Two sided efficient frontiers at multiple time horizons

Abstract

Two price economy principles motivate measuring risk by the cost of acquiring the opposite of the centered or pure risk position at its upper price. Asymmetry in returns leads to differences in risk charges for short and long positions. Short risk charges dominate long ones when the upper tail dominates the comparable lower tail for charges based on distorted expectations. Positive mean return targets acquire long positions with negative mean return targets taking short positions. In each case the appropriate risk charge is minimized to construct two frontiers, one for the positive, and the other for negative, mean return targets. Multivariate return distributions reflect limit laws given by Q self-decomposable laws displaying decay rates in skewness and excess kurtosis slower than those for processes of independent and identically distributed returns. Frontiers at longer horizons display greater efficiency reflected by lower risk charges for comparable mean return targets. The short side frontiers also display greater risk charges than their long side counterparts. All efficient portfolios deliver asset pricing equations whereby required returns in excess of a reference rate are a market price of risk times a risk gradient evaluated at the efficient portfolio. Variations in frontiers and points on the frontier induce differences in reference rates, risk gradients, and the market prices of risk that can yet lead to comparable required returns.