Risk capital allocation problems have been widely discussed in the academic literature. We consider a set of independent subunits collaborating in order to reduce risk: that is, when subunit portfolios are merged a diversification benefit arises and the risk of the group as a whole is smaller than the sum of the risks of the individual subunits. The question is how to allocate the risk capital of the group among the subunits in a fair way. In this paper we propose to use the Lorenz set as an allocation method. We show that the Lorenz set is operational and coherent. Moreover, we propose three fairness tests related directly to the problem of risk capital allocation and show that the Lorenz set satisfies all three tests in contrast to other well-known coherent methods. Finally, we discuss how to deal with non-uniqueness of the Lorenz set.