{
"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": 10,
"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": 11,
"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": 12,
"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": 13,
"metadata": {},
"outputs": [],
"source": [
"investment_start = \"2016-03-18\"\n",
"investment_end = \"2021-03-18\""
]
},
{
"cell_type": "code",
"execution_count": 14,
"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": [
"First we generate return scenario data using Monte Carlo method based on the moments of historical prices. "
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [],
"source": [
"# Number of scenarios\n",
"T = 99999\n",
"\n",
"# Mean and covariance of historical log-returns. \n",
"m_log, S_log = compute_inputs(df_prices, return_log=True)\n",
"\n",
"# Generate logarithmic return scenarios assuming normal distribution\n",
"scenarios_log = np.random.default_rng().multivariate_normal(m_log, S_log, T)\n",
" \n",
"# Convert logarithmic return scenarios to linear return scenarios \n",
"scenarios_lin = np.exp(scenarios_log) - 1\n",
"\n",
"# Scenario probabilities\n",
"p = np.ones(T) / T"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We would like to optimize the 95% EVaR of the portfolio loss distribution."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"scrolled": true
},
"outputs": [],
"source": [
"# Confidence level\n",
"alpha = 0.95"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [],
"source": [
"# Primal CVaR formula (for testing)\n",
"def CVaR(alpha, p, q):\n",
" # We need to be careful that math index starts from 1 but numpy starts from 0 (matters in formulas like ceil(alpha * T))\n",
" T = q.shape[0]\n",
" \n",
" # Starting index \n",
" i_alpha = np.sort(np.nonzero(np.cumsum(p) >= alpha)[0])[0]\n",
"\n",
" # Weight of VaR component in CVaR\n",
" lambda_alpha = (sum(p[:(i_alpha + 1)]) - alpha) / (1 - alpha) \n",
" \n",
" # CVaR\n",
" sort_idx = np.argsort(q)\n",
" sorted_q = q[sort_idx]\n",
" sorted_p = p[sort_idx]\n",
" cvar = lambda_alpha * sorted_q[i_alpha] + np.dot(sorted_p[(i_alpha + 1):], sorted_q[(i_alpha + 1):]) / (1 - alpha)\n",
" \n",
" return cvar"
]
},
{
"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": 18,
"metadata": {},
"outputs": [],
"source": [
"# s * log(sum_i(p_i * exp(x_i / s))) <= t\n",
"def persplogsumexp(M, x, s, p, t):\n",
" n = int(x.getSize())\n",
" u = M.variable(n)\n",
" M.constraint(Expr.hstack(u, Var.repeat(s, n), Expr.sub(x, Var.repeat(t, n))), Domain.inPExpCone())\n",
" M.constraint(Expr.sub(Expr.dot(p, u), s), Domain.lessThan(0.0))\n",
" \n",
"\n",
"def EfficientFrontier(N, T, m, R, p, alpha, 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 constrained to take only positive values. \n",
" x = M.variable(\"x\", N, Domain.greaterThan(0.0))\n",
" \n",
" # Budget constraint\n",
" M.constraint('budget', Expr.sum(x), Domain.equalsTo(1.0))\n",
" \n",
" # Auxiliary variables.\n",
" z = M.variable(\"z\", 1, Domain.unbounded())\n",
" t = M.variable(\"t\", 1, Domain.greaterThan(0.0))\n",
" \n",
" # Constraint modeling perspective of log-sum-exp\n",
