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

manyam (1998) obtain a related result when investors have HARA-Utility
and face background risk.This risk destroys linearity of the sharing rules.A
poor investor suffers more from a given level of background risk than a rich
investor and, therefore, tends to buy portfolio insurance.

5 The Pricing Kernel and Investor Hetero-
geneity

5.1 General Risk Functions

In this section we analyze the pricing kernel in a risk-value equilibrium. We
first investigate how in equilibrium the price for state-contingent claims is
related to the aggregate payoff. Proposition 7 provides the result. Note that
proposition 7 is true also if some investors have endogeneous and the others
have exogenous benchmarks.

Proposition 7 : In a risk-value equilibrium the probability-deflated price for
state-contingent claims, r , is decreasing and convex in the aggregate payoff
ε.⁹

Proof: The sum of the payoffs of individual investors is the aggregate
payoff. Hence proposition 1 implies that the aggregate payoff is decreasing
and convex in π. Therefore π must be decreasing and convex in the aggregate
payoff ■

The result of proposition 7 does not come as a surprise. It is also obtained
in an EU-equilibrium if every investor has a von Neumann-Morgenstern util-
ity function with positive prudence. The differences between the risk-value-
⁹For very high aggregate payoffs the market price for State-Coiitiiigeiit claims could be
negative. This would violate the assumption of arbitrage-free markets. A negative price
for some state-contingent claims requires, however, that every investor is beyond satiation
in that state, a possibility that can be safely ignored (see also footnote 7).

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