should increase (decrease) as the default boundary decreases (increase) when the value of the elasticity parameter is negative (positive). Further, we assume a fixed proportional bankruptcy cost of about 50%, implying that the recovery rate for the bond holders is Minð100%; ð1  50%ÞK=PÞ, very close to the historical average from Moody’s of 51.31% used in HH, where K and P denote, respectively, the default boundary and the face value of the debt. Since the endogenous default boundaries introduce variations in recovery rates, the impact of state-dependent volatility on the credit spreads depends on both the change in default probability as well as in the recovery rate. For short-term debts with negative elasticity parameters the reduction of default probabilities dominates the reduction of recovery rates because the endogenous default boundaries are high, usually above the face value of the debt, suggesting a decrease of credit spreads compared to the case with constant asset volatility, and vice versa for long-term debts. 4.3. Optimal capital structure Under an endogenous default boundary and a pre-determined debt structure, the optimal capital structure that maximizes the total firm value can be achieved by altering the leverage ratio,27 the ratio of the total outstanding debt value over the total firm value, D=v . Fig. 3 shows the relationship between total firm value and leverage ratio for bonds with average maturities from 1 to 20 years, and Table 2 reports the optimal leverage ratios and the values of key 25 In fact the default boundary cannot be close to zero in real life cases because of protective covenants on bonds or loans. In our empirical analysis we use an exogenous default boundary for our parameter estimations to avoid such extreme cases and show the corresponding endogenous boundaries. 26 In the empirical data the elasticity parameter is about 0.5. We also incorporate the cases, b ¼ 1 and b ¼ 1, to consider more general scenarios. 27 The detailed procedure first selects a starting value for the face value of the debt and calculates the corresponding debt, equity and total firm values with endogenous default boundary according to the base case calibrations. Last, it chooses the face value of the debt that maximizes the total firm value.

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Endogenous Default Trigger

T= 5 80

60

40

20

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.6

0.7

0.8

0.9

Leverage Ratio

Endogenous Default Trigger

T = 20 80

60

40

20

0

0

0.1

0.2

0.3

0.4

0.5

Leverage Ratio Fig. 2. Endogenous default trigger as a function of leverage ratio. This figure depicts the value of the endogenous default trigger with respect to the leverage ratio under the Leland (1994b) model (solid line) and CEV structural models with b ¼ 1 (bolded dashed lines), b ¼ 0:5 (bolded dotted lines), b ¼ 0:5 (dotted lines) and b ¼ 1 (dashed lines). It is assumed that current asset value V ¼ 100, risk free rate r ¼ 0:08, the firm’s payout rate q ¼ 0:06, tax rate w ¼ 0:35, and proportional bankruptcy cost a ¼ 0:5. The coupon rate is determined by making the debt issued at par value under the endogenous default boundary. The initial volatilities are the same for these scenarios and r0 ¼ 20%. For each given b ¼ under the CEV diffusion process, h ¼ r0 =V b .

Table 1 Credit spreads. P

1-year 40

5-year

10-year

60

40

50

60

40

50

60

default boundaries 15.86 83.31 17.08 88.29 18.78 91.86 21.30 95.86

327.99 340.07 346.16 352.05

16.30 22.92 33.36 46.13

39.58 54.60 67.79 81.42

88.73 114.95 130.16 144.14

12.54 22.81 39.99 59.09

25.78 46.33 66.09 85.23

51.36 85.35 106.60 125.17

Panel B: Endogenous default boundaries b¼1 16.01 86.87 b¼0 17.08 88.29 b ¼ 0:5 17.86 88.80 b ¼ 1 18.72 89.06

358.64 340.07 334.84 330.35

16.46 22.92 28.35 33.57

42.82 54.60 59.63 62.89

105.76 114.95 116.78 116.44

12.68 22.81 32.04 41.99

28.70 46.32 54.75 60.30

66.67 85.35 90.51 91.98

Panel A: Exogenous b¼1 b¼0 b ¼ 0:5 b ¼ 1

50

This table reports the credit spreads with varying elasticity parameter b and face value of debt P under both exogenous (Panel A) and endogenous (Panel B) default boundaries. The coupon payments are assumed to be the same as the coupon that makes the debt issued at par when the elasticity parameter equals zero. It is assumed that current asset value V ¼ 100 dollars, risk free rate r ¼ 0:08, the firm’s payout rate q ¼ 0:06, tax rate w ¼ 0:35, and proportional bankruptcy cost a ¼ 0:5. The initial volatilities are the same for these scenarios and r0 ¼ 20%. For each given b under the CEV diffusion process, h ¼ r0 =V b .

endogenous variables at optimal leverage. As the average maturity increases, the optimal leverage ratio increases and the optimal total firm value increases as well. Based on the results in Figs. 1 and 2, we anticipate that the optimal leverage ratios are affected by the value of the elasticity parameter b for given debt characteristics. We also expect that the negative values of the elasticity parameter are more leverage-friendly. These indeed turn out to be the cases for intermediate or long term debt structures, with the optimal leverage ratios increasing (decreasing) with the absolute value of the elasticity parameter when the elasticity parameter is negative (positive). The dependence of the leverage on b is weaker for short-term debt structures, less than or equal to 1-year, with the optimal leverage ratios decreasing significantly when b rises from 1 to 0, but then staying approximately constant when the elasticity parameter

increases from 0 to 1. Similarly, the total firm value and the total debt value are both increasing functions while the total equity value is a decreasing function of the average maturity of the debt under each scenario considered in Table 2. This effect is consistent with the trade-off theory, which balances the tax benefits and bankruptcy costs of the firm in order to maximize the total firm value. For the long term debt, the present value of anticipated tax benefits that accumulate over time should dominate the present value of anticipated bankruptcy costs because of the lower endogenous default boundaries compared to those of short-term debt, thus increasing both optimal leverage and firm value. Given the intermediate- or long-term debt structure, we observe that the total firm value decreases as the value of the elasticity parameter increases. As discussed in the previous section, the negative elasticity parameter implies a relatively higher anticipated default

S. Perrakis, R. Zhong / Journal of Banking & Finance 55 (2015) 215–231

4.4. Empirical fitting of the historical default probabilities, leverage ratios and equity premia Huang and Huang (2012) first document the so-called ‘‘credit spread puzzle’’, the failure of structural models to explain the observed corporate yield spreads after fitting the empirical datasets. In this subsection, we use the CEV structural model to conduct this exercise again and report the results in Table 3. We use the same calibrations as HH to fit the rating classes from Aaa to B, respectively, for 10- and 4-year maturities.28 We set the elasticity parameter to b ¼ 0:54, as estimated in Section 5. For bonds of each maturity and rating class we find the model-implied initial asset value, asset volatility and asset risk premium that match the observed leverage ratio, equity premium and cumulative default probabilities shown in columns 2–4 of Table 3. Comparing the percentage of the model-implied credit spreads due to default from the CEV model, shown in the last column of Table 3, with the equivalent results from the Collin-Dufresne and Goldstein (2001) model shown in Table 6 of HH, we see that CEV achieves a much better approximation to the observed average yield spreads for all debt ratings greater than or equal to Baa; it only fails for the two lowest debt categories.29 Chen et al. (2009) note the importance of systematic risk and a time-varying Sharpe ratio in explaining the credit spread puzzle. As our calibration essentially incorporates the equity Sharpe ratio 28 The detailed calibrations are described in Section 3.1 in Huang and Huang (2012, pp. 167–171). 29 Note also that Huang and Zhou (2008) judge the Collin-Dufresne and Goldstein (2001) model as the best structural model in approximating observed reality.

