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CAPM-like formulae and good deal absence with ambiguous setting and coherent risk measure

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  • Alejandro Balbás

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  • Beatriz Balbás

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  • Raquel Balbás

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    Abstract

    Risk measures beyond the variance have shown theoretical advantages when addressing some classical problems of Financial Economics, at least if asymmetries and/or heavy tails are involved. Nevertheless, in portfolio selection they have provoked several caveats such as the existence of good deals in most of the arbitrage free pricing models. In other words, models such as Black and Scholes or Heston allow investors to build sequences of strategies whose expected return tends to in nite and whose risk remains bounded or tends to minus in nite. This paper studies whether this drawback still holds if the investor is facing the presence of multiple priors, as well as the properties of optimal portfolios in a good deal free ambiguous framework. With respect to the rst objective, we show that there are four possible results. If the investor uncertainty is too high he/she has no incentives to buy risky assets. As the uncertainty (set of priors) decreases the interest in risky securities increases. If her/his uncertainty becomes too low then two types of good deal may arise. Consequently, there is a very important di¤erence between the ambiguous and the non ambiguous setting. Under ambiguity the investor uncertainty may increase in such a manner that the model becomes good deal free and presents a market price of risk as close as possible to that re ected by the investor empirical evidence. Hence, ambiguity may help to overcome some meaningless ndings in asset pricing. With respect to our second objective, good deal free ambiguous models imply the existence of a benchmark generating a robust capital market line. The robust (worst-case) risk of every strategy may be divided into systemic and speci c, and no robust return is paid by the speci c robust risk. A couple of betas may be associated with every strategy, and extensions of the CAPM most important formulas will be proved.

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    Bibliographic Info

    Paper provided by Universidad Carlos III, Instituto sobre Desarrollo Empresarial "Carmen Vidal Ballester" in its series Business Economics Working Papers with number id-11-04.

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    Date of creation: Apr 2011
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    Handle: RePEc:cte:idrepe:id-11-04

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    Keywords: Ambiguity; Conditional value at risk; Good deal; CAPM;

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    1. Peter Bossaerts & Paolo Ghirardato & Serena Guarnaschelli & William R. Zame, 2010. "Ambiguity in Asset Markets: Theory and Experiment," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 23(4), pages 1325-1359, April.
    2. Dean Foster & Sergiu Hart, 2007. "An Operational Measure of Riskiness," Levine's Bibliography 843644000000000095, UCLA Department of Economics.
    3. Larry Epstein & Martin Schneider, 2005. "Ambiguity, Information Quality and Asset Pricing," RCER Working Papers, University of Rochester - Center for Economic Research (RCER) 519, University of Rochester - Center for Economic Research (RCER).
    4. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 9(3), pages 203-228.
    5. Lorenzo Garlappi & Raman Uppal & Tan Wang, 2007. "Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 20(1), pages 41-81, January.
    6. Turan G. Bali & Nusret Cakici & Fousseni Chabi-Yo, 2011. "A Generalized Measure of Riskiness," Management Science, INFORMS, INFORMS, vol. 57(8), pages 1406-1423, August.
    7. David B. Brown & Melvyn Sim, 2009. "Satisficing Measures for Analysis of Risky Positions," Management Science, INFORMS, INFORMS, vol. 55(1), pages 71-84, January.
    8. Vikas Agarwal, 2004. "Risks and Portfolio Decisions Involving Hedge Funds," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 17(1), pages 63-98.
    9. H. Henry Cao & Tan Wang & Harold H. Zhang, 2005. "Model Uncertainty, Limited Market Participation, and Asset Prices," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 18(4), pages 1219-1251.
    10. Goovaerts, Marc J. & Kaas, Rob & Dhaene, Jan & Tang, Qihe, 2004. "Some new classes of consistent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 34(3), pages 505-516, June.
    11. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, Elsevier, vol. 18(2), pages 141-153, April.
    12. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2004. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Carlo Alberto Notebooks, Collegio Carlo Alberto 12, Collegio Carlo Alberto, revised 2006.
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