Markowitz principles for multi-period portfolio selection problems with moments of any order

The multi-period portfolio selection problem is formulated as a Markowitz mean variance optimization problem in terms of time-varying means, covariances and higher-order and intertemporal moments of the asset prices. The crux lies in expressing the number of shares of any particular asset to be transacted on any future trading date, which is a non-anticipative process, as a polynomial of the changes in the discounted prices of all the risky assets. This results in the expected return of the portfolio being dependent on not only the means of the asset prices, but also the higher-order and intertemporal moments, and its variance being dependent on not only the second-order moments, but also the higher-order and intertemporal moments. As illustrations, we study the portfolio selection problems including the discrete version of the Merton problem. It is shown numerically that the efficient frontier obtained from Markowitz's discrete multi-period formulation coincides with that from Merton's continuous-time formulation when the number of rebalancing periods is 'large'. The effects of dynamic trading, in particular volatility pumping, in comparison with a static single-period model are measured by a non-dimensional number, Dyn(P) (P, number of trading periods), which quantifies the relative gain due to dynamic trading. It is sufficient to rebalance the portfolio a few times in order to get more than 95% of the gain due to continuous trading.