Firm Value

T= 1 110 105 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.5 0.6 Leverage

0.7

0.8

0.9

1

T = 10 Firm Value

probability. At the same time, it also implies a relatively lower bankruptcy costs because of the lower endogenous default boundary. These mixed effects indicate a decrease of the value of outstanding debts compared to the case with constant volatility. As we select the coupon payment that makes the debt issued at par for each scenario, the anticipated tax benefit should be much higher for the scenarios with negative elasticity parameters because of the relatively large coupon payments compared to the case with constant asset volatility. Table 2 examines the risk characteristics of the optimally levered firm with the equity risk measured by equity volatility and the debt risk measured by debt volatility and credit spread. For a given debt maturity, both equity volatility and debt volatility monotonically increase when the value of the elasticity parameter decreases; the same is true for the credit spread. For all the debt maturities considered in Table 2 the largest equity risk and debt risk measured by both risk metrics is always reached when the elasticity parameter equals 1, the smallest one in our calibrations. On the other hand, for any given value of the elasticity parameter the volatility of debt increases monotonically with the average debt maturity. The volatility of equity and the credit spread, however, first increase and then decrease with maturity for the elasticity parameter equals to 1 and 1, while they increase with maturity for the elasticity parameter equals to 0. Apparently, the impact of the magnitude of the elasticity parameter on optimal debt financing is major. In Table A2 of our online appendix we also verified the ability of the CEV model to approximate the observed CDP for various bond risk categories. As noted in Leland (2004), the constant volatility assumption does not hold under realistic parameter values, especially for low maturities. We find that the CEV model’s b parameter can be calibrated to yield a satisfactory fit to the observed CDPs for all rating categories, but that the values are all negative and unrealistically high in absolute values. We revisit this issue in the following subsection and also in Section 5.

120 110 100

0

0.1

0.2

0.3

0.4

0.5 T = 20

Firm Value

222

120 110 100

0

0.1

0.2

0.3

0.4

Fig. 3. Total firm value as a function of leverage ratio. This figure depicts the total firm value with respect to the leverage ratio under the Leland (1994b) model (solid line) and CEV structure models with b ¼ 0:5 (dashed lines) and b ¼ 0:5 (dotted lines). Three scenarios of average debt maturity are considered, T ¼ 1, T ¼ 10 and T ¼ 20. It is assumed that current asset value V ¼ 100, risk free rate r ¼ 0:08, the firm’s payout rate q ¼ 0:06, tax rate w ¼ 0:35, and proportional bankruptcy cost a ¼ 0:5. The coupon rate is determined by making the debt issued at par value under the endogenous default boundary. The initial volatilities are the same for these scenarios and r0 ¼ 20%. For the given b under the CEV diffusion process, h ¼ r0 =V b .

implicitly by fitting the equity premium explicitly, we calculate the implied Sharpe ratio indirectly and report the results in Table 3 as well.30 Compared to the average equity Sharpe ratio of about 22% for Baa rating, calibrated in Chen et al. (2009), the range of our Sharpe ratios are from 19.78% to 29.92%, which are almost consistent with the empirical evidence. 5. Empirical evidence from the credit default swap market 5.1. Data description In this section, we recognize the characteristics of individual firms and estimate the L model with firm-level information from the financial statements, equity market, debt market, option market and Credit Default Swap (CDS) markets. The credit spreads are obtained from Markit during the period from January, 2001 to December, 2011. We limited our sample to United States firms. We select single name contracts denominated in US dollars with senior unsecured debts and modified restructure (MR) clause. We only keep the single name contracts which have at least 60 consecutive months’ observations. As the reported frequency of CDS database is daily, the CDS spread on the last Wednesday in each month is extracted as the CDS spread in that month. The accounting and equity information are extracted from the COMPUSTAT and CRSP data bases respectively. We calculate the total assets as the sum of book value of debt and market value of equity. The firms’ payout ratio is represented by the sum of cash dividends and interest payments divided by the total assets. As the frequency of accounting information is quarterly, we convert it into monthly by assuming that the values are constant within each quarter. We consider option implied volatilities31 that are 30 HH (2012) use the same approach and find a ratio of 19.4% for Baa bonds with 10year maturity. 31 Cao et al. (2010) document that individual firms’ put option-implied volatility dominates historical volatility in explaining the time-series variation in CDS spreads since the implied volatility is a more efficient forecast for future realized volatility than historical volatility and also the volatility risk premium embedded in option prices co-varies with the CDS spread.

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42.41 34.50 31.80 30.45

35.63 30.88 25.94 27.99

0.71 0.19 3.24E4 3.12E2

39.34 45.45 67.69 54.88

80.10 70.01 46.64 58.12

58.40 50.81 30.24 40.82

8.90 5.94 0.14 2.69

286.30 205.30 6.45 100.51

123.37 119.40 116.83 116.63

36.49 39.40 61.05 46.99

86.89 80.00 55.78 69.64

55.50 53.66 33.60 45.69

9.93 8.32 0.53 4.92

245.62 225.12 20.98 147.46

130.43 126.85 120.95 124.43

29.43 31.34 59.60 36.61

101.00 95.51 61.35 87.82

52.03 53.08 33.52 49.53

11.20 10.43 0.39 7.69

165.33 176.50 12.36 153.83

Optimal leverage (percent)

Firm value (dollars)

Equity value (dollars)

1-year average maturity b ¼ 1 3.59 b ¼ 0:5 2.80 b¼1 2.54 L(b ¼ 0) 2.44

37.52 35.95 38.76 35.67

39.05 32.23 28.90 28.44

108.59 107.05 110.00 107.06

66.18 72.55 78.21 76.61

5-year average maturity b ¼ 1 8.70 b ¼ 0:5 7.04 b¼1 3.76 L(b ¼ 0) 5.23

51.83 50.54 41.29 46.36

67.06 60.64 40.79 51.43

119.44 115.46 114.33 112.99

10-years average maturity b ¼ 1 9.09 46.95 b ¼ 0:5 8.20 50.16 b¼1 4.58 43.27 L(b ¼ 0) 6.60 48.09

70.43 67.00 47.74 59.71

Infinite average maturity b ¼ 1 9.75 36.23 b ¼ 0:5 9.33 43.78 b¼1 4.98 36.19 L(b ¼ 0) 8.38 45.37

77.44 75.29 50.73 70.58

Coupon (dollars)

Credit spread (basis points)

Equity volatility (Percent)

Bankruptcy trigger (dollars)

Models

Total debt value (dollars)

48.31 13.07 0.03 2.3

This table exhibits the characteristics of optimally levered firms under the Leland (1994b) and CEV structural models with 1-year’s, 5-years’, and 10-years’ average maturity. It is assumed that current asset value V ¼ 100 dollars, risk free rate r ¼ 0:08, the firm’s payout rate q ¼ 0:06, tax rate w ¼ 0:35, and proportional bankruptcy cost a ¼ 0:5. The leverage is chosen by maximizing total firm value and the coupon rate is determined by making the debt issued at par value under the endogenous default boundary. The initial volatilities are the same for these scenarios and r0 ¼ 20%. For each given b under the CEV diffusion process, h ¼ r0 =V b .

