Heterogeneity of Investors and Asset Pricing in a Risk-Value World

to take more risk. Now suppose, she would choose a sharing rule which is
linear relative to that of the other investor but has higher slope (such as in
EU-portfolio analysis). This would raise her risk dramatically because her
return would fall strongly in the low states and the strict concavity of her
marginal risk function would reinforce the impact on risk. Therefore, she
rebalances her portfolio by buying more claims in the low states yielding a
convex sharing rule relative to investor j. This result may be counterintuitive
since one may feel that investors who demand a higher expected return are
more aggressive and, therefore, should sell portfolio insurance.

Proposition 6 b2) states that, given the same expected return, the more
risk sensitive investor chooses a portfolio such that her return is higher in
the very low and in the very high states, but lower in between. Thus, the
more risk sensitive investor cuts back large negative deviations of her return
from the benchmark return in the very low states and, thereby, reduces her
risk. Moreover, she raises positive deviations in the very high states which
also reduces her risk. In order to obtain the same expected return, she has
to accept lower returns in the states in between in which the deviations are
small anyway.

Nonlinear sharing rules are also obtained in EU - equilibria based on
HARA utility with distortions. Grossman and Zhou (1996) analyze the
equilibrium for two investors who maximize expected utility. Suppose that
both optimization problems are identical except that the second investor also
makes sure that his random wealth never falls below a given floor. Then this
investor buys options from the other one. Benninga and Mayshar (2000)
prove a related result for an equilibrium with two investors who maximize
their expected utility without a floor. Both have constant relative risk aver-
sion, but at different levels. Then the investor with higher relative risk aver-
sion buys options from the other investor. Franke, Stapleton and Subrah-
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