Fund Certification, Performance Prediction and Learnings from the Past
for Chapter 2 1 Proof of Lemma 1 Proof. The average reputation after certifying a PEF has to be positive for the certiﬁer to remain in business, i. e. the reputation change from investors x ·RI,g − y ·RI,b has to be geater or equal to 0. In the long run, the certiﬁer wants to keep his reputation for any x and y. In consequence x ·RI,g − y ·RI,b ≥ 0 x ·RI,g ≥ y ·RI,b With RI,b ≥ RI,g and x > 0 the inequality returns 1 ≥ RI,g RI,b ≥ y x x ≥ y . For x = 0 we get −y · RI,b ≥ 0, so y would have to be 0. In this case the certiﬁer would not issue any certiﬁcates. 2 Proof of Lemma 2 Proof. If a certiﬁer blindly certiﬁes with z > 0 funds his reputation gain will be z · γ · RI,g − z · (1 − γ) · RI,b which has to be at least 0 to keep his reputation level. Rewritten it follows RI,g RI,b ≥ 1− γ γ . 238 Appendices for Chapter 2 Due to Equation (2.2) the reputation ratio is less than 1. This is only true for γ ≥ 0.5 which is excluded by assumption (2.1). 3 Proof of Theorem 3 Proof. To proof the condition p ≥ 1 − γ, I consider the contrary case p 0 p ≥ (1− p) · 1− γ γ RI,b RI,g p · (1 + 1− γ γ RI,b RI,g ) ≥ 1− γ γ RI,b RI,g p ≥ 1−γ γ RI,b...
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