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1

Performance of robust portfolio optimization in crisis periods

Balcilar M., Ozun A.

Control and Cybernetics

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2013

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Vol. 42, no. 4

855--871

EN

We examin empirical performances of two alterna- tive robust optimization models, namely the worst-case conditional value-at-risk (worst-case CVaR) model and the nominal conditional value-at-risk (CVaR) model in crisis periods. Both models are based on historical value-at-risk methodology. These performances are compared by using a portfolio constructed on the basis of daily clos- ing values of different stock indices in developed markets using data from 1990 to 2013. An empirical evidence is produced with Ro- bustRisk software application. Both a Monte-Carlo simulation and an out-of-sample test show that robust optimization with worst-case CVaR model outperforms the nominal CVaR model in the crisis peri- ods. However, the trade-off between model misspecification risk and return maximization depending on the market movements should be optimized in a robust model selection.

2

Bi-Criteria Portfolio Optimization Models with Percentile and Symmetric Risk Measures by Mathematical Programming

Sawik B.

Przegląd Elektrotechniczny

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2012

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R. 88, nr 10b

176-180

EN

The purpose of this paper is to compare three different bi-criteria portfolio optimization models. The first model is constructed with the use of percentile risk measure Value-at-Risk and solved by mixed integer programming. The second one is constructed with the use of percentile risk measure Conditional Value-at-Risk and solved by linear programming. The third one is constructed with the use of a symmetric measure of risk - variance of return - as in the Markowitz portfolio and solved by quadratic programming. Computational experiments are conducted for bi-criteria portfolio stock exchange investments. The results obtained prove, that the bi-objective portfolio optimization models with Value-at-Risk and Conditional Value-at-Risk could be used to shape the distribution of portfolio returns. The decision maker can assess the value of portfolio return and the risk level, and can decide how to invest in a real life situation comparing with ideal (optimal) portfolio solutions. The proposed scenario-based portfolio optimization problems under uncertainty, formulated as a bi-objective linear, mixed integer or quadratic program are solved using commercially available software (AMPL/CPLEX) for mathematical programming.

PL

W pracy zaprezentowano problem portfelowy, dla którego zaproponowane zostały trzy dwukryterialne modele optymalizacji. W kazdym z zaimplementowanych modeli funkcja celu jest wazona suma dwóch kryteriów, które podlegają optymalizacji dla konkretnego portfela. W pierwszym modelu z warunkową wartoscią zagrozoną zwrotu (Conditional Value-at-Risk) kryteria decyzyjne to maksymalizacja CVaR i przewidywanego zwrotu portfela. W drugim modelu z wartoscią zagrozoną zwrotu (Value-at-Risk) minimalizowane jest prawdopodobienstwo ryzyka straty oraz maksymalizacja oczekiwanego zwrotu z portfela. Trzeci model to omawiany szeroko w literaturze model portfela Markowitza zmodyfikowany tak, by funkcja kryterialna była wazona sumą kryteriów i minimalizowała ryzyko portfela zdefiniowanego jako macierz kowariancji historycznych zwrotów oraz maksymalizowała przewidywany zwrot portfela. Modele te zostały zaimplementowane: w pierwszym - uzywając metody programowania mieszanego całkowitoliczbowego, a w drugi i trzeci z uzyciem programowania liniowego i kwadratowego. Efektywnosc zaproponowanych modeli została zweryfikowana eksperymentalnie, zwracając szczególną uwagę na czas obliczen oraz przewidywany zwrot portfela. Zamieszczono wyniki eksperymentów obliczeniowych przeprowadzonych z zastosowaniem optymizatora CPLEX i jezyka modelowania algebraicznego AMPL.

3

Metaheuristic optimization of marginal risk constrained long - short portfolios

Vijayalakshmi Pai G. A., Michel T.

