Cmbs Efficiency Ree

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Commercial Mortgage Backed Securities

(CMBS) and Market Efficiency with respect to

Costly InformationA. Christopoulos, R. Jarrow, and Y. Yildirim

September 4, 2007

Abstract

Commercial mortgage backed securities (CMBS) are complex assetbacked securities trading in markets that do not currently use derivativespricing technology. This lack of usage is due to the complexity of themodeling exercise, and only the recent and costly availability of historicaldata. As such, CMBS markets provide a natural environment for the test-ing of market efficiency with respect to this "costly" information. Usingthis information, this paper develops a CMBS pricing model to providea joint test of the model and market efficiency. Backtesting our pricingmodel for 4 years, although there is some evidence of abnormal tradingprofits, we cannot reject the efficiency of the CMBS markets.

Introduction

The commercial mortgage backed securities (CMBS) market is a relatively newmarket, jump started by the Resolution Trust Corporation working out thecommercial loan portfolios of many failed Thrifts and Savings & Loans in theearly 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 cashflow modelingand structuring software tools for the generation of scenario specific changes inCMBS yields were first available in the late 1990s as well. At present, however,there is still no vendor that provides derivatives based CMBS pricing modelsfor industry usage.2 This is in contrast to the residential mortgage backed

Chief Executive Officer WOTN, Ithaca, N.Y. 14850.J h G d t S h l f M g t C ll U i it Ith N Y k 14853


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securities market, where derivatives pricing technology has been in significantusage for over a decade (see Fabozzi (2000)). This lack of usage is due to thehistorical evolution of the market (being real estate based), the complexity ofthe modeling exercise, and only the recent (but costly) availability of relevanthistorical data.

The complexity of CMBS modeling is due to the simultaneous inclusion offour significant risks - market, credit, prepayment and liquidity. In addition,

CMBS are quite complex. To value a CMBS, one must first understand thecash flows to the underlying CMBS loan pools, the cash flow allocation rulesto the various bond tranches, the prepayment restrictions, and the prepaymentpenalties. These provisions can, and often do, differ across the different CMBStrusts. Also, implementation of a model requires significant computational effortdue to the quantity of the outstanding loans in any particular CMBS trust, thedimension of the valuation problem (due to the number of risks present), andthe number of outstanding CMBS trusts. Finally, estimation of the relevantparameters is itself a non-trivial problem, given the sparsity and the diversity

of historical CMBS data, especially with respect to the history and currentcomposition of the existing CMBS loan pools. Only recently have vendors madeavailable (at a significant fee) the relevant historical data and computationaltools, but in a form 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. Assuch, the CMBS markets provide a natural environment for the testing of marketefficiency with respect to this "costly" risk management information. Every such

test of market efficiency, however, necessarily involves a joint test with a pricingmodel. Formulating a new model for pricing CMBS, this paper provides such a

joint test of CMBS market efficiency and our new pricing model.Hence, a secondary contribution of this paper is the development, implemen-

tation, and testing of a risk management model for pricing CMBS. Of the fourrelevant risks discussed previously, we model interest rate risk using a multi-factor Heath, Jarrow, Morton (1992) model. Credit risk is modeled using thereduced form methodology introduced by Jarrow and Turnbull (1992), (1995).Because we are valuing CMBS from the markets perspective, following the re-

cent insights of Duffie and Lando (2001) and Cetin, Jarrow, Protter, Yildirim(2004), an intensity process is used to incorporate prepayment risk with regionalproperty value indices included as explanatory variables. Lastly, liquidity riskis incorporated into both the estimation of CMBS fair values and the testingof various trading strategies, motivated by the recent insights of Cetin, Jarrow,Protter (2004) in this regard.

In total, our model includes 64 correlated factors generating the randomness


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We fit our CMBS model using daily forward rate curves, National Council ofReal Estate Investment fiduciaries (NCREIF) regional property value indices,various Bloombergs Real Estate Investment Trust (REIT) stock price indices,and Trepps 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

competing risk hazard rate procedure was used to fit both default and pre-payment risk, consistent with the observation that the occurrence of defaultprecludes prepayment, and conversely.

Market quotes obtained from Trepps CMBS Pricing ServiceTM were used forthe joint testing of the model and market efficiency. Trepps market quotes forthe entire universe of investment grade CMBS are updated daily and they repre-sent an average of end of day marked-to-market prices, contributed by multipledealers and institutional investors.3 Because CMBS trade over-the-counter,Trepps CMBS Pricing Service provides the best historical pricing database for

analysis. It is important to note that non IO, investment grade CMBS trade inrelatively liquid markets, and the Trepp database is comparable to other fixedincome databases frequently used in empirical investigations of credit risk mod-els (see, e.g. Duffee (1999)). To the extent that these quotes do not representprices at which actual trades could have be executed, our conclusions must beaccordingly 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 equivalentto computing the bonds option adjusted spreads (OAS). We observe statisti-

cally significant 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 themarket 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 portfolios,matched by risk (both credit and interest rate) from July 2001 to April 2005.In these tests, for all risk categories, the undervalued portfolios significantly

outperformed the overvalued portfolios. For example, in comparing cumula-tive returns from the undervalued short tenor AAA portfolio (33.26%) to theovervalued short tenor AAA (17.60%), we see a cumulative outperformance of15.65% over the testing period. We also compared the undervalued and theovervalued portfolios to those of matching equal risk CMBS indices over thetesting period. Consistent with the under- versus overvalued comparison, theundervalued portfolios outperformed the matched CMBS index and the over-


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We analyzed these abnormal returns for omitted risk factors related to in-terest rates, property value, and market risk. We found that the overvaluedand undervalued portfolio returns are correlated with various interest rate riskfactors, providing some evidence that these abnormal returns may be due toomitted risk premia. Omitted risk premia is consistent with an efficient CMBSmarket. Alternatively, we also provide a behavioral finance explanation forthese correlations, that is inconsistent with an efficient market. Which expla-

nation is correct awaits subsequent research. Since we cannot unambiguouslyreject an efficient CMBS market, we also provide a simple adjustment to ourmodel that generates unbiased estimates for CMBS prices, useful for hedging ormarking-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, thepapers related to our methodology include Schwartz and Torous (1992),(1989),Deng, Quigley, Van Order (2000), McConnell and Singh (1994), Stanton (1995),Kau, Keenan and Smurov (2004), and Goncharov (2004). With respect to

CMBS, Snyderman (1991) and Esaki, LHeureux and Snyderman (1999) pro-vide summary statistics on commercial mortgage defaults and loss severitiesfrom 1972 - 1997. Archer, Elmer, Harrison and Ling (2004) study a defaultmodel for multifamily commercial mortgages. Ambrose and Sanders (2003),Ciochetti, Lee, Shilling and Yao (2004), and Yildirim (2005) estimate com-peting risk hazard and prepayment models for CMBS. These studies do notinvestigate valuation or hedging of CMBS. Titman and Torous (1989) studythe valuation of a class of CMBS called bullet bonds. Childs, Ott and Rid-diough (1996) simulate a structural model for CMBS valuation to determine

the models implications for tranche values. They do not empirically test theirmodel.

An outline for this paper is as follows. Section 2 provides a brief descriptionof CMBS. The model is described in section 3, the empirical processes in section4, and the estimates are provided in section 5. Section 6 provides the joint testsof market efficiency, and section 7 concludes the paper.

