CommercialMortgage-BackedSecurities (CMBS ... · CommercialMortgage-BackedSecurities (CMBS)andMarketEfﬁciencywith RespecttoCostlyInformation Andreas D. Christopoulos, ... (IO) bonds,

2008 V36 3: pp. 441–498

REAL ESTATE

ECONOMICS

Commercial Mortgage-Backed Securities(CMBS) and Market Efficiency withRespect to Costly InformationAndreas D. Christopoulos,∗ Robert A. Jarrow∗∗ and Yildiray Yildirim∗∗∗

Commercial mortgage-backed securities (CMBS) are complex asset-backedsecurities trading in markets that do not currently use derivatives pricing tech-nology. This lack of usage is due to the complexity of the modeling exercise,and only the recent and costly availability of historical data. As such, CMBSmarkets provide a natural environment for the testing of market efficiency withrespect to this costly information. Using this information, this article develops aCMBS pricing model to provide a joint test of the model and market efficiency.Backtesting our pricing model for 4 years, although there is some evidence ofabnormal trading profits, we cannot reject the efficiency of the CMBS markets.

The commercial mortgage-backed securities (CMBS) market is a relativelynew market, jump started by the Resolution Trust Corporation working outthe commercial loan portfolios of many failed thrifts and savings and loans inthe early 1990s. The annual issuances of new CMBS first exceeded 50 billiondollars in 1998, and the current outstanding balance of CMBS now exceeds$800 billion dollars.1 Third-party vendors providing rudimentary cash flowmodeling and structuring software tools for the generation of scenario-specificchanges in CMBS yields were first available in the late 1990s as well. At present,however, there is still no vendor that provides derivatives-based CMBS pricingmodels for industry usage.2 This is in contrast to the residential mortgage-backed securities market, where derivatives pricing technology has been insignificant usage for over a decade (see Fabozzi 2000). This lack of usageis due to the historical evolution of the market (being real estate based), the

∗Chief Executive Officer WOTN, Ithaca, NY 14850 or [email protected].∗∗Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853 or

[email protected].∗∗∗Martin J. Whitman School of Management, Syracuse University, Syracuse, NY 13244

or [email protected].

1 From the Commercial Mortgage Securities Association, www.cmbs.org/statistics.2 In fact, a private survey we conducted of the major investment banks indicated nousage of derivatives pricing tools for CMBS by either the investment bank’s trading orrisk managment departments.

C© 2008 American Real Estate and Urban Economics Association

442 Christopoulos, Jarrow and Yildirim

complexity of the modeling exercise and only the recent (but costly) availabilityof relevant historical data.

The complexity of CMBS modeling is due to the simultaneous inclusion of foursignificant risks: market, credit, prepayment and liquidity. In addition, CMBSare quite complex. To value a CMBS, one must first understand the cash flowsto the underlying CMBS loan pools, the cash flow allocation rules to the variousbond tranches, the prepayment restrictions and the prepayment penalties. Theseprovisions can, and often do, differ across the different CMBS trusts. Also,implementation of a model requires significant computational effort due to thequantity of the outstanding loans in any particular CMBS trust, the dimensionof the valuation problem (due to the number of risks present) and the numberof outstanding CMBS trusts. Finally, estimation of the relevant parameters isitself a nontrivial problem, given the sparsity and the diversity of historicalCMBS data, especially with respect to the history and current composition ofthe existing CMBS loan pools. Only recently have vendors made available (ata significant fee) the relevant historical data and computational tools, but in aform not conducive to risk-management analysis.

Even though the relevant risk-management technology is publicly availableinformation, as argued above, its implementation requires significant expertiseand out-of-pocket costs for computing facilities and historical information.As such, the CMBS markets provide a natural environment for the testingof market efficiency with respect to this costly risk-management information.Every such test of market efficiency, however, necessarily involves a joint testwith a pricing model. Formulating a new model for pricing CMBS, this articleprovides such a joint test of CMBS market efficiency and our new pricingmodel.

Hence, a secondary contribution of this article is the development, imple-mentation and testing of a risk-management model for pricing CMBS. Ofthe four relevant risks discussed previously, we model interest rate risk us-ing a multi-factor Heath, Jarrow and Morton (HJM, 1992) model. Credit riskis modeled using the reduced form methodology introduced by Jarrow andTurnbull (1992, 1995). Because we are valuing CMBS from the market’sperspective, following the recent insights of Duffie and Lando (2001) andCetin, Jarrow, Protter and Yildirim (2004), an intensity process is used to in-corporate prepayment risk with regional property value indices included asexplanatory variables. Lastly, liquidity risk is incorporated into both the es-timation of CMBS fair values and the testing of various trading strategies,motivated by the recent insights of Cetin, Jarrow and Protter (2004) in thisregard.

Commercial Mortgage Backed Securities 443

In total, our model includes 64 correlated factors generating the randomnessinherent in the CMBS loan pool cash flows. This represents four interest ratefactors and 60 property value indices. In addition, each commercial mortgageloan (CML) has three independent random variables generating its delinquencystatus, default and prepayment risk. As such, the complexity of the modelingstructure necessitates using Monte Carlo simulation for the computation of fairvalues.

We fit our CMBS model using daily forward rate curves, National Council ofReal Estate Investment Fiduciaries’ (NCREIF) regional property value indices,various Bloomberg’s Real Estate Investment Trust (REIT) stock price indicesand Trepp’s comprehensive historical commercial loan database that includesinformation on loan characteristics, defaults, prepayments and recovery rates(see Reilly and Golub 2000, Trepp and Savitsky 2000). Standard procedureswere used to parameterize the term structure model (see Jarrow 2002). A com-peting risk hazard rate procedure was used to fit both default and prepaymentrisk, consistent with the observation that the occurrence of default precludesprepayment, and conversely.

Market quotes obtained from Trepp’s CMBS Pricing ServiceTM were used forthe joint testing of the model and market efficiency. Trepp’s market quotes forthe entire universe of investment-grade CMBS are updated daily and they rep-resent an average of end-of-day marked-to-market prices, contributed by multi-ple dealers and institutional investors.3 Because CMBS trade over-the-counter,Trepp’s CMBS Pricing Service provides the best historical pricing databasefor analysis. It is important to note that non-interest-only investment-gradeCMBS trade in relatively liquid markets, and the Trepp database is comparableto other fixed-income databases frequently used in empirical investigations ofcredit risk models (see, e.g., Duffee 1999). To the extent that these quotes do notrepresent prices at which actual trades could have be executed, our conclusionsmust be accordingly tempered.

We compared the estimated model prices to market prices from July 2001 toApril 2005. If we had used spreads instead of prices, this would be equivalent tocomputing the bonds’ option-adjusted spreads (OAS). We observe statisticallysignificant price differences, rejecting the joint hypothesis of the model andmarket efficiency. To study whether the source of this rejection could be due toCMBS market inefficiency, we back tested our model to see if it could generateabnormal returns. In this regard, respecting the information available to the

3 Many of the largest institutions in the country rely on Trepp CMBS PricingTM fordaily and month-end valuations. See http://trepp.com/m/ovr_pricing.cgi?whichTrepp=m&whichLoan= for more information.


market at the relevant times, we formed trading strategies to take advantageof the mispricings as identified by the model. Using these trading strategies,we compared the performance of undervalued and overvalued CMBS portfo-lios, matched by risk (both credit and interest rate) from July 2001 to April2005. In these tests, for all risk categories, the undervalued portfolios signif-icantly outperformed the overvalued portfolios. For example, in comparingcumulative returns from the undervalued short tenor AAA portfolio (33.26%)to the overvalued short tenor AAA (17.60%), we see a cumulative outperfor-mance of 15.65% over the testing period. We also compared the undervaluedand the overvalued portfolios to those of matching equal-risk CMBS indicesover the testing period. Consistent with the under- versus overvalued compar-ison, the undervalued portfolios outperformed the matched CMBS index andthe overvalued underperformed the matched CMBS index for all risk categoriesover the testing horizon.

We analyzed these abnormal returns for omitted risk factors related to interestrates, property value and market risk. We found that the overvalued and un-dervalued portfolio returns are correlated with various interest rate risk factors,providing some evidence that these abnormal returns may be due to omittedrisk premia. Omitted risk premia is consistent with an efficient CMBS mar-ket. Alternatively, we also provide a behavioral finance explanation for thesecorrelations that is inconsistent with an efficient market. Which explanationis correct awaits subsequent research. Because we cannot unambiguously re-ject an efficient CMBS market, we also provide a simple adjustment to ourmodel that generates unbiased estimates for CMBS prices, useful for hedgingor marking-to-market, in the circumstance that the CMBS market is efficient.

There is an enormous literature related to mortgage-backed securities, bothresidential and commercial. For residential mortgage-backed securities, the pa-pers related to our methodology include Schwartz and Torous (1992, 1989),Deng, Quigley and Van Order (2000), McConnell and Singh (1994), Stanton(1995), Kau, Keenan and Smurov (2004) and Goncharov (2004). With re-spect to CMBS, Snyderman (1991) and Esaki, L’Heureux and Snyderman(1999) provide summary statistics on commercial mortgage defaults and lossseverities from 1972 to 1997. Archer et al. (2002) study a default modelfor multifamily commercial mortgages. Ambrose and Sanders (2003), Cio-chetti et al. (2004) and Yildirim (2008) estimate competing risk hazardand prepayment models for CMBS. These studies do not investigate val-uation or hedging of CMBS. Titman and Torous (1989) study the valua-tion of a class of CMBS called “bullet” bonds. Childs, Ott and Riddiough(1996) simulate a structural model for CMBS valuation to determine themodel’s implications for tranche values. They do not empirically test theirmodel.


An outline for this article is as follows. The next section provides a briefdescription of CMBS. The model is described in the third section, the empiricalprocesses in the fourth section and the estimates in the fifth section. The sixthsection provides the joint tests of market efficiency, and the seventh sectionconcludes the article.

A Brief Description of CMBS

CMBS are a subset of a class of financial securities known as asset-backedsecurities. CMBS are bonds of various seniorities, whose payments are madefrom the cash flows obtained from a CMBS trust. A CMBS trust is a legalentity that consists of a collection of CMLs, called the underlying loan pool.These CMLs are usually fixed-rate loans of a particular maturity, althoughthey can include floating-rate loans as well. The properties underlying theseloans can be diverse, both geographically and economically. Issued against thecash flows of the CMBS loan pool are a collection of bonds. These bondsare usually fixed rate with a given maturity. Different classes of bonds areissued (called bond tranches) with different seniorities with respect to the cashflows from the loan pool. While all bonds receive interest payments (subjectto availability), the outstanding principal balance of the bonds are reduced inthe order of seniority by scheduled amortization and prepayment of principal.Principal repayment can occur due to voluntary prepayments or recoveries inthe event of default. In reverse order to seniority, the least senior bonds lose theirunderlying principal first from defaults. In addition, most CMBS trusts issue aclass of bonds called interest-only (IO) bonds, whose cash flows come solelyfrom the loan pool interest payments, but only after all the senior bond couponsare paid. IO bonds make no principal payments. A typical conduit transactionis structured using a CMBS trust that contain anywhere from one to severalhundred underlying loans (mostly first liens) and issues about 10 different bondtranches including at least one IO bond. There are over 650 CMBS trusts tradingover-the-counter across both fixed-rate and floating-rate collateral includingboth U.S. and foreign collateral. For a more detailed description, see Sanders(2000).

The Model

This section constructs the CMBS valuation model. The model presented isnot the most complex formulation possible. Extensions and generalizationswill be discussed in footnotes. However, given the current limitations inthe availability of accurate and consistent data, more complex formulationswould provide little, if any, additional explanatory power in the subsequentimplementation.


CMBS face market (interest rate), credit, prepayment and liquidity risks.4

In the initial formulation, we abstract from liquidity risk. That is, we as-sume frictionless and competitive bond markets. Liquidity risk is only ad-dressed in the empirical implementation. This decomposition is similar tothe logic underlying the computation of option-adjusted spreads for residen-tial mortgage-backed securities. OAS measure all market impacts (includingmispricings and liquidity risk),5 after controlling for interest rate and pre-payment risk via the use of a model. We concentrate on modeling the in-terest rate, credit and prepayment risks inherent in CMBS. The interest raterisk is handled using a multi-factor Heath, Jarrow and Morton (1992) modelfor the term-structure evolution. To model the credit risk component, we uti-lize the reduced-form credit risk methodology first introduced by Jarrow andTurnbull (1992, 1995). This is done because our goal is to value CMBS fromthe market’s perspective and not the borrower’s. As shown by Duffie andLando (2001) and Cetin et al. (2004), given the market’s information set—a subset of the borrower’s—default times are inaccessible. Lastly, we modelprepayment risk using an intensity process. This is also done because fromthe market’s perspective prepayment often appears as a surprise. As withcredit risk, this is due to asymmetric information with respect to the prop-erty’s economic condition, transaction costs and the borrower’s personal sit-uation (see Stanton 1995 for a justification of this approach for residentialmortgages).

We are given a filtered probability space (�, �, (�t )t∈[0,T ),P) satisfying theusual conditions (see Protter 1990) with P the statistical probability measure.The trading interval is [0, T ]. Traded are default-free bonds of all maturitiesT ∈ [0, T ], with time t prices denoted p(t, T), and various properties, CMLsand CMBS bonds introduced below. The spot rate of interest at time t is denotedr t . Let (Xt )t∈[0,T ] represent a vector of state variables, adapted to the filtration,describing the relevant economic state of the economy. For example, the spotrate of interest could be included in this set of state variables.

We assume that markets are complete and arbitrage free so that there exists aunique equivalent martingale probability measure Q under which discountedprices are martingales. The discount factor at time t is e

∫ t

0 rs ds . Because we areinterested in valuing CMBS, most of the model formulation will be under theprobability measure Q.

4 There is often a fifth type of risk discussed with respect to CMBS called extensionrisk. Extension risk is discussed below.5 See Jarrow (2007) for an explanation of OAS spreads in the context of an HJM model.


CMLs

CMLs are issued against commercial properties. These CMLs can be fixed-rateor floating. For this discussion, we concentrate on fixed-rate loans. Floating-rate notes can be handled in a similar fashion. Our computer implementation, tobe discussed below, handles these explicitly. These mortgage loans are issuedto borrowers based on the quality (economic earning power) of the underlyingproperty. If the property loses value, the borrower may decide to default on theloan. As such, CMLs face both market (interest rate) and credit risk.

Description. Fixed-rate CMLs are similar to straight corporate bonds withthe exception that the loan’s principal is partly amortized over the life of theloan. Typical CMLs have a (T/n) balloon payment structure. In the (T/n) balloonpayment structure, the loan has a fixed maturity date T , a principal paymentF, scheduled payments P paid at equally spaced intervals over the life of theloan (usually monthly) and a coupon rate per payment period c = C/F whereC is the dollar coupon payment. The payments P are determined as if the loanwould be completely amortized in n periods. But, instead of lasting n periods,a balloon payment occurs at time T < n representing the remaining principalbalance at that time, denoted BT . The payment per period is

P = cF

[(1 + c)n

(1 + c)n − 1

]

and balloon payment is

BT = F (1 + c)T − P

[(1 + c)T − 1

c

].

For analysis, one can think of CMLs as an ordinary coupon bond with a facevalue of BT and a coupon payment of P.

As common to residential mortgages, CMLs have an embedded prepaymentoption. Unlike residential mortgages, CMLs cannot be prepaid during a lockoutperiod, denoted [0, T L]. After the lockout period, the loan can be prepaid, butthere is a time-dependent prepayment penalty. These prepayment penalties cantake various forms (see Trepp and Savitsky 2000), and they are designed tomake prepayment unattractive based on the changing level of interest rates. Atpresent, roughly 85% of all new-issue CML borrowers are permitted to prepaythe CML only by incurring the costs of defeasance. Defeasance loans requirethe borrower to replace the mortgage loan payments P with a collection ofU.S. Treasury STRIPS that match the remaining payments on the loan (foran analysis of the defeasance option in CMBS, see Dierker, Quan and Torous2005). As such, prepayments under defeasance actually increase the value tothe lender as the U.S. Treasury will neither default nor prepay such cash flows in


the future. In the case of CMLs in CMBS, the lender is an institutional investorwho is entitled to cash flows from such defeased loans that are included in thetrust backing their CMBS.

