Multiperiod mean absolute deviation uncertain portfolio selection with real constraints

Abstract

Absolute deviation is a commonly used risk measure, which has attracted more attentions in portfolio optimization. Most of existing mean–absolute deviation models are devoted to stochastic single-period portfolio optimization. However, practical investment decision problems often involve the uncertain dynamic information. Considering transaction costs, borrowing constraints, threshold constraints, cardinality constraints and risk control, we present a novel multiperiod mean absolute deviation uncertain portfolio selection model, which an optimal investment policy can be generated to help investors not only achieve an optimal return, but also have a good risk control. In proposed model, the return rate of asset and the risk are quantified by uncertain expected value and uncertain absolute deviation, respectively. Cardinality constraints limit the number of risky assets in the optimal portfolio. Threshold constraints limit the amount of capital to be invested in each asset and prevent very small investments in any asset. Based on uncertainty theories, the model is transformed into a crisp dynamic optimization problem. Because of the transaction costs and cardinality constraints, the multiperiod portfolio selection is a mix integer dynamic optimization problem with path dependence, which is “NP hard” problem that is very difficult to solve. The proposed model is approximated to a mix integer dynamic programming model. A novel discrete iteration method is designed to obtain the optimal portfolio strategy and is proved linearly convergent. Finally, an example is given to illustrate the behavior of the proposed model and the designed algorithm using real data from the Shanghai Stock Exchange.

Appendices

Appendix A

Table 6 The uncertain return rates on assets of five periods investment

Table 7 The uncertain return rates on assets of five periods investment

Table 8 The uncertain return rates on assets of five periods investment

Table 9 The uncertain return rates on assets of five periods investment

Table 10 The uncertain return rates on assets of five periods investment

Table 11 The uncertain return rates on assets of five periods investment

Table 12 The uncertain return rates on assets of five periods investment

Table 13 The uncertain return rates on assets of five periods investment

Table 14 The uncertain return rates on assets of five periods investment

Table 15 The uncertain absolute deviation of assets of five periods investment

The codes of thirty stocks are, respectively, $\hbox {S}_{1}$ (600,000), $\hbox {S}_{2}$ (600,005), $\hbox {S}_{3}$ (600,015), $\hbox {S}_{4}$ (600,016), $\hbox {S}_{5}$ (600,019), $\hbox {S}_{6}$ (600,028), $\hbox {S}_{7}$ (600,030), $\hbox {S}_{8}$ (600,036), $\hbox {S}_{9}$ (600,048), $\hbox {S}_{10}$ (600,050), $\hbox {S}_{11}$ (600,104), $\hbox {S}_{12}$ (600,362), $\hbox {S}_{13}$ (600,519), $\hbox {S}_{14}$ (600,900), $\hbox {S}_{15}$ (601,088), $\hbox {S}_{16}$ (601,111), $\hbox {S}_{17}$ (601,166), $\hbox {S}_{18}$ (601,168), $\hbox {S}_{19}$ (601,318), $\hbox {S}_{20}$ (601,328), $\hbox {S}_{21}$ (601,390), $\hbox {S}_{22}$ (601,398), $\hbox {S}_{23}$ (601,600), $\hbox {S}_{24}$ (601,601), $\hbox {S}_{25}$ (601,628), $\hbox {S}_{26}$ (601,857), $\hbox {S}_{27}$ (601,919), $\hbox {S}_{28}$ (601,939), $\hbox {S}_{29}$ (601,988), $\hbox {S}_{30}$ (601,998). The triangle uncertain distributions, $\xi _{it }=(a_{it}, \alpha _{it}, \beta _{it})$, of the return rates of assets at each period can be obtained as shown in Tables 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14.

Appendix B

According Tables 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14, and Eq. (25), $\mathrm{AD}_{t}(R_{it}) (i = 1,{\ldots },30; {t} = 1,{\ldots },5)$ can be obtained as shown in Tables 15, 16, 17 and Table 18.

Table 16 The credibilistic absolute deviation of assets of five periods investment

Table 17 The credibilistic absolute deviation of assets of five periods investment

Table 18 The credibilistic absolute deviation of assets of five periods investment