A New Class of Time-Consistent Dynamic Risk Measures and its Application

Rui Gao, Zhiping Chen

School of Mathematics and statistics, Xi’an Jiaotong University, Xi’an, China.

DOI: 10.4236/ti.2013.41B008                

Abstract

We construct a new time consistent dynamic convex cash-subadditive risk measure in this paper. Different from exist-ing measures, both potential loss and volatility of risky objects are considered. Based on a one-period measure that dis-torts financial values, punishes downside risk yet rewards upside potential, a dynamic time consistent version is con-structed recursively through a modified translation property. We then establish a portfolio selection model and give its optimal condition.

Keywords

Dynamic Risk Measure; Time-Consistent; Cash-Subadditive; Portfolio Optimization; Stochastic Programming

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Konno, Hiroshi, and Hiroaki Yamazaki. "Mean-absolute deviation portfolio optimization model and its applica-tions to Tokyo stock market." Management science 37, no. 5 (1991): 519-531. doi:10.1287/mnsc.37.5.519
[2]	Bawa, Vijay S., and Eric B. Lindenberg. "Capital market equilibrium in a mean-lower partial moment framework." Journal of Financial Economics 5, no. 2 (1977): 189-200. doi:10.1016/0304-405X(77)90017-4
[3]	Artzner, Philippe, Freddy Delbaen, Jean‐Marc Eber, and David Heath. "Coherent measures of risk." Mathematical finance 9, no. 3 (1999): 203-228. doi:10.1111/1467-9965.00068
[4]	F?llmer, Hans, and Alexander Schied. "Convex measures of risk and trading constraints." Finance and Stochastics 6, no. 4 (2002): 429-447. doi:10.1007/s007800200072
[5]	El Karoui, Ni-cole, and Claudia Ravanelli. "Cash subadditive risk measures and interest rate ambiguity." Mathematical Finance 19, no. 4 (2009): 561-590. doi:10.1111/j.1467-9965.2009.00380.x
[6]	Rachev, Svet-lozar, Sergio Ortobelli, Stoyan Stoyanov, J. FABOZZI FRANK, and Almira Biglova. "Desirable properties of an ideal risk measure in portfolio theory." International Journal of Theoretical and Applied Finance 11, no. 01 (2008): 19-54. doi:10.1142/S0219024908004713
[7]	Cheridito, Patrick, Freddy Delbaen, and Michael Kupper. "Dynamic mone-tary risk measures for bounded discrete-time processes." Electronic Journal of Probability 11, no. 3 (2006): 57-106. doi:10.1214/EJP.v11-302
[8]	Chen Zhiping, Li Yang, Daobao Xu, and Qianhui Hu. "Tail nonlinearly transformed risk measure and its application." OR Spec-trum (2011): 1-44. doi:10.1007/s00291-011-0271-2
[9]	Shapiro A., Dentche-va D., Ruszczynski A., Lectures on stochastic program-ming: modeling and theory. Society for Industrial Ma-thematics, Vol.9, 2009. doi:10.1137/1.9780898718751
[10]	Denneberg, Dieter. "Premium Calculation." Astin Bulletin 20, no. 2 (1990): 181-190.doi:10.2143/AST.20.2.2005441
[11]	Rockafellar, R. Tyrrell, and Stanislav Uryasev. "Optimization of con-ditional value-at-risk." Journal of risk 2 (2000): 21-42.
[12]	Ruszczyński, Andrzej. "Risk-averse dy-namic programming for Markov decision processes." Mathematical programming 125, no. 2 (2010): 235-261. doi:10.1007/s10107-010-0393-3
[13]	Alexander Shapiro, Wajdi Tekaya, Joari Paulo da Costa, Murilo Pereira Soares, "Risk neutral and risk averse Stochastic Dual Dynamic Programming method", European Journal of Operations Research, vol. 224, pp. 375-391, 2013. doi:10.1016/j.ejor.2012.08.022