Fair risk allocation in illiquid markets

Let us consider a financially constrained leveraged financial firm having some divisions which have invested into some risky assets. Using coherent measures of risk the sum of the capital requirements of the divisions is larger than the capital requirement of the firm itself, there is some diversification benefit that should be allocated somehow for proper performance evaluation of the divisions. In this paper we use cooperative game theory and simulation to assess the possibility to jointly satisfy three natural fairness requirements for allocating risk capital in illiquid markets: Core Compatibility, Equal Treatment Property and Strong Monotonicity. Core Compatibility can be viewed as the allocated risk to each coalition (subset) of divisions should be at least as much as the risk increment the coalition causes by joining the rest of the divisions. Equal Treatment Property guarantees that if two divisions have the same stand-alone risk and also they contribute the same risk to all the subsets of divisions not containing them, then the same risk capital should be allocated to them. Strong Monotonicity requires that if a division weakly reduces its stand-alone risk and also its risk contribution to all the subsets of the other divisions, then as an incentive its allocated risk capital should not increase. Analyzing the simulation results we conclude that in most of the cases it is not possible to allocate risk in illiquid markets satisfying the three fairness notions at the same time, one has to give up at least one of them.