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To manage the risk of insurance companies, a reinsurance transaction is among the myriad risk management mechanisms the top ranked choice. In this paper, we study the design of optimal reinsurance contracts within a risk measure minimization framework and subject to the Vajda condition. The Vajda condition requires the reinsurer to take an increasing proportion of the loss when it increases and therefore imposes constraints on the indemnity function. The distortion-risk-measure-based objective function is very generic, and allows for, for example, an objective to minimize the risk-adjusted value of the insurer’s liability, and for heterogeneous beliefs regarding the loss distribution by the insurer and reinsurer. Under a mild condition, we propose a backward-forward optimization method that is based on a marginal indemnity function formulation. To show the applicability and simplicity of our strategy, we provide three concrete examples with the Value-at-Risk: one with the risk-adjusted value of the insurer’s liability, one with an objective function that follows from imposing Pareto optimality, and one with heterogeneous beliefs. We conclude this paper with an empirical application with Danish fire insurance losses and the Value-at-Risk and the Tail Value-at-Risk.

In this paper we propose a quantile-based risk measure which is defined using the modified loss distribution according to the decision maker’s risk and loss aversion. The properties related to different classes of disutility functions are established. A portfolio selection model in the Mean-Risk framework is proposed and equivalent formulations of the model generating the same efficient frontier are given. The advantages of this approach are investigated using real world data from NYSE. The differences between the efficient frontier of the proposed model and the classical Mean-Variance and Mean-CVaR are quantified and interpreted. Extensive experiments show that the efficient portfolios obtained by using the proposed model exhibit lower risk levels and an increased satisfaction compared to the other two Mean-Risk models.

In general, a portfolio problem minimizes risk (or negative utility) of a portfolio of financial assets with respect to portfolio weights subject to a budget constraint. The inverse portfolio problem then arises when an investor assumes that his/her risk preferences have a numerical representation in the form of a certain class of functionals, e.g. in the form of expected utility, coherent risk measure or mean-deviation functional, and aims to identify such a functional, whose minimization results in a portfolio, e.g. a market index, that he/she is most satisfied with. In this work, the portfolio risk is determined by a coherent risk measure, and the rate of return of investor’s preferred portfolio is assumed to be known. The inverse portfolio problem then recovers investor’s coherent risk measure either through finding a convex set of feasible probability measures (risk envelope) or in the form of either mixed CVaR or negative Yaari’s dual utility. It is solved in single-period and multi-period formulations and is demonstrated in a case study with the FTSE 100 index.

Model risk has a huge impact on any risk measurement procedure and its quantification is therefore a crucial step. In this paper, we introduce three quantitative measures of model risk when choosing a particular reference model within a given class: the absolute measure of model risk, the relative measure of model risk and the local measure of model risk. Each of the measures has a specific purpose and so allows for flexibility. We illustrate the various notions by studying some relevant examples, so as to emphasize the practicability and tractability of our approach., Comment: 23 pages, 6 figures

The conditions under which the classical measures of risk like the mean, the linear correlation coefficient and VaR can be used are discussed. The definition of risk measure and the main recently proposed risk measures are presented. The problems connected with co-dependence are outlined.

When stating a risk measure, the attention has to be concentrated on the amount of money, which is required to guarantee the feasibility of the final value for the position in exam. But this has to be done in relative terms, with respect to a lower bounding threshold if the aim is safety, with respect to a stimulating target, if the objective is to improve the rentability. Moreover, the attitude towards potential losses from the demand and offer sides have to be modelled in different ways. Aim of this paper is to review the current literature to show how one-side and intertemporal elements have to be explicitly included in the definition of a risk measure, to give a flexible policy instrument to risk managers. A new one-side dynamic measure is then defined according to suitable coherence conditions.

One of the basic problems of applied finance is the optimal selection of stocks, with the aim of maximizing future returns and constraining risks by an appropriate measure. Here, the problem is formulated by finding the portfolio that maximizes the expected return, with risks constrained by the worst conditional expectation. This model is a straightforward extension of the classic Markovitz mean–variance approach, where the original risk measure, variance, is replaced by the worst conditional expectation. The worst conditional expectation with a threshold a of a risk X , in brief WCEaðX Þ, is a function that belongs to the class of coherent risk measures. These are measures that satisfy a set of properties, such as subadditivity and monotonicity, that are introduced to prevent some of the drawbacks that affect some other common measures. This paper shows that the optimal portfolio selection problem can be formulated as a linear programming instance, but with an exponential number of constraints. It can be solved efficiently by an appropriate generation constraint subroutine, so that only a small number of inequalities are actually needed. This method is applied to the optimal selection of stocks in the Italian financial market and some computational results suggest that the optimal portfolios are better than the market index. � 2003 Elsevier B.V. All rights reserved.

This paper presents a portfolio selection model based on the idea of approximation. The model describes a portfolio by its decumulative distribution curve and a preference structure by a family of convex indifference curves. It prescribes the optimal portfolio as the one whose decumulative curve has the highest tangent indifference curve. The model extends the mean–variance model in the sense that it does not restrict the return distributions of assets to be normal. While under the assumption of normality, the model simplifies to the mean–variance model. The model has a measure of risk attitudes that resembles the Arrow–Pratt measure while combining both wealth and probability attitudes. Using this measure, we show that the smaller the curvature of a value function and the larger the curvature of a weighting function, the more risk averse an agent.