extracted from the Optionmetrics data using at-the-money call options as proxies for equity volatility. The risk free rates are interpolated from the observed 6-month Libor rates and 1, 2, 3, 5, 7, 10 years interest rate swap rates.32 The final sample consists of 104 firms whose detailed characteristics are reported in Table 4. Table 5 reports the distribution of individual firms in term of industries and ratings.33 Approximately half of the firms belong to consumer goods and industrials. There are only four firms in technology and telecommunication services, while the rest of the firms are almost equally distributed in basic materials, energy, healthcare and consumer services. The average payout ratios are equal in all the industries, about 1%, which is much lower than the calibration values used in Leland and Toft (1996). The average leverage ratio across all the industries is around 38%. The highest implied volatility from the option markets occurs in energy, around 31%, while the average implied volatility in the full sample is around 27%.34 In term of credit default swap spreads, the firms in consumer services have the highest spreads and these in healthcare have the lowest spreads across all the maturities. We also note that the highest credit spreads do not coincide with the highest leverage ratios, indicating that industry is potentially an important factor in the determination of credit spreads, most probably because of the recovery rates in case of default, that enter into the CDS spread determination.35 Panel B reports the rating distribution of individual firms. Most of the sample firms are rated as A and BBB, approximately 82%, while the numbers of firms in the highest and lowest ratings are relatively small. As there is only 1 firm in the B and CCC ratings respectively, we attribute the abnormal behavior of leverage, 32

See Longstaff and Schwartz (1995) and Collin-Dufresne and Goldstein (2001). Both industries and ratings classifications are extracted from the Markit database. The implied volatility of Telecommunication services is only 22%. As there is only one firm in this industry which is not enough to represent the whole industry, we leave this outside our discussion. 35 Acharya et al. (2007) studied the impact of the industry factor on credit spreads through the recovery channel and documented that creditors of defaulted firms recover significantly lower amounts in present-value terms when the industry of the defaulted firms is itself in distress. 33 34

implied volatility and credit spreads in these ratings to firm specific characteristics. A comparison of our sample’s characteristics with corresponding data from other studies summarized in HH (2012, p. 165) shows that for both the A and BBB classes that form the overwhelming majority of our sample the average leverage matches almost exactly the historical data. By contrast, the observed yield spreads are less than half as large for our sample than for the equivalent HH data for both 4-year and 10-year maturities, reflecting the much different financial markets in the approximately 20 years that separate our data from theirs. Similarly, the estimated recovery rates in the Markit data base shown in Table 4 are all much lower for our sample than the average value of 51.31% in HH. 5.2. Methodology For the parameter estimation we use the Generalized Method of Moments (GMM) method. Denote the observed and model implied Obs equity volatility by rObs and rIm and EIm , leverE E , equity value by E

age ratio by Lev Obs and Lev Im , the estimation parameter set as w1 ¼ ðr0 ; bÞ, respectively. This approach obviously nests the constant volatility case by setting b ¼ 0. At each time point t ¼ 1; . . . ; T of our data base we have the following vector f ðw; tÞ, a function of the parameter set

8 9 > CDSObs ðt; T 1 Þ  CDSIm ðt; T 1 Þ > > > > > > > > > < = .. . f ðw; tÞ ¼ > > CDSObs ðt; T 10 Þ  CDSIm ðt; T 10 Þ > > > > > > > > : ; Im rObs ðtÞ  r ðtÞ E E

ð5:1Þ

where CDSIm ðt; T j Þ; j ¼ 1; 2; 3; 5; 7; 10 denote the CDS spreads for denotes the implied equity volatility. various maturities and rIm E Given the unlevered asset value and estimation parameters, the implied volatility rIm E can be calculated by,

rIm E ¼

@EðVÞ V b hV @V E

ð5:2Þ

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Table 3 Calibrated yield spread by fitting leverage ratios, equity premia and cumulative default probabilities. Credit rating

Target

Aaa Aa A Baa Ba B Aaa Aa A Baa Ba B

Implied

Leverage ratio (%)

Equity prem (%)

Cum default prob (%)

Asset vol (%)

Asset risk prem (%)

Sharpe ratio (%)

13.1 21.2 32.0 43.3 53.5 65.7 13.1 21.2 32.0 43.3 53.5 65.7

5.38 5.60 5.99 6.55 7.30 8.76 5.38 5.60 5.99 6.55 7.30 8.76

0.77 0.99 1.55 4.39 20.63 43.91 0.04 0.23 0.35 1.24 8.51 23.32

20.01 19.46 19.28 21.36 30.11 41.31 21.26 22.41 21.19 22.39 29.33 37.47

5.27 5.42 5.74 6.39 8.09 11.79 5.24 5.39 5.64 6.15 7.45 10.46

26.34 27.85 29.77 29.92 26.87 28.54 24.65 24.05 26.62 27.47 25.40 27.92

Model credit spread (bp)

Avg yield spread (bp)

% of Spread due to default

57 68 93 180 536 1221 25 48 59 113 355 847

63 91 123 194 320 470 55 65 96 158 320 470

90.48% 74.73% 75.61% 92.78% 167.5% 259.8% 45.45% 73.85% 61.46% 71.52% 110.9% 180.2%

This table reports the CEV model’s calculated credit spreads for b ¼ 0:54, by fitting the leverage ratios, equity premia and cumulative default probabilities for various rating categories. It is assumed that the risk free rate is 8%; payout rate is 6%; exogenous default boundary is 60% of the face value of debt. We set the coupon rate for risk-free bonds with semi-annual coupon payments when the continuously compounded interest rate is 8%. We calibrate the initial asset value, asset volatility and asset risk premium to fit the observed leverage ratio, equity premium and cumulative default probabilities. Panels A and B are for 10- and 4-year bond maturities respectively.

Setting g ¼ T 1 everywhere the CDS spreads cj are given by the folj lowing expression in our continuous time notation, with R denoting the estimated recovery rates, whose estimates are also available in the CDS data base

cj ¼

RT ð1  RÞ t j f ðs; V:KÞers ds RT ½1  AðsÞers ds t

ð5:3Þ

database. Under these assumptions, the L model has only one variable, asset volatility, to estimate, while the CEV structural model has one extra parameter, the elasticity of variance, besides the initial volatility level. Since we can only observe the levered asset value, we convert the levered asset value into the unlevered asset value by solving the following equation based on the trade-off model, with the two extra terms representing the tax benefits of debt and the bankruptcy costs respectively

where

AðTÞ ¼

Z

v¼Vþ

tþT

f ðs; V:KÞds

Discretizing this expression in terms of quarters si ; i ¼ 1; :::; 4T j , and setting Dð0; si Þ and Q ð0; si Þ for the discount factor and survival probabilities respectively in the time interval ½0; si , we have for CDSð0; T j Þ, the total spread paid by the default protection buyers in ½0; T j 

ð1  RÞ

P4T j

i¼1 Dð0; si Þ½Q ð0; si1 Þ  Q ð0; si Þ P4T j i¼1 Dð0; si ÞQð0; si Þ=4

ð5:4Þ

As discussed in Section 3, the cumulative first passage time to default probability distribution for the risk neutral state dependent volatility processes is given in terms of its Laplace transform by the following expression

AðT j Þ ¼

ð5:6Þ

Since the GMM estimates w by minimizing E½f ðw; tÞ, we set

t

CDSð0; T j Þ ¼

wC ½1  Y r ðV; KÞ  aKY r ðV; KÞ r

 1 1 /k ðVÞ  k /k ðKÞ

ð5:5Þ

with the expressions in parentheses given by (3.4) in the case of the CEV distribution; k is equivalent to r or r þ g in equations (3.3), (3.4). We use the KMV method for the default boundary K. To determine the model implied moments numerically, we need both accounting and equity information. It is assumed that the time-varying exogenous default boundary equals the value of current debt plus one-half of the long term debt36 for both candidate models. The asset values are the sum of book value of debt plus the market value of equity; the latter equals the product of stock price and outstanding shares. The firm’s total payouts are the sum of cash dividends to shareholders and interest payments to debt holders. The corporate tax rate is assumed to be 35% and the recovery rate is equal to the estimated recovery rate in the Markit 36 This exogenous default boundary was introduced by the KMV group and is widely used.

