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Portfolio Optimization Problem for Stock Portfolio Construction

Student : Lee, Dah-Sheng

Professor: Lee, Hahn-Ming

Date: 9 July 2004

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Outline

Portfolio Definition Property of Portfolio Related work Discussion References

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Portfolio Definition 1/2

The investor considers k different stocks S

1,... , Sk and wishes to invest some xi dollars in each stock Si for a certain period of time, where and xi > 0 for all i. The vector is called a portfolio.

Effective portfolio optimization involves simultaneously maximizing the portfolio return and minimizing the portfolio risk

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k

i ix k

kii xxxxx ,...,, 211

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Portfolio Definition 2/2

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Property of Portfolio

Number of Securities

Market Risk

Unique Risk

1

)2,1(2

21

2122

22

21

21

2211

ww

Covwwww

ww

p

p

Where p is the return of portfolio 1 and 2 are the returns of security 1 and 2 w1 and w2 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

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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.

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Related work 1/4

Year Author Abstract

2000 Ming-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.

R

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Related work 2/4

Year Author Abstract

1994 Lowe [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.

1995 Wendt [5]Used a GA to build a portfolio efficient frontier. The underlying data consisted of 250 scenarios of annual returns for eight asset classes.

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Related work 3/4

Year Author Abstract

1997 Jackson [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.

2004 Ravi 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 apositive 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.

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Related work 4/4

Year Author Abstract

2004 Se-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.

2004 Shu-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 begenerated 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.

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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”

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Discussion 2/2

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References 1/3 [1] “Risk Control Over Bankruptcy in Dynamic Portfolio Selection: A Generalized Mea

n-Variance Formulation” Shu-shang Zhu, Dual Li, and Shou-Yang Wang IEEE TRANSACTIONS ON AUTOMATIC CONTROL Vol. 49, No. 3 MARCH 2004 page(s): 447-457

[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): 131-139

[3]”The value of active portfolio management” Ravi Shukla Journal of Economics and Business vol. 56, 2004 page(s): 331-346

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References 2/3

[4] “Genetic Algorithms for Use in Financial Problems” Jackson, A. AFIR Vol 2, 1997 page(s): 481-503

[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 Jul

y 1994 page(s): 3623-3628

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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): 228-234.