Credit Risk Model

University of Florence Faculty of Economy Master’s Degree in Bank, Insurance and Financial Markets Thesis in Applied Statistics for Banks and Insurances

Credit Risk Models: Single Firm Default and Contagion Default Analysis

Supervisor: P rof essor Fabrizio Cipollini Student: Marco Gambacciani

Academic Year 2009/2010

Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Structural Models 1.1 Terminal Default . . . . . . . . . . . . 1.2 First Passage Models . . . . . . . . . . 1.2.1 The Black and Cox’s Model . . 1.2.2 Longstaff and Schwartz’s Model 1.2.3 Leland and Toft’s Model . . . . 1.2.4 Zhou’s Model . . . . . . . . . . 1.2.5 Random Threshold Model . . . 2 5 5 11 11 15 19 24 30 35 36 39 41 45 48 50 51 56 67 76 77 79 79 82 83 84 94 114

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Modelli reduced form 2.1 Approach With An Homogenous Poisson Process . . 2.2 Approach With a Non-Homogenous Poisson Process 2.3 Approach with a Cox’s Process . . . . . . . . . . . . 2.4 Bond and Spread Valuation . . . . . . . . . . . . . . Models For The Correlation Between Defaults 3.1 Bottom-Up Models . . . . . . . . . . . . . 3.1.1 Structural Apporach . . . . . . . . 3.1.2 Intensity Models Approaches . . . 3.1.3 Approaches with Copulas . . . . . 3.2 Top-Down Models . . . . . . . . . . . . .

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Credit Risk Derivates 4.1 The Credit Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 CDSs Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Collateral Debt Obbligation . . . . . . . . . . . . . . . . . . . . . . Empirical Implementation 5.1 Analysis of the Trend for the iTraxx Europe . . . . . . . . . . . . . . . . 5.2 Factorial Model for the Intensity of Default . . . . . . . . . . . . . . . . Conclusion

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Introduction
The starting point of this thesis is the risk of default, which is one of the main sources of risk faced by banks, ﬁnancial institutions, insurance companies and businesses in general. In recent years we have been seeing a high pace development of models and ﬁnancial instruments aimed at achieving the goal of adeguate management of the risk of default. In this context, among the instruments which could be used to hedge against the risk of default, there are the credit derivatives, which have assumed a central role in national and international markets. These assets are especially useful for banks and issuers or buyers of risky bonds who are subject to extensive exposure to credit risk, ie the risk that a counterparty does not fulﬁll an obligation, both for the payment of one or more coupons and, more seriously, for the occurence of its default. In particular, with the credit derivatives, the underlying credit risk corresponding to a given credit exposure is effectively eliminated, or more properly transferred to a third subject not directly exposed. However, the credit derivative holder has the potential to earn from a possible default or unfulﬁllment. These contracts do transfer the underlying credit risk without physically transfer the credit or instead use the traditional collaterals or guarantees for the lowering the credit risk. In my thesis, the attention is focused particularly on the assessment of the problem of default correlation: we must, in fact, examine the structure of dependence between the companies underlying credit derivative instruments on credit risk. In order to estimate the joint probability, since the common1 measures of dependence
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Such as the coefﬁcient of linear correlation

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are not adequate in this context, we avail ourselves of the copula, the use of which in recent years has been becoming more and more increasingly popular in the framework of multivariate risk of default, especially for its relative easiness of implementation. For an analysis of the structure of dependence of the default risk between several companies, we analyze a particular credit iTraxx Europe index, which measures the spreads relative to 125 European companies with high credit merit. Finally we estimate the probability of default for some ﬁrms in the European ﬁnancial sector, almost all of them belonging to the iTraxx Europe index, by constructing a factor model for the intensity of default, which is the most popular among the academics, since it constitutes a link between the use of observable variables in the economic scenario in which ﬁrms operate, and which is quite easy to implement for the assessment of the default probability at the univariate and at the multivariate level. My work is organized as follows: In the ﬁrst chapter and in the second chapter, respectively, are studied the structural models and the reduced form models for the assessment of univariate default risk, focusing on the estimation of the probability of default in these models and the determination of the spreads. In the third chapter the attention is shifted to the models for the analysis of the structure of dependence in default risk, in fact the chapetr is focused on the introduction and the use of copula functions, which are by far the most appropriate tool to tackle the multivariate default estimation, because they are of immediate application and their use has been widespread in the recent years in describing the joint distribution probability of default of two or more ﬁrms in terms of their marginal distributions. In the fourth chapter, is given a precise deﬁnition of what is meant by credit risk derivatives as well as the various elements that characterize them together with a detailed description of a Credit Default Swap (CDS) and of an index of CDSs. Finally in the ﬁfth chapter, we undertake an analysis of the CDSs index, the iTraxx Europe, in the light of the results of which we proceed to build a factor model for the intensity of default, applied to

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a sample of banks and insurance companies, for which it is estimated the probability of default at univariate and multivariate level, analyzing by the use of a Gumbel copula the level of infection or contagion risk in the probability of default among ﬁrms.

Chapter 1 Structural Models
1.1 Terminal Default

The ﬁrst sempliﬁed structural model was developed by Merton (1974), on the basis of the Black and Scholes framework for the option pricing. Formulating that model Merton makes some assumptions, some of the them are for sempliﬁcation purpose while other are the fundamental ones 1 : 1. Assets divisibility, no transaction costs and no taxes. 2. Existence of a market where borrowing and lending is allowed at the same interest rate. 3. All the agents holds a wealth which is compatible with their intetions to buy or to sell. 4. Short sales are allowed. 5. Assets negotiation takes place continously.
From the Merton (1974)’s paper the sempliﬁcatory assumptions are the ﬁrst four ones regarding market efﬁciency, the sixth is veriﬁed by the model and the sixth allows to split the effect on the pricing in the credit risk effect and the interest risk effect. Those assumptions are introduced to semplify the model construction. While the fundamental ones for the deﬁnition of the model are the ﬁth and the eigth.
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6. Validity of the Modigliani and Miller theorem, that is the ﬁrm valuation doesn’t depend on the capital structure (Miller (1988)), hence this is assumed to be stationary2 . 7. Flat term structure of the interest rates. 8. The Assets value obeys to a Geometric Brownian Motion. We assume that ﬁrm ﬁnancial structure is composed by debt and equity capital, hence for the balance sheet constraint we get:

V (t) = D(t) + S(t)

where V (t), D(t), S(t) are respictively the assets value, the debt value and the capital value. Furthermore we know that the debt is represented by a risky Zero Coupon Bond (ZCB, from now on) with maturity T and nominal value of L. The V (t) dynamic, rispectively under the real measure P and under risk neutral measure Q, can be described by the following stochastic differential equation (SDE, from now on):

dV (t) = (µ − q)V (t)dt + σV (t)dW (t) dV (t) = (r − q)V (t)dt + σV (t)dW (t)

(1.1) (1.2)

where r is the deterministic and constant risk-free rate, which takes the place of µ, the expected/mean rate of return for V under P, q (i.e. which is equal to C/V 3 for Merton (1974)) is the payout-ratio, σ the volatility of the assets value and W the standard brownian motion.
The capital structure is deﬁned as the composition of the capital which comes out from the liabilities reported on the balance sheet of a ﬁrm. The capital structure represents the relation among the several ﬁnancial instruments (i.e. such as debt, risk capital, or other ﬁnancial assets obtained as a mixing of the previous ones) with which the ﬁrm buys its assets hence ﬁnancing its activity. A stationary capital structure represents a condition in which the ﬁrm mantains constant the ratio between its liabilities, hence the leverage (i.e. the ratio between Debt and Equity Capital) doesn’t change. 3 The value C corresponds to the amount per unit of time of the payments the ﬁrm owes to its shareholders and debts holders.
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CHAPTER 1. STRUCTURAL MODELS This equation has an unique solution under Q, which is given by:

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V (t) = V (0) exp (r − q − 0.5σ 2 )t + σW (t)

(1.3)

The assets value follows a lognormal evolution as it is for the underlying in the option pricing framework accordingly to the Black and Scholes model. If at the debt’s expiring date the assets value V (T ) is lower than the debt nominal value L, default takes place and the bondholders riceive V (T ); otherwise there is no default and the shareholders receive what is left out of the assets after the debt is reimboursed, that is S(T ) = V (T ) − L. Hence, the ﬁrm default is linked with the capability to pay back the debt at its maturity T , from which this model is also called Terminal Default model. Accordingly to the Merton scheme for the default, we can consider the following relations:

D(T ) = min(L, V (T )) S(T ) = max(0, V (T ) − L)

(1.4) (1.5)

Given in t the value of an unit risk-free ZCB with maturity T , P (t, T ) = exp (−r(T − t)), for t < T we obtain

D(t) = EQ [P (t, T ) min (V (T ), L)] = = EQ [P (t, T ) [L − max (L − V (T ), 0)]] = = P (t, T )L − P ut(t, T ; V (t), σ 2 , L) (1.6)

For the balance sheet parity condition between assets and liabilities and for the call-put parity we get the following relation for the equity capital:

S(t) = Call(t, T ; V (t), σ 2 , L)

(1.7)

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In t the debt value for the bondholders can be split in a long position on a default free bond with nominal value L and a short position on a put option4 written on the assets value with strike L. Conversely, in t the assets value for the shareholders corresponds to a long position on a call option on the assets value with strike L. Despite the underlying of the options, that is V (t), is not a traded asset while the equity value S(t) is a traded asset 5 , assuming no arbitrage oppurtunities Merton shows that there exists an Equivalent Martingale Measure under which V (t) once discounted is a martingale. Given the lognormality of the underlying Black and Scholes (1973) plain vanilla 6 option pricing formulas then can be applied in this case, obtaining

D(t) = S(t) =

V (t)Φ(−d1 ) + P (t, T )LΦ(d2 ) V (t)Φ(d1 ) − P (t, T )LΦ(d2 )

(1.8) (1.9)

where
(t) ln( V L ) + (r − q + 0.5σ 2 )(T − t) √ = σ T −t √ = d1 − σ T − t

d1 d2

(1.10) (1.11)

and Φ(•) is the Cumulative density function (Cdf) of the Standard Normal. The actual probability that there is default at T is equal to the actual probability that the call option is not exercited by the shareholders. Deﬁning with τ 7 the time of default, in t the default
The call option and the put option are two basic types of derivative instruments (vanilla options). The ﬁrst gives to the buyer the option the right to buy the underlying asset at a speciﬁed price by a certain date. The second one gives the ption the right to sell the underlying asset at a speciﬁed price by a certain date. The price speciﬁed in the contract is deﬁned exercise price or strike sprice, while the date of expiry of the right is deﬁned exercise date or maturity of the option. 5 The equity capital is traded over the ﬁnancial markets through stocks which are representitive of the ﬁrm risk capital. 6 Plain vanilla option are deﬁned as the basic ones, such as the call option and the put option. 7 Now we introduce the notation for the time of default τ , since this notation will be used also in the rest of the thesis in order to get an homogenous view of the different models. Anyway we remember that under this model’s assumptions the default may occur only at the debt maturity T , hence we get that P (τ < T ) = 0.
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CHAPTER 1. STRUCTURAL MODELS probability is:

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Pr (τ = T ) = Pr(V (T ) < L) = = Pr log(V (t)) + (µ − q − 0.5σ 2 )(T − t) + σ(W (T ) − W (t)) < log(L) = = Pr W (T ) − W (t) < ln( V L ) − (µ − q − 0.5σ 2 )(T − t) (t) σ (1.12)

Since W (t) is normally distribuited with zero mean and variance t we then obtain by substituting in the previous ln( V L ) − (µ − q − 0.5σ 2 )(T − t) (t) √ σ T −t

Pr (τ = T ) = Φ

(1.13)

Replacing µ with r we get the following risk-neutral probability of default

Pr (τ = T ) = Φ(−d2 )

(1.14)

Can be trivially proved that8 the implicit credit spread9 derived from the risky debt value, at time t is given by − ln Φ(d2 ) + s(t) =
V (t) Φ(−d1 ) LP (t,T )

T −t

(1.15)

With this model it is possible to strive out the term structure of the probability of default just letting T vary. The main drawback of this model is that it allows for the occurence of the default only at the debt maturity T , without considering the possibility that τ < T when there could be a ﬁrm’s capital restructuration or ﬁrm reorganization. Therefore we should consider a model which allows a non zero probability of default also before T , as this is the case as we can observe by quotes of the Credit Default Swaps (CDS) for which the reference
For a formal demonstration the intersted reader can see at Hull et al. (2003). The credit spread or yield spread is deﬁned as the difference between the rate of return of the risky bond and the on of the risk-free bond with the same maturity and same coupons rate.
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CHAPTER 1. STRUCTURAL MODELS entity can experience default before the debt maturity10 .

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For more explanation of the CDS see chapter four.

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1.2
1.2.1

First Passage Models
The Black and Cox’s Model

The Black and Cox (1976) model extends the previous framework allowing for the time of default to be also before the maturity of the debt. In their model the default occurs when the process followed by the assets value V (t) hits from above a ﬁxed exogenous barrier. For these reasons the model is the ﬁrst of the class of the default models which are the so called First Time Passage Models. In their model Black and Cox assume a lognormal distribution for the assets value and that the shareholders receive a continous dividend payment proportional to the assets valueof the ﬁrm. Analogously to the the Merton model, under the risk-neutral probability the process of V follows a geometric browian motion deﬁned by the following SDE

dV (t) = (r − q)V (t)dt + σV (t)dW (t)

(1.16)

where the deﬁnition for r, q, σ, W are the same previously seen in (1.1). Since also in this model the interest rate r is still constant and deterministic, hence also the Black and Cox (1976) model does not include the interest rate risk. The barrier value, which determines the default everytime it is reached from above by the assets value, is completely determinited by the variable C1 (t) that is ﬁxed exogenously and dipendent with both the passing time t and the debt maturity T , while in the Merton model the latter one was the only critical variable for the default determination. Such a barrier to trigger the default represents the level of the safety covenants imposed in the contract by the bondholders, through which the latters may ask for a reorganization or impose the ﬁrm to ﬁle for bankruptcy when the ﬁrm is in a very bad both economic and ﬁnancial condition, that is when in the model the assets value hits the barrier. The level

CHAPTER 1. STRUCTURAL MODELS of the barrier C1 (t) is then a reasonable exponential function of the time11 :

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C1 (t) = C exp(−γ(T − t))

(1.17)

where: • C is the value of the reorganized ﬁrm at the maturity T which value should be lower or equal than the nominal of the debt L • γ is implied as discount factor and is to be applied to the barrier value at the maturity T , hence it has a role similar to the one of the risk free rate r to discount the future cash amounts • C exp(−γ(T − t)) ≤ L exp(−r(T − t)) = P (t, T )L In particular Black and Cox assume a precise form for the barrier , C exp(−γ(T − t)) = ρP (t, T )L with the costant 0 ≤ ρ ≤ 1. Thereafter the time of default is deﬁned as following:

τ = inf(t > 0 : V (t) ≤ C(t))

(1.18)

When the default occurs, we can consider the following relations for the payoff D(T, V (T )) of the debt with maturity T and which depends on the assets value, for the net equity value as a function of the time t and ﬁnally for the assets value V (t) when respectively either we reach the maturity without default before or eventually the default event is veriﬁed
The use of an exponential function form is adeguated since the expected value of the debt also assume such a functional form.
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CHAPTER 1. STRUCTURAL MODELS before the date of the debt maturity:

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D(T, V (T )) = min(L, V (T )) S(T, V (T )) = max(V (T ) − L, 0) D(t, C1 (t)) = C exp(−γ(T − t)) S(t, C1 (t)) = 0

(1.19) (1.20) (1.21) (1.22)

The ﬁrst two relations replicates the ones present also in Merton’s framework in case the default can realize only at the maturity T ; whereas the last two relations deﬁne the payoff when the barrier is touched from above at time t, having the default before the maturity. In order to estimate the actual value of both the payoffs it is necessary to evaluate the risk neutral ditribution of the assets assets conditioning on its present value. Under Black and Cox assumptions as in Merton we obtain a lognormal distribution for the process of V (t) adding an absorbing barrier represented by the safety covenant barrier12 . In this context we may apply the approach developed in Cox and Ross (1976) to attack the barrier option pricing problem, hence Black and Cox obtain a closed formula for the expected present value of the debt D(t). Furthermore in this model the distribution of the risk neutral probability of default before τ is therefore:
V ln( C1(t)) ) + (r − q + 0.5σ 2 )(τ − t) (τ √ σ τ −t 1−(2−(r−q−γ)/σ 2

Pr(τ ≤ T ) =1 − Φ + Φ

V (t) C1 (t)

V 2 ln(C1 (t)) − ln( C1(t)) ) + (r − q + 0.5σ 2 )(τ − t) (τ √ σ τ −t

(1.23)

Black and Cox introduce also the possibility for the ﬁrm to have a debt structure with different seniority 13 . Under this context, after having computed the total value of the debt,
For an analytical formulation of the barrier and the explicit solution to the pricing problem for corporate bond the reader can refer to Black and Cox (1976) 13 The seniority is deﬁned as the order of repayment in case of ﬁrm bankruptcy. The senior debt is the saftiest, while the junior debt is repayed only after all the other debts are satisﬁed hence it is the riskiest
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we ﬁnd the value of the senior debt without considering the presence of the junior debt, whose value is then computed as the difference between the the total and the senior debt. Despite the hypothesis of subordination is valid for CDOs product and for securitized products, it is not respected rigorously during the the bankruptcy procedures. Then should be necessary to consider a recovery rate for the debt which is indipendent with respect to the barrier value, whilst this is somehow correlated with seniority of the debt issued by the ﬁrm. The model allows for a default event to happen before the maturity, thus accounting for the contractual strenghts of the bondholders and as well as for the necessity of an optimal reorganization of the ﬁrm run by the shareholders against the present unefﬁcient management. Notwithstanding such extension, this model is still based on constant and deterministic interest rate. Furthermore the model is not even realistic when we consider the presence of corporate bonds with different seniority to which is not completely applied the strict absolute priority14 in case a default is veriﬁed and the ﬁrm is then reorganized.

class of debt 14 The strict absolute priority principle is deﬁned as the rule mainly followed by the USA courts when they settle on how the creditors issues should be fullﬁll during a bankruptcy or a reorganization procedure. The interested reader could refer to Eberhart et al. (1990) which deals with this subject.

