Macroeconomic conditions, systematic risk factors, and the time series dynamics of commercial mortgage credit risk

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Content MACROECONOMIC CONDITIONS, SYSTEMATIC RISK FACTORS, AND THE TIME SERIES DYNAMICS OF COMMERCIAL MORTGAGE CREDIT RISK by Xudong An A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PLANNING) August 2007 Copyright 2007 Xudong An Acknowledgments To me, the happiest part of writing this dissertation is to write the thank you note, not just because it symbolizes the near completion of an endeavor, but because it brings me back sweet memories – the encouragement, guidance, support, friendship and love I’ve received from my mentors, my friends and my family. I’d like to first thank my dissertation committee members Dr. Raphael Bostic, Dr. Yongheng Deng, Dr. Stuart Gabriel, Dr. Chris Jones and Dr. Fernando Zapatero. As outside members, Fernando and Chris have provided a whole lot more help than I would expect. They have helped make my ideas clear, provided many constructive suggestions, assisted me to dig references and dataset, and even helped me prepare my presentations. Stuart has been very supportive not only on my dissertation, but also on my academic and career development. He always provides critical thinking on my work, no matter it’s my dissertation, or it’s our joint research project. Raphael is a great mentor, and a nice friend. He has shown me what is scholarship through our joint work, helped me shape my way of thinking, assisted me to develop my presentation skills and encouraged me all the way through my Ph.D. study. Of course, my greatest appreciation goes to my ii advisor Dr. Yongheng Deng, who has provided enormous guidance, encouragement and support during my years at USC. He led me to the real estate finance world, provided me opportunities to work on a dozen of research projects, helped me find my concentration, and assisted me in every step of my study. I still remember the days when Yongheng taught me hand in hand on how to run the Fortran programs under Unix, and I will be missing the hours and hours of intellectual discussions with him. Special thanks are due to Dr. Tony Sanders, who has kindly provided me the data for my dissertation study. I’d also like to thank my qualifying committee members Dr. Cheng Hsiao and Dr. Gary Painter for helping me develop my dissertation research ques- tions. I also have benefited from conversations with Paul Calem, Steve Bardzik, Sally Gordon, Weifei Li, Steve Ott, Tim Riddiough, Tony Sanders and Antonios Sangvinatsos on my dissertation work. I would also like to thank those people who have contributed in some way to the success of my academic endeavors. Giving an even modest list is an impossible mission. Some of them are Dr. John Clapp, Dr. Delores Conway, Dr. Genevieve Giuliano, Dr. Dowell Meyers, Ehud Mouchly, June Muranaka, Dr. Juliet Musso, Dr. Chris Redfearn, Dr. Tsur Somerville and Nina Tibayan. And I would like to pay my heartiest thanks to my colleagues and friends for their great help and invaluable discussions throughout my graduate study. A far from com- plete list includes Lihong Yang, Zhou Yu, Liang Wei, Duan Zhuang, Della Zheng, Xuey- ong Zhan, Huanghai Li, Mingye Zhang, Peng Fei, Weifei Li, Xingdong Zhang and Zizhe Cai. iii I would not be here without the love, devotion, and encouragement of my family. My thanks go to my wife, Haiyan Mao, who has given me the greatest love and has sacrificed a lot for me; my parents, Ziqiu An and Xiaohong Ding, who have been blessing and supporting me silently; my brother, Jianjun An, who have always encouraged me to aim high, and have provided immeasurable support along my way. I’d also like to present my dissertation as a gift to my little daughter Janet Yuyan An, who brings me joy and happiness. Finally, I’d like to thank the Lusk Center for Real Estate at USC, Real Estate Research Institute, the Urban Land Institute, Pension Real Estate Association and the Alpha Asso- ciation of Southern California for their generous funding support of my dissertation research. iv Table of Contents Acknowledgments ii List of Figures vii Abstract viii 1 Introduction 1 1.1 Credit Risk and Commercial Mortgage Default . . . . . . . . . . . . . 1 1.2 Research Questions and Approach . . . . . . . . . . . . . . . . . . . . 3 1.3 Summary of Results and Expected Contribution of the Dissertation . . . 6 2 Literature Review 11 2.1 Credit Risk Dynamics and Credit Cycles . . . . . . . . . . . . . . . . . 11 2.2 Credit Risk Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Default Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Theoretical Model 26 3.1 The First-passage Model . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 A First-Passage Model for Commercial Mortgage Default . . . . . . . . 29 3.2.1 The Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2 The Income Producing Property . . . . . . . . . . . . . . . . . 34 3.2.3 Commercial Mortgage Default Boundary . . . . . . . . . . . . 36 3.2.4 First-Passage Time Density . . . . . . . . . . . . . . . . . . . 38 3.2.5 Default Hazard Rate (Default Intensity) . . . . . . . . . . . . . 43 4 Empirical Methodology and Results 48 4.1 The State Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 The Cox Proportional Hazard Model . . . . . . . . . . . . . . . . . . . 51 4.3 Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Data and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5 Conclusions and Discussions 86 v References 89 Appendices 99 vi List of Figures 3.1 An Illustration of the First-passage Model. . . . . . . . . . . . . . . . . 27 4.1 Annual Default Rates of Commercial Mortgage Loans, 1994–2003. . . 66 4.2 Raw Plots of the Commercial Mortgage Default Seasoning. . . . . . . . 77 4.3 Estimated Seasoning Effect of Commercial Mortgage Loan Default. . . 78 4.4 Estimated Hazard Rate Time Series of a Representative Mortgage. . . . 79 4.5 Estimated Seasoning Effect for different property type Commercial Mort- gages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.6 Estimated Hazard Rate Time Series for different property type Commer- cial Mortgages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.7 The Time Series of the 5-year Pure Discount Bond Yield. This figure plots the Fama-Bliss (artificial) 5-year pure discount bond yields. . . . . 82 4.8 Simulated Relationship between Hazard Rate and the Macroeconomy. . 83 4.9 Simulated Relationship between Hazard Rate and the Property Market Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.10 Simulated Relationship between Hazard Rate and the Instantaneous Inter- est Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 vii Abstract I study the time series dynamics of commercial mortgage credit risk and the unob- servable systematic risk factors underlying those dynamics. The research is conducted within a structural model framework. I modify the first-passage model of Black and Cox (1976) and Longstaff and Schwartz (1995) to capture unique features of commercial mortgage default. First, the first-passage condition is imposed on net operating income (NOI) rather than on property value, which reflects the fact that commercial mortgages mainly rely on the income from underlying properties to service the debt. Second, equi- librium macroeconomic dynamics are linked to commercial mortgage default through the NOI function, based on the observation that commercial property cash flow is largely affected by macroeconomic conditions. Third, I solve the default hazard rate of a rep- resentative commercial mortgage as a function of two unobservable state variables, the expected macroeconomic growth and expected property market-specific growth. The solutions of the model provide an estimable system that includes the relationship between systematic risk factors and commercial mortgage credit risk, as well as the dynamics of the risk factors. viii My empirical work aims at extracting information about how the systematic risk factors drive the evolutions of commercial mortgage credit risk based on the observed default events of individual loans. I first use a Cox proportional hazard model to control the heterogeneity of commercial mortgage loans and to estimate the default hazard rate time series of a representative mortgage, which is the systematic component of default risk in the commercial mortgage market. The hazard rate time series are then used to estimate my first-passage model in state space form. I estimate the nonlinear state space model using extended Kalman filter. Results show large variations of default proba- bility over time in the commercial mortgage market, and that these variations are well explained by the two risk factors. Both factors are mean-reverting, and the elasticity of commercial mortgage default probability with respect to the expected macroeconomic growth increases when the economy deteriorates. The estimates also reveal a negative relationship between the instantaneous risk free interest rate and commercial mortgage default probability. My model can be used in predicting long term dynamics of com- mercial mortgage credit risk. ix Chapter 1 Introduction 1.1 Credit Risk and Commercial Mortgage Default As put by Caouette, Altman and Narayanan (1998, page 1), “(i)f credit can be defined as ‘nothing but the expectation of a sum of money within some limited time,’ then credit risk is the chance that this expectation will not be met.” To define it more formally, credit risk is the risk of changes in value of securities and contracts caused by “default” or “changes in credit quality of issuers or counterparties” (Duffie and Singleton 2003). The Russian bond default in August 1998 and the downgrade of GM’s and Ford’s corporate bonds to junk status in May 2005 are examples of credit risk realization. In addition to corporate bonds, secured or non-secured loans bear credit risk. Commercial mortgages are secured loans to finance commercial real estate investments. Typically, when an investor buys a commercial property such as an apartment complex, a retail center or an office building, he or she (the borrower) gets a loan from a commercial 1 bank or a mortgage bank (the lender), and the property is used as collateral for the loan. In case of default, the lender can seize the collateral and partially recover payment loss. Although secured with the property, the loan may still be defaulted during the period of repayment if the borrower finds it beneficial to do so, e.g. when the value of the property is less than the value of the remaining mortgage payment. In fact, commercial mortgages have high default risk. The annual default rate of a commercial mortgage is usually between 0.1 to 2.4 percent, which is about 5 to 6 times as high as residential mortgage default rate and is comparable to BB corporate bond default rate. The loss ratio (loss given default) is typically between 40 and 60 percent. As a result, commercial mortgages usually have credit premiums of 200 to 300 basis points (bps). The past decades’ strong growth of commercial mortgage-backed securities (CMBS) market 1 reflects the needs of credit risk management for commercial mortgages. CMBS typically have the senior-subordinated structure, in which various seniority CMBS bonds represent different rights to receive cash flows from the underlying commercial mortgage pool. Prepayments of principal are often distributed first to the senior tranches while losses due to default are allocated first to the subordinated tranches. The central idea of this structure is to protect senior tranche investors from commercial mortgage default risk and to compensate the subordinated bond investors with high expected return for taking extraordinary default risk. 1 The annual CMBS issuance grows from $42 billion in 1995 to $412 billion in 2005. 2 Therefore, no matter in commercial mortgage loan origination, or in the sale of com- mercial mortgage loans to CMBS pools, or in the issuance and investment of CMBS bonds, understanding the default risk of commercial mortgages is the key to success. 1.2 Research Questions and Approach Credit risk of corporate bonds or commercial mortgages has two dimensions. One is the cross sectional dimension – it varies across different firms or borrowers. The other dimension of credit risk is the time series dynamics – it changes over time. A “good” credit may become “bad” when the market environment deteriorates, and the aggregate default rates in the market fluctuate over time. Understanding the time series dynamics of credit risk is equally, if not more, important than knowing the cross sectional variations in credit risk. For example, portfolio risk managers want to have the ability of distinguishing “good” times from “bad” times in addition to telling “good” assets from “bad” assets; regulators and policy makers desire to know how fluctuations of credit risk in the financial markets are related to changes in macroeconomic conditions and whether there are credit cycles, so that they can take policy instruments to stabilize the market; and financial economists seek to incorporate 3 the dynamics of systematic risk factors into their credit risk pricing and measurement models 2 . In this dissertation, I study the time series dynamics of commercial mortgage credit risk. My research questions are: 1) How does credit risk in the commercial mortgage market fluctuate over time? Specifically, how can we quantify the changes of default probabilities that are due to the changes of macroeconomic environment and commercial real estate market conditions? 2) How are the credit risk changes driven by systematic risk factors? Alternatively, what’s the relationship between the systematic component of commercial mortgage default probabilities and systematic risk factors, and how do the risk factors evolve over time? The research tasks are carried out within a structural model framework. I generally follow the first-passage model framework of Black and Cox (1976) and Longstaff and Schwartz (1995), which assumes that a firm defaults once the firm asset value falls below a certain threshold. Given the incomes of commercial properties sustaining commercial mortgages are largely affected by the macro economy, I modify the first-passage model of Black and Cox (1976) and Longstaff and Schwartz (1995) to incorporate equilib- rium macroeconomic dynamics. Aggregate economic output and commercial property cash flows are the primitive processes in my model. The economy has some business 2 See Wilson (1998), Jarrow and Turnbull (2000), Allen and Saunders (2003), Koopman and Lucas (2005), and Duffie, Wang and Saita (2006) for elaborations. 4 cycle characteristics, as modeled in Goldstein and Zapatero (1996), in that the expected growth rate of aggregate output is mean-reverting. I model the commercial property cash flow as determined by two unobservable variables, the state of the economy and a commercial property market-specific factor, similar to the approach by Tang and Yan (2006) in modelling firm cash flow. Commercial mortgage default occurs when the cash flow of the underlying property falls below an exogeneously given default boundary. I solve the first-passage time density and default hazard rate of a representative commer- cial mortgage, which gives the relationship between mortgage credit risk and the two unobservable state variables. My empirical work takes advantage of a large commercial mortgage performance dataset available and focuses on default probabilities. The main objective is to extract information about how the systematic risk factors drive the evolutions of commercial mortgage credit risk based on the observed default events of individual loans. I first use a Cox proportional hazard model to control the heterogeneity of commercial mortgage loans and to estimate the default hazard rate time series of a representative mortgage, which can be understand as the systematic component of default risk in the commercial mortgage market. The hazard rate time series are then used to estimate the first-passage model in state space form, in which the risk free bond yield and the mortgage default hazard rate are the two variables treated as observed. I estimate the nonlinear state space model using extended Kalman filter. Parameter estimates reveal the relationships 5 between default probabilities and the risk factors, and state variable estimates capture the dynamics of the unobservable systematic risk factors. 