Statistical Methods in Credit Risk Modeling - Deep …

1 Statistical Methods in Credit Risk ModelingbyAijun ZhangA dissertation submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy(Statistics)in The University of Michigan2009 Doctoral Committee:Professor Vijayan N. Nair, Co-ChairAgus Sudjianto, Co-Chair, Bank of AmericaProfessor Tailen HsingAssociate Professor Jionghua JinAssociate Professor Ji Zhuc Aijun Zhang2009 All Rights ReservedTo my elementary school, high school and university teachersiiACKNOWLEDGEMENTSF irst of all, I would express my gratitude to my advisor Prof. Vijay Nair for guidingme during the entire PhD research. I appreciate his inspiration, encouragement andprotection through these valuable years at the University of Michigan. I am thankfulto Julian Faraway for his encouragement during the first years of my PhD journey. Iwould also like to thank Ji Zhu, Judy Jin and Tailen Hsing for serving on my doctoralcommittee and helpful discussions on this thesis and other research am grateful to Dr.

2 Agus Sudjianto, my co-advisor from Bank of America, forgiving me the opportunity to work with him during the summers of 2006 and 2007and for offering me a full-time position. I appreciate his guidance, active supportand his many illuminating ideas. I would also like to thank Tony Nobili, Mike Bonn,Ruilong He, Shelly Ennis, Xuejun Zhou, Arun Pinto, and others I first met in 2006at the Bank. They all persuaded me to jump into the area of Credit risk research; Idid it a year later and finally came up with this thesis within two more would extend my acknowledgement to Prof. Kai-Tai Fang for his consistentencouragement ever since I graduated from his class in Hong Kong 5 years thesis is impossible without the love and remote support of my parents inChina. To them, I am most OF CONTENTSDEDICATION..iiACKNOWLEDGEMENTS.. iiiLIST OF FIGURES..viLIST OF TABLES..ixABSTRACT..xCHAPTERI. An Introduction to Credit Risk Modeling .

3 Two Worlds of Credit Risk .. Spread Puzzle .. Defaults .. Credit Risk Models .. Approach .. Approach .. Survival Models .. Default Intensity .. Covariates .. Credit Defaults .. Scope of the Thesis ..18II. Adaptive Smoothing Spline.. Introduction .. AdaSS: Adaptive Smoothing Spline .. Kernels .. Penalty Adaptation .. AdaSS Properties .. Experimental Results .. Summary .. Technical Proofs ..46 III. Vintage Data Analysis.. Introduction .. MEV Decomposition Framework .. Gaussian Process Models .. Kernels .. , Spline and Kernel Methods .. Backfitting Algorithm .. Semiparametric Regression .. Segment .. Segments .. Applications in Credit Risk Modeling .. Study .. Default Rates .. Loan Loss Rates .. Discussion .. Technical Proofs ..86IV. Dual-time Survival Analysis.. Introduction .. Nonparametric Methods .

4 Hazards .. Estimator .. Decomposition .. Structural Models .. Parameterization .. of Covariate Effects .. Dual-time Cox Regression .. Cox Models .. Likelihood Estimation .. Vintage Effects .. Applications in Retail Credit Risk Modeling .. Card Portfolios .. Competing risks .. Summary .. Supplementary Materials .. 130 BIBLIOGRAPHY..137vLIST OF s-rated corporate default rates, bond spreads and NBER-dated recessions. Data sources: a) Moody s Baa & Aaa corpo-rate bond yields ( );b) Moody s Special Comment on Corporate Default and RecoveryRates, 1920-2008 ( ); c) NBER-dated reces-sions ( ).. drifted Wiener process, first-passage-time and hazard of retail Credit portfolios and vintage diagram.. s speculative-grade default rates for annual cohorts 1970-2008:projection views in lifetime, calendar and vintage origination time.. road map of thesis developments of Statistical Methods in creditrisk Modeling .

5 Smooth functions: (1)Dopplerfunction simulated withnoise, (2)HeaviSinefunction simulated with noise, (3)Motorcycle-Accidentexperimental data, and (4) Credit -Risktweaked sample.. function estimate and its 2nd-order derivative (upon stan-dardization): scaled signal fromf(t) = sin( t) (upper panel) andf(t) = sin( t4) (lower panel), = 10 ,n= 100 andsnr= 7. Thesin( t4) signal resembles the Doppler function in Figure ; bothhave time-varying frequency.. curve fitting withm= 2 (shown by solid lines). The dashedlines represent 95% confidence intervals. In the Credit -risk case, thelog loss rates are considered as the responses, and the time-dependentweights are specified proportional to the number of replicates.. study ofDopplerandHeaviSinefunctions: OrdSS (blue),AdaSS (red) and the heterogeneous truth (light background).. OrdSS and AdaSS for Motorcycle-Accident Data.. , non-stationary OrdSS and AdaSS: performance in comparison.

