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2 CENTRE FRANCO - VIETNAMIEN DE FORMATION a LA GESTION OUVERTURE internationale International vision N o 11 EDITIONs DE STATISTIQUEs

3 Comité de rédaction: Jérôme BON NguyÔn ch n Patrick GOUGEON Joël MÉTAIS Comité de lecture: Jérôme BON Joël BROUSTAIL NguyÔn ch n Vò TrÝ dòng TrÇn Thä ¹t Patrick GOUGEON Bïi ThÞ lan h ng Joël MÉTAIS Vò Hoµng ng n Lª V n PHI NguyÔn ThÞ TuyÕt mai

4 prðambule Après s être ouvert à l économie de marché, le Vietnam doit aujourd hui faire face au nouveau défi que représente son entrée dans l OMC. Son ouverture internationale fait naître de nombreuses interrogations dans les domaines de l économie, du management, de la sociologie, des sciences juridiques et de l éthique. Autant de questions qui mobilisent les chercheurs de ces disciplines et dont des éléments de réponse doivent être alimentés par les contributions du monde académique et de la recherche. L objectif de cette revue est de diffuser les travaux des chercheurs vietnamiens et européens qui par leurs réflexions et leurs recherches dans ces domaines peuvent contribuer au développement du Vietnam en aidant les entreprises à s intégrer dans la concurrence internationale. Cette ouverture internationale concerne tous les pans de la société vietnamienne et nous avons naturellement voulu créer une revue qui respecte les standards internationaux en matière académique. Il s agit donc d une revue dont le comité de rédaction est international et dont les articles sont sélectionnés à travers une procédure de double lecture aveugle. Le lancement de cette revue n aurait pas été possible sans le support financier de l Ambassade de France et sans l implication des membres du comité de rédaction et du comité de lecture. Je les remercie chaleureusement pour leur soutien. Prof. Dr. Jérôme BON Directeur du CFVG Ouverture internationale - N O 11 3

5 table des matieres / contents Page Préambule 3 Credit Risk, an Overview - Three Essais with Jump Diffusion Processes Structural Approach of Credit Risk Dao Thanh BINH 6 Promotion Commerciale des Petites Entreprises Agro-Alimentaires: le Rôle de la Coopération Internationale au Vietnam Nguyen CHAN & Philippe RÉGNIER 21 Le Commerce et la Croissance Economique du Viet-Nam Dinh Xuan CUONG 35 From Traditional to Modern Market: the Dynamics of Vietnamese Consumer Behavior and its Implications on the Development Strategy of Modern Retailers Bui Thi LAN HUONG 48 La Microfinance au Vietnam: Situation et Enjeux Michel LELART 67 Management Competencies of Corporate Managers: The Case of Vietnamese Companies Nguyen CHAN & Truong Thi NAM THANG 81 Management in Public Hospital Sector: the Case of Dong Da General Hospital Vu Hoang NGAN & Phan Thanh NGA 105 Les Jeunes Entrepreneurs Vietnamiens Sont - Ils Opportunistes? 116 Ouverture internationale - N O 11 4

6 Le QUAN Impacts of Business Factors on Informal Activities in Vietnam s SMEs Nguyen Dinh TAI 128 Prise en Compte des Facteurs Personnels, Contextuels et Socio-Culturels dans les Pratiques de Motivation au Travail: Cas des Cadres Vietnamiens et Français Vu Thi THU HANG 150 An Investigation into Factors Influencing Impulse Buying Behaviors of Urban Vietnamese Consumers Nguyen Thi TUYET MAI, Jung KWON, Sandra LOEB, Garold LANTZ 161 Mécanisme d Apprentissage dans les JVI: une Étude Exploratoire au Secteur de Télécommunucations et Agro-Alimentaire Tran VAN NHU 171 International vision - N O 11 4