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1.2.2

Longstaff and Schwartz’s Model

In their framework Longstaff and Schwartz introduce two main extensions with respect to the models we have analyzed so far: the credit risk and the interest risk are now considered together and is assumed that deviations from the strict absolute priority are allowed. In order to introduce their model in the framework ﬁrst proposed by Merton (1974) and later extended by Black and Cox (1976), they consider some assumptions somehow strongly linked with the ones present in the models previuosly considered15 . In particular in this section we will deal only with some assumption strictly necessary for an analytical presentation of the model, staring the attention at the features which differentiate this model with respect to the previous framework. For the ﬁrst assumption, as in Merton (1974), the assets value V follows a geometric brownian motion deﬁned by the SDE:

dV (t) = µV (t)dt + σV (t)dZ1 (t)

(1.24)

The second assumption states that instantaneous risk free rate r follows, as in Vasicek (1977), a mean reverting stochastic process deﬁned by the SDE:

dr(t) = (ζ − βr(t))dt + ηdZ2 (t)

(1.25)

where ζ, β, η are costants, Z2 is a standard brownian motion such that dZ1 = ρdtdZ2 , with ρdt istantaneous correlation coefﬁcient between the two motions Z1 (t) and Z2 (t). To model the interest rate is used the Vasicek model because, even if it could bring the presence of negative interest rates, with an adeguate choice of the parameters for the which push the probability of negative interest rate very close to zero. The latter allow to get a postive expected value of r(t), thus obtaining a coherent analysis of its inﬂuence over the spread. Furthermore from this choice we get formulas which are less analytically
15 For a complete discussion of the arguments of this model the reader could refer to Longstaff and Schwartz (1995)

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complex with respect to what we would have get if we had chosen a model generally applied for r(t), such that the model of Cox et al. (1985). As in the previous models and in accordance to Modigliani and Miller (1958), the capital structure is assumed stationary and can be composed by a variety of liabilities, including debt contracts with coupon with different rates, seniority and maturity. The default occurs when the assets value reach the level of the barrier K < V (0), which is exogenous, deterministic and constant. In the case a ﬁrm experiences a default event restricted for a single debt instrument (i.e. the ﬁrm misses to meet its obbligations which arise from that instrument), then such a situation impose the default extended to the ﬁrm and hence to all the others issued debt instruments not already matured. The use of a constant barrier is justiﬁed by the fact that in their model K inﬂuences the default model only through the relation V (t)/K with the assets value and this choice is also consistent with the fact the insolvency could be either ﬂow-based or stock-based16 . If the default occurs before the maturity, the debt holders receive a payoff given by a prespeciﬁed and constant fraction 1 − w of the nominal value of the debt; this fraction is also called recovery rate. Hence w ∈ (0, 1) correspond to the percentage constant loss over the debt when the ﬁrm is reorganized. Assume

D(r, T ) = exp(A(T ) − B(T )r)

(1.26)

to be the value of a risk free ZCB17 in accordance to the Vasicek (1977)’s model. We consider P (V, r, T ) as the value of a risky ZCB with maturity T and nominal value one. Its payoff is: 1 − wIτ ≤T
The interested reader could refer to Wruck (1990) and Kim et al. (1992) for a formal deﬁnition of this distinction. 17 The interested reader could look at Longstaff and Schwartz (1995) and Vasicek (1977) for furher details.
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with I indicator function and τ ﬁrst passage time over the barrier. Given X = V (t)/K, as prooved by Longstaff and Schwartz (1995) we get:

P (X, r, T ) = D(r, T ) − wD(r, T )Q(X, r, T )

(1.27)

where Q(X, r, T ) is equal to the risk neutral probability of default computed as the solution of a recursive system with n variables which converge to an unique value for n = 20018 . Thus the bond value depends on the assets value and the barrier value only through the ratio V (t)/K, which for this reason could be considered as a default risk measure and as a proxy the rating of the ﬁrm. We can observe how the bond value is driven by three main components: its default free value D(r, T ); the present value of the loss over the bond in case of default, wD(r, T ); the probability of default Q(X, r, T ). The bond value depends on the choice of the parameters which govern the dynamics of the assets value, µ σ, and the risk free rate, ζ, β, η. The price P (X, r, T ) is a function of X, r , T ; is increasing in K and decreasing in other two components. With a capital structure with several debt instruments, the main fact that the default depends only on the common variable X, let us apply the additivity principle in the pricing of risky coupon bonds, rather through the composition of a hedging portfolio that mimics the bond payoff and risk and which contains some risky ZCBs. From the explicit formula for the solution of the risky ZCB’s value, reported above, we can also determine the credit spreads. At this point is quite important to underline that in the Longstaff and Schwartz (1995) model, differently from Merton (1974) and Black and Cox (1976), the spread term structure is coherent with empirical data. In particular we obtain that the term structure is such that: • it is monotonically increasing for the bonds with an high rating, that is with an high value for X; • it shows hump shaped curve for the bonds with low rating, that is with a low X;
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See Longstaff and Schwartz (1995) for the analytical details.

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• it shows different levels for different debt seniorities, that is for different values of w19 ; • it is a decreasing function of the interest rate r, which inﬂuences the drift of the process of V (t) and hence also the default probability; • it is an increasing function of the correlation coefﬁcient ρ, in fact the volatility level of V depends also on the covariance with interest rate. Furthermore Longstaff and Schwartz (1995) ﬁnd also an explicit solution for the value of the bond with not ﬁxed interest rate. Although, this model extends the precedent framework, relaxing the assumptions made in the two models analyzed so far, it still shows some cons and it does not take into account some necessary extensions. In fact, this model allows to ﬁnd out only approximated formulas for the default probability, and needs to be impletend with simulations; under this model can also happen that the liquidation value of the assets is higher than their own values, for instance the latter at the default time could be less than 1 − w multiplied by the debt nominal value; it does not solve the problem of obtaining zero short-term spread; it considers an exogenous threshold, that is not facing the optimization problem for tyhe capital structure. In their paper Wei and Guo (1997) show that such a model is by far difﬁcult to be estimated and its performance is less accurate than the Merton’s model one, this happen when for the latter we consider a volatility that varies with time, as it has been proposed by Hull et al. (2003). Nevertheless, as addressed by Wei and Guo (1997), the difﬁcult and less performant calibration of the model is mainly due to exogenously given recovery rate 1 − w, that is of the barrier level.

We could consider w as a proxy for the debt seniority. Thus, high levels for w imply a low value recovery rate, to which usually correspond a low level of seniority.

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1.2.3

Leland and Toft’s Model

In their model, Leland and Toft (1996), examine the optimal capital structure and its inﬂuence on the debt value. In their analysis they consider the fact that the ﬁrm management can optimally determine the total value and the maturity for each debt instrument to be issued, evaluating the trade-off between agency costs, default costs and tax concessions for the debt. In their model the default is endogenously determine, in fact the barrier is deﬁned in an endogenous way through considerations cencerning the capital optimization objective problem. Thus, both the debt value and spread level are inﬂuenced by the ﬁnancing choices and, for this reason, they depend on the inﬂuence that the leverage level has over them. In such a model is not taken into account a stochastic process for the risk free interest rate, contrary to Longstaff and Schwartz (1995), inasmuch, as asserted in the Leland and Toft (1996)’s paper, some academics have shown that this assumption acquit a realitvely low variation in the spreads with respect to a model with a deterministic rate, while it complicates a lot the research for analytical formulas and cloesd formulas. In a previous paper, Leland (1994) had analyzed the same problem of optimizing the capital structure, resulting in a closed form solution for the value of risky long-term debt and the spread. In the formulation of his model, he assumed a debt maturity T , tending to inﬁnity, which is incompatible with the possibility for companies to freely determine the optimal amount and maturity of each debt instrument issued from time to time. Leland and Toft (1996) deviate from this limiting assumption, analyzing the relationship between debt maturity and its price, then between the spread and the optimal amount of debt. They obtain a model consistent with the empirical debt maturity structure, according to the different effect of the trade-off between costs and beneﬁts in the choice of debt maturity, then the choice to issue debt in the short term or long term. Assume a stationary debt structure, in which till default the ﬁrm issues a constant amount of debt at the parity, with maturity T from the issuance moment and with a nominal an-

CHAPTER 1. STRUCTURAL MODELS

20

nual rate equal to p = P/T , with P total nominal value of debt issued not yet matured, and that the same amount p of capital will be withdrawn when the previously issued bonds will be matured. Then for each time s, before the moment of default, the value of total debt not accrued will always be equal to P and with uniform distribution with respect to the maturity in the time interval (s, s + T ) . We assume that the assets value follows a geometric brownian motion, deﬁned by the SDE dV (t) = (µ(V (t), t) − δ)V (t)dt + σV (t)dW (t) where the drift factor is composed by • µ(V (t), t) expected yield for V (t) which depends on the assets value and on the time, and which under the risk neutral measure is replaced by the constant risk-free rate r; • δ constant payout factor proportional to the value of the assets. The barrier is represented by the variable VB determined endogenously, according to the assumptions of the optimization of capital structure. In the event that it is not touched by the process V (t), debt holders receive the value of the barrier VB multiplied by the recovery rate ρ(t). Consider a bond with maturity t, par value p(t) and a coupon that (1.28)

pays a constant c(t). We put V = V (t) to simplify the notation in the following formulas. Suppose initially that the barrier is determined exogenously, then the dynamics of the value of the bond will be deﬁned by the following process: c(t) r c(t) ] [1 r c(t) r

dB(V ; VB , t) =

+ exp(−rt)[p(t) −

− F (t)] + p(t)VB −

G(t) (1.29)

CHAPTER 1. STRUCTURAL MODELS where: V VB
−2a

21

F (t) = Φ(h1 (t)) + t Φ(h2 (t))

G(t) =
0

f (s)ds

V − ln( VB ) − (r − δ − 0.5σ 2 )t √ h1 (t) = σ t V − ln( VB ) + (r − δ − 0.5σ 2 )t √ h2 (t) = σ t

with Φ(•), the CDF of the Normal, f (s) and F (s), respectively, pdf (probability density function) and the CDF of the distribution of ﬁrst passage time. Considering now the

set of bonds with different maturity, t in[0, T ], we have that the set of annual payments is given by the value (C + (P/T )) independent of t. Also for simplicity we assume that ρ(t) to be equal to a constant annual rate ρ/T for each t20 ; this implies that ρ = (1 − α) is the sum of all the recovery rates on all the bonds not yet matured, where α corresponds to the loss the ﬁrm would suffer in case of default, expressed as a fraction of the value of the assets. So we get:

T

D(V ; VB , T ) = t=0 dB(V ; VB , t)dt C C + (P − ) r r zσ T 1 √ − V VB 1 − exp(−rt) − G(T ) + exp(−rt)F (T ) rT
−a+z

=

+ (1 − α)VB −

C r

Φ(q1 (T ))q1 (T ) +

V VB

−a−z

Φ(q2 (T ))q2 (T ) (1.30)

Leland and Toft (1996) stated in the notes that in their framework we can use a different formulation for ρ(t), for example, by proposing a decaying exponential form.

20

CHAPTER 1. STRUCTURAL MODELS where: (r − δ − 0.5σ 2 )2 + 2rσ 2 )t √ q1 (t) = σ t V − ln( VB ) + ( (r − δ − 0.5σ 2 )2 + 2rσ 2 )t √ q2 (t) = σ t (r − δ − 0.5σ 2 )2 + 2rσ 2 z = σ2
V − ln( VB ) − (

22

Once determined the ﬁrm’s market value, v(V ; VB ), equals to the value of the assets plus the taxation beneﬁts, the equity value is given by S(V ; VB , T ) = v(V ; VB )−D(V ; VB , T ). The endogenous equilibrium value for the barrier21 is obtained by maximizing S(V ; VB , T ) with respect to V , that is as a solution to ∂S(V ; VB , T ) |V =VB = 0 ∂V . Also from this model the probability distribution of actual default is given by
V − ln( V )−(µ−δ−0.5σ 2 ) B √ σ s V − ln( V )−(µ−δ−0.5σ 2 )s B √ σ s

Φ

V + exp (−2(µ − δ − 0.5σ 2 )) ln( VB )Φ

(1.31)

As shown by Leland and Toft (1996), in the empirical analysis, the probability of default depends mainly from the value of the drift µ, the volatility σ of the value of the assets, as well as the level of endogenous barrier VB . Moreover, the path of the term structure of credit spreads is increasing with respect to interest risk-free rate r, similar to what occurred in Longstaff and Schwartz (1995). For higher maturity we will have higher spreads, conﬁrming the tendency of ﬁrms to issue more debt in the short term than long-term, as the ﬁrst dramatically reduces agency costs. If cash ﬂows are expressed as a function

of the value of assets V , then considering the ratios C/δV and (P/T )/∆V where ∆V is a proxy for cash ﬂows, default probabilities and spreads can be expressed in terms of
21

For the explicit solution, please refer to Leland and Toft (1996).

CHAPTER 1. STRUCTURAL MODELS

23

these relationships. For this reason the model can be interpreted as a formalization of the traditional rating models, which are based on ﬁnancial ratios equivalent to those described above.

Even in their model, however, the presence of spread remains virtually zero for short-term debt. This situation is due to the fact that the stochastic process that governs the value of the company, for reasonable levels of volatility, will rarely touch the optimum barrier in the short term, implying low probability of default in the short term.

CHAPTER 1. STRUCTURAL MODELS

24

1.2.4

Zhou’s Model

The model of Zhou (1997) develops and generalizes the jump model with binomial amplitude previously proposed by Mason and Bhattacharya (1981). To apply the model, the more classical assumptions of structural models are used: the invariance of the value of the company compared to its capital structure, according to Modigliani and Miller (1958); the simplifying assumption of a risk-free interest rate constant and the presence of a perfect market with continuous trading activities and the absence of arbitrage opportunities, which implies the existence of a risk-neutral measure Q. The process followed by the value of the assets has two stochastic components: a continuous diffusive and a discontinuous jump, where the amplitude of the jump follows a lognormal distribution. The ﬁrst component represents the trend of the market value of the assets, determined by the ﬂow of continuous and normal information. The second jumping component determines the effect of suddenly important information, which can also cause immediate collapse of the ﬁrm’s market value. We have, therefore, that the value of assets V (t) evolves according to the following SDE dV (t) = (µ − λν)dt + σdW1 (t) + (Π − 1)dY (t) V (t) where: • µ, λ, ν, σ are positive constants; • W1 is a standard brownian motion; • Y is a Poisson process with parameter λ; it is the speciﬁc or idiosyncratic component of the ﬁrm, and which is also indipendent from the systematic default component (i.e. the one triggered by the market);
2 • Π > 0 ∼ LogN orm(µπ , σπ ) is the amplitude of the jump sach that E[Π] = ν − 1 2 • ν = E[Π] − 1 = exp(µπ + 0.5σπ ) − 1

CHAPTER 1. STRUCTURAL MODELS • W1 , Y, Π are indipendent.

25

We observe a default when the process V (t) touches or falls below the constant barrier K. In this situation the default extends to all bonds and there begins a phase of reorganization of the company, where the bondholders receive a portion 1 − w(Xs ) of the par value at maturity T of the bond, with X =
V (t) , K

s = min(τ, T ) 22 and 1 − w(Xs ) recovery rate on

the bond is is not decreasing function in Xs , depending on the rating class of bonds and the value of the company. We set Xs = X to simplify the notation in the following expressions. Then, under

the above assumptions and given some constraint conditions, we obtain the value H of a generic credit risk derivative , dependent on X, as a solution, when it exists, to the following partial differential equation (PDE, in the rest of the thesis):

0.5σ 2 X 2 HXX + (r − λν)XHX − rH + λEt [H(XΠ, T ) − H(X, T )] − HT = 0 (1.32)

where HX and HXX rispectively are the ﬁrst and the second derivative of H with respect to X. In the event that the contingent claim H considered is a bond, then we have that H = B must satisfy the PDE (1.28). In this approach the model of Merton (1974) for the value of assets is extended assuming the jumps in the process, thus solving the longstanding problem of zero spreads in the short term. Now we see that the default is no longer a predictable stoppig-time
23

,

in fact, the dynamic process of the value of the assets can present jumps which may not
The time of default τ is deﬁned, as in precedence, as the ﬁrst instant at which the barrier is crossed, that is: τ = inf{t > 0|V (t) ≤ K} . In structural models the concept of predictablity is referred to us as the ability for investors to observe the phenomenon that in this type of models is the default event, given the complete information on the value of the assets and the value of the barrier. In fact, one can determine the evolution of the process V (t), approaching or moving away from the barrier level, it determines the forecast for a default time to be more or less remote from the instant of observation. Therefore, we have that the estimated probability of default is made by continuously assessing the quantity, the process of the assets and the barrier, empirically detectable, checking from time to time any deviations from the predictions or noting any increasing in default likelihood due to the bad conditions of the company dectuable from the balance sheet. In this type of models are impossible any sudden default, which would lead to the realization of stoppig-time in the short term, with estimated probability almost zero. For a formal deﬁnition refer to Sch¨ nbucher (2003). o
23 22

CHAPTER 1. STRUCTURAL MODELS

26

have been foreseen before and which are triggered by a certain jump process. Therefore, we have that the value of the assets evolves according to a diffusion process with jumps, jump-diffusion process, where the jumps are leading to periodic shocks in ﬁrm value. This implies the possibility to observe instantaneous and sudden default event, thus to get instantaneous positive spreads and spreads high even for companies which are not in an advanced state of ﬁnancial distress.

While in the model of Merton (1974) is stated that the yield curve for the spread starts from zero and it is monotonically increasing, the model of Zhou (1997) admits that this can take several forms: ﬂat, decreasing and hump-shaped for any ﬁnancial status of companies. In addition, we also consider a recovery rate modeled endogenously and which depends on the assets value at default, unlike most of the structural models in which it is a constant value at default, except in the model of Merton (1974), but where the default can occur only at the maturity of the debt. Zhou considered two approaches: a simpliﬁed one in which the default occurs only at maturity T of the only instrument of debt issued and the recovery rate is a linear function of X; in the second, more general, is admitted the possibility of default to occur at any time before the maturity and w(X) to be any continuous function. For the ﬁrst he derives a closed formula, while for the second Monte Carlo simulations are needed.

Sempliﬁed Approach

Given the bond with an unitary nominal value and the indicator

function I, we have that the price of the bond will be:

B(X, T ) = exp(−rT )E Q [IXT >1 + (1 − w(XT ))IXT ≤1 ]
Q = exp(−rT ) − exp(−rT )E Q [w(XT |XT ≤ 1]FT (1|X)

(1.33)

CHAPTER 1. STRUCTURAL MODELS with w(X) = w0 − w1 X, linear function of X, we then obtain
Q B(X, T ) = exp(−rT )1 − w0 FT (1|X) ∞

27

+w1 X i=0 exp(−λT )(λT )i 2 exp(µi + 0.5σi ) i!
2 ln(X) + µi + σi σi

1−Φ

(1.34)

with:

µi = r − 0.5σ 2 − λν + iµπ σi =
2 σ 2 T + i!σπ

Q and the probability of default under Q conditioned on the current value of X, FT (1|X) =

P rob(XT ≤ 1|X)

24

. Zhou also obtains the value of the bond in case we require via w

a constraint for limited responsability for the bond holders, with w(X) = min(0, w0 − w1 X). It is noted however that this constraint is not crucial, since the probability of obtaining negative value of 1 − w is very low and does not affect much the value of the spreads.

Generalized Approach time

Now we consider that the default is deﬁned by ﬁrst-passage

τ = inf(t|Xt ≤ 1, t ≥ 0)

(1.35)

where the payoff for the bondholders will be 1 − w(Xτ ) for τ ≤ T . Xτ will be dependent on a particular distribution of τ , in contrast to Longstaff and Schwartz (1995) where X assumes the value 1. The value of the bond that solves the PDE (1.28) will be:

B(X, T ) = exp(−rT ) − exp(−rT )E Q [w(Xτ )|Iτ ≤T ]
24

(1.36)

Refer to Zhou (1997) for the formulation and demonstration.

