Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due

MATH 11158 : Optimization Methods in Finance
Assignment 2

Question 1 (Sortino ratio, (10 marks) ). Consider a portfolio selection problem where feasible portfolios are required to belong to a compact convex set X ⊂ R n. Furthermore, the portfolios are also required to achieve an expected return of at least the target return rate R. Let ST R∗ denote the maximum Sortino ratio of all such portfolios.

Show how you can compute a lower bound on ST R∗ by solving a convex optimization problem. Clearly justify why the problem you are solving is convex.

Question 2 ( (10 marks) ). Suppose that the asset returns vector r ∈ R n + follows a discrete distribution with its values in N scenarios given by {r(ω1), r(ω2), . . . , r(ωN )} whose probabilities are p1, p2, . . . , pN . Let R > 0 and β ∈ (0, 1) be given parameters.

For the questions below, your formulations should be in a tractable form so that they can be

2. Formulate the problem of finding the optimal portfolio that minimises the probability that the portfolio return is less than or equal to R.

Hint: You may find useful the fact from optimization that minimising any function f(x) is equivalent to adding a new variable t and minimising t subject to the constraint that f(x) ≤ t.

Question 3 (Semidefinite relaxations, (20 marks) ). Consider the 4th question from Tutorial 4. All else being the same, modify the requirement that instead of investing in at least 4 securities you must now invest in at most 4 securities.

1. Provide 3 semidefinite relaxations of this problem. (6 marks)

2. Solve the above SDPs and compare their bounds for different target returns. (4 marks)

3. Devise a heuristic method that uses any of the semidefinite solutions to obtain a feasible portfolio. (5 marks)

4. Apply your heuristic to the three SDPs and compare the portfolios that you get. (5 marks)

Question 4 ( (15 marks) ). Consider the Markowtiz tradeoff model with risk-aversion δ > 0,

Here, Σ is a given covariance matrix, but the mean returns µ is an uncertain parameter.

Question 5 (Portfolio computations, (15 marks) ). TBD on Tuesday 1st April at noon.