RiskTranchesCreatedfromMortgages-HullWhite

54 www.cfapubs.org 2010 CFA Institute

Financial Analysts JournalVolume 66 Number 52010 CFA Institute

The Risk of Tranches Created from Mortgages

John Hull and Alan White

Using the criteria of the rating agencies, the authors tested how wide the AAA tranches created fromresidential mortgages can be. They found that the AAA ratings assigned to ABSs were not totallyunreasonable but that the AAA ratings assigned to tranches of Mezz ABS CDOs cannot be justified.

ating agencies have come under a greatdeal of criticism since the start of the sub-prime crisis in July 2007. Some commenta-tors have argued that the AAA ratings

assigned to the structured products created fromresidential mortgages convinced investors that thenew products were almost completely free of risk.As a result, some investors were lulled into a falsesense of confidence and did not evaluate the prod-ucts for themselves. Recent research by Adelino(2009) supports this view. He tested whether yieldspreads on the AAA rated products at issuancecontained information, in addition to that in theirratings, that would be useful in predicting perfor-mance. He concluded that they did not.1

The traditional business of rating agencies isthe rating of corporate and sovereign bonds, whichis based on a combination of analysis and judg-ment. Thus, the rating of structured products wasa departure from their traditional business. Itinvolved the application of a model rather thananalysis and judgment. The rating agencies werequite open about the models they used. Standard& Poors (S&P) and Fitch Ratings based theirratings on the probability of loss given by theirmodels. If the calculated probability of loss on astructured product corresponded to the probabilityof loss on a AAA rated bond, the structured prod-uct was rated AAA. In contrast, Moodys InvestorsService based its ratings on the expected loss as apercentage of principal. If the expected loss on astructured product corresponded to the expectedloss on a Aaa rated bond, the structured productwas rated Aaa. When a structured product wasdesigned, its creators wanted to achieve their target

ratings for tranches by meeting the model require-ments of the rating agencies.2 Typically, their objec-tive was to make the total principal of the AAAtranches that they created as large as possible. Theyoften received advance rulings on ratings beforefinishing the products design.

We evaluated whether the AAA ratingsassigned to structured products by rating agencieswere reasonable. We looked at both asset-backedsecurities (ABSs), which were created from pools ofmortgages, and ABS collateralized debt obligations(ABS CDOs), which were created from the tranchesof several ABSs. Coval, Jurek, and Stafford (2008)also evaluated ratings for these types of products.They assumed, however, that the asset poolsunderlying ABS CDOs have zero default correla-tion with each other. We did not make that assump-tion because the risks of the tranches in an ABSCDO are critically dependent on the correlationbetween different asset pools.

The Structured ProductsFrom 1999 to 2007, financial institutions foundsecuritization increasingly attractive for a numberof reasons. Securitization was a way to reduce reg-ulatory capital. It was profitable because theweighted average interest paid on the securitizedproducts was less than the weighted average inter-est earned on the underlying assets. This profit,referred to as spread arbitrage, was an essentialaspect of the originate-to-distribute model thatwas used by many banks. Rather than keep on theirbalance sheets assets that they originated, financialinstitutions could pass the credit risk on to inves-tors. Many types of assets were securitized: corpo-rate debt, credit card receivables, car loans, and soon. In our study, we focused on the securitizationof subprime residential mortgages.

John Hull and Alan White are professors of finance atthe Joseph L. Rotman School of Management, Universityof Toronto, Ontario, Canada.

R

September/October 2010 www.cfapubs.org 55


ABSs. Gorton (2008) discussed in some detailthe nature of the ABSs created from subprimeresidential mortgages. Figure 1 illustrates a typicalstructure. The subprime mortgage portfolio mightconsist of 1,000 mortgages. This underlying collat-eral was allocated to (1) one or more seniortranches rated AAA, (2) a number of mezzaninetranches rated AA, A, or BBB, and (3) subordinatedtranches that were either unrated or rated BB.Principal was allocated to each tranche. Some-times, the total principal of the mortgages equaledthe total principal of the tranches. In other cases,there was overcollateralization, so the total princi-pal of the mortgage portfolio exceeded the totalprincipal of the tranches.

A key aspect of the structures design was theamount of principal allocated to each rating cate-gory. Typically, 7585 percent of the mortgageprincipal was allocated to AAA tranches. The prin-cipal allocated to other rating categories was muchsmaller. For example, the BBB tranches, takentogether, typically accounted for 3 percent or lessof the mortgage principal. One of the goals of theABS designers was to create value from spreadarbitrage, as previously mentioned. The greatestvalue was created when the credit quality (as mea-sured by the credit rating) of the tranches wasmaximized. First, the AAA tranches were made aswide as possible, and then the AA tranches weremade as wide as possible, and so on.

The rules for allocating cash flows from mort-gages to tranches were defined by what was knownas a waterfall. The interest payments from mort-gages were typically allocated to tranches in orderof seniority. Thus, the AAA rated tranches receivedpromised interest payments on their outstandingprincipal first, and then the AA rated tranchesreceived promised interest payments on their out-standing principal, and so on.

The principal payments (both scheduled pay-ments and prepayments) were handled separatelyfrom interest payments, and the rules were rela-tively complicated. There was usually a lockoutperiod during which principal payments weresequential, which meant that all principal pay-ments went first to the most senior tranche. Whenthat tranche was completely amortized, paymentswent to the next most senior tranche, and so on.After the lockout period, if certain performancetargets were met, principal payments were allo-cated to tranches in proportion to their outstandingprincipal. If, however, there was a cumulative losstrigger event (where cumulative losses on the mort-gages were higher than a certain level) or a delin-quency event (where the rate of delinquency over athree-month period was above a certain level),principal payments became sequential again.

When the default rate was high, the repaymentof principal was, to a reasonable approximation,entirely sequential; thus, tranches bore losses inorder of reverse seniority. The unrated tranchesabsorbed losses first. Once their principal was lost,the BB rated tranches bore losses, and so on.