To address the powerful credit and prepayment aspect of defeasance loans, inour model we treat loans in the interval characterized by defeasance in theprepayment restriction schedules as fully locked out until such loans emergefrom the defeasance period. Despite the prevalence of defeasance and otherprepayment restrictions such as yield maintenance and fixed penalties, prepay-ment can and does occur, primarily due to cashout refinancings. In a cashoutrefinancing, CML borrowers incur the costs related to prepayment penalties ontheir loans if such costs are outweighed by the benefits of the rise in the valueof the property securing the loans.

Because our historical database includes many loans that are not defeasance,and because a significant portion of new-issue loans in the CMBS markettoday are not exclusively defeasance loans, it is essential that our methodologyincludes a general procedure for accommodating the risks of prepayments inCMBS. Letting Bt denote the remaining principal balance of the loan at timet, we represent the total prepayment amount as Bt (1 + Y t ), where Y t is thetime t prepayment cost as a percentage of the remaining principal balance. Y t

is determined by the specific mortgage loan’s prepayment penalities and candiffer across mortgages.

Valuation. To value a CML, we fix a particular loan. Let U t be a vectorof deterministic loan-specific characteristics. There are four possible states ofthe loan (current c, delinquent l, defaulted d and prepaid p). Both default andprepayment are absorbing states, and current/delinquency are repeating statesfor the loan.

Let C(t) be a process taking values in the set {0, 1, 2, 3, . . .}, counting thenumber of times up to and including time t that the loan switches betweenthe current and the delinquent states. Note that C(0) = 0. Let the loan beinitialized in the current state (we could have initialized in the delinquent stateinstead). We assume that C(t) follows a Cox process with an intensity ν (t) ={1{C(t−) is odd} λc(t , U t , Xt ) + 1{C(t−) is even} λl(t , U t , Xt )} under the martingalemeasure.6 If C(t−) is odd, then the intensity governing switching to current

6 This intensity process and others introduced below are assumed to satisfy the necessarymeasurability and integrability conditions required to guarantee that Expression (3)exists and is well defined; see Lando (1998).


is λc(t , U t , Xt ). If C(t−) is even, then the intensity governing switching todelinquent is λl(t , U t , Xt ). We can define a point process N (t) = {1 if C(t)is even, 0 if C(t) is odd} = {1 if delinquent, 0 if current} with N (0) = 0. Therelationship between the point process N(t) and the counting process C(t) isgiven by

dN(t) = −1{N(t−)=1}dC(t) + 1{N(t−)=0}dC(t)

where dN(t) = N (t) − N (t−). Note that N(t) also follows a Cox process withthe same intensity ν (t) = {1{N(t−)=1} λc(t , U t , Xt ) + 1{N(t−)=0} λl(t , U t , Xt )}under the martingale measure.

There is an implicit assumption here that needs to be made explicit. Thisprocess supposes that the probability of current or delinquent is independentof being current or delinquent. This assumption is equivalent to supposingthat current or delinquency are uninformative with respect to the next period’scurrent or delinquency status. An analogy might be that of a household payingan electric bill. The households are usually current, but in some months, theyare delinquent. Being delinquent does not mean the household will be morelikely to be delinquent in the next month. However, the electric company doesnotice that those households that are delinquent in a month are more likelyto default then are those that are current. This is reflected in the process nextdescribed.

Let τ d be the random default time on this loan and denote its point processby Nd (t) ≡ 1{τd≤t}. We assume that the point process follows a Cox processwith an intensity λd (t) = λd (t , Nt , U t , Xt ) under the martingale measure.The Cox process assumption implies that conditional upon the information setgenerated by (Nt,Xt )t∈[0,T ] up to time T , Nd (t) behaves like a Poisson process.If default occurs, the recovery on the loan is assumed to be δτd

· (Bτd+ P ). The

trust receives δτdpercent of the remaining principal balance plus the prorated

scheduled principal payment.7 We assume that δt = δ (t , U t , Xt ) is a functiondepending upon the state of the loan and the economy, implying a stochasticrecovery rate.

Under this intensity process, the probability that a default will occur on theloan’s balloon payment date [T − dt , T ] is approximately λd (T , NT , U T ,

7 For notational simplicity, we assume that the prorated portion of the loan payment isthe entire payment P. However, in the subsequent valuation software, the exact proratedportion of the loan payment is computed.


XT ) dt .8 Allowing for default on the balloon payment date captures what isoften called “extension risk” in the CMBS literature. Extension risk is the riskthat, on the balloon payment date, the borrower will not be able (or willing) tomake the balloon payment but will be able (or willing) to continue making thecoupon and amortization payments P. The belief is that, by extending the loan,the balloon payment will be made at a later date. The trustee of the CMBS trustdecides whether or not to extend the loan and the conditions of the extension.The balloon date could be extended (usually less than 3 years) and the under-lying coupon rate, tenor and leverage of the loan may be altererd in workoutproceedings, though generally with no alternation to the trust documentation.If extension occurs, one can think of this situation as being equivalent to theoccurrence of default, but the extension process is initiated to increase therecovery rate on the loan.

Let τ p be the random prepayment time on this loan and denote its pointprocess by Np(t) ≡ 1{τp≤t}. Again, we let the prepayment point process bea Cox process with intensity λp(t) = λp(t , Nt , U t , Xt ) under the martin-gale measure. If the loan is prepaid, the trustee receives Bτp

(1 + Yτp

) dol-lars. This represents principal plus the prepayment penalty. For analytic con-venience, we assume that conditional upon the information set generatedby (Nt,Xt )t∈[0,T ] up to time T , Nd (t) and Np(t) are independent Poissonprocesses.

Given the previous notation, as viewed from time t, the cash flow to a CML attime T is

T∑j=t+1

P 1{j<τd }1{j<τp}e∫ T

jrs ds + (BT + P )1{T <τd }1{T <τp}

+ δτd(Bτd

+ P )1{τd≤T }1{τd≤τp}e∫ T

τdrs ds

+Bτp(1 + Yτp

)1{TL<t}1{τp<T }1{τp<τd }e∫ T

τprs ds

. (1)

The first two terms in Expression (1) give the promised payments on the CMLif there is no default and no prepayment. The third term gives the accumulatedpayment up to time T if a default occurs prior to a prepayment, and the fourthterm gives the accumulated payment up to time T if a prepayment occurs priorto a default.

8 The default process could be generalized to allow a positive probability of a defaultat time T . Unfortunately, our database does not have sufficient defaults on the balloonpayment date to allow estimation of this more general model.


Given the martingale measure Q, the time t present value of these cash flows is

Et

{T∑

j=t+1

P 1{j<τd }1{j<τp}e− ∫ j

trs ds + (BT + P )1{T <τd }1{T <τp}e− ∫ T

trs ds

+ δτd(Bτd

+ P )1{τd≤T }1{τd≤τp}e− ∫ τdt

rs ds

+Bτp(1 + Yτp

)1{TL<t}1{τp<T }1{τp<τd }e− ∫ τpt

rs ds

}(2)

where Et (·) denotes expectation under the martingale measure. Under the Coxprocesses, standard techniques yield

Et

{T∑

j=t+1

Pe− ∫ j

t(rs+λd (s)+λp(s)) ds + (BT + P )e− ∫ T

t(rs+λd (s)+λp(s)) ds

+∫ T

t

δs(Bs + P )λd (s)e− ∫ s

t(ru+λd (u)) du ds · e− ∫ T

tλp(s) ds

+∫ T

t

∫ k

t


t(ru+λd (u)) du ds · λp(k)e− ∫ k

tλp(s) ds dk

+∫ T

t

Bs(1 + Ys)λp(s)e− ∫ s

t(ru+λp(u)) du ds · e− ∫ T

tλd (s) ds

+∫ T

t

∫ k

t

Bs(1 + Ys)λp(s)e− ∫ s

t(ru+λp(u)) du ds · λd (k)e− ∫ k

tλd (s) ds dk

}. (3)

This valuation expression is proved in the appendix. For the implementation,we calculate the expectation in Expression (3) using Monte Carlo simulation.

CMBS Bonds

A CMBS trust’s assets consist of a pool of loans whose values were modeled inthe previous section. A typical trust, in turn, issues a collection of i = 1, . . . , mordinary coupon bonds and at least one IO bond.

Description. The ordinary coupon bonds have coupon rates ci , face valuesF i and stated maturity dates T i at issuance with increasing maturities T 1 ≤T 2 ≤ · · · ≤ T m. In a typical sequential pay, senior/subordinated structure, theprincipal cashflows from the trust are allocated based on the bond’s stated matu-rities at issuance. The most senior bonds (maturity T 1) receive their scheduledprincipal payments in any month first, until their principal balance is reducedto zero. Then, the next most senior bonds (maturity T 2) receive their scheduledprincipal payments (if any cash flows remain) in any given month until their


principal balance is reduced to zero, and so forth to the least senior bonds (ma-turity T m). Coupon cash flows are allocated pro rata subject to the availabilityof such coupon cash flows. In the case of interest shortfalls, coupon cash flowsare allocated in order of seniority and subject to the documentation of the trust.Also, any loan prepayments and/or default recovery payments received are al-located according to seniority as well. These prepayments or defaults on theloans result in prepayment of the principal on the CMBS bonds prior to theirmaturity. In reverse fashion, any losses in default are subtracted from the leastsenior bond’s (maturity T m) principal first, then working backwards up to themost senior bond’s (maturity T 1) principal.

The outstanding principal balance of the IOs at all times is a reflection of thenotional amount of some collection of principal-paying bonds secured by thetrust’s principal cash flows. While the IO bonds have a stated maturity dateT 0, implying no principal repayment and only the payment of coupons, theirnotional balance declines in lockstep with the principal-paying bonds fromwhich they are stripped. The interest payments received on the IOs are thecumulative interest payments from the loan pool, less the cumulative couponspaid to the principal-paying CMBS coupon bonds.

Valuation. For the purposes of this section, we let the random cash flows attime t to bonds i = 0, 1, . . . , m be denoted vi(t). We let bond i = 0 correspondto the IO bond. Then, the time t value of these bonds is given by the followingexpression

bi(t) = Et

{Ti∑

j=t+1

vi(j )e− ∫ j

trs ds

}if t < Ti. (4)

Because of the complexity of the valuation problem, the expectation in thisexpression will be evaluated using Monte Carlo simulation, described in asubsequent section.

The Empirical Processes

This section describes the stochastic processes for the term structure of inter-est rates and state variables used in the empirical implementation. We use amultiple-factor HJM model for interest rate risk and standard diffusion pro-cesses for the state variables.

The HJM Model

We can specify the evolution of the term structure using forward rates underthe martingale measure.


The Stochastic Process. Let f (t, T ) = − ∂ log p(t,T )∂T

be the instantaneous (con-tinuously compounded) forward rate at time t for the future date T . We use a Kfactor model HJM model.

df (t, T ) = α(t, T ) dt +K∑

j=1

σj (t, T ) dWj (t) (5)

where K is a positive integer, α(t, T ) = ∑Kj=1 σj (t, T )

∫ T

tσj (t, u) du,

σj (t, T ) ≡ min[σrj (T )f (t, T ),M] for M a large positive constant, σ rj (T ) aredeterministic functions of T for j = 1, . . . , K , and W j (t) for j = 1, . . . , Kare uncorrelated Brownian motions initialized at zero. Under this evolution,forward rates are almost lognormally distributed.

The spot rate process, used for valuation, can be deduced from the forward rateevolution. Let r t ≡ f (t , t).

drt = [∂f (t, t)/∂T ] dt + α(t, t) dt +K∑

j=1

σj (t, t) dWj (t).

But α(t, t) = ∑Kj=1 σj (t, t)

∫ t

tσj (t, t) du = 0, so

drt = [∂f (t, t)/∂T ] dt +K∑

j=1

min[σrj (T )rt ,M] dWj (t). (6)

For the subsequent analysis, we will need to know the evolution of constantmaturity zero-coupon bonds. It is shown in the Appendix that

dp(t, t + T )

p(t, t + T )= (rt − f (t, t + T )) dt −

K∑j=1

(∫ t+T

t

σj (t, u) du

)dWj (t). (7)

The Empirical Methodology. To estimate the forward rate process givenin Expression (5), we employ a principal component analysis as discussedin Jarrow (2002). Given is a time series of discretized forward rate curves{f (t, T1), f (t, T2), . . . , f (t, TNr

)}mt=1 where Nr is the number of discreteforward rates observed, the interval between sequential time observationsis and m is the number of observations. Then, percentage changes arecomputed { f (t+,T1)−f (t,T1)

f (t,T1) , . . . ,f (t+,TN )−f (t,TNr )

f (t,TNr ) }mt=1. From the percentagechanges, the Nr × Nr covariance matrix (from the different maturity for-ward rates) is computed, and its eigenvalue/eigenvector decomposition is cal-culated. The normalized eigenvectors give the discretized volatility vectors{σrj (T1)

√, . . . , σrj (TNr

)√

} for j = 1, . . . , Nr .


The State Variables

The state variables in our model correspond to various indices related to theproperty values underlying the CMBS trusts. All indices are assumed to corre-spond to the prices of actively traded assets (i.e., values of different portfolios ofproperties). To the extent that these indices do not correspond to actively tradedassets, the resulting property prices will contain more noise, and the price pro-cesses’ parameters will be estimated with larger sampling errors. Nonetheless,under the plausible assumption that this additional noise has zero mean, theresulting estimates will still be unbiased. Because our model only uses theseproperty value indices to determine fair value, and not to hedge, this assumptionwill not have a significant effect on the subsequent analysis.

There are three levels of property value indices. The first set of state variablescorrespond to the price of a particular type of property located in a particularregion of the country, for example, hotels in New York City. The second set ofstate variables correspond to an index for a particular property type (but acrossthe entire country), for example, hotels. Lastly, the third state variable is anindex across all property types across the entire country, for example, a REITgeneral stock price index. The idea underlying this decomposition comes fromportfolio theory, where the first state variable is an individual stock price, thesecond state variable is an industry index and the third state variable is the mar-ket index. This construction is formulated to facilitate simulation of the statevariable processes in a subsequent section.

The Stochastic Process. We specify the stochastic processes for these statevariables in reverse order. All stochastic processes are specified under themartingale measure. The evolution for the economy-wide property index andthe regional property index are

dH̄ (t) = rt H̄ (t) dt + σ (H̄ )H̄ (t) dZH̄ (t) (8)

dHi(t) = rtHi(t) dt + σi(H )Hi(t) dZHi (t) for i = 1, . . . , nH (9)

where ZH̄ (t), ZHi (t) are Brownian motions for all i, σ (H̄ ), σi(H ) are con-

stants for all i, dZH̄ (t) dZHi (t) = ρHH̄

i dt, dZHj (t) dZH

i (t) = ρHHji dt , and the

state variable Brownian motions are all correlated with the forward rate Brow-nian motions: dZH̄ (t) dW i(t) = ηH̄

i dt and dZHj (t) dW i(t) = ηH

ji dt .

The property×region index satisfies

dhi(t)

hi(t)= rt dt +

K∑k=1

αik

(dp(t, t + Tk)

p(t, t + Tk)− rt dt

)+ βij (i)

(dHj (i)(t)

Hj (i)(t)− rt dt

)

+ γi

(dH̄ (t)

H̄ (t)− rt dt

)+ σi(h) dZh

i (t) for i = 1, . . . , nh (10)


where αik , βij (i), γ i , σ i(h) are constants for all i, k, j(i), the property×regionindex represented by hi(t) is the property type corresponding to the prop-erty index H j (i)(t),9 and Zh

i (t) are Brownian motions independent ofZH̄ (t), ZH

i (t),Wk(t) for all i, k. The K maturity bonds T k are chosen to bedistinct. Note that in Expression (10), the zero coupon bond price returns cor-respond to constant maturity bonds, that is, their maturity is always T k fork = 1, . . . K .

