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

Appendix E: Proof of Proposition 6

The proof will be shown for 7 > -∞. It is the same for 7 = -∞.

a) First, we prove statement a). Consider the minimisation of the objec-
tive function (19) s.t. E[Rπ] = 1 and E(R) > R*.

Hence for two investors i and j with d∩, H ∩, = Aq^∕Wq₃- and R* = R* it fol-
lows that the efficient portfolio returns are the same. By Lemma 1, investor
i’s sharing constant decreases when her initial endowment increases or when
she demands a higher expected portfolio return. Then, by Proposition 4, her
sharing rule is strictly convex relative to that of investor j.

bl) Now we prove statement bl). As R_i(R₃■) is strictly convex, the curves
R_i (R₃■) and R₃-(R₃■) can intersect at most twice. They have to intersect at
least once, otherwise the budget constraint E[Rπ] = 1 cannot hold for both.
Hence we have to show that two intersections are impossible. Equation (28)
yields in the HARA-case, starting from objective function (19),

A* i ⅞ ∖
w_0i ⁱ 1—7

A* R ∙ I
____3 i ⁿ-3 I

W₀₃ ' l-7 /

At an intersection R_i = R₃ so that W∣_i, .( = W∩,/A* implies that the
bracketed term equals 1. Hence the slope dR_i∕dR₃ at an intersection is
unique. Therefore convexity of R_i(R₃■) rules out two intersections. Given one
intersection at Rj = R¹, a higher expected portfolio return can be obtained
only if in the states Rj < R¹ relatively expensive claims are sold and in the
states Rj > R¹ relatively cheap claims are bought. Thus dR_i∕dRj > 1 for
R_j = R¹.

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