AA T OP - DOWN M ODEL FOR C ASH
CLO
Yadong Li and Ziyu Zheng ∗ April 19, 2010
Abstract
We propose a top-down model for cash CLO. This model can consistently price cash CLO tranchesboth within the same deal and across different deals. Meaningful risk measures for cash CLO tranchescan also be defined and computed. This method is self-consistent, easy to implement and computation-ally efficient. It has the potential to bring much pricing transparency to the cash CLO markets; and itcould also greatly improve the risk management of cash instruments.
Throughout the credit crisis since 2007, cash structured finance instruments such as cash CDO, CLO, MBSand CMBS have suffered much larger losses and write-downs than the synthetic instruments such as syn-thetic CDO/CLOs. The cause of the disparity in losses is mainly due to the difference in their risk man-agement practices. In the synthetic CDO/CLO market, the market participants are much more accustomedto hedging the market risks via credit indices, index tranches and single name CDS/LCDS contracts. Thebase correlation model is the market standard model for the synthetic CDO/CLOs, which have played a keyrole in the risk management of synthetic CDO/CLOs. Having a market standard model like base correla-tion has nurtured and encouraged proper and sophisticated risk management practice across both investorsand market makers in the synthetic market. In comparison, the risk management in the cash CDO/CLOmarket is much less sophisticated, which is mainly due to the limited availability of hedging instruments aswell as the lack of a standard risk-neutral model. The model development for cash instruments has laggedfar behind comparing to the modeling capabilities of the synthetic CDO/CLOs. The modeling efforts ofcash instruments have been mostly focused on the cashflow modeling, and there are few attempts to pricecash CDO/CLOs under the proper risk-neutral framework. Risk management of cash instruments without aproper risk-neutral model is inevitably difficult.In this paper, we present a simple and practical risk-neutral model for cash CLO (Collateralized loanobligation), which is one of the most important type of cash CDOs. The total outstanding notional amountof cash CLO assets has grown to over $400B in 2007 right before the credit crisis. The same methodologycan also be applied to other types of cash CDOs. The proposed methodology brings the modeling capabilityof cash instruments much closer to the modeling capability of the synthetic instruments. ∗ Li: Barclays Capital, [email protected]. Zheng: Morgan Stanley, [email protected]. This paper is based the authors’research between May and Sep 2008 while both were employed by Lehman Brothers. The views expressed in this paper are theauthors’ own and do not necessarily reflect those of Lehman Brothers or their current employers. The authors thank Greg Xue forIntex analytics support, Lorraine Fan, Marco Naldi, Ariye Shater, Gaurav Tejwani for many helpful comments and discussions, andthe Lehman CLO trading desk for many valuable inputs. a r X i v : . [ q -f i n . P R ] A p r his paper is organized as follows, we first give a brief overview of the important differences betweenthe cash and synthetic CLO markets in section 2, then we review the existing pricing method for cash CLOsand its limitations in section 3. In section 4, we introduce the top-down method for CLOs. In the rest ofthis paper, the unqualified term “CLO” always refers to the cash CLO, while the “synthetic CLO” is alwaysfully qualified to avoid confusion. A cash CLO and a synthetic CLO share very little in common besides their names. A cash CLO deal isusually managed by a manager who can actively trade and hedge the underlying loan portfolio, while asynthetic CLO usually trades in single tranche format referencing a static portfolio. A cash CLO often hascomplex cashflow waterfall structure so that the cashflow of its tranches are highly path-dependent, whilethe cashflows of synthetic CLO tranche are very simple and not path-dependent. The cash CLO typicallyattracts institutional buy-and-hold investors whose investment decisions are mainly based on ratings andyields, while the synthetic CLO index (i.e.: the LCDX) and tranches are mostly traded by the correlationdesks of hedge funds and banks who are required to mark their positions to the market.The market dynamics are totally different between the cash and synthetic CLO. In the synthetic CLOmarket, the participants can easily take long or short positions in the credit indices (e.g. LCDX), indextranches and the underlying single name LCDS, therefore there are strong arbitrage relationships amongthem. It is easy to construct a profitable basis trade if any of the basis becomes very large. The basis betweenindices, tranches or single names in the synthetic market therefore tends to stay