Three Solutions to the Pricing Kernel Puzzle

The pricing kernel puzzle is the observation that the pricing kernel might be increasing in some range of the market returns. This paper analyzes the pricing kernel in a financial market equilibrium. If mar- kets are complete and investors are risk-averse and have common and true beliefs, the pricing kernel is a decreasing function of aggregate resources. If at least one of these assumptions is violated, the pricing kernel is not necessarily decreasing. Thus, incomplete markets, risk- seeking behaviour and incorrect beliefs can induce increasing parts in the pricing kernel and can be seen as potential solutions for the pricing kernel puzzle. We construct examples to illustrate the three explana- tions. We verify the robustness of the explanations under aggregation and compare the phenomena with the findings in the empirical liter- ature. The results are used to reveal strengths and weaknesses of the three solutions. Risk-seeking behaviour is a fragile explanation that can only work in a model with atomic state space. Biased beliefs are robust under aggregation and consistent with the empirical findings. In incomplete markets, it is easy to find a pricing kernel with increasing parts. In order to get situations where all pricing kernels have increas- ing parts, we need extreme assumptions on the wealth distribution.