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

For the exponential risk function (7 = -∞) equation (21) holds with

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[A + e_εI(1 — 7)] ¹ being replaced by ∑ 1∕B₇∙

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Proof. See Appendix G.

Proposition 9 reveals the impact of investor heterogeneity on the convex-
ity of the pricing kernel. By equation (21), the heterogeneity is reflected in
the variance measure of the sharing constants which varies with ε. If all shar-
ing constants are equal, then V (ε) = 0. Hence the convexity of the pricing
kernel is minimal. The higher the variance is, the higher is the convexity of
the pricing kernel.

Alternatively, the heterogeneity between investors is reflected in the con-
vexities of their absolute sharing rules. The second part of equation (21)
shows that the convexity of the pricing kernel is minimal if all sharing
rules are linear. The sum ^Sjd²ej∕dε² is a weighted sum of the convexi-
ties d²e_i∕dε². If the weights s_i were the same across investors, then the sum
would be zero. Hence it is the difference in weights which makes the sum
nonzero. Proposition 5 tells us that the sharing constant s_i is determined
by investor i’s risk sensitivity and her required portfolio return. The higher
these parameters are, the lower is her sharing constant, the more convex is
her sharing rule. Consider, for example, an economy with two investors only.
Investor i (R) has low (high) risk sensitivity, both demand the same portfolio
return R*. Then investor i (R) buys a concave (convex) sharing rule. Since

0 < s_h < s_i and — d²e_i∕dε² = d²e_h∕dε²,       ¾d²e_j∙∕dε² < 0, Vε.

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In general, the inverse relationship between s_i and d²e_i∕dε² across in-
vestors renders the sum ∑ s_id²e_i∕dε² negative. Hence, divided by π^z(ε), it is
positive.

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