Credit Risk Modeling

David Lando

Abstract The chapter gives a broad outline of the central themes of creditrisk modeling starting with the modeling of default probabilities, ratings andrecovery. We present the two main frameworks for pricing credit risky instru-ments and credit derivatives. The key credit derivative - the Credit DefaultSwap - is introduced. The premium on this contract provides a meausure ofthe credit spread of the reference issuer. We then provide some key empiricalworks looking at credit spreads thorugh CDS contracts and bonds and finishwith a description of the role of correlation in credit risk modeling.

1 Introduction

Credit risk modeling is a rapidly growing area of financial economics andfinancial engineering. Banks and other financial institutions are applying in-creasingly sophisticated methods for assessing the risk of their loan portfo-lios and their counterparty exposure on derivatives contracts. The market forcredit derivatives is growing at an extreme pace with the credit default swapand the CDO markets as the primary sources of growth. These new mar-kets and better data availability on the traditional corporate bond markethave provided new laboratories for financial economists to test asset pricingtheories, to look at capital structure decisions, and to understand financialinnovation.

Classical credit risk analysis is concerned with deciding whether a loanshould be granted and, after a loan has been granted, trying to assess the riskof default. Modern credit risk analysis still addresses these issues but thereis more focus on pricing of loans and corporate bonds in secondary markets

David LandoCopenhagen Business School, Department of Finance, Solbjerg Plads 3, DK-2000 Fred-eriksberg, Denmark, e-mail: dl.fi@cbs.dk

T.G. Anderson et al., Handbook of Financial Time Series, 787DOI: 10.1007/978-3-540-71297-8_35, © Springer-Verlag Berlin Heidelberg 2009

788 D. Lando

and on the pricing and hedging of derivative contracts whose pay-offs dependon the performance of a single bond or loan or on a portfolio of bonds orloans. This survey will focus on the modeling of corporate bond prices andcredit spreads and on some implications for modeling credit derivatives. Theseare areas of credit risk modeling where rich data sets and well-developedpricing models allow for extensive econometric analysis. There are interestingeconometric challenges in more traditional credit scoring but we only havetime to touch briefly upon this topic. Other topics will not be covered at all,including the role of credit risk in the macro-economy (credit crunches, therole of credit in the propagation of business cycles), implications for optimalcapital structure decisions of firms and capital adequacy requirements forbanks.

2 Modeling the Probability of Default and Recovery

The decision theoretic problem of whether to grant a loan or not has beenattacked for a long time using financial ratios but the systematic attemptsof looking at which predictors perform well was started in Altman (1968)and Beaver (1966) who mainly rely on discriminant analysis, an attempt ofclassifying firms into defaulting or non-defaulting groups based on companycharacteristics. The outcome of the analysis is a discriminant function whichmaps the relevant predictors into a single score and classifies the companyas a defaulter or a non-defaulter based on whether the score is above orbelow a certain threshold. The derivation of the discriminant function andthe threshold can be based on a likelihood approach or on a decision theoreticapproach, in which a cost is assigned to misclassification. For the precisearguments, see for example Anderson (1984). This method of analysis wasin part chosen for computational convenience and it is not easily adaptedto a dynamic framework with time-varying covariates. The method is alsonot well-suited for handling the effects of covariates that are common to allfirms such as business cycle indicators. The use of computers have greatlyfacilitated the use of logistic regression methods, see for example Shumway(2001) and survival analytic methods based on hazard regressions, see forexample Duffie et al. (2004) which both deal easily with dynamic features ofthe model, common covariates and which offer default probability predictionsuseful for risk management.

As an alternative to developing full models for default probabilities of in-dividual firms, investors in corporate bonds often use the ratings which areassigned to bond issuers by the major rating agencies. The ratings are basedon analysis of the company’s key ratios and on meetings with the issuers. Rat-ings play important roles in a regulatory context when assigning risk-weightsto assets, in defining loan covenants and investment guidelines and as a sig-nalling device to less informed investors. Given these important roles, there

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is a substantial literature on the statistical behavior of ratings focusing onvarious forms of non-Markov behavior, business cycle dependence, the abilityof ratings to forecast default and their information content measured throughprice reactions in markets around rating changes. A number of references tothis literature can be found in Lando (2005)

The recovery rates on defaulted bonds play an important role in pricingboth bonds and derivatives but systematic research into what determinesrecovery rates is still relatively new. For a comprehensive recent study, seeAcharya et al. (2007) which also contains a number of references.

