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Since risky positions in multivariate portfolios can be offset by variouschoices of capital requirements that depend on the exchange rules and relatedtransaction costs, it is natural to assume that the risk measures of randomvectors are set-valued. Furthermore, it is reasonable to include the exchangerules in the argument of the risk measure and so consider risk measures ofset-valued portfolios. This situation includes the classical Kabanov'stransaction costs model, where the set-valued portfolio is given by the sum ofa random vector and an exchange cone, but also a number of further cases ofadditional liquidity constraints.  We suggest a definition of the risk measure based on calling a set-valuedportfolio acceptable if it possesses a selection with all individuallyacceptable marginals. The obtained selection risk measure is coherent (orconvex), law invariant and has values being upper convex closed sets. Wedescribe the dual representation of the selection risk measure and suggestefficient ways of approximating it from below and from above. In case ofKabanov's exchange cone model, it is shown how the selection risk measurerelates to the set-valued risk measures considered by Kulikov (2008), Hamel andHeyde (2010), and Hamel, Heyde and Rudloff (2013).

Multivariate risk measures: a constructive approach based on selections