within a reasonable range. Itis important for a synthetic CDO model to maintain the consistency between the tranches and its underlyingportfolio because of this strong arbitrage relationship. The cashflows of synthetic CLO tranches are verysimple and not path-dependent; therefore it is not only necessary but also feasible to model synthetic CLOsas derivatives of the underlying LCDS contracts. This approach is commonly called “bottom-up” becausethe model drills down to the individual constituents in the collateral portfolio. Base correlation is the mostcommon synthetic CDO/CLOs model in practice, readers are referred to O’Kane and Livesey (2004) fora description of the base correlation model. By modeling the synthetic CDO/CLO tranches as derivativesof the underlying CDS/LCDS, the total protection value of the tranches automatically adds up to the totalprotection value of the underlying CDS/LCDS.In contrast, the cash CLOs behave very differently. A cash CLO is very similar to a regular companyas it has a manager, some assets (the loan collateral) that are funded with various classes of debt (tranches)and equity (tranche). Unlike the synthetic market, it is almost impossible to construct a basis trade for cashCLOs even if there exists large basis between the market value of CLO assets (underlying collateral loans)and CLO liabilities (including tranches, management fees and other expenses). There are several reasonsfor this: 1) cash CLO managers typically only report the underlying loan positions once a month, thus theexact loan positions are unknown between reports since the manager may trade the underlying loans at anytime 2) the individual loans are often illiquid and rarely traded 3) taking short positions in underlying loansor cash CLO tranches is very difficult in the cash market. Therefore the basis trades for cash CLOs haverarely (if ever) been attempted in practice. As a result, there is no market force that brings the values of cashCLO assets and liabilities together quickly. The values of a cash CLO’s assets and liabilities can divergesignificantly for an extended time even though they eventually have to converge . The cash CLO tranchesand the underlying loans can behave like unrelated instruments in short term even though the cashflowsof cash CLO tranches are derived from the cashflows of the underlying loans. It is not uncommon for At the maturity of the CLO deal to the latest since all the loans will either mature or liquidate at maturity.
Even though the cash CLO tranches trade in the upfront price format just like ordinary corporate bonds, amore structured quoting/pricing convention is required for cash CLO investors to compare relative valuesamong different cash CLO tranches. The current market standard for quoting and pricing cash CLO tranchesis based on a single pricing scenario where the overall underlying loan collateral is assumed to have aconstant annualized default rate (CADR), constant annualized prepayment rate (CAPR), and a constantrecovery rate (CRR) over the whole life of the CLO. The CADR of the single pricing scenario is usuallyvery low, for example 3% CADR is often used. The CLO tranche cashflows from this single pricing scenarioof CADR, CAPR and CRR are then discounted using the riskfree rate plus a discount margin (DM) spreadto produce the tranche PVs. Different DMs are required for different tranches to re-produce their marketprices from the single pricing scenario.Intex is a standard software package used by most market participants to compute the cashflows fromCLO tranches. Intex has modeled the majority of outstanding cash CLO deals in the market and it cancompute the cashflows of (almost) any CLO tranches under any CADR, CAPR and CRR scenarios. Thewide adoption of Intex tool is important for the transparency of the cash CLO market because it providesa consensus on the CLO tranche cashflow among market participants. Different market participants thuscan reach the same conversion between the DM and CLO tranche prices as long as they all use Intex forthe cashflow calculation. The implied DM from the market tranche prices can then be used by the investorsto compare the relative values of different cash CLO tranches. We can view the DM as a similar measureas the credit spreads for corporate bonds: the higher the DM, the more likely the cashflow from the singlelow-CADR pricing scenario would not be paid due to the increase in loan default rates.The cash CLO market makers often maintain a matrix of DMs based on recent market transactions ofvarious ratings, vintages and deal types; and the DM matrix is used to price similar cash CLO trancheswhose prices are not observable in the market. Due to the lack of liquidity, the DM matrix is only updated It is more difficult to construct proper models for cash CLOs than for synthetic CLOs due to the opaquenessof the