1.3 Summary of Results and Expected Contribution of the Dissertation Empirical results show substantial variations in commercial mortgage credit risk over time. The default hazard rate in “bad” years is 14 times as high as that in “good” years. However, the variations are largely explained by the dynamics of the two mean- reverting unobservable risk factors. The expected economic growth is a significant factor with long term mean of 1.85%, volatility of 2.05% and annual mean-reversion speed of 0.33. In addition, there is a significant commercial property market-specific factor with long term growth rate of 1.50%, volatility of 2.08% and mean-reversion speed of 0.59. Changes of these two factors explain over 80 percent of the variations in default hazard rates. Simulations show that commercial mortgage default risk becomes more sensitive when the economy turns down. Based on the model estimates, the long term mean of annual default hazard rate is 0.01% for a newly originated loan and 2.31% for a five-year seasoned loan. Several other findings verify the conventional wisdom. For example, empirical estimates reveal that the insolvency threshold for default is 0.86 6 assuming an underwriting debt-service-coverage ratio (DSCR) of 1.3. This estimate is consistent with the notion that mortgage borrowers do not default ruthlessly possibly because of high transaction cost. Moreover, significant seasoning effect has been found. My estimates show that default hazard rate reaches its peak after four and a half years of loan origination. The main contribution of this research is to exploit the time series perspective of credit risk in the commercial mortgage market. This is largely motivated by recent discussions of dynamic credit risk modelling. For example, Wilson (1998) stresses the importance of understanding how credit risk changes over business cycles. Jarrow and Turnbull (2000) argue for incorporating dynamics of “common economic influence” into credit risk pricing models. Allen and Saunders (2003) suggest capital regulators to consider the banking pro-cyclicality. Recently, Koopman and Lucas (2005) and Pesaran, et al (2005) have developed time series models to study credit risk cycles. Duffie, Wang and Saita (2006) estimate corporate default risk dynamics, incorporating autoregressive macroeconomic and firm-specific covariates. My research follows this line of thinking, but uses a structural model to examine the time series dynamics of credit risk. The esti- mated model is useful in predicting long term dynamics of credit risk. It also provides hedging implications for portfolio credit risk. Moreover, pricing of commercial mort- gages can be conducted with the estimated default probabilities and the transformation of real measure into risk neutral measure included in my structural model. 7 The mortgage default risk literature has long incorporated interest rate and property market growth rate as systematic factors in the mortgage valuation and default prediction models (See, e.g. Kau et al 1987, Titman and Torous 1989, and Deng, Quigley and Van Order 2000). However, the emphasis has not been put on the time series properties of default risk. Further, there is a gap between theoretical default risk models and empirical default estimations. On one hand, theoretical pricing models incorporate the dynamics of these risk factors, but on the other hand empirical estimations only apply reduced- form approach to link these factors to default risk. Here I bridge the gap by solving a structural model and keep the structure of the model in my empirical estimation. The dynamics of the risk factors is also estimated within my integrated model. In my theoretical model, I make substantial modifications to the original first-passage model to reflect unique features of commercial mortgage default. First, the first-passage condition is imposed on the NOI of the commercial property rather than the value of the property. This is to reflect the fact that commercial mortgages mainly rely on the income from the property to make monthly payment, and empirically the debt service coverage condition is found to be the most relevant for default 3 . Second, since the income of commercial properties are largely affected by the macroeconomic conditions, I incorporate equilibrium macroeconomic dynamics into my model and build the link between macroeconomic conditions and commercial mortgage default. Finally, given 3 See Moody’s 2005 structured finance special report “US CMBS: DSCR Migration and Contempora- neous Probability of Default”. 8 default hazard rate has been an important variable in the mortgage default risk literature, and that I have real world data on individual loan default events, I solve my model for commercial mortgage default hazard rate in closed-form and estimate my model based on this solution. Another important feature of the current study is that it studies unobservable risk fac- tors. Recent research has found that there are a small number of latent variables playing fundamental roles in the economy, and that many of the observable variables are deter- mined by these latent factors (Bai and Ng 2004). In fact, economic theories have long found it useful to explain observed economic data by some fundamental unobservable factors 4 . My empirical approach is similar to that of Duffee (1999), who uses extended Kalman Filter to estimate the default intensity process of corporate debt from corporate and trea- sury bond yields. The main difference is that Duffee (1999) uses a reduced-form credit risk pricing model, which directly assumes the default intensity to follow an exoge- nously given process. I pay more attention to the economics of default by using a struc- tural model. 4 For example, the arbitrage pricing theory (APT) of Ross (1976) assumes the existence of a set of theoretically unidentified common factors underlying all asset returns. Most term structure models of interest rate including Vasicek (1977) and Cox, Ingsoll and Ross (1985) assume an unobservable “short” rate. In fact, Stambaugh (1988) derives the conditions under which an affine term structure model implies a latent-variable structure for bond returns. 9 While the research mainly helps us understand the long term dynamics of systematic credit risk, it also considers the interactive impact of idiosyncratic risk factors and sys- tematic risk factors on individual commercial mortgage loan default. The Cox propor- tional hazard model used in the study helps identify the effects of loan specific charac- teristics on default. For example, estimates show substantial variations in default hazard rate across geographic regions; maturity and amortization terms are found to be nega- tively correlated with default hazard rate; and certain property types such as hotel and healthcare show significantly elevated default risk. These estimates add to the existing literature additional information on cross sectional properties of default risk. They are useful in industrial applications. 10 Chapter 2 Literature Review My research questions are largely motivated by recent discussions in the literature about the time series dynamics of credit risk in corporate debt and bank loans. I merge the two branches of credit risk literature, credit risk pricing studies and default prediction studies, in forming an integrated default risk model for commercial mortgages. 2.1 Credit Risk Dynamics and Credit Cycles How credit risk changes over time has been an important topic in corporate credit risk literature. Early research focuses on the correlations between changes in macroeco- nomic variables and changes of corporate default rates or credit spreads. For example, Altman (1990) regresses changes of US business failure rates on changes of macroe- conomic variables such as real GNP, money supply and stock market index, and new business formation. He finds a negative relationship between changes in those variables 11 and changes in the aggregate business failure rate. Longstaff and Schwartz (1995) and Duffee (1998) examine how changes in treasury yields affect changes in corporate credit spreads also using linear regressions. Both studies find a negative relationship. Later, researchers explore the characteristics of credit risk changes, especially the cycli- cal patterns. The emphasis is on the credit risk series itself, rather than the correlation of credit risk and macroeconomic variables as in Altman (1990), Longstaff and Schwartz (1995) and Duffee (1998). For example, Carey (1998) documents significant differences in default rates for “good” years, as compared to “bad” years. Erlenmaier and Gersbach (2001) find both the levels and standard deviations of default rates vary throughout the business cycle. Wilson (1998), in calculating portfolio loss distribution, stresses the importance of cyclical default. He sees the portfolio default risk as mainly determined by “systematic risk factors” and loss distributions are “driven by the state of the econ- omy”. He also argues that, for credit risk management at portfolio level, “the traditional binary classification of credits into ‘good’ credits and ‘bad’ credits is not sufficient all credits can potentially become ‘bad’ over time given a particular economic scenario.” Nickell, Perraudin and Varotto (2000) also consider the fact that credit risk could be very different for different “regimes”. Therefore, they use GDP growth to classify indi- vidual quarters as high-, medium-, or low-growth periods and compute separate default and rating transition probabilities for each of these regimes. Bangia et al (2002) take a similar approach, although they use NBER’s classification of months as recessions and expansions rather than GDP growth to define different “regimes”. Blume, Keim and 12 Patel (1991) and Blume and Keim (1991) conjecture that the seasoning (aging) effect found in corporate default might be a manifestation of business cycle effect. Berger and Udell (2002) test the “institutional memory hypothesis” of bank lending and sug- gest that credit risk of bank loans is amplified by banks’ loosing standard of lending during economic expansion. Allen and Saunders (2003) provide a comprehensive sur- vey of cyclical effects of default probability, of loss given default and of exposure at default, which includes other empirical evidence that cyclical default are linked to busi- ness cycles. They further argue that capital regulations must take into account the cycli- cal fluctuations in credit risk to avoid amplification of the pro-cyclicality of credit risk. Kent and D’Arcy (2001) identify four major credit cycles in Australia over the past 150 years, with the most recent one in early 1990s. Credit cycles are featured with large increase and then steep fall in the ratios of credit to GDP, as well as financial system instability, like bank failures and substantial losses of banks. In addition, they analyze the relations between credit cycles and business cycles and find them to be intertwined and that credit cycles amplify business cycles. Based on these observations, researchers have recently developed models to try to cap- ture the time series dynamics of credit risk. Gordy (2003) presents a theoretical risk- factor model for bank portfolio default risk in the spirit of the APT model of Ross (1976). Building on Gordy’s theoretical framework, Koopman, Lucas and Klaassen (2005) analyze US business failure rates using an unobserved components time series model. In their model, default rate is a function of an unobservable ”surplus variable, 13 and the “surplus variable” is composed of a trend and a cycle depicted by a sin – cos function. Using US business failure rates during 1927 and 1997, they show evidence of 10-year cycles for default rates. This study is among the first to formally model time series properties of credit risk. Using a similar modeling approach, Koopman and Lucas (2005) analyze cycles in credit spreads and business failures, as well as their potential interactions with general business cycles. A 6-year short term credit cycle is identified, which is not significantly related to general business cycle. A longer period cycle of around 11 years is also found, which has a close relationship with general business cycle. Pesaran, et al (2005) develop a vector autoregressive model to study credit risk profile of commercial banks and macroeconomic dynamics. Hose and V ogl (2005) directly esti- mate an autoregressive model with exogenous macroeconomic input for default proba- bilities. They argue for the importance of taking credit cycle into account when forecast- ing default probabilities. Duffie, Wang and Saita (2006) call for the needs of a dynamic approach of credit risk modeling. The argument is: in order to make multi-period ahead predictions of corporate default, we need to know not only the relationship between credit risk and risk factors, but also the dynamics of risk factors themselves. They esti- mate a hazard model for conditional probabilities of corporate default, incorporating autoregressive macroeconomic and firm-specific covariates. They show that their model outperforms other static models in out-of-sample predictions. 14 In the mortgage default risk literature, limited work has been done on the time series dimension of credit risk. Although time-varying interest rate and property value appre- ciation have long been incorporated as systematic risk factors in the mortgage default risk models, the emphasis of those studies is not on the time series properties of default risk. As will be discussed in the next section, Cunningham and Hendershott (1984), Kau et al (1987), Titman and Torous (1989), Schwartz and Torous (1992), Childs, Ott and Riddiough (1996) and others develop structural models for mortgage default risk. In their models, mortgages are contingent claims on interest rate and property value and the Merton (1974) model is applied to price mortgage default risk. Model implied mortgage rates are compared to those observed in the marketplace. The purpose of those studies is on the ability of the pricing models to explain observed credit premium. On the other hand, Vandell et al (1993), Deng, Quigley and Van Order (2000) and Ambrose and Sanders (2002) apply econometric models to analyze the relationship between default probabilities and variables related to interest rate and property value growth, such as the equity position and prepayment incentive. The focus is on how different values of those variables explain the cross sectional differences in borrower default behaviors. Nang Neo and Ong (2003) and Tompaidis and Tsyplakov (2006) use regression analysis to study the determinants of mortgage default rates and credit premium respectively. The emphasis is on the correlation between credit risk and macroeconomic variables. Rid- diough (2004) runs regression of CMBS AAA and BBB bond credit premium on vari- ables representing CMBS characteristics and macroeconomic environments, and further examine the autocorrelations of the residual from the regression. Partly based on the 15 autocorrelations he find in the regression residuals, he argues that CMBS issuers and investors is “learning by doing” in setting up CMBS prices. Therefore, I take the time series perspective to study changes of credit risk in the com- mercial mortgage market and the underlying factors governing those changes. It follows the corporate and bank loan credit risk literature and fill the gap in the mortgage default risk literature. 