CHAPTER 1. STRUCTURAL MODELS Then for X > 1: n 28

B(X, T ) = exp(−rT ) − exp(−rT ) lim

n→∞

E Q [w(Xt∗i )|Ωi ]Qi i=1 (1.37)

where: i T n Xt∗i > 1 ∀j < 1

ti =

Ωi = Xt∗i ≤ 1 Qi = P rob(Ωi ) Xt∗0 = X

Xt∗i = Xti−1 exp(xi + yi + πi ) i = 1, . . . , n

with xi , yi , πi indipendent and with the following distributions: (r − 0.5σ 2 − λν)T σ 2 T , ) n n

xi = ∼ N (

(1.38) (1.39) (1.40)

2 πi = ∼ N (µπ , σπ    1 λT  n yi =   0 1 − λT  n

In this case there are no closed-form formulas, while to approximate the value of the bond we must carry out some Monte Carlo simulations. The risk arising from the jump of process leads to an increase in spreads, especially for bonds with low maturity holding constant the volatility process of the assets value. Zhou (1997) notes that for high values of the jump intensity λ and lower values of σπ , you have more continuity in V (t) associated with low spreads for short-term bonds and higher spreads for long-term bonds. This phenomenon is due to the relationship between the process of jumping and the probabilities of default which decrease with X. Zhou shows

CHAPTER 1. STRUCTURAL MODELS

29

that the jump process is more likely to cause default in the short term than in the long term compared to a diffusion process. In the paper Zhou (1997) presents a model that considers the Vasicek (1977)’s stochastic process for the risk-free interest rate. Although there are numerous possible extensions to the model, this represents a simpliﬁed framework to model the probability of default and spreads, when you consider the possibility that there may be sudden and otherwise unpredictable defaults, according to a common structural model with a conventional diffusion process for the stochastic ﬁrm value. It is therefore a model capable of capturing non-systematic shocks to the value of the company and to obtain non-zero spread in the short-term, unlike what happens for other structural models.

CHAPTER 1. STRUCTURAL MODELS

30

1.2.5

Random Threshold Model

The ﬁrst to introduce a stochastic model that includes a barrier and a Vasicek dynamic for the interest rate were Nielsen et al. (1993). Later this model was generalized by Sa´ a Requejo and Santa-Clara (1999) admitting any choice for the dynamics of the interest rate. This framework was then revised in Hsu et al. (2004). In the latter it is propose a model in which the value of the company without default V (t) and the value of the barrier K(t) affect the probability of default only through their relationship, not imposing an analysis of the capital structure with which the ﬁrm is ﬁnanced. This ratio is deﬁned as an indicator of the solvibility of the ﬁrm (solvency ratio) in respect of the commitments with the bondholders. In this approach the authors consider that the value of the barrier has a stochastic dynamic which is inﬂuenced by shocks related to the market value of assets and interest rate trends. So the two processes of the ﬁrm’s market value V (t) and the barrier K(t) determine, trough their relationship, the value of corporate bonds, the values of spreads and the probability of default. In their model the barrier K(t) is deﬁned as the value of the company in case of bankruptcy25 . Are then analyzed the equilibrium conditions of the default, rather than, as in other structural models, the causes that lead to ﬁnancial difﬁculties. In this context, the bankruptcy code, the control of corporate governance and its mechanisms must be adequate and functional for the market, given the capital structure and actual governance, it should be able to efﬁciently assess the value of the company compared with the value after a reorganization. For these reasons the value of the barrier follows a stochastic process which covariates with V , rather than being deterministic or constant, reﬂecting both the value of the commitments and the effect of disputes between shareholders and creditors. In their model the ﬁrm value and the value of the barrier follow the following dynamics under the
Hsu et al. (2004) specify that this value depends on what happens in case of default. Then K can be either: the liquidation value of assets or the value of the recapitalized ﬁrm which continues its business.
25

CHAPTER 1. STRUCTURAL MODELS martingale or risk neutral measure Q:

31

dV (t) = [r(t) − δe (t, V, K, r) − δd (t, V, K, r)] dt + γv dZv (t) + γr dZr (t)(1.41) V (t) dK(t) = [r(t) − δe (t, V, K, r) − δd (t, V, K, r)] dt + βv dZv (t) + βr dZr (t)(1.42) K(t) where: • δe (t, V, K, r), δd (t, V, K, r) represent the pay-out rate respectively to shareholders and to bondholders; • Zv (t), Zr (t) are Brownian motions expressing shocks, respectively, relative to the value of the fundamentals V (t) and the level of risk-free interest rate; • instantaneous correlation between the Brownian motions is equal to ρrv dt • γv , γr , βv , βr are their respective loads to the erratic part of the Brownian motions. The barrier is constructed such that if the market is efﬁcient, the default will occur the ﬁrst time that the process V (t) crosses the barrier K(t) from above. As a proxy for K(t) we can use a fraction (1 − w) of the assets value 26 . To obtain a model for default that does not depend directly on the structure of the ﬁrm’s
V (t) capital, Hsu et al. (2004) choose to model the log-solvency ratio X(t) = log( K(t) ). X(t)

evolves according to a Brownian motion deﬁned as follows:

dX(t) = µx dt + σx dZx (t)
26

(1.43)

For the formal speciﬁcation of the value of K(t), see Hsu et al. (2004).

CHAPTER 1. STRUCTURAL MODELS where:

32

(σx dZx )2 = [(γr − βr )2 + (γv − βv )2 + 2ρrv (γr − βr )(γv − βv )]dt
2 2 µx = 0.5(σk − σv ) 2 2 2 σk = βr + βv + 2ρrv βr βv 2 2 2 σv = γr + γv + 2ρrv γr γv

corr < dZx , dZr > = ρxr dt =

ρrv (γv − βv ) − (γr − βr ) dt σx

The generalized SDE for the risk-free interest rate is as follows:

dr(t) = µr (r, t)dt + σr (r, t)dZr (t)

According to the speciﬁcation of a model for r, we can obtain closed-form formulas or analytical approximations for the pricing of coupon bonds. The default time is τ = inf{t : X = 0} simultaneously for all debt instruments issued and not expired. The bondholders receive a fraction (1 − w) of the value of an equivalent risk-free bond 27 , according to the so-called principle of recovery treasury. The recovery rate (1−w) depends on the seniority of the speciﬁc bond held by the investor. In this case, then we have that for a coupon bond is valid the additivity principle28 , will therefore be sufﬁcient to determine the price of individual corporate ZCB to get the price of corporate bond coupons. Then the debt of the company which is not yet expiered can be considerate as a portfolio of ZCBs with imposing the same date of default. Furthermore, the price of each coupon bond will be calculated individually, regardless of the speciﬁcation of the ﬁrm’s capital structure. To qualify for this pricing is needed the speciﬁcation of the term structure of interest rates and the probability of ﬁrst passage time τ .
With the same maturity and same nominal value of corporate bonds. According to the additivity principle the value of a coupon bond corresponds to the value of a replication portfolio composed of ZCBs, and its value is the sum of the ZCBs prices.
28 27

CHAPTER 1. STRUCTURAL MODELS

33

For simplicity Hsu et al. (2004) take as numeraire the price in t of a ZCB risk-free P (t, T ) with maturity T . Then they consider as a probability measure QT , called Forward Measure. Under this measure the process of log-solvency rate is as follows:

dX(t) = (µx + ρxr σx s(t, T ))dt + σx dZx (t)

with Zx Brownian motion under QT and s(t, T ) volatility of P (t, T ) adjusted for interest rate risk. Then the price of a ZCB corporate under the forward martingale measure is given by the following equation:
T

C(t, T ) = P (t, T )(1 − wEtQ [Iτ ≤ T ]) = = P (t, T )(1 − wΠ(t, T ))

(1.44) (1.45)

where Π(t, T ) is the probability that there is default between t and T , deﬁned forward default probability or risk-adjusted default probability. Π(t, T ) can be calculated with some simulations, or, with appropriate restrictions for certain parameters of the model, we can obtain closed-form formulas or formulas with analytical approximation. In particular, Hsu et al. (2004) showed that if: • ρxr = 0, we can ﬁnd closed-form formulas given any dynamics of r; • s(t, T ) is a deterministic function of the time29 , we obtain analytical approximation. Following its deﬁnition, the spread is: 1 P (t, T ) 1 1 ln( )= ln( ) T −t C(t, T ) T −t 1 − wΠ(t, T )

y(t, T ) =

The spread is increasing in w and in Π(t, T ), and then decreasing with respect to seniority and increasing with respect to the risk of default. In their paper they also show the trend of the spread in relation to other variables.
29

As it happens in model of Vasicek (1977) and in the other models for the interest rates.

CHAPTER 1. STRUCTURAL MODELS

34

• For low initial values of leverage X we get a curve hump shaped for the term structure, while for high values a monotonically increasing curve. • With the use of a stochastic interest rate we note higher spreads. • The spreads are increasing with respect to ρxr . In particular, one can compare the effects on the spread for non-cyclical companies, with ρxr < 0, and cyclical business, with ρxr > 0, which belong to the same rating. • The spread is increasing with respect to the volatility σx . • Coherently with what stated and demonstrated by Longstaff and Schwartz (1995), the spread narrows as the rate r grows, but this time this relation is due to the fact that the process of K may be more sensitive compared to the process V to the shocks caused by the interest rate, that is βr > γr . This model, unlike those with a constant barrier, where the drift of the solvency rate is positive, show higher expected probability of default and higher spreads for corporate bonds, in accordance with what is the empirical evidence. Another strong perk of this model is that it admits that the rating of the ﬁrms may vary, having the the credit merit reduced as to the conditional probability of default decreses, while it is constant in the reduced form models. So you can see a hump in the curve for the term structure of spreads for ﬁrms with a low level of X, associated to the decreasing conditional probabilities of default. This model is suitable to capture adequately the default of a single ﬁrm.

The reduced form models are more appropriate with respect to this model to analyze the asset swap spread or the spread of a portfolio of bonds belonging to the same rating class. In addition, like all the structural models considered previously, it shows that spreads tend to zero for shorter maturities. Empirical evidence shows that this does not happen. As observed by Dufﬁe and Lando (2001), this problem is coming from having assumed a market where there is perfect information of the value of the assets and liabilities among investors, and thus balance sheets that are transparent and correct.

Chapter 2 Modelli reduced form
The reduced form models or intensity models, assume that the default is exogenously determined by a Poisson-type process governed by an intensity λ(t), jump, or a unit increase of the stochastic process underlying the process of Poisson, described in (2.1); we assume, therefore, that the default occurs the instant when we observe the ﬁrst jump of a poisson process. Depending on the choice made for the intensity the models can be characterized by homogeneous or non-homogeneous processes of Poisson. For instance, with respect to the above statment should be noted that although most of the reduced form models admit the existence of an intensity process for default, there are some who do not admit it 1 . These fall into the category of reduced form models, but can not be deﬁned intensity models, while the latter is considered for most of the reduced form models.

Among the reduced form models we can also distinquish four types of them: basic models, where the default process is independent of the majority of economic factors or even independent from changes in the underlying; models that are based on the rating of the Bonds and their transition probabilities from one class to another, the so-called ratingbased models; some hybrid models that consider some aspects of structural models, in particular the performance of the value of the company; the market models that parallel
See as an example of reduced form model that does not admit an intensity, the model of Brody et al. (2005).
1

35

CHAPTER 2. MODELLI REDUCED FORM

36

to those developed for the pricing of derivatives on interest rates and stocks, seek to build models and transparent standards for the pricing of derivatives on credit risk.

As stated above, the default in these models is treated as a sudden event, and sometimes independent of economic events. In this approach we can model the default time as the time of the ﬁrst jump of a Poisson-type process with a particular form for the intensity. The time that elapses between the instant of observation or of evaluation and the time of default, then, is a random variable with exponential distribution with parameter equal to the value of the intensity of the jump process.

2.1

Approach With An Homogenous Poisson Process

In a ﬁrst approach to the model we consider a Poisson process Nt with constant intensity, λ(t) = λ > 0. If we consider an ordered series of stopping-times or jump times τ1 , τ2 , ..., τn , ..., the unitary Poisson process of jump can be expressed by the following stochastic process:
∞

Nt = i=1 Iτi ≤t

(2.1)

where I is an indicator function, N (0) = 0 and the increments are stationary and indipendent, such that for 0 ≤ s ≤ t: exp(λ(t − s))(λ(t − s))k k!

P (Nt − Ns = k) =

(2.2)

where the increment Nt − Ns ∼ Po(λ(t − s)). Furthermore, for such a Poisson process we have that the time lags between the jumps are i.i.d. random variables with, τj − τi ∼ EXP(λ). We set that the instant of default τ is equal to time of of the ﬁrst jump τ1 ∼ EXP(λ), with

CHAPTER 2. MODELLI REDUCED FORM τ1 = {t ∈
+

37

|Nt > 0}. From this we obtain that the probability of survival up to t is P (τ > t) = e−λt

(2.3)

Which gives that the probability of default in a inﬁnitesimal interval [t, t+dt), conditioned by the fact that there has been no default up to time t is P (τ ∈ [t, t + dt) ∩ τ ≥ t) P (τ > t) P (τ > t) − P (τ > t + dt) = P (τ > t) exp(−λt) − exp(−λ(t + dt)) = exp(−λt) (2.4)

P (τ ∈ [t, t + dt)|τ ≥ t) =

= λdt

The factor lambda is also called hazard rate. Its generalized form is of the form h(t, T ), depending on the maturity T and on the instant t of evaluation. The conclusion therefore is that in this case this function is equal to the constant λ. For the properties of the exponential distribution we have that the expected time of default is 1/λ and variance 1/λ2 and that the probability of two jumps, that is two defaults, at the same point in time t is null. From (2.3) we observe that in the probability of survival the role of the intensity is comparable to that performed by the interest rate in the discount factors. For these reasons, in reduced form models default probabilities can be considered such as discount factors and therefore the intensities of default as credit spreads. This basic framework for the reduced form models, for example, allows us to use the binomial tree model of Litterman and Iben (1991) to estimate the probability of default by the quoted credit default swaps and then determine the intensity calibrated to the market prices for CDS spreads. The assumption of a constant intensity, however, does not allow to model a realistic and ﬂexible term structure of spreads as it is observed on the market. In fact, the model considered in this section for h(t, T ) implies a constant term structure of spreads that

CHAPTER 2. MODELLI REDUCED FORM does not vary neither with t nor with T , hence a ﬂat constant term structure.

38

CHAPTER 2. MODELLI REDUCED FORM

39

2.2

Approach With a Non-Homogenous Poisson Process

A possible solution to obtain a ﬂexible model that can be calibrated to the term structure of the credit spreads is the assumption of an intensity λ(t) as a function of time t. In this way, the process of default is deﬁned by the ﬁrst jump of a Poisson process not homogenous over time. We assume λ(t) to be positive and deﬁne the hazard function or the cumulative hazard rate or cumulative intensity as: t Γ(t) =
0

λ(s)ds

Setting M (t) as a standard Poisson process with unitary intensity, we deﬁne the nonhomogeneous Poisson process in time with unit jump as:

Nt = MΓ(t)

(2.5)

In other words, a non-homogeneous Poisson process is equivalent to a Poisson process with standard time ”distorted” by the cumulative intensity of Γ(t); thereafter the increments of Nt are independent as in the model with constant intensity, but not identically distributed, since their distribution depends on Γ(t), which varies over time. For (2.5), we have that the instant τ of the ﬁrst jump of Nt corresponds to the time Γ(τ ) of the ﬁrst jump of Mt . It is assumed that the time lag between the jumps of Mt , with unit intensity, is distributed as a standard exponential ξ, then we have:

Γ(τ ) = ξ ∼ EXP(1)

from which, with Γ(t) invertible, we obtain:

τ = Γ−1 (ξ)

CHAPTER 2. MODELLI REDUCED FORM From this we deduce that the probability of survive till t is:

40

P (τ > t) = P (Γ(τ ) > Γ(t)) = P (EXP(1) > Γ(t)) = e−Γ(t) = e− t 0

λ(s)ds

(2.6)

From which the probability of default in a inﬁnitesimal interval [t, t + dt) conditioned on the survival up to t, similarly to (2.4), is: exp(− t 0

P (τ ∈ [t, t + dt)|τ ≥ t) =

λ(s)ds) − exp(− exp(− t 0

t+dt 0

λ(s)ds) (2.7)

λ(s)ds)

= λ(t)dt

The conclusion therefore is that the hazard rate h(t, T ) = λ(T ) only depends on the maturity of T , thus admitting a curve that is not ﬂat for credit spreads, which remains unchanged with respect to time t of observation. Still, like the previous approach, we observe that the probability of survival can be considered as a discount factor, where the cumulative intensity once again plays the role of discount rate and therefore can be considered a credit spread. By a suitable choice of

λ(t), we can obtain any desired shape for the term curve of spreads. One negative aspect of the model lies in the fact that the standard exponential random variable ξ, with which is modeled the process of default, represents an exogenous source of uncertainty, which is independent of any observable variable on the market, such as the rate of interest, thus making the model incomplete with respect to information available to investors. Also the use of a deterministic function of the intensity does not consider the volatility of credit spreads.

CHAPTER 2. MODELLI REDUCED FORM

41

2.3

Approach with a Cox’s Process

A possible generalization of the non-homogeneous Poisson process with stochastic intensity is given by Cox’s process. Considering the stochastic intensity λ(t) ≥ 0, the ﬁltration {Gt }t≥0 it generates and which corresponds to the information on trends of the deafultfree market, will be given by the sequence of σ-algebras Gt = σ{λs : 0 ≤ s ≤ t}. Given this ﬁltration, the Cox process conditioned to the ﬂow of updated information available, Gt , can be seen as a non-homogeneous Poisson process with stochastic intensity λ(t). For this reason, to the Cox process can be applied all the results obtained in the case of a time-dependent intensity, provided we reason by conditioning to the information available. The default will occur, as usual, in the instant of the ﬁrst jump: having two elements of uncertainty, the process of the intensity and the jump time, in fact the process is also called doubly stochastic Cox process. As mentioned above, Nt is a Cox process if, given a standard Poisson process Mt independent of the stochastic process of the intensity, it satisﬁes the relation

Nt = M

t 0

λ(s)ds

(2.8)

; it is therefore a Poisson process with a random change of time conditional on Gt . For this process it holds that: 1 ( k! t t s

P (Nt − Ns = k|Gt ) = E

λ(u)du)k e− s λ(u)du

(2.9)

With the hazard function:
T

Λ(T ) =
0

λ(s)ds

(2.10)

which is a random variable. Then given the standard exponential random variable ξ, the time of default can be expressed by

τ = Λ−1 (ξ)

(2.11)

CHAPTER 2. MODELLI REDUCED FORM

42

Applying the results seen for the previous approach in chapter (2.2), this time condtioning on Gt , we get for s > t, τ > t:

P (τ ≥ s) = P (Λ(τ ) ≥ Λ(s)) = P (ξ ≥ Λ(s)) = E [P (ξ ≥ Λ(s))|Gt ] = E e− s 0

λ(u)du

(2.12)

and P (τ ∈ [t, t + dt)|τ > t, Gt ) = λ(t)dt Assume that λ(t) is a sthocastic process described by the SDE (2.13)

λ(t) = µ(Xt )dt + σ(Xt )dW (t)

(2.14)

with µ, σ conveniently chosen so that λ(t) is positive and Xt is the process of the economy state variables, but without including also a variable which represents the series of default realizations, for instance the jump times of the process Nt .

The models that are based on the framework of a Cox process for modelling the time of default, base their effectiveness on the fact that, considering the conditional probabilities, not all the information is needed to be known, i.e. the one which also contains the information about the realizations of past defaults is not compoulsary for the estimates. It is sufﬁcient to know the evolution of state variables, which determine the realization of λ(t), in order to estimate the probability of default and make the pricing without having to observe the realizations of the process Nt . In synthesis, we obtain that, through the use of conditional expected values, the probability depends only on the information available to investors, dynamically represented by the ﬁltration {Gt }t≥0 .

