The instantaneous and forward default intensity of structural models

Transcription

1 The instantaneous and forward default intensity of structural models Cho-Jieh Chen Department of Mathematical and Statistical Sciences University of Alberta Edmonton, AB First version: 2004 Current version: 2007 ABSTRACT If investors are able to observe the value of a firm, it has been a belief that the two main credit risk models: the structural model and the intensity model, cannot be generally unified. By showing the intensity paradox, we prove that this is not totally correct. We extend the definition of the classic default intensity, which we call it the instantaneous default intensity, to the forward default intensity in two different ways. For the instantaneous default intensity, we concur with Duffie and Lando (2001) and Giesecke (2005) that the instantaneous default intensity does not generally exist unless the default time is totally inaccessible. We extend this result to firm-value processes that are stochastically continuous with independent and stationary increments. However, we prove that forward default intensities always exist. Based on the firm-value information from the past to current time, the conditional probability of default is a function of the forward default intensity, not the instantaneous default intensity. As a result, we are able to obtain the yield spread and the forward spread in terms of forward intensities. For a stopping time that is neither accessible nor totally inaccessible, we show that the forward default intensity can be decomposed into the forward default intensity caused by an accessible process and the forward default intensity caused by a totally inaccessible process. We conclude that the structural model

2 and the intensity model can be unified through the forward default intensity, although the instantaneous default intensity does not always exist. Key words and phrases: Credit risk; Structural model; Intensity model; Hazard rate; Diffusion process; Jump-diffusion process. JEL: G100, G320, G330 Acknowledgement: This research is supported by the Natural Sciences and Engineering Research Council of Canada. The major results of this paper were first presented at the National Centre for Theoretical Sciences of Taiwan on April 22, This paper was presented in the 14th Conference on Pacific Basin Finance, Economics, Accounting, and Business on July 14, 2006 and was submitted in I would like to thank Dr. Harry H. Panjer for his precious suggestions on this paper. 1

3 1 Introduction It has been a belief that the two main credit risk models: the structural model and the intensity model, cannot be generally unified when investors are able to observe the firm value. Duffie and Lando (2001) and Giesecke (2005) claim that, when the value of a firm is observable, the default intensity of the structural model does not generally exist unless the time of default is totally inaccessible. When the firm value follows a diffusion process or a jump-diffusion process, the probability that an announcing sequence approaches the default time strictly from below is not zero. Therefore, the default intensity does not generally exist. They conclude that the structural approach does not lay the theoretical groundwork for hazard-rate based estimation of default intensities. In this paper, we refer to this default intensity as the instantaneous default intensity (see section 3). The instantaneous default intensity can be defined in two different ways. It can be defined as the expected rate of default occurrence in the immediate next moment, given the information available up to the current time. It can also be defined as the compensator of the number of default. Definitions of instantaneous default intensities and further discussions on their existence are given in section 3 and section 4. For the instantaneous default intensity, we concur with Duffie and Lando (2001) and Giesecke (2005) that the instantaneous default intensity exists only if the default time is totally inaccessible. We extend this result to firm-value processes that are stochastically continuous with independent and stationary increments. When the firm value follows a diffusion process with a constant default threshold, it is well known that the time of default follows an inverse Gaussian distribution. The probability density function of the default time divided by the survival function, represents the expected rate of default occurrence. However, the default time of this diffusion process is accessible. Therefore, there should not be an associated intensity that leads to the conditional default probability. We call this paradox the intensity paradox (see section 5). The answer to the intensity paradox is the existence of the forward default intensity. We define the forward default intensity as the expected rate of default occurrence in the future, given the information available up to the current time. Alternatively, we define the forward default intensity as the compensator of a submartingale (see section 6). For firm-value processes that are stochastically continuous with independent and stationary increments, when the default threshold is a constant, we prove that the time of default is a continuous random 2

4 variable. The conditional hazard rate exists and equals the probability density function of the default time divided by its survival function. Based on the relationship between the conditional hazard rate and the forward intensity, we prove that the forward intensity exists. In section 7, we examine the existence of forward intensities based on accessibility. We prove that forward intensities exist for the accessible stopping time, the totally inaccessible stopping time, and the stopping time that is neither accessible nor totally inaccessible. For a stopping time that is neither accessible nor totally inaccessible, we prove that the forward default intensity can be decomposed into the forward default intensity caused by an accessible process and the forward default intensity caused by a totally inaccessible process (see section 7.3). Specifically for a jump-diffusion process, we show that the forward default intensity can be decomposed into the forward default intensity caused by diffusion and the forward default intensity caused by a jump (see section 8.2). Given firm-value information available up to current time, the conditional probability of default is a function the forward intensity, not the instantaneous intensity. In section 8, we show how to calculate the conditional default probability in terms of the forward intensity. We obtain the yield spread and the forward spread in terms of forward intensities for the diffusion process and the jump-diffusion process under the recovery-of-treasury-value scheme. Although the instantaneous default intensity does not always exist, we conclude that the structural model and the intensity model can be unified through the forward default intensity. 2 Model setup We consider a stochastic basis (Ω, F, (F t ) t 0, Q), which is a probability space (Ω, F, Q) endowed with a filtration, (F t ) t 0, where Ω is the set of outcomes of the world, F is the sigma-algebra generated by Ω, (F t ) t 0 is an increasing sub-sigma-algebra with F s F t for 0 s < t and F t = F t+, and Q is the probability measure of the set of events in F. Q is a countably additive non-negative normalized set function, with Q( i B i ) = i Q(B i ), if B i F and B i B j = for i j, 0 Q(B) 1, B F, and Q(Ω) = 1. We assume that F 0 contains all Q-null sets. At time t > 0, F t can be regarded as the sigma-algebra of events observable on the time interval [0, t]. We assume that investors are able to observe the value of a public firm from the past to the current time. Duffie and Lando (2001) and Giesecke (2005) refer to this assump- 3

