In this paper we survey some recent developments on risk measures for port¬folio vectors and on the allocation of risk problem. The main purpose to study risk measures for portfolio vectors X = (X1, …, Xd) is to measure not only the risk of the marginals separately but to measure the joint risk of Xcaused by the variation of the components and their possible dependence.
Thus an important property of risk measures for portfolio vectors is con¬sistency with respect to various classes of convex and dependence orderings. It turns out that axiomatically defined convex risk measures are consistent w.r.t. multivariate convex ordering. Two types of examples of risk measures for portfolio measures are introduced and their consistency properties are in¬vestigated w.r.t. various types of convex resp. dependence orderings.
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References
P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. Finance and Stochastics, 9:203–228, 1998
P. Barrieu and N. El Karoui. Inf-convolution of risk measures and optimal risk transfer. Finance and Stochastics, 9:269–298, 2005
K. Borch. Reciprocal reinsurance treaties. ASTIN Bulletin, 1:170–191, 1960a
K. Borch. The safety loading of reinsurance premiums. Skand. Aktuari-etidskr., 1:163–184, 1960b
K. Borch. Equilibrium in a reinsurance market. Econometrica, 30: 424–444, 1962
H. Bühlmann and W. S. Jewell. Optimal risk exchanges. ASTIN Bulletin, 10:243–263, 1979
C. Burgert and L. Rüschendorf. Allocations of risks and equilibrium in markets with finitely many traders. Preprint, University Freiburg, 2005
C. Burgert and L. Rüschendorf. Consistent risk measures for portfolio vectors. Insurance: Mathematics and Economics, 38:289–297, 2006
C. Burgert and L. Rüschendorf. On the optimal risk allocation problem. Statistics & Decisions, 24(1), 2006, 153–172
E. Chevallier and H. H. Müller. Risk allocation in capital markets: Port¬folio insurance tactical asset allocation and collar strategies. ASTIN Bulletin, 24:5–18, 1994
C. Christofides and E. Vaggelatou. A connection between supermod-ular ordering and positive, negative association. Journal Multivariate Analysis, 88:138–151, 2004
F. Delbaen. Coherent risk measures. Cattedra Galileiana. Scuola Normale Superiore, Classe di Scienze, Pisa, 2000
F. Delbaen. Coherent risk measures on general probability spaces. In Klaus Sandmann et al., editors, Advances in Finance and Stochastics. Essays in Honour of Dieter Sondermann, pages 1–37. Springer, 2002
D. Filipovic and M. Kupper. Optimal capital and risk transfers for group diversification. Preprint, 2006
H. Föllmer and A. Schied. Stochastic Finance. de Gruyter, 2nd edition, 2004
D. Heath and H. Ku. Pareto equilibria with coherent measures of risk. Mathematical Finance, 14:163–172, 2004
E. Jouini, M. Meddeb, and N. Touzi. Vector-valued coherent risk measures. Finance and Stochastics, 4:531–552, 2004
S. Kusuoka. On law-invariant coherent risk measures. Advances in Math¬ematical Economics, 3:83–95, 2001
I. Meilijson and A. Nadas. Convex majorization with an application to the length of critical paths. Journal of Applied Probability, 16:671–677, 1979
D. Müller and D. Stoyan. Comparison Methods for Stochastic Models and Risks. Wiley, 2002
Risk Measures and Their Applications. Special volume, L. Rüschendorf (ed.). Statistics & Decisions, vol. 24(1), 2006
L. Rüschendorf. Inequalities for the expectation of Δ-monotone func¬tions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 54:341–349, 1980
L. Rüschendorf. Solution of statistical optimization problem by rear¬rangement methods. Metrika, 30:55–61, 1983
L. Rüschendorf. Comparison of multivariate risks and positive depen¬dence. J. Appl. Probab., 41:391–406, 2004
L. Rüschendorf. Stochastic ordering of risks, influence of dependence and a.s. constructions. In N. Balakrishnan, I. G. Bairamov, and O. L. Gebi-zlioglu, editors, Advances in Models, Characterizations and Applications, volume 180 of Statistics: Textbooks and Monographs, pages 19–56. CRC Press, 2005
L. Rüschendorf. Law invariant risk measures for portfolio vectors. Statistics & Decisions, 24(1), 2006, 97–108
L. Rüschendorf and S. T. Rachev. A characterization of random variables with minimum L2-distance. Journal of Multivariate Analysis, 1:48–54, 1990
A. Schied. On the Neyman—Person problem for law invariant risk mea¬sures and robust utility functionals. Ann. Appl. Prob., 3:1398–1423, 2004
A. H. Tchen. Inequalities for distributions with given marginals. Ann. Prob., 8:814–827, 1980
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Rüschendorf, L. (2009). Risk Measures for Portfolio Vectors and Allocation of Risks. In: Bol, G., Rachev, S.T., Würth, R. (eds) Risk Assessment. Contributions to Economics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2050-8_7
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