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

b2) Finally we prove statement b2). The strict convexity of R₁ (R₃) implies
that R_i(R₃) and R₃-(R₃) can intersect at most twice. R* = R* requires at
least one intersection. Suppose there exists one intersection only. As π(ε) is
monotonically decreasing, purchasing claims in some low states ε < ε⁺ and
selling claims in some high states ε > ε⁺ such that this transaction is self-
financing implies a lower expected return. Thus, one intersection contradicts
R* = R* so that two intersections are implied.                           ■

Appendix F: Proof of Proposition 8

Franke/Stapleton/Subrahmanyan (1999) have shown the following. Con-
sider two pricing kernels π_i(ε) and π₂(ε) with E[π_i(ε)] = E[π₂(ε)] = 1 and
E[επ₁ (ε)] = E[επ₂(ε)]. Suppose that these kernels intersect twice such that
πj. (ε) > π₂ (ε) for low and high of levels of ε and π_i (ε) < π₂ (ε) in between.
Then all European options on ε are more expensive under πχ (ε).

Hence in order to prove Proposition 8 it suffices to show: If π₁ (ε) is more
convex than π₂ (ε) everywhere, then these pricing kernels must intersect twice
such that for low and high ε-levels π₁ (ε) > π₂ (ε) and vice versa in between.
Franke/Stapleton/Subrahmanyan (1999) have shown that the pricing kernels
must intersect at least twice to produce the same forward price of the un-
derlying asset. More than two intersections will be shown to be impossible.
Ci (ε) > c₂ (ε) Vε implies

dZn [—π^z₁ (ε)]          dZn [—π^z₂(ε)]

dε                 dε

_{or —} ^»^—H (ε)/ ^— π^z₂ (ε)] > θ

dε

41

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