This formulation is chosen because in the subsequent simulation, under thissystem, there are only (1 + nH + K) correlated Brownian motions to besimulated. The remaining nh Brownian motions associated with Expression(10) are independent. This substantially reduces the size of the simulation fromone where (1 + nH + K + nh) correlated Brownian motions need to begenerated.

Because all three classes of stochastic processes represent the prices of tradedassets, their drifts are set equal to the spot rate of interest under the martingalemeasure. The drift of a traded asset equals the spot rate of interest, however, onlyif there are no dividends nor storage costs from holding the asset. In the case ofcommerical properties, there are two offsetting considerations in this regard.First, there are rental fees from owning commerical property and, second,there are maintenance costs. On average, one would expect that the rental feesexceed the maintenance costs, making the drift rate for our commercial propertyindices lower than the spot rate of interest. This is only true on average, however,because not all commerical properties are profitable based on cash flows alone.Unfortunately, we could not find any data sources to capture these rental andmaintenance fees. As a first approximation, therefore, we set their drifts equal tothe spot rate of interest. Although the subsequent estimation procedures do notdepend on the specification of the drift process, our simulation methodologydoes. This introduces a potential model misspecification into our valuationmethodology.

The Empirical Methodology. To compute the parameters of Expressions (8)and (9), we use the quadratic variation, which is invariant under a change ofequivalent probability measures. This invariance implies that our estimationprocedure does not depend on the drifts of the commercial property indices.We illustrate this estimation with respect to Expression (9). The procedure isidentical for Expression (8) as well.

9 This is the reason for the double indexing of j(i). i is the property×region index, andj(i) is the property index corresponding to i.


Given is a time series of {H i(t)}mt=1 where the interval between sequential

time observations is and m is the number of observations. Define H i(t) ≡[H i(t + ) − H i(t)]. We compute

m∑t=1

(Hi(t)

Hi(t)

)2 1

mgiving an estimate of σi(H )2. (11)

Next we calculatem∑

t=1

(Hj (t)

Hj (t)

Hi(t)

Hi(t)

)1

mgiving an estimate of σj (H )σi(H )ρHH

ji . (12)

To obtain the correlation between the forward rates and the regional propertyindex ηH

ji for j = 1, . . . , K we compute

m∑t=1

(f (t, t + Tk)

f (t, t + Tk)

Hi(t)

Hi(t)

)1

mgiving an estimate of

K∑j=1

σrj (Tk)σi(H )ηHji.

(13)

This is computed for k = 1, . . . , K for distinct T 1, . . . , T K yielding K equationsin K unknowns {ηH

1i , . . . , ηHKi}. Solving this system gives the estimates. This is

done for all i = 1, . . . , nH .

To estimate the parameters of Expression (10), we discretize this expression,and change to the statistical measure.

hi(t)

hi(t)− rt = χi +

K∑k=1

αik

(p(t, t + Tk)

p(t, t + Tk)− rt

)

+βij (i)

(Hj (i)(t)

Hj (i)(t)− rt

)

+ γi

(H̄ (t)

H̄ (t)− rt

)+ εi(t) for i = 1, . . . , nh

(14)

where χ i is a constant and εi(t) ≡ σi(h)Z̃hi (t).10 In this expression Z̃h

i (t) is theBrownian motion under the empirical measure and χ i is the adjustment in thedrift due to the market price of risk associated with the change of measure.11 Weassume that the market price of risk is a constant. Standard regression analysisprovides the estimates for the parameters in Expression (14). This time-seriesregression is run nh times.

10 To the extent that commerical properties have a positive cash flow, the constant χ i

will capture this consideration as well.11 Formally, dZh

i (t)= dZ̃hi (t) + χi

σi (h)dt .


The Default and Prepayment Intensity Processes

Each commercial loan i has a default and prepayment intensity process thatdepends on its delinquency status, the state variable vector Xt and a vectorof loan-specific characteristics U i

t that are deterministic (nonrandom), for ex-ample, the net operating income (NOI) of the underlying property at the loanorigination.

The Stochastic Processes. The current and delinquent intensity processes foreach loan have the same functional form, differing only in the loan-specificvariables used. We let

λc

(t, U i

t , Xt

) = eϕc+φcUit +ψcXt and (15)

λl

(t, U i

t , Xt

) = eϕl+φlUit +ψlXt for loan i (16)

where ϕc, φc, ψc, ϕl , φl , ψl are vectors of constants. The default and prepay-ment intensity processes are similar, but they include the delinquency status ofthe loan N(t) as an additional state variable.

λd

(t, Nt , U

it , Xt

) = eϕd+θdNt+φdUit +ψdXt and (17)

λp

(t, Nt , U

it , Xt

) = eϕp+θpNt+φpUit +ψpXt for loan i (18)

where ϕd , θ d , φd , ψd , ϕp, θ p, φp, ψp are vectors of constants.

These intensity processes are given under the martingale measure for inclusionin the valuation equations. However, estimation of these intensities is under thestatistical measure. Fortunately, given the assumption that delinquency, defaultand prepayment risk are conditionally diversifiable, these intensity functionswill be equivalent under both the empirical and martingale measures, see Jarrow,Lando and Yu (2005). This assumption is reasonable if the intensity processes,through the state variables employed, include all relevant systematic risks in theeconomy. This inclusion leaves only idiosyncratic risk to determine the actualoccurrence of delinquency, default and prepayment. The alternative approachis to modify Expressions (15) and (16) by estimates of the market prices ofdelinquency, default and prepayment risk.12

The Empirical Methodology. Given are commercial loan payment historiesincluding scheduled principal payments, defaults, losses, prepayments, loancharacteristics and time-series observations for the state variables (Nt , Xt )Tt=1where the interval between sequential time observations is .

12 Let κ a denote the random processes representing the market prices of delinquency,default or prepayment risk, respectively. The transformation is κ a λa(t , U i

t , Xt ); seeJarrow, Lando and Yu (2005) for details.


For empirical estimation, Expressions (15) and (16) are replaced with theirdiscrete approximations

λc

(t, U i

t , Xt

) = 1/

(1 + e−(ϕc+φcU

it +ψcXt )

)(19)

λl

(t, U i

t , Xt

) = 1/

(1 + e−(ϕl+φlU

it +ψlXt )

). (20)

In the estimation, we have λl(t) = 11+eηt

for ηt = ϕc + φcUit + ψcXt and

λc(t) = 11+e−ηt

. Note that the parameter(s) within ηt in the estimation areequal and opposite in sign for λl(t) and λc(t).13 Next,

λd

(t, Nt , U

it , Xt

) = 1/

(1 + e−(ϕd+θdNt+φdUi

t +ψdXt ))

(21)

λp

(t, Nt , U

it , Xt

) = 1/

(1 + e−(ϕp+θpNt+φpUi

t +ψpXt )). (22)

These discrete approximations have the interpretation of being the probability ofdefault and prepayment over [t , t + ], respectively, conditional on no defaultor prepayment prior to time t. These expressions are estimated in a competing-risk paradigm (see Deng, Quigley and Van Order 2000, Ambrose and Sanders2003, Ciochetti et al. 2004) using maximum likelihood estimation. Competingrisk means that the occurrence of default precludes prepayment, and conversely.The estimation methodology explicitly incorporates this interdependence.

The Estimates

This section discusses the data and the results from the estimations.

The Term Structure of Interest Rates

The term structure data were obtained from the Federal Reserve Board.14 Itconsists of daily constant maturity yields from 3 months up to 20 years. Thedata set starts on January 4, 1982, and goes to May 16, 2005. We convertthese constant maturity yields into a term structure of (smoothed) continuouslycompounded forward rates with maturities 3 months, 6 months and 1, 2, 3, 5, 7,10 and 20 years. In our sample period, spot rates have ranged from a maximumof approximately 12% to a low of approximately 1.5%. Table 1 provides thevolatility coefficients for the 3-month, 6-month and 1-, 2-, 3-, 5-, 7-, 10- and20-year forward rates and the percentage variance explained by the first four

13 This can be seen as follows. Because λl(t) + λc(t) = 1,dλl (t)

dη= − dλc (t)

dη.

14 See www.federalreserve.gov/releases/h15.


Tab

le1

�Fo

rwar

dra

tevo

latil

ityfu

nctio

ns.

Mat

uriti

es\F

acto

rs1

23

45

67

89

3m

onth

s−0

.172

30.

0303

−0.1

129

0.08

13−0

.056

00.

0185

−0.0

133

0.03

630.

0154

6m

onth

s−0

.256

80.

0264

−0.1

019

0.03

560.

0190

−0.0

124

0.02

62−0

.051

3−0

.020

71

year

−0.2

924

−0.0

205

−0.0

255

−0.0

959

−0.0

280

−0.0

421

−0.0

476

0.00

080.

0016

2ye

ars

−0.2

922

0.01

400.

0161

−0.0

238

0.08

620.

0138

0.02

890.

0223

0.03

253

year

s−0

.205

7−0

.010

80.

0665

−0.0

424

−0.0

688

0.03

240.

0491

0.01

38−0

.026

75

year

s−0

.162

4−0

.003

80.

0626

0.04

670.

0503

0.02

91−0

.046

70.

0130

−0.0

419

7ye

ars

−0.1

329

−0.0

545

0.08

690.

0288

−0.0

335

0.04

29−0

.023

0−0

.042

40.

0338

10ye

ars

−0.0

860

0.18

580.

1090

0.04

84−0

.018

6−0

.051

80.

0037

0.00

000.

0056

20ye

ars

−0.0

792

−0.2

017

0.04

560.

0631

−0.0

025

−0.0

552

0.01

610.

0107

−0.0

001

%V

ar62

.330

017

.272

58.

8499

4.19

652.

9405

1.88

381.

3456

0.79

560.

3856

Not

es:

The

estim

ates

corr

espo

ndto

the

fact

orvo

latil

ities

σj(

T)

for

j=

1,..

.,9

and

T=

3m

onth

s,..

.,20

year

sob

tain

edfr

oma

prin

cipa

lco

mpo

nent

san

alys

isus

ing

mon

thly

perc

enta

gech

ange

sin

forw

ard

rate

s.%

Var

isth

epe

rcen

tage

ofth

eva

rian

ceex

plai

ned

byea

chfa

ctor

.The

estim

atio

nis

over

the

time

peri

odJa

nuar

y4,

1982

toM

ay16

,200

5.


factors based on monthly observation intervals ( = 1/12).15 As indicated,the first four factors explain 93% of the variation in monthly forward ratemovements. For the subsequent analysis, we set K = 4 in Expression (5).

The State Variables

The REIT stock price index, H̄ (t), is obtained from Bloomberg. The Bloombergreal estate investment trust index is a capitalization-weighted index of REITs,excluding mortgage-related REITs. To be included in the index, the trusts musthave a capitalization of at least $15 million. We have monthly observations ofthis index over the time period June 1998 to May 2005. To confirm the appro-priateness of the Bloomberg REIT index as a measure of national commercialreal estate values, we compared it to the general NCREIF property value index,using quarterly observation intervals. The correlation between the returns onthe two indices is 0.92, confirming its appropriateness.

The property value indices, H i(t), are obtained from Bloomberg as well. Anal-ogous to the REIT stock price index, we have monthly observations over thesame time period.16 There are six property types considered (nH = 6): indus-trial (IN), lodging (LO), multifamily (MF), office (OF), retail (RT) and other(OT). Lastly, the property×region indices, hi(t), are obtained from NCREIF.These indices are obtained quarterly and they correspond to nine regions: EastNorth Central, Mideast, Mountain, Northeast, Pacific, Southeast, Southwest,West North and Other. This gives a total of 54 different indices (nh = 54). Forthe constant maturity zero-coupon bond prices in Expression (14), we used thematurities 1 year, 2 years, 5 years and 7 years. Note that, for the property×region indices, LO (lodging) and OT (other) have the same index across allregions due to the absence of a property×region index for these property types.The indices were high in 1998, declined and then increased again to theirmaximum values in 2005.

The estimated values for the state variable parameters are given in Tables 2and 3. As indicated, all the property value indices are negatively correlated

15 Daily observation intervals were not used because daily variations in rates are partlycaused by the smoothing procedure. Monthly observation intervals reduces the impor-tance of this smoothing noise in the estimated coefficients.16 Longer time histories are available, but the default, loss and prepayment data providedby Trepp begins in 1998 and, for consistency in the statistical analysis of the propertyvalue state variables, their estimation was chosen to begin in 1998 as well. Futureresearch could relax this restriction and use the longer data set for the estimation ofthese property value parameters.


Table 2 � State variable parameter estimates using monthly observations over the timeperiod June 1998 to May 2005.

Panel A: The Monthly Standard Deviation σ (H̄ ) of the Economy-Wide REIT StockPrice Index H̄ .

σ (H̄ ) 0.1221

Panel B: The Monthly Standard Deviations σ (Hi) of the Property Type REIT StockPrice Indices Hi for the Property Types Industry (IN), Lodging (LO), Multifamily (MF),Office (OF), Other (OT) and Retail (RT).

Index i IN LO MF OF OT RT

σ (Hi) 0.1260 0.2180 0.1150 0.1272 0.2180 0.1597

Panel C: The Monthly Correlations ηH̄j Between the Economy-Wide REIT Stock Price

Index H̄ and the Four Zero-Coupon Bond Prices p(t, T) with Maturities T = 1 Year,2 years, 5 years and 7 years.

Maturity 1 year 2 years 5 years 7 years

ηH̄j −0.2424 0.1697 −0.1705 0.2681

Panel D: The Monthly Correlations ρHH̄i Between the Economy-Wide REIT Stock Price

Index H̄ and the Property Type Stock Price Indices Hi for the Property Types Industry(IN), Lodging (LO), Multifamily (MF), Office (OF), Other (OT) and Retail (RT).

Index i IN LO MF OF OT RT

ρHH̄i 0.9239 0.7062 0.9038 0.9541 0.7062 0.8891

Panel E: The Monthly Correlations ηHij Between the Property Type REIT Stock Price

Indices Hi for the Property Types Industry (IN), Lodging (LO), Multifamily (MF),Office (OF), Other (OT) and Retail (RT) and the Zero-Coupon Bond Prices p(t, T) withMaturities 1 Year, 2 years, 5 years and 7 years.

ηHij IN LO MF OF OT RT

1 year −0.1426 −0.3074 −0.3396 −0.2727 −0.3074 −0.13842 years 0.1892 0.1435 0.1119 0.2093 0.1435 0.14165 years −0.1654 −0.0280 −0.1920 −0.1559 −0.0280 −0.14337 years 0.3182 0.0803 0.2703 0.2695 0.0803 0.2474

Panel F: The Monthly Correlations ρHHij Across the Various Property Type REIT Stock

Price Indices Hi for the Property Types Industry (IN), Lodging (LO), Multifamily (MF),Office (OF), Other (OT) and Retail (RT).

ρHHij IN LO MF OF OT RT

IN 1 0.5961 0.8172 0.8365 0.5961 0.8527LO 0.5961 1 0.6280 0.6491 0.9881 0.5871MF 0.8172 0.6280 1 0.8590 0.6280 0.7101OF 0.8365 0.6491 0.8590 1 0.6491 0.7746OT 0.5961 0.9881 0.6280 0.6491 1 0.5871RT 0.8527 0.5871 0.7101 0.7746 0.5871 1


Tab

le3

�T

heN

CR

EIF

regi

onal

×pr

oper

tyva

lue

indi

ces

regr

essi

onpa

ram

eter

estim

ates

usin

gm

onth

lyob

serv

atio

nsov

erth

etim

epe

riod

June

1998

toM

ay20

05.