2.2 Credit Risk Pricing A vast literature has been developed for credit risk pricing, which includes two strands of models - the structural model and the reduced-form model. The structural models emphasize the reason why firms default. The Merton (1974) type model assumes debt holders default when asset values fall under certain default thresholds at debt maturity date. Then, the equity is similar to a call option in the sense if asset value is higher than the debt value at debt maturity date, the equity holder can ”call” the firm with a strike price of the debt, while if the firm value is less than the debt value, the equity holder will not exercise the ”call option” resulting with an zero value in equity. With a specification of asset value process, and by applying the Merton-Black-Scholes option pricing technique, the equity value can be easily found out, and the debt value is just 16 the asset value minus the equity value. The original Merton (1974) model specifies a constant volatility diffusion process for asset value and assumes interest rate is constant. He gives a closed form solution of the term structure of pure discount bond yield, and shows that the value of the debt is an increasing function of the firm market value and promised debt payment at maturity, and a decreasing function of the time to maturity, the risk-free interest rate and the volatility of firm value. The Merton (1974) model assumes that default only happens at debt maturity. However in real world, corporate bond, bank loans and mortgages can default at any time before maturity. Black and Cox (1976) relax this assumption and introduce the first-passage model, in which default happens at the first time the asset value falls under a thresh- old (default boundary). Given an asset value process and a default boundary, it is easy to find out the probabilities of asset value hitting the default boundary at each point in time, as well as the debt payment outcome at various states, and the debt value is just the expected discount value of the payments under risk neutral measure 1 . Black and Cox (1976) give a closed-form solution of discount bond based on the assumptions that asset value follows a square root diffusion process, interest rate is constant and default bound- ary is a pre-specified time varying function. Longstaff and Schwartz (1995) generalize the first-passage model of Black and Cox (1976) to allow stochastic risk free interest rate, and Leland (1994) endogenizes the default boundary based on the argument that 1 Since the key of this approach is to find out the first passage time distribution, Black and Cox (1976) call this a probabilistic approach. 17 firm equity holder chooses the optimal default policy to maximize its equity position. Other extensions of the structural model include: Kim, Ramaswamy and Sundaresan (1993) change the firm default trigger from asset value to cash flows and applies a CIR model for the risk-free term structure, Leland and Toft (1996), Anderson and Sundare- san (1996) model firm bankruptcy as the result of an extensive form game in which debtholder and equity holder behave non-cooperatively, and they study the implications of their model on debt pricing and debt contract design. Zhou (1997) model the firm value to follow a diffusion process with jumps, Duffie and Lando (2001) incorporate bond investors’ imperfect information about firm asset value in the endogenous default credit risk pricing model of Leland (1994) and Leland and Toft (1996) to explain the observed high credit premium near the lower end of the term structure, Collin-Dufresne and Goldstein (2001) consider firms mean-reverting leverage ratio policy and that firms with good credit quality are likely to issue more debt resulting in high yield spreads of their bonds. Huang and Huang (2003) consider a credit risk model with a counter- cyclical market risk premium to capture the effects of business cycles on credit risk premium. Eom, Helwege and Huang (2004) conduct a comprehensive study on the abil- ity of various structural models in explaining observed corporate bond yield spreads and find the results to be mixed. In a structural model, default probabilities are determined within the model in the sense that it is derived based on the assumed state variable processes and the default condi- tions. In contrast, the reduced form approach on credit risk directly assumes that there 18 exists an exogenously given default intensity. For example, Jarrow and Turnbull (1995) assume default follows a Poission process to arrive at a constant rate at each point in time 2 . The advantage of this approach is that it greatly simplifies the debt valuation. Lando (1998) shows that with this approach, the zero-recovery defaultable bond price is just the expected discount payoff with a discount rate ofr+¸ instead of the risk free rate r, where¸ is the default intensity under risk neutral measure. Madan and Unal (1996) generalize the Jarrow and Turnbull (1995) model to allow the default hazard rate to be a parametric function of the firm equity level. Jarrow, Lando and Turnbull (1997) model default as the first time a continuous time Markov chain with K states hits the absorbing state (default state). Duffie and Singleton (1998) introduce a slightly different type of reduced form model where default intensity follows an exogenously given stochastic process and show that a defaultable bond can be priced by replacing the usual short term interest rate r with a default adjusted short-rate r +hL as the discount rate, where h is the default intensity and L is loss given default under risk neutral measure. Further extensions on reduced form credit risk pricing include Jarrow and Turnbull (2000) and Madan and Unal (2000). Duffee (1999) empirical studies the ability of the reduced form model in fitting observed corporate credit spreads. He assumes a firm’s instantaneous probability of default to follow a translated square root diffusion process and is corre- lated with defaulr free interest rate. He uses the historical firm credit spreads as well as the treasury bond yields to empirically estimate the risk free short rate process and default processes with extended Kalman filter. He finds that the model has a reasonable 2 Equivalent to a constant hazard rate and an exponentially distributed duration time. 19 fitting for corporate bond yields and the parameter estimates captures the features of the term structure of yield spreads. Comparing to the structural model, the reduced form models pay less attention to the economics of why firms or borrowers default. The mortgage credit risk valuation literature generally follows the Merton (1974) model. For example, Cunningham and Hendershott (1984) value mortgage default insurance using the contingent claims pricing approach. Assuming a diffusion process for house value, they construct the Merton-Black-Scholes pricing PDE, specify the boundary con- ditions, and use numerical methods to obtain the fair premiums for different FHA insur- ance contracts. Kau et al (1987) generalize the model to have two factors, the stochastic property value and risk-free interest rate. Titman and Torous (1989) and Schwartz and Torous (1992) take the same approach to study whether the options pricing model with some reasonable parameter values generates default premiums consistent with those observed in the marketplace. Further generalizations of mortgage default risk models include Giliberto and Ling (1992), Childs, Otts and Riddiough (1996), Capozza, Kazar- ian and Thomson (1998), and Downing, Stanton and Wallace (2005), among many oth- ers. Recently, Kau, Keenan and Smurov (2006) apply the reduced-from model of credit risk to value residential mortgages with default (and prepayment) risks. They assume an exogenously given default intensity and use particle filter to estimate the process from 20 mortgage performance data, and then calibrate their model to observed mortgage rates 3 . In fact, Riddiough and Thompson (1993) take a similar reduced form approach of spec- ifying exogenously given default probabilities for their commercial mortgage pricing model, but they use deterministic rather than stochastic functions. My study builds on the first-passage model of Black and Cox (1976) and Longstaff and Schwartz (1995). It pays close attention to the economics of why commercial mortgage borrowers default. However, different from most structural models that use a calibration approach for empirical investigations, I follow Duffee (1999) to empirically estimate my structural model using extended Kalman filter. In addition, I take advantage of the data availability to estimate my model based on relatively large number of real default events observed in the marketplace, rather than to make inference based on the pricing (credit premium) information. 2.3 Default Prediction Predicting default probabilities is important. On one hand, value of the debt is deter- mined by the probability of default together with loss given default. On the other hand, 3 Given the reduced form modeling nature, this approach has the difficulty of adjusting the real default probabilities estimated from default data into risk neutral probabilities for pricing. 21 creditors and rating agencies make decisions of loan origination and bond rating directly based on predicted default probabilities. A wide variety of econometric models have been used in the literature for default probability predictions. Early research such as Altman (1968) and Ohlson (1980) use dscriminant analysis and probit model to form the “Z-score” and “O-score” based on accounting ratios such as working capital to total asset, retained earnings to total asset, earnings to total asset, market equity to book value of debt and sales to total asset. The “Z-score” and “O-score” later become widely accepted financial distress measures. Later studies such as Martin (1977), West (1985) and Smith and Lawrence (1995) use logit models to empirically investigate the factors affecting bank failure and loan default. In a logit model, the firms’ default and non-default decisions are seen as discrete choices associated with certain utility functions determined by some risk factors. Lennox (1999) compares discrete choice models with the discriminant analysis and conclude that a well-specified probit or logit model has better performance in out-of-sample predictions. Recently, Campbell, Hischer and Szilagyi (2006) also apply a logit model to construct a measure of financial distress to study stock returns. Recent research on bankruptcy predictions also applies duration analysis, which treats the time from loan origination to loan default (duration) as a random variable and usually makes econometric inference based on the assumptions of the distribution of duration. 22 For example, Lee and Urrutia (1996) use a duration model based on a Weibull distribu- tion of failure time. McDonald and Van de Gucht (1999) apply a special type of duration models, the Cox proportional hazard model, which assumes the firm faces a hazard of ending its life (go bankruptcy) at each point in time and the that the hazard is determined by the duration of the loan as well as some covariates (exogenous variables). Empirical estimates tell relationship between default hazard rate and the risk factors. Shumway (2001) also estimates a hazard model of bankruptcy with both accounting ratios and “market-driven” variables and finds the time-varying “market-driven” variables to play important roles in bankruptcy predictions. In a hazard model, default path dependency is explicitly modeled given the hazard rate is defined as unit time conditional default prob- ability during [t, t+s] given the firm has not defaulted upon time t. Shumway (2001) argues that the hazard model produces more accurate predictions than the static logit model does 4 . Mortgage default prediction studies have been explosive in the past decades possibly because of the wide availability of mortgage performance data. Early studies apply lin- ear regression or logit models to estimate default determinants, and only use snapshot variables such as borrower characteristics, origination loan-to-value ratio (LTV) or con- temporaneous LTV , etc. For example, von Furstenberg (1969, 1970a, 1970b) developed 4 Many more studies are reviewed in Altman, Resti and Sironi (2004) and Duffie, Wang and Saita (2006). 23 the first academic default risk model, which show that home equity at the time of origina- tion was an important predictor of residential mortgage default. Jackson and Kaserman (1980), Campbell and Dietrich (1983) and Vandell and Thibodeau (1985) hypothesize borrowers’ equity maximization and use multinomial logit models to test their hypoth- esis with both FHA and conventional loan data. They find that equity position (con- temporaneous LTV) has the dominant effect on mortgage default. Viewing default as a put option, Foster and Van Order (1984, 1985), Quigley and Van Order (1991) use linear regressions with aggregate mortgage default and loss rates to test implications of the put option proposition, and find that default is significantly related to put option ”in the money”. Archer et al (2002) and Ambrose and Sanders (2002) apply a multinomial logit model to commercial mortgage default (Other examples include Herzog and Ear- ley 1970, von Furstenberg and Green 1974, Williams, Beranek and Kenkel 1974, Sandor and Sosin 1975, Follain and Struyk 1977, Vandell 1978, Webb 1982, Cunningham and Capone 1980, Lekkas, Quigley and Van Order 1993, Capozza Kazarian and Thomson 1997, Clapp et al 2001, Goldberg and Harding 2003 and others.). Later, researchers use proportional hazard model to account for the path dependence of default behaviors. For example, Vandell et al (1993) apply a proportional hazard model to individual commercial mortgage loan default data and find property value trends, loan terms and property types to have significant impact on commercial mortgage default probability. Quigley and Van Order (1995) explicitly test the ”frictionless” models and validate the importance of transaction costs besides equity position with a hazard model. 