Risk Perception and Loan Underwriting in Securitized Commercial Mortgages

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Risk Perception and Loan Underwriting in Securitized Commercial Mortgages Simon Firestone, Nathan Godin, Akos Horvath, Jacob Sagi 2024-019 Please cite this paper as: Firestone, Simon, Nathan Godin, Akos Horvath, and Jacob Sagi (2024). “Risk Perception and Loan Underwriting in Securitized Commercial Mortgages,” Finance and Economics DiscussionSeries2024-019. Washington: BoardofGovernorsoftheFederalReserveSystem, https://doi.org/10.17016/FEDS.2024.019. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Risk Perception and Loan Underwriting in Securitized Commercial Mortgages By Simon Firestone, Nathan Godin, Akos Horvath, Jacob Sagi∗ Draft: April 2024 We use model-implied volatility to proxy for property risk perceptions in the commercial real estate lending market. Although loan-to-value ratios (LTVs) unconditionally decreased following the Global Financial Crisis, LTVs conditioned on implied volatility and other theoretically motivated fundamental determinants of optimal leverage show no conclusive trend before or after the crisis. Taking reported property and loan attributes at face value, we find no clear pattern of unwarranted credit being extended to commercial real estate assets. We conclude that systematically higher LTV decisions pre-crisis would have primarily stemmed from risk misperceptions rather than imprudent practices. Our findings suggest that the aggregate LTV level should be interpreted as a proxy for lending standards only after controlling for aggregate risk perceptions, among a host of asset and lending market factors. Ourfindingsalsohighlighttheimportanceofmeasuringandtracking aggregate risk perceptions in informing regulators and policymakers. JEL: C22, D80, G01, G10, G18, G21, R38 Keywords: Loan underwriting, Lending standards, Global Financial Crisis, Mortgages, Real estate finance, Implied volatility ∗ Firestone: GroIntelligence, NewYork, NY(simonbfirestone@gmail.com); Godin: HaasSchoolof Business, University of California, Berkeley (nathan godin@berkeley.edu); Horvath: Federal Reserve Board of Governors, Washington, DC (akos.horvath@frb.gov); Sagi: Kenan-Flagler Business School, University of North Carolina at Chapel Hill (jacob sagi@kenan-flagler.unc.edu). We thank Elizabeth DuncanandCarlosWardlowfortheirhelpfulassistance. Wearegratefulfortheinsightfuldiscussion ofChesterSpattatthe2023WesternFinancialAssociationConferenceandthehelpfulcommentsfrom participantsatthe2023FDICBankResearchConference. Wealsothankseminarparticipantsatthe FederalReserveBoardofGovernors,FederalReserveBankofCleveland,UniversityofWisconsinSchoolof Business,CUNYBaruchCollege,UCBerkeleyHaasSchoolofBusiness,andtheSecuritiesandExchange Commissionfortheircomments. Theviewsexpressedinthispaperarethoseoftheauthorsanddonot necessarilyreflecttheviewsoftheFederalReserveBoardortheFederalReserveSystem.

I. Introduction Against which of these assets should one extend more credit in 2023: a suburban office or a warehouse facility leased to an e-commerce company? With the relative uncertainty surrounding work-from-home trends, the answer seems clear— but it may have been a tossup twenty years ago. This example illustrates the core motivationofourpaper. Loan-to-valueratios(LTVs)areoftenviewedasaprimary measure of underwriting standards in commercial real estate (CRE) lending.1 We argue that variation in observed LTVs should be interpreted through the lens of the lender’s (originator’s) risk perceptions, which not only vary across collateral (property) types but also over time. What might be interpreted as “aggressively” high LTV, may, in fact, be optimal or justifiable given the property’s riskiness and the business cycle. Theoretically, this idea is not new: Jaffee and Russell (1976) demonstrate that lenders may limit credit (i.e., require more “skin in the game”) when it is hard to tell which borrowers are riskier, and Leland and Pyle (1977) provide a framework of firm financing with greater equity accompanying greater risk. Lenders’ willingness to extend more credit should reflect perceived property risk and not only “lax” or “tight” credit conditions (which may also reflect lenders’ risk tolerance or cost of capital). Despite the clear role of risk perceptions, aggregate changes in LTV are often interpreted as changes in underwriting standards when it comes to measuring the aggressiveness of real estate lending. For instance, a 2010 Congressional OversightPanelreportbyElizabethWarrenetal.(2010)aftertheGlobalFinancial Crisis (GFC) came to the following conclusion: “The commercial real estate bubble [...] resulted in the origination of a significant amount of commercial real estate loans based on dramatically weakened underwriting standards. These loans were based on overly aggressive rental or cash flow projections [...], had higher levels of allowable leverage, and were not soundly underwritten.” The culprits identified in the above quote are unrealistic cash flow forecasts andoverlyhighLTVs,consistentwithotherassessments(e.g.,LevitinandWachter, 1 Althoughtheincome-to-debtratioandthedebtservicecoverageratiosarealsocommonqualifiers ofCREloanunderwriting,wefindthattheempiricalrelationshipbetweenperceivedpropertyriskand LTVisstrongerthantheempiricalrelationshipbetweenpropertyriskandthesealternativeratios. 2

2013). Although there is evidence to suggest that property income measures are, at times, inflated by commercial mortgage-backed securities (CMBS) originators (Griffin and Priest, 2023), the diagnosis of aggressive LTVs is a narrative that is more difficult to test, though it may often be conjectured. Jacob and Manzi (2005) describe what they believe to be lenders pushing the limit on LTVs in a trend toward “weaker lending standards,” and Fabozzi, McBride and Clancy (2015) claim that this tendency was especially egregious in 2006 and 2007. Meanwhile, Wilcox (2012) and Wilcox (2018) argue that aggregate LTVs may not provide a faithful portrayal of underwriting standards. Weprovideempiricalevidencethat, controllingforimpliedexanteperceptions of property risk, as proxied by the implied volatility (IV) of individual properties, the average LTVs of securitized CRE loans in the period 2000–04 were only about 1.5 percentage points higher than the average LTVs in the post risk retention rule period of 2016–20. Likewise, average LTVs in 2005–07 were similar to those in 2008–15. Differencesamongepochsshrinkfurtherwhenwecontrolforpropertycap rate (cash yield) spreads over the 10-year U.S. Treasury yield. Indeed, we find that credit rationing “frontiers” (i.e., maximum LTV thresholds) were most permissive in 2000–04, and they were most restrictive in 2005–07, coinciding with the peak of collateralized debt obligation (CDO) issuances. Importantly, credit rationing frontiers explain only a negligible fraction of LTV variation across epochs, while perceived property risk explains the lion’s share. Our main contribution is demonstrating that LTVs for securitized CRE loans, throughout different economic epochs from 2000 to 2020, were largely driven by perceived property risk and market fundamentals. We calculate implied volatility using a two-factor derivative asset pricing model, which allows for standard CRE mortgage contract provisions. In the model, IV is the asset’s diffusion risk that rationalizes the loan’s interest rate given its LTV, maturity, and amortization schedule, as well as the property’s cap rate, the term structure of U.S. Treasury yields, and the mortgage market liquidity premium.2 Our findings are consistent with tradeoff theories of optimal leverage (e.g., Leland, 1994), which imply that observedLTVsshoulddeclinewithIVandcaprates. Onitsown,IVexplainsabout 2/3 of the cross-sectional and time series variation in LTV. Controlling for the cap 2 WeestimateliquiditypremiumontheCREmortgagemarketastheeffectiveyieldspreadbetween short-termAAA-ratedtranchesofCMBSsandU.S.Treasurysecuritieswithequivalentmaturities. 3

rate and various (such as, property type and location) fixed effects helps explain an additional 10 percent of LTV variation. The residual time series variation seems to be random. Our results are in line with Driessen and Van Hemert (2012) and Stanton and Wallace (2018), who find no evidence that underwriting practices in the CRE mortgage market deteriorated in the way that they did in the residential real estate mortgage market before the GFC. Our results are also consistent with the position in Wilcox (2012) and Wilcox (2018) that LTVs, on their own, may not be informative about aggregate loan underwriting standards. FromthecollapseoftheCREmarketinthewakeoftheGFC,itistemptingto conclude that CRE loan leverage was overly aggressive before the crisis. However, after controlling for implied volatility, we find no evidence for abnormally high LTVs. What we do find is that risk perceptions were lower in 2003–07 than in any other epoch from 2000 to 2020. Hence, our findings suggest systematic shifts in perceived property risk as a compelling explanation for the growth in CRE lending in 2003–07, which fueled the subsequent CRE market decline. Moreover, to the extent that there was a failure in the CRE mortgage market in the run-up to the GFC, our findings may also indicate aggregate risk misperceptions.3 Indeed, systematic misperceptions of risk would have led to more credit extended but also to CRE loan underpricing (i.e., low interest rates). The paper is organized as follows. Section II provides theoretical background forconceptualizingtheaggregateLTVasadynamicvariablethatcaptureschanges in systematic risk perceptions as well as other property and capital market attributes—even in the presence of frictions due to taxes and costs of default. Section II reviews the related academic literature and provides institutional details on the CMBS market. Section III describes our data. Section IV analyzes our impliedvolatilityestimatesandtheirrelationshiptoothermeasuresofpropertyrisk. SectionVpresentsourmainresults, examiningCREloanLTVsovertimewithand without controls for determinants of optimal leverage, including (and especially) implied volatility. Section VI concludes. 3 There is an important distinction between aggressive loan underwriting and loan underpricing. Intheformercase,lendersknowinglyundertakemoreriskthanwarrantedbyprudentpractices. Inthe lattercase,lendersfalselybelievethattheyfollowbestunderwritingpractices. 4

II. Conceptual motivation and methodological overview Mortgage provision in the primary market depends on the price of liquidity in the secondary mortgage market, competition among lenders, and the availability of capital in the credit market, all of which are held fixed. Figure 1 shows stylized mortgage offer curves for properties with different perceived risk.4 The mortgage offer curves are truncated beyond a certain LTV because of credit rationing due to asymmetric information (Jaffee and Russell, 1976; Leland and Pyle, 1977) and dead-weight costs of default (Leland, 1994). Figure 1. Stylized mortgage offer curves for properties with different risk Thisfigureshowsstylizedfive-yearzero-couponmortgageoffercurvesforpropertieswithdifferentlevels ofperceivedrisk. Propertyriskisproxiedbyannualizedassetvolatility(“vol”). Theverticalaxisshows the mortgage spread over a zero-coupon five-year U.S. Treasury at which lenders would be willing to issuethemortgageloangiventheloan-to-valueratioandtheannualizedassetvolatility. Thecurvesare computedusingtheMerton(1974)modelandincorporatealiquidityspread,whichrepresentstheprice ofliquidityinthemortgage-backedsecuritiesmarket. 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 50% 55% 60% 65% 70% 75% 80% 85% 90% )pp( daerpS vol = 21% vol = 17% vol = 13% "Liquidity spread" Loan-to-value ratio Given the mortgage offer curve specific to the property, the borrower’s choice reduces to picking the LTV. In the Leland (1994) model, for example, the optimal borrower’s choice is decreasing in the property’s cap rate, increasing in the owner’s 4 In this stylized chart, we ignore other mortgage contract terms available to the borrower and commensurately priced by the lender, including maturity, interim coupon payments, and prepayment options. Wedotakesuchloanfeaturesintoaccountinourempiricalmethodology. 5