Gðw; TÞ ¼

T 1X f ðw; tÞ T 1

ð5:7Þ

and estimate w by the relation

^ ¼ arg min Gðw; TÞ0 WGðw; TÞ w

ð5:8Þ

where W is a matrix of weights that is computed by successive approximations.37 5.3. Empirical results We carry out the GMM estimation with seven moments including the information from financial statements, equity, option and CDS markets, whose average values of parameters are shown in Tables 4 and 5.38 For the entire sample, the average values of the elasticity parameter, b and initial asset volatility, r0 are around 0.54 and 20%, respectively, and both of them are significantly different from zero at the highest conventional confidence level. Over 72% of the firms in the sample have negative elasticity parameters, indicating a negative relationship between asset value and asset volatility. In other words, we document a significant skewness in the asset value distribution after incorporating the information from the CDS market. As reported in Table 6, most of the elasticity parameters are significantly different from zero at the conventional level, except the firm rated at ‘‘CCC’’ and the firm belonging to ‘‘Telecom Services’’. For the firms with negative elasticity parameters, the value of the 37 We choose W 1 by setting W 1 1 as the covariance matrix of moments. See page 443–447 in Green’s Econometrics Analysis (Sixth Edition). 38 Diamond Offshore Drilling Inc. cannot be calculated for both positive and negative betas. Therefore, we only have 103 firms in Table 4.

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S. Perrakis, R. Zhong / Journal of Banking & Finance 55 (2015) 215–231 Table 4 Characteristics of the individual firms. Company name

Sector

Rating

Begin date

End date

Total asset (billion)

Payout ratio

Leverage ratio

Recovery rate

Implied volatility

3M Co Abbott Labs Air Prods & Chems Inc Alcoa Inc. AmerisourceBergen Corp Anadarko Pete Corp Anheuser Busch Cos Inc APACHE CORP Archer Daniels Midland Arrow Electric Inc Autozone Inc Avon Prods Inc Baker Hughes Inc Baxter Intl Inc Black & Decker Corp Boeing Co BorgWarner Inc Bristol Myers Squibb Co Campbell Soup Co Caterpillar Inc CenturyTel Inc Clorox Co Coca Cola Enterprises Inc Colgate Palmolive Co ConAgra Foods Inc ConocoPhillips Costco Whsl Corp CSX Corp Cytec Inds Inc Danaher Corp Diamond Offshore Drilling Dover Corp Dow Chem Co Eastman Chem Co FedEx Corp Gen Dynamics Corp Gen Mls Inc Goodrich Corp Halliburton Co H J HEINZ CO Home Depot Inc Honeywell Intl Inc Intl Business Machs Corp Intl Paper Co Johnson & Johnson Kellogg Co Kimberly Clark Corp The Kroger Co. Eli Lilly & Co Ltd Brands Inc Lockheed Martin Corp Lowes Cos Inc Marriott Intl Inc Masco Corp Medtronic Inc Merck & Co Inc Mohawk Inds Inc Molson Coors Brewing Monsanto Co Motorola Inc Newell Rubbermaid Inc Nordstrom Inc Norfolk Sthn Corp Northrop Grumman Corp OCCIDENTAL PETRO Omnicare Inc Omnicom Gp Inc ONEOK Partners LP J C Penney Co Inc Pepsico Inc Pfizer Inc Pitney Bowes Inc PPG Inds Inc Praxair Inc

IN HC BM BM CS EN CG EN CG CG CS CG EN HC CG IN CG HC CG IN TS CG CG CG CG EN CS IN BM IN EN IN BM BM IN IN CG IN EN CG CS IN TE BM HC CG CG CS HC CS IN CS CS CG HC HC CG CG BM TE CG CS IN IN EN CS CS EN CS CG HC TE BM BM

AA AA A BBB BBB BBB A A A BBB BBB BBB A A BBB A BBB A A A BBB BBB A AA BBB A A BBB BBB A A A BBB BBB BBB A BBB BBB A BBB A A AA BBB AAA BBB A BBB A BB A A BBB BB A AA BBB BBB A BBB BBB A BBB BBB A BB BBB BBB BB A AA BBB BBB A

04/2003 10/2003 04/2003 08/2001 02/2004 01/2003 06/2003 03/2003 06/2003 11/2001 03/2003 01/2003 11/2001 02/2002 05/2002 04/2001 11/2001 04/2003 06/2002 04/2001 03/2003 07/2004 06/2003 08/2003 08/2001 01/2003 07/2004 01/2003 02/2004 01/2004 07/2003 12/2004 01/2002 01/2003 08/2002 11/2004 04/2002 09/2001 02/2003 04/2001 02/2002 11/2001 04/2001 04/2001 03/2003 03/2003 02/2004 08/2006 06/2003 03/2003 04/2001 01/2003 05/2002 07/2002 09/2003 03/2004 12/2004 10/2005 04/2003 08/2002 05/2001 11/2001 04/2001 04/2003 09/2002 11/2004 05/2002 05/2006 06/2001 06/2004 10/2003 11/2003 07/2001 10/2003

12/2011 12/2011 09/2008 09/2008 12/2011 09/2008 10/2008 09/2008 09/2008 12/2011 07/2011 12/2011 09/2008 12/2011 01/2010 09/2008 09/2008 12/2011 10/2011 09/2008 04/2008 07/2009 09/2008 12/2011 07/2011 09/2008 07/2011 09/2008 12/2011 12/2011 09/2008 12/2011 09/2008 09/2008 07/2011 12/2011 07/2011 09/2008 09/2008 10/2011 09/2008 12/2011 12/2011 09/2008 12/2011 12/2011 12/2011 10/2011 12/2011 09/2008 12/2011 09/2008 09/2008 09/2008 10/2011 10/2009 12/2011 12/2011 09/2008 09/2008 02/2009 09/2008 09/2008 03/2011 09/2008 02/2011 12/2011 12/2011 09/2008 12/2011 12/2011 12/2011 12/2011 09/2008

69.56 98.95 20.78 45.79 17.37 40.52 51.64 31.58 31.83 7.43 12.58 19.04 20.82 36.91 8.50 92.46 5.21 64.12 17.43 67.27 8.99 13.16 30.04 41.92 20.08 152.40 35.93 29.17 4.30 29.13 10.89 12.74 65.87 8.57 36.34 41.54 31.57 9.19 34.94 21.96 95.07 54.86 233.09 39.13 207.15 27.45 38.91 33.61 70.15 12.76 52.81 54.34 18.00 18.45 62.35 105.48 7.91 12.93 31.34 57.09 11.75 10.30 29.32 36.98 45.32 7.65 25.77 7.79 20.37 124.03 234.37 15.76 17.96 25.16

0.01 0.01 0.01 0.01 0.00 0.01 0.01 0.00 0.01 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.03 0.01 0.00 0.01 0.00 0.00 0.02 0.01 0.02 0.01 0.00 0.01 0.02 0.01 0.01 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.01 0.02 0.01 0.01 0.00 0.00 0.01 0.01 0.02 0.00 0.01 0.01 0.01 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.04 0.01 0.01 0.02 0.02 0.01 0.01