Journal of Artificial Intelligence and Soft Computing Research

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2012

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Vol. 2, No. 3

259--274

EN

The problem of portfolio optimization with its twin objectives of maximizing expected portfolio return and minimizing portfolio risk renders itself difficult for direct solving using traditional methods when constraints reflective of investor preferences, risk management and market conditions are imposed on the underlying mathematical model. Marginal risk that represents the risk contributed by an asset to the total portfolio risk is an important criterion during portfolio selection and risk management. However, the inclusion of the constraint turns the problem model into a notorious non-convex quadratic constrained quadratic programming problem that seeks acceptable solutions using metaheuristic methods. In this work, two metaheuristic methods, viz., Evolution Strategy with Hall of Fame and Differential Evolution (rand/1/bin) with Hall of Fame have been evolved to solve the complex problem and compare the quality of the solutions obtained. The experimental studies have been undertaken on the Bombay Stock Exchange (BSE200) data set for the period March 1999-March 2009. The efficiency of the portfolios obtained by the two metaheuristic methods have been analyzed using Data Envelopment Analysis.

4

Conditional value-at-risk and value-at-risk for portfolio optimization model with weighting approach

Sawik B.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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2011

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T. 15, z. 2

429-434

EN

This paper presents a multi-objective portfolio models with the expected return as a performance measure and the expected worst-case return as a risk measure. The problem objective is to allocate the wealth on different securities to optimize the portfolio expected return. This portfolio approach has allowed the two popular in financial engineering percentile measures of risk, value-at-risk (VaR) and conditional value-at-risk (CVaR) to be applied. Numerical examples based on historical daily input data from the Warsaw Stock Exchange are presented and selected computational results are provided.

PL

W artykule przedstawiono model wielokryterialnej optymalizacji portfelowej z ważoną funkcją celu. Celem optymalizacji jest wyznaczenie portfela o maksymalnej oczekiwanej stopie zwrotu przy ryzyku wyznaczonym za pomocą miar CVaR oraz VaR. Przedstawiono wyniki eksperymentów obliczeniowych z użyciem danych z GPW w Warszawie.

5

A bi-objective portfolio optimization with conditional value-at-risk

Sawik B.

Decision Making in Manufacturing and Services

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2010

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Vol. 4, no. 1-2

47-69

EN

This paper presents a bi-objective portfolio model with the expected return as a performance measure and the expected worst-case return as a risk measure. The problems are formulated as a bi-objective linear program. Numerical examples based on 1000, 3500 and 4020 historical daily input data from the Warsaw Stock Exchange are presented and selected computational results are provided. The computational experiments prove that the proposed linear programming approach provides the decision maker with a simple tool for evaluating the relationship between the expected and the worst-case portfolio return.

6

A three stage lexicographic approach for multi-criteria portfolio optimization by mixed integer programming

Sawik B.

Przegląd Elektrotechniczny

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2008

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R. 84, nr 9

108-112

EN

The portfolio optimization problem is formulated as multi-objective mixed integer program. The problem considered is based on a single period model of investment. The problem objective is to allocate wealth on different assets to maximize the weighted difference of portfolio expected return, the threshold of the probability that the return is not less than required level and the amount of wealth to be invested. The results of some computational experiments modeled after a real data from the Warsaw Stock Exchange are reported.

PL

W artykule przedstawiono wielokryterialny model optymalizacji portfelowej wraz z trójetapowym podejściem lksykalno-graficznym. Celem optymalizacji jest wyznaczenie portfela o maksymalnej oczekiwanej stopie zwrotu, dla której prawdopodobieństwo wartości zagrożonej zwrotu (VaR) przy ryzyku mniejszym od zadanej wartości będzie nie większe od zadanego lub minimalizowanego progu. Przedstawiono wyniki eksperymentów obliczeniowych dla danych zaczerpniętych z GPW w Warszawie.

7

Information pricing for portfolio optimization

Banek T., Kulikowski R.

Control and Cybernetics

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2003

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Vol. 32, no 4

867-882

EN

We consider the following problem: is there a rational or fair price for the reports made by analysts, experts, investor advisers concerning the rate of return (RR) of investments? We define the notion of the value of information included in the family of probability distributions of the RR. Next, we illustrate this notion for a linear-quadratic utility function.

8

Value of Information for Portfolio Optimization

Banek T., Kulikowski R.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2003

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Vol. 51, nr 3

245-258

EN

We consider the following problem: is there a rational or fair price for the reports made by analysts, experts, investor advisers concerning the rate of return (RR) of investments? We define the notion of the value of information included in the family of probability distributions of the RR. Next, we illustrate this notion for a linear-quadratic utility function.