tax rate, and decreasing in asset volatility. All else being equal, a model with rational agents yields equilibrium LTVs that decrease in asset volatility. There are several key takeaways from the conceptual considerations above. First, time variation in aggregate LTVs may be entirely attributable to changing market fundamentals, rather than the tension between regulators and lenders or agency frictions within lending institutions. Second, in those instances where LTV limits are below the optimal leverage point for a sufficiently large number of borrowers, there would be clustering at the credit rationing frontier. Third, the frontier should decline with perceived property risk. Finally, although lenders and borrowers do not directly express or observe asset volatility, their risk assessment is implicit in the loan spread at which the contract is originated. In Figure 1 above, it is sufficient to know that a loan with a 67% LTV is priced at 2 percentage points above the five-year U.S. Treasury yield to conclude that the perceived property risk was roughly 17% in implied volatility. A. Overview of empirical approach and results Our null hypothesis is that the cross section of LTVs results from borrower demand in response to rational mortgage offer curves, akin to those depicted in Figure1. Underthenullhypothesis,itisconsistentwithprudentlendingtoprovide an infinitely elastic supply of credit at any point on the curves. Risk misperception corresponds to lending using offer curves that are systematically lower or higher than the true asset volatilities, which could be detected ex post. By contrast, aggressive lending manifests ex ante as loans that would not normally be made (e.g., an 80% LTV loan with asset volatility of 21% in Figure 1). Hence, one could test for aggressive lending in a period like 2005–07 by examining whether the credit rationing frontier was higher during that period than at other times.5 In order to test the null hypothesis described above, and look for evidence of aggressive lending practices, we first estimate the implied volatility of each propertyunderlyingasampleofsecuritizedCREmortgages. Themortgagepricing model that we use to estimate IV captures various ex ante features of the deal, including the property’s cap rate, the loan’s term and amortization schedule, 5 Notethatchangesinthecreditrationingfrontiercouldresultfromtheinfluenceoffundamental factors,suchasasystematicchangeindead-weightcostsofdefault. Therefore,identifyinganepochwith ahigherrationingfrontierdoesnotconstituteevidenceforaggressivelendingpractices. 6

default and interest rate risk, and mortgage market liquidity.6 Under the null hypothesisdefinedabove,IVmeasuresthelender’sperceivedpropertyrisk. Second, we identify the credit rationing frontier, as a function of IV, for four time periods in our sample. We confirm that, consistent with the null hypothesis, the frontier gradually declines with implied volatility in each of the four periods. Next, we examine the rationing frontiers and empirically reject the narrative that lenders were more aggressive in the run-up to the GFC. We use ex ante fundamentals to explain the variation of LTVs across loans and over time, such as implied volatility, cap rate, and other property and market features that can influence mortgage offer curves and borrower demand for loans. We fit a censored (tobit) regression model for LTV, using the rationing frontier as upper bound, and conduct a counterfactual analysis. In particular, we fix the frontier over time and compare actual and estimated counterfactual LTVs to analyze the effect of changing frontiers. We find little evidence that shifts in the rationing frontier explain LTVs, which indicates that, even if such shifts are driven by changing underwriting standards, they have little effect on the distribution of LTVs. What we do find is that the leading determinant of credit provision is implied volatility(perceivedpropertyrisk). Importantly,thisrelationshipisnotmechanical. If borrowers randomly selected LTVs from the mortgage offer curves in Figure 1, then the only link between LTVs and risk perceptions would be through credit rationing, because the rationing frontier is more likely to be binding at higher IVs. This is not consistent with our result that shifts in the rationing frontier explain little of the distribution of LTVs. By contrast, if borrowers optimally choose LTVs to trade off costly default against the benefits of debt (e.g., lower taxes), then, consistent with our empirical findings, LTVs vary with property risk even if the rationing frontier is not a binding constraint. B. Literature on commercial real estate risk and mortgage implied volatility We contribute to an evolving understanding of CRE asset volatility. Previous analyses use aggregate data to study CRE price dynamics. Ciochetti et al. (2002) create a property value volatility index at the property type-census district level. Plazzi, Torous and Valkanov (2010) use quarterly averages at the metropolitan 6 We also examine the possibility that prepayment options affect our results. We confirm that, becauseofthepresenceoflargeprepaymentpenaltiesinthismarketsegment,theydonot. 7

statistical area level for broad property types and apply the Campbell and Shiller (1988) price-dividend decomposition to better understand the characteristics of CRE rents, cap rates, and asset returns. They find that CRE returns are related to the local regulatory environment and population density, and that expected returns are related to factors such as local population, employment, and income growth as well as construction costs. Using property-level data, we find that many of these factors are important determinants of asset volatility. Studies using property-level data have shed light on the magnitude of idiosyncratic asset volatility, that is, how much higher asset volatility is than can be inferred from indexes or other area averages. Plazzi, Torous and Valkanov (2010) calculate that, aggregated at the metropolitan statistical area-level, the standard deviation of CRE excess returns ranges from 3.7% to 6.1%, depending on the property type. By contrast, Downing, Stanton and Wallace (2008) estimate the asset volatility of CMBS loans using a two-factor Titman and Torous (1989) model. They find implied volatilities in excess of 20%—higher than our estimates for their sample period but similar to our post-GFC calculations. Sagi (2021) uses property-level data from the National Council of Real Estate Investment Fiduciaries to measure price appreciation volatility. He finds that the standard deviation of annual price appreciation volatility is about 13%.7 The mortgage pricing model we use to estimate implied volatility builds on an extensive body of literature that applies option theoretical methods for pricing mortgage debt. Some models stipulate a partial differential equation for property value that is solved using finite difference methods (Titman and Torous, 1989; Kau et al., 1995). Another popular method, and the one we employ, uses a binomial model for property valuation (Leung and Sirmans, 1990; Giliberto and Ling, 1992; Hilliard, Kau and Slawson, 1998; Ciochetti and Vandell, 1999). Similar to our pricing approach, many of these models incorporate default and prepayment options. However, while other models assume a single stochastic mean-reverting interest rate process similar to Cox, Ingersoll and Ross (1985), we model interest rates using multiple competing models. Furthermore, we include contractual 7 AssetpricingmodelstypicallyassumethattheassetpricefollowsageometricBrownianmotion, and thus the variance of cumulative price appreciation is linear in the length of the holding period. Consequently,astheholdingperiodapproacheszero,returnvolatilityalsoapproacheszero. Bycontrast, Sagi(2021)findsthatvolatilityremainshighevenforshortholdingperiods. Hetracesthisphenomenon totransactionrisk,findingthatthereturndatafitwelltopredictionsfromasearchmodel. 8

characteristics, such as interest-only versus amortizing payment schedules, and property attributes, such as the cap rate. Our analysis is closest to that of Downing, Stanton and Wallace (2008). They also use a two-factor pricing model that prices the mortgage at par, albeit with the goal to examine the relationship between IVs and CMBS ratings. Similar toourapproach, mortgagevalueintheirmodelisafunctionofshortratedynamics and the property value process. By contrast, our model incorporates a richer set of loan and property characteristics, including property income and the length of the interest-only period. Finally, their analysis ends in 2006, while we also examine developments right before and after the GFC. C. Evolution of the commercial mortgage-backed securities market The CRE loans in our data set are CMBS loans: loans originated to be pooled within Real Estate Mortgage Investment Conduit trusts that issue MBSs. CMBSs allocate risk among different tranches: the tranches least exposed to credit risk typically receive investment-grade ratings, while the tranches that absorb credit losses first are often unrated. In the past two decades, various changes in the CMBS market affected both the cost of funding and the market for riskier tranches. In particular, the investor base of riskier tranches changed because of the rise and fall of CDOs as well as regulatory changes. Before 2005, unrated tranches were usually held by a set of special (“B-piece”) investors, who were involved in security design, performed due diligence, and selected the servicer responsible for handling delinquencies. In the time period between 2005 and 2008, it became common practice among CMBS issuers to repackage such tranches in CRE CDOs. Rating agencies, which made (overly) optimistic assumptions about the benefits of diversification, assigned favorable ratings to many CDO tranches. Another factor affecting CMBS markets before the GFC was a reduction in regulatory capital requirements, as both commercial bank and investment bank capital requirements for CMBS were reduced in 2004. Duca and Ling (2020) calculate that commercial bank capital requirements were reduced from 8% to 2%, while investment bank capital requirements were reduced from 6% to 3.7%, permitting much higher levels of leverage. As a result, the cost of funding decreased for both commercial and investment banks, making it easier for CMBS issuers to sell riskier tranches. 9

After the GFC, CMBS issuance stopped for several years and the CRE CDO market disappeared. In response to the crisis, regulators increased capital requirements for commercial banks at the end of 2010, by which time the major investment banks had been merged with commercial banks. Moreover, U.S. regulatory agencies proposed Regulation RR, requiring issuers of asset-backed securities to retain at least five percent of the credit risk, with the intention of ensuring that issuer incentives are aligned with those of investors.8 Issuers may satisfy the risk retention requirement by holding a “vertical” piece of the issued security, which includes a portion of all tranches, a “horizontal” piece of the riskiest tranche, or a combination of the two approaches. Although “qualified” CMBS issuances are exempt from Regulation RR, they are defined in a relatively conservative manner.9 Consequently, the risk retention requirement is often binding for issuers (Flynn Jr, Ghent and Tchistyi, 2020). Motivated by the substantive variation in the CMBS market and regulatory environment, we subdivide our sample period into the following four epochs. 1) 2000–04: “B-piece” investors retain the riskiest tranches of CMBSs. 2) 2005–07: CMBS issuers repackage portions of riskier tranches as CRE CDOs, part of which receive investment-grade ratings. Regulatory capital requirements associated with CMBS holdings decrease. 3) 2008–15: CMBS issuances plummet then gradually recover to pre-2005 levels. 4) 2016–20: The risk retention rule takes effect in December 2016. These epoch boundaries are also consistent with the empirical distribution of CRE loan originations over time. Indeed, as Figure 2 shows, the number of loan originations exhibits clear cutoffs at the epoch boundaries we use. 8 TheriskretentionrulewasfinalizedinOctober2014andcameintofulleffectinDecember2016. Seehttps://www.sec.gov/news/pressrelease/2014-236.htmlformoreinformation. 9 Regulation RR defines a qualifying CRE loan as a fixed-rate loan with a minimum maturity of 10 years and a maximum amortization period of 25 years. Lenders must document property income foratleasttheprevioustwoyears. Theborrower’sdebtserviceratiomustexceed1.25formultifamily properties,1.5forleasedproperties,and1.7forallotherloans. Furthermore,thecombinedLTVsofall loansonthepropertycannotexceed70%,andtheLTVofthefirstlienloancannotexceed65%. 10

III. Data construction and summary statistics A. Securitized mortgage loan data collection Our data consist of 58,127 securitized CRE loans from the year 2000.10 The data are provided by Morningstar, which gathers information from public CMBS disclosures, including a rich set of loan and property characteristics. The Morningstar data include loans originated by a variety of institutions and are not dominated by a single underwriting approach. Many loans are originated by large U.S. banks, such as Bank of America and Citibank. The top ten originators also include large foreign banks, such as Deutsche Bank, Credit Suisse, and UBS. Non-depository institutions are a substantial part of the market, but no single such institution has a large market share. The complete data set consists of 111,465 loans. For the purposes of our analysis, we drop loans missing key variables needed for our analysis and loans with problematic observations (see Appendix A for more details). Specifically, we drop loans that have missing or wrong data for key inputs such as the date of origination, the loan interest rate, whether the loan is interest-only or amortizing, and the date of maturity. We also drop non-fixed-rate and pari passu loans, which our model does not price, as well as agency CMBS loans because the agency guarantees would distort our implied volatility estimates. Finally, for analytical and expositional simplicity, we restrict our sample to single-property loans, which constitute the overwhelming majority of observations. The data collection process described above results in a sample of 58,127 CRE loans. We present summary statistics for these loans in Table 1 and their corresponding cross-sectional distributions in Figure F2 of Appendix F. Loans vary widely in size, from $2 million to over $2.5 billion. LTVs are generally around 70%. The debt yield, the ratio of net operating income (NOI) to loan amount at origination, varies between 7% and 15%. The debt service coverage ratio (DSCR), theratioofNOItodebtservicingamountatorigination, fallsgenerallybetween1.2 and 2.4. The vast majority (more than 80%) of loans are 10-year loans. 10 SecuritizedCREloansareonlypartoftheoverallCREloanmarket. Blacketal.(2017)compare CMBS loans in the Morningstar data with portfolio CRE loans reported by large U.S. banks on the FR Y-14 form. They find that these banks are more likely to hold riskier loans, such as construction loans,intheirportfolio. Portfolioloansarealsomorelikelytohavefloatinginterestrates,shorterterms, andlowerLTVsthanCMBSloans. Fromtheirresults,itisnotclearthatsecuritizedandportfolioCRE loansdifferwhenonesetsasideconstructionloans,whichweexcludefromourempiricalanalysis. 11