Now, following Lando (2004), it is possible to show how it is suffcient enough to know

CHAPTER 2. MODELLI REDUCED FORM

43

such a reduced information, rather than the complete evolution of information, for the pricing of contingent claims in a non-default-free framework. Consider an economy arbitrage-free deﬁned by the probabilistic space (Ω, F, Q) with Q martingale measure, and with F = d Ft being the complete information. Still assume Xt , with values in t≥0 , to be the process of state variables of the economy and the stochastic process for the

intensity to be a function λ(Xt ), non negative and measurable with respect to the ﬁltration {Gt }t≥0 generated by the process Xt . Deﬁned by {Ht }t≥0 the ﬁltration generated by process of Cox Nt , we have that Ft = Gt ∪ Ht . Then, we have that for s > t, s 0

P (τ > s|Gt ) = P (Iτ >s |Gt ) = E e− = e− t 0

λ(Xu )du

|Gt s t

(2.15) λ(Xu )du

λ(Xu )du

E e−

|Gt

(2.16)

; while for s = t, t 0

P (τ > t|Gt ) = P (Iτ >t |Gt ) = e−

λ(Xu )du

(2.17)

. Given Z ∈ F such that E | Z |< ∞. Then can be proved that there exists a random variable Yt adapted to the ﬁltration {Gt } such that

Iτ >t E [Z|Ft ] = Iτ >t Yt = Iτ >t E [ZIτ >t |Gt ] E [Iτ >t |Gt ] (2.18)

. From this we get that dynamic evolution of the default probability, for s = T > t, is E [Iτ >T |Ft ] E [Iτ >t |Gt ]
T t

P (τ > T |Ft ) = Iτ >t

= Iτ >t E e−

λ(Xu )du

|Gt

(2.19)

CHAPTER 2. MODELLI REDUCED FORM Finally from this can be shown that t 44

St = Nt −
0

λ(Xu )Iτ >u du

(2.20)

it is a martingale with respect to the information G.

With the martingale property and with the only available information about the state of the economy we can make the pricing of defaultable ZCB, the contingent claims involving a pay-off with either a payment only at the maturity, either with continous payments until the default or until the maturity either a payment exactly at the moment of default; using combinations of these we can manage to price more complicated instruments dependent on defaultable assets

CHAPTER 2. MODELLI REDUCED FORM

45

2.4

Bond and Spread Valuation

Now we present some formulas for the pricing of defaultable bonds 2 , for the value of some contingent claim with basic payoffs3 and for spreads.

Deterministic Hazard Function
We initially consider the case of non-homogeneous poisson process with deterministic intensity. In this case we assume that the default depends on the time of the ﬁrst jump of the process Nt as in the above equation (2.5). Let P (0, T ) be the price at t = 0 for a unit risk-free ZCB with maturity T . Assuming that the process of the risk-free interest rate r(t) is independent of Nt and having a recovery rate of zero, then the price of a corporate unit ZCB B(0, T ) at t = 0 will be:
T

B(0, T ) = E exp(−
0 T

r(s)ds)INT =0 r(s)ds) E [INT =0 ]
0 T

= E exp(− = P (0, T ) exp(−

λ(s)ds)
0

(2.21)

Denote by X the unit payoff of a contingent claim. Assume that the payments provided by the structure of the contingent claim occur in intervals of width ∆k = Tk+1 − Tk , which corresponds to discrete variables that can be considered constant or to vary with the parameter k. In the event of experience a default in the time interval [Tk , Tk+1 ], the value at t = 0 of X with unit payoff at the end of the
We deﬁne defaultable bonds or corporate bonds, the bonds issued by ﬁrms, which are differentiate from the risk-free bonds, as they are subject to the risk of issuer’s default. 3 For the type of payoffs considered, see below in the paragraph.
2

CHAPTER 2. MODELLI REDUCED FORM period, Tk+1 , is

46

X(0, Tk , Tk+1 ) = P (0, Tk+1 )E INTk =0 − INTk+1 =0
Tk Tk+1

(2.22) λ(s)ds))
0 Tk+1

= P (0, Tk+1 )(exp(−
0 Tk

λ(s)ds) exp(− λ(s)ds))(1 − exp(−
0

= P (0, Tk+1 )(exp(−

λ(s)ds))
Tk

So if we consider the following positions: Tk = T and Tk+1 = T + ∆t, we obtain the following payoff in case of default at time T : X(0, T, T + ∆t) = ∆t→0 ∆t lim B(0, T )λ(T ) = X(0, T ) (2.23)

From the above, we have obtained that the credit spread is given by the average value of the hazard function, that is 1 T −t
T

λ(s)ds t (2.24)

Hence we observe that this value depends only on the time t. In fact, for a future date s > t, the value of the spread is already deﬁned in t by the intensity λ(t), determined by calibrating the hazard function the at initial time t. As noted in section (2.2), we obtain a term curve for the spread deterministic and quite regular, which is a version that is neither realistic nor consistent with what is observed by the trend of prices of corporate bonds and CDSs. From these observations it is assumed most appropriate a model that considers a stochastic intensity, thus involving a ﬂexible stochastic trend for the term structure of spreads.

Cox’s Model
In the framework of Cox are used iterated conditional expected values to obtain the prices of different types of contingent claim. We now present brieﬂy the results obtained for

CHAPTER 2. MODELLI REDUCED FORM

47

the price of contingent claims and ZCB with generic payoffs, respectively, X(T ), g(t) = X(t)Iτ >t , X(Tk+1 ) in case of default in the time interval [Tk , Tk+1 ]:
T

B(0, T ) = E exp(−
0 T

r(s) + λ(s)ds) r(s) + λ(s)ds)X(T )
0 T

(2.25)

X(0, T ) = E exp(− g(0, T ) = E exp(−
0

r(s) + λ(s)ds)λ(t)
Tk+1

X(0, Tk , Tk+1 ) = E exp(−
0

r(s) + λ(s)ds)P (Tk , Tk+1 ) − B(0, Tk+1 )

Can be noted that, unlike the previous situation with deterministic intensity, now we could take out of the expectation the value of the risk-free ZCB, exp(− in which processes r(t) and λ(t) are independent. Using a model speciﬁcation for the process of risk-free interest rate and for the intensity dynamic, we can obtain closed formulas or numerical approximations to the PDEs which should be satisﬁed by the corrispondent values of risky ﬁnancial instruments. Specify a model for such processes is equivalent to deﬁne the multivariate process Xt for the variables of the state of the economy and how this affects the dynamics of interest rate and of default intensity.
T 0

r(s)ds), only in cases

For these models you can get the most suitable forms for the spread term curve, being able to model also the volatility. We therefore have a very ﬂexible and apropriate model to be calibrated to market conditions. Still remains the problem of exponential random variable independent of driving factors of the economy, from which depends through (2.11) and (2.12) the likelihood of default, making, then, the default time also inﬂuenced by an exogenous source of uncertainty.

Chapter 3 Models For The Correlation Between Defaults
After having analyzed in the previous chapters the main approaches to the univariate analysis of defaults and their use for the pricing of the main ﬁnancial instruments, in this chapter we will build the models for multivariate analysis of the default. The theory which involves the construction of models that can capture the dependence between the defaults of several companies is constantly changing. This development is especially crucial after the recent ﬁnancial crisis, which has infected also most of the real economy, causing a succession of businesses’ crises concentrated over the past two years, and which sometimes led to the default of the companies involved. The ﬁrms were to be somehow related to each other, through commercial or ﬁnancial relationship, or belonging to the same industry.

To introduce a dependence structure in default of several companies, can be considered two frameworks, respectively, deﬁned bottom-up and top-down. In the ﬁrst is modeled the correlation between the default, specifying the default dependence between the models of individual ﬁrms. So ﬁrst, during the ”bottom” phase, univariate models are constructed for the default and then during the ”up” phase it is introduced a dependence structure

48

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

49

between individual defaults. In the second framework it is constructed directly a model for the default of the companies in the ”top” phase, and then afterwards in the ”down” phase, verify that the model calibrated to the market is also consistent with the prices of instruments dependent from the default of individual ﬁrms involved in the ﬁrst phase analysis. The channels leading to a correlation between the spread of default, according to most models that have been developed, can be considered to be the following: • comovements related to risk factors that cause changes in the joint probability of default; • contagion effect on the ﬁnancial condition of a company arising from the default of another entity; • review of the prospects of a company, necessary for the detection of previously hidden information that becomes known after the default of another. We present some approaches that have developed within the two frameworks, focusing mainly on the latest models for the risk of contagion and the so-called cluster of defaults. We should also point out the existence of a model recently presented by Dufﬁe et al. (2009) and then reviseted by Azizpour et al. (2010), which is a model based on the socalled ”frialty” effect, where the correlation between the defaults is inﬂuenced by latent risk factors, whose presence after being detected involves jumps in the default probability of ﬁrms, whose intensity depends on those latent variables.

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

50

3.1

Bottom-Up Models

The bottom-up models trigger the correlation between the defaults, introducing a certain dependence structure among the variables that deﬁne the default processes of individual ﬁrms. Even in this case there can be considered different methodologies to approach the study of multivariate default, in fact they may differ in the type of framework which is used for both univariate default and for the dependence between the defaults. With regard to univariate modelling, the approaches are distinguished based on structural models, reduced form models, or on those models which seek to combine the two approaches, taking advantage of their strengths. On the side of the structure of dependence, this can be induced by the correlation of the processes that deﬁne the default (process assets, barriers, indicator functions of default, the default time, intensity or others) or by the use of a copula, which describes the dependence between the marginal univariate models. By analyzing the theory on the construction of models for the correlation of defaults and empirical studies on the results of some multivariate models, we could verify that the use of copulas is the most widely accepted in the theory of credit risk models, as well as the most used in practice. As for the univariate case, we initially treat the introduction of dependence according to the approach based on a structural model, then move on to analyze the reduced form approach, and ﬁnally treat the argument of copulas. For what it concerns the study of the correlation in the framework of structural models, another important approach, in addition to those presented in this thesis, we surely can include Hull and White (2001).

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

51

3.1.1

Structural Apporach

Reconsidering the ﬁrst passage structural models, already seen in the ﬁrst chapter, we have that their immediate multivariate extension might be possibly given by assuming that the processes of the assets value are correlated through the correlation between the sources of uncertainty introduced in their dynamic behaviour. If we consider a diffusion process of the type seen in (1.13) for the default of an individual ﬁrm, then there would be that the dependence between the defaults arises from the correlation between the standard Brownian motions.

Multivariate Model of Zhou Zhou (2001) introduces a bivariate model for the default of this type, so we get an analytical formula for the correlation of default and for the joint default probability distribution of a couple of ﬁrms. In the paper are initially presented the deﬁnitions of correlation between the defaults, of joint default probability and of disjoint probability of default event for two companies. So given the default indicator function for the two companies Di (t) with i = 1, 2, and given that for this function is E[Di (t)] = P (Di (t) = 1) and V ar[Di (t)) = P (Di (t) = 1)(1 − P (Di (t) = 1)), we have the following expressions for the correlation and the two probabilities, as deﬁned above, as a function of the correlation: E[D1 (t)D2 (t)] − E[D1 (t)]E[D2 (t)]

V ar[D1 (t)]V ar[D2 (t)] P (D1 (t) = 1 ∩ D2 (t) = 1) = E[D1 (t)]E[D2 (t)] + Corr[D1 (t), D2 (t)] V ar[D1 (t)]V ar[D2 (t)] P (D1 (t) = 1 ∨ D2 (t) = 1) = E[D1 (t)] + E[D2 (t)] −P (D1 (t) = 1 ∩ D2 (t) = 1)

Corr[D1 (t), D2 (t)] =

(3.1)

(3.2)

(3.3)

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS Then is deﬁned the related processes as follows:      

52

 dln(V1 )   µ1   dZ1    =   dt + Ω   dln(V2 ) µ2 dZ2

(3.4)

with µ1 , µ2 average returns of the assets, dZ1 , dZ2 independent standard Brownian motions and   Ω×Ω =
2 σ1

 ρσ1 σ2 ρσ1 σ2   2 σ2

where Ω is a square matrix 2×2 of variances-covariances of the returns of the assets of two companies, in fact, ρ = Corr[d ln(V1 ), d ln(V1 )] is the correlation coefﬁcient between the processes of the value of the assets. The univariate default for the two ﬁrms is deﬁned by the passage of the exponential barrier Ci (t) = exp(−λi (t))Ki , similar to the one deﬁned in Black and Cox (1976). Depending on the choice made for the value taken by λi , Zhou (2001) considers two situations: a simpliﬁed one in which λi = µi , so we have that the value of assets and debt grow at the same rate, so we are in the presence of a constant leverage ratio; a second one in which is considered that the two growth rates differ. For the ﬁrst case we obtain analytical formulas, which are computationally tractable, for both for the correlation and for the bivariate default probability. In the second case we obtain analytical formulas also depending from a double integral of the Bessel function, which, as noted by the author, makes them computationally very complex1 . Moreover, with the implementation of the model, Zhou shows that the assumption of λi = µi has a negligible effect on the correlation of default, which is the reason why is sufﬁcient to consider the simpliﬁed case. In Zhou (2001) is presented a theoretical and also an empirical analysis for the model. In particular is shown that there is a positive relationship over time between the correlation of returns on assets and the correlation of default to occur. This type of models with the diffusion process of the value of assets already had many
1

See the appendix of Zhou (2001) for a presentation and formal proofs of these formulas for both cases.

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

53

problems in the univariate framework, where for instance we could have observed null spreads in the short term. One possible remedy for this problem was proposed in the univariate context by Zhou (1997) with the introduction of a jump process, which described the sudden shocks in the value of the assets. By introducing jump processes correlated, it would result in a multivariate structural model that would make the default not predictable and at the same time would introduce within the multivariate framework the effects of the risk of contagion2 . This extension will, however, presents some complexity in the calibration phase of the jump components. Furthermore the model of Zhou (2001) considers only the cyclical component of the default dependence between ﬁrms.

See below in the thesis for an analysis of the phenomenon of contagion risk and the of the cluster of default.

2

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

54

Giesecke and Goldberg’s Model In the event of a default of a ﬁrm, the news of this event could lead to a contagion effect on other businesses directly related to that by commercial or ﬁnancial relations. The contagion effect can result in a deterioration in the creditworthiness of companies, resulting in a change in the probability of default, which in some cases may even run out in default of other ﬁrms. Moreover, the effect of contagion also leads to the tendency to concentrate of default in certain periods, in which all the joint probability of default increase, and this phenomenon is called default clustering. Giesecke and Goldberg (2004b) extend the models of Giesecke (2004) and Giesecke and Goldberg (2004a), introducing the effect of contagion in multivariate structural models, assuming that investors do not have complete information3 neither on the processes of the value of ﬁrm’s assets nor on the default’s barriers. The default correlation is introduced by the correlation between the processes of the value of the assets and the dependence structure between the levels of barriers4 : the ﬁrst is the cyclical component, while the second is the effect of the contagion. The model5 assumes that investors form a hypothetical joint distribution a priori, based on assumptions for the levels of the barriers and the values of the assets. When a default occurs suddenly, information about the true values of the assets and of the barrier allow investors to update, according to the formulas of Bayes (1763), the a priori distributions, resulting in an effect of contagion in the estimation of the joint probabilities of default. The main problem of this model concerns the estimation and calibration of the copula between the barriers.

The incomplete information in the structural univariate models was introduced by Dufﬁe and Lando (2001), to cope with the default predictability. 4 This dependence structure between the barriers is introduced through the use of a copula. 5 See the papers Giesecke (2004), Giesecke and Goldberg (2004a), Giesecke and Goldberg (2004b) for the presentation of the model and the resulting analytical formulas.

3

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS Approach of Duan, Gauthier, Simonato and Zaanoun

55

In their approach Duan et al.

(2002) deal with the problem of estimating parameters of structural models, which are used for the portfolio analysis of credit risk, when sometimes is assumed the presence of stochastic interest rates. This problem arises from the fact that the necessary parameters for the implementation of models are usually not directly stripped from the market. In fact, neither processes of the values of the assets, from which we should get the drift, volatilities and correlations, nor the the parameters of the barriers nor the parameters of stochastic process of the interest rates are observable.

Duan et al. (2002) after building a likelihood function based on a transformation6 of observable values for the stocks, to estimate the parameters they apply the maximimum lieklihood method in the context of credit risk7 . The authors formalize the likelihood function8 and then the log-likelihood both in the framework of Merton (1974) with constant interest rates, and within the framework of Longstaff and Schwartz (1995) with interest deterministic or stochastic rates. After obtaining the maximum likelihood estimated parameters, with these values we could run Monte Carlo simulations to verify the performance in a ﬁnite sample inference, then using the asymptotic properties of estimators of maximum likelihood, such as the consistency and convergence in probability to a normal, to comment also on their behaviour in the limit. Moreover with this approach we obtain estimates of the parameters directly from market prices and therefore under the real or physical probability measure.

This transformation consists in the use of the formula (1.8) of Merton (1974) or Longstaff and Schwartz (1995) for the pricing of the equity value as a function of the value of the assets. 7 See also Ericsson and Reneby (2001) for maximum likelihood estimation applied to models for credit spreads. See Duan (1994) for an application to models for the pricing of insurance reserves. 8 Please refer to Duan et al. (2002) for explicit formulas of the likelihood, and for estimation and simulation procedures.

6

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

56

3.1.2

Intensity Models Approaches

Conditionally Indipendent Models The ﬁrst multivariate approach for modeling the default in the intensity models framework consists in the so called conditionally independent models. The dependence between the defaults is induced by the correlation between the intensities, which in turn depends on a process of the economy state variables related to each other and on an idiosyncratic factor different for each company. These models are deﬁned conditionally independent models because when we condition on a realization of the process of the state variables, the intensities and default times are independent, as all the dependence structure is derived only from the correlations between the variables of the state of the economy.

Duffee’s Model

In the model proposed by Duffee (1999) to estimate the yield curve of

the defaultable bond, we have that the interest rate follows a stochastic diffusion process of the so called ”translated two factor square-root diffusion” type, while the process of intensity is deﬁned by a so called single factor translated square-root diffusion process, modiﬁed so that the default processes depend also on the term structure of risk-free rates. Then we get that the risk-free interest rate follows the stochastic process

r(t) = αr + s1,t + s2,t

(3.5)

where αr is a constant and the two factors s1,t , s2,t , representing the state variables of the economy, follow two indipendent sthocastic process deﬁned by the SDE: √ dsi,t = ki (θi − si,t )dt + σi si,t dZi,t

(3.6)

where the deterministic component converges on average to a constant θi with a rate of convergence equals to ki and where Zi,t , i = 1, 2 are indipendent standard Brownian motions.

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

57

From these relations we can be obtain closed-form formulas for the prices of risk-free ZCBs, whose yields are linear with respect to the two-term factors s1,t , s2,t , which can be interpreted respectively as the slope and intercept of the level in t of the term curve of the risk-free rate. From the yield curve obtained from the quotations of the risk-free ZCBs, we can estimate the values of the two state variables and their sample means s1,t , s1,t . With these estimates are constructed, together with a stochastic ﬁrm-speciﬁc factor h∗ , the default intensity j,t processes. We have that the diffusive stochastic process followed by the intensity hj,t of the j-th ﬁrm is

hj,t = αj + h∗ + β1,j (s1,t − s1,t ) + β2,j (s2,t − s2,t ) j,t

(3.7)

where αj is a constant, and the coefﬁcients β1,t , β2,t capture all the correlation between the intensity and r, while the idiosyncratic component follows a stochastic process independent of r and deﬁned by the SDE

dh∗ = kj (θj − h∗ )dt + σj j,t j,t

h∗ dZj,t j,t

(3.8)

with Zi,t , i = 1, 2 indipendent standard Brownian motions. In this model we have that the two common state variables, which appeare in the default intensity processes, are deﬁned by the two factors that inﬂuence the dynamic of risk-free rate. In particular, since these two factors are latent, i.e. can not be directly observed, their time series must be estimated from the yield curve of risk-free bonds before they can be used to build up the intensity of the default processes. Also the only independent factor between companies is present in every different process of intensity h∗ , which represents j,t the speciﬁc component of the randomness of the default for a speciﬁc ﬁrm. So we get a model in which the dependence structure of defaults is given by the different relationship that each intensity has with respect to the common factors, which deﬁne the term structure of interest rates, while the idiosyncratic component of default is governed by the speciﬁc

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS independent factor different for each ﬁrm’s default intensity.

