A Semiparametric Stochastic Spline Model as a Managerial Tool for Potential Insolvency

This study introduces a flexible nonlinear semiparametric spline model, new to solvency studies, as a tool for managerial discretion and regulatory oversight. The model has a linear component and a nonlinear component that uses stochastic splines. The study focuses on the functional relationship between regressors and the probability of financial distress as an object for managerial action. Leverage plots are provided to analyze the potential effect of decisions to modify firm levels of financial variables. If the true relationship between regressors and the response is not linear, then managerial efforts to rectify deteriorating financial conditions can be misinformed by reliance on a linear solvency model. The leverage plots adjust to the firm’s position within the industry and its specific levels of various financial variables. A five-regressor semiparametric spline model is shown to yield insights into the behavior of the risk of financial distress probabilities that linear parametric models suppress. The model also classifies and validates well in comparison with recent insolvency studies and as well as parametric logit and probit models on the same data. INTRODUCTION This article explores the suitability of linear modeling in a test application to prediction of financial distress for firms in the life insurance industry. Financial distress is a slightly broader category than insolvency, as distress also includes confidential supervision by state regulatory agencies. The study introduces a flexible, nonlinear, semiparametric stochastic spline model, new to solvency studies, to compare with logit and probit models.1 The advantage of using a semiparametric spline model is Etti G. Baranoff is assistant professor of insurance at Virginia Commonwealth University, School of Business, Richmond, Va. Thomas W. Sager is professor of statistics and is director— Center for Statistical Sciences, Department of Management Science and Information Systems, The University of Texas at Austin. Thomas S. Shively is professor of statistics in the Department of Management Science and Information Systems, the University of Texas at Austin. 1 The terms “linear” and “parametric” are sometimes confused. A linear model contains a linear combination of predictor variables at its core. “Parametric,” “semiparametric,” and “nonparametric” represent rough ranges on a continuum describing the degrees of freedom permitted the functional form of a model, from few to moderate to many, respectively. Thus, linear models are usually parametric, and nonparametric models are usually nonlinear. See Hastie and Tibshirani (1990) for an excellent introduction to semiparametric and nonparametric models. 370 THE JOURNAL OF RISK AND INSURANCE that no strong assumptions (e.g., linearity) are made regarding the functional forms used to model the relationships between the probability of distress and the explanatory variables. The data are allowed to specify the appropriate functional forms. The semiparametric model that is used reduces to the standard linear probit model if the latter model is appropriate. Although thesemiparametric spline model performs as well as logit and probit models at overall classification and validation of distress, improvement of the rate of correct classification is not the primary objective of the article. Rather, the focus is on the functional relationship between the financial regressors and the probability of distress as a tool for managerial discretion. If the true relationship between regressors and response is not linear, then managerial efforts to rectify deteriorating financial conditions can be badly misinformed by reliance on a linear model. Consider as a simple example a distress response ( ) 1, 2 X X q 2 depending on two financial predictors, X1 and X2, both subject to managerial control. In a linear model, the partial derivatives (slopes) of q are constant, whatever the level of X1 and X2. In a nonlinear model, these partial derivatives are not constant.3 Thus, in linear models, managers may be indifferent to the firm’s levels of X1 and X2 in influencing q . But in nonlinear models, the more efficient variable to manage and the most efficacious actions to reduce q may be highly dependent on the firm’s levels of X1 and X2. This will be true even if the linear and nonlinear models have roughly equivalent distress prediction success rates. Therefore, the levels of X1 and X2 are very important in the semiparametric spline models. This point is the focal contribution of this article. Most statistical models in insurance solvency research are fundamentally linear and parametric, such as logistic regression, probit regression, and discriminant analysis. Yet the suitability of linear assumptions is not often examined. The first parametric insurance solvency models were created for the property-liability industry by Trieschmann and Pinches (1973) and Pinches and Trieschmann (1974) and for the life insurance industry by BarNiv and Hershbarger (1990). For a complete summary of