A Comparison of Credit Risk Models

Transcription

1 CARLOS III UNIVERSITY IN MADRID DEPARTMENT OF BUSINESS ADMINISTRATION A Comparison of Credit Risk Models Risk Theory Enrique Benito, Silviu Glavan & Peter Jacko March 2005

2 Abstract In this paper we present the most important models of credit risk. We begin by a description of classical structural models that make use of an analogy of option pricing models. Then we proceed by reduced-form (or, intensity) models, which are market-based and in general show better prediction possibilities. A very recent stream of reconciliation models that tries to incorporate both classical approaches is explained afterwards. Finally, we define and theoretically analyse a new, behavioral model of credit risk. 1

3 1. Structural Models Structural models try to determine the time of default by using the evolution of the firm s structural variables, such as the value of the firm s assets. Hence, any structural default model will make explicit assumptions about the dynamics of the firm s structural variables and the situation that will trigger the default event. This approach originated with the seminal paper of Merton (1974) where the firm defaults if the value of its assets is below its outstanding debt at the time of servicing it. However, several problems arise in Merton s model due to the restrictive assumptions that it introduces in order to allow the straightforward application of the Black-Scholes (1973) formula to price defaultable debt. Hence, several extensions of this model have been developed. These extensions basically try to make Merton s framework more flexible by assuming more realistic assumptions. For instance, Geske (1977) departs from Merton s model by assuming that the firm s debt is a coupon bond instead of a zero-coupon one. Another departure is the model introduced by Black and Cox (1976) that pioneered the so-called First Passage Models (FPM) and in which the firm s default occurs as soon as the firm s asset value falls below a certain threshold and can happen at any time instead of occurring just at the maturity date of the debt. Other improvements of the original model of Merton have introduced stochastic interest rates, callable-bond debt, the existence of bankruptcy costs, or jumps in the process that underlies the firm s assets value. It is important to notice that as far as model assumptions depart from those that Merton assumed its complexity significantly grows. Thus, the final selection of a structural model will imply a trade-off between realism and complexity. In the following pages, we briefly review the main existing developments in this field and analyse the existing empirical evidence. Merton s Model In the classic model of Merton (1974), the capital structure of the firm is comprised by equity and a zero-coupon bond with maturity T and face value of D. Under this structure, we can consider that the equity owners possess a European call option on the firm s assets with maturity T and strike price D. 2

4 Figure 1. Firm s default in Merton s approach. Default will occur at the maturity date of debt in the event that the value of firm s assets is less than the face value of the debt. In this case, the debtholders will take the control of the firm and the shareholders will not receive anything. In this framework, Merton also assumes that there are no transaction costs, taxes or problems with indivisibilities of assets. He assumes that there exist a sufficient number of investors with comparable wealth levels and an exchange market with a competitive interest rate. He also allows for short-sales of assets and assumes that the value of the firm is a traded asset and is invariant to its capital structure (i.e. M-M theorem holds). Finally, the value of the firm is assumed to follow the following diffusion process dv t = (µ γ)dt + σdw t where µ is the mean rate of return on assets, γ is the proportional cash payout rate, σ is the asset volatility, and W is a standard Brownian motion. At maturity, the payoff of the bondholders will be given by min(d, V t ) and the payoff of the shareholders will be given by max(0, V t D). Hence, under these settings we can apply Black-Scholes pricing formula in a straightforward manner. The value of the equity at time t is given by [ ] E t (V t, σv, T t) = e r(t t) e r(t t) V t φ(d 1 ) Dφ(d 2 ) 3

5 where φ is the distribution function of a standard normal random variable and d 1 and d 2 are given by d 1 = ( ) ln e r(t t) V t D + σ2 V 2 (T t) σ V T t d 2 = d 1 σ V T t The probability of default at time T is given by P [V T < D] = φ( d 2 ). In order to implement the model, estimates of the firm s asset value V t and its volatility σ V are needed. Several procedures have been suggested in the literature to estimate them both. See for example Duan (1994). First Passage Models The Merton s model introduces restrictive assumptions in order to allow the straightforward application of the Black-Scholes formula to price defaultable debt. Hence, several extensions have appeared. One of these extensions is the Black and Cox (1976) model, in which the firm s default occurs as soon as the firm s asset value drops to a sufficiently low default threshold and thus, can happen at any time instead of occurring just at the maturity date of the debt. In this model, the default threshold will be chosen by the firm in order to maximize the market value of equity. Concretely, if we consider a constant default threshold K > 0, and V t > K, then the time of default τ is given by τ = inf{s t V s K} Then, we can infer the default probability from time t to time T : { ( ) ( P [τ T τ > t] = N(h 1 ) + exp 2 r σ2 V K ln 2 V t ) 1 σ 2 V } N(h 2 ) where h 1 = ( ) K ln e r(t t) V t + σ2 V 2 (T t) σ V T t h 2 = h 1 σ V T t 4