5 tion as the complete information. In practice, the value of a public firm can possibly be obtained by adding up the market price of the equity and the liability if these data are available. We treat the firm value as if it were a traded instrument. Let {V s } s 0 be a stochastic process where V s denotes the firm value at time s. We assume that the firm-value process has independent and stationary increments and is stochastically continuous, i.e. for all c > 0 and for all s > 0, lim t s Q{ V (t) V (s) > c} = 0. At current time t, investors can observe firm values V s = v s, 0 s t while {V s } s>t remains unknown because no one can foresee the future trajectory of the firm value. When a firm is not able to fulfill its financial obligations, we call it default. Let the time of default of an event ω Ω be denoted by τ(ω) or τ. τ is also referred to as the first passage default time. It is possible that default will never occur. Thus, τ is a random variable taking values in R + { }. We call τ a continuous random variable if F τ (t) is continuous in t for t <. If τ is a continuous random variable, the probability density function f τ (t) = F τ (t) exists for t < and f t τ (t) < for t <. Equivalently, we can Q{t<τ t+h} say lim h 0 = c, 0 c <, or Q{t < τ t + h} = O(h) for all t <, as h 0. h If {ω : τ(ω) t} F t, t 0, τ is called a Markov time or a stopping time, which means that we know whether or not default has occurred at every time t. The predictable σ algebra of subsets of Ω R + is the σ algebra generated by all the adapted processes Y = Y (ω, t), t R +, ω Ω, with left-continuous trajectories. A process X is called predictable with respect to a filtration if, as a mapping from Ω R + to R, it is measurable with respect to the predictable σ algebra generated by that filtration. A predictable firm-value process can be regarded as a process whose behavior at time t is determined by information on [0, t). When the firm value follows a diffusion process, it is a predictable process. When the firm value follows a jump-diffusion process or a pure jump process, it is not a predictable process. A stopping time is predictable if {(ω, t) : t R +, 0 t < τ(ω)} is a predictable set. For the structural model, we define the time of default in a standard way, as the first time the firm value hits its default threshold [Black and Cox (1976), Leland (1994), Longstaff and Schwartz (1995), or Zhou (1997)] 1. It is defined by τ = inf{t > 0 : V t D} where D 1 If we define the default time, as the first time the firm value falls below the default threshold, it does not change the probability of default and the distribution function of τ because we assume the firm value 4

6 is a known default threshold with a trivial condition V 0 > D. Consider that lim F τ (t + h) F τ (t) = lim Q{t < τ t + h} 0 = lim Q{V s > D, 0 s t, V t+h D} h 0 lim Q{V t > D, V t+h D} h 0 = lim h 0 D+ Q{V t+h D V t = x}f Vt (x)dx. (1) Because V t+h converges to V t in probability almost surely as h 0, Q{V t+h D V t = x} = 0 for almost all x unless x = D+. Therefore, equation (1) equals zero. Because F τ (t) is a nondecreasing function in t for t > 0, lim h 0 F τ (t + h) F τ (t) 0. From equation (1), we have lim h 0 F τ (t + h) F τ (t) 0. We can conclude that lim F τ (t + h) F τ (t) = 0, t > 0. h 0 Because F τ (t) is nondecreasing in t with F τ (0) = 0 and lim h 0 F τ (t + h) F τ (t) = 0, F τ (t) is continuous in t. Therefore, τ is a continuous random variable. We assume that the firm is liquidated at or after the time of default to pay off its financial obligations if default occurs. The solvency ratio process is defined by { U(t) = ln ( V t )}. The default time can D t 0 also be defined by τ = inf{t > 0, U(t) 0}. Let L(t 1, t 2 ] = inf {U(t), t 1 < t t 2, t 1 < t 2 } be the minimum solvency ratio during the time interval (t 1, t 2 ]. The corresponding model filtration is F t = σ{v s, 0 s t}, t 0. Because {ω : τ(ω) t} F t, τ is a stopping time. If a stopping time is predictable, then there exists a nondecreasing sequence of stopping times τ 1 τ 2 with τ n < τ and lim n τ n = τ for ω Ω with {τ(ω) > 0}. This sequence is called an announcing sequence for τ. The graph of a stopping time is defined by [τ] = {(ω, t), t R +, t = τ(ω)}. A stopping time is called accessible if there is a sequence {τ n } n 1 of predictable times such that [τ] n [τ n ]. A stopping time is called totally inaccessible if Q{τ = τ < } = 0 for each predictable stopping time τ. Therefore, no predictable stopping time can give any information about the totally inaccessible stopping time. Some stopping times are neither accessible nor totally inaccessible. When default will never occur, the default time is denoted by τ =. For each stopping time τ <, there exists one and only one (to within Q-negligibility) pair of an accessible stopping time τ a and a totally inaccessible stopping time τ i such that [τ] = [τ a ] [τ i ] is stochastically continuous. 5

7 and [τ a ] [τ i ] =. (see Liptser and Shiryaev, 1998, theorem 3.3). One can say that the classes of accessible stopping times and totally inaccessible stopping times are orthogonal. Consider a structural default model. Each default time τ < can be decomposed into τ a and τ i. Because [τ a ] and [τ i ] are disjoint, the default model can be regarded as a doubledecrement context (see Bowers, N.L. eds., 1997, Ch 10). If τ <, then the default time follows τ = min{τ a, τ i } = τ a τ i. If default is caused by a predictable process, then τ a < and τ i =. Otherwise, τ i < and τ a =. Accessibility and the decomposition property play important roles in determining whether the default intensity exists. This issue is explained in the next section. 3 The instantaneous default intensities of the first passage default time 3.1 Hazard rate The hazard rate function of τ is defined by h(t ) = f τ(t ) 1 F τ (T ), T 0. Given that the firm has not defaulted by time t with information F t, the conditional hazard rate function is given by h(t t) = f τ(t t) 1 F τ (T t) = f τ(t t), T t > 0, (2) S τ (T t) where F τ (T t) = Q{τ T τ > t, F t }, S τ (T t) = 1 F τ (T t), and f τ (T t) = conditional hazard rate function equals h(t t) = f τ (T t) Q{τ > T τ > t, F t } Fτ (T t). The T = Q[T < τ T + h τ > T, τ > t, F t ] lim = Q[T < τ T + h τ > T, F t ] lim (3) = Q[T < τ T + h τ > T, V t = v t > D] lim, 0 < t T. (4) Equation (3) equals equation (4) because the firm value evolution has independent and stationary increments. 6

