Quantitative Model for US Issuer Default: Logistic Regression vs. Other Models, Essays (university) of Statistics

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Credit Risk Modeling: Default Probabilities

Jaime Frade

December 28, 2008

Contents

• 1 Introduction
  • 1.1 Credit Risk Methodology
• 2 Preliminaries
  • 2.1 Financial Definitions
• 3 Methods of Modeling Credit Risk
  • 3.1 Existing Methods of Quantitatively Scoring
  • 3.2 Two Classes of Quantitative Models
    • 3.2.1 Statistical Models
    • 3.2.2 Structural Models
• 4 Methodology
  • 4.1 A Statistical Model: Logistic Regression
  • 4.2 Choosing a Significant Model
• 5 Data Description
  • 5.1 Defaulted Bonds
  • 5.2 Data
  • 5.3 Predictors for Logistic Regression Model
• 6 Results
  • 6.1 Saturated Logistic Regression Model - agnostics 6.1.1 Saturated Logistic Regression Model 1: Output and Di-
    • 6.1.2 Saturated Logistic Regression Model 1 Interpretation
  • 6.2 Logistic Regression Reduced Model
    • 6.2.1 Logistic Regression Model 2: Output and Diagnostics
    • 6.2.2 Logistic Regression Model 2: Interpretation
  • 6.3 Logistic Regression Reduced Model
    • 6.3.1 Logistic Regression Model 3: Output and Diagnostics
    • 6.3.2 Logistic Regression Model 3: Interpretation
    • 6.3.3 Logistic Regression Model: Excel Output
• 7 Validation Techniques
  • 7.1 Bootstrapping Confidence Intervals for Accuracy Ratios
    • 7.1.1 Bootstrapping Confidence Intervals: Output
    • 7.1.2 Accuracy Ratios Confidence Intervals: Interpretation

List of Figures

6.1 Correlation Matrix.......................... 12 6.2 SAS Output: Saturated Logistic Model.............. 13 6.3 SAS Output: Reduced Logistic Model 2.............. 14 6.4 SAS Output: Reduced Logistic Model 3.............. 16 6.5 Excel Output: Logistic Models................... 17

7.1 Confidence Intervals for Accuracy Ratios of Model......... 19

8.1 Sorted Comparison of Prediction Results: By Reduced Logistic Model 3................................ 22 8.2 Sorted Comparison of Prediction Results: By Reduced CRE Model 23 8.3 Sorted Comparison of Prediction Results: By Reduced CDP Model 24 8.4 Sorted Comparison of Prediction Results: By Reduced Altman’s Z-Score................................ 25

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Acknowledgments

I would like to thank Dr. Niu for helping me learn the capabilities to incor- porate statistical models and computational techniques in the real world. The foundations and principals in this paper are largely due to his instruction and guidance. I would also like to thank my family and friends for supporting me when I my hard drive crashed and I lost all my data and this paper. They helped me to stay focus and build this from scratch. I would also like to thank my supervisor, Kevin Ceurvorst, and colleague, Ken Hill, at Florida State Board of Administration. They helped me to combine my statistical and computational capabilities with their financial knowledge to produce the models in this paper. They provided guidance and critical ideas that helped me in this paper’s model building process.

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The source language used to compile the model was SAS and Microsoft Ex- cel. Both programs were used for a confirmation of results. Excel was also incorporated so that future usage by other analysts can be easily updated be- cause of the lack of a SAS license. Macros and programs in Excel were created and used in this paper that go along with the SAS results. The goal of this paper was to build a statistical model that incorporate financial balance sheet items, which has been done in the past, but also to seek other market equity related variables. The drive in this direction was mainly due to the recent high volatility in late 2008 of the credit arena. The months in late 2008 has experienced many companies defaulting on loans and disappear from the markets, such as Bear Stearns and Lehman Brothers. Analyzing the market as a whole at this current time at the end of 2008, it is projected that many other companies may follow. As of the date of this paper, General Motors and Chrysler Automotive companies are at headline risk. In this paper, it was found that not all market variables are significant in a logistic regression model. The two main models found that either Market Equity over Total Assets (a measurement of Leverage ), Sales over Total Assets, (a measurement of Competitiveness), Earnings before Income Tax Adjusted over Total Assets (a ratio that measures Profitability), and the last traded price of an issuer (the only market indicator)for that date were significant variables to predicting probabilities of default within a 1 year time horizon. In this paper, the predications of the models are compared to other models that are currently accepted in the financial market as credit risk indicators. Overall, there are high correlations between the statistical models and the models of this paper. The logistic model can be used to predict default rates, however, more technical validation techniques need to me incorporated. Also, depending on availability, more issuers that have defaulted may be added to the model.