6 Black and Cox, consider the default threshold as a safety covenant, which acts as a protection mechanism for the bondholders against an unsatisfactory performance. This will allow them to take the control of the firm if the asset value reaches the default threshold. First Passage Models have been extended to account for stochastic interest rates, bankruptcy costs, taxes, debt subordination, jumps in the asset value process, etc For instance, Longstaff and Schwartz (1995) extend Black and Cox model by incorporating both default and interest rate risk and provide close-form valuation expressions for a variety of defaultable debt securities. The assumptions of the model are similar to those of Black and Cox, but they incorporate the dynamics of the short-term riskless interest rate by the following formula dr = (ζ βr)dt + ηdz 2 where ζ, β, and η are constants and Z 2 is a standard Wiener process. If we consider that Z 1 is the standard Wiener process that control for the dynamics of the firm s assets value, then the instantaneous correlation between Z 1 and Z 2 is ρdt. In this model, the constant default threshold K is given exogenously. One of the main implications of Longstaff and Schwartz model is that credit spreads for firms with similar default risk can vary significantly if the assets of the firms have different correlations with changes in interest rates. Hence, the model is able to explain why bonds with similar credit ratings but in different sectors or industries have widely differing credit spreads. Other interesting extensions are the models by Leland (1994) and Leland and Toft (1996) that consider the optimal capital structure of the firm. While in the former, the debt is considered to have an infinite life, in the latter, this assumption is eliminated. Hence, the firm will choose not only the amount but also the maturity of debt. In the case of Leland and Toft model the optimal capital structure and hence the debt maturity represents a trade-off between tax advantages, bankruptcy costs and agency costs. The latter, which are not included in the original model of Leland, will provide further insights on why firms choose to finance with shortterm debt. Since long-term debt generates higher firm value, then the authors argue that short-term debt reduces agency costs and moral hazard, and that is the reason why it is often chosen. Both Leland and Leland & Toft models determine the equilibrium bankrupt- 5

7 cy-triggering asset value V B endogenously. Concretely, Leland and Toft derive the following expression V B = C r ( A rt B) AP rt 1 + αx (1 α)b τcx r where as far as T, V B tends to the value derived in Leland (1994). Notice that V B depends on the maturity of debt chosen by the firm for any values of bond principal P and coupon rate C. Finally, it is worthy of comment that Leland and Toft model does not allow for stochastic interest rates, arguing that previous studies have found that models with stochastic interest rates reduce the estimated credit spreads and this problem of small credit spreads is precisely one of the drawbacks of previous structural models. Empirical Evidence Empirical testing of structural default models is somewhat limited. One of the most extensive empirical analyses of these models is found in Jones, Mason and Rosenfeld (1984). They apply the Merton model to a sample of firms with simple capital structures and secondary market bond prices during the period One of their results is that the predicted prices of the model are too high (by 4.5% on average). Ogden (1987) also finds that Merton model generally underpredicts spreads and argues that Merton models suffers for the lackness of stochastic interest rates. Lyden and Saraniti (2000) compared Merton and Longstaff and Schwartz models using individual bond prices and they found that both models underestimate yield spreads, though they argue that allowing for stochastic interest rates has little impact on the underestimation. One of the most recent conducted studies is Eom, Helwege and Huan (2003). They implement and compare the models of Merton (1974), Geske (1977), Longstaff and Schwartz (1995), Leland and Toft (1996), and Collin-Dufresne and Goldstein (2001) by using a sample of 182 bond prices from firms with simple capital structures during the period In general they find that the models tend to underpredict the spreads. However, they find that the Leland and Toft model overpredict spreads. This contrasts with the argument of Leland and Toft for not including stochastic interest rates in their model. The authors conclude that future research on structural default models should direct efforts toward raising spreads on the safer bonds without raising them too much for riskiest bonds. 6