7 Credit Risk, an overview Three essais with Jump Diffusion Processes Structural Approach of Credit Risk Dao Thanh BINH, Hanoi University Abstract In this article we present first the three majors approaches in Credit Risk: the structural approach, intensity approach and rating based approach. The majors points of the three essais with different kinds of jump diffusion processes as double exponential jumps, uniform jumps and normal jumps in three different kinds of capital structure as perpetual coupon debt, roll-over coupon and principal debt and maturity debt are also presented here. Key words: Credit risk, jump diffusion process, structural approach, default risk. I. Credit Risk Oview I.1. Credit Risk Modelling Credit risk is a research field that has been actively developed in the past decades, both by academics and practitioners. However, the first article in this field goes back to the year 1959, to Fisher's paper on risk factors influencing credit premium. The valuation of defaultable securities in continuous time finance has, as its benchmark, the two pioneering papers of Black and Scholes (1973) and of Merton (1974). Credit risk, by definition, refers to the risk that a firm (a debtor, a borrower) would fail to service or repay a debt. Default occurs when a firm cannot fulfill key engagements, such as making an interest payment, or a principal payment to a lender. Credit spread (or yield spread) is the difference between the interest rate (yield) a firm pays for its debt (defaultable bond) and the defaultfree interest rate (Treasury bonds). The credit spread is generally considered as the compensation to the lender due to credit risk. Credit risk affects both parties, debtor and lender. In order to hedge against the credit risk, new financial instruments called credit derivatives have been developed. The interest in credit risk has been renewed recently for several reasons. Firstly, credit risk has become the determining key for prices on bond markets. Secondly, several important defaults which happened in the early 2000s, and also institutional regulations, called for a reasonable model of credit risk. Thirdly, derivatives are becoming a regular International vision - N O 11 6

8 and desirable part of investment strategies. The pricing and management of these credit derivatives require increasingly flexible and sophisticated credit risk and derivative models. The recent interest in credit risk pricing has resulted in many model variations. However, all these models rely on three basic building blocks: the interest rate process, the default (or rating transition) process, and the asset recovery process. One important additional theoretical modelling element is the correlation factor amongst these three processes. Nevertheless, the question of specifying realistic correlation factors is rather difficult due to limited empirical data. The mathematical modelling of the pure interest rate process (defaultfree interest rate risk) specification is an important component for credit risk modelling. The term structure of interest rates defines how interest rates evolve over time maturity. The best known "short rates" models are those of Vasicek (1977), Cox, Ingersoll and Ross (1985), Ho and Lee (1986), Hull and White (1990), Nelson and Siegel (1987) and Sandmann and Sondermann (1997). It is also acknowledged, from an academic point of view, that the market based models of the "forward rates" such as those of Health, Jarrow and Morton - HJM (1992), Miltersen, Sandmann and Sondermann (1997), Brace, Gatarek and Musiela - BGM (1997) or Jamshidian (1997), are coming closer to a generally accepted benchmark. The interest rate process used in the credit risk modelling can be a deterministic or stochastic process. The default event is usually considered as the ultimate outcome, however other credit events, such as financial distress, reorganization, rating migration or risk grade migration, can have significant impact on the pricing of credit risk. Most models consider default to be the only credit event but some incorporate the full range of rating transitions. Like other credit model elements, the associated default process can be a diffusion, jump diffusion or pure jumps process. The recovery process, in the event of default, may be the most complex of the three building blocks processes. Many elements of the recovery process such as the level of violation of the absolute priority rule (APR), the direct and indirect costs in distress, bankruptcy or reconstruction and reorganization costs are difficult to quantify and lack empirical research. The recovery process can be also a deterministic fraction of defaultfree bond, of value prior to default or a stochastic process. As a result, the last two processes (default and recovery) are key elements in determining the credit spread. In the next subsection, we shall see how credit derivatives play a considerable role in the development of credit risk modelling. I.2. Credit Derivative Motivation Nowadays, we are in a world where credit derivatives grow very quickly each day. These markets are very developed in order to meet the needs of market investors and to make the markets complete in an economical sense. The valuation of credit derivatives Ouverture internationale - N O 11 7

9 in the coming years will have a colossal impact on the balance sheets of firms and particularly, of banks. Examples of several important credit derivatives, are namely, convertible bonds, collateralized debt obligations (CDO), credit default swaps (CDS), options on credit spreads, brady bond options, basket protection, tranche insurance, first n to default, etc. I.3. The Three Credit Risk Pricing Approaches As mentioned previously, credit risk modelling is subject to have the three basic building blocks: the interest rate process, the default process, and the asset recovery process. These blocks can be pieced together in many different ways depending on the credit risk modelling approach. There are three main approaches in credit risk modelling: the structural approach, intensity approach and rating approach. The structural approach or firm's value approach, as its name indicates, takes the firm's assets (or firm's value) as a state variable. Defaultable security is regarded as a contingent claim on the value of the firm's assets and is valued according to the option pricing theory. In these models, the firm's asset value is assumed to follow either a diffusion process, a mixed jump diffusion process or a pure jump process and the default is modelled as the first stopping time when the firm's value hits a boundary. This boundary can be determined exogenously or endogenously. This approach is represented by the models of, among others, Black and Scholes (1973), Merton (1974), Black and Cox (1976), Geske (1977, 1979), Leland (1994a), Leland and Toft (1996), Ericsson (1997), Anderson and Sundaresan (1996) and Mella-Barral and Perraudin (1997), Ericsson (1997), Longstaff and Schwartz (1993, 1995), Das (1995), Briys and Varenne (1997), Zhou (1997), Hilberink and Rogers (2000) and others. The "reduced form" or the intensity approach ignores the firm's value but directly models the time of default as a totally inaccessible stopping time with intensity. This approach is studied by (among others), Artzner and Delbaen (1992, 1994), Duffie (1994), Duffie, Schroder and Skiadas (1994), Jarrow and Turnbull (1995), Jarrow and Yu (2001), Lando (1998), Madan and Unal (1998), Duffie and Huang (1996), Duffie and Singleton (1997, 1999), Jeanblanc and Rutkowski (1997), Bielecki and Rutkowski (1999), Belanger, Shreve and Wong (2004) and Bielecki, Jeanblanc and Rutkowski (2004). The rating based approach or the rating transition approach takes the rating as the state variable. This approach was first considered a direction of intensity model however given recent developments, it now constitutes a stable approach. In general, credit ratings can provide a good proxy for the default risk of a firm. This approach follows the papers of, among others, Lando (1994), Jarrow, Lando and Turnbull (1997), Lando (1998), Das and Tufano (1998), Schönbucher (2000), Crouhy, Im and Nudelman (2001), Hull-White (2000), Avellaneda and Zhu (2001), Douady and Jeanblanc (2002). International vision - N O 11 8