There were usually several tranches corre-sponding to each rating category. For example, theStructured Asset Investment Loan Trust (SAIL),issued in 2006, had four AAA tranches (with equalseniority), accounting for 83.25 percent of the col-lateral; two AA rated tranches (with unequalseniority), accounting for 8.2 percent of the collat-eral; three A rated tranches (with unequal senior-ity), accounting for 4.1 percent of the principal; andthree BBB tranches (with unequal seniority),accounting for 2.8 percent of the principal.

The BBB tranches were often very thin.Although the total of all BBB tranches mightaccount for about 3 percent of the total underlyingprincipal, each individual BBB tranche was oftenonly about 1 percent wide. If the macroeconomicenvironment was relatively benign and there werelarge repayments of principal, the AAA tranchecould be expected to shrink and the proportion ofthe remaining mortgage principal accounted for bythe BBB tranches could be expected to increase. Butif default rates were high, a thin BBB tranche could

Figure 1. Creation of Tranches from a Portfolio of Subprime Mortgages

AAA Tranches

Portfolio ofSubprimeMortgages

AA Tranches

A Tranches

BBB Tranches

BB Tranches

Unrated Tranches


Financial Analysts Journal

easily lose its entire principal. In the SAIL structure,the most senior BBB tranche (rated Baa1/BBB+)was 1.1 percent wide, with 3.25 percent subordina-tion; the next BBB tranche (rated Baa2/BBB) was0.85 percent wide, with 2.40 percent subordination;and the most junior BBB tranche (rated Baa3/BBB)was 0.8 percent wide, with 1.60 percent subordina-tion. Assuming that principal payments weresequential, the three tranches would lose theirentire principal if losses on the subprime mortgageportfolio were greater than 4.35 percent, 3.25 per-cent, and 2.40 percent, respectively.

ABS CDOs. In the second level of securitiza-tion, ABS CDOs were formed by creating tranchesfrom tranches. Two types were common: a HighGrade ABS CDO, created from the AAA, AA, and Atranches of ABSs, and a Mezz ABS CDO, createdfrom the BBB tranches of ABSs.

We focused on the Mezz ABS CDO, whosecreation is illustrated in Figure 2. Although theAAA rated tranche in Figure 2 was usually lesswide than the one in Figure 1, it typically accountedfor more than 50 percent of the ABS CDO principal.(In an example in Gorton [2008, p. 35], taken froma UBS publication, the AAA rated tranche of theABS CDO accounts for 76 percent of the principal.)

Many ABS CDOs were managed, and thus thetranches forming the collateral did not remain fixedover time. A portfolio manager was allowed totrade a certain percentage of the underlying collat-eral each year subject to restrictions with respect to

measures involving the ratings of the collateral,correlation, and the weighted average life of theunderlying assets.

ABS CDOs were like ABSs in that the waterfallwas complex. Losses tended to be allocated to themost junior tranches first. Coverage tests and trig-gers caused amortizations to be sequential anddiverted cash flows from junior to senior tranches.In certain circumstances, the senior tranche holderscould liquidate the assets.

The Default ModelIn our study, we focused on the criteria used by therating agencies. Under various assumptions, wetested (1) what the attachment point for a AAArated tranche should be if it is to have the sameprobability of experiencing losses as a AAA ratedcorporate bond and (2) what the attachment pointshould be if it is to have the same expected loss ofprincipal as a AAA rated corporate bond. Ourdefault model had three components:1. An estimate of the expected default rate

(EDR) for the mortgages in the underlyingportfolio (i.e., an estimate of the expectedproportion of the mortgages in the portfoliothat would default)

2. A correlation model that converted the EDRto a probability distribution for the actualdefault rate

3. A specification of the expected loss givendefault (ELGD) as a percentage of the initialmortgage principal

Figure 2. A Mezz ABS CDO Created from the BBB Tranches of an ABS

AAA Tranche

Portfolio ofBBB Tranches

Each BBB tranche iscreated from a different

pool of subprimemortgages.

AA Tranche

A Tranche

BBB Tranche

BB Tranche

Unrated Tranche



Our model was relatively simple in that it didnot incorporate a model of prepayment behavioror the timing of defaults. A more elaborate modelwould be essential for valuation, but the modelwas sufficient for our purposes because our objec-tive was to test the reasonableness of what ratingagencies did, not to value the securities. The ratingagency criteria did not consider the timing ofdefaults (although arguably they should).3 Model-ing prepayments explicitly would be more accu-rate but would also involve a big increase in thecomplexity of our model. Therefore, we assumedthat EDR and ELGD estimates incorporated theeffect of prepayments.

We did not explicitly consider overcollateral-ization. In high-default-rate situations, x percentof overcollateralization can be thought of as adummy junior tranche that absorbs the first xpercent of losses. The attachment points that wereport reflect the total subordination, includingovercollateralization.

We assumed that principal payments wereallocated to tranches sequentially, so losses wereallocated in reverse order of seniority.4 As men-tioned earlier, this approach corresponds to theway ABS CDOs usually work. It also correspondsto the way ABSs usually work for the first few yearsand to the way they usually work in subsequentyears when the default rate is high. In assigningratings, we were interested in observing howtranches fare in high-default-rate situations. Theassumption that principal is always allocatedsequentially is therefore reasonable for both ABSsand ABS CDOs.

We assumed that the mortgages in the poolhave equal principal and the same probability ofdefault. We also assumed that a mortgage pool issufficiently large that a large portfolio assumptionappliesthat is, the actual proportion of mortgagesdefaulting in the portfolio equals the probability ofeach mortgage defaulting, which we called thedefault rate. In practice, a pool has about 1,000 mort-gages. Our tests showed that the large portfolioassumption (which reduces computation time con-siderably) had only a small effect on our results.