underlying collateral and the complex cashflow waterfall features. There have been prior attemptsto price cash CLOs tranches by computing their cashflows from a Monte Carlo simulation of the defaulttimes and recovery rates of underlying loans. The default time and recovery simulation could be driven bya default time copula (e.g.: Gaussian Copula). This bottom-up approach achieved little success because ofthe uncertainties in the underlying loan positions and prices, as well as the prohibitive computational cost toobtain the tranche cashflows from a large number of simulated scenarios. Custom implementations of thecash CLO’s cashflow waterfalls are often required to achieve reasonable simulation speeds as the standardIntex tool may not be fast enough to support a large number of simulated scenarios. In our view, it is notonly a costly, but also an almost useless exercise to build a bottom-up cash CLO model in practice. Themain benefit of a bottom-up model is the ability to produce risks to individual underlying loans, but it is notfeasible to hedge cash CLO tranches by trading individual underlying loans anyway (as discussed in section2). 4ecognizing the drawbacks of the current DM based method and the practical hedging restrictions inthe cash CLO market, we hereby propose a simple but practical top-down model for cash CLO. Top-downmodels were originally developed for exotic synthetic instruments. In a top-down model, the collateralportfolio is modeled as a whole instead of drilling down to individual constituents. The benefit of a top-downapproach is its simplicity as a result of not having to model the individual constituents of the underlyingportfolio. The adoption of top-down models in synthetic CDO/CLOs has been limited so far because theprices of synthetic tranches do move with its underlying CDS/LCDS due to the strong arbitrage relationship.Ignoring the single name risk is considered a drawback for synthetic instruments since it is critical to hedgesynthetic tranches by trading the underlying single name CDS/LCDS contracts.However, a top-down approach is ideal for cash CLOs since there is no strong arbitrage relationshipbetween the CLO assets and liabilities in the cash market, and the cash CLO tranche prices do not neces-sarily move consistently with individual underlying loans. Therefore, ignoring the individual single nameinformation, being a vice in the synthetic CLOs modeling, becomes a virtue in the cash CLO modeling be-cause it is closer to the market reality and it greatly simplifies the model setup. Note that a top-down modelcan produce risk measures to the overall average loan price in the cash CLO portfolio, thus it is possible tomacro-hedge the overall loan price movements using a top-down model. It is just not possible to producehedge ratios to the individual loans with a top-down approach, which is useless in practice anyway.
Fundamentally, the objective of a pricing model is to find the prices of less liquid instruments from the pricesof liquid instruments. For example, when pricing a bespoke synthetic CDO using the base correlation model,we first extracts the correlation information from the liquid index tranches by calibrating a base correlationsurface, which is then mapped to the bespoke portfolios to produce the bespoke tranche prices. The pricingconsistency among different bespoke tranches are maintained because all of them are priced from the sameset of liquid index tranches. This procedure allows us to compute risk sensitivities of bespoke tranches to theliquid index tranches; and we can hedge the illiquid bespoke tranches by trading the liquid index tranches.Given the success of synthetic CDO/CLO models, it is a natural idea trying to apply the same pricingmethod to cash CLOs. However, a practical challenge is that there is no standard liquid index for cash CLOs.As discussed before, the LCDX tranches can’t be used to price cash CLOs because of the fundamentaldifferences between the cash and synthetic markets. To get around this, we have to assume that there is arepresentative cash CLO deal whose market price is somewhat transparent, which can be used as an “index”to price other cash CLO deals. In practice, cash CLO market participants can choose a representative CLOthat are reasonably liquid as the CLO “index”. Once we identified a cash CLO index and its tranche prices,we then can carry out the calibration and mapping procedure for cash CLOs in a similar manner as in thesynthetic CLO models. Figure 1 showed the deal information and tranche prices of an actual cash CLOdeal, whose price marks are provided by the Lehman CLO trading desk as of Aug. 12, 2008. We will usethis CLO deal as the “index” for the following discussion. This “index” deal is subsequently referred asCLO-IDX.The traditional DM method only uses a single pricing scenario of CADR, CAPR and CRR for all thetranches, which is an overly simplified assumption since the future default, prepay and recovery rates are byno means deterministic. It is much more realistic to assume that there are a set of possible market scenariosof (CADR, CAPR and CRR), and each scenario has certain risk-neutral probability of realization. Figure2 is a set of representative scenarios that are provided by the Lehman CLO research based on the marketcondition as of mid 2008. The CAPR and CRR are chosen to be decreasing with the CADR based onhistorical observations. Even though the default, prepay and recovery rates can be time dependent, we kept5igure 1: CLO-IDX Deal Information