A Brief Description of CMBS

CMBS are a subset of a class offinancial securities known as asset backed secu-rities. CMBS are bonds of various seniorities, whose payments are made fromthe cash flows obtained from a CMBS trust. A CMBS trust is a legal entitythat consists of a collection of commercial mortgage loans, called the underlyingloan pool. These commercial mortgage loans are usually fixed rate loans of aparticular maturity, although they can include floating rate loans as well. The


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terest payments (subject to availability), the outstanding principal balance ofthe bonds are reduced in the order of seniority by scheduled amortization andprepayment of principal. Principal repayment can occur due to voluntary pre-payments or recoveries in the event of default. In reverse order to seniority, theleast senior bonds lose their underlying principal first from defaults. In addition,most CMBS trusts issue a class of bonds call interest only (IO) bonds, whosecash flows come solely from the loan pool interest payments, but only after

all the senior bond coupons are paid. IO bonds make no principal payments.A typical conduit transaction is structured using a CMBS trust that containanywhere from one to sever hundred underlying loans (mostly first liens), andissues about 10 different bond tranches including at least one IO bond. Thereare over 650 CMBS trusts trading over-the-counter across both fixed-rate andfloating-rate collateral including both US and foreign collateral. For a moredetailed description see Sanders (2000).

The ModelThis section constructs the CMBS valuation model. The model presented is notthe most complex formulation possible. Extensions and generalizations will bediscussed in footnotes. However, given the current limitations in the availabilityof accurate and consistent data, more complex formulations would provide little,if any, additional explanatory power in the subsequent implementation.

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

In the initial formulation, we abstract from liquidity risk. That is, we assume

frictionless and competitive bond markets. Liquidity risk is only addressed inthe empirical implementation. This decomposition is similar to the logic un-derlying the computation of option adjusted spreads for residential mortgagebacked securities. OAS measure all market impacts (including mispricings andliquidity risk)5 , after controlling for interest rate and prepayment risk via theuse of a model. We concentrate on modeling the interest rate, credit and pre-payment risks inherent in CMBS. The interest rate risk is handled using amulti-factor Heath, Jarrow, Morton (1992) model for the term-structure evolu-tion. To model the credit risk component, we utilize the reduced form credit

risk methodologyfi

rst introduced by Jarrow and Turnbull (1992),(1995). Thisis done because our goal is to value CMBS from the markets perspective, andnot the borrowers. As shown by Duffie and Lando 2001) and Cetin, Jarrow,Protter, Yildirim (2004), given the markets information set - a subset of theborrowers- default times are inaccessible. Lastly, we model prepayment riskusing an intensity process. This is also done because from the markets perspec-tive, prepayment often appears as a surprise. As with credit risk, this is due


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a justification of this approach for residential mortgages).We are given a filtered probability space (,z, (zt)t[0,T),P) satisfying the

usual 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 denotedp(t, T), and various properties, commercialmortgage loans, and CMBS bonds introduced below. The spot rate of interestat time t is denoted rt. 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 spot rate of interest could be included in this set of statevariables.

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

t0 rsds. Since we are

interested in valuing CMBS, most of the model formulation will be under theprobability measure Q.

Commercial Mortgage Loans

Commercial mortgage loans (CMLs) are issued against commercial properties.These CMLs can be fixed rate or floating. For this discussion, we concentrateon fixed rate loans. Floating rate notes can be handled in a similar fashion.Our computer implementation, to be discussed below, handles these explicitly.These mortgage loans are issued to borrowers based on the quality (economicearning power) of the underlying property. If the property loses value, theborrower may decide to default on the loan. As such, CMLs face both market

(interest rate) and credit risk.

Description

Fixed rate CMLs are similar to straight corporate bonds with the exceptionthat the loans principal is partly amortized over the life of the loan. TypicalCMLs have a (T /n) balloon payment structure. In the (T /n) balloon pay-ment structure, the loan has a fixed maturity date T, a principal payment F,scheduled payments P paid at equally spaced intervals over the life of the loan

(usually monthly), and a coupon rate per payment period c = C/F where Cis 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 ) 1


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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, TL]. 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 to

make 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 of USTreasury Strips that match the remaining payments on the loan (for an analysisof the defeasance option in CMBS, see Dierker, Quan and Torous (2005)). Assuch, prepayments under defeasance actually increase the value to the lender asthe US Treasury will neither default nor prepay such cashflows in the future.In the case of CMLs in CMBS, the lender is an institutional investor who is

entitled to cashflows from such defeased loans that are included in the trustbacking their CMBS.

To address the powerful credit and prepayment aspect of defeasance loans,in our 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 cash-out refinancings. In a cash-outrefinancing, CML borrowers incur the costs related to prepayment penalties on

their loans, if such costs are outweighed by the benefits of the rise in the valueof the property securing the loans.

Since our historical database includes many loans that are not defeasance,and since a significant portion of new issue loans in the CMBS market today arenot exclusively defeasance loans, it is essential that our methodology includesa general procedure for accommodating the risks of prepayments in CMBS.Letting Bt denote the remaining principal balance of the loan at time t, werepresent the total prepayment amount as Bt(1 + Yt) where Yt is the time tprepayment cost as a percentage of the remaining principal balance. Yt is

determined by the specific mortgage loans prepayment penalities and can differacross mortgages.

Valuation

To value a CML, we fix a particular loan. Let Ut be a vector of deterministicloan specific characteristics. There are 4 possible states of the loan (current c,


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initialized in the current state (we could have initialized in the delinquent stateinstead). We assume that C(t) follows a Cox process with an intensity (t) ={1{C(t) is odd}c(t, Ut, Xt) + 1{C(t) is even}l(t, Ut, Xt)} under the martingalemeasure.6 If C(t) is odd, then the intensity governing switching to currentis c(t, Ut, Xt). If C(t) is even, then the intensity governing switching todelinquent is l(t, Ut, Xt). We can define a point process N(t) = {1 if C(t) iseven, 0 if C(t) is odd} = {1 if delinquent, 0 if current} with N(0) = 0. The

relationship 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 with thesame intensity (t) = {1{N(t)=1}c(t, Ut, Xt) + 1{N(t)=0}l(t, Ut, Xt)} underthe martingale measure.

There is an implicit assumption here, that needs to be made explicit. Thisprocess supposes that the probability of current or delinquent is independent ofbeing current or delinquent. This assumption is equivalent to supposing thatcurrent or delinquency are uninformative with respect to next periods currentor delinquency status. An analogy might be that of a household paying anelectric bill. The households are usually current, but in some months, they aredelinquent. Being delinquent does not mean the household will be more likelyto be delinquent in the next month. However, the electric company does noticethat those households that are delinquent in a month, are more likely to defaultthen are those that are current. This is reflected in the process next described.

Let d be the random default time on this loan and denote its point processby Nd(t)

1{

dt}. We assume that the point process follows a Cox process

with an intensity d(t) = d(t, Nt, Ut, Xt) under the martingale measure. TheCox process assumption implies that conditional upon the information set gen-erated by (Nt, Xt)t[0,T] up to time T, Nd(t) behaves like a Poisson process. Ifdefault occurs, the recovery on the loan is assumed to be d (Bd + P). Thetrust receives d percent of the remaining principal balance plus the proratedscheduled principal payment.7 We assume that t = (t, Ut, 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 theloans balloon payment date [Tdt,T] is approximately d(T, NT, UT, XT)dt.8Allowing for default on the balloon payment date captures what is often calledextension risk in the CMBS literature. Extension risk is the risk that, on the

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

7 F i l i li i h h d i f h l i h


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balloon payment date, the borrower will not be able (or willing) to make theballoon payment, but is able (or willing) to continue making the coupon andamortization payments P. The belief is that by extending the loan, the balloonpayment will be made at a later date. The trustee of the CMBS trust decideswhether or not to extend the loan and the conditions of the extension. Theballoon date could be extended (usually less than 3 years) and the underlyingcoupon rate, tenor, and leverage of the loan may be altererd in workout pro-

ceedings, though generally with no alternation to the trust documentation. Ifextension occurs, one can think of this situation as being equivalent to the occur-rence of default, but the extension process is initiated to increase the recoveryrate on the loan.