8 3.2 Conditional intensity In this paper, we do not treat default as a recurrent event. The number of default is defined by N(t) = 1 {τ t}. This one-jump process is a step function which is right continuous with left limits. Default can occur only if default has not occurred in the past. The expected rate of default occurrence, given the firm has not defaulted with information F t, is given by E[N(t + h) N(t) τ > t, F t ] µ(t F t ) = lim. (5) We call it the instantaneous default intensity conditional on survival 2. Because N(t) can take values 0 or 1, so µ(t F t ) = 1 Q[N(t + h) N(t) = 1 τ > t, F t ] + 0 Q[N(t + h) N(t) = 0] τ > t, F t ] lim = Q[N(t + h) N(t) = 1 τ > t, F t ] lim = Q[t < τ t + h τ > t, F t ] lim. When the firm value is observable, µ(t F t ) is a time-dependent function. When the firm value is not observable, µ(t F t ) is a stochastic process depending on the realization of the event ω. In this paper, we are working on the case when the firm value is observable. Therefore, µ(t F t ) is a time-dependent function. The relationship between the instantaneous default intensity conditional on survival and the conditional hazard rate is given by µ(t F t ) = h(t t), t Intensity of default arrival {N(t)} t 0 is adapted to (F t ) t 0 with E[N(t)] <, for t <. Because E[N(t + s) F t ] N(t) a.s. for s 0, t 0, {N(t)} t 0 is a right-continuous nonnegative submartingale. The Doob-Meyer decomposition theorem states that there exists a unique (to within indistinguishability) nondecreasing right-continuous predictable process {A(t)} t 0 such that M(t) = N(t) A(t), 2 If we treat default as a recurrent event, it does not have to be conditional on survival. 7

9 {M(t)} t 0 is a martingale. {A(t)} t 0 is called the predictable compensator of {N(t)} t 0. A(t+h) A(t) If A(t) is finite and differentiable (lim h 0 exists and is finite) with A (t) = λ(t), h then {λ(t)} t 0 is called the intensity process for {N(t)} t 0. The process A(t) = t 0 λ(s)ds, is called the cumulative intensity process for {N(t)} t 0. The da(t) = lim h 0 [A(t+h) A(t)] can be regarded as the average rate of growth of N(t) with information available up to time t. If µ(t F t ) exists and µ(t F t ) < for 0 t <, we consider For t 0, M(t) = N(t) t E[M(t)] = E [ 1 {τ t} ] E [ t 0 1 {τ>s} µ(s F s )ds, t 0. = Q{τ t} Q{τ t} = 0. 0 ] 1 {τ>s} µ(s F s )ds We can see that M(t) is a martingale and t 0 1 {τ>s}µ(s F s )ds is F t -predictable. If t 0 1 {τ>s}µ(s s)ds exists, by the uniqueness (to within indistinguishability) of the predictable compensator, we have λ(t) = 1 {τ>t} µ(t F t ) a.s.. (6) The intensity can also be defined as the expected rate of default arrival by with λ(t) = lim h 0 E[N(t + h) N(t) F t ] h [ t ] E λ(s)ds = Q{τ t}. 0 Q{t < τ < t + h F t } = lim, This intensity 3 is the rate of default arrival in the immediate next moment, given information available up to time t. Therefore, in this paper, we call it instantaneous default intensity. 3 E[N(t+h) N(t) τ t,f On {ω : τ(ω) t}, λ(t) = lim t ] E[1 1 τ t,f = lim t ] = 0. On {ω : τ(ω) > t}, E[N(t+h) N(t) τ>t,f λ(t) = lim t] = µ(t F t ). 8

10 When the N(t) is not a one-jump process and default can recur, then 1 {τ>t} will drop out of equation (6). For example, when N(t) is a homogeneous Poisson process with parameter λ, then lim h 0 Q{t<τ<t+h F t } h = λ. λ(t) represents the expected rate of the growth of N(t), given information available up to time t, while µ(t F t ) in equation (5) represents the expected rate of growth of N(t) given default has not occurred with information available up to time t. The existence of the intensity is examined in the next section. 4 The existence of instantaneous default intensity 4.1 Accessible stopping times When the stopping time is accessible, N(t) = 1 {τ t} is predictable. When N(t) is predictable, A(t) = N(t) (see Liptser and Shiryaev, 1998, theorem 3.12). Thus, the derivative of A(t) is A (t) = 0 for t < τ and A (t) = for t = τ. This violates the necessary conditions for the existence of the default intensity that A(t) is differentiable for t 0 and is finite. Therefore, the instantaneous default intensity process {λ(t)} t 0 does not exist. Given F t at time t, investors know V s = v s for 0 s t. For V s, s > t, investors know its probability density function but not exactly what will occur. Because the firm value evolution has independent and stationary increments, the instantaneous default intensity conditional on survival equals µ(t F t ) = Q{τ < t + h τ > t, F t } lim = Q[V t+h D L[0, t] > 0, V t = v t ] lim = Q[V t+h D V t = v t > D] lim. Because the stopping time is accessible, there exists an announcing sequence, such as τ n = inf{t > 0, V t D + 1/n}, giving early warning signals of default with lim n τ n = τ. By looking at the latest τ n, investors can be certain of whether or not default will occur in the immediate next moment. If default will not occur, then µ(t F t ) = 0. If default will occur, then µ(t F t ) =. Because of the accessibility, given that the firm has not defaulted at time t, the range of the firm value at time t can be classified into V D = {v t : lim h 0 Q{V t+h 9

11 D V t = v t } = 1, t 0} and V D = {v t : lim h 0 Q{V t+h D V t = v t } = 0, t 0}. { 0, v t V µ(t F t ) = D, v t V D. Consider the case when the firm value follows a diffusion process with a constant default threshold D. If v t > D, the firm value will not to diffuse D in the immediate next moment. If v t = D+, the firm value will diffuse to D with probability one. Therefore, { 0, v t > D µ(t F t ) =, v t = D +. The instantaneous default intensity conditional on survival does not exist when the stopping time is accessible. 4.2 Totally inaccessible stopping times Duffie and Lando (2001) state that for a stopping time to have an associated intensity, it must (among other properties) be totally inaccessible, meaning that, for any sequence of stopping time, the probability that the sequence approaches τ strictly from below is zero. Based on F t, no matter what firm value V t is, investors cannot assert that default will occur (µ(t F t ) = ), nor can they assert that default will not occur (µ(t F t ) = 0) in the immediate next moment. It must be the case 0 < µ(t F t ) <. If this is violated (µ(t F t ) = 0 or ) for some specific firm values, then there exists a sequence of stopping times such that the probability that the sequence approaches time τ strictly from below is larger than zero. Mathematically speaking, a totally inaccessible stopping must satisfy Q{τ < t + h τ > t, F t } = O(h), t 0, as h 0. The instantaneous default intensity conditional on survival equals the probability density function that dv t is larger or equals to the distance to default V t D as µ(t F t ) = Q{τ < t + h τ > t, F t } lim = Q[V t+h D L[0, t] > 0, V t = v t ] lim, = Q[V t+h V t D V t L[0, t] > 0, V t = v t ] lim. Because we assume that the firm value evolution has independent and stationary increments, µ(t F t ) = lim h 0 Q[V h D v t V 0 = 0] h 10