" persplogsumexp(M, Expr.mul(-R, x), t, p, z)\n",
" \n",
" # Objective\n",
" delta = M.parameter()\n",
" evar_term = Expr.sub(z, Expr.mul(t, np.log(1.0 - alpha)))\n",
" M.objective('obj', ObjectiveSense.Maximize, Expr.sub(Expr.dot(m, x), Expr.mul(delta, evar_term)))\n",
" \n",
" # Create DataFrame to store the results.\n",
" columns = [\"delta\", \"obj\", \"return\", \"risk\", \"evar\"] + 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",
" # 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 = z.level()[0] - t.level()[0] * np.log(1.0 - alpha)\n",
" cvar = CVaR(alpha, p, -R @ x.level())\n",
" row = pd.Series([d, M.primalObjValue(), portfolio_return, portfolio_risk, cvar] + list(x.level()), index=columns)\n",
" df_result = pd.concat([df_result, pd.DataFrame([row])], ignore_index=True)\n",
" \n",
" # Check CVaR value using primal formula\n",
" print(f\"Relative difference between EVaR and CVaR (%): {(cvar - portfolio_risk) / portfolio_risk * 100}\")\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": 19,
"metadata": {
"scrolled": true
},
"outputs": [],
"source": [
"# Number of securities\n",
"N = df_prices.shape[1] \n",
"\n",
"# Get optimization parameters\n",
"m, _ = compute_inputs(df_prices)"
]
},
{
"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: $\\delta = 10^{-1},\\dots,10^{2}$. We compute the efficient frontier this way both with and without using shrinkage estimation. "
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"scrolled": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Relative difference between EVaR and CVaR (%): -33.42378204398512\n",
"Relative difference between EVaR and CVaR (%): -33.45646972266584\n",
"Relative difference between EVaR and CVaR (%): -33.50204787771225\n",
"Relative difference between EVaR and CVaR (%): -33.56488606024007\n",
"Relative difference between EVaR and CVaR (%): -33.65252039086718\n",
"Relative difference between EVaR and CVaR (%): -33.77210072922806\n",
"Relative difference between EVaR and CVaR (%): -33.90833145658367\n",
"Relative difference between EVaR and CVaR (%): -34.06647761773668\n",
"Relative difference between EVaR and CVaR (%): -34.228864613910865\n",
"Relative difference between EVaR and CVaR (%): -34.32076597469258\n",
"Relative difference between EVaR and CVaR (%): -34.26903206118584\n",
"Relative difference between EVaR and CVaR (%): -34.0913142099972\n",
"Relative difference between EVaR and CVaR (%): -33.706820776915826\n",
"Relative difference between EVaR and CVaR (%): -32.87550621993\n",
"Relative difference between EVaR and CVaR (%): -32.13821514072266\n",
"Relative difference between EVaR and CVaR (%): -31.414840069754284\n",
"Relative difference between EVaR and CVaR (%): -29.94969115976413\n",
"Relative difference between EVaR and CVaR (%): -27.216047490936095\n",
"Relative difference between EVaR and CVaR (%): -23.044884017560292\n",
"Relative difference between EVaR and CVaR (%): -20.845617388377192\n"
]
}
],
"source": [
"# Compute efficient frontier with and without shrinkage\n",
"deltas = np.logspace(start=-1, stop=2, num=20)[::-1]\n",