58

This family of conditionally independent models, as stated by Yu (2002), results in levels of default correlation which are not sufﬁciently high. This fact stems from the impossibility of an accurate and speciﬁc choice of state variables involved in the process of intensity, for which in Duffee (1999) we consider only two factors that govern the evolution of the yield curve of risk-free rates .

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59

Approach of Dufﬁe and Singletone Within the conditionally independent models Dufﬁe and Singleton (1999) consider the problem of low levels of correlation as seen in Duffee (1999), proposing two possible extensions: the possibility that there are joint defaults caused by the same event or the presence of joint jumps in the intensity of default9 . In the ﬁrst approach we model the intensity of arrival of joint events relevant to the situation of the credit merit of a ﬁrm, which might cause the default of several companies simultaneously. These events are deﬁned as credit events, for which in Dufﬁe and Singleton (1999) there are some examples. Upon the occurrence of certain combined events the intensity of some ﬁrms may vary, leading in some cases even to the default of one or more ﬁrms or otherwise to a change in the probability of this oppurtunity. Then the default intensity for each ﬁrm i is

hi,t = pi,t Jt + Hi,t

(3.9)

where Hi,t is the speciﬁc intensity of the i-th ﬁrm, Jt and pi,t are respectively the intensity of the arrival of credit events and the joint probability that they cause the default of the ﬁrm i. In this case the correlation between the defaults is triggered either through the joint variations of the default intensities or through the probability that certain joint events, common to all ﬁrms, even cause simultaneous defaults. In the second approach is assumed that the intensities of default of individual ﬁrms follow a mean-reverting process, to which is added a jump process with a component differently related to the various companies. So we have that the intensity of default hi of the i-th ﬁrm has a diffusion component deﬁned by a mean-reverting process, to which is added a process of jump ji with constant intensity λi , independent jump size distributed as an exponential variable with mean J. Then we have dhi,t = ki (θi − hi,t )dt + dji (t) (3.10)

where the diffusion component converges on average to a constant θi with a rate of con9

Other extensions have also been proposed by Kijima (2000) and Kijima and Muromachi (2000)

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

60

vergence ki . The jump process of each ﬁrm is comprised of two components: an idiosyncratic jump process independent among ﬁrms with an intensity γi and a joint jump component common to all the ﬁrms with intensity γc and differently related to the ﬁrms from using the probability pi , which inﬂuences the intensity λi . Moreover, to obtain that the intensity of the jump is λi ∼ exp(J), there must result that λi = γi + pi γc and that all the intensities have an exponential distribution with mean J. We have, therefore, that the jump times have a multivariate exponential distribution and that they are related through the likelihood that the intensities are inﬂuenced by the effects of a joint jump. The dependence structure between default is induced by correlation in changes in intensities, due to the occurence with different probabilities of joint jumps, which, together with the diffusion component, governs the processes of the intensity of default of individual ﬁrms. These two alternative approaches, in which the correlation of default is conditional on the occurrence of credit events or joint jumps in the intensities of default, they came under criticism due to the fact that both admit either that more than one default can occur exactly at the same instant either that the default of some ﬁrm does not make the intensity of the associated companies to vary signiﬁcantly. In addition, the calibration and the model implementation is quite complex.

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS Contagion Models

61

Another approach to model the correlation of defaults within the intensity models are the so-called contagion models. These models extend the conditionlly independent models including the risk of contagion of default and the risk of cluster of default, i.e. we consider the possibility that the default of a company may cause the default of related companies and that the defaults tend to be concentrated in certain periods, where the probability of default of any business increases. Now we present two models: in the ﬁrst Davis and Lo (1999) is considered the contagion of defaults among ﬁrms, and in the second of Jarrow and Yu (2001) is included the possibility to observe clusters.

Approach of Davis and Lo

In Davis and Lo (1999) and Davis and Lo (2001) we ana-

lyze the inﬂuence that the risk of contagion has on the estimation of default correlation, considering two classes of probabilistic models: a static and a dynamic one. In the ﬁrst class is constructed the probability distribution of the number of defaults, so that it incorporates the effects of interaction between the defaults. The default of an entity, a bond or generalizing a company belonging to a speciﬁc sector, it activates a sort of mechanism of ”infection” of the default within the same sector, so there might be an increased likelihood of default. This model evaluates the diffusion of the so-called ”infectious defaults” between companies or bonds belonging to the same sector within a given time interval. It is therefore a static model to estimate the number of defaults in a portfolio of entities belonging to different sectors10 .

In the second class of models is revisited the static approach, extending it to obtain a dynamic version of the effect of ”infection” of the default. In this model, is constructed a stochastic process continuous in time called ”enhanced-risk model”, as the occurrence of a default results at the beginning of a period of increased risk of default with random duration, in which all the probabilities of default increase. While in the static model is
It extends to the BET model(Binomial Expansion Technique) developed by the agency for investment services Moody’s, see Moody’s (1997)
10

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

62

estimated the number of defaults, with this dynamic model we can try to determine when these take place, building a stochastic process that provides a sequence of random times of default for multiple entities, for which we consider some mechanism of infection . Initially we assume that each ﬁrm i has a given process λi (t) for the default intensity. When there is the default for i, the intensities of all other companies j = i, which have not experienced a default, increase by a certain multiplicative factor a > 1, deﬁned enhanced factor. This factor continues to remain at a for a period of time distributed as an exponential with a given mean µ. Elapsed this time period, in which all ﬁrms are more likely to default than what is observed in normal conditions, then the factor a becomes equal to 1 until another default occurs, with which begins another period called enhanced-risk period, in which the default probabilities reﬂect the risk of infection resulted from the occurrence of default.

Approach of Jarrow and Yu With the use of conditionally independent models we have that the correlation, depending on joint inﬂuence of risk factors, does not take into account the presence of the risk of contagion and the cluster of defaults in certain periods, such as in particular situations of economic and ﬁnancial crisis and global downturn. Jarrow and Yu (2001) generalize these models including the risk that derives from the structure of the counterparts of a ﬁrm. This risk, depending on the set of relationships that a company has with its counterparties, is deﬁned counterparty risk. The risk of default of a counterparty, together with that arising from changes in macroeconomic variables relevant to the default, they affect the probability of default of a company, making it more prone to default in times of crisis among all ﬁrms. Thus we have that the counterparty risk tends to concentrate defaults in times of recession or when ﬁnancial crisis spreads across multiple businesses. In this way Jarrow and Yu (2001) consider in their model for related defaults the presence of clusters of defaults, due to the tight web of interdependencies among ﬁrms, so that also the default of a company can trigger a cascade of defaults. In their paper they present some examples of historical events, in which the default of a

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

63

counterparty has caused many other companies linked together to default. The type of relationship between counterparties may be of ﬁnancial type (e.g. possession of the assets of another company) or of commercial type (e.g. suppliers of the company). With the counterparty risk we introduce an additional source of dependence between the defaults with respect to that found in the conditionally independent models, incorporating the contagion and the cluster of defaults. Their model extends the reduced form models, making the process of the intensity to be a stochastic process that depends on the state variables of the economy11 and on jump processes, which capture the interdependencies between ﬁrms, including in this way the risk of the counterparty in the model. Taking the framework of a Cox process for the default, presented in chapter (2.3), we assume that the uncertainty of the economy is represented by the probabilistic space Ω, F, {Ft }T , P , where F = FT and P is an equivalent martingale measure, unt=0 der which the discounted values of bonds are martingales. Furthermore given the process of the d state variables Xt which takes values in d and Cox processes Ni , i = 1, 2, . . . , I,

whose ﬁrst unit jump represents the default of the I ﬁrms considered. The complete information on the economy is represented by the ﬁltration Ft = FtX ∨ Ft1 ∨ Ft2 ∨ · · · ∨ FtI , where FtX = σ{Xs , 0 ≤ s ≤ t} and Fti = σ{Ni (s), 0 ≤ s ≤ t} represent the information relative to the state variables and to the processes of default of the ﬁrms. In addition, we
−i X introduce the ﬁltration Gti = Fti ∨ FT ∨ FT representing the information to an i-th ﬁrm

at time t about the realizations of its default process given the knowledge untill time T of the series of the default for other businesses and of the state variables of the economy, in which Ft−i = Ft1 ∨ · · · ∨ Fti−1 ∨ Fti+1 ∨ · · · ∨ FtI is the ﬁltration generated by the processes of default of the other companies j = i. We get, therefore, that at the initial i time t = 0, G0 contains the complete information on the state variables of the economy

and on the realiztions of the default processes of other companies. Then we deﬁne with Ni (t) the Cox process for the default of ﬁrm i, and with λi (t) the intensity of the jump
11

See Section (2.3) for a deﬁnition of these variables.

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS process adapted to the ﬁltration Gti , such that the condition t 0

64

λi (s)ds < ∞ is satisﬁed

and that the intensity is deﬁned by a process that depends on the state variables and the default of other ﬁrms. Then we have that the conditional probability and the unconditional probability of default of the i-th ﬁrm at time τi (deﬁned as the instant of the ﬁrst jump of Ni ), respectively are: t i t|G0 )

P (τi >

= exp(−
0

λi (s)ds), t ∈ [0, T ] t (3.11) (3.12)

P (τi > t) = E exp(−
0

λi (s)ds) , t ∈ [0, T ]

We obtain a series of Cox processes for the default of companies, where the default intensity may also depend on the state of all ﬁrms of the economy. In this general framework, as noted by the Jarrow and Yu (2001), it may happen that the complexity of the model becomes unmanageable when there are ”loops” in the structure of the counterparties, making it impossible or extremely complicated to construct a joint default probabilities model. This situation with a possible loop refers also to the symmetric dependence between companies, where the intensity of default of a company depends on the state of default or non default of other ﬁrms and vice versa. This situation occurs for example when you have cross-holdings of debt, or when each ﬁrm has a market share of debt issued by other ﬁrms. To overcome this problem Jarrow and Yu (2001) introduce a restriction on the structure of the counterparts in the model. Then they distinguish primary businesses deﬁned by the subset S1 ⊂ I, from secondary ﬁrms deﬁned by the subset S2 ⊂ I complement of S1 . The intensities of the primary business depend only on the state of the economy FtX , while the second group ones are dependent both on the state of the economy and on the state of the other companies and therefore their default processes. Given a set of exponential random variables {ξ i , 1 ≤ i ≤ S1 } independent of each other and with respect to the process Xt , then we deﬁne the default time τi of the i-th primary

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS ﬁrm as t 65

τi = {t :
0

λi (s)ds ≥ ξ i }, 1 ≤ i ≤ S1

(3.13)

which is distribuited as t X P (τi > t|FT ) = exp(− 0

λi (s)ds), t ∈ [0, T ]

(3.14)

where λi (t) is measurable with respect to FtX . Then we introduce a further set of exponential random variables {ξ j , S1 + 1 ≤ j ≤ I} independent of each other and with respect to both the process FtX and to set of the default times of the primary ﬁrms {τi , 1 ≤ i ≤ S1 }. Then the time of default τj of the j-th secondary ﬁrm is t τj = {t :
0

λj (s)ds ≥ ξ j }, S1 + 1 ≤ j ≤ I

(3.15)

Whose distribution of probability is t S X 1 P (τi > t|FT ∨ FT ∨ · · · ∨ FT 1 ) = exp(− 0

λj (s)ds), t ∈ [0, T ]

(3.16)

where now the intensity of default λj (t) of the secondary ﬁrm is adapted to FtX ∨ Ft1 ∨ · · · ∨ FtS1 . Among the secondary ﬁrms the counterparty risk structure is introduced trough the intensity λj (t), which is assumed to have the following form
S1

λj (t) =

aj 0,t

+ k=1 aj It≥τk , S1 + 1 ≤ j ≤ I k,t

(3.17)

where the indicator function of default It≥τk = 1 − Nk (t), ∀k, aj is adapted to FtX , aj 0,t k,t which represents the component of the intensity indipendent of the state of default of the other ﬁrms and it is adapted to FtX ; moreover we assume that the ﬁrm j holds only assets of the primary ﬁrms i.

In this model the primary companies have independent default, while the asymmetric

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

66

counterpart structure between the primary and secondary ﬁrms deﬁnes the dependence structure of defaults among secondary ﬁrms and between secondary and primary ﬁrms. This dependency is deﬁned according to the sign given by aj , which binds the default k,t intensities of the secondary ﬁrms with indicator functions of default of the primary ﬁrms. This model, as observed by Yu (2002), despite it introduces counterparty risk as a new source of correlation between the defaults, allowing the analysis of default probability to take into account also the risk of contagion and the cluster of defaults, it still keeps the conditionally independent model setting to model the default probabilities of primary businesses.

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

67

3.1.3

Approaches with Copulas

The third approach in the multivariate models to introduce a structure of dependence between the defaults is the use of copulas. According to this approach, contrary to what we have already discussed above for multivariate models both structural and based on the intensity, we separate the deﬁnition of the univariate models of the default from the structuring of the multivariate model for defaults. The formulation of the joint distribution of default occurs in two separate steps: at ﬁrst we construct the marginal distributions of default times of individual ﬁrms, while in the next step we introduce a copula that links the marginal distributions, thus forming the joint distribution of default with a speciﬁc dependence structure deﬁned by the copula introduced. This division allows us to calibrate and imply any model at the univariate level and then with the introduction of the copula calibrate the model to multivariate level. This makes the modeling and calibration of the joint distribution to be less complex than the multivariate models in which the dependence structure should be introduced already in default models for the single entities. The main and essential step of the approach is now in the careful choice of the copula, in its formulation and calibration, making it the focal point of the approach and therefore the most attention demanding and delicate to be addressed in the joint modeling of default. The copula takes as input estimates of marginal probability distributions of default times, transforming them into a joint probability distribution of default times, in which the dependence structure is completely introduced by the copula by linking the marginals. The copulas have been introduced in the probabilistic ﬁeld by Sklar (1959)12 , and their use has been growing strongly, especially in recent years. The use of copulas was introduced in the actuarial ﬁeld for estimating the probability of survival by Frees and Valdez (1998) and in ﬁnance by Embrechts et al. (1999), while in the models for the dependence of credit risk, among others, we remind Li (2000), Frey and McNeil (2001), Sch¨ nbucher o and Schubert (2001) and Schmidt and Ward (2002). Initially we introduce the use of copFor more theoretical treatment about copula, see also Sklar (1996), Nelsen (1999), Sch¨ nbucher (2003) o and McNeil et al. (2005).
12

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

68

ulas to tie univariate structural models, then afterwards we analyze their introduction in the univariate models for the intensity.

Inroduction to the Use of Copulas in the Credit Risk Models. Given the probabilistic space {Ω, F, P }. Consider n ﬁrms whose default times are deﬁned as τi , i = 1, . . . , n and whose default indicator functions at time t are Ii (t), which are also deﬁned as    1 τ ≤t  i   0 τ >t  i It is deﬁned with Gi = {Gi (t), t ≥ 0} the ﬁltration generated by the stochastic process of default of individual ﬁrms, so we have the sigma algebra Gi (t) = σ{Gi (s), 0 ≤ s ≤ t}. Then the marginal probability of default at time t, calculated at time 0, for the i-th ﬁrm is equal to Fi (t) = P (Ii (t) = 1|Gi (0)), while the corresponding marginal probability of survival up to time t is Si (t) = 1 − Fi (t). The joint distributions of default probabilities and survival, respectively, are

Ii (t) =

(3.18)

F (t) = P (I1 (t) = 1, . . . , In (t) = 1|G(0)) S(t) = P (I1 (t) = 0, . . . , In (t) = 0|G(0)) n (3.19) (3.20)

where G =

Gi . Assuming the continuity of the marginal, we admit, according to i=1 Sklar’s theorem, that these joint distributions can be expressed by means of n-dimensional copulas, which in this case exist and are unique, and of the marginals Fi (t) and Si (t) ∀i = 1, . . . , n. Then we have the following expressions for the joint distributions of the probabilities of default and survival:

F (t) = C(F1 (t), . . . , Fn (t)) S(t) = Cs (S1 (t), . . . , Sn (t))

(3.21) (3.22)

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69

where C, Cs are two copulas wich describe the dependence structure of the probability of default and survival. We can note that by the deﬁnition of the copula and of the time of default, we have the following links between the marginal probabilities and the joint distributions expressed through the copula:

Fi (ti ) = F (∞, . . . , ti . . . , ∞) = C(1, . . . , ti . . . , 1) Si (ti ) = S(0, . . . , ti . . . , 0) = Cs (1, . . . , ti . . . , 1)

(3.23) (3.24)

In general we have that the copulas are parametrized by measures of dependence, including the most known coefﬁcient of linear correlation ρ, which alone characterizes the Gaussian copula, while together with the degrees of freedom g they deﬁne the Student’s-t copula13 .

13

See Nelsen (1999) for more details.

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS Copulas in the Structural Framework

70

In the structural framework the copula is introduced by ﬁrst deﬁning the process followed by the value of assets for each ﬁrm and the relative marginal probability that it is touched or crossed the barrier, depending on the type of process followed by the value of the assets, then in the next step bind the marginal distributions by means of a copula, thus obtaining the joint distribution of the probability of default. Within the framework of Merton (1974) for univariate default, we assume that the default can only occur at debt maturity. Assuming that this date is equal to t equal for all ﬁrms i = 1, . . . , n, we have that the marginal probability of default for the i-th company is

Fi (t) = P (Vi (t) ≤ Ki (t))

(3.25)

with Vi (t), Ki (t) respectively, the value at time t of the process of the value of the assets of the ﬁrm i and the value of the respective barrier. As discussed in Chapter (1.1), in the model of Merton (1974) we have that the marginal probability is equal to

Fi (t) = Φ(−di,2 (t))

(3.26)

where still Φ is the Standard Normal CDF, while we have
V ln( Kii(0) ) + (r − q − 0.5σ 2 )(t) (t) √ di,2 (t) = σ t

(3.27)

with the same interpretations for r and q already discussed in the chapter (1.1). Then the joint distribution of default probabilities can be expressed, through the use of a given copula, as follows

F (t) = P (V1 (t) ≤ K1 (t), . . . , Vn (t) ≤ Kn (t)) = C(Φ(−d1,2 (t)), . . . , Φ(−dn,2 (t))) (3.28)

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71

In the model of Merton ln(Vi (t)) have an univariate normal distribution derived from the fact that Vi (t) follows a geometric Brownian motion. If we assume that they are also jointly normal, to obtain the joint distribution function we have to introduce the Gaussian copula ΦR with R matrix of correlations. Setting ui = Fi , we obtain

F (t) = ΦR (Φ−1 (u1 ), . . . , Φ−1 (un )) = ΦR (−d1,2 (t), . . . , −dn,2 (t))

(3.29)

If, however, keeping the marginal as above normal, we introduce the Student’s-t copula instead of the Gaussian one, thus considering another dependence structure, we then obtain the following joint probability distribution for the default

F (t) = tR,g (t−1 (u1 ), . . . , t−1 (un )) g g

(3.30)

where t−1 is the quantile function of the Student’s-t standard, tR,g is the CDF of the g multivariate standard t-Student, and g are the degrees of freedom. If instead the model of Merton (1974) we consider an univariate approach for the default, which admits the default to occur also before the debt maturity, such as Black and Cox (1976)’s model, then we would get that the marginal probability would be

Fi (t) = P ( min (Vi (t) ≤ Ki (t)))
0≤s≤t

(3.31)

Hence now we have that the probability would depend not only on the process Vi (t) but would depend as well on the minimum process, making it impossible to obtain normal distributions at the univariate level. We have other cases in which the assumption of normality is not fulﬁlled, for example when is introduced stochastic dynamics for the interest rates, as it is assumed in the model of Longstaff and Schwartz (1995). These extensions make no more acceptable the assumption of multivariate normal joint distribution. At the multivariate level, we can still introduce structures of dependency also described by

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

72

the Gaussian copula, as well as other elliptical copula, such as the Student’s-t. We may also consider additional functions for the copula, such as those belonging to the family of Archimedean copulas, chosen only by the type of dependency that you want to introduce regardless of the estimated marginals.