10 It is obvious that the way the default process (default event, default time) is modelled, makes a difference to the three approaches of valuing or hedging defaultable securities. We consider the endogeneity degree of the default process as the dependence degree of the default process on the firm's asset value or the firm's related values (such as rating data, EBIT, firm's accounting data...). In other words, the more the default time is defined in terms of firm's value, the higher endogeneity degree of the default time. In what follows, we shall present three approaches to valuing defaultable bonds and credit derivatives, in an increasing order of the endogeneity degree of the default event, i.e., firstly, the reduced form approach, secondly, the rating based approach and finally, the structural approach. However, in doing so, we shall still see the interactions among these approaches. I.3.1. The Reduced Form Approach The "reduced form" or the intensity approach proposes the modelling of default process directly. This approach models the time of default (default event), as the time of the first jump of a Poisson process with constant or random intensity (a Cox process). As a result, this time of default is a totally inaccessible stopping time (a surprise default). The intensity models have been implemented into a commercial software package, called Credit Risk + and "other Credit". This was developed by Credit Suisse Financial Products as a tool for the portfolio management of credit risk. In this model, default is triggered by the first jump of a Poisson process whose intensity is randomly drawn for each debtor class. The intensity approach begins with a series of papers by Duffie et al., such as Duffie (1994), Duffie and Huang(1994), Duffie, Schroder and Skiadas (1994), Duffie and Singleton (1997), Duffie and Singleton (1999). These models proposed the modelling of payoff at default (recovery process) in terms of cash (default here is liquidation), as a fraction of the value of the defaultable security just before default. Lando (1998) was the first author who developed and formalized the Cox process (intensity stochastic) methodology with the iterated conditional expectations making it easy to price the credit derivatives. His model had a default payoff in terms of a fraction of defaultfree bonds and he applied his results to a Markov chain model of credit rating transitions. There have been a variety of other models in the class of intensity based models. They are the models by Artzner and Delbaen (1992, 1994), Jarrow and Turnbull (1997), Schönbucher (2001) and Jeanblanc et al. (1999, 2000, 2003, 2004). Further development in this approach can be seen in Madan and Unal (1998) who modelled the intensity of the default, driven by an underlying stochastic process (a firm's value process). The payoff in default was a random variable, unpredictable before default. Duffie and Lando (1997) presented a setup with asymmetric information. The models of both Madan and Unal (1998) and Duffie and Lando (1997) aimed to have a close link between the firm's value approach and the intensity approach. Ouverture internationale - N O 11 9

