An Analytical Evaluation Method of the Operational Risk Using Fast Wavelet Expansion Techniques

A financial institution that adopts an advanced measurement approach (AMA) as a method of computing operational risk capital has to measure 99.9\%value-at-risk (VaR) as the amount of an operational risk. The most popular method to satisfy the AMA standards requires the evaluation of aggregate (compound) loss distribution, which is called the loss distribution approach (LDA). The Monte Carlo (MC) method is a well known method for calculating VaR under the LDA. However, when using the MC method to calculate VaR, the statistical error of VaR for the fat-tailed distribution increases and the computation time increases in proportion to the expected value of frequency distribution. Since the MC method has these problems, this paper presents a new methodology to compute VaR under the LDA using fast wavelet expansion techniques. The key features of our algorithm are follows: (i) Scale transformation technique for loss distributions (ii) Double exponential (DE) transformation for oscillatory integrals (iii) Finite series expansion of the wavelet scaling coefficients (iv) Wynn's epsilon algorithm to accelerate the convergence of those series (v) Efficient cubic spline interpolation method to calculate the moment generating function. We illustrate the effectiveness of our algorithms through numerical examples