where the superscript T stands for transposed (i.e., transforms a column vector into a row vector) and reciprocally, e denotes the column vector of expected returns to the N assets, 1 represents the column vector of ones, and A, y are Lagrange multipliers. Short sales are permitted (no nonnegativity constraints are present). The solution to this problem can be characterized as the solution to min L, where L is the Lagrangian: 1 L=wVw+ X(E– w" e) + (1 – w"1) (8.8) Under these assumptions, the optimal w,ps A and y must satisfy Eqs. (8.9) through (8.11), which are the necessary and sufficient first-order conditions (FOCS): = Vw - Xe -y1 = 0 (8.9) ƏL = E – w'e = 0 (8.10) = 1- w'1 = 0 (8.11)
where the superscript T stands for transposed (i.e., transforms a column vector into a row vector) and reciprocally, e denotes the column vector of expected returns to the N assets, 1 represents the column vector of ones, and A, y are Lagrange multipliers. Short sales are permitted (no nonnegativity constraints are present). The solution to this problem can be characterized as the solution to min L, where L is the Lagrangian: 1 L=wVw+ X(E– w" e) + (1 – w"1) (8.8) Under these assumptions, the optimal w,ps A and y must satisfy Eqs. (8.9) through (8.11), which are the necessary and sufficient first-order conditions (FOCS): = Vw - Xe -y1 = 0 (8.9) ƏL = E – w'e = 0 (8.10) = 1- w'1 = 0 (8.11)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Can you show the step by step process of the first order conditions in order to get equation 8.9 , 8.10 and 8.11
Lagrange multipliers.
L = Lagrangian
Lagrange multipliers: Gamma and Lambda
w_p = Portfolio weight
Transcribed Image Text:where the superscript T stands for transposed (i.e., transforms a column vector into a row
vector) and reciprocally, e denotes the column vector of expected returns to the N assets,
1 represents the column vector of ones, and A, y are Lagrange multipliers. Short sales are
permitted (no nonnegativity constraints are present). The solution to this problem can be
characterized as the solution to min L, where L is the Lagrangian:
1
L=w"Vw+ X(E– w" e) + (1 – w"1)
(8.8)
Under these assumptions, the optimal w,ps A and y must satisfy Eqs. (8.9) through (8.11),
which are the necessary and sufficient first-order conditions (FOCS):
= Vw – Xe -y1 = 0
Əw
(8.9)
= E – w'e = 0
(8.10)
= 1- w'1=0
(8.11)
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