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S0927-5398(16)30078-0 doi: 10.1016/j.jempfin.2016.08.001 EMPFIN 930

To appear in:

Journal of Empirical Finance

Received date: Revised date: Accepted date:

11 October 2015 26 July 2016 2 August 2016

Please cite this article as: Rhee, Yuna, Jang, Bong-Gyu, Yoon, Ji Hee, Business Cycle and Credit Risk Modeling with Jump Risks, Journal of Empirical Finance (2016), doi: 10.1016/j.jempﬁn.2016.08.001

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, and Ji Hee Yoon

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Yuna Rhee , Bong-Gyu Jang

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Business Cycle and Credit Risk Modeling with Jump Risks

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Fri Aug 19 18:22:26 2016

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We would like to thank Hyeng Keun Koo, Bong Soo Lee, and Gyoocheol Shim for helpful discussions and insightful comments. A preliminary version of the paper was presented at the 2012 conferences of CAFM, KFA, and KORMS. This research in the paper is supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2012R1A1A2038735, NRF-2013R1A2A2A03068890, NRF-2014S1A3A2036037) and the POSTECH Basic Science Research Institute Grant (No. 4.0006572.01). Department of Risk Management, Korea Development Bank, E-mail: [email protected]

Department of Industrial and Management Engineering, POSTECH, Korea, E-mail: [email protected] Economics, University of Wisconsin - Madison, WI, USA, [email protected]

E-mail:

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Business Cycle and Credit Risk Modeling with Jump Risks

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Abstract

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We develop a structural model that incorporates both macroeconomic risks and firm-specific jump risks. We derive analytic formulas for default probability, equity price, and

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CDS spreads. Based on reasonably calibrated parameters, we find that our model could predict

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actual default probabilities and overcome the underestimation of credit risks, especially for firms with high credit ratings, which has been one of the major limitations of the currently available

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structural models. The structural model highlights that macroeconomic factors are important in modeling credit risks and that default probabilities and CDS spreads could be dependent on the current economic state.

ACCEPTED MANUSCRIPT 1 Introduction

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Analysis on credit risks has gained significant importance in finance over the past decades, and theoretical and empirical models have been developed in a variety of ways. There are three

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well-known approaches for modeling firm’s credit risks: reduced-form approach, structural approach, and incomplete information approach. The reduced-form approach is originally

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developed by Duffie and Singleton (1999) and Jarrow and Turnbull (1995) and is based on the assumption that information flows can be reduced to the observations of random time representing

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a firm’s default events. On the other hand, the structural approach refers directly to the capital structure of individual firms and the firm’s asset value process. Merton (1974) develops a

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structural model exploiting the valuation method of contingent claims. The incomplete information approach combines the features of both the structural and reduced-form approaches (Duffie and Lando, 2001).

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The structural models reflect the relationship between changes in firms’ fundamental

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values such as total amounts of debt and capital and changes in their equity prices observed in financial markets. These models generally assume that a firm’s default is triggered by an event in

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which asset value becomes lower than debt at maturity and occasionally the total asset value of the firm is restored from the market price of its equity by employing option pricing theories. Following the pioneering work by Merton (1974), Black and Cox (1976) introduces a structural model that enables default events to occur before debt maturity. They apply the famous Black-Scholes formula in option pricing theory to bonds with indenture provisions and show that the model can be used to analyze the effects of indenture provisions on a firm’s default events. Extended studies of the first-passage time model are Longstaff and Schwartz (1995), Collin-Dufresne and Goldstein (2001), Leland and Toft (1996) among others. We build our study on a structural approach and link credit risks to macroeconomic and firm-specific jump risks. The observation that firm default rates are counter-cyclical is now widely accepted by both researchers and practitioners. Many studies conclude that macroeconomic conditions influence credit risks as well as the firm-specific variables (see Cremers et al., 2008; Hackbarth et al., 2006; Carling et al., 2007). Moreover, some commonly used credit risk models in practice such as KMV model of Moody’s and CrediMetrics of J.P. Morgan also importantly consider macroeconomic variables. On the other hand, credit risk models with a jump

ACCEPTED MANUSCRIPT component, such as Zhou (2001), show that adding jumps on a firm’s asset process allows us to have various structures of credit spread and default probability. Our study is motivated by recent research on credit risk models with macroeconomic and

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jump risks. In this paper, we aim to develop a structural model that simultaneously incorporates business cycle and jump risks and try to show that combining these two provides a better

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explanation for the firm default rates observed in the real world. Many empirical studies including

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Cremers et al. (2008), Carling et al. (2007), Pesaran et al. (2003), Nickell et al. (2000), Koopman and Lucas (2005), Zhang et al. (2009), Tang and Yang (2010), Alexander and Kaeck

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(2008), Chen et al. (2009) and Bonfim (2009) focus on investigating the determinants of the credit spread, credit default swap (CDS) spread, or default rates. There is little literature in the

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credit modeling context involving macroeconomic risks and jump risks simultaneously. Barndorff-Nielsen and Shephard (2004), Todorv (2010), Tauchen and Zhou (2011), and Bollerslev and Todorv (2011) identify realized jumps and investigate that rare and large jumps

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explain market prices and puzzles such as variance and equity risk premia. Based on

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high-frequency data, Tauchen and Zhou (2011) propose a strategy for estimating parameters in the jump diffusion process and apply it to financial markets. They show that realized jumps have

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important implications for accounting for credit spread indices; jump volatilities can predict the higher spread variation than standard models especially for high investment grade credit spreads, which is consistent to our empirical results. Moreover, some recent theoretical works such as Hanckbarth et al. (2006), Bhamra et al. (2010), Chen (2010), and Elkamhi and Jiang (2011) have considered macroeconomic risks in modeling asset prices dynamically; however, their models do not consider firm-specific jump risks. Although Bhamra et al. (2010), Chen (2010), and Elkamhi and Jiang (2011) include not only systematic but also idiosyncratic volatilities of the firm’s earnings growth rate in the firm’s cash flow process, idiosyncratic volatilities are assumed to be constant over time.1 To the best of our knowledge, thus far, no studies have considered the effect of macroeconomic factors and firm-specific jumps directly on the firm’s asset process. This paper contributes to the literature in the structural approach both theoretically and empirically. We develop a structural model including macroeconomic risks and firm-specific jump risks, and derive new analytic formulas for default probability, equity price, and CDS spreads under the model. Moreover, we analyze the characteristics of our model in terms of default 1

We thank reviewer for suggesting to clarify this point.

ACCEPTED MANUSCRIPT probabilities and CDS spreads by exploiting parameters calibrated with historical credit ratings, equity prices and high-frequency transaction data of individual firms. We find that the structural model proposed in the paper emphasizes the importance of macroeconomic factors and

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firm-specific jumps in modeling credit risks and that the default probabilities and CDS spreads could be dependent on the current economic state. Furthermore, our model could better predict

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actual default probabilities and overcome the underestimation of credit risks, especially for firms

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with high credit ratings, which has been one of the major limitations of conventional structural models.

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Researchers have a consensus that conventional structural models have an intuitive framework; a firm’s default is declared upon the first occurrence of an event in which the firm’s

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asset value reaches its debt level. Nonetheless, the empirical literature virtually indicates that their performances are somewhat unsuccessful. Jones et al. (1984) and Elton et al. (2001) point out that theoretical credit spreads derived from conventional structural models are smaller than actual

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credit spreads. Several studies such as Franks and Torous (1989), Campbell and Taksler (2003),

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and Eom et al. (2004) also provide empirical evidence demonstrating that conventional structural models underestimate credit risks especially of high credit-rated firms having low leverage and

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volatility and are therefore, difficult to align with actual credit spreads. Various theoretical and empirical efforts have been made in order to overcome these shortcomings.

Some studies incorporate jump risks to the conventional models and others link macroeconomic risks to a firm’s credit risk. For example, Zhou (2001) includes a jump risk in a firm’s asset process and shows that the jump diffusion model can explain a number of empirical regularities. Moreover, Zhang et al. (2009) incorporate jump events into a stochastic volatility model for firm’s asset process and find that the resultant model could account for variations in CDS spreads better than the original model. Cremers et al. (2008) develop a structural model with both firm-specific jumps and market-wide common jumps. Collin-Dufresne et al. (2001) investigate the determinants of credit spread changes and show that one of the principal components of credit spread changes is local supply or demand shocks. The empirical literature examines the effects of macroeconomic variables on the credit risks of bonds (Cremers et al., 2008; Carling et al., 2007; Pesaran et al., 2003; Nickell et al., 2000; Koopman and Lucas, 2005), credit default swaps (Zhang et al., 2009; Tang and Yang,

ACCEPTED MANUSCRIPT 2010; Alexander and Kaeck, 2008), and firm default rates (Bonfim, 2009; Koopman and Lucas, 2005). The conclusions of these studies show that macroeconomic factors could play a significantly and economically important role as well as firm-specific factors in explaining credit

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risks.

