Estimation of Default Probabilities with Support Vector Machines

Predicting default probabilities is important for firms and banks to operate successfully and to estimate their specific risks. There are many reasons to use nonlinear techniques for predicting bankruptcy from financial ratios. Here we propose the so called Support Vector Machine (SVM) to estimate default probabilities of German firms. Our analysis is based on the Creditreform database. The results reveal that the most important eight predictors related to bankruptcy for these German firms belong to the ratios of activity, profitability, liquidity, leverage and the percentage of incremental inventories. Based on the performance measures, the SVM tool can predict a firms default risk and identify the insolvent firm more accurately than the benchmark logit model. The sensitivity investigation and a corresponding visualization tool reveal that the classifying ability of SVM appears to be superior over a wide range of the SVM parameters. Based on the nonparametric Nadaraya-Watson estimator, the expected returns predicted by the SVM for regression have a significant positive linear relationship with the risk scores obtained for classification. This evidence is stronger than empirical results for the CAPM based on a linear regression and confirms that higher risks need to be compensated by higher potential returns.

[1]  E. Altman,et al.  ZETATM analysis A new model to identify bankruptcy risk of corporations , 1977 .

[2]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.

[3]  Marjorie B. Platt,et al.  BANKRUPTCY DISCRIMINATION WITH REAL VARIABLES , 1994 .

[4]  S. Ross.  The arbitrage theory of capital asset pricing , 1976 .

[5]  Pavel V. Gapeev,et al.  Discounted Optimal Stopping for Maxima of Some Jump-Diffusion Processes , 2007, Journal of Applied Probability.

[6]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments. , 1960 .

[7]  W. Sharpe.  CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* , 1964 .

[8]  E. Fama,et al.  The CAPM is Wanted, Dead or Alive , 1996 .

[9]  Edward I. Altman,et al.  FINANCIAL RATIOS, DISCRIMINANT ANALYSIS AND THE PREDICTION OF CORPORATE BANKRUPTCY , 1968 .

[10]  S. Myers.  Determinants of corporate borrowing , 1977 .

[11]  D. Hernández-Hernández,et al.  A control approach to robust utility maximization with logarithmic utility and time-consistent penalties , 2007 .

[12]  Robert A. Lordo,et al.  Nonparametric and Semiparametric Models , 2005, Technometrics.

[13]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[14]  Kaisa Sere,et al.  Choosing Bankruptcy Predictors Using Discriminant Analysis, Logit Analysis, and Genetic Algorithms , 1996 .

[15]  Clive S. Lennox,et al.  Identifying failing companies: a re-evaluation of the logit, probit and DA approaches , 1999 .

[16]  H. Markowitz.  Portfolio Selection: Efficient Diversification of Investments , 1971 .

[17]  The Division of Ownership in New Ventures , 2004 .

[18]  Charles L. Merwin,et al.  Financing Small Corporations in Five Manufacturing Industries, 1926-36. , 1944 .

[19]  Enno Mammen,et al.  Estimating a Smooth Monotone Regression Function , 1991 .

[20]  Lea V. Carty,et al.  MOODY'S RISKCALC™ FOR PRIVATE COMPANIES: , 2001 .

[21]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments , 1959 .

[22]  Charles E. Heckler,et al.  Applied Multivariate Statistical Analysis , 2005, Technometrics.

[23]  P. Bolton,et al.  An incomplete contracts approach to financial contracting , 1992 .

[24]  Richard Roll,et al.  A Critique of the Asset Pricing Theory''s Tests: Part I , 1977 .

[25]  J. Lintner.  THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS , 1965 .

[26]  W. Beaver.  Financial Ratios As Predictors Of Failure , 1966 .

[27]  The Source of Historical Economic Fluctuations: An Analysis Using Long-Run Restrictions , 2004 .

[28]  W. Härdle,et al.  Statistical Tools for Finance and Insurance , 2003 .

[29]  P. Gapeev.  Perpetual barrier options in jump-diffusion models , 2007 .

[30]  J. Kalbfleisch.  Statistical Inference Under Order Restrictions , 1975 .

[31]  Wolfgang Härdle,et al.  On the appropriateness of inappropriate VaR models , 2006 .

[32]  Lea V. Carty,et al.  Riskcalc for Private Companies: Moody's Default Model , 2000 .

[33]  Pavel V. Gapeev,et al.  On Maximal Inequalities for some Jump Processes , 2006 .

[34]  R. C. Merton,et al.  AN INTERTEMPORAL CAPITAL ASSET PRICING MODEL , 1973 .

[35]  Roger M. Stein,et al.  Benchmarking Quantitative Default Risk Models: A Validation Methodology , 2000 .

[36]  Harald Uhlig,et al.  The Welfare Enhancing Effects of a Selfish Government in the Presence of Uninsurable, Idiosyncratic Risk , 2006 .

[37]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[38]  Pavel V. Gapeev,et al.  Integral Options in Models with Jumps , 2006 .

[39]  E. Fama,et al.  Risk, Return, and Equilibrium: Empirical Tests , 1973, Journal of Political Economy.

[40]  E. Fama,et al.  Common risk factors in the returns on stocks and bonds , 1993 .

[41]  J. Swart,et al.  Financial Aspects of Corporate Net Worth.@@@Changes in the Financial Structure of Unsuccessful Industrial Corporations. , 1936 .

[42]  Pavel V. Gapeev,et al.  Discounted Optimal Stopping for Maxima in Diffusion Models with Finite Horizon , 2006 .

[43]  Wolfgang Härdle,et al.  Graphical Data Representation in Bankruptcy Analysis , 2006 .

[44]  F. Black,et al.  The Capital Asset Pricing Model: Some Empirical Tests , 2006 .

[45]  V. Vapnik.  Estimation of Dependences Based on Empirical Data , 2006 .

[46]  James A. Ohlson.  FINANCIAL RATIOS AND THE PROBABILISTIC PREDICTION OF BANKRUPTCY , 1980 .

[47]  J. Nazuno.  Haykin, Simon. Neural networks: A comprehensive foundation, Prentice Hall, Inc. Segunda Edición, 1999 , 2000 .

[48]  Harald Uhlig,et al.  How Far are We from the Slippery Slope? The Laffer Curve Revisited , 2006, SSRN Electronic Journal.

[49]  Douglas T. Breeden.  An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities , 1979 .

[50]  Andrew W. Lo,et al.  Logit versus discriminant analysis : A specification test and application to corporate bankruptcies , 1986 .

[51]  H. D. Brunk,et al.  AN EMPIRICAL DISTRIBUTION FUNCTION FOR SAMPLING WITH INCOMPLETE INFORMATION , 1955 .

[52]  M. Subrahmanyam,et al.  When are Options Overpriced? The Black-Scholes Model and Alternative Characterizations of the Pricing Kernel , 1999 .

[53]  J. L. Bicksler,et al.  Studies in the Theory of Capital Markets. , 1973 .

[54]  Ramesh Sharda,et al.  Bankruptcy prediction using neural networks , 1994, Decis. Support Syst..