Abstract
Optimality conditions are derived for problems of minimizing a general measure of deviation of a random variable, with special attention to situations where the random variable could be the rate of return from a portfolio of financial instruments. General measures of deviation go beyond standard deviation in satisfying axioms that do not demand symmetry between ups and downs. The optimality conditions are applied to characterize the generalized ``master funds'' which elsewhere have been developed in extending classical portfolio theory beyond the case of standard deviation. The consequences are worked out for deviation based on conditional value-at-risk and its variants, in particular.
Similar content being viewed by others
References
Acerbi, C.: Spectral Measures of Risk: a Coherent Representation of Subjective Risk Aversion. Journal of Banking and Finance 26, 1505–1518 (2002)
Acerbi, C., Simonetti, P.: Portfolio Optimization with Spectral Measures of Risk. preprint (2002)
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent Measures of Risk. Mathematical Finance 9, 203–227 (1999)
Bawa, V.S., Lindenberg, E.B.: Capital Market Equilibrium in a Mean-Lower Partial Moment Framework. Journal of Financial Economics 5, 189–200 (1977)
Bertsimas, D., Lauprete, G.J., Samarov, A.: Shortfall as a Risk Measure: Properties Optimization and Applications. Journal of Economic Dynamics and Control 28, 1227–1480 (2004)
Feinstein, C.D., Thapa, M.N.: A Reformulation of a Mean-Absolute Deviation Portfolio Optimization Model. Management Science 39, 1552–1553 (1993)
Föllmer, H., Schied, A.: Stochastic Finance. Walter deGruyter Inc., 2002, p. 400
Grauer, R.R.: Introduction to Asset Pricing Theory and Tests. In: R. Roll, (ed.) The International Library of Critical Writings in Financial Economics (Edward Elgar Publishing Inc., 2001)
Konno, H., Shirakawa, H.: Equilibrium Relations in the Mean-Absolute Deviation Capital Market. Asia-Pacific Financial Markets 1, 21–35 (1994)
Luenberger, D.G.: Investment Science. Oxford University Press, Oxford, New York, 1998, p. 494
Malevergne, Y., Sornette, D.: Multi-Moments Method for Portfolio Management: Generalized Capital Asset Pricing Model in Homogeneous and Heterogeneous markets, preprint 2002
Markowitz, H.M.: Portfolio Selection: Efficient Diversification of Investments. Yale University Press, 1971, p. 368
Nielsen, L.T.: Existence of Equilibrium in CAPM. The Journal of Economic Theory 52, 223–231 (1990)
Pflug, G.C.: Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk. In: S.P. Uryasev, (ed.) Probabilistic Constrained Optimization: Methodology and Applications. Kluwer, 2000, pp. 278–287
Rockafellar, R.T.: Convex Analysis. Princeton University Press, 1970
Rockafellar, R.T.: Conjugate Duality and Optimization monograph No. 16 in Conference Board of Math. Sciences Series. SIAM Publications, (1974)
Rockafellar, R.J., Uryasev, S.P.: Optimization of Conditional Value-at-Risk. Journal of Risk 2, 21–42 (2000)
Rockafellar, R.T., Uryasev, S.P.: Conditional Value-at-Risk for General Loss Distributions. Journal of Banking and Finance 26, 1443–1471 (2002)
Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Deviation Measures in Risk Analysis and Optimization, Research Report 2002-7, Dept. of Industrial and Systems Engineering. University of Florida (2002)
Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Generalized Deviations in Risk Analysis. Finance and Stochastics 10, 51–74 (2006)
Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Master Funds in Portfolio Analysis with General Deviation Measures. Journal of Banking and Finance 30 (2), 743–778 (2006)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer-Verlag, Berlin, 1998
Ruszczynski, A., Shapiro, A.: Optimization of Convex Risk Functions, preprint (2004)
Samuelson, P.A.: The Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances and Higher Moments. The Review of Economic Studies 37, 537–542 (1970)
Sharpe, W.F.: Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk. The Journal of Finance 19, 425–442 (1964)
Sharpe, W.F.: Capital Asset Prices With and Without Negative Holdings. The Journal of Finance 46, 489–509 (1991)
Tasche, D.: Modern Portfolio Theory with Homogeneous Risk Measures, Zentrum Mathematik, Technische Universitat, Munich, preprint (2001)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rockafellar, R., Uryasev, S. & Zabarankin, M. Optimality conditions in portfolio analysis with general deviation measures. Math. Program. 108, 515–540 (2006). https://doi.org/10.1007/s10107-006-0721-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-006-0721-9
Keywords
- General deviation measures
- Portfolio analysis
- Generalized master funds
- CAPM-like relations
- Optimality conditions
- Risk envelopes
- Risk identifiers
- Conditional value-at-risk
- Risk management
- Stochastic optimization