python/notebooks/ch9_risk_budgeting_convex.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "LaTeX macros (hidden cell)\n",
    "$\n",
    "\\newcommand{\\Q}{\\mathcal{Q}}\n",
    "\\newcommand{\\ECov}{\\boldsymbol{\\Sigma}}\n",
    "\\newcommand{\\EMean}{\\boldsymbol{\\mu}}\n",
    "\\newcommand{\\EAlpha}{\\boldsymbol{\\alpha}}\n",
    "\\newcommand{\\EBeta}{\\boldsymbol{\\beta}}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Imports and configuration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "import sys\n",
    "import os\n",
    "import re\n",
    "import datetime as dt\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "%matplotlib inline\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.colors import LinearSegmentedColormap\n",
    "\n",
    "from mosek.fusion import *\n",
    "import mosek.fusion.pythonic       # Requires MOSEK >= 10.2\n",
    "\n",
    "from notebook.services.config import ConfigManager\n",
    "\n",
    "from portfolio_tools import data_download, DataReader, compute_inputs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.9.7 (default, Sep 16 2021, 13:09:58) \n",
      "[GCC 7.5.0]\n",
      "matplotlib: 3.7.2\n"
     ]
    }
   ],
   "source": [
    "# Version checks\n",
    "print(sys.version)\n",
    "print('matplotlib: {}'.format(matplotlib.__version__))\n",
    "\n",
    "# Jupyter configuration\n",
    "c = ConfigManager()\n",
    "c.update('notebook', {\"CodeCell\": {\"cm_config\": {\"autoCloseBrackets\": False}}})  \n",
    "\n",
    "# Numpy options\n",
    "np.set_printoptions(precision=5, linewidth=120, suppress=True)\n",
    "\n",
    "# Pandas options\n",
    "pd.set_option('display.max_rows', None)\n",
    "\n",
    "# Matplotlib options\n",
    "plt.rcParams['figure.figsize'] = [12, 8]\n",
    "plt.rcParams['figure.dpi'] = 200"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Prepare input data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we load the raw data that will be used to compute the optimization input variables, the vector $\\EMean$ of expected returns and the covariance matrix $\\ECov$. The data consists of daily stock prices of $8$ stocks from the US market. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Download data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Data downloading:\n",
    "# If the user has an API key for alphavantage.co, then this code part will download the data. \n",
    "# The code can be modified to download from other sources. To be able to run the examples, \n",
    "# and reproduce results in the cookbook, the files have to have the following format and content:\n",
    "# - File name pattern: \"daily_adjusted_[TICKER].csv\", where TICKER is the symbol of a stock. \n",
    "# - The file contains at least columns \"timestamp\", \"adjusted_close\", and \"volume\".\n",
    "# - The data is daily price/volume, covering at least the period from 2016-03-18 until 2021-03-18, \n",
    "# - Files are for the stocks PM, LMT, MCD, MMM, AAPL, MSFT, TXN, CSCO.\n",
    "list_stocks = [\"PM\", \"LMT\", \"MCD\", \"MMM\", \"AAPL\", \"MSFT\", \"TXN\", \"CSCO\"]\n",
    "list_factors = []\n",
    "alphaToken = None\n",
    " \n",
    "list_tickers = list_stocks + list_factors\n",
    "if alphaToken is not None:\n",
    "    data_download(list_tickers, alphaToken)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Read data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We load the daily stock price data from the downloaded CSV files. The data is adjusted for splits and dividends. Then a selected time period is taken from the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "investment_start = \"2016-03-18\"\n",
    "investment_end = \"2021-03-18\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found data files: \n",
      "stock_data/daily_adjusted_AAPL.csv\n",
      "stock_data/daily_adjusted_PM.csv\n",
      "stock_data/daily_adjusted_CSCO.csv\n",