0.18 0.23 0.30 0.42 0.55 0.48 0.25 0.30 0.44 0.57 0.35 0.25 0.17 0.26 0.47 0.50 0.43 0.26 0.32 0.54 0.50 0.32 0.66 0.19 0.42 0.45 0.29 0.58 0.49 0.23 0.19 0.31 0.44 0.52 0.30 0.36 0.39 0.52 0.30 0.38 0.22 0.41 0.34 0.56 0.15 0.33 0.29 0.53 0.22 0.29 0.45 0.22 0.31 0.39 0.16 0.23 0.44 0.41 0.22 0.35 0.43 0.35 0.54 0.46 0.29 0.50 0.47 0.53 0.51 0.19 0.28 0.53 0.42 0.26

0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.41 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.39 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.41 0.41 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.41 0.40 0.40 0.40 0.40 0.40 0.39 0.41 0.41 0.39 0.40 0.40 0.26 0.40 0.40 0.38 0.40 0.40 0.40 0.40 0.40

0.22 0.21 0.22 0.34 0.27 0.31 0.18 0.31 0.30 0.38 0.27 0.31 0.34 0.26 0.32 0.28 0.31 0.25 0.21 0.28 0.22 0.21 0.23 0.20 0.22 0.25 0.25 0.29 0.36 0.25 0.37 0.29 0.28 0.25 0.28 0.24 0.19 0.33 0.33 0.21 0.28 0.30 0.25 0.27 0.17 0.18 0.18 0.29 0.24 0.32 0.26 0.28 0.30 0.31 0.24 0.27 0.38 0.27 0.32 0.38 0.30 0.37 0.32 0.22 0.29 0.40 0.29 0.22 0.39 0.18 0.24 0.24 0.27 0.23 (continued on next page)

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S. Perrakis, R. Zhong / Journal of Banking & Finance 55 (2015) 215–231

Table 4 (continued) Company name

Sector

Rating

Begin date

End date

Total asset (billion)

Payout ratio

Leverage ratio

Recovery rate

Implied volatility

Pride Intl Inc Procter & Gamble Co Quest Diagnostics Inc Raytheon Co Rep Services Inc Reynolds American Inc Rohm & Haas Co Ryder Sys Inc Safeway Inc Schering Plough Corp Sealed Air Corp US Sherwin Williams Co Smithfield Foods Inc Southwest Airls Co Sunoco Inc SUPERVALU INC Sysco Corp Target Corp Textron Inc Un Pac Corp Utd Parcel Svc Inc Utd Tech Corp Unvl Health Svcs Inc UST Inc. V F Corp Wal Mart Stores Inc Waste Mgmt Inc Whirlpool Corp Wyeth

EN CG HC IN IN CG BM IN CS HC IN CG CG IN EN CS CS CS IN IN IN IN HC CG CG CS IN CG HC

BBB AA BBB A BBB BBB BBB BBB BBB A B A BB BBB BB CCC A A BBB BBB AA A BB BBB A AA BBB BBB A

06/2003 04/2001 09/2005 06/2003 09/2004 11/2004 05/2001 01/2003 07/2005 04/2003 02/2006 06/2002 07/2003 06/2003 07/2003 03/2003 03/2005 04/2002 10/2002 09/2003 08/2004 06/2003 03/2004 04/2003 09/2004 01/2001 01/2004 04/2001 02/2003

09/2008 12/2011 12/2011 12/2011 12/2011 12/2011 11/2008 09/2008 12/2011 09/2008 12/2011 12/2011 08/2008 12/2011 09/2008 09/2008 12/2011 09/2008 09/2008 09/2008 12/2011 09/2008 12/2011 10/2008 12/2011 10/2011 08/2009 09/2008 07/2009

6.49 213.09 14.15 32.39 14.30 26.53 15.81 7.26 20.99 40.65 7.05 9.60 7.53 18.62 14.55 15.13 24.48 64.06 23.69 45.93 68.52 85.24 5.10 8.99 11.07 296.93 31.50 12.73 81.20

0.00 0.01 0.01 0.01 0.01 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.01 0.01 0.01 0.00 0.01 0.01 0.02 0.01 0.01 0.03 0.01 0.01 0.01 0.01 0.01

0.35 0.25 0.30 0.42 0.43 0.39 0.39 0.62 0.50 0.23 0.47 0.31 0.57 0.45 0.51 0.60 0.26 0.35 0.58 0.49 0.32 0.33 0.42 0.17 0.26 0.28 0.46 0.59 0.28

0.40 0.40 0.40 0.40 0.40 0.40 0.41 0.39 0.40 0.40 0.40 0.40 0.39 0.39 0.40 0.40 0.40 0.40 0.39 0.39 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0.38 0.19 0.24 0.22 0.27 0.23 0.27 0.29 0.31 0.28 0.31 0.29 0.29 0.35 0.34 0.28 0.23 0.30 0.28 0.24 0.23 0.20 0.31 0.22 0.28 0.23 0.24 0.33 0.26

This table reports the summary statistics of individual firms. We have 104 firms in our total sample. Note that in sector column, BM, CG, CS, EN, HC, IN, TE, TS are abbreviations of Basic Materials, Consumer Goods, Consumer Services, Energy, Healthcare, Industrials, Technology and Telecommunications Services, respectively. The payout ratio is the sum of cash dividend and interest expense divided by the total asset. The recovery rates are the estimated recovery rates reported in Markit datasets. The implied volatilities are extracted from Optionmetrics for the at the money call options.

Table 5 Summary statistics of the variables. Rating/Industry Panel A: Rating distribution AAA AA A BBB BB B CCC Panel B: Industry distribution Basic materials Consumer goods Consumer services Energy Healthcare Industrials Technology Telecommunications services Total

N

1 year spread

2 year spread

3 year spread

5 year spread

7 year spread

10 year spread

Implied volatility

Leverage

1 8 39 47 7 1 1

11.51 15.60 17.30 37.94 83.09 72.47 54.54

14.05 19.39 21.85 46.28 104.23 91.65 71.96

16.57 23.09 26.42 54.70 121.94 111.10 89.48

22.33 30.72 35.80 71.91 154.16 150.17 124.22

26.19 35.54 41.81 80.68 166.00 163.88 140.64

30.76 41.26 48.93 90.29 177.62 176.62 157.33

0.1743 0.2262 0.2643 0.2797 0.3377 0.3115 0.2807

0.1547 0.2478 0.3123 0.4420 0.4571 0.4728 0.6001

10 27 17 10 12 24 3 1 104

26.85 32.96 45.53 30.21 20.89 29.49 37.66 24.76 31.90

33.17 40.62 56.66 37.82 26.64 35.94 45.74 34.88 39.49

39.11 48.07 67.02 44.73 32.54 42.60 54.12 46.52 46.90

51.92 62.81 87.08 58.86 44.76 56.27 68.55 71.38 61.76

59.30 70.40 95.90 66.96 51.22 63.71 76.46 86.84 69.49

68.51 78.63 105.87 76.00 58.01 71.82 85.50 102.93 78.07

0.2807 0.2499 0.3000 0.3134 0.2486 0.2740 0.2869 0.2224 0.2731

0.4022 0.3661 0.3937 0.3576 0.2516 0.4388 0.4077 0.4962 0.3791

This table reports the industry and rating distribution in Panel A and B, respectively. The average values of all the variables are reported. The 1, 3, 5, 7, 10 years credit default swap spreads are reported as basis points (bps).

elasticity parameter decreases as the rating decreases from AA to BB on average, while there is no clear trend for the movement of the initial asset volatilities. We also note that most of the firms falling into the basic materials, energy, healthcare, industrials and technology sectors have negative betas, and the highest negative beta occurs in the technology sector on average. On the other hand, the firms with positive elasticity parameters are mainly found in consumer goods and consumer services. Compared to the Leland (1994b) structural model with constant volatility, the CEV structural model shows a clearly superior fitting of its estimates to the observed data, especially with respect to the

CDS spreads. Table 7 reports the means of absolute errors (MAE) for both models and for different CDS maturity cases, as well as for the equity volatility. For the in-sample CDS spreads, the presence of the state dependent volatility reduces significantly the fittings’ MAE for all maturities, but especially for the longer ones, with an average improvement of approximately 50%. The intertemporal accuracy of the two competing models can be observed visually in Figs. 4 and 5. The structural model with constant volatility consistently underestimates the CDS spreads with 1-year, 5-year and 10-year maturities, while the CEV model estimates lie much closer to the observed values.