Let p be the random prepayment time on this loan and denote its pointprocess by Np(t) 1{pt}. Again, we let the prepayment point process bea Cox process with intensity p(t) = p(t, Nt, Ut, Xt) under the martingalemeasure. If the loan is prepaid, the trustee receives Bp(1 + Yp ) dollars. Thisrepresents principal plus the prepayment penalty. For analytic convenience, we

assume that conditional upon the information set generated by (Nt, Xt)t[0,T]up to time T, Nd(t) and Np(t) are independent Poisson processes.

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

TXj=t+1

P1{j


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+

ZTt

s(Bs + P)d(s)e

st(ru+d(u))duds e

Ttp(s)ds (3)

+

ZTt

Zkt

s(Bs + P)d(s)e

st(ru+d(u))duds p(k)e

ktp(s)dsdk

+ZT

t Bs(1 + Ys)p(s)e

st(ru+p(u))du

ds e

Ttd(s)ds

+

ZTt

Zkt

Bs(1 + Ys)p(s)e

st(ru+p(u))duds d(k)e

ktd(s)dsdk}.

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 trusts 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 values Fi , and statedmaturity dates Ti at issuance with increasing maturities T1 T2 ... Tm. Ina typical sequential pay, senior/subordinated structure, the principal cashflows

from the trust are allocated based on the bonds stated maturities at issuance.The most senior bonds (maturity T1) receive their scheduled principal paymentsin any month first, until their principal balance is reduced to zero. Then, thenext most senior bonds (maturity T2) receive their scheduled principal payments(if any cash flows remain) in any given month until their principal balance isreduced to zero, and so forth to the least senior bonds (maturity Tm). Couponcash flows are allocated pro-rata subject to the availability of such coupon cash-flows. In the case of interest shortfalls, coupon cash flows are allocated in orderof seniority and subject to the documentation of the trust. Also, any loan

prepayments and/or default recovery payments received are allocated accord-ing to seniority as well. These prepayments or defaults on the loans result inprepayment of the principal on the CMBS bonds prior to their maturity. Inreverse fashion, any losses in default are subtracted from the least senior bonds(maturity Tm) principal first, then working backwards up to the most seniorbonds (maturity T1) principal.

The outstanding principal balance of the IOs at all times is a reflection of


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cumulative interest payments from the loan pool, less the cumulative couponspaid to the principal paying CMBS coupon bonds.

Valuation

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

bi(t) = Et{TiX

j=t+1

vi(j)e

jtrsds} 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 a sub-sequent section.

The Empirical Processes

This section describes the stochastic processes for the term structure of interestrates and state variables used in the empirical implementation. We use a mul-tiple factor HJM model for interest rate risk and standard diffusion processesfor 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) = logp(t,T)T

be the instantaneous (continuously compounded) for-ward rate at time t for the future date T. We use a K factor model HJMmodel.

df(t, T) = (t, T)dt +KXj=1

j(t, T)dWj(t) (5)

where K is a positive integer, (t, T) =PK

j=1 j(t, T)RTt

j(t, u)du, j(t, T) min[rj(T)f(t, T), M] for M a large positive constant, rj(T) are determinis-tic functions of T for j = 1,...,K , and Wj(t) for j = 1,...,K are uncorrelatedBrownian motions initialized at zero Under this evolution forward rates are


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The spot rate process, used for valuation, can be deduced from the forwardrate evolution. Let rt f(t, t).

drt = [f(t, t)/T]dt + (t, t)dt +KXj=1

j(t, t)dWj(t).

But (t, t) = PKj=1 j(t, t) Rtt j(t, t)du = 0, sodrt = [f(t, t)/T]dt +

KXj=1

min[rj(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 an appendix that

dp(t, t + T)

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

K

Xj=1(Zt+T

t

j(t, u)du)dWj(t). (7)

The Empirical Methodology

To estimate the forward rate process given in expression (5), we employ a prin-cipal component analysis as discussed in Jarrow (2002). Given is a time seriesof discretized forward rate curves {f(t, T1), f(t, T2),...,f(t, TNr)}

mt=1 where Nr

is the number of discrete forward rates observed, the interval between sequentialtime observations is and m is the number of observations. Then, percentage

changes are computed {

f(t+,T1)f(t,T1)

f(t,T1) , ...,

f(t+,TN)f(t,TNr )

f(t,TNr ) }

m

t=1. From thepercentage changes, the Nr Nr covariance matrix (from the different matu-rity forward rates) is computed, and its eigenvalue/eigenvector decompositioncalculated. The normalized eigenvectors give the discretized volatility vectors{rj(T1)

,...,rj(TNr)

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

The State Variables

The state variables in our model correspond to various indices related to the

property values underlying the CMBS trusts. All indices are assumed to cor-respond to the prices of actively traded assets (i.e. values of different portfoliosof properties). To the extent that these indices do not correspond to activelytraded assets, the resulting property prices will contain more noise, and theprice processes parameters estimated with larger sampling errors. Nonethe-less, under the plausible assumption that this additional noise has zero mean,the resulting estimates will still be unbiased. Since our model only uses these


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set of state variables correspond to an index for a particular property type(but across the entire country), e.g. hotels. Lastly, the third state variableis an index across all property types across the entire country, e.g. 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 themarket index. This construction is formulated to facilitate simulation of the

state variable processes in a subsequent section.

The Stochastic Process

We specify the stochastic processes for these state variables in reverse order. Allstochastic processes are specified under the martingale measure. The evolutionfor the economy-wide property index, and the regional property index are

dH(t) = rtH(t)dt + (H)H(t)dZH(t) (8)

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

where ZH(t), ZHi (t) are Brownian motions for all i, (H), i(H) are constants

for all i, dZH(t)dZHi (t) = HHi dt, dZ

Hj (t)dZ

Hi (t) =

HHji dt, and the state

variable Brownian motions are all correlated with the forward rate Brownianmotions: dZH(t)dWi(t) =

Hi dt and dZ

Hj (t)dWi(t) =

Hjidt.

The propertyregion index satisfies

dhi(t)

hi(t) = rtdt +

K

Xk=1

ik(

dp(t, t + Tk)

p(t, t + Tk) rtdt) + ij(i)(dHj(i)(t)

Hj(i)(t) rtdt) (10)

+i(dH(t)

H(t) rtdt) + i(h)dZhi (t) for i = 1,...,nh

where ik, ij(i), i, i(h) are constants for all i,k,j(i), the propertyregion in-dex represented by hi(t) is the property type corresponding to the property index

Hj(i)(t),9 and Zhi (t) are Brownian motions independent of Z

H(t), ZHi (t), Wk(t)for all i, k. The K maturity bonds Tk are chosen to be distinct. Note that

in expression (10), the zero coupon bond price returns correspond to constantmaturity bonds, i.e. their maturity is always Tk for k = 1,...K.

This formulation is chosen because in the subsequent simulation, under thissystem, there are only (1 + nH + K) correlated Brownian motions to be sim-ulated. The remaining nh Brownian motions associated with expression (10)are independent. This substantially reduces the size of the simulation from one

h (1 + + K + ) l t d B i ti d t b t d


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measure. The drift of a traded asset equals the spot rate of interest, however,only if there are no dividends nor storage costs from holding the asset. In thecase of commerical properties, there are two offsetting considerations in this re-gard. 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 prop-erty indices lower than the spot rate of interest. This is only true on average,

however, since not all commerical properties are profitable based on cash flowsalone. Unfortunately, we could not find any data sources to capture these rentaland maintenance fees. As a first approximation, therefore, we set their driftsequal to the spot rate of interest. Although the subsequent estimation pro-cedures do not depend on the specification of the drift process, our simulationmethodology does. This introduces a potential model misspecification into ourvaluation methodology.