Table 1—Characteristics of sample commercial real estate loans This table shows summary statistics for key characteristics of commercial real estate mortgage loans in our sample, containing fixed-rate, single-property loans securitized in non-agency commercial mortgage-backedsecurities(CMBSs). Fromlefttoright,thecolumnsshowthenumberofobservations and the sample mean, standard deviation, as well as the 10th and 90th percentiles of variables in the cross section of sample loans. At the bottom of the table, the respective spread measures represent the percentage point yield spreads of sample loans over the 10-year zero-coupon U.S. Treasury yield and the value-weighted effective yield of the securities constituting the ICE BofA 0-to-3-year AAA U.S.Fixed-RateCMBSIndex. (Source: ICEDataIndices,LLC,usedwithpermission.) Count Mean SD P10 P90 Loan amount ($1,000) 58,127 11,465 16,694 2,010 24,260 Loan term (months) 58,127 113 24 83 120 Amortization period (months) 58,127 312 110 0 360 Interest-only period (months) 58,127 21 35 0 60 Loan-to-value ratio 58,127 0.68 0.11 0.55 0.79 Debt yield 58,127 0.11 0.04 0.07 0.15 Debt service coverage ratio 58,127 1.69 0.71 1.18 2.32 Spread over 10-yr U.S. Treasury (pp) 58,127 1.85 0.75 0.92 2.78 Spread over 0-3-yr AAA CMBS (pp) 58,127 1.84 1.17 0.36 3.32 Figure 2 and Table 2 show the distribution of observations over time and across property types. The volume of loan originations steadily increased until the GFC, fell to almost zero in 2008, and gradually recovered after 2010. The most common property types are retail and multifamily, and also a large number of loans belong to the “other” category.11 Hotel and industrial properties have the smallest frequency share in the sample. We present summary statistics for the properties used as collateral for sample CREloansinTable3. Themedianpropertyisnineyearsoldandnearlyfullyleased. Properties vary widely in size, from 16 thousand to 24 million sqft. Some propertylevel variables are unevenly populated, mostly due to heterogeneous measurement and reporting standards across property types. For instance, information on the lead tenant is not collected for multifamily properties because they have many small units, each leased to a different tenant. 11 The“other”categoryconsistsofmini-storageandmixed-usepropertiesrepresentingacombination ofpropertytypessuchasacomplexwithbothmultifamilyandretailproperty. 12

Figure 2. Annual number of sample commercial real estate loan originations over time This figure shows the annual number of commercial real estate mortgage loan originations in our sample, color coded by time period (epoch). The sample contains fixed-rate, single-property loans securitizedinnon-agencycommercialmortgage-backedsecurities. EpochchoiceisexplainedinSectionII.C. 8,000 6,000 4,000 2,000 0 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 Number of Observations Over Time Table 2—Distribution of sample commercial real estate loans across property types Thistableshowstheabsoluteandrelativefrequenciesofcommercialrealestatemortgageloansinour sampleacrossdifferentcollateralpropertytypes. Thesamplecontainsfixed-rate,single-propertyloans securitizedinnon-agencycommercialmortgage-backedsecurities. Property type Count Share Hotel 4,808 8.3% Industrial 3,052 5.3% Multifamily 18,972 32.6% Office 9,308 16.0% Other 5,903 10.2% Retail 16,084 27.7% Total 58,127 100.0% In our data filtering, we drop loans with a debt yield less than 0.07 and DSCR less than 1.25. These lower bounds correspond to standard underwriting limits (i.e., lenders are reluctant to lend if the debt yield or DSCR is too low) and fall around the 10th percentile in our full sample of CMBS loans. When the debt yield and DSCR are very low, it may suggest that the property is not currently 13

Table 3—Characteristics of sample commercial real estate loan properties This table shows summary statistics for key characteristics of the properties used as collateral for commercialrealestatemortgageloansinoursampleatthetimeofloanorigination. Thesamplecontains fixed-rate, single-property loans securitized in non-agency commercial mortgage-backed securities. From left to right, the columns show the number of observations and the sample mean, standard deviation,aswellasthe10th and90th percentilesofvariablesinthecrosssectionofsampleproperties. Theareaofthepropertyinsquarefeet,aswellasthederivedvariablesoccupancyrateandleadtenant share,areavailableonlyforindustrial,office,retail,andmost“other”propertytypes. Bycommercialreal estatemarketconvention,thesizeofhotelandmultifamilypropertiesismeasuredbythenumberofunits. Count Mean SD P10 P90 Property value ($1,000) 58,127 17,524 28,014 3,070 36,000 Net operating income ($1,000) 58,127 1,158 1,771 214 2,335 Area (1,000 sqft) 33,562 112.45 151.08 15.99 240.02 Age (years) 52,569 14.26 15.77 1.00 35.00 Occupancy rate 31,929 0.94 0.10 0.83 1.00 Lead tenant area share 29,333 0.42 0.29 0.12 1.00 Lead tenant lease length (years) 29,832 15.98 240.45 2.17 16.25 stabilized—even if it may be anticipated to be shortly. Since our model uses the property’s underwritten cap rate as an input, including non-stabilized properties would distort our implied risk estimate. We are left with 48,468 observations, which we use throughout our empirical analysis in the paper. B. Adjustment for capital market liquidity dynamics One limitation of our mortgage valuation model is that it incorporates only twodynamicfactors: theshortinterestrateprocessandthepropertyvalueprocess. However, in practice, Christopoulos (2017) shows that mortgage pricing is also affected by a time-varying liquidity premium. Therefore, without appropriate correction, our model would attribute an increase in primary CRE mortgage rates due to a higher liquidity premium to increased credit risk, which would cause an upward bias in our volatility estimates. We take the liquidity premium into account by adjusting loan rates before the model-based property valuation step. Specifically, we create a monthly time series of the CMBS liquidity spread by taking the value-weighted effective yield of securities in the ICE BofA 0-to-3-Year AAA U.S. Fixed-Rate CMBS Index minus the yield of zero-coupon U.S. Treasury securities with the corresponding effective 14

(i.e., option-adjusted) duration.12 We then adjust the mortgage rate for each loan by the prevailing liquidity spread as follows: (1) r = r −(CMBS spread−120bp), adj observed where 120 basis points is the median value of the CMBS yield spread defined above. Figure 3 shows the yield spread and the number and market value of the CMBSs with the shortest duration. Although this adjustment leaves a constant baseline level of liquidity premium embedded in mortgage rates, the remaining upward bias should permit relative comparisons of perceived property risk over time based on our model-implied volatilities. Figure 3. Statistics for 0-to-3-year AAA fixed-rate commercial mortgage-backed securities This figure shows quarterly aggregates from 1998 to 2022 for the commercial mortgage-backed securities(CMBSs)constitutingtheICEBofA0-to-3-YearAAAU.S.Fixed-RateCMBSIndex. (Source: ICE Data Indices, LLC, used with permission.) UST yield spread is defined as the basis point (bp) difference between the value-weighted mean effective yield of the index constituents over the yield of thecorrespondingzero-couponU.S.Treasurysecuritywithmaturityequaltothevalue-weightedmean effectivedurationoftheindexconstituents. 250 1,024 200 512 150 256 100 128 50 64 0 32 8991 9991 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 1202 2202 Number of securities (x10) Market value ($ billion) UST yield spread (bp, right) 12 Source: ICEDataIndices,LLC,usedwithpermission. 15

IV. Implied volatility estimation and diagnostics We use a two-factor model (with disaster risk) to estimate implied asset volatility, which we then use as a proxy measure for perceived property risk. The model ignores correlations between U.S. Treasury yields and property values and is described in Appendix B. Although there is no standard way to measure property risk in the presence of market frictions and incompleteness, our implied volatility estimate is a reasonable measure of risk perception.13 Figure 4 shows different implied volatility estimates based on our model. The first IV estimate makes no liquidity adjustments to mortgage rates and does not consider the effect of prepayment options. The second IV estimate allows for optimal prepayment in the presence of contractual penalties. Save for 2012, the presence of prepayment penalties makes little difference in implied volatility. This is because prepayment penalties, which are ubiquitous in the CRE mortgage market, are usually sufficiently punitive to render the value of a prepayment option second-order in mortgage valuation. Notably, given missing data problems (see Appendix A for further details), ignoring prepayment options in our model has the advantage of permitting a larger data set. The third IV estimate applies the liquidity adjustment to mortgage rates discussed in Section III.B and ignores prepayment options. Adjusting rates for mortgage market liquidity has a profound effect on the implied measure of property risk. Therefore, because of concerns raised earlier about risk mismeasurement, we use the liquidity-adjusted implied volatilities throughout our empirical analysis in the paper. Figure5depictsthedistributionofimpliedvolatilitiescalculatedusingliquidity adjustments (and no prepayment options). The time series mean (median) is 20% (19%) and the standard deviation is 7.5%. The time variation in IVs is pronounced and corresponds to shifts in the entire distribution, which suggests that perceived property risk changes systematically over time. It is tempting to expect this time series variation to coincide with property market cycles, but that need not be the case because risk perceptions and liquidity on the credit market also affect the 13 The risk-neutral valuation methodology is based on the assumption that contingent claims can bereplicatedthroughself-financingportfoliostrategies. Clearly,thisassumptiondoesnotholdinthe illiquidrealestatemarket. Hence,insomesense,relyingontherisk-neutralvaluationmethodologyis similartoassumingthenormalityofunobservedshocksinalinearfilteringproblem. Weacknowledgethe limitationsofthisapproachandprovidevalidityandrobustnesstests,butourIVestimateisultimatelya proxymeasurefor,ratherthananexactidentificationof,thepropertyriskperceivedbylenders. 16

Figure 4. Sample means of implied volatility estimates over time Thisfigureshowsthecross-sectionalmeansoftheestimatedmodel-impliedvolatilitiesofcommercialreal estatemortgageloansinoursampleovertime. Thesamplecontainsfixed-rate, single-propertyloans, withdebtyieldsover7%anddebtservicecoverageratiosover1.25,securitizedinnon-agencycommercial mortgage-backedsecurities. Theimpliedvolatilitiesareestimatedusingthetwo-factormodeldescribed in Appendix B. There are three batches of estimates: a baseline batch without prepayment penalties ormarketliquidityadjustment,abatchwithprepaymentpenalties,andabatchwithmarketliquidity adjustment. ThemarketliquidityadjustmentprocessisexplainedinSectionIII.B. 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 Baseline Prepayment Liquidity property market equilibrium. For example, during times of low perceived risk and high liquidity in credit markets, more properties meet lenders’ and borrowers’ criteria for financing. Hence, the effect of credit market cycles is also reflected in CRE loan terms and, ultimately, in our volatility estimates. Indeed, the period with the lowest average IV is 2003–07, which coincides with the period of the greatest number of CMBS loan originations (Figure 2) and liquid credit markets. Meanwhile, IVs in 2008–10 are likely biased downward because lenders extended credit only to the safest properties as credit markets dried up (Figure 3). By contrast, the highest IVs come from 2001 and 2017–19, which are periods characterized by relative liquidity in credit markets. Such high IVs are a function of higher-than-average perceived property risk in the aggregate as well as an increased willingness by lenders and borrowers to finance riskier assets. 17

