It makes sense to stochastic discount factor as a function of impatience and marginal utility of consumption, but what is the rationale to project it into the vector space spanned by the payoff vectors?
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The SDF should be viewed as the Radon-Nikodym derivative of the risk-neutral measure with respect to the objective measure. When you're pricing a derivative security using the risk-neutral measure, the part of the SDF that is orthogonal to the payoff vectors is not seen. If the market is complete, of course then projected SDF is the original SDF. – Michael May 11 '16 at 20:31