12 D vt = lim f Vh (x)dx, h 0 where f V h (x)dx = 1 4. For the one-jump process, we can conclude that 0 < µ(t F t ) < 1, t 0, meaning that the expected rate of default arrival is less than one. The instantaneous default intensity equals The compensator for N(t), A(t) = t 0 D vt λ(t) = 1 {τ>t} f Vh (x)dx. 1 {τ>t} µ(s F s )ds < t <, 0 t <. A(t) is bounded by t. For example, the stopping time is totally inaccessible when the firm value follows a compound Poisson process as V t = N t i=1 X i, where N t is a Poisson counting process representing the number of jumps up to time t with parameter λ, and X i is the jump size of the firm value of the i th jump and is independent of N t with CDF F X (x) and PDF f X (x) for < x <. We have Q{τ < t + h τ > t, F t } = λhf X (D v t ) + o(h) = O(h), t 0. Thus, the instantaneous default intensity conditional on survival equals µ(t F t ) = λhf X (D v t ) lim = λf X (D v t ). 4.3 Stopping times that are neither accessible nor totally inaccessible There are stopping times that are neither accessible nor totally inaccessible, meaning that there exists at least a sequence of stopping times approaching τ strictly from below with 4 For an accessible stopping, { such as a diffusion type firm-value model, lim h 0 f Vh (x) = 0, x D v t if v t > D. lim h 0 f Vh (x) = 0, x < D v t, x = D v t, if v t = D+. Therefore, the instantaneous default intensity does not exist. 11

13 probability larger than 0 and smaller than 1. Because every stopping time can be decomposed into an accessible stopping time and a totally inaccessible stopping time, the one-jump process N(t) follows N(t) = 1 {(τa τ i ) t} = 1 {τa t} + 1 {τi t}. 1 {τi t} has an associated instantaneous intensity process, however, 1 {τa t} does not. Therefore, we cannot find an associated intensity process for a stopping time that is neither totally inaccessible nor accessible. When the firm value follows a jump-diffusion process, such as Zhou (1997), the stopping time is neither accessible nor totally inaccessible. When the firm value is away from the default threshold, default can occur only caused by a jump. Investors cannot assert whether or not default will occur. The instantaneous intensity is finite. When the firm value is about to cross the default threshold at V t = D+, the firm will diffuse to default threshold with probability 1. The intensity is. Therefore, we cannot find an associated intensity process for a stopping time that is neither totally inaccessible nor accessible. 5 Intensity paradox? In this section, we consider the case when the default threshold is a constant D and the firm value follows a diffusion process as dv t = rv t dt + σv t dz t, (7) where the rate of return r equals the risk-less rate, σ 2 is the variance, and Z t is a standard Brownian motion under the Q measure with respect to {F t } t 0. The default time τ = inf{t > 0, V t D} is an accessible stopping time. Therefore, there does [ not exist an t associated instantaneous intensity process {λ(t)} t 0 so that Q{τ t} = E ]. λ(s)ds 0 Consider that at current time t, the investors know the firm value {V s = v s } {s t}. Investors do not know the firm values V s for s > t but know the distribution of the firm value evolution in the future. If the firm has not defaulted in the past, the distribution function of τ given F t can be obtained using the result of Harrison (1990) as ( F τ (T t) = Φ ln ( ) v t σ2 ) + (r )(T t) D 2 σ T t ( + exp 2 ln ( v t σ2 ) (r ) ) ( D 2 Φ ln ( ) v t σ2 ) (r )(T t) D 2 σ 2 σ T t 12

14 ( = Φ ln ( ) v t σ2 ) + (r )(T t) D 2 σ T t ( ( vt ) 1 2r σ + 2 Φ ln ( ) v t σ2 ) (r )(T t) D 2 D σ, t < T <. (8) T t The probability mass function of the event that default will never occur is ( Q{τ = F t } = 1 exp 2 ln ( v t σ2 ) (r ) ) ( D 2 vt ) 1 2r σ = 1 2. σ 2 D The probability density function of τ is the derivative of F τ (T t) as [ ] 2 ln ( v t D f τ (T t) = ) ln ( v t σ2 ) + (r )(T t) exp 2πσ2 (T t) 3 2 D 2 2σ 2 (T t), τ <. Conditional on τ <, τ follows an inverse Gaussian distribution 5. function can be obtained by The hazard rate h(t t) = = f τ (T t) 1 F τ (T t) ( ln( Φ v t ln( v t D ) 2πσ 2 (T t) 3 2 D )+(r σ2 )(T t) 2 σ T t exp ) { [ ln( v t D )+(r σ2 2 )(T t) ] 2 2σ 2 (T t) ( ( ) v t 1 2r σ 2 Φ ln( v t D } D ) (r σ2 )(T t) 2 σ T t ), (9) which represents the expected default occurrence at time T given that the firm survives time T with information available up to time t, where 0 t < T. The probability of default, given that default has not occurred with information available up to current time t, is Q{t < τ T τ > t, F t } = T t e s t h(z t)dz h(s t)ds. We can define a process {λ(s t)} s t, which is similar to the default intensity process, by λ(s t) = 1 {τ>s Ft}h(s t), s > t. 5 Conditional on τ <, the probability density function of τ is f τ (T τ <, F t ) = exp f τ (T t) Q(τ < F t ) = ln ( vt D ) 2πσ2 (T t) 3 2 ] 2 2 )(T t) 2σ 2 (T t), [ ln ( v t D ) (r σ2 which is an inverse Gaussian probability density function. 13

15 [ T ] E λ(s t)ds t = T t Q{τ > s F t }h(s t)ds = Q{t < τ T τ > t, F t }. Because the time of default is an accessible stopping time, there [ does not exist an associated ] T instantaneous default intensity process {λ(t)} t 0 so that E λ(s)ds = Q{τ T }, T > 0 [ ] T 0. Why is there a process {λ(s 0)} s>0 so that E λ(s 0)ds = Q{τ T F 0 0 }? 6 The forward intensity In this section, we show that the answer to the intensity paradox is the existence of the forward default intensity. At current time t, investors concern about the probability of default occurring in (t, T ] based on the information up to current time t, given that the firm has not defaulted by time t. The conditional default probability Q{t < τ T τ > t, F t } is assessed based on F t for the fixed t, but not based on the progressively filtration (F s ) t<s T. To calculate the conditional default probability, we can use the conditional hazard rate. Alternatively, we can define the forward default intensities, conditional on {τ > T } and {τ > t} respectively and use them to calculate the conditional default probability. Given firm-value information available up to time t and that the firm will survive the future time T, the probability of default at the immediate next moment after time T is defined by equations (2) and (3) as h(t t) = f τ(t t) 1 F τ (T t) = lim h 0 Q{T < τ T + h τ > T, F t }. h Because τ is a continuous random variable, both probability density function and cumulative distribution function exist. As a result, the conditional hazard rate h(t t) exists. To assess the rate of default occurrence in (t, T ], investors can use firm-value information available up to time t, but not progressively in (t, T ]. First, we consider the rate of default occurrence at the future time T if the firm will survive time T. 6.1 The forward default intensity conditional on future survival The expected rate of default occurrence at time T > t, given that the firm will survive by time T with firm-value information available up to time t, is defined by E[N(T + h) N(T ) τ > T, F t ] µ(t F t ) = lim, 0 t < T <. 14