Single Pool Correlation Model. Supposethat Q is the fraction of original mortgages in thepool that are expected to default within T years. Ifall the mortgages have similar risk, then Q is theprobability of default for any individual mortgage.A natural model to use is the one-factor Gaussiancopula model. Originally suggested by Li (2000),this model has become the standard market modelfor valuing synthetic CDOs as a result of researchby Gregory and Laurent (2005) and others. The one-

factor Gaussian copula model has both a factorcommon to all mortgages, which we denote by M,and a factor specific to mortgage i, which we denoteby Zi. The factors M and Zi are assumed to haveindependent standard normal distributions. In themodel, mortgage i defaults within T years if

where parameter K determines the expecteddefault rate and is the correlation between thetransformed times to default of any two mortgages.Under the assumptions of standard normal distri-butions, the probability of default is N(K), where Nis the cumulative normal distribution function. Themodel is calibrated to the expected default rate bysetting N(K) = Q.

The ith mortgage, therefore, defaults if

or

The realized default rate, P, conditional on M, is thus

(1)

Hull and White (2004) showed that any zeromean unit variance distributions can be chosen forM and Zi. They found that the double-t copulamodel, where both M and Zi have t-distributionswith four degrees of freedom (scaled so that thevariance is 1), fits market data on synthetic CDOswell. This model has considerably more tail defaultcorrelation (i.e., it has a higher probability ofextreme clustering of defaults) than the Gaussiancopula model.

In the double-t copula model, the ith mortgagedefaults if

where F is the cumulative probability distribu-tion5 of

The realized default rate, conditional on M, is

(2)

where H is the cumulative probability distribu-tion of a scaled t-distribution with four degreesof freedom.

M Z Ki+



Later in the article, we will present results fromour tests with both the Gaussian copula model andthe double-t copula model.

The Multi-Pool Correlation Model. Whenseveral pools are considered simultaneously, onemust define a between-pool factor, Mbp, andwithin-pool factors, Mwp, j. The factor Mbp affectsthe probability of default for all mortgages,whereas for any given j, Mwp, j affects the probabilityof default only for mortgages in pool j. In the multi-pool correlation model, the ith mortgage in the jthpool defaults if

where Zij is a variable that affects only the ithmortgage in the jth pool and is the cumulativeprobability distribution of

The variable Zij and the factors are independent ofeach other.

As before, the parameter is the total within-pool correlation. The parameter is the proportionof the default correlation that comes from a factorcommon to all pools. When = 0, the default ratesof different pools are independent of each other.(As noted earlier, Coval et al. [2008] used the multi-pool correlation model for normally distributedvariables.) At the other extreme, when = 1, asingle factor affects all mortgage defaults and thedefault rates in all mortgage pools are the same.

A two-factor model is useful when consider-ing ABS CDOs. One of the potential advantages ofABS CDOs over ABSs is that investors benefit fromboth between-pool and within-pool diversifica-tion. Suppose that half the underlying pools of anABS CDO consist entirely of mortgages on homesin Florida and the other half consist entirely ofmortgages on homes in California. If the defaultrate in California is less than perfectly correlatedwith the default rate in Florida, investors receive adiversification benefit. The parameter measuresthis benefit. If is low, this extra diversification isvaluable to investors, but if is high, it has verylittle value. Research suggests that correlationsincrease in stressed market conditions. For exam-ple, de Servigny and Renault (2002), who looked athistorical data on defaults and ratings transitionsto estimate default correlations, found that suchcorrelations are higher in recessions than in expan-sions. Das, Freed, Geng, and Kapadia (2006) useda reduced-form approach and computed the corre-lations between default intensities. They con-cluded that default correlations increase when

default rates are high. Ang and Chen (2002) foundthat the correlations between equity returns arehigher during a market downturn. Given that rat-ing agencies are most interested in what happensduring stressed market conditions, this researchsuggests that rating agencies should have used arelatively high value for .6 Note that if ABS mort-gage pools are already well diversified across theUnited States and thus very little diversificationbenefit can be derived from forming an ABS CDO,then should be close to 1.

The realized default rate for pool j, conditionalon Mbp and Mwp, j, is

(3)

where is the cumulative probability distributionof Zij. The simplest version of the model is whereall the M and Z variables have standard normaldistributions. We also considered the case wherethey all have t-distributions with four degrees offreedom (scaled so that the variance is 1), which wecalled the triple-t copula model.

Recovery Rate Model. We defined recoveryrate as the amount recovered in the event of adefault as a percentage of the initial principal, so itwould equal 1 minus ELGD. Credit derivativesmodels often assume that the recovery rate realizedin the event of a default is constant. This assump-tion is less than ideal. As the default rate increases,the recovery rate for a particular asset class can beexpected to decline because a high default rateleads to more of the assets coming on the marketand thus a reduction in price.7

As is now well known, this argument is partic-ularly true for residential mortgages. In a normalmarket, a recovery rate of about 75 percent is oftenassumed for this asset class. If this percentage isassumed to be the recovery rate in all situations, theworst possible loss on a portfolio of residentialmortgages under the model would be 25 percent;thus, the 25100 percent senior tranche of an ABScreated from the mortgages could reasonably beconsidered safe. (In fact, recovery rates on mort-gages have declined sharply in the high-default-rate environment since 2007.)

We defined the recovery rate when the defaultrate equals the expected default rate as R*, themaximum recovery rate (which occurs when thedefault rate is very low) as Rmax, and the minimumrecovery rate (which occurs when the default rateis very high) as Rmin. We used the following simplerecovery rate model for the actual recovery rate, R:8

(4)

M M Z

Q

bp wp j ij+ ( ) + < ( )

1 11

,

,

M M Zbp wp j ij+ ( ) + 1 1, .