11 In empirical studies, Duffee (1995), considering individual bonds, showed that the model fitted market prices well but had difficulty in simultaneously explaining the level and slope of the credit curve for investment grade bonds Applying to the certificates of deposit data, Madan and Unal (1999) showed that default probabilities were negatively correlated with the level of equity excess returns but positively correlated with the volatility of returns. They also found that the estimated spreads from their model were too low when the company was far from the default boundary and too high when it was close to the boundary. Bakshi, Madan and Zhang (2001) presented a framework to study the role of recovery on defaultable debt prices. They found that the recovery concept specifying recovery as a fraction of the discounted par value (zero coupon bond) has broader empirical support and that parametric debt valuation models could provide a useful assessment of recovery rates embedded in bond prices. In summary, the reduced form models appear to be suited to value bonds and credit derivatives. They can be easily adapted and flexile enough to be calibrated to arbitrary market data. The criticism of the reduced form model is its reliance on the existence of traded defaultable claims, market prices and spreads. As a result, the intensity approach is difficult to apply to corporate (private) debt and commercial industrial loans (the inverse of the structural approach, that we shall see below). Moreover, most of sector data about default rates, the default term curve, the recovery rate are only average values. Thus, firm risk is not evaluated directly and financial fundamentals are essentially ignored. I.3.2. The Rating Based Approach The second approach in credit risk modelling is the rating based approach. A general measure of a firm's credit risk is its credit rating which is evaluated by one of the three most important rating agencies, such as Standard and Poor, Moody or Fitch 1. The rating transition approach takes the rating as a state variable. As the rating is part of the firm's fundamental information, this approach bring it closer to the firm's related value in terms of endogeneity degree of the default. It addresses the issue that credit spreads may change without default occurring and that the payoff of certain credit derivatives depends on the rating or the occurrence of other credit events. It is mathematically similar to the default based approach except for the specification of multiple rating categories rather than a single state of default. The rating based approach has been also implemented into a commercial software package called CreditMetrics by JP Morgan. In this model of the rating based approach, the process of rating transition serves as a state variable. 1 Two of the major independent credit rating services are Standard & Poor's and Moody's. The investment grade bonds are of rating classes AAA, AA, A, BBB or AAA, Aa, A, Baa and the junk bonds (speculative bonds) are of rating classes BB, B, CCC, CC, D or Ba, B, Caa, Ca, C, respectively. International vision - N O 11 10

12 Lando (1994) and Jarrow, Lando and Turnbull (1997) modelled the rating by the Markov chain dynamics, however, they did not allow for stochastic spread dynamics within the rating classes. This weakness of constant spread was corrected by the paper of Lando (1998). He proposed a model incorporating a stochastic multiplier in front of the transition matrix in order to have stochastic credit spreads. However, this did not fully allow for general spread dynamics in all classes, because the credit spreads of all the rating classes are driven by the same factor. Das and Tufano (1998) extended the Jarrow, Lando and Turnbull (1997) model to incorporate stochastic recovery rates. Thus they had stochastic credit spreads within the individual classes although the default intensities remained constant. In Schönbucher (2000), the credit rating transition models were extended to an HJM model which could fully incorporate stochastic dynamics for the credit spreads of all credit classes. The term structure of credit spreads of all rating classes was modelled and joined with a default model in a consistent and arbitrage free framework. Avellanedand and Zhu (2001) introduced the original idea from KMV, of a "risk neutral distance-to-default process" of a firm. They characterized risk neutrality by the fact that the default index satisfied a parabolic partial differential equation. In the same direction, Douady and Jeanblanc (2002) modelled a term structure of defaultfree interest rate, and a term structure of spread. This spread was modelled as a function of the rating process and of the derivative spread process (both are stochastic processes). The drift of the rating process was calculated in a risk neutral probability. This model had features which in a unique framework, covered the various rating-based models. The paper also incorporated most famous credit models of the other two approaches such as Merton (1974), Jarrow and Turnbull (1995) and Duffie and Singleton (1997). These models can be seen either as particular cases or as limited cases of this model. However, this model was still too general for an application in reality. The rating based approach presented further explanations of the default intensity in connection with the firm's fundamental parameters. Rating is an imprecise measure of an issuer's credit or default risk, as reported by the empirical studies of Hite and Warga (1997). Altman and Kao (1992) have also documented that the propensity for rating changes varied by sector and issue types. Most models assumed that the rating transition between credit classes was Markovian, however, as shown in Altman and Kao (1992), a rating transition was likely to be a non-stationary process. They also found that serial correlation existed for rating downgrades. Kao (1997) showed that, for most rating categories, the variance of rating transition matrixes was significantly large over a short time interval. I.3.3. The Structural Approach The structural approach is also called the fundamental approach along with the name, firm's value approach. As its name indicates, this approach takes the firm's asset value Ouverture internationale - N O 11 11