To compute the parameters of expressions (8) and (9), we use the quadraticvariation, which is invariant under a change of equivalent probability measures.This invariance implies that our estimation procedure does not depend on thedrifts of the commercial property indices. We illustrate this estimation withrespect to expression (9). The procedure is identical for expression (8) as well.

Given is a time series of {Hi(t)}mt=1 where the interval between sequentialtime observations is and m is the number of observations. Define Hi(t) [Hi(t +)Hi(t)]. We compute

mXt=1

Hi(t)Hi(t)

2 1m

giving an estimate of i(H)2. (11)

Next we calculatemXt=1

Hj(t)

Hj(t)

Hi(t)

Hi(t)

1

mgiving an estimate of j(H)i(H)

HHji . (12)

To obtain the correlation between the forward rates and the regional propertyindex Hji for j = 1,...,K we compute

mXt=1

f(t, t + Tk)f(t, t + Tk)

Hi(t)

Hi(t)

1m

giving an estimate ofKXj=1

rj(Tk)i(H)Hji.

(13)This is computed for k = 1,...,K for distinct T1,...,TK yielding K equationsin K unknowns {H1i,...,

HKi}. Solving this system gives the estimates. This is

done for all i = 1 nH


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+i(H(t)

H(t) rt) + i(t) for i = 1,...,nh (14)

where i is a constant and i(t) i(h) eZhi (t).10 In this expression eZhi (t) is theBrownian motion under the empirical measure and i is the adjustment in thedrift due to the market price of risk associated with the change of measure.11

We assume that the market price of risk is a constant. Standard regressionanalysis provides the estimates for the parameters in expression (14). Thistime series regression is run nh times.

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 vector ofloan specific characteristics Uit that are deterministic (non-random), e.g. thenet operating income of the underlying property at the loan origination.

The Stochastic Processes

The current and delinquent intensity processes for each loan have the samefunctional form, differing only in the loan specific variables used. We let

c(t, Uit , Xt) = e

c+cUit+cXt and (15)

l(t, Uit , Xt) = e

l+lUit+lXt for loan i (16)

where c

, c

, c,

l,

l,

lare vectors of constants. The default and prepayment

intensity processes are similar, but they include the delinquency status of theloan N(t) as an additional state variable.

d(t, Nt, Uit , Xt) = e

d+dNt+dUit+dXt and (17)

p(t, Nt, Uit , Xt) = e

p+pNt+pUit+pXt for loan i (18)

where d, d, d, d, p, p, p, p are vectors of constants.These intensity processes are given under the martingale measure for in-

clusion in the valuation equations. However, estimation of these intensities isunder the statistical measure. Fortunately, given the assumption that delin-quency, default and prepayment risk are conditionally diversifiable, these in-tensity functions will be equivalent under both the empirical and martingalemeasures, see Jarrow, Lando, Yu (2005). This assumption is reasonable if theintensity processes, through the state variables employed, include all relevantsystematic risks in the economy. This inclusion leaves only idiosyncratic risk to


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alternative approach is to modify expressions (15) and (16) by estimates of themarket prices of delinquency, default and prepayment risk.12


Given are commercial loan payment histories including scheduled principal pay-ments, defaults, losses, prepayments, loan characteristics; and time series obser-vations for the state variables (Nt, Xt)T

t=1

where the interval between sequentialtime observations is .

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

c(t, Uit , Xt) = 1/(1 + e

(c+cUit+cXt)) (19)

l(t, Uit , Xt) = 1/(1 + e

(l+lUit+lXt)). (20)

In the estimation, we have l(t) =1

1+etfor t = c + cU

it + cXt and

c(t) =1

1+et . Note that the parameter(s) within t in the estimation areequal and opposite in sign for l(t) and c(t).

13 Next,

d(t, Nt, Uit , Xt) = 1/(1 + e

(d+dNt+dUit+dXt)) (21)

p(t, Nt, Uit , Xt) = 1/(1 + e

(p+pNt+pUit+pXt)). (22)

These discrete approximations have the interpretation of being the probabil-ity of default and prepayment over [t, t + ], respectively, conditional on nodefault or prepayment prior to time t. These expressions are estimated ina competing risk paradigm (see Deng, Quigley, Van Order (2000), Ambroseand Sanders (2003), Ciochetti, Lee, Shilling and Yao (2004)) using maximumlikelihood estimation. Competing risk means that the occurrence of defaultprecludes prepayment, and conversely. The estimation methodology explicitlyincorporates 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 was 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 convert


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these constant maturity yields into a term structure of (smoothed) continuouslycompounded forward rates with maturities 3 month, 6 month, 1, 2, 3, 5, 7, 10,and 20 years. In our sample period, spot rates have ranged from a maximumof approximately 12 percent to a low of approximately 1.5 percent. Table 1provides the volatility coefficients for the 3 month, 6 month, 1, 2, 3, 5, 7, 10, and20 year forward rates and the percentage variance explained by the first fourfactors based on monthly observation intervals ( = 1/12).15 As indicated,

thefi

rst four factors explain 93 percent 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, H(t), is obtained from Bloomberg. The Bloombergreal estate investment trust index is a capitalization-weighted index of real estateinvestment trusts, excluding mortgage related REITs. To be included in theindex, the trusts must have a capitalization of at least $15 million. We have

monthly observations of this index over the time period June 1998 to May 2005.To confirm the appropriateness of the Bloomberg REIT index as a measure ofnational commercial real estate values, we compared it to the general NCREIFproperty value index, using quarterly observation intervals. The correlationbetween the returns on the two indices is .92, confirming its appropriateness.

The property value indices, Hi(t), are obtained from Bloomberg as well.Analogous 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 propertyregion indices, hi(t), are obtained from NCREIF.These indices are obtained quarterly and they correspond to 9 regions: eastnorth central, mideast, mountain, northeast, pacific, southeast, southwest, westnorth, and other. This gives a total of 54 different indices (nh = 54). For theconstant maturity zero-coupon bond prices in expression (14), we used the ma-turities 1 year, 2 years, 5 years and 7 years. Note that for the propertyregionindices, LO (lodging) and OT (other) have the same index across all regionsdue to the absence of a propertyregion index for these property types. Theindices were high in 1998, declined and then increased again to their maximum

values in 2005.The estimated values for the state variable parameters are given in Tables2 and 3. As indicated, all the property value indices are negatively correlatedwith the first and third interest rate factors, but positively correlated to thesecond and fourth. The property value indices are positively correlated with

15 Daily observation intervals were not used because daily variations in rates are partlycaused by the smoothing procedure Monthly observation intervals reduces the importance


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each other. Table 3 contains the coefficients, standard errors, and R2 fromtime series regression equation (10) based on monthly observations. Note thatthe R2, ranging between 2 to 68 percent, are similar in magnitude to thoseobserved in the empirical asset pricing literature for stock returns. Althoughwhen considered individually, most coefficients are insignificantly different fromzero (due to multicollinearity issues), their collective influence is statisticallysignificant for most regressions (as indicated by the high R2s).

Delinquency, Default and Prepayment Intensity Processes

The loan history database - including defaults, prepayments, and loan charac-teristics - was provided by Trepp.17 This database contains information on over100,000 commercial loans. The data provides monthly observations of the rel-evant variables over the time period June 1998 to May 2005. In this database,the loans are classified as current, 30-59 days delinquent, 60-89 days delinquent,90 plus days delinquent and defaulted. Loans exhibiting REO or Foreclosure

status are considered to be in default. Defaults are distinct from delinquencies.Since our model has only three classifications (current, delinquent, or default),not five, we needed to determine a coarser partitioning of the classification. Astatistical analysis was done to see if 30-59 days delinquent should be classifiedas delinquent or current, and 90 plus days delinquent should be classified asdelinquent or default.