24 Deng, Quigley and Van Order (2000) introduced the influential full maximum likeli- hood estimation of competing risks hazard model, which provide a solution to empirical modeling of competing risks of prepayment and default with proportional hazard model. Clapp et al (2001) apply logit models with event-history data to mimic the borrower’s discrete choice in each month of loan life rather than only in the loan ending (default or fall out of sample) point. It also takes into account time-varying covariates within a logit model framework. He argues that logit models with event-history data is a good alterna- tive to hazard model in estimating and predicting commercial mortgage default probabil- ities. Recent advancements in mortgage default predictions include models accounting for unobserved heterogeneity (Deng, Quigley and Van Order 2000, Deng and Quigley 2002, Deng, Pavlov and Yang 2005, Clapp, Deng and An 2005) and models to further explore borrowers’ choice set with respect to mortgage termination within a competing risks framework (Clapp et al 2001, An, Clapp and Deng 2005). It is noteworthy that the hazard model and logit models with event-history approach take a dynamic approach that acknowledges the changes of default risk over time. However, the emphasis is still on cross sectional differences of default risk. Further, the dynamics of risk factors are not integrated into the reduced-form econometric models. 25 Chapter 3 Theoretical Model A commercial mortgage is a debt with income producing property(ies) such as retail space, office, hotel or multifamily building as collateral. The mortgage borrower is obligated to make monthly repayment of principal and interest 1 . However, in each month the borrower may choose to default, usually because of the inability to make the monthly payment. A first-passage model is ideal to study commercial mortgage default. 3.1 The First-passage Model Black and Cox (1976) introduced the idea that default could occur at the first time that assets falls below a sufficiently low default boundary. The central idea of this first- passage model can be illustrated using a simple setting as shown in figure 3.1. 1 Based on pre-specified mortgage terms in the contract such as coupon rate and amortization. Some loans are interest-only during early period of contracts. 26 0 T 0 V Time Asset value s At D Figure 3.1: An Illustration of the First-passage Model. This figure illustrate the basic idea of the first-passage model. The firm asset value follows some stochastic process A t , and there is a default thresholdD. Default happens at times(s·T) when the asset value hits the default boundary. Given A t and D, we can calculate the probability that A t reaches D at t. Given we know at each point in time what’s the payoff if default happen and what’s the payoff if default does not happen, the value of the discount bond is just the expected value of the discounted payoff under the risk neutral measure. 27 Assume the firm issues a pure discount bond which matures at time T , the firm asset value follows some stochastic process A t , and there is a default threshold D, which needs not to be the face value of the debt. Default happens at time s(s · T) when the asset value hits the default boundary. Given A t and D, we can calculate the prob- ability that A t reaches D at t, Pr(t · s < t+¢t). For the pricing purpose, we can also calculate the probability under risk neutral measure Pr ¤ (t · s < t+¢t). The transformation from the physical measure to the risk neutral measure is determined by the market price of risk, which is assumed in the model set up. Further assume the bond pays 1 at maturity if default does not happen att and pays1¡w if default does happen. Then the payoff function of the debt is 1¡wI s·T . I is the indicator function which equals 1 if the first passage time is less than or equal to T . The value of the discount bond is just the expected value of the discounted payoff, which is Y(A;D;r;T)=D(r;T)(1¡wF ¤ (s<T)) (3.1) whereD(r;T) is the value of the risk-free discount bond,F ¤ (s < T) is the cumulative default probability under the risk-neutral measure. Black and Cox (1976) assume asset valueA t follows a square root diffusion process, the instantaneous risk free interest rate is constant and the default boundaryD is a constant fraction of the present value of the promised final debt payment (a deterministic time- dependent function), and gives the closed form solution of the bond value (see page 28 356). Longstaff and Schwartz (1995) show that assuming a Vasicek (1977) short rate process, a constant volatility asset value process and a constant default boundary, the first-passage time density and the values of both fixed rate discount bond and floating rate coupon bond can be solved in closed-form. 3.2 A First-Passage Model for Commercial Mortgage Default I modify the fist-passage model of Black and Cox (1976) and Longstaff and Schwartz (1995) to reflect unique feathers of commercial mortgage default. Since commercial mortgage borrowers mainly rely on the net operating income (NOI) from the underlying property to make monthly mortgage payment, I directly model the cash flow process, rather than the value process of the property. Default occurs when the NOI of the under- lying property falls under an exogenously specified default boundary. The income of a commercial property is closely related to the macroeconomic condi- tions. Therefore, I model the property NOI as depending on the state of the economy and I start with an equilibrium characterization of the economy. 29 3.2.1 The Economy Consider a continuous-time version of a Lucas (1978) type pure exchange econ- omy studied by Goldstein and Zapatero (1996). The single productive technology is described by an aggregate output process de t e t =¹ t dt+¾ e dW e t (3.2) and there is a risky security (stock) whose dividend comes exactly from the above aggre- gate output. The drift of the aggregate output is stochastic and follows an Ornstein- Uhlenbeck process d¹ t =·(¹¡¹ t )dt+¾ ¹ dW e t (3.3) where· is a positive constant representing the mean-reversion rate of the drift towards the long-term mean ¹. ¾ e and ¾ ¹ are positive constants for volatilities of aggregate output and its drift. W e t is a standard Brownian motion under real measure. Intuitively, we can think of det et as the GDP growth rate we observe. Given the noise, however, we cannot observe the expected growth rate ¹ t . Notice that in this setting, a shock affects both the realized return in the current period and expected return in the next period. This is consistent with what happens in the real world, e.g. when there is over-investment, it takes time for people to make corrections. Further, the economy has 30 some business cycle characteristics, in that the expected growth rate is mean-reverting. The agent can invest in the stock and gain both appreciation of the stock value and the dividend paid by the stock. Denote the price of the stock by S t , then the return of the stock is given by dG t S t = dS t S t + e t S t dt (3.4) There is also a (locally) risk-free bond accessible to the agent with zero net supply. The bond priceB t at timet follows the following process dB t =B t r t dt; B 0 =1 (3.5) where r t is the instantaneous risk-free rate, which is determined endogenously as an equilibrium outcome. In this economy, there is a single representative agent who is a price taker and has a power utility (CRRA) over consumption with the relative risk aversion parameter°. U(c t )= c 1¡° t 1¡° (3.6) 31 The agent’s problem is to choose the optimal consumption and investment to maximize his expected life-time utility max fct;¼tg E ½Z T 0 exp(¡±t) c 1¡° t 1¡° dt ¾ s:t: dX t =¼ t dG t S t +(X t ¡¼ t ) dB t B t ¡c t dt (3.7) Here, ¼ t is the amount invested in the stock, X t represents the wealth process of the representative agent, and± is the subjective discount rate. The equilibrium of the economy is characterized by c ¤ t =e t 8 t2[0;T] (3.8) As shown in Goldstein and Zapatero (1996), the equilibrium instantaneous risk-free interest rate satisfies r t =±+°¹ t ¡ 1 2 °(1+°)¾ 2 e (3.9) or dr t =·(r¡r t )dt+°¾ ¹ dW e t where r =°¹+±¡ 1 2 °(1+°)¾ 2 e 32 which is the well-known Vasicek (1977) spot rate process. The market price of risk is constant µ : ¹ t ¡r t ¾ e =°¾ e (3.10) Again, following Vasicek (1977) and Goldstein and Zapatero (1996), the equilibrium price of risk-free discount bond is P(t;T;r t )=E Q ½ exp µ ¡ Z T t r s ds ¶¯ ¯ ¯ ¯ F t ¾ =expfA(t;T)¡B · (t;T)r t g (3.11) where A(t;T) = · ¡r+ 1 2 ° 2 µ 2¾ e ¾ ¹ · + ¾ 2 ¹ · 2 ¶¸ (T ¡t) + · r¡° 2 µ ¾ e ¾ ¹ · + ¾ 2 ¹ · 2 ¶¸ B · (t;T)¡ 1 2 ° 2 ¾ 2 ¹ · 2 B 2 ·(t;T) B · (t;T) = 1¡expf¡·(T ¡t)g · It can be easily shown that after some simple formula manipulation, the risk-free dis- count bond yield depends on the state variable¹ t as R(t;T;¹ t ) =¡ 1 T ¡t logP(t;T;¹ t ) (3.12) = ° T ¡t B · (t;T)¹ t ¡ 1 T ¡t · A(t;T)¡±B · (t;T)+ °(°+1) 2 ¾ 2 e B · (t;T) ¸ 33 3.2.2 The Income Producing Property The NOI of the commercial property also depends on certain conditions specific to the commercial property market, e.g. whether there is over-supply of commercial spaces, whether the operating expenses systematically elevates during certain periods. For parsi- mony purposes, I assume all these effects can be summarized by a single factor indepen- dent of the macroeconomic factor. Therefore, the NOI of the income producing property is determined by the state of the economy¹ t and a commercial property market-specific factor» t 2 . This approach is similar to Tang and Yan (2006) in modelling firm cash flow. dK t K t =(¯¹ t +» t )dt+¾ K ½dW e t +¾ K p 1¡½ 2 dW K t (3.13) The drift´ t =¯¹ t +» t is the expected NOI growth rate, which is composed of the aggre- gate economy expected growth rate¹ t and a property market-specific expected growth rate » t independent of ¹ t . ¯ < 1 is the correlation coefficient of expected commercial property market growth rate and aggregate economy expected growth rate. ¾ K is a pos- itive constant representing the volatility of the property cash flow. ½ is the correlation coefficient between the property cash flow process and the aggregate output process, and½=¯ ¾e ¾ K . W K t is a standard Brownian motion processes independent ofW e t . 2 I assume the existence of commercial mortgages does not affect equilibrium pricing of securities given commercial mortgages only account for a tiny portion of the macro economy 34 The commercial property market-specific expected growth rate is also stochastic with the mean-reverting speed of¸, long term mean» and constant volatility¾ » . d» t =¸(»¡» t )dt+¾ » dW K t (3.14) The economics of this mean-reversion specification is: people in the commercial prop- erty market usually make adjustments in supply and operating expenses in response to the observed return, which drives the expected return towards a long term mean. We can see that at each point in time, property NOI is determined by four components: expected growth rate of the whole economy, commercial property market expected growth rate, economy-wide shock and commercial property market-specific shock. The primitive processes in my model are economic growth and commercial property NOI growth processes rather than interest rate and property value processes, as in most existing structural models. Here, interest rate process is derived from the general equi- librium of the economy rather than exogenously given. Regarding NOI and property value, we know that commercial property value is largely determined by its NOI, so directly modelling NOI is potentially a better choice. In the following subsection, I’ll come back to this when I discuss the so-called “double-trigger” of commercial mortgage default. 35 3.2.3 Commercial Mortgage Default Boundary A typical commercial mortgage contract specifies the principal M, a fixed mortgage rate (coupon rate) R and an amortization term T . The monthly principal and interest paymentC is determined implicitly by M = Z T 0 Cexp(¡Rt)dt (3.15) Every month before maturity, the mortgage borrower is obligated to make the monthly payment. Usually commercial mortgages are balloon loans, with a maturity termS <T . At balloon date, the remaining principal is M S = Z T S Cexp(¡Rt)dt (3.16) A commercial mortgage borrower usually default because he is unable or unwilling to make the monthly payment. A natural candidate of default boundary is the monthly payment amount. However, given high transaction cost of default, I follow Huang and Huang (2003) and Leland (2004) to specify a slightly different default boundary K B =ÁC (3.17) 36 where C is the monthly mortgage payment defined above. This boundary condition specifies that a commercial mortgage will be defaulted if the NOI of the underlying property falls under a certain proportion of the monthly mortgage payment. The param- eterÁ·1 is a control of transaction cost of mortgage default. In the commercial mortgage credit risk literature, an important concern is the “double trigger” of default, which means the insolvency condition K B = C and the net worth condition V B = H are both important for default (see Riddiough 1991, Goldberg and Capone 2002 and Tu and Eppli 2002). Here, I only consider the insolvency condition. This is partly for the tractability of the model. However, there are other good reasons to focus on the insolvency condition. First, commercial property value is calculated as its cash flow divided by the market capitalization rate and is largely determined by its NOI. Therefore, the insolvency condition and the net worth condition are closely related. Second, from a practical point of view, the tangible monthly cash flow is well reflected on the mortgage holders accounting book, and thus it is easy to believe that mortgage holders make their default decisions based on observable cash flow conditions. This is consistent with the fact that insolvency condition expressed as debt-service coverage ratio (DSCR) is found to be the most relevant for commercial mortgage default event according to Moody’s. The default boundary here is assumed to be exogenously given. A more elegant treat- ment should specify an endogenously determined default boundary as in Leland (1994), 37 Leland and Toft (1996) and Duffie and Lando (2001). In that setting, the equity value of the firm or property is S =sup ¿ E t ½Z ¿ t exp(¡rs)[±V s ¡(1¡µ s )C s ]ds ¾ (3.18) where ¿ is stopping time, ±V t is cash flow of the firm or property 3 , C t is monthly debt payment and µ t is tax benefit associated with interest payment. Equity is just the accumulation of the cash inflow minus outflow. Borrower chooses an optimal stopping time ^ ¿ to maximize his equity. However, in the above mentioned endogenous default boundary studies, a constant default boundary proportional to the debt coupon payment is usually derived based on assumptions such as constant interest rate, debt roll over and constant asset risk premium (see, for example, Duffie and Lando 2001, page 638). Therefore, I stick to the parsimonious specification of an exogenously given constant default boundary. 