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS Copulas in the Framework of Intensity Models

73

As part of the intensity models for the univariate default, we have the default time for the company i can be expressed as follows t τi = inf{t :
0

λi (s)ds ≥ ξi }

(3.32)

where ξi ∼ Exp(1) for each i independent of each other and of the information on the state of the economy available to investors. These exponential random variables represent a kind of barrier for the stochastic hazard function, which if crossed it involves the default of the ﬁrm. Making a change of variable by the position ξi = − ln(Ui ), with Ui ∼ U (0, 1), uniformly distribuited in [0.1] for each i independent of each other and of the information available. Then we obtain: t τi = inf{t : exp(−
0

λi (s)ds) ≥ Ui }

(3.33)

Recalling what has been obtained in Chapter (2) for the marginal probability of default and survival in the case of deterministic intensity, we get t Fi (t) = 1 − exp(−
0 t

λ(s)ds)

(3.34) (3.35)

Si (t) = exp(−
0

λ(s)ds)

While in case of a stochastic intensity, conditioning with respect to information available in t = 0, deﬁned by G0 14 , we get t Fi (t) = 1 − E exp(−
0 t

λ(s)ds)

(3.36) (3.37)

Si (t) =
14

exp(−
0

λ(s)ds)

See Section 2.3 for its deﬁnition.

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS Li’s model

74

Li (2000) can be considered one of the ﬁrst to use a copula to introduce the

structure of dependence between the defaults, whose marginals are estimated by means of the formulation of univariate models based on the intensity of default. In this approach the joint distribution is built up using a copula as dependence link between the marginal probability of survival and of default of ﬁrms. As noted in paragraph ”Introduction to the copoulas in credit risk model”, this procedure is considered analogous to the consideration of a copula between the marginal joint probability of default. According to this approach, we consider the stochastic independent barriers, obtaining thus a conditionally independent model for the default. For the estimation of correlated default times, Li (2000) use the Gaussian copula with correlation matrix R to bind the marginals of the joint probability of survival. Thus in order to obtain a realization for the vector, (τ1 , τ2 , . . . , τn ), of default times for ﬁrms, will be sufﬁcient to simulate a multivariate normal vector, (Y1 , Y2 , . . . , Yn ), with the same matrix of coefﬁcients of correlation R; then using the relation τi = Fi−1 Φ(Yi ), we retrieve a simulation of the vector of default times with the the given correlation matrix R.

Schonbucher e Schubert’s model In the approach of Sch¨ nbucher and Schubert (2001) o are admitted not conditionally indipendent default, assuming that the stochastic barriers, either ξi or Ui , are mutually dependent through the introduction of a copula between their marginals, but they still are independent from the information of the state variables of the economy. In this case, the default time of the i-th ﬁrm is deﬁned from (3.33) as t τi = inf{t : exp(−
0

λi (s)ds) ≥ ui }

(3.38)

where ui is a realization of the uniform Ui in [0.1]. Sch¨ nbucher and Schubert (2001) o consider the intensities λi , i = 1, . . . , n, adapted to the respective ﬁltrations Fi (t), representing the information available on both state variables of the economy and on the default state of the ﬁrm i, and where the intensities are called ”pseudo default intensi-

CHAPTER 3. MODELS FOR THE CORRELATION BETWEEN DEFAULTS

75

ties”. Anyhow investors also observe the status of other companies, thus such intensities are not adapted to information actually available to investors, which is represented by the ﬁltration F(t). The information on the status of other companies in the model of Sch¨ nbucher and Schubert (2001) is introduced by the copula linking the barriers which o triggers defaults. For the estimation of default probabilities based on all information available, would be necessary to ﬁnd the real intensities hi (t), i = 1, . . . , n adapted to the ﬁltration F(t), by combining pseudo-intensities with the speciﬁcation of the copula between the barriers. For the simulation of a vector of default times for the n companies in the economy, Sch¨ nbucher and Schubert (2001) apply the same procedure described o above for the model of Li (2000), except that in this case we use the actual intensity to estimate the marginals of the joint probability of default. This approach by using the copula between the barriers can also introduce a level of risk of contagion in default, but differently from what we found in Davis and Lo (1999) and Jarrow and Yu (2001), we ﬁnd that the risk of contagion derives endogenously from the use of the copula between the barriers. In their approach is formalized the general expression of the real intensity for Archimedean copulas and in particular for the Gumbel and Clayton copula. They also present the results of the calibration of their model by introducing a Gaussian copula on the barriers in the presence of a univariate conditionally independent model for the intensity, however allowing for a different speciﬁcation for the univariate intensity model. In a further extension to this approach, Rogge and Sch¨ nbucher (2003) use an Archimedean o copula instead of the Gaussian, since they state that the elliptical copulas are not appropriate to calibrate a real joint dynamic of the intensity of default.

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76

3.2

Top-Down Models

Instead of deﬁning the dependence structure, introducing it after having deﬁned in a ﬁrst step the univariate models for default by one of the techniques discussed above, we could initially ignore the univariate distributions, and formulate directly the multivariate model for the default of ﬁrms. We therefore consider a model for the total loss and for the counting processes of the number of defaults, building a model for the aggregate loss, associated with the same level of loss or with any particular rate of loss. In this type of approach, a model after being calibrated with derivatives on multivariate credit risk, will also be implemented for the pricing of products dependent on the default of a single entity. The latter is needed to verify the consistency of estimates even with respect to the univariate risk of default. In general this phase, called ”down”, it is crucial for the overall adequacy of a top-down model. For this reason it is extremely difﬁcult to analyze the an ex post consistency of estimates for the univariate default, derived from a model initially constructed at a multivariate level. The ﬁrst top-down models, among others, have been proposed by Sch¨ nbucher (2005), Giesecke and Goldberg (2005) and o Errais et al. (2006), while the ﬁrst who perform a calibration to quotes of the indices and other derivatives on multivariate credit risk have been Brigo et al. (2007). Regarding the problem of the ”down” phase there have been some proposals, among which we point up the technique of the random thinning, which is used to decompose the intensity of default at portfolio level in the sum of the intensities of the components of the portfolio. This technique was initially proposed by Errais et al. (2006) and later extended by Ding et al. (2009), Halperin and Tomecek (2008), Zhou (2009) and Giesecke et al. (2010), who analyse the relationship between portfolio risk and the risk of its individual components, whose analysis is needed to evaluate the sensitivity of the value of a credit risk portfolio with respect to the value of a derivative on the univariate credit risk, in order to asses the correct hedging strategy for the portfolio positions, without further speciﬁcation for the top-down model at which it applies random thinning.

Chapter 4 Credit Risk Derivates
The credit risk derivative instruments have been developed especially in the last ten years, infact those products have been established themselves among investors, as they respond to the need to acquire a hedge against the risk of default or to another credit event, such as voluntary restructuring of the capital of a company, which could detoriate the solvency or the credit rating of a security held by the investor, such as a corporate bond or a government bond. With regard to the credit events against whose occurrence can be purchased a hedging derivative contract, there has been formalized a deﬁnition by the ISDA (International Swaps and Derivatives Association), under whose rules are deﬁned the characteristics of most derivative contracts on the credit risk, both those traded on regulated markets, ant those negotiated OTC (over the Counter) or through private contracts. In the thesis we use the term default to deﬁne the credit event that determines the complete non-fulﬁllment of the obligations with respect to the creditors of a company, in particular as regards of corporate bonds holders. The deﬁnition of default considered can be related to the deﬁnition of events relating to credit risk, which are proposed by ISDA and used for the settlement of derivatives, if at least such a credit event occurs prior to maturity. Using the deﬁnition of the default proposed in the report Moody’s (2001), which summarizes the deﬁnitions of the credit risk event according to the ISDA and its links with the deﬁnition of default as considered by Moody’s, to which we refer to for the deﬁni-

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tion of the models presented for the default risk valuation. The ﬁrst derivatives on credit risk that have been introduced in the early 1990’s were bilateral contracts, through which they allow for the negotiation of the risk associated with the creditworthiness of a third party, the debtor to whose solvibility is referred the hedging purpose of the derivative contract. The settlment of such contracts (their pay off), depends on the occurrence of speciﬁc credit events relating to the debtor to which the derivative contract is referred. In those contracts the underlying consists of an asset issued by the same debtor of reference of the contract. The regulation of the contract may be done in cash (cash settlement) or physically (physical settlement), with the delivery of the underlying security done by the buyer of the derivative to the seller in exchange of the pair value of the same underlying ﬁnancial instrument. The main credit risk derivatives are the follows: 1. Asset Swaps 2. Credit Default Swaps (CDS) 3. Credit-Linked Notes 4. Total Return Swaps 5. Credit Spread Options 6. Basket Instruments For a deﬁnition of all those listed here, please refer to books or specialized sites, such as the ISDA website which is a good source for their detailed deﬁnition; while now we introduce the features of the CDS and also of CDSs Index, as these represent the types of credit risk derivatives most liquid, since liquitity is also necessary to consistently reﬂect the sentiment of the market participants on the probability of default of a speciﬁc ﬁrm.

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4.1

The Credit Default Swaps

Credit Default Swaps are bilateral contracts in which the contractor (protection buyer) who wants to acquire an hedge against the credit risk relating to a ﬁnancial underlying (reference obligation), he pays a periodic premium (CDS premium or simply fair spread), applied on the notional of the contract, to the seller of the contract (protection seller), which in turn is committed to a ﬁnal payoff in the event of default by the issuer of the reference obligation. The ﬁnal payoff can be done by the protection seller through a payment in cash, cash settlement, or through the purchase at par of the ﬁnancial underlying delivered by the protection buyer, physical settlement. In the cash settlement procedure, the most common for derivatives traded on regulated markets, the protection seller reimburse the protection buyer by an amount equal to the notional of the contract multiplied by the loss on the ﬁnancial underlying asset, computed as the difference between the par value of the underlying and its market or realizable value after the default or as a quote of the par value speciﬁed in the contract, where this proportion is also deﬁned as Loss Given to Default (LGD). Below, in Figure (4.1), we present a graphical scheme of the contract with cash settlement or physical settlement.

4.2

CDSs Index

In order to negotiate and transfer the credit risk on a basket of ﬁnancial instruments issued by several companies which also belong not necessarily to homogeneous areas, without incurring in an excessive cost due to the purchasing of a portfolio of CDSs on the OTC market or negotiated on the regulated market. For this main reason the CDSs indices products have been constructed, since they are by far less costly than buying separately each CDSs, and with which we can be covered against the risk of default of counterparties belonging to the index. These indeces also provide the agent both a measure of the portfolio spread of the CDSs and even a useful tool for the negotiation of the CDSs. In particular, the CDS market indices, hugely developed in recent years, has had further success

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Figure 4.1: CDS Scheme in 2004, when there was a consolidation in the sector through agreements between the companies that had set up the indeces. In particular, the most popular and liquid indices, belonging to the two main economic regions, Europe and North America, respectively, are the indices: • iTraxx Europe, regarding the 5 and 10 years spreads of a basket of 125 European companies from different sectors and with high credit worthiness; • CDX NA IG, relating to 5 and 10 years spreads of a basket of 125 companies in North America (NA) with high credit quality, i.e. investment grade (IG), as deﬁned in the index name.

iTraxx Europe Index

This index is the most liquid iTraxx indix, owned, managed,

compiled and published by International Index Company (IIC), which provides the licenses for the market makers. The iTraxx Europe index is composed of the 125 most liquid CDSs with reference credit instruments issued by investment grade European companies, subject to certain rules of the sector as determined by the IIC and the SEC (U.S.

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Securities and Exchange Commission). The iTraxx index composition is divided in ﬁnance sector, in turn divided into Sub Financials and Senior Financials, and non-ﬁnancial sector, to which belong the ﬁeld of technology, media and telecommunications (TMT: Technology, Media, Telecommunications), the auto and industry sector (Autos and Industrials), the sector of consumer goods (Consumers) and the energy sector (Energy). In addition, each CDS has the same weight, equal to 0.8% of the total value of the index. There is also a signiﬁcant volume, in terms of liquidity, in the negotiation of the indeces HiVol and Crossover, which are part of the iTraxx indices. The HiVol index is a subset of the main index, the iTraxx Europe, consisting of what are considered the thirty most risky components of the index at the time it is updated. The Crossover index is constructed in a similar way, but is composed of ﬁfty sub-investment grade ﬁrms references. The components of the indeces are updated every six months, through a process known as rolling index. The rolling or rotation dates are March 20 and September 20 of each year. For example, the index series number 13 was launched on March 20, 2010, with maturity on June 20th, 2015 for 5 years contract. The maturities for the iTraxx Europe index and the HiVol index are at three, ﬁve, seven and ten 10 years, while for the Crossover index are only at ﬁve and at ten years. These indices are instruments traded at a predetermined ﬁxed rate, and the prices of which, corresponding to the fair spreads on the indices, are determined by supply and demand of the market. The ofﬁcial quotation of the prices are made available on behalf of the IIC by Markit Group Limited on a daily basis, formed by the collection of information on prices, provided by the trading desks of the banks which are licensed as market makers.

Currently, the iTraxx CDS contract negotiation is limited to the OTC market. A bank, acting as market maker, quotes on its trading desk the bid and ask spread for the index on the basis of the requests received from investors, according to these two quotations, investors may decide to participate in the contract as counterparties of the market maker, selling or by buying protection on the index for a given notional amount on which is

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applied the quoted spread of the index. In case do not verify any credit event for any of the companies issuing the reference securities of the CDSs that belong to the index, the protection buyer continue to perform the due payments until the maturity, computed as the spread multiplied by the notional value deﬁned at the beginning of the contract. However, in case a credit event occurs for one or more index reference ﬁrms, from that time the subsequent payments done by the purchaser of the contract would be computed on the notional reduced by an amount equal to the percentage of the index that those ﬁrms represent. Thus in this iTraxx index case there would be a reduction equal to 0.8n%, with n equal to the number of the ﬁrms for which a credit event occurs. Alongside the market maker or the other part who is entitled as the protection seller, for each company for which a credit event occurs, he must make a payment calculated as the product of the notional amount and a certain percentage, which differs according to the settlement procedure set in the contract: in the case of cash settlement, the percentage due for each ﬁrm will be equal to (1 − R)p, with p equals to the percentage represented by the ﬁrm in the index and R representing the recovery rate on the reference ﬁnancial instrument of the CDS and issued by the same speciﬁc ﬁrm; while in the case of physical settlement the buyer pays the seller a percentage p of the notional, in exchange of the delivery of the ﬁnancial instruments issued by the ﬁrm of reference with a nominal value equal to the same amount paid.

4.3

The Collateral Debt Obbligation

For instance, another very common type of securities dependent on credit risk are the Credit Debt Obbligations (CDOs). These securities are made to share the credit risk on a portfolio of securities underlying the CDO, whose portfolio is divided into several classes of securities, in which to each class is assigned a different regulation of insolvency. A particular type of CDO is represented by the synthetic CDO whose underlying portfolio consists of CDSs contracts.

Chapter 5 Empirical Implementation
After having analyzed and discussed the different ways to tackle the problem of constructing a model for estimating the probability of default in univariate and multivariate setting, as well as having presented some of the most common types of derivatives dependent on the risk of default of a single ﬁrm, or more companies, now we present the construction of a model1 for the default intensity based on the framework of factor models. Under this type of models, we have that the intensity of occurrence of the default event, for each company, will be a function of some observable factors, for which we can observe historical realizations. For the determination of the coefﬁcients, which determine the relationships between these factors and the levels of intensity, we estimate them by running some regression applied to reasonably motivated models and whose construction is also well established among the academics. Before estimating the coefﬁcients of the univariate models of the intensity of default of individual ﬁrms considered, we deem to be compoulsary to perform a preliminary analysis of what are the factors affecting, and, without loss of generality, determine the trend in iTraxx Europe CDS index spreads. This is done in view of a use of the model in a multivariate framework, in which the relationship between the intensities derived from different dependencies that every company shows in relation to macroeconomic factors that represent the general economic trend.
For a discussion of estimation methods and tests carried out in the chapter, see Hamilton (1994), Fox (2008) and Gallo and Pacini (2002).
1

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The companies considered for the construction of the factor models for the intensity are some2 of the companies in the ﬁnancial sector, whose CDSs are part of the reference components of series 13 of the more liquid index of European CDSs presented in chapter (4.2), the iTraxx Europe. The data used for the analysis and construction of models have been fully downloaded from the Datastream database, while the estimates and analysis were performed using the freeware R-labstat.

5.1

Analysis of the Trend for the iTraxx Europe

In this section we construct a model for estimating the determinants of logarithmic changes, or logarithmic yields, of the spreads of iTraxx Europe index. Bystrom (2005) was the ﬁrst to propose a linear regression model to study the link between the spread of a set of CDSs, related to several ﬁrms, and the stock market, verifying the presence of a signiﬁcant positive autocorrelation between the yield of the spread of the iTraxx indices, i.e. that they are autoregressive (AR). Other authors, including Alexander and Kaeck (2008), have shown that there is a negative relationship with risk-free interest rates, and the presence of a positive link with the volatility of stock returns, which in turn is positively associated with the volatility of the logarithmic changes of the value of the ﬁrms’ assets. All these models assume, however, that the error terms of the estimated linear regressions have a normal distribution, when in fact, as noted in the literature for ﬁnancial time series of stock returns, the returns on the CDS show different trends: in fact can be observed the presence of the phenomenon of volatility cluster, which had already been addressed by the Autoregressive Conditional Heteroskedasticity(ARCH) model proposed by Engle (1982) and the Generalized Autoregressive Conditional Heteroskedasticity(GARCH) model proposed by Bollerslev (1986). The presence of volatility cluster implies large tail (heavy tail)3 in the empirical distribution of the residuals, which is incompatible with the use of
I have not been able to build models for all ﬁnancial ﬁrms in the index iTraxx Europe, because for some of them I did not have the necessary data. 3 An indicator of the presence of the so-called fat tail or heavy tail in a distribution is the kurtosis (Kurtosis).
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a theoretical normal distribution for the errors, but which makes more appropriate the use of a Student’s-t distribution, which permits the presence of thick tails in the distribution of residuals. Furthermore, empirical evidence shows the presence of high asymmetry, that is a level of the skewness4 very different from 0, whise level, in order to be considered in the analysis of the residuals, make necessary the use of a distribution that admits a ﬂexible choice for the skewness level, such as the asymmetric Student’s-t distribution proposed by Hansen (1994). The residuals analysis concludes with the presentation of the main tests that verify the ”robustness” of the considerations made above. Finally, in addition to an implementation on the entire sample period of data available in the database, we present also comparison with two subsamples, in fact we split the data in the period before the crisis and in the crisis period.

