Nonparametric Estimation Of The Density Function Of Recovery Rate

The probability of default (PD) has always been the focus of professionals. As theassumptions go from simple to more pragmatic, and with the change of the attentions ofdifferent entities, Structural model, Reduced form model and Credit VaR model wereadvanced. The ultimate purpose of these models is to develop a scientific and systematicmodel to measure or predict PD in order to make prices or prevent losses in the future.However, when refereed to another credit risk factor RR, people always think it belongsto a statistics category and relatively haven’t paid much attention until recently whennon-performing assets and default events increase. Researches on RR basically involvestatistics, distribution, influencers and so on. The distribution research may generally bedivided into two types, one is to utilize statistical methods like histogram to observe thedistribution, the other is to parametrically estimate its density function which is setinitially. The most common distribution assumption is Beta distribution. But what isinteresting is that sometimes those two results present a contradiction. Some empiricalresearches find out that multimodal rather than sloped or unimodal accrues in someranks of bonds and loans. This is because of the same micro-economic and seniority thatfinancial assets are facing with, but this phenomenon results in the limitation of usingBeta distribution.So in this paper, we research on a nonparametric estimation method called kernelestimation which is able to settle the “boundary problem” and be utilized in the compactsupport where LGD lies in. Then we do an empirical analysis on the LGDs of defaultedloans and bonds calculated by Moody’s.Firstly, the principle and categories of nonparametric estimation are overviewedsynoptically. Then several indices are introduced to evaluate the performances ofkernels and the way to choose an optimal bandwidth based on these indices. In practice,we use a method combining DPI and interactive algorithm called “Solve the Equation”,which is brought out in2008, to choose the optimal bandwidth.Then, the paper analyzes the differences of statistical properties between compactand unbounded support of common symmetric kernels and find out the boundaryproblem occurring in boundary region. It shows that the estimator is no longer ofasymptotic consistency and unbiasedness. To solve this, the thought of importing asymmetric or boundary kernel in to boundary region is inspired and two representativekernels, Beta kernel and boundary kernel, are proposed. Their statistical properties likevariance, bias, MISE and the results of Monte Carlo simulation show that the two bothsolve the boundary problem properly, in which boundary kernel is better. In practicalapplication, we also propose a method to choose the variable bandwidth in the boundaryregion, which helps to improve the bad performance where samples are sparse.Finally, the paper empirically estimates the density function of RRs of defaultedloans and bonds published by Moody’s using the chosen boundary kernel and Betadistribution. In order to distinguish between normal years and downturn, we separatethe samples into two categories. The empirical results show that boundary kernel has animmense advantage no matter in figure curves or MISE. Then a chi-square test isapplied and the result confirms our conclusion. After that a bootstrap sampling is doneto evaluate how one differs from another.In conclusion, to use a boundary kernel to estimate the density function of RRmakes it possible that visualizing the recovery function without assuming a specificparametric specification. It lays a foundation for a precise modeling of RR and creditmodel in the future and supplies basis for allocating capital and supervising.