3.2.4 First-Passage Time Density We want to know default probabilities of the commercial mortgage at each point in time before its maturity. Given the above set up, it is straightforward to find out the default time density, which is the first-passage time density of the stochastic property cash flow 3 Or ”service flow” or imputed rent. 38 K t falling under the constant default boundaryK B . The following proposition gives the first-passage time density under real measure. Proposition 1: The first-passage time densityq(¿) is defined implicitly in the following integral equation N µ ¡lnX 0 ¡L(t) (t) ¶ = Z t 0 q(¿)N µ L(¿)¡L(t) (t)¡(¿) ¶ d¿ (3.19) whereN(¢) denotes the cumulative standard normal distribution function, and X t = K t K B L(t) = µ ¯¹ 0 e ¡·t +» 0 e ¡¸t +¯¹+»¡ ¾ 2 K 2 ¶ t+ ¯¹ · ¡ e ¡·t ¡1 ¢ + » ¸ ¡ e ¡¸t ¡1 ¢ (t) = ¯ 2 ¾ 2 ¹ · 2 µ t¡2 1¡e ¡·t · + 1¡e ¡2·t 2· ¶ + 2¾ K ½¯¾ ¹ · µ t¡ 1¡e ¡·t · ¶ + ¾ 2 » ¸ 2 µ t¡2 1¡e ¡¸t ¸ + 1¡e ¡2¸t 2¸ ¶ + 2¾ K p 1¡½ 2 ¾ » ¸ µ t¡ 1¡e ¡¸t ¸ ¶ +2¾ 2 K t Proof: We have the set up in equations 3.20, 3.3 and 3.14 as follows dK t K t =(¯¹ t +» t )dt+¾ K ½dW e t +¾ K p 1¡½ 2 dW K t (3.20) d¹ t =·(¹¡¹ t )dt+¾ ¹ dW e t (3.21) 39 d» t =¸(»¡» t )dt+¾ » dW K t (3.22) DefineX t = Kt K B , then dXt Xt =(¯¹ t +» t )dt+¾ K ½dW e t +¾ K p 1¡½ 2 dW K t (3.23) lnX t ¡lnX 0 = Z t 0 · ¯¹ s +» s ¡ ¾ 2 K 2 ¸ ds+ Z t 0 h ¾ K ½dW e s +¾ K p 1¡½ 2 dW K s i Rewrite equation 3.21 and 3.22 as ¹ t =e ¡·t · ¹ 0 + Z t 0 ·¹e ·s ds+e ¡·t Z t 0 ¾ ¹ e ·s dW e s ¸ (3.24) » t =e ¡¸t · » 0 + Z t 0 ¸»e ¸s ds+e ¡¸t Z t 0 ¾ » e ¸s dW K s ¸ Substitute¹ s and» s in equation 3.23 by the above, and rearrange items, we have lnX t ¡lnX 0 = Z t 0 · ¯¹ 0 e ¡·t +» 0 e ¡¸t ¡ ¾ 2 K 2 ¸ ds (3.25) + Z t 0 · ¯e ¡·s Z s 0 ·¹e ·q dq+e ¡¸s Z s 0 ¸»e ¸q dq ¸ ds + Z t 0 · ¯e ¡·s Z s 0 ¾ ¹ e ·q dW e q +e ¡¸s Z s 0 ¾ » e ¸q dW K q ¸ ds + Z t 0 h ¾ K ½dW e s +¾ K p 1¡½ 2 dW K s i 40 Notice R t 0 e ¡·s R s 0 e ·q dW e q ds = Z t 0 dW e q Z t q e ·(q¡s) ds (3.26) = Z t 0 dW e q e ·q e ¡·q ¡e ¡·t · = Z t 0 1¡e ·(q¡t) · dW e q Hence, equation 3.25 can be further written as lnX t ¡lnX 0 = µ ¯¹ 0 e ¡·t +» 0 e ¡¸t +¯¹+»¡ ¾ 2 K 2 ¶ t (3.27) + ¯¹ · ¡ e ¡·t ¡1 ¢ + » ¸ ¡ e ¡¸t ¡1 ¢ + Z t 0 · ¯¾ ¹ · ¡ 1¡e ·(q¡t) ¢ dW e q + ¾ » ¸ ¡ 1¡e ¸(q¡t) ¢ dW K q ¸ + Z t 0 h ¾ K ½dW e s +¾ K p 1¡½ 2 dW K s i =L(t)+ Z t 0 · ¯¾ ¹ · ¡ 1¡e ·(q¡t) ¢ dW e q + ¾ » ¸ ¡ 1¡e ¸(q¡t) ¢ dW K q ¸ + Z t 0 h ¾ K ½dW e s +¾ K p 1¡½ 2 dW K s i Therefore, E(lnX t )=lnX 0 +L(t) (3.28) 41 and Var(lnX t ) = Z t 0 · ¯¾ ¹ · ¡ 1¡e ·(q¡t) ¢ +¾ K ½ ¸ 2 dq (3.29) + Z t 0 h ¾ » ¸ ¡ 1¡e ¸(q¡t) ¢ +¾ K p 1¡½ 2 i 2 dq+¾ 2 K t = Z t 0 · ¯ 2 ¾ 2 ¹ · 2 ¡ 1¡e ·(q¡t) ¢ 2 + 2¾ K ½¯¾ ¹ · ¡ 1¡e ·(q¡t) ¢ ¸ dq + Z t 0 " ¾ 2 » ¸ 2 ¡ 1¡e ¸(q¡t) ¢ 2 + 2¾ K p 1¡½ 2 ¾ » ¸ ¡ 1¡e ¸(q¡t) ¢ # dq+2¾ 2 K t which can be further written as Var(lnX t ) = ¯ 2 ¾ 2 ¹ · 2 µ t¡2 1¡e ¡·t · + 1¡e ¡2·t 2· ¶ (3.30) + 2¾ K ½¯¾ ¹ · µ t¡ 1¡e ¡·t · ¶ + ¾ 2 » ¸ 2 µ t¡2 1¡e ¡¸t ¸ + 1¡e ¡2¸t 2¸ ¶ + 2¾ K p 1¡½ 2 ¾ » ¸ µ t¡ 1¡e ¡¸t ¸ ¶ +2¾ 2 K t=(t) So,lnX t » N(lnX 0 +L(t);(t)), and according to Buonocore, Nobile and Ricciardi (1987, 2.2a), the first-passage time density q(¿) is defined implicitly in the following integral equation 4 N µ ¡lnX 0 ¡L(t) (t) ¶ = Z t 0 q(¿)N µ L(¿)¡L(t) (t)¡(¿) ¶ d¿ (3.31) 4 A minor issue is that I assumet can be infinity here. However, if we impose the upper bound oft as the maturity term of the mortgage, the only treatment needed is to add boundary conditions. 42 whereN(¢) denotes the cumulative standard normal distribution function. Dividing the period from time zero to time T inton equal sub-periods and discretizing the above integral equation give the following system of linear equations, which can be solved recursively to find out the default time density. N(a i ) = i X j=1 q i N(b i;j ); i=1;2;¢¢¢ ;n (3.32) q i =q µ i T n ¶ ¢ T n 3.2.5 Default Hazard Rate (Default Intensity) Default hazard rate is a key variable studied in the mortgage default prediction liter- ature. It is the mean arrival rate of default during [t;s] conditioning on the mortgage not defaulted at time t 5 . In real world, we observe default hazard rate based on pools of mortgages. We want to solve the relationship between default hazard rate and the risk factors in our model, so that we can use the real world data to estimate parameters of the model and learn the dynamics of risk factors and hazard rates. The following proposition gives the closed-form solution of default hazard rate in my model. 5 In discrete time, it is just the conditional default probability. 43 Proposition 2: The default hazard rate under real measure is h t = Pr(t·T <t+1;¹ t ;» t ) 1¡Pr(T <t;¹ t ;» t ) (3.33) where Pr(t·T <t+1;¹ t ;» t ) = Z +1 1 Z 1 0 jlnx t +a t j p 2¼s 3 ¾ K exp ½ ¡ (lnx t +a t ) 2 2s¾ 2 K ¾ ds 1 p 2¼t¾ K exp ½ ¡ (lnx t ¡lnx 0 ¡ P t i=1 a i ) 2 2t¾ 2 K ¾ dx t Pr(T <t;¹ t ;» t )=Pr(T <1)+Pr(1·T <2)+¢¢¢+Pr(t¡1·T <t) and a t =¯¹ t +» t ¡ ¾ 2 K 2 Proof: DefineX t = Kt K B , again we have dXt Xt =(¯¹ t +» t )dt+¾ K ½dW e t +¾ K p 1¡½ 2 dW K t (3.34) lnX t ¡lnX 0 = Z t 0 · ¯¹ s +» s ¡ ¾ 2 K 2 ¸ ds+ Z t 0 h ¾ K ½dW e s +¾ K p 1¡½ 2 dW K s i 44 Further defining a t =¯¹ t +» t ¡ ¾ 2 K 2 (3.35) and orthogonizing the Brownian motions in the last integration term, we get lnX t ¡lnX 0 = Z t 0 a s ds+ Z t 0 ¾ K dW s (3.36) So, lnX t+1 ¡lnX t =a t +¾ K W(1) (3.37) or lnX t+1 ¡lnX t ¡a t ¾ K =W(1)»N(0;1) SinceX t+1 = 1 is the boundary condition, according to Grimmett and Stirzaker (2001, p. 526), the first-passage time probability in the time interval[t;t+1) is Pr(t·T <t+1jX t =x t ;¹ t ;» t )= Z 1 0 jlnx t +a t j p 2¼s 3 ¾ K exp ½ ¡ (lnx t +a t ) 2 2s¾ 2 K ¾ ds (3.38) 45 The density ofX t is f Xt (x t ;¹ t ;» t )= 1 p 2¼t¾ K exp ½ ¡ (lnx t ¡lnx 0 ¡ P t i=1 a i ) 2 2t¾ 2 K ¾ (3.39) Therefore, Pr(t·T <t+1;¹ t ;» t ) = Z +1 1 P(t·T <t+1jX t )f Xt (X t =x t )dx t (3.40) = Z +1 1 Z 1 0 jlnx t +a t j p 2¼s 3 ¾ K exp ½ ¡ (lnx t +a t ) 2 2s¾ 2 K ¾ ds 1 p 2¼t¾ K exp ½ ¡ (lnx t ¡lnx 0 ¡ P t i=1 a i ) 2 2t¾ 2 K ¾ dx t Finally, the hazard rate is h t = lim ¢t¡!0 + Pr(t·T <t+¢tjT ¸t;¹ t ;» t ) ¢t (3.41) In discrete time, we have h t =Pr(t·T <t+1jT ¸t;¹ t ;» t ) (3.42) = Pr(t·T <t+1;¹ t ;» t ) Pr(T ¸t;¹ t ;» t ) = Pr(t·T <t+1;¹ t ;» t ) 1¡Pr(T <t;¹ t ;» t ) 46 For pricing purposes, we can solve default probabilities under risk neutral measure in my model. The only thing we need to do is to transform the expressions of primitive processes from real measure into risk neutral measure using change of measure based on the result of market price of risk in equation 3.10. Then we can apply the same approach as above. Here I only give the hazard rate solutions in real measure because in the following empirical estimations, I will focus on default probabilities observed from real world commercial mortgage default experience. 47 Chapter 4 Empirical Methodology and Results The objectives are to learn the evolutions of default risk over time based on observed commercial mortgage defaults, and to extract information about the latent systematic factors driving the default risk dynamics. Solutions of the structural model in the pre- vious section are put into a state space form and the model is estimated using extended Kalman filter. To help identify the model, I use risk-free bond yield as another measure- ment variable in addition to default hazard rate. 4.1 The State Space Model At time t we observe commercial mortgage default hazard rate h t and risk-free bond yieldR t . These two observable variables are functions of the two latent state variables ¹ t and» t . 48 Rewrite the two state variable processes (equations 3.3 and 3.14) in discrete time ¹ t =·¹+(1¡·)¹ t +¾ ¹ ² e t (4.1) » t =¸»+(1¡¸)» t +¾ » ² xi t (4.2) and the solutions of the above structural model are R t =g(¹ t ) (4.3) h t =f(¹ t ;» t ) (4.4) whereg is the nonlinear function mapping¹ t into the risk-free discount bond yield R t as given in equation 3.12, and f is the nonlinear function mapping ¹ t and » t into the commercial mortgage hazard rateh t as given in equation 3.41. We can express the above structural model in a state-space form. DenoteZ t =(R t ;h t ) 0 and S t = (¹ t ;» t ) 0 , and suppress the dependence of the model on the parameters to be estimated, the measurement and transition equations are Z t =z(S t )+² t ; E t¡1 (² t ² t 0 )=§ (4.5) S t =®+TS t¡1 +´ t ; E t¡1 (´ t ´ t 0 )=© (4.6) 49 The transition equation 4.6 depicts the dynamics of the two state variables given in equa- tions 4.1 and 4.2. Notice S t is not directly observable. The non-linear function z(S t ) in the measurement equation 4.5 maps the two latent state variables into the observable bond yield and mortgage hazard rate (equations 4.3 and 4.4. Therefore, we can infer the dynamics of the latent variables from the time series of the measurement variables. The innovations in the transition equation are normally distributed based on our model set up in equations 3.3 and 3.14. ©= 0 B @ ¾ 2 ¹ 0 0 ¾ 2 » 1 C A I further assume the innovations in the measurement equation are also normally dis- tributed, reflecting the random observation errors of these two variables. §= 0 B @ § 11 0 0 § 22 1 C A The model is estimated with extended Kalman Filter as discussed in below. To apply the extended Kalman filter, I use the first-order Taylor expansion to linearize the mea- surement equation 4.5 z(S t )tz(S ¤ t )¡L t S ¤ t +L t S t (4.7) 50 where L t = @z(S t ) @S t ¯ ¯ ¯ ¯ St=S ¤ t andS ¤ t is the 1-month ahead forecast ofS t . 4.2 The Cox Proportional Hazard Model To apply the state space model, we need a time series of default hazard rateh t of a rep- resentative commercial mortgage as a measurement variable. In real world, commercial mortgage loans are heterogeneous, e.g. they are in different region, backed by different properties, and have different mortgage terms. Existing studies have found loan charac- teristics to be important determinants of commercial mortgage default. Therefore, we need to first obtain a representative default hazard rate time series from individual loan default events we observe in the marketplace. To achive this objective, I use the Cox proportional hazard model to control for covariates of individual mortgage loans. The Cox proportional hazard model is widely accepted in the literature as an effective tool to study mortgage prepayment and default (See, for example, Green and Shoven 1987, Deng, Quigley and Van Order 2000). In the Cox model and other duration models, 51 the life of a mortgage loan (durationT ) is the random variable which is usually assumed to follow a certain distribution. In each month, the mortgage loan has a risk of ending its life, which is represented by the hazard function. The hazard function is defined as h t = lim ¢t!0 Pr(t·T <t+¢tjT ¸t) ¢t (4.8) It specifies the instantaneous rate at which default occurs for loans that are surviving at time t. The functional form of the hazard function depends on the duration distribution assumption. The Cox model starts with the specification of the hazard function h i (t;T)=h 0 (T)exp(X i;t ¯) (4.9) where t is calendar time and T is duration. h 0 (T) is the baseline hazard function rep- resenting represents the default rate of a typical loan during the interval[T;T +1] con- ditioning on the loan has not been defaulted at duration T . I is a function of duration T reflecting different hazard rates of a loan at different stage of its life. From a cross sectional perspective, it is shared by all loans. X i;t are covariates representing the het- erogeneity of individual mortgage loans. They shift the baseline hazard up or down depending on their values. Covariates can be both time constant and time-varying. 52 In each month, the hazard rate of a specific mortgage loan is determined by economic fundamentals, loan specific characteristics and seasoning. Therefore, we can further separate time varying covariates from time constant covariates, and write the hazard function as h i (t;T)=h 0 (T)exp(D i;t ®+Z i ¯) (4.10) whereZ is a vector representing time constant idiosyncratic risk factors such as property type, region, maturity term, amortization term, etc. D i;t =(d i;1 ;d i;2 ;:::;d i;t ) is a series of time varying dummies and thus¯ captures systematic risk, which evolves over time. h 0 (T) represents the seasoning effect. Following Schwartz and Torous (1989), I assume the conditional (on the covariates) distribution of the duration time T follows a log- logistic distribution. More specifically, assume logT = ® +¾W where ® and ¾ are constants andW has a logistic density: f(W =w)= exp(w) (1+exp(w)) 2 (4.11) Accordingly, the baseline hazard function is h 0 (T)= º!(ºT) !¡1 1+(ºT) ! (4.12) 53 4.3 Estimation Methods I estimate the Cox proportional hazard model using the maximum likelihood estimation method presented in Kalbfleisch and Prentice (1983/2002) and implemented by Deng, Quigley and Van Order (2000) 1 . The MLE involves the survival function, which is V i (t;T)=exp ½ ¡ Z T 0 h i (t;s)ds ¾ (4.13) The likelihood function is L= T Y l=0 f Y i2D i (V i (t;T +1)¡V i (t;T)) Y i2C i V i (t;T +1)g (4.14) withD i be the set of labels associated with individuals failing att l , andC i be the set of labels associated with individuals censored in [t l ;t l+1 ). The asymptotic covariance of the MLE estimates is calculated with the information matrix using the BHHH method (Greene 2000, p.132). 1 Deng, Quigley and Van Order (2000) have competing risks (prepayment and default) in their model and they estimate a flexible baseline using a semi-parametric method. 54 The hazard model estimation gives the time series of default hazard rate of a represen- tative commercial mortgage c H t = ( b h 1 ; b h 2 ;:::; b h t ), which is the measurement variable h t in the state space model. The state space model is estimated using the extended Kalman filter. The nonlinear functionz(S t ) is linearized with equation 4.7 at the 1-month ahead forecast ofS t and© is evaluated at the contemporaneous prediction ofS t . After linearization, the following standard Kalman filter recursion is used. Consider a linear Gaussian state space model: Z t =L t s t +" t ; " t »N(0;H t ); (4.15) s t+1 =T t s t +´ t ; ´ t »N(0;Q t ); t=1;:::;n (4.16) ® 1 »N(a 1 ;P 1 ) (4.17) Denote Z t¡1 as the set of past observations z 1 ;z 2 ;:::;z t¡1 . Assume that s t given Z t¡1 is N(a t ;P t ), or a t+1 = E(s t+1 jZ t ) and P t+1 = Var(s t+1 jZ t ). Let v t be the one-step 55 forecast error ofz t givenZ t¡1 , orv t = z t ¡E(z t jZ t¡1 ) andF t = Var(z t jZ t¡1 ). Then the Kalman filter recursion is: v t =z t ¡L t a t ; F t =L t P t L 0 t +H t ; (4.18) K t =T t P t L 0 t F ¡1 t ; M t =T t ¡K t L t ; (4.19) a t+1 =T t a t +K t v t ; P t+1 =T t P t M 0 t +Q t ; t=1;:::;n (4.20) with a 1 and P 1 as the mean vector and variance matrix of the initial state vector s 1 , respectively. The Kalman filter requires initialization to start the recursion. Since we don’t know the exact distribution of the initial state vector S 1 , I apply a diffuse initial- ization, which assumes arbitrary means and infinite variances of the initial state vector, a 1 =c andP 1 =1. Parameters of the state space model are estimated using maximum likelihood estimation method. The log-likelihood function with diffuse initialization is LogL d (Z)=¡ np 2 log2¼¡ 1 2 logjF 1;t j¡ 1 2 n X t=2 (logjF t j+v 0 t F ¡1 t v t ) (4.21) where n is the number of time series observations, p is the measurement variable dimension, which is 2 in my model. F 1;t = L t P 1;t L 0 t . F t is the prediction variance Var(Z t jF t¡1 ) withF t¡1 denoting information up tot¡1, andv t is the prediction error 56 Z t ¡L t S t . P 1;t = T t T 0 t . Again, the asymptotic covariance of the MLE estimates is calculated using the BHHH method 2 . 4.4 Data and Results I access a large commercial mortgage loan performance database maintained by Intex. Information are collected for nearly 60,000 commercial mortgage loans, which all underlie commercial mortgage-backed securities (CMBS). The data collecting point is June 31, 2003. After excluding adjustable rate mortgages (ARMs) and mortgages in Canada, the final sample contains 49,389 fixed rate commercial mortgage loans. These loans are originated between 1992 and 2003, and are from 355 MSAs across 54 US states and territories. The data contains detailed information on loan characteristics and performance, such as origination date, original balance, original loan-to-value ratio (LTV), maturity date, amortization term, coupon rate, lender, property type, geography, 60-day and 90-day delinquency dates, pay down date, losses, etc.. In order to utilize the simple Affine term structure results in equation 3.12, we need yields of risk-free pure discount bond. I use the Fama-Bliss 5-year pure discount bond 2 See Durbin and Koopman (2001) for more detailed discussions on state space model estimation. 