      "stock_data/daily_adjusted_TXN.csv\n",
      "stock_data/daily_adjusted_MMM.csv\n",
      "stock_data/daily_adjusted_IWM.csv\n",
      "stock_data/daily_adjusted_MCD.csv\n",
      "stock_data/daily_adjusted_SPY.csv\n",
      "stock_data/daily_adjusted_MSFT.csv\n",
      "stock_data/daily_adjusted_LMT.csv\n",
      "\n",
      "Using data files: \n",
      "stock_data/daily_adjusted_PM.csv\n",
      "stock_data/daily_adjusted_LMT.csv\n",
      "stock_data/daily_adjusted_MCD.csv\n",
      "stock_data/daily_adjusted_MMM.csv\n",
      "stock_data/daily_adjusted_AAPL.csv\n",
      "stock_data/daily_adjusted_MSFT.csv\n",
      "stock_data/daily_adjusted_TXN.csv\n",
      "stock_data/daily_adjusted_CSCO.csv\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# The files are in \"stock_data\" folder, named as \"daily_adjusted_[TICKER].csv\"\n",
    "dr = DataReader(folder_path=\"stock_data\", symbol_list=list_tickers)\n",
    "dr.read_data()\n",
    "df_prices, _ = dr.get_period(start_date=investment_start, end_date=investment_end)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Run the optimization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define the optimization model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below we implement the optimization model in Fusion API. We create it inside a function so we can call it later."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def RiskBudgeting(N, G, b, z, a):\n",
    "    \n",
    "    with Model('Risk budgeting') as M:\n",
    "        # Settings\n",
    "        M.setLogHandler(sys.stdout)\n",
    "        \n",
    "        # Portfolio weights\n",
    "        x = M.variable(\"x\", N, Domain.unbounded())\n",
    "      \n",
    "        # Auxiliary variables\n",
    "        t = M.variable(\"t\", N, Domain.unbounded())\n",
    "        s = M.variable(\"s\", 1, Domain.unbounded())\n",
    "    \n",
    "        # Objective function: 1/2 * x'Sx - a * b'log(z*x) becomes s - a * b't\n",
    "        M.objective(ObjectiveSense.Minimize, s - a * (t.T @ b))\n",
    "    \n",
    "        # Bound on risk term\n",
    "        M.constraint(Expr.vstack(s, 1, G.T @ x), Domain.inRotatedQCone())\n",
    "    \n",
    "        # Bound on log term t <= log(z*x) becomes (z*x, 1, t) in K_exp\n",
    "        M.constraint(Expr.hstack(Expr.mulElm(z, x), Expr.constTerm(N, 1.0), t), Domain.inPExpCone())\n",
    "    \n",
    "        # Create DataFrame to store the results.\n",
    "        columns = [\"obj\", \"risk\", \"xsum\", \"bsum\"] + df_prices.columns.tolist()\n",
    "        df_result = pd.DataFrame(columns=columns) \n",
    "    \n",
    "        # Solve optimization\n",
    "        M.solve()\n",
    "        # Check if the solution is an optimal point\n",
    "        solsta = M.getPrimalSolutionStatus()\n",
    "        if (solsta != SolutionStatus.Optimal):\n",
    "            # See https://docs.mosek.com/latest/pythonfusion/accessing-solution.html about handling solution statuses.\n",
    "            raise Exception(\"Unexpected solution status!\")\n",
    "    \n",
    "        # Save results\n",
    "        xv = x.level()\n",
    "                   \n",
    "        # Check solution quality\n",
    "        risk_budgets = xv * np.dot(G @ G.T, xv)\n",
    "       \n",
    "        # Renormalize to gross exposure = 1\n",
    "        xv = xv / np.abs(xv).sum()\n",
    "        \n",
    "        # Compute portfolio metrics\n",
    "        Gx = np.dot(G.T, xv)\n",
    "        portfolio_risk = np.sqrt(np.dot(Gx, Gx))\n",
    "               \n",
    "        row = pd.Series([M.primalObjValue(), portfolio_risk, np.sum(z * xv), np.sum(risk_budgets)] + list(xv), index=columns)\n",
    "        df_result = pd.concat([df_result, pd.DataFrame([row])], ignore_index=True)\n",
    "        row = pd.Series([None] * 4 + list(risk_budgets), index=columns)\n",
    "        df_result = pd.concat([df_result, pd.DataFrame([row])], ignore_index=True)\n",