Figure 5. Sample quartiles of implied volatilities over time Thisfigureshowsthecross-sectionalquartilesoftheestimatedmodel-impliedvolatilitiesofthecommercial realestatemortgageloansinoursampleovertime. Thesamplecontainsfixed-rate,single-propertyloans, withdebtyieldsover7%anddebtservicecoverageratiosover1.25,securitizedinnon-agencycommercial mortgage-backedsecurities. Theimpliedvolatilitiesareestimatedusingthetwo-factormodeldescribedin AppendixB,applyingthemarketliquidityadjustmentexplainedinSectionIII.B.Forlackofobservations, thequartilescannotbeestimatedin2009,whenthecommercialrealestatemortgagemarketdriedup. 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 1st Quartile Median 3st Quartile A. Structural determinants of implied volatility One potential critique of our use of implied volatility as a proxy for perceived property risk is the claim that lenders were risk insensitive when setting CRE loan spreads pre-GFC. In particular, our IVs might capture something other than property risk in the run-up to the GFC. For instance, pressure to originate for fees during the height of CDO issuances could have spurred competition for originating CMBS loans, resulting in exceptionally low mortgage rates, which do not accurately reflect the true risk of the underlying properties. We address this critique, and validate the conjecture that implied volatility is related to structural determinants of property risk, by investigating the preand post-GFC drivers of IV and verifying whether relevant macroeconomic and property-level risk indicators contributed similarly to risk perceptions over time. Table 4 examines the pre- and post-crisis relationship between IV (the dependent 18

variable) and various structural variables, such as state GDP, real estate sector GDP, unemployment rate, and income per capita as well as property size and age. Property and interacted state and time fixed effects are included. Property age, state GDP, and state employment rates are positively correlated with risk, consistent with the findings in Fisher et al. (2022) that urban density is associated with higher property market risk. Controlling for these variables, we find that property size and state income levels are negatively related to risk. Importantly, almost every coefficient that is significant post-GFC is also significant pre-GFC and has the same sign. If anything, the marginal effects of these variables on IV are stronger before the crisis than after it. Table 4—Marginal effects of structural variables on implied volatility (%) Thistableshowstheestimatedmarginaleffectsofrelevantlocalmacroeconomicandproperty-specific variables on the estimated model-implied volatilities of commercial real estate mortgage loans in our sample. Thesampleconsistsoffixed-rate,single-propertyloans,withdebtyieldsover7%anddebtservice coverageratiosover1.25,securitizedinnon-agencycommercialmortgage-backedsecurities. Themarginal effectsareestimatedonsubsamplesbeforeandaftertheGlobalFinancialCrisis(GFC),usingalinear regression model with the logarithm of implied volatility as dependent variable. The model includes loanoriginator,propertystate,andpropertytype-quarteroforiginationfixedeffects. Standarderrors aredoubleclusteredbystateandquarter. Theimpliedvolatilitiesareestimatedusingthetwo-factor model described in Appendix B, applying the market liquidity adjustment explained in Section III.B. LocalmacroeconomicvariablesaremeasuredataquarterlyfrequencyandobtainedfromtheBureauof EconomicAnalysis. “GDPinsector”standsforthegrossdomesticproductoftherealestateindustry. Pre-GFC Post-GFC 100 × Log of state real GDP (USD mm) 0.008∗ 0.009∗∗ 100 × Log of state real GDP in sector (USD mm) 0.012 −0.002 100 × Log of state income per capita (USD) −0.045∗∗ −0.027∗∗∗ State unemployment rate (%) −0.128 −0.126∗∗∗ Property Age (years) 0.022∗∗∗ 0.008∗∗∗ 100 × Log of property size (sqft) −0.010∗∗∗ −0.004∗∗∗ 100 × Log of property size (units) −0.110∗∗∗ −0.043∗∗∗ Number of observations 25,493 15,869 ∗ p<0.1,∗∗ p<0.05,∗∗∗ p<0.01 Additional analysis, not reported here, suggests no significant difference across IVs based on whether CRE loans were issued by large U.S. banks, smaller U.S. banks, foreign banks, nonbank lenders, or ex-post acquired or failed lenders. Overall, we find no empirical evidence that IVs were decoupled from property fundamentals before the GFC, as compared to the post-GFC period. 19

V. Loan-to-value ratios and risk perceptions Consistent with a tradeoff theory of optimal firm leverage, Figure 6 shows that LTV exhibits a strong inverse relationship with implied volatility. Indeed, a linear regression of IV on LTV yields a far superior fit than a regression on the other two common CRE mortgage metrics (i.e., DSCR and debt yield). At any level of perceived property risk, it appears that LTVs were generally highest in Epoch 1 (2000–04). A notable exception is low-risk loans (with IVs below 15%), for which the most aggressive period was Epoch 2 (2005–07), when the issuance of CDOs became prevalent. Interestingly, the same epoch has relatively conservative LTVs for high-risk loans (with IVs above 20%), while the risk-retention Epoch 4 features slightly higher LTVs than Epoch 3 (2008–15). Figure 6. Mean loan-to-value ratios across implied volatility bins and epochs This figure shows the sample means of the loan-to-value ratios (LTVs) of commercial real estate loans that fall into a given integer bin of model-implied volatility and were originated in a given time period (epoch). The sample contains fixed-rate, single-property loans, with debt yields over 7% and debt service coverage ratios over 1.25, securitized in non-agency commercial mortgage-backed securities. EpochchoiceisexplainedinSectionII.C.TheimpliedvolatilitiesareestimatedusingthetwofactormodeldescribedinAppendixB,applyingthemarketliquidityadjustmentexplainedinSectionIII.B. 0.8 0.7 0.6 0.5 0.4 0.3 0.10 0.15 0.20 0.25 0.30 0.35 Implied Volatility 2000-04 2005-07 2008-15 2016-20 It is important to emphasize that the inverse relationship in Figure 6 is not tautological. In a frictionless setting, a so-called Modigliani-Miller world, LTV 20

would be arbitrary, and plotting LTVs against IVs would yield no (or a random) pattern. By contrast, the theories of credit rationing (Jaffee and Russell, 1976; Leland and Pyle, 1977) and tax benefit–bankruptcy tradeoff (Leland, 1994) predict a downward-sloping relationship, which we observe empirically. Figure 6 suggests that there is time variation in leverage choice, even after we control for perceived property risk. This variation may arise simply from shifts in the credit rationing frontier, but it may also be due to differences in cap rates or other, potentially unobserved, property characteristics. In principle, any variable that affects optimal firm leverage can confound the observed relationship between LTVandIV.Suchvariablesmayincludethelocalcreditenvironment, themarginal tax rate of investors, and capital expenditure expectations not reflected in cap rates. In our analysis, we control for such factors and examine their effect on LTV to gain insight into the evolution of the supply and demand of CRE credit. A. Credit rationing frontier estimation and diagnostics In this subsection, we estimate the credit rationing frontier and examine changes in it over time. Since applying a rationing frontier results in a truncated distribution of observed LTVs, the observed mean of LTVs moves monotonically with the truncation point, as long as the distribution of borrower demand is unchanged. Therefore, a natural question is how much variation in LTV is driven by changes in the frontier. Our conjecture is that more aggressive lending practices would primarily be manifested as increases in the rationing frontier. We define and estimate the rationing frontier as a function of IV, which shows the maximum amount of leverage that lenders are willing to accept at a certain level of perceived property risk. Given the scarcity of observations at the extremes of the distribution, we estimate the frontier only for IVs between 5% and 40%. The existence of a rationing frontier is clearly visible (see Figure F3 in the Appendix), with clustering at around 80% LTV for IVs between 5% and 20%. Using a quantile regression, we estimate the frontier as the 95th percentile of LTV within each 1 percent IV interval (“IV bin”) for each of the four epochs.14 14 Inunreportednon-parametricanalysis,weverifythattherationingfrontierestimateisrobustto thechoiceofthespecificLTVpercentile. Furthermore,the95th percentilepassesdensitydiscontinuity testsandfallswithintheclusterofobservationslyingonthefrontierinallfourepochs. 21

Figure 7 shows the rationing frontier estimates as a function of IV by epoch. The 2005–07 epoch stands out in being generally lower than the others. Using the standard error estimates from our quantile regression, a pairwise mean comparison across rationing frontier estimates across epochs (see Table 5) shows that the frontier for this period is, on average, about 4 percentage points lower than the frontier for the 2000–04 epoch, and about 2 to 3 percentage points lower than the frontiers after the GFC. This result is contrary to the common belief that lending standards were lax in the period leading up to the crisis. Figure 7. Credit rationing frontier estimates by epoch This figure shows our credit rationing frontier estimates across periods (epochs). The frontiers are estimatedbyfittingaquantileregressionmodelforthe95thpercentileoftheloan-to-valueratios(LTVs)of commercialrealestateloansthatfallintoagivenintegerbinofmodel-impliedvolatilityandwereoriginated inagivenepoch. Theestimationsamplecontainsfixed-rate,single-propertyloans,withdebtyieldsover 7%anddebtservicecoverageratiosover1.25,securitizedinnon-agencycommercialmortgage-backed securities. EpochchoiceisexplainedinSectionII.C.TheimpliedvolatilitiesareestimatedusingthetwofactormodeldescribedinAppendixB,applyingthemarketliquidityadjustmentexplainedinSectionIII.B. 0.8 0.7 0.6 0.5 LTV0.4 0.3 0.2 0.1 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Implied Volatility 2000-04 2005-07 2008-15 2016-20 Overall, our analysis of rationing frontiers does not support a narrative that CRE lending was more aggressive in the 2005–07 epoch. One possible resolution for the paradox of low rationing frontier in this epoch is that perceived property risk was systematically biased downward. Such misperceptions may have led to excessive supply of credit (e.g., an 80% LTV loan for a property deemed 22

Table 5—Testing mean differences between rationing frontier estimates across epochs Thistableshowstheresultsofstatisticallytestingmeandifferencesbetweenrationingfrontierestimates across different periods (epochs). The frontiers are estimated by fitting a quantile regression model forthe95th percentileoftheloan-to-valueratiosofcommercialrealestateloansthatfallintoagiven integer model-implied volatility bin and were originated in a given epoch. The estimation sample containsfixed-rate,single-propertyloans,withdebtyieldsover7%anddebtservicecoverageratiosover 1.25, securitized in non-agency commercial mortgage-backed securities. Epoch choice is explained in SectionII.C.Theimpliedvolatilitiesareestimatedusingthetwo-factormodeldescribedinAppendixB, applying the market liquidity adjustment explained in Section III.B. From left to right, the columns showthemeandifferences(Diff.),theirstandarderrors(Std. err.),t-statistics,andcorrespondingp-values. Frontier pair Diff. Std. err. t-stat p-value 2005–2007 vs. 2000–2004 −0.041 0.0022 −18.64 0.0000 2008–2015 vs. 2000–2004 −0.012 0.0016 −7.86 0.0000 2016–2020 vs. 2000–2004 −0.025 0.0016 −15.22 0.0000 2008–2015 vs. 2005–2007 0.029 0.0023 12.36 0.0000 2016–2020 vs. 2005–2007 0.017 0.0023 7.09 0.0000 2016–2020 vs. 2008–2015 −0.012 0.0018 −6.86 0.0000 to have a volatility of 13% when its volatility is in fact 21%). An alternative explanation is that loan originators were less risk sensitive when setting mortgage rates because property risk would affect tranches that were subsequently placed in CDOs. Although plausible at face value, this alternative is not supported by our empirical analysis of IV determinants in Section IV, leaving open the question why credit rationing was tighter in 2005–07. B. Quantification of loan-to-value ratio determinants Although we find no empirical support for a laxer credit rationing frontier before the crisis, it is still useful to examine if shifts in the frontier, which may be partly driven by imprudent lending practices, have a substantive effect on the distribution of LTVs. Indeed, based on the analysis of the rationing frontier, one might argue that the 2000–04 epoch was characterized by lending that was too permissive. In this section, we analyze the influence of such frontier shifts and, more broadly, investigate how much of the variation in LTV can be explained by changes in risk perceptions and property fundamentals. For our LTV analysis, we introduce the following notation. The demand for credit by the borrower of loan i is LTV . The amount of credit that is observed to i be extended is cLTV = min{R(c ,b(IV )),LTV }, where R(c,k) is the rationing i i i i 23