16 We call it the forward default intensity conditional on future survival. For the one-jump process, this intensity equals µ(t F t ) 1 Q{N(T + h) N(T ) = 1 τ > T, F t } + 0 Q{N(T + h) N(T ) = 0 τ > T, F t } = lim Q{N(T + h) N(T ) = 1 τ > T, F t } = lim Q{T < τ T + h τ > T, F t } = lim = h(t t), 0 t < T <. Therefore, the forward default intensity conditional on future survival equals the conditional hazard rate. Because the conditional hazard rate exists, this forward intensity exists. Here, we show a simple example. Assume that the riskless rate is 4.7%. Investors are able to observe the value of a firm from time 0 to current time 3 with V 3 = 13 (in millions), D = 8 (in millions), and σ = 0.2. Using equation (8), we can find that the one-year conditional default probability equals Table 1: The forward default intensity from time 3 to time 4 T µ(t F 3 ) 3.52E E At time 3, the firm value is away from default threshold so the instantaneous default intensity µ(3 F 3 ) = 0. We are not able to express the probability of default in terms of instantaneous default intensities. However, the conditional default probability is a function of the forward default intensity. Next, we consider the rate of default occurrence at future time T if the firm has not defaulted at current time. 6.2 The forward default intensity conditional on current survival We consider the following process, a process that is assessed based on F t for fixed t, given that the firm has not defaulted by time t, M(T τ > t, F t ) = N(T τ > t, F t ) A(T τ > t, F t ) 15

17 T = 1 {t<τ T τ>t,ft } 1 {τ>s τ>t,ft }µ(s F t )ds, (10) t for 0 t < T <. This process is assessed based on a fixed filtration F t for a fixed t, given that the firm has not defaulted by time t. This is not the Doob-Meyer decomposition, which is a decomposition progressively based on (F t ) t 0. Taking the expected value of equation (10), we have E[M(T τ > t, F t )] = E [ [ ] T ] 1 {t<τ T τ>t,ft } E 1 {τ>s τ>t,ft }µ(s F t )ds t = Q{t < τ T τ > t, F t } Q{t < τ T τ > t, F t } = 0, which is a F t -martingale for fixed t. If A (T τ > t, F t ) exists and is finite, we call it the forward default intensity conditional on current survival and denote it by λ(t τ > t, F t ). It equals λ(t τ > t, F t ) = A (T τ > t, F t ) = 1 {τ>t τ>t,ft}µ(t F t ) E[N(T + h) N(T ) τ > t, F t ] = lim, 0 t < T <. Because µ(s F t ) = h(s t) exists for continuous random variable τ, the forward default intensity conditional on current survival, λ(t τ > t, F t ) = 1 {τ>t τ>t,ft }µ(t F t ), also exists. The instantaneous increment of A(T τ > t, F t ) is da(t τ > t, F t ) = E[N(T + dt) N(T ) τ > t, F t ]. It represents the instantaneous increment of the expected rate of default arrival in the future, given that the firm has not defaulted by time t with information available up to time t. 6.3 The forward default intensity conditional on current information We reconsider the equation (10) without conditioning it on {τ > t} as, M(T F t ) = N(T F t ) A(T F t ) = 1 {t<τ T Ft } T t 1 {τ>s Ft }µ(s F t )ds, (11) 16

18 for 0 t < T <. On {ω : τ(ω) t}, N(T τ t, F t ) = 1 {t<τ T τ t,ft } = 0 and 1 {τ>s τ t,ft} = 0, for s > t. On {ω : τ(ω) t}, M(T F t ) = 0. Taking the expected value of equation (11), we have { E[M(T F t )] = E [ [ ] T ]} 1 {t<τ T τ>t,ft} E 1 {τ>s τ>t,ft}µ(s F t )ds Q{τ > t} t = [Q{t < τ T τ > t, F t } Q{t < τ T τ > t, F t }] Q{τ > t} = 0, which is a F t -martingale for fixed t. If A (T F t ) exists and is finite, we call it the forward default intensity conditional on current information and denote it by λ(t F t ). It equals λ(t F t ) = A (T F t ) = 1 {τ>t Ft }µ(t F t ) E[N(T + h) N(T ) F t ] = lim, 0 t < T <. Because µ(t F t ) = h(t t) exists for the continuous random variable τ, the forward default intensity conditional on current information, λ(t F t ) = 1 {τ>t Ft }µ(t F t ), also exists. 7 Discussion on the existence of the forward intensity In the previous section, we have shown the existence of the forward intensity. In this section, we examine the existence of the forward intensity based on the accessibility of the stopping time. 7.1 The accessible stopping time When a stopping time is accessible, investors can assert whether default will occur at time t + dt, given F t, t 0. Given that the firm has not defaulted by time t, the set of all firm values at time t can be separated into V D = {v t : lim h 0 Q{V t+h D V t = v t } = 1, t 0} and V D = {v t : lim h 0 Q{V t+h D V t = v t } = 0, t 0}. The forward default intensity conditional on future survival is Q{V T +h D V t = v t, L[0, T ] > 0} µ(t F t ) = lim, 0 t < T <. Due to the accessibility of the stopping time, investors can assert whether default will occur or not at time T + dt, if the firm value V T is given. Let W (T F t ) denotes the set of events that the firm will survive time T, given that the firm has not defaulted by time t with firmvalue information available up to time t. Let W D (T F t ) = {ω : lim h 0 Q{V T +h D V t = 17

