Edward Furman

Weighted Premium Calculation Principles

A prominent problem in actuarial science is to define, or describe, premium calculation principles (pcps) that satisfy certain properties. A frequently used resolution of the problem is achieved via distorting (e.g., lifting) the decumulative distribution function, and then calculating the expectation with respect to it. This leads to coherent pcps. Not every pcp can be arrived at in this way. Hence, in this paper we suggest and investigate a broad class of pcps, which we call weighted premiums, that are based on weighted loss distributions. Different weight functions lead to different pcps: any constant weight function leads to the net premium, an exponential weight function leads to the Esscher premium, and an indicator function leads to the conditional tail expectation. We investigate properties of weighted premiums such as ordering (and in particular loading), invariance. In addition, we derive explicit formulas for weighted premiums for several important classes of loss distributions, thus facilitating parametric statistical inference. We also provide hints and references on non-parametric statistical inferential tools in the area.

Tail Variance Premium with Applications for Elliptical Portfolio of Risks

In this paper we consider the important circumstances involved when risk managers are concerned with risks that exceed a certain threshold. Such conditions are well-known to insurance professionals, for instance in the context of policies involving deductibles and reinsurance contracts. We propose a new premium called tail variance premium (TVP) which answers the demands of these circumstances. In addition, we suggest a number of risk measures associated with TVP. While the well-known tail conditional expectation risk measure provides a risk manager with information about the average of the tail of the loss distribution, tail variance risk measure (TV) estimates the variability along such a tail. Furthermore, given a multivariate setup, we offer a number of allocation techniques which preserve different desirable properties (sub-additivity and fulladditivity, for instance). We are able to derive explicit expressions for TV and TVP, and risk capital decomposition rules based on them, in the general framework of multivariate elliptical distributions. This class is very popular among actuaries and risk managers because it contains distributions with marginals whose tails are heavier than those of normal distributions. This distinctive feature is desirable when modeling financial datasets. Moreover, according to our results, in some cases there exists an optimal threshold, such that by choosing it, an insurance company minimizes its risk.

Weighted Pricing Functionals With Applications to Insurance

Abstract We explore the role of weighted distributions in pricing insurance risks. In particular, we relate the distributions to actuarial and economic premium calculation principles and in this way provide a unifying methodology for constructing new principles and analyzing known ones.

Economic Capital Allocations for Non-negative Portfolios of Dependent Risks

In this paper we explore the problem of economic capital allocations in the context of non-negative multivariate (insurance) risks possessing a dependence structure. We derive a general result and illustrate it with a number of useful examples. One such example, for instance, develops explicit expressions for the discussed economic capital decomposition rule when the underlying portfolio consists of dependent compound Poisson risks.

A form of multivariate Pareto distribution with applications to financial risk measurement

A new multivariate distribution possessing arbitrarily parametrized and positively dependent univariate Pareto margins is introduced. Unlike the probability law of Asimit et al. (2010) [Asimit, V., Furman, E. and Vernic, R. (2010) On a multivariate Pareto distribution. Insurance: Mathematics and Economics 46(2), 308-316], the structure in this paper is absolutely continuous with respect to the corresponding Lebesgue measure. The distribution is of importance to actuaries through its connections to the popular frailty models, as well as because of the capacity to describe dependent heavy-tailed risks. The genesis of the new distribution is linked to a number of existing probability models, and useful characteristic results are proved. Expressions for, e.g., the decumulative distribution and probability density functions, (joint) moments and regressions are developed. The distributions of minima and maxima, as well as, some weighted risk measures are employed to exemplify possible applications of the distribution in insurance.

Statistical Inference for a New Class of Multivariate Pareto Distributions

Various solutions to the parameter estimation problem of a recently introduced multivariate Pareto distribution are developed and exemplified numerically. Namely, a density of the aforementioned multivariate Pareto distribution with respect to a dominating measure, rather than the corresponding Lebesgue measure, is specified and then employed to investigate the maximum likelihood estimation (MLE) approach. Also, in an attempt to fully enjoy the common shock origins of the multivariate model of interest, an adapted variant of the expectation-maximization (EM) algorithm is formulated and studied. The method of moments is discussed as a convenient way to obtain starting values for the numerical optimization procedures associated with the MLE and EM methods.

Beyond the Pearson Correlation: Heavy-Tailed Risks, Weighted Gini Correlations, and a Gini-Type Weighted Insurance Pricing Model

Gini-type correlation coefficients have become increasingly important in a variety of research areas, including economics, insurance and finance, where modelling with heavy-tailed distributions is of pivotal importance. In such situations, naturally, the classical Pearson correlation coefficient is of little use. On the other hand, it has been observed that when light-tailed situations are of interest, and hence when both the Gini-type and Pearson correlation coefficients are well-defined and finite, then these coefficients are related and sometimes even coincide. In general, understanding how the correlation coefficients above are related has been an illusive task. In this paper we put forward arguments that establish such a connection via certain regression-type equations. This, in turn, allows us to introduce a Gini-type Weighted Insurance Pricing Model that works in heavy-tailed situation and thus provides a natural alternative to the classical Capital Asset Pricing Model. We illustrate our theoretical considerations using several bivariate distributions, such as elliptical and those with heavy-tailed Pareto margins.

Weighted Pricing Functionals

We explore the concept of weighted distributions and their role in various phenomena occurring in insurance and finance. In particular, we relate weighted distributions to actuarial and economic premium calculation principles, and also to the capital asset pricing model (CAPM). Imitating the latter, we propose a weighted insurance pricing model (WIPM). Although general in formulation, we show that the WIPM can successfully be evaluated in a variety of situations, which we illustrate with a number of examples.

Paths and indices of maximal tail dependence

We demonstrate both analytically and numerically that the existing methods for measuring tail dependence in copulas may sometimes underestimate the extent of extreme co-movements of dependent risks and, therefore, may not always comply with the new paradigm of prudent risk management. This phenomenon holds in the context of both symmetric and asymmetric copulas with and without singularities. As a remedy, we introduce a notion of paths of maximal (tail) dependence and utilize it to propose several new indices of tail dependence. The suggested new indices are conservative, conform with the basic concepts of modern quantitative risk management, and are able to distinguish between distinct risky positions in situations when the existing indices fail to do so.

On Some Layer-Based Risk Measures with Applications to Exponential Dispersion Models

Layer-based counterparts of a number of well-known risk measures have been proposed and studied. Namely, some motivations and elementary properties have been discussed, and the analytic tractability has been demonstrated by developing closed-form expressions in the general framework of exponential dispersion models.