CO 372: Portfolio Optimization Models Winter 2024
Problem Set 4
H. Wolkowicz
Handed out: Saturday 2024-02-17.
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.
Collaboration policy.
1. Students are allowed to discuss question with each other in general terms including helping each other on Piazza. Do not post solutions or partial solutions on Piazza.
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.
3. Students who work together privately on homework should list their teammates in their submission. Teams of size up to 5 are allowed.
Please explain/justify your answers. Please document your codes.
Contents
1 Maximum Expected Return Portfolio, (20 marks) 1
2 KKT Optimality Conditions (10 points) 3
3 Active Set of Constraints, 10 points 3
1 Maximum Expected Return Portfolio, (20 marks)
Consider the problem of maximizing the expected return of a portfolio with n = 25 securities where all but one are risky, i.e., the 25th security is risk-free. The portfolio has the following constraints:
1. the standard budget constraints;
2. 40% of the portfolio is invested in technology securities 5 − 10;
3. 30% of the portfolio is invested in real estate securities 15 − 20;
4. for each dollar invested in the risk-free security, exactly 2 must be invested in security 21;
5. at most 13 risky securities can be included in the portfolio;
6. upper bound on risk tolerance is σp2 (as in Pr2).
Let:
• V ∈ Sn−1, n − 1 = 24, be the covariance matrix for the risky securities; +
• r ̄ ∈ R24 be the vector of expected returns for the risky securities; +
• rf ∈ R+ be the return of the risk-free investment.
1. (10 points) State the program of maximizing the expected return and include the con- straints given above. Carefully indicate the sizes of all vectors and matrices and be sure to include the risk-free security. Let p∗ denote the optimal value and x∗ the optimal portfolio.
2. (10 points) Write (and run) a code to find the optimal portfolio but with no shorting. Include the seed you used in the output you hand in. What was the amount (propor- tion) of the portfolio in the risk-free security (i) when rf = .5; (ii) when rf = .9? Can you comment on the technology and real estate sectors results? Use the following data; and find the covariance matrix V and expected returns r ̄ from the m samples matrix R0.
clear
seed = randi(300,1);
rng(seed);
fprintf('seed is %i\n',seed)
sigmasqp = 1.8; % bound on variance
temp = rand(100,2);
R510 = temp*rand(2,6); % returns 5-10 rank 2
temp = rand(100,2);
R1520 = temp*rand(2,6); % returns 15-20 rank 2
R0 = [rand(100,4) R510 rand(100,4) R1520 rand(100,4)]; rf = .5; % return on the risk-free security
%rf = .9; % re-run with this new rf;
3. (BONUS 5 points) Suppose that the objective function is changed to max x(n) instead of maximizing expected return and the optimal portfolio is xˆ and the optimal value is xˆ(n). Recall the optimal value of maximizing expected return is denoted p∗ with the optimal portfolio x∗.
What are possible conditions on the data (such as magnitude of rf , values in V , etc...) that results in:
(a) x∗(n) = xˆ(n)? (The maximum investment in the risk-free security was obtained by maximizing the expected return.)
(b) x∗(n) = 0? (Zero investment in the risk-free security was obtained by maximizing the expected return.)
2 KKT Optimality Conditions (10 points)
Consider the convex optimization problem
P∗ = min f(x)
s.t. A1x−b1 =0∈RmE
Prove the useful lemma, i.e.: suppose that g is differentiable and g(y) < g(x); then the directional derivative is negative, g′(x; y − x) = ∇g(x)T (y − x) < 0.
State the Lagrangian function for (CP).
State the definition of strict feasibility (the Slater constraint qualification).
StateandprovethesufficientKarush-Kuhn-Tucker,KKT,optimalitytheorem,i.e.,state the conditions for optimality of a point x ̄ ∈ Rn and prove the statement.
Active Set of Constraints, 10 points
(CP)
A2x−b2 ≥0∈RmE g(x) ≤ 0 ∈ Rmg ,
wheref,g:Rn →Rareconvexfunctionsandp∗ ∈R.
Consider the variance (risk) minimization problem (Pr1) in the notes with the definitions of Fa, xmin, Rmin, Vmin, Rmax.
1. Prove that Fa is a convex set.
2. Assume that Fa ̸= ∅. What are the conditions that guarantee that the feasible set of
(Pr1) is not empty.
3. AssumethatFa ̸=∅andthatV ≻0.
(a) Prove that the optimal value p∗Pr1 is finite, and is attained, and the optimum is unique.
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.)
