Publication: CAPM-like formulae and good deal absence with ambiguous setting and coherent risk measure
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2011
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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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Ambiguity, Conditional value at risk, Good deal, CAPM