frontierinEpochcandimpliedvolatilitybink, asidentifiedintheprevioussection. We fit a censored linear regression (tobit) model to cLTV of the form: (cid:2) (cid:8) cLTV =max 0,min R(c ,b(IV )), i i i (2) µ +µ +o +α(c )IV +β CRS +β CRS 2+mmm γγγ +ε (cid:9)(cid:3) = type s,q i i i 1 i 2 i q i (cid:2) (cid:8) (cid:9)(cid:3) =max 0,min R(c ,b(IV )),x β +ε , i i i i where µ are property type fixed effects, µ are quarterly time fixed effects and type s,q state/county fixed effects, o are originator fixed effects, IV is implied volatility i (with epoch-specific coefficients), CRS is the cap rate spread over the 10-year U.S. Treasury yield, and ε ∼ N(0,σ2) is the model noise term.15 Additionally, ε mmm is a vector of quarterly macro-level variables that we include in model q specifications without time fixed effects. We present the model coefficient estimates in Table 6, which indicate a strong negative relationship between LTV and IV. Notably, strictly on its own, and with a fixed slope coefficient, IV explains two-thirds of LTV variation over time. When we control for time and property type fixed effects, 2005–07 emerges as the epoch with the highest sensitivity to IV. This result is also robust to the presence of fixed effects for the 111 originators in our sample, which suggests that the idiosyncratic characteristics of individual originators—including those that potentially overstate property fundamentals, as documented in Griffin and Priest (2023)—do not drive the estimated relationship between LTV and IV. The coefficient of the cap rate spread is significant and takes the negative sign predicted in Leland (1994). The CMBS yield spread of short-term AAA CMBS bonds over U.S. Treasury securities, which measures illiquidity in the mortgage market, accounts for a large portion of the time series variation in the data. Construed as a cost of financing,marketilliquidityshouldnegativelyimpactthechoiceofoptimalleverage, consistent with the sign of its estimated coefficient. Using the model coefficient estimates, we conduct a counterfactual analysis, investigating the effect of LTV determinants over time. In particular, we examine 15 Thecapratespreadoverthe10-yearyieldisameasureofthecapratenetoftherisk-freerate. Leland(1994)showsthatoptimalLTVshoulddeclinewithcaprate. Theintuitionisthat,underthe risk-neutralmeasure,allassetsgrowatthesamerate(therisk-freerate),soanassetthatreinvestsincome willgrowmorethananassetthatdistributesincome. Correspondingly,aslower-growingassetismore likelytodefaultatloanmaturity. Weusenetcapratebecauseinterestratesexperiencedaseculardecline between2000and2020,accompaniedbyacommensuratelydecliningpropertycaprate. 24

Table 6—Estimation results of censored linear regression for the loan-to-value ratio ThistableshowstheestimationresultsofthecensoredlinearregressionmodeldefinedinEquation(2). Theestimationsamplecontainsfixed-rate,single-propertyloans,withdebtyieldsover7%anddebtservice coverageratiosover1.25,securitizedinnon-agencycommercialmortgage-backedsecurities. Thedependent variableistheloan-to-valueratio,andIVstandsforthemodel-impliedvolatilityestimateforsampleloans. Theimpliedvolatilitiesareestimatedusingthetwo-factormodeldescribedinAppendixB,applyingthe marketliquidityadjustmentexplainedinSectionIII.B.Thecolumnsshowdifferentmodelspecifications withanexpandingsetofexplanatoryvariablesandfixedeffectsincluded. Standarderrorsareclustered bythequarterofloanoriginationandreportedunderthecorrespondingcoefficientestimatesinparentheses. (1) (2) (3) (4) (5) (6) (7) (8) IV -1.28 (0.05) 2000–2004#IV -1.23 -1.21 -1.46 -1.40 -1.39 -1.38 -1.16 (0.05) (0.05) (0.07) (0.06) (0.06) (0.06) (0.04) 2005–2007#IV -1.37 -1.32 -1.68 -1.64 -1.61 -1.64 -1.42 (0.05) (0.07) (0.03) (0.03) (0.03) (0.03) (0.06) 2008–2015#IV -1.37 -1.34 -1.16 -1.29 -1.27 -1.26 -1.17 (0.05) (0.06) (0.03) (0.03) (0.03) (0.03) (0.04) 2016–2020#IV -1.27 -1.25 -1.23 -1.31 -1.29 -1.28 -1.25 (0.04) (0.05) (0.03) (0.03) (0.03) (0.03) (0.04) Capratespread 0.15 -0.07 -0.20 -0.40 -0.43 -0.14 (0.19) (0.13) (0.12) (0.12) (0.12) (0.16) CMBSyieldspread -4.62 (0.73) UST10yryield 0.31 (0.37) Time(quarterly) x x x Propertytype x x x x x Loanoriginator x x x x Propertystate x Propertystate×Time x Propertycounty x GeneralizedR2 0.61 0.62 0.63 0.71 0.73 0.74 0.76 0.71 Numberofobservations 45,917 45,917 45,917 45,917 45,917 45,878 45,878 45,779 the effect of shifts in the rationing frontier across epochs on LTVs. To this end, we estimate counterfactual LTVs, denoted as cLTV∗, setting certain independent variables in the model constant over time. Formally, we estimate (cid:110) (cid:111) (3) cLTV∗ = E max (cid:2) 0,min (cid:8) R(c∗,b(IV∗)),LTV∗(cid:9)(cid:3) | cLTV ,βββ ˆ ,σˆ2,xxx∗ , i i i i ε i where cLTV is the observable, cLTV∗ is the censored counterfactual, and LTV∗ is the latent counterfactual LTV of the loan, βββ ˆ and σˆ2 are the vector of coefficient ε estimates and the noise variance estimate from the 8th model specification in Table 6, and xxx∗ is the vector of independent variable values we use for a specific i 25

counterfactual scenario. Depending on the observed relation of cLTV and the rationing frontier R, the expression in Equation (3) becomes  (4) cLTV∗ = max (cid:2) 0,min (cid:8) R i ∗,cLTV i +(xxx∗ i βββ ˆ −xxx i βββ ˆ ) (cid:9)(cid:3) if cLTV i < R i , i E (cid:110) max (cid:2) 0,min (cid:8) R∗,LTV∗(cid:9)(cid:3) (cid:111) if cLTV >= R , i i i i where R = R(c ,b(IV )) is the original rationing frontier, R∗ = R(c∗,b(IV∗)) i i i i i i is the rationing frontier in the counterfactual scenario, and LTV∗ is the latent i counterfactual LTV of the loan, which follows the truncated normal distribution N (xxx∗βββ ˆ ,σˆ2) with lower bound R +(xxx∗βββ ˆ −xxxβββ ˆ ). TR i ε i i i Figure 8 shows the mean LTV in each year of the original data set (Panel A) and for various counterfactual data sets (Panels B through F). The original data clearly exhibit a secular decline of LTVs over the sample period. However, this trend disappears in Panel B, using a counterfactual data set where IV for each loan is fixed at its sample mean (20%). Further setting the cap rate spread to its sample mean of 3.7% (Panel C) does not make much difference (consistent with the estimate in Column 8 of Figure 8). Fixing the CMBS yield spread does appear to reduce the time series variation (Panel D), while fixing the U.S. Treasury yield has little effect (Panel E). Importantly, fixing the rationing frontier to correspond to the first epoch (2000–04) also has little effect on the time series means (Panel F). Recalling that the first epoch features the most permissive rationing frontier and thesecondepochtheleast, onemightexpectalargechangeinLTVsinthe2005–07 epoch when moving from Panel E to Panel F. Our analysis suggests that shifts in the rationing frontier have little effect on the distribution of LTVs. 26

Figure 8. Means of actual and counterfactual loan-to-value ratios over time Thisfigureshowsthemeansoftheactual(PanelA)andcounterfactual(restofthepanels)loan-to-value ratiosofcommercialleanestateloansinthesampleovertime,with99%confidenceintervals. Thesample containsfixed-rate,single-propertyloans,withdebtyieldsover7%anddebtservicecoverageratiosover 1.25,securitizedinnon-agencycommercialmortgage-backedsecurities. Thecounterfactualloan-to-value ratiosareestimatedbyapplyingEquation(3)andusingthe8th censoredlinearmodelspecificationin Table 6. Panels B to F incrementally fix the values of various explanatory variables. UST stands for the10-yearzero-couponU.S.Treasuryyield,whileIVandCRSstandformodel-impliedvolatilityand capitalizationratespreadovertheUST,respectively. (a) Actual sample means 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 (b) Setting IVs to 20% Mean LTVs with 99% CIs (Unconditional) 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 Mean LTVs with 99% CIs (Conditional on IV = 20%) (c) Setting IVs to 20% and CRSs to 370bp 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 (d) Setting IVs to 20%, CRSs to 370bp, and CMBS spread to 120bp Mean LTVs with 99% CIs (Conditional on IV = 20% and CRS = 370bp) 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 Mean LTVs with 99% CIs (Conditional on IV = 20%, CRS = 370bp, and CMBS = 120bp) (e) Setting IVs to 20%, CRSs to 370bp, CMBS spread to 120bp, and UST to 3.2% 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 (f) Setting IVs to 20%, CRSs to 370bp, CMBSspreadto120bp,USTto3.2%,andfixing rationing frontier at Epoch 1 level Mean LTVs with 99% CIs (Conditional on IV = 20%, CRS = 370bp, CMBS = 120bp, UST10 = 3.2%) 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 Mean LTVs with 99% CIs (Conditional on IV = 20%, CRS = 370bp, CMBS = 120bp, UST10 = 3.2%, and Static Frontier) 27

Table 7 compares the annual counterfactual LTV means across the four epochs and shows that they are statistically distinct. This result means that there is still statistically significant residual variation across epochs, even after controlling for IV, cap rate, CMBS spread, U.S. Treasury yield, and changes in the rationing frontier. That said, the residual time variation does not seem to coincide with macro events and may rather correspond to shifts in the demand for CRE loans. Moreover, the residual differences in mean LTV are economically small: less than three percentage points across all three epochs in Column 6 of Table 7. Notably, the residual mean differences between Epochs 1 and 4 as well as the difference between Epochs 2 and 3 are less than one percentage point. Table 7—Actual and counterfactual means of loan-to-value ratios across epochs Thistableshowsthemeansoftheactual(Column1)andcounterfactual(restofthecolumns)loan-to-value ratios of commercial lean estate loans in the sample across periods (epochs). The sample contains fixed-rate, single-property loans, with debt yields over 7% and debt service coverage ratios over 1.25, securitizedinnon-agencycommercialmortgage-backedsecurities. Thecounterfactualloan-to-valueratios areestimatedbyapplyingEquation(3)andusingthe8th censoredlinearmodelspecificationinTable6. Eachregression,(2)-(6)incrementallyfixesthevaluesofvariousexplanatoryvariables. US10standsfor the10-yearzero-couponU.S.Treasuryyield,whileIVandCRSstandformodel-impliedvolatilityand capitalization rate spread over the US10, respectively. CMBS stands for the market liquidity spread definedinSectionIII.B.Robuststandarderrorsarereportedinparentheses. Atthebottom,F-statistics arereportedforthejointmeanequalitytestsacrossEpochs1to4andEpochs2to4,respectively. Epoch (1) (2) (3) (4) (5) (6) 2000–2004 68.8 69.4 69.8 69.7 69.1 69.1 (0.1) (0.1) (0.1) (0.1) (0.1) (0.1) 2005–2007 68.7 67.1 68.1 67.1 66.5 66.8 (0.1) (0.0) (0.0) (0.0) (0.0) (0.0) 2008–2015 66.8 66.2 66.6 67.3 67.6 67.8 (0.1) (0.1) (0.1) (0.1) (0.1) (0.1) 2016–2020 62.8 67.6 68.2 67.8 68.1 68.4 (0.1) (0.1) (0.1) (0.1) (0.1) (0.1) IV=20% x x x x x CRS=370bp x x x x CMBS=120bp x x x US10=3.2% x x FrontiersettoEpoch1level x WaldFStat. of1–4equality 587.7 430.4 438.9 403.2 368.1 300.1 WaldFStat. of2–4equality 711.4 109.1 224.1 20.9 175.9 160.9 Numberofobservations 47,616 45,779 45,779 45,779 45,779 45,779 It is clear from the generalized R2 coefficient in Table 6, as well as the counterfactual LTVs in Figure 8, that the first-order determinant of LTV in 28

