A risky design equilibrium problem is an equilibrium system that involves N designers who invest in risky assets, such as production plants, evaluate these using convex or coherent risk measures, and also trade financial securities in order to manage their risk. Our main finding is that in a complete risk market - when all uncertainties can be replicated by financial products - a risky design equilibrium problem collapses to what we call a risky design game, i.e., a stochastic Nash game in which the original design agents act as risk neutral and there emerges an additional system risk agent. The system risk agent simultaneously prices risk and determines the probability density used by the other agents for their risk neutral evaluations. This situation is stochastic-endogenous: the probability density used by agents to value uncertain investments is endogenous to the risky design equilibrium problem. This result is most striking when design agents use coherent risk measures in which case the intersection of their risk sets turns out to be a risk set for the system risk agent, thereby extending existing results for risk markets. We also investigate existence of equilibria in both the complete and incomplete cases.