Zinoviy Landsman

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.

Tail Conditional Expectations for Exponential Dispersion Models

There is a growing interest in the use of the tail conditional expectation as a measure of risk. For an institution faced with a random loss, the tail conditional expectation represents the conditional average amount of loss that can be incurred in a fixed period, given that the loss exceeds a specified value. This value is typically based on the quantile of the loss distribution, the so-called value-at-risk. The tail conditional expectation can therefore provide a measure of the amount of capital needed due to exposure to loss. This paper examines this risk measure for “exponential dispersion models”, a wide and popular class of distributions to actuaries which, on one hand, generalizes the Normal and shares some of its many important properties, but on the other hand, contains many distributions of nonnegative random variables like the Gamma and the Inverse Gaussian.

Multivariate Pareto portfolios: TCE-based capital allocation and divided differences

Determination of risk capital is a subject of active interest to researchers, regulators of financial institutions, and commercial vendors of financial products and services. Recently, there has been growing concentration among the insurance companies and regulators on the use of tail conditional expectation (TCE) as measure of risk. The present study examines the TCE-based portfolio allocation for multivariate dependent Pareto risks. This family is broadly popular in actuarial sciences, mostly because of modeling heavy-tailed dependent losses. We show that the tool of divided differences, actually important in numerical analysis and polynomials approximations, is quite convenient in the problem of capital asset allocation.

Risk measures and insurance premium principles

Abstract Risk measures based on distorted probabilities have been recently developed in actuarial science and applied to insurance rate making. We propose a risk measure that has the properties of risk aversion and diversification, is additive for losses and consistent in its treatment of insurance and investment risks. We show that the risk measure based on distorted probabilities is not consistent in its ordering of insurance and investment risks.

Exponential dispersion models and credibility

Abstract The Exponential Dispersion Family is a rich family of distributions, comprised of several distributions, some of which are heavy-tailed and as such could be of significant relevance to actuarial science. The family draws its richness from a dispersion parameter σ 2 = 1/λ which is equal to 1 in the case of the Natural Exponential Family. We consider three cases. In the first λ is assumed known, in the second a prior distribution for λ is given, and in the third the prior distribution of λ is not known and is derived by means of the maximum entropy principle, assuming the prior mean of λ can be specified. For these cases, a conjugate prior distribution for the risk parameter is assumed and credibility formulae are derived for the estimation of the fair premium.

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.

Asymptotic behavior of sample mean location for manifolds

We investigate some asymptotic properties of empirical mean location on compact smooth submanifolds of Euclidean space. Thus our results provide the framework for asymptotic least-squares statistics inference regarding mean location in a rather general situation.

Credibility evaluation for the exponential dispersion family

Abstract It has long been established that under regularity conditions, the linear credibility formula with an appropriate credibility factor produces exact fair premium for claims or losses whose distribution is a member of the natural exponential family. Recently, this result has been extended to a richer family of distribution, the exponential dispersion family which comprised of several distributions, some of which are heavy-tailed and as such could be of significant relevance to actuarial science. The family draws its richness from a dispersion parameter σ2=1/λ which is equal to 1 in the case of the natural exponential family. In this paper neither λ is regarded known, nor a fully specified prior distribution for λ is assumed. Instead, by establishing a link between the m.s.e. of the linear credibility and Fisher information we derive optimal credibility for the case where only the mean and variance of λ are specified.

Industrial job-shop scheduling with random operations and different priorities

Abstract A classical job-shop scheduling problem with n jobs (orders) and m machines is considered. Each job-operation Oil (the l− th operation of job i, l = 1,…,m, i = 1, 2,…,n) has a random time duration til with the average value t il and the variance Vil. Each job Ji has its due date Di and its priority index ϱi. Given p i ∗ , the desired probability for job Ji to be accomplished on time, and p i ∗ ∗ , the least permissible probability for the job to meet its due date on time, the problem is to determine starting time values Sil for each job-operation Oil. Those values are not calculated beforehand and are values conditioned on our decisions. Decision-making, i.e., determining values Sil is carried out at the moments when at least one of the machines is free for service and at least one job is ready to be processed on that machine. If at a certain moment t more than one job is ready to be processed, these jobs are compared pairwise. The winner of the first pair will be compared with the third job, etc., until only one job will be left. The latter has to be chosen for the machine. The competition is carried out by calculating the jobs delivery performance, i.e., the probability for a certain job to meet its due date on time. Such a calculation is carried out by determining the probability to meet the deadline for the chain of random operations. Two different heuristics for choosing a job from the line will be imbedded in the problem. The first one is based on examining delivery performance values together with priority indices ϱi. The second one deals with examining confidence possibilities p i ∗ and p i ∗ ∗ and does not take into account priority indices. A numerical example is presented. Both heuristics are examined via extensive simulation in order to evaluate their comparative efficiency for practical industrial problems.

Robustness via a mixture of exponential power distributions

A mixture of exponential power distributions (EPD) is suggested and is shown to possess robust qualities. The Gibbs sampler is applied for estimating the unknown parameters of the model and its algorithm is devised in order to allow a wide range of prior distributions for the unknown parameters. Numerical studies, using real financial data, demonstrate the effectiveness of the proposed model and a theoretical study explains the superiority of the EPD mixture over a normal mixture.