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Impact of error in parameter estimations on large scale portfolio optimization
Mathematical programming is often concerned with determining the optimum of some real-world objective. One of the main questions in this field is the way of addressing uncertainty. Whereas deterministic optimization models are formulated with known parameters, real life issues almost invariably include uncertain parameters which are unknown at the time when a decision should be made.
An example of this is the area of financial optimization, since there are various uncertainties, such as prices of goods, economic factors, asset returns, turnover constraints, etc. One of the most known problems in the field is portfolio optimization (PO). The objective is to distribute capital between available investment instruments in the financial market. Examples of these instruments are stocks, bonds, options, and bank deposits. The aim is to maximize or keep at the desired level the wealth resulting from the investment, but at the same time minimize the involved risks. The major progress started after publication of Harry Markowitz’s seminal work [14]. Since the late 1950s, PO has been an active field in finance. An extensive overview of modern portfolio theory, concepts, and mathematical models for financial markets can be found in the work of Lyuu [13]. The author presents a variety of algorithms
for computational techniques in pricing, risk management, and portfolio selection, analyzes their efficiency, and offers thorough theoretical grounding for the proposed models.
One of the ways to deal with uncertainty is to estimate the unknown parameters. Since some optimization models require statistical estimators of parameters, an adequate choice of estimator can substantially improve optimization performance. On the other hand, many estimators contain estimation error or bias that most likely
to perturb an optimizer. Consequently, it is not enough to wisely choose a model for the considered problem. Another crucial aspect of success is to determine a proper way to estimate model parameters. Use of improper estimation technique and insufficiency of information may lead to that selected portfolios will have poor out-of-sample performance.
The effect of misspecification and estimation errors on optimal portfolio selection has been a point of interest in the scientific community for many years. For a comprehensive overview of the topic and the modern trends in mean–variance portfolio optimization the reader may refer to the paper of Kolm et al. [9]. The main purpose of this study is to investigate how estimators of uncertain parameters can affect the portfolio optimization results in the case of large scale portfolio.