{
"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": 58,
"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",
"\n",
"from notebook.services.config import ConfigManager\n",
"\n",
"from portfolio_tools import data_download, DataReader, compute_inputs"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"3.6.9 (default, Jan 26 2021, 15:33:00) \n",
"[GCC 8.4.0]\n",
"matplotlib: 3.3.4\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": 60,
"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": 61,
"metadata": {},
"outputs": [],
"source": [
"investment_start = \"2016-03-18\"\n",
"investment_end = \"2021-03-18\""
]
},
{
"cell_type": "code",
"execution_count": 62,
"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": 63,
"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",
" # Orthant specifier constraint\n",
" M.constraint(\"orthant\", Expr.mulElm(z, x), Domain.greaterThan(0.0))\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, Expr.sub(s, Expr.mul(a, Expr.dot(b, t))))\n",
" \n",
" # Bound on risk term\n",
" M.constraint(Expr.vstack(s, 1, Expr.mul(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": 64,
"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": 65,
"metadata": {},
"outputs": [],
"source": [
"G = np.linalg.cholesky(S) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Call the optimizer function"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Problem\n",
" Name : Risk budgeting \n",
" Objective sense : min \n",
" Type : CONIC (conic optimization problem)\n",
" Constraints : 42 \n",
" Cones : 9 \n",
" Scalar variables : 52 \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. - number : 0 \n",
"Presolve terminated. Time: 0.00 \n",
"Problem\n",
" Name : Risk budgeting \n",
" Objective sense : min \n",
" Type : CONIC (conic optimization problem)\n",
" Constraints : 42 \n",
" Cones : 9 \n",
" Scalar variables : 52 \n",
" Matrix variables : 0 \n",
" Integer variables : 0 \n",
"\n",
"Optimizer - threads : 20 \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 dense det. time : 0.00 \n",
"Factor - ML order 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 0.01 \n",
"1 2.4e-01 2.3e-01 3.9e-01 5.11e-01 2.860275193e+00 1.099946215e+00 1.8e-01 0.02 \n",
"2 3.1e-02 3.0e-02 1.3e-02 1.33e+00 1.294944866e+00 1.109238006e+00 2.3e-02 0.02 \n",
"3 1.6e-03 1.5e-03 1.6e-04 1.15e+00 1.052570526e+00 1.043915847e+00 1.2e-03 0.02 \n",
"4 1.8e-04 1.7e-04 6.1e-06 1.01e+00 1.043789450e+00 1.042801879e+00 1.3e-04 0.02 \n",
"5 2.6e-05 2.5e-05 3.4e-07 1.00e+00 1.042814807e+00 1.042672665e+00 1.9e-05 0.02 \n",
"6 5.1e-06 4.8e-06 3.1e-08 1.00e+00 1.042685135e+00 1.042657303e+00 3.8e-06 0.02 \n",
"7 1.4e-06 1.3e-06 4.5e-09 1.00e+00 1.042662348e+00 1.042655043e+00 9.9e-07 0.02 \n",
"8 3.3e-07 3.1e-07 6.6e-10 1.00e+00 1.042656394e+00 1.042654634e+00 2.4e-07 0.02 \n",
"9 3.6e-08 3.4e-08 3.0e-11 1.00e+00 1.042654983e+00 1.042654798e+00 2.6e-08 0.02 \n",
"10 4.4e-09 4.2e-09 1.4e-12 1.00e+00 1.042654892e+00 1.042654869e+00 3.3e-09 0.02 \n",
"11 1.5e-08 1.9e-09 5.2e-14 1.00e+00 1.042654881e+00 1.042654879e+00 3.6e-10 0.02 \n",
"Optimizer terminated. Time: 0.03 \n",
"\n",
"\n",
"Interior-point solution summary\n",
" Problem status : PRIMAL_AND_DUAL_FEASIBLE\n",
" Solution status : OPTIMAL\n",
" Primal. obj: 1.0426548813e+00 nrm: 1e+00 Viol. con: 2e-09 var: 0e+00 cones: 5e-09 \n",
" Dual. obj: 1.0426548788e+00 nrm: 1e+00 Viol. con: 0e+00 var: 1e-09 cones: 0e+00 \n"
]
}
],
"source": [
"df_result = RiskBudgeting(N, G, b, z, a)"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
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