Login | Register

Capm and Apt Like Models with Risk Measures


Capm and Apt Like Models with Risk Measures

Balbás, Alejandro, Balbás, Beatriz and Balbás, Raquel (2009) Capm and Apt Like Models with Risk Measures. Technical Report. Concordia University. Department of Mathematics & Statistics, Montreal, Quebec.

Text (application/pdf)


The paper deals with optimal portfolio choice problems when risk levels are given by coherent risk measures, expectation bounded risk measures or general deviations. Both static and dynamic pricing models may be involved.
Unbounded problems are characterized by new notions such as compatibility and strong compatibility between pricing rules and risk measures. Surprisingly, it is
pointed out that the lack of bounded optimal risk and/or return levels arises in practice for very important pricing models (for instance, the Black and Scholes model)
and risk measures (V aR, CV aR, absolute deviation and downside semi-deviation, etc.).

Bounded problems will present a Market Price of Risk and generate a pair of benchmarks. From these benchmarks we will introduce APT and CAPM like analyses, in the sense that the level of correlation between every available security and some economic factors will expalin the security expected return. On the contray, the risk level non correlated with these factors will have no influence on any return, despite we are dealing with very general risk functions that are beyond the standard deviation.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Monograph (Technical Report)
Authors:Balbás, Alejandro and Balbás, Beatriz and Balbás, Raquel
Series Name:Department of Mathematics & Statistics. Technical Report No. 1/09
Corporate Authors:Concordia University. Department of Mathematics & Statistics
Institution:Concordia University
Date:June 2009
Keywords:Risk Measure, Compatibility between Prices and Risks, Efficient Portfolio, APT and CAPM like models
ID Code:6721
Deposited On:23 Jun 2010 14:48
Last Modified:18 Jan 2018 17:29


[1] Alexander, S., T.F. Coleman and Y. Li, 2006. Minimizing CV aR and V aR for a portfolio of derivatives. Journal of Banking & Finance, 30, 538-605.
[2] Artzner, P., F. Delbaen, J.M. Eber and D. Heath, 1999. Coherent measures of risk. Mathematical Finance, 9, 203-228.
[3] Balbás, A. and R. Balbás 2009. Compatibility between pricing rules and risk measures: The CCV aR. Revista de la Real Academia de Ciencias, RACSAM, forthcoming.
[4] Balbás, A., R. Balbás and J. Garrido, 2009a. Extending pricing rules with general risk functions. European Journal of Operational Research, forthcoming, doi:.10.1016/j,ejor.2009.02.015.
[5] Balbás, A., B. Balbás and A. Heras, 2009b. Optimal reinsurance with general risk measures. Insurance: Mathematics and Economics, 44, 374 - 384.
[6] Barbarin, J. and P. Devolder, 2005. Risk measure and fair valuation of an investment guarantee in life insurance. Insurance: Mathematics and Economics, 37, 2, 297-323.
[7] Brown, D. and M. Sim, 2009. Satisfying measures for analysis of risky positions. Management Science, forthcoming, doi: 10.1287mnsc.1080.0929.
[8] Calafiore, G.C., 2007. Ambiguous risk measures and optimal robust portfolios. SIAM Journal of Optimization, 18, 3. 853-877.
[9] Chamberlain, G. and M. Rothschild, 1983. Arbitrage, factor structure and meanvariance analysis of large assets. Econometrica, 51, 1281-1304.
[10] Duffie, D., 1988. Security markets: Stochastic models. Academic Press.
[11] Föllmer, H. and A. Schied, 2002. Convex measures of risk and trading constraints. Finance & Stochastics, 6, 429-447.
[12] Goovaerts, M., R. Kaas, J. Dhaene and Q. Tang, 2004. A new classes of consistent risk measures. Insurance: Mathematics and Economics, 34, 505-516.
[13] Hamada, M. and M. Sherris, 2003. Contingent claim pricing using probability distortion operators: Method from insurance risk pricing and their relationship to financial
theory. Applied Mathematical Finance, 10, 19-47.
[14] Horvàth, J., 1966. Topological vector spaces and distributions, vol I. Addison Wesley, Reading, MA.
[15] Luenberger, D.G.,1969. Optimization by vector spaces methods. John Wiley & Sons, New York.
[16] Maurin, K., 1967. Methods of Hilbert spaces. PWN-Polish Scientific Publishers.
[17] Miller, N. and A. Ruszczynski, 2008. Risk-adjusted probability measures in portfolio optimization with coherent measures of risk. European Journal of Operational Research, 191, 193-206.
[18] Nakano, Y., 2004. Efficient hedging with coherent risk measure. Journal of Mathematical Analysis and Applications, 293, 345-354.
[19] Ogryczak, W. and A. Ruszczynski, 1999. From stochastic dominance to mean risk models: Semideviations and risk measures. European Journal of Operational Research, 116, 33-50.
[20] Ogryczak, W. and A. Ruszczynski, 2002. Dual stochastic dominance and related mean risk models. SIAM Journal of Optimization, 13, 60-78.
[21] Rockafellar, R.T., S. Uryasev and M. Zabarankin, 2006a. Generalized deviations in risk analysis. Finance & Stochastics, 10, 51-74.
[22] Rockafellar, R.T., S. Uryasev and M. Zabarankin, 2006b. Optimality conditions in portfolio analysis with general deviations measures. Mathematical Programming, Ser.
B, 108, 515-540.
[23] Rockafellar, R.T., S. Uryasev and M. Zabarankin, 2006c. Master funds in portfolio analysis with general deviation measures. Journal of Banking and Finance, 30, 743-
[24] Rockafellar, R.T., S. Uryasev and M. Zabarankin, 2007. Equilibrium with investors using a diversity of deviation measures. Journal of Banking and Finance, 31, 3251-
[25] Ruszczynski, A. and A. Shapiro, 2006. Optimization of convex risk functions. Mathematics of Operations Research, 31, 3, 433-452.
[26] Schied, A., 2007. Optimal investments for risk- and ambiguity-averse preferences: A duality approach. Finance & Stochastics, 11, 107-129.
[27] Staum, J., 2004. Fundamental theorems of asset pricing for good deal bounds. Mathematical Finance, 14, 141-161.
[28] Stoyanov, S.V., S.T. Rachev and F.J. Fabozzi, 2007. Optimal financial portfolios. Applied Mathematical Finance, 14, 401-436.
[29] Wang, S.S., 2000. A class of distortion operators for pricing financial and insurance risks. Journal of Risk and Insurance, 67, 15-36.
[30] Zalinescu, C., 2002. Convex analysis in general vector spaces. World Scientific Publishing Co.
All items in Spectrum are protected by copyright, with all rights reserved. The use of items is governed by Spectrum's terms of access.

Repository Staff Only: item control page

Downloads per month over past year

Back to top Back to top