We conducted a 6 year study of delinquency transitions of more than 2.3million loan life observations. Table 4 shows the transitions over all loans fromhealthy to worse or conversely over the period 6/1998 to 6/2004. A healthy

state is defi

ned as current (0 days delinquent). A worse state is defi

ned 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, a

multiple month state) are not transitioned until they migrate to either healthy(0 days 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 of


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days 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 net operating incomeat origination divided by the loan balance at origination (noi), (5) the prepay-

ment restriction (normalized, monthly) (pen), (6) the logarithm of the originalloan balance (origloanbal), (7) the debt service coverage ratio at origination(dscr), (8) the loan to value ratio at origination (ltv), (9) the weighted averagecoupon at origination (wac), (10) the loan spread at securitization (only for fixedrate loans) (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

In addition to the property value state variables {H(t), Hi(t), hi(t) for all i}discussed above, included in the set of state variables is the spot rate of interest{rt} and a measure of the slope of the yield curve {f(t, t + 10 years)rt}. Thisyields a total of 62 state variables. This number of state variables is consistentwith both default and prepayment risk being conditionally diversifiable (seethe discussion 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 guaranteed

by a parent company (e.g. WalMart). 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 companys identity is unknown.If the parent companys 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 companys 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 fixed rate is (2, 153) and (130)for the floating rate. The analogous number of prepayments are (8, 989) fixedrate and (2, 960) floating, respectively. Prepayments are more numerous in oursample then are defaults. The number of defaulting floating rate loans in our

18


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database 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.

Table 6A contains the current and delinquency intensity process coefficientsfor fixed and floating rate loans. Note that the coefficients are equal and op-

posite in sign for current and delinquency. We concentrate on explaining theintensity of going from current to delinquent. For fixed rate loans: (i) as theage of the loan increases, the likelihood of delinquency increases, (ii) as histori-cal foreclosures increase (fore index), the likelihood of delinquency declines, (iii)the net operating income (noi) is insignificantly different from zero20 , (iv) thehigher the prepayment penalties (pen), the higher the likelihood of delinquency,(v) the larger the original loan balance, the higher the likelihood of delinquency,(vi) the higher the debt service coverage ratio (dscr), the higher the likelihoodof delinquency, (vii) the higher the loan to value at securitization (ltv), the

lower the likelihood of delinquency21

, and (viii) the higher the weighted averagecoupon (wac) and coupon spread, the lower the likelihood of delinquency. Asthe property indices increase (hi(t), Hi(t), H(t)), the likelihood of delinquencydeclines. As either the spot rate (spot) or the slope of the forward rate curve(f(t, 10) r(t)) increases, the likelihood of delinquency increases. All of thesecomparative statics are as expected. For floating rate loans, similar interpre-tations follow for many of the coefficients. The descriptions for floating rateloans are omitted for brevity and because of the smaller sample size.

Table 6B contain the default and prepayment processes. The signs of these

coeffi

cients are mostly as expected. Forfi

xed 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) appearsto have no impact on either the likelihood of default or prepayment, (v) thehigher the prepayment penalties (pen), the higher the likelihood of default andthe lower 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) the

higher the debt service coverage ratio, the less likely to prepay or default, (viii)the higher the loan to value at securitization (ltv), the higher the likelihood ofdefault and prepayment, (ix) the higher the weighted average coupon (wac), thehigher the probability of default and lower the probability of prepayment, and(x) the higher the coupon spread at origination, the higher the probability ofprepayment and the lower the probability of default. Continuing with fixedrate loans, as the spot rate (rt) increases or the term structure becomes more


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steep (f(t, t + 10) rt), default and prepayment are more likely. Lastly, asthe propertyregion index (hi(t)) increases, prepayment increases but defaultis unchanged. As the property index (Hi(t)) increases, the likelihood of defaultdeclines, and the likelihood of prepayment increases. Finally, as the REIT indexincreases (H(t)) both default and prepayment decline. For floating rate loans,similar interpretations follow for many of the coefficients. However, becauseof the smaller sample size (see Table 5), less of the coefficients are significantly

diff

erent from zero.A standard method for measuring out-of-sample performance is the areaunder the ROC curve. For comparison across models, a value of 0.5 for theROC measure indicates a random model with no predictive ability, while avalue of 1.0 indicates perfect forecasting. Table 7 contains the ROC accuracyratios for the different intensity processes estimated. As indicated, the ROCratios, ranging from 72% to 95%, are quite high for all models. These numbersare comparable to those obtained in the estimation of corporate 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 alinear regression model to our loss severity database consisting of 919 loans.23

1 it = 0 + 1Uit + 3Xtwhere 1 is a constant and 2, 3 are vectors of constants. The regressionestimates are contained in Table 8. We see that as the age of the loan or

the original loan balance or the loan to value ratio increases, the loss severityincreases. Second, as the spot rate or slope of the forward rate curve increases,the loss severity declines. Many of the coefficients are insignificant from 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 CTLs name, we obtain

default 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(see Lando (1998, p. 207)

cds(T) = (1 )d


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obtain the recovery rates for specific industries for senior secured debt, we useMoodys (2005, exhibit 23, p. 21. Then,

d =cds(T)

1 .

If there is no credit default swap data, we instead use the firms Moodys creditrating, and from Moodys (2005) we obtain the default intensity for that credit

rating.

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

on the history of the previous payments for each of the existing CMBS trusts.To circumvent 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 theuser to input a scenario for each of the underlying loans. The outputs arethe cash flows to the various bond tranches (including changes in yields due

to the scenario modification). As such, this provides the intermediate step inour simulation algorithm. Using the above modeling structure and parameterestimates, we are able to generate, using variance reduction techniques, theequivalent of 10,000 different scenarios24 to input into the Trepp software. Adescription of the simulation algorithm and variance reduction techniques areprovided in the appendix. The resulting cash flows to the loans and bonds,enable us to compute expressions (3) and (4) as discounted averages over theset of simulated scenarios.

A Joint Test of Market Efficiency

This section jointly tests our model and CMBS market efficiency using historicaldata and market prices for the CMBS bonds from July 2001 to April 2005. 25 Foreach month over this observation period, starting with July 2001, we estimatethe models parameters using only data available to the market at that time.


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With these estimates, we compute the CMBS bond prices (denoted bi(t) fromexpression (4)) as explained above, and compare them to the market prices atthat date (denoted mi(t)).

The market prices were obtained from Trepps CMBS Pricing Service TM .These prices are the average of the bid and offer quotes obtained by pollingmultiple dealers and institutional investors. These prices are updated daily andthey are widely used in the CMBS industry for marking-to-market positions.

These prices are not matrix prices. Because CMBS trade over-the-counter,Trepps CMBS Pricing Service provides the best historical pricing database foranalysis. It is important to note that non IO, investment grade CMBS tradein relatively liquid markets, and the Trepp database is comparable to otherfixed income databases frequently used in empirical investigations of credit riskmodels (see, e.g. Duffee (1999)). To the extent that these quotes still might notrepresent prices at which actual trades could have be executed, we modified ourtesting procedure to make this possibility less likely. First, to further avoid stalequotes, we restricted our sample to include only investment grade bonds that

are more actively traded in the market and, therefore, more frequently updatedby Trepp. In this regard, we omitted all bonds rated below baa3 and the IObonds.26 Finally, to further reduce the impact of using market quotes, insteadof transaction prices, we used only monthly trading intervals. For monthlytrading intervals, the differences between transaction prices and market quotesis expected to be minimal.

The comparison between model and market prices is repeated for everymonth in the back-testing period. For this back-testing, we used more than3000 bonds from 336 distinct trusts (the bond index is i and the monthly time

index is t). In addition to computing theoretical CMBS bond prices, we alsocomputed the CMBS bonds weighted average lives (WAL) and the option ad-justed weighted average lives (OAWAL). The formulas are at time 0:

W ALi =1

principal

TXt=1

t promised principal paymentst

!,

OAWALi =1

principal X T

Xt=1t principal paymentst()!

