1 Portfolio Optimization Problem for Stock Portfolio Construction Student : Lee, Dah-Sheng Professor: Lee, Hahn-Ming Date: 9 July 2004

2 Outline  Portfolio Definition  Property of Portfolio  Related work  Discussion  References

3 Portfolio Definition 1/2  The investor considers k different stocks S 1,..., S k and wishes to invest some x i dollars in each stock S i for a certain period of time, where and x i > 0 for all i. The vector is called a portfolio.  Effective portfolio optimization involves simultaneously maximizing the portfolio return and minimizing the portfolio risk

4 Portfolio Definition 2/2

5 Property of Portfolio Number of Securities Market Risk Unique Risk Where  p is the return of portfolio  1 and  2 are the returns of security 1 and 2 w 1 and w 2 are the weight of security 1 and 2 in the portfolio  1 and  2 are the Standard Deviation of security 1 and 2  NP-Complete Problem

6 Property of Portfolio  Time Series  Trade-off Problem of risk and return –For a risk-averse investor, minimizing loss is more important than maximizing win, while an aggressive investor has the opposite priority.

7 Related work 1/4 YearAuthorAbstract 2000Ming- Yang Kao et. al. [7] They describe an approximation algorithm, that solves the problem of determining the worst case probability for a given portfolio within a given error in polynomial time. Additionally, they describe an important, non-trivial special case, where the problem can be solved exactly in polynomial time.

8 Related work 2/4 YearAuthorAbstract 1994Lowe [6]Demonstrated the use of NNs in effective portfolio optimization. His goal was to find an approximating portfolio that minimized the "risk," defined in terms of the mean squared error between the market portfolio and the approximating portfolio. 1995Wendt [5]Used a GA to build a portfolio efficient frontier. The underlying data consisted of 250 scenarios of annual returns for eight asset classes.

9 Related work 3/4 YearAuthorAbstract 1997Jackso n [4]He compared the performance of GAs with Newton's method of portfolio optimization and found that the portfolio compositions were similar for both the Newton method and the GA, but that the GA took considerably longer to optimize the portfolio. 2004Ravi Shukla [3]They calculate the value of interim portfolio revision. The results show that excess returns from interim portfolio revision do not cover the incremental trading costs. Across mutual funds, they find evidence of a positive relationship between the excess returns and mutual fund expense ratios suggesting that those managers who generate higher excess returns charge higher fees from the stockholders.

10 Related work 4/4 YearAuthorAbstract 2004Se-Hak Chun and Steven H. Kim [2]A series of case studies indicated that superior returns can be obtained by coupling learning systems(ex. NNs) with active trading strategies. An outcome was the hefty margin by which a multi-market portfolio can outperform a collection of isolated markets. 2004Shu- shang et, al [1]An integration of bankruptcy control and dynamic portfolio selection has been considered in this note. They have proposed a generalized mean-variance formulation from which an optimal investment policy can be generated to help investors not only achieve an optimal return in the sense of a mean- variance tradeoff, but also have a good risk control over bankruptcy.

11 Discussion 1/2  Approximate result of NP-Complete Problem can be obtained faster by “pre-process unit”  Dynamic portfolio selection and management policy is proposed recently for time series property of portfolio. We can improve them in “portfolio construction unit”

12 Discussion 2/2

13 References 1/3 [1] “Risk Control Over Bankruptcy in Dynamic Portfolio Selection: A Generalized Mean-Variance Formulation” Shu-shang Zhu, Dual Li, and Shou-Yang Wang IEEE TRANSACTIONS ON AUTOMATIC CONTROL Vol. 49, No. 3 MARCH 2004 page(s): [2] “Data mining for financial prediction and trading: application to single and multiple markets” Se-Hak Chun and Steven H. Kim Expert System with Application, vol. 26, 2004 page(s): [3]”The value of active portfolio management” Ravi Shukla Journal of Economics and Business vol. 56, 2004 page(s):

14 References 2/3 [4] “Genetic Algorithms for Use in Financial Problems” Jackson, A. AFIR Vol 2, 1997 page(s): [5] “Build Your own GA Efficient Frontier” Wendt, R. Q. Risks and Rewards, December 1995 page(s): 1-5 [6]”Novel Exploitation of Neural Network Methods in Financial Markets” Lowe, D. IEEE International Conference on Neural Networks 6, 27 June-2 July 1994 page(s):

15 References 3/3 [7]"The Risk Profile Problem for Stock Portfolio Optimization" M.-Y. Kao, A. Nolte, and S. R. Tate. Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), 2000, page(s):