Closed-form approximations for constant continuous annuities.

Abstract: For a series of cash flows, its stochastically discounted or compounded value is often a key quantity of interest in finance and actuarial science. Unfortunately, even for most realistic rate of return models, it may be too difficult to obtain analytic expressions for the risk measures involving this discounted sum. Some recent research has demonstrated that in the case where the return process follows a Brownian motion, the so-called comonotonic approximations usually provide excellent and robust estimates of risk measures associated with discounted sums of cash flows involving log-normal returns. In this paper, we derive analytic approximations for risk measures in case one considers the continuous counterpart of a discounted sum of log-normal returns. Although one may consider the discrete sums as providing a more realistic situation than its continuous counterpart, considering in this case, the continuous setting leads to more tractable explicit formulas and may therefore provide further insight necessary to expand the theory and to exploit new ideas for later developments. Moreover, the closed-form approximations we derive in this continuous set-up can then be compared more effectively with some exact results, thereby facilitating a discussion about the accuracy of the approximations. Indeed, in the discrete setting, one always must compare approximations with results from simulation procedures which always give rise to room of debate. Our numerical comparisons reveal that the comonotonic 'maximal variance' lower bound approximation provides an excellent fit for several risk measures associated with integrals involving log-normal returns. Similar results as we derive here for continuous annuities can also be obtained in case of continuously compounding which therefore opens a roadmap for deriving closed-form approximations for the prices of Asian options. Future research will also focus on optimal portfolio slection problems.Approximation; Choice; Comonotonicity; Criteria; Decision; Distribution;

The hurdle-race problem.

We consider the problem of how to determine the required level of the current provision in order to be able to meet a series of future deterministic payment obligations, in case the provision is invested according to a given random return process. Approximate solutions are derived, taking into account imposed minimum levels of the future random values of the reserve. The paper ends with numerical examples illustrating the presented approximations.Processes; Value;

Bounds for sums of random variables when the marginal distributions and the variance of the sum are given

postprin

On the computation of Wasserstein barycenters

The Wasserstein barycenter is an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics, and in particular to clustering and image processing. In this paper, we state a general version of the equivalence of the Wasserstein barycenter problem to the n-coupling problem. As a consequence, the coupling to the sum principle (characterizing solutions to the n-coupling problem) provides a novel criterion for the explicit characterization of barycenters. Based on this criterion, we provide as a main contribution the simple to implement iterative swapping algorithm (ISA) for computing barycenters. The ISA is a completely non-parametric algorithm which provides a sharp image of the support of the barycenter and has a quadratic time complexity which is comparable to other well established algorithms designed to compute barycenters. The algorithm can also be applied to more complex optimization problems like the k-barycenter problem

An interview with Roger Cooke

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How to Determine the Capital Requirement for a Portfolio of Annuity Liabilities

This paper illustrates an analytic method that can be used to determine the total capital requirements necessary to properly provide for the future obligations of a portfolio of annuity liabilities and to protect the enterprise from the related risks it faces. This example is based on the work of Kaas, Dhaene and Goovaerts (2000).

Stat Trek : an interview with Christian Genest

Christian Genest is Professor and Canada Research Chair in Stochastic Dependence Modeling at McGill University, Montr\ue9al, Canada. He studied mathematics and statistics at the Universit\ue9 du Qu\ue9bec \ue0 Chicoutimi (BSpSc, 1974), the Universit\ue9 de Montr\ue9al (MSc, 1978), and The University of British Columbia (PhD, 1983). Before joining McGill in 2010, he held academic posts at Carnegie Mellon University (1983\u201384), the University of Waterloo (1984\u201387), and Universit\ue9 Laval (1987\u2013 2010). Over the years, he also held visiting positions in Belgium, France, Germany, and Switzerland. Christian\u2019s primary research focus lies in multivariate analysis, nonparametric statistics, and extreme-value theory. He also collaborates regularly with researchers in insurance, nance, and hydrology. He has published extensively and earned various distinctions for his seminal and widely cited work in dependence modeling. In particular, he received the Statistical Society of Canada Gold Medal for Research in 2011 and was elected a Fellow of the Royal Society of Canada in 2015. He has also served the profession in various capacities, e.g., as Director of the Institut des sciences math\ue9matiques du Qu\ue9bec, President of the Statistical Society of Canada, and Editor-in- Chief of The Canadian Journal of Statistics (1998\u20132000). He is the current Editor-in-Chief of the Journal of Multivariate Analysis

The Extraction of 3D Shape from Texture and Shading in the Human Brain

We used functional magnetic resonance imaging to investigate the human cortical areas involved in processing 3-dimensional (3D) shape from texture (SfT) and shading. The stimuli included monocular images of randomly shaped 3D surfaces and a wide variety of 2-dimensional (2D) controls. The results of both passive and active experiments reveal that the extraction of 3D SfT involves the bilateral caudal inferior temporal gyrus (caudal ITG), lateral occipital sulcus (LOS) and several bilateral sites along the intraparietal sulcus. These areas are largely consistent with those involved in the processing of 3D shape from motion and stereo. The experiments also demonstrate, however, that the analysis of 3D shape from shading is primarily restricted to the caudal ITG areas. Additional results from psychophysical experiments reveal that this difference in neuronal substrate cannot be explained by a difference in strength between the 2 cues. These results underscore the importance of the posterior part of the lateral occipital complex for the extraction of visual 3D shape information from all depth cues, and they suggest strongly that the importance of shading is diminished relative to other cues for the analysis of 3D shape in parietal regions