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For lifelong health insurance covers, medical inflation not incorporated in the level premiums determined at policy issue requires an appropriate increase of these premiums and/or the corresponding reserves during the term of the contract. In this paper, we investigate appropriate premium indexing mechanisms, based on a given medical inflation index. First, we consider a general relation between benefit, premium and reserve increases, which can be used on a yearly basis to restore the actuarial equivalence that is broken due to observed medical inflation over the past year. Next, we consider an individual premium indexing mechanism, depending on the age at policy issue, which makes the relative premium increases above the observed medical inflation more stable over time. Finally, we consider an aggregate premium indexing mechanism for a portfolio of new entrants, where the relative premium increase above the observed inflation is independent of age-at-entry, introducing intergenerational solidarity.

Using a standard reduction argument based on conditional expectations, this paper argues that risk sharing is always beneficial (with respect to convex order or second degree stochastic dominance) provided the risk-averse agents share the total losses appropriately (whatever the distribution of the losses, their correlation structure and individual degrees of risk aversion). Specifically, all agents hand their individual losses over to a pool and each of them is liable for the conditional expectation of his own loss given the total loss of the pool. We call this risk sharing mechanism the conditional mean risk sharing. If all the conditional expectations involved are non-decreasing functions of the total loss then the conditional mean risk sharing is shown to be Pareto-optimal. Explicit expressions for the individual contributions to the pool are derived in some special cases of interest: independent and identically distributed losses, comonotonic losses, and mutually exclusive losses. In particular, conditions under which this payment rule leads to a comonotonic risk sharing are examined.

In Dhaene et al. (2005), multiperiod portfolio selection problems are discussed, using an analytical approach to find optimal constant mix investment strategies in a provisioning or savings context. In this paper we extend some of these results, investigating some specific, real-life situations. The problems that we consider in the …rst section of this paper are general in the sense that they allow for liabilities that can be both positive or negative, as opposed to Dhaene et al. (2005), where all liabilities have to be of the same sign. Secondly, we generalize portfolio selection problems to the case where a minimal return requirement is imposed. We derive an intuitive formula that can be used in provisioning and terminal wealth problems as a constraint on the admissable investment portfolios, in order to guarantee a minimal annualized return. We always apply our results to optimal portfolio selection.

Modern Actuarial Risk Theory -- Using R contains what every actuary needs to know about non-life insurance mathematics. It starts with the standard material like utility theory, individual and collective model and basic ruin theory. Other topics are risk measures and premium principles, bonus-malus systems, ordering of risks and credibility theory. It also contains some chapters about Generalized Linear Models, applied to rating and IBNR problems. As to the level of the mathematics, the book would fit in a bachelors or masters program in quantitative economics or mathematical statistics. This second and much expanded edition emphasizes the implementation of these techniques through the use of R. This free but incredibly powerful software is rapidly developing into the de facto standard for statistical computation, not just in academic circles but also in practice. With R, one can do simulations, find maximum likelihood estimators, compute distributions by inverting transforms, and much more.

A premium principle is an economic decision rule used by the insurer in order to determine the amount of the net premium for each risk in his portfolio. In this paper we investigate the problem how to determine the premium principle to be used. In Goovaerts & Dhaene (1997), DTEW Research Report 9740, K.U.Leuven, we can see some desirable properties of a premium principle. We consider a premium principle for risks of any sign, and prove a representation of premium principle without some property which involves the distribution of a risk. Later we introduce this property as a corollary.

In this paper we analyze and evaluate a standard approach financial institutions use to calculate their so-called total economic capital. If we consider a business that faces a total random loss S over a given one-year horizon then economic capital is traditionally defined as the difference between 99.97% percentile of S and its expectation. The standard approach essentially assumes that the different components (risks) of S are multivariate normally distributed and this highly facilitates the computation of the total aggregated economic capital. In this paper we show that this approach also holds for a more general framework which encompasses as a special case the multivariate normal (and elliptical) setting. We question also the assumption of multivariate normality since for many risks one often assumes other than normal distributions (e.g. a lognormal distribution for insurqnce risk). Assuming that risks are either normal or lognormal distributed we propose, using the concept of comonotonicity, an alternative aggregation approach.

The Basel Accords represent landmark financial agreements for the regulation of commercial banks. The main purpose of the accords was to strenghten the soundness and stability of the international banking system by providing a minimum standard for capital requirements. In 2004, the Basel Committee proposed new guidelines, which have become known as 'Basel II'. We give a short overview of the Basel II framework and present the different approaches which can be used to determine the amount of regulatory capital needed for equity exposures. These methods vary from simple, rather rule of thump methods, to more sophisticated and economic-oriented approaches. We compare the regulatory capital consumption of two equity portfolios using the different Basel II-compliant methods. We provide evidence that, as far as regulatory capital consumption for equity exposures is concerned, there is no real incentive for banks to use the more sophisticated and economic-oriented models such as VaR or EVT models. Keywords: Basel II, Regulatory Capital, Value at Risk, Extreme Value Theory.

In this paper we examine and summarize properties of several well-known risk measures, with special attention given to the class of distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. We also consider the problem of evaluating risk measures for sums of non-independent random variables and propose approximations based on the concept of comonotonicity.

One of the key parameters in the computation of Credit Economic Capital is the so called capital multiplier. In the light of an existing variance-covariance framework we propose a methodology rather than using a benchmark number. The paper provides an algorithm in doing so.

In this pedagogical note, it is shown how extremal values of classical measures of association like Pearson's correlation coefficient, Kendall's τ, Spearman's ρ and Gini's γ, characterize comonotonicity and countermonotonicity. The link between zero-correlation and mutual independence is also examined.

Actuaries usually compute multiple life premiums based on the unrealistic assumption of independence of the lifelengths of insured persons. Many clinical studies, however, have demonstrated dependence of the lifetimes of paired lives such as husband and wife. In this respect, the present article tries to give an answer to the following question: does this simplifying hypothesis constitute a real financial danger for the insurance company? The answer turns out to be affirmative: this dependence materially affects the values of annuities and insurances involving multiple lives. In order to quantify the impact of a possible dependence on the amount of premium charged for annuities, insurances and widow's pension, we resort here on the Fréchet bounds, Markov processes and some copula models. These techniques are applied to classical insurance contracts issued to married couples and illustrated on NIS data as well as on observations from Brussels city.