( ) ( )

1 1

1

Q M Mbp wp j

, ,

R R R R aP= + ( ) ( )min max min exp ,



where

(5)

As before, P is the actual default rate and Qis the expected default rate. As P increases from0 to 100 percent, the recovery rate decreases fromRmax to close to Rmin in such a way that when P =Q, R = R*.9 Using Equation 1 or Equation 2, wecan express R as a function of M. This model isillustrated in Figure 3.

Subprime Default ExperienceSubprime mortgages first became common in theUnited States in 1999. Thus, in 2006 and 2007,rating agencies had relatively little experiencewith their performance.

Figure 4 shows statistics on subprime mort-gages collected by Moodys in March 2007.10 Forsubprime mortgages originated in a given year, itshows the cumulative percentage that was delin-quent after a given number of months. For thispurpose, delinquent mortgages are defined as thetotal of those mortgages (1) whose payments aremore than 60 days overdue, (2) in foreclosure, or (3)whose properties are being sold by the lender.

Moodys had more than five years experience withmortgages originated between 1999 and 2003. Thecumulative default rate for mortgages originatedfour or more years ago was between 2 percent and4 percent. Note that the percentage of delinquentloans in Figure 4 does not increase monotonicallywith time because borrowers who become delin-quent sometimes catch up on late payments, refi-nance, or sell their house.

Panel A of Figure 4 shows that there were signsthat mortgages originated in 2006 were performingworse than mortgages originated in the four previ-ous years. In March 2007, however, they appearedto be performing similarly to mortgages originatedbetween 1999 and 2001 (Panel B). After 11 months,the percentages of delinquent mortgages for 1999,2000, and 2001 were 6.10 percent, 7.63 percent, and7.15 percent, respectively. The percentage for the2006 mortgages was similar.11

In March 2007, investors in the AAA tranchesof ABSs could draw some comfort from the AAAABX indices, which indicated no serious impair-ment. The TABX index, which tracks the value ofAAA tranches formed from the BBB (BBB)tranches of ABSs, stood at 92.75 (84.00) at the endof March 2007.

aR R / R R

Q= ( ) ( ) ln * .maxmin min

Figure 3. The Recovery Rate Model

Note: The maximum recovery rate, Rmax, is 100 percent; the minimum recovery rate, Rmin, is 50 percent;the average recovery rate, R*, is 75 percent; and the expected default rate is 10 percent.

Recovery Rate, R (%)

100

75

500 4010 20 30 355 15 25

Realized Default Rate, P (%)



Of course, there were a number of warningsignals. The S&P/Case-Shiller Composite of 10Home Price Index, which was set at 100 in January2000, reached more than 225 in mid-2006 butstarted to decline by the beginning of 2007.Although few people anticipated the full extent ofthe fall in house prices that took place over the nexttwo years, there was general agreement that somedecline would occur. For obvious reasons, home-owners are much more likely to default whenhouse prices are falling than when they are rising.Therefore, the mortgage default experience from1999 to 2006 should have been treated with caution.

The evaluation of ABSs depends on (1) theexpected default rate, Q, for mortgages in theunderlying pool, (2) the default correlation, , for

mortgages in the pool, and (3) the recovery rate, R.Data for 19992006 suggest a value for Q of lessthan 5 percent (assuming an average mortgage lifeof five years). As previously mentioned, however,a different macroeconomic environment could beanticipated over the next few years. An estimate of10 percent, or even higher, would seem more pru-dent to use. (Later in the article, we report resultsfor values of Q equal to 5 percent, 10 percent, and20 percent.) The Basel II capital requirements arebased on a copula correlation of 0.15 for residentialmortgages.12 (We present results for values of between 0.05 and 0.30.) As already mentioned, arecovery rate of 75 percent is often assumed forresidential mortgages, but this percentage is prob-ably too optimistic in a high-default-rate environ-ment. (We report results for a recovery rate of 75percent and for the previously discussed recoveryrate model with R* = 75 percent, Rmin = 50 percent,and Rmax = 100 percent.)

ABS CDOs also depend on the parameter .Roughly speaking, this parameter measures theproportion of the default correlation that comesfrom a factor common to all pools. If is close tozero, investors can obtain good diversification ben-efits from the ABS CDO structure. In adverse mar-ket conditions, some mezzanine tranches can beexpected to suffer 100 percent losses, while otherswill incur no losses. But if is close to 1, all mezza-nine tranches will tend to sink or swim together.Because we do not know what estimates therating agencies made (ex post, of course, we knowthey were high), we present results that are basedon a wide range of values for this parameter.

ResultsAlthough mortgages are amortized over manyyears, prepayments lead to a weighted average lifeof about five years. Therefore, when determiningthe ratings of instruments created from mort-gages, their losses are compared with the losses onbonds over a five-year period. Table 1 reportsstatistics from Moodys for 19702007 concerningthe cumulative five-year probability of default forAAA and BBB bonds. The expected loss is calcu-lated from the probability of default, assuming arecovery rate of 40 percent (a typical recovery ratefor a corporate bond).

The Probability of Loss Criterion for ABSs.Suppose that the attachment point for the AAAtranche of an ABS is X percent, so the tranche isresponsible for losses between X percent and 100percent. The probability of the tranche experienc-ing losses is the probability that losses on the under-lying portfolio are greater than X percent. Given

Figure 4. Subprime Loans Delinquent 60 Days or More, in Foreclosure, or Held for Sale, 19992006

Percent of Original Balance

A. Loans Originated in 20022006

8

7

6

5

4

3

2

1

03 4815 27 3912 24 369 21 33 45426 18 30

Months since Issuance

Percent of Original Balance

B. Loans Originated in 19992001 and 2006

10

9

8

7

6

5

4

3

23 4815 27 3912 24 369 21 33 45426 18 30

Months since Issuance

20061999 2000 2001

2002 2003 2004

20062005



our large portfolio assumption that the proportionof defaulting mortgages equals the default rate, thetranche experiences losses when the default rate isgreater than

where R is the recovery rate for the mortgages.Equation 1 shows that this happens in the case ofthe Gaussian copula model when

From Table 1, the minimum attachment pointis the value of X for which the probability of lossis 0.1 percent. It follows that the minimum attach-ment point is

(6)

The variable R is the recovery rate when M =N1(0.001).

