Anna Śleszyńska-Połomska

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The article concerns portfolio analysis where risk is measured by value at risk (VaR) assuming that the returns are normally distributed and short-selling of risky securities has no limitations. Two models have been analysed: one with the assumption that no risk-free security exists in the economy, another assuming just one risk-free security. An analysis of Black Model and Modified Tobin Model has been carried out using VaR as the measure of risk. Models with VaR as a measure of risk and standard deviation as measure of risk have been compared. It is shown that, depending on the confidence level used to calculate VaR, an efficient portfolio might not exist. The results for the model with one risk-free asset and VaR as a measure of risk have not been published before and are the original contribution of this paper.


Value at Risk (VaR), portfolio analysis, E-VaR model with no risk-free assets, E-VaR model with risk-free asset


[1] Alexander G.J., Baptista A.M., (2002), Economic Implications of Using a Mean-Var Model For Portfolio Selection: A Comparison With Mean-Variance Analysis, Journal of Economic Dynamics & Control, 26 (7-8), 1159-1193.

[2] Alexander G.J., Francis J.C., (1986), Portfolio Analysis, PRENTICE-HALL, 1159-1193.

[3] Baumol W.J., (1963), An Expected Gain-Confidence Limit Criterion For Portfolio Selection, Management Science, 10 (1), 174-182.

[4] Black F., (1972), Capital Market Equilibrium With Restricted Borrowing, The Journal of Business 45 (3), 444-455.

[5] Charpentier A., Oulidi A., (2009), Estimating Allocations For Value-At-Risk Portfolio Optimization, Mathematical Methods of Operations Research.

[6] Consigli G., (2002), Tail Estimation And Mean-VaR Portfolio Selection In Markets Subject To Financial Instability, Journal of Banking & Finance, 26, 1355-1382.

[7] Egert B., Koubaa Y., (2004), Modeling Stock Returns In The G-7 And In Selected CEE Economies: A Non-linear GARCH Approach, William Davidson Institute Working Paper Number 663.

[8] Fiszeder P., Bruzda J., (2001), Badanie zaleznosci pomiedzy indeksami giełdowymi na GPW w Warszawie – analiza wielorównaniowych modeli GARCH, Prace Naukowe Akademii Ekonomicznej we Wrocławiu, 890, 110-122.

[9] Fiszeder P., (2003), Zastosowanie modeli GARCH w analizie procesów obserwowanych na GPW w Warszawie oraz wybranych rynkach akcji na swiecie. Metody ilosciowe w naukach ekonomicznych, Trzecie Warsztaty Doktorskie z Zakresu Ekonometrii i Statystyki, 11-33.

[10] Gaivoronski A.A., Pflug G., (2004-2005), Value-at-Risk in Portfolio Optimization: Properties and Computational Approach, Journal of Risk, 7, 1-31.

[11] De Giorgi E., (2002), A Note on Portfolio Selection under Various Risk Measures, Institute for Empirical Research in Economics – IEW Working Papers, 122.

[12] Markowitz H., (1952), Portfolio Selection, The Journal of Finance, 7, 77-91.

[13] Markowitz H., (1959), Portfolio Selection: Efficient Diversification of Investments, John Wiley, New York.

[14] Merton R.C., (1972), An Analytic Derivation Of The Efficient Portfolio Frontier, Journal of Financial and Quantitative Analysis, 7 (4), 1851-1872.

[15] Rockafellar R.T., Uryasev S., (1999), Optimization of Conditional Value-at-Risk, Working Paper. University of Washington and University of Florida.

[16] Shields K.K., (1997), Stock Return Volatility On Emerging Eastern European Markets, University of Leicester. The Manchester School Supplement, 118-138.

[17] Steinbach M.C., (2001), Markowitz Revisited: Mean-Variance Models in Financial Portfolio Analysis, Society for Industrial and Applied Mathematics, 43 (1), 31-85.

[18] Sleszynska A., (2006), Asymetryczne modele GARCH dla stóp zwrotu indeksu WIG w latach 1995- 2005, Praca licencjacka pod kierunkiem dr Wojciecha Grabowskiego. WNE UW.

[19] Sleszynska A., (2007), Wykorzystanie wielorównaniowych modeli GARCH-BEKK do analizy efektów przenikania wystepujacych pomiedzy polskimi indeksami giełdowymi, Praca magisterska pod kierunkiem dr Wojciecha Grabowskiego. WNE UW.

[20] Sleszynska-Połomska A., (2008), O analizie portfelowej wartosc srednia – wartosc zagrozona, Praca magisterska pod kierunkiem dr hab. Karola Krzyzewskiego. WMIM UW.

[21] Tobin J.E., (1958), Liquidity Preference as Behaviour Towards Risk, Review of Economic Studies, 25, 65-86.

[22] Tobin J.E., (1965), The Theory of Portfolio Selection w Hahn F.H. and Brechling F. P. R., The Theory of Interest Rates, Macmillan, Londyn, 3-51.

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