Class Coupon Rate Notional OC Trigger(%) IC Trigger(%) S&P Rating Prices (%)
A Libor+69.5bp 506,250,000 118.8 120.0 AAA 92.97B Libor+110.0bp 61,875,000 AA 82.16C Libor+200.0bp 43,125,000 111.2 112.5 A 78.83D Libor+3.25% 30,000,000 107.2 107.5 BBB 72.13E Libor+5.00% 33,750,000 104.4 100.1 BB 63.77SUBORD - 75,000,000 - - NA 44.89
Figure 2: Loan Market Scenarios
CADR (%) CAPR (%) CRR (%) CADR (%) CAPR (%) CRR (%) them constant in this study for simplicity. It is sensible to add time varying default scenarios if default islikely to be front or back loaded. The distribution of these market scenarios can be calibrated to the marketprices of the cash CLO tranches.More formally, we use S i to represent the i-th (CADR, CAPR, CRR) scenario in Figure 2, and we use v j ( S i ) to represent the PV of the j-th CLO tranche under the scenario S i , which can be computed by simplydiscounting the cashflow from Intex using the risk free rates without any additional DM. We also use V j torepresent the market price of the j-th CLO tranche as shown in Figure 1, then the calibration reduces to aproblem of finding a discrete distribution of { p i } for the given set of scenarios so that for every CLO tranche j : ∑ i p i v j ( S i ) = V j (1)We call the { p i } that solves (1) the market implied scenario distribution (MISD). This approach is similar inspirit to the Brigo, Pallavicini and Torresetti (2007) for the synthetic CDOs, the main difference here is thatthere is no constraints from the underlying collateral loan prices, which is a conscious choice because thereis no strong arbitrage relationship between the cash CLO assets and liabilities. The tranche prices across thefull capital structures have to be used in the MISD calibration otherwise the overall risk of the underlyingportfolio cannot be determined. In this approach, the CLO tranche cashflows are computed by Intex usingonly the aggregated CADR, CAPR and CRR of the whole loan collateral portfolio, this is effectively atop-down approach since it does not drill down to the individual collateral loans.Since the number of scenarios in Figure 2 is much greater than the number of tranches in Figure 1, there6igure 3: CLO-IDX Tranche Prices under Market Scenario CADR A B C D E SUB COL are infinitely many distributions that can reprice all the index CLO tranches. Therefore, certain objectivefunction has to be exogenously chosen so that we can find a unique distribution using an optimizationmethod. The maximum entropy method is well suited for such under-determined optimization problems inderivative pricing as it finds a distribution with the most uncertainty and the least bias. Readers are referredto Avellaneda et al. (2001) for an introduction to the maximum entropy optimization method.Figure 3 showed the CLO-IDX tranche prices computed by Intex for every scenario in Figure 2. Notethat these tranche prices are computed without any additional DM above the risk free rate. Given thedata in Figure 3, we can easily find the MISD that reproduces the market CLO tranche prices in Figure 1via the maximum entropy method. The calibrated distribution is shown in the left side of Figure 4. Thefitting quality of the MISD is excellent, the market tranche prices are matched almost exactly, which is notsurprising because the number of tranches is much less than the number of market scenarios.By finding the MISD, we have moved from the traditional pricing method of a single pricing scenariowith different DMs for different tranches, to a more consistent pricing method of a single MISD and a singleset of risk-free discount factors for all tranches. It seems to be a small step to replace multiple DMs withmultiple market scenarios in the MISD, however this is a significant step forward since it is not only more7igure 4: Calibrated MISDCLO-IDX P r obab ili t y LCDX10 P r obab ili t y Figure 5: CLO-IDX Calibration with Bumped AA Tranche Prices P r obab ili t y realistic, but also addresses the first two limitations of the traditional DM method listed in section 3. Sincethe market scenarios in Figure 2 cover a wide range of CADR from 0% to 90%, every structural feature in acash CLO deal is expected to be triggered under