Analysis of the Data for the Model The daily data used for my analysis are: 1. the spreads of the iTraxx Europe index, expressed in basic points (BP, 10−4 ), with which it is calculated both the time series of the logarithmic return (spread return), which represents the dependent variable of the models, and the time series of lagged spread return (spread lag-return) used as independent variable; 2. the quote of the Stoxx Europe 600, which is an European stock index of 600 leading European companies; 3. quotations, expressed in percentage points (10−2 ), of the Volatility Index, Vstoxx, which is an index of the implied volatility derived from options on the shares of the main European stock index, Stoxx Europe 600. The Vstoxx index is a useful tool to represent the forecast of the market concerning the short-term volatility of stock returns. This index is used as an independent variable in the ﬁnal model;
The degree of skewness of a distribution is deﬁned as the value of the parameter λ, which is an indicator of the level of symmetry or asymmetry of the distribution considered. In particular, with λ = 0 we have perfect symmetry, as is the case for the Normal distribution and the Student’s-t distribution, while with λ > 0 and λ < 0 we have respectively positive asymmetry and negative asymmetry, or equivalently if the mode is shifted towards the minimum or the maximum of the distribution.
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4. the quotes of the EMU Benchmark index, which is an index of government bonds with ﬁve year maturity, used as an independent variable in the model; 5. quotes for the 12, 6 and 3 months Euribor, but the 12 months one loses signiﬁcance when is introduced in the models the independent variable EMU Benchmark; 6. the quotes of the EONIA rate, not signiﬁcant as an independent variable 7. the quotes of other indicators for the risk-free rates, for the trend of stock prices and other macroeconomic indicators, which result not to be signiﬁcant as independent variables of the model. The complete time period considered for the analysis is from 2005-03-23 to 2010-11-11 for a total of 1472 daily observations available, and the period for the the pre-crisis analysis goes from 2005-03-23 to 2007-03-07 and for the the crisis period the observations range from 2007-03-08 to 11/11/2010 for a total of 961 and 511 daily data considered. The macroeconomic factors, included as independent variables in the multivariate model, are all resulted signiﬁcant as determinants of the CDS spread returns at the univariate level. The variables 12-months Euribor and one year EONIA swap index, although they are found to be signiﬁcant at the univariate level, when it is introduced as an independent variable representing the performance of risk-free interest rates, the EMU benchmark index of government bonds with maturity 5 years, both the other variables become insigniﬁcant or at least not very signiﬁcant, but in any case highly non-signiﬁcant compared with the index of government bonds, which are linked by a negative relationship characterized by a slight negative correlation. The variables considered as determinants in the ﬁnal model for estimating the dependent variable corresponding to the logarithmic returns of the iTraxx Europe index spreads, are logarithmic returns/changes of: 1. index Stoxx Europe 600 2. Vstoxx Volatility Index 3. index EMU for the benchmark government bond with maturity 5 years

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The table (5.1) presents some descriptive statistics of the dependent variable and independent variables. Figure (5.1) show the trends of the variables. iTraxx EU Stoxx EU 600 Vstoxx Vol. Ind. Emu Bench. 5YR -0.32586 -0.07930 -0.24919 -0.01143 -0.01518 -0.00553 -0.03384 -0.00118 -0.00037 0.00046 -0.00365 0.0000 0.00069 0.00003 0.00036 0.00006 0.01679 0.00606 0.027038 0.00128 0.25536 0.09410 0.32767 0.00905 0.03929 0.01372 0.05785 0.00211 -0.28267 -0.07873 0.94745 -0.09333 9.79360 7.41218 3.88415 2.07370 Table 5.1: Descriptive Statistcs of the Logarithmic Returns Afterwards in the table (5.2) we consider both the test statistics that verify the absence of a Normal distribution behaviour of the empirical distributions of the variables, the JarqueBera normality test (JB test), and those that show the absence of a mean-reverting trend in the returns, the Augmented Dickey-Fuller test (ADF test). From the results of the p-values of the test statistics can be observed that all variables have a distribution signiﬁcantly different from the Normal, and that trends of the variables are signiﬁcantly mean-reverting, that is they can now be considered stationary. Figure (5.2) shows the possible ﬁt to empirical distributions through the use of theoretical distributions, Normal and Student’s-t, noting that the Student’s-t (green curve) has a more appropriate ﬁt than the Normal (red curve) for the empirical distributions of all variables, except for the performance of the Benchmark Emu 5YR, so only in the latter case it seems appropriate a Normal. iTraxx EU Stoxx EU 600 Vstoxx Vol. Ind. Emu Bench. 5YR 5923.33 3384.09 1150.46 267.65 0.00 0.00 0.00 0.00 greater greater greater greater -10.32 -11.09 -11.41 -11.10 stationary stationary stationary stationary 0.01 0.01 0.01 0.01 Table 5.2: Normality test and Mean Reverting test

Min. 1st Qu. Median Mean 3rd Qu. Max. Std.Dev Skewness Kurtosis

J-B Test p-value Alternative ADF Test Alternative p-value≤

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Analysis and estimation of the model In Figure (5.3) can be observed the graphs representing the relationships between independent variables and the dependent variable, graphically checking the presence of a particular relation. In particular, we note that, as stated previously, the trend of the dependent variable is positively associated with both the returns of the volatility index and with the returns of the index of government bonds, as these are negatively linked to the performance of risk-free interest rate, while the relationship with the returns on the stock turns out to be negative. This link between the independent variable and the dependent variable suggested in the data analysis and veriﬁed graphically in Figure (5.3), turns out to be conﬁrmed by the linear regression estimation, whose results are shown in Table (5.3), in which are also included as independent variable one day lagged returns (lag1-return) of the CDSs index spread. The addition of this variable in the regression was carried out because, by testing the Durbin-Watson test for autocorrelation of residuals in the table (5.4), we noted the presence of ﬁrst-order autocorrelation in the residuals, while after that is added the variable lag1-return spread, we do not note further signiﬁcant autocorrelation in the residuals. Finally, the model considered is the following:

Yt = α1 X1,t + α2 X2,t + α3 Yt−1 + α4 X3,t + εt

(5.1)

where the dependent variable Yt is the log-return of the iTraxx Europe CDSs index, the independent variables X1,t , X2,t , Yt−1 , X3,t are respectively the Stoxx Europe 600 Index log-returns, the log-returns of the Vstoxx Volatility Index, the one day lagged log-return of the dependent variable and the log-return on the EMU benchmark for the government bonds with 5 years of maturity, while α1 , α2 , α3 , α4 are the linear coefﬁcients of the respective independent variables and εt is the error term of the model, i.e. the residuals that will be analyzed at the end of chapter. From the the p-values, all the independent variables included in the model are signiﬁcant at a 0.1% signiﬁcance level, conﬁrming to be highly determinants with respect to the the pattern of the iTraxx Europe index spread.

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The value of R2 -adjusted, which represents the degree of explanation of the time series of the dependent variable captured by the estimated model, is 0.39. This value for R2 adjusted is sufﬁciently high for this type of models, in which it is estimated the trend of a portfolio of CDS spreads, in addition, this value is consistent with what is found by the academics for other estimation models of multivariate spreads, which may include other determinants of macroeconomic factors as independent variables. Estimate Std. Error t value Pr(> |t|) (Intercept) 0.0003 0.0006 0.5534 0.5800 Stoxx Europe 600 -1.0688 0.0697 -15.34 0.0000 Vstoxx Volatility Index 0.0686 0.0156 4.39 0.0000 Lag iTraxx Europe 0.1394 0.0148 9.39 0.0000 Emu Benchmark 5YR 3.0079 0.3192 9.42 0.0000 Table 5.3: Estimated regression model with robust Standard Errors Model with Lag spread return Model without Lag spread return statistic 1.88 2.10 method Durbin-Watson test Durbin-Watson test alternative true autocorrelation is greater than 0 true autocorrelation is greater than 0 p.value 0.01 0.97 Table 5.4: Autocorrelation Test of residuals Finally, with the residual of the model we verify that, along with ﬁrst-order autocorrelation, there is also heteroscedasticity in the variance (volatility heteroskedasticity) of the residuals of the ﬁnal model. For this analysis, using the test of Breusch and Pagan (1979) for the null hypothesis of homoskedasticity, we observe that this hypothesis can be rejected because the p-value of the test statistic is equal to 1.678e−10 , making acceptable the assumption of heteroscedasticity in the variance of the residuals of the model, as conﬁrmed also by the signiﬁcant parameters of the GARCH(1,1) estimated over the residuals, for which we have not reported the results. The model estimates were also carried out in the two subperiods of the sample of the available data. The ﬁrst period, called pre-crisis, in which the ﬁnancial crisis had not yet unleashed in all its drama, but it was only the beginning of the series of write-downs

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and failures induced by the increasing difﬁculty of the subprime mortgage market, to be passed up well in adverse impact on the entire ﬁnancial sector, starting from Wall Street. This period covers the time period between the initial date of the sample, the 2005-03-23 until the date 2007-03-07. The latter date was chosen because in the following weeks of March, 2007, there were the ﬁrst signs of the sharp deterioration of the crisis, which began to be felt heavily on quotes on major markets, as stated in Repubblica (2007). Moreover, in the same month, the ﬁrst U.S. real estate companies to succumb to the crisis, due to the huge rescaling properties values in the U.S.A., were New Century Financial and Accredited Home Lenders Holding, and ﬁnally there was a weak increase in retail sales in February, below expectations, which has contributed to a loss of 1.98% on the U.S. Dow Jones stock index. The second period covers the time period from 2007-08-03 to 2010-11-11, when the ﬁnancial crisis has made us feel its worst effects, derailing many banks, companies, without excluding even sovereign debt, we can recall the bankruptcy of Lehman Brothers and Iceland among all the events that followed, till have a also negative impact on real economy. The table (5.5) presents the results of the estimates in the periods described above, noting that even in this case, all coefﬁcients are signiﬁcant now at the 5% for both periods. The value of R2 -adjusted amounted respectively to a lower value of 0.181 in pre-crisis period, and to an higher value of 0.42 during the crisis. Furthermore, may be observed that: • the negative inﬂuence of the trend of the Stoxx Europe index is more than halved in absolute terms in the pre-crisis period than both the period of the crisis and the entire sample of observations;

• the coefﬁcient of returns of the volatility index, Vstoxx Volatility Index, is higher, as expected, during the crisis period compared to the pre-crisis period, while the coefﬁcient in the pre-crisis period is quite close to the value for the entire period;

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• the trend of European government bonds index, Emu Benchmark 5YR, has a positive effect on the spreads in a more pronounced way during the crisis and in the complete sample, compared to that observed for the pre-crisis period;

• the positive coefﬁcient of one-day lag-returns of the spread of the index, Lag iTraxx Europe, conﬁrms the presence of positive ﬁrst order autocorrelation in the spreads, which appear to be more sensitive to this phenomenon in the pre-crisis period, rather than during the crisis and during the entire sample. Pre-crisis Model Estimate Std. Error t value Pr(> |t|) (Intercept) -0.0003 0.0006 -0.47 0.6400 Stoxx Europe 600 -0.4581 0.1318 -3.47 0.0006 Vstoxx Volatility Index 0.0651 0.0191 3.41 0.0007 Lag iTraxx Europe 0.2448 0.0280 8.73 0.0000 Emu Benchmark 5YR 1.1788 0.4058 2.91 0.0000 Crisis Model Estimate Std. Error t value Pr(> |t|) (Intercept) -0.0007 0.0009 -0.79 0.4310 Stoxx Europe 600 -0.9934 0.0952 -10.44 0.0000 Vstoxx Volatility Index 0.1193 0.0236 5.06 0.0000 Lag iTraxx Europe 0.1257 0.0196 6.40 0.0000 Emu Benchmark 5YR 3.2851 0.4576 7.18 0.0000 Table 5.5: Comparison between pre-crisis and crisis model After having veriﬁed that the residuals of the estimated model for the evolution of the returns of the spreads of the CDSs index show ﬁrst order autocorrelation, AR(1), and having found that the conditional variance of the estimates do follow a GARCH(1,1) process, we estimate the AR(1)-GARCH(1,1) models for the entire period. The estimate of the latter, having been implemented using the usual normal distribution for the residuals of the regression, is tested with some test-statistcs, which we will discuss later, for the presence of a different distribution for the residuals. For these reasons, it is initially considered a standard Student’s-t distribution, then we consider a more suitable asymmetric distribution, the Skewed Student’s-t proposed by Hansen (1994), whose higher performance was

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From Table (5.8) can be observed that the coefﬁcients of the ﬁrst order autocorrelation of the conditional mean of the spreads returns, ar1, of ﬁrst-order ARCH effect, α1, and of the ﬁrst-order GARCH effect, β1, of the conditional variance are all signiﬁcant at the 1% level and positive. As is usual for ﬁnancial time series, we observe that the sum of the estimated parameters for the ARCH and GARCH effects, equals to 0.9989, is close to the unit and we also have that the parameter of the GARCH effect, that is the inﬂuence of lagged conditional variance on the conditional variance, is much higher than the ARCH effect parameter, namely the effect of lagged squared residuals, this implies the presence of the phenomenon of persistence in the volatility of the CDS index spreads returns, i.e. there is presence of the so-called clusters of volatility, for which certain levels of volatility persistently affect its levels in subsequent periods. All the three periods models AR(1)-GARCH(1,1) estimated for the residuals of the model for the index CDSs spreads are well speciﬁed, as both the Ljung-Box tests for the presence of additional autocorrelation in the conditional mean and in the conditional variance, and the test ARCH-LM for the presence of additional heteroscedasticity, are highly signiﬁcant at a level of 5%, as shown in Tables (5.6), (5.7), (5.8). Concerning the more appropriate distribution to be used for the regression AR(1)-GARCH(1,1), as can be seen from the results of the highly signiﬁcant level both for the estimates of the degrees of freedom, ν (shape), which with a value of 6.03 reveals the presence of large tails in the empirical distribution, and for the estimate in the case of Skewed Students’s-t distribution of the skewness value λ (skew), which with a value of 1.05 shows a positive asymmetric tendency of the residuals distribution. For this reason we state that between the use of either a Normal or a standard Student’s-t or a asymmetric Student’s-t distribution, the use of the latter is the more appropriate for the estimation.

The above statement is conﬁrmed by the other tests carried out. From the tables (5.6), (5.7), (5.8), we can observe that the test diagnostic Pearson goodness-of-ﬁt (GOF) to ver-

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ify that the theoretical distribution used is consistent with the empirical distribution of standardized residuals, is signiﬁcant only in the case of Skewed Student’s-t. Furthermore, the values of the Akiko Information Criteria (AIC) and of the Lof-Likelihood (LogLik) to test the goodness of the ﬁt of the model, conﬁrm that the model with Skewed Student’s-t is the best, while the model with the Normal is the worst in terms of performance.

Finally, we perform some tests on the residuals of the AR(1)-GARCH(1,1) model in order to test the adequacy of the choice of the theoretical distribution. In the table (5.9) are reported the results of the Jarque-Bera test to verify the normality of the distribution of the residuals of the model, in which the theoretical distribution used is the Normal, and the corresponding Kolmogorov- Smirnov (KS) test to verify both the use of Standard Student’s-t and the use of the Skewed Student’s-t. From the results of the tests is conﬁrmed, once again, that the use of the Skewed Student’s-t is the more appropriate for the construction of a model AR(1)-GARCH(1,1), which is able to capture the cluster in the volatility and the autocorrelation in the residuals of the model for the trend of the spread.

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5.2

Factorial Model for the Intensity of Default
After having analyzed the factors determining the behavior of

Model Presentation

iTraxx Europe index spreads, we build up a model at the univariate level for the intensity of default of certain ﬁnancial companies, which are part of the CDSs reference portfolio of the index, to which are added two Italian banking ﬁrms, Mediobanca and Banca Popolare di Milano. Those intensities are considered to be determined by both time-varying macroeconomic factors, which are in accordance with what is found in the previous chapter (5.1), and by a speciﬁc factor of the ﬁrm’s risk which varies over time.

The model considered for the intensity of default is a model with an exponential acceleration factor, which is determined by macroeconomic factors common to all ﬁrms and by a speciﬁc factor for each ﬁrm. The acceleration factor determines the positive or negative shocks to the normal trend (or base value), where the latter is the average trend observed for the intensity. The use of an exponential form for the acceleration factor can prevent the intensity from taking negative values, and this assumption for the construction of estimation models for the intensity of default is also very common among the academics, in fact this functional form was used, among the others, by Azizpour et al. (2010). Therefore, we have that the intensity λi for the i-th ﬁrm at time t is t λi = λi Υt t Υt = exp αi + γi Sti + θi Ct + εi (5.2)

where Υt corresponds to the acceleration factor that determines the deviations at time t of the intensity of default with respect to its base value of λi . In addition, this acceleration factor is assumed to be an exponential function, depending on: • the speciﬁc factor for the ﬁrm i, Sti • the macroeconomic factor that affects all the ﬁrms, Ct , which is constant in time,

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• the values of the coefﬁcients αi , γi and θi , estimated for each ﬁrm at the initial evaluation time t. Thus we assume that the acceleration factor varies with changing conditions of the environment in which the companies operate, and that the inﬂuence that those changes have on the value of the acceleration factor remain constant over time, since the coefﬁcients in the model are assumed to be constant, this implies that the acceleration factor maintains constant its exponential functional form. For these reasons, we can consider Υt as a random variable, whose distribution depends on the distribution of the factors Sti , Ct . What we are interested in building a model for the default intensity concerns the estimation of a probability distribution for the default of a company or of a portfolio of companies, for which we may want to evaluate a risky bond or a credit risk derivative. Therefore we want to try to forecast what could be the future default probability over a given time horizon. Considering the reduced form models presented in chapter (2), we assume an exponential form for the probability of default that has ”no memory”, which means that it should not depend on the time passed without the company is bankrupt, that is independent of the ”operational seniority” of the ﬁrm. From equation (2.6) for the probability of survival, we obtain the probability of default before time τ to be equal to

P (τ ≤ t) = 1 − P (τ > t) = = 1 − e− t 0

λ(s)ds

(5.3) t 0

Assuming that there has been no default up to t and that Γ(t) =

λ(s)ds, we obtain that

the conditional probability of default in the interval [t, t + ∆t] is5 P (τ ∈ [t, t + ∆t]) P (τ ≥ t) t+∆t P (τ ∈ [t, t + ∆t]|τ ≥ t) =

= 1 − exp(− t 5

λ(s)ds)

(5.4)

We remind that in (2.7) we found the conditional default probability relative to an inﬁnitesimal interval, by the use of the formula for ﬁrst order approximation of Taylor.

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This exponential form for the probability of default is with ”no memory” if and only if it does not depend on time t, i.e. if and only if Γ(t) does not depend on the time passed over the interval, hence considering that the intensity remains constant in the interval ∆t considered. Therefore, we have that the intensity will have a particular form piecewise constant for each time horizon [t, T ], for which the estimated probability of default within a speciﬁed maturity T , will change depending on the value assumed by the intensity at time t, which in turn, after having set a base value, λ for the level of the intensity during ”normal” conditions of the economy, will depend on the value assumed in t by the acceleration factor of the state economy Υt deﬁned above.

