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

risk function which is determined up to a linear positive transformation.
Hence we need to standardize these marginal risk increases to make them
comparable across investors. This is done by dividing - η_i through the ex-
pected slope of the risk function based on the efficient portfolio, E[f_i (e_i)].
Hence the inverse sharing constant measures the standardized efficient risk
increase due to a marginal reduction in the initial endowment.

In order to gain some insight into the mechanics of the risk-value model,
we analyse the impact of changes in initial endowment and in the required
expected return on an investor’s sharing constant. For simplicity of notation,
we drop the index i in Lemma 1.

Lemma 1: Consider risk-value efficient portfolios under the condition
λ > 0 and η < 0. Then, given the prices of state-contingent claims, the
sharing constant s declines when

- the initial endowment Hq increases, or

- the required expected return R* increases.

Proof. See Appendix A.

From Lemma 1 it is apparent that the sharing constants differ across
investors. The sharing constants, in fact, prohibit linear sharing rules. This
is illustrated by proposition 2 which does not constrain f ^z(e) to be positive.
It should be noted that each investor has a linear absolute sharing rule if all
relative sharing rules are linear, and vice versa.

Proposition 2 : Let f ^z(e) be unconstrained in sign. Then in an equilibrium
with risk-value models every investor has a linear (absolute) sharing rule if
and only if every investor uses a quadratic function F(ê).

Proof. See Appendix B.

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