3 Two Modeling Frameworks

Models for default probabilities and recovery are important inputs to pricingmodels for loans, bonds and derivatives but we need more structure in themodels than what we get from a statistical model to actually perform thepricing. A statistical model typically gives us estimates of default probabil-ities as a function of certain explanatory variables, but the dynamics of theexplanatory variables are not modeled. A pricing model needs to include thefull dynamics of the explanatory variables.

There are broadly speaking two approaches to pricing corporate bondsand these approaches are outlined in this section. One approach - sometimesreferred to as the structural approach - views equity and bonds (and otherclaims to the firm’s assets) as derivative contracts on the market value of afirm’s assets. It then uses option pricing theory to price these claims. Thisapproach is systematically carried out in Merton (1974) in a Black-Scholessetting. To briefly summarize the approach, assume that the market value ofa firm’s assets follows a geometric Brownian motion

dVt = μVtdt+ σVtdWt

where W is a standard Browninan motion and that the firm has issued azero coupon bond promising to pay the principal D at the maturity dateT. Assume that the actual pay-off of the bond at maturity is min(VT , D).This means that debt holders receive their payment in full if the firm hassufficient assets to pay, but if assets are insufficient the bondholders takeover the remaining assets. In the setting of a Black-Scholes market with aconstant interest rate r the price of this bond is give at time zero as

B0 = D exp(−rT ) − PBS(V0, D, σ, r, T )

where PBS denotes the value of a European put option in the Black-Scholessetting. We can translate this bond price into a (promised) yield

790 D. Lando

y(T ) = − 1T

logB0

D

and by varying T in this expression we obtain what is known as the risk struc-ture of interest rates. We obtain the credit spread by subtracting the risklessrate r from the promised yield. The most important observation from thisrelationship is that yield spreads increase when leverage, i.e. the ratio of D toV, increases and when asset volatility increases. Note that credit spreads areincreasing in volatility and do not distinguish between whether the volatilityis systematic or non-systematic. The expected return of the bond is sensi-tive to whether volatility risk is systematic or not. So one should carefullydistinguish between credit spreads and expected returns on corporate bonds.

There are many modifications and extensions of the fundamental setupoutlined above. Bonds may pay continuous or lumpy coupons, there may bea lower boundary which when reached by the asset value of the firm causes thefirm to default. This lower boundary could represent bond safety covenants orliquidity problems of the firm and it could represent future capital structuredecisions of the firm trying to maintain a stationary leverage ratio. Dynamicsof key inputs may be changed as well, allowing for example for stochasticinterest rates or jumps in firm asset value. For a survey, see Lando (2005). Thestructural models differ in how literally the option approach is taken. Severalof the models impose default boundaries exogenously and obtain prices oncoupon bonds by summing the prices of zero-coupon bonds obtained fromthe risk structure of interest rates - an approach which does not make sensein the Merton model, see Lando (2005) for a simple explanation.

A special branch of the literature endogenizes the default boundary, seeLeland (1994) and Leland and Toft (1996), or takes into account the abilityof owners to act strategically by exploiting that the costs of bankruptcy areborne by the debt holders, see for example Anderson and Sundaresan (1996).

Applying contingent claims analysis to corporate bonds differs from con-tingent claims pricing in many important respects. Most notably, the marketvalue of the underlying assets is not observable. In applications the asset valuemust therefore be estimated and for companies with liquid stocks trading apopular approach has been to estimate the underlying asset value and thedrift and volatility of the asset value by looking at equity prices as transfor-mations (by the Black-Scholes call option formula) of the asset value. It istherefore possible using the transformation theorem from statistics to writedown the likelihood function for the observed equity prices and to estimatethe volatility and the drift along with the value of the underlying assets. Formore on this, see Duan (1994).

The option-based approach is essential for looking at relative prices ofdifferent liabilities in a firm’s capital structure, for discussing optimal capitalstructure in a dynamic setting and for defining market-based predictors ofdefault.

The second approach to spread modeling - sometimes referred to as thereduced-form approach or the intensity-based approach - takes as given a

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stochastic intensity of default which plays the same role for default risk asthe short rate plays in government term structure modeling. The advantage ofthis approach is precisely that it integrates the modeling of corporate bondswith modeling of the default-free term structure of interest rates, and thismakes it particularly suitable for econometric specification of the evolutionof credit spreads and for pricing credit derivatives.

