CVXPY library

  1. Constrained linear least squares Solve the following problem with cvxpy library.

    \[\begin{split} &\|X \theta - y\|^2_2 \to \min\limits_{\theta \in \mathbb{R}^{n} } \\ \text{s.t. } & 0_n \leq \theta \leq 1_n \end{split}\]
  2. Linear programming A linear program is an optimization problem with a linear objective and affine inequality constraints. A common standard form is the following:

    \[\begin{array}{ll} \text{minimize} & c^Tx \\ \text{subject to} & Ax \leq b. \end{array}\]

    Here $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^m$, and $c \in \mathbb{R}^n$ are problem data and $x \in \mathbb{R}^{n}$ is the optimization variable. The inequality constraint $Ax \leq b$ is elementwise. Solve this problem with cvxpy library.

  3. List the installed solvers in cvxpy using cp.installed_solvers() method.
  4. Solve the following optimization problem using CVXPY:

    \[\begin{array}{ll} \text{minimize} & |x| - 2\sqrt{y}\\ \text{subject to} & 2 \geq e^x \\ & x + y = 5, \end{array}\]

    where $x,y \in \mathbb{R}$ are variables. Find the optimal values of $x$ and $y$.

  5. Risk budget allocation Suppose an amount $x_i>0$ is invested in $n$ assets, labeled $i=1,…, n$, with asset return covariance matrix $\Sigma \in \mathcal{S}_{++}^n$. We define the risk of the investments as the standard deviation of the total return

    \[R(x) = (x^T\Sigma x)^{1/2}.\]

    We define the (relative) risk contribution of asset $i$ (in the portfolio $x$) as

    \[\rho_i = \frac{\partial \log R(x)}{\partial \log x_i} = \frac{\partial R(x)}{R(x)} \frac{x_i}{\partial x_i}, \quad i=1, \ldots, n.\]

    Thus $\rho_i$ gives the fractional increase in risk per fractional increase in investment $i$. We can express the risk contributions as

    \[\rho_i = \frac{x_i (\Sigma x)_i} {x^T\Sigma x}, \quad i=1, \ldots, n,\]

    from which we see that $\sum_{i=1}^n \rho_i = 1$. For general $x$, we can have $\rho_i <0$, which means that a small increase in investment $i$ decreases the risk. Desirable investment choices have $\rho_i>0$, in which case we can interpret $\rho_i$ as the fraction of the total risk contributed by the investment in asset $i$. Note that the risk contributions are homogeneous, i.e., scaling $x$ by a positive constant does not affect $\rho_i$.

    • Problem statement: In the risk budget allocation problem, we are given $\Sigma$ and a set of desired risk contributions $\rho_i^\mathrm{des}>0$ with $\bf{1}^T \rho^\mathrm{des}=1$; the goal is to find an investment mix $x\succ 0$, $\bf{1}^Tx =1$, with these risk contributions. When $\rho^\mathrm{des} = (1/n)\bf{1}$, the problem is to find an investment mix that achieves so-called risk parity.
    • a) Explain how to solve the risk budget allocation problem using convex optimization. Hint. Minimize $(1/2)x^T\Sigma x - \sum_{i=1}^n \rho_i^\mathrm{des} \log x_i$.
    • b) Find the investment mix that achieves risk parity for the return covariance matrix $\Sigma$ below.
        import numpy as np
        import cvxpy as cp
        Sigma = np.array(np.matrix("""6.1  2.9  -0.8  0.1;
                             2.9  4.3  -0.3  0.9;
                            -0.8 -0.3   1.2 -0.7;
                             0.1  0.9  -0.7  2.3"""))
        rho = np.ones(4)/4