insurance solvency studies up to 1992, see BarNiv and McDonald (1992). More recent studies that used linear classical modeling include Baranoff, Sager, and Witt (1999); BarNiv et al. (1999); Pottier (1998); Grace, Harrington, and Klein (1998); LammTennant, Starks, and Stokes (1996); Carson and Hoyt (1995); Cummins, Harrington, and Klein (1995); and Ambrose and Carroll (1994). Some newer studies also include nonparametric solvency models. These studies include the neural network model by Brockett et al. (1994); the recursive partitioning model (CART) in Carson and Hoyt (1995); and the hazard models in Lee and Urrutia (1996). Table 5 summarizes the results of these articles. It is well known in the statistics profession that nonparametric models may overfit the existing data and fail to validate on new data; in addition, their statistical effi2 The distress response parameter varies among models. (See the Methodology section.) 3 Consider the model ( ) 2 1 2 1 1 2 1 2 , x x x x x q a b b = + + . Although this model is still parametric and has the form of a linear combination, the slope of q with respect to x1 is ( ) 1 2 1 1 2 2 1 , 2 x x x x x q b b ∂ = + ∂ , which varies with the levels of both x1 and x2. A SEMIPARAMETRIC STOCHASTIC SPLINE MODEL AS A MANAGERIAL TOOL FOR POTENTIAL INSOLVENCY 371 ciency may be impaired (see Härdle, 1990, Chapter 1). Semiparametric models can correct the shortcomings of both parametric and nonparametric procedures by adopting the best features of each. The semiparametric spline model includes terms representing a linear component and a nonlinear component. If the nonlinear component is insignificant, then the model reduces to the classical linear form, which is then deemed adequate. If the nonlinear component is significant, then the linear form alone is inadequate, and the model estimates the nonlinear form from the data. The initial version of the model was introduced into the statistical literature by Wahba (1978). Recent articles that discuss extensions and new developments adapted for this article include Wong and Kohn (1996) and Shively, Kohn, and Wood (1999). The results of the analyses suggest that the relationships between critical financial variables and distress may be highly nonlinear. The authors’ primary contribution is the leverage plot, which can be used as a tool for both managers and regulators who seek to control and understand the likelihood of severe financial distress. The authors’ goal is not to improve prediction rates in solvency studies. It is enough that the model be competitive at forecasting distress. Nevertheless, the semiparametric spline model is very competitive on measures of prediction success, with as few as five predictors, relative to logit, probit, cubic logit, and cubic probit models with the same data. The spline model shows modest improvements in expected costs of misclassification. Following this introduction, the semiparametric methodology is presented in the Methodology section. The Data section describes the data, and the Results section provides the leverage plots for the semiparametric spline, logistic, and probit models on the same data. In the section entitled “Validation of the Semiparametric Model and Comparison With Alternatives,” the authors verify that the semiparametric model performs satisfactorily at the traditional solvency model task of classifying firms correctly. In that section the authors also offer a summary comparison to the results of prior nonparametric solvency studies, and the Summary and Conclusion section provides some closing remarks. Appendix 1 contains an intuitive explanation of Bayesian models and their application to semiparametric stochastic spline models. Appendix 2 provides a heuristic derivation of the stochastic spline model. METHODOLOGY The basic form of most regression models (univariate case) is ( ) y f x e = + , (1) where f(x) is the mean of y at x and ε is the error term. Interest focuses on the form of the unknown function f(x), which specifies the nature of the relationship between y and x. In classical models used for solvency studies, f(x) contains a linear function 0 1x b b + at its core. For example, in logistic regression, the log-odds ratio ( ) ( ) 0 1 log / 1 – x p p b b = + (hence, 0 1 1 ( ) exp 1 exp( ) f x x b b   =   + − −   ), and in probit regression the normal probability quantile is ( ) 1 0 1x f p b b − = + (hence, ( ) ( ) 0 1 f x x f b b = + ), where π represents the probability of insolvency at x, i.e., ( ) | E y x p = . 372 THE JOURNAL OF RISK AND INSURANCE To model distress status in this article, the authors use a semiparametric probit regression4 model embedded in a hierarchical Bayes framework that is explained in the appendices. The following notation will be used: Let yj be a 0-1 binary variable that takes on the value 1 if the jth company is in distress; let j p be the conditional probability that the jth company is in distress given the regressors; let xij be the value of the ith regressor for company j; and let p be the number of regressors in the model. Using this notation, the semiparametric probit model is