the sample is perceived property risk. We assess the individual contribution of explanatory variables to the cross-sectional variation of LTV∗ by sequentially decomposing its sample variance into components of the following form: (cid:16) (cid:17) Cov LTV∗,LTV∗| −LTV∗| (5) VC n,t = t t (x1,...,xn−1)=(x¯1 (cid:0) ,...,x¯n−1 (cid:1) ) t (x1,...,xn)=(x¯1,...,x¯n) , Var LTV∗ t where LTV∗| denotes the counterfactual latent LTVs in year t, t (x1,...,xn)=(x¯1,...,x¯n) obtainedbyfixingvariablesx tox attheirsamplemeans.16 Accordingly,thefirst 1 n variance component is defined as (cid:16) (cid:17) Cov LTV t ∗,LTV t ∗−LTV t ∗|x1=x¯1 (5a) VC 1,t = (cid:0) (cid:1) , Var LTV∗ t which simplifies to Var(β x )/Var(LTV∗), which in turn is akin to the R2 1 1 coefficient of a simple linear regression of LTV∗ on x . Analogously, the residual 1 variance component is defined as (cid:16) (cid:17) Cov LTV∗,LTV∗| (5b) VC ε,t = t t (cid:0) (x1,...,x (cid:1) n)=(x¯1,...,x¯n) , Var LTV∗ t which is the part of variance that cannot be explained with variables x ,...,x . 1 n InPanelAofFigure9, wepresentavariancedecompositioninwhichx stands 1 for IV and x corresponds to all remaining explanatory variables. IV explains 2 between 40% and 65% of LTV variance in any given year. The incremental contribution of all other explanatory variables to LTV variance is less than 20% and averages around 10%. Of these variables, cap rate spread, originator fixed effects, and geographic fixed effects are the most important determinants of LTV. We examine their variance contributions in Panel B of Figure 9, which are most pronounced before the GFC. While originator fixed effects play a relatively more importantrolebeforethecrisis, theircontributiondeclinessubstantiallyin2005-07. 16Inessence,wedecomposeLTVvariationusingatelescopicsumofcounterfactualestimates. Notably, thisdecompositiondoesnotensurethatallvariancecomponents(VC)arenon-negative. Importantly,the orderofthecomponentsdoesnotchangetheirvaluesbecauseofthelinearcoreofourtobitmodel. 29

VI. Conclusion Theory (e.g., Leland, 1994) suggests that while the choice of optimal firm leverage depends on both asset and market factors, one of the most important factors is the risk of the underlying asset. In our empirical anlysis, we demonstrate that the single most important determinant of LTVs in securitized CRE loans is perceived property risk, as measured by implied volatility. Indeed, by itself, IV explains about two-thirds of the variation in LTV. We confirm that other market-level and asset-specific fundamentals also drive observed choices of LTVs, albeit explaining less than an additional 10% of LTV variation. Although LTVs have gradually declined throughout our sample period from 2000 to 2020, the decline disappears once we control for these fundamental factors. The residual variation does not appear to reflect any market trends. This finding is relevant because LTV is commonly seen as an important measure of lending standards and often used in financial regulations (e.g., DiSalvo and Johnston, 2018). We find some evidence that LTV contains information about lending standards through shifting maximum LTV criteria (i.e., credit rationing frontiers). However, the shifts we identify do not support the narrative that lending standards were lax in the run-up to the GFC. Moreover, we find that changes in frontiers have had little effect on the distribution of LTVs over time. Finally, we find that IV was lower in the years before the GFC than at any other period in the past two decades. A plausible interpretation of this result is that lenders and borrowers systematically underestimated property risk in this period, leading to excessive CRE lending and exacerbating the subsequent market downturn. Totheextentthatexanteriskperceptionsdetermineexpostloanperformance, our work motivates the use of measures of market-specific risk perceptions, such as implied volatility, by both investors and policymakers. 30

Figure 9. Model-based variance decomposition of loan-to-value ratios over time This figure shows the model-based variance decomposition of the loan-to-value ratios (LTVs) of commercialleanestateloansinthesampleovertime. Thesamplecontainsfixed-rate,single-property loans, with debt yields over 7% and debt service coverage ratios over 1.25, securitized in non-agency commercialmortgage-backedsecurities. Thevariancedecompositionappliesthemethodologydescribed inthetext(seeEquation(5)),measuringthecontributionofeachvariabletothevarianceofLTVinthe contextofthecensoredlinearmodeldefinedinEquation(2). Morespecifically,LTVismodeledusingthe 8th censoredlinearmodelspecificationpresentedinTable6. PanelAdecomposesthevarianceofLTV into the contribution from implied volatility, other model variables, and residual variation. Panel B presentsthevariancecontributionofthethreemostrelevantmodelvariablesotherthanimpliedvolatility. (a) Variance attributable to implied volatility and other model variables 100 80 60 40 20 0 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 Variance Decompositon of LTVs 1) Contribution of Implied Volatility Variable 2) Incremental Contribution of Other Model Variables 3) Residual Variation (b) Variance attributable to selected other model variables 10 8 6 4 2 0 -2 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 7102 8102 9102 0202 Variance Decompositon of LTVs (Contribution of Other Relevant Model Variables) 1) Caprate Spread 2) Loan Originator 3) Property Location 31

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Appendix A: Data construction process The complete data set consists of 111,465 loans. • We remove loans in deals originated by Freddie Mac or Fannie Mae, many of which finance affordable housing, senior housing, and other subsidized projects. The pricing of such loans may not fully reflect property risk perceptions in the commercial real estate mortgage market, partly because of the embedded guarantee, which our valuation model does not accommodate. • We remove loans with missing key variables such as the dates of origination and maturity, coupon rate, loan amount, net operating income (NOI), and loan-to-value ratio (LTV). We also drop loans with unrealistic features.17 • We only keep loans with debt service coverage ratios above 1.25 and debt yields above 7%. Our valuation model assumes that dividends, relative to property value, are constant over the lifetime of the loan. In reality, some loans finance renovations or other projects that result in NOI increases over time. Such loans may have projected cash shortfalls at origination, which would be inconsistent with our structural approach. • Wehaveanumberofloansforwhichthecallprotectionlengthsplusseasoning do not add up to the loan term. We have 326 such loans that undershoot the loan term and 6,908 that overshoot the loan term. When they undershoot, we extend the last call protection period. When they overshoot, we start subtracting from the last call protection type, then the second to last, and so on. For 3,506 loans, the last call protection type is completely removed. • Weremoveasmallnumberofloansthathavebothdefeasanceandyieldmaintenance penalties, more than three call protection options, or ambiguous call protection designations, which our valuation model does not accommodate. • We drop 1,850 multi-property and pari passu loans, which our valuation model does not accommodate. 17 Werequirecouponratestobebetween1%and25%,LTVstobeatmost90%,theloantermtobe atmost12years,andNOInottoexceedtheloanamount(i.e.,thedebtyieldtobeatmost100%). 35

Appendix B: Implied volatility model B1. Interest rate process Gupta and Subrahmanyam (2005) hold a horse race among several pricing models and find that the accuracy of one-factor models is comparable with that of other,morecomplicated,models. Weusetwoofthemodelstheyexamine: theHull and White (1990) (HW) and the Black and Karasinski (1991) (BK) models, which are among the most commonly used term structure models in practice. We modify both the HW and BK models so that no more than one tree branch can be above 10% or below zero.18 This way, we ensure that risk neutral probabilities for the property price model are positive at property diffusion volatilities as low as 3%. We note that, during our sample period, forward rates for a one-year zero-coupon U.S. Treasury bond never exceed 7.5% or fall below 0%. Hence, the bounds we impose likely reflect market perceptions for the possible range of interest rates over the lifetime of the mortgages in our data set. We calibrate each month’s term structure model using yield data obtained for nominal zero-coupon bonds with maturities ranging from one to twelve years.19 We obtain data on swaptions with exercise maturity of one year (the most liquid contracts) from Eikon for tenors (underlying swap maturities) of 1, 5, and 10 years. Each month, we fit a HW and a BK model to the data and select the one that best fits the swaptions data.20 Table B1 summarizes the percentage price accuracy across the fitted monthly term structure models. Our fitted term structure models are generally accurate. The BK model seems to be the better performer in roughly 2/3 of cases, and nearly exclusively so between 2007 and 2015. 18 WemodifytheHWandBKmodelsasfollows. IftheconventionalHWorBKtreeisconsistent withthebounds,weemployit. Otherwise,wetruncateallbranchesbeyondthefirstonethatcrossesthe boundbysettingtheirtransitionprobabilitiestozero. Atanynodeforwhichbranchprobabilitiesareset tozero,wesolvefortheremainingbranchprobabilitiesbyenforcingthenode’sexpectedinterestrate toequalthequantityimpliedbytheunderlyingmean-revertingprocess. Generally, theresultingrate volatilityatedgenodesdiffersfromtheconstantvolatilitypresentelsewhereinthetree. 19 We obtain data on yields from the Federal Reserve: https://www.federalreserve.gov/data/ nominal-yield-curve.htm. Fortheshortendofthetermstructure,weusethethree-monthU.S.Treasury constantmaturityyieldscalculatedbytheFederalReserveofSt. Louis. 20BoththeHWandBKmodelscanbemadetofitanyarbitrarytermstructureofzero-couponbonds. 36

Table B1—Term Structure Model Precision This table shows the accuracy of 277 “best-fitting” term structure models, estimated for each month fromJune1997toJune2020. ThetwomodelsusedforestimationaretheHullandWhite(1990)and theBlackandKarasinski(1991)models. Eachdatapointconstitutestherootoftheweightedmeanof squaredpricingerrors(i.e.,percentageaccuracy)fromasinglemonth’stermstructuremodel. Statistic Model precision Mean 0.0202 SD 0.0439 P1 0.0001 P5 0.0003 P10 0.0007 P25 0.0015 P50 0.0042 P75 0.0149 P90 0.0534 P95 0.1146 P99 0.2322 B2. Property value process We model property values using a binomial process similar to that in Cox, Ross and Rubinstein (1979), but modified to incorporate time-varying short-term interest rates as well as a “catastrophic drop” in value that triggers immediate default. The latter modification is motivated by the actual distribution of creditor losses. Without the possibility of a sudden drop (discontinuous downward jump) in property value, the optimal exercise of the default option tends to predict relatively small loan losses to what we observe in practice. The catastrophic property-level event is Poisson distributed and arrives with annual intensity of λ. In our model, the event permanently reduces the property’s value to zero, and the rate λ is calibrated to our loan pool to match historical commercial real estate (CRE) loss given default (LGD) rates of 30–35%.21 Let σ be the annual volatility of the property’s value. We incorporate catastrophic hazard in the model of Cox, Ross and Rubinstein (1979) by dividing 21 ThisrangeisbasedonestimatesbyEsaki,L’HeureuxandSnyderman(1999),Ciochetti(1997),and Curry,BlalockandCole(1991). Onecouldpotentiallymodeladistributionofcatastrophiclosses,butit isuncleartowhatextentthiswouldmakeourresultsmorereflectiveoflenderperceptionsversusafixed LGDvalue. SeeAppendixDforfurtherelaborationonmethodology. 37