Prop

erty

Reg

ion

(α0,s

tder

r)(α

1,s

tder

r)(α

2,s

tder

r)(α

3,s

tder

r)(α

4,s

tder

r)(β

,std

err)

(γ,s

tder

r)R

2N

INE

ast–

Nor

th–

(−0.

1732

,0.1

571)

(2.8

09,4

4.64

)(−

11.2

,60.

24)

(−1.

457,

38.4

2)(8

.414

,23.

38)

(−0.

3208

,5.9

22)

(2.4

39,6

.157

)0.

0578

83C

entr

alIN

Mid

east

(−0.

002,

0.06

825)

(−30

.32,

19.4

)(3

2.11

,26.

17)

(13.

76,1

6.69

)(−

15.1

6,0.

16)

(0.9

718,

2.57

3)(−

0.47

4,2.

675)

0.22

3683

INM

ount

ain

(0.1

304,

0.09

33)

(5.4

43,2

6.53

)(6

.969

,35.

81)

(−21

.43,

22.8

4)(1

0.25

,13.

9)(1

.552

,3.5

2)(−

1.53

,3.6

6)0.

2405

83IN

Nor

thea

st(0

.042

1,0.

0356

)(1

0.91

,10.

12)

(−11

.64,

13.6

6)(−

2.04

7,8.

71)

(3.9

57,5

.301

)(−

0.71

24,1

.343

)(0

.622

1,1.

396)

0.59

3083

INPa

cific

(0.0

274,

0.05

18)

(8.6

15,1

4.73

)(−

8.38

6,19

.88)

(−4.

649,

12.6

8)(5

.663

,7.7

18)

(0.0

774,

1.95

5)(−

0.28

8,2.

032)

0.37

6883

INSo

uthe

ast

(−0.

0904

,0.1

146)

(3.8

85,3

2.56

)(−

12.2

7,43

.94)

(−2.

988,

28.0

2)(1

0.86

,17.

05)

(0.0

6418

,4.3

19)

(1.3

6,4.

491)

0.14

2583

INSo

uthw

est

(0.1

304,

0.10

62)

(10.

81,3

0.18

)(−

14.3

3,40

.73)

(−3.

186,

25.9

8)(6

.067

,15.

81)

(3.7

78,4

.004

)(−

1.81

6,4.

163)

0.22

8883

INW

est–

Nor

th–

(−0.

5825

,0.3

372)

(32.

44,9

5.82

)(−

53.7

9,12

9.3)

(−35

.26,

82.4

8)(5

2.15

,50.

19)

(−9.

74,1

2.71

)(1

4,13

.22)

0.06

2383

Cen

tral

INO

ther

(0.0

425,

0.04

08)

(6.9

36,1

1.6)

(−3.

414,

15.6

6)(−

10.3

5,9.

986)

(7.7

84,6

.077

)(0

.197

4,1.

539)

(−0.

0385

7,1.

6)0.

5385

83

LO

Eas

t–N

orth

–(−

0.33

03,0

.485

9)(3

2.14

,139

.7)

(−37

.1,1

87.6

)(−

17.4

7,12

0.7)

(17.

54,7

3.82

)(1

0.23

,5.9

38)

(−5.

805,

10.2

2)0.

0620

83C

entr

alL

OM

idea

st(−

0.33

03,0

.485

9)(3

2.14

,139

.7)

(−37

.1,1

87.6

)(−

17.4

7,12

0.7)

(17.

54,7

3.82

)(1

0.23

,5.9

38)

(−5.

805,

10.2

2)0.

0620

83L

OM

ount

ain

( −0.

3303

,0.4

859)

(32.

14,1

39.7

)(−

37.1

,187

.6)

(−17

.47,

120.

7)(1

7.54

,73.

82)

(10.

23,5

.938

)(−

5.80

5,10

.22)

0.06

2083

LO

Nor

thea

st(−

0.33

03,0

.485

9)(3

2.14

,139

.7)

(−37

.1,1

87.6

)(−

17.4

7,12

0.7)

(17.

54,7

3.82

)(1

0.23

,5.9

38)

(−5.

805,

10.2

2)0.

0620

83L

OPa

cific

(−0.

3303

,0.4

859)

(32.

14,1

39.7

)(−

37.1

,187

.6)

(−17

.47,

120.

7)(1

7.54

,73.

82)

(10.

23,5

.938

)(−

5.80

5,10

.22)

0.06

2083

LO

Sout

heas

t(−

0.33

03,0

.485

9)(3

2.14

,139

.7)

(−37

.1,1

87.6

)(−

17.4

7,12

0.7)

(17.

54,7

3.82

)(1

0.23

,5.9

38)

(−5.

805,

10.2

2)0.

0620

83L

OSo

uthw

est

(−0.

3303

,0.4

859)

(32.

14,1

39.7

)(−

37.1

,187

.6)

(−17

.47,

120.

7)(1

7.54

,73.

82)

(10.

23,5

.938

)(−

5.80

5,10

.22)

0.06

2083

LO

Wes

t–N

orth

–(−

0.33

03,0

.485

9)(3

2.14

,139

.7)

(−37

.1,1

87.6

)(−

17.4

7,12

0.7)

(17.

54,7

3.82

)(1

0.23

,5.9

38)

(−5.

805,

10.2

2)0.

0620

83C

entr

alL

OO

ther

(−0.

3303

,0.4

859)

(32.

14,1

39.7

)(−

37.1

,187

.6)

(−17

.47,

120.

7)(1

7.54

,73.

82)

(10.

23,5

.938

)(−

5.80

5,10

.22)

0.06

2083


Ta

ble

3�

cont

inue

d

Prop

erty

Reg

ion

(α0,s

tder

r)(α

1,s

tder

r)(α

2,s

tder

r)(α

3,s

tder

r)(α

4,s

tder

r)(β

,std

err)

(γ,s

tder

r)R

2N

MF

Eas

t–N

orth

–(0

.027

4,0.

0904

)(1

9.34

,25.

33)

(−21

.24,

34.2

)(−

13.9

4,21

.84)

(14.

86,1

3.35

)(5

.969

,3.4

45)

(−3.

82,3

.217

)0.

2347

83C

entr

alM

FM

idea

st(0

.095

5,0.

0378

)(1

4.35

,10.

61)

(−15

.26,

14.3

3)(3

.239

,9.1

48)

(0.2

17,5

.592

)(4

.834

,1.4

43)

(−6.

113,

1.34

7)0.

6170

83M

FM

ount

ain

(0.6

241,

0.78

96)

(103

.6,2

21.2

)(−

38.6

,298

.6)

(−29

.2,1

90.7

)(−

19.5

,116

.6)

(−65

.37,

30.0

7)(5

0.16

,28.

08)

0.10

7783

MF

Nor

thea

st(0

.035

3,0.

0626

)(1

0.05

,17.

54)

(−12

.78,

23.6

8)(4

.764

,15.

12)

(−1.

633,

.244

)(−

2.68

8,2.

385)

(3.2

86,2

.227

)0.

3397

83M

FPa

cific

(0.0

088,

0.04

85)

(19.

72,1

3.61

)(−

20.6

2,18

.37)

(−4.

752,

11.7

3)(7

.279

,7.1

71)

(−3.

001,

1.85

)(2

.318

,1.7

28)

0.43

3683

MF

Sout

heas

t(0

.056

4,0.

0438

)(−

2.86

,12.

27)

(6.2

84,1

6.57

)(−

2.75

6,10

.58)

(−0.

025,

6.47

)(−

1.15

8,1.

669)

(1.6

4,1.

559)

0.52

6383

MF

Sout

hwes

t(0

.017

8,0.

0285

)(−

4.78

9,7.

989)

(5.3

62,1

0.79

)(1

.211

,6.8

88)

(−1.

041,

4.21

1)(−

1.05

4,1.

086)

(1.3

56,1

.014

)0.

6864

83M

FW

est–

Nor

th–

(0.2

044,

0.20

72)

(7.2

41,5

8.05

)(9

.869

,78.

38)

(1.7

59,5

0.05

)(−

14.1

5,30

.6)

(−17

.85,

7.89

4)(1

4.27

,7.3

71)

0.15

3783

Cen

tral

MF

Oth

er(0

.017

7,0.

0288

)(5

.475

,8.0

76)

(−4.

536,

10.9

)(−

3.37

6,6.

964)

(3.1

98,4

.257

)(−

0.54

56,1

.098

)(0

.814

5,1.

025)

0.67

6883

OF

Eas

t–N

orth

–(0

.955

,1.3

62)

(−72

.91,

393.

6)(2

15.4

,528

.9)

(−15

3.6,

322.

7)(1

1.53

,195

.7)

(41.

58,6

5.73

)(−

39.0

9,68

.52)

0.05

2083

Cen

tral

OF

Mid

east

(0.0

891,

0.05

74)

(14.

25,1

6.6)

(−12

.91,

22.3

1)(−

4.97

7,13

.61)

(6.5

56,8

.253

)(0

.031

46,2

.772

)(−

1.72

8,2.

89)

0.42

6683

OF

Mou

ntai

n(−

0.21

53,0

.448

5)(−

65.2

2,12

9.6)

(69.

03,1

74.1

)(−

5.72

,106

.2)

(−4.

30,6

4.42

)(−

3.36

2,21

.64)

(10.

26,2

2.56

)0.

0233

83O

FN

orth

east

(0.3

464,

0.28

82)

(22.

2,83

.26)

(0.1

90,1

11.9

)(−

24.3

4,68

.26)

(5.6

06,4

1.39

)(5

.31,

13.9

)(−

7.28

4,14

.49)

0.08

1083

OF

Paci

fic(0

.099

1,0.

1786

)(−

23.4

1,51

.61)

(49.

28,6

9.34

)(−

64.0

9,42

.31)

(39.

42,2

5.66

)(1

2.28

,8.6

18)

(−12

.1,8

.984

)0.

1165

83O

FSo

uthe

ast

(0.1

53,0

.075

47)

(−18

.77,

21.8

)(2

3.61

,29.

29)

(−7.

197,

17.8

7)(3

.475

,10.

84)

(2.5

83,3

.641

)(−

2.27

3,3.

795)

0.35

3183

OF

Sout

hwes

t(−

0.42

11,0

.298

5)(6

5.41

,86.

23)

(−90

.47,

115.

9)(3

7.76

,70.

69)

(−17

.46,

2.87

)(−

0.18

99,1

4.4)

(4.9

81,1

5.01

)0.

0464

83O

FW

est–

Nor

th–

(−0.

4328

,0.2

368)

(−7.

092,

68.4

)(1

7.13

,91.

9)(−

53,5

6.07

)(3

7.12

,34)

(22.

33,1

1.42

)(−

16.3

,11.

91)

0.10

8083

Cen

tral

OF

Oth

er(−

0.62

08,0

.65)

(−47

.09,

187.

8)(1

5.83

,252

.3)

(2.1

88,1

54)

(24.

39,9

3.36

)(−

10.7

1,31

.36)

(15.

34,3

2.69

)0.

0412

83


Tab

le3

�co

ntin

ued

Prop

erty

Reg

ion

(α0,s

tder

r)(α

1,s

tder

r)(α

2,s

tder

r)(α

3,s

tder

r)(α

4,s

tder

r)(β

,std

err)

(γ,s

tder

r)R

2N

OT

Eas

t–N

orth

–(0

.010

9,0.

1871

)(9

5.09

,53.

8)(−

177,

72.2

4)(1

66.6

,46.

47)

(−85

.46,

8.43

)(3

.675

,2.2

87)

(−2.

023,

3.93

5)0.

2065

83C

entr

alO

TM

idea

st(0

.126

,0.0

6249

)(2

3.86

,17.

96)

(−25

.26,

24.1

2)(7

.204

,15.

52)

(−1.

101,

.492

)(1

.484

,0.7

635)

(−4.

942,

1.31

4)0.

4320

83O

TM

ount

ain

(0.1

035,

0.06

19)

(19.

97,1

7.82

)(−

20.6

4,23

.93)

(−2.

38,1

5.39

)(5

.183

,9.4

17)

(0.7

127,

0.75

74)

(−1.

634,

1.30

3)0.

3858

83O

TN

orth

east

(0.0

381,

0.04

28)

(18.

61,1

2.3)

(−26

.32,

16.5

2)(7

.942

,10.

63)

(0.5

753,

.502

)(0

.623

2,0.

523)

(−0.

371,

0.89

9)0.

5141

83O

TPa

cific

(0.0

551,

0.03

68)

(16.

98,1

0.58

)(−

20.1

,14.

21)

(−0.

332,

9.14

2)(5

.032

,5.5

92)

(1.2

55,0

.449

8)(−

1.71

6,0.

774)

0.62

2183

OT

Sout

heas

t(0

.082

3,0.

0504

)(−

2.67

3,14

.52)

(4.4

57,1

9.49

)(−

1.98

6,12

.54)

(0.8

211,

7.67

)(1

.148

,0.6

17)

(−0.

567,

1.06

2)0.

4926

83O

TSo

uthw

est

(0.0

312,

0.03

07)

(16.

39,8

.84)

(−20

.91,

11.8

7)(1

.736

,7.6

36)

(2.8

34,4

.671

)(0

.348

7,0.

3757

)(0

.691

,0.6

465)

0.69

1283

OT

Wes

t–N

orth

–(0

.692

7,0.

4831

)(−

118.

9,13

8.9)

(142

.8,1

86.5

)(5

1.18

,120

)(−

74.0

9,73

.4)

(2.4

66,5

.904

)(−

0.97

6,10

.16)

0.06

6983

Cen

tral

OT

Oth

er(0

.060

5,0.

0381

)(1

6.87

,10.

97)

(−18

.26,

14.7

2)(−

0.92

0,9.

472)

(3.7

41,5

.794

)(1

.162

,0.4

661)

(−1.

47,0

.802

)0.

6087

83

RT

Eas

t–N

orth

–(0

.953

3,1.

422)

(−37

6.6,

376)

(471

,505

.9)

(−12

2.2,

320)

(15.

37,1

94.6

)(4

5.25

,34.

85)

(−28

.9,4

3.81

)0.

0757

83C

entr

alR

TM

idea

st(0

.142

9,0.

0653

)(3

6.51

,17.

29)

(−45

.06,

23.2

6)(1

0.48

,14.

71)

(0.7

06,8

.946

)(−

2.43

2,1.

603)

(1.0

96,2

.014

)0.

4240

83R

TM

ount

ain

(0.0

569,

0.06

05)

(0.1

952,

16)

(−5.

563,

21.5

2)(9

.777

,13.

61)

(−3.

65,8

.278

)(−

0.07

84,1

.483

)(0

.422

8,1.

864)

0.37

8383

RT

Nor

thea

st(0

.053

4,0.

0558

)(2

3.57

,14.

76)

(−31

.44,

19.8

5)(9

.043

,12.

56)

(−0.

259,

7.63

)(0

.170

1,1.

368)

(0.0

187,

1.71

9)0.

4359

83R

TPa

cific

(0.1

63,0

.066

46)

(14.

29,1

7.57

)(−

26.5

5,23

.64)

(18.

34,1

4.96

)( −

3.73

1,9.

09)

(−3.

219,

1.62

9)(2

.242

,2.0

48)

0.44

0183

RT

Sout

heas

t(0

.049

9,0.

0323

)(−

2.35

6,8.

556)

(4.5

18,1

1.51

)(−

1.88

4,7.

281)

(0.5

91,4

.428

)(−

0.22

75,0

.793

)(0

.513

,0.9

969)

0.68

9283

RT

Sout

hwes

t(0

.100

7,0.

0576

)(1

3.44

,15.

25)

(−19

,20.

52)

(16.

61,1

2.98

)(−

9.35

,7.8

92)

(−0.

7918

,1.4

14)

(0.2

998,

1.77

7)0.

4293

83R

TW

est–

Nor

th–

(0.1

364,

0.14

26)

(−9.

245,

37.7

1)(8

.446

,50.

74)

(16.

33,3

2.09

)(−

12.5

,19.

