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

These conditions are related to those obtained by Leland (1980) in an EU
equilibrium.

4.2 HARA-Based Risk Functions

In this section, we derive more specific results assuming that the risk function
F belongs to the negative HARA-class. F(e) is given by equation (2) with
7 > 2 or 7 < 1. 7 is assumed to be the same for all investors.

A necessary condition for the existence of an equilibrium in the case of a
non-exponential risk function (7 > -∞) is

This condition must hold because A_i + e_iε /(1 — 7) > 0 Vi, ε is required
by the FOC (11) resp. (14). Let e ≡ ∑_i e_i, then e_ε = ε — e. Hence an
equilibrium requires A + (ε — e) / (1 — 7) > 0. For the more important case
7 < 1 this implies ε ≥ e — A(1 — 7), V ε. If Wq ≡ ∑_i Wq_i, then in equilibrium
Wq is the forward market value of the aggregate payoff ε, i.e. Wq = U(επ(ε)).

We first show that investor i’s sharing rule relative to that of investor j
is either concave, linear or convex.

Proposition 4 ; Consider two investors i and j who measure risk by a neg-
ative HARA-function with 7 being the same for both. Then the following
statements are equivalent;

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

- Investor i^,s sharing constant is smaller than [equal to][greater than] that
of investor j.

Proof. See Appendix D.

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