13 as a state and fundamental variable. Defaultable securities are considered as contingent claims on the value of the firm's asset and are valued according to option pricing theory. In these models, the firm's asset value, usually interpreted as the total value of the assets of the equivalent unleveraged firm, is assumed to follow a diffusion process (in recently development, a mixed jump-diffusion process or a pure jump process). The time of default is modelled as the first time the firm's asset value hits a boundary (a barrier). This boundary can be zero, constant, time dependent or stochastic. Depending on the models, this boundary can be determined exogenously or endogenously. The firm's value approach is historically the oldest for pricing defaultable securities in modern continuous time finance. It was first proposed by Black and Scholes (1973) in their article "The Pricing of Options and Corporate Liabilities" which already explicitly refers to corporate bond pricing in its title. Merton (1974) expanded on this idea and presented the first rigorous dynamic theory for pricing corporate debt. In the model, a default could only occur at maturity of the debt and the exogenous barrier was the principal of the debt. The payoff of the firm's shares in the model is like a European call on the firm's value. The firm's value approach has also been implemented into a commercial model package which is marketed by KMV 2. The KMV model is mostly based on the original Merton (1974) approach, but its main strength came from its use of a large database of historical defaults (an extensive proprietary database). The KMV technique for determining default probabilities differed in two ways. Firstly, it introduced a measure "Distance to Default" (DD) that, as in the Merton model, determined the probability that the asset value exceeds the boundary only at the horizon time t, and not the probability that the asset value exceeds the boundary at any time up until time t. Secondly, it used an empirically estimated relationship between DD and the expected default frequency (EDF), rather than the one implied by a diffusion process of asset values. Details on the KMV approach can be seen in Crosbie and Bohn (2002). The structural approach has several strengths as well as weaknesses. The first strength is that the firm's value models for defaultable bonds are suited to fundamentals, such as corporate bonds, and convertible bonds or callable bonds that can be converted into shares when called by the issuer. The firm's value models could also well fit for collateralized loans with traded collateral or for the commercial mortgage. The second strength is the foundation on fundamentals makes models in this approach also fit well for the analysis of questions from corporate finance like the relative powers of shareholders and creditors or questions of optimal capital structure. However, this fundamental orientation is also one of the model's weaknesses. It is not easy to define a meaningful process for the firm's value and to observe it continuously. The firm value is difficult to measure, especially if the company's securities are private 2 KMV are the initials of its founders: Steven Kealhofer, John McQuown, and Oldrich Vasicek. International vision - N O 11 12

14 or thinly traded. A second weakness of the firm's value models is the unrealistic short term credit spreads implied by the model. These spreads are very low and tend towards zero as the maturity of the debt approaches zero. The latter result comes mostly from the fact that the time of default is a predictable stopping time under the firm' value diffusion process. These results comes from the studies of Jones, Mason and Rosenfeld (1984), Weinstein (1981), Titman and Torous (1989) and Delianedis and Geske (1998). Fortunately, recent papers that have modelled the firm's asset value with jump diffusion process, as we shall present hereafter, have remedied this second weakness and have made it a strong point. Considering for jumps in asset value has shown to generate higher credit spread by making the default time an unpredictable time. Huang and Huang (2003) calibrated the historical default data and estimated how much of the corporate Treasury yield spread was due to credit risk. They found that the credit risk accounted for only a small portion of the yield spread for investment grade bonds of all maturities (less than 20% for grade A, Aaa), but accounted for a much higher fraction of the yield spread for junk bonds (more than 90% for grade B). This papers has shown encouraging results for the use of structural models in general, in terms of credit spread, hedge ratio, and default probability, and of the endogenous default boundary model with jump diffusion process in particular. Hereafter, we shall present several major models in the structural approach. We can categorize these models into the following: the exogenous default boundary model of Black and Cox (1976), the stochastic interest rate model of Longstaff and Schwartz (1995), the debt strategic service model of Anderson and Sundaresan (1996), the endogenous default boundary model of Leland (1994) and Leland and Toft (1996) and the jump diffusion model of Zhou (1997). In terms of the default boundary, we can simply divide it into two categories: the exogenous default models and the endogenous default models. Black and Cox (1976) extended the model of Merton (1974) to allow for defaults before maturity of the debt when the firm's value hit a lower boundary. This model had more similarity with a barrier option model. Black and Cox showed how to value a variety of corporate bonds and bond covenants in this framework. Further papers using this approach in a defaultfree interest rate setup were Merton (1977) and Geske (1977, 1979). Geske models considered the defaultable debt as a compound option on the firm's value, but this approach had the problem of limited dimensions. Longstaff and Schwartz (1993) took into account stochastic interest rates of a Vasicek type (1977) in the structural approach. Kim, Ramaswamy, and Sundaresan (1993) and Nielsen, Sa-Requejo and Santa-Clara (1993) also allowed for stochastic defaultfree interest rates. Briys and de Varenne (1997) assumed the model of Longstaff and Schwartz (1993) and proposed a simpler solution to this problem. They supposed that the default barrier was defined as a fixed quantity discounted at the defaultfree interest rate up to the maturity date of the defaultable corporate bond. Ouverture internationale - N O 11 13