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S. Perrakis, R. Zhong / Journal of Banking & Finance 55 (2015) 215–231 Table 6 GMM estimation results. All

Negative betas

Positive betas

N

Beta

Sigma

N

Beta

Sigma

Panel A: Rating distribution AAA

1 8

A

38

BBB

47

BB

7

B

1

0.9688 (0.0606) 0.6761 (0.0471) 0.8940 (0.0499) 0.9766 (0.0559) 1.3240 (0.0701) 1.3682 (0.0693)

0.1639 (0.0057) 0.1860 (0.0055) 0.2336 (0.0072) 0.1973 (0.0064) 0.2281 (0.0053) 0.1213 (0.0061)

CCC

1

Total

103

0.1639 (0.0057) 0.1899 (0.0060) 0.2198 (0.0075) 0.1858 (0.0073) 0.2175 (0.0108) 0.1213 (0.0061) 0.1261 (0.0203) 0.1994 (0.0076)

1

AA

0.9688 (0.0606) 0.4820 (0.0589) 0.5197 (0.0660) 0.5090 (0.1723) 0.7865 (0.2223) 1.3682 (0.0693) 0.1005 (0.2103) 0.5366 (0.1260)

Panel B: Industry distribution Basic materials

10

9

Consumer goods

27

Consumer services

17

Energy

10

Healthcare

12

Industrials

24

Technology

3

Telecom services

1

0.2291 (0.0098) 0.1796 (0.0064) 0.2016 (0.0084) 0.2683 (0.0091) 0.1965 (0.0097) 0.1925 (0.0063) 0.1492 (0.0040) 0.1282 (0.0106) 5% 98.06% 97.07%

10% 100% 98.06%

Panel C: Summary of J-statistics (df = 4 for Significance level Non-rejection rate of CEV: Non-rejection rate of Leland:

0.8618 (0.2668) 0.5385 (0.1445) 0.2254 (0.1153) 0.6084 (0.0497) 0.2919 (0.1491) 0.7108 (0.0698) 0.7947 (0.0731) 0.1393 (0.3129) CEV and 5 for L) 1% 43.69% 29.13%

6 27 34 5 1

1.0733 (0.0578) 1.0906 (0.0620) 0.8863 (0.0503) 0.7045 (0.0490) 0.7378 (0.0440) 0.9347 (0.0518) 1.3271 (0.0659)

19 8 9 8 20 2

0.2350 (0.0094) 0.1911 (0.0051) 0.2282 (0.0068) 0.2718 (0.0091) 0.1989 (0.0055) 0.1992 (0.0061) 0.1189 (0.0045)

N

Beta

Sigma

2

0.1000 (0.0945) 0.3989 (0.1053) 0.7139 (0.4768) 0.5574 (0.6029)

0.2016 (0.0073) 0.1858 (0.0081) 0.1555 (0.0098) 0.1911 (0.0246)

1

0.1005 (0.2103)

0.1261 (0.0203)

1

1.0414 (2.1474) 0.7725 (0.3403) 0.3621 (0.1730) 0.1605 (0.0554) 0.5998 (0.3595) 0.4084 (0.1601) 0.2702 (0.0874) 0.1393 (0.3129)

0.1751 (0.0131) 0.1524 (0.0095) 0.1780 (0.0097) 0.2403 (0.0091) 0.1917 (0.0180) 0.1590 (0.0075) 0.2099 (0.0030) 0.1282 (0.0106)

11 13 2

8 9 1 4 4 1 1

This table reports the distribution of GMM estimation results based on rating (Panel A) and industry (Panel B), respectively. The average standard errors are reported in the parentheses. Panel C reports the summary results for the J-statistics.

Table 7 Means of absolute errors of the empirical fitting. 1-year CDS

2-year CDS

3-year CDS

5-year CDS

7-year CDS

10-year CDS

Equity Volatility

Panel A: Full sample MAE (L) MAE(CEV) Fitting Improvement

0.0039 0.0032 17.47%

0.0046 0.0024 46.75%

0.0051 0.0021 59.06%

0.0059 0.0024 59.39%

0.0058 0.0026 54.73%

0.0055 0.0028 49.56%

0.0583 0.0536 7.99%

Panel B: Investment rated MAE (L) MAE (CEV) Fitting Improvement

0.0021 0.0017 18.87%

0.0026 0.0020 22.65%

0.0029 0.0025 15.14%

0.0035 0.0031 13.33%

0.0035 0.0031 12.79%

0.0034 0.0031 6.45%

0.0559 0.0535 4.25%

Panel C: Junk rated MAE (L) MAE (CEV) Fitting improvement

0.0054 0.0046 14.91%

0.0064 0.0033 47.82%

0.0071 0.0023 67.48%

0.0081 0.0025 68.84%

0.0078 0.0028 64.65%

0.0074 0.0031 58.48%

0.0607 0.0518 14.79%

This table reports the means of the absolute errors (MAE) of the empirical fitting for the daily average CDS spreads and equity volatilities, respectively. The fitting improvements are calculated by ð1  MAECEV =MAEL Þ.

Panel C in Table 6 shows the summary results of the J-statistics for all firms, including the percentage not rejected for various significance levels; these values are slightly but consistently higher for the CEV.39 Both L and CEV models also show much higher

39 Table A3 of our online appendix shows the values of the J-statistics for each firm for both L and CEV with time-varying exogenous default boundaries for both models, which fits the empirical data much better compared to the corresponding endogenous default boundaries where available.

non-rejection values of the J-statistics than the GMM exercises done by Huang and Zhou (2008) for five constant volatility structural models that use, however, different assumptions and are not nested in the same test. Hence, in term of fitting errors the CEV structural model shows a superior performance compared to structural models with constant asset volatility. We also verified the dependence of the CEV fitting on the underlying bond rating, by dividing the full sample into two subsamples: investment-rated class that includes all the ratings

4/1/2001 8/1/2001 12/1/2001 4/1/2002 8/1/2002 12/1/2002 4/1/2003 8/1/2003 12/1/2003 4/1/2004 8/1/2004 12/1/2004 4/1/2005 8/1/2005 12/1/2005 4/1/2006 8/1/2006 12/1/2006 4/1/2007 8/1/2007 12/1/2007 4/1/2008 8/1/2008 12/1/2008 4/1/2009 8/1/2009 12/1/2009 4/1/2010 8/1/2010 12/1/2010 4/1/2011 8/1/2011 12/1/2011