57 yields from CRSP, and apply the seasonally adjusted CPI to make the inflation indexed yields. Table 1 presents the statistics of original balance, original LTV coupon rate of the 49,403 commercial mortgage loans in my sample. These are large-amount while low-LTV loans, comparing to residential mortgages. The average original loan amount is $6.14 million and average LTV is only 68.22%. I also calculate the spreads of the mortgage rates over comparable maturity treasury bond rates. Results show that these loans have spreads ranging from 90 bps to 469 bps, with an average of 234 bps. Table 1 Descriptive Statistics of the Commercial Mortgage Loans This table presents the statistics of loan amount, loan-to-value ratio (LTV) and spreads of 49,403 commercial mortgage loans in my study. All loans are in Commercial Mortgage- backed Securities (CMBS) pools. Original balances are in thousands (000s) and LTVs are in percent. Spread is defined as the difference between mortgage gross coupon rate and comparable maturity treasury bond yield. Variable Mean Stand. Dev. Minimum Median Maximum Original balance 6,142 17,323 14 2,925 800,000 Original LTV 68.22 12.48 1.01 70.61 100.00 Gross coupon 7.96 1.04 1.85 7.85 17.50 Spread 2.34 0.68 0.90 2.29 4.69 Number of loans 49,403 58 Table 2 Origination Timing of the Commercial Mortgage Loans This table shows the origination year distribution of the 49,403 commercial mort- gage loans. Origination Number Percentage in Total loan Percentage in year of loans # of loans amount($m) loan amount 1992 673 1.36 2,065 0.68 1993 1,214 2.46 5,324 1.75 1994 1,482 3.00 6,837 2.25 1995 2,157 4.37 9,713 3.20 1996 4,258 8.62 18,755 6.18 1997 6,473 13.10 36,553 12.05 1998 13,260 26.84 66,913 22.05 1999 5,660 11.46 36,001 11.87 2000 4,422 8.95 29,258 9.64 2001 4,906 9.93 45,439 14.98 2002 3,637 7.36 34,042 11.22 2003 1,261 2.55 12,515 4.12 All years 49,403 100.00 303,415 100.00 Table 2 shows the origination year distribution of the sample with respect to both number of loans and original balance. The year 1998 sees a large number of loan origination. 59 Table 3 Originators of the Commercial Mortgage Loans This table lists the major originators of the 49,403 commercial mortgage loans. Name of Number Percentage in Total loan Percentage in Originator of loans # of loans amount($m) loan amount Bank of America 4,428 8.96 17,670 5.82 Citi Corporation 1,435 2.90 7,615 2.51 Column 4,403 8.91 28,024 9.24 GE Capital 2,364 4.79 13,325 4.39 GMAC 1,927 3.90 13,099 4.32 JPMorgan Chase 2,161 4.37 16,972 5.59 Lehman 2,006 4.06 17,337 5.71 Merrill Lynch 1,310 2.65 8,867 2.92 Midland 1,466 2.97 5,536 1.82 Morgan Stanley 1,256 2.54 11,147 3.67 Nomura 1,635 3.31 18,368 6.05 Wachovia 2,729 5.52 15,448 5.09 Wells Fargo 1,840 3.72 7,857 2.59 Other 20,443 41.38 122,148 40.26 Total 49,403 100.00 303,415 100.00 60 Table 3 lists the “big” originators of commercial mortgage loans, among them are Bank of America, Column, and Wachovia, each having a market share of over 5% in terms of number of origination. In terms of loan amount, Nomura, Lehman and JPMorgan Chase also have over 5% market shares. Table 4 reports the distributions of the sample with respect to different loan characteris- tics. I identify 1,892 defaults, which is 3.83% of the whole sample 3 . Table 4 also shows the proportion of loans defaulted in each group, as shown in the third column of the table. The most populated property type in my sample is multifamily, followed by retail and office. Hotel and healthcare property loans have substantially higher default rates. The loans in my sample are from 10 US regions, with those in the west coast to have lower default rates. Loans are mostly from 334 MSAs, among which are the nine big ones such as Los Angeles, CA, New York, NY , Dallas, TX, Houston, TX, Washington, DC, Atlanta, GA, Phoenix, AZ, Chicago, IL, and Boston, MA. Most loans have amorti- zation terms between 20 and 30 years. Less than 6% of loans have very short (less than 10 years) or very long (over 30 years) amortization, and those loans have lower default rates. Different from residential mortgages, most commercial mortgages are balloon loans. Nearly 75% of loans in my sample mature in 5 to 10 years. Average LTV of these loans are also lower than residential mortgages. 3 Also excluded from the sample are 164 balloon default loans, since balloon default is not modelled in my model. Causes of balloon defaults could be very different from those of term defaults, e.g. some borrowers default at maturity date because of the inability to refinance the current loan at the balloon date given a substantially higher interest rate environment than that at origination. 61 Table 4 Characteristics of the Commercial Mortgage Loans This table shows the compositions of the commercial mortgage loans in my study by different loan characteristics, and percentage of loans defaulted in each category. Other property type includes manufactured housing, self-storage, mixed-use, etc. Charact- Number Percentage in Proportion eristics Category of loans # of loans defaulted Property Multifamily 16,060 32.51 2.73 type Retail 13,184 26.69 3.88 Office 7,115 14.40 2.46 Industrial 4,472 9.05 2.80 Hotel 2,381 4.82 17.56 Healthcare 977 1.98 13.00 Other 5,214 10.55 1.84 MSA Big (¸ 1,000 loans) 13,398 27.12 2.84 scale Median (100 – 1000 loans) 25,514 51.64 4.05 Small (< 100 loans) 7,814 15.82 5.16 Non-MSAs 2,677 5.42 2.84 62 Table 4, continued Characteristics of the Commercial Mortgage Loans Charact- Number Percentage in Proportion eristics Category of loans # of loans defaulted Region Midwest/Eastern 5,177 10.48 4.70 Midwest/Western 1,950 3.95 4.92 Northeast/Mid-Atlantic 5,964 12.07 3.71 Northeast/New-England 2,806 5.68 3.24 Southern/Atlantic 9,606 19.44 4.49 Southern/East-coast 1,806 3.66 6.98 Southern/West-coast 6,268 12.69 4.87 Western/Mountain 4,645 9.40 3.83 Western/Northern Pacific 4,785 9.69 2.13 Western/Southern Pacific 6,396 12.95 1.55 Original less than $0.35m 2,469 5.00 2.27 balance $0.35m-$18.25m 44,463 90.00 4.00 greater than $18.25m 2,471 5.00 2.23 63 Table 4, continued Characteristics of the Commercial Mortgage Loans Charact- Number Percentage in Proportion eristics Category of loans # of loans defaulted Loan-to-value less than 40% 1,849 3.74 1.46 ratio 40-60% 6,999 14.17 4.13 60-80% 38,494 77.92 3.89 greater than 80% 2,061 4.17 3.89 Amortization less than 10 years 1,310 2.65 2.37 term 10-20 years 7,261 14.70 5.47 20-30 years 39,327 79.60 3.65 greater than 30 years 1,505 3.05 2.00 Maturity less than 5 years 2,565 5.19 2.77 term 5-10 years 36,793 74.48 3.65 10-15 years 5,319 10.77 4.72 greater than 15 years 4,726 9.57 4.82 All loans 49,403 100.00 3.83 64 The majority of loans have LTV in the 60-80% range. Loans with very low LTV (lower than 40%) have significantly lower default rates. The average original balance is $6.14 million, and the top 5 percent of the sample have balances over $18 million. The simple tabulation in table 4 tends to suggest substantial variations in default rates with respect to loan characteristics. Commercial mortgage loans typically have prepay- ment constraints, such as lock out, yield maintenance and prepayment penalty. Over 91% of loans in our sample have at least one type of prepayment constraint. Figure 4.1 plots the actual annual default rates during 1994 and 2003. There is no single loan default in 1992 and 1993 in my sample. From the figure, we see high default rates during the 2001-2002 economic recession. Figure 4.2 plots the default rates of commercial mortgage loans by duration, which tends to show a humped-shape seasoning effect. 65 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Year Default rate 0.32 0.27 0.27 1.15 1.20 0.61 0.70 1.41 1.18 0.35 Figure 4.1: Annual Default Rates of Commercial Mortgage Loans, 1994–2003. This figure plots the raw annual default rates observed from my commercial mortgage sam- ple. The default rate is calculated as the number of loans defaulted in the year divided by number of loans outstanding at the beginning of the year. 66 Table 5 presents the maximum likelihood estimation results of the Cox proportional haz- ard model. Most of the results are conforming to expectation. For example, hotel and healthcare loans are more risky than the reference group - retail loans. Loans in Southern California (Western/Pacific) have the lowest risk across regions, while those in the South have higher risk comparing to the reference group - Northeast/Mid-Atlantic loans. The maturity term is negatively related with default risk. The negative relationship applies to amortization. These two relationships are consistent with the notion that properties with lower risk usually get more favorable loan terms. Default hazard rate has no significant correlation with LTV , which is possibly due to the endogeneity of LTV to default risk suggested by previous studies (Archer et al 2001, Ambrose and Sanders 2003 and Cio- chetti et al 2003). It is surprising that multifamily loans tends to have higher risk than retail properties, every other things being equal. This may be attributed to the over-built of multifamily properties during the study period due to declining interest rate 4 . The other surprise comes from the significant lower risk of non-MSA loans. Appendix table 1 reports the estimation results with originator dummies, while appendix tables 2 and 3 reports the coefficients of the hazard model estimated with stratified samples. 4 I also experiment on adding interactions of property type and other variables such as LTV and contract terms, and the coefficients of those interaction terms are not significant. 67 Table 5 Maximum Likelihood Estimates of the Hazard Model This table presents the maximum likelihood estimates of the hazard model h i (t;¿) = h 0 (¿)exp(D i;t ® +Z i ¯). The baseline function h 0 (T) = º!(ºT) !¡1 1+(ºT) ! is estimated jointly with the covariate parameters, and is plotted in figure 3. The ® coefficients of time dummies are presented in figure 4. The reference group are loans backed by retail property, with original balance 0.35-18.25 million and loan-to-value ratio (LTV) 60-80%, within Northeast/Mid-Atlantic region and small MSAs, having amortization term between 20 and 30 years and maturity term between 5 and 10 years. The asymptotic covariance is calculated with the infor- mation matrix using the BHHH method. Covariates Estimate Asy. S.E. T stat. Multifamily 0.224 0.049 4.591 Office -0.025 0.071 -0.352 Industrial 0.041 0.076 0.544 Hotel 1.148 0.062 18.602 Healthcare 0.943 0.084 11.194 Other 0.053 0.075 0.710 Midwest/Eastern 0.210 0.067 3.147 Midwest/Western 0.034 0.103 0.328 Northeast/New-England -0.076 0.081 -0.932 Southern/Atlantic 0.084 0.062 1.350 Southern/East-coast 0.251 0.093 2.701 Southern/West-coast 0.246 0.067 3.665 Western/Mountain 0.044 0.076 0.580 68 Table 5, continued Maximum Likelihood Estimates of the Hazard Model Covariates Estimate Asy. S.E. T stat. Western/Northern Pacific -0.349 0.086 -4.076 Western/Southern Pacific -0.523 0.087 -5.984 Small MSA 0.082 0.049 1.687 Non-MSA -0.456 0.069 -6.600 Loan-to-value ratio (LTV)· 40% -0.074 0.092 -0.800 Loan-to-value ratio (LTV) 40-60% 0.040 0.050 0.815 Loan-to-value ratio (LTV)> 80% 0.141 0.082 1.717 Amortization term· 10 years 0.609 0.080 7.583 Amortization term 10-20 years 0.416 0.051 8.141 Amortization term> 30 years -0.035 0.169 -0.209 Maturity term· 5 years 2.588 0.058 44.314 Maturity term 10-15 years -0.624 0.074 -8.426 Maturity term> 15 years -0.861 0.077 -11.172 Original balance· 0.35 m 0.455 0.070 6.492 Original balance> 18.25 m -0.276 0.108 -2.546 Baseline parameter! 2.630 0.057 46.386 Baseline parameterº 0.065 0.002 26.125 Calendar time dummies In figure 4 69 Figure 4.3 plots the estimated seasoning effect, which shows a humped shape. Default hazard rate increases rapidly in the first 3 (duration) years, reaches its maximum in four and a half years, and then declines gradually. This pattern is generally consistent with findings of Esaki (2002), who find commercial mortgage loan default rate to increase with duration, reach the highest level in year 4, and then have small changes until year 7. Interestingly, the decline of commercial mortgage hazard rate after the peak year is much slower than that of residential mortgage. The magnitude of commercial mortgage hazard rate is apparently much higher than that of residential mortgage. The figure shows that for mortgages in a fair year, the peak default hazard rate (with respect to seasoning) is about 2.0%, while the 100% SDA has a peak hazard rate of only 0.6%. The 5 year and 7 year cumulative default probabilities are 2.30% and 4.99% based on these estimates. As mentioned earlier, the focus of this paper is really on the time series dynamics of commercial mortgage default risk. Figure 4.4 shows the estimated hazard rate time series of a representative mortgage. At least four points can be taken from these results: first, there are large variations in default risk over time. The annual default hazard rate of a five-year seasoned commercial mortgage could be as low as 0.40% (16.47% of sample average) in good years, and could be as high as 5.68% (232.72% of sample average) in bad years. Second, the changes in default hazard rate is generally consistent with changes in macroeconomic conditions, e.g. default risk increase substantially during 2001 and 2002, when the economy enters a recession. Third, the observed high default 70 hazard rate during 1997-1998 when the economy is strong points to the importance of the market-specific factor. In fact, large scales of foreign investment enters the US com- mercial real estate market in the 1990s, and the Asian financial crisis and the Russian financial crisis cause a lot of problems in those properties. Last, default hazard rate tends to have a cyclical movement, going down in 1994-1996, rising up in 1997-1998, and going down again in 1999-2000 and rising up again in 2001-2002. Figure 4.5 and Figure 4.6 plots the estimated seasoning effect and the estimated hazard rate time series, respectively, based on stratified samples. Figure 4.7 plots the 5-year risk-free pure discount bond yields. The bond yields tend to move oppositely with the default hazard rates shown in figure 5. Notice that in my model low bond yield implies lower expected economic growth rate. Therefore, this is a sensible pattern. Table 6 reports the core results of the paper. For identification purposes, I do not esti- mate all the free parameters in my model. Instead, I choose to pre-specify some of the parameters, e.g. the risk aversion parameter° is chosen to be 2 and the subjective dis- count factor± is 4% according to existing literature. The volatilities of economic growth ¾ e is fixed at 8%, which is close to the volatility of industrial output we observe 5 . The 5 In appendix table 4, I report estimation results of the state space model based on different¾ e value assumptions. 