risk-return-portfolios-with-loss-constraint.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "94f82970",
   "metadata": {},
   "outputs": [],
   "source": [
    "import cvxpy as cvx\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a7f9b2a9",
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 4\n",
    "\n",
    "# initialize vector for price changes pbar and covariance matrix Sigma \n",
    "pbar = np.array([.12, .1, .07, .03])\n",
    "Sigma = np.array([[.0064, .0008, -.0011, .0],\n",
    "       [.0008, .0025, .0, .0],\n",
    "       [-.0011, .0, .0004, .0],\n",
    "       [.0, .0, .0, .0]])\n",
    "\n",
    "# Initialize quadratic objective function -pbar.T @ x + x.T @ Sigma @ x\n",
    "x = cvx.Variable(shape=(n, 1))\n",
    "mu_vals = np.logspace(start = 1, stop = 7, num = 100)\n",
    "cp_quad_sigma = cvx.quad_form(x, Sigma)\n",
    "\n",
    "runs = 100\n",
    "\n",
    "returns = list()\n",
    "std = list()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "fd3cbbd5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'returns')"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Markowitz optimization approach by minimizing negative return and risk measure \n",
    "\n",
    "for i in range(runs):\n",
    "    \n",
    "    # generate different combinations of portfolios (std vs returns) varied by mu\n",
    "    \n",
    "    mu = mu_vals[i]\n",
    "    \n",
    "    obj_marko = -pbar.T @ x + mu * cvx.quad_form(x, Sigma)\n",
    "\n",
    "    const = [cvx.sum(x) == 1]\n",
    "    const += [x >= 0]\n",
    "\n",
    "    obj_marko = cvx.Problem(cvx.Minimize(obj_marko), const).solve()\n",
    "    \n",
    "    returns.append(pbar.T @ x.value)\n",
    "    std.append(np.sqrt((x.value).T @ Sigma @ x.value))\n",
    "    \n",
    "  \n",
    "plt.plot(np.ndarray.flatten(np.array(std)), np.ndarray.flatten(np.array(returns)))\n",
    "plt.xlabel('std')\n",
    "plt.ylabel('returns')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "605625d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'returns')"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# portfolio optimization with loss constraint \n",
    "\n",
    "# solve:\n",
    "# max pbar.T x s.t.\n",
    "# constraint that probability for negative return is lower then beta -> calculate for beta\n",
    "# Phi((beta - pbar.T x) / (sqrt(x.T Sigma x))) <= eta,\n",
    "# sum(x) == 1 and x >= 0\n",
    "# We can isolate beta by using Phi^-1 inverse mapping \n",
    "\n",
    "# load package for standard normal distribution \n",
    "from scipy.stats import norm\n",
    "from numpy import linalg as la\n",
    "\n",
    "y = cvx.Variable(shape=(n, 1))\n",
    "\n",
    "# eta in log scale from 10-4 to 10-1\n",
    "eta_vals = np.logspace(start = -4, stop = -1, num = 100)\n",
    "beta = 0.01\n",
    "\n",
    "#eigenvector decomposition of Sigma because cvxpy is not able to take sqrt of quadform sqrt(x.T Sigma x)\n",
    "V, D = la.eigh(Sigma)\n",
    "\n",
    "sqrtSigma = D @ np.sqrt(np.diag(V)) @ D.T\n",
    "\n",
    "returns_set = [0] * 100\n",
    "\n",
    "for i in range(runs):\n",
    "    \n",
    "    # generate different combinations of portfolios (std vs returns) varied by mu\n",
    "    \n",
    "    eta = eta_vals[i]\n",
    "    \n",
    "    # max return \n",
    "    \n",
    "    obj_max_return = pbar.T @ y\n",
    "    \n",
    "    # reformulated constraints (isolated beta)\n",
    "    \n",
    "    const_1 = [cvx.sum(y) == 1]\n",
    "    const_1 += [y >= 0]\n",
    "    const_1 += [norm.ppf(eta) * cvx.norm(sqrtSigma @ y, 2) + pbar.T @ y >= beta]\n",
    "\n",
    "    obj_return = cvx.Problem(cvx.Maximize(obj_max_return), const_1).solve()\n",
    "    \n",
    "    returns_set[i] = obj_return\n",
    "\n",
    "\n",
    "plt.plot(eta_vals, np.array(returns_set))\n",
    "plt.xlabel('eta')\n",
    "plt.ylabel('returns')    "
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.7"
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 },
 "nbformat": 4,
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}