代写CO 372: Portfolio Optimization Models Fall 2024 Problem Set 1帮做Matlab程序

2024-09-20

CO 372: Portfolio Optimization Models

Fall 2024

Problem Set 1

Due: Friday 2024-09-20 at 4 pm EDT. Papers must be handed in on-line using the labeled 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.

Your papers may be handed in up to 24 hours late, in which case there is a 10% late penalty. Please use the late-paper dropbox on Crowdmark if you are handing in a late paper. You may hand in some questions on time and others late. In this case, use the on-time dropbox for the on-time questions and the late dropbox for the late questions. However, you may not split the parts of a question (i.e., (a), (b), etc.) between the two dropboxes. .

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.

3. Students who work together privately on homework should list their teammates in their submission. Teams of size up to 5 are allowed.

1. (a) Show that for any A ∈ Rm ×n , rank(A) = rank(AT A). Suggested approach: Show that A and AT A have the same null space, in which case the result follows from the rank-nullity theorem. It is straightforward to show that Null(A) ⊆ Null(AT A) by considering a test vector x ∈ Rn. To show then that Null(AT A) ⊆ Null(A), again consider a test vector x ∈ Rn and use the fact that Ax = 0 if and only if ⅡAxⅡ2 = 0.

(b) Construct three 30 × 60 random matrices in Matlab using the randn function like this: A=randn(30,60); For each such matrix, say A, ask Matlab for rank(A) and rank(A’*A). Do the answers come out equal?

Note: For part (b) and part (c), please hand in a copy of your interaction with the Matlab command window. In Windows, if you right-click on the command window and select “Print”, and then “Microsoft Print to PDF” as the printer, you can make a PDF copy of the command window. Alternatively, you can use screenshots to capture the command window.

(c) In principle, one could iterate the process in (b) indefinitely, and the rank should come out the same. In other words, starting from a random 30 × 60 matrix A, if B := AT A (Matlab: B=A’*A;), C := BT B , D := CT C, and so on, all these matrices A,B,C, . . . should have the same rank. Try this for 12 iterations in Matlab, and report on the results. Note: It is not required to provide an explanation of this behavior from Matlab.

2. Let U be an n × n upper triangular matrix with nonzeros on the diagonal. Consider solving the system of linear equations Ux = b, where b ∈ Rn is given and x ∈ Rn is the unknown.

(a) Write down the last (nth) equation of the system Ux = b explicitly, and argue that this equation uniquely determines x(n) (last entry of n).

(b) Assuming x(n) has been computed as in part (a), argue that x(n − 1) is uniquely determined by the second-to-last ((n − 1)st) equation of the system.

(d) Does the argument in (a)–(c) still work in the case that U has one or more zeros on the diagonal?

3. Suppose by accident that a firm lists the same security twice in its catalog: say in the universe of n securities that securities 1 and 2 are actually the same. In this case, a portfolio with x(1) = 20, x(2) = 80 is equivalent to one with x(1) = 90, x(2) = 10, assuming that the remaining x(i)’s (i = 3, 4,..., n) are equal.

(a) In this situation, one might regard portfolios as consisting of only n−1 securities, say in amounts y(1), y(3), y(4),..., y(n) where y(1) := x(1) + x(2) and y(i) = x(i) for i = 3, . . . ,n. Write down an (n−1)×n matrix M that maps the vector [x(1);x(2); ··· ;x(n)] to [y(1);y(3); ··· ;y(n)].

(b) Argue that rank(M) = n − 1, where M is as in (a).

(c) Suppose x1 , x2 are portfolios that equivalent in the sense of this question. Char- acterize the difference d := x1 − x2 . Show that the set of such differences d is a 1-dimensional subspace of Rn , and find a basis for this subspace.

4. The following is awell known theorem that will be used in this course. Let C ⊆ Rn be

closed, bounded, and nonempty. Let f, g be two continuous functions C → R. Then

min{f(x) + g(x) : x ∈ C} ≥ min{f(x) : x ∈ C} + min{g(x) : x ∈ C}.

(a) Prove this theorem.

(b) Take n = 1 and C = [0, 1] (the unit interval). Come up with an example f,g where the inequality in the theorem is strict.

(c) Again with n = 1, C = [0, 1], come up with an example f,g where the inequality in the theorem is satisfied as an equation.

5. Suppose in a universe of n securities that x1 , x2 ∈ Rn are two portfolios. Let r(¯) ∈ Rn denote the expected return vector, so that the expected return of x1 is r(¯)Tx1 while the expected return of x2 is r(¯)Tx2 .

(a) Show the following fact: if x′ is a convex combination of x1, x2 , i.e., x′ = (1 − λ)x1 + λx2 for some λ ∈ [0, 1], then r(¯)Tx′ ≤ max(r(¯)Tx1 , r(¯)Tx2 ).