Similarly, Equation 2 shows that for thedouble-t copula model, the minimum attachmentpoint is

where, as before, H is the cumulative probabilitydistribution for a t-distribution with four degreesof freedom (scaled so that the variance is 1). In thiscase, R is the recovery rate when M = H1(0.001).

Table 2 shows results for various values of theexpected default rate, Q, and the copula correla-tion, . Four different models are considered: 1. The Gaussian copula model with a recovery

rate of 75 percent on the underlying mortgages2. The double-t copula model with a recovery

rate of 75 percent on the underlying mortgages3. The Gaussian copula model with the stochas-

tic recovery rate model in Equations 4 and 5,with R* = 75 percent, Rmax = 100 percent, andRmin = 50 percent

4. The double-t copula model with the stochas-tic recovery rate model in Equations 4 and 5,with R* = 75 percent, Rmax = 100 percent, andRmin = 50 percentAs might be expected, the minimum attach-

ment point increases as we move from the Gauss-ian copula to the double-t copula and from theconstant recovery rate model to the stochasticrecovery rate model. As mentioned earlier, theattachment point for AAA rated tranches was

Table 1. Cumulative Five-Year Probability of Default for AAA and BBB Bonds, 19702007

Probability of Loss Expected Loss

AAA 0.1% 0.06%BBB 1.8% 1.08%

Source: Moodys Investors Service.

XR1

,

11

1( ) ( )

>

R NN Q M

X

.

10 001

1

1 1( ) ( ) ( )

R NN Q N

.

.

10 001

1

1 1( ) ( ) ( )

R HF Q H

.

,

Table 2. Minimum Attachment Points for AAA Tranche of ABS with Probability of Loss under 0.1 Percent

Expected Default Rate

5% 10% 20%

Gaussian copula = 0.05 4.1% 6.8% 11.0%(constant recovery) = 0.10 6.0 9.4 13.9

= 0.20 9.6 13.6 18.2 = 0.30 13.1 17.2 21.1

Double-t copula = 0.05 7.6% 13.0% 18.2%(constant recovery) = 0.10 13.6 18.7 21.9

= 0.20 21.1 23.2 24.1 = 0.30 23.7 24.4 24.7

Gaussian copula = 0.05 7.3% 11.6% 17.1%(stochastic recovery) = 0.10 11.6 17.3 23.8

= 0.20 19.1 26.6 33.4 = 0.30 26.1 34.1 40.0

Double-t copula = 0.05 15.0% 25.3% 33.4%(stochastic recovery) = 0.10 27.2 37.2 41.8

= 0.20 42.2 46.3 46.6 = 0.30 47.4 48.7 47.8



typically 15 percent to 25 percent. There are someindications that attachment points were raised in2006. According to Moodys Investors Service(2007), Moodys AaA rated bonds issued in 2006were designed to withstand a total loss on theunderlying mortgage pool of approximately 26percent to 30 percent without defaulting.

Table 2 shows that when a 20 percent defaultrate is combined with a high default correlation anda stochastic recovery rate model, the AAA ratingsseem a little high. The ratings are also difficult tojustify when the most extreme model (double-tcopula, stochastic recovery rate) is used. But over-all, given the published criteria of rating agencies,the results in Table 2 suggest that the AAA ratingswere not totally unreasonable.

The Expected Loss Criterion for ABSs.If L(M) is the proportional loss on the mortgageportfolio for a particular value of M, the expectedproportional loss on the ABS when the attachmentpoint for the senior tranche is X is

(7)

where M* is the value of M that leads to a loss onthe portfolio equal to X and is the probabilitydensity of M. Because L(M) is always less than 1 Rmin, L(M) X is also less than 1 Rmin. It followsthat the expected loss is always less than 1 Rmintimes the probability of a loss. Assuming thatRmin, the minimum recovery rate for mortgages,is greater than the assumed recovery rate forbonds, it follows that a value of X that satisfies theprobability of loss criterion must also satisfy theexpected loss criterion.

Put another way, the minimum attachmentpoint under the expected loss criterion must be lessthan the minimum attachment point under theprobability of loss criterion.13 Table 3 confirmsthis result for the double-t copula model with sto-chastic recovery. Table 3 shows that even whenthis exacting model is used, the expected loss cri-terion would lead to a 7075 percent wide AAArated senior tranche that would be deemed reason-able when = 0.1.

The expected loss from a tranche equals theprobability of loss multiplied by the expected lossgiven default. ELGD is typically quite low for themost senior tranche, which means that expectedloss is relatively low for this tranche and explainswhy obtaining a AAA rating is relatively easy whenthe expected loss measure is used. For more juniortranches, which tend to be quite thin, ELGD is high.(As a tranche becomes infinitesimally thin, ELGDbecomes 1.) Thus, expected loss is relatively highfor these tranches and tends to produce more con-servative ratings than probability of loss.14

The Creation of BBB Tranches. BBBtranches must usually satisfy both the Moodysand the S&P/Fitch criteria. Interestingly, the S&P/Fitch criterion depends on only the attachmentpoint, whereas the Moodys criterion depends onboth the attachment point and the tranche width.In practice, the minimum attachment point waslikely determined by using the S&P/Fitch criterionand the minimum tranche width was likely deter-mined by using the Moodys criterion.