52)

(3.2

84,3

.496

)(−

4.95

,4.3

94)

0.15

5483

Cen

tral

RT

Oth

er(0

.123

,0.0

5491

)(2

3.72

,14.

52)

(−32

.5,1

9.54

)(1

5.95

,12.

36)

(−5.

02,7

.514

)(−

0.95

69,1

.346

)(0

.106

6,1.

692)

0.49

5083

Not

es:

The

regr

essi

ones

timat

edis

:

h

i(t

)

hi(t

)−

r t

=α

0+

4 ∑ k=1

αk

( p

(t,t+

Tk)

p(t

,t+

Tk)

−r t

) +β

( H

j(i

)(t)

Hj

(i)(t)

−r t

) +γ

( H̄

(t)

H̄(t

)−

r t

) +εi(t

).

The

regi

ons

are

spec

ified

inco

lum

n2.

Prop

erty

type

sar

ein

dust

ry(I

N),

lodg

ing

(LO

),m

ultif

amily

(MF

),of

fice

(OF

),ot

her

(OT

)an

dre

tail

(RT

).G

iven

are

the

coef

ficie

nts,

the

stan

dard

erro

rs,t

heR

2an

dth

enu

mbe

rof

obse

rvat

ions

.


Table 4 � Monthly transition frequencies.

Transition to the Transition to aSame or Better State Worse State

Current (0 days delinquent) 99% 1%Delinquent (30–59 days) 62% 38%Delinquent (60–89 days) 36% 64%Delinquent (90+ days) 61% 39%

Notes: This table gives the monthly transition frequencies of moving from the presentstate (column 1) to the same or better state (column 2) versus a worse state (column 3),for all commercial mortgage loans over the time period June 1988 to June 2004.

with the first and third interest rate factors but positively correlated to thesecond and fourth. The property value indices are positively correlated witheach other. Table 3 contains the coefficients, standard errors and R2 from time-series regression Equation (10) based on monthly observations. Note that the R2,ranging between 2% and 68%, are similar in magnitude to those observed in theempirical asset pricing literature for stock returns. Although when consideredindividually, most coefficients are insignificantly different from zero (due tomulticollinearity issues), their collective influence is statistically significant formost regressions (as indicated by the high R2s).

Delinquency, Default and Prepayment Intensity Processes

The loan history database—including defaults, prepayments and loancharacteristics—was provided by Trepp.17 This database contains informationon over 100,000 commercial loans. The data provides monthly observationsof the relevant variables over the time period June 1998 to May 2005. In thisdatabase, the loans are classified as current, 30-59 days delinquent, 60-89 daysdelinquent, 90 plus days delinquent and defaulted. Loans exhibiting REO orForeclosure status are considered to be in default. Defaults are distinct fromdelinquencies. Because our model has only three classifications (current, delin-quent or default), not five, we needed to determine a coarser partitioning of theclassification. A statistical analysis was done to see if 30-59 days delinquentshould be classified as delinquent or current and if 90 plus days delinquentshould be classified as delinquent or default.

We conducted a 6-year study of delinquency transitions of more than 2.3 millionloan life observations. Table 4 shows the transitions over all loans from healthyto worse or conversely over the period June 1998 to June 2004. A healthy

17 See Reilly and Golub (2000) and Trepp and Savitsky (2000).


state is defined as current (0 days delinquent). A worse state is defined asthe next higher delinquency status. So, for example, a loan that is current inmonth 1 is characterized as having transitioned to a worse state in month 2if its delinquency status in month 2 is 30-59 days delinquent. Similarly, if aloan in month 1 is 90 plus days delinquent, it is said to have transitioned to ahealthy state if it becomes 0 days delinquent in month 2. Loans that persist innon-transition for multiple months either due to aberrations in the data (foundin loans exhibiting 30-59 or 60-89 loan delinquency status for multiple monthsin a row) or due to categorization (90 plus days delinquent is, by definition, amultiple month state) are not transitioned until they migrate to either healthy (0days delinquent) or a worse delinquency or defaulted (REO, Foreclosure) state.

Historically, more loans that were 30-59 days delinquent went to current thenon to a further delinquent status, hence they were so classified as current. Incontrast, more loans that were observed in 60-89 days delinquent migrated to aworse state and were therefore classified as delinquent Finally, the majority of90 days plus delinquent loans did not default. Hence, they too were classifiedas delinquent. In summary, in our model current loans are defined as actuallycurrent and 30-59 days delinquent, while delinquencies are classified as 60-89days delinquent and 90 days plus delinquent. Defaulted loans are those loansthat are classified as either REO or in foreclosure.

For the intensity process estimation, the loan-specific characteristics includedare: (1) age of the loan (as a percent of the life of the loan), (2) the delinquencystatus of the loan (dlqstatus), (3) an American Council of Life Insurers (ACLI)foreclosure survivor bias variable (fore index),18 (4) the NOI at originationdivided by the loan balance at origination (noi), (5) the prepayment restriction(normalized, monthly) (pen), (6) the logarithm of the original loan balance(origloanbal), (7) the debt service coverage ratio at origination (dscr), (8)the loan-to-value ratio at origination (ltv), (9) the weighted average couponat origination (wac), (10) the loan spread at securitization (only for fixed-rateloans) (coupon spread), (11) a dummy variable for property type (IN, LO,NF, OF, OT) and (12) a dummy variable for geographical location (R1-R8).The choice of many of the variables were dictated by data availability. Ourdatabase contained reliable data on loan characteristics at origination, but notafterwards.19

18 This is the aveage foreclosure rate over the past 14 years for each property× region,constructed from the ACLI foreclosure database.19 For example, some but not all of the loans had data on NOI after origination. Thesparsity of the updated NOI observations made this variable inappropriate to use. Inaddition, the updated NOI information is self reported and not reliable. Whereas, at theorigination date, the information is audited by the originator.


Table 5 � The number of commercial loans in the database from June 1988 to June2004.

Non-CTLs CTLs

Fixed Floating Fixed Floating

Prepaid 8989 2960 102 4Default 2153 130 56 1Total 94011 7198 1358 10

Notes: The loans are partitioned into those that are credit tenant leases (CTLs), fixed- ver-sus floating-rate, prepaid and defaulted. There are a total of 102,577 loans in the sample.

In addition to the property value state variables {H̄ (t),Hi(t), hi(t) for all i}discussed above, included in the set of state variables is the spot rate of interest{r t} and a measure of the slope of the yield curve {f (t , t + 10 years) − r t}. Thisyields a total of 62 state variables. This number of state variables is consistentwith both default and prepayment risk being conditionally diversifiable (see thediscussion following Expression (16)), an assumption we impose to facilitateestimation.

The loans are first split into two categories: those that are credit tenant leases(CTLs) and those that are not. CTLs are loans whose payments are guaranteedby a parent company (e.g., Wal-Mart). For non-CTLs and CTLs, separateintensity processes are estimated. Due to the small sample size of CTLs, weuse the prepayment intensity process of the non-CTLs and the default intensityprocess of the non-CTLs but only if the parent company’s identity is unknown.If the parent company’s identification is available, the default intensity from thehazard rate estimation is replaced with the default rate intensity of the parentcompany. The estimation procedure for the parent company’s default intensityis described at the end of this section.

The hazard rate estimation is also done separately for fixed-rate and floating-rate loans. Table 5 contains a summary of the loans contained in the estimation.As indicated, for non-CTLs there are 94,011 fixed-rate loans and 7,198 floating-rate loans. The number of defaults for the fixed-rate loans is 2,153 and 130 forthe floating-rate loans. The analogous number of prepayments are 8,989 fixed-rate and 2,960 floating, respectively. Prepayments are more numerous in oursample then are defaults. The number of defaulting floating-rate loans in ourdatabase is quite small which makes the estimates for floating-rate loans lessreliable. The number of CTL loans is smaller, totaling (1,368). A statisticalanalysis was not done on CTLs given the small sample size. The parameterestimates for the intensity processes are given in Tables 6A and 6B for non-CTLs.


Tab

le6

�In

tens

itypr

oces

spa

ram

eter

estim

ates

base

don

mon

thly

obse

rvat

ions

over

the

time

peri

odJu

ne19

98to

May

2005

.

Pane

lA

Fixe

dR

ates

Floa

ting

Rat

es

Cur

rent

Del

inqu

ent

Cur

rent

Del

inqu

ent

Var

iabl

eE

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.

Inte

rcep

t8.

3819

50.

1584

9−8

.381

950.

1584

9−0

.446

350.

4491

10.

4463

50.

4491

1h i

(t)

−0.0

0037

0.00

116

0.00

037

0.00

116

−0.0

1870

0.00

401

0.01

870

0.00

401

P1

1.57

494

0.03

986

−1.5

7494

0.03

986

2.70

221

0.17

415

−2.7

0221

0.17

415

P2

−0.9

4843

0.03

908

0.94

843

0.03

908

−2.4

8318

0.17

822

2.48

318

0.17

822

P3

0.94

830

0.02

458

−0.9

4830

0.02

458

0.48

016

0.09

555

−0.4

8016

0.09

555

P4

1.31

866

0.03

398

−1.3

1866

0.03

398

1.76

591

0.12

039

−1.7

6591

0.12

039

P5

−1.3

9621

0.02

957

1.39

621

0.02

957

−0.4

3966

0.17

860

0.43

966

0.17

860

R1

−3.7

0456

0.07

946

3.70

456

0.07

946

−1.1

3054

0.13

042

1.13

054

0.13

042

R2

−3.0

3486

0.08

014

3.03

486

0.08

014

0.44

784

0.13

584

−0.4

4784

0.13

584

R3

−3.3

0865

0.08

093

3.30

865

0.08

093

1.08

380

0.17

257

−1.0

8380

0.17

257

R4

−2.6

1916

0.08

049

2.61

916

0.08

049

1.41

365

0.15

064

−1.4

1365

0.15

064

R5

−2.2

8950

0.08

231

2.28

950

0.08

231

1.33

022

0.14

691

−1.3

3022

0.14

691

R6

−3.8

0976

0.07

855

3.80

976

0.07

855

−0.6

5106

0.12

932

0.65

106

0.12

932


Tab

le6

�co

ntin

ued

Pane

lA

Fixe

dR

ates

Floa

ting

Rat

es

Cur

rent

Del

inqu

ent

Cur

rent

Del

inqu

ent

Var

iabl

eE

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.

R7

−3.1

4409

0.07

914

3.14

409

0.07

914

−0.1

3140

0.15

191

0.13

140

0.15

191

R8

−3.7

6712

0.07

952

3.76

712

0.07

952

−0.5

0560

0.16

802

0.50

560

0.16

802

age

0.45

991

0.03

154

−0.4

5991

0.03

154

1.75

788

0.10

395

−1.7

5788

0.10

395

fore

inde

x−7

.715

300.

7559

97.

7153

00.

7559

9−1

4.45

546

3.65

702

14.4

5546

3.65

702

noi

0.00

000

0.00

000

0.00

000

0.00

000

0.19

173

0.02

133

−0.1

9173

0.02

133

Pen

0.00

036

0.00

001

−0.0

0036

0.00

001

0.00

119

0.00

014

−0.0

0119

0.00

014

orig

loan

bal

0.33

835

0.00

622

−0.3

3835

0.00

622

0.36

036

0.02

245

−0.3

6036

0.02

245

Hi(

t)−0

.004

550.

0002

30.

0045

50.

0002

3−0

.031

350.

0011

80.

0313

50.

0011

8ds

cr0.

0059

10.

0030

8−0

.005

910.

0030

80.

0299

30.

0057

2−0

.029

930.

0057

2lt

v−0

.019

760.

0004

40.

0197

60.

0004

4−0

.034

150.

0017

30.

0341

50.

0017

3sp

ot0.

6633

70.

0112

0−0

.663

370.

0112

00.

3598

10.

0382

3−0

.359

810.

0382

3H̄

(t)

−0.0

0994

0.00

051

0.00

994

0.00

051

0.03

837

0.00

296

−0.0

3837

0.00

296

f(t,

10)−

r(t)

0.27

295

0.01

442

−0.2

7295

0.01

442

0.00

856

0.04

985

−0.0

0856

0.04

985

wac

−0.3

8494

0.00

897

0.38

494

0.00

897

0.00

000

0.00

000

coup

on_s

prea

d−0

.459

250.

0107

40.

4592

50.

0107

40.

0000

00.

0000

0

470 Christopoulos, Jarrow and YildirimT

ab

le6

�co

ntin

ued

Pane

lB

Fixe

dR

ates

Floa

ting

Rat

es

Prep

aym

ent

Def

ault

Prep

aym

ent

Def

ault

Var

iabl

eE

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.

Inte

rcep

t−1

0.17

425

0.26

556

−10.

9694

50.

6217

9−5

.965

890.

3723

1−1

0.56

163

1.81

182

h i(t

)0.

0099

90.

0030

30.

0012

00.

0036

10.

0063

50.

0049

50.

0130

50.

0144

9P

1−0

.766

900.

0455

80.

6182

90.

1233

5−0

.298

730.

0984

90.

7746

50.

5400

3P

21.

0310

00.

0895

6−0

.318

150.

1302

9−0

.009

070.

1460

50.

5109

70.

6062

3P

30.

7602

50.

0359

9−0

.054

650.

0746

70.

4352

90.

0596

90.

9802

60.

3460

9P

4−0

.667

980.

0473

50.

9902

20.

1027

80.

1455

90.

0979

31.

4088

10.

4178

8P

52.

6107

60.

0709

4−1

.840

970.

1484

61.

1307

30.

1104

90.

7340

30.

5953

1R

1−0

.118

770.

0660

10.

3801

10.

2137

3−0

.906

550.

1177

00.

5540

40.

4685

9R

20.

0862

20.

0634

50.

5758

00.

2133

1−2

.540

440.

1443

9−0

.314

330.

5701

6R

30.

0162

90.

0635

70.

2748

30.

2186

9−1

.215

790.

0978

7−0

.278

540.

5515

8R

4−0

.305

670.

0585

90.

1062

20.

2127

9−2

.728

880.

0949

10.

4017

60.

4219

0R

5−0

.272

440.

0553

0−0

.292

270.

2189

7−1

.428

980.

0818

2−0

.794

310.

4645

4R

6−0

.049

610.

0614

60.

4125

60.

2110

8−1

.367

910.

0987

80.

1060

80.

4536

4R

7−0

.134

500.

0630

00.

7214

40.

2103

0−1

.378

450.

1033

2−0

.079

690.

4846

3R

80.

0161

60.

0840

20.

6911

60.

2232

5−1

.769

930.

2489

11.

1781

70.

5649

6ag

e12

.121

350.

1018

53.

2445

80.

1149

15.

6539

80.

0926

02.

5381

80.

3638

2dl

qsta

tus

−2.7

3125

0.23

559

5.38

979

0.05

646

−12.

3720

497

.270

804.

1213

00.

2119

8fo

rein

dex

1.57

502

1.73

865

−6.5

4448

2.40

202

−37.

0178

23.

4608

9−1

3.54

602

8.32

882

noi

0.00

000

0.00

000

0.00

000

0.00

000

−0.0

0103

0.00

140

−0.0

0965

0.01

657

Pen

−0.0

0099

0.00

005

0.00

109

0.00

006

−0.0

0047

0.00

021

0.00

068

0.00

036

orig

loan

bal

−0.3

0611

0.01

013

0.05

524

0.02

320

−0.2

5369

0.01

434

0.02

793

0.06

852

Hi(

t)0.

0102

30.

0004

9−0

.009

540.

0010

7−0

.001

190.

0010

0−0

.008

160.

0046

0


Ta

ble

6�

cont

inue

d

Pane

lB

Fixe

dR

ates

Floa

ting

Rat

es

Prep

aym

ent

Def

ault

Prep

aym

ent

Def

ault

Var

iabl

eE

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.E

stim

ate

Std.