Some notable theoretical attempts are made to associate business cycle with the structural

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models. Kim et al. (2008) combine a two-state Markov regime-switching market condition to the

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first-passage-time structural model and suggest a valuation method for credit derivatives. Some articles such as Hanckbarth et al. (2006), Bhamra et al. (2010), Chen (2010), and Elkamhi and

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Jiang (2011) attempt to incorporate business cycle into a structural (partial) equilibrium model and examine the impact of the economic states on credit risks. Contrarily, we build our model over a

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traditional structural model to directly incorporate both macroeconomic factors, representing business cycle, and idiosyncratic factors, representing firm-specific jump events. We also conduct empirical investigations based on financial market data and accounting data of individual firms.

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The paper proceeds as follows. Section 2 specifies our model and provides analytic

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formulas for default probability, equity price, and CDS spreads. Section 3 presents a calibration method for the model parameters and empirical results for firms with different credit rates. Section

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4 discusses properties of the proposed model and Section 5 concludes the paper.

2 The Model

2.1 Model Description

We begin developing our model by generalizing Merton (1974)’s model. We assume that a firm’s asset value (or equivalently, the firm value), which is defined as the expected sum of discounted future cash flows, evolves according to an exogenously-given stochastic process. Moreover, we assume that there exists a frictionless financial market in which there are no taxes, no transaction costs, no short-sale restrictions, and no differences between borrowing and lending interest rates. Following Merton (1974), we assume that the firm issues an equity and a zero-coupon bond with a certain maturity. At the bond maturity, the firm should pay the pre-fixed principal to the bond holders and the remaining firm value would be distributed to the equity

ACCEPTED MANUSCRIPT holders. If the firm value is not sufficient to clear the debt to the bond holders, the firm would default and its equity would lose value. Under this set-up, it is well-known that the equity of the firm is priced as an European call option written on the asset process, whereas the firm’s default

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probability is as an European digital (or binomial) option.

In particular, we assume that the firm is financed by an equity and a zero coupon bond;

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thus, the sum of these two becomes the firm’s asset value V () . The zero-coupon bond has a face

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value K with maturity T , and K is determined at the bond issue date and remains unchanged until T . If V (T ) is lower than K at T , the firm defaults on its debt. The default status of the

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firm is defined as:

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1, if a firm defaults at time T , D(T ) = I{V (T )< K } = 0, otherwise, where IA is the indicator function on a set A . The equity holders receive a nonnegative profit

(V (T ) K ) if the firm does not default, and they get nothing otherwise. Thus, the profit function

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of the equity holders at the bond’s maturity is

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V (T ) K (1 D(T )) = V (T ) K .

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Merton (1974) assumes that firm value simply follows a log-normal distribution at the maturity of its bond. We extend Merton’s set-up to incorporate business cycles and firm-specific jump risks. Technically, a two-state Markov chain is utilized as a hidden process representing the regime of the real market, and we consider the two regimes to be the two states of the business cycle. We refer to them as the

high (“ H ") and

low (“ L ") regimes. The high regime

corresponds to a economic recession period where the real market is riskier than usual, causing firm value to be more volatile, whereas a low regime represents an economic expansion period, thereby causing firm value to be less volatile.2 Jumps are also incorporated into the firm value process to describe sudden changes in value caused by significant idiosyncratic shocks and thus, they are assumed to occur independently of changes in business cycle. More specifically, we take (,F,{Ft},P ) as a filtered probability space and the process of total information set, {Ft }t 0 , is assumed to satisfy the usual conditions. B(t ) is a standard Brownian motion, and H (t ) , L (t ) , and ZV (t ) are Poisson processes on [0,) . The three Because equity price can be considered as the option value written on a firm’s asset, this assumption implies that a firm’s equity price shifts at the exact time when the value of its asset shifts. 2

ACCEPTED MANUSCRIPT Poisson processes are assumed to be independent of each other and also independent of the Brownian motion BV (t ) . The intensities of H (t ), L (t ) , and ZV (t ) are denoted by H , L , and

V , respectively. An alternating renewal process (or a hidden Markov chain) defined by two

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Poisson processes H and L , represents the business cycle. If the current market is in regime

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k {H , L}, the market environment would change when the first jump time of k arrives. The

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Poisson process ZV (t ) represents the jump risks of each firm. We assume the jump size LV (t ) is normally distributed with mean mV and variance V which are constant over time and market

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regime. LV (t ) is also assumed to be independent with BV (t ) and ZV (t ) . The firm value V (t ) is specified as the following stochastic differential equation:

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dV (t ) = ( V (t ) V mV )dt V (t )dBV (t ) LV (t )dZV (t ), V (0) > 0, V (t )

(1)

where is a constant asset payout ratio, V (t ) is the expected return on the firm’s assets, and

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V (t ) is the volatility of the firm’s asset return. The expected return and volatility are constant in each regime, but have different values across the two regimes:

H ,V

V (t ) =

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L ,V

in in

regime H H ,V , V (t ) = regime L L ,V

in in

regime H . regime L

We provide analytic formulas for the default probability, equity price, and CDS spreads of firm in the following subsection. For that purpose, we need to use more functions related to the Poisson processes which are derived in existing literature such as Jang et al. (2011). For any time interval [s, s t ] , the transition densities from regime k to regime l of the hidden Markov chain is defined as Pkl (t ) = P{ V (s t ) = l ,V | V (s) = k ,V } for k , l {H , L}, and s,t 0,

satisfying

H L ( H L ) t = 1 PHL (t ), PHH (t ) = e H L H L L H ( ) t PLL (t ) = e H L = 1 PLH (t ). H L H L Also, the occupation time, Tt , for which the firm value process is in regime H from time 0 to time t , is defined as

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Tt := I{ ( s )= }ds, for t [0, ). H

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If we let f k (t , u ) be a probability density function of Tt for a fixed time 0 < t < and regime

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k {H , L} at time 0 , then f k (t , u ) satisfies

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f H (t , u ) := e L (t u ) H u ((H L ut u )1/2 I1 (2(H Lu (t u ))1/2 ) H I 0 (2(H Lu (t u ))1/2 )), 0 < u < t, L ( t u ) H u 1/2 1/2 ((H L (t u )u ) I1 (2(H Lu (t u )) ) f L (t , u ) := e L I 0 (2(H Lu (t u ))1/2 )), 0 < u < t,

f H (t ,0) = 0, f H (t , t ) = e

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and H t

, f L (t ,0) = e

Lt

, f L (t , t ) = 0,

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where I a (z ) is the modified Bessel function defined by I a ( z ) := z 2

a

( z/2)

2n

n!(a n 1).

n=0

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On the other hand, the firm value process in equation (1) is a generalized affine process

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which includes regime-dependent drifts and volatilities. The Fourier transform method for affine processes introduced by Carr and Madan (1999) and Duffie et al. (2000) can be applied to this

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model with modifications. We derive the conditional characteristic function of terminal density of

log (V (T )/V (0)) when TT is given by u [0,T ] :

1 2

( w; u ) := E[e wlog(V (T )/V (0)) | TT = u ] = exp w 2 w2 V (e where

1 T

1 mV w V w2 2

1) T ,

(2)

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= {( H ,V u L,V (T u )) ( H2 ,V u L2,V (T u )) ( V mV )T } and

1 T

2 = { H2 ,V u L2,V (T u )}. Notice that the expectation E[] in the above equation is taken under the physical probability ~ measure P . We define E[] as the expectation under a risk-neutral probability measure Q ~ and ( w; u ) as the conditional characteristic function under Q . If we denote the risk-free interest ~ rate by r , ( w; u ) has the following form similar to the right hand side of equation (2):

ACCEPTED MANUSCRIPT ~

1 2

~

1 2

~

~ 2 ) w 2 w 2 (e ( w; u ) := exp (r V m V V

~ ~ w 1 m w2 V 2 V

1) T ,

(3)

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~ where the Poisson jump process has intensity V and the jump size is normally distributed with

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~ ~ and volatility mean m V under Q . V

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2.2 Default Probability, Equity Price, and CDS spreads

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We show analytic expressions for the default probability and the firm’s equity price in Theorem 2.1 and Theorem 2.2 respectively. All proofs are in Appendix.

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Theorem 2.1 (Default Probability) When the current regime is l {H , L} , the default probability of the individual firm with face value K is

PV (T ) K < 0 =

T

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1 (ln ( K/V (0)); u ) f l (T , u )du 0

LT

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L (l )e

(ln ( K/V (0));0) H (l )e

H T

(ln ( K/V (0));T ),

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where l (k ) is the indicator function I{l = k } , and

( iy ; u ) . iy

( x; u ) = e xF 1

Here, F 1 is the Fourier-inversion operator and is the decay rate.