Early intensity models are in Jarrow and Turnbull (1995) and Madan andUnal (1998) and the full integration of stochastic intensities in term structuremodels is found in Lando (1994, 1998) and Duffie and Singleton (1999). Theintegration using a Cox process setup proceeds as follows. A non-negativestochastic process λ describes the instantaneous probability of default undera risk-neutral measure Q, i.e.

Q(τ ∈ (t, t+Δt]|Ft) = 1{τ>t}λtΔt

where Ft contains information at time t including whether the default timeτ is smaller than t. Assume that the intensity process λ and a process forthe riskless short rate r are adapted to a filtration (Gt) where Gt ⊂ Ft, andassume that the default time τ of a firm is modeled as

τ = inf{

t :∫ t

0

λ(s)ds ≥ E1

}

.

where E1 is an exponentially distributed random variable with mean 1 whichis independent of the filtration (Gt). In accordance with the formal defini-tion of an intensity, it can be shown that 1{τ≤t} −

∫ t

0λ(s)1{τ≥s}ds is an Ft-

martingale. The key link to term structure modeling can be seen most easilyfrom the price v(0, T ) of a zero coupon bond maturing at T with zero recov-ery in default (in contrast with B0 defined earlier) and issued by a companywith default intensity λ under the risk-neutral measure Q :

v(0, T ) = EQ

[

exp

(

−∫ T

0

rs ds

)

1{τ>T}

]

= EQ

[

exp

(

−∫ T

0

(rs + λs)ds

)]

.

A key advantage of the intensity models is that the functional form for theprice of the defaultable bond is the same as that of a default-free zero-couponbond in term structure modeling, and therefore the machinery of affine pro-cesses can be applied. This is true also when the formula is extended to allowfor recovery by bond holders in the event of default, see for example Lando(2005).

In the reduced-form setting prices of coupon bonds are typically computedby summing the prices of zero-coupon bonds - an approach which clearlyshould be applied with caution. It is suitable for valuing credit default swaps

792 D. Lando

(see below) in which the contract has no influence on the capital structureof the firm or for assessing the default risk of small additional exposures of afirm for example in the context of counterparty risk in derivatives contracts.Pricing large new corporate bond issues will require an intensity which takesinto account the effects on leverage of the new issue.

The division between the two approaches should not be taken too literally.A structural model with incomplete information under certain assumptionscan be recast as an intensity model as shown in Duffie and Lando (2001),and nothing precludes an intensity model from letting the default intensitydepend on the asset value of the firm and other firm specific variables.

Estimation of intensity models is performed in Duffee (1999) and Driessen(2005) using a Kalman filter approach. Duffie and Singleton (1997) focuson swap rates in an intensity-based setting. The default intensity process istreated as a latent process similar to the short-rate process in classical termstructure modeling. This means that estimation proceeds in close analoguewith estimation of term structure models for default-free bonds. There isone important difference, however, in how risk premia are specified for de-fault intensities and for the riskless rate. To understand the difference in asingle factor setting, think of a diffusion-driven short rate depending on asingle Brownian motion, as for example in the Cox-Ingersoll-Ross setting.The change of measure between the physical measure and the risk-neutralmeasure corresponds to a change of drift of the Brownian motion. A similarchange of measure can be made for the intensity controlling the default timeof a defaultable bond issuer. However, it is also possible to have an intensityprocess which is changed by a strictly positive, multiplicative factor whengoing from the physical to the risk-neutral measure. This gives rise to animportant distinction between compensation for variation in default risk andcompensation for jump-event risk. For more on this distinction, see Jarrowet al. (2005) and for a paper estimating the two risk premia componentsseparately, see Driessen (2005).

4 Credit Default Swap Spreads

The first empirical work on estimating intensity models for credit spreadsemployed corporate bond data, but this is rapidly changing with the explosivegrowth of the credit default swap (CDS) market. The CDS contracts havebecome the benchmark for measuring credit spreads, at least for the largestcorporate issuers. A CDS is a contract between two parties: one who buys andone who sells protection against default of a particular bond issuer which wecall the reference issuer. The protection buyer pays a periodic premium, theCDS premium, until whichever comes first, the default event of the referenceissuer or the maturity date of the contract. In the event of default of thereference issuer before the maturity of the CDS contract, the protection seller

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compensates the protection buyer for the loss on a corporate bond issuedby the reference firm. The compensation is made either by paying a cashamount equivalent to the difference between the face value and the postdefault market value of the defaulted bond or by paying the face value whiletaking physical delivery of the defaulted bond. In practice, there is a deliveryoption allowing the protection buyer to deliver one of several defaulted bondswith pre-specified characteristics in terms of seniority and maturity. The CDSpremium is set such that the initial value of the contract is zero.