1

||.

The percentage difference [W AL 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 analy-ses.

26 Although the IO bonds are rated investment grade, we also exclude them from subse-


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Goodness of Fit

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

i(t) bi(t)mi(t) for all i and t. (23)If we had used spreads instead of prices, this would be equivalent to computingthe bonds option adjusted spreads (OAS). These price differences are sum-

marized in Tables 9A and 9B. Table 9A groups the average i(t) across timebased on ratings, while Table 9B groups the average i(t) across time based onthe weighted average life (WAL). Since the par value of the bond is 100, theseerrors can be interpreted as percentages. For example, for a1 in Table 9A, thepercentage pricing error is 7.1538. As indicated, for either grouping of thebonds, there is a statistically significant bias in the models price. The modelappears to overvalue the bonds. Tables 9A and 9B reject the hypothesis thatthe model prices equal the market prices. This rejection could be due to eithermodel error or market mispricing. This is the content of the next section.

Trading Strategy Profits

This section generates trading strategies based on i(t) to investigate if theCMBS market is inefficient. The analogy of the pricing difference i(t) toan OAS facilitates the intuition underlying this strategy. The ideal tradingstrategy for this determination is to construct an arbitrage opportunity thatinvolves going long undervalued bonds (highest OAS) and shorting overvaluedbonds (lowest OAS). Unfortunately, there is no well-organized market for short-ing cash CMBS.28 Consequently, we investigate the relative performance ofundervalued and overvalued portfolios, pairwise matched to control for risk. Inthis matching, credit risk is measured by the bonds credit rating and interestrate risk by the bonds weighted average life (WAL).

It is difficult to directly control for property value risk in this comparison.However, since 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 weighted average lives, ofall the CMBS bonds in our database in an attempt to control for property valuerisk premia. The indices are equally weighted portfolios of all the bonds in ourCMBS universe falling into the appropriate classification. In total we have sixCMBS indices corresponding to the ratings/WAL classifications selected. Table10 contains a listing of the total number of CMBS in each risk category, for eachmonth, over our testing horizon. As indicated, the average number of bonds ineach risk category exceed 120, with the exception of the long tenor aaa bondswhich averaged only 96. Consequently, there are sufficient bonds in each risk


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category to justify the belief that these indices reflect returns to well-diversifiedportfolios of CMBS, reflecting the markets compensation for property valuerisks.

Procedure

This section describes the procedure utilized to construct the over- and under-valued CMBS bond portfolios. At time 0 (July 2001), we first group the bonds

into roughly equal risk categories based on credit rating (for credit risk) andWAL (for interest rate risk). IOs and bonds with credit classes ba1, ba2,ba3, and Unrated are excluded because Trepp does not provide reliable marketquotes for these infrequently traded bonds. The groupings selected were:

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 i(0) across i from highestto lowest. Those with the highest i(0) represent undervalued bonds and thosewith the lowest 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. Al-

though this implies that the upper and lower decile portfolios may have diff

erentdollar investments, this will not effect the subsequent return calculations.These 10 portfolios are held until time t = 1 (one month later). The returns

on the 10 portfolios (marking-to-market) were computed taking into account allcash flows obtained over the first time period.

At time 1, we liquidate both portfolios, and 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

explicit transaction costs. However, because we are comparing two matchedportfolios, both exhibiting similar rebalancing across time, the transaction costswould be roughly equivalent for both portfolios. This implies that, as a firstapproximation, the relative performance differential between the two portfoliosshould be unaffected by the exclusion of explicit transaction costs.

Results


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The average monthly returns are contained in Table 11a. As indicated, thereturns on the undervalued portfolios exceed those for the overvalued portfoliosfor all 5 risk groupings. For example, for the short tenor aaa the difference is0.29% per month. A control for property value risk is obtained by a pairwisecomparison of the undervalued and overvalued portfolios with the CMBS indexportfolios. As seen therein, the undervalued portfolio returns exceed thosefor the equal risk CMBS indices, and the equal risk CMBS indices exceed thereturns on the overvalued portfolios, for all 5 risk categories. Only the short aaaundervalued less overvalued difference is significant at the 95 percent confidencelevel as indicated by the p-values given in column 6. However, the likelihoodthat all 5 risk grouping undervalued portfolios would outperform the overvaluedportfolios over this time period, under the null hypothesis that the model cannotidentify mispricings, is (1/2)5 = 0.0156. This is unlikely. Note also that allthe differences to the index are significant at the 95 percent confidence level.Combined, this evidence is consistent with a CMBS market inefficiency.

These same observations are confirmed by the cumulative returns in Table11b. The cumulative returns on the undervalued portfolios outperform the over-valued portfolios for all 5 risk groupings. For example, the undervalued shortAAA bonds outperformed the overvalued short AAA bonds by 15.65% over thistime period, while the long aaa outperformed the overvalued by 7.85%. Again,the likelihood that all 5 risk grouping undervalued portfolios would outper-form the overvalued portfolios over this time period, under the null hypothesisthat the model cannot identify mispricings, is 0.0156. Controlling for propertyvalue risk, the undervalued portfolios also outperform the CMBS index returnsfor every risk grouping considered. Yet, the possibility still exists that theseabnormal returns are due to unaccounted for risk premia, as compensation foromitted risk factors. The next subsection of the paper explores this possibility.

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 theover- and under-valued portfolios can be written using a multi-beta model as

ERpt = rt +

MXi=1

pi(ERit rt)

where Rpt is the p portfolios return over [t, t + 1], Rit is the return over [t, t + 1]

on a portfolio perfectly correlated to the ith systematic risk component, andpi is the beta of portfolio p to the i

th risk component portfolio. There are Mpossible risk factors Using the relation


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where the jt have zero means and are independent across t and j, we can rewritethe multi-beta model as

Rpt = rt +MXi=1

pi(Rit rt) + t

where the t =M

Xi=1 piit have zero means and are independent across t.To construct our regression model, it is reasonable to assume that one of the

systematic risk factors is a CMBS portfolio of equal credit and WAL risk as theover- or undervalued portfolio under consideration. Letting the return on theindex CMBS portfolio be denoted by i = 1, we can write this last expression as:

Rpt = rt + p1(R1t rt) +

MXi=2

pi(Rit rt) + t.

And, it is reasonable to also assume that the beta of the over- or under- valuedportfolio with respect to the index is unity, i.e.

p1 = 1,

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

Rpt R1t = +M

Xi=2pi(R

it rt) + t. (24)

In this specification, we include a constant to capture the abnormal returns.To estimate this model, we use the following assets (portfolios) to capture

various 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 premium29 . Tables 12A and 12B contain the regression results,using monthly returns, for the various risk categories. These two regression

tables differ only by the inclusion or not of both SMB and HML.Because the two regressions are similar, we concentrate on the results pre-

sented in Panel B that includes both the SMB and HML indices. First, wesee that all of the abnormal returns for both the overvalued and undervaluedportfolios (as measured by the ) are insignificantly different from zero, exceptfor the short aaa grouping, and this difference is negative (it should be positive).


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be significant. All of the property value and market indices have coefficientsinsignificantly different from zero (except one). These regressions support thehypothesis that all of the abnormal trading profits associated with the overval-ued and undervalued portfolios can be attributed to omitted interest rate riskpremia. Consequently, we cannot unambiguously reject the hypothesis that theCMBS market is efficiently priced.

A Corrected Pricing ModelIf the CMBS market is efficient, then it is essential to modify the original modelto remove the model errors. Let us recall the specifics of the model structure toidentify the possible misspecifications. First, all of the CMBS risks, includingthe embedded options (default and prepayment), depend on the specificationof the interest rate and property value processes. If these processes are mis-specified, then the model prices will be in error. Second, although the modelstructure explicitly incorporates interest rate, credit, and prepayment risk, dueto the frictionless and competitive market assumption, it excludes liquidity risk.

