A considerable number of equivalent formulas defining conditional value-at-risk and expected shortfall are gathered together. Then, we present a simple method to bound the conditional value-at-risk of compound Poisson loss distributions under incomplete information about its severity...

The exact probability distribution of the first digit of integer powers up to an arbitrary but fixed number of digits is derived. Based on its asymptotic distribution, it is shown that it approaches Benford's law very closely for sufficiently high powers

Given integer-valued and more general real-valued wagers, Feller(1968), Ethier and Khoshnevisan(2002) have established upper and lower bounds on the probability of ruin, which often turn out to be very close to each other. However, the exact calculation of these bounds depends on the unique...

Building on a new theory of parametric risk models initiated in Hürlimann(1998), it is shown how mean scaled individual risk models can be constructed. The approximate computation of their distributions and related quantities can be done in the author's (1990) mathematical framework of...

Based on a non-parametric criterion of independence derived from a generalized version of the Schweizer-Wolff non-parametric measure of dependence, we calculate rates of convergence to asymptotic independence between the order statistics in a general setting. Three different rates of convergence...

A copula based statistical method for fitting joint cumulative returns between a market index and a single stock to daily data is proposed. Modifying the method of inference functions for margins (IFM method), one performs two separate maximum likelihood estimations of the univariate marginal...

The class of bivariate extreme value copulas, which satisfies the monotone regression positive dependence property or equivalently the stochastic increasing property, is considered. A variational calculus proof of the Hutchinson-Lai conjecture about Kendall's tau and Spearman's rho for this...

A formula for the conditional value-at-risk of classical portfolio insurance is derived and shown to be constant for sufficiently small loss probabilities. As illustrations, we discuss portfolio insurance for an equity market index using empirical data, and analyze the more general multivariate...

Based on the method of copulas, we construct a parametric family of multivariate distributions using mixtures of independent conditional distributions. The new family of multivariate copulas is a convex combination of products of independent and comonotone subcopulas. It fulfills the four most...

Based on the three-parameter solution in natural numbers of the four squares equation by Sierpinski(1962) and an identity by Catalan(1885), we determine in two different ways the set of all primitive solutions to this equation. The obtained algorithms are illustrated with lists of the first...