some of the scenarios; thus they are fully priced in by theMISD method. Also, it is obvious that all the tranches from the same CLO deal are priced consistently toeach other because the same MISD and risk-free discount factors are used.For comparison purposes, we also calibrated the MISD to the synthetic LCDX10 tranches, the resultsare shown on the right side of Figure 4. The tranche PVs of LCDX10 under each market scenario canbe directly computed without using Intex since the LCDX tranche cashflow is a simple function of theaggregated portfolio loss. At first glance, the MISD from the CLO-IDX and the LCDX10 are quite different:the MISD from LCDX10 is roughly uni-modal while the MISD from CLO-IDX is obviously multi-modal.We have calibrated the MISD to many cash CLO deals and found that the multi-modality of MISD is acommon feature among almost all cash CLOs in mid 2008, whereas it is not present in the MISD fromsynthetic CLOs such as LCDX10. The multi-modality of CLOs is caused by the strong market demand forthe very safe AAA rated assets during the severe market stress of mid 2008. As shown in Figure 1, the AA8igure 6: Tranche Implied Expectations for CLO-IDX CADR
CAPR
CRR
Average Collateral Loan Price tranche is priced more than 10 points cheaper than the AAA tranche, such a steep price drop from the AAAtranche to AA tranche is mostly caused by market technicals instead of fundamentals. If we bump the AAtranche price in Figure 1 up by 5 points and re-calibrate the MISD to the bumped CLO prices, the resultingMISD (Figure 5) becomes closer to the uni-modal MISD from the LCDX10. This exercise showed that theMISD method allows us to meaningfully compare and identify discrepancies between cash and syntheticCLO markets, it also demonstrated the huge differences in dynamics and technicalities between cash andsynthetic CLO markets.From the calibrated MISD, we can easily compute expectations of various quantities, such as expectedCADR, CAPR and CRR rates, as shown in Figure 6. Since the MISD are calibrated to cash CLO trancheprices, we call them the tranche implied CADR, CAPR and CRR. The tranche implied CRR is much lowerthan the historical loan recovery rates (usually above 70%), which is a sign of stress in the cash CLO tranchemarket.Using the average underlying collateral loan price (last column of Figure 3) computed by Intex, wecan also obtain the tranche implied average collateral loan price, which is the average collateral loan pricethat makes the total asset value equals the total liability value for the given cash CLO deal. Comparingthe tranche implied average loan price against the average of actual market loan prices gives us the basisbetween the cash CLO tranche market and the underlying loan market. This basis cannot be obtained bysimply comparing the notional weighted average of the CLO tranche prices against the average price of theunderlying loans since 1) this does not take into account the cash CLO manager’s fee and other expensesthat are taken from the collateral loan cashflows 2) the total notional amount of the cash CLO tranches isusually not the same as the total notional amount of the underlying loans. The COL column in Figure 3reported by Intex, on the other hand, is the PV of the total underlying collateral loan (inclusive of any feesand expenses) normalized by the outstanding notional of the loan collateral, therefore it is the right quantityfor computing the tranche implied average loan prices.In the case of CLO-IDX, the average market price of its underlying loans is around 89.51 and the trancheimplied average collateral loan price is 84.42 as shown in Figure 6, i.e., there is a very large negative basisof -5 points, which implies that a investor would lose 5 points instantly if he creates a cash CLO out ofa pool of loans. At the mid of 2008, almost all cash CLO deals showed large negative basis between thetranche implied loan price and the average market loan price. In a normal market environment, this basisshould be positive otherwise there is no economic incentive to package individual loans to CLOs in the firstplace . However, during market stress of mid 2008, the cash CLO tranches are severely depressed amid thewide-spread fear of complex structured finance products, therefore the cash CLO tranches traded at deepdiscounts comparing to the underlying loans.It is another important advantage for the top-down MISD method to be able to compute the tranche im-plied quantities and compare them against those from the underlying loan market. This offers a meaningfulrelative value comparison between the cash CLO market and the underlying loan market. These tranche Since some of the underlying loans are illiquid, the average loan prices are actually computed only from loans with observablemarket prices Release of capital is another reason to create cash CLO from loans.
