The optimal risk allocation problem, equivalently the optimal risk sharing problem, in a market with n traders endowed with risk measure ϱ1,..., ϱn is a classical problem in insurance and mathematical finance. This problem however makes only sense under a condition motivated from game theory which is called Pareto equilibrium. There are many situations of practical interest, where this condition does not hold. This is the case if the risk measures are based on essential different views towards risk. In this paper we introduce and analyze a meaningful extension of the optimal risk allocation (risk sharing) problem without assuming the equilibrium condition. The main point of this is to introduce a suitable and well motivated restriction on the class of admissible allocations which prevents effects of artificial ‘risk arbitrage’. As a result we obtain a new coherent risk measure which describes the inherent risk which remains after using admissible risk exchange in an optimal way. 1

The optimal risk allocation problem or equivalently the problem of risk sharing is the problem to allocate a risk in an optimal way to n traders endowed with risk measures ϱ1,..., ϱn. This problem has a long history in mathematical economics and insurance. In the first part of the paper we review some mathematical tools and discuss their applications to various problems on risk measures related to the allocation problem like to monotonicity properties of optimal allocations, to optimal investment problems or to an appropriate definition of the conditional value at risk. We then consider the risk allocation problem for convex risk measures ϱi. In general the optimal risk allocation problem is well defined only under an equilibrium condition. This condition can be characterized by the existence of a common scenario measure. We formulate a meaningful modification of the optimal risk allocation problem also for markets without assuming the equilibrium condition and characterize optimal solutions. The basic idea is to restrict the class of admissible allocations in a proper way. 1

Summary. In this paper we survey some recent developments on risk measures for portfolio vectors and on the allocation of risk problem. The main purpose to study risk measures for portfolio vectors X = (X1,..., Xd) is to measure not only the risk of the marginals separately but to measure the joint risk of X caused by the variation of the components and their possible dependence. Thus an important property of risk measures for portfolio vectors is consistency with respect to various classes of convex and dependence orderings. It turns out that axiomatically defined convex risk measures are consistent w.r.t. multivariate convex ordering. Two types of examples of risk measures for portfolio measures are introduced and their consistency properties are investigated w.r.t. various types of convex resp. dependence orderings. We introduce the general class of convex risk measures for portfolio vectors. These have a representation result based on penalized scenario measures. It turns out that maximal correlation risk measures play in the portfolio case the same role that average value at risk measures have in one dimensional case. The second part is concerned with applications of risk measures to the optimal risk allocation problem. The optimal risk allocation problem or, equivalently, the problem of risk sharing is the problem to allocate a risk in an optimal way to n traders endowed with risk measures ϱ1,..., ϱn. This problem has a long history in mathematical economics and insurance. We show that the optimal risk allocation problem is well defined only under an equilibrium condition. This condition can be characterized by the existence of a common scenario measure. A meaningful modification of the optimal risk allocation problem can be given also for markets without assuming the equilibrium condition. Optimal solutions are characterized by a suitable dual formulation. The basic idea of this extension is to restrict the class of admissible allocations in a proper way. We also discuss briefly some variants of the risk allocation problem as the capital allocation problem. Key words: risk measures, portfolio vector, allocation of risks

This paper examines possible barriers to securitization, focusing on behavioral responses to such novel instruments. These barriers include the difficulties of conveying the associated risks, even to investors who are sophisticated about finance (but still uncertain about model risk and structural uncertainties). Our analyses will draw on results in behavioral decision making and psychology. They will lead to proposals for empirical research and general strategies for making securities design more consonant with investor behavior.

Abstract. Risk exchange is considered here as a cooperative game with transferable utility. The set-up fits markets for insurance, securities and contingent endowments. When convoluted payoff is concave at the aggregate endowment, there is a price-supported core solution. Under variance aversion the latter mirrors the two-fund separation in allocating to each agent some sure holding plus a fraction of the aggregate.

Abstract. In this article we present a few of the results obtained on optimal reinsurance, since the pioneer work by Bruno de Finetti in 1940. As literature on the subject increased substantially in the last decade, a particular attention was given to these more recent results. Reaseguro óptimo Resumen. Este artículo presenta algunos resultados importanes de reaseguro óptimo, desde el trabajo pionero de Bruno de Finetti en 1940. Ya que la literatura sobre este tema ha aumentado de forma sustancial en la última década, le damos una atención particular a los resultados más recientes. 1 Basics of Reinsurance 1.1 Insurance and Reinsurance Under an insurance contract, the insurer accepts to pay the policyholder’s loss, (or part of it), on the occurrence of an uncertain specified event, and the policyholder accepts to pay the premium. This also happens in reinsurance contracts. Reinsurance is a form of insurance, with some differences that result from the fact that it is insurance for insurers. Reinsurance contracts are celebrated between a direct insurer and a reinsurer, with the purpose of transferring part of the risks assumed by the insurer in its business. In this way, improved conditions