In the context of regime-switching model, it is natural to provide two solutions depending on the current state of regime. Related literature is referred to Kim et al. (2008) and Jang et al. (2007) among others. A single solution could be obtained by using weighted average of the one regime state: for example, PDH and PDL are default probabilities when the current regime is high and low, respectively, and pH is the probability when the current regime state is high, then the default probability could be inferred to pH PDH (1 pH ) PDL . Theorem 2.2 (Equity Price) When the current regime is l {H , L} , the 0 -time equity price

Sl (0) is

ACCEPTED MANUSCRIPT T~ Sl (0) = V (0)e rT (ln ( K/V (0)); u ) f l (T , u )du 0 ( r L )T ~ ( r H )T ~ L (l )V (0)e (ln ( K/V (0));0) H (l )V (0)e (ln ( K/V (0));T ),

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where l (k ) is the indicator function I{l = k } , and

~

(1 iy; u ) . 2 2 y i (2 1) y

~

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( x; u ) = e xF 1

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Here, F 1 is the Fourier-inversion operator and is the decay rate.

Using the default probability in Theorem 2.1, we can calculate CDS spreads. In general,

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the protection seller of a CDS will compensate the protection buyer when default of its reference entity (or equivalently, the firm described in this section) occurs, and the amount of compensation depends on the recovery rate. The protection buyer should periodically pay a premium for each

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unit amount of nominal. We denote the recovery rate as R and assume that the maturity of the

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CDS is same as the maturity of the firm’s debt, T . We also assume the premium payments occur at ti = i, i = 1,, (T/) , where is the length of time between two adjacent payment dates. If

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the CDS spreads are determined at its issue date, time 0 and constant over all payments and the protection buyer pays accrual premium upon default, they are calculated as follows:

cdsT =

T /

(1 R)(Qt i =1

i 1

Qt ) B(0, ti ) i

Qt B(0, ti ) (Qt Qt ) B(0, ti ) , i i 1 i 2 i =1 T /

(4)

where Qt is the survival probability from 0 to t , B(0,t ) is a default-free discount bond price at time 0 with maturity t and principal amount 1 . The average accrual from ti 1 to t i is assumed to be

for the principal amount. 2

3 Empirical Analysis 3.1 Data Description

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We employ historical credit rating data from non-financial US firms provided by Capital IQ Compustat for the period from January 2001 to December 2011, and select firms included in

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three credit rating categories of Standard and Poor’s: A, BBB, and BB. Although Standard and Poor’s has detailed rating scales with positive or negative signs, known as notch signs, we regard

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firms rated A , A and A as being in the same category as A-rated firms; similarly, BBB , BBB

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and BBB (BB , BB and BB ) rated firms are regarded to have BBB ratings (BB ratings). According to the business cycle reference dates for the US provided by the National

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Bureau of Economic Research (NBER), our sample period comprises 26 months of recession (high regime) period, and 106 months of expansion (low regime) period. Specifically, the sample

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period comprises two high regimes (March 2001 to October 2001 and December 2007 to May 2009) and three low regimes (January 2001 to February 2001, November 2001 to November 2007, and June 2009 to December 2011).

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We nonparametrically detect daily jumps by using five-minute high-frequency data on

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equity prices obtained from Tick Data and estimate the jump parameters at the first stage under the following assumptions: there is at most one jump per day, jumps are large, and the jump size

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dominates the signs of daily return.3 Subsequently, we calibrate the equity and asset parameters for the high and low regimes based on monthly equity return data from Datastream. Instead of detecting jumps for the same period as the equity return data covering 11 years (2001 – 2011), we use three years of historical intraday transaction data on individual stocks to estimate jump parameters for each firm. Specifically, we select the period from August 2005 to July 2008 for the high-frequency data set. These three years comprise eight months of high regime and 28 months of low regime. Although jumps are firm-specific and independent of state of regime in our model setting, we attempted to carefully and reasonably select the period for the high-frequency data in order to preserve the proportion of the duration of both regimes in the original set. The number of months in the high regime constitutes approximately 20% of the high-frequency data set (8 months

/ 3 years 0.2 ) and of the entire data set (28 months / 11 years 0.2 ). Detailed calibration methodologies are introduced in Section 3.2.

[Insert Table 1 here.] 3

These assumptions are in Zhang et al. (2009) and Tauchen and Zhou (2011).

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After selecting firms with more than 70 transactions per day from the lists of the three credit rating cohorts, 198 firms (62 A-rated firms, 100 BBB-rated firms, and 36 BB-rated firms)

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are left in the data set. Table 1 presents the summary statistics including means, standard deviations (Std), and the 0%, 25%, 50%, 75%, and 100% quantiles (Min, 0.25Q, Median, 0.75Q,

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and Max, respectively) for equity returns provided by Datastream and the industrial classification

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of the firms in each credit rating category. All nine industry sectors are classified on the basis of the Global Industrial Classification Standard (GICS) developed by Standard and Poor’s and the

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data are obtained from Capital IQ Compustat.

Procedures for calibrating regime-dependent asset parameters based on the target variables

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such as regime-dependent equity parameters for each credit rating category are described in the following subsections.

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3.2 Target Parameters

We calibrate the regime-dependent and regime-independent asset parameters from the

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target parameters, including equity parameters for the three credit rating classes from both the 11 years of equity price data and three years of high-frequency data on equity price. The logarithmic monthly equity price of i -th firm, pi (t ) , has the following approximate relationship:

pi (t ) k ,S (t )t k ,S (t )BS (t ) LS (t )Z S (t ), k = H , L, i

i

i

i

(5)

i

where Si (t ) is the equity price of i -th firm at time t , k , S , k , S , and k , S are the expected i

i

i

equity return, volatility, and intensity of i -th firm, respectively. The jump process LS (t ) of i -th i

firm is normally distributed with mean mS and volatility S , when Z S (t ) is as Poisson i

i

i

process with intensity S . LS (t ) , BS (t ) , and Z S (t ) are assumed to be mutually independent. i

i

i

i

The equity return parameters are estimated as follows: Step 1. Estimation of the regime-independent jump parameters Step 2. Estimation of the regime-dependent parameters. Parameter values other than the target equity return parameters need to be determined. The

ACCEPTED MANUSCRIPT risk-free interest rate r is 4%, which is estimated according to 3-month US treasury bills from 1985 to 2011. Mostly we set the same values as those assumed in Huang and Huang (2003) and Zhang et al. (2009). The initial asset value V (0) is set to 100 and the asset payout ratio is 2%. i

i

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The equity risk premium k , S ( k ,S = k ,S r ) are 5.99%, 6.55%, and 7.30% for A-, BBB- and i

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BB-rated firms, respectively. Moreover, the leverage ratios are set to 43.29% for A-rated firms, 48.02% for BBB-rated firms, and 58.63% for BB-rated firms, which are used for the face value of

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each credit rating.

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3.2.1 Estimation Approach

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Step 1. Regime-Independent Jump Parameters.

First, we focus on estimating the jump parameters, which are assumed to be regime-independent. We use the nonparametric jump identification method introduced in Zhang

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et al. (2009) to detect daily equity jumps for individual firm and to estimate three jump parameters:

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jump mean, volatility, and intensity.4 Similar to Zhang et al. (2009), we also make the following assumptions: there is at most one jump per day, the jumps are large, and the jump size dominates

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the signs of daily returns. Three-year high-frequency data on equity prices from August 2005 to July 2008 are utilized. After detecting daily jumps, we obtain the three jump-related parameters of each firm, mS , S , and S in equation (5). i

i

i

Step 2. Regime-Dependent Parameters. We additionally include the regime-switching environment within the model. We believe that the jump risks are firm-specific rather than at the macroeconomic level. Then, the jump component in the equity return process is assumed to be independent of the regime shift. This implies we can divide the equation (5) into two parts: the regime-dependent part (drift and volatility terms) and the regime-independent part (jump term). Using the equity jump parameters estimated in Step 1, we generate jumps in each month t by simulation to separate the jump term in equation (5) from the monthly equity returns of individual firms, for the period from January 2001 to December 2011. Replacing the subscript S i 4

Similar nonparametric jump detection methods are also proposed in Andersen et al. (2007) and Tauchen and Zhou (2011).

ACCEPTED MANUSCRIPT by S for simplicity, we have jump-separated (JS) monthly equity prices, p JS (t ) , for the 11 corresponding years. From equation (5),

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p JS (t ) = p(t ) LS (t )Z S (t ) k ,S (t )t k ,S (t )BS (t ), k = H , L.

