Before revision:  We show that the excess rate of return of a risky asset in a pure-jump model is given by an integral of the product of a term representing the riskiness of the asset and a term representing the level of market risk aversion. The integral is over the state space of the Poisson random measure and is taken with respect to the L\'evy measure associated with the n-dimensional L\'evy process. The resulting framework is applied to a variety of different asset classes, allowing one to construct new models as well as non-trivial generalizations of familiar models.
After revision:  We show that the excess rate of return above the interest rate can be calculated for risky assets in such models, thus showing the relationship between risk and return when asset prices have jumps. The framework is applied to a variety of different asset classes, allowing one to construct new models as well as non-trivial generalizations of familiar models.