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

Interestingly, heterogeneity of investors always raises the convexity of the
pricing kernel. This is intuitively appealing since investors being very risk
sensitive and/or demanding a high portfolio return have a strong appetite
for portfolio insurance, i.e. they have a strong appetite for claims in the very
low and in the very high aggregate payoff-states. This makes these claims
more expensive relative to claims in the intermediate states.

In order to illustrate this, consider the case 1 > 7 > -∞ and let investor
h have the lowest sharing constant. He is very risk sensitive so that his
Wq/A* is high and he demands a high portfolio return R*. Since his Wq/A*
is high, — A* /Wq is high. Hence in a state with a very low aggregate payoff,
this investor has to buy enough claims so as to assure R > — A* (1 — 7)/Wo.
Therefore he has to buy a substantial fraction of the available claims driving
up the price of these claims. Also in the very high payoff-states he buys a
very high fraction of the available claims. This can be seen by analyzing g⅛_ε.
g_hε may be interpreted as investor h’s fractional purchase of claims on the
aggregate payoff, distorted by the constants A_h and A.

Proposition 10 ; Assume 1 > 7 > -∞. Investor h is the unique investor
with the lowest sharing constant. Then g_hε increases monotonically in ε and
approaches 1 for ε → ∞.

Proof. See Appendix H.

Proposition 10 shows that investor h being very risk sensitive and de-
manding a high portfolio return buys a high fraction of all claims in the very
high-payoff states. In these states claims are relatively cheap enabling the
investor to earn a high return.

Since g_hε → 1 for ε → ∞, V (ε) → 0 for ε → ∞. Investor h dominates
pricing in the high payoff-states so that the measure of investor heterogeneity

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