| We generalize two risk-reward optimization schemes extant in the finance literature, and examine if CAPM-like asset pricing relations obtain for these schemes via the classic Sharpe argument. First, a natural generalization of the Markowitz portfolio risk measure is proposed, and a corresponding beta-pricing model closely related to the Sharpe-Lintner CAPM is obtained. The associated stochastic discount factor is expressed in closed form as a nonlinear function of the market excess return, and an equivalent non-EUT value function is provided. Explicit analytical solutions are obtained in the complete markets case for optimal portfolios, and the maximum portfolio performance ratio in an economy, as functions of the economy's unique complete markets discount factor. Finally, econometric issues are considered, and correspondence with the Sharpe-Litner CAPM is discussed. A second class of risk-reward optimization schemes is obtained by generalizing the gain-loss measures of Bernardo and Ledoit. Again, CAPM-like asset-pricing relations are derived, and equivalent non-EUT value functions are provided. It is shown that in the appropriate parameter regime, the non-EUT value function corresponding to the generalized gain-loss measures is akin to the S-shaped value function used by Kahneman and Tversky to describe loss-averse agents. An explicit closed-form expression is obtained for the stochastic discount factor as a function of the market excess return. The discount factor is shown to be non-negative in general, and strictly positive in a complete markets setting. Explicit analytical solutions are obtained in the complete markets case for optimal portfolios, and the maximum portfolio performance ratio in a (generalized) gain-loss economy, as functions of the economy's unique complete markets discount factor. It is shown that a representative agent's optimal zero-cost portfolio payoff in a complete markets setting is strictly positive in all future states of nature except one, where it is strictly negative. The asymmetry in positive and negative state-specific payoffs is found to decrease as the market becomes more incomplete. Our CAPM-like models are in a sufficient state of development to be tested directly against market data. |
