代写CO 372Portfolio Optimization Models

Problem Set 4

Due: Tuesday 2024-03-05 at 9:00 PM EDT. Papers must be handed in on-line using the labelled dropbox on Crowdmark. Each question is handed in as a separate upload. You can either prepare your solutions electronically using, e.g., LaTeX, or else you can hand-write them and submit a scan. In the latter case, please take care that the scan is of good quality with a white background.

2. No student should hand in work that entirely represents someone else’s effort. Coding should be done entirely on one’s own. Please hand in your codes properly documented so that a marker can run the code if verification is desired.

Contents

3 Active Set of Constraints, 10 points 3

   1. the standard budget constraints;

4. for each dollar invested in the risk-free security, exactly 2 must be invested in security 21;

Let:

• rf ∈ R+ be the return of the risk-free investment.

clear

fprintf('seed is %i\n',seed)

R510 = temp*rand(2,6); % returns 5-10 rank 2

R0 = [rand(100,4) R510 rand(100,4) R1520 rand(100,4)]; rf = .5; % return on the risk-free security

What are possible conditions on the data (such as magnitude of rf , values in V , etc...) that results in:

2 KKT Optimality Conditions (10 points)

s.t. A1x−b1 =0∈RmE

2. 3. 4.

State the definition of strict feasibility (the Slater constraint qualification).

(CP)

Consider the variance (risk) minimization problem (Pr1) in the notes with the definitions of Fa, xmin, Rmin, Vmin, Rmax.

(Pr1) is not empty.

T∗ (b) Provethattheexpectedreturntargetconstraintisactiveattheoptimum,i.e.,r ̄ x =rp. (Recall that an inequality constraint is active at a point if it holds with equality at that point.)