{
"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",
"\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": [
"3.6.9 (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": 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 EfficientFrontier(N, m, G, deltas):\n",
"\n",
" with Model(\"Case study\") as M:\n",
" # Settings\n",
" #M.setLogHandler(sys.stdout)\n",
" \n",
" # Variables \n",
" # The variable x is the fraction of holdings in each security. \n",
" # It is restricted to be positive, which imposes the constraint of no short-selling. \n",
" x = M.variable(\"x\", N, Domain.greaterThan(0.0))\n",
" \n",
" # The variable s models the portfolio variance term in the objective.\n",
" s = M.variable(\"s\", 1, Domain.unbounded())\n",
" \n",
" # Budget constraint\n",
" M.constraint('budget', Expr.sum(x), Domain.equalsTo(1.0))\n",
" \n",
" # Objective (quadratic utility version)\n",
" delta = M.parameter()\n",
" M.objective('obj', ObjectiveSense.Maximize, Expr.sub(Expr.dot(m, x), Expr.mul(delta, s)))\n",
"\n",
" # Conic constraint for the portfolio variance\n",
" M.constraint('risk', Expr.vstack(s, Expr.mul(G.transpose(), x)), Domain.inQCone())\n",
" \n",
" # Create DataFrame to store the results. Last security name (the SPY) is removed.\n",
" columns = [\"delta\", \"obj\", \"return\", \"risk\"] + df_prices.columns.tolist()\n",
" df_result = pd.DataFrame(columns=columns)\n",
" for d in deltas:\n",
" # Update parameter\n",
" delta.setValue(d);\n",
" \n",
" # Solve optimization\n",
" M.solve()\n",
" \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",
" portfolio_return = m @ x.level()\n",
" portfolio_risk = s.level()[0]\n",
" row = pd.Series([d, M.primalObjValue(), portfolio_return, portfolio_risk] + list(x.level()), 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",
"m_orig, S_orig = compute_inputs(df_prices, shrinkage=False)\n",
"m_shrunk, S_shrunk = compute_inputs(df_prices, shrinkage=True)"
]
},
{
"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_orig = np.linalg.cholesky(S_orig) \n",
"G_shrunk = np.linalg.cholesky(S_shrunk)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Call the optimizer function"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We run the optimization for a range of risk aversion parameter values: $\\delta = 10^{-1},\\dots,10^{1.5}$. We compute the efficient frontier this way both with and without using shrinkage estimation. "
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"scrolled": true
},
"outputs": [],
"source": [
"# Compute efficient frontier with and without shrinkage\n",
"deltas = np.logspace(start=-1, stop=1.5, num=20)[::-1]\n",
"df_result_orig = EfficientFrontier(N, m_orig, G_orig, deltas)\n",
"df_result_shrunk = EfficientFrontier(N, m_shrunk, G_shrunk, deltas)\n",
"\n",
"# Set small negatives to zero to make plotting work\n",
"mask = df_result_orig < 0\n",
"mask.iloc[:, :-8] = False\n",
"df_result_orig[mask] = 0\n",
"\n",
"# Set small negatives to zero to make plotting work\n",
