Selfsimilarity in Long Horizon Asset Returns

Abstract

Daily return distributions are modeled by pure jump limit laws that are selfdecomposable laws. The returns may be seen as composed of a sum of independent and identically distributed increments or as a selfsimilar law scaling the sum of exponentially weighted past shocks or a combination thereof. To the extent the selfsimilar component is present and the scaling coefficient is above a half it is shown that long horizon returns may not converge to a normal distribution. Estimations conducted on 214 equity underliers over the period January 2007 to February 2017 support this lack of convergence to normality at very long horizons. An analysis of distributions embedded in option data shows that the convergence to normality is also halted risk neutrally. Selfsimilar components are estimated to have a physical half life between one or two days and a risk neutral half life around a year. In the long run markets are in an equilibrium state of motion engineered to avoid the evolution of good deals. The associated equilibrium solutions are illustrated. The implications of a selfsimilar scaling component for the equity bias, volatility and desirability of returns across horizons, horizon effects on expected returns, and Sharpe ratios are developed. Additionally long horizon return modeling is employed to construct alternative long horizon risk free rates using volatility targets.