8 2. Reduced-Form Models In the reduced-form models, default is treated as an unexpected event whose likelihood is governed by a default-intensity process. Some authors (see for example Jarrow and Protter (2004)) stress that the key aspect that distinguishes reducedform models from structural models is that the former use only public (market) information which is fully observable by everybody. Indeed, reduced-form models do not consider the relation between default and (the true) firm value in an explicit manner. Therefore, it is argued that reduced-form models are much more useful for investors who use them for pricing and hedging, whereas the clasical structural models are more appropriate for managers and for regulatory needs. Modeling the Intensity of Default Reduced-form models are sometimes also called intensity models, although some authors do not agree those are exactly the same. We could say that the question Why the firm defaults? is in these models answered as Due to market. The time of default is true surprise it is modeles as the first jump of an exogenously given jump process providing an intensity of default, based on market information. One characteristic of reduced-form models is that it is assumed that the liability structure of the firm is usually not continuously observable, whereas the resulting recovery rate process is. The first reduced-form model was introduced by Jarrow and Turnbull (1992) and the field grew mostly at the late 1990 s. After making a brief review of the approach to intensity modeling, we will describe the most important family of reduced-from models - affine intensity models. Drawing on the analogy between survival probabilities and discounts, several of this kind of models are based on term-structure models of short rates. More generally, one can formulate the intensity as a function of observable firm-specific and macroeconomic variables. Using a default intensity λ, the default time is generated by a stochastic process with intensity λ depending on state variable X t (which may be one or more dimensional). A more mathematical statement of the information framework follows. We assume that economic uncertainty is modelled with the specification of a filtered probability space Π = (Ω, F, (F t ), P ), where Ω is the set of possible states 7

9 of the economic world, and P is a probability measure that we assume we can uniquelly fix at the beginning. The filtration (F t ) represents the flow of information over time and F is a σ-algebra of events at which we can asign probabilities in a consistent way. As usual, σ-algebra is a model for information and filtration a model for flows of information. On the probability space Π we assume that there exists an R J -valued Markov process X t = (X 1,t,..., X J,t ) we shall refer to as a background process, that represents J economy-wide variables, either state (observable) or latent (not observable). There also exist I counting processes N i,t, i = 1,..., I, initialized at 0, that represent the default processes of the I firms in the economy such that the default of the ith firm occurs when N i,t jumps from 0 to 1. Finally, we present the definition of the filtration (F t ). Let (G X,t ) be the filtration generated on the σ-algebra G X,t = σ(x s, 0 s t) by the process X t, so that it represents information about the development of general market variables and all the background information. Similarly, let (G i,t ), i = 1,..., I be the filtrations generated by processes N i,t, that is, they contain information about the default status of each firm i. Then, the filtration (F i ) contains the information generated by both the information contained in the state variables and the default processes: (F t ) = (G X,t ) (G i,t ) (G I,t ) The model for the default-free term structure of interest rates is given by a nonnegative, bounded and (F t )-adapted default-free short-rate process r t. The money market account value process is given by ( t ) β t = exp r s ds 0 For bond pricing, we assume a perfect and arbitrage-free capital market. The price at time t of a default-free zero coupon bond with maturity T and face value 1 is given by [ ( P (t, T ) = E exp 8 T t r s ds ) F t ]

10 Similarly, the survival probability during the period between t and T, modelled by a Poisson jump process, can be expressed as [ ( s(t, T ) = E exp T t λ s ds ) F t ] Two examples when the formula of survival probability significantly simplifies come up when we suppose λ t is a deterministic function of time t. Then, default occurs as a Poisson process with intensity function λ t, stopped at its first jump. The survival probability is given by ( s(t, T ) = exp T t ) λ s ds Moreover, if λ is a constant, the default time τ is exponentially distributed with parameter λ and the survival probability simplifies to s(t, T ) = exp ( λ(t t)) It is straightforward to price a defaultable zero coupon bond issued by firm i with maturity T and face value of M that, in case of default at time τ < T, generates a recovery payment of R τ, given by an (F t )-adapted stochastic process R t with R t = 0 for all t > T. Then, the price of this bond at time t < T, provided that τ > t (and assuming the expectations are finite), is Q(t, T ) = E [ exp ( + E T t [ T t (r s + λ s )ds ( R s λ s exp ) M F t ] s t + ) ] (r u + λ u )du ds F t It is noteworthy the case when the face value M = 1 and the recovery rate at default is zero (R t = 0 for all t), when the expression simplifies to [ ( Q 0 (t, T ) = E exp T 9 t (r s + λ s )ds ) F t ]

