Multiperiod dynamic investment for a generalized situation

Abstract

This study aims to demonstrate the optimal multiperiod dynamic asset allocation for a generalized situation and enable the investor to maximize his expected terminal wealth utility. Previous researches solved this problem constrained by the investor's utility function, the asset return distributions, the completeness of the market, the lack of transaction costs and other factors. Accordingly, this study considers a generalized situation where all the constraints are relaxed and provides a calculation process for solving this problem.

Notes

¹ The Markowitz's theory has been extended numerously. For example, Levy and Sarnat (1970) provided a graphical exposition of efficiency criteria and suggested a simple modification of the mean-variance rule. Sengupta and Sfeir (1985) examined the mean-variance efficiencies among some limited diversification portfolios. Moreno et al. (2005) replaced the historical means and variances with a forecasting model in the emerging security markets and determined the optimal mean-variance efficiency portfolio. Ulucan (2007) investigated the optimal holding period (investment horizon) for the classical mean-variance portfolio optimization problem.

² If two funds separation theorem holds, investors only require two funds for investment. The two funds generally include a riskless asset and a risky assets portfolio. The risky assets portfolio generally consists of equities, long-term bonds and other risky securities. Most investors use a broader range of money instruments as the riskless asset. Additionally, most money market funds hold, for the most part, three types of securities: treasury bills, bank certificates of deposit and commercial paper (Bodie et al., 2005, Chap. 7).

³ For each node occurring, in Fig. 1, there are four possible scenarios at next period. However, even though the numbers of the possible scenarios for each node are not completely the same, the analysis process remains unchanged. Additionally, since the number of possible scenarios at next period (=4) exceeds the number of assets in the market (=2), the market is incomplete.

⁴ For example, if the one-period interest rate of return on asset 2 from node 1 (at t = 0) to node 4 (at t = 1) is 5% and the one-period interest rate of return on asset 2 from node 4 to maturity (at t = 2) given state s = 10 is 8%, then R _2,1,10 (1 + 5%)(1 + 8%) − 1 = 13.4%.

⁵ A self-financing strategy means that investors have no other income or capital input during the investment horizon. That is, in addition to initial endowment, the final total wealth is raised only from the investment on market assets.

⁶ The detailed proof can be obtained from the corresponding author. Although the proof is not difficult, it is tedious and hence omitted.

⁷ For further details about these numerical methods, the readers can refer to Judd (1998, Chap. 4) and Lindfield and Penny (2000, Chap. 8).

⁸ In Taiwan Stock Exchange Corporation, the brokerage is 0.1425% per dollar traded (bought or sold) and the transaction tax is 0.3% per dollar sold and free for bought. Accordingly, the average transaction cost per dollar traded equals (0.1425% × 2 + 0.3%) ÷ 2 = 0.2925% ≈ 0.003.