15 In Goldstein, Ju and Leland (1998, 2001), the firm's value model was replaced by the firm's cashflow or EBIT (Earnings Before Interest and Taxes) was the stochastic state variable. Anderson and Sundaresan (1996) analyzed the pricing of strategic debt renegotiations in a discrete time framework. They introduced the "strategic debt service" by equity holders once the generated cash flow was insufficient to pay for bondholders. The liquidation or bankruptcy costs forced creditors to accept the strategic service or to accept the deviation from the absolute priority rule (APR). Mella-Barral and Perraudin (1997) also examined the pricing of strategic debt service in a continuous time model, with infinite debt maturity. They modelled the state variable as an output price of the firm's product and assumed fixed production costs. The power of shareholders for strategic debt service negotiations was based on the assumption that new owners could only generate less earnings after bankruptcy (inefficient management). The firm could, at any time, be liquidated with a constant value. Hege and Mella-Barral (2000) developed the model of Mella-Barral and Perraudin (1997) and Mella-Barral (1999) in integrating strategic service to address the issue of security design. Anderson, Pan and Sundaresan (2000) followed the Anderson and Sundaresan (1996) debt renegotiation model but added a stochastic (mean reverting) interest rate process in the model. All these models have the same feature in that the default boundary is fixed exogenously, thus they are belonging to the first class of the structural approach named "exogenous default barrier" class. Now, we turn our attention to the second important class of this approach named "endogenous default barrier". The models that extend the Black and Scholes model by introducing taxes and bankruptcy costs to consider the optimality of capital structure, shareholder's value, endogenous default boundary, can be called Leland's approach by their benchmark. This direction of credit risk structural models focuses on the analysis of complex relations between credit risk, risk premium, and firm's financing decisions. These models try to combine the problem of corporate finance literature into a valuation theory of defaultable claims (option pricing theory). The default event or default boundary is determined endogenously by the decision-making processes of the firm. This class begins with the article of Green and Talmore (1986) that studied the question of agency costs of debt for optimal corporate debt policy. They examined the incentive for asset substitution by endogenously solving the optimal risk policy. Their results supported the idea that the more the debt value decreases, the more the shareholders' incentives are to increase the firm risk. Leland (1994a) developed and formalized this idea in a trade-off model with different factors such as firm risk, taxes, bankruptcy costs, risk free interest rates, payout rates and bond covenants in the optimality of capital structure, firm value, debt value, equity value and leverage. In this model, the value of the leveraged firm was not equal to the value of the assets, which can be interpreted as the value of an identical but unleveraged International vision - N O 11 14

16 firm. The value of the firm increased by the value of tax benefits and decreased by the value of bankruptcy costs. Thus, minimizing the debt value was no longer the optimal strategy for the equity holders when they choose the default barrier that maximized their claim value (i.e., equity value). Leland (1994b) extended the Leland (1994a) perpetual coupon paying debt structure, to consider a roll-over debt structure with regular repayments and renewals of principal and of coupon. Thanks to this special debt structure, this model was flexible enough to examine coupon paying bonds with arbitrary maturity, while remaining in a time homogeneous environment. Leland and Toft (1996) extended Leland (1994b) by considering finite maturity debt rather than infinite debt. Thus they could study the optimal maturity of debt as well as the optimal amount of debt. The trade-off between short term debt and long term debt depended on the balancing of tax benefits, bankruptcy costs, and asset substitution costs. Ericsson (1997) was along the line of Leland and Toft (1997) by examining in more detail the problem of asset substitution in the optimality of debt structure. Ericsson (1997) proposed also the pricing of defaultable claims by three building blocs: a down and out call, a heaviside down and out and a down and in asset. Chesney and Gibson- Asner (1999, 2001) also studied the question of reducing asset substitution by using several special kinds of debt such as warrant and convertible. All these above models used a diffusion process to model the firm's asset value, thus leading to a critical problem of low short term credit spread in the structural approach. Although with very appealing, interesting and intuitive ideas, the results of Leland's models had also the same critical problem of low short term credit spread. In order to solve the problem of low credit spread, Zhou (1997) extended the model of Merton (1973) by allowing jumps in the firm's value process, thus introducing a jumpdiffusion process for the firm's value. This mixed process helped to solve the problem of low short term credit spread. Zhou (2001) and Delianedis and Geske (2001) also considered an asset value process mixed jump diffusion. Hilberink and Rogers (2002) have used a spectrally negative Lévy process in the framework of the models of Leland (1994b) and Leland and Toft (1996). The main contribution of this doctoral dissertation is found in the research direction opened by Leland, studying a structural model with a default barrier endogenously determined in a trade-off approach of the optimal capital structure. In our models, we follow the trade-off theory, thus emphasizing the effect of taxes and financial distress (default or reorganization costs). Therefore, the optimal capital structure results from the balancing between the tax advantages and the financial distress costs. In the next section, we shall summarize the main points of this thesis, using jump diffusion processes to model the firm's asset value. Ouverture internationale - N O 11 15

