Online Appendices for “ The Benchmark Inclusion Subsidy ” by

In this appendix we explore the robustness of our model to an alternative specification where a manager’s compensation is tied to the per-dollar returns on the fund and on the benchmark portfolio as opposed to the performance measure used in the main text. Define Ri = Di/(x̄iSi), i = 1, . . . , n, and let R = (R1, . . . , Rn) be the vector of (perdollar) returns. It is distributed normally with mean μR = (μ1/(x̄1S1), . . . , μn/(x̄nSn)) and variance ΣR, where (ΣR)ij = ρijσiσj/(x̄iSix̄jSj), i = 1, . . . , n, j = 1, . . . , n. It is now more convenient to specify investors’ portfolio optimization problem in terms of fractions φi of wealth under management invested in stock i, i = 1, . . . , n, with the remaining fraction 1− ∑n i=1 φi invested in the bond. Denote φ = (φ1, . . . , φn) >. Let us start by considering the problem of a direct investor. Let W 0 denote the initial wealth of each direct investor. Let 1 = (1, . . . , 1)> be a vector of ones. As in main model, CARA preferences with normal returns are equivalent to mean-variance preferences. Then the direct investor’s problem can be written as maxφ ( φμR + 1− 1>φ ) W 0 − (γ/2)φΣRφ ( W 0 )2 . The optimal solution is φW 0 = Σ −1 R μR − 1/γ. Now consider fund managers. Suppose each manager is given W 0 amount of money to manage, which is all or part of the fund investor’s initial wealth. The manager’s compensation is w = [aRφ+b(Rφ−Rb)]W 0 +c, where Rφ = φ>R+1−1>φ is the return on the manager’s portfolio, and Rb = ω>R is the benchmark return. The benchmark weights (defined as in Lemma 4 in Appendix A) are ωi = 1ix̄iSi/ ∑n j=1 1jx̄jSj, and ω = (ω1, . . . , ωn) >. Then the manager’s compensation can be written as w = [ (a+ b)(φ>R + 1− 1>φ)− bω>R ] W 0 + c, and the manager’s problem is