Evolutionary Strategies for Building Risk-Optimal Portfolios
This chapter describes an evolutionary approach to portfolio optimization. It rejects some assumptions from classic models, introduces alternative risk measures and proposes three evolutionary algorithms to solve the optimization problem. In order to vali
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4.1 Introduction Evolutionary Algorithms have been successfully incorporated into many fields of science and technology, notably including the domains of economics and finance (5), (7), (10), (14). This chapter presents another application of evolutionary algorithms in this domain, namely an evolutionary approach to the problem of portfolio optimization. Although some analytical methods are well-known for classic versions of the problem (1), (4), an extension of the problem by introducing more complex risk measures and loosening several artificial assumptions requires a new efficient approach, which cannot be developed on the basis of classic methods due to the irregularity of the objective function and the search space. However, the opportunities provided by evolutionary algorithms (2), (12) may lead to an efficient optimization of portfolio structures. Moreover, apart from theoretical constraints, the approach presented in this chapter focuses also on a few practical constraints such as budget constraints, which means that the user of the system has only finite amount of money, as well as investor capabilities and preferences, which means that the user has to obey various market regulations and pay transaction fees. Moreover, an important constraint is constituted by time restrictions and hardware limits.
4.2 Classic Approach to Portfolio Optimization In this chapter, we focus on the main goal of investors, which is to optimally allocate their capital among various financial assets. Searching for an optimal portfolio of P. Lipinski: Evolutionary Strategies for Building Risk-Optimal Portfolios, Studies in Computational Intelligence (SCI) 100, 53–65 (2008) c Springer-Verlag Berlin Heidelberg 2008 www.springerlink.com
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stocks, characterized by random future returns, seems to be a difficult task and is usually formalized as a risk-minimization problem under a constraint of expected portfolio return (4). The risk of portfolio is often measured as the variance of returns, but many other risk criteria have been proposed in the financial literature (1) (3). Portfolio theory may be traced back to the seminal Markowitz paper (11) and is presented in an elegant way in (4). Consider a financial market on which n risky assets are traded. Let (4.1) R = (R1 , R2 , . . . , Rn ) be the square-integrable random vector of random variables representing their return rates. Denote as r = (r1 , r2 , . . . , rn ) ∈ Rn the vector of their expected return rates r = (E[R1 ], E[R2 ], . . . , E[Rn ]) = E[R]
(4.2)
and as V the corresponding covariance matrix which is assumed positive definite. A portfolio is a vector x = (x1 , x2 , . . . , xn ) ∈ Rn verifying x1 + x2 + . . . + xn = 1.
(4.3)
Hence xi is the proportion of capital invested in the i-th asset. Denote as X the set of all portfolios. For each portfolio x ∈ X, we define Rx = x1 R1 + x2 R2 + . . . + xn Rn = x R
(4.4)
as the random variable representing the portfolio return rate and then E[Rx ] = x1 r1 + x2 r2 + . . . + xn rn = x r
(4.5)
is the portfolio expected r
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