In this paper, we explore the Loss Distribution Approach (LDA) for computing the capital charge of a bank for operational risk where LDA refers to statistical/actuarial methods for modelling the loss distribution. In this framework, the capital charge is calculated using a Value-at-Risk measure. In the first part of the paper, we give a detailed description of the LDA implementation and we explain how it could be used for economic capital allocation. In particular, we show how to compute the aggregate loss distribution by compounding the loss severity distribution and the loss frequency distribution, how to compute the total Capital-at-Risk using copulas, how to control the upper tail of the loss severity distribution with order statistics. In the second part of the paper, we compare LDA with the Internal Measurement Approach (IMA)proposed by the Basel Committee on Banking Supervision to calculate regulatory capital for operational risk. LDA and IMA are bottom - up internal measurement models which are apparently different. Nevertheless, we could map LDA into IMA and give then some justifications about the choice done by regulators to define IMA. Finally, we provide alternative ways of mapping both methods together.

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Loss Distribution Approach for Operational Risk

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Combinatorial methods in the theory of stochastic processes

Publisher Summary The main concern in this chapter is the simultaneous adaptive choice of the shape parameter and the number of extremes for both of the two classes by means of minimizing estimators of the asymptotic mean-square errors It considers two methods for each of the two classes: one takes the estimated bias from the median of all estimators in the class, the other from the average of all estimators in the class While presenting the main ideas, a substantial part of the literature is also reviewed

Estimating the tail index

The Laplace transform \psi(t)=E[{\rm exp}(-tX)] of a random variable X with exponential density u exp( m u x ), x S 0, satisfies the equation (\lambda+t)\psi(t)-\lambda=0 , t S 0. We study the behavior of a class of consistent tests for exponentiality based on a suitably weighted integral of [({\hat\lambda}_n+t)\psi_n(t)-{\hat\lambda}_n]^2 , where {\hat\lambda}_n is the maximum-likelihood estimate of u , and  n is the empirical Laplace transform, each based on an i.i.d. sample X 1 , , X n . As the decay of the weight function tends to infinity, the test statistic approaches the square of the first nonzero component of Neyman's smooth test for exponentiality. The new tests are compared with other omnibus tests for exponentiality.

Tests of Fit for Exponentiality based on the Empirical Laplace Transform

A non-parametric estimator of the Gerber–Shiu function is proposed for a risk process with a compound Poisson claim process plus a diffusion perturbation; the Wiener–Poisson risk model. The estimator is based on a regularized inversion of an empirical-type estimator of the Laplace transform of the Gerber–Shiu function. We show the weak consistency of the estimator in the sense of an integrated squared error with the rate of convergence.

Non-parametric estimation of the Gerber–Shiu function for the Wiener–Poisson risk model

Let (Xn) be a sequence of non-negative random variables with distribution function F and Laplace transform L, and let N be an integer independent of the sequence. In many applications one knows that for y→∞ and a function φ P{Σ i=1 N X i >y}∼φ(y,τ,L(τ),L'(τ),...) where in turn τ is the solution of an equation ψ(τ,L(τ),...)=0. On the basis of a sample of size n we derive an estimator τ n for τ by solving ψ(τ n ,L n (τ n ),L' n (τ n ),...)=0 where L n is the empirical version of L. This estimator is then used to derive the asymptotic behaviour of φ(y,τ n ,L n (τ n ),L' n (τ n ),...). We include examples from insurance mathematics

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Empirical Laplace transform and approximation of compound distributions

Let (Xi} be i.i.d. non-negative random variables with d.f. F and Laplace transform L. Let N be integer valued and independent of (X, }i. In many applications one knows that for y - oc and a function (p

Sandor Csorgo

Papers

Empirical Laplace transform and approximation of compound distributions

And approximation of compound distributions