Err

.

dscr

−0.0

1227

0.00

937

−0.4

2866

0.06

687

−0.0

4519

0.01

466

−0.6

4949

0.21

575

ltv

0.00

892

0.00

069

0.01

134

0.00

123

0.00

337

0.00

124

0.01

509

0.00

550

spot

0.16

979

0.02

056

0.21

948

0.04

018

0.19

869

0.03

339

−0.0

9297

0.15

982

H̄(t

)−0

.005

210.

0009

4−0

.009

240.

0019

40.

0163

40.

0018

20.

0160

10.

0082

4f(

t,10

)−

r(t)

0.17

895

0.02

689

0.32

873

0.05

438

0.12

641

0.04

374

−0.2

1663

0.21

081

wac

−0.2

6759

0.01

247

0.11

301

0.03

162

0.00

000

0.00

000

0.00

000

0.00

000

coup

on_s

prea

d0.

1135

00.

0086

6−0

.052

130.

0307

40.

0000

00.

0000

00.

0000

00.

0000

0

Not

es:

The

estim

ates

are

base

don

the

equa

tion:

inte

nsit

y=

1(1

+e∑ i

coef

ficie

nti·va

riab

lei).S

epar

ate

estim

ates

are

done

for

fixed

vers

usflo

atin

g-ra

telo

ans.

Pane

lA:T

hepa

ram

eter

estim

ates

for

aco

mpe

ting

risk

curr

entv

ersu

sde

linqu

entp

oint

proc

ess.

Pane

lB:T

hepa

ram

eter

estim

ates

for

aco

mpe

ting

risk

defa

ultv

ersu

spr

epay

men

tpoi

ntpr

oces

s.T

hefir

stco

lum

nco

ntai

nsth

eva

riab

les:

h i(t

)=

ithpr

oper

ty×

regi

onst

ock

pric

ein

dex

attim

et;

R1

=in

dust

rial

prop

erty

dum

my

vari

able

;R

2=

lodg

ing

prop

erty

dum

my

vari

able

;R

3=

mul

tifam

ilypr

oper

tydu

mm

yva

riab

le;

R4

=of

fice

prop

erty

dum

my

vari

able

;R

5=

othe

rpr

oper

tydu

mm

yva

riab

le;R

6=

reta

ilpr

oper

tydu

mm

yva

riab

le(o

mitt

ed);

R1

=E

ast–

Nor

th–C

entr

alre

gion

dum

my

vari

able

;R2

=M

idea

stre

gion

dum

my

vari

able

;R

3=

Mou

ntai

nre

gion

dum

my

vari

able

;R

4=

Nor

thea

stre

gion

dum

my

vari

able

;R

5=

Paci

ficre

gion

dum

my

vari

able

;R

6=

Sout

heas

tre

gion

dum

my

vari

able

;R7

=So

uthw

estr

egio

ndu

mm

yva

riab

le;R

8=

Wes

t–N

orth

–Cen

tral

dum

my

vari

able

;R9

=ot

her

regi

ondu

mm

yva

riab

le(o

mitt

ed);

age

=(1

−re

mai

ning

term

/ori

gina

lte

rm);

dlqS

tatu

s=

delin

quen

cyst

atus

;fo

rein

dex

=A

CL

Ifo

recl

osur

ein

dex;

noi=

net

oper

atin

gin

com

eat

orig

inat

ion

divi

ded

byth

eor

igin

allo

anba

lanc

e;pe

n=

pena

lties

divi

ded

byou

tsta

ndin

gba

lanc

eat

time

t;or

iglo

anba

l=

loga

rith

mof

the

orig

inal

loan

bala

nce;

Hi(

t)=

ithpr

oper

tyst

ock

pric

ein

dex

attim

et;

dscr

=de

btse

rvic

eco

vera

gera

tioat

orig

inat

ion;

ltv

=lo

anto

valu

era

tioat

orig

inat

ion;

r(t)

=sp

otra

teof

inte

rest

attim

et;

H̄(t

)=

Rei

tsto

ckpr

ice

inde

xat

time

t;f(

t,10

)−

r(t)

=10

year

forw

ard

rate

min

usth

esp

otra

teat

time

t;w

ac=

wei

ghte

dav

erag

eco

upon

ator

igin

atio

n;co

upon

_spr

ead

=co

upon

min

ustr

easu

ryra

tesp

read

atlo

anor

igin

atio

n;T

here

mai

ning

colu

mns

give

the

coef

ficie

nts

and

stan

dard

erro

rs.


Table 6A contains the current and delinquency intensity process coefficients forfixed- and floating-rate loans. Note that the coefficients are equal and opposite insign for current and delinquency. We concentrate on explaining the intensity ofgoing from current to delinquent. For fixed-rate loans: (i) as the age of the loanincreases, the likelihood of delinquency increases, (ii) as historical foreclosuresincrease (fore index), the likelihood of delinquency declines, (iii) the NOI (noi)is insignificantly different from zero,20 (iv) the higher the prepayment penalties(pen), the higher the likelihood of delinquency, (v) the larger the original loanbalance, the higher the likelihood of delinquency, (vi) the higher the debtservice coverage ratio (dscr), the higher the likelihood of delinquency, (vii) thehigher the loan-to-value ratio at securitization (ltv), the lower the likelihoodof delinquency,21 and (viii) the higher the weighted average coupon (wac) andcoupon spread, the lower the likelihood of delinquency. As the property indicesincrease (hi(t),Hi(t), H̄ (t)), the likelihood of delinquency declines. As eitherthe spot rate (spot) or the slope of the forward rate curve (f (t , 10) − r(t))increases, the likelihood of delinquency increases. All of these comparativestatics are as expected. For floating-rate loans, similar interpretations followfor many of the coefficients. The descriptions for floating-rate loans are omittedfor brevity and because of the smaller sample size.

Table 6B contains the default and prepayment processes. The signs of thesecoefficients are mostly as expected. For fixed-rate loans: (i) the larger the age ofthe loan, the more likely to prepay and default, (ii) if the loan is delinquent, thenthe probability of prepayment declines and the probability of default increases,(iii) as historical foreclosures increase (fore index), the likelihood of defaultdecreases and prepayment increases, (iv) net operating income (noi) appears tohave no impact on either the likelihood of default or prepayment, (v) the higherthe prepayment penalties (pen), the higher the likelihood of default and thelower the likelihood of prepayment, (vi) the larger the original loan balance,the less likely the loan is to prepay but the more likely it is to default, (vii) thehigher the debt service coverage ratio, the less likely to prepay or default, (viii)the higher the loan-to-value ratio at securitization (ltv), the higher the likelihoodof default and prepayment, (ix) the higher the weighted average coupon (wac),the higher the probability of default and lower the probability of prepaymentand (x) the higher the coupon spread at origination, the higher the probability ofprepayment and the lower the probability of default. Continuing with fixed-rateloans, as the spot rate (r t ) increases or the term structure becomes more steep

20 This is probably due to the endogeneity of the origination process. The terms of theloan contract are set to reflect the NOI of the given property, making its explanatorypower zero.21 Again, this is probably due to the origination process. Those loans that have highinitial loan-to-value ratios are probably viewed as having less default risk at origination.Otherwise, the originators would have reduced the loan-to-value ratio of the borrowingentity.


Table 7 � Receiver operating characteristic (ROC) accuracy ratios using monthlyobservations over the time period June 1998 to May 2005.

Intensity Processes ROC Accuracy Ratio

Fixed ratesPrepayment 0.952Default 0.830Current 0.886Delinquent 0.886

Floating ratesPrepayment 0.879Default 0.720Current 0.882Delinquent 0.882

Note: The ROC ratio measures forecasting ability, with a score of 0.5 indicating randomselection. Separate ROC ratios are provided for forecasting prepayment, default,current or delinquency for both fixed- and floating-rate loans.

(f (t , t + 10) − r t ), default and prepayment are more likely. Lastly, as theproperty×region index (H i(t)) increases, prepayment increases but default isunchanged. As the property index (hi(t)) increases, the likelihood of defaultdeclines, and the likelihood of prepayment increases. Finally, as the REIT indexincreases (H̄ (t)) both default and repayment decline. For floating-rate loans,similar interpretations follow for many of the coefficients. However, because ofthe smaller sample size (see Table 5), fewer of the coefficients are significantlydifferent from zero.

A standard method for measuring out-of-sample performance is the area un-der the receiver operating characteristic (ROC) curve. For comparison acrossmodels, a value of 0.5 for the ROC measure indicates a random model withno predictive ability, while a value of 1.0 indicates perfect forecasting. Table 7contains the ROC accuracy ratios for the different intensity processes estimated.As indicated, the ROC ratios, ranging from 72% to 95%, are quite high for allmodels. These numbers are comparable to those obtained in the estimation ofcorporate bankruptcies (see Chava and Jarrow 2004).

The Loss Severity Process

The final process to estimate is the loss severity rate for loan i.22 We fit a linearregression model to our loss severity database consisting of 919 loans.23

22 The Trepp software also requires a time to recovery; we set this as 12 months. Thisimplies that the actual loss severity used in the valuation was (1 − δ) × p(τ d , τ d + 12months).23 We do not constrain our regression equation to lie between zero and one.


1 − δit = α0 + α1U

it + α3Xt

where α1 is a constant and α2, α3 are vectors of constants. The regressionestimates are contained in Table 8. We see that, as the age of the loan orthe original loan balance or the loan-to-value ratio increases, the loss severity

Table 8 � The parameter estimates for the loss severity regression using monthlyobservations over the time period June 1998–May 2005.

Estimate Std. Err. R2

Intercept −0.08135 0.21166 0.1680hi(t) 0.00239 0.00155P1 0.08041 0.05162P2 0.11096 0.06412P3 0.02349 0.03664P4 0.04434 0.03809P5 0.08982 0.06518R1 0.03269 0.12901R2 −0.10908 0.12899R3 −0.02175 0.1303R4 −0.05972 0.1297R5 −0.0777 0.1314R6 0.02644 0.12782R7 −0.05022 0.12789R8 0.20105 0.13071age 0.05325 0.05995origloanbal 0.02438 0.00918Hi(t) −0.00004 0.00047ltv 0.00090 0.00032spot −0.07397 0.01867H̄ (t) 0.00098 0.00085f (t, 10) − r(t) −0.01359 0.02500

Notes: The regression equation is loss = ∑i coefficienti · variablei.

The variables in the first column are hi(t) = ith property x region stock price indexat time t; R1 = industrial property dummy variable; R2 = lodging property dummyvariable; R3 = multifamily property dummy variable; R4 = office property dummyvariable; R5 = other property dummy variable; R6 = retail property dummy variable(omitted); R1 = East north central region dummy variable; R2 = Mideast region dummyvariable; R3 = Mountain region dummy variable; R4 = Northeast region dummyvariable; R5 = Pacific region dummy variable; R6 = Southeast region dummy variable;R7 = Southwest region dummy variable; R8 = West North Central dummy variable;R9 = other region dummy variable (omitted); age = (1 − remaining term/originalterm); origloanbal = logarithm of the original loan balance; Hi(t) = ith property stockprice index at time t; ltv = loan-to-value ratio at origination; spot = spot rate of interestat time t; H̄ (t) = Reit stock price index at time t; f (t, 10) − r(t) = 10 year forward rateminus the spot rate at time t; The remaining columns give the coefficients and theirstandard errors. The R2 is also provided.


increases. Second, as the spot rate or slope of the forward rate curve increases,the loss severity declines. Many of the coefficients are insignificantly differentfrom zero, however, including all the property value indices.

Known CTLs Default Intensity Estimation

To estimate the default intensity process for known CTLs, we use credit defaultswap data or credit rating data. First, given the known CTL’s name, we obtaindefault swap quotes from Bloomberg for tenors T (this could be 1, 2 or 3 years,depending upon the name of the CTL). We use the longest tenor available inorder to better match the maturity of the CMBS. The default swap quote is (seeLando 1998, p. 207)

cds(T ) = (1 − δ)λd

where δ is the recovery rate on the firm’s debt and λd is the default rate on thefirm’s debt, assuming both a constant recovery rate and default intensity. Toobtain the recovery rates for specific industries for senior secured debt, we useMoody’s (2005, exhibit 23, p. 21). Then,

λd = cds(T )

1 − δ.

If there is no credit default swap data, we instead use the firm’s Moody’s creditrating, and from Moody’s (2005) we obtain the default intensity for that creditrating.

The Simulation

Given the parameter estimates from the previous section, the next step is tocompute the values of the CML and the CMBS bonds using Expressions (3)and (4). The state contingent cash flow allocations to the various CMBS bondtranches are quite complicated, and they differ, depending upon the CMBStrust. Furthermore, at any given date, the future cash flows will also depend onthe history of the previous payments for each of the existing CMBS trusts. Tocircumvent these obstacles, we obtained permission to use the Trepp scenariogenerating professional software service through WOTN and Trepp. The Treppservice tracks all CMBS trusts (archiving the loan pool history) and it providessoftware for modifying the loan pool payments for events of default, loss andprepayment in order to determine its influence on the CMBS bond cash flows.

In particular, for each existing CMBS trust, the Trepp engine enables the user toinput a scenario for each of the underlying loans. The outputs are the cash flowsto the various bond tranches (including changes in yields due to the scenariomodification). As such, this provides the intermediate step in our simulationalgorithm. Using the above modeling structure and parameter estimates, we


are able to generate, using variance reduction techniques, the equivalent of10,000 different scenarios24 to input into the Trepp software. A descriptionof the simulation algorithm and variance reduction techniques are providedin the Appendix. The resulting cash flows to the loans and bonds enable usto compute Expressions (3) and (4) as discounted averages over the set ofsimulated scenarios.

A Joint Test of Market Efficiency

This section jointly tests our model and CMBS market efficiency using his-torical data and market prices for the CMBS bonds from July 2001 to April2005.25 For each month over this observation period, starting with July 2001,we estimate the model’s parameters using only data available to the market atthat time. With these estimates, we compute the CMBS bond prices (denotedbi(t) from Expression (4)) as explained above and compare them to the marketprices at that date (denoted mi(t)).

The market prices were obtained from Trepp’s CMBS Pricing ServiceTM. Theseprices are the average of the bid and offer quotes obtained by polling multi-ple dealers and institutional investors. These prices are updated daily and theyare widely used in the CMBS industry for marking-to-market positions. Theseprices are not matrix prices. Because CMBS trade over-the-counter, Trepp’sCMBS Pricing Service provides the best historical pricing database for analysis.It is important to note that non-IO, investment-grade CMBS trade in relativelyliquid markets, and the Trepp database is comparable to other fixed-incomedatabases frequently used in empirical investigations of credit risk models (see,e.g., Duffee 1999). To the extent that these quotes still might not representprices at which actual trades could have be executed, we modified our testingprocedure to make this possibility less likely. First, to further avoid stale quotes,we restricted our sample to include only investment-grade bonds that are moreactively traded in the market and, therefore, more frequently updated by Trepp.In this regard, we omitted all bonds rated below baa3 and the IO bonds.26 Fi-nally, to further reduce the impact of using market quotes, instead of transaction

24 10,000 scenarios was selected because the standard error of a simulation convergesto zero at the rate 1/

√N . For N = 10, 000, the error will be approximately 1% of the

resulting values. The number of scenarios actually run using control variates was 2,500.25 We begin with July 2001, and not earlier, because that is the first date for which wehave a large enough sample of bonds priced by Trepp’s pricing service.26 Although the IO bonds are rated investment grade, we also exclude them fromsubsequent consideration because: (a) they tend to be traded less frequently than areinvestment-grade, principal-bearing bonds, and (b) because they are typically strippedacross the entire capital structure and thus share default exposure attributes similar tobelow-investment-grade securities.


prices, we used only monthly trading intervals. For monthly trading intervals,the differences between transaction prices and market quotes is expected to beminimal.

The comparison between model and market prices is repeated for every monthin the back-testing period. For this back testing, we used more than 3,000 bondsfrom 336 distinct trusts (the bond index is i, and the monthly time index is t).In addition to computing theoretical CMBS bond prices, we also computed theCMBS bonds’ weighted average lives (WAL) and the option adjusted weightedaverage lives (OAWAL). The formulas are at time 0:

WALi = 1

principal

(T∑

t=1

t · promised principal paymentst

),

OAWALi = 1

principal

∑ω

(T∑

t=1

t · principal paymentst (ω)

)1

|ω| .