Class Coupon Rate Notional OC Trigger(%) IC Trigger(%) S&P Rating
A Libor+25.0bp 375,000,000 111.7 111.7 AAAB Libor+37.0bp 22,500,000 AAC Libor+65.0bp 17,500,000 108.9 108.9 AD Libor+140.0bp 30,000,000 103.5 103.5 BBBE Libor+3.650% 15,000,000 102.1 - BBSUBORD - 40,000,000 - - NA implied quantities cannot be computed from the traditional DM method, nor can they be obtained fromany bottom-up models because bottom-up models enforces the value equality between assets and liabili-ties. Therefore, the top-down MISD approach is actually more useful and closer to market reality than thebottom-up approach for cash CLOs.
After calibrating the MISD to an “index” CLO, we now investigate how to price tranches of other CLOdeals. Borrowing the terminology from synthetic CLO, we refer the cash CLO deal we want to price asthe “bespoke” CLO. In this section, we use another actual CLO as a sample bespoke CLO, which wesubsequently refer to as CLO-BSPK. The CLO-BSPK is of the same vintage and has similar structuralfeatures as the CLO-IDX, therefore it is quite sensible to price CLO-BSPK from CLO-IDX. Figure 7 showssome basic information on the CLO-BSPK deal, and Figure 8 shows the tranche prices of CLO-BSPKcalculated by Intex under the market scenarios of Figure 2.The key to price a “bespoke” CLO is to perturb the index MISD to incorporate the bespoke specificinformation. Using terminology from synthetic CLO, we need to find a mapping methodology betweenthe index MISD and the bespoke MISD. The cross entropy method is an ideal method for such mappingoperation between two distributions since it gives a distribution that is closest to the prior distribution (i.e.,the calibrated index MISD) while satisfying additional linear constraints that accounts for important bespokespecific features. Readers are referred to Avellaneda et al. (2001) for an introduction to the cross entropy(also known as Kullback-Leibler relative entropy) method. Among all the features of the “bespoke” CLO,two of them are the most important:1. The average price of the underlying loan collateral: This is important because it adjusts for the loanquality difference between the index and bespoke CLO. It also allows us to compute tranche sensitiv-ities to the average loan prices, which can be used for macro-hedging.2. The AAA-rated tranche: Since the AAA tranches are the most liquid, and all the AAA CLO tranchesare priced very similarly to each other with minor adjustments for coupons and the underlying loanquality. The market participants can accurately determine the bespoke AAA tranche prices fromsimilar AAA transactions on the market.Both of these features can be easily added as linear constraints in the cross entropy optimization. To incor-porate the underlying loan price, we need an assumption on the size of the basis between the tranche impliedand market loan prices for the bespoke CLO. Here we assume that the basis is constant and we use K I todenote the basis between the tranche implied loan price and average market loan price for the index CLOand M B to denote the average market price of the underlying loans in the bespoke portfolio, then the linear10igure 8: CLO-BSPK Tranche Prices under Market Scenarios CADR A B C D E SUB COL
Figure 9: Mapping from Index to Bespoke P r obab ili t y IndexBespoke
Class S&P Rating Model Price Model Delta
CLO-BSPK A AAA 89.35 0.30CLO-BSPK B AA 78.24 1.52CLO-BSPK C A 70.78 2.38CLO-BSPK D BBB 55.90 3.55CLO-BSPK E BB 60.95 3.77CLO-BSPK SUBORD - 47.63 5.21 constraint for the market loan price of the bespoke CLO becomes: ∑ i q i C ( S i ) = K I + M B (2)where q i is the MISD of the bespoke cash CLO we want to price; C j ( S i ) is the average collateral loan pricesfrom Intex as shown in the last column of Figure 8, which already accounts for the average underlying loanfeatures such as coupon rate, payment schedule and amortization etc. Similarly, the constraints for the AAAtranche can be expressed as: ∑ i q i v B AAA ( S i ) = V B AAA (3)where v B AAA is the tranche PVs for the AAA tranche in Figure 8, and the V B AAA is the expected bespoke AAAtranche price.Other adjustments can also be included in the cross entropy mapping method. For example, CLO dealsmanaged by a reputable manager often command a sizable premium comparing to those managed by amediocre manager. This management quality factor can also be included as an adjustment to (2). In thisexample, we assume there is no difference in management quality between the CLO-BSPK and CLO-IDX.With these two linear constraints in (2) and (3), it is easy to find the MISD for the bespoke cash CLO viathe cross entropy method. Figure 9 showed both the MISD