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

Differentiate equation (15) with respect to θ. Recall, f_i(e⅛) > 0, f∙ (e⅛) <
0, and f^z(¾) > 0. Hence it follows that e_i is an increasing and convex
function in θ. As θ = 1 — π, eɪ is a decreasing, convex function in π. This
proves proposition 1 ■

An absolute sharing rule relates investor i’s payoff e_i to the aggregate
payoff ε. Proposition 1 states de_i∕dπ < 0. Aggregation across investors
implies dε∕dπ < 0. Hence it follows that de_i∕dε > 0. The positive slope of
the absolute sharing rule does not come as a surprise. This is still in line with
EU-theory. The major difference between the risk-value and the EU-model
is reflected in the shapes of the sharing rules. Equation (15) implies for two
investors i and j in the risk-value model

^{— E[f}i (^e¾)] + ^f (^e⅛)

^—di

Define s_i := E[f_i(e⅛)]/ — η_i to be investor i ^,s sharing constant. Equation
(16) can be written then as

—s_i + ^f⅛^- = — ¾ + ^f '   ; ∀ ε.            (17)

^—η            ^— η_j

The new element in risk-value models as compared to expected utility
models are the sharing constants. They are generated in the case of an ex-
ogenous benchmark by the payoff constraint and, in the case of an endogenous
benchmark, by the impact of a payoff change on the benchmark. In order to
grasp the intuition behind the sharing constants, consider the inverse sharing
constant - η_i∕E[f (e⅛)]. — η_i is the efficient increase in risk due to a marginal
reduction in the initial endowment available for buying claims, holding the
required expected payoff WqR* constant. This risk increase depends on the

19

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