As an example of how this might work, sup-pose that we use the Gaussian copula model with aconstant recovery rate. Suppose further that the

L M X M dMM ( ) ( ) * ,

Table 3. Minimum Attachment Points for AAA Tranche of ABS under Expected Loss and Probability of Loss Criteria

Expected Default Rate

5% 10% 20%

Expected loss criterion = 0.05 3.9% 10.9% 19.7% = 0.10 10.5 21.2 28.9 = 0.20 24.7 33.2 37.3 = 0.30 33.4 39.0 41.1

Probability of loss criterion = 0.05 15.0% 25.3% 33.4% = 0.10 27.2 37.2 41.8 = 0.20 42.2 46.3 46.6 = 0.30 47.4 48.7 47.8

Notes: This table compares minimum attachment points for an ABSs AAA rated tranche when (1) theexpected loss criterion is used with the result that a AAA tranche can achieve an expected loss of lessthan 0.06 percent and (2) the probability of loss criterion is used with the result that a AAA tranche canachieve a probability of loss of less than 0.1 percent. The model is the double-t copula model with astochastic recovery rate. The recovery rate depends on the default rate and ranges from a high of 100percent to a low of 50 percent.



expected default rate, copula correlation, and recov-ery rate are 7 percent, 0.1, and 75 percent. The min-imum attachment point is the attachment point thatgives 1.8 percent in Equation 6, which we find to be4.90 percent. The expected loss in Equation 7 can becalculated numerically. When the attachment pointis 4.90 percent, the minimum detachment point isthe detachment point that gives 1.08 percent inEquation 7. Numerical analysis reveals that thisdetachment point is 5.93 percent. Therefore, a4.905.93 percent tranche just satisfies the criteria ofall three rating agencies. This type of analysis mayexplain why BBB tranches were so thin.

Data in Stanton and Wallace (2008) and otherdata we obtained by browsing the U.S. SEC web-site suggest that the average subordination of BBBtranches created in 2006 was about 4 percent andthe average tranche width was about 1 percent.Therefore, we considered a benchmark ABS CDOwhose underlying BBB tranches are responsiblefor losses of 45 percent on the underlying mort-gage portfolio.

The Probability of Loss Criterion for ABSCDOs. The probability distribution of losses for anABS CDO can be determined by using Monte Carlosimulation.15 Values for Mbp and Mbp, j are simulatedto determine the default rate and the loss rate for themortgages in each pool. If the average loss rate isless than the attachment point, the loss on the ABS

CDO tranche is zero. If the average loss rate isgreater than the detachment point, the loss on theABS CDO is 100 percent. If the average loss rate isbetween the attachment point and the detachmentpoint, the ABS CDO tranche suffers a partial loss.

We obtained results for an ABS CDO createdfrom 100 BBB tranches of CDSs, with each trancheresponsible for losses in the range of 4 percent to 5percent on the underlying portfolio.16 We consid-ered a number of different values for the and parameters. We also considered expected defaultrates of 5 percent and 10 percent on the underlyingmortgages. As before, we used four models:1. The two-factor Gaussian copula model with a

recovery rate of 75 percent on the underlyingmortgages

2. The two-factor triple-t copula model with arecovery rate of 75 percent on the underlyingmortgages

3. The two-factor Gaussian copula model with thestochastic recovery rate model in Equations 4and 5, with R* = 75 percent, Rmax = 100 percent,and Rmin = 50 percent

4. The two-factor triple-t copula model with thestochastic recovery rate model in Equations 4and 5, with R* = 75 percent, Rmax = 100 percent,and Rmin = 50 percent

Table 4 presents our results for models 1 and 4. Asexpected, the results for models 2 and 3 are betweenthese two extreme cases.

Table 4. Minimum Attachment Points for AAA Tranche of ABS CDO Formedfrom ABS Tranches Responsible for Losses of 45 Percent

= 0.05 = 0.25 = 0.50 = 0.75 = 0.95

Gaussian copula = 0.05 17.1% 42.7% 73.5% 96.2% 99.9%(constant recovery; = 0.10 29.7 62.3 89.7 99.8 99.9EDR = 10%) = 0.20 39.7 73.6 95.4 99.9 99.9

= 0.30 43.5 77.2 96.7 99.9 99.9


= 0.30 20.5 48.8 80.2 98.7 99.9

Triple-t copula = 0.05 95.9% 100.0% 100.0% 100.0% 100.0%(stochastic recovery; = 0.10 93.8 100.0 100.0 100.0 100.0EDR = 10%) = 0.20 92.0 100.0 100.0 100.0 100.0

= 0.30 90.3 100.0 100.0 100.0 100.0


= 0.30 80.0 99.0 100.0 100.0 100.0Note: The parameters and are so defined that the between-pool copula correlation is and thewithin-pool correlation is .



The pattern of results in Table 4 is differentfrom that in Table 2. Clearly, the attachment pointmust be quite high for a wide range of assumptions.In some cases, the attachment point is so high thata AAA rating for even a very thin senior tranche isunwarranted (i.e., the minimum attachment pointis 100 percent).

Table 5 and Table 6 explore the impact ofincreasing the width of the underlying BBBtranches. In Table 5, all the tranches are responsiblefor losses of 4 percent to 7 percent. In Table 6, allthe tranches are responsible for losses of 4 percentto 9 percent. Although the minimum attachmentpoint does decrease as the tranche is widened, inall cases where one moves away from a low-Gaussian copula model, an attachment point below50 percent becomes difficult to justify.

In practice, the underlying BBB tranches havesome heterogeneity. Table 7 tests the effect of thisheterogeneity by considering the case where theattachment point has a uniform distributionbetween 2 percent and 6 percent and the tranchewidth has a uniform distribution (independent ofthe first uniform distribution) between 1 percentand 5 percent. The average attachment point andtranche width are 4 percent and 3 percent, respec-tively, as before. Our results clearly show that theyare not driven by the homogeneity assumption forthe BBB tranches.

Note that a CDO created from the BBBtranches of ABSs is quite different from a CDO

created from BBB bonds, even when the BBBtranches have been chosen so that their probabili-ties of default and expected loss are consistent withtheir BBB rating. The reason is that the probabilitydistribution of the loss from a BBB tranche is quitedifferent from the probability distribution of theloss from a BBB bond.