17 II. Thesis Summary II.1. First Essai: Perpetual Coupon Paying Debt 3... In this chapter, we present a structural model which is in line with Leland's approach (1994a), a model of endogenous default boundary in the structural approach framework. As mentioned previously, the model of Leland, using a geometric Brownian motion to model the firm's asset value, generates as a consequence, a low yield spread, as well as a predictable default time. In order to correct these weaknesses, we propose jump diffusion processes with two different kinds of jump distributions to replace the geometric Brownian motion. The first type of jumps has the double exponential density distribution and was proposed earlier by Kou (2002). The second type of jumps is the uniform density distribution which is new and is one of the original results in our thesis. Therefore, the modelling of double exponential and uniform jump diffusion is used for the first time in the structural approach with endogenous default barrier. The debt structure in the model is the perpetual coupon-paying debt. This structure helps us to write the debt value, the firm value and also the equity value as a timeindependent function. The default barrier is determined endogenously by maximizing the equity value. We find a close connection between the perpetual American put and the equity value. That is, in our model of perpetual coupon-paying debt structure, the question of maximizing the equity value can be considered as a question of evaluation of a perpetual American put. Hence, the optimal default barrier can be considered as the exercise boundary of the perpetual American put. This brings out another interesting point in our thesis. We use this relationship between the equity value and the perpetual American put to find the results of Leland (1994a) in a continuous geometric Brownian motion model. In the double exponential jump diffusion model, applying the result to the value of perpetual American put as well as to the exercise boundary, as in Kou and Wang (2004), we obtain the value of equity, debt and firm in closed form formulae. In the negative uniform jump diffusion model, we derive the optimal exercise boundary from the Laplace Transform and the value of perpetual American put from recurrent formulae. Thus, we obtain values for the equity, the debt and the firm in quasi-closed form formulae. Analysis of comparative results is done among three models (geometric Brownian motion, uniform jump diffusion and double exponential jump diffusion) on the debt value, the yield spreads and the firm value as a function of the coupon and of the leverage. The influence of jumps on the debt, equity and firm values in a structural model with endogenous default barrier has been tested using the proposed approach. It confirms that 3 This is the chapter 2 in Binh DAO thesis (2005). International vision - N O 11 16

18 the yield spreads are higher with the jump diffusion model. This corrects one of the weaknesses of the structural approach, where the yield spread is much lower than the observed level in the market, especially for junk bonds. One of the original results of adding jumps into the diffusion process while modelling the firm's asset is that the default time is no longer predictable. II.2. Second Essai: Debt "Roll-over" Structure 4 In this chapter, as in chapter 2, we also examine the debt value subject to default risk in the structural approach. We propose a continuous time framework with double exponential jump diffusion process (two-sided jumps). As in Leland (1994b), we consider a roll-over debt structure with regular repayments and renewals of principal and coupon. Thanks to this special debt structure, we are able to examine coupon paying bonds with arbitrary maturity, while remaining in a time-homogeneous environment. This chapter extends the results of the second chapter to a much wider class of possible debt structures and offers an analysis of debt value and yield spreads with arbitrary maturity. In this chapter, we consider again the tax benefit of coupon payments and the reorganization costs at default as well as other firm's parameters such as firm risk, riskfree interest rate and payout rate. We also consider two additional factors, usually observed in financial markets, such as the violation of the absolute priority rule (APR) and the tax rebate (tax deductibility can be lost). We observe that the values of debt, equity and firm and optimal default barrier depend on the Laplace transforms of the first passage time as well as of this first passage time and its firm's value. Based on the results of the Laplace transforms derived by Kou and Wang (2003) for the passage time of an upward barrier, we obtain the Laplace transforms for a downward barrier. Therefore, we obtain values for the equity, the debt and the firm in closed form formulae. The analysis of a coupon-paying bond with arbitrary maturity has been a long-standing, difficult question. The main problem is that, in general cases, bond values must satisfy a non-homogeneous stochastic differential equation, whose closed-form solution is unknown. Up to now, only the models of Merton (1974), Leland (1994a,b) and Leland and Toft (1996), in the structural approach, using a geometric Brownian motion, have obtained the closed-form formulae for the value of debt, equity and firm as well as for the optimal default boundary. Our model with the double exponential jump diffusion process offers an example of another process to obtain the closed form formulae. II.3. Essai 3: Special Default Stopping Time... In the two previous chapters, we proposed the modelling of jump diffusion processes 4 This is the chapter 3 in Binh DAO thesis (2005). Ouverture internationale - N O 11 17

