Coherent and convex loss-based risk measures for portfolio vectors

In this paper, we introduce two new classes of risk measures, named coherent and convex loss-based risk measures for portfolio vectors. These new risk measures can be considered as a multivariate extension of univariate loss-based risk measures introduced by Cont et al. (Stat Risk Model 30:133–167, 2013). Representation results for these new introduced risk measures are provided. The links between convex loss-based risk measures for portfolios and convex risk measures for portfolios introduced by Burgert and Rüschendorf (Insur Math Econ 38:289–297, 2006) or Wei and Hu (Stat Probab Lett 90:114–120, 2014) are stated. Finally, applications to the multi-period coherent and convex loss-based risk measures are addressed.

[1]  Walter Schachermayer,et al.  Law invariant risk measures on L∞ (ℝd) , 2011 .

[2]  Patrick Cheridito,et al.  RISK MEASURES ON ORLICZ HEARTS , 2009 .

[3]  Nizar Touzi,et al.  Vector-valued coherent risk measures , 2002, Finance Stochastics.

[4]  Marc Henry,et al.  COMONOTONIC MEASURES OF MULTIVARIATE RISKS , 2009, 2102.04175.

[5]  Andreas H. Hamel,et al.  Set-valued average value at risk and its computation , 2012 .

[6]  Andreas H. Hamel,et al.  A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory , 2009 .

[7]  Cosimo Munari,et al.  Measuring risk with multiple eligible assets , 2013, 1308.3331.

[8]  Marco Frittelli,et al.  Law Invariant Risk Measures , 2005 .

[9]  C. Labuschagne,et al.  Representations of set-valued risk measures defined on the $$l$$l-tensor product of Banach lattices , 2014 .

[10]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[11]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[12]  N. El Karoui,et al.  CASH SUBADDITIVE RISK MEASURES AND INTEREST RATE AMBIGUITY , 2007, 0710.4106.

[13]  M. Frittelli,et al.  Putting order in risk measures , 2002 .

[14]  L. Rüschendorf.  Mathematical Risk Analysis , 2013 .

[15]  Andreas H. Hamel,et al.  Set-valued risk measures for conical market models , 2010, 1011.5986.

[16]  Romain Deguest,et al.  Loss-Based Risk Measures , 2011 .

[17]  M. Frittelli,et al.  RISK MEASURES AND CAPITAL REQUIREMENTS FOR PROCESSES , 2006 .

[18]  Ilya Molchanov,et al.  MULTIVARIATE RISK MEASURES: A CONSTRUCTIVE APPROACH BASED ON SELECTIONS , 2013, 1301.1496.

[19]  Andreas H. Hamel,et al.  Duality for Set-Valued Measures of Risk , 2010, SIAM J. Financial Math..

[20]  Yijun Hu,et al.  Coherent and convex risk measures for portfolios with applications , 2014 .

[21]  Ludger Rüschendorf,et al.  Consistent risk measures for portfolio vectors , 2006 .