We can gain an insight into the characteristicsof the loss distribution of BBB rated tranches byconsidering an extreme case. Suppose that tranchesare infinitesimally thin and that = 1, so the losseson tranches are perfectly correlated with eachother. Therefore, either the BBB tranches lose noneof their principal or each BBB tranche loses its entireprincipal. An ABS CDO consisting of a portfolio ofthese tranches suffers either zero loss or 100 percentloss. Thus, every tranche of the ABS CDO also loseseither everything or nothing. In sum, there shouldbe no differences between the ratings of thetranchesindeed, they should all be rated BBB.

As explained earlier, the BBB tranches wereoften very thin. Furthermore, inspecting publiclyavailable data on ABSs, we found that the underly-ing mortgages are from various parts of the UnitedStates rather than one area, which suggests that is quite high.

The Expected Loss Criterion for ABS CDOs.Although we were able to show theoretically thatthe expected loss criterion for the senior ABStranche always leads to lower minimum attachment

Table 5. Minimum Attachment Points for AAA Tranche of ABS CDO Formedfrom ABS Tranches Responsible for Losses of 47 Percent

= 0.05 = 0.25 = 0.50 = 0.75 = 0.95


= 0.30 35.2 68.5 92.8 99.9 99.9


= 0.30 15.2 39.3 70.0 94.9 99.9


= 0.30 88.0 99.4 100.0 100.0 100.0





points than does the probability of loss criterion, wewere unable to produce a similar theoretical resultfor the senior ABS CDO tranche. Our numericalresults, however, indicate that this finding is true inall the cases that we considered.

ConclusionContrary to many of the opinions that have beenexpressed in the popular press, the AAA ratings forthe senior tranches of ABSs were not totally unrea-sonable. Given that the weighted average life ofmortgages is about five years, for many of theassumptions that rating agencies might reasonablyhave made, expected loss and probability of loss forthe AAA rated tranches were not markedly differ-ent from those of AAA rated five-year bonds.

The AAA ratings for the Mezz ABS CDOs aremuch less defensible. Scenarios in which all theunderlying BBB tranches lose virtually all theirprincipal were sufficiently probable that assigninga AAA rating to even a very thin senior tranchewas unreasonable. The risks in Mezz ABS CDOsdepended critically on (1) the correlation betweenpools, (2) the tail default correlation, and (3) therelationship between the recovery rate and thedefault rate. The very thin BBB tranches accentu-ated the risks, but making the tranches widerwould not have made the AAA ratings any moredefensible. An important point is that the BBBtranche of an ABS cannot be considered similar toa BBB bond for purposes of determining the risksin ABS CDO tranches.

Table 6. Minimum Attachment Points for AAA Tranche of ABS CDO Formed from ABS Tranches Responsible for Losses of 49 Percent

= 0.05 = 0.25 = 0.50 = 0.75 = 0.95


= 0.30 28.7 60.1 87.2 99.3 99.9


= 0.30 11.7 31.8 59.9 87.9 99.9


= 0.30 85.9 99.0 100.0 100.0 100.0



Table 7. Effect of Heterogeneity on Minimum Attachment Points for AAA Senior Tranche of ABS CDO

= 0.05 = 0.25 = 0.50 = 0.75 = 0.95

Gaussian copula = 0.05 5.2% 4.9% 2.8% 1.7% 12.2%(constant recovery) = 0.10 3.1 0.7 3.4 5.4 2.3

= 0.20 2.0 3.9 2.4 2.2 0.0 = 0.30 0.2 0.2 0.6 0.5 0.0

Triple-t copula = 0.05 0.6% 0.1% 0.0% 0.0% 0.0%(stochastic recovery) = 0.10 3.5 0.5 0.0 0.0 0.0

= 0.20 1.4 0.6 0.0 0.0 0.0 = 0.30 2.6 0.0 0.0 0.0 0.0

Notes: The parameters and are so defined that the between-pool copula correlation is and thewithin-pool correlation is . The expected default rate is 10 percent.



In practice, Mezz ABS CDOs accounted forabout 3 percent of all mortgage securitizations, butthey were a more prominent feature of financialmarkets than this statistic indicates. Market partic-ipants frequently used the AAA tranches of ABSCDOs to create synthetic CDOs. Also, the purchas-ers of the tranches often bought protection againstlosses on them from third parties. The TABX indexshows that ABS CDO tranches originally ratedAAA became worthless by mid-2009. An importantimplication of our research is that when the securi-tization market becomes active again, both regula-

tors and market participants should be wary ofresecuritizations (i.e., the formation of tranchesfrom other tranches). Resecuritizations are difficultto analyze and can serve no useful purpose if theunderlying asset pool is well diversified. To createmore robust ABS products, constructors shouldlook for ways to increase diversification in theunderlying asset poolfor example, by includingseveral different asset classes in the same pool.

This article qualifies for 1 CE credit.

Notes1. Interestingly, the yield spreads did improve predictions for

products with ratings below AAA.2. See Brennan, Hein, and Poon (2008) for a discussion of

this point.3. The timing of defaults is particularly important for the val-

uation of lower-rated tranches because interest paymentsform a larger component of the return for such tranches.

4. We did not consider the allocation of interest because therating agency models are concerned with only the impair-ment of principal.

5. In general, this distribution must be determined numerically.6. The factor copula model could be modified to make corre-

lation parameters dependent on the default rate. This mod-ification was suggested by Andersen and Sidenius (2004).

7. The negative relationship between recovery rates anddefault rates has been documented for bonds by Altman,Brady, Resti, and Sironi (2005) and Moodys InvestorsService (2008).

8. We ran tests that showed our results were not very sensitiveto the choice of recovery rate model.

9. For convenience, we refer to R*, the recovery rate observedwhen the realized default rate equals the expected defaultrate, as the average recovery rate and the loss rate associatedwith it as the average loss rate. R* is not, however, themathematical expected recovery rate.