9

Application of malliavin calculus to the portfolio theory

Krajna L.

Prace Naukowe Instytutu Matematyki Politechniki Wrocławskiej. Konferencje

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2001

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Vol. 24, nr 3

67-75

EN

We present general solutions of portfolio optimization problems received for a huge subclass of diffusion models of financial markets with stochastic coefficients, defined by systems of stochastic differential equation. The idea of proposed methodology is based on the application of stochastic variational calculus to the construction of replicating portfolios (see also [6]). Derived closed formula for optimal portfolios beside traditional mean-variance optimization factor (see Markovitz Theory) also include intertemporal hedging factors which were postulated by Merton as early as in the 70s. Received results help to better understand the role of stochastic volatility and other stochastic parameters of the financial market model. Computer experiments allow to describe quantitatively the dynamics of optimal portfolios as well as the importance of all factors.

10

Multiobjective duality for the Markowitz portfolio optimization problem

Wanka G.

Control and Cybernetics

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1999

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Vol. 28, no 4

691-702

EN

The classical Markowitz approach to portfolio selection leads to a biobjective optimization problem where the objectives are the expected return and the variance of a portfolio. In this paper a biobjective dual optimization problem to the Markowitz portfolio optimization problem is introduced and analyzed. For the Markowitz problem and its dual, weak and strong vector duality assertions are derived. The optimality conditions are also verified.

11

Application of modern portfolio theory to the Russian state bond market

Pervozvanskij A., Barinov V., Kozlova O.

Control and Cybernetics

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1999

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Vol. 28, no 4

799-810

EN

The behaviour of the Russian state bond market is analyzed. Attention is mainly paid to short-term fluctuations and efficiency of short-term investments. Analysis of return time series has shown that there exists a significant autocorrelation, and that distribution of random fluctuations is non-Gaussian. It predetermines a choice of forecasting schemes. The most efficient ones appear to be non-linear. The efficiency was checked not only by the traditional statistical indices by direct numerical experiments where various types of predictors were used as basic elements of decision rules. The decision algorithms have included the solution to the modified optimal portfolio problem where the forecasts were used as expected returns and the covariance matrix was estimated via forecasting errors.

12

A recursive procedure for selecting optimal portfolio according to the MAD model

Michałowski W., Ogryczak W.

Control and Cybernetics

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1999

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Vol. 28, no 4

725-738

EN

The mathematical model of portfolio optimization is usually represented as a bicriteria optimization problem where a reasonable trade-off between expected rate of return and risk is sought. Im a classical Markowitz model the risk is measured by a variance, thus resulting in a quadratic programming model. As an alternative, the MAD model was proposed where risk is measured by (mean) absolute deviation instead of a variance. The MAD model is computationally attractive, since it is transformed into an easy to solve linear programming program. In this paper we poesent a recursive procedure which allows to identify optimal portfolio of the MAD model depending on investor's downside risk aversion.

13

Two factors utility approach

Kulikowski R.

Control and Cybernetics

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1998

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Vol. 27, no 3

417-428

EN

This paper deals with optimization of portfolios composed of securities (equities). The drawbacks of existing methodologies, based on a single factor utility function, are indicated. The two-factor utility function introduced takes into account the expected excess return and expected worst case return (both in monetary units). Assuming that utility is "risk averse" and "constant returns to scale", a theorem on existence of optimum strategy of investments is proven. The optimum strategy is derived in an explicit form. A numerical example is also given.

14

Portfolio optimization - two rules approach

Kulikowski R.

Control and Cybernetics

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1998

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Vol. 27, no 3

429-446

EN

The new approach to the portfolio optimization, based on the concept of two-factor utility function, is proposed. The first factor describes the expected average profit, while the second - the worse case profit. Then, two rules enabling one to compose an optimum portfolio are formulated. The first rule determines the level of acceptance for all assets with given risk/return ratio. The second rule enables one to allocate the investment fund among all the accepted assets. The methodology proposed does not require to specify the individual utility function in an explicit form. It can be used to optimize portfolios composed of equities as well as bond and other securities, using a passive or - active management strategy.