Mean-variance-time: An extension of Markowitz's mean-variance portfolio theory

•We extend the classical mean-variance (MV) portfolio theory to a time dimension that allows ex-post trading.•Our extension validates MV theory by proving the uniqueness of a well-behaved utility representation.•Our extension is also consistent with the short-run reversal and long-run momentum of observed stock returns. In finance, investment decisions are commonly based on Markowitz's ex-ante mean-variance (MV) portfolio problem. The static ex-post trading problem, however, is completely absent. In this paper, we propose a theoretical extension of the MV framework by adding a time dimension so that the construction of a portfolio is thought of as an activity that consists of n monetary outcomes, i.e., rates of return on n risk assets, and the portfolio duration time t, which is the investor's optimal trading strategy time. Under a set of axioms, we show the existence and uniqueness of a utility function that represents investors’ preference over different time horizons. The analytical solution over the extended field yields an expression where optimal portfolio duration time depends explicitly on various sources of uncertainty; a key result that distinguishes this paper from the existing literature. We demonstrate empirically that our proposed model can explain many of the observed time-related anomalies of stock returns. Finally, we show that long-term trading strategies are more profitable for rational investors under perfect information.