10. See Moodys Investors Service (2007).11. Note that the 11-month percentage calculated in March

2007 reflects only loans originated early in 2006. The portionof all loans originated in 2006 that became delinquent after11 months (calculated at the end of 2007) was 12.13 percent.

12. See Bank for International Settlements (2006, p. 77) and Hull(2009). Basel II uses essentially the same copula model thatwe did, with M and Zi normally distributed.

13. This point is discussed further in Das and Stein (2010).14. For a discussion of this point, see Moodys Investors

Service (2007).15. We obtained results similar to the Monte Carlo results by

using the following analytic approximate approach. Wecalculated the mean and standard deviation of the loss onone BBB tranche of an ABS conditional on Mbp. We thenused the central limit theorem to estimate the conditionalprobability distribution of the average loss across alltranches. Finally, we integrated over Mbp to calculate theunconditional distribution.

16. Finding the AAA tranches attachment point is equivalentto determining the value at risk for a portfolio. In both cases,we are seeking the level of loss that is exceeded only 0.1percent of the time. Our estimates are based on 2.5 millionsimulations. The standard errors are fairly small, usuallyless than 0.5 percent.

ReferencesAdelino, Manuel. 2009. Do Investors Rely Only on Ratings?The Case of Mortgage-Backed Securities. Working paper, MITSloan School of Management (November).

Altman, Edward I., Brooks Brady, Andrea Resti, and AndreaSironi. 2005. The Link between Default and Recovery Rates:Theory, Empirical Evidence, and Implications. Journal ofBusiness, vol. 78, no. 6 (November):22032228.

Andersen, Leif, and Jakob Sidenius. 2004. Extensions to theGaussian Copula: Random Recovery and Random FactorLoadings. Journal of Credit Risk, vol. 1, no. 1 (Winter):2070.

Ang, Andrew, and Joseph Chen. 2002. Asymmetric Correla-tions of Equity Portfolios. Journal of Financial Economics, vol. 63,no. 3 (March):443494.

Bank for International Settlements. 2006. International Conver-gence of Capital Measurement and Capital Standards. Basel,Switzerland: Bank for International Settlements.

Brennan, Michael J., Julia Hein, and Ser-Huang Poon. 2008.Tranching and Rating. Working paper (March).

Coval, Joshua D., Jakub Jurek, and Erik Stafford. 2008. TheEconomics of Structured Finance. Working Paper 09-060,Harvard Business School.

Das, Ashis, and Roger M. Stein. 2010. Some Intuition andMathematics of Tranching. Working paper, MoodysInvestors Service.

Das, Sanjiv R., Laurence Freed, Gary Geng, and Nikunj Kapa-dia. 2006. Correlated Default Risk. Journal of Fixed Income,vol. 16, no. 2 (September):732.

de Servigny, Arnaud, and Olivier Renault. 2002. DefaultCorrelation: Empirical Evidence. Working paper, Standard &Poors (October).

Gorton, Gary B. 2008. The Panic of 2007. Working paper, YaleSchool of Management (August).



CFA INSTITUTE BOARD OF GOVERNORS 20102011

ChairMargaret E. Franklin, CFAKinsale Private Wealth Inc.Toronto, Ontario, Canada

Vice ChairDaniel S. Meader, CFATrinity Private Equity GroupSouthlake, Texas

CFA Institute President and CEOJohn D. Rogers, CFACFA InstituteCharlottesville, Virginia

Saeed M. Al-Hajeri, CFAAbu Dhabi Investment AuthorityAbu Dhabi, United Arab Emirates

Mark J.P. Anson, CFAOak Hill InvestmentsMenlo Park, California

Giuseppe Ballocchi, CFAPictet & CieGeneva, Switzerland

Kay Ryan BoothHarrison, New York

Pierre Cardon, CFABank for International SettlementsBasel, Switzerland

Beth Hamilton-Keen, CFAMawer Investment

Management Ltd.Calgary, AB, Canada

James G. Jones, CFASterling Investment Advisors, LLCBolivar, Missouri

Stanley G. Lee, CFANeuberger BermanThe Greene GroupNew York, New York

Jeffrey D. Lorenzen, CFAAmerican Equity Investment

Life Insurance Co.West Des Moines, Iowa

Aaron Low, CFALumen AdvisorsSingapore

Alan M. Meder, CFADuff & Phelps Investment

Management Co.Chicago, Illinois

Jane Shao, CFALumiereBeijing, China

Roger UrwinMSCI BarraSurrey, United Kingdom

Thomas B. Welch, CFAWells Capital ManagementMinneapolis, Minnesota

Charles J. Yang, CFAT&D Asset ManagementTokyo, Japan

Gregory, Jon, and Jean-Paul Laurent. 2005. Basket DefaultSwaps, CDOs and Factor Copulas. Journal of Risk, vol. 7, no. 4(Summer):103122.

Hull, John. 2009. Risk Management and Financial Institutions.2nd ed. Upper Saddle River, NJ: Pearson.

Hull, John, and Alan White. 2004. Valuation of a CDO andan nth to Default CDS without Monte Carlo Simulation.Journal of Derivatives, vol. 12, no. 2 (Winter):823.

Li, David X. 2000. On Default Correlation: A Copula FunctionApproach. Journal of Fixed Income, vol. 9, no. 4 (March):4354.

Moodys Investors Service. 2007. Challenging Times for theUS Subprime Mortgage Market. Research report (7 March).

. 2008. Corporate Default and Recovery Rates, 19202007. Research report (February).

Stanton, Richard, and Nancy Wallace. 2008. ABX.HE IndexedCredit Default Swaps and the Valuation of Subprime MBS.Working paper, Fisher Center for Real Estate and UrbanEconomics, University of California, Berkeley (February).

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