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Chapter 1

Introduction

1.1 Credit Risk Methodology

In recent months more emphasis has shifted to the modeling and evaluation of credit risk. There are several forces behind this trend. First, credit markets have grown steadily and the credit derivatives market (CDS) has grown expo- nentially. Second, credit risk is still a developing field. The understanding and the methodologies in managing and measuring other market risks, including in- terest rate risk, currency risk etc., have matured and are now assessed based on widely accepted principles. Credit markets are implying significant levels of of default over the next twelve months, but rating agencies, such as Moody’s or Standard and Poors, downgrades and realized defaults have yet to follow suit. This is apparent in the latest sub prime bubble bursts and the more recent defaults such as Lehman Brothers that have surprised the economy. Therefore, in order to stay afloat in the financial, one needs to become more active in credit risk management to avoid substantial losses in any position. Any approach to credit risk management has o balance many often com- peting requirements. Objectivity, accuracy, and stability are all important, but not at the expense of timeliness. Last, but not clear, coverage is important for consistent decision-making across large portfolios. Given these requirements, the approach to credit risk is to steer a course between market signals and fun- damental creditworthiness, but mixing financial statement and equity market data together within a credit scoring framework. By using financial statements, which will focus on the internal factors that drive company credit risk, and the inclusion of equity market signals adds timeliness, recognizing (as Merton did in 1974) that equity prices and equity volatility will be useful factors for the a measurement of riskiness of an issuer’s debt. In this paper, have collected a database of risk factors and risk measures of individual companies to iden- tify a either company level, or an industry level, probabilities of default using statistical models and analysis.

Chapter 3

Methods of Modeling

Credit Risk

3.1 Existing Methods of Quantitatively Scoring

Many of the common types of qualitative methods for evaluating risk target to explain the following general categories of a company. Overall, predictors should target explaining/correlating with this following framework The most commonly used typology of risk factors for issuers is the one regulators uses the CAMEL framework. The focus in this paper, is to determine predictors that targeted the following general areas which affect default concerns.

• Capital Adequacy: Amount of core capital to a issuer’s risk-weighted assets. This area is designed to relate the amount of core capital (less intangibles and other items) to a bank’s risk-weighted assets. Capital adequacy can be used to define insolvency itself, as well as to predict future insolvency. The key to avoiding confusion is that capital ratios used as a predictor (along with all other predictors) must be measured before the institution’s insolvency. That is, the question is whether poorly capitalized institutions later tend to strengthen their capital positions or to become yet more poorly capitalized to the point that they are so undercapitalized as to be defined as insolvent.
• Asset quality: Chosen on the basis that poor asset quality should increase the risk of failure. A common one is the ratio of nonperforming assets to total assets
• Management quality. A qualitative measurement on the past actions and strength of management. While an important part of qualitative analysis of a issuer’s risk profile, is not measurable quantitatively.
• Earnings: most common index of earnings is the ratio of net income to total assets. Research has shown that growth in either of these two quan- tities likewise should predict a reduced risk of insolvency. [5]
• Asset Liquidity: Issuers with assets that are more liquid should have lower failure risk, all else equal.

Each of the above areas may not pertain to all issuers, however, in the ultimate determination in deciding the stability of a company or entity the areas are basic tools to help focus attention in the right area. It is up the analyst in determining the methods/models of how to accurately and precisely target the general areas. The assessment made can either be qualitative, using the areas above, or quantitative. A quantitative credit risk scoring model can complement all the sources of qualitative judgment. Furthermore, a quantitative model offers breadth of coverage, including firms that are too small or that issue securities too infrequently to substantiate ongoing qualitative coverage. Furthermore, a quantitative model can be updated far more frequently than any other type of evaluation, and timely evaluations are more likely to be accurate. Finally, a quantitative model is valuable in the context of an integrated analytic approach because it provides a systematic algorithm for scoring risk that can be used as a check and starting point for in-depth qualitative analysis.

