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

Proposition 2 provides a strong result about the shape of the sharing rules.
In a risk-value world the sharing rules are linear if and only if every investor
uses variance or a related quadratic risk measure (which both behaviorally
are not appropriate), or, equivalently, if and only if the sharing constant
disappears. This is true only for quadratic risk functions. Then f (e) = a + be
so that f(ê) — E(f(ê)) = b(e — E(e)) = be. In contrast, for EU- models,
Rubinstein (1974) has shown that linear sharing rules are obtained whenever
all investors have a HARA-utility function with the same 7.

In the following, we require again f^z(e) > 0 and analyse the sharing rule of
investor i relative to that of investor j, i.e. the relative sharing rule e_i(e₃■). In
analogy to Leland (1980), we say that investor i purchases portfolio insurance
from investor j if his sharing rule e_i is strictly convex in e_j■. Proposition 3
provides conditions for trading portfolio insurance.

Proposition 3 ; In a risk-value equilibrium the following statements are
equivalent:

- Investor i^,s sharing rule is strictly convex [linear] [strictly concave] relative
to that of investor j.

- The coefficient of absolute prudence of investor i ^,s risk function, —fl∖e_i )/f (Çê) ),
multiplied by dci/'de -, is everywhere greater than [equal to] [smaller than] the
coefficient of absolute prudence of investor j ^,s risk function.

■  > _l=_ιw. _nJ2⅛h           ₍₁₈₎

' [f_i'(⅛)l² > ^{1 IM %} [f(⅛)∣²∙                 ⁽⁸⁾

Proof. See Appendix C.

Proposition 3 provides necessary and sufficient conditions for the shape of
an investor’s sharing rule relative to that of another one. The shape depends
on the investors’ coefficients of absolute prudence given efficient portfolios.

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