the usual “up” and “down” states by (1−λ∆t) for each ∆t time increment: √ √ eσ ∆t e−σ ∆t u = d = . 1−λ∆t 1−λ∆t The property value changes by a factor of u or d. This specification keeps the expected price appreciation of the property, under the risk-neutral measure, independent of the value of λ and thus independent of the idiosyncratic event. It also has the virtue of setting the Arrow-Debreu prices of “up” and “down” non-disaster states equal to (1−λ∆t) times their usual values in the Cox model: (u) er k,t ∆t−d(1−λ∆t) (d) u(1−λ∆t)−er k,t ∆t π = π = , j,t,k u−d j,t,k u−d where we denote property state j, time t, and continuously compounded short interest rate state k obtained from the term structure model. We assume that a one-period binomial “up” or “down” move in the property price process is uncorrelated with the one-period short-term interest rate process. CRE properties generate income for their owners, which we incorporate by assumingthatthepropertypaysaconstantannual“dividend”rateδ corresponding t to the property’s net cash-flow-to-value ratio (NCF) at the time of mortgage origination.22 We include the dividend in our property value process, except for the time of origination, when the property’s value is equal to the appraised value. We define “up” and “down” cum-dividend property value V for all j,t+1 non-origination periods t ∈ {1,...,T −1} recursively as follows: V = uV (1−δ ∆t) V = dV (1−δ ∆t). ju,t+1 j,t t j d ,t+1 j,t t B3. Valuation of commercial real estate mortgages We employ a structural mortgage valuation model to estimate the property’s annual volatility parameter σ, which is otherwise not observable. Mortgage terms comprise the LTV (or, equivalently, the amount borrowed), the time to maturity, and the amortization schedule. Together with a complete specification of the property and interest rate model parameters, the mortgage terms imply a fair 22 Inthenumerator,weuseNCFinsteadofNOIbecausenetcashflowtakescapitalexpendituresand capitalexpenditurereservesintoaccountandisarguablyabettermeasureofpropertycashflows. 38

market mortgage rate that can be calculated by setting the present value of the mortgage obligation equal to the amount borrowed. Using the model, we solve for the implied volatility that sets the present value of the mortgage obligation to the amount borrowed given the observed mortgage rate. We denote property value V (which is V because property value is j,t,k j,t independent of interest rate movement), as well as corresponding equity and debt values, E and D , respectively. We allow for interest-only or amortizing j,t,k j,t,k mortgage paymentschedules (ora combination of these). We denotethe remaining mortgagebalanceB andthefixedmortgagepaymentorcouponc . AsinCox,Ross t t and Rubinstein (1979), we specify our model working backwards from maturity. Similar to Merton (1974), we define borrower equity and debt at maturity T: E = max(0, V −(B +c )) j,T,k j,T T T D = min(V , B +c ). j,T,k j,T T T These formulas capture the notion of “mechanical” default: the borrower defaults if the property value falls below the debt value. Notably, the Modigliani– Miller value additivity holds: D +E = V , as we prove in Appendix C, j,t,k j,t,k j,t,k whichimpliesthatthereisnodead-weightcosttodefault.23 Thereisnoprepayment or dividend payment at maturity. Consequently, for each non-maturity and non-origination period t ∈ {1,...,T −1}, the following equations determine the borrower’s value of equity and debt: (cid:18) (cid:19) (cid:104) (cid:105) E = max 0, δ ∆tV −c + (cid:0) e−r k,t ∆t(cid:1) E E˜ , V −c −B −P j,t,k t j,t,k t t+1 j,t,k t t j,t,k j,t,k (cid:18) (cid:19) (cid:104) (cid:105) D = min V , c + (cid:0) e−r k,t ∆t(cid:1) E D˜ , c +B +P , j,t,k j,t,k t t+1 t t j,t,k j,t,k where E [E˜ ] and E [D˜ ] respectively stand for the risk-neutral expected j,t,k t+1 j,t,k t+1 values for equity and debt, and P is the prepayment penalty. The terms in j,t,k each equation represent values for default, continuation, and prepayment options. 23 We opt to ignore the dead-weight cost of default because we do not correspondingly model tax benefitsofdebtorinvestorswithheterogeneousprivatevalues. Ifmortgagedebtcameonlywithcosts andnobenefits,thennorationalinvestorwouldfinanceapropertywithmortgagedebt. 39

For simplicity, we define the risk-neutral expected values of X ∈ {E,D}: (cid:104) (cid:105) (cid:104) (cid:105) E X˜ = π (u) i (u) X +i (m) X +i (d) X + t+1 j,t,k k,t ju,t+1,ku k,t ju,t+1,k k,t ju,t+1,k d j,t,k (cid:104) (cid:105) (d) (u) (m) (d) π i X +i X +i X , j,t,k k,t j d ,t+1,ku k,t j d ,t+1,k k,t j d ,t+1,k d (u) (m) (d) with i , i , and i being interest rate up, middle, and down state probabilities, k,t k,t k,t respectively. At origination (i.e., t = 0), the values of equity and debt are equal to their respective continuation values, E 0 = (e−r0∆t)E 0,0,0 [E˜ t+1 ] and D 0 = (e−r0∆t)E 0,0,0 [D˜ t+1 ], based on the assumption that there are no coupon payments or dividends at origination. Given the other model parameters, we estimate the property’s annual implied volatility by computing the volatility parameter σ that sets D equal to the contractual loan amount. 0 Prepayment rules are specified in the mortgage covenant and usually vary by mortgage period. For instance, a common feature is a prepayment lockout of several months until prepayment is allowed, followed by a lengthy period when prepaymentisallowedbutwithpenalties(usuallydefeasanceoryieldmaintenance), followed by another, shorter period when prepayment is allowed without penalties (the “open” prepayment period). These periods sum to the length of the mortgage. Accordingly, as described Appendix E in more detail, we model prepayments as follows: weremovetheprepaymentoptionduringthelockoutperiod,thenexplicitly model defeasance or yield maintenance during penalty period, and finally set P j,t,k equal to zero during the open prepayment period. 40

Appendix C: Proof of Modigliani–Miller additivity We would like to show that the Modigliani–Miller additivity E +D = j,t,k j,t,k V holds for all t ∈ {0,...,T}. j,t,k Part 1 of Proof We begin by demonstrating that V j,t,k = δ t ∆tV j,t,k + (e−r k,t ∆t)E j,t,k [V˜ t+1 ] for t ∈ {1,...,T −1}. We redefine the following (note that V = V since the j,t j,t,k property value is independent of interest rate movements):  V ju,t+1,k = uV j,t,k (1−δ t ∆t)  for t ∈ {1,...,T −1}. V = dV (1−δ ∆t)  j ,t+1,k j,t,k t d (cid:104) (cid:105) We use these to show that V j,t,k = δ t ∆tV j,t,k + (cid:0) e−r k,t ∆t(cid:1) E j,t,k V˜ t+1 : (cid:104) (cid:105) δ ∆tV + (cid:0) e−r k,t ∆t(cid:1) E V˜ t j,t,k t+1 j,t,k (cid:104) (cid:105) = δ ∆tV + (cid:0) e−r k,t ∆t(cid:1) π (u) V + π (d) V t j,t,k j,t,k ju,t+1,k j,t,k j d ,t+1,k (cid:104) (cid:105) = δ ∆tV + (cid:0) e−r k,t ∆t(cid:1) π (u) uV (1−δ ∆t) + π (d) dV (1−δ ∆t) t j,t,k j,t,k j,t,k t j,t,k j,t,k t    (cid:16) (cid:17) = V δ ∆t + (1−δ ∆t) (cid:0) e−r k,t ∆t(cid:1) π (u) u + π (d) d  j,t,k  t t j,t,k j,t,k    (cid:124) (cid:123)(cid:122) (cid:125) =erk,t∆t (NoArbitrage) = V (cid:2) δ ∆t + (1−δ ∆t) (cid:0) e−r k,t ∆t)(er k,t ∆t(cid:1)(cid:3) j,t,k t t = V [δ ∆t + 1−δ ∆t] = V . j,t,k t t j,t,k Part 2 of Proof At maturity T, the Modigliani–Miller additivity clearly holds because: E = max(0, V −(B +c )), j,T,k j,T T T D = min(V , B +c ). j,T,k j,T T T 41

We show that M&M holds at any arbitrary time t ∈ {1,...,T −1}. Assuming E +D = V (true for t+1 = T) and using induction, we get: j,t+1,k j,t+1,k j,t+1,k (cid:18) (cid:19) (cid:104) (cid:105) E = max 0,δ ∆tV −c + (cid:0) e−r k,t ∆t(cid:1) E E˜ ,V −c −B −P , j,t,k t j,t,k t t+1 j,t,k t t j,t,k j,t,k (cid:18) (cid:19) (cid:104) (cid:105) D = min V , c + (cid:0) e−r k,t ∆t(cid:1) E D˜ , c +B +P . j,t,k j,t,k t t+1 t t j,t,k j,t,k By hypothesis, E +D = V . Therefore, the continuation state j,t+1,k j,t+1,k j,t+1,k value of date t equity equals: (cid:104) (cid:105) δ ∆tV −c + (cid:0) e−r k,t ∆t(cid:1) E E˜ t j,t,k t t+1 j,t,k (cid:18) (cid:19) (cid:104) (cid:105) (cid:104) (cid:105) = δ ∆tV + (cid:0) e−r k,t ∆t(cid:1) E V˜ −c − (cid:0) e−r k,t ∆t(cid:1) E D˜ t j,t,k t+1 t t+1 j,t,k j,t,k (cid:124) (cid:123)(cid:122) (cid:125) =V byResult1 j,t,k (cid:104) (cid:105) = V −c − (cid:0) e−r k,t ∆t(cid:1) E D˜ , j,t,k t t+1 j,t,k which implies that: (cid:18) (cid:19) (cid:104) (cid:105) E = max 0, V −c − (cid:0) e−r k,t ∆t(cid:1) E D˜ , V −c −B −P j,t,k j,t,k t t+1 j,t,k t t j,t,k j,t,k    (cid:104) (cid:105)  = V + max−V , −c − (cid:0) e−r k,t ∆t(cid:1) E D˜ , −c −B −P . j,t,k  j,t,k t t+1 t t j,t,k (cid:124) (cid:123)(cid:122) (cid:125) j,t,k (cid:124) (cid:123)(cid:122) (cid:125) −x j,t,k (cid:124) (cid:123)(cid:122) (cid:125) −z j,t,k −y j,t,k In conclusion: E +D = V +max(−x , −y , −z )+ j,t,k j,t,k j,t,k j,t,k j,t,k j,t,k +min(x , y , z ) = V . j,t,k j,t,k j,t,k j,t,k 42

We can easily show that the additivity holds at t = 0 as well. Taking the appraised property value at origination S , we divide by (1−δ ∆t) to get V . 0 t 0 E = (e−r0∆t) E [E˜ ], 0 t+1 0,0,0 D = (e−r0∆t) E [D˜ ], 0 t+1 0,0,0 S 0 V = ⇔ S = V −δ ∆tV . 0 0 0 t 0 (1−δ ∆t) t Usingthecontinuationvaluesofequityanddebtfort ∈ {1,...,T−1}referenced earlier and removing the dividend δ ∆tV and coupon c , we get: t 0 t E = S −(e−r0∆t) E [D˜ ], 0 0 t+1 0,0,0 E +D = S −(e−r0∆t) E [D˜ ]+(e−r0∆t) E [D˜ ] = S . 0 0 0 t+1 t+1 0 0,0,0 0,0,0 Appendix D: Loss given default We define LGD at each default node j,t,k as follows: V j,t,k LGD = 1− . j,t,k c +B t t For each loan in our sample, we obtain an expected LGD figure using 10,000 Monte Carlo simulations. We do this by generating random property and interest rate trajectories and weighting them by their respective risk-neutral probabilities. Uponreachingadefaultnode, thesimulationstopsandrecordsLGDforsimulation number i as LGD = LGD . If no default occurs, LGD = 0. With probability i j,t,k i λ∆t, a catastrophic property loss happens (the risk-neutral “up” and “down” property probabilities sum to 1 − λ∆t) and default occurs with LGD = 1. i Therefore, the expected LGD for each loan l is: eLGD = (cid:0) Σ10000LGD (cid:1) /10000. l i=1 i 43