3.2 Two Classes of Quantitative Models

Existing quantitative models for scoring issuers’ credit risk fall into two broad classes: statistical and structural.

3.2.1 Statistical Models

Statistical Models: Use historical data on characteristics of issuer (for ex- ample, measures of earnings or liquidity) to determine the set of characteristics that best predict the occurrence of the selected outcome. The precise form of the relationship between the inputs and the outcome is specified by the partic- ulars of the statistical model used. These models are historically specific: the model parameters depend on the data used to create the model.

3.2.2 Structural Models

Structural Models: models are form and parameters are, in principle, speci- fied by a theory of economic structure. In this model, the common framework is to us a contingent claims models, which is based on the concept that the equity of a firm as a call option on the assets of a firm. In this approach, it uses the Black-Scholes (1973) framework and applies it to a debt pricing model by Mer- ton Approach (1974). In this model, the strike price of the option represented by the firm’s equity is a function of the firm’s debt level, and if the value of the firm’s assets falls below that point the equity holders decline to exercise the call, turning over the assets of the firm to the debt holders. This is the point of de- fault. The probability of that event is determined by the difference between the firm’s asset value and default point, along with the volatility of the firm’s asset value. The asset value is not observed, these quantities are generally estimated as a function of the firm’s equity value and volatility. [5] There are challenges of applying contingent claims models to banks specif- ically, the models more generally share certain limitations. Through various studies, the default probabilities implied by Merton-type contingent claims models are inconsistent with historically observed default rates [3] (Falkenstein,

need not be normally distributed and that are nonlinear in their effects on the probability of an event, and for dependent variables with different sized groups.

Chapter 4

Methodology

4.1 A Statistical Model: Logistic Regression

I apply a statistical model to historical data on issuers’ characteristics. The particular form of statistical model is a discrete-time event history model. This model is designed to predict the risk of an event occuring, as a function of specified variables measured before the event occurs. The linear regression (a discrete time model) can be used to predict the risk of an event within a certain time period. This is equalvent and estimated by applying a logistic regression to issuer-year of data. The logistic regression takes the following form

log

p 1 − p

∑^ K

k=

βk xk (4.1)

where p is the probability of the even occuring, and K independent variables, x, are each weighted by a coeffiecent, β. From 4.1, this different from the conventional linear regression model

y =

∑^ K

k=

βk xk (4.2)

From 4.2 differs from 4.1, because does not predict the value of the dependent variable, y, but rather the natural logarithm of the ratio of the probability of the event occurring to the probability of the event not occurring. This logit function preferable to the linear regression function because it limits p, the probability of the event, to be between 0 and 1 (or zero and 100%), where applying the conventional linear regression to dichotomous out- comes would instead allow nonsensical results like probabilities greater than 1 or less than zero. A transformation of 4.1, obtain the logistic model which gives the probability of the event as

p = e

∑ ∑K

k=1 βk^ xk

1 + e

∑K

k=1 βk^ xk

Chapter 5

Data Description

In this paper, I obtained quarterly financial and market information from differ- ent issuers (ranging from publicly-traded US bank holding companies to indus- trial industries, etc) during 1996-2008. Issuers consisted of myriad of companies that have either defaulted or are currently stable. The data for the predictors was obtained from the financial statements from each issuer. There were 186 individual issuers that were collected. Each issuer reported 4 data points of each variable for one year.

5.1 Defaulted Bonds

To provide a encompass a wider selection of issuers, defaulted bonds were chosen industry and time independent. Issuers that defaulted most (but not all) of these subsequently facing regulatory intervention as well. In total there were 37 issuers that have defaulted in the time span of this model’s data collection.