"mask = df_result_shrunk < 0\n",
"mask.iloc[:, :-8] = False\n",
"df_result_shrunk[mask] = 0"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Check the results without shrinkage."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" delta | \n",
" obj | \n",
" return | \n",
" risk | \n",
" PM | \n",
" LMT | \n",
" MCD | \n",
" MMM | \n",
" AAPL | \n",
" MSFT | \n",
" TXN | \n",
" CSCO | \n",
"
\n",
" \n",
" \n",
" \n",
" | 0 | \n",
" 31.622777 | \n",
" -6.271623 | \n",
" 0.175479 | \n",
" 0.203875 | \n",
" 1.028265e-01 | \n",
" 1.130694e-01 | \n",
" 3.006178e-01 | \n",
" 1.649833e-01 | \n",
" 1.859269e-09 | \n",
" 7.935826e-02 | \n",
" 5.704450e-02 | \n",
" 1.821003e-01 | \n",
"
\n",
" \n",
" | 1 | \n",
" 23.357215 | \n",
" -4.586052 | \n",
" 0.178755 | \n",
" 0.203997 | \n",
" 9.909669e-02 | \n",
" 1.126088e-01 | \n",
" 3.000666e-01 | \n",
" 1.591047e-01 | \n",
" 4.990948e-09 | \n",
" 8.752508e-02 | \n",
" 6.125147e-02 | \n",
" 1.803467e-01 | \n",
"
\n",
" \n",
" | 2 | \n",
" 17.252105 | \n",
" -3.340046 | \n",
" 0.183200 | \n",
" 0.204221 | \n",
" 9.403792e-02 | \n",
" 1.119876e-01 | \n",
" 2.993067e-01 | \n",
" 1.511269e-01 | \n",
" 1.573623e-08 | \n",
" 9.860880e-02 | \n",
" 6.696294e-02 | \n",
" 1.779691e-01 | \n",
"
\n",
" \n",
" | 3 | \n",
" 12.742750 | \n",
" -2.418350 | \n",
" 0.189246 | \n",
" 0.204634 | \n",
" 8.715666e-02 | \n",
" 1.111420e-01 | \n",
" 2.982765e-01 | \n",
" 1.402759e-01 | \n",
" 1.023495e-08 | \n",
" 1.136858e-01 | \n",
" 7.473031e-02 | \n",
" 1.747329e-01 | \n",
"
\n",
" \n",
" | 4 | \n",
" 9.412050 | \n",
" -1.735700 | \n",
" 0.197502 | \n",
" 0.205396 | \n",
" 7.775956e-02 | \n",
" 1.099847e-01 | \n",
" 2.968750e-01 | \n",
" 1.254593e-01 | \n",
" 7.721570e-09 | \n",
" 1.342709e-01 | \n",
" 8.533623e-02 | \n",
" 1.703143e-01 | \n",
"
\n",
" \n",
" | 5 | \n",
" 6.951928 | \n",
" -1.228921 | \n",
" 0.208856 | \n",
" 0.206817 | \n",
" 6.483614e-02 | \n",
" 1.083978e-01 | \n",
" 2.949460e-01 | \n",
" 1.050807e-01 | \n",
" 2.072306e-08 | \n",
" 1.625818e-01 | \n",
" 9.991932e-02 | \n",
" 1.642383e-01 | \n",
"
\n",
" \n",
" | 6 | \n",
" 5.134833 | \n",
" -0.851059 | \n",
" 0.224685 | \n",
" 0.209499 | \n",
" 4.681659e-02 | \n",
" 1.061779e-01 | \n",
" 2.922631e-01 | \n",
" 7.667409e-02 | \n",
" 1.105620e-08 | \n",
" 2.020499e-01 | \n",
" 1.202504e-01 | \n",
" 1.557680e-01 | \n",
"
\n",
" \n",
" | 7 | \n",
" 3.792690 | \n",
" -0.566941 | \n",
" 0.247845 | \n",
" 0.214831 | \n",
" 1.987726e-02 | \n",
" 1.033740e-01 | \n",
" 2.885322e-01 | \n",
" 3.622298e-02 | \n",
" 4.707309e-03 | \n",
" 2.559452e-01 | \n",
" 1.478726e-01 | \n",
" 1.434684e-01 | \n",
"
\n",
" \n",
" | 8 | \n",
" 2.801357 | \n",
" -0.349680 | \n",
" 0.277674 | \n",
" 0.223947 | \n",
" 3.664982e-09 | \n",
" 9.021648e-02 | \n",
" 2.674048e-01 | \n",
" 9.155908e-09 | \n",
" 2.610527e-02 | \n",
" 3.244756e-01 | \n",
" 1.773813e-01 | \n",
" 1.144165e-01 | \n",
"
\n",
" \n",
" | 9 | \n",
" 2.069138 | \n",
" -0.181326 | \n",
