Multi-period Portfolio Optimization With Liability And Quadratic Transaction Costs

In the financial market,how to get the maximum investment income with the smallest risk in a long-term investment process has always been a problem that investors are trying to solve.In the early 21st century,multi-period portfolio optimization research has a breakthrough development,but there are many assumptions such as no transaction costs and no liabilities.Thus there are many limitations in strategies of past studies when they are applied to the actual investment process.Transaction costs is the most common friction factor,and ignoring this factor will have a significant impact on investment outcomes.At the same time,the issue of debt is common in institutional investors,which is a real problem.Therefore,it is necessary to consider transaction costs and liabilities in investment decision-making.However,it is difficult to obtain the optimal analytical solution and the corresponding frontier expression considering both the liability and transaction costs.Based on the multi-period mean-variance framework,this paper considers the liability and the quadratic transaction cost which is more in line with the actual situation,and obtains the analytical solution of the optimal investment decision.Firstly,we construct a multi-period mean variance portfolio models with a riskless asset or without the riskless asset,considering quadratic transaction costs and liabilities.Then,in each model,the analytic expressions of the pre-commitment strategy and the time consistency strategy are solved by the embedding method and the backward strategy respectively.And the expectation and variance of the final wealth and the total expected transaction cost under each strategy are also obtained.Finally,the numerical examples are used to compare the frontiers under different strategies.It is found that the analytical solutions of this paper is universal and general,which not only summarize the investment strategy under the condition of no debt,but also summarize the investment strategy without considering transaction costs.At the same time,the simulation results show that the mean variance frontier of the pre-commitment strategy is better than the frontier of the time consistency strategy in the case with(or without)the riskless asset;under the same strategy,the frontier of containing a riskless asset is better than that of no riskless assets.