19 v t, L(t, T ] > 0} = 1} and W D (T F t ) = {ω : lim h 0 Q{V T +h D V t = v t, L(t, T ] > 0} = 0}. Therefore, W (T F t ) = W D (T F t ) W D (T F t ). The forward intensity can be obtained using Bayes rule as µ(t F t ) = W D (T F t ) f V T F t (x)dx Q{W D (T F t ) W D (T F t )} = f τ(t t) S τ (T t) <, 0 t < T <. The forward intensity conditional on future survival exists. Because λ(t τ > t, F t ) = 1 {τ>t τ>t,ft} µ(t F t ) and λ(t F t ) = 1 {τ>t Ft}µ(T F t ), both of them exist. We can conclude that Q{T < τ T + h τ > T, F t } = O(h), as h 0, because we do not know the exact value of V T based on information available up to current time t. For a diffusion firm-value process with a constant default threshold D, the forward intensity conditional on future survival, µ(t F t ), is the conditional probability density function of the firm value at time T equal D+, given that the firm will survive time T. 7.2 Totally inaccessible stopping time When a stopping time τ is totally inaccessible, for any stopping time τ, Q{τ = τ } = 0. For a structural model, investors cannot assert whether or not the default event {t < τ t + h V t = v t }, as h 0, will occur. For T > t, neither can investors predict whether or not the event {T < τ T + h τ > T, V t = v t }, as h 0, will occur. Conditional on {ω : τ > T, V t = v t }, no matter what value v t is, the µ(t F t ) = lim h 0 Q{T <τ T +h τ>t,f t } h should follow 0 < µ(t F t ) <. 6 Because λ(t τ > t, F t ) = 1 {τ>t τ>t,ft } µ(t F t ) and λ(t F t ) = 1 {τ>t Ft }µ(t F t ), both of them exist. 7.3 Stopping times that are neither accessible nor totally inaccessible When the stopping time is neither accessible nor totally inaccessible, the stopping τ can be decomposed as τ = τ a τ i. Default can be caused by either an accessible process or a totally inaccessible process. The one-jump process N(T τ > t, F t ) = 1 {t<τ T τ>t,ft } can be 6 µ(t F t ) = 0 means that default will not occur with certainty. µ(t F t ) = means that default will occur with some certainty. Both are not properties of a totally inaccessible stopping time. 18

20 decomposed as N(T τ > t, F t ) = 1 {t<τ T τ>t,ft} (12) = 1 {t<τa T τ>t,f t} + 1 {t<τi T τ>t,f t}. (13) = N a (T t) + N i (T t), where N a (T t) = 1 {t<τa T τ>t,f t} and N i (T t) = 1 {t<τi T τ>t,f t}. Default events can be classified into two disjoint groups, default events caused by an accessible process and default events caused by a totally inaccessible process. We can consider the default model as a double-decrement model. Let J be the cause of default (see Bowers, N.L., eds., 1997, Ch 10) where J = { a, τ = τ a i, τ = τ i. Taking the expectations of equation (12) and (13), we have Q{t < τ T τ > t, F t } = Q{t < τ a T τ > t, F t } + Q{t < τ i T τ > t, F t } = Q{t < τ T, J = a τ > t, F t } + Q{t < τ T, J = i τ > t, F t }. By total probability, the forward default intensity conditional on future survival can be written as Q{T < τ T + h τ > T, F t } µ(t F t ) = lim [ Q{T < τ T + h, J = a τ > T, Ft } = lim h 0 h + Q{T < τ T + h, J = i τ > T, F ] t} h E[N a (T + h) N a (T ) τ > T, F t ] E[N i (T + h) N i (T ) τ > T, F t ] = lim + lim = µ (a) (T F t ) + µ (i) (T F t ), 0 t < T <, where the µ (a) E[N (T F t ) = lim a (T +h) N a (T ) τ>t,f t ] h 0 represents the forward intensity of h default caused by an accessible process, conditional on future survival, while the µ (i) (T F t ) = lim h 0 E[N i (T +h) N i (T ) τ>t,f t ] represents the forward intensity of default caused by a totally inaccessible process, conditional on future h survival. The forward default intensity conditional on current survival is [ λ(t τ > t, F t ) = 1 {τ>t τ>t,ft }µ(t F t ) = 1 {τ>t τ>t,ft } µ (a) (T F t ) + µ (i) (T F t ) ]. The forward default intensity conditional on current information is [ λ(t F t ) = 1 {τ>t Ft }µ(t F t ) = 1 {τ>t Ft } µ (a) (T F t ) + µ (i) (T F t ) ]. 19

21 The probability of default given that default has not occurred by time t with firm-value information available up to time t, is [ T Q{t < τ T τ > t, F t } = E = T t t ] λ(s t)ds e s t [µ (a) (z F t)+µ (i) (z F t)]dz [ µ (a) (s F t ) + µ (i) (s F t ) ] ds. The conditional default probability is a function of the forward intensity, not the instantaneous intensity. Some authors state that the structural approach does not lay the theoretical groundwork for hazard-rate based estimation of default intensities. This is not correct because we should use the forward intensity to calculate the conditional default probability, not the instantaneous default intensity. Although instantaneous default intensities do not exist for the accessible stopping time and the stoping time that is neither accessible nor totally inaccessible, the structural model does define an intensity process, the forward default intensity. 8 The credit spread and the forward intensity In this section, we show the credit spread of a risky bond as a function of the forward intensity. We do not consider the transaction costs nor the liquidation cost. We assume that there exists a risk neutral probability measure. Under this risk-neutral probability measure, the time-t price of a risky bond of one-dollar face amount is the time-t expected value of a contingent claim paying one dollar if default does not occurs and a fractional amount of one dollar if default occurs. No matter whether or not default occurs, let the amount received for one-dollar face amount be denoted by δ. On the space {t < τ T }, the amount received is called the recovery rate and is denoted by δ(τ). Let V (t, T ) represent the time-t price of a defaultable corporate zero-coupon bond paying one dollar at maturity time T for 0 t T <, with a yield to maturity of R(t, T ). The risky bond price can be written as V (t, T ) = e R(t,T )(T t). Let P (t, T ) represent the time-t price of a default-free zero-coupon bond paying one dollar at maturity time T, with a yield to maturity r(t, T ). We have P (t, T ) = e r(t,t )(T t). A time-t value of a bank account process, which is the value of one dollar accumulated from time 0 to time t, is defined by B(t) = e t 0 r(s)ds, where r(s) is the short rate at time s. When the short rate is assumed to be flat, we have B(t) = e rt. We assume that the recovery rate is the ratio of the market value of the firm at default 20

