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

a more aggressive investor, i.e. an investor, who uses a higher exogenous
benchmark return or, in the case of an endogenous benchmark, demands a
higher expected portfolio return.

Proposition 6 characterizes the sharing rules of investors.

Proposition 6 ; Consider two investors i and j who have HARA-risk func-
tions with the same 7.

a) Suppose that the risk sensitivity, ∣1∩ ∕A* for 7 > -∞ resp. BWq for
7 = -∞, is higher for investor i than for investor j and/or she demands
a higher expected portfolio return. Then investor i^,s sharing rule is strictly
convex relative to that of investor j.

bl) Suppose that for both investors the risk sensitivity is the same, but
investor i demands a higher expected portfolio return. Then there exists some
portfolio return R¹ such that

R_i < [=][>]R_j   for   R_j < [=][>]R¹.

b2) Suppose that the risk sensitivity is higher for investor i but both in-
vestors demand the same expected portfolio return. Then there exist R⁰ and
R°° with R⁰ < R⁰⁰ such that

R_i > R_j for R_j < R⁰ and R_j > R⁰⁰,

R_i < R_j   for R⁰ < R_j < R⁰⁰.

Proof. See Appendix E.

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