11 which is precisely the expression for pricing a zero coupon riskless bond using a default-adjusted discount rate r s + λ s instead of the risk-free rate. This parallel between pricing formulas is one of the best features of reduced form models. Moreover, this means that credit spreads are given by the intensity λ, which should be contrasted with the structural models, where the spread goes to zero with time to maturity going to zero. As we see, this framework requires to specify recovery rates exogenously, not by the level of assets and liabilities at default. Apart of the trivial case that recovery rate is zero, there are three basic possibilities that have been adopted in the literature. The first type used, recovery of face value considers that the recovery rate is a fraction of the face value of the defaulable bond. Recovery of treasury considers instead a fraction of the value of an equivalent default-free bond (introduced by Jarrow, Lando and Turnbull (1997)). And market-value recovery fixes the recovery rate equal to a fraction of the market value of the bond just before default (Duffie and Singleton, 1999). The approach of market-value recovery rates gained a great deal of attention in the literature, because of good tractability, where the pricing formula simplifies to a form similar to zero-recovery rate, and acceptable preciseness and reasonability. It moreover allows one to include the case in which a firm reorganizes itself and continues with its activity. Another advantage of this framework is that it allows one to consider liquidity risk by introducing a stochastic process as a liquidity spread in the adjusted discount process. Intensity Models It is natural to think of time dependent intensities λ t, which are usually modeled as either a constant, linear or quadratic polynomial of the time to maturity. The treatment of default-free interest rates, the recovery rate and the intensity process differentiates each intensity model. Jump intensity models utilize concept of mean-reverting processes with jumps. That is, between jumps, intensity λ reverts at rate κ to mean γ. We will see that the analogy between intensity-based default risk models and interest rate models, that we showed in the previous section, allows us to apply well known short-rate term models to the modeling of default intensities. 10

12 A very rich class of models can be developed using affine transformations in the models (see Duffie and Kan (1996)). Consider X j,t be a basic affine process with parameters (κ j, θ j, σ j, µ j, δ j ) given by dx j,t = κ j (θ j X j,t )dt + σ j Xj,t dw j,t + dq j,t for j = 1,..., J, where W j,t is an ((F t ), P )-Brownian motion. The constants κ j and θ j represent the mean revertion rate and level of the process, and σ j is a constant affecting the volatility of the process. Term dq j,t denotes any jump that occurs at time t of a pure jump process q j,t, independent of W j,t, whose jump sizes are exponentially distributed with mean µ j (and thus positive) and whose jump times are independent Poisson random variables with intensity of arrival γ j. One can show that if we use basic affine processes for the common factors X t, mathematical results yield closed form solutions for bond pricing and survival probability we defined in the previous section. Therefore, there are clear gains in terms of tractability achieved by the use of affine processes in the modeling of the default term structure. Moreover, the affine intensity models are simply extendable to multiple jump types and are also the basis for tractable modeling of a correlation in default times for multiple firms. Simplifying the expression of basic affine process, we can obtain two wellknown processes. First, if we eliminate the jump component from the process of X j,t, we get the CIR process (Cox, Ingersoll and Ross (1985)): dx j,t = κ j (θ j X j,t )dt + σ j Xj,t dw j,t Furthermore, eliminating the square root of X j,t, we end up with a Vasicek model dx j,t = κ j (θ j X j,t )dt + σ j dw j,t Various intensity models differ from each other in the choices of the state variables and the processes they follow. For example, one can consider the following expressions r t = a 0,r (t) + a 1,r (t)x 1,t + + a J,r (t)x J,t λ t = a 0,λ (t) + a 1,λ (t)x 1,t + + a J,λ (t)x J,t 11

13 for some deterministic (possibly time-dependent) coefficients a j,r and a j,λ, j = 1,..., J. This type of models allows us to treat r t and λ t as stochastic processes, to introduce correlations between them, and to have analytically tractable expressions for pricing. Elizalde (2003a) shows a simple example of this model, where the state variables are r t and λ t themselves, whose Brownian motions are correlated: dr t = κ r (θ r r t )dt + σ j rt dw r,t dλ t = κ λ (θ λ λ t )dt + σ λ λt dw λ,t + dq λ,t dw r,t dw λ,t = ρdt After seeing the case of one firm, one could ask: What is the probability that n 1 different firms default before time T?, or What is the probability that they all survive until time T? We will slighly touch the field of default correlation intensity models (for a broader description of these models see Elizalde 2003a). There are three different approaches in the literature. The first approach introduces correlation in the firms default intensities, making them dependent on a set of common variables X t and on a firm-specific factor. They are usually called conditionally independent defaults models, because the firms default intensities are mutually independent, provided a realization of the state variables X t. The main drawback here is that it is extremely difficult to choose perfectly the state variables, and so, the default correlations generated are not sufficiently high. An incorporation of joint jumps in the default intensities or default-event triggers that cause joint defaults slightly improve the results. The second approach to model default correlation, contagion models, is based on the idea of default contagion in which, when a firm defaults, the default intensities of related firms jump upwards. And the las approach makes use of copula functions, which link univariate marginal distributions to the joint multivariate distribution with auxiliary correlating variables. 12