" 0.312261 | \n",
" 0.238547 | \n",
" 2.982481e-09 | \n",
" 5.163382e-02 | \n",
" 2.062153e-01 | \n",
" 2.615126e-10 | \n",
" 5.362076e-02 | \n",
" 4.289136e-01 | \n",
" 2.091390e-01 | \n",
" 5.047755e-02 | \n",
"
\n",
" \n",
" | 10 | \n",
" 1.528307 | \n",
" -0.045407 | \n",
" 0.361694 | \n",
" 0.266374 | \n",
" 4.009003e-09 | \n",
" 2.089985e-07 | \n",
" 8.067627e-02 | \n",
" 3.986740e-09 | \n",
" 9.755740e-02 | \n",
" 5.804044e-01 | \n",
" 2.413616e-01 | \n",
" 5.340562e-08 | \n",
"
\n",
" \n",
" | 11 | \n",
" 1.128838 | \n",
" 0.065444 | \n",
" 0.384750 | \n",
" 0.282863 | \n",
" 5.519963e-10 | \n",
" 3.570274e-09 | \n",
" 4.261622e-09 | \n",
" 0.000000e+00 | \n",
" 1.486028e-01 | \n",
" 6.614851e-01 | \n",
" 1.899121e-01 | \n",
" 1.702059e-09 | \n",
"
\n",
" \n",
" | 12 | \n",
" 0.833782 | \n",
" 0.150242 | \n",
" 0.395189 | \n",
" 0.293778 | \n",
" 1.080010e-09 | \n",
" 1.517056e-08 | \n",
" 1.099099e-08 | \n",
" 0.000000e+00 | \n",
" 2.138700e-01 | \n",
" 7.097808e-01 | \n",
" 7.634917e-02 | \n",
" 5.430428e-10 | \n",
"
\n",
" \n",
" | 13 | \n",
" 0.615848 | \n",
" 0.215749 | \n",
" 0.403111 | \n",
" 0.304233 | \n",
" 7.329334e-10 | \n",
" 1.595340e-08 | \n",
" 1.294185e-09 | \n",
" 0.000000e+00 | \n",
" 2.828077e-01 | \n",
" 7.171919e-01 | \n",
" 3.424213e-07 | \n",
" 0.000000e+00 | \n",
"
\n",
" \n",
" | 14 | \n",
" 0.454878 | \n",
" 0.265016 | \n",
" 0.405375 | \n",
" 0.308566 | \n",
" 0.000000e+00 | \n",
" 1.240693e-09 | \n",
" 0.000000e+00 | \n",
" 0.000000e+00 | \n",
" 3.456455e-01 | \n",
" 6.543544e-01 | \n",
" 3.395459e-08 | \n",
" 0.000000e+00 | \n",
"
\n",
" \n",
" | 15 | \n",
" 0.335982 | \n",
" 0.302123 | \n",
" 0.408638 | \n",
" 0.317024 | \n",
" 0.000000e+00 | \n",
" 7.753166e-09 | \n",
" 0.000000e+00 | \n",
" 0.000000e+00 | \n",
" 4.361732e-01 | \n",
" 5.638266e-01 | \n",
" 1.814609e-07 | \n",
" 0.000000e+00 | \n",
"
\n",
" \n",
" | 16 | \n",
" 0.248163 | \n",
" 0.330596 | \n",
" 0.413612 | \n",
" 0.334520 | \n",
" 0.000000e+00 | \n",
" 2.820040e-09 | \n",
" 0.000000e+00 | \n",
" 0.000000e+00 | \n",
" 5.741893e-01 | \n",
" 4.258107e-01 | \n",
" 3.321413e-08 | \n",
" 0.000000e+00 | \n",
"
\n",
" \n",
" | 17 | \n",
" 0.183298 | \n",
" 0.353357 | \n",
" 0.422248 | \n",
" 0.375837 | \n",
" 8.334403e-13 | \n",
" 9.239579e-11 | \n",
" 0.000000e+00 | \n",
" 0.000000e+00 | \n",
" 8.138078e-01 | \n",
" 1.861922e-01 | \n",
" 8.563839e-10 | \n",
" 0.000000e+00 | \n",
"
\n",
" \n",
" | 18 | \n",
" 0.135388 | \n",
" 0.372738 | \n",
" 0.428958 | \n",
" 0.415251 | \n",
" 0.000000e+00 | \n",
" 1.845514e-09 | \n",
" 0.000000e+00 | \n",
" 0.000000e+00 | \n",
" 9.999997e-01 | \n",
" 2.564263e-07 | \n",
" 1.665269e-08 | \n",
" 0.000000e+00 | \n",
"
\n",
" \n",
" | 19 | \n",
" 0.100000 | \n",
" 0.387433 | \n",
" 0.428958 | \n",
" 0.415251 | \n",
" 0.000000e+00 | \n",
" 6.340395e-09 | \n",
" 0.000000e+00 | \n",
" 0.000000e+00 | \n",
" 9.999991e-01 | \n",
" 8.996339e-07 | \n",
" 5.941383e-08 | \n",
" 0.000000e+00 | \n",
"
\n",
" \n",
"
\n",
"