Starting from the reward-risk model for portfolio selection introduced in De Giorgi (2005), we derive the reward-risk Capital Asset Pricing Model (CAPM) analogously to the classical mean-variance CAPM. In contrast to the mean-variance model, reward-risk portfolio selection arises from an axiomatic definition of reward and risk measures based on a few basic principles, including consistency with second-order stochastic dominance. With complete markets, we show that at any financial market equilibrium, reward-risk investors' optimal allocations are comonotonic and, therefore, our model reduces to a representative investor model. Moreover, the pricing kernel is an explicitly given, non-increasing function of the market portfolio return, reflecting the representative investor's risk attitude. Finally, an empirical application shows that the reward-risk CAPM captures the cross section of U.S. stock returns better than the mean-variance CAPM does.

We analyze if the value-weighted stock market portfolio is stochastic dominance (SD) efficient relative to benchmark portfolios formed on size, value, and momentum. In the process, we also develop several methodological improvements to the existing tests for SD efficiency. Interestingly, the market portfolio seems third-order SD (TSD) efficient relative to all benchmark sets. By contrast, the market portfolio is inefficient if we replace the TSD criterion with the traditional mean–variance criterion. Combined these results suggest that the mean–variance inefficiency of the market portfolio is caused by the omission of return moments other than variance. Especially downside risk seems to be important for explaining the high average returns of small/value/winner stocks.

We provide theoretical and empirical arguments in favor of a diminishing marginal premium for market risk. In capital market equilibrium with binding portfolio restrictions, investors with different risk aversion levels generally hold different sets of risky securities. Whereas the traditional linear relation breaks down, equilibrium can be described or approximated by a concave relation between expected return and market beta, and a concave relationship between market alpha and market beta. An empirical analysis of U.S. stock market data confirms the existence of a significant concave cross-sectional relation between average return and estimated market beta. We estimate that the market risk premium is at least four to six percent per annum, substantially above traditional estimates. A practical implication for active portfolio managers is that the alpha of “betting against beta” strategies seems dominated by the medium-minus-high-beta spread rather than the low-minus-medium-beta spread. The success of such strategies thus largely depends on underweighting or short selling high-beta stocks.