Generalized Numeraire Portfolios

Given a set of assets, a numeraire portfolio (Long, 1990) is a self-financing portfolio with positive value and whose return process is a stochastic discount factors process. By relaxing the self-financing constraint, we define the generalized numeraire portfolios, and state necessary and sufficient conditions for their existence. We show that a set of assets admits generalized numeraire portfolios if and only if it is arbitrage free and at least one trading strategy has positive value. We also show that generalized numeraire portfolios are solutions to the optimal growth problem under the weaker constraint that the self-financing condition holds in conditional discounted expected value. Since the numeraire portfolio is unique (up to a scale factor), it generates only one admissible stochastic discount factor process. Generalized numeraire portfolios generate instead an infinite subset of, and under some conditions, all the admissible one-period stochastic discount factors. Finally, we propose some interesting tests that exploit the notion of generalized numeraire portfolios and provide preliminary empirical evidence.