19 for the firm's asset value, considering our approach mainly from an empirical finance standpoint (cf. empirical papers of Bates (1996)). However, in this chapter we consider our model of jump diffusion from another standpoint: behavioral (comportemental) finance. Two-sided jump diffusion modelling can be interpreted as the market's response to news. More precisely, in the absence of news, the firm's asset value can be considered following only a geometric Brownian motion. From time to time, good or bad news arrives as a Poisson process (the counting jump process), and the firm's asset value changes in response to the jump size distribution. Two-sided jumps correspond to the market's reaction towards good news or bad news. One original modelling feature in this chapter, different from previous chapters, is the way we define a special default stopping time. This special default stopping time can also be called "anticipated" default stopping time. We propose a simplified measure accounting for the impact of important bad news that may influence the firm's contingent claim values. The reason that we are interested in important bad news only (significant negative jumps) is the common remark that crashes (negative jumps, bad news) occur more often and are much more likely than booms (positive jumps, good news). Furthermore, in this framework of being interested in credit risk (default risk), it is much more important to take into account significant negative jumps than positive jumps. In further detail, we firstly define the default time as the first stopping time when the firm's asset value drops by a relatively important negative jump. This means that the variation of the firm's asset value before and after jump time is higher than a fixed percentage (we are interested in percentages, as we usually see in the market, for instance, that an arbitrary share drops by 10% of its value). Secondly, we define the default time as the first stopping time after the firm's asset value drops by a relatively important jump over a fixed number of times or over a cascade fall (for example, the firm is considered to default after three significant successive drops). The second definition is based on the fact that a cascade fall in the firm's asset value can lead the firm directly to default. In this chapter, we propose a structural approach to two different kinds of debt structures as well as to two different kinds of jump diffusion processes. The debt structures of the firm proposed here are two completely different debt structures, in order to obtain contrasting and complementary settings. The first debt structure is the zero-coupon debt structure (or discount debt structure) as first proposed in Merton (1974). The second debt structure is a perpetual coupon-paying debt structure as in Leland (1994a) (and also as in chapter 2 of this thesis). The modelling of the two different jump size distributions here, are the normal jumps and (asymmetric) double exponential jumps. Both the normal jump diffusion and the double exponential jump diffusion offer two-sized jumps, therefore they can generate reactions to both bad and good news with ease. While the normal jump diffusion model helps obtain a more International vision - N O 11 18

20 compact formulae, the double exponential jump diffusion model has both a high peak and heavy tails and can generate over-reactions and under-reactions towards news. In this chapter, we obtain two main results. Firstly, we obtain the characteristics of the proposed special default stopping times or in other terms, we obtain a special measure accounting for important bad news. Secondly, we establish general value formulae of the building blocks that can lead to obtaining the values of equity, debt and firm as well as a higher yield spread. Our interest is principally on the effect of a "cascade" fall in firm values to the values of the debt and its yield spread. The results of this research show that in general, the yield spreads generated by our model are higher than those of Merton due to accounting for the correlation of the default stopping time. The framework proposed has a quasi-closed form of the European call (in terms of the wellknown Back Scholes European call conditional to jumps occurrence). In the two particular models such as the double exponential jump diffusion model and the normal jump diffusion model, we present more detailed calculations. References (Majors) Bates, D., (1996), "Jumps and Stochastic Volatility: Exchange Rate Process Implicit in Deutsche Mark Option", Review of Financial Studies, vol.9, No.1, p Bertoin, J. (1996), "Some elements on Lévy processes", Cambridge University press. Black, F., J. Cox, (1976), "Valuing Corporate Securities: Some Effects of Bond Indenture Provisions", Journal of Finance, June, vol.31, iss.2, p Black, F., M. Scholes, (1973), "The Pricing of Options and Corporate Liabilities", Journal of Politicial Economy, vol.81, iss.3, p Brennan, M., E.. Schwartz, (1984), "Optimal Financial Policy and Firm Valuation", The Journal of Finance, vol.39, iss.3, p Chesney M., M. Jeanblanc, (2004), "Pricing American currency options in an exponential Lévy model", Applied Mathematical Finance, vol.11, p Collin-Dufresne, P., R. Goldstein, S. J. Martin (2001), "The Determinants of Credit Spreads", The Journal of Finance, vol.66, n.6, p Dao, B, (2005), Jump Diffusion Process in a Structural Approach of Credit Risk, thesis Dauphine University, 230 pages. Hilberink, B., C. Rogers, (2002), "Optimal capital structure and endogenous default", Finance and Stochastics, n.6, p Ouverture internationale - N O 11 19

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