of the index cash CLO and the mapped MISD ofthe bespoke CLO. The bespoke cash CLO tranche prices are easy to compute from the mapped MISD andthe PV scenarios in Figure 8. Figure 10 showed the resulting bespoke cash CLO tranche prices.It is interesting to note that the E tranche is priced higher than the D tranche for CLO-BSPK as shownin Figure 10. A careful examination of the tranche PVs in Figure 8 reveals the reason being that E trancheworths much more than the D tranche under most high CADR scenarios due to a structural features calledCERT trigger in the CLO-BSPK deal, which diverts the cashflow to the E tranche instead of the more seniorC and D tranches under certain high default rate scenarios. As shown in the cashflow table in Figure 3, asimilar but less prominent CERT trigger also exists in the CLO-IDX deal. The purpose of the CERT triggerwas to boost the rating of the E tranche. If we use the traditional DM method to price the CLO-BSPK, the Etranche would certainly be priced less than the D tranche since the DM method only uses the single pricingscenario of 3% CADR, under which the D and E tranches of CLO-BSPK behave very similarly to the Dand E tranches of CLO-IDX since the CERT trigger is only active under high default rate scenarios. Thisexample showed that the top-down MISD method automatically prices in the structural differences betweenthe index and bespoke cash CLO tranches, thus being able to identify potential mis-prices in the traditionalDM method.The proposed calibration and mapping procedure is a one-period model without any term structure.Unlike synthetic CDO/CLOs which can trade at multiple maturities, each cash CLO only have a singlepre-determined maturity, therefore a one-period model is adequate if the index and bespoke CLOs are fromsimilar vintage and have similar reinvestment period.12igure 11: CLO-BSPK Tranche01 Risk
Bespoke Tranche AAA AA A BBB BB NA
CLO-BSPK A -0.20 0.01 -0.01 0.00CLO-BSPK B 0.11 -0.08 0.01CLO-BSPK D -0.02 -0.01 -0.02 -0.06CLO-BSPK E 0.04 0.12 0.12 -0.08
Under the traditional DM method, it is very difficult to quantify the risk of a cash CLO book since thereare no meaningful risk measures. The most common view of the risk is the aggregated tranche notional foreach rating bucket, which is a very crude estimate of the overall risk as there is no indication of the relativeriskiness between different rating buckets. No concrete hedging strategy can be devised from the aggregatedcash CLO notional amounts by the rating bucket.With the cross-entropy mapping method between the MISD of the index and bespoke CLO, we caneasily define and compute a set of consistent risk measures for the CLO tranches. For example, it is easyto compute the cash CLO tranche sensitivities to the underlying loan price movements via a simple bump-remap-reprice procedure. As shown in Figure 10, the tranche deltas computed this way are quite reasonableas the deltas are positive and decreasing with the tranche seniority. The aggregated tranche deltas can beused to predict the P&L change of the whole book for a given movement of the average underlying loanprice. This is a very precise risk measure which can be used to macro-hedge the cash CLO book.Similarly, we can define and compute the sensitivities to the index cash CLO tranches via the bump-remap-reprice procedure, as shown in Figure 11. This sensitivity is commonly called tranche01 in thesynthetic CLO terminology. The tranche01 risk of CLO-BSPK showed that the CLO-BSPK tranches arethe most sensitive to the index tranches of the same rating with some spillover to the next junior tranche.The tranche01 risk is a measure of correlation risk, which is very useful in practice because we can breakdown the risk of a cash CLO trading book into corresponding tranches of the index CLO, thus allowing usto understand and manage the risk exposure to different parts of the capital structure. The tranche01 risk is amuch better measure than the aggregated tranche notional by rating bucket. For example, Figure 11 showedthat the BBB-rated D tranche of the CLO-BSPK behaves like a 50-50 mix of the D(rated BBB) and E(ratedBB) tranches of the CLO-IDX; which is mainly due to the fact that the D tranche of CLO-BSPK has 3.5%less subordination than the D tranche of CLO-IDX even though they are both BBB rated. These structuraldifferences between cash CLO deals are automatically captured by the tranche01 risk from the top-downMISD method, thus providing a much more coherent view of the correlation risk of a cash CLO book.Other risk measures such as interest rate risk and theta risk can be similarly defined and computed fromthe top-down method.