"df_result = EfficientFrontier(N, T, m, scenarios_lin, p, alpha, deltas)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Check the results."
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" delta | \n",
" obj | \n",
" return | \n",
" risk | \n",
" evar | \n",
" PM | \n",
" LMT | \n",
" MCD | \n",
" MMM | \n",
" AAPL | \n",
" MSFT | \n",
" TXN | \n",
" CSCO | \n",
"
\n",
" \n",
" \n",
" \n",
" | 0 | \n",
" 100.000000 | \n",
" -14.809388 | \n",
" 0.347111 | \n",
" 0.151565 | \n",
" 0.100906 | \n",
" -2.712381e-10 | \n",
" 1.494921e-02 | \n",
" 1.363686e-01 | \n",
" 2.281447e-10 | \n",
" 0.090220 | \n",
" 5.360870e-01 | \n",
" 2.223747e-01 | \n",
" 3.692416e-08 | \n",
"
\n",
" \n",
" | 1 | \n",
" 69.519280 | \n",
" -10.189479 | \n",
" 0.347760 | \n",
" 0.151573 | \n",
" 0.100862 | \n",
" -1.494799e-10 | \n",
" 1.380761e-02 | \n",
" 1.347412e-01 | \n",
" 8.227581e-10 | \n",
" 0.090979 | \n",
" 5.380318e-01 | \n",
" 2.224405e-01 | \n",
" 2.026158e-08 | \n",
"
\n",
" \n",
" | 2 | \n",
" 48.329302 | \n",
" -6.977512 | \n",
" 0.348694 | \n",
" 0.151589 | \n",
" 0.100804 | \n",
" -2.945664e-10 | \n",
" 1.216126e-02 | \n",
" 1.323962e-01 | \n",
" -2.890110e-10 | \n",
" 0.092075 | \n",
" 5.408365e-01 | \n",
" 2.225308e-01 | \n",
" 5.209527e-10 | \n",
"
\n",
" \n",
" | 3 | \n",
" 33.598183 | \n",
" -4.744225 | \n",
" 0.350042 | \n",
" 0.151623 | \n",
" 0.100731 | \n",
" -1.268907e-10 | \n",
" 9.786780e-03 | \n",
" 1.290179e-01 | \n",
" -1.236874e-10 | \n",
" 0.093662 | \n",
" 5.448811e-01 | \n",
" 2.226519e-01 | \n",
" -5.141085e-11 | \n",
"
\n",
" \n",
" | 4 | \n",
" 23.357215 | \n",
" -3.191162 | \n",
" 0.351989 | \n",
" 0.151694 | \n",
" 0.100645 | \n",
" 9.499372e-11 | \n",
" 6.357014e-03 | \n",
" 1.241467e-01 | \n",
" 1.313828e-09 | \n",
" 0.095965 | \n",
" 5.507230e-01 | \n",
" 2.228084e-01 | \n",
" 7.211346e-08 | \n",
"
\n",
" \n",
" | 5 | \n",
" 16.237767 | \n",
" -2.110756 | \n",
" 0.354809 | \n",
" 0.151841 | \n",
" 0.100561 | \n",
" -3.079113e-10 | \n",
" 1.390277e-03 | \n",
" 1.171098e-01 | \n",
" -2.952811e-10 | \n",
" 0.099325 | \n",
" 5.591803e-01 | \n",
" 2.229945e-01 | \n",
" -1.388656e-10 | \n",
"
\n",
" \n",
" | 6 | \n",
" 11.288379 | \n",
" -1.358693 | \n",
" 0.358261 | \n",
" 0.152099 | \n",
" 0.100525 | \n",
" -4.289182e-10 | \n",
" 1.324001e-07 | \n",
" 1.038537e-01 | \n",
" -3.300871e-10 | \n",
" 0.104016 | \n",
" 5.700917e-01 | \n",
" 2.220389e-01 | \n",
" 4.122753e-09 | \n",
"
\n",
" \n",
" | 7 | \n",
" 7.847600 | \n",
" -0.834635 | \n",
" 0.363036 | \n",
" 0.152616 | \n",
" 0.100625 | \n",
" -7.075282e-10 | \n",
" 1.695273e-08 | \n",
" 8.366961e-02 | \n",
" -2.133383e-10 | \n",
" 0.110821 | \n",
" 5.854145e-01 | \n",
" 2.200944e-01 | \n",
" 4.496131e-09 | \n",
"
\n",
" \n",
" | 8 | \n",
" 5.455595 | \n",
" -0.468526 | \n",
" 0.369963 | \n",
" 0.153693 | \n",
" 0.101086 | \n",
" -6.042409e-10 | \n",
" 5.839406e-08 | \n",
" 5.453391e-02 | \n",
" 1.396929e-10 | \n",
" 0.120911 | \n",
" 6.076218e-01 | \n",
" 2.169333e-01 | \n",
" 1.057438e-08 | \n",
"
\n",
" \n",
" | 9 | \n",
" 3.792690 | \n",
" -0.211414 | \n",
" 0.380061 | \n",
" 0.155952 | \n",