As a ﬁnal result we obtain a model in which once deﬁned the probability distribution of the acceleration factor and its determinants (macroeconomic factors and ﬁrm-speciﬁc factors), we obtain the probability distribution for the future value of the intensity, which gives us the probability of default for a speciﬁed time horizon, assuming that during this period the intensity remains constant. This procedure ensures that at any point in time where we know the value of the macroeconomic factors and ﬁrm-speciﬁc factors, we are able to determine either an hypothetical probability of default present at that moment, either its expected value for the next instant, obtained from the forecasted probability distribution of future values for the intensity.

Model Estimation The data used to estimate the model have a daily basis frequence and cover the period from 2007-12-176 until 2010-11-11, for a total of 759 daily observations. As macroeconomic variable we consider the index of European government bonds with maturity 5 years, Emu Benchmark 5YR, Ct , already used in chapter (5.1), while has been excluded with respect to the model of the trend of iTraxx Eurpoe index spreads, the trend
We have taken out of the sample the ﬁrst observation of 2007-12-14, as this was used to estimate the GARCH(1,1) one-step-ahead volatility of the return of the ﬁrms’ stocks.
6

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of the Stoxx Europe 600 index for stock prices, which in this case resulted not signiﬁcant at both univariate and multi-level, the index of the European equity market volatility, Vstoxx, since in some cases it was not performing well as an independent variable of the model. As a speciﬁc factor, Sti , we use the index of the riskiness of the ﬁrm-speciﬁc returns, represented by the implied annual volatility, which we estimate with a GARCH(1,1) applied to the residuals of the Capital Asset Pricing Model (CAPM)7 for the expected excess return for the company in relation to the risk-free return. The model for the CAPM is estimated using the following linear regression for each ﬁrm i

R i − R f = a + β i R m − R f + ui

(5.5)

where • Ri is the daily return for the ﬁrm i, equal to the daily logarithmic changes in the prices quoted • Rf is the daily logarithmic change for the risk-free rate, for which we used the risk-free rate EON IA • Rm is the daily return of the reference stock index for the ﬁrm i • ui are the residuals of the regression for the ﬁrm i As can be seen from equation (5.5), the residuals of the CAPM can be considered as an adequate proxy for ﬁrm-speciﬁc return, as they capture the share of performance of the excess of return not captured by the model, for instance by the Market Beta and the beta Market Premium. After having obtained the residuals we checked that their variance presented autocorrelation and heteroscedasticity, that is statistically signiﬁcant parameters for both the ARCH and GARCH factor of the estimated GARCH (1,1). Then with this model we compoute the one-step-ahead forecasts for the conditional variance of
7

See Fama and MacBeth (1973) for a presentation of the model.

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the residuals of the CAPM, from which we obtain the annual implied volatility for each ﬁrm. The model presented in equation (5.2) can then be rewritten as follows λi t λi

ln

= αi + γi Sti + θi Ct + εi

(5.6) λi t , λi

The dependent variable of the model appears to be the logarithm of the ratio

that

can be calculated from the time series of the prices of CDSs spreads for each ﬁrm i. This statement comes from the fact that, as in Walker (2005), we are in the context of the simpliﬁed case of piecewise constant hazard rate, which is approximated at the ﬁrst order of the Taylor expansion formula for a very small values of the spreads. The latter approximation is admissible since the spreads are usually quoted in bp, hence very close to zero. Then without considering the counterparty risk we get that the spread at the time t for a time to maturity T , can be expressed as a function of λt , through the following equation st = (1 − ρ)(λt T ) (5.7)

where ρ is the recovery rate. From this relation, assuming a constant recovery rate for each company i in the observed time period, then the dependent variable of our model can be estimated from time series of spreads with time to maturity of ﬁve years, since the following relation is valid for the initial intensity at time t = 0 λi 0 = λi ⇓ λi si 0 = 0 λi si (5.8) si 0 (1−ρ)T si (1−ρ)T

where si is the value of the spread for the ﬁrm i, stripped from quotes at the time t = 0 0 of the CDSs, while si is the average spread for the ﬁrm i in the sample period. The CDSs considered have a 5 years maturity, coherently with the maturity of the spread for the index

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of CDSs analysed in the chapter (5.1). In the table (5.10) are presented in alphabetical order the ﬁrms of the ﬁnancial sector, with the estimates of the parameter γi for the ﬁrmspeciﬁc factor, the parameter θi for the macroeconomic factor, with the respective standard error and t-statistic, and also the R2 -adjusted as an indicator of the level of explanation of the dependent variable captured by the estimated model. From the estimation results we may observe that all the parameters of the independent variables, which determine the course of the acceleration factor, are positive. With regard to macroeconomic factors, this result is consistent with what is found in section (5.1) for the multivariate case, while for the ﬁrm-speciﬁc factor is logical to expect that an increase in the volatility, i.e. the risk of a company, involves an acceleration of the intensity of default, thus an higher probability of default. Almost all the parameters are signiﬁcant at a level of 0.01%, with the exception of the parameter of the speciﬁc factor γi for Credit Agricole, which it is still signiﬁcant at a satisfactory level of 5%, generally accepted as discriminant signiﬁcance level in the inference of linear regression models. We also note that the values of R2 -adjusted, are on average high, with an average value for it of 0.408 and a maximum value for Aegon of 0.676, while we also ﬁnd for Commerzbank and Hannover Ruck, respectively, 0157 and 0.162, which are fairly low level compared to the rest of the sample, but even not too much low for this type of model. After having performed the estimation for the model of the acceleration factor of the intensity, we want to forecast the distribution probability of default for a speciﬁed time to maturity or to default. For example, suppose we want to estimate the probability of default within a year valid at a future date. Assuming that the intensity does not change until that date, then the future probability is estimated using a simulation of the possible values that the acceleration factor could take. The acceleration factor’s distribution in turn depend on the multivariate distribution of the factors Sti and Ct , for which usually no one knows neither the univariate nor the joint distribution. To get their univariate distributions, we have made tests using some theoretical distributions to ﬁt the empirical univariate distribution, for which no satisfactory results have been obtained, we therefore

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decided to use the empirical distributions from the hystoric series. From the other hand to estimate the joint distribution, we chose to ﬁt a theoretical copula to their univariate empirical distributions. The copula that shows the best ﬁt among those considered8 is the Gumbel copula with estimated parameter s = 1.8413. After the estimation of the copula and having simulated with this a series of 1000 realizations, then from this series by applying the univariate empirical quantiles of the 199 variables of the model, we obtain a series of realizations for the acceleration factor. Then we derive the realizations of the intensity with which we calculate as many values for the probability of default within one year for each company i. We are therefore able to forecast the probability distribution of a future default, for which the graphs for each company sorted in alphabetical order, as shown in the Table (5.10), can be found in Figure (5.5) and are comparable with the historical distribution plots shown in Figure (5.4). In Table (5.11) we show for each ﬁrm, the average value of the probability distribution of the default, with their standard error and, for comparison, the value of the current default probabilities, calculated with the data availble up to 2010 − 11 − 11.

Finally, we show in Figure (5.6) the graph of the joint distribution for three of the major players in the insurance industry, Assicurazioni Generali, Allianz, Swiss Re. This distribution can be compared with the joint distribution of the estimated probability of future default, whose graph is shown in Figure (5.7), in which may note common trend for the default probabilities of the ﬁrms, which also unveils high likelihood for low values of the marginal probability of default of individual ﬁrms. Furthermore, by comparing these graphs with the graph (5.8) of a simulation of the threevariate ﬁtted theoretical copula, it can be conﬁrmed that we have correctly specify the copula to represent the trend of the multivariate probability of default. We also note that the joint probabilities of default tend
We have used the following copulas: from the family of the Archimedean copulas the Gumbel, the Clayton and the Frank, while among the elliptical copulas we have considered the Gaussian and the Student’s-t. 9 We consider the 18 speciﬁc factors, and the only macroeconomic common factor.
8

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to dispose along a diagonal with positive slope with respect to the univariate probabilities, even if compared to the Gumbel the joint probabilities tend to take more frequently low values. The latter observations represent the possibility of a contagion among ﬁrms, also for low levels of default risk. In Figures (5.9), (5.10) and (5.11) are reported the graphs of the bivariate joint densities, estimated through the copula. We observe a trend compatible with the assumption of the Gumbel copula to model the dependence of the variables that determine the acceleration factor and then the joint probabilities of the default of the ﬁrms. Finally we, once again, observe that the joint probability density, represented by the coloured surface, assumes high values also for low values of the univariate default probabilities of ﬁrms, still conﬁrming the so far addressed contagion of the risk of default among ﬁrms.

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Figure 5.1: Graphs for the variables’ trends

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Figure 5.2: Graphs of the ﬁtting distributions

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Figure 5.3: Graphs of the relations with dependent variable

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Conditional Dist. mu ar1 omega alpha1 beta1 LogLik AIC Q-Statistics SR Lag10 Lag15 Lag20 Q-Statistics SSR Lag10 Lag15 Lag20 ARCH LM Tests ARCH Lag[2] ARCH Lag[5] ARCH Lag[10] GOF Test: 1 2 3 4

Normal Estimate Std. Error -0.001443 0.000677 0.189487 0.032946 0.000023 0.000012 0.189852 0.037856 0.807865 0.034321 3110.309 -4.2192 statistic 8.218 17.694 21.138 statistic 9.317 9.917 13.627 Statistic 6.599 6.857 9.350 group 20 30 40 50 p-value 0.6075 0.2791 0.3890 p-value 0.5023 0.8250 0.8489 DoF 2 5 10

t value -2.1330 5.7515 2.0075 5.0151 23.5383

Pr(> |t|) 0.032926 0.000000 0.044692 0.000001 0.000000

P-Value 0.03689 0.23150 0.49928

statistic p-value(g-1 ) 83.03 5.562e-10 82.17 5.548e-07 91.37 4.346e-06 104.43 6.923e-06

Table 5.6: Normal AR(1)-GARCH(1,1)

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Conditional Dist. t-Studend standard Estimate Std. Error t value mu -0.001639 0.000538 -3.0474 ar1 0.188104 0.029967 6.2770 omega 0.000009 0.000004 2.1625 alpha1 0.186755 0.031528 5.9234 beta1 0.812245 0.034538 23.5172 shape 6.028991 0.942216 6.3987 LogLik 3188.953 AIC -4.3247 Q-Statistics SR statistic p-value Lag10 7.246 0.7021 Lag15 16.120 0.3741 Lag20 19.731 0.4748 Q-Statistics SSR statistic p-value Lag10 6.098 0.8070 Lag15 6.700 0.9654 Lag20 8.755 0.9856 ARCH LM Tests Statistic DoF P-Value ARCH Lag[2] 3.481 2 0.1754 ARCH Lag[5] 4.055 5 0.5415 ARCH Lag[10] 6.011 10 0.8144 GOF Test: group statistic p-value(g-1 ) 1 20 52.84 4.954e-05 2 30 49.07 1.134e-02 3 40 64.58 6.158e-03 4 50 72.02 1.777e-02 Table 5.7: Standard Student’s t AR(1)-GARCH(1,1)

Pr(> |t|) 0.002309 0.000000 0.030578 0.000000 0.000000 0.000000

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Conditional Dist. t-Studend skewed Estimate Std. Error t value mu -0.001381 0.000590 -2.3423 ar1 0.186734 0.029377 6.3564 omega 0.000009 0.000004 2.1370 alpha1 0.184933 0.031926 5.7925 beta1 0.814067 0.034817 23.3815 skew 1.046347 0.034503 30.3266 shape 6.030570 0.959776 6.2833 LogLik 3189.885 AIC -4.3246 Q-Statistics SR statistic p-value Lag10 7.249 0.7017 Lag15 16.265 0.3647 Lag20 19.864 0.4664 Q-Statistics SSR statistic p-value Lag10 6.172 0.8006 Lag15 6.760 0.9640 Lag20 8.858 0.9845 ARCH LM Tests Statistic DoF P-Value ARCH Lag[2] 3.600 2 0.1653 ARCH Lag[5] 4.152 5 0.5278 ARCH Lag[10] 6.082 10 0.8083 GOF Test: group statistic p-value(g-1 ) 1 20 40.47 0.002835 2 30 40.76 0.072283 3 40 51.86 0.081544 4 50 64.96 0.063030 Table 5.8: Skewed Student’s-t AR(1)-GARCH(1,1)

Pr(> |t|) 0.019165 0.000000 0.032601 0.000000 0.000000 0.000000 0.000000

Normal Standard Student’s-t Skewed Student’s-t method Jarque-Bera Normality Test Kolmogorov-Smirnov Test Kolmogorov-Smirnov Test statistic 8762.89 17.13 17.48 p.value 0.00 0.00 0.45 Table 5.9: Test for the consistency of the theoretical distribution

CHAPTER 5. EMPIRICAL IMPLEMENTATION
Firm Aegon N.V. Assicurazioni Generali S.p.a. Allianz SE Axa Banca Monte dei Paschi di Siena Banco Popolare di Milano Banco Santander Barclays Bank PLC BNP Paribas Commerzbank Credit Agricole Credit Suisse Group Ltd Deutsche Bank Hannover Ruck AG Mediobanca Societe Generale Swiss Reinsurance Comapny Ltd Zurich Financials γi 0.0135 0.0195 0.0144 0.0162 0.0102 0.0087 0.0118 0.0048 0.0027 0.0011 0.0013 0.0107 0.0067 0.0059 0.0207 0.0037 0.0162 0.0154 Std.Error 0.0003 0.0015 0.0006 0.0006 0.0017 0.0012 0.0012 0.0003 0.0005 0.0003 0.0007 0.0005 0.0004 0.0008 0.0013 0.0004 0.0006 0.0007 = 0.5 t-stat 38.84 12.58 25.82 25.33 6.12 7.03 9.50 18.57 5.70 3.751 1.918 22.89 15.08 7.61 15.77 8.25 19.69 22.85 = 0.01 θi 0.0331 0.0462 0.0173 0.0169 0.0606 0.0488 0.0560 0.02509 0.0456 0.0225 0.0373 0.0263 0.0322 0.0265 0.0515 0.0383 0.0283 0.0096 = 0.001 Std.Error 0.0021 0.0021 0.0016 0.0021 0.0022 0.0024 0.0018 0.0022 0.0018 0.0019 0.0019 0.0021 0.0017 0.0024 0.0024 0.0018 0.0035 0.0016 t-stat 16.07 22.10 10.54 7.88 28.03 20.62 30.51 11.46 25.19 11.90 19.69 12.31 19.03 11.028 21.86 21.55 8.19 5.88 R2 adjusted 0.676 0.426 0.475 0.461 0.508 0.361 0.556 0.358 0.460 0.157 0.349 0.430 0.413 0.162 0.462 0.391 0.360 0.407

108

Signiﬁcance Level

Table 5.10: Estimates for the intensity of default’s model Firm Average probability Standard Error Present probability Aegon N.V. 0.02101 0.01353 0.01577 Assicurazioni Generali s.p.a. 0.00903 0.00300 0.01127 Allianz SE 0.00795 0.00253 0.00741 Axa 0.01174 0.00454 0.01007 Banca Monte dei Paschi di Siena 0.00901 0.00280 0.01145 Banco Popolare di Milano 0.00923 0.00277 0.01164 Banco Santander 0.01114 0.00370 0.01499 Barclays Bank PLC 0.01185 0.00440 0.01172 BNP Paribas 0.00724 0.00183 0.00910 Commerzbank 0.00904 0.00119 0.01027 Credit Agricole 0.00958 0.00180 0.01188 Credit Suisse Group Ltd 0.01061 0.00409 0.01020 Deutsche Bank 0.01045 0.00274 0.01147 Hannover Ruck AG 0.00719 0.00144 0.00795 Mediobanca 0.00957 0.00412 0.01207 Societe Generale 0.00961 0.00227 0.01157 Swiss Reinsurance Compant Ltd 0.01656 0.01278 0.01535 Zurich Financials 0.00978 0.00235 0.00852 Table 5.11: Estimated default probability

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Figure 5.4: Empirical distribution of the default probability

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Figure 5.5: Distribution of forecasted future default probability

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Figure 5.6: Joint historical probability distribution of default

Figure 5.7: Joint forecasted probability distribution of default

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Figure 5.8: Simulated Gumbel Copula with estimated parameter s = 1.8413

Figure 5.9: Joint Density for Generali-Allianz

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Figure 5.10: Joint Density for Generali-Swiss RE

Figure 5.11: Joint Density for Allianz-Swiss RE

Chapter 6 Conclusion
In the thesis we discussed the use of models for estimating the risk of default, with a focus on the spread, which expresses the implied probability of default for a company. The quoted spread is, therefore, the market assessment for the probability of default of a risky asset issued by a company, thus the risky asset requires a certain level of return higher than the risk-free return. The aim of the thesis is to identify some methods for building up a model which could be imply to estimate the probability of default for single entity and multiple entities, based on statistical techniques widely accepted in literature, which include the use of a default intensity model for a single ﬁrm and copulas to consider a particular dependence structure of defaults by several ﬁrms. The approach eventually applied in the last part of dissertation is considered by some recent studies on the credit risk subject to be the most appropriate, even for its ease of implementation. In the ﬁrst part of the dissertation we have discussed the theoretical framework of default models for the valutation of single default risk and single ﬁrm spread, explaining the reasons for the choice made for the estimation model and proposing possible alternatives, for which we have analyzed the strengths and drawbacks that arise in their implementation. Moreover we have analyzed some approaches to model the dependence between the defaults of several ﬁrms. In fact the joint default risk is reported to be widespread interest and fundamental study, especially in light of the events over the last years, when a single ﬁrm default, namely the

114

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115

increased risk of this, triggers a default contagion effect over several ﬁrms, leading to the phenomenon of default clustering. In recent years the use of copulas to model dependence is also stated in the credit risk models, because of the ﬂexibility of the copula, which is explained by the consideration that we can build models for multiple default risk in two separated steps: ﬁrst building single ﬁrm models; secondly introduce a certain structure of dependence through the copula. Thereafter we can obtain a multivariate model for the credit risk of a portfolio of ﬁrms, which is consistent both with the estimation of individual default risks and with the the presence of contagion between different levels of default risk. In the last part of the dissertation we have built a model for estimating the intensity of default, which in my opinion is the most appropriate amongst the multiple entities default’s models. The model is based on a factor model with an acceleration factor, the intensity of default, which is consistent with the market sentiment originated from the credit risk derivatives quotations. For this purpose We have used two different types of credit risk derivatives: the single ﬁrm CDSs of some European companies in the ﬁnancial sector as a proxy for the single ﬁrm default; and the index of CDSs, the iTraxx Europe from whose spread we bootstrap the implicit multiple default risk. In particular We have tried to detect the presence of some trend in critical periods, selected with ﬁnancial and economic criterion. We have implied the model to estimate the probability of default of a group of European banks and insurance companies. First we have estimated and forecast the single entities default of the ﬁrms, then in a second moment we have introduced an estimated and calibrated copula to model the joint probability of default. The result of my empirical application then veriﬁed the presence of contagion between the probability of default, resulting in a trend of signiﬁcant joint default risk for several ﬁrms. The model outcomes also conﬁrm what I had found also for the iTraxx index analysis, previously performed in the dissertation. Finally, from the arguments treated in the dissertation and the ﬁnal empirical application, we can state that the cluster of default brings about persistent trends of spread changes in some critical periods, when a cluster is due to the variation of the joint probability of default of a group or a index of reference entities, furthermore the

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multiple default probability is also linked to the single entity default probabilities, which tend to jointly change because of contagion effects which last over some time, depending on the ﬁnancial and economic enviroment condition.

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