The percentage difference [WAL − OAWAL]/WAL is a measure of the op-tionality present in a CMBS; the larger the difference, the larger the embeddedoptionality.27 These computations generate the data for the subsequent analyses.

Goodness of Fit

For evaluating the model performance, we compute a time series of the pricingdifferences

θi(t) ≡ bi(t) − mi(t) for all i and t. (23)

If we had used spreads instead of prices, this would be equivalent to com-puting the bonds’ option adjusted spreads (OAS). These price differences aresummarized in Tables 9A and 9B. Table 9A groups the average θ i(t) acrosstime based on ratings, while Table 9B groups the average θ i(t) across timebased on the weighted average life (WAL). Because the par value of the bondis 100, these errors can be interpreted as percentages. For example, for a1in Table 9A, the percentage pricing error is 7.1538. As indicated, for eithergrouping of the bonds, there is a statistically significant bias in the model’sprice. The model appears to overvalue the bonds. Tables 9A and 9B reject thehypothesis that the model prices equal the market prices. This rejection couldbe due to either model error or market mispricing. This is the content of the nextsection.

27 An alternative, and equally usable, measure would be the percentage differencebetween the CMBS model price with and without the embedded options.


Table 9 � Summary statistics for the CMBS pricing errors.

Panel A: Pricing Errors Grouped by Rating Categories

Rating N avg(θ ) stdev(θ )

a1 46 7.1538∗ 2.1888a2 46 6.1022∗ 2.0055a3 46 6.5469 3.9829aa1 46 4.9759∗ 1.0774aa2 46 5.5188∗ 1.6151aa3 46 6.2538∗ 2.7423aaa 46 2.6231∗ 0.7151baa1 46 7.3284∗ 5.0916baa2 46 8.6955∗ 3.5805baa3 46 10.9503∗ 4.8909

Panel B: Pricing Errors Grouped by Weighted Average Life (WAL)

WAL N avg(θ ) stdev(θ )

2.5< 46 1.3518∗ 0.39202.5–5.5 46 3.8053∗ 0.35405.5–7.5 46 6.8029∗ 1.26497.5–11 46 8.1070∗ 3.7777>11 45 17.4494 9.1612

Notes: ∗Denotes significance at the 95% confidence level.The pricing errors are given by θ = [model price − market price], for all CMBS bondsgrouped by rating categories (Panel A) and weighted average life (Panel B) usingmonthly observations over the time period July 2001–April 2005.

Trading Strategy Profits

This section generates trading strategies based on θ i(t) to investigate if theCMBS market is inefficient. The analogy of the pricing difference θ i(t) to anOAS facilitates the intuition underlying this strategy. The ideal trading strategyfor this determination is to construct an arbitrage opportunity that involves go-ing long on undervalued bonds (highest OAS) and shorting overvalued bonds(lowest OAS). Unfortunately, there is no well-organized market for shortingcash CMBS.28 Consequently, we investigate the relative performance of under-valued and overvalued portfolios, pairwise matched to control for risk. In thismatching, credit risk is measured by the bond’s credit rating and interest raterisk by the bond’s WAL.

28 Over our sample period, total return swaps on CMBS indices were traded sporadically,while credit default swaps on CMBS were not actively traded. Today, both CMBS creditdefault swaps and total rate of return swaps on several market indices are actively traded.These new securities effectively enable one to short many risks embedded in CMBS.This is a vibrant subject for future academic inquiry.


It is difficult to directly control for property value risk in this comparison.However, because the CMBS trusts usually are diversified across property typesand geographic locations, property value risk is less of a concern. Nonetheless,we also formed indices, stratified by credit ratings and WALs, of all the CMBSbonds in our database in an attempt to control for property value risk premia.The indices are equally weighted portfolios of all the bonds in our CMBSuniverse falling into the appropriate classification. In total we have six CMBSindices corresponding to the ratings/WAL classifications selected. Table 10contains a listing of the total number of CMBS in each risk category, for each

Table 10 � The numbers of bonds in the CMBS indices, per month, over the timeperiod July 2001 to April 2005.

Short aaa Long aaa aa a bbb(2–6 WAL) (>6 WAL) (>2 WAL) (>2 WAL) (>2 WAL)

Jul-01 126 115 124 126 121Aug-01 129 117 127 129 124Sep-01 138 123 136 138 129Oct-01 133 116 133 135 127Nov-01 136 119 136 138 128Dec-01 138 120 137 139 129Jan-02 148 126 147 149 135Feb-02 147 124 146 148 135Mar-02 147 123 146 148 139Apr-02 148 119 147 149 141May-02 148 114 148 150 141Jun-02 143 105 142 144 138Jul-02 148 111 148 150 142Aug-02 148 107 148 150 144Sep-02 150 100 150 152 145Oct-02 151 97 150 152 145Nov-02 150 92 150 152 145Dec-02 151 91 150 152 145Jan-03 149 86 148 150 143Feb-03 153 89 152 154 147Mar-03 149 84 149 151 144Apr-03 149 83 149 152 145May-03 147 84 148 150 145Jun-03 148 80 148 150 147Jul-03 146 75 146 149 144Aug-03 146 70 145 147 142Sep-03 148 69 147 149 144Oct-03 149 67 147 150 144Nov-03 148 61 146 149 144Dec-03 150 62 148 151 145Jan-04 151 60 149 152 146Feb-04 153 62 151 153 148


Table 10 � continued

Short aaa Long aaa aa a bbb(2–6 WAL) (>6 WAL) (>2 WAL) (>2 WAL) (>2 WAL)

Mar-04 150 58 149 152 146Apr-04 154 54 152 155 149May-04 152 54 151 154 148Jun-04 152 50 151 153 148Jul-04 152 50 151 153 148Aug-04 152 48 152 154 149Sep-04 250 129 254 256 253Oct-04 278 142 285 284 280Nov-04 274 136 281 281 277Dec-04 271 135 282 280 276Jan-05 269 129 280 277 276Feb-05 265 126 279 276 275Mar-05 265 121 276 276 274

Average 166 96 167 168 162

Total 7,449 4,283 7,481 7,559 7,280

GRAND Total 39,590

Notes: The bonds are partitioned into the five buckets {aaa, 2 < WAL < 6}, {aaa,WAL > 6}, {(aa1, aa2, aa3), WAL > 2}, {a1, a2, a3), WAL > 2}, {(baa1, baa2, baa3),WAL > 2}, where the first argument is the credit rating and the second argument is theweighted average life.

month, over our testing horizon. As indicated, the average number of bonds ineach risk category exceed 120, with the exception of the long-tenor aaa bonds,which averaged only 96. Consequently, there are sufficient bonds in each riskcategory to justify the belief that these indices reflect returns to well-diversifiedportfolios of CMBS, reflecting the market’s compensation for property valuerisks.

Procedure. This section describes the procedure utilized to construct the over-and undervalued CMBS bond portfolios. At time 0 (July 2001), we first groupthe bonds into roughly equal-risk categories based on credit rating (for creditrisk) and WAL (for interest rate risk). IOs and bonds with credit classes ba1,ba2, ba3 and Unrated are excluded because Trepp does not provide reliablemarket quotes for these infrequently traded bonds. The groupings selectedwere:

1. {aaa, 2 ≤ WAL < 6},

2. {aaa, WAL > 6},

3. {(aa1, aa2, aa3), WAL > 2},


4. {(a1, a2, a3), WAL > 2},

5. {(baa1, baa2, baa3), WAL > 2}.

For each of these equal-risk groupings, we sort θ i(0) across i from highest tolowest. Those with the highest θ i(0) represent undervalued bonds and thosewith the lowest θ i(0) represent overvalued bonds. We invest $1 million in eachbond in the top decile and $1 million in each bond in the bottom decile. Althoughthis implies that the upper- and lower-decile portfolios may have different dollarinvestments, this will not affect the subsequent return calculations.

These 10 portfolios are held until time t = 1 (1 month later). The returns on the10 portfolios (marking-to-market) were computed taking into account all cashflows obtained over the first time period.

At time 1, we liquidate both portfolios, and we repeat the same procedure thatwe employed at time 0. We continue in this fashion until the end of the sampleperiod time T (April 2005).

It is important to emphasize that in this computation we do not include explicittransaction costs. However, because we are comparing two matched portfolios,both exhibiting similar rebalancing across time, the transaction costs would beroughly equivalent for both portfolios. This implies that, as a first approxima-tion, the relative performance differential between the two portfolios should beunaffected by the exclusion of explicit transaction costs.

Results. Tables 11A and 11B contain the average monthly and the cumu-lative returns, respectively, on the overvalued, undervalued and CMBS indexportfolios over the time period July 2001-April 2005.

The average monthly returns are contained in Table 11A. As indicated, the re-turns on the undervalued portfolios exceed those for the overvalued portfoliosfor all five risk groupings. For example, for the short-tenor aaa the differenceis 0.29% per month. A control for property value risk is obtained by a pair-wise comparison of the undervalued and overvalued portfolios with the CMBSindex portfolios. As seen therein, the undervalued portfolio returns exceedthose for the equal-risk CMBS indices, and the equal-risk CMBS indices ex-ceed the returns on the overvalued portfolios for all five risk categories. Onlythe short aaa undervalued less overvalued difference is significant at the 95%confidence level as indicated by the p values given in column 6. However,the likelihood that all five risk grouping undervalued portfolios would outper-form the overvalued portfolios over this time period, under the null hypothesisthat the model cannot identify mispricings, is (1/2)5 = 0.0156. This is un-likely. Note also that all the differences to the index are significant at the 95%


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confidence level. Combined, this evidence is consistent with a CMBS marketinefficiency.

These same observations are confirmed by the cumulative returns in Table 11B.The cumulative returns on the undervalued portfolios outperform the overval-ued portfolios for all five risk groupings. For example, the undervalued shortAAA bonds outperformed the overvalued short AAA bonds by 15.65% overthis time period, while the long aaa outperformed the overvalued by 7.85%.Again, the likelihood that all five risk grouping undervalued portfolios wouldoutperform the overvalued portfolios over this time period, under the nullhypothesis that the model cannot identify mispricings, is 0.0156. Control-ling for property-value risk, the undervalued portfolios also outperform theCMBS index returns for every risk grouping considered. Yet, the possibilitystill exists that these abnormal returns are due to unaccounted for risk premia,as compensation for omitted risk factors. The next subsection explores thispossibility.

Omitted Risk Premia

To test for the possibility of omitted risk premia, we utilize a standard intertem-poral CAPM (see Merton 1990, p. 511), where the expected return on the over-and undervalued portfolios can be written using a multi-beta model as

ERpt = rt +

M∑i=1

βpi

(ERi

t − rt

)

where Rpt is the p portfolio’s return over [t , t + 1], Ri

t is the return over [t , t +1] on a portfolio perfectly correlated to the ith systematic risk component, andβpi is the beta of portfolio p to the ith risk component portfolio. There are Mpossible risk factors. Using the relation

Rjt = ERj

t + εjt

where the εjt have zero means and are independent across t and j, we can rewrite

the multi-beta model as

Rpt = rt +

M∑i=1

βpi

(Ri

t − rt

) + εt

where the εt = ∑Mi=1 βpiε

it have zero means and are independent across t.

To construct our regression model, it is reasonable to assume that one of thesystematic risk factors is a CMBS portfolio of equal credit and WAL risk asthe over- or undervalued portfolio under consideration. Letting the return on


the index CMBS portfolio be denoted by i = 1, we can write this last expressionas:

Rpt = rt + βp1

(R1

t − rt

) +M∑i=2

βpi

(Ri

t − rt

) + εt .

And, it is reasonable to also assume that the beta of the over- or undervaluedportfolio with respect to the index is unity, that is,

βp1 = 1,

yielding our final regression model to test for omitted risk premia:

Rpt − R1

t = α +M∑i=2

βpi

(Ri

t − rt

) + εt . (24)

In this specification, we include a constant α to capture the abnormal returns.

To estimate this model, we use the following assets (portfolios) to capturevarious risk premia: (i) the REIT stock price index to capture property valuerisk premium, (ii) the 1-year, 2-year, 5-year and 7-year zero-coupon bond pricesto capture interest rate risk premium and (iii) a stock market index, the SMBindex (small minus big) and the HML index (high minus low) to capture equitymarket risk premium.29 Tables 12A and 12B contain the regression results,using monthly returns, for the various risk categories. These two regressiontables differ only by the inclusion or not of both SMB and HML.

Because the two regressions are similar, we concentrate on the results presentedin Panel B that includes both the SMB and HML indices. First, we see that allof the abnormal returns for both the overvalued and undervalued portfolios (asmeasured by the α) are insignificantly different from zero, except for the shortaaa grouping, and this difference is negative (it should be positive). Controllingfor omitted risk premia appears to remove all of the abnormal trading profits.In this risk adjustment, only the interest rate factors appear to be significant.All of the property value and market indices have coefficients insignificantlydifferent from zero (except one). These regressions support the hypothesisthat all of the abnormal trading profits associated with the overvalued andundervalued portfolios can be attributed to omitted interest rate risk premia.Consequently, we cannot unambiguously reject the hypothesis that the CMBSmarket is efficiently priced.

29 The market portfolio consists of all NYSE, AMEX and NASDAQ firms, and theSML and HML indices are obtained from Ken French’s Web site: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french.


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A Corrected Pricing Model

If the CMBS market is efficient, then it is essential to modify the originalmodel to remove the model errors. Let us recall the specifics of the modelstructure to identify the possible misspecifications. First, all of the CMBSrisks, including the embedded options (default and prepayment), depend on thespecification of the interest rate and property value processes. If these processesare misspecified, then the model prices will be in error. Second, although themodel structure explicitly incorporates interest rate, credit and prepayment risk,due to the frictionless and competitive market assumption, it excludes liquidityrisk. Consequently, the θ i(t) differences could be due to this omission. Asshown in Cetin, Jarrow and Protter (2004), liquidity risk can generate such abias between model prices, based on the classical model with no liquidity riskand market prices.

In addition, we have one remaining potential source of model error relatedto the default intensity. In valuing the CMBS bond’s cash flows, we used theintensity process estimated under the statistical measure (and not the intensityprocess under the martingale measure). If default risk is not diversifiable (seeJarrow, Lando and Yu 1995), then this would introduce a positive pricing errorin θ i(t).

Given the complexity of the model construction and the inherent limitationsof the CMBS database, we employ the procedure discussed in Jacquier andJarrow (2000) to generate an extended pricing and hedging model to overcomeany model error related to stochastic process misspecification, liquidity or non-diversifiable risk. After this extension, we will perform an additional test tosee if the interest rate or property value processes could be the cause of the(residual) model error.

The extended model, denoted b̂i(t), is obtained via the expression

b̂i(t) ≡ bi(t) + θ̄rating(t) (25)

where θ̄rating(t) ≡

( ∑j∈ rating class

θj (t))

N (t)

and N(t) equals the number of bonds at time t in the same rating class to whichbond i belongs.

The intuition underlying this extended model is that any model error due toliquidity risk and/or nondiversifiable default risk is captured by the averagemispricing with respect to the bond’s rating. It is conjectured that the bond’s


rating will be correlated with all of these model misspecifications. Indeed,market liquidity is related to a bond’s rating because institutions often restricttheir investments to investment-grade bonds. Furthermore, empirical studieshave documented a correlation between a bond’s rating and its default risk(see Moody’s 2005), and default risk has been correlated to recovery rates (seeAltman et al. 2003, Acharya, Bharath, and Srinivasan 2004).

Using this extended model, the relevant pricing differences for model perfor-mance are now given by

εi(t) = b̂i(t) − mi(t) for all i and t. (26)

By construction, the average pricing error εi(t) for a rating class is zero, thatis, the new model prices the rating classes correctly.