22 to its debt obligation and this payment is payable to the debt holders at maturity. This recovery scheme is usually referred to as the recovery-of-treasury-value (RTV) scheme. If default has not occurred, 1 {τ>t} = 1. The time-t price of the risky bond of one-dollar face amount can be written as [ ] B(t) V (t, T ) = E t B(T ) δ [ B(t) [ ] ] = E t δ 1{τ>t} B(T ) [ B(t) [ ] ] = E t δ(τ)1{τ T τ>t} + 1 {τ>t τ>t}, B(T ) where E t ( ) denotes the conditional expectation E t ( F t ) under the probability measure Q. If the riskless rate is independent of the default process, with information available up to time t, equation (14) can be written as V (t, T ) = P (t, T ) E t { Et [ δ 1{τ T τ>t} ]} = P (t, T ) { E t [ δ 1{τ T τ>t} = 0 ] Q{1 {τ T τ>t,ft} = 0} + E t [ δ 1{τ T τ>t} = 1 ] Q{1 {τ T τ>t,ft} = 1} } = P (t, T ) (1 Q{τ > T τ > t, F t } + E t [δ t < τ T ] Q{τ T τ > t, F t }) = P (t, T ) (1 Q{τ T τ > t, F t } + E t [δ(τ)] Q{τ T τ > t, F t }) (14) = P (t, T ) (1 {1 E t [δ(τ)]} Q{τ T τ > t, F t }). (15) Taking the log value of equation (15) and dividing it by (T t), we can obtain the yield spread as R(t, T ) r(t, T ) = 1 T t ln [1 {1 E t[δ(τ)]}q{τ T τ > t, F t }]. (16) To link equation (16) with the default intensity, the yield spread equals R(t, T ) r(t, T ) = 1 ( { T T t ln 1 {1 E t [δ(τ)]}e t = 1 = 1 T t ln }) λ(s τ > t, F t )ds T t ln (1 {1 E t[δ(τ)]}e[a(t τ > t, F t )]) ( T 1 {1 E t [δ(τ)]} t e s t µ(z F t)dz µ(s F t )ds (17) ). (18) Let the forward rate for a riskless zero-coupon bond be defined by f(t, T ) ln P (t, T ) T and the forward rate for a risky zero-coupon bond be defined by F (t, T ) ln V (t, T ). T The forward spread for a reduced-form model is obtained by Chen (2005). The forward spread for a structural model is shown in the following theorem: 21

23 Theorem 8.1 If the default-free spot rates and the default process are independent and E t [δ(τ)] is not a function of T in a RTV scheme, then the forward spread is { F (t, T ) f(t, T ) = µ(t F t ) 1 E t [δ(τ)] P (t, T ) }. (19) V (t, T ) Proof: From equation (15), we have F (t, T ) = T ln V (t, T ) = T ln {P (t, T ) [1 {1 E t[δ(τ)]} Q{τ T τ > t, F t }]} = f(t, T ) T ln [1 {1 E t[δ(τ)]} Q{τ T τ > t, F t }] T = f(t, T ) [1 {1 E t[δ(τ)]} Q{τ T τ > t, F t }] 1 {1 E t [δ(τ)]} Q{τ T τ > t, F t } T = f(t, T ) + [{1 E t[δ(τ)]} Q{τ T τ > t, F t }] 1 {1 E t [δ(τ)]} Q{τ T τ > t, F t } = f(t, T ) + {1 E t[δ(τ)]} [Q{τ T τ > t, F T t}] 1 {1 E t [δ(τ)]} Q{τ T τ > t, F t } = f(t, T ) + {1 E t[δ(τ)]} e T t µ(z F t )dz µ(t F t ) 1 {1 E t [δ(τ)]} Q{τ T τ > t, F t } = f(t, T ) + µ(t F t ) {1 E t[δ(τ)]} [1 Q{τ T τ > t, F t }] 1 {1 E t [δ(τ)]} Q{τ T τ > t, F t } = f(t, T ) + µ(t F t ) 1 E t[δ(τ)] {1 E t [δ(τ)]} Q{τ T τ > t, F t } 1 {1 E t [δ(τ)]} Q{τ T τ > t, F t } { E t [δ(τ)] = f(t, T ) + µ(t F t ) 1 1 {1 E t [δ(τ)]} Q{τ T τ > t, F t } { = f(t, T ) + µ(t F t ) 1 E t [δ(τ)] P (t, T ) V (t, T ) }. }. When t T, the forward spread becomes the short-rate spread of Duffie and Singleton (1999) as R(t) r(t) = µ(t F t ) {1 E t [δ(τ)]}. However, the instantaneous default intensity µ(t F t ) exists only for the totally inaccessible stopping time such as the compound Poisson process. For diffusion processes and the jump-diffusion processes, the instantaneous default intensity and short-rate spread does not exist. 22

24 8.1 When the firm value follows a diffusion process When the firm value follows a diffusion process, default occurs once the firm value hits the default threshold D. The ratio V τ equals one so a partial recovery rate is impossible. In D order to link the structural model with the credit spread, we have to assume that there exist liquidation and transaction costs. We assume that the recovery rate δ(τ) is an exogenous random variable. We further assume that the recovery rate is not a function of the maturity T. Assume that the firm value evolution is governed by equation (7) as dv t = rv t dt + σv t dz t. The yield spread follows equation (17) and equation (18) with λ(t τ > t, F t ) = 1 {τ>t τ>t,ft }µ(t F t ) and µ(t F t ) = ln( v [ ] t D ) ln ( v 2 t 2πσ 2 (T t) 2 3 exp D ) +(r σ2 2 )(T t) 2σ 2 (T t) ( ln( v ) ( t D )+(r σ2 2 )(T t) Φ σ T t ( v t 1 2r D ) σ 2 Φ ln ( v t D ) (r σ2 2 )(T t) σ T t ) from equation (9). The forward spread follows equation (19). The short-rate spread does not exist because investors can assert that default will not occur in the immediate next moment if the current firm value is away from the default threshold and default will occur in the immediate next moment if the current firm value is at the default threshold. 8.2 When the firm value follows a jump-diffusion process Merton (1976) models a risky asset using a jump-diffusion process. The diffusion process represents the systematic risk and the jump process represents the nonsystematic risk or the firm-specific risk, which can be hedged. Merton (1976) assumes that the capital asset pricing model (CAPM) holds for equilibrium returns. Thus, the expected instantaneous return of the risky asset equals riskless rate r. The jump process does not affect the expected return of the asset but increases its volatility. Zhou (1997) uses a jump-diffusion process to model the firm value evolution. The jump process follows a compound Poisson process whose occurrence follows a Poisson process and the amplitude is Lognormally distributed. When default can occur only at maturity, Zhou (1997) obtains the closed-form solution for the probability of default. When default can occur continuously in time, the closed-form solution is not obtained. A Monte Carlo approach is used to obtain the default probability. Chen and Panjer (2006) assume that the firm value evolution follows a jump-diffusion process whose jump process is a Poisson-Logexponential compound Poisson process. Assuming that default can occur continuously in time, Chen and Panjer (2006) obtain the closed-form solution for the mean recovery rate and obtain the implied default probability from the credit spread. In this section, we show the forward intensity of the jump-diffusion model of Chen and Panjer (2006) and link it to credit spread. We assume the firm value 23

