Dynamic portfolio frontier in a mean–variance framework

Abstract

The dynamic portfolio frontier theory in a mean–variance framework previously developed by scholars suffers some limitations. Specifically, the theory assumes the use of the martingale approach, the assumption of a complete market and particular probability distribution of asset returns. Accordingly, under relaxing these limitations, this study develops a calculation process for explicitly deriving the dynamic portfolio frontier and the corresponding dynamic asset allocation. Finally, for comparison, this study provides a numerical example and then draws the dynamic and static portfolio frontiers on the same graph.

Keywords:

• portfolio choice
• dynamic portfolio frontier
• dynamic asset allocation
• mean–variance frontier

JEL Classification:

• G11
• G61

Notes

¹ Static asset allocation can only be made in the beginning of the investment horizon. Nevertheless, dynamic asset allocation can be adjusted at the beginning of the investment horizon and at any time during the investment horizon. Consequently, the traditional portfolio frontier theory in the mean–variance framework, proposed by Markowitz (1952), is a static asset allocation theory. In short, static asset allocation is a passive strategy while dynamic asset allocation is an active strategy.

² Assuming a complete market, the martingale approach separates the maximizing expected terminal wealth utility problem into two steps. First, investors determine terminal wealth allocation based on each state subject to their initial endowments or budget constraints. Second, investors find a dynamic trading strategy for achieving their terminal wealth allocations. See Pliska (1986) and Cox and Huang (1989) for detail.

³ Figure 1 shows that there are four, four, five, four or five possible scenarios at the next period when entering node 1, 2, 3, 4 or 5. However, the market consists of only three assets for the entire investment horizon (t = 0–2). Since all the possible scenarios starting from each node exceed the number of assets, by definition, the market is incomplete.

⁴ A self-financing strategy means that investors have no additional income or capital input during the investment horizon.