problems of Credit Risk Modeling
Ursulenko Anna
PhD student,
Department of Economic Cybernetics,
Faculty of Economic,
Taras Shevchenko National University of Kiev
Problems of Credit Risk Modeling
Structural models of credit risk represent a convenient framework for modeling valuation of risky debt. Those models make explicit assumptions about the dynamics of a firm’s assets and its capital structure, which are then used to determine the occurrence of default. All those models are based on the assumption that the firm’s dynamic is regulated by a diffusion process and that the value of the firm can be observed directly. Since diffusion processes have continuous sample paths and default is the first hitting time of a barrier, then default is a predictable stopping time. This leads to an underestimation of the short-term credit spread, which are by far lower than those observed in the market. This major flaw of structural models has given rise to alternative approaches to credit risk modeling. One possibility is to extend structural models, for example by including jumps. Another very popular method is the so-called intensity based approach, also known as reduced form approach. The intensity based approach does not model default in terms of assets and liabilities of the firm, but defines the time of default as the first jump-time of an exogenously given counting process. The advantage is that the default event becomes an inaccessible stopping time, thereby removing the disturbing feature of strong underestimation of short-term credit spreads. Intensity-based models, however, are often criticized because they lose the micro-economic interpretation of the default time. There are also hybrid models which combine the best features of both approaches [1].
We consider a probability space ) with the following system of stochastic difference equations, a generalized version of a Hidden Markov Model :
where 
– describes the evolution of the asset value of the firm and is modeled as a discredited geometric Brownian motion;
The interpretation is as follows. We can think of θ as a random variable which designates the report model used by the manager of the firm to release the observations to the investors. Such report model affects both the future evolution of the actual asset value (see Equation (1)), and the value released to the outsiders by the manager of the firm (see Equation (4)). More precisely, the released log-asset value Zk depends on the report model θk-1 in place during the time interval [tk-1 tk ] through the function which models the amount of misreporting associated with a given report model [2]. Depending on whether h(θk-1) is positive or negative, an overstatement or an understatement of the actual performance of the firm will occur when the report model θk-1 is selected by the manager. The situation of no distortion occurring can be modeled by having h(θk-1)= 0. The parameter captures the variance of accounting noise associated with the report model V(θk-1).
We apply the above methodology to estimate h,µ ,ô ,v2 for the case of Joint Stock Commercial Bank “Transbank”, Ukrainian commercial bank which experienced a crisis during the years 2008-2010 and resulted in his bankruptcy in 2011.
We collected the daily stock data from January 1, 2008 to March 1, 2011 and computed the daily value of equity multiplying the number of outstanding shares by the stock price. At each time ti, we assume that the outstanding debt Ki has five-year maturity and proxy it with the long term debt recovered from the balance sheet statements. We use the five-year treasury yield as the discount factor. We run the calibration procedure using our method and compare the parameter estimates with the ones obtained by running the maximum likelihood estimation procedure on the standard Merton model [3]. The estimates show that the hidden parameter h plays an important role in the specification of the model. If it were omitted, as it happens for the Merton model, then the reduced value of the firm due to accounting misreporting would simply be explained by an increase of the asset volatility which rises to 17 %. Our estimate of asset volatility of 11 % matches closely with the estimates of asset volatility found by different calibration procedures. Our estimate 0.2052 of the distortion factor recalls the findings of M. Morini [4], who perform a historical analysis of risk neutral probabilities of fraud via a modification of Merton model and extract an implied probability of fraud of about 0.2. Although there are differences in the assumptions and analysis between our work and theirs, making it hard to compare, using the simplest version of our model we find qualitatively similar results to the ones they obtain with the full power of their model.
References
1. Duffie D. Term structure of credit spreads with incomplete accounting information / D. Duffie, D. Lando. // Econometrica. – 2001. – №63. – р. 633–664.
2. Brody D. Information-Based Asset Pricing / D. Brody, L. Hughston, A. Macrina. // International Journal of Theoretical and Applied Finance. – 2008. – №11. – р. 107–142.
3. Coculescu D. Valuation of Default Sensitive Claims Under Imperfect Information / D. Coculescu, H. Geman, M. Jeanblanc. // Finance and Stochastics. – 2008. – №12. – р. 195-218.
4. Brigo D. CDS Calibration with tractable structural models under uncertain credit quality / D. Brigo, М. Morini. // Risk Magazine. – 2006. – №19. – р. 43-45.