25 evolution follows dv t = rv t dt + σv t dz t + (J Nt dn t λµ J dt)v t, using the notation: σ 2 : instantaneous variance of return, Z t : a standard Brownian motion, N t : total number of jumps of firm value up to time t, J Nt : jump size as a proportion of V t of the N t -th jump, where (N t ) t 0 is a counting process which follows a Poisson process with parameter λ and is independent of Z t. The jump amplitudes J 1, J 2, are assumed to be independent of (N t ) t 0 and independently and identically distributed. It is also assumed that their moment generating function exists [ ] with mean µ J and variance σj 2. The instantaneous return dv of the value of a firm is E t V t = rdt. Based on Itô s lemma, we have d ln V t = ) (r σ2 2 λµ J dt + σdz t + ln(j Nt + 1)dN t. The percentage change dvt V t is bounded below by 1 and can go up to infinity. The range of the jump amplitude J i thus can only take values in [ 1, ). Assume that Y i = J i + 1 for i = 1 N t. Then Y i is a nonnegative impulse function which takes value 1 when there is no jump and takes values from 0 to other than 1 if there is a jump. Let Y (t) = 1 when N t = 0 and N t N t Y (t) = Y i = (J i + 1), when N t 0. The firm value at time t equals ) ] V t = V 0 exp [(r σ2 2 λµ J t + σz t Y (t). i=1 Default occurs once the firm value hits or falls below the default threshold D. When the firm value is away from the threshold D, default can occur only caused by a jump. When the firm value is at default threshold at D+, the firm value will diffuse to D with probability one. We say that default is caused by diffusion if for every ɛ > 0, there is a δ > 0 such that U(τ δ) U(τ) < ɛ. Otherwise, we say default is caused by a jump. Therefore, {τ = t} = {L[0, t) > D, V t D} = {L[0, t) > D, V t = D+, V t D} {L[0, t) > D, V t > D+, V t D}. For a jump-diffusion model, let τ d denote the time of default caused by diffusion and τ j denote the time of default caused by a jump. Let the cause of the default 24 i=1

26 event be J = { d, τ = τ d j, τ = τ j. We do not consider the transaction and liquidation costs in this jump-diffusion model. If default is caused by diffusion, the recovery rate δ(τ d ) is 100%. If default is caused by a jump, the recovery rate δ(τ j ) is less than or equal to 100%. The forward default intensity caused by diffusion, conditional on future survival, is denoted by µ (d) (T F t ). The forward default intensity caused by a jump, conditional on future survival, is denoted by µ (j) (T F t ). Chen and Panjer (2006) show that the credit spread follows the following theorem. Theorem 8.2 Chen and Panjer (2006) R(t, T ) r(t, T ) = 1 T t ln (1 {1 E t[δ(τ s )]}Q{τ T, J = j τ > t, F t }). (20) An equivalent equation to equation (20), in the form of bond prices, is given by V (t, T ) = P (t, T ) (1 {1 E t [δ(τ s )]}Q{τ T, J = j τ > t, F t }). (21) Chen and Panjer (2006) assume that X = ln Y i is Exponentially distributed with probability density function f X (x) = αe αx, α > 0. This implies that f J (x) = α(x + 1) α 1, 1 < x < 0. The log function of J is Exponentially distributed so J follows a Logexponential distribution. Chen and Panjer (2006) find that f δ(τs)(x) = αx α 1 0 < x < 1, which is a beta distribution with parameters (α, 1) or a power distribution. recovery rate for default caused by a jump is E t [δ(τ s )] = α α + 1. The mean The recovery rate for default caused by a jump is not a function a maturity T. The forward spread follows Theorem 8.3 When X = ln(j + 1) is Exponentially distributed with pdf f X (x) = αe αx, α, x > 0, the forward spread is T F (t, T ) f(t, T ) = e t µ(z F t)dz P (t, T ) µ (j) (T F t ). (22) (α + 1)V (t, T ) 25

27 Proof: Taking the derivative of the log value of equation (21), we have F (t, T ) = ln V (t, T ) T = T ln {P (t, T ) (1 {1 E t[δ(τ s )]}Q{τ T, J = j τ > t, F t })} = f(t, T ) [( { T ln 1 1 α } )] Q{τ T, J = j τ > t, F t } α + 1 ( T 1 1 α+1 = f(t, T ) Q{τ T, J = j τ > t, F t} ) 1 {1 E t [δ(τ s )]}Q{τ T, J = j τ > t, F t } 1 α+1 = f(t, T ) + Q{τ T, J = j τ > t, F T t} = f(t, T ) + 1 α+1 T V (t, T )/P (t, T ) T t e s t µ(z F t)dz µ (j) (s F t )ds V (t, T )/P (t, T ) = 1 α+1 f(t, T ) + T e t µ(z F t )dz µ (j) (T F t ) V (t, T )/P (t, T ) = T f(t, T ) + e t µ(z F t )dz P (t, T ) µ (j) (T F t ). (α + 1)V (t, T ) 9 Conclusion In this paper, we find the general relationship between the two main credit risk models: the structural model and the intensity model. It has been a belief that the structural model does not generally define an intensity process. In this paper, we prove that this is not totally correct. We classify default intensities into two different categories, instantaneous and forward default intensities. We show that both intensities can be expressed as expected rates of default arrival and compensators of submartingales. For firm-value processes that are stochastically continuous with independent and stationary increments, we prove that the instantaneous default intensity exists only if the default time is totally inaccessible. This result is consistent with those of Duffie and Lando (2001) and Giesecke (2005). However, when we extend our consideration in the forward measure, we find that the structural generally defines a forward intensity process. For the firm-value process mentioned in this paper, we find that the time of default is a continuous random variable. We can use the probability density function and survival function of the default time to obtain the conditional hazard rate h(t t). Using equations µ(t F t ) = h(t t), λ(t τ > t, F t ) = 1 {τ>t τ>t,ft}µ(t F t ), and λ(t F t ) = 1 {τ>t Ft}µ(T F t ), we prove that forward default intensities generally exist. Alternatively, based on accessibility, we prove 26