We compare capital requirements derived by tail conditional expectation (TCE) with those derived by tail conditional median (TCM) and find that there is no clear-cut relationship between these two measures in empirical data. Our results highlight the relevance of TCM as a robust alternative to TCE, especially for regulatory control. JEL Classification: G10, G11, G23, G29

We compare capital requirements derived by tail conditional expectation (TCE) with those derived by tail conditional median (TCM) and find that there is no clear-cut relationship between these two measures in empirical data. Our results highlight the relevance of TCM as a robust alternative to TCE, especially for regulatory control.

The purpose of the present note is in part to extend results of Sidák [3] to cases where the measure is not finite, perhaps not even <r-finite. In particular, the conditional expectation of a random variable in Lx is obtained without the use of the Radon-Nikodym Theorem, which, indeed, does not apply with the required generality. It was found possible to extend at the same time results of [l] s...

Let (Ω,F , μ) be a probability space and let T = P1P2 · · ·Pd be a finite product of conditional expectations with respect to the sub σ-algebras F1,F2, . . . ,Fd. Since conditional expectations are contractions of all Lp(μ) spaces, p ∈ [1,∞], so is T . When d = 2, Burkholder and Chow [2] proved that for every f ∈ L2(μ) the iterates T f converge a.s. (and thus also in L2-norm) to the conditional...

This paper deals with the problem of estimating the Multivariate version of the Conditional-TailExpectation, proposed by Cousin and Di Bernardino (2012). We propose a new non-parametric estimator for this multivariate risk-measure, which is essentially based on the Kendall’s process (see Genest and Rivest, 1993). Using the Central Limit Theorem for the Kendall’s process, proved by Barbe et al. ...

This paper proposes a new method to introduce coherent risk measures for risks with infinite expectation, such as those characterized by some Pareto distributions. Extensions of the conditional value at risk, the weighted conditional value at risk and other examples are given. Actuarial applications are analyzed, such as extensions of the expected value premium principle when expected losses ar...

Bias and variance for small-sample error estimation are typically posed in terms of statistics for the distributions of the true and estimated errors. On the other hand, a salient practical issue asks, given an error estimate, what can be said about the true error? This question relates to the joint distribution of the true and estimated errors, specifically, the conditional expectation of the ...

An investigation of the limiting behavior of a risk capital allocation rule based on the Conditional Tail Expectation (CTE) risk measure is carried out. More specifically, with the help of general notions of Extreme Value Theory (EVT), the aforementioned risk capital allocation is shown to be asymptotically proportional to the corresponding Value-at-Risk (VaR) risk measure. The existing methodo...

We develop a new notion of approximation of labelled Markov processes based on the use of conditional expectations. The key idea is to approximate a system by a coarse-graining of the state space and using averages of the transition probabilities. This is unlike any of the previous notions based on the use of simulation. The resulting approximations are customizable, more accurate and stay with...

This paper proposes novel lattice algorithms to compute tail conditional expectation of European calls and puts in linear time. We incorporate the technique of prefix-sum into tilting, trinomial, and extrapolation algorithms as well as some syntheses of these algorithms. Furthermore, we introduce fractional-step lattices to help reduce interpolation error in the extrapolation algorithms. We dem...