Haar Wavelets-Based Methods for Credit Risk Portfolio Modeling
Authors:   , Ortiz Gracia Luis 
Published: October 2023

In this dissertation we have investigated the credit risk measurement of a credit portfolio by means of the wavelets theory. Banks became subject to
regulatory capital requirements under Basel Accords and also to the supervisory review process of capital adequacy, this is the economic capital.
Concentration risks in credit portfolios arise from an unequal distribution of loans to single borrowers (name concentration) or different industry or
regional sectors (sector concentration) and may lead banks to face bankruptcy.
The Merton model is the basis of the Basel II approach, it is a Gaussian one-factor model such that default events are driven by a latent common factor
that is assumed to follow the Gaussian distribution. Under this model, loss only occurs when an obligor defaults in a fixed time horizon. If we assume
certain homogeneity conditions, this one-factor model leads to a simple analytical asymptotic approximation of the loss distribution function and VaR.
The VaR value at a high confidence level is the measure chosen in Basel II to calculate regulatory capital. This approximation, usually called Asymptotic
Single Risk Factor model (ASRF), works well for a large number of small exposures but can underestimates risks in the presence of exposure
concentrations. Then, the ASRF model does not provide an appropriate quantitative framework for the computation of economic capital. Monte Carlo
simulation is a standard method for measuring credit portfolio risk in order to deal with concentration risks. However, this method is very time consuming
when the size of the portfolio increases, making the computation unworkable in many situations. In summary, credit risk managers are interested in how
can concentration risk be quantified in short times and how can the contributions of individual transactions to the total risk be computed. Since the loss
variable can take only a finite number of discrete values, the cumulative distribution function (CDF) is discontinuous and then the Haar wavelets are
particularly well-suited for this stepped-shape functions. For this reason, we have developed a new method for numerically inverting the Laplace
transform of the density function, once we have approximated the CDF by a finite sum of Haar wavelet basis functions. Wavelets are used in
mathematical analysis to denote a kind of orthonormal basis with remarkable approximation properties. The difference between the usual sine wave and
a wavelet may be described by the localization property, while the sine wave is localized in frequency domain but not in time domain, a wavelet is
localized in both, frequency and time domain. Once the CDF has been computed, we are able to calculate the VaR at a high loss level. Furthermore, we
have computed also the Expected Shortfall (ES), since VaR is not a coherent risk measure in the sense that it is not sub-additive. We have shown that,
in a wide variety of portfolios, these measures are fast and accurately computed with a relative error lower than 1% when compared with Monte Carlo.
We have also extended this methodology to the estimation of the risk contributions to the VaR and the ES, by taking partial derivatives with respect to
the exposures, obtaining again high accuracy. Some technical improvements have also been implemented in the computation of the Gauss-Hermite
integration formula in order to get the coefficients of the approximation, making the method faster while the accuracy remains. Finally, we have extended
the wavelet approximation method to the multi-factor setting by means of Monte Carlo and quasi-Monte Carlo methods.