8 Credit Risk Management
Credit risk is the risk that the value of a portfolio changes due to unexpected changes in the credit quality of issuers or trading partners. This subsumes both losses due to defaults and losses caused by changes in credit quality, such as the downgrading of a counterparty in an internal or external rating system. Credit risk is omnipresent in the portfolio of a typical financial institution. To begin with, the lending and cor- porate bond portfolios are obviously affected by credit risk. Perhaps less obviously, credit risk accompanies any OTC (over-the-counter, i.e. non-exchange-guaranteed) derivative transaction such as a swap, because the default of one of the parties involved may substantially affect the actual pay-off of the transaction. Moreover, in recent years a specialized market for credit derivatives has emerged in which financial institutions are active players (see Section 9.1 for details).
This brief list should convince the reader that credit risk is a highly relevant risk category indeed, as it relates to the core activities of most banks. Credit risk is also at the heart of many recent developments on the regulatory side, such as the new I Capital Accord discussed in Chapter 1. We devote two chapters to this important risk category. In the present chapter we focus on static models and credit risk management; dynamic models and credit derivatives are discussed in Chapter 9.
8.1 Introduction to Credit Risk Modelling
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In this section we provide a brief overview of the various model types that are used in credit risk before discussing some of the main challenges that are encountered in credit risk management.
8.1.1 Credit Risk Models
The development of the market for credit derivatives and the I process has generated a lot of interest in quantitative credit risk models in industry, academia and among regulators, so that credit risk modelling is at present a very active subfield of quantitative finance and risk management. In this context it is interesting that parts of the new minimum capital requirements for credit risk are closely linked to the structure of existing credit portfolio models, as will be explained in more detail in Section 8.4.5.
There are two main areas of application for quantitative credit risk models: credit risk management and the analysis of credit-risky securities. Credit risk management models are used to determine the loss distribution of a loan or bond portfolio over
328 8. CreditRiskManagement
a fixed time period (typically at least one year), and to compute loss-distribution- based risk measures or to make risk-capital allocations of the kind discussed in Section 6.3. Hence these models are typically static, meaning that the focus is on the loss distribution for the fixed time period rather than a stochastic process describing the evolution of risk in time.
For the analysis of credit-risky securities, on the other hand, dynamic models (generally in continuous time) are needed, because the pay-off of most products depends on the exact timing of default. Moreover, in building a pricing model one often works directly under an equivalent martingale or risk-neutral probability mea- sure (as opposed to the real-world probability measure). Issues related to dynamic credit risk models and risk-neutral and real-world measures will be studied in detail in Chapter 9.
Depending on their formulation, credit risk models can be divided into structural or firm-value models on the one hand and reduced-form models on the other; this division cuts across that of dynamic and static models. The progenitor of all firm- value models is the model of Merton (1974), which postulates a mechanism for the default of a firm in terms of the relationship between its assets and the liabilities that it faces at the end of a given time period. More generally, in firm-value models default occurs whenever a stochastic variable (or in dynamic models a stochastic process) generally representing an asset value falls below a threshold representing liabilities. For this reason static structural models are referred to in this book as threshold models, particularly when applied at portfolio level. The general structural model approach is discussed in Section 8.2 (where the emphasis is on modelling the default of a single firm). In Section 8.3 we look at threshold models for portfolios; in particular we show that copulas play an important role in understanding the multivariate nature of these models.
In reduced-form models the precise mechanism leading to default is left unspec- ified. The default time of a firm is modelled as a non-negative rv, whose distribution typically depends on economic covariables. The mixture models that we treat in Section 8.4 can be thought of as static portfolio versions of reduced-form models. More specifically, a mixture model assumes conditional independence of defaults given common underlying stochastic factors.
It is important to realize that mixture models are not a new class of models; on the contrary, the most useful static threshold models all have mixture model representations, as will be shown in Section 8.4.4. In continuous time a similar mapping between firm-value and reduced-form models is also possible if one makes the realistic assumption that assets and/or liabilities are not perfectly observable (see Notes and Comments).
From a practical point of view, mixture models represent perhaps the most use- ful way of analysing and comparing one-period portfolio credit risk models. For these models, Monte Carlo techniques from the area of importance sampling can be used to approximate risk measures for the portfolio loss distribution, and to cal- culate associated capital allocations, as will be shown in Section 8.5. Moreover, it is possible to devise efficient methods of statistical inference for portfolio models
8.1. IntroductiontoCreditRiskModelling 329
using historical default data. These models exploit the connection between mixture models and the well-known class of generalized linear mixed models in statistics; this is the topic of Section 8.6.
8.1.2 The Nature of the Challenge
Credit risk management poses certain specific challenges for quantitative modelling, which are less relevant in the context of market risk.
Lack of public information and data. Publicly available information regarding the credit quality of corporations is typically scarce. This creates problems for corporate lending, as the management of a firm is usually better informed about the true economic prospects of the firm and hence about default risk than are prospective lenders. The implications of this informational asymmetry are widely discussed in the microeconomics literature (see Notes and Comments). The lack of publicly available credit data is also a substantial obstacle to the use of statistical methods in credit risk, a problem that is compounded by the fact that in credit risk the risk- management horizon is usually at least one year. It is fair to say that data problems are the main obstacle to the reliable calibration of credit models.
Skewed loss distributions. Typical credit loss distributions are strongly skewed with a relatively heavy upper tail. Over the years a typical credit portfolio will produce frequent small profits accompanied by occasional large losses. A fairly large amount of risk capital is therefore required to sustain such a portfolio: the economic capital required for a loan portfolio (the risk capital deemed necessary by shareholders and the board of directors of a financial institution, independent of the regulatory environment) is often equated to the 99.97% quantile of the loss distribution (see Section 1.4.3).
The role of dependence modelling. A major cause for concern in managing the credit risk in a given loan or bond portfolio is the occurrence in a particular time period of disproportionately many defaults of different counterparties. This risk is directly linked to the dependence structure of the default events. In fact, default dependence has a crucial impact on the upper tail of a credit loss distribution for a large portfolio. This is illustrated in Figure 8.1, where we compare the loss distri- bution for a portfolio of 1000 firms that default independently (portfolio 1) with a more realistic portfolio of the same size where defaults are dependent (portfolio 2). In portfolio 2 defaults are weakly dependent, in the sense that the correlation between default events (see Section 8.3.1) is approximately 0.5%. In both cases the default probability is approximately 1% so that on average we expect 10 defaults. As will be seen in Section 8.6, portfolio 2 can be viewed as a realistic model for the loss distribution generated by a homogeneous portfolio of 1000 loans with a Standard & Poor’s rating of BB. We clearly see from Figure 8.1 that the loss distribution of portfolio 2 is skewed and that its right tail is substantially heavier than the right tail of the loss distribution of portfolio 1, illustrating the drastic impact of default dependence on credit loss distributions. Typically, more dependence is reflected in the loss distribution by a shift of the mode to the left and a longer right tail. For this
8. CreditRiskManagement
Dependent defaults Independent defaults
0.12 0.10 0.08 0.06 0.04 0.02
0 10 20 30 40 50 60
Number of losses
Figure 8.1. Comparison of the loss distribution of two homogeneous portfolios of 1000 loans with a default probability of 1% and different dependence structure. In port- folio 1 defaults are assumed to be independent; in portfolio 2 we assume a default correlation of 0.5%. Portfolio 2 can be considered as representative for BB-rated loans. We clearly see that the default dependence generates a loss distribution with a heavier right tail.
reason we devote a large part of our exposition to the analysis of credit portfolio models and dependent defaults.
There are sound economic reasons for expecting default dependence. To begin with, the financial health of a firm varies with randomly fluctuating macroeconomic factors, such as changes in economic growth. Since different firms are affected by common macroeconomic factors, we have dependence between their defaults. Moreover, default dependence is caused by direct economic links between firms, such as a strong borrower–lender relationship. Given the enormous size of typical loan portfolios it can be argued that, in credit risk management, direct business relations play a less prominent role in explaining default dependence. Dependence due to common factors, on the other hand, is of crucial importance and will be a recurring theme in our analysis. In the pricing of portfolio credit derivatives, the portfolios of interest are smaller, so modelling direct business relationships becomes more relevant (see Section 9.8 for models of this kind).
Notes and Comments
Chapter 2 of Duffie and Singleton (2003) contains a good discussion of the economic principles of credit risk management, elaborating on some of the issues discussed above. For a microeconomic analysis of the functioning of credit markets in the presence of informational asymmetries between borrowers and lenders we refer to the seminal paper by Stiglitz and Weiss (1981).
Duffie and Lando (2001) established a relationship between firm-value models and reduced-form models in continuous time. Essentially, they showed that, from the perspective of investors with incomplete accounting information (i.e. incomplete information about assets or liabilities of a firm), a firm-value model becomes a reduced-form model. A less technical discussion of these issues can be found in Jarrow and Protter (2004).
Probability
8.2. StructuralModelsofDefault 331
The available empirical evidence for the existence of macroeconomic common factors is surveyed in Section 3.1 of Duffie and Singleton (2003). Without going into details, it seems that a substantial amount of the variation over time in empirical default rates (the proportion of firms with a given credit rating that actually defaulted in a given year) can be explained by fluctuations in GDP growth rates, with empirical default rates going up in recessions and down in periods of economic recovery.
8.2 Structural Models of Default
A model of default is known as a structural or firm-value model when it attempts to explain the mechanism by which default takes place. Because the kind of thinking embodied in these models has been so influential in the development of the study of credit risk and the emergence of industry solutions (like the KMV model discussed in Section 8.2.3), we consider this to be the best starting point for a treatment of credit risk models.
From now on we denote a generic stochastic process in continuous time by (Xt ); the value of the process at time t 0 is given by the rv Xt.
8.2.1 The Merton Model
The model proposed in Merton (1974) is the prototype of all firm-value models. Many extensions of this model have been developed over the years, but Merton’s original model remains an influential benchmark and is still popular with practition- ers in credit risk analysis.
Consider a firm whose asset value follows some stochastic process (Vt ). The firm finances itself by equity (i.e. by issuing shares) and by debt. In Merton’s model debt has a very simple structure: it consists of one single debt obligation or zero- coupon bond with face value B and maturity T . The value at time t of equity and debt is denoted by St and Bt and, if we assume that markets are frictionless (no taxes or transaction costs), the value of the firm’s assets is simply the sum of these, i.e.Vt = St +Bt, 0 t T.IntheMertonmodelitisassumedthatthefirmcannot pay out dividends or issue new debt. Default occurs if the firm misses a payment to its debt holders, which in the Merton model can occur only at the maturity T of the bond. At maturity we have to distinguish between two cases.
(i) VT > B: the value of the firm’s assets exceeds the liabilities. In that case the debtholders receive B, the shareholders receive the residual value ST = VT − B, and there is no default.
(ii) VT B: the value of the firm’s assets is less than its liabilities and the firm cannot meet its financial obligations. In that case shareholders have no interest in providing new equity capital, which would go immediately to the bond- holders. Instead they “exercise their limited-liability option” and hand over control of the firm to the bondholders, who liquidate the firm and distribute the proceeds among themselves. Shareholders pay and receive nothing, so thatwehaveBT =VT,ST =0.
332 8. CreditRiskManagement Summarizing, we have the relations
ST =max(VT −B,0)=(VT −B)+, (8.1) BT =min(VT,B)=B−(B−VT)+. (8.2)
Equation (8.1) implies that the value of the firm’s equity at time T equals the pay- off of a European call option on VT , while (8.2) implies that the value of the firm’s debt at maturity equals the nominal value of the liabilities minus the pay-off of a EuropeanputoptiononVT withexercisepriceequaltoB.
The above model is of course a stylized description of default. In reality the structure of a company’s debt is much more complex, so that default can occur on many different dates. Moreover, under modern bankruptcy code, default does not automatically imply bankruptcy, i.e. liquidation of a firm. Nonetheless, Merton’s model is a useful starting point for modelling credit risk and for pricing securities subject to default.