Forming a portfolio of a credit risky bond and a CDS contract with thesame maturity protecting against default on that bond, one has a positionclose to a riskless bond and this gives an intuitive argument for why the creditdefault swap premium ought to be close to a par bond spread on a corporatebond. For a more rigorous argument, see Duffie (1999).

It is illustrative to use the intensity setting to compute the fair premiumon a CDS, i.e. the premium which gives the contract value 0 at initiation. Inpractice, this relationship is primarily used to infer the default intensity fromobserved CDS prices and then price other derivatives from that intensity,or to analyze risk premia of default by comparing market implied defaultintensities with actual default intensities.

To clearly illustrate the principle, consider a stylized CDS with maturity Tyears, where premium payments are made annually. As above, let the defaultintensity of the reference issuer be denoted λ under a risk-neutral measure Q.Then the present value of the CDS premium payments made by the protectionbuyer before the issuer defaults (or maturity) is simply

π̃pb = c

T∑

t=1

EQ exp(−∫ t

0

rs ds)1{τ>t} = c

T∑

t=1

v(0, t)

where v(0, t) is the value of a zero recovery bond issued by the reference firmwhich we used as the basic example above. If we write the probability ofsurviving past t under the risk neutral measure as

S(0, t) = EQ[

exp(−∫ t

0

λs ds)]

and we assume (as is commonly done when pricing CDS contracts) that theriskless rate and the default intensity are independent, then we can expressthe value of the protection payment made before default occurs as

π̃pb = c

T∑

t=1

p(0, t)S(0, t)

where p(0, t) is the value of a zero-coupon riskless bond maturing at date t.This formula ignores the fact that if a default occurs between two paymentdates, the protection buyer pays a premium determined by the fraction of a

794 D. Lando

period that elapsed from the last premium payment to the default time. Ifwe assume that default happens in the middle between two coupon dates,then the value of this extra payment should be added to get the value of theprotection payment:

πpb = π̃pb +c

2

T∑

t=1

p(0, t− 12)(

S(0, t) − S(0, t− 1))

.

The value of the protection seller’s obligation is more complicated (but cer-tainly manageable) if we insist on taking into account the exact timing ofdefault. To simplify, however, we again assume that if a default occurs be-tween two default dates, it occurs in the middle, and the settlement paymentis made at that date. Furthermore, we assume a recovery per unit of principalequal to δ, so that the protection seller has to pay 1 − δ to the protectionbuyer per unit of notional. Still assuming independence of the riskless rateand the default intensity, we obtain

πps = (1 − δ)T∑

t=1

p(0, t− 12)(

S(0, t) − S(0, t− 1))

.

The fair CDS premium c can now be found by equating πpb and πps. Inpractice payments are often made quarterly in rates equal to one fourth ofthe quoted annual CDS premium.

There are several advantages of using CDS contracts for default studies:First of all, they trade in a variety of maturities thus automatically providinga term structure for each underlying name. They are becoming very liquidfor large corporate bond issuers, their documentation is becoming standard-ized and unlike corporate bonds they do not require a benchmark bond forextracting credit spreads.

Currently, there is an explosion of papers using CDS data. One early con-tribution is Blanco et al. (2005) who among other things study lead/lagrelationships between CDS premia and corporate bond spreads on a sampleof European investment grade names. They find evidence that CDS con-tracts lead corporate bonds. Longstaff et al. (2005) study the size of the CDSspread compared to corporate bond spreads measured with respect to differ-ent benchmark riskless rates. Assuming that the CDS premium representspure credit risk they are then able to take out the credit risk component ofcorporate bonds and ask to what extent liquidity-related measures influencethe residual spread on corporate bonds.