Appendix E: Prepayment penalties We use the standard concepts of defeasance and yield maintenance, with small modifications to accommodate our term structure models. The basic principle of defeasance is that the lender charges a penalty for the spread that it loses when the borrower refinances the loan. We model this spread by using our term structure calculations to create a portfolio of risk-free assets (zero-coupon bonds, ZCB) with the same cash flows. The calculation for any interest rate state k,t is as follows: (cid:32)T−t (cid:33) (cid:88) Def = m ZCB −B , k,t t+i k,t,t+i t i=1 where m is the mortgage payment at date t+i, ZCB is the value at date t t+i k,t,t+i ofazero-couponbondwithmaturityatdatet+i,andB istheremainingmortgage t balance.24 ZCB is calculated by creating a sub-tree M of all continuation k,t,t+i states starting at node k,t of the trinomial interest rate tree. The final column of the tree, which represents time t+i, has payoffs of 1 (M = 1 ∀k). We then k,t+i+1 determine the zero-coupon bond price by iterating backwards to the original node (k,t) so that the following recursive formula holds: (cid:104) (cid:105) M = (cid:0) e−r k,t ∆t(cid:1) i (u) M +i (m) M +i (d) M . k,t k,t ku,t+1 k,t km,t+1 k,t k d ,t+1 Yield maintenance differs from defeasance in that it replaces the lost spread with a risk-free asset of the same remaining term as the original loan. We model yield maintenance by using the zero-coupon bond rates calculated above. First, we calculate the “risk-free” par bond rate for the appropriate maturity: (cid:32)T−t (cid:33) (cid:88) rf = (1−ZCB ) / ZCB . k,t k,t,T k,t,t+i i=1 24 For simplicity, we calculate defeasance and yield maintenance up to maturity T. In practice, thereisheterogeneityacrosslenders,withsomecalculatingthepenaltyuptothebeginningoftheopen prepayment period instead. Although we do not observe the exact lender method in our data, the differencesbetweenthebeginningoftheopenprepaymentperiodandmaturityaretypicallyverysmall. 44

Then, we calculate an annual “present value factor” f: (cid:16) (cid:17) f = (1−(1+rf ))−(T−t)/∆t / rf . k,t k,t k,t Finally, the yield maintenance penalty is calculated: YM = (rm−rf ) f B , k,t k,t k,t t where rm is the mortgage rate. 45

Appendix F: Additional figures and tables Table F1—Distribution of CRE Loans by Property Type ThistableshowstheabsolutefrequenciesofcommercialrealestatemortgageloansintheMorningstar datasetacrossdifferentcollateralpropertytypes,withanemphasisonthedistinctionsbetweensingle and multi-property loan frequencies. The sample contains fixed-rate loans, with debt yields over 7% anddebtservicecoverageratiosover1.25,securitizedinnon-agencycommercialmortgage-backedsecurities. Single-Property Loans Multi-Property Loans Total Hotel 4,808 194 5,002 Industrial 3,052 133 3,185 Mixed 0 174 174 Multifamily 18,972 491 19,463 Office 9,308 215 9,523 Other 5,903 331 6,234 Retail 16,084 292 16,376 Total 58,127 1,830 59,957 Table F2—Extreme Percentiles of Implied Volatility Estimates by Epoch This table shows the lowest and highest three percentiles of implied volatility estimates by period (epoch). Epoch choice is explained in Section II.C. The implied volatilities are estimated using the two-factor model described in Appendix B, applying the market liquidity adjustment explained in Section III.B. The sample contains fixed-rate, single-property loans, with debt yields over 7% anddebtservicecoverageratiosover1.25,securitizedinnon-agencycommercialmortgage-backedsecurities. P1 P2 P3 P97 P98 P99 2000–2004 0.06 0.08 0.09 0.36 0.39 0.44 2005–2007 0.08 0.09 0.10 0.32 0.34 0.39 2008–2015 0.06 0.08 0.09 0.32 0.34 0.37 2016–2020 0.10 0.12 0.13 0.36 0.38 0.40 Entire sample 0.07 0.08 0.09 0.34 0.36 0.40 46

Figure F1. Distribution of Implied Volatility by Epoch This figure shows the distribution of implied volatility estimates by period (epoch). Epoch choice is explainedinSectionII.C.Theimpliedvolatilitiesareestimatedusingthetwo-factormodeldescribedin AppendixB,applyingthemarketliquidityadjustmentexplainedinSectionIII.B.Thesamplecontains fixed-rate, single-property loans, with debt yields over 7% and debt service coverage ratios over 1.25, securitizedinnon-agencycommercialmortgage-backedsecurities. (a) 2000–2004 (b) 2005–2007 10 100 10 100 8 80 8 80 6 60 6 60 4 40 4 40 2 20 2 20 0 0 0 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Implied Volatility (%) Implied Volatility (%) (c) 2008–2015 (d) 2016–2020 10 100 10 100 8 80 8 80 6 60 6 60 4 40 4 40 2 20 2 20 0 0 0 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Implied Volatility (%) Implied Volatility (%) 47

Figure F2. Distribution of Selected CRE Loan Characteristics Thisfigureshowsthedistributionofdebtservicecoverageratios,debtyield,loan-to-valueratios,andloan termlengthsintheMorningstardatasetoffixed-rate,single-propertyloanssecuritizedinnon-agency commercialmortgage-backedsecurities. Thesampleislatercutforfurtheranalysisbyremovingdebt yieldsunder7%anddebtservicecoverageratiosunder1.25forreasonsexplainedinSectionIII.A. (a) DSCR (b) Debt Yield 30 20 15 20 10 10 5 0 0 0 1 2 3 4 5 0.00 0.05 0.10 0.15 0.20 0.25 (c) Loan-to-Value Ratio (d) Loan Term (Month) 30 80 60 20 40 10 20 0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 12 24 36 48 60 72 84 96 108 120 132 144 48

Figure F3. Association Between the LTVs and IVs of CMBS Loans Over Different Epochs Thisfigureshowstherelationshipbetweenloan-levelloan-to-valueratiosandimpliedvolatilityestimates byperiod(epoch). EpochchoiceisexplainedinSectionII.C.Theimpliedvolatilitiesareestimatedusing thetwo-factormodeldescribedinAppendixB,applyingthemarketliquidityadjustmentexplainedin Section III.B. The overlaid frontiers are estimated by fitting a quantile regression model for the 95th percentileoftheloan-to-valueratios(LTVs)ofcommercialrealestateloansthatfallintoagiveninteger bin of model-implied volatility and were originated in a given epoch. The samples contain fixed-rate, single-propertyloans,withdebtyieldsover7%anddebtservicecoverageratiosover1.25,securitizedin non-agencycommercialmortgage-backedsecurities. (a) 2000–2004 (b) 2005–2007 (c) 2008–2015 (d) 2016–2020 49

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)%( ytilitaloV deilpmI no selbairaV yrotanalpxE fo stceffE lanigraM—4F elbaT gnisu detamitse era seitilitalov deilpmi ehT .ytilitalov deilpmi egatnecrep no selbairav yrotanalpxe eseht fo stceffe lanigram eht swohs elbat sihT ,etar-dexfi niatnoc selpmas ehT .B.III noitceS ni denialpxe tnemtsujda ytidiuqil tekram eht gniylppa ,B xidneppA ni debircsed ledom rotcaf-owt eht .seitirucesdekcab-egagtromlaicremmocycnega-nonnidezitiruces,52.1revosoitaregarevocecivrestbeddna%7revosdleiytbedhtiw,snaolytreporp-elgnis selbairavesehtforehtieezilituew,noisrevnocatpmettanahtrehtaR .ratsgninroMyb”stinu“dnateeferauqsniylevitanretladerusaemsiezisytreporP dna,ylimafitlum,letohedulcxednaedulcniotsugnitavitom,sepytytreporpgnomadetalupopylsuoenegoreteheraselbairavniatreC .elbaliavasitinehw ”rotceS“.)AEB(sisylanAcimonocEfouaeruBehtmorfdeniatboeraatadtnemyolpmenudna,emocni,PDG .snmulocsuoiravehtnisepytytreporp”rehto“ .srosivdAcirtemonocEERBCmorfdeniatboeraatadetarycnacavdnaezistekramehT.AEBehtybdenfiedsa”yrtsudnietatselaer“ehtotsrefer )5( )4( )3( )2( )1( 100.0- 310.0 400.0 )mm DSU( PDG laeR etatS fo goL × 001 200.0- 600.0- 900.0 )mm DSU( rotceS ni PDG laeR etatS fo goL × 001 110.0- 020.0- ∗730.0- )DSU( atipaC rep emocnI etatS fo goL × 001 ∗∗122.0- ∗322.0- ∗∗352.0- )%( etaR tnemyolpmenU etatS ∗∗010.0 ∗010.0 ∗∗∗410.0 ∗∗∗510.0 ∗∗∗410.0 )sraeY( egA ytreporP ∗∗∗600.0- ∗∗∗700.0- ∗∗∗700.0- ∗∗∗700.0- ∗∗∗700.0- )tfqS( eziS ytreporP fo goL × 001 ∗∗∗170.0- ∗∗∗770.0- ∗∗∗770.0- )stinU( eziS ytreporP fo goL × 001 ∗∗∗920.0- ∗∗∗910.0- )%( ycnapuccO ytreporP ∗∗∗900.0 ∗∗∗010.0 )%( erahS tnaneT daeL 200.0 ∗∗∗500.0 )tfqS( eziS tekraM fo goL × 001 ∗∗∗050.0 500.0- )%( etaR ycnacaV tekraM x x x dedulcnI letoH x x x dedulcnI ylimafitluM x x x dedulcnI rehtO x x x x x rotanigirO naoL x x x etatS ytreporP x x x x x )ylretrauQ( emiT × epyT ytreporP x x )ylretrauQ( emiT × ytnuoC ytreporP 902,12 581,31 334,62 626,91 263,14 snoitavresbO fo rebmuN .noitanigiro fo retrauq eht dna etats ytreporp eht yb deretsulc elbuod era srorre dradnatS 10.0 < p ∗∗∗ ,50.0 < p ∗∗ ,1.0 < p ∗ 51

Table F5—Marginal Effects of Explanatory Variables on IV (%) This table shows the marginal effects of these explanatory variables on percentage implied volatility. The implied volatilities are estimated using the two-factor model described in Appendix B, applying the market liquidity adjustment explained in Section III.B. This sample contain fixed-rate, singleproperty loans, with debt yields over 7% and debt service coverage ratios over 1.25, securitized in non-agencycommercialmortgage-backedsecurities. Thissampleexcludeshotelandmultifamilyproperties. Pre-GFC Post-GFC Property Age (Years) 0.009∗ 0.012 100 × Log of Property Size (Sqft) -0.008∗∗∗ -0.004∗∗∗ Property Occupancy (%) -0.025∗∗∗ -0.072∗∗∗ Lead Tenant Share (%) 0.006∗∗ 0.027∗∗∗ Number of Observations 10,573 2,612 ∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01 Table F6—Marginal Effects of Explanatory Variables on IV (%) Thistableshowsthemarginaleffectsoftheseexplanatoryvariablesonpercentageimpliedvolatility. The implied volatilities are estimated using the two-factor model described in Appendix B, applying the marketliquidityadjustmentexplainedinSectionIII.B.Thissamplecontainfixed-rate,single-property loans, with debt yields over 7% and debt service coverage ratios over 1.25, securitized in non-agency commercialmortgage-backedsecurities. Thissampleexcludesthe“other”propertytype. Themarket vacancyrateisobtainedfromCBRE. Pre-GFC Post-GFC Property Age (Years) 0.010∗∗∗ 0.009∗ 100 × Log of Property Size (Sqft) -0.006∗∗∗ -0.006∗∗∗ 100 × Log of Property Size (Units) -0.062∗∗∗ -0.070∗∗∗ 100 × Log of Market Size (Sqft) 0.002 0.002 Market Vacancy Rate (%) 0.051∗ 0.050 Number of Observations 13,814 7,395 ∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01 52