• Airlines: Delta Air Lines Inc, Northwest Airlines Corporation, FLYi, Inc,
• Financial Insitutions: Lehman Brothers, Bear Stearns, WAMU
• Delphi Corporation, Levitz Home Furnishings, Inc, Winn-Dixie Stores Inc, Calpine Generating Co LLC

5.2 Data

As mentioned in an early section, predictors should target explaining/correlating with the following CAMEL framework. This is the most commonly used typol- ogy of risk factors for banks is the one regulators use to summarize the results of their on-site examinations. These framework is setup to qualitatively or quan- titatively represent the amount of cash the company has on hand, incoming cash flow of a company, the strength of the corporate structure, and how eas- ily the company can liquidate the assets. The financial variables listed below were chosen to target the stability of the company. A subset of the variables listed below are used in common practice in calculation of the Altman Z-score (Altman 1968) [1]. Altman, using Multiple Discriminant Analysis, combined a

set of financial ratios to come up with the Z-Score. This score uses statistical techniques to predict a company’s probability of failure. The predictors will use information from the financial and balance sheet of an issuer. Listed below are the definitions and/or and explanation of the param- eters. Also to incorporate market behavior, there are equity-driven measures added to the possible set of parameters.

• Working Capital: Operating liquidity available to a business, calculated as current assets minus current liabilities
• Retained Earnings: The percentage of net earnings not paid out as div- idends, but retained by the company to be reinvested in its core business or to pay debt.
• Earnings Before Debit and Interest (EBIT): A company’s earning power, represents the earnings which the company has achieved. Measure of a issuer’s profitability.
• Market Value of Equity: The price at which investors buy or sell a issue.
• Equity Market Volatility: movements in the market in which it is traded
• Sales: Total dollar amount collected for goods and services provided
• Total Liabilities: The liabilities found by adding current liabilities to long-term debts.
• Total Assets: All the property owned by a corporation
• Net Income: The company’s total earnings, minus expenses and taxes

Equity market indicators may serve as vital flags of risk that may not be reflected in the company’s financial statements. Equity markets identify those believed to be taking on greater risk, by a higher total equity volatility. Com- panies with greater equity volatility should be more likely to become insolvent. For the model in this paper, ratios of the above financial parameters were used so that the predictors can not only encapsulate the standard CAMEL framework, but also incorporate market behaviors. For example A company’s debt ratio is a leverage ratio calculated by dividing total liabilities by total assets. This ratio measures the extent to which total assets have been financed with debt.

5.3 Predictors for Logistic Regression Model

The following are the ratios and equity market information that were calculated from the collected parameters for each issuer. From financial analysis, when analyzing certain ratios each targets certain areas of credit risk fundamentals

• Working Capital/Total Assets: Measures the short-term liquidity of the issuer

Chapter 6

Results

6.1 Saturated Logistic Regression Model 1

Preliminary analysis began by creating a ”saturated” model including all pre- dictors (ratio) described in the previous section. There were concerns of that several ratios are correlated. The correlation matrix of the variables is provided below. A decision was made to to include only parameters with insignificant correlation in the model. It should be noted, however, that in the aggregate the results would have been essentially identical with either variable, given that the two ratios are empirically virtually indistinguishable. A correlation matrix is provided. In preliminary analysis, the correlations between EBITA/TA and EBIT/TA was high. Running the model with each variable, it was determined to use only EBITA/TA. Also, NI/TA, also showed higher correlations across several other parameters. The variable was dropped from the model possibilities, and furthermore, the variable was found to be insignificant, which will be illustrated in the outputs.

Figure 6.1: Correlation Matrix

6.1.1 Saturated Logistic Regression Model 1: Output and

Diagnostics

Using SAS and Excel, I obtained the following outputs. SAS is listed below and Excel is provided later. Both programs confirmed the results. In the saturated model it was found that EQY SH OUT (Equity Share Values), WC/TA (Working Capital/Total Assets), RE/TA (Retained Earnings/Total Assets). and EBIT/TA to be insignificant to the model. These parameters were taken out and the likelihood ratio will be calculated with the second model. Although the p-values were insignificant on EBITA/TA and PX LAST (Last trading price), each was left in the analysis to determine a reduced model.

Figure 6.2: SAS Output: Saturated Logistic Model

6.1.2 Saturated Logistic Regression Model 1 Interpreta-

tion

• Then if ME/TL changes increases by 1, the odds that the default takes the value 1 increase e−^0.^2396 = 76.69%