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 Ri(R3■) is strictly convex, the curves
Ri (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),
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di
d
dRi
dR^
A* i ⅞ ∖
w0i i 1—7
A* R ∙ I
____3 i n-3 I
W03 ' l-7 /
At an intersection Ri = R3 so that W∣i, .( = W∩,/A* implies that the
bracketed term equals 1. Hence the slope dRi∕dR3 at an intersection is
unique. Therefore convexity of Ri(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 dRi∕dRj > 1 for
Rj = R1.
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