71 estimates show the existence of two significant mean-reverting unobservable state vari- ables underlying commercial mortgage default. The expected economic growth has a long term mean of 1.85%, volatility of 2.05% and annual mean-reversion speed of 0.33. The commercial property market-specific factor has a long term growth rate of 1.50% and mean-reversion speed of 0.591. The volatility of this property market-specific fac- tor is 2.08%. As shown in the table, dynamics of default hazard rate can be largely explained by the dynamics of these two mean-reverting latent variables. The root mean square error (RMSE) is less than 20% of the sample mean, which implies that changes of these two factors explain over 80 percent of the variations in default hazard rate. 72 Table 6 Estimates of the State Space Model This table presents the maximum likelihood estimates of the state space model. The processes of the two unobservable state variables, expected economic growth rate and expected commercial property market-specific growth rate, are d¹ t =·(¹¡¹ t )dt+¾ ¹ dW e t d» t =¸(»¡» t )dt+¾ » dW K t The relationship between these two state variables and the two measurement vari- ables, bond yieldR t and default hazard rateh t are given in equations (11) and (19). For identification purposes, I pre-specify the risk aversion parameter° and the sub- jective discount factor± to be 2 and 4% based on the existing literature. Here the volatility of economic growth ¾ e is pre-specified at 8%. In appendix tables 4 and 5, I report the model estimates assuming¾ e to be 7% and 9% respectively. For the discount bond yield, observation error p § 11 is assumed to be zero. The parameters and state variables are jointly estimated with the extended Kalman Filter, and the asymptotic covariance is calculated with the information matrix using the BHHH method. The root-mean-square error (RMSE) is between the fitted hazard rates and the actual hazard rates. Parameters Estimates Estimate Asy. S.E. T stat. Macroeconomic factor ¹ 0.0185 0.0083 2.2289 · 0.3288 0.1130 2.9097 ¾ ¹ 0.0205 0.0045 4.5556 73 Table 6, continued Estimates of the State Space Model Parameters Estimates Estimate Asy. S.E. T stat. Macroeconomic factor ¹ 0.0185 0.0083 2.2289 · 0.3288 0.1130 2.9097 ¾ ¹ 0.0205 0.0045 4.5556 Property market factor » 0.0150 0.0063 2.3810 ¸ 0.5910 0.3041 1.9434 ¾ » 0.0208 0.0093 2.2366 ¾ · 0.0932 0.0232 4.0172 x 0 1.5080 0.3941 3.8264 ¯ 0.5001 0.2420 2.0665 100¢ p § 22 0.0029 0.0013 2.2308 Default series fitting 100¢RMSE 0.0079 100¢Samplemean 0.0424 Fitting of the series 81.37% Based on the model estimates, the long term mean of spot interest rate is 4.4%, and the long term mean of annual default hazard rate is 0.01% for a newly originated loan 74 and 2.31% for a five-year seasoned loan. Table 7 also shows default hazard rates of a representative mortgage with different seasoning under a favorable environment and under a pessimistic environment. Table 7 Default Hazard Rates Under Different Scenarios This table presents annual default hazard rates (%) of a representative commercial mortgage with various seasoning under different scenarios based on my model esti- mates. The long term mean default hazard rate is calculated by assuming the two state variables are at their long term means. The optimistic (pessimistic) scenario assumes the two state variables are two standard errors above (below) their long term means. Seasoning Optimistic Long term mean Pessimistic 1 month 0.0047 0.0087 0.0159 1 year 0.2609 0.4878 0.8880 3 year 1.0588 1.9792 3.6032 5 year 1.2363 2.3111 4.2075 7 year 1.0986 2.0535 3.7387 Simulations shown in figures 4.8 and 4.9 tells that commercial mortgage default risk is more sensitive when the economy is in downturn than at a time when the economic growth is high. There is also a negative relationship between default risk and interest rate (figure 4.10), conforming to the existing literature (Duffee 1998 and Longstaff and Schwartz 1995). This relationship provides hedging implications for portfolio credit risk. 75 The final point to take from the empirical estimates is that the insolvency threshold for default is 86% of the monthly mortgage payment, assuming an underwriting debt- service-coverage ratio (DSCR) of 1.3. This is calculated from the relationship x 0 = K 0 ÁC = DSCR Á . Since the estimated x 0 is 1.51, Á is 0.86. This result is consistent with the observation that mortgage borrowers do not default ruthlessly possibly because of substantial transaction cost of default. 76 1 2 3 4 5 6 7 8 9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Duration Year Default hazard rate 0.28 0.73 1.07 1.19 1.04 0.92 0.95 0.96 0.70 Figure 4.2: Raw Plots of the Commercial Mortgage Default Seasoning. This figure plots the raw annual default hazard rates of commercial mortgages with different seasoning. 77 1 12 24 36 48 60 72 84 96 108 0 0.5 1 1.5 2 2.5 3 Duration month Annual hazard rate (%) Max at month 54 Figure 4.3: Estimated Seasoning Effect of Commercial Mortgage Loan Default. This figure plots the hazard rates of a commercial mortgage over duration time, which shows the seasoning effect of default. The baseline parametersb º andb ! are 0.065 and 2.630 respectively, and the covariates are at sample mean. The hazard rate reaches its maxi- mum at 54 month (4.5 years). The 5-year cumulative default probability is 2.30%, and the 7-year cumulative default probability is 4.99% based on these estimates. 78 6/1994 6/1995 6/1996 6/1997 6/1998 6/1999 6/2000 6/2001 6/2002 6/2003 0 1 2 3 4 5 6 7 8 Calendar month Annual hazard rate (%) Mean = 2.44% STD = 1.24% Figure 4.4: Estimated Hazard Rate Time Series of a Representative Mortgage. The fig- ure plots the hazard rates of a 4.5-year seasoned representative commercial mortgage over calendar time, which shows the time series dynamics of default. The hazard rate varies substantially over time from 16.47% of the sample mean to 232.72% of the sam- ple mean. 79 1 12 24 36 48 60 72 84 96 108 0 0.5 1 1.5 2 2.5 3 Duration month Annual hazard rate (%) All loans Multifamily loans Retail loans Figure 4.5: Estimated Seasoning Effect for different property type Commercial Mort- gages. This figure plots the estimated hazard rates over duration time for multifamily commercial mortgages and retail commercial mortgages in addition to that for all com- mercial mortgages pooled together. 80 6/1994 6/1995 6/1996 6/1997 6/1998 6/1999 6/2000 6/2001 6/2002 6/2003 0 1 2 3 4 5 6 7 8 Calendar month Annual hazard rate (%) All loans Multifamily loans Retail loans Figure 4.6: Estimated Hazard Rate Time Series for different property type Commercial Mortgages. The figure plots the estimated hazard rates over calendar time for 4.5-year seasoned commercial mortgages backed by different type of properties. 81 6/1994 6/1995 6/1996 6/1997 6/1998 6/1999 6/2000 6/2001 6/2002 6/2003 0 2 4 6 8 10 12 Calendar month Bond Yield (%) Real Nominal Figure 4.7: The Time Series of the 5-year Pure Discount Bond Yield. This figure plots the Fama-Bliss (artificial) 5-year pure discount bond yields. The nominal yields are from CRSP, and CPI is used to make adjustments to obtain the real yields. 82 mulo mubar muhi 0 0.005 0.01 0.015 0.02 0.025 Expected economic growth 1−month default hazard rate (%) Xibar Xilo Xihi Figure 4.8: Simulated Relationship between Hazard Rate and the Macroeconomy. 83 xilo xibar xihi 0 0.005 0.01 0.015 0.02 0.025 Expected property market−specific growth 1−month default hazard rate (%) Mubar Mulo Muhi Figure 4.9: Simulated Relationship between Hazard Rate and the Property Market Fac- tor. 84 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 Short rate 1−month default hazard rate (%) Xibar Xilo Xihi Figure 4.10: Simulated Relationship between Hazard Rate and the Instantaneous Inter- est Rate. 85 Chapter 5 Conclusions and Discussions Credit risk not only varies across firms or borrowers, but also changes over time depend- ing on market environments. Understanding the time series dynamics of credit risk and their underlying risk factors is important to portfolio risk management, hedging and pricing. The mortgage credit risk literature has extensively examined the cross sectional differ- ences in default risk, but has limited work on the time series properties of default risk. I study the time series dynamics of commercial mortgage credit risk and the unobserv- able systematic risk factors underlying those dynamics by developing and estimating a structural model. I present a first-passage model for commercial mortgage credit risk and solve the default hazard rate under real measure. The model is then estimated with commercial mortgage 86 default data using extended Kalman filter. The estimates show the existence of two sig- nificant latent factors underlying commercial mortgage default. The expected macroe- conomic growth is one factor and the commercial property market-specific growth is the other factor. Both factors are mean-reverting, and the elasticity of commercial mortgage default probability with respect to the expected macroeconomic growth increases when the economy deteriorates. The estimates also reveal a negative relationship between the instantaneous risk free interest rate and commercial mortgage default probability. The variations in default hazard rate are well explained by the dynamics of the two mean-reverting latent factors. Empirical estimates also confirm the notion that there is substantial transaction cost of default. The estimated model is useful in predicting long term dynamics of credit risk. It also provides hedging implications for portfolio credit risk. From a methodological per- spective, it bridges the gap between theoretical credit risk models and empirical default estimations by providing an integrated credit risk model. Future research can go several directions. First, more measurement variables can be brought in to identify all the free parameters in the model, e.g. incorporating the dynam- ics of the whole term structure of interest rate will provide additional degrees of freedom. Second, more elements can be incorporated into the current theoretical framework, e.g. an exogenously given inflation process as modelled in Wachter (2006) is desirable, and 87 mean-reverting asset risk premium (Huang and Huang, 2003) might also better approx- imate real world situations. Third, pricing is definitely an interesting topic. Given the current structure of the model, we can extend the study into commercial mortgage pric- ing once data is available. In addition, an important application of my model will be in the residential mortgage market. In fact, with much longer data series in residential mortgage market, we can perform empirical tests of credit cycles. 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Multifamily 0.226 0.049 4.616 Office -0.019 0.071 -0.272 Industrial 0.050 0.076 0.656 Hotel 1.153 0.063 18.443 Healthcare 0.943 0.084 11.190 Other 0.062 0.076 0.815 Midwest/Eastern 0.223 0.067 3.325 Midwest/Western 0.045 0.103 0.432 Northeast/New-England -0.070 0.081 -0.870 Southern/Atlantic 0.095 0.062 1.515 Southern/East-coast 0.264 0.093 2.854 Southern/West-coast 0.255 0.067 3.791 Western/Mountain 0.055 0.077 0.704 Western/Northern Pacific -0.340 0.088 -3.876 Western/Southern Pacific -0.514 0.089 -5.770 99 Appendix table 1, continued Estimates of the Model with Originator Dummies Covariates Estimate Asy. S.E. T stat. Small MSA 0.083 0.049 1.707 Non-MSA -0.461 0.069 -6.662 Loan-to-value ratio (LTV)· 40% -0.072 0.090 -0.800 Loan-to-value ratio (LTV) 40-60% 0.040 0.050 0.814 Loan-to-value ratio (LTV)> 80% 0.142 0.082 1.732 Amortization term· 10 years 0.607 0.080 7.617 Amortization term 10-20 years 0.416 0.052 8.067 Amortization term> 30 years -0.038 0.170 -0.222 Maturity term· 5 years 2.599 0.060 43.078 Maturity term 10-15 years -0.621 0.077 -8.035 Maturity term> 15 years -0.863 0.081 -10.692 Original balance· 0.35 m 0.454 0.070 6.522 Original balance> 18.25 m -0.270 0.109 -2.473 100 Appendix table 1, continued Estimates of the Model with Originator Dummies Covariates Estimate Asy. S.E. T stat. Bank of America -0.019 0.085 -0.226 Citi Corporation -0.006 0.116 -0.049 Column 0.008 0.069 0.113 GE Capital -0.040 0.149 -0.266 GMAC 0.006 0.110 0.055 JPMorgan Chase -0.003 0.114 -0.025 Lehman -0.018 0.100 -0.178 Merrill Lynch 0.008 0.110 0.069 Midland 0.003 0.121 0.027 Morgan Stanley -0.018 0.144 -0.122 Nomura -0.001 0.101 -0.005 Wachovia 0.003 0.092 0.032 Wells Fargo -0.035 0.182 -0.191 Baseline parameter 2.615 0.057 45.527 Baseline parameter 0.065 0.003 24.342 101 Appendix Table 2 Estimates of the Hazard Model for Multifamily Loans This table presents the maximum likelihood estimates of the hazard model for multifam- ily loans only. Covariates Estimate Asy. S.E. T stat. Midwest/Eastern 0.263 0.116 2.270 Midwest/Western 0.019 0.215 0.087 Northeast/New-England -0.037 0.143 -0.256 Southern/Atlantic 0.134 0.114 1.179 Southern/East-coast 0.337 0.170 1.983 Southern/West-coast 0.287 0.115 2.502 Western/Mountain 0.094 0.143 0.657 Western/Northern Pacific -0.366 0.199 -1.835 Western/Southern Pacific -0.510 0.194 -2.634 Small MSA 0.073 0.092 0.799 Non-MSA -0.480 0.151 -3.183 102 Appendix Table 2, continued Estimates of the Model for Multifamily Loans Covariates Estimate Asy. S.E. T stat. Loan-to-value ratio (LTV)· 40% -0.094 0.194 -0.486 Loan-to-value ratio (LTV) 40-60% 0.046 0.105 0.440 Loan-to-value ratio (LTV)> 80% 0.134 0.150 0.894 Amortization term· 10 years 0.617 0.177 3.493 Amortization term 10-20 years 0.404 0.138 2.918 Amortization term> 30 years 0.007 0.304 0.024 Maturity term· 5 years 2.671 0.120 22.166 Maturity term 10-15 years -0.722 0.224 -3.229 Maturity term> 15 years -0.937 0.208 -4.509 Original balance· 0.35 m 0.446 0.128 3.483 Original balance> 18.25 m -0.296 0.356 -0.832 Baseline parameter 2.550 0.098 25.933 Baseline parameter 0.065 0.005 14.066 Calendar time dummies In app. fig. 2 103 Appendix Table 3 Estimates of the Hazard Model for Retail Loans This table presents the maximum likelihood estimates of the hazard model for retail loans only. Covariates Estimate Asy. S.E. T stat. Midwest/Eastern 0.213 0.146 1.452 Midwest/Western 0.076 0.200 0.380 Northeast/New-England -0.076 0.192 -0.397 Southern/Atlantic 0.087 0.133 0.657 Southern/East-coast 0.248 0.196 1.268 Southern/West-coast 0.271 0.144 1.885 Western/Mountain -0.070 0.184 -0.381 Western/Northern Pacific -0.359 0.201 -1.790 Western/Southern Pacific -0.501 0.178 -2.812 Small MSA 0.122 0.104 1.171 Non-MSA -0.428 0.156 -2.747 104 Appendix Table 3, continued Estimates of the Hazard Model for Retail Loans Covariates Estimate Asy. S.E. T stat. Loan-to-value ratio (LTV)· 40% -0.030 0.241 -0.127 Loan-to-value ratio (LTV) 40-60% 0.083 0.112 0.741 Loan-to-value ratio (LTV)> 80% 0.132 0.165 0.797 Amortization term· 10 years 0.565 0.173 3.274 Amortization term 10-20 years 0.391 0.107 3.656 Amortization term> 30 years -0.048 0.427 -0.113 Maturity term· 5 years 2.702 0.146 18.556 Maturity term 10-15 years -0.495 0.143 -3.463 Maturity term> 15 years -0.832 0.148 -5.622 Original balance· 0.35 m 0.495 0.193 2.566 Original balance> 18.25 m -0.185 0.194 -0.953 Baseline parameter 2.582 0.158 16.325 Baseline parameter 0.065 0.006 10.559 Calendar time dummies In app. fig. 2 105 Appendix table 4 Estimates of the State Space Model This table presents the maximum likelihood estimates of the state space model as shown in table 6 under different assumptions of the economic growth volatility parameter¾ e . Panel A:¾ e =7% Parameters Estimates Estimate Asy. S.E. T stat. ¹ 0.0155 0.0081 1.9136 · 0.3357 0.1146 2.9293 ¾ ¹ 0.0207 0.0046 4.5000 » 0.0167 0.0065 2.5692 ¸ 0.5890 0.3051 1.9305 ¾ » 0.0217 0.0101 2.1485 ¾ · 0.0946 0.0237 3.9916 x 0 1.5082 0.3951 3.8173 ¯ 0.5003 0.2420 2.0674 100¢ p § 22 0.0031 0.0013 2.3846 Default series fitting 100¢RMSE 0.0083 100¢Samplemean 0.0424 Fitting of the series 80.42% 106 Appendix table 4, continued Estimates of the State Space Model Panel B:¾ e =9% Parameters Estimates Estimate Asy. S.E. T stat. ¹ 0.0219 0.0085 2.5765 · 0.3209 0.1108 2.8962 ¾ ¹ 0.0202 0.0044 4.5909 » 0.0138 0.0056 2.4643 ¸ 0.5732 0.3023 1.8961 ¾ » 0.0232 0.0115 2.0174 ¾ · 0.0895 0.0230 3.8913 x 0 1.5079 0.3951 3.8165 ¯ 0.4911 0.2417 2.0319 100¢ p § 22 0.0032 0.0014 2.2857 Default series fitting 100¢RMSE 0.0084 100¢Samplemean 0.0424 Fitting of the series 80.19% 107