14 3. Reconciliation Models We saw up to now two main streams of credit risk models: structural and reduced form ones. We will quickly compare their advantages and weak points and we will look at the new stream reconciliation models, a hybrid between the previous two streams and also for possible ways to think for improvements. In this section we will analyze the models from an information perspective, in the style of Jarrow and Protter (2004). This will allow us to make easier the comparison and it is also a good source of creation for hybrid models. First of all, to anticipate a common point of view, we can notice that in both previous streams, the main objective was the default prediction; hence the debate amongst models should naturally be concentrated on it. We propose, as mentioned above, a unitary vision, considering the two streams of models as being actually the same, but using different assumptions about information available to the modeler. From that point of view, structural models assume complete knowledge (very detailed and continuous information), closed to that of the managers visibility. The main implication is that this type of model is very appropriate for internal scopes and also for regulatory ones (in case of commercial banks). To clarify this, we have to look closer at this affirmation: it is clear that for internal needs the most appropriate models should use as much as possible from the information available for the managers; the question remains Why the regulators could (and should) use structural models?. The answer is Because they have the right, by law, (and access) to almost the same information as insiders (managers), so they should benefit from this situation. Also, when analyzing structural models, one should also take into account a weak point: If no jumps are allowed, default time is predictable, or in other words, they theoretically suggest a zero-short term spread, which contradicts the empirically observed behavior. On the other hand, the reduced form models assume knowledge of a less detailed information set, like that (mimicking) observed by the market (the investors). The main implication, a desirable one, is that default time is unpredictable. One should ask here the question: Why should be taken into account the market? (Hence why reduced models are important?). Besides the natural expla- 13

15 nation coming from realistic reasons, one should also note that, when prices hence firms evolution are determined, in equilibrium, in the market, they are drawn by the actors knowing (majority of them) only public info. For example, assets value process is not observable to the outsiders, more exactly is discontinuously observed; when firm disclose accounting info and other relevant issues. This construction makes the reduced form models very appropriate for pricing and hedging credit risk. For a correct analysis of these models one should also take into account a big problem that they induce, comparing with the first stream: now the default arises exogenously, not endogenously, like in the structural models (called for this characteristic cause-effect models). This is also contradicted by reality, because reduced form models assumed that default is not linked at all with firms characteristics, which is clearly a short coming of these models. The intuition of the reconciliation models arises now as a way to improve the previous two ones, and also to take into account their weak points and to find some way to deal with them. Basically, the target is the following: one should want a model more realistic from the general outsider point of view (with incomplete information or at least with relaxed assumptions about complete information), where default should be unpredictable, but also endogenously influenced. The task is not so easy and the price is more technical models to deal with. Also, the very recent development of the topic can be an explanation of their complexity and importance but also can suggest that many things about these models were not revealed up to now. More exactly, the next models analyzed are just relaxing complete information knowledge assumption, such that from structural models with predictable defaults to obtain hazard rate models with inaccessible default. We present here 3 different approaches of the reconciliation models, but of course any variation assuming realistic knowledge of information and producing desired results has to be taken into account. Approaches for Reconciliation Models In the next framework, in general, the information set available to the modeler G is a complete filtered probability space, while the asset value process A is a subfiltration of G. 14

16 1. The first model we look at is Duffie and Lando (2001) model: It is very similar with first passage models (the default time is fixed by the managers for maximization of firms equity value). The investors are receiving periodic and imperfect accounting reports, and they make inferences about the firm s evolution based on these reports, and adding, obviously, their beliefs (noise). Mathematically, the process A is only discontinuously observed (for the rest of the time it is obscured) and it is added an independent noise, such that the new observed process is Z t = A t + Y t. More rigorously, A t = e Z(t), where Z(t) = Z(0) + mt + σw t, with a standard Brownian motion W. The filtered probability space (Ω, F, (F t ) t=0,t, Q) of information available for managers is the embedding world. The authors derive the asset value conditional to information available for outsiders and from it the intensity of default in terms of conditional asset distribution and default threshold. A problem appears if one is involved in this kind of natural extensions: the investor judges, according to its available information and observes the historical default times, and makes this way his inferences. The issue is that our investor is analyzing real default times using its information, altered by its noise. Put in paper s words, the real default times τ = inf{t > 0; A t L t } are not stopping times (i.e. not measurable) for the new filtration, induced by the process Z t. In this paper, the authors showed that t is still a stopping time, related with the intensity process. The information available to the market (secondary market for firm s bond and credit risk) is L t = σ({y (t 1 ), Y (t 2 ),..., Y (t n ), 1 {t s} : 0 s t}), with Y = log(z), and the stopping times are judged accordingly. The intensity obtained λ t is appearing in formula: P (τ (t, t + dt] L t ) = λ t dt. The main merit of the paper is that, even it starts more as a structural model, it can assure an unpredictable default, like the reduced form models. The explanation of this major changing of the stopping time τ from predictable to unpredictable is that between the observation times, the investor cannot see the evolution of assets. 2. The second referred model is due to Giesecke and Goldberg (2003) (and related findings of the same authors): 15