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Now, we are ready to estimate the regime-dependent equity return parameters, k , S and

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k , S , on the basis of the time series of jump-separated monthly equity returns of individual firms. We apply the EM (Expectation-Maximization) algorithm to estimate six regime-dependent

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parameters, k , S , k , S , and k , S based on the jump-separated equity prices data of 198 firms for

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11 years.5

3.2.2 Estimation Results

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We estimate the target equity parameters of each firm in the three credit rating cohorts.

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Besides the proposed model that includes regime-shifting and jumps (henceforth RS-Jump), we include three more models for comparison: a model with regime-shifting and no jump (henceforth

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RS-noJump), one with no regime-shifting and jumps (henceforth noRS-Jump), and one with no regime-shifting and no jump (henceforth noRS-noJump). Notice that the noRS-noJump model is the same as the one presented in Merton (1974).

[Insert Table 2 here.]

Table 2 summarizes the average parameter estimates across A- (62), BBB- (100) and BB-rated (36) firms. Panel A in Table 2 shows the annualized, average jump parameters for each credit rating category with the sample standard deviation in parentheses. The second column in panel B in Table 2 represents the estimation results of each credit rating category for the regime-dependent equity return parameters of our RS-Jump model. Estimation results of the RS-noJump, noRS-Jump, and noRS-noJump models are displayed in the third, fourth, and fifth columns, respectively, for each credit rating in panel B. These estimates are utilized for the subsequent asset parameter calibration. 5

Related literature is referred to Hamilton (1989), Ang and Bekaert (2002a), and Dempster et al. (1977).

ACCEPTED MANUSCRIPT The means and volatilities of jumps increase as the credit ratings decreases. It is also found that jump events occur more frequently for speculative-grade firms than for investment-grade firms. The noRS-Jump model has the same volatility values across high and low regimes and as

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the noRS-noJump model. For the noRS-Jump and noRS-noJump models, the volatility of the noRS-noJump model has higher values than that of the noRS-Jump model. This results from the

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assumption that jumps are separated from equity return data in the noRS-Jump model.

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As expected, in both the RS-Jump and RS-noJump model, the volatility values are higher in the high regime and lower in the low regime across all the credit rating groups. Moreover, the

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high regime volatility values of the RS-noJump model are higher than those of the RS-Jump model; the volatility values of the RS-Jump and RS-noJump models are 0.4471 and 0.4530, 0.5497

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and 0.5631, and 0.7446 and 0.7556 for A, BBB, and BB ratings, respectively. On the other hand, the low regime volatility values are higher in the RS-Jump model (0.2241, 0.2665 and 0.3491 for A, BBB and BB ratings, respectively) than in the RS-noJump model (0.2232, 0.2631 and 0.3453

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for A, BBB and BB ratings, respectively) across all the credit ratings. However, it can be observed

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that the average volatilities in the two regimes continue to be higher in the RS-noJump model than in the RS-Jump model: the volatility values of the RS-Jump model and the RS-noJump model for

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the A, BBB and BB ratings are 0.3356 and 0.3381, 0.4081 and 0.4131, and 0.5469 and 0.5504, respectively. This indicates a similar relationship in terms of volatility between the noRS-noJump and noRS-Jump models.

It is noteworthy that the values of regime-shifting intensities, H and L , are mostly the same across the credit ratings: approximately 1.2 for intensity H and approximately 0.4 for intensity L . This would possibly imply the existence of fundamental macroeconomic factors to broadly affect the values of assets. We discuss importance of the business cycle in modeling credit risks in detail in Section 4.

3.3 Calibration for Asset Return Parameters 3.3.1 Calibration Approach We obtain an approximate formula for the monthly return of a firm’s asset return process

ACCEPTED MANUSCRIPT RV (t ) from equation (1): i

RV (t ) (k ,V V mV )t k ,V BV (t ) LV (t )ZV (t ), k = H , L.

(6)

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Moreover, under the risk-neutral measure Q , the return process is written as

~ ~ ~ ~ ~ RV (t ) (r V m V )t k ,V BV (t ) LV (t )ZV (t ), k = H , L,

(7)

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~ ~ where BV is the Brownian motion and ZV is the Poisson jump process under Q measure with

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~ ~ ~ ~ and volatility intensity V . LV (t ) is the jump size normally distributed with mean m V V under Q measure.

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We essentially match a firm’s asset value parameters in three credit cohorts with estimated target parameters in the previous subsection. There are nine asset parameters to be calibrated for

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each credit rating category: H ,V , H ,V , and H for a high regime, L,V , L,V , and L for a low regime, and mV , V , and V for regime-independent jumps. We assume that all three

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intensities among the asset parameters are the same as the corresponding intensities obtained from

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the equity data. In other words, the regime-dependent intensities of assets, H and L , are assumed to be equal to those of equity, H , S and L, S , and the jump intensity V is set to be

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same as S . The remainder of the asset parameters to be calibrated are regime-dependent expected returns, H ,V and L,V , regime-dependent volatilities, H ,V and L,V , and regime-independent jump mean and volatility, mV and V , respectively. We initiate with parameter calibration for the simple noRS-Jump model, and subsequently, determine the parameters of the RS-Jump model by using jump parameters calibrated in the noRS-Jump model. Three parameters are calibrated in the noRS-Jump model: asset volatility, jump mean, and jump intensity. Because regime shifting is not reflected in the noRS-Jump model,

H ,V and L,V have same values and H and L are zero. To calibrate asset parameters, we adopt the equity price formula in Theorem 2.2. Moreover, we employ the following two relationships between equity and asset in terms of jump and volatility:

LS (t ) = log(S J (t; )) log(S (t; )), and

(8)

ACCEPTED MANUSCRIPT S = V

S (t ) V (t ) , V (t ) S (t )

(9)

where is the parameter vector and S J (t; ) is the equity price at time t , when the jump event

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to the firm’s asset value occurred at time t . 6 We estimate jump parameters from the high-frequency data on equity price in Table 2, which implies that we can obtain asset jump

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parameters by simulating equity jumps LS (t ) fitted to their jump means, volatilities and

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intensities by using equation (8).

The regime-dependent expected asset returns, H ,V and L,V , are determined by

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matching the Sharpe ratio of a firm’s equity to that of its assets. Therefore, the Sharpe ratio enables us to relate the asset risk premium k ,V ( k ,V = k ,V r ) to the equity risk premium k , S .

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~ Following Zhou (2001), the jump intensity risk spreads, V V , and the jump size risk spreads, ~ , if the jump risk ~ , are assumed to be zero. Zhou (2001) stated that = ~ and m = m mV m V V V V V

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~ ~ , even if the jump risk is completely systematic. is diversifiable, and that V V and mV m V The jump risks in our model are firm-specific and estimated by individual firms.

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For the next, we discuss parameter calibration for the RS-Jump model. In this case, we focus on the volatilities in both regimes. We regard the values of jump parameters in the RS-Jump model as being the same as the jump parameter values in the previous calibration of the noRS-Jump model. This procedure can be appropriate because of the independence assumption between regime-shifting and jumps. Similar to the calibration approach in the noRS-Jump models, we utilize the relationship between equity and asset volatilities for each group of rated firms as follows:

k , S = k ,V

S (t ) V (t ) , (k = H , L). V (t ) S (t )

Consequently, throughout the process, we can achieve the calibration of all parameters in the regime-dependent asset return process with jumps.

3.3.2 Calibration Results 6

As a matter of fact,

= {Vt , T , K , r, H , L , H ,V , L,V , mV , V , V } , which includes all parameters used in the equity price

calculation. If the jump event is occurred, the underling firm value at time Zhang et al (2009).

t

is

Vt (1 LV (t )) . The relationship of this kind is appeared in

ACCEPTED MANUSCRIPT

We obtain parameters for four asset return models, RS-Jump, RS-noJump, noRS-Jump,

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and noRS-noJump, across the A, BBB, and BB ratings.

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[Insert Table 3 here.]

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The results in Table 3 present the calibrated asset parameters of our RS-Jump, RS-noJump, noRS-Jump, and noRS-noJump models for the A, BBB, and BB credit rating groups.7 Panel A in

models across the three rating categories.

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Table 3 shows the jump parameters and panel B represents the calibration results for the four

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As compared to the values of the equity return volatilities, the asset volatilities for all the credit rating groups have lower values. The asset volatility values are higher in the high regime than in the low regime for all the credit rating groups for the regime-dependent models. As

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observed in the estimation results for the equity return parameters in Table 2, the asset returns are

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more volatile in lower credit grades regardless of the current regime state; for the RS-Jump model, they are 0.2916, 0.3471, and 0.4611 for A, BBB, and BB ratings, respectively in the high regime,

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and 0.1462, 0.1683, and 0.2162 for A, BBB, and BB ratings, respectively in the low regime. The average volatilities between the two regimes are higher in the RS-noJump model than in the RS-Jump model except for BB rating; the values in the RS-Jump and RS-noJump models are 0.2189 and 0.2195, 0.2577 and 0.2587, and 0.3387 and 0.3312 for A, BBB, and BB ratings, respectively. These features are similar to the volatility estimates of the target parameters.