Table 13 contains summary statistics for εi(t) grouped by WAL. As indicated,the percentage pricing error differs from that in Table 5B (sign and magnitude),and it is insignificantly different from zero for all WAL groupings except WALless than 2.5 years.

Given the extended models error are now unbiased (by construction), we testfor model misspecification with respect to the interest rate and property valueprocesses. If the adjusted CMBS model is not properly specified, one wouldexpect that the pricing errors would be correlated with interest rates or propertyvalues. To test for this sort of misspecification, we ran the following time-seriesregression on the pricing errors for each bond i:

εi(t) = c0 + c1rt + c2[f (t, 10) − rt ] + c3H̄ (t) (27)

where r t is the time t spot rate, (f (t , 10) − r t ) measures the slope of the forwardrate curve at time t, and H̄ (t) is the REIT stock price index. The spot interest

Table 13 � Summary statistics for the adjusted model’s CMBS pricing errors over thetime period July 2001–April 2005.

WAL N avg(ε) stdev(ε)

2.5< 46 −1.7813∗ 0.52252.5–5.5 46 −0.6333 0.60745.5–7.5 46 0.4132 0.83167.5–11 46 0.8974 1.1943>11 45 9.1800 6.8717

Notes: ∗denotes significance at the 95% confidence level.The pricing errors ε = [adjusted model price − market price]. The pricing errors arepartitioned by weighted average life (WAL). Given is the sample size, average pricingerror and the standard deviation of the pricing error.


Table 14 � Regression equation coefficients for CMBS pricing error biases.

Rating N Intercept spot f(t,10)-spot H̄ (t) R2

a1 35 1.8284 −0.1240 0.2202 −0.0085 0.51687.1987 1.9454 1.9154 0.0350

a2 303 0.4114 −0.0860 −0.0109 −0.0014 0.58534.3097 1.0781 1.0778 0.0198

a3 201 0.3626 −0.1145 0.0793 −0.0016 0.52616.2513 1.6220 1.6010 0.0294

aa1 25 0.3038 −0.6570 −0.2005 0.0067 0.63714.1590 1.1014 1.0970 0.0200

aa2 300 −0.3006 0.0576 0.0705 0.0002 0.59503.3457 0.8231 0.8228 0.0152

aa3 86 1.6475 −0.1532 −0.1048 −0.0062 0.50724.7273 1.3009 1.2727 0.0232

aaa 316 −0.8484 0.2738 0.1362 −0.0004 0.10987.7001 2.0746 1.9933 0.0356

baa1 121 1.3836 −0.2842 −0.0966 −0.0031 0.51447.8066 2.1117 2.0747 0.0380

baa2 296 0.6305 −0.0460 0.2513 −0.0072 0.60767.7014 1.9830 1.9713 0.0363

baa3 281 0.1002 0.0652 0.1641 −0.0030 0.56419.8400 2.6172 2.5770 0.0470

Notes: The regression equation estimated is errort = c0 + c1spot + c2(f (t, 10) −spot) + c3H̄ (t), where f (t, 10) is the 10-year forward rate, spot is the three-monthTreasury bill yield, and H̄ (t) is the economy-wide REIT stock price index. In all cases,the coefficients are not significant at 95% level. Standard errors are given below theparameter estimates.

rate and the slope of the forward rate curve reflect interest rate risk while theREIT stock price index reflect property value risk.

Table 14 contains the average coefficients, the standard errors of the estimatesand the average R2 from these regressions, grouped by rating categories. Allcoefficients have insignificant t scores (less than two). It appears that the re-maining pricing errors are not correlated with interest rates or property values.This evidence indicates that our simple adjustment for model misspecificationas given in Expression (25) yields an unbiased pricing model for CMBS usefulfor hedging or marking-to-market if the CMBS market is efficient.

A Behavioral Finance Explanation

In contrast to omitted risk factors, there is a behavioral finance explanationfor the abnormal trading strategy returns, that is inconsistent with market effi-ciency. Due to the complexity of the CMBS trust structure and the interaction of


the various embedded options, the optionality cannot be accurately measuredwithout a model. As documented next, the undervalued portfolios consist ofhigher coupon bonds with low optionality relative to the overvalued portfo-lios in the same equal-risk category. This observation is consistent with themarket misestimating the risk of the embedded options in a manner corre-lated with interest rates. Note that, over our sample period, the market experi-enced a period of declining interest rates and increasing commercial propertyvalues.

To document these facts, let us consider the short-tenor aaa bonds. The returnsfor the short aaa bonds (as for all bonds) are due to four sources: (a) inter-est/coupon payments, (b) gains/losses on the sales of securities after the 1 monthholding period, (c) prepayment penalties allocated to the securities if prepaidand (d) gains/losses on any principal paid down (expected or unexpected due toamortization, prepayment or default recoveries). Table 15 provides this break-down in dollar profits for the trading strategies. For convenience, we combinesources (b) and (c) into one category. As indicated, the undervalued portfoliosprovide larger returns on all four sources. The largest difference (in percentageterms) between the undervalued and overvalued deciles is attributable to prin-cipal paydowns. Further supporting this statement, as documented in Table 15,we see that the undervalued portfolios experienced less principal paydowns (32versus 459). Second, the undervalued portfolio exhibits less embedded optionrisk as measured by the percentage difference (WAL-OAWAL)/WAL, (1.90%versus 19.02%). And, the undervalued portfolios had higher average coupons(7.01 versus 6.23) and consequently, higher average prices as well. Based onthe differences in coupons between these two portfolios, one would expect the

Table 15 � Monthly cash flows of the short aaa overvalued and undervalued portfoliosover the time period July 2001–May 2005.

Short aaa Undervalued Overvalued

(a) Cumulative interest $5,730,000.00 $5,000,000.00(b)+(c) Cumulative gains/losses + $134,378.00 ($1,171,582.00)

prepayment penalties(d) Cumulative pay downs at par ($11,000.00) ($645,100.00)

Cumulative total dollar profits $5,853,378.00 $3,183,318.00

Number of pay downs at par 32 459Average dollar price 110.4864 106.2513Average coupon 7.01 6.23Average WAL 5.34 2.60Average OAWAL 5.24 2.11(WAL − OAWAL)/WAL 1.90% 19.02%


undervalued bonds to have more embedded optionality, but this is not the case.A model for the embedded options is necessary to obtain this understanding.

These same conclusions also apply to the remaining bond portfolios as docu-mented in Table 16, where the undervalued portfolios are seen to have higher

Table 16 � Summary statistics of the overvalued and undervalued bond portfoliosusing monthly observations over the time period July 2001–May 2005.

Short aaa Profile (2–6 WAL) Undervalued Overvalued

Average dollar price 110.4864 106.2513Average coupon 7.01 6.23Average WAL 5.34 2.60Average OAWAL 5.24 2.11Diff. (WAL − OAWAL)/WAL 1.90% 19.02%

Long aaa Profile (>6 WAL) Undervalued Overvalued


All aa Profile (>2 WAL) Undervalued Overvalued


All a Profile (>2 WAL) Undervalued Overvalued


All bbb Profile (>2 WAL) Undervalued Overvalued

Average dollar price 101.6154 107.8686Average coupon 7.49 7.19Average WAL 8.57 6.08Average OAWAL 8.64 5.28Diff. (WAL − OAWAL)/WAL −0.79% 13.25%

Notes: WAL is the weighted average life. OAWAL is the option-adjusted weightedaverage life that is adjusted through simulation for the embedded default andprepayment options in the CMBS bonds.


average coupons (except perhaps the aa) and uniformly lower embedded op-tionality. The shorter average tenor of the overvalued portfolios relative to theundervalued portfolios seems to indicate that the risks of the CMBS withinequal-risk categories are not being accurately priced. And, these mispricingswould be correlated with the interest rate risk factors. Only subsequent researchcan distinguish between these two competing hypotheses for the abnormal trad-ing strategy returns documented previously in Table 11.

Conclusion

This article provides a joint test of our CMBS pricing model and marketinefficiency. The CMBS pricing model includes interest rate, credit, prepaymentand liquidity risk. The model’s parameters are estimated using a comprehensiveCMBS database, and the model’s prices are compared to market prices over a5-year period. The joint test is rejected. To investigate if the joint test’s rejectionis due to market inefficiency, the model’s mispricings are back tested for tradingprofits over a 46-month period. The back testing results are consistent withthe existence of abnormal trading profits. However, controlling for omittedrisk premia appears to remove these abnormal trading profits. These omittedrisk premia are also consistent with a behavioral finance explanation in aninefficient market. At present, we cannot unambiguously reject the efficiencyof CMBS markets. In light of this ambiguity, a simple adjustment to our modelis also provided to generate an unbiased pricing model useful for hedging andmarking-to-market, for those who believe that the CMBS market is efficient.

Thanks are expressed to WOTN, LLC for providing financial support, to the Intel Cor-porate Early Access Program for allowing use of their Xeon computational cluster, toTrepp, LLC for the use of their software and database and to Joshua Barratt for helpfulcomments and suggestions throughout the execution of this project.

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Appendix

Derivation of Expression (3)

Given the information set up to {NT , XT }, Nd (t) and Np(t) are independentPoisson processes with

Qt (τd > T | NT ,XT ) = e− ∫ T

tλd (s) ds .

The density is

dQt (τd = s | NT ,XT ) = λd (s)e− ∫ s

tλd (u) du ds

Similar expressions hold for Np(t). From Expression (2), we have terms similarto

Et

(Et

{1{T <τd }1{T <τp}e− ∫ T

trs ds

∣∣NT ,XT

})= Et

(e− ∫ T

trs dsEt {1{T <τd }

∣∣NT ,XT }Et {1{T <τp} | NT ,XT })= Et

(e− ∫ T

trs dse− ∫ T

tλd (s) dse− ∫ T

tλp(s) ds

).


We also have terms like

Et

(Et

{δτd

(Bτd+ P )1{τd≤T }1{τd≤τp}e− ∫ τd

trs ds

∣∣NT ,XT

})= Et

(Et

{δτd

(Bτd+ P )1{τd≤T }[1{τd≤τp<T } + 1{T ≤τp}]e− ∫ τd

trs ds

∣∣ NT ,XT

})= Et

(Et

{δτd

(Bτd+ P )1{τd≤τp<T }e− ∫ τd

trs ds

∣∣NT ,XT

})+Et

(Et

{δτd

(Bτd+ P )1{τd≤T }e− ∫ τd

trs ds

∣∣NT ,XT }Et {1{T ≤τp} | NT ,XT

})The first term can be written

Et

(Et

{∫ T

t

∫ k

t

δs(Bs + P )1{τd=s,τp=k}e− ∫ s

tru du ds · dk | NT ,XT

})

= Et

(∫ T

t

∫ k

t


tλd (u) due− ∫ s

tru du ds · λp(k)e− ∫ k

tλp(u) du dk

).

The second term can be written

Et

(Et

{∫ T

t

1{τd=s}e− ∫ s

tru du ds | NT ,XT }Et {1{T ≤τp} | NT ,XT

})

= Et

(∫ T

t

λd (s)e− ∫ s

tλd (u) due− ∫ s

tru du ds · e− ∫ T

tλp(s) ds

).

This (and similar terms) gives Expression (3).

Derivation of Expression (7)

In Heath Jarrow Morton (1992), the equation for the bond price process is

dp(t, T )

p(t, T )= rt dt −

K∑j=1

(∫ t+T

t

σj (t, u) du

)dWj (t).

Define q(t , T ) = p(t , t + T ). Then,

dq(t, T ) = ∂p(t, t + T )

∂Tdt + dtp(t, t + T )

= ∂p(t, t + T )

∂Tdt

+⎡⎣rt dt −

K∑j=1

∫ t+T

t

(σj (t, u) du) dWj (t)

⎤⎦p(t, t + T )

But, ∂p(t,t+T )∂T

= ∂e− ∫ t+Tt f (t,u) du

∂T− p(t, t + T )f (t, t + T ). Substitution gives the

result.


Simulation Algorithm

The simulation algorithm is now described in more detail.

Step 1: Generate discrete time observations of the sample paths forW1(t), . . . ,W4(t), ZH̄ (t), ZH

1 (t), . . . , ZH6 (t), Zh

1(t), . . . , Zh54(t) over a time pe-

riod t = 0, . . . , T representing 30 years. The discrete observation period isweekly. These sample paths are generated under the martingale measure.

Step 2: Given the Brownian motion paths, generate sample paths for the termstructure of interest rates using Expressions (5)-(7) and the state variables usingExpressions (8)-(10).

Step 3: Given the state variables and term structure of interest rate paths,compute λc(t , U i

t , Xt ), λl(t , U it , Xt ). Check to see if the loan is current or

delinquent at time 0.

Step 4: Generate Ei for i = 1, . . . , 2 × (number of loans in database) in-dependent unit exponential random variables. If current at time 0, computeτ il = inf{t > 0 :

∫ t

0 λl(s, Uis , Xs) ds ≥ Ei} for i = 1, . . . , (number of loans in

database). If τ il > T , stop. If τ i

l ≤ T , then starting at time τ il , generate Ei

for i = 1, . . . , 2× (number of loans in database) independent unit exponen-tial random variables. Compute τ i

c = inf{t > 0 :∫ t

τlλc(s, Ui

s , Xs) ds ≥ Ei} fori = 1, . . . , (number of loans in database). If τ i

c > T , stop. If τ ic ≤ T , repeat

the process to obtain the next switch to delinquency or until time T is reached.Instead, if the loan is delinquent at time 0, then perform the previous stepswith the obvious changes. This step generates the state variables N i

t for eachloan i.

Step 5: Given N it for each loan i, compute λp(s, N i

t , U is , Xs), λd (s, N i

t , U is ,

Xs) and δit = δi(t , U i

t , Xt ). Note that the computation of the default intensityand the recovery rate is for non-CTL loans. For CTL loans, λd and δ areassumed to be constants and are computed independently of the above process.λd is computed from credit default swap data or Moody’s rating data, and δ iscomputed from Moody’s recovery rate data for senior secured debt. For CTLloans, the process proceeds as for non-CTL loans, except for the substitutionof the default intensity and recovery rate process as explained above.

Generate Ei for i = 1, . . . , 2 × (number of loans in database) in-dependent unit exponential random variables. Compute τ i

d = inf{t > 0 :∫ t

0 λd (s,Nit , U

is , Xs) ds ≥ Ei} for i = 1, . . . , (number of loans in database)

and


τ ip = inf{t > 0 :

∫ t

0 λp(s,Nit , U

is , Xs) ds ≥ Ei} for i = (number of loans in

database) + 1, . . . , 2 × (number of loans in database). Compute δit = δi(t , U i

t ,Xt ) and set the time to recovery at 12 months.

Step 6: Input (r t )Tt=1, min (τ id , τ i

p) and δi into the Trepp software.

Step 7: Collect the cash flows in Expression (1) for the CMLs and Expression(4) for the CMBS bonds. Denote the cash flows for any of these securities by(V t (ω))Tt=1 for the scenario ω.

Step 8: Repeat steps (1)-(7) 10,000 times. Compute time 0 values of the secu-rities using

∑ω

(T∑

t=1

Vt (ω)e− ∫ t

0 rs (ω) ds

)1

|ω| .

Given that the above simulation is done for each loan in a CMBS loan pool,aggregating to generate the loan pool’s cash flow (approximately 100 loansper pool), then generating the all bond tranches cash flows (approximately 10bonds per trust) across all CMBS trusts (for approximately 650 trusts), thissimulation results in a large computational exercise.

To facilitate this computation, we perform this simulation on a cluster of PCsusing parallel processing and variance reduction techniques. The variance re-duction techniques reduced the simulation calls to the Trepp engine from 10,000to 2,500, maintaining the same accuracy. Two techniques used are stratifiedsampling and a control variate (see Glasserman 2004). The control variate isthe sum of the discounted promised cash flows, but not including any defaultsor prepayments, representing an otherwise riskless bond.

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