In the first part of the presentation we introduce a class of systemic risk measures that are determined in terms of sufficient, possibly random capital allocation to individual banks before aggregation of their risks. In particular, we focus on the question how to allocate the corresponding total systemic risk among institutions in a fair way. We show that the dual problem of the minimization problem that identifies the systemic risk measure provides a valuation of the random capital allocation which is fair both from the point of view of the society/regulator and from the individual financial institutions. This leads to a new systemic equilibrium concept that we denote by Systemic Optimal Risk Transfer Equilibrium (SORTE). In the second part we then analyze the concept of a SORTE, which is inspired by Buhlmann's classical notion of an Equilibrium Risk Exchange. We provide sufficient general assumptions that guarantee existence, uniqueness, and Pareto optimality of a SORTE. In both the Buhlmann and the SORTE definition, each agent is behaving rationally by maximizing her expected utility given a budget constraint. However, while in Buhlmann's definition the budget constraint is given a priori, in the SORTE approach the budget constraint is endogenously determined by solving a systemic utility maximization. SORTE gives priority to the systemic aspects of the problem, in order to optimize the overall systemic performance, rather than to individual rationality.