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Cooperative investment problem with an authoritative risk determined by Central Bank

Almualim Anwar

Abstract: In this paper, we are interested to provide an analytic solution for cooperative investment risk with an authoritative risk determined by the central Bank. This problem plays an important role in solving cooperative investment problems in an investment sector such as insurance companies or banks etc and keeping in our mind the effect of a risk determined by the central Bank which has not been done before. We reformulate cooperative investment risk by writing dual representation for each risk preference (Coherent risk measure) for each agent (investor). Finding an analytic solution for this problem for both cases individual and cooperative investment problem by using dual representation for each risk preference has a strong effect on the financial market. Moreover, we find the equilibrium allocation in terms of an equilibrium price by formulating the optimization problem in the case of equilibrium with an initial endowment for each agent’s ’investor’. In addition, formulate a problem that covers the risk minimization problem with an expected return constraint and expected return maximization problem with risk constraint, in both individual and cooperative investment cases, for the general case of an arbitrary joint distribution for the asset return under certain conditions and assuming that all coherent risk measure is continuous from below. Thus, the optimal portfolio is written as the optimal Lagrange multiplier associated with an equality-constrained dual problem. Furthermore, a unique equilibrium allocation as a fair optimal allocation solution in terms of equilibrium price density function for each agent (investor) is also shown.
Keywords: optimal / cooperative investment / return / analytic solution / equilibrium price / risk measure / coherent risk / equilibrium allocation

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