A Multi-Factor Approach for Systematic Default and Recovery Risk

The present paper develops a simultaneous multi-factor model for defaults and recoveries. Applying this model, risk parameters can be forecast using systematic and idiosyncratic risk factors and their implied correlations. The theoretical framework is accompanied by an empirical analysis in which a negative correlation between defaults and recoveries over the business cycle is observed. In the study, default and recovery rates are modeled by business cycle indicators and the properties of the economic and regulatory capital given these risk drivers are shown. 3 Modeling Default and Recovery Risk Today’s banks face the challenge of forecasting losses and loss distributions in relation to their credit risk exposures. It can be observed that most banks choose a modular approach which is in line with the current proposals of the Basel Committee on Banking Supervision [2004], where selected risk parameters such as default probabilities, exposures at default and recoveries given default are modeled in independent modules. However, the assumption of independence is questionable. Previous studies have shown that default probabilities and recovery rates given default are negatively correlated (e.g., Carey [1998], Hu/Perraudin [2002], Frye [2003], Altman et al. [2003] or Cantor/Varma [2005]). A failure to take these dependencies into account will lead to incorrect forecasts of the loss distribution and the derived capital allocation. The present paper extends a model introduced by Frye [2000]. Modifications of the approach can be found in Pykhtin [2003] and Düllmann/Trapp [2004]. Our contribution is original with regard to the following three aspects. First, we develop a theoretical model for the default probabilities and recovery rates and show how to combine observable information with random risk factors. In comparison to the above mentioned models, our approach explains the default and the recovery rate by risk factors which can be observed at the time of the risk assessment. According to the current Basel proposal banks can opt to provide their own recovery rate forecasts for the regulatory capital calculation. Thus, there is an immediate industry need for modeling. Second, we show a framework for estimating the joint processes of all variables in the model. Particularly, the simultaneous model allows the measurement of the correlation between the defaults and recoveries given the information. In this model statistical tests for the variables and correlations can easily be conducted. An empirical study reveals additional evidence on the correlations between risk drivers of default and recovery. Note that Cantor/Varma [2003] essentially analyze the same dataset and identify seniority and security as the main risk factors explaining recovery rates. The present paper extends their approach by developing a framework for modeling correlations between factor-based models for default and recovery rates. 4 Third, the implications of our results on economic and regulatory capital are shown. Note that according to the current proposals of the Basel Committee only the forecast default probabilities and recovery rates but no correlation estimates enter the calculation of the latter. We demonstrate the effects of spuriously neglecting correlations in practical application. The rest of the paper is organized as follows. The theoretical framework is introduced in the second section (‘Model and Estimation’) for a model using historic averages as forecasts and a model taking time-varying risk factors into account. The third section (‘Data and Results’) includes an empirical analysis based on default and recovery rates published by Moody’s rating agency and macroeconomic indices from the Conference Board. Section four (‘Implications for Economic and Regulatory Capital’) shows the implications of the different models on the economic capital derived from the loss distribution and the regulatory capital proposed by the Basel Committee. Section five (‘Discussion’) concludes with a summary and discussion of the findings. Model and Estimation The Model for the Default Process Our basic framework follows the approach taken by Frye [2000]. We assume that t n firms of one risk segment are observed during the time periods t (t=1,...,T). For simplicity these firms are assumed to be homogenous with regard to the relevant parameters and a latent variable describes each obligor i’s (i=1,..., t n ) credit quality it t it U w F w S ⋅ − + ⋅ = 2 1 (1) ( [ ] 1 , 0 ∈ w ). ( ) 1 0, ~ N Ft and ( ) 1 0, ~ N Uit are independent systematic and idiosyncratic standard normally distributed risk factors. The Gaussian random variable it S may be interpreted as the return on a firm’s assets and therefore 2 w is often called ‘asset correlation’. A default event occurs if the latent variable crosses a threshold c