Consequently, the i(t) differences could be due to this omission. As shown inCetin, Jarrow, Protter (2004), liquidity risk can generate such a bias betweenmodel prices, based on the classical model with no liquidity risk and marketprices.

In addition, we have one remaining potential source of model error relatedto the default intensity. In valuing the CMBS bonds 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, Yu (1995)), then this would introduce a positive pricing errorin 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, denotedbbi(t), is obtained via the expressionbbi(t) bi(t) + rating(t) (25) Pj rating class

j(t)

!


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The intuition underlying this extended model is that any model error dueto liquidity risk and/or non-diversifiable default risk is captured by the aver-age mispricing with respect to the bonds rating. It is conjectured that thebonds rating will be correlated with all of these model misspecifications. In-deed, market liquidity is related to a bonds rating because institutions oftenrestrict their investments to investment grade bonds. Furthermore, empiricalstudies have documented a correlation between a bonds rating and its defaultrisk (see Moodys (2005)), and default risk has been correlated to recovery rates(see Altman, Brady, Resti and Sironi (2003) and Acharya, Bharath, Srinivasan(2004)).

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

i(t) =bbi(t)mi(t) for all i and t. (26)By construction, the average pricing error i(t) for a rating class is zero, i.e. thenew model prices the rating classes correctly.

Table 13 contains summary statistics for i(t) grouped by WAL. As in-dicated, the percentage pricing error differs from that in Table 5B (sign andmagnitude), and it is insignificantly different from zero for all WAL groupingsexcept WAL less than 2.5 years.

Given the extended models error are now unbiased (by construction), wetest for model misspecification with respect to the interest rate and propertyvalue processes. If the adjusted CMBS model is not properly specified, onewould expect that the pricing errors would be correlated with interest rates orproperty values. To test for this sort of misspecification, we ran the following

time series regression on the pricing errors for each bond i:

i(t) = c0 + c1rt + c2[f(t, 10) rt] + c3H(t) (27)

where rt is the time t spot rate, (f(t, 10)rt) measures the slope of the forwardrate curve at time t, and H(t) is the REIT stock price index. The spot interestrate 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 esti-

mates, and the average R

2

from these regressions, grouped by rating categories.All coefficients have insignificant t-scores (less than 2). 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.


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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 portfolios inthe same "equal risk" category. This observation is consistent with the marketmis-estimating the risk of the embedded options in a manner correlated withinterest rates. Note that over our sample period, the market experienced aperiod of declining interest rates and increasing commercial property values.

To document these facts, lets consider the short tenor aaa bonds. Thereturns for the short aaa bonds (as for all bonds) are due to four sources: (a)interest/coupon payments, (b) gains/losses on the sales of securities after theone month holding period, (c) prepayment penalties allocated to the securi-ties if prepaid, and (d) gains/losses on any principal paid down (expected orunexpected due to amortization, prepayment or default recoveries). Table 15provides this breakdown in dollar profits for the trading strategies. For con-venience, we combine sources (b) and (c) into one category. As indicated, theundervalued portfolios provide larger returns on all four sources. The largestdifference (in percentage terms) between the undervalued and overvalued decilesis attributable to principal paydowns. Further supporting this statement, asdocumented in Table 15, we see that the undervalued portfolios experiencedless principal paydowns (32 versus 459). Second, the undervalued portfolioexhibits less embedded option risk as measured by the percentage difference(WAL-OAWAL)/WAL, (1.90% versus 19.02%). And, the undervalued port-folios had higher average coupons (7.01 versus 6.23) and consequently, higheraverage prices as well. Based on the differences in coupons between these twoportfolios, one would expect the undervalued bonds to have more embeddedoptionality, but this is not the case. A model for the embedded options isnecessary to obtain this understanding.

These same conclusions also apply to the remaining bond portfolios as doc-umented in Table 16, where the undervalued portfolios are seen to have higheraverage 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 within equalrisk categories are not being accurately priced. And, these mispricings wouldbe correlated with the interest rate risk factors. Only subsequent research candistinguish between these two competing hypotheses for the abnormal tradingstrategy returns documented previously in Table 11.

Conclusion

This paper provides a joint test of our CMBS pricing model and market ineffi-


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for trading profits over a 46 month period. The back testing results are con-sistent with the existence of abnormal trading profits. However, controlling foromitted risk premia appears to remove these abnormal trading profits. Theseomitted risk premia are also consistent with a behavioral finance explanation inan inefficient 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 Corporate Early Access Program for allowing use of their Xeon computa-

tional cluster, to Trepp, LLC for the use of their software and database, and to

Joshua Barratt for helpful comments and suggestions throughout the execution

of this project.

References

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W. Archer, P. Elmer, D. Harrison and D. Ling. 2002. Determinants ofMultifamily Mortgage Default. Real Estate Economics 30(3), 445-473.

E. Altman, B. Brady, A. Resti and A. Sironi. 2003. The Link betweenDefault and Recoveyry Rates: Theory, Empirical Evidence and Implications.working paper, New York University.

B. Ambrose and A. Sanders. 2003. Commercial Mortgage Backed Securities:Prepayment and Default. Journal of Real Estate Finance and Economics 26,March-May.

U. Cetin, R. Jarrow, P. Protter. 2004. Liquidity Risk and Arbitrage PricingTheory. Finance and Stochastics 8, 311-341.

U. Cetin, R. Jarrow, P. Protter and Y. Yildirim. 2004. Modeling CreditRisk with Partial Information. Annals of Applied Probability 14(3), 1168-1178.

S. Chava and R. Jarrow. 2004. Bankruptcy Prediction with Industry Effects.Review of Finance 8, 537-569.

P. Childs, S. Ott and T. Riddiough. 1996. The Pricing of Multiclass Com-mercial Mortgage Backed Securities. Journal of Financial and QuantitativeAnalysis 31 (4), 581 - 603.

B. Ciochetti, G. Lee, J. Shilling and R.Yao. A Proportional Hazards Modelof Commercial Mortgage Default with Originator Bias. Journal of Real EstateFinance and Economics 27(1), 5 23.

Y. Deng, J. Quigley, and R. Van Order. 2000. Mortgage Terminations,


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D. Duffie and D. Lando. 2001. Term Structure of Credit Spreads withIncomplete Accounting Information. Econometrica 69, 633-664.

G. Duffee. 1999. Estimating the Price of Default Risk. The Review ofFinancial Studies 12 (1), 197 226.

H. Esaki, S. LHeureux and M. Snyderman. 1999. Commercial MortgageDefaults: An Update. Real Estate Finance Spring, 80 - 86.

F. Fabozzi. 2000. Bond Markets, Analysis and Strategies. 4th edition,Prentice Hall.

P. Glasserman. 2004. Monte Carlo Methods in Financial Engineering.Springer Verlag.

Y. Goncharov. 2004. An Intensity-Based Approach to the Valuation ofMortgage Contracts and Computation of the Endogenous Mortgage Rate. toappear in the International Journal of Theoretical and Applied Finance.

D. Heath, R. Jarrow, and A. Morton. 1992. Bond Pricing and the TermStructure of Interest Rates: A New Methodology for Contingent Claims Valua-tion. Econometrica 60, 77 - 105.

E. Jacquier and R. Jarrow. 2000. Bayesian Analysis of Contingent ClaimModel Error. Journal of Econometrics 94, 145 - 180.

R. Jarrow. 2002. Modeling Fixed Income Securities and Interest Rate Op-tions. 2nd edition, Stanford University Press: Palo Alto, CA.

R. Jarrow. 2007. A Tutorial on Zero Volatility and Option Adjusted Spreads.Advances in Mathematical Finance, Birkhauser, Boston MA.