17 As a starting point, and showing very much similarity with the structural models this can be a critique to this type of model they assume continuously observed asset value. In this approach the noise is also introduced, but the manner is quite different: default barrier is a random curve, more exactly beta distributed, with height expressed in terms of firm leverage. The explanation behind it is quite intuitive: when the leverage ratio is at a recent high, then the short-term uncertainty is high also. The modeler cannot see the curve, which is independent of the underlying structural model, so the default time depends on an unobservable curve, hence is inaccessible. That way it is solved the predictability problem from the structural models. 3. As the third approach we look also at Çetin et al. (2004) model: This can be viewed as an alternative approach to previous two described papers. The authors, instead of adding noise to obscure information as in Duffie and Lando (2001), they start also with a structural model but with modeler filtration G to be a strict sub-filtration of that available to the managers (incomplete, but correct information). For analyzing asset evolution, they redefine the asset value to be the firm s cash flows, and the barrier is now L t = 0, for all t. They used the filtered probability space (Ω, F, (F t ) t=0,t, Q). The modeler (and implicitly market) only observes whether the cash flow is positive, zero, or negative, i.e. only observes the states of a variable as being positive or negative, a very coarse filtration of the manager s information set. The default time τ d = inf{t > 0 : X t = 2X τα, X s < 0, for s (τ α, t)} is the first time, after the cash flows are below zero, when cash flow remains below zero for a certain time and then doubles in absolute magnitude. This way is obtained a totally inaccessible default time, and the point process has intensity, so this is an intensity based hazard model. Mathematical Tools After seeing these approaches and their performance, we look in detail at some key concepts involved more or less explicitly in the reconciliation literature, 16

18 and we remark their importance and their role for different assumptions about the models. Note that F t here represents the information available to the modeler (manager or market, depending on assumptions). First of all, we consider a complete definition of an intensity process as: Default stopping time has an intensity process λ with respect to the filtration F t, if λ is a non-negative progressively measurable process satisfying t { 0 λ sds < a.s. for all t, subject to 1 τ t } t 0 λ sds : t 0 is an F t -martingale. For t < τ, λ is the conditional rate of default just after time t, given F t : Pr[τ (t, t + dt) F t ] = λ t dt As we saw in the chapter about reduced-form models, also the following holds: [ ( ) ] T Pr[τ T F t ] = E exp λ s ds F t The default indicator process N generated by τ is given by N t = 1 τ t. Now we will adapt an approach similar to Giesecke and Goldberg (2003), to realize the link between models starting from structural approaches and reduced ones, assuming incomplete info about assets and/or barrier. For that reason there is need for use of the compensator of the default process the cumulative rate of default of the structural model (the pricing trend), that characterizes the intensity of the default time, when it exists, and providing the link. The definition of a compensator follows. A process C is called the F t compensator of the process N, if the following two conditions are satisfied: C is a F t predictable increasing process, with C 0 = 0 the process N C, called the compensated process, follows a F t martingale. A very important result should also be remarked: there is a unique F t compensator C for the process N. Also, we introduce another concept, as Giesecke and Goldberg (2003) is using it: a process Γ (pricing trend) associated with the compensator C, s.t. C t = Γ t τ = Γ min{t,τ}. 17 t

19 The importance of this new process Γ is obvious: with it we can describe the default time s distribution: Pr[τ T F t ] = E [ e Γ t Γ T F t ] Also, if one considers a defaultable security which pays X at time T if not default up to T and 0 otherwise, the security value at time t < T is also priced with the help of the process Γ: [ E Xe Γ T ] t Γ T r s ds t F t We can remark here that the last two expressions were similar with a process with constant intensity lambda and cumulative default probability t 0 λ sds. One can conclude that the main role of the pricing trend Γ is that, even if it only admits an intensity representation when it is differentiable, we can use it for describing stopping times or for pricing defaultable securities. In the particular case when it is differentiable, there exists a process lambda such that Γ t = t 0 λ s ds which is the intensity of the counting process N, i.e. intensity of arrival of the stopping time. The pricing trend is the cumulative default rate. As we remarked before Γ is not always differentiable. Here is the difference between the reduced and intensity type models: reduced approach does not assume differentiability (default is a stopping time but there is no intensity process), while intensity models are like reduced ones, but intensity existence is taken for granted. The inexistence of an intensity lambda (reduced models), does not mean we can t compute default probabilities or price defaultable securities, the pricing trend is helping. Unitary Vision: Importance of Information As we seen, the pricing trend Γ is characterized by a compensator process C, which is chosen such that the difference between the default process N and the compensator follows a F t martingale. An influence on F t assumptions on 18