6 Estimate of maturation curve for Credit -Risk sample: piecewise-constant 1(t) (upper panel) and piecewise-linear 1(t) (lower panel). diagram upon truncation and exemplified prototypes.. vintage data analysis: (top) underlying true marginal ef-fects; (2nd) simulation with noise; (3nd) MEV Backfitting algorithmupon convergence; (bottom) Estimation compared to the underlyingtruth.. vintage data analysis: (top) projection views of fitted val-ues; (bottom) data smoothing on vintage diagram.. of MEV Backfitting algorithm upon convergence (right panel)based on the squared exponential kernels. Shown in the left panel isthe GCV selection of smoothing and structural parameters.. fitted values: Moody s-rated Corporate Default Rates .. data analysis of retail loan loss rates: (top) projection viewsof emprical loss rates in lifetimem, calendartand vintage originationtimev; (bottom) MEV decomposition effects f(m), g(t) and h(v) (atlog scale).

7 : (left) Lexis diagram of sub-sampled simula-tions; (right) empirical hazard rates in either lifetime or calendar time. estimator vs one-way nonparametric estimator using thedata of (top) both pre-2005 and post-2005 vintages; (middle) pre-2005 vintages only; (bottom) post-2005 vintages only.. Modeling of empirical hazard rates, based on the dual-time-to-default data of (top) both pre-2005 and post-2005 vintages; (middle)pre-2005 vintages only; (bottom) post-2005 vintages only.. analysis of Credit card risk: (top) one-way empiricalhazards, (middle) DtBrewlow estimation, (bottom) MEV decompo-sition.. decomposition of Credit card risk for low, medium and highFICO buckets: Segment 1 (top panel); Segment 2 (bottom panel).. prices and unemployment rate of California state: decomposition of mortgage hazards: default vs. prepayment . time-varying covariates into little time segments, illustrated.

8 133viiiLIST OF s speculative-grade default rates. Data source: Moody s spe-cial comment (release: February 2009) on corporate default and re-covery rates, 1920-2008 ( ) and author s cal-culations.. parameters inDopplerandHeaviSinesimulation study.. vintage data analysis: MEV Modeling exercise with GCV-selected structural and smoothing parameters.. data format of pooled Credit card loans .. covariates considered in mortgage Credit risk Modeling ,where NoteRate could be dynamic and others are static upon origi-nation.. partial likelihood estimation of mortgage covariate effectsin dual-time Cox regression models.. 128ixABSTRACTThis research deals with some Statistical Modeling problems that are motivatedby Credit risk analysis. Credit risk Modeling has been the subject of considerableresearch interest in finance and has recently drawn the attention of Statistical re-searchers. In the first chapter, we provide an up-to-date review of Credit risk modelsand demonstrate their close connection to survival first Statistical problem considered is the development of adaptive smooth-ing spline (AdaSS) for heterogeneously smooth function estimation.

9 Two challengingissues that arise in this context are evaluation of reproducing kernel and determi-nation of local penalty, for which we derive an explicit solution based on piecewisetype of local adaptation. Our nonparametric AdaSS technique is capable of fitting adiverse set of smooth functions including possible jumps, and it plays a key role insubsequent work in the second topic is the development of dual-time analytics for observations in-volving both lifetime and calendar timescale. It includes vintage data analysis (VDA) for continuous type of responses in the third chapter, and dual-time survivalanalysis (DtSA) in the fourth chapter. We propose a maturation-exogenous-vintage(MEV) decomposition strategy in order to understand the risk determinants in termsof self-maturation in lifetime, exogenous influence by macroeconomic conditions, andheterogeneity induced from vintage originations. The intrinsic identification problemis discussed for both VDA and DtSA.

10 Specifically, we consider VDA under Gaus-sian process models, provide an efficient MEV backfitting algorithm and assess itsperformance with both simulation and real on Lexis diagram is of particular importance in Credit risk Modeling wherethe default events could be triggered by both endogenous and exogenous consider nonparametric estimators, first-passage-time parameterization and semi-parametric Cox regression. These developments extend the family of models for bothcredit risk Modeling and survival analysis. We demonstrate the application of DtSAto Credit card and mortgage risk analysis in retail banking, and shed some light onunderstanding the ongoing Credit IAn Introduction to Credit Risk ModelingCredit risk is a critical area in banking and is of concern to a variety of stakehold-ers: institutions, consumers and regulators. It has been the subject of considerableresearch interest in banking and finance communities, and has recently drawn theattention of Statistical researchers.