Remark 8.1. The option interpretation of equity and debt is useful in explaining potential conflicts of interest between shareholders and debtholders of a company. It is well known that the value of an option increases if the volatility of the underlying security is increased, provided of course that the mean is not adversely affected. Hence shareholders have an interest in the firm taking on very risky projects. Bond- holders, on the other hand, have a short position in a put option on the firm’s assets and would therefore like to see the volatility of the asset value reduced.
In the Merton model it is assumed that under the real-world or physical probability measure P the process (Vt ) follows a diffusion model (known as Black–Scholes model or geometric Brownian motion) of the form
dVt =μVVt dt+σVVt dWt (8.3)
forconstantsμV ∈R,σV >0,andastandardBrownianmotion(Wt).Equation(8.3) implies that VT = V0 exp((μV − 21 σV2 )T + σV WT ), and, in particular, that ln VT ∼ N (ln V0 + (μV − 21 σV2 )T , σV2 T ). Under the dynamics (8.3) the default probability of our firm is readily computed. We have
ln(B/V0)−(μV −21σV2)T
P(VT B)=P(lnVT lnB)=Φ σV√T . (8.4)
It is immediately seen from (8.4) that the default probability is increasing in B, decreasing in V0 and μV and, for V0 > B, increasing in σV , which is all perfectly in line with economic intuition.
8.2.2 Pricing in Merton’s Model
In the context of Merton’s model we can price securities whose pay-off depends on the value VT of the firm’s assets at T . Prime examples are the firm’s debt (or, equivalently, zero-coupon bonds issued by the firm) and the firm’s equity. We briefly explain the main results, since we need them in our treatment of the KMV model
8.2. StructuralModelsofDefault 333
in Section 8.2.3. The derivation of pricing formulas uses basic results from finan- cial mathematics. Readers not familiar with these results should simply accept the valuation formulas we present in the remainder of this section as facts and proceed quickly to Section 8.2.3; references to useful texts in financial mathematics are given in Notes and Comments.
We make the following assumptions.
Assumption 8.2.
(i) We have frictionless markets with continuous trading.
(ii) The risk-free interest rate is deterministic and equal to r 0.
(iii) The firm’s asset-value process (Vt) is independent of the way the firm is financed, and in particular it is independent of the debt level B. Moreover, (Vt ) is a traded security with dynamics given in (8.3).
Assumption (iii) merits some comment. First, the independence of (Vt ) from the financial structure of the firm is questionable, because a very high debt level and hence a high default probability may adversely affect the capability of a firm to generate business and hence affect the value of its assets. This is a special case of the indirect bankruptcy costs discussed in Section 1.4.2. Second, while there are many firms with traded equity, the value of the assets of a firm is usually neither completely observable nor traded. We come back to this issue in Section 8.2.3 below.
General pricing results. Consider a claim on the value of the firm with maturity T and pay-off h(VT ), such as the firm’s equity and debt in (8.1) and (8.2), and suppose that Assumption 8.2 holds. Standard derivative pricing theory offers two ways for computing the fair value f (t , Vt ) of this claim at time t T . Under the partial differential equation (PDE) approach the function f (t, v) is computed by solving the PDE (subscripts denote partial derivatives)
ft(t,v)+ 21σV2v2fvv(t,v)+rvfv(t,v)=rf(t,v) fort ∈[0,T), (8.5)
with terminal condition f (T , v) = h(v) reflecting the exact form of the claim to be priced. Equation (8.5) is the famous Black–Scholes PDE for terminal-value claims. Alternatively, the value f (t , Vt ) can be computed as the expectation of the dis- counted pay-off under the risk-neutral measure Q (the so-called risk-neutral pricing approach). Under Q the process (Vt ) satisfies the stochastic differential equation (SDE) dVt = r Vt dt + σV Vt dW ̃ t for a standard Q-Brownian motion W ̃ ; in par- ticular, the drift μV in (8.3) has been replaced by the risk-free interest rate r. The
risk-neutral pricing rule now states that
f (t, Vt ) = EQ(e−r(T −t)h(VT ) | Ft ), (8.6)
where EQ denotes expectation with respect to Q. For details we refer to the text- books on financial mathematics listed in Notes and Comments;
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