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5 Corporate Bond Spreads and Bond Returns

The current price of a corporate bond in the Merton model - even under riskneutrality - is smaller than the price of a default-free bond. Therefore, theyield spread is positive even when there is risk neutrality. This part of thecorporate bond spread is the expected loss component. Empirically, corporatespreads are found to be larger than the expected loss component, and in factthey are apparently so much larger that it is questioned whether reasonableassumptions on risk premia for default can explain the remainder. This isthe essence of the credit risk puzzle. How can actual spreads be so muchlarger than the expected loss component? Roughly, potential contributionsto the corporate spread can be divided into the expected loss component,components due to priced market risk factors in corporate bonds and otherfactors, such as taxes (in the US coupons on corporate bonds are taxed at thestate level whereas Treasury bond coupons are not), a liquidity premium oncorporate bonds, and the choice of riskless benchmark (is the Treasury ratethe appropriate benchmark to use?). Early papers pointing to the difficulty ofstructural models in explaining spreads are Jones et al. (1984) and Sarig andWarga (1989). Newer papers attempting to decompose spreads in structuralmodel settings along the dimensions mentioned above include Huang andHuang (2003) and Eom et al. (2003). Many papers study spreads in timeseries and/or regression frameworks where inspiration on relevant covariatesto include come from form structural models, including Elton et al. (2001)and Collin-Dufresne et al. (2001). A consensus seems to be building that theTreasury rate is not an appropriate benchmark in US markets for defining ariskless rate and using a rate closer to the swap rate at least removes part ofthe credit risk puzzle. Using the swap rate as riskless benchmark also bringscredit spreads measured from CDS contracts closer to spreads measured fromcorporate bonds as shown in Longstaff et al. (2005). Still, there is no definitiveanswer yet to what fraction of credit spreads are in fact explained by default-related factors.

6 Credit Risk Correlation

Dependence of default events is a key concern of regulators who are lookinginto issues of financial stability. Performing empirical studies on dependencebased solely on events, similar to the basic default studies, is difficult sincethe number of defaults is fairly limited. Fortunately, a very large market ofCollateralized Debt Obligations has emerged and the pricing of securitiesin this market is intimately linked to correlation. This therefore allows usto analyze market implied correlation. A CDO is an example of an assetbacked security in which the collateral consists of loans or bonds, and wheresecurities issued against the collateral are prioritized claims to the cash flow of

796 D. Lando

the collateral. The lowest priority claim, the so-called equity tranche, is mostsensitive to default in the underlying pool and the highest priority claim, thesenior tranche, is strongly protected against default. This closely resembleshow equity, junior debt and senior debt are prioritized claims to a firm’sassets, but there are often 5-7 different ’layers’ in the CDO tranches.

There are roughly speaking three approaches to pricing CDOs.

1. In the copula based approach one takes as given the marginal default in-tensities of the individual loans in the collateral pool and defines a jointdistribution of default by applying a copula function, i.e. a device forcollecting a family of marginal distributions into a joint distribution pre-serving the marginal distributions. Often the loans are actually given thesame default intensity and the copula function is then chosen from a para-metric class of copulas attempting to fit the model to observed trancheprices. A first paper in this area is Li (2000) but see also Schönbucher(2003).

2. A full modeling approach is taken in Duffie and Gârleanu (2001) wherethe default intensities of individual issuers are given a factor structure,such that the individual intensities are a sum of a common ’factor’ inten-sity and idiosyncratic intensities. While one can facilitate computationsby working with affine intensity processes, the model still requires manyparameters as inputs, and this is one reason why it is not adapted asquickly by market participants despite its more appealing structure. Onemay also question the implicit assumption of conditional independencewhereby the only dependence that the defaults have is through their com-mon dependence on the aggregate factor. Conditionally on this factor thedefaults are independent. This rules out direct contagion among namesin the portfolio.

3. In the third approach, the process of cumulative losses is modeled directlywithout focus on the individual issues. One can impose simple jumpsto accommodate one default at a time or one can choose to work withpossibilities of multiple defaults.

As noted, the choice of modeling here is linked to the issue of the extentto which defaults are correlated only through their common dependence onthe economic environment or whether there is true contagion. This contagioncould take several forms: The most direct is that defaults cause other defaultsbut it might also just be the case that defaults cause an increasing likelihoodof default for other issuers. Das et al. (2004) test whether the correlationfound in US corporate defaults can be attributed to common variation onthe estimated intensity of default for all issuers. By transforming the time-scale, they test the joint hypothesis that the intensities of default of individualfirms are estimated correctly and that the defaults are conditionally indepen-dent given the realized intensities, and they reject the joint hypothesis. Thisis an indication that contagion events could be at play but the reason for therejection could also lie in a mis-specification of the intensities, neglecting for

Credit Risk Modeling 797

example an unobserved common default risk factor, sometimes referred to asa frailty factor in line with the survival analysis literature. An inaccuratelyspecified underlying intensity could also be the problem. A proper under-standing and parsimonious modeling of dependence effects in CDO pricingremains an important challenge.

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