R. Jarrow, D. Lando, and F. Yu. 2005. Default Risk and Diversification:Theory and Applications. Mathematical Finance 15 (1), 1-26.

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R. Jarrow and S. Turnbull. 1995. Pricing Derivatives on Financial SecuritiesSubject to Credit Risk. Journal of Finance 50 (1), 53-85.

J. Kau, D. Keenan and A. Smurov. 2004. Reduced Form Mortgage Valua-tion. working paper, University of Georgia.

D. Lando. 1998. On Cox Processes and Credit Risky Securities. Review ofDerivatives Research 2, 99-120.

D. Lando. 2004. Credit Risk Modeling, Princeton U. Press.J. McConnell and M. Singh. 1994. Rational Prepayments and the Valuation

of Collateralized Mortgage Obligations. Journal of Finance 49 (3), 891 - 921.R.C. Merton. 1990. Continuous Time Finance. Blackwell Pub.Moodys. 2005.Default and Recovery Rates of Corporate Bond Issuers,

1920-2004. Moodys Investors Service, N.Y.P. Protter. 1990. Stochastic Integration and Differential Equations: A New

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E. Schwartz and W. Torous. 1989. Prepayment and the Valuation of Mort-gage Backed Securities. Journal of Finance 44 (2), 375 - 392.

E. Schwartz and W. Torous. 1992. Prepayment, Default, and the Valuationof Mortgage Pass-Through Securities. Journal of Business 65 (2), 221 - 239.

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R. Stanton. 1995. Rational Prepayment and the Valuation of Mortgage-backed Securities. Review of Financial Studies 8 (3), 677-708.

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Y. Yildirim. 2005. Estimating Default Probabilities of CMBS loans withClustering and Heavy Censoring. forthcoming, Journal of Real Estate Financeand Economics.

Appendix

Derivation of Expression (3)

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

Qt(d > T | NT, XT) = e

Ttd(s)ds.

The density is

dQt(d = s | NT, XT) = d(s)e

std(u)duds

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

Et(Et{1{T


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The first term can be written

Et(Et{

ZTt

Zkt

s(Bs + P)1{d=s,p=k}e

strududs dk | NT, XT})

= Et(

ZTt

Zkt

s(Bs + P)d(s)e

std(u)due

strududs p(k)e

ktp(u)dudk).

The second term can be written

Et(Et{

ZTt

1{d=s}e

strududs | NT, XT}Et{1{Tp} | NT, XT})

= Et(

ZTt

d(s)e

std(u)due

strududs e

Ttp(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)= rtdt

KXj=1

(

Zt+Tt

j(t, u)du)dWj(t).

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

dq(t, T) = p(t, t + T)T

dt + dtp(t, t + T)

=p(t, t + T)

Tdt + [rtdt

KXj=1

Zt+Tt

(j(t, u)du)dWj(t)]p(t, t + T)

But, p(t,t+T)T

= e

t+Tt f(t,u)du

Tp(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 for

W1(t),...,W4(t), ZH(t), ZH1 (t),...,ZH6 (t), Z

h1 (t),...,Z

h54(t) over a time period

t = 0 T representing 30 years The discrete observation period is weekly


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(step 3) Given the state variables and term structure of interest rate paths,compute c(t, U

it , Xt), l(t, U

it , Xt). Check to see if the loan is current or delin-

quent 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, computeil = inf{t > 0 :

Rt0

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

database). If il > T, stop. Ifil T, then starting at time il, generate Ei

for i = 1, ..., 2 (number of loans in database) independent unit exponential

random variables. Compute ic = inf{t > 0 : Rtl c(s, Uis, Xs)ds Ei} fori = 1, ..., (number of loans in database). If ic > T, stop. If

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 steps withthe obvious changes. This step generates the state variables Nit for each loan i.

(step 5) Given Nit for each loan i, compute p(s, Nit , U

is, Xs), d(s, N

it , U

is, Xs)

and it = i(t, Uit , Xt). Note that the computation of the default intensity and

the recovery rate is for non-CTL loans. For CTL loans, d and are assumed

to be constants, and computed independently of the above process. d is com-puted from credit default swap data or Moodys rating data, and is computedfrom Moodys recovery rate data for senior secured debt. For CTL loans, theprocess proceeds as for non-CTL loans, except for the substitution of the defaultintensity and recovery rate process as explained above.

Generate Ei for i = 1, ..., 2(number of loans in database) independent unit

exponential random variables. Compute id = inf{t > 0 :Rt0

d(s, Nit , U

is, Xs)ds

Ei} for i = 1, ..., (number of loans in database) and

ip = inf{t > 0 : Rt

0p(s, Nit , U

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

database) + 1, ..., 2 (number of loans in database). Compute it =

i

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

(step 6) Input (rt)Tt=1, min(

id,

ip), and

i 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(Vt())

Tt=1 for the scenario .

(step 8) Repeat steps (1)-(7) 10,000 times. Compute time 0 values of thesecurities using

X

T

Xt=1

Vt()e

t0 rs()ds! 1|| .

Given that the above simulation is done for each loan in a CMBS loan pool,aggregating to generate the loan pools cash flow (approximately 100 loans perpool), then generating the all bond tranches cash flows (approximately 10 bondsper trust), across all CMBS trusts (for approximately 650 trusts), this simulation


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stratified sampling and a control variate (see Glasserman (2004)). The controlvariate is the sum of the discounted promised cash flows, but not including anydefaults or prepayments, representing an otherwise riskless bond.


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Maturities\Factors 1 2 3 4 5 6 7 8 9

3 months -0.1723 0.0303 -0.1129 0.0813 -0.0560 0.0185 -0.0133 0.0363 0.0154

6 months -0.2568 0.0264 -0.1019 0.0356 0.0190 -0.0124 0.0262 -0.0513 -0.0207

1 year -0.2924 -0.0205 -0.0255 -0.0959 -0.0280 -0.0421 -0.0476 0.0008 0.0016

2 year -0.2922 0.0140 0.0161 -0.0238 0.0862 0.0138 0.0289 0.0223 0.0325

3 year -0.2057 -0.0108 0.0665 -0.0424 -0.0688 0.0324 0.0491 0.0138 -0.0267

5 year -0.1624 -0.0038 0.0626 0.0467 0.0503 0.0291 -0.0467 0.0130 -0.0419

7 year -0.1329 -0.0545 0.0869 0.0288 -0.0335 0.0429 -0.0230 -0.0424 0.0338

10 year -0.0860 0.1858 0.1090 0.0484 -0.0186 -0.0518 0.0037 0.0000 0.0056

20 year -0.0792 -0.2017 0.0456 0.0631 -0.0025 -0.0552 0.0161 0.0107 -0.0001

% Var 62.3300 17.2725 8.8499 4.1965 2.9405 1.8838 1.3456 0.7956 0.3856

TABLE 1: Forward rate volatility functions.

The estimates correspond to the factor volatilities )(Tj forj = 1,,9 and T = 3 months, , 20 years obtained from a principal components analysis using monthly

percentage changes in forward rates. % Var is the percentage of the variance explained by each factor. The estimation is over the time period 04-Jan-1982 to16-May-2005.


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)(

__

H 0.1221

Panel A: The monthly standard deviation )(__

H of the economy wide REIT stock price index__

H.

index i IN LO MF OF OT RT

)( iH 0.1260 0.2180 0.1150 0.1272 0.2180 0.1597

Panel B: The monthly standard deviations )( iH of the property type REIT stock price indices iH for the property types industry (IN), lodging (LO), multifamily

(MF), office (OF), other (OT), and retail (RT).

maturity 1 year 2 years 5 years 7 years__

Hj -0.2424 0.1697 -0.1705 0.2681

Panel C: The monthly correlations__

Hj between the economy wide REIT stock price

Documents

Cmbs Efficiency Ree