4 Equilibrium: Investors’ Sharing Rules
4.1 General Risk Functions
Next equilibrium in the capital market will be investigated. We assume the
existence of an equilibrium. If there exist multiple equilibria, we analyse
anyone of them. Every investor chooses a risk-value efficient portfolio. All
investors are assumed to have homogeneous expectations. First, individual
sharing rules will be analysed in equilibrium (Section 4). After considering
general risk functions (Section 4.1), we restrict ourselves to HARA-based
risk functions (Section 4.2). Second, the equilibrium pricing kernel will be
derived and analyzed (Section 5).
Individual investors are indexed by i. Hence all investor-dependent vari-
ables have to be indexed by i. Then condition (11) for an endogenous bench-
mark reads
-E[fi (A)] + fi(⅛) = ηiθe ; ∀ ε,i. (15)
Condition (14) for an exogenous benchmark is the same. Unless stated
otherwise, the following results hold for efficient portfolios with an endoge-
nous and with an exogenous benchmark. λi> 0 and ηi < 0, will be assumed
throughout. Then we can derive a proposition which relates investors’ port-
folio choices to the pricing of state-contingent claims.
Proposition 1 ; For every investor, his/her optimal payoff is decreasing
and convex in the probability-deflated price for state-contingent claims.
Proof. 8
8By the implicit function theorem, ei is twice continuously differentiable since fi
(êi) and θε = θ(ε) are twice continuously differentiable.
18
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