" 0.102428 | \n",
" 4.562780e-09 | \n",
" 1.048043e-07 | \n",
" 1.233776e-02 | \n",
" 9.746963e-09 | \n",
" 0.135994 | \n",
" 6.399893e-01 | \n",
" 2.116789e-01 | \n",
" 7.122438e-08 | \n",
"
\n",
" \n",
" | 10 | \n",
" 2.636651 | \n",
" -0.030093 | \n",
" 0.384901 | \n",
" 0.157394 | \n",
" 0.103457 | \n",
" -2.675437e-10 | \n",
" 8.275791e-09 | \n",
" 1.358343e-07 | \n",
" 7.311167e-10 | \n",
" 0.153269 | \n",
" 6.565735e-01 | \n",
" 1.901573e-01 | \n",
" 7.955828e-09 | \n",
"
\n",
" \n",
" | 11 | \n",
" 1.832981 | \n",
" 0.096914 | \n",
" 0.388271 | \n",
" 0.158953 | \n",
" 0.104764 | \n",
" 1.299553e-09 | \n",
" 4.993040e-08 | \n",
" 4.729194e-07 | \n",
" 2.645259e-09 | \n",
" 0.176696 | \n",
" 6.686228e-01 | \n",
" 1.546808e-01 | \n",
" 3.657430e-08 | \n",
"
\n",
" \n",
" | 12 | \n",
" 1.274275 | \n",
" 0.186450 | \n",
" 0.393112 | \n",
" 0.162180 | \n",
" 0.107514 | \n",
" -2.211452e-10 | \n",
" 6.756127e-10 | \n",
" 4.488115e-08 | \n",
" 9.512924e-10 | \n",
" 0.211567 | \n",
" 6.840754e-01 | \n",
" 1.043575e-01 | \n",
" 4.037568e-09 | \n",
"
\n",
" \n",
" | 13 | \n",
" 0.885867 | \n",
" 0.250502 | \n",
" 0.400035 | \n",
" 0.168798 | \n",
" 0.113305 | \n",
" 2.755646e-09 | \n",
" -1.266690e-11 | \n",
" 2.424045e-08 | \n",
" 3.777747e-10 | \n",
" 0.264047 | \n",
" 7.022571e-01 | \n",
" 3.369572e-02 | \n",
" 2.769004e-09 | \n",
"
\n",
" \n",
" | 14 | \n",
" 0.615848 | \n",
" 0.297040 | \n",
" 0.404439 | \n",
" 0.174391 | \n",
" 0.118345 | \n",
" -2.052848e-11 | \n",
" -1.011557e-11 | \n",
" 1.175816e-11 | \n",
" -1.748460e-11 | \n",
" 0.319658 | \n",
" 6.803416e-01 | \n",
" 8.585560e-10 | \n",
" -6.242794e-12 | \n",
"
\n",
" \n",
" | 15 | \n",
" 0.428133 | \n",
" 0.330133 | \n",
" 0.406807 | \n",
" 0.179091 | \n",
" 0.122830 | \n",
" 5.774546e-10 | \n",
" 3.875773e-10 | \n",
" 2.384531e-08 | \n",
" 1.150304e-09 | \n",
" 0.385380 | \n",
" 6.146204e-01 | \n",
" 2.126529e-08 | \n",
" 2.074802e-09 | \n",
"
\n",
" \n",
" | 16 | \n",
" 0.297635 | \n",
" 0.354048 | \n",
" 0.410417 | \n",
" 0.189392 | \n",
" 0.132669 | \n",
" 6.395015e-10 | \n",
" 6.525280e-10 | \n",
" 1.226104e-09 | \n",
" 2.599322e-09 | \n",
" 0.485545 | \n",
" 5.144548e-01 | \n",
" 3.869882e-08 | \n",
" 3.423996e-09 | \n",
"
\n",
" \n",
" | 17 | \n",
" 0.206914 | \n",
" 0.372056 | \n",
" 0.415926 | \n",
" 0.212022 | \n",
" 0.154318 | \n",
" 6.981106e-10 | \n",
" 1.975800e-09 | \n",
" 5.781582e-09 | \n",
" 3.504131e-10 | \n",
" 0.638413 | \n",
" 3.615865e-01 | \n",
" 1.405329e-08 | \n",
" 2.153144e-09 | \n",
"
\n",
" \n",
" | 18 | \n",
" 0.143845 | \n",
" 0.386679 | \n",
" 0.424172 | \n",
" 0.260645 | \n",
" 0.200580 | \n",
" 2.219064e-09 | \n",
" 6.181270e-09 | \n",
" 1.355847e-08 | \n",
" 4.917751e-09 | \n",
" 0.867205 | \n",
" 1.327948e-01 | \n",
" 1.368773e-08 | \n",
" 2.201932e-09 | \n",
"
\n",
" \n",
" | 19 | \n",
" 0.100000 | \n",
" 0.399278 | \n",
" 0.428958 | \n",
" 0.296801 | \n",
" 0.234931 | \n",
" -1.725104e-12 | \n",
" 1.257411e-11 | \n",
" 2.114564e-11 | \n",
" -4.253230e-13 | \n",
" 1.000000 | \n",
" 1.343903e-08 | \n",
" 1.405113e-10 | \n",
" 4.463059e-12 | \n",
"
\n",
" \n",
"
\n",
"