We investigate the structure of the pricing kernels in a general dynamic investment setting by making use of their duality with the self financing portfolios. We generalize the variance bound on the intertemporal marginal rate of substitution introduced in Hansen and Jagannathan (1991) along two dimensions, first by looking at the variance of the pricing kernels over several trading periods, and second by studying the restrictions imposed by the market prices of a set of securities. The variance bound is the square of the optimal Sharpe ratio which can be achieved through a dynamic self financing strategy. This Sharpe ratio may be further enhanced by investing dynamically in some additional securities. We exhibit the kernel which yields the smallest possible increase in optimal dynamic Sharpe ratio while agreeing with the current market quotes of the additional instruments.
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