Consider the problem of choosing a portfolio of risky assets, a proportion b 0 of

Consider the problem of choosing a portfolio π of risky assets, a proportion

φb ≥ 0 of initial wealthto borrow, and a proportionφ ≥ 0 of initial wealthto lend to maximize the expected return π

μ + φR − φbRb subject to the constraints

(1/2)π

π ≤ k and ι



π +φ −φb = 1. Assume B/C > Rb > R, where B and C are defined in (5.6). Define πb = 1 ι
−1(μ− Rbι)
−1 (μ− Rbι), π = 1 ι
−1(μ−Rι)
−1 (μ−Rι).
Using the Kuhn-Tucker conditions, show that the solution is either (i) π = (1−
φ)π for 0 ≤ φ ≤ 1, (ii) π = λπ + (1 − λ)πb for 0 ≤ λ ≤ 1, or (iii) π = (1 +φb)πb for φb ≥ 0.