    "\n",
    "        return df_result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Compute optimization input variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we use the loaded daily price data to compute the corresponding yearly mean return and covariance matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# Number of securities\n",
    "N = df_prices.shape[1]\n",
    "\n",
    "# Get optimization parameters\n",
    "_, S = compute_inputs(df_prices)\n",
    "\n",
    "# Risk budget\n",
    "b = np.ones(N) / N\n",
    "\n",
    "# Orthant selector\n",
    "z = np.ones(N)\n",
    "\n",
    "# Global setting for sum of b\n",
    "a = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we compute the matrix $G$ such that $\\ECov=GG^\\mathsf{T}$, this is the input of the conic form of the optimization problem. Here we use Cholesky factorization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "G = np.linalg.cholesky(S)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Call the optimizer function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Problem\n",
      "  Name                   : Risk budgeting  \n",
      "  Objective sense        : minimize        \n",
      "  Type                   : CONIC (conic optimization problem)\n",
      "  Constraints            : 8               \n",
      "  Affine conic cons.     : 9 (34 rows)\n",
      "  Disjunctive cons.      : 0               \n",
      "  Cones                  : 0               \n",
      "  Scalar variables       : 18              \n",
      "  Matrix variables       : 0               \n",
      "  Integer variables      : 0               \n",
      "\n",
      "Optimizer started.\n",
      "Presolve started.\n",
      "Linear dependency checker started.\n",
      "Linear dependency checker terminated.\n",
      "Eliminator started.\n",
      "Freed constraints in eliminator : 0\n",
      "Eliminator terminated.\n",
      "Eliminator - tries                  : 1                 time                   : 0.00            \n",
      "Lin. dep.  - tries                  : 1                 time                   : 0.00            \n",
      "Lin. dep.  - primal attempts        : 1                 successes              : 1               \n",
      "Lin. dep.  - dual attempts          : 0                 successes              : 0               \n",
      "Lin. dep.  - primal deps.           : 0                 dual deps.             : 0               \n",
      "Presolve terminated. Time: 0.00    \n",
      "Optimizer  - threads                : 64              \n",
      "Optimizer  - solved problem         : the primal      \n",
      "Optimizer  - Constraints            : 8               \n",
      "Optimizer  - Cones                  : 9               \n",
      "Optimizer  - Scalar variables       : 34                conic                  : 34              \n",
      "Optimizer  - Semi-definite variables: 0                 scalarized             : 0               \n",
      "Factor     - setup time             : 0.00            \n",
      "Factor     - dense det. time        : 0.00              GP order time          : 0.00            \n",
      "Factor     - nonzeros before factor : 36                after factor           : 36              \n",
      "Factor     - dense dim.             : 0                 flops                  : 7.74e+02        \n",
      "ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  \n",
      "0   1.4e+00  1.3e+00  9.7e+00  0.00e+00   1.534945180e+00   -7.147922794e+00  1.0e+00  0.00  \n",
      "1   2.4e-01  2.3e-01  3.9e-01  5.11e-01   2.860275193e+00   1.099946215e+00   1.8e-01  0.01  \n",
      "2   3.1e-02  3.0e-02  1.3e-02  1.33e+00   1.294944866e+00   1.109238006e+00   2.3e-02  0.01  \n",
      "3   1.6e-03  1.5e-03  1.6e-04  1.15e+00   1.052570526e+00   1.043915847e+00   1.2e-03  0.01  \n",
