as a portfolio policy which leads to a convex absolute sharing rule. The in-
tuition behind his concept is that a convex absolute sharing rule gives the
investor a higher payoff when the aggregate payoff is low (= low state) as
compared to a linear or concave sharing rule. Our results show that an in-
vestor with a risk function of the HARA-type who demands a higher expected
portfolio return or has a higher initial endowment than another investor buys
a sharing rule which is strictly convex relative to that of the other investor.
Thus, the first investor buys portfolio insurance from the second investor.
Second, our equilibrium analysis yields a pricing kernel which is declin-
ing and convex in aggregate consumption. Hence, as Dybvig (1988) points
out, there exists a von Neumann-Morgenstern utility function such that a
representative investor with this function would imply the same pricing ker-
nel. Hence there would be no need to talk about risk-value models. The
important contribution of this paper is, however, to derive the impact of
investor heterogeneity on this pricing kernel. It will be shown for HARA-
based risk functions that the convexity of the kernel increases monotonically
in a simple measure of heterogeneity. More heterogeneity is reflected in more
convexity∕concavity of relative sharing rules. The more convex∕concave rel-
ative sharing rules are, the stronger is the need of investors to trade options,
the higher option prices will be relative to the price of the underlying asset.
Hence the implied volatility derived from the Black-Scholes model increases
with heterogeneity. This might explain the observation that stock index
options appear to be more expensive than suggested by the Black∕Scholes
model [Christensen and Prabhala 1998]. Also, our results could help to ex-
plain why the pricing kernels estimated from stock index option prices are
leptokurtic and negatively skewed [Longstaff 1995, Brenner and Eom 1996,
Jackwerth 2000].
The paper is organized as follows. In section 2, we will review some of