\chapter{Methods of Modeling Credit Risk}
\label{ch:hist}

\section{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.
\vspace{.25cm}

\begin{itemize}
\item \textbf{C}apital 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.
\item \textbf{A}sset 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
\item \textbf{M}anagement 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.
\item \textbf{E}arnings: most common index of earnings is the ratio of net income to total assets.  Research has shown that growth in either of these two quantities likewise should predict a reduced risk of insolvency.  \cite{merton74}
\item \textbf{A}sset Liquidity:  Issuers with assets that are more liquid should have lower failure risk, all else equal.
\end{itemize}



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.


\section{Two Classes of Quantitative Models}

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

\subsection{Statistical Models}

\textbf{Statistical Models:} Use historical data on characteristics of issuer (for example, 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 particulars of the statistical model used.  These models are historically specific: the model parameters depend on the data used to create the model.  


\subsection{Structural Models}

\textbf{Structural Models:} models are form and parameters are, in principle, specified 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 \textit{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 Merton 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 default. 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. ~\cite{merton74}

There are challenges of applying contingent claims models to banks specifically, 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 \cite{Falken00} (Falkenstein, Carty, and Boral: 2000).  Although there are models that do attempt to compensate for this, the underlying fact is that exists market inefficiencies for many of the firms that make market valuation different.  The main concern today, because of the deteriorating conditions is that market prices may no longer possess the desirable properties they exhibit under perfect and complete markets.  (Beaver, Datar, and Wilson 1992: 261). If market prices were fully efficient and complete summaries of all relevant information, no additional factors could improve the prediction of firm riskiness. \cite{beav1} But as Krainer and Lopez (2003) show in the case of banks, financial ratios are strong predictors of subsequent regulatory ratings for bank holding companies, and remain so even with the inclusion of equity variables as potential predictors. \cite{Krainer03}

The Merton model is not statistically estimated. It is a theoretically specified mathematical statement of relationships among the market value of a firm’s equity, the value of its liabilities, and the volatility of its equity value. Statistical models specify only their component variables and the family of forms to which the relationships among them must belong, with the data being used to estimate the
exact nature of those relationships. Statistical default models estimate those relationships using variants of regression-based techniques


Overall, there are several challenges make contingent claims models difficult to successfully apply to certain industries, such as banks: a lack of any single dominant model for banks, data requirements that severely limit the scope of application for some models, and dependence of some models' results on distributional assumptions. Furthermore, that market imperfections exist implies that nonmarket measures may add significant power in
predicting default beyond that possible with even an ideal contingent claims model.

The importance note is that contingent claims model uses, but not limited to, equity markets information as a source of evaluations for the current conditions of any firm.  In this paper, I will address the need to encompass not only equity data, but to incorporate financial ratios to build a statistical model that best predicts an issuer's insolvency.  The goal is to determine a logistic regression model that includes financial ratios in addition to equity-driven measures.  While this model strategy attempts to explicitly predict default probabilities, this or any model, is not guaranteed from modifications due to the environment.  It is a useful tool in examining credit risk, and provides another indicator to a broad spectrum of qualitative and quantitative tools that identify risk factors that are emerging and/or difficult to quantify.  

\subsubsection{Altman’s Z-score}

The earliest, Altman’s Z-score (1968), used discriminant analysis. \cite{merton74} Discriminant analysis shares with linear regression
several assumptions (and adds assumptions of its own): predictors with multivariate
normal distributions, normally distributed errors, linear relationships between
independent and dependent variables, and a dependent variable with roughly similarsize
groups. Violating those assumptions, as default data and many other kinds of
data often do, can produce biased results. 

In the decades since Altman published his Z-score model, logistic regression has
replaced discriminant analysis as the preferred tool for dichotomous outcomes,
largely because logistic regression has assumptions that are less restrictive, and so
less frequently violated.  This model allows for predictors that 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.