3.4 Default Probability Based on the asset parameter sets in Table 3, we calculate the real default probabilities obtained in Theorem 2.1 for each credit-rated firm.

[Insert Figure 1, Figure 2, and Figure 3 here.]

7

We also calibrate the parameters for the four models across each firm by using equity parameter estimates. We find that averages of calibrated parameters of firms in each credit rating group are similar to the results reported in this section. Although we do not include the results here, they are available upon request.

ACCEPTED MANUSCRIPT Figures 1, 2, and 3 depict the default probabilities of the four models across the A, BBB, and BB rating groups, respectively. For the regime-independent models (noRS-Jump and noRS-noJump), jumps increase the probabilities of default. This is also observed in the

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regime-dependent models (RS-Jump and RS-noJump), and default probabilities have higher values when the current regime is high rather than when it is low, which indicates that the two

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models are strongly dependent on the current state of regime. Overall, the default probabilities of

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the regime-dependent models are higher than those of the regime-independent models for all rating groups; this difference indicates that the two regime-independent models could underestimate

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default risks and that it might be important to employ regime changes in modeling credit risk. More importantly, considering regime change in the structural model framework is especially

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advantageous for predicting the default probabilities of firms with high credit ratings. For example, when the current regime is high (low), the ten-year default probability for A-rated firms is 3.88% (3.00%) in the proposed RS-Jump model, but only 2.02% in the noRS-noJump model. In

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reality, the average ten-year cumulative default rates reported by Moody’s for the A credit rating

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group are 2.09%, 2.04%, and 3.30% for 1983 – 2009, 1970 – 2009, and 1920 – 2009, respectively (Emery et al., 2010).

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The ten-year default probability given by the RS-Jump model for the credit rating BBB (BB) is 8.53% (23.32%) when the current regime is high and it is 6.96% (20.47%) when the current regime is low. On the other hand, the ten-year default probability from the noRS-noJump model for the credit rating BBB (BB) is 5.16% (17.17%). The Moody’s historical cumulative default probabilities for BBB (BB) firms are 4.82% (21.13%), 4.85% (19.96%), and 7.12% (19.23%) for the period of 1983 – 2009, 1970 – 2009, and 1920 – 2009, respectively. The proposed RS-Jump model provides a better fit to the historical default probabilities than the model without regime-shifting or jump risks.

[Insert Figure 4 and Figure 5 here.]

The four models are used to predict one-year (Figure 4) and five-year (Figure 5) default probabilities of each firm across three credit ratings. For each firm, parameters of the four models are obtained by applying the calibration method introduced in the previous section. The models that include business cycle features are likely to predict higher default probabilities than other

ACCEPTED MANUSCRIPT models such as the jump diffusion model or the traditional Merton (1974) model. The difference of predicted default probabilities of models with and without regime-switching is greater for short-term default probabilities than for long-term: for example in Figure 4, most one-year default

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probabilities of the regime-independent models are close to zero.

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3.5 CDS Spreads

The CDS spreads described in equation (4) with maturity T are presented in Figures 6, 7,

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and 8 for A, BBB, and BB rating groups, respectively. The calibration results of the four models in Table 3 are used to calculate the survival probability under the risk-neutral measure. The CDS is

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assumed to be paid semi-annually and the recovery rate R is set to 51.31% (Eom et al., 2004; Huang and Huang, 2003).

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[Insert Figure 6, Figure 7, and Figure 8 here.] As observed in the results associated with the probability of default (Figures 1 – 3), CDS

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spreads are lowest for the Merton model (noRS-noJump) and highest for the models that includes regime-shifting and jumps (RS-Jump). For the regime-dependent models, the CDS spreads are dependent on the current regime state. For each model, five-year CDS spreads range from 17.17 to 41.47 basis points (bps) for A-rated firms, 58.15 to 108.11 bps for BBB-rated firms, and 234.84.62 to 352.27 bps for BB-rated firms. 8 According to Table 5 in Das et al. (2009), the average five-year CDS spreads excluding the financial sector in 2001 – 2005 for A, BBB, and BB credit ratings are 48.77 bps, 109.22 bps, and 277.55 bps, respectively. For A-rated firms, the regime-dependent models account for up to 85% of the observed CDS spreads. The regime-independent models have less than 50% of the observed spreads for high rating, and tend to have smaller divergence from the observations for lower credit rating than for high rating.

8

Admittedly, because our model is not a first-passage type, defaults in the proposed model can be decided at every discrete time-step (

ti = i, i = 1, ... , T/, : time interval, T : maturity). In the analysis, we set the time interval to be the same as the semi-annual payment period and calculate CDS spreads. However, this limitation can be relieved by considering smaller time intervals. If the reference obligation is decided to default four times per month, or 1/48 year, with semi-annual payment and 51.31% of the recovery rate, then five-year CDS spreads for the RS-Jump model are 40.32, 105.03, and 341.91 bps for A, BBB, and BB rating groups, respectively, when the current regime is high. In that case, five-year CDS spreads for the noRS-noJump model are 16.70, 56.52, and 228.10 bps for A, BBB, and BB rating groups, respectively.

ACCEPTED MANUSCRIPT 3.6 Sensitivity Analysis We analyze how default probabilities and CDS spreads are sensitive to parameter choices

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such as the payout ratio, leverage ratio, equity risk premium, and recovery rate. We calculate five-year default probabilities and CDS spreads from the RS-Jump, RS-noJump, noRS-Jump, and

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noRS-noJump models by varying 20% above and below initial payout ratio and leverage ratio for

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each credit rating. We also report five-year default probabilities when equity risk premiums for each credit rating vary 20% above and below initially-assumed values, and CDS spreads for 30%

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recovery rate. Table 4 and Table 5 summarize the results.

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[Insert Table 4 and Table 5 here.]

We report default probabilities and CDS spreads on the basis of initial parameter choices

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for each model across three credit ratings as benchmarks in the second column in Tables 4 and 5.

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Low (high) payout ratios are associated with high (low) level of firm values, and low (high) leverage ratios are associated with low (high) level of default barriers; thus, both might lead to low

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(high) default probabilities and CDS spreads. In the third and fourth columns in Tables 4 and 5, default probabilities and CDS spreads are not sensitive to changes in payout ratios; although they are more sensitive to different leverage ratios than to payout ratios especially in the regime-independent models, they are relatively robust to changes in leverage ratio in the regime-dependent models (the fifth and sixth columns). Moreover, the equity risk premiums allow us to calculate real default probabilities. According to our model assumption, higher equity risk premiums are related to both higher asset risk premiums and higher firm values for same parameter values of volatilities of equity and asset. Changes in default probabilities in the last two columns in Table 4 compared to benchmarks imply results for different values of equity risk premium. In Table 5, five-year CDS spreads firms given by four models range from 24.69 to 59.62 bps for A-rated firms, 83.59 to 155.42 bps for BBB-rated firms, and 337.62 to 506.45 bps for BB-rated when the recovery rate is varied from 51.31% to 30%.9

9

Hull and White (2000) assume the recovery rate to 30%.

ACCEPTED MANUSCRIPT 4 Discussions

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4.1 Comparisons to Other Structural Models

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In the empirical analysis in the previous section, we investigate the properties of different types of structural models, such as the noRS-noJump, noRS-Jump, RS-noJump, and RS-Jump

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models, and the proposed RS-Jump model is the most generalized of these models. In this section, we compare the RS-Jump model to other existing structural models proposed by Black and Cox

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(1976, henceforth BC), Longstaff and Schwartz (1995, henceforth LS), Leland and Toft (1996, henceforth LT), and Collin-Dufresne and Goldstein (2001, henceforth CDG). The BC model

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extends Merton (1974), or equivalently the noRS-noJump model, and considers that default occurs at the first time that the firm value drops below a default boundary, whether or not at the maturity date of a debt.10 The LS and CDG models are extended versions of the BC model. The LS model

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allows the risk-free interest rate to be stochastic and follows the Vasicek (1977) process. The CDG

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model extends the LS model and considers mean-reverting leverage ratios with stochastic interest rates. Whereas the BC, LS, and CDG models assume an exogenously specified default boundary,

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the LT model assumes an endogenous default boundary which is decided by equity holders to maximize the value of equity.

Parameter values used in the analysis are explained in detail in Appendix 8. We mostly choose the same values for the initial asset value, asset payout ratio, leverage ratios, and face values for A, BBB, and BB credit ratings as in the previous section except for the asset volatility. The asset volatilities for all models across three credit ratings are estimated from historical equity volatility adjusted for each credit rating’s leverage ratio as mentioned in Eom et al. (2004), that is, V = (1 l ) S , where l is the leverage ratio. For the BC, LS, CDG, and LT models, the asset volatilities of A, BBB, and BB credit ratings are 0.16, 0.18, and 0.19, respectively. For the RS-Jump model, asset volatilities in the high regime ( V , H = (1 l ) S , H ) are 0.25, 0.28, and 0.30 for A, BBB, BB credit ratings, respectively, and those in the low regime ( V , L = (1 l ) S , L ) are

10

K =0

Black and Cox (1976) consider a bond with the time-dependent safety covenant and the default boundary

K = Ce

( T t )

, where

is a constant and

level is assumed to be exogenous and constant.

C

is the safety covenant. However, in our analysis,

takes an exponential form; and the specified bankruptcy

ACCEPTED MANUSCRIPT 0.12, 0.13, and 0.14 for A, BBB, BB credit ratings, respectively. Moreover, two intensities and three jump parameters in the RS-Jump model are assumed to be same as those in the equity. The LS and CDG models assume the stochastic interest rate and parameter values of the interest rate

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process are chosen as follows: = 0.226 , = 0.06 , = 0.0468 , and = 0.25 . Moreover, the CDG model assumes the dynamic log default threshold and the parameter values are D = 0.18 ,

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= 0.6 , and = 2.8 . Additionally, for the LT model, the recovery and tax rates are assumed to be

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51.31% and 0%, respectively (Leland, 2004).

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[Insert Figure 9 here.]

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Risk-neutral default probabilities for three credit ratings (Figure 9) are calculated for the RS-Jump model when the current regime is high and for the first-passage type models (the BC, LS, CDG, and LT models). Among the four first-passage type models, the CDG model predicts the

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highest five-year default probabilities followed by the LS, BC, and LT models for BBB- and

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BB-rated firms. For A-rated firms, the CDG model still predicts the highest default probabilities, followed by the LS model and the BC model predicts the lowest default probability. The RS-Jump

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model mostly predicts the second-highest default probabilities for maturities longer than four years. Notably, the RS-Jump model predicts the highest default probabilities for short maturities across all credit ratings, whereas other first-passage type models predict default probabilities close to zero. This difference may indicate the benefits of a credit risk model that considers business cycle and firm-specific risks together; even though the proposed model is based on the simple Merton (1974) structural model, in which the default occurs only at maturity time, the proposed model can capture risks better than the first-passage type models for short maturities.

4.2 Importance of Business Cycle We estimate equity return parameters of the regime-dependent models in Section 3.2 and find that the values of regime-shifting intensities, H and L , are mostly the same across the credit ratings: they are approximately 1.2 for intensity H and approximately 0.4 for intensity

L (Table 2). This would possibly imply the existence of fundamental macroeconomic factors to

ACCEPTED MANUSCRIPT broadly affect the values of assets.

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[Insert Table 6 here.]

According to the NBER announcement, there were two peaks and two troughs during our

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analysis period of 2001 to 2011. Panel A in Table 6 includes the classification of the economic

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recession periods provided by NBER as a benchmark. It also identifies the two high regimes obtained from the two models, RS-Jump and RS-noJump, in a particular quarter (Qtr.), including

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the time of each peak and trough. The current state of regime is determined by the smoothed probability t (k ), k = H , L, and the decision rule ( t (k ) 0.5 ), as provided in Hamilton (1989):

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t ( H ) = P( V (t ) = V , H | p(T ), p(T 1)..., p(1)), t ( L) = 1 t ( H ), where { p(T ), p(T 1)..., p(1)} represent all observed equity data. Therefore, the current state of

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regime is classified as H when t ( H ) 0.5 . In panel A of Table 6, our results show that

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classification of peak time is similar to the NBER benchmark and that of trough time is two quarters later than the benchmark for the period Recession 1. For the period Recession 2, the

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classification of the peak time in our model is two to three quarters later than that of the benchmark. The time of trough classified in our model is similar to that of the benchmark. Panel B in Table 6 reports the values of the regime classification measure (RCM) in Ang and Bekaert (2002b). Ang and Bekaert (2002b) propose RCM = 400 *

1 T t ( H )(1 t ( H )), T i =1

where t (H ) is the smoothed probability of the high regime. RCM is a statistic that captures regime classification quality ranging from 0 to 100; a lower value of RCM implies better regime classification performance. In our result, the proposed RS-Jump model leads to slightly better classifications than the RS-noJump model.

5 Conclusion We develop a structural model that simultaneously incorporates macroeconomic risks and

ACCEPTED MANUSCRIPT firm-specific jump risks and show that the consideration of these two types of risks could better explain the firm default rates and CDS spreads that are observed in reality. We model the macroeconomic risks by employing a two-state Markov chain and consider these two states as

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being the states of the business cycle and also individual jump risks by including additional Poisson process. Our study contributes to the credit risk literature by analyzing the effects of

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macroeconomic risk factors and individual jumps, and by providing an intuitive model that

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incorporates both macroeconomic and jump risks, which is useful for predicting the default probability of individual firms.

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Under our model set-up, we find new analytic formulas for default probabilities, equity price, and CDS spreads. We also calibrate model parameter obtained from financial data for

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individual US firms, analyze model properties, and compare model performances with other existing structural models. We find that our model could better predict actual default probabilities and CDS spreads, and thus overcome the underestimation of credit risks, especially for firms with

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high credit ratings, which has been one of the major limitations of conventional structural models.

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Moreover, our analysis emphasizes the importance of macroeconomic risk factors in modeling credit risks and shows the dependence of default probabilities and CDS spreads on the current

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economic state.

ACCEPTED MANUSCRIPT Appendix

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Proof of Theorem 2.1

1 = ( H u L (T u )) ( H2 u L2 (T u )) ( V mV )T 2 T u u L2 (1 ) BV (T ) LV (t )dZV (t ) 0 T T T = T 2 BV (T ) LV (t )dZV (t )

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H2

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V (T ) ln V (0) TT =u

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For a given value of occupation time, TT = u [0, T ] , the asset price at maturity T is

0

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We define a new process X u (t ) on [0,) by letting t

X u (t ) := t 2 BV (t ) LV (s)dZV (s), X u (0) = 0,

D

0

in probability. Notice that X u (t ) is an affine

which satisfies X u (T ) = ln (V (T )/V (0)) |T

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T =u

jump-diffusion process, and, for t < T , X u (T ) ln (V (t )/V (0)) |T

T =u

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and the conditional return ln (V (t )/V (0)) |T

T =u

(10)

. In that, the process X u (t )

are probabilistically equivalent only if t = T .

As we can notice in equation (10), the process X u (t ) is an affine process with constant drift, volatility, and intensity of the Poission jump. Duffie, Pan, and Singleton (2000) introduce a system of ordinary differential equations to derive the characteristic function of such processes. Following them, we define the (conditional) characteristic function of X u (t ) for given u [0,T ] ;

( w; u ) := E[e w X

u (T )

1 m w w2 1 ] = exp w 2 w2 V (e V 2 V 1) T . 2

The above equation is same as equation (2) in the main body of this paper. Now, consider the default probability of a firm,

P(V (T ) < K TT = u) = P( X u (T ) < ln ( K/V (0))). If we let k = ln ( K/V (0)) and define a function

f ( x; u) = ex P( X u (T ) x), x (, ),

ACCEPTED MANUSCRIPT the default probability becomes 1 e k f (ln (

K ); u ). V (0)

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If we take the Fourier transform of f ( x; u) with respect to x , we get

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X u (T ) ( iy ) x F{ f ( x; u )} = eiyx ex P( X u (T ) x)dx = E e dx u (T ) ( iy ) X e ( iy ; u ) = E . = iy iy

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Therefore, the conditional default probability for given TT = u is

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( iy; u ) P(V (T ) < K TT = u ) = 1 e T F 1 . iy x =ln ( K/V (0))

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Proof of Theorem 2.2

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Q.E.D.

We use the similar argument to that in the proof in Appendix 6; (1) fix the occupation time, (2) define an affine process which has a same distribution with ln (V (T )/(V (0))) at maturity T , (3) use the characteristic function and the Fourier transform method, and (4) integrate it over the level of occupation time. However, we start with the firm’s asset value process under the risk-neutral probability Q . Under Q -measure, the process follows dV (t ) ~ ~ ~ ~ ~ = (r V m V ) dt (t )dBV (t ) LV (t )dZV (t ), V (0) > 0, V (t )

which is the same form except replacing the drift parameter by the risk-free interest rate r . Therefore, the conditional characteristic function under Q is expressed as in equation (3) with

~ ~ 1 2 ~ (r V m V ) instead of . We denote it by ( w; u ) as seen in (3). 2 For any given level of occupation time u [0,T ] , the equity price can be calculated as a call option on the firm’s asset value;

S (0) |T

T =u

u ~ ~ = e rT E[(V (T ) K ) TT = u] = e rT E[(V (0)e X (T ) K )I

{ X u (T ) > ln ( K/V (0))}

].

ACCEPTED MANUSCRIPT If we define a function u ~ g ( x; u ) = ex E[(e X (T ) e x )I

{ X u ( T ) > x}

],

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then the (conditional) equity price is

V (0) exp (rT ) exp (k ) g (k; u) with k = ln ( K/V (0)).

u ~ F g ( x; u ) = eiyx ex E[(e X (T ) e x )I u

]dx

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X (T ) > x

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We can get the function g ( x; u ) by using the Fourier transform method:

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~ X u (T ) ( iy ) x X u (T ) (1 iy ) x = E e e dx ~ (1 iy ) X u (T ) (1 iy ) X u (T ) e (1 iy ; u ) ~ e = E . = 1 iy 2 y 2 i (2 1) y iy

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Q.E.D.

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Parameter Values in Section 4.1 The firm value process under the risk-neutral measure Q in the RS-Jump model is

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assumed as follows:

dV (t ) ~ ~ ~ ~ ~ = (r (t ) V m V )dt V , k dBV (t ) LV (t )dZV (t ), k = H , L, V (t )

(11)

where we explain the parameters in equation (7). For the BC, LS, LT, and CDG model, V ,k has

~ ~ same value across the state of regime, that is V ,k = V , and the jump components V m V in the

~ ~ drift term and LV (t )dZV (t ) in the jump term are zero. Parameters of the interest rate process and the leverage ratio process need to be chosen in the LS and CDG models. Firstly, the stochastic interest rate is considered in the LS and CDG models. The dynamics of interest rate r under the risk-neutral measure Q are driven by the Vasicek (1997) model: ~ dr (t ) = ( r (t ))dt dBr (t ),

~ where , , and are constants, Br is a standard Brownian motion, and the instantaneous

~ ~ correlation between dBV (t ) and dBr (t ) is dt . The mean reversion speed of the short term

ACCEPTED MANUSCRIPT interest rate = 0.226 , its long-run mean = 0.06 , its standard deviation = 0.0468 , and the correlation between the interest rate process and the firm value process = 0.25 . The parameter values are from those in Huang and Huang (2003). Moreover, the CDG model assumes the log

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where y(t ) = log V (t ) and the parameter values are assumed to D = 0.18 , = 0.6 , and = 2.8

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(Collin-Dufresne and Goldstein, 2001). Additionally, for the LT model, the recovery and tax rates are assumed to be 51.31% and 0%, respectively (Leland, 2004). We use an endogenous default

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boundary for infinite maturity debt for consistency with other models.

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Figure 1: Default probabilities for credit rating A

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The figures present the probability of default for A-rated firms based on the four models. The left side of the figure plots the results when the current state of regime is high and the right side of the figure plots the results when the current state of regime is low. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks.

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Figure 2: Default probability for credit rating BBB

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The figures present the probability of default for BBB-rated firms based on the four models. The left side of the figure plots the results when the current state of regime is high and the right side of the figure plots the results when the current state of regime is low. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks.

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Figure 3: Default probability for credit rating BB

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The figures present the probability of default for BB-rated firms based on the four models. The left side of the figure plots the results when the current state of regime is high and the right side of the figure plots the results when the current state of regime is low. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks.

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Figure 4: One-year default probabilities

The figures present one-year default probabilities of the four models (the RS-Jump, RS-noJump, noRS-Jump, and noRS-noJump models in the first to fourth rows, respectively) across three credit rating groups (A, BBB, and BB credit ratings in the first to third columns, respectively). The x-axis is default probabilities in percentage and the y-axis is the number of frequency. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks.

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Figure 5: Five-year default probabilities

The figures present five-year default probabilities of the four models (the RS-Jump, RS-noJump, noRS-Jump, and noRS-noJump models in the first to fourth rows, respectively) across three credit rating groups (A, BBB, and BB credit ratings in the first to third columns, respectively). The x-axis is default probabilities in percentage and the y-axis is the number of frequency. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks.

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Figure 6: CDS spreads for credit rating A

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The figures present CDS spreads for A-rated firms based on the four models. The left side of the figure plots the results when the current state of regime is high and the right side of the figure plots the results when the current state of regime is low. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks.

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Figure 7: CDS spreads for credit rating BBB

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The figures present CDS spreads for BBB-rated firms based on the four models. The left side of the figure plots the results when the current state of regime is high and the right side of the figure plots the results when the current state of regime is low. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks.

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Figure 8: CDS spreads for credit rating BB

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The figures present CDS spreads for BB-rated firms based on the four models. The left side of the figure plots the results when the current state of regime is high and the right side of the figure plots the results when the current state of regime is low. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks.

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Figure 9: Risk-neutral default probabilities

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The first to third figures present risk-neutral default probabilities for A, BBB, BB credit ratings, respectively, based on the RS-Jump, BC, LS, LT, and CDG models. The RS-Jump is the proposed regime-switching with jump model, BC is Black and Cox (1976) model, LS is Longstaff and Schwartz (1995) model, LT is Leland and Toft (1996) model, and CDG is Collin-Dufresne and Goldstein (2001) model. Used parameter values are in Section 4.1.

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A 5 5 15 6 12 4 6 1 8 62

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Energy Materials Industrials Consumer discretionary Consumer Staples Health care Information technology Telecommunication services Utilities (Total observations)

Credit rating BBB 11 16 23 15 7 6 5 0 17 100

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Industry

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B. Industry classification

Median 0.0631 0.0642 0.0863

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A. Means, standard deviations and quantiles Credit rating Mean Std Min A 0.0552 0.2893 -0.0781 BBB 0.0636 0.3463 -0.1011 BB 0.0724 0.4615 -0.1421

0.75Q 0.0947 0.1046 0.1391

Max 0.1440 0.2268 0.2606

BB 5 6 8 6 3 3 5 0 0 36

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The table presents summary statistics for monthly equity price returns in our analysis. Panel A repots mean, standard deviation (Std), and 0%, 25%, 50%, 75% and 100% quantiles (Min, 0.25Q, Median, 0.75Q, Max, respectively) information of three credit rating classes. Panel B reports the industrial constituents of firms in the analysis based on the Global Industrial Classification Standard (GICS). The data cover the period from January 2001 to December 2011.

ACCEPTED MANUSCRIPT Table 2: Estimated target parameters

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Intensity 0.1284 (0.0451) 0.1417 (0.0449) 0.1855 (0.0804)

noRS-noJump

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RS-noJump -0.1501 (0.1431) 0.1086 (0.0642) 0.4530 (0.1622) 0.2232 (0.0595) 1.1582 (0.2279) 0.3843 (0.0529) 152.7891

0.0388 (0.0504) 0.2871 (0.0837) -

0.0552 (0.0508) 0.2893 (0.0838) -

-0.1749 (0.1926) 0.1222 (0.0804) 0.5631 (0.1997) 0.2631 (0.0641) 1.1986 (0.2050) 0.3816 (0.0547) 129.6447

0.0398 (0.0729) 0.3427 (0.0919) -

0.0636 (0.0667) 0.3463 (0.0922) -

-0.1886 (0.3444) 0.1342 (0.0837) 0.7556 (0.2318) 0.3453 (0.1125) 1.2111 (0.2677) 0.3713 (0.0383) 93.1949

0.0977 (0.0831) 0.4587 (0.1314) -

0.0724 (0.0857) 0.4615 (0.1312) -

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B. Means, volatilities, and intensities RS-Jump B-1. Credit rating A Mean - High -0.1630 (0.1387) Mean - Low 0.0900 (0.0638) Volatility - High 0.4471 (0.1608) Volatility - Low 0.2241 (0.0592) Intensity - High 1.1737 (0.2314) Intensity - Low 0.3803 (0.0499) Log-likelihood 153.3772 B-2. Credit rating BBB Mean - High -0.1909 (0.2065) Mean - Low 0.0953 (0.0873) Volatility - High 0.5497 (0.1989) Volatility - Low 0.2665 (0.0623) Intensity - High 1.2330 (0.2033) Intensity - Low 0.3768 (0.0537) Log-likelihood 129.7145 B-3. Credit rating BB Mean - High -0.1504 (0.3323) Mean - Low 0.1553 (0.0835) Volatility - High 0.7446 (0.2343) Volatility - Low 0.3491 (0.1103) Intensity - High 1.2359 (0.2498) Intensity - Low 0.3687 (0.0380) Log-likelihood 93.0758

Volatility 0.0680 (0.0493) 0.0776 (0.0328) 0.0965 (0.0269)

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Mean 0.0006 (0.0479) 0.0050 (0.0475) 0.0206 (0.0680)

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A. Jump parameters Credit rating A BBB BB

The table reports the estimation results for the target parameters in four different models over the three credit rated firms. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks. Panel A reports the jump parameter estimates from the five-minute high-frequency data on equity price from August 2005 to July 2008. Panel B reports regime-dependent mean, volatility and intensity estimates of high and low regime of four models, over the three credit rating classes. The data cover the period from January 2001 to December 2011. The sample standard deviations are in parentheses. Table 3: Calibrated parameters

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Mean

Volatility

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0.0725 0.0725 0.2941 0.1450 1.1582 0.3843

0.1284 0.1417 0.1855

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0.1332 0.1486 0.1541

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B. Means, volatilities, and intensities RS-Jump B-1. Credit rating A Mean - High 0.0726 Mean - Low 0.0726 Volatility - High 0.2916 Volatility - Low 0.1462 Intensity - High 1.1737 Intensity - Low 0.3803 B-2. Credit rating BBB Mean - High 0.0779 Mean - Low 0.0779 Volatility - High 0.3471 Volatility - Low 0.1683 Intensity - High 1.2330 Intensity - Low 0.3768 B-3. Credit rating BB Mean - High 0.0833 Mean - Low 0.0834 Volatility - High 0.4611 Volatility - Low 0.2162 Intensity - High 1.2359 Intensity - Low 0.3687

Intensity

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0.0105 0.0219 0.0268

0.0717 0.1820 -

0.0716 0.1830 -

0.0776 0.0776 0.3526 0.1648 1.1986 0.3816

0.0764 0.2079 -

0.0762 0.2091 -

0.0822 0.0822 0.4547 0.2078 1.2111 0.3713

0.0803 0.2639 -

0.0799 0.2628 -

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A. Jump parameters Credit rating A BBB BB

The table reports the calibration results of four different models over the three groups of credit rated firms based on the target parameter estimates. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks. Panel A reports the calibration results of the jump parameters and Panel B reports the calibration results of regime-dependent mean, volatility and intensity of high and low regime of four models, over the three credit rating classes.

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A. RS-Jump Credit rating Benchmark Payout (L) Payout (H) Lvrg. (L) Lvrg. (H) Prem. (L) Prem. (H) (Current regime is high) A 2.16 1.99 2.28 1.36 3.11 2.46 1.85 BBB 5.91 5.49 6.14 4.14 7.92 6.56 5.11 BB 19.78 18.40 19.83 15.37 23.65 20.72 17.56 (Current regime is low) A 1.25 1.14 1.34 0.73 1.93 1.45 1.05 BBB 3.85 3.54 4.04 2.52 5.55 4.35 3.27 BB 15.29 14.04 15.37 11.11 19.38 16.20 13.28

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B. RS-noJump Credit rating Benchmark Payout (L) Payout (H) Lvrg. (L) Lvrg. (H) Prem. (L) Prem. (H) (Current regime is high) A 2.04 1.91 2.18 1.36 2.82 2.35 1.76 BBB 5.79 5.49 6.10 4.31 7.37 6.54 5.11 BB 18.73 18.04 19.45 15.45 22.14 20.34 17.21 (Current regime is low) A 1.14 1.06 1.23 0.71 1.66 1.34 0.97 BBB 3.62 3.40 3.85 2.55 4.85 4.17 3.14 BB 14.01 13.39 14.67 10.94 17.43 15.49 12.64

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C. noRS-Jump Credit rating Benchmark Payout (L) Payout (H) Lvrg. (L) Lvrg. (H) Prem. (L) Prem. (H) A 0.81 0.70 0.89 0.39 1.45 0.99 0.66 BBB 2.96 2.73 3.21 1.76 4.73 3.51 2.49 BB 13.85 13.17 14.56 10.14 18.71 15.36 12.43 D. noRS-noJump Credit rating Benchmark Payout (L) Payout (H) Lvrg. (L) Lvrg. (H) Prem. (L) Prem. (H) A 0.67 0.60 0.74 0.34 1.12 0.83 0.54 BBB 2.64 2.44 2.85 1.65 3.82 3.15 2.20 BB 13.05 12.39 13.76 9.90 16.48 14.55 11.67 The table presents sensitivity analysis results for five-year default probabilities of four different models over the three groups of credit rated firms by varying initial parameter choices. Benchmark in the first column in each panel reports default probabilities on the basis of initial parameters for each model. In each panel, columns of Payout (L), Lvrg. (L), and Prem. (L) report default probabilities by varying 20% below initial parameter values of payout ratio, leverage ratio, and equity risk premium, respectively. Also, columns of Payout (H), Lvrg. (H), and Prem. (H) report default probabilities by varying 20% above initial parameter values. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks. Table 5: Sensitivity analysis for five-year CDS spreads

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Payout (L)

Payout (H)

Lvrg. (L)

Lvrg. (H)

Recovery

43.70 111.84 351.40

26.66 76.29 262.03

58.72 144.36 443.98

59.62 155.42 506.45

23.70 68.83 245.23

27.41 77.65 267.48

15.43 49.44 190.40

38.83 105.16 349.25

37.08 107.08 383.59

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38.56 101.03 235.53

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54.04 135.68 417.07

56.65 151.99 478.27

25.32 73.95 256.20

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Payout (L) Payout (H) Lvrg. (L) Lvrg. (H) Recovery 18.03 21.52 10.30 32.81 28.27 59.02 67.80 39.88 95.37 90.86 235.53 258.32 180.26 341.65 354.63

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A. RS-Jump Credit rating Benchmark (Current regime is high) A 41.47 BBB 108.11 BB 352.27 (Current regime is low) A 25.79 BBB 74.48 BB 266.81 B. RS-noJump Credit rating Benchmark (Current regime is high) A 39.41 BBB 105.72 BB 332.67 (Current regime is low) A 23.61 BBB 69.84 BB 245.10 C. noRS-Jump Credit rating Benchmark A 19.67 BBB 63.20 BB 246.67 D. noRS-noJump Credit rating Benchmark A 17.17 BBB 58.15 BB 234.84

Payout (L) Payout (H) Lvrg. (L) Lvrg. (H) Recovery 15.68 18.76 9.38 27.08 24.69 54.32 62.14 38.09 81.35 83.59 224.09 246.52 176.92 305.45 337.62

The table presents sensitivity analysis results for five-year CDS spreads of four different models over the three groups of credit rated firms by varying initial parameter choices. Benchmark in the first column in each panel reports CDS spreads on the basis of initial parameters for each model. In each panel, columns of Payout (L) and Lvrg. (L) report CDS spreads by varying 20% below initial parameter values of payout ratio, leverage ratio, and equity risk premium, respectively. Also, columns of Payout (H) and Lvrg. (H) report CDS spreads by varying 20% above initial parameter values. Recovery in the last column reports CDS spreads when recovery rate is 30%. The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Moreover, the noRS-Jump model considers no regime-shifting and jump risks and the noRS-Jump model does not consider regime-shifting nor jump risks. Table 6: Regime classification

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RS-Jump 33.6352 32.5517 30.0636

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Recession 2 Peak Trough 2007 4Qtr. 2009 2Qtr. 2008 3Qtr. 2009 2Qtr. 2008 2Qtr. 2009 2Qtr. 2008 3Qtr. 2009 2Qtr. 2008 3Qtr. 2009 2Qtr. 2008 2Qtr. 2009 2Qtr. 2008 3Qtr. 2009 2Qtr.

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Recession 1 Peak Trough 2001 1Qtr. 2001 4Qtr. 2001 1Qtr. 2001 2Qtr. 2001 1Qtr. 2001 2Qtr. 2001 1Qtr. 2001 2Qtr. 2001 1Qtr. 2001 2Qtr. 2001 1Qtr. 2001 2Qtr. 2001 1Qtr. 2001 2Qtr.

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RS-noJump 33.6203 32.3816 29.9793

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In the table, Panel A reports the classification of peak and trough quarters (Qtr.) from the RS-Jump and RS-noJump models, comparing to the business cycle classification by the National Bureau of Economic Research (NBER). The RS-Jump represents the model with both regime-shifting and jump risks and the RS-noJump model is the model only with regime-shifting risks. Panel B reports the value of the regime classification measure (RCM) in Ang and Bekaert (2002b), where the lower values of RCM implies better regime classification.

ACCEPTED MANUSCRIPT Highlights A structural model incorporating macroeconomic and firm-specific jump risks is developed.

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Analytic formulas for default probability, equity price, and CDS spreads are derived. Model parameters are calibrated with historical equity prices and high-frequency data.

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Proposed model overcomes underestimation problems of credit risks for high-rated firms.