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

benchmark can be exogenous (any target or aspiration level) or endogenous,
e.g. the expected outcome. The transformation is achieved by a risk func-
tion. Empirical work suggests that the risk function can be a monotonically
decreasing convex function, with positive deviations from the benchmark re-
ducing risk and negative deviations increasing risk. Such a risk function
reflects the practitioners’ view that negative deviations constitute ’’risk” and
positive deviations ’chances”. At some point in our analysis, we will re-
strict the risk function to be of the negative HARA (hyperbolic absolute risk
aversion)-type. This measure still captures major empirical findings on risk
perception.

Using this behavioral framework we get two classes of results. First, we
determine the shape of the investors’ sharing rules in equilibrium. Two defi-
nitions of sharing rules will be used. The absolute sharing rule is the function
which relates the investor’s portfolio payoff to the aggregate payoff, i.e. the
exogenously given payoff to all investors. The relative sharing rule is the
function which relates an investor’s payoff to the payoff of another investor.
In the EU context, Cass and Stiglitz (1970) and Rubinstein (1974) showed
that for expected utility maximizers using a utility function of the HARA-
class with the same exponent, relative sharing rules are linear. In risk-value
models where risk is measured by a negative HARA function, relative shar-
ing rules are linear if and only if this risk function is quadratic. Otherwise
strictly convex or concave relative sharing rules are obtained. More het-
erogeneity among investors translates into more concavity or convexity of
relative sharing rules.

Nonlinear relative sharing rules relate our model to the literature on
portfolio insurance [Leland 1980, Brennan and Solanki 1981, Benninga and
Blume 1985, Franke, Stapleton and Subrahmanyam 1998, Grossman and
Zhou 1996, Benninga and Mayshar 2000]. Leland defines portfolio insurance

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