20 properties of the process Γ and hence on all implied computations is clear here. Also, an important feature has to be remarked: F t is the investor information in time, hence different F t will imply different compensator processes and then different pricing trends! So, in a natural implication, taking into account also the observed default times, pricing trends are determined by the specification of a stopping time and the information framework, in particular also the differentiability of the pricing trend is influenced by these assumptions, i.e. the existence of a default intensity. For our reconciliation models the assumptions about information are mirrored in assumptions about properties of the compensators, and these compensators are involved, not necessarily implicitly, in the findings of the models. One can see how assumptions on information available to modeler are decisive in determining the model! To clarify this, we provide here some findings of Giesecke and Goldberg (2003) by playing with these instruments. The following findings are important for correctly designing any possible future reconciliation model, by taking into account some statistical properties it can fulfill, according to the main assumptions on information. Any structural model with incomplete information admits a pricing trend, but not all admit intensity. Unpredictability of default is a necessary, but not sufficient, condition for the pricing trend to admit intensity. Also, information level determines whether the model admits intensity: when there is an certainty, there is no intensity. Then, by starting with a structural model, and relaxing (as all the reconciliation models we saw until now) in different degrees the information assumptions, authors noticed the followings (by considering as starting point the most complete in terms of quantity and quality of information available the structural model with complete info about assets and barrier): If it is assumed incomplete information about barrier, it can be calculated the pricing trend in terms of distribution function for the barrier and the observable historical asset value. Here pricing trend does not admit an intensity of default. If it is assumed incomplete info for both assets and barrier, the pricing trend admits an intensity representation. 19

21 One important finding in this work is that, in some particular conditions, regardless that the barrier is observable or not, a structural model with incomplete asset info admits an intensity representation! So there is not gain for a reconciliation modeler to relax on barrier part; also the existence of intensity, if it is needed in a particular model, is assured. To conclude and make a prediction about the future of the reconciliation models, it seems like compensators (and other related statistical instruments) can open new perspectives for developing such models. Possible extensions One idea for developing a desirable reconciliation model can be to start the other way around, i.e. with reduced models approach. Here the modeler only knows discrete and perfect pieces of information (accounting reports at the moment when revealed, public information about the firm, also in short term after revealing). He has to make inferences for the periods he does not have information. For that reason, the modeler can use the theory and formulas developed for reduced form models, but has to weight differently information, relying more on what he perfectly knows (for short period of time) and less on the inferred part, because it is altered with its own beliefs and inferences. However, we found such a model too difficult to be developed. In the next chapter we propose a second variant, which we call a behavioral model. 20

22 4. A Behavioral Model In this section we attempt to introduce a new model that we call a behavioral model. We believe that an ideal model of credit risk should incorporate three components: (1) business truth the true value of the difference between assets and liabilities, (2) external (market) shocks, and (3) human panic factor. The first component of our model captures the business opportunities that can be made use of by the firm s assets, given the stable market environment. The second component models the abrupt jumps in the expected cashflows the firm is to generate in future. The third element takes into account that people do not allways behave in a rational way, in particular, it covers the inability to evaluate quickly and precisely the reality when speculators enter the market. The classical structural models show that credit risk modeling can be done considering the internal information about the value of firm s assets and the value of liabilities. Simulations and empirical research demonstrate that these models capture the realistic process of firm s value evolution, although they often underestimate the probability of failure (see for example Eom, Helwegge, and Huang 2003). We interpret this as a reason that an internal factor should be present in any model, because outside information is not the only determinant of the cause of default. There is a growing number of papers in the literature investigating correlated defaults. It is not difficult to see that default of a firm influences the probability of default of some other companies. Usually, this comes as an external shock to the capability of firm s assets to produce cashflows in the future. The value of the firm then suddenly changes. If the bancrupted firm was an important competitor, the value of the other firms in the sector increases. However, it may also decrease, if the failure shows up that this sector is in a crisis. The value of a firm also jumps down when its commercial partner, for example an important supplier or demander, gets bancrupted. A list of many other events may in a similar way cause a sudden jump in the firm s value. For example, development of a new technology in the market influences the value of a firm: in a positive way, if the firm is supposed to incorporate it in the producion process, or in a negative way, if the firm has an implementation 21

