On the Distortion of a Copula and its Margins

This article examines the notion of distortion of copulas, a natural extension of distortion within the univariate framework. We study three approaches to this extension: (1) distortion of the margins alone while keeping the original copula structure, (2) distortion of the margins while simultaneously altering the copula structure, and (3) synchronized distortion of the copula and its margins. When applying distortion within the multivariate framework, it is important to preserve the properties of a copula function. For the first two approaches, this is a rather straightforward result, however for the third approach, the proof has been exquisitely constructed in Morillas (2005). These three approaches of multivariate distortion unify the different types of multivariate distortion that have scarcely scattered in the literature. Our contribution in this paper is to further consider this unifying framework: we give numerous examples to illustrate and we examine their properties particularly with some aspects of ordering multivariate risks. The extension of multivariate distortion can be practically implemented in risk management where there is a need to perform aggregation and attribution of portfolios of correlated risks. Furthermore, ancillary to the results discussed in this article, we are able to generalize the formula developed by Genest and Rivest (2001) for computing the distribution of the probability integral transformation of a random vector and extend it to the case within the distortion framework.Multivariate distortion; Ordering of risks; Probability integral transformation

On the Distortion of a Copula and its Margins

This article examines the notion of distortion of copulas, a natural extension of distortion within the univariate framework. We study three approaches to this extension: (1) distortion of the margins alone while keeping the original copula structure, (2) distortion of the margins while simultaneously altering the copula structure, and (3) synchronized distortion of the copula and its margins. When applying distortion within the multivariate framework, it is important to preserve the properties of a copula function. For the first two approaches, this is a rather straightforward result, however for the third approach, the proof has been exquisitely constructed in Morillas (2005). These three approaches of multivariate distortion unify the different types of multivariate distortion that have scarcely scattered in the literature. Our contribution in this paper is to further consider this unifying framework: we give numerous examples to illustrate and we examine their properties particularly with some aspects of ordering multivariate risks. The extension of multivariate distortion can be practically implemented in risk management where there is a need to perform aggregation and attribution of portfolios of correlated risks. Furthermore, ancillary to the results discussed in this article, we are able to generalize the formula developed by Genest and Rivest (2001) for computing the distribution of the probability integral transformation of a random vector and extend it to the case within the distortion framework

On the Distortion of a Copula and its Margins

This article examines the notion of distortion of copulas, a natural extension of distortion within the univariate framework. We study three approaches to this extension: (1) distortion of the margins alone while keeping the original copula structure, (2) distortion of the margins while simultaneously altering the copula structure, and (3) synchronized distortion of the copula and its margins. When applying distortion within the multivariate framework, it is important to preserve the properties of a copula function. For the first two approaches, this is a rather straightforward result, however for the third approach, the proof has been exquisitely constructed in Morillas (2005). These three approaches of multivariate distortion unify the different types of multivariate distortion that have scarcely scattered in the literature. Our contribution in this paper is to further consider this unifying framework: we give numerous examples to illustrate and we examine their properties particularly with some aspects of ordering multivariate risks. The extension of multivariate distortion can be practically implemented in risk management where there is a need to perform aggregation and attribution of portfolios of correlated risks. Furthermore, ancillary to the results discussed in this article, we are able to generalize the formula developed by Genest and Rivest (2001) for computing the distribution of the probability integral transformation of a random vector and extend it to the case within the distortion framework

Applications of Clustering with Mixed Type Data in Life Insurance

Death benefits are generally the largest cash flow item that affects financial statements of life insurers where some still do not have a systematic process to track and monitor death claims experience. In this article, we explore data clustering to examine and understand how actual death claims differ from expected, an early stage of developing a monitoring system crucial for risk management. We extend the $k$-prototypes clustering algorithm to draw inference from a life insurance dataset using only the insured's characteristics and policy information without regard to known mortality. This clustering has the feature to efficiently handle categorical, numerical, and spatial attributes. Using gap statistics, the optimal clusters obtained from the algorithm are then used to compare actual to expected death claims experience of the life insurance portfolio. Our empirical data contains observations, during 2014, of approximately 1.14 million policies with a total insured amount of over 650 billion dollars. For this portfolio, the algorithm produced three natural clusters, with each cluster having a lower actual to expected death claims but with differing variability. The analytical results provide management a process to identify policyholders' attributes that dominate significant mortality deviations, and thereby enhance decision making for taking necessary actions.Comment: 25 pages, 6 figures, 5 table

Actuarial Analysis of Retirement Income Replacement Ratios

A measure of level of post-retirement standard of living is the replacement ratio, i.e., percentage of final salary received as annual retirement income derived from savings. The replacement ratio depends on many factors including salary, salary increases, investment returns, and post-retirement mortality. Elementary life contingencies techniques are used to develop a replacement ratio formula and analyze its sensitivity to these factors