1/1/2001 5/1/2001 9/1/2001 1/1/2002 5/1/2002 9/1/2002 1/1/2003 5/1/2003 9/1/2003 1/1/2004 5/1/2004 9/1/2004 1/1/2005 5/1/2005 9/1/2005 1/1/2006 5/1/2006 9/1/2006 1/1/2007 5/1/2007 9/1/2007 1/1/2008 5/1/2008 9/1/2008 1/1/2009 5/1/2009 9/1/2009 1/1/2010 5/1/2010 9/1/2010 1/1/2011 5/1/2011 9/1/2011

1/1/2001 5/1/2001 9/1/2001 1/1/2002 5/1/2002 9/1/2002 1/1/2003 5/1/2003 9/1/2003 1/1/2004 5/1/2004 9/1/2004 1/1/2005 5/1/2005 9/1/2005 1/1/2006 5/1/2006 9/1/2006 1/1/2007 5/1/2007 9/1/2007 1/1/2008 5/1/2008 9/1/2008 1/1/2009 5/1/2009 9/1/2009 1/1/2010 5/1/2010 9/1/2010 1/1/2011 5/1/2011 9/1/2011

1/1/2001 5/1/2001 9/1/2001 1/1/2002 5/1/2002 9/1/2002 1/1/2003 5/1/2003 9/1/2003 1/1/2004 5/1/2004 9/1/2004 1/1/2005 5/1/2005 9/1/2005 1/1/2006 5/1/2006 9/1/2006 1/1/2007 5/1/2007 9/1/2007 1/1/2008 5/1/2008 9/1/2008 1/1/2009 5/1/2009 9/1/2009 1/1/2010 5/1/2010 9/1/2010 1/1/2011 5/1/2011 9/1/2011

228 S. Perrakis, R. Zhong / Journal of Banking & Finance 55 (2015) 215–231

0.014 0.012 0.01 0.008 0.006 0.004 0.002 0

Panel A:1-year CDS

L

L

L

CEV

CEV

CEV

CEV

CEV

Leland

Leland

Obs

Panel B: 5-year CDS

0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0

Obs

0.03

Panel C: 10-year CDS

0.025

0.015 0.02

0.005 0.01

0

Obs

Fig. 4. Time series of CDS spreads.

0.02

Panel A: Invest Ratings

0.015

0.01

0.005

0 Obs

0.03

Panel B: Junk Ratings

0.025

0.015

0.02

0.005

0.01

0

Obs

Fig. 5. Time series of 5-year CDS spreads for different ratings. This figure exhibits the time series of daily average of 5-year CDS spreads for Invest (Panel A) and Junk (Panel B) ratings, respectively. The invest class includes the letter rating that is greater than or equal to ‘‘A’’. The junk class includes the letter ratings that are below ‘‘A’’.

S. Perrakis, R. Zhong / Journal of Banking & Finance 55 (2015) 215–231

greater than or equal to ‘‘A’’, and junk-rated class with all the ratings below ‘‘A’’.40 While there are improvements from the CEV model for both classes, the difference is major for the junk rated class, where the constant asset volatility model exhibits a poor fitting performance for the CDS spreads. Allowing for CEV time-varying volatility brings major reductions in the value of MAEs, especially for the intermediate and long-term contracts in the junk-rated class, which can be visualized in Fig. 5. As for the equity volatility, the CEV structural model also reduces the in-sample fitting errors by about 7.99% for the full sample, even though the improvement is not visually obvious.41 We also notice that the fitting improvement of equity volatility in the junk-rated contract, approximately 15%, is much greater than that in the investment-rated contract, which verifies the conjecture that the asset volatilities of the junk-rated firms are more likely to be state-dependent. We also verified the out-of-sample prediction errors and report the results in Tables 8 and 9. Since the in-sample fitting is based on the empirical data up to December, 2011, we used the information in the following three months, January, February and March in 2012, to compare the out-of-sample prediction performances between the two structural models. The CEV structural model is able to reduce the out-of-sample prediction errors by about 10% across all the maturities while maintaining the predictive power of equity volatilities. The superiority of the CEV model is approximately the same for all three months, even though for both models the predictions in March are relatively more accurate than those in January and February. The reason is that we only have quarterly data from the firm’s financial reports and assume that the information they contain is time invariant during the entire quarter. We also found similar out-of-sample prediction improvements for both investment and junk rated firms, reported in Table 9. Last, we consider the unobservable default probabilities and endogenous default boundaries implied by our estimated models from observable firm data. For the CDP we need the objective asset value growth rates l in equation (2.1). These are derived from the average of the time series of the asset value V for each firm, filtered from our estimation process. Once l is known the cumulative default probability is found from equation (5.5), setting r ¼ l in the asset dynamics. Fig. A4, Panels A and B, of our online appendix shows the average CDP for the investment and speculative classes, respectively. A comparison of these figures with the historical data shown in Fig. A1 of our online appendix42 shows that our estimates tend to over predict the historical CDP data for the investment class, but is consistent with it for the speculative class. Nonetheless, the range of our estimates, shown in Table A4, is very wide and contains the Figure A1 averages for each rating and maturity for which there is more than one firm in our sample. For our GMM estimation of the model’s parameters we chose to fix the default boundary to the exogenous one based on the KMV method. An alternative approach would be to estimate the parameters under an endogenous boundary, found by solving equations (3.10), (3.11). Unfortunately the numerical issues involved in solving numerically (3.10), (3.11) for unknown parameters and then using the derived boundary as an input to GMM require a separate and entirely new computational method that transcends the realm of this study; such an exercise must also 40 The common practice uses ‘‘BBB’’ as a cut off for investment and speculative grades. Since the majority of the firms are rated ‘‘A’’ and ‘‘BBB’’, we use ‘‘A’’ as a cut off to separate the investment and speculative samples, which is slightly different from the common practice but does not affect our results. 41 The in-sample fitting for the equity volatilities are shown in Fig. A4 of the online appendix. 42 The 1983–2010 CDP data in our online appendix is more current than the one in Huang and Huang (2012, p. 165), which is based on earlier evidence that reports much lower CDPs in most cases.

229

deal with technical issues that arise when the boundary is very close to zero, as shown in Fig. 2. An indication of the divergence of the implied endogenous boundary from the assumed KMV one is provided by Fig. A5 in our online appendix, which compares the average endogenous boundary implied by our KMV-based estimated parameters to the corresponding boundary extracted endogenously from Leland (1994b) and to the exogenous KMV. It turns out that on average the endogenous boundary corresponding to the b that was estimated under the assumed KMV boundary always lies above (below) that assumed boundary when the maturity is less(more) than 5 years. Further, the endogenous CEV boundary is always lower than the L-model’s boundary, close to it for low maturities but increasingly diverging for maturities above 3 years. In conclusion, we note that the one extra parameter, the elasticity of variance b, under the CEV structural model compared to the structural model with constant volatility, improves dramatically both the in-sample fitting and the out-of-sample predictions of the time series of CDS spreads across all the maturities, while maintaining the fitting of the implied equity volatilities time series at comparable levels of accuracy to those of constant volatility. Note that the introduction of the state-dependent volatilities does not bring additional sources of uncertainty into the structural model. To improve the fitting of equity volatilities and credit spreads simultaneously, the stochastic volatility models that incorporate the uncertainty in the volatility dynamics might help to solve this problem. Nonetheless, the numerical problems involved in empirical research for such models preclude the derivation of many of the results available for CEV.43