23 barrier and its competitors gain a competitive advantage using the new technology. Unpredictable political and economical changes may also provide a shock, e.g. sudden political change, revelation of a (relevant) negative information or terrorist attacks often influence the business possibilities of firms. Finally, one can observe the presence of speculators in the market. Their objective is to make a price arbitrage, and they typically do not possess any private information nor are interested in taking control of the firms. Speculators usually invest into high-risk firms. The causes of their riskiness include high volatility of market price (i.e. high beta), high probability of default, or exceptionally fast changes in market price (e.g. due to market shocks). Obviously, the actions of speculators move the market price of the firm and so, this price gets further from its true value more easily. If this behavior is observed by the market, people tend to cease to believe in the more-less correct firm s evaluation by the market and start to behave in a less predictable way. Following the ideal model, we propose to design it as a sum of three stochastic processes: (1) [business truth] continuous process of the true value of the difference between assets and liabilities, given the stable (that is, fully predictable) market environment; (2) [market shocks] system of jump processes; (3) [panic] continuous process, whose magnitude negatively depends on the observed difference between assets and liabilities (i.e. market price), negatively depends on the actual change of this difference, and positively depends on the firm s beta. Depending on how we calibrate these processes we can get variety of models. To use the ideal model, a lot of information is necessary, which is usually not accessible to the modeler. If we relax model s information needs to the publicly observable information (that is, the view of investors), we have to use proxy variables for the business truth and for the firm s beta. The true value can be approximated by the market price in the periods when it is not volatile more than usually and is far from the bankruptcy barrier. However, in the other periods more sophisticated proxies must be invented, so that they do not incorporate market panic behavior. For example, evolution process of the market price of a similar (but unrelated) firm can be used instead. The firm s beta can be estimated from the past volatility of the market price of the firm or sector. The view of managers, who possess internal information about the true value 22

24 of the assets, is neither simple to model. The insiders are not able to predict market shocks, nor the panic factor in the market, as they come randomly. Some estimations can be assessed form the past market data, althought it does not guarantee much that the same kind of events will happen in future, since the financial market is growing and evolving very fast. One could go even further and suggest that in our behavioral model a relaxation is needed to mimic also the managers knowledge, because they are unable to evaluate (or predict) the evolution of their own company perfectly, just because of their imperfect personal capacity. We can interpret classical structural and reduced-form models in the light of this behavioral model. The Merton s model, for example, considers only the business truth. Other structural models also incorporate jumps, that is, they try to design external shocks, although they do not explain much why these shock occur. We believe that some authors have in mind the panic factor as well, although they do not talk about it explicitely, and consider the panic factor as included in the jump processes. On the other hand, intensity models focus on market shocks and human factor in the market. They usually do not separate the two components and moreover, they model the business truth as a constant (the long-run average). We believe that they damage the preciseness of the model by doing this assumption, which is not very realistic. In the universe of our behavioral model, we think that the reconciliation framework has an important role to play, (also the future seems opened to such models, as we noted before). First we must remark that the relaxation about information knowledge for the modeler, which is dominant in this stream of models, guided for complying with reality in the first stage, has another significance. Up to now we enforced this relaxation in order to mimic the market (the outsiders), as opposite to insiders (managers). From that point of view we can equalise the relaxation of information by allowing for type 2 and 3 events to occur. To see better the good integration of reconciliation models in the behavioral model, one has to notice the following. A main benefit of the reconciliation models (endogenously influenced default, as opposite to reduced form models) is preserved also as a strong point under the behavioral model, as allowing for type 1 process ( business truth ) to play its 23

25 important role in determining the default. For such reasons we consider that there is a parallel between the mathematical language (in general quite complicated) and the intuitive way we described the behavioral model. However, we do not elaborate the model mathematically, due to time and our knowledge constraints. We believe that this model can be in general very efficient and precise in the prediction of firms default. References Black, F. and Cox, J. C. (1976): Valuing Corporate Securities: Some Effects of Bond Indenture Provisions, Journal of Finance 31, Black, F. and Scholes, M. (1973): The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81, Çetin U., Jarrow R., and Protter P. (2002): Modeling Credit Risk with Partial Information, Working Paper, Cornell University. Collin-Dufresne, P. and Goldstein, R. (2001): Do Credit Spreads Reflect Stationary Leverage Ratios?, Journal of Finance 56, Cox, J. C., Ingersoll, J., and Ross, S. (1985): A Theory of the Term Structure of Interest Rates, Econometrica 53, Duan, J. C. (1994): Maximum Likelihood Estimation Using Price Data of the Derivative Contract, Mathematical Finance 4, Duffie, D. and Kan, R. (1996): A Yield-Factor Model of Interest Rates, Mathematical Finance 6, Duffie, D. and Lando, D. (2001): Term Structures of Credit Spreads with Incomplete Accounting Information, Econometrica 69 (3), Duffie, D. and Singleton, K. J. (1999): Modeling Term Structures of Defaultable Bonds, Review of Financial Studies 12, Elizalde, A. (2003a): Credit Risk Models I: Default Correlation in Intensity Models, Working Paper, CEMFI. 24

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