Multi-stage Tracking Error Portfolio Model And Empirical Research

Markowitz(1952) published Portfolio selection on The Journal of Finance and he explained how to reduce risk by diversification and choose the optimal portfolio by efficient frontier for the first time from the perspective of mathematical economics. The paper laid the core status of the finance portfolio theory. Tracking-error portfolio model is more scientific than the mean- variance model in evaluating the performance of investment manager. Classical portfolio optimization model only considered single stage portfolio optimization problem, however institutional investors in the real financial markets tend to choose a long-term investment. Therefore, an investment manager has to adjust continuously portfolio positions in time according to the market environment so that it is necessary to research dynamic portfolio optimization problem. This paper expand the tracking-error model from single-stage to multi-stage combined with the following three types of risk control indicators.1. Mean absolute deviation: it is the average of the absolute value of all the individual observations and the arithmetic mean deviation. The average absolute deviation won’t appear that the positive and negative offset because the deviation is absolute value. Therefore, mean absolute deviation can reflect the actual situation of the prediction error better than the mean deviation.2. Negative semi-variance: mean variance model abandoned both high yield and low yield in order to control risks which does not conform to investors’ real psychological in reality. Because the variance risk measure to eliminate high and low income, investors lose the opportunity to obtain high returns. However, investors tend to only focus on the risks when portfolio income is lower than the average income in the process of actual investment, so the negative semi-variance variance model conforms to the investors better than mean variance model.3. Worst-Case Conditional VaR: the traditional VaR(Value at Risk) can’t study the downside risk which exceed the quantile, satisfy the additivity and convexity and conform to the consistency of risk measurement. CVaR(Conditional Value-at-Risk) makes up for the above defects,but it is only appropriate for the situation that the distribution of random variable is known. In order to remedy the defect, this paper choose WCVaR(Worst-Case Conditional VaR) to measure the total risk.Besides, there are several situations in real financial market as follows. At first, transaction costs for the investing activities is essential and investors have to pay the corresponding cost in every transaction to Exchange and the broker such as fees and taxes. Then, investors will not always invest all property in risky assets and their property may be partially used for risk-free asset。At last,Investors often restrict the amount of risk assets and the proportion of corresponding asset in their portfolios. In consequence, this paper establishes a multi-stage portfolio tracking error model considering short-sale constraints, transaction costs, risk-free assets and cardinality constraints at the same time.This paper makes numerical analysis based on SSE 50 constituent stock’s data in China stock market and use Matlab software to operate the SQP( Sequential quadratic programming) algorithm and genetic algorithm for empirical analysis. The results show that the portfolio model established in this paper can obtain more incomes and better performance than the benchmark portfolio both inside and outside the sample.