Proposition 4 illustrates our earlier claim that the sharing constants in
the optimality condition (11) resp. (14) generate room for a larger variety
of sharing rules. While in the EU-model with HARA-utility all investors
have linear sharing rules, in the risk-value model one investor’s sharing rule
relative to another one’s is concave, linear or convex. It is the difference
between the sharing constants of both investors which determines convexity,
linearity and concavity. If all sharing constants are the same, then linearity
obtains as in the EU- model, and a representative investor exists. Conversely,
heterogeneity of investors may be measured by differences in the sharing
constants.
Corollary: The convexity of investor i,s sharing rule relative to that of
investor j grows with the difference in the sharing constants (sj — s,).
Proof. Consider a sequence of investors such that Sj — Sj+ɪ = Δ with
Δ being a small positive number; j = 1,...,J — 1.. Then the convexity of
ej+1(ej) is positive. Hence the convexity of èj^ffèj) must increase in k, and,
hence, in Sj — Sj+в B
More insight can be gained by analysing the investor’s sharing rule in
terms of the gross return R rather than the payoff e. Since R := è/W$, the
risk function can also be written as [R := R — R with R = E(R) for an
endogenous benchmark resp. R being the exogenous benchmark]
E[F R ’-yE f A + ɪʌ
y y W0 1 — y J
except for the case of the exponential risk function (y = — œ). In this
case E[F(R)] = E[èxp(—BWq-R)].
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