{
"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": [
"3.9.7 (default, Sep 16 2021, 13:09:58) \n",
"[GCC 7.5.0]\n",
"matplotlib: 3.4.3\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": [
"# |x| <= t\n",
"def absval(M, x, t):\n",
" M.constraint(Expr.add(t, x), Domain.greaterThan(0.0))\n",
" M.constraint(Expr.sub(t, x), Domain.greaterThan(0.0))\n",
"\n",
"# ||x||_1 <= t\n",
"def norm1(M, x, t):\n",
" u = M.variable(x.getShape(), Domain.unbounded())\n",
" absval(M, x, u)\n",
" M.constraint(Expr.sub(Expr.sum(u), t), Domain.equalsTo(0.0))\n",
"\n",
"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 relative to the initial capital. \n",
" # It is a free variable, allowing long and short positions.\n",
" x = M.variable(\"x\", N, Domain.unbounded())\n",
" \n",
" # The variable s models the portfolio variance term in the objective.\n",
" s = M.variable(\"s\", 1, Domain.unbounded())\n",
" \n",
" # Gross exposure constraint (allows 2 times the initial capital)\n",
" norm1(M, x, 2.0)\n",
" \n",
" # Dollar neutrality constraint\n",
" M.constraint('neutrality', Expr.sum(x), Domain.equalsTo(0.0))\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, 0.5, Expr.mul(G.transpose(), x)), Domain.inRotatedQCone())\n",
" \n",
" # Create DataFrame to store the results. Last security name (the SPY) is removed.\n",
" columns = [\"delta\", \"obj\", \"return\", \"risk\", \"g. exp.\"] + 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 = np.sqrt(s.level()[0])\n",
" gross_exp = sum(np.absolute(x.level()))\n",
" row = pd.Series([d, M.primalObjValue(), portfolio_return, portfolio_risk, gross_exp] + 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, S = compute_inputs(df_prices)"
]
},
{
"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": "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 = EfficientFrontier(N, m, G, deltas)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Check the results."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
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" -3.239002e-01 | \n",
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" 3.788759e-01 | \n",
" 1.145439e-09 | \n",
" -6.673766e-10 | \n",
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\n",
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" 0.306991 | \n",
" 0.344282 | \n",
" 0.333155 | \n",
" 2.000000 | \n",
" -0.766835 | \n",
" -2.036140e-10 | \n",
" -1.237167e-10 | \n",
" -2.331652e-01 | \n",
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" 2.370640e-01 | \n",
" 1.872415e-10 | \n",
" -1.152393e-10 | \n",
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\n",
" \n",
" | 16 | \n",
" 0.248163 | \n",
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" 0.353408 | \n",
" 0.378110 | \n",
" 2.000000 | \n",
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" -2.863282e-08 | \n",
" -1.101740e-01 | \n",
" 0.955296 | \n",
" 4.470383e-02 | \n",
" 4.271174e-08 | \n",
" -2.692994e-08 | \n",
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\n",
" \n",
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" 0.183298 | \n",
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" -3.563641e-09 | \n",
"
\n",
" \n",
" | 18 | \n",
" 0.135388 | \n",
" 0.335509 | \n",
" 0.356984 | \n",
" 0.398270 | \n",
" 2.000000 | \n",
" -1.000000 | \n",
" -6.606806e-09 | \n",
" -4.545752e-09 | \n",
" -4.995802e-08 | \n",
" 1.000000 | \n",
" 2.714103e-08 | \n",
" 4.472662e-09 | \n",
" -4.067312e-09 | \n",
"
\n",
" \n",
" | 19 | \n",
" 0.100000 | \n",
" 0.341122 | \n",
" 0.356984 | \n",
" 0.398270 | \n",
" 2.000000 | \n",
" -1.000000 | \n",
" -1.669964e-09 | \n",
" -1.164704e-09 | \n",
" -9.496848e-09 | \n",
" 1.000000 | \n",
" 9.899558e-09 | \n",
" 1.368620e-09 | \n",
" -1.014214e-09 | \n",
"
\n",
" \n",
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