      "4   1.8e-04  1.7e-04  6.1e-06  1.01e+00   1.043789450e+00   1.042801879e+00   1.3e-04  0.01  \n",
      "5   2.6e-05  2.5e-05  3.4e-07  1.00e+00   1.042814807e+00   1.042672665e+00   1.9e-05  0.01  \n",
      "6   5.1e-06  4.8e-06  3.1e-08  1.00e+00   1.042685135e+00   1.042657303e+00   3.8e-06  0.01  \n",
      "7   1.4e-06  1.3e-06  4.5e-09  1.00e+00   1.042662348e+00   1.042655043e+00   9.9e-07  0.01  \n",
      "8   3.3e-07  3.1e-07  6.6e-10  1.00e+00   1.042656394e+00   1.042654634e+00   2.4e-07  0.01  \n",
      "9   3.6e-08  3.4e-08  3.0e-11  1.00e+00   1.042654983e+00   1.042654798e+00   2.6e-08  0.01  \n",
      "10  4.6e-09  4.2e-09  1.4e-12  1.00e+00   1.042654892e+00   1.042654869e+00   3.3e-09  0.01  \n",
      "11  6.5e-09  7.3e-10  5.2e-14  1.00e+00   1.042654882e+00   1.042654879e+00   3.6e-10  0.01  \n",
      "Optimizer terminated. Time: 0.02    \n",
      "\n",
      "\n",
      "Interior-point solution summary\n",
      "  Problem status  : PRIMAL_AND_DUAL_FEASIBLE\n",
      "  Solution status : OPTIMAL\n",
      "  Primal.  obj: 1.0426548817e+00    nrm: 1e+00    Viol.  con: 0e+00    var: 0e+00    acc: 2e-09  \n",
      "  Dual.    obj: 1.0426548792e+00    nrm: 1e+00    Viol.  con: 0e+00    var: 6e-10    acc: 0e+00  \n"
     ]
    }
   ],
   "source": [
    "df_result = RiskBudgeting(N, G, b, z, a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>obj</th>\n",
       "      <th>risk</th>\n",
       "      <th>xsum</th>\n",
       "      <th>bsum</th>\n",
       "      <th>PM</th>\n",
       "      <th>LMT</th>\n",
       "      <th>MCD</th>\n",
       "      <th>MMM</th>\n",
       "      <th>AAPL</th>\n",
       "      <th>MSFT</th>\n",
       "      <th>TXN</th>\n",
       "      <th>CSCO</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.042655</td>\n",
       "      <td>0.212983</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.000007</td>\n",
       "      <td>0.128198</td>\n",
       "      <td>0.134368</td>\n",
       "      <td>0.147567</td>\n",
       "      <td>0.137782</td>\n",
       "      <td>0.090373</td>\n",
       "      <td>0.114948</td>\n",
       "      <td>0.114974</td>\n",
       "      <td>0.131791</td>\n",
       "    </tr>\n",
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       "      <th>1</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.125001</td>\n",
       "      <td>0.125001</td>\n",
       "      <td>0.125001</td>\n",
       "      <td>0.125001</td>\n",
       "      <td>0.125001</td>\n",
       "      <td>0.125001</td>\n",
       "      <td>0.125001</td>\n",
       "      <td>0.125001</td>\n",
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      ],
      "text/plain": [
       "        obj      risk  xsum      bsum        PM       LMT       MCD       MMM  \\\n",
       "0  1.042655  0.212983   1.0  1.000007  0.128198  0.134368  0.147567  0.137782   \n",
       "1       NaN       NaN   NaN       NaN  0.125001  0.125001  0.125001  0.125001   \n",
       "\n",
       "       AAPL      MSFT       TXN      CSCO  \n",
       "0  0.090373  0.114948  0.114974  0.131791  \n",
       "1  0.125001  0.125001  0.125001  0.125001  "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualize the results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the portfolio components."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 2400x1600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = df_result.iloc[0, 4:].T.plot.bar(xlabel=\"securities\", ylabel=\"x\", grid=True, rot=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the risk budgets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 2400x1600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = df_result.iloc[1, 4:].T.plot.bar(xlabel=\"securities\", ylabel=\"risk budget\", grid=True, rot=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "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"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}