Abstract (if available)

Abstract I study the time series dynamics of commercial mortgage credit risk and the unobservable systematic risk factors underlying those dynamics. The research is conducted within a structural model framework. I modify the first-passage model of Black and Cox (1976) and Longstaff and Schwartz (1995) to capture unique features of commercial mortgage default. First, the first-passage condition is imposed on net operating income (NOI) rather than on property value, which reflects the fact that commercial mortgages mainly rely on the income from underlying properties to service the debt. Second, equilibrium macroeconomic dynamics are linked to commercial mortgage default through the NOI function, based on the observation that commercial property cash flow is largely affected by macroeconomic conditions. Third, I solve the default hazard rate of a representative commercial mortgage as a function of two unobservable state variables, the expected macroeconomic growth and expected property market-specific growth. The solutions of the model provide an estimable system that includes the relationship between systematic risk factors and commercial mortgage credit risk, as well as the dynamics of the risk factors.

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University of Southern California Dissertations and Theses

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Housing consumption-based asset pricing and residential mortgage default risk

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Risks, returns, and regulations in real estate markets

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Creator An, Xudong (author)

Core Title Macroeconomic conditions, systematic risk factors, and the time series dynamics of commercial mortgage credit risk

School School of Policy, Planning, and Development

Degree Doctor of Philosophy

Degree Program Planning

Publication Date 07/05/2009

Defense Date 05/08/2007

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag commercial mortgage,Cox proportional hazard,credit risk,default hazard rate,extended Kalman filter,first-passage model,OAI-PMH Harvest,state space model

Language English

Advisor Deng, Yongheng (committee chair), Bostic, Raphael W. (committee member), Gabriel, Stuart A. (committee member), Jones, Christopher S. (committee member), Zapatero, Fernando (committee member)

Creator Email xudongan@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m586

Unique identifier UC1411519

Identifier etd-An-20070705 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-515224 (legacy record id),usctheses-m586 (legacy record id)

Legacy Identifier etd-An-20070705.pdf

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Document Type Dissertation

Rights An, Xudong

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu