Risk and Valuation of Collateralized Debt ObligationsRisk and Valuation of Collateralized Debt Obligations Darrell Duffie and Nicolae Gârleanu In this discussion of risk analysis

January/February 2001

41

Risk and Valuation of Collateralized Debt Obligations

Darrell Duffie and Nicolae Gârleanu

In this discussion of risk analysis and market valuation of collateralizeddebt obligations, we illustrate the effects of correlation and prioritizationon valuation and discuss the “diversity score” (a measure of the risk of theCDO collateral pool that has been used for CDO risk analysis by ratingagencies) in a simple jump diffusion setting for correlated default

intensities.

collateralized debt obligation

is an asset-backed security whose underlying collat-eral is typically a portfolio of (corporateor sovereign) bonds or bank loans. A CDO

cash flow structure allocates interest income andprincipal repayments from a collateral pool of dif-ferent debt instruments to a prioritized collectionof CDO securities, which we call “tranches.”Although many forms of prioritization exist, a stan-dard prioritization scheme is simple subordination:Senior CDO notes are paid before mezzanine andlower subordinated notes are paid, with any resid-ual cash flow paid to an equity piece. We providesome examples of prioritization later in this article.

A

cash flow

CDO is one for which the collateralportfolio is not subjected to active trading by theCDO manager, which implies that the uncertaintyregarding interest and principal payments to theCDO tranches is induced mainly by the numberand timing of defaults of the collateral securities. A

market-value

CDO is one in which the CDO tranchesreceive payments based essentially on the marked-to-market returns of the collateral pool, as deter-mined largely by the trading performance of theCDO manager. We concentrate here on cash flowCDOs, thus avoiding an analysis of the tradingbehavior of CDO managers.

A generic example of the contractual relation-ships involved in a CDO is shown in

Figure 1

. Thecollateral manager is charged with the selectionand purchase of collateral assets for the CDO spe-cial-purpose vehicle (SPV). The trustee of the CDOis responsible for monitoring the contractual provi-sions of the CDO. Our analysis assumes perfectadherence to these contractual provisions. The

main issue we address is the impact of the jointdistribution of default risk of the underlying collat-eral securities on the risk and valuation of the CDOtranches. We are also interested in the efficacy ofalternative computational methods and the role of“diversity scores,” a measure of the risk of the CDOcollateral pool that has been used for CDO riskanalysis by rating agencies.

CDO Design and Valuation

In perfect capital markets, CDOs would serve nopurpose; the costs of constructing and marketing aCDO would inhibit its creation. In practice, CDOsaddress some important market imperfections.First, banks and certain other financial institutionshave regulatory capital requirements that makevaluable the securitizing and selling of some por-tion of their assets; the value lies in reducing theamount of (expensive) regulatory capital they musthold. Second, individual bonds or loans may beilliquid, which reduces their market values; if secu-ritization improves their liquidity, it raises the totalvaluation to the issuer of the CDO structure.

In light of these market imperfections, at leasttwo classes of CDOs are popular. The

balance sheet

CDO, typically in the form of a collateralized loanobligation (CLO), is designed to remove loans fromthe balance sheets of banks, thereby providing cap-ital relief and perhaps also increasing the valuationof the assets through an increase in liquidity.

1

An

arbitrage

CDO, often underwritten by an invest-ment bank, is designed to capture some fraction ofthe likely difference between the total cost ofacquiring collateral assets in the secondary marketand the value received from management fees andthe sale of the associated CDO structure. Balancesheet CDOs are normally of the cash flow type.Arbitrage CDOs may be collateralized bond obli-gations and have either cash flow or market-valuestructures.

Darrell Duffie is James I. Miller Professor of Finance andNicolae Gârleanu is a graduate student

at the StanfordUniversity Graduate School of Business.

A

Financial Analysts Journal

42

©2001,

Association for Investment Management and Research

®

Among the sources of illiquidity that promote,or limit, the use of CDOs are adverse selection,trading costs, and moral hazard.

With regard to adverse selection, enough pri-vate information may exist about the credit qualityof a junk bond or a bank loan that an investor isconcerned about being “picked off” when tradingsuch an instrument. For instance, a better-informedseller has an option to trade or not at the givenprice. The value of this option is related to thequality of the seller’s private information. Giventhe risk of being picked off, the buyer offers a pricethat, on average, is below the price at which theasset would be sold in a setting of symmetric infor-mation. This reduction in price as a result ofadverse selection is sometimes called a “lemon’spremium” (Akerlof 1970).

In general, adverse selection cannot be elimi-nated by securitization of assets in a CDO, but it canbe mitigated. The seller achieves a higher totalvaluation (for what is sold and what is retained) bydesigning the CDO structure so as to concentrateinto small subordinate tranches the majority of therisk that may be cause for adverse selection. Forexample, a large senior tranche, relatively immuneto the effects of adverse selection, can be sold at asmall lemon’s premium. The issuer can retain, onaverage, significant fractions of the smaller subor-dinate tranches, which are more subject to adverseselection. For models supporting this design andretention behavior, see DeMarzo (1999) andDeMarzo and Duffie (1999).

For a relatively small junk bond or a singlebank loan to a relatively obscure borrower, themarket of potential buyers and sellers may besmall; thus, trading may be costly.

2

Searching forsuch buyers can be expensive, for example, and tosell such an illiquid asset quickly, one may beforced to sell to the highest bidder among the rela-tively few buyers with whom one can negotiate onshort notice. One’s negotiating position may alsobe poorer than it would be in an active market. Thevalue of the asset is correspondingly reduced.Potential buyers recognize that they are placingthemselves at the risk of facing the same situationwhen they try to resell in the future, which resultsin yet lower valuations. The net cost of bearingthese costs may be reduced through securitizationinto relatively large homogeneous senior CDOtranches, perhaps with significant retention ofsmaller and less easily traded junior tranches.

Moral hazard, in the context of CDOs, bears onthe issuer’s or CDO manager’s incentives to selecthigh-quality assets for the CDO and to engage incostly enforcement of covenants and other restric-tions on the behavior of obligors. Securitizing andselling a significant portion of the cash flows of theunderlying assets dilute these incentives. Reduc-tions in value through lack of effort are borne tosome extent by investors. Also, the opportunitymay arise for “cherry picking” (sorting assets intothe issuer’s own portfolio or into the SPV portfoliobased on the issuer’s private information). In addi-tion, opportunities may arise for front running, inwhich a CDO manager trades on its own account

Figure 1. Typical CDO Contractual Relationships

Source

: Morgan Stanley, from Schorin and Weinreich (1998).

OngoingCommunication

CollateralManager

Trustee

Hedge Provider(if needed)

CDOSpecial-Purpose

Vehicle

UnderlyingSecurities(collateral)

Senior Fixed/Floating-Rate

Notes

MezzanineFixed/Floating-

Rate Notes

SubordinatedNotes/Equity


January/February 2001

43

in advance of trades on behalf of the CDO. Thesemoral hazards act

against

the creation of CDOs,because the incentives to select and monitor assetspromote greater efficiency and higher valuation ifthe issuer retains a 100 percent interest in the assetcash flows.

The opportunity to reduce the other marketimperfections through a CDO may be sufficientlylarge—especially in light of the advantages inbuilding and maintaining a reputation for notexploiting CDO investors—to offset the effects ofmoral hazard. In this case, the issuer has an incen-tive to design the CDO so that the issuer retains asignificant portion of one or more subordinatetranches that would be among the first to sufferlosses stemming from poor monitoring or poorasset selection. Doing so demonstrates a degree ofcommitment to diligent efforts to manage the issue.Similarly, for arbitrage CDOs, a significant portionof the management fees may be subordinated to theissued tranches (Schorin and Weinreich 1998). Inlight of this commitment, investors may be willingto pay more for tranches, and the total valuation tothe issuer will be higher than in an unprioritizedstructure, such as a straight equity pass-throughsecurity. Innes (1990) described a model that sup-ports this motive for security design.

An example of a CLO structure consistent withthis theory is shown in

Figure 2

. The figure illus-trates one of a pair of CLO cash flow structuresissued by NationsBank in 1997. A senior tranche of$2 billion in face value, whose credit rating is AAA,is followed by successively lower subordination

tranches, with their ratings. The bulk of the under-lying assets are floating-rate NationsBank loansrated BBB or BB. Any fixed-rate loans were hedgedin terms of interest rate risk by fixed-to-floatinginterest rate swaps. As predicted by theory,NationsBank retained the majority of the (unrated)lowest tranche.

Our valuation model does not deal directlywith the effects of market imperfections. It takes asgiven the default risk of the underlying loans andassumes that investors are symmetricallyinformed. Although this approach is not perfectlyrealistic, it is not necessarily inconsistent with theroles of moral hazard or adverse selection in theoriginal security design. For example, DeMarzoand Duffie demonstrated a “fully separating equi-librium,” in which the sale price of the security orthe amount retained by the seller signals to allinvestors any of the seller’s privately held value-relevant information. Moral hazard can beaddressed by the model because the diligence of theissuer or manager is, to a large extent, determinedby the security design and the fractions retained bythe issuer. Once these factors are known, thedefault risk of the underlying debt is also known.Our simple model does not, however, account forthe valuation effects of many other forms of marketimperfections. Moreover, inferring separate riskpremiums for default timing and default recoveryfrom the prices of the underlying debt and marketrisk-free interest rates is generally difficult (seeDuffie and Singleton 1999.) These risk premiums

Figure 2. NationsBank CLO Tranches for Three-Year Floating Rate Notes

Source

: Fitch.

Obligors$2,164 million

Issuer Interest Rate Swaps

A. $2,000 million(AAA) B. $43 million

(A)C. $54 million

(BBB)D. $64 million

(not rated)


44

©2001,

Association for Investment Management and Research

®

play separate roles in the valuation of CDOtranches. We simply take these risk premiums asgiven in the form of “risk-neutral” parametric mod-els for default timing and recovery distributionsunder an equivalent martingale measure.

Default Risk Model

We lay out some of the basic default modeling forthe underlying collateral. First, we propose a sim-ple model for the default risk of one obligor. Then,we turn to the multi-issuer setting. Throughout, wework under risk-neutral probabilities, so value isgiven by expectations of discounted future cashflows. If objective likelihoods or variances are ofinterest, however, the results should be interpretedas if the probabilities were actual, not risk neutral.

Obligor Default Intensity.

We suppose thateach underlying obligor defaults at some expectedarrival rate. The idea is that at each time

t

before thedefault time

τ

of the given obligor, the defaultarrives at some “intensity”

λ

(

t

), given all currentlyavailable information. We thus have the approxi-mation

(1)

for the conditional probability at the time

t

ofdefault within a “small” time interval

∆

t

> 0.

3

Forexample, if time is measured in years (as we dohere), a current default intensity of 0.04 implies thatthe conditional probability of default within thenext three months is approximately 0.01. Immedi-ately after default, the intensity drops to zero. Sto-chastic variation in the intensity over time, as newinformation becomes available, reflects anychanges in perceived credit quality. Correlationacross obligors in the changes over time of theircredit qualities is reflected by correlation in thechanges of those obligors’ default intensities.Indeed, our model has the property that

all

correla-tion in default timing arises in this manner.

4

We calla stochastic process

λ

a

pre-intensity

for a stoppingtime

τ

if whenever

t

<

τ

, first, the current intensityis

λ

t

and, second,

,

s

>0, (2)

where

E

t

denotes conditional expectation given allinformation at time

t

and

s

is the length of theperiod over which survival (no default) is consid-ered. A pre-intensity need not fall to zero afterdefault. For example, default at a constant pre-intensity of 0.04 means that the intensity itself is0.04 until default and is zero thereafter.

We adopt a pre-intensity model that is a specialcase of the “affine” family of processes that havebeen used for this purpose and for modeling short-term interest rates.

5

Specifically, we suppose thateach obligor’s default time has some pre-intensityprocess

λ

solving a stochastic differential equationof the form

, (3)

where

W

is a standard Brownian motion and

∆

J

(

t

)denotes any jump that occurs at time

t

of a pure-jump process

J

, independent of

W

, whose jumpsizes are independent and exponentially distrib-uted with mean

µ

and whose jump times are thoseof an independent Poisson process with mean jumparrival rate

l

. (Jump times and jump sizes are alsoindependent.

6

) We call a process

λ

of this form(Equation 3) a

basic affine process with parameters

(

κ

,

θ

,

σ

,

µ

,

l

). These parameters can be adjusted inseveral ways to control the manner in which defaultrisk changes over time. For example, we can varythe mean-reversion rate

κ

, the long-run mean, or the relative contributions to

the total variance of

λ

t

that are attributed to jumprisk and to diffusive volatility.

We can also vary the relative contributions tojump risk of the mean jump size

µ

and the meanjump arrival rate

l

. A special case is the no-jump(

l

= 0) model of Feller (1951), which was used byCox, Ingersoll, and Ross (1985) to model interestrates. From the results of Duffie and Kan (1996),we can calculate that for any

t

and any

s

≥

0,

(4)

where explicit solutions for the coefficients

α

(

s

) and

β

(

s

) are provided in Appendix A. Together, Equa-tion 2 and Equation 4 give a simple, reasonably rich,and tractable model for the default time probabilitydistribution and how it varies at random over timeas information arrives in the market.

Multi-Issuer Default Model.

To study theimplications of changing the correlation in thedefault times of the various participations (collat-eralizing bonds or loans) in a CDO while holdingconstant the default risk model of each underlyingobligor, we will exploit the following result. Westate that a basic affine model can be written as thesum of independent basic affine models as long asthe parameters

κ

,

σ

, and

µ

governing, respectively,the mean reversion rate, diffusive volatility, andmean jump size are common to the underlying pairof independent basic affine processes.

Proposition 1:

Suppose

X

and

Y

are indepen-dent basic affine processes with respective

Pt τ t ∆t+<( ) λ t( )∆t≅

Pt τ t s+>( ) Et exp λu–tt s+∫ ud( )[ ]=

dλ t( ) κ θ λ t( )–[ ]dt σ λ t( ) dW t( )+= ∆J t( )+

m θ l µ( ) κ⁄+=

Et exp λu–tt s+∫ ud( )[ ] e

α s( ) β s( )λ t( )+ ,=


January/February 2001 45

parameters (κ, θX, σ, µ, l X) and (κ, θY, σ, µ, l Y).Then, X + Y is a basic affine process withparameters (κ, θ, σ, µ, l ), where l = l X + l Y andθ = θX + θY.7 This result allows us to maintain a fixed, par-

simonious, and tractable one-factor Markov modelfor each obligor’s default probabilities while vary-ing the correlations between different obligors’default times, as explained in the following discus-sion.

Suppose that the N participations in the collat-eral pool have default times τ1, . . ., τN with pre-intensity processes λ1, . . ., λN, respectively, that arebasic affine processes.8 To introduce correlation ina simple way, we suppose that

λi = Xc + Xi, (5)

where Xc and Xi are basic affine processes with,respectively, parameters (κ, θc, σ, µ, l c) and (κ, θi, σ,µ, l i) and where X1, . . ., XN and Xc are independent.By Proposition 1, λi is itself a basic affine processwith parameters (κ, θ, σ, µ, l ), where θ = θc + θi andl = l c + l i. One may view Xc as a state processgoverning common aspects of economic perfor-mance in an industry, sector, or currency region andXi as a state variable governing the idiosyncraticdefault risk specific to obligor i. The parameter

, (6)

is the long-run fraction of jumps to a given obligor’sintensity that are common to all (surviving) obli-gors’ intensities. One can also see that ρ is theprobability that the pre-intensity λj jumps at time tgiven that λi jumps at time t for any time t and anydistinct i and j. We also suppose that θc = ρθ.

Sectoral, Regional, and Global Risk. Exten-sions of the model to handle multifactor risk(regional, sectoral, and other sources) could easilybe incorporated with repeated use of Proposition 1.Suppose, for example, that we have S differentsectors. We denote by c(i) the sector to which the ithobligor belongs; consequently, c(1), c(2), . . ., c(S) aredisjoint and exhaustive subsets of (1, 2, . . ., N).Suppose that the default time τi of the ith obligorhas a pre-intensity λi = Xi + Yc(i) + Z, where thesector factor Yc(i) is common to all issuers in the“sector” c(i), where Z is common to all issuers, andwhere (X1, . . ., XN, Y1, . . .,YS, Z) are independentbasic affine processes.

If we do not restrict the parameters of theunderlying basic affine processes, then an individ-ual obligor’s pre-intensity need not itself be a basicaffine process, but calculations are nevertheless

easy. We can use the independence of the underly-ing state variables to see that

(7)

where α(s) = αi(s) + αc(i)(s) + αZ(s) and all the α andβ coefficients are obtained explicitly from Appen-dix A, from the respective parameters of the under-lying basic affine processes Xi, Yc(i), and Z.

Even more generally, we can adopt multifactoraffine models in which the underlying state vari-ables are not independent. Interest rates that arejointly determined by an underlying multifactoraffine jump diffusion model can also be accommo-dated. We refer the interested reader to the Duffieand Gârleanu 1999 working paper.

Recovery Risk. We suppose that, at default,any given piece of debt in the collateral pool maybe sold for a fraction of its face value whose risk-neutral conditional expectation, given all informa-tion available at any time t before default, is aconstant that does not depend on t. Therecovery fractions of the underlying participationsare assumed to be independently distributed andindependent of default times and interest rates.(Here again, we are referring to risk-neutral behav-ior.) For simplicity, we assume throughout that therecovered fraction of face value is uniformly dis-tributed on (0, 1).

Collateral Credit Spreads. We suppose forsimplicity that changes in default intensities andchanges in interest rates are (risk-neutrally) inde-pendent.9 Combined with the preceding assump-tions, this implies that for an issuer whose defaulttime τ has a basic affine pre-intensity process λ, azero-coupon bond maturing at time t has an initialmarket value of

, (8)

where δ(t) denotes the default-free zero-coupondiscount to time t and

(9)

is the (risk-neutral) probability density at time u ofthe default time.

The first term of Equation 8 is the market valueof a claim that pays 1 at maturity in the event ofsurvival. The second term is the market value of aclaim to any default recovery between times 0 andt. The integral is computed numerically by use of

ρl c

l-----=

Et exp λi u( )–tt s+∫ ud[ ]

exp[α s( ) βi s( )Xi t( ) βc i( ) s( )Yc i( ) t( ) βz s( )Z t( ) ] ,+ + +=

f 0 1,( )∈

p t λ 0( ),[ ] δ t( )eα t( ) β t( )λ 0( )+

f δ u( )π u( ) ud0t∫+=

π u( ) ddu------P τ u>( )–=

eα u( ) β u( )λ 0( )+ α′ u( ) β′ u( )λ 0( )+[ ]–=


46 ©2001, Association for Investment Management and Research®

our explicit solutions from Appendix A for α(t) andβ(t).10

Using this defaultable discount function, p(•),we can value any straight coupon bond or deter-mine par coupon rates. For example, for quarterlycoupon periods, the (annualized) par coupon rate,c(s), for maturity in s years (for an integer s > 0) isdetermined at any time t by the identity

, (10)

which is easily solved for c(s).We develop our numerical example with con-

stant interest rates r, for which the risk-free dis-count δ has the simple form

δ(t) = e–rt. (11)

Diversity Scores. A key measure of collateraldiversity developed by Moody’s Investors Servicefor CDO risk analysis is the diversity score. Thediversity score of a given pool of participations isthe number, n, of bonds in an idealized comparisonportfolio that meets the following criteria:• The total face value of the comparison portfolio

is the same as the total face value of the collat-eral pool.

• The bonds of the comparison portfolio haveequal face values.

• The comparison bonds are equally likely todefault, and their default is independent.

• The comparison bonds are, in some sense, ofthe same average default probability as theparticipations of the collateral pool.

• The comparison portfolio has the same totalloss risk, according to some measure of risk, asdoes the collateral pool.At least in terms of publicly available informa-

tion, it is not clear how the (equal) probability p ofdefault of the bonds of the comparison portfolio isdetermined. A method discussed by Schorin andWeinreich for this purpose is to assign a defaultprobability corresponding to the weighted-averagerating score of the collateral pool by using pre-determined rating scores and weights that are pro-portional to face value. Given the average ratingscore, one can assign a default probability to theresulting “average” rating. For the choice of prob-ability p, Schorin and Winreich discussed the use ofthe historical default frequency for that rating.

A diversity score of n and a comparison-bonddefault probability of p imply, under the indepen-dence assumption for the comparison portfolio,that the probability of k defaults out of n bonds ofthe comparison portfolio is

, (12)

where n! (read “n factorial”) is the product 1 × 2 ×… × n. From this “binomial-expansion” formula, arisk analysis of the CDO can be conducted byassuming that the performance of the collateralpool is sufficiently well approximated by the per-formance of the comparison portfolio.

Table 1 shows the diversity score thatMoody’s would apply to a collateral pool ofequally sized bonds of different companies in thesame industry.11 Table 1 also shows the impliedprobability of default of one participation giventhe default of another, as well as the implied cor-relations of the 0–1, survival–default, random vari-ables associated with any two participations fortwo levels of individual default probability,namely, p = 0.5 and p = 0.05.

Pricing ExamplesIn this section, we apply a standard risk-neutralderivative valuation approach to the pricing ofCDO tranches. In the absence of any tractable alter-native, we use Monte Carlo simulation of thedefault times. Essentially, any intensity modelcould be substituted for the basic affine model thatwe have adopted here. An advantage of the affinemodel is the ability one has to quickly calibrate themodel to the underlying participations, in terms ofgiven correlations, default probabilities, yieldspreads, and so on.

We study various alternative CDO cash flowstructures and default risk parameters. The basic

1 p s λt,( ) c s( )4---------- p

j4-- λ t,

j 1=

4s

∑+=

Table 1. Diversity Scores, Conditional Default Probabilities, and Default Correlations

Number of Companies

DiversityScore

Conditional Default Probability

Default Correlation

(p = 0.5) (p = 0.05) (p = 0.5) (p = 0.05)

1 1.002 1.50 0.78 0.48 0.56 0.453 2.00 0.71 0.37 0.42 0.344 2.33 0.70 0.36 0.40 0.325 2.67 0.68 0.33 0.36 0.306 3.00 0.67 0.31 0.33 0.277 3.25 0.66 0.30 0.32 0.268 3.50 0.65 0.29 0.31 0.259 3.75 0.65 0.27 0.29 0.24

10 4.00 0.64 0.26 0.28 0.23

Note: Situations in which the number of companies is greater than 10 are evaluated on a case-by-case basis.

q k n,( ) n!n k–( )!k!

----------------------- pk 1 p–( )n k–

=



CDO structure consists of a special-purpose vehiclethat acquires a collateral portfolio of participations(debt instruments of various obligors) and allocatesinterest, principal, and default recovery cash flowsfrom the collateral pool to the CDO tranches—andperhaps to a manager.

Collateral. The collateral pool has N participa-tions. Each participation pays quarterly cash flowsto the SPV at its coupon rate until maturity ordefault. At default, a participation is sold for itsrecovery value and the proceeds from the sale arealso made available to the SPV.

Let A(k) denote the subset of {1, . . ., N} contain-ing the surviving participations at the kth couponperiod. The total interest income in coupon periodk is then

, (13)

where Mi is the face value of participation i and Ciis the coupon rate on participation i.

If B(k) denotes the set of participations default-ing between coupon periods k – 1 and k,12 the totalcash flow in period k is

, (14)

where Li is the loss of face value at the default ofparticipation i.

For our example, the initial pool of collateralavailable to the CDO structure consists of N = 100participations that are straight quarterly coupon10-year par bonds of equal face value. Without lossof generality, we take the face value of each bondto be 1.

We initiate the common and the idiosyncraticrisk factors, Xc and Xi, at their long-run means,θc+ l c µ/κ and θi + l i µ/κ, respectively. Thus, theinitial condition (and long-run mean) for each obli-gor’s (risk-neutral) default pre-intensity is 5.33 per-cent. Our base-case default-risk model is definedby Parameter Set Number 1 in Table 2 and byletting ρ = 0.5 determine the degree of diversifica-tion. The three other parameter sets shown in Table2 are designed to illustrate the effects of replacingsome or all of the diffusive volatility with jumpvolatility or the effects of reducing the mean jumpsize and increasing the mean jump arrival fre-quency l .

All the parameter sets have the same long-runmean θ + µl/κ. The parameters θ, σ, l , and µ wereadjusted so as to maintain essentially the same termstructure of zero-coupon yields, as illustrated inFigure 3.13 Table 2 provides, for each parameter set,

the 10-year par-coupon spread (in basis points) andthe long-run variance of λi(t). To illustrate the qual-itative differences between parameter sets, Figure4 shows sample paths of new 10-year par spreadsfor two issuers, one with the base-case parameters(Set 1) and the other with pure-jump intensity (Set2) calibrated to the same initial spread curve.

With di denoting the event of default by the ithparticipation, Table 3 shows for each parameter setand each of three levels of the correlation parameterρ, the unconditional probability of default and theconditional probability of default by one participa-tion given default by another. Table 3 also showsthe diversity score of the collateral pool that isimplied by matching the (risk-neutral) variance ofthe total loss of principal of the collateral portfolioto that of a comparison portfolio of bonds of thesame individual default probabilities.14

For our basic examples, we suppose, first, thatany cash in the SPV reserve account is invested atthe default-free short-term rate. We later considerinvestment of SPV free cash flows in additionalrisky participations.

Sinking-Fund Tranches. We consider a CDOstructure that pays SPV cash flows to a prioritizedsequence of sinking-fund bonds and a junior sub-ordinated residual.

In general, a sinking-fund bond with n couponperiods per year has some remaining principal,F(k), at coupon period k, some annualized couponrate c, and a scheduled interest payment at couponperiod k of F(k)c/n. In the event that the actualinterest paid, Y(k), is less than the scheduled inter-est payment, any difference F(k)c/n – Y(k) isaccrued at the bond’s own coupon rate, c, so as togenerate an accrued unpaid interest at period k ofU(k), where U(0) = 0 and

. (15)

To prioritize payments in light of the default andrecovery history of the collateral pool, some pre-payment of principal, D(k), and some contractual

W k( ) MiCi

n-----

i A k( )∈∑=

Z k( ) W k( ) Mi Li–( )i B k( )∈∑+=

Table 2. Risk-Neutral Default Parameter Sets

Set κ θ σ l µSpread(bps) var∞

1 0.6 0.0200 0.141 0.2000 0.1000 254 0.422 0.6 0.0156 0.000 0.2000 0.1132 254 0.433 0.6 0.0373 0.141 0.0384 0.2500 253 0.494 0.6 0.0005 0.141 0.5280 0.0600 254 0.41

U k( ) 1 cn---+

U k 1–( ) cn---F k( ) Y k( )–+=



unpaid reduction in principal, J(k), may also occurin period k. By contract, we have D(k) + J(k) ≤ F(k –1), so the remaining principal at quarter k is

F(k) = F(k – 1) – D(k) – J(k). (16)

At maturity (coupon period number K), anyunpaid accrued interest and unpaid principal, U(K)and F(K), respectively, are paid to the extent pro-vided in the CDO contract. (A shortfall does notconstitute default so long as the contractual priori-tization scheme is maintained.15) The total actualpayment in any coupon period k is Y(k) + D(k).

The par coupon rate on a given sinking-fundbond is the scheduled coupon rate c with the prop-erty that the initial market value of the bond isequal to its initial face value, F(0). If the default-freeshort rate r(k) is constant, as in our results, anysinking-fund bond that pays all remaining princi-pal and all accrued unpaid interest by or at itsmaturity date has a par coupon rate equal to thedefault-free coupon rate, no matter the timing ofthe interest and principal payments.

We illustrate our initial valuation results interms of the par coupon spreads of the respectivetranches, which are the excess of the par coupon

rates of the tranches over the default-free par cou-pon rate.

Prioritization Schemes. We experiment withthe relative sizes and prioritization of two CDObond tranches, one 10-year senior sinking-fundbond with some initial principal F1(0) = P1 and one10-year mezzanine sinking-fund bond with initialprincipal F2(0) = P2. The residual junior tranchereceives any cash flow remaining at the end of the10-year structure. Because the base-case couponrates on the senior and mezzanine CDO tranchesare, by design, par rates, the base-case initial mar-ket value of the residual tranche is P3 = 100 – P1 – P2.

At the kth coupon period, tranche j has a facevalue of Fj(k) and accrued unpaid interest Uj(k)calculated at its own coupon rate, cj. Any excesscash flows from the collateral pool (interest incomeand default recoveries) are deposited in a reserveaccount. To begin, we suppose that the reserveaccount earns interest at the default-free one-period interest rate, denoted r(k) at the kth coupondate. At maturity, coupon period K, any remainingfunds in the reserve account after payments atquarter K to the two tranches are paid to the

Figure 3. Zero-Coupon Yield Spreads with and without Diffusion

Zero-Coupon Yield Spread(basis points)

280

260

240

220

200

180

160

140

1200 1 2 3 4 5 6 7 8 9

Maturity (years)

10

Base Case Zero Diffusion



subordinated residual tranche. (Later, we investi-gate the effects of investing the reserve account inadditional participations that are added to the col-lateral pool.) We ignore management fees.

This example investigates valuation for twoprioritization schemes: Given the definition of thesinking funds in the previous subsection, completespecification of cash flows to all tranches requiresonly that we define for the senior sinking fund andfor the mezzanine sinking fund, respectively, theactual interest payments, Y1(k) and Y2(k); any pay-ments of principal, D1(k) and D2(k); and any con-tractual reductions in principal, J1(k) and J2(k).

Under our uniform prioritization scheme, inter-est W(k) collected from the surviving participationsis allocated in priority order, with the seniortranche getting Y1(k) = min[U1(k), W(k)] and themezzanine getting Y2(k) = min[U2(k), W(k) – Y1(k)].

The reserve available before payments atperiod k, R(k), is thus defined by [recall that Z(k) isthe total cash flow from the participations inperiod k]

(17)

Unpaid reductions in principal from defaultlosses occur in reverse priority order, so the juniorresidual tranche suffers the reduction

J3(k) = min[F3(k – 1), H(k)], (18)

where

, (19)

is the total of default losses since the previous cou-pon date less collected and undistributed interestincome. Then, the mezzanine and senior tranchesare successively reduced in principal by, respec-tively,

J2(k) = min[F2(k – 1), H(k) – J3(k)] (20a)

and

J1(k) = min[F1(k – 1), H(k) – J3(k) – J2(k)]. (20b)

Under uniform prioritization, no early pay-ments of principal are made, so D1(k) = D2(k) = 0 fork < K. At maturity, the remaining reserve is paid inpriority order, and principal and accrued interestare treated identically, so without loss of generalityfor purposes of valuation, we take

Y1(K) = Y2(K) = 0, (21a)

D1(K) = min[F1(K) + U1(K), R(K)], (21b)

and

D2(K) = min[F2(K) + U2(K), R(K) – D1(K)]. (21c)

The residual tranche finally collects

D3(K) = R(K) – Y1(K) – D1(K) – Y2(K) – D2(K). (21d)

For the alternative fast-prioritization scheme,the senior tranche is allocated interest and principal

Figure 4. New 10-Year Par-Coupon Spreads for the Base-Case Parameters and for the Pure-Jump Intensity Parameters

600

550

500

450

400

350

300

250

200

Coupon Spread(basis points)

0

Time (years)

2 4 6 8 10

Pure JumpsBase Case

R k( ) 1 r k( )4

----------+=

a R k 1–( ) Y1 k 1–( )– Y2 k 1–( )–[ ]×

z Z k( ) .+

H k( ) max Li W k( ) Y1 k( )– Y2 k( )–[ ] 0,–i B k( )∈∑

=

Table 3. Conditional Probabilities of Default and Diversity Scoresρ = 0.1 ρ = 0.5 ρ = 0.9

Set P(di) P(di|dj)Diversity

Score P(di|dj)Diversity

Score P(di|dj)Diversity

Score

1 0.386 0.393 58.5 0.420 21.8 0.449 13.22 0.386 0.393 59.1 0.420 22.2 0.447 13.53 0.386 0.392 63.3 0.414 25.2 0.437 15.84 0.386 0.393 56.7 0.423 20.5 0.454 12.4



payments as quickly as possible until maturity oruntil its principal remaining is reduced to zero,whichever is first. Until the senior tranche is paidin full, the mezzanine tranche accrues unpaid inter-est at its coupon rate. Then, the mezzanine trancheis paid interest and principal as quickly as possibleuntil maturity or until the mezzanine tranche isretired. Finally, any remaining cash flows are allo-cated to the residual tranche. Specifically, in cou-pon period k, the senior tranche is allocated theinterest payment,

Y1(k) = min[U1(k), Z(k)] (22a)

and the principal payment

D1(k) = min[F1(k – 1), Z(k) – Y1(k)], (22b)

where the total cash generated by the collateralpool, Z(k), is again defined by Equation 14. Themezzanine receives the interest payments,Y2(k) = min[U2(k), Z(k) – Y1(k) – D1(k) – Y2(k)] (23a)

and principal paymentsD2(k) = min[F2(k – 1), Z(k) – Y1(k) – D1(k) – Y2(k)]. (23b)

Finally, any residual cash flows are paid to thejunior subordinated tranche. For this scheme, nocontractual reductions in principal occur [that is,Ji(k) = 0].

In practice, many other types of prioritizationschemes are possible. For example, during the lifeof a CDO, failure to meet certain contractual over-collateralization ratios in many cases triggers a shiftto some version of fast prioritization. For our exam-ples, the CDO yield spreads for uniform and fastprioritization would provide upper and lowerbounds, respectively, on the senior spreads thatwould apply if one were to add such a feature to theuniform prioritization scheme that we illustrated.

Simulation Methodology. Our computa-tional approach consists of simulating piecewise lin-ear approximations of the paths of Xc and X1, . . ., XNfor time intervals of some relatively small fixed-length ∆t. (We use a ∆t interval of one week.)Defaults during one of these intervals are simulatedat the corresponding discretization of the totalarrival intensity,

, (24)

where the variable 1A(i,t) equals 1 if issuer i has notdefaulted by t and equals 0 otherwise. (That is, onlythe intensities of the participations still alive aresummed.) When a default arrives, the identity of thedefaulter is drawn at random, with the probabilitythat i is selected as the defaulter given by the discret-ization approximation of λi(t)1A(i,t)/Λ(t). The prob-

ability that more than one default will occur duringone time step is very small; hence, we ignore it.

Based on experimentation, we chose to simu-late 10,000 pseudo-independent scenarios.16

Results for Par CDO Spreads. The esti-mated par spreads of the senior (s1) and mezzanine(s2) CDO tranches for the four-parameter sets areshown in Table 4 for various levels of overcollater-alization and for the two prioritization schemes. Toillustrate the accuracy of the simulation methodol-ogy, we show in parentheses estimates of the stan-dard deviation of these estimated spreads resultingfrom “Monte Carlo noise.” Table 5 and Table 6show estimated par spreads for the case of, respec-tively, “low” (ρ = 0.1) and “high” (ρ = 0.9) defaultcorrelations. In all these examples, the risk-free rateis 0.06 and no management fees are considered.

Figure 5 and Figure 6 illustrate the impacts onthe market values of the three tranches of a givenCDO structure of changing the correlation param-eter ρ (with uniform prioritization). The base-caseCDO structures used for this illustration weredetermined by uniform prioritization of senior andmezzanine tranches whose coupon rates are at parfor the base-case Parameter Set 1 and with thecorrelation parameter ρ = 0.5. For example, supposethis correlation parameter is moved from the base

Λ t( ) λii∑ t( )1A i t,( )=

Table 4. Par Spreads (ρ = 0.5)(estimated standard deviation in parentheses)

Uniform Scheme Fast Scheme

Set s1 s2 s1 s2

Principal: P1 = 92.5; P2 = 5

1 18.7 bps 636 bps 13.5 bps 292 bps(1) (16) (0.4) (1.6)

2 17.9 589 13.5 270(1) (15) (0.5) (1.6)

3 15.3 574 11.2 220(1) (14) (0.5) (1.5)

4 19.1 681 12.7 329(1) (17) (0.4) (1.6)

Principal: P1 = 80; P2 = 10

1 1.64 bps 67.4 bps 0.92 bps 38.9 bps(0.1) (2.2) (0.1) (0.6)

2 1.69 66.3 0.94 39.5(0.1) (2.2) (0.01) (0.6)

3 2.08 51.6 1.70 32.4(0.2) (2.0) (0.2) (0.6)

4 1.15 68.1 0.37 34.6(0.1) (2.0) (0.2) (0.6)



case of 0.5 to 0.9. Figure 6, which treats the case ofrelatively little subordination available to thesenior tranche (P1 = 92.5), shows that the loss indiversification reduces the market value of thesenior tranche from 92.5 to about 91.9. The marketvalue of the residual tranche, which benefits fromvolatility in the manner of a call option, increasesfrom 2.5 to approximately 3.2, a dramatic relativechange. Although a precise statement of convexityis complicated by the timing of the prioritizationeffects, the effect illustrated by Figures 5 and 6 isalong the lines of Jensen’s inequality; an increase incorrelation also increases the (risk-neutral) vari-ance of the total loss of principal. These opposingreactions to diversification of the senior and juniortranches also show that the residual tranche mayoffer some benefits to certain investors as a volatil-ity hedge against default risk for the senior tranche.

The mezzanine tranche absorbs the net effectof the impacts of correlation changes on the marketvalues of the senior and junior residual tranches (inthis example, the result is a decline in market valueof the mezzanine from 5.0 to approximately 4.9),which it must do because the total market value of

the collateral portfolio is not affected by the corre-lation of default risk. These effects can be comparedwith the impact of correlation on the par spreads ofthe senior and mezzanine tranches that are shownin Tables 4, 5, and 6. Clearly, given the relatively

Table 5. Par Spreads (ρ = 0.1)Uniform Scheme Fast Scheme

Set s1 s2 s1 s2

Principal: P1 = 92.5; P2 = 5

1 6.7 bps 487 bps 2.7 bps 122.bps2 6.7 492 2.9 1173 6.3 473 2.5 1024 7.0 507 2.7 137


1 0.27 bps 17.6 bps 0.13 bps 7.28 bps

2 0.31 17.5 0.15 7.92

3 0.45 14.9 0.40 6.89

4 0.16 19.0 0.05 6.69

Table 6. Par Spreads (ρ = 0.9)Uniform Scheme Fast Scheme

Set s1 s2 s1 s2

Principal: P1 = 92.5; P2 = 5

1 30.7 bps 778 bps 23.9 bps 420.bps2 29.5 687 23.7 3973 25.3 684 20.6 3254 32.1 896 23.1 479


1 3.17 bps 113.bps 1.87 bps 68.8 bps2 3.28 112 1.95 70.03 4.03 90 3.27 60.44 2.52 117 1.06 65.4

Figure 5. Impact on Tranche Value of Correla-tion: High Overcollateralization (P1 = 80)

Note: Uniform prioritization. The premium is the market valuenet of par.

Figure 6. Impact on Tranche Value of Correla-tion: Low Overcollateralization(P1 = 92.5)

Note: Uniform prioritization. The premium is the market valuenet of par.

0.5

0

–0.5

Premium(percentage of face value)

Initial Correlation

ρ = 0.1 ρ = 0.5 ρ = 0.9

Mezzanine JuniorSenior

0.8

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

–0.8

Premium(percentage of face value)

Initial Correlation

ρ = 0.1 ρ = 0.5 ρ = 0.9

Mezzanine JuniorSenior



small size of the mezzanine principal, the mezza-nine par spreads can be dramatically influenced bycorrelation. Moreover, experimenting with variousmezzanine overcollateralization values shows thatthe effect is ambiguous: Increasing default correla-tion may raise or lower mezzanine spreads.

Prioritization Scheme. For our example,ensuring faster payments to the senior and then tothe mezzanine tranches makes these tranches safer;that is, their par coupon spreads are smaller whenthe fast prioritization scheme is adopted. For thesenior tranche, this statement holds for two rea-sons: This tranche risks no contractual loss of prin-cipal (and hence no contractual loss of futureinterest) and it loses no interest payments to themezzanine tranche. For the mezzanine tranche, thestatement holds in our example because the effectof experiencing no contractual principal losses isstronger than that of postponing the interest pay-ments, which carries the risk of not receiving themin full. Tables 4–6 provide the computed spreads.

Risky Reinvestment. This method can alsobe used to allow for collateral assets that maturebefore the termination of the CDO. The defaultintensity of each new collateral asset is of the typegiven by Equation 5, where Xi is initialized at thetime of the purchase at the initial base-case level(long-run mean of λi). Figure 7 and Figure 8 showthe effect of changing from safe to risky reinvest-ment (with uniform prioritization). Given the“short-option” aspect of the senior tranche, itbecomes less valuable when reinvestment becomesrisky. Note that the mezzanine tranche benefitsfrom the increased variance in this case.

Analytical ResultsIn this section, exploiting the symmetry assump-tions of our special example, we provide analyticalresults for the probability distribution for the num-ber of defaulting participations and the total ofdefault losses of principal, including the effects ofrandom recovery.

The key to calculating these probability distri-butions exactly is the ability to explicitly computethe probability of survival of all participations in anychosen subgroup of obligors. These explicit proba-bilities must be evaluated with extremely highnumerical accuracy, however, because of the numer-ous combinations of subgroups to be considered.

For a given time horizon T, let dj denote theevent that obligor j defaults by T. That is, dj ={τj < T}. Also, let dj

c denote the event complemen-tary to dj, namely, dj

c = {τj > T}. Let M denote thenumber of defaults. Assuming symmetry (invari-

ance under permutation) in the unconditional jointdistribution of default times,

(25)

where

. (26)

We let q(k, N) = P(d1 ∩ . . . ∩ dk ∩ dck+1 ∩ . . . ∩ dN

c ).The probability pj = P(d1 ∪ . . . ∪dj) that at least one

Figure 7. Risky Reinvestment: Senior Tranche Spreads

Figure 8. Risky Reinvestment: Mezzanine Tranche Spreads

30

25

20

15

10

5

0

Spread (bps)

Parameter Set

1 2 3 4

Risky ReinvestmentSafe Reinvestment

700

600

500

400

300

200

100

0

Spread (bps)

Parameter Set

1 2 3 4

Risky ReinvestmentSafe Reinvestment

P M k=( ) =

N

k P d1 … dk d

ck 1+ … d

cN∩ ∩ ∩ ∩ ∩( ),

N

k N!

N k–( )!k!------------------------=



of the first j names will default by T is computedlater; for now, we take this calculation as given.

Proposition 2: We have17

. (27)

A proof, found in the Duffie and Gârleanu 1999working paper, is based on a careful counting of thenumber of scenarios in which k participationsdefault. Using the fact that the pre-intensity of thefirst-to-arrive τ(j) = min(τ1, . . ., τj) of the stoppingtimes τ1, . . ., τj is λ1 + . . . + λj and using theindependence of X1, . . ., XN and Xc, we have

(28)

where αc(T) and βc(T) are given explicitly in Appen-dix A as the solutions of the ordinary differential

Equations A2 and A3 for the case in which n = –κ,p = σ2, q = –j, l = lc, and m = κθc; αi(T) and βi(T) arethe explicitly given solutions of Equations A2 andA3 for the case in which n = –κ, p = σ2, q = –1, l = li,and m = κθi.

It is not hard to see how to generalize Proposi-tion 2 so as to accommodate more than one type ofintensity—that is, how to treat a case with severalinternally symmetric pools. Introducing each suchgroup increases by one the dimensions of the arrayp, however, and the summation. Thus, given therelatively lengthy computation required to obtainadequate accuracy for even two subgroups of issu-ers, one might prefer simulation to this analyticalapproach for multiple types of issuers.

Based on this analytical method, Figure 9shows the probability q(k, 100) of k defaults within10 years out of the original group of 100 issuers forParameter Set 1 for a correlation-determiningparameter ρ that is high (0.9) or low (0.1). Figure 10shows the associated cumulative probability func-tions, including the base case of ρ = 0.5. For exam-ple, from Figure 9, the likelihood of at least 60defaults out of the original 100 participations in 10

q k N,( ) 1–( ) j k N 1+ + +( )

j 1=

N

∑ k

N j– pj=

pj 1 P τ j( )T>[ ]–=

1 E exp λi t( )i 1=

j

∑ td0T∫–

–=

1 eαc

T( ) βc T( )Xc 0( ) jαi T( ) jβi T( )Xi 0( )+ + +,–=

Figure 9. Probability of k Defaults for High and Low Correlation

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

Probability of k Defaults

0

Number of Defaults, k

10 20 40 50 60 7030 80 90

ρ = 0.1

ρ = 0.9

100



years is on the order of 1 percent for the low-correlation case, whereas it is roughly 12 percentfor the high-correlation case. One can use thismethod to compare the probability q(k, 100) of kdefaults across all four parameter sets.18

One can also compute analytically the likeli-hood of a total loss of principal of a given amountx.19 The approach is to add up, over k, the proba-bilities q(k, 100) of k defaults multiplied by the prob-abilities that the total loss of principal from kdefaults is at least x.20 A sample of the resulting lossdistributions is illustrated in Figure 11.

One can also analytically compute the varianceof total loss of principal, from which come diversityscores, as tabulated for our example in Table 3. Adescription of the computation of diversity scoresfor a general pool of collateral, not necessarily withsymmetric default risk, is in Appendix B. Given a(risk-neutral) diversity score of n, one can estimateCDO yield spreads by a much simpler algorithm,which approximates by substituting the compari-son portfolio of n independently defaulting partic-ipations for the actual collateral portfolio.

The default times can be independently simu-lated directly from an explicit unconditional distri-

bution rather than through a much more arduoussimulation of the pre-intensity processes. The algo-rithm is roughly as follows: • Simulate n draws from the explicit distribution

of the default time of a participation in thecomparison portfolio. Record those defaulttimes that are before T—say, M in number—and ignore the others.

• Simulate M fractional losses of principal.• Allocate cash flows to the CDO tranches,

period by period, according to the desired pri-oritization scheme.

• Discount the cash flows of each CDO trancheto a present value at risk-free rates.

• For each tranche, average the discounted cashflows over independently generated scenarios.A comparison of the resulting approximation

of CDO spreads with those computed earlier isprovided for certain cases in Figures 12–14. Forwell-collateralized tranches, the diversity-basedestimates of spreads are reasonably accurate (asshown in Figures 12 and 14), at least relative to theuncertainty that one would in any case haveregarding the actual degree of diversification in thecollateral pool. For highly subordinated tranchesand with moderate or large default correlations, the

Figure 10. Cumulative Probability of Number of Defaults: High, Low, and Base-Case Correlation

ρ = 0.1

1.0

0.8

0.6

0.4

0.2

0

Probability of k or Fewer Defaults

0

Number of Defaults, k

10 20 40 50 60 7030 80 90 100

ρ = 0.9 ρ = 0.5



Figure 11. Probability Density of Total Losses of Principal through Default: High Correlation

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

Probability Density

0

Total Losses (percentage of face value)

10 20 40 50 6030

Set 3Set 4

Set 1Set 2

Figure 12. Spread Comparisons for Senior Tranches: High Correlation (ρ = 0.9), Low Subordination (P1 = 92.5)

40

30

20

10

0

Spread (bps)

Parameter Set

2 3 41

Diversity-Score SpreadFull-Simulation Spread

Figure 13. Spread Comparisons for Mezzanine Tranches: Moderate Correlation (ρ = 0.5), Low Subordination (P1 = 92.5, P2 = 5)

1,200

1,000

800

600

400

200

0

Spread (bps)

Parameter Set

2 3 41




diversity-based spreads can be rather inaccurate, ascan be seen in Figure 13.

Another source of approximation error fromdiversity-score-based calculations is concentrationrisk in the original portfolio. Suppose, for example,that the notional amount of one obligor is a largefraction of the total notional amount of the collat-eral pool. The hypothetical comparison portfolio,with equal notional amounts to each obligor, mayhave tail risk of default losses that is markedlysmaller than the tail risk of the actual portfolio,despite having the same mean and variance ofdefault losses. One might allow for concentrationrisk by construction of a comparison portfolio withsimilar concentration but still assuming no correla-tion of default losses.

Figure 15 shows the likelihood of a total loss ofprincipal of at least 24.3 percent of the original facevalue as the correlation-determining parameter ρ isvaried. The figure also illustrates the calculation ofthe probability of failing to meet an overcollateral-ization target.

ConclusionsDefault-time correlation has a significant impact onthe market values of individual tranches. The pri-ority of the senior tranche, by which it is effectively“short a call option” on the performance of theunderlying collateral pool, causes its market valueto decrease with the risk-neutral default-time cor-relation, fixing the (risk-neutral) distribution ofindividual default times. The value of the equitypiece, which resembles a call option, increases withcorrelation. Intermediate tranches exhibit no clearJensen effect. With sufficient overcollateralization,the option “written” (to the lower tranches) domi-nates, but it is the other way around for sufficientlylow levels of overcollateralization.

Spreads, at least for mezzanine and seniortranches, are not especially sensitive to the “lump-iness” in the arrival of information about creditquality, in that replacing the contribution of diffu-sion with jump risks (of various types), while hold-ing constant the degree of mean reversion and theterm structure of credit spreads, plays a relativelysmall role in the determination of spreads.

Regarding alternative computational meth-ods, if (risk-neutral) diversity scores can be evalu-ated accurately, which is computationally simplein the framework we proposed, these scores can beused to obtain good approximate market valua-tions for reasonably well-collateralized tranches.

Currently, the weakest link in the chain ofCDO analysis is the availability of empirical datathat would bear on the correlation, actual or riskneutral, of default.

Figure 14. Spread Comparisons for Mezzanine Tranches: Moderate Correlation (ρ = 0.5), High Subordination (P1 = 80, P2 = 10)

Figure 15. Likelihood of Total Default Losses of at Least 24.3 Percent of Principal within 10 Years

This work was supported in part by a grant from theGifford Fong Associates Fund at the Graduate School ofBusiness, Stanford University. We are grateful for dis-cussions with Ken Singleton of Stanford University,Reza Bahar of Standard and Poor’s, and Sergio Kostekof Morgan Stanley Dean Witter. Helpful comments andresearch assistance were provided by Jun Pan. We arealso grateful for several helpful suggestions for improve-ment by Kazuhisa Uehara of the Fuji Research InstituteCorporation.

80

60

40

20

0

Spread (bps)

Parameter Set

2 3 41


��

��

��

��

��

��

�� !"

��!��# ��!�

�$ �� %�� &



Appendix A. Solution for the Basic Affine ModelFor the basic affine model, the expectation

(A1)

is given by coefficient functions α(s) and β(s) solv-ing the following differential equations:

(A2)

and

, (A3)

with the general boundary conditions α(0) = v andβ(0) = u. The survival probability (Equation 4) isobtained as a special case, with u = v = 0, n = –κ,p = σ2, q = –1, and m = κθ. Barring degenerate cases,which may require separate treatment, the solutionsare given by

(A4)

and

, (A5)

where a1 = (d1 + c1)u–1

b2 = b1

With these explicit solutions, we can compute thecharacteristic function ϕ(•) of λ(s) given λ(0),defined by

(A6)

by taking q to equal zero and u to equal iz andtreating the coefficients α(s) and β(s) as complex.The Laplace transform L(•) of λ(s) is computedanalogously:

(A7)

In this case, we take u = –z. For details and exten-sions to a general affine jump diffusion setting, seeDuffie, Pan, and Singleton (2000).

Appendix B. Computation of Diversity ScoresWe define diversity score S associated with a “tar-get” portfolio of bonds of total principal F to be thenumber of identically and independently default-ing bonds, each with principal F/S, whose totaldefault losses have the same variance as the targetportfolio’s default losses. The computation of Sentails computation of the variance of losses on thetarget portfolio of N bonds, which we address inthis appendix.

For simplicity, we ignore any interest rateeffects on losses in the calculation of diversity,although such effects could be tractably incorpo-rated—by the discounting of losses, for example.(With correlation between interest rates anddefault losses, the meaningfulness of direct dis-counting is questionable.) Also, we do not sepa-rately consider lost-coupon effects in diversityscores. These effects, which could be particularlyimportant for high-premium bonds, can be cap-tured along the lines of the following calculations.Finally, diversity scores do not account directly forthe replacement of defaulted collateral or newinvestment in defaultable securities during the lifeof a product, except insofar as covenants or ratingsrequirements stipulate a minimum diversity scorethat is to be maintained for the current collateralportfolio for the life of the CDO structure.

Letting di denote the indicator of the event thatparticipation i defaults by a given time T and lettingLi denote the random loss of principal when thisevent occurs, the variance of total default losses is

E exp qλ t( ) td0s∫[ ]e

v uλ s( )+

eα s( ) β s( )+

=

α′s mβs lµβs

1 µβs–------------------+=

β′s nβs12-- pβs

2q+ +=

αs vm a1c1 d1–( )

b1c1d1--------------------------------

c1 d1eb1s

+

c1 d1+--------------------------

mc1-----

s+log+=

l a2c2 d2–( )b2c2d2

------------------------------+c2 d2e

b2s+

c2d2--------------------------

lc2----- l–

s+log

βs1 a1e

b1s+

c1 d1eb1s

+--------------------------=

b1

d1 n 2qc1+( ) a1 nc1 p+( )+

a1c1 d1–-------------------------------------------------------------------=

c1– n n

2 2pq–+2q

---------------------------------------=

d1 1 c1u–( )n pu n pu+( )2p pu

2 2nu 2q+ +( )–+ +2nu pu

2 2q+ +-----------------------------------------------------------------------------------------------------=

a2d1c1-----=

c2 1 uc1-----–=

d2d1 µa1–

c1--------------------=

ϕ z( ) E eizλ s( )[ ]=

eα s( ) β s( )λ s( )+ ,=

L z( ) E ezλ s( )–[ ]=

eα s( ) β s( )λ s( )+ .=



(B1)

Given an affine intensity model, one can com-pute all terms in Equation B1. In the symmetriccase, letting p(1) denote the marginal probability ofdefault of a bond and p(2) denote the joint probabil-ity of default of any two bonds, Equation B1reduces to

(B2)

Equating the variance of the original pool to that ofthe comparison pool yields

(B3)

Solving Equation B3 for the diversity score, S, onegets

. (B4)

To end the computation, one uses the identitiesp(1) = p1 and p(2) = 2p1 – p2, where p1 and p2 arecomputed according to Equation 28 in the text, andthe fact that E(Li

2 ) = 1/3 and [E(Li)]2 = 1/4 if we

assume losses are uniformly distributed on (0,1).(Other assumptions about the distribution of Li,even allowing for correlation here, can be accom-modated.)

More generally, suppose λi(t) = bi[X(t)] andλj(t) = bj[X(t)], where X is a multivariate affineprocess of the general type considered in AppendixA and bi and bj are coefficient vectors. Even in theabsence of symmetry, for any times t(i) and t(j),assuming without loss of generality that t(i) ≤ t(j),the probability of default by i before t(i) and of jbefore t(j) is

(B5)

where b(t) = bi + bj for t < t(i) and b(t) = bj fort(i) ≤ t ≤ t(j). Each of the terms in Equation B5 isanalytically explicit in an affine setting, as can begathered from Appendix A.

Beginning with Equation B5, the covariance ofdefault losses between any pair of participationsduring any pair of respective time windows maybe calculated, and from that result, the total vari-ance of default losses on a portfolio and, finally,diversity score S can be calculated. With lack ofsymmetry, however, one must take a stand on thedefinition of “average” default risk to be applied toeach of the S independently defaulting issues of thecomparison portfolio. We do not address that issuehere. A pragmatic decision could be based on fur-ther investigation, perhaps accompanied by addi-tional empirical work.

Notes

1. A synthetic CLO differs from a conventional CLO in thatfor a synthetic CLO, the bank originating the loans does notactually transfer ownership of the loans to the SPV but,instead, uses credit derivatives to transfer the default riskto the SPV. A synthetic CLO may be preferred when thedirect sale of loans to SPVs might compromise client rela-tionships or secrecy or would be costly because of contrac-tual restrictions on transferring the underlying loans.

Unfortunately, regulations do not always provide the samecapital relief for a synthetic CLO as for a standard balancesheet CLO (see Punjabi and Tierney 1999).

2. This issue is related to the effects of adverse selection, butit also depends on the total size of the issue.

3. Supporting technical details are provided in the Duffie andGârleanu 1999 working paper.

var Lidii 1=

N

∑

E Lidii 1=

N

∑ 2

E Lidii 1=

N

∑ 2

–=

E(Li2 ) E (di

2 )i 1=

N

∑=

E LiLj( )E didj( )i 1≠∑+

E Li( )[ ]2E di( )[ ]2 .

i 1=

N

∑–

var Lidii 1=

N

∑

Np 1( )E Li2( )=

N N 1–( )p 2( ) E Li( )[ ]2+

N2p 1( )

2E Li( )[ ]2 .–

NS---- p 1( )E Li

2( ) p 1( )2

E Li( )[ ]2–

p 1( )E(Li2 )=

N 1–( )p 2( ) E Li( )[ ]2+

Np 1( )2 E Li( )[ ]2 .–

S

N p 1( )E L12( ) p 1( )

2E Li( )[ ]2

–

p 1( )E L12( ) N 1–( )p 2( ) E Li( )[ ]2

Np 1( )2

E Li( )[ ]2–+

-------------------------------------------------------------------------------------------------------------------------=

E didj( ) 1 E eλi u( ) ud

0

t i( )∫––=

E eλj u( ) ud

0

t j( )∫––

E eb t( )X t( ) ud

0

t j( )∫–,+



4. An alternative would be a model in which simultaneousdefaults could be caused by certain common credit events.An example is a multivariate exponential model (see Duffieand Singleton 1998). Another alternative is a contagionmodel, such as the static “infectious default” model ofDavis and Lo (1999).

5. An affine interest-rate process is one in which zero-couponyields are linear with respect to underlying state variables.For a more precise definition, see Duffie and Kan (1996).

6. A technical condition that is sufficient for the existence of astrictly positive solution to Equation 3 is that κθ ≥ σ2/2. Wedo not require it because none of our results depends onstrict positivity.

7. A proof can be found in the Duffie and Gârleanu 1999working paper. The method of the proof is to verify that theLaplace transform of Xt + Yt is that of an affine process withinitial condition X0 + Y0 and parameters (κ, θ, σ, µ, l ).

8. Our model is “doubly stochastic,” in the sense that, condi-tional on the processes λ1, . . ., λN, the default times τ1, . . .,τn are independent and are the first jump times of countingprocesses with these respective intensities. Thus, the onlysource of default time correlation in our model is throughcorrelation in the intensity processes.

9. A modeling alternative allowing for the treatment of corre-lated interest rate risk, and technical details, is in the 1999Duffie and Gârleanu working paper.

10. This pricing approach was developed by Lando (1998) forslightly different default intensity models. Lando alsoallowed for correlation between interest rates and defaultintensity.

11. Moody’s would not rely exclusively on the diversity scorein rating CDO tranches.

12. That is, B(k) is the subset of A(k – 1) that is not in A(k).

13. Because all parameter sets have the same κ parameter,maintaining the same term structure of zero-coupon yieldswas a rather straightforward numerical exercise, withEquation 4 used for risk-neutral default probabilities.

14. This calculation is based on the analytical methodsdescribed in the next section, “Analytical Results.”

15. As a practical matter, if the investors’ losses from default inthe underlying collateral pool are sufficiently severe,Moody’s may assign a “default” to a CDO tranche even ifit meets its contractual payments.

16. The basis for this approach and other multi-obligor default-time simulation approaches is discussed in Duffie and Sin-gleton (1998).

17. We use the convention that if l < 0 or m < l .

18. We have not included the graphs in this article, but we notethat low-correlation distributions are rather more similaracross the various parameter sets than are high-correlationdistributions. The full set of graphs is in the Duffie andGârleanu 1999 working paper.

19. We are referring to the risk-neutral likelihood, unless themodel under the objective probability is used.

20. For computation, we did not use the actual distribution ofthe total fractional loss of principal of a given number k ofdefaulting participations. For ease of computation, we sub-stituted with a central limit (normal) approximation for thedistribution of the sum of k identically and independentlydistributed uniform (0, 1) random variables, which ismerely the distribution of a normally distributed variablewith the same mean and variance. We were interested inthis calculation for moderate to large levels of x, corre-sponding to, for example, estimating the probability offailure to meet an overcollateralization target. We haveverified that, even for relatively few defaults, the centrallimit approximation is adequate for our purposes.

ReferencesAkerlof, G. 1970. “The Market for ‘Lemons’: QualitativeUncertainty and the Market Mechanism.” Quarterly Journal ofEconomics, vol. 84, no. 3:488–500.

Cox, J.C., J. Ingersoll, and S. Ross. 1985. “A Theory of the TermStructure of Interest Rates.” Econometrica, vol. 53, no. 2(March):385–407.

Davis, M., and V. Lo. 1999. “Infectious Default.” Working paper,Research and Product Development, Tokyo-MitsubishiInternational PLC.

DeMarzo, P. 1999. “Asset Sales and Security Design with PrivateInformation.” Working paper, Haas School of Business,University of California at Berkeley.

DeMarzo, P., and D. Duffie. 1999. “A Liquidity-Based Model ofSecurity Design.” Econometrica, vol. 67, no. 1:65–99.

Duffie, D., and N. Gârleanu. 1999. “Risk and Valuation ofCollateralized Debt Obligations, Unabridged Version.”Working paper, Graduate School of Business, StanfordUniversity (www.Stanford.edu/~duffie/working.htm).

Duffie, D., and R. Kan. 1996. “A Yield Factor Model of InterestRates.” Mathematical Finance, vol. 6, no. 4 (October):379–406.

Duffie, D., and K. Singleton. 1998. “Simulating CorrelatedDefaults.” Working paper, Graduate School of Business,Stanford University.

———. 1999. “Modeling Term Structures of DefaultableBonds.” Review of Financial Studies, vol. 12, no. 4 (Special):687–720.

Duffie, D., J. Pan, and K. Singleton. 2000. “Transform Analysisand Asset Pricing for Affine Jump Diffusions.” Econometrica, vol.68, no. 6 (November): 1343–76.

Feller, W. 1951. “Two Singular Diffusion Problems.” Annals ofMathematics, vol. 54, no. 1:173–182.

Innes, R. 1990. “Limited Liability and Incentive Contractingwith Ex-Ante Choices.” Journal of Economic Theory, vol. 52, no.1:45–67.

Lando, D. 1998. “On Cox Processes and Credit-RiskySecurities.” Review of Derivatives Research, vol. 2, nos. 2–3:99–120.

Punjabi, S., and J. Tierney. 1999. “Synthetic CLOs and Their Rolein Bank Balance Sheet Management.” Working paper, FixedIncome Research, Deutsche Bank.

Schorin, C., and S. Weinreich. 1998. “Collateralized DebtObligation Handbook.” Working paper, Fixed IncomeResearch, Morgan Stanley Dean Witter.

ml

0=

Documents

Risk and Valuation of Collateralized Debt ObligationsRisk and Valuation of Collateralized Debt Obligations Darrell Duffie and Nicolae Gârleanu In this discussion of risk analysis