A Random Matrix Approach to Credit Risk

Citation: Mu nnix MC, Schafer R, Guhr T ( A Random Matrix Approach to Credit Risk Michael C. Mu nnix 0 Rudi Scha fer. 0 Thomas Guhr 0 Renaud Lambiotte, University of Namur, Belgium 0 Faculty of Physics, University of Duisburg-Essen , Essen , Germany We estimate generic statistical properties of a structural credit risk model by considering an ensemble of correlation matrices. This ensemble is set up by Random Matrix Theory. We demonstrate analytically that the presence of correlations severely limits the effect of diversification in a credit portfolio if the correlations are not identically zero. The existence of correlations alters the tails of the loss distribution considerably, even if their average is zero. Under the assumption of randomly fluctuating correlations, a lower bound for the estimation of the loss distribution is provided. - The financial crisis of 20082009 clearly revealed that an improper estimation of credit risk can lead to dramatic effects on the worlds economy. The vast underestimation of risks embedded in credits for the subprime housing markets induced a chain reaction that propagated into the worldwide economy. A better estimation of credit risk (see, eg, [1,2,3,4,5]) is therefore of vital interest. We can distinguish two fundamentally different approaches to credit risk modeling (see, eg, [6]): the structural and the reducedform approach. Structural models have a long history, going back to the work of Black and Scholes [7] and Merton [8]. The Merton model assumes a zerocoupon debt structure with a fixed time to maturity. The value of the companys assets is modeled by a stochastic process. The risk of default and the associated recovery rate, the residual payment in case of a loss, are directly determined by the companys asset value at maturity. Reducedform models attempt to capture the dependence of default and recovery rates on macroeconomic risk factors. Both quantities are modeled as independent stochastic variables. Some well known reducedform model approaches can be found in [9,10,11,12,13]. First passage models were first introduced by Black and Cox [14] and they fall somewhat in between the two modeling approaches. Similar to Mertons model, the market value of a company is modeled by a stochastic process. However, in the first passage models a default occurs whenever this market value hits a certain threshold for the first time. The recovery rates are typically modeled independently, for example, by a reducedform model, see [15,16], or are even assumed to be constant, see, eg, [6]. Recent approaches aim at improving first passage models by including the chance of full recovery, even if a companys market value is below the threshold, see [17], and estimating correlations between default probabilities of industry sectors, see [18]. Reducedform and firstpassage models are implemented in commercial software solutions, for example, CreditMetrics initially developed by JP Morgan [19], CreditPortfolioView by McKinsey & Company [20] or CreditRisk+ by Credit Suisse [21]. As there can be a strong connection between default risks and recovery rates, the chances of large losses are often underestimated in the reducedform and first passage models, see [22,23]. The Merton model does not require this separation and is, for example, adopted by Moodys KMV. Structural models provide a "microscopical tool to study credit risk as the defaults and recoveries are traced back to stochastic processes modeling the state of individual obligors. For a portfolio of credits, such as collateralized debt obligations (CDOs), correlations represent a key factor that influences its risk. The benefit of diversification, ie, the reduction of risk by increasing the portfolio size, is severely limited by the presence of even weak correlations. This has been demonstrated for the case of constant positive correlations, both in the first passage model with constant recovery [24,25] and in the Merton model [26,22]. The key problem in estimating the credit risk of a realistic portfolio is of course the huge number of parameters involved. This is precisely where approaches from statistical physics can be most helpful: the state of a system with many degrees of freedom is, under certain conditions, described by few macroscopic observables. In the thermodynamic equilibrium, these are energy, temperature, pressure, etc. Ergodicity holds, ie, time and ensemble average yield the same results. A somewhat similar situation exists for spectral statistics in quantum chaotic systems, see [27]. A moving average over one long spectrum equals an ensemble average over random matrices, if the number of levels is very large. Originally, random matrix theory was developed in the 1950s to describe the spectra of heavy nuclei, see [28]. Here we transfer this idea to large credit portfolios involving correlated assets. In the case of a great many contracts, we expect a selfaveraging property which then should allow to average over an ensemble of random correlation matrices. We manage to carry out this approach largely analytically. We obtain estimates for the distribution of asset values and the portfolio loss distribution in which the complicated effects of all correlations are indeed reduced to a single parameter measuring the correlation strength. A Structural Credit Risk Model Our model is based on Mertons original model, assuming a zero-coupon bond for the debt structure of the obligor. Our aim is to analytically describe the impact of correlations on the losses of a credit portfolio. Even though the Merton model makes many simplifying assumptions, it can provide more than just qualitative insights into credit risk. Indeed we demonstrated recently that empirical credit data are in accordance with analytical results derived from the Merton model [29]. The cash flow of the zero-coupon bond is limited to two dates: the date of issue t~0 and maturity t~T . At the issue date the creditor lends a specified amount of money to the obligor. At maturity, the obligor has to repay the face value of the bond. The face value is the amount borrowed plus interest and risk premium. A default occurs if the asset value Vk of company k is below the face value Fk at maturity time T . The size of the loss then depends on how far Vk is below the face value Fk. We assume that the asset values in a portfolio of K companies follow a geometric Brownian motion. An overview of the models input parameters is given in Table 1. Average distribution of asset values For the sake of simplicity, let us first consider the case of a Brownian motion for the asset values. Later on this can be easily mapped to the geometric Brownian motion by a simple substitution. For a Brownian motion, the probability density function (pdf) of the vector V of K asset values at maturity T is described by 1 1 p(mv)(V ,S)~ pffiffiffiffiffiffiffiffiffiK pffidffiffieffiffitffiffi(ffiSffiffiffiffi)ffi 2pT (V {mT ){S{1(V {mT ) Here, S (...truncated)