6. Conclusions We have presented a new structural model of the firm, which generalizes the asset dynamics assumptions of the well-known models of Leland (1994b) and Leland and Toft (1996), by allowing for state-varying volatility of the assets. We derive closed form expressions for all the variables of interest and show with simulated data that the relaxation of the constant volatility assumption of the earlier models brings major modifications to several of the results. In particular, the more important negative elasticity of variance case results in a relatively lower endogenous default boundary and widens the credit spreads for all debt maturities, especially when the boundary is exogenous. We verify the relevance of our model by empirically fitting our derived expressions to data from a sample of firms that participate in the CDS market. We find that the elasticity of variance parameter is negative and significantly different from zero for more than two thirds of the sample. More to the point, the structural model with state-varying asset volatility clearly dominates the constant volatility case in goodness-of-fit tests of CDS spreads, both in- and out-of-sample. The dominance is clear in all cases, but is especially pronounced for medium and long term debt and for junk rated debt. Thus, the CEV model emerges as the prime CDS-pricing model for firms with finite maturity debt that allow default at any time and not only at maturity. The CEV model can also in principle be extended to allow for jumps in the asset dynamics under some restrictions on the jump amplitude, but the resulting asset value distribution does not yield a closed form solution for the first passage to default 43 Elkamhi et al. (2011) incorporate stochastic volatility into the structural model and examine the fitting of observed credit spreads and equity volatility within the context of the Leland (1994a) infinite maturity debt. Since the first passage time for an exogenous default boundary is found numerically, the determination of the optimal endogenous boundary presents major computational challenges and, to our knowledge, has not been done yet.

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Table 8 Out-of-sample predictions for varying time horizon. No. of firms

1-year CDS (bps)

2-year CDS (bps)

3-year CDS (bps)

5-year CDS (bps)

7-year CDS (bps)

10-year CDS (bps)

Equity volatility

Panel A: First month prediction MAE(CEV) 89 37 MAE(L) 89 41 Improvements 89 11.58%

72 77 6.91%

103 112 7.64%

126 143 11.28%

142 159 10.92%

154 173 11.14%

8.94% 8.94% 0%

Panel B: Second month prediction MAE(CEV) 89 32 MAE(L) 89 36 Improvements 89 11.76%

64 67 4.87%

95 102 6.84%

119 134 10.78%

135 151 10.70%

147 164 10.94%

9.11% 9.11% 0%

Panel C: Third month prediction MAE(CEV) 89 30 MAE(L) 89 35 Improvements 89 12.92%

61 65 6.21%

91 100 9.02%

115 131 12.48%

131 149 12.02%

143 162 11.98%

9.70% 9.70% 0%

This table reports the out-of-sample prediction improvements of the CEV structural model comparing to the Leland (1994b) model for one, two and three months ahead, respectively. The parameters are extracted from the GMM estimations. The out-of-sample prediction periods are January (Panel A: first month), February (Panel B: second month) and March (Panel C: third month) in 2012. The prediction improvement is calculated by (1  MAECEV =MAEL ), where MAE denotes the mean of the absolute errors between the model predicted and observed CDS spreads and equity volatilities.

Table 9 Out-of-sample predictions for varying rating classes. No. of firms

1-year CDS (bps)

2-year CDS (bps)

3-year CDS (bps)

5-year CDS (bps)

7-year CDS (bps)

10-year CDS (bps)

Equity Volatility

Panel A: Investment rating MAE(CEV) 89 MAE(L) 89 Improvements 89

37 41 11.58%

72 77 6.91%

103 112 7.64%

126 143 11.28%

142 159 10.92%

154 173 11.14%

8.94% 8.94% 0%

Panel B: Junk rating MAE(CEV) 89 MAE(L) 89 Improvements 89

32 36 11.76%

64 67 4.87%

95 102 6.84%

119 134 10.78%

135 151 10.70%

147 164 10.94%

9.11% 9.11% 0%

This table reports the out-of-sample prediction improvements of the CEV structural model compared to the Leland (1994b) model three months ahead (March 2012) for investment (Panel A) and junk (Panel B) rated classes, respectively. The parameters are extracted from the GMM estimations. The prediction improvement is equal to (1  MAECEV =MAEL ), where MAE denotes the mean of the absolute errors between the model predicted and observed CDS spreads and equity volatilities.

probability. This problem also exists under constant volatility diffusion, in which the closed form solution exists only if the jump amplitude is restricted to the double exponential.44 Although numerical approximation methods are available, the resulting expressions are computationally heavy and do not lend themselves easily to empirical estimations. This is clearly an area of future research. Appendix A. Proofs of Lemmas

1

ers f ðg; V; KÞds ¼

0

/rþg ðVÞ /rþg ðKÞ

ðA:1Þ

Proof of Lemma 2. According to equation (3.1), the Feynman-Kac representation of the debt value is

Z 0

1

Z s

ers ðegs c þ gegs pÞds þ ers gegs ð1  aÞK



0

 f ðs; V; KÞds

  Z 1 c þ gp 1 eðrþgÞt f ðs; V; KÞdt rþg Z0 1 eðrþgÞt f ðs; V; KÞdt þ gð1  aÞK

Using the auxiliary parameters in (3.2) to replace the integral term in (A.3) and re-arranging terms, we get equation (3.5), QED.

We conjecture the solution for this differential equation is,

EðVÞ ¼ V þ

ðA:2Þ

wC ½1  F 0   aKF 0  ½A0 þ A1 F 00  r

See Zhou (2001) and Kou and Wang (2003).

ðA:4Þ

where,

/rþg ðVÞ /r ðVÞ ; F 00 ¼ ; /r ðKÞ /rþg ðKÞ   C þ gP ¼ ð1  aÞK  rþg

A0 ¼

C þ gP ; rþg

A1 ðA:5Þ

It is straightforward to show that EðKÞ ¼ 0. When V ! 1, we have, F 0 ¼ 0 ¼ F 00 ¼ 0. Similarly, if V ! 1, we have EðVÞ ¼ V. We also have,

EV ¼ 1  wC F 0V  aKF 0V  A1 F 00V r 44

ðA:3Þ

0

F0 ¼

where, /r ð  Þ is defined in (3.4), QED.

dðg; V; KÞ ¼

dðg; V; KÞ ¼

Proof of Lemma 3. The equity value has to satisfy the differential equation (3.7) under time-varying volatility, rðVÞ, with the boundary condition EðKÞ ¼ 0; EðVÞ  V when V ! 1. h

Proof of Lemma 1. According to Proposition 1 and Proposition 2 in Davydov and Linetsky (2001), given that the asset dynamics follow the CEV diffusion process, we have

Z

Integrating (A.2), we have,

EVV ¼  wC F 0VV  aKF 0VV  A1 F 00VV r

ðA:6Þ

S. Perrakis, R. Zhong / Journal of Banking & Finance 55 (2015) 215–231

Substituting the conjectured solution (A.4) and the corresponding derivatives (A.6) into the differential equation (3.8), it is straightforward to show that the equality holds, QED. Proof of Proposition 1. The equity value is given by equation (3.8) with time-varying volatility. The first partial derivative of equity value with respect to the asset value is

   @EðV; KÞ wC 1 @/r ðVÞ ¼1 þ aK @V r /r ðKÞ @V   @/rþg ðVÞ C þ gP 1 þ  ð1  aÞK @V rþg /rþg ðKÞ

ðA:7Þ

Applying the smooth pasting condition that sets the RHS of (A.7) to 0 when the asset value equals to default boundary, we have equation (3.10). We set for the Whittaker functions

@W k;m ðxÞ ¼ W 0; @x

@M k;m ðxÞ ¼ M0 ; @x

@x ¼ x0 @V

ðA:8Þ

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