The proposed top-down method is ideal for cash CLOs as it produces consistent prices and risks across cashCLO deals while being very simple, intuitive and computationally efficient. Drilling down to the individualcollateral loans provides very little practical benefits because of the lack of liquidity in individual loans and13he lack of strong arbitrage relationship between cash CLO assets and liabilities.In practice, this top-down model allows a cash CLO trading desk to only mark the prices of a fewrepresentative cash CLO deals as the indices for different vintages and deal types, then the rest of the cashCLO tranches in the book can be automatically priced via the cross entropy mapping method. This allows thecash CLO tranches to be priced consistently using the same calibrate-and-mapping procedure as in syntheticCDO/CLOs, making it much more difficult to manipulate the price marks and book P&L.All the structural features of a cash CLO are automatically taken into consideration by the proposed top-down method, which is a big improvement over the traditional DM based method. Although the consistencybetween the value of cash CLO assets and liability is not enforced during the calibration because of thelack of strong arbitrage relationship, the average loan price is a valuable piece of market information andit is used by the cross entropy mapping to adjust for the underlying loan quality difference between cashCLO deals. Therefore this top-down MISD method can be a very effective method in finding relative valuetrading opportunities between cash CLO tranches, especially when most market participants are still usingthe traditional DM method.This top-down method also produces a full set of risk measures. It is feasible to attribute the P&Lmovement of a cash CLO trading book using the change of the average underlying loan prices and the indexcash CLO tranche prices. Even though it is not feasible to hedge CLO tranches by trading individual loans,it is certainly possible to macro hedges the risk of overall loan market movement based on the cash CLOdeltas from the top-down model. If the market develops and certain “index” cash CLO deals becomes easierto short (for example, via TRS), then it is also possible to hedge the correlation risk of a cash CLO bookvia the tranche01 risk from the model. Being able to meaningfully define risk measures and devise theirhedging strategies for a cash CLO book is certainly another big improvement over the DM based method.This method is also computationally efficient since there is only a limited number of scenarios (Figure2) to run for each deal. The calibration, pricing and risk measures of cash CLO tranches can be computedvery efficiently using the the standard Intex tool, and this is no need to build any custom cashflow waterfallengine. This top-down method is very easy to implement and operate in practice as most cash CLO marketparticipants already use the Intex tool. Using this top-down method, different market participants will reachthe same CLO tranche prices if they can agree on a standard set of market scenarios like those listed inFigure 2, and if they can establish a poll to determine the prices of a small set of representative “index” cashCLO tranches. Both of these two steps are well within reach therefore this method has the potential to bringmuch more pricing transparency to the cash CLO market.
References
Avellaneda, M., R. Buff, C. Friedman, N. Grandchamp, N. Gr, L. Kruk and J. Newman. 2001. “WeightedMonte Carlo: A New Technique for Calibrating Asset-Pricing Models.”
International Journal of Theo-retical and Applied Finance defaultrisk.com .O’Kane, D. and M. Livesey. 2004. “Base Correlation Explained.”