{
"cells": [
{
"cell_type": "markdown",
"id": "86e024bf",
"metadata": {},
"source": [
"# Leverage by Borrowing Cash\n",
"\n",
"The *standard mean-variance (Markowitz) portfolio selection model* determines an optimal investment portfolio that balances risk and expected return. In this notebook, we maximize the portfolio's expected return while constraining the admissible variance (risk) to a given maximum level. Please refer to the [annotated list of references](../literature.rst#portfolio-optimization) for more background information on portfolio optimization.\n",
"\n",
"To this basic model, we add *leverage*. Leverage means borrowing capital from a third party to buy more assets (and paying interest on the borrowed capital). This magnifies both the potential upside and downside."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "256b3463",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-03T11:04:06.493025Z",
"iopub.status.busy": "2026-07-03T11:04:06.492763Z",
"iopub.status.idle": "2026-07-03T11:04:07.075053Z",
"shell.execute_reply": "2026-07-03T11:04:07.073938Z"
},
"nbsphinx": "hidden"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Requirement already satisfied: numpy in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (2.4.6)\r\n",
"Requirement already satisfied: scipy in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (1.17.1)\r\n",
"Requirement already satisfied: gurobipy in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (13.0.2)\r\n",
"Requirement already satisfied: pandas in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (3.0.3)\r\n",
"Requirement already satisfied: matplotlib in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (3.11.0)\r\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Requirement already satisfied: python-dateutil>=2.8.2 in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (from pandas) (2.9.0.post0)\r\n",
"Requirement already satisfied: contourpy>=1.0.1 in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (from matplotlib) (1.3.3)\r\n",
"Requirement already satisfied: cycler>=0.10 in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (from matplotlib) (0.12.1)\r\n",
"Requirement already satisfied: fonttools>=4.22.0 in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (from matplotlib) (4.63.0)\r\n",
"Requirement already satisfied: kiwisolver>=1.3.1 in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (from matplotlib) (1.5.0)\r\n",
"Requirement already satisfied: packaging>=20.0 in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (from matplotlib) (26.2)\r\n",
"Requirement already satisfied: pillow>=9 in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (from matplotlib) (12.3.0)\r\n",
"Requirement already satisfied: pyparsing>=3 in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (from matplotlib) (3.3.2)\r\n",
"Requirement already satisfied: six>=1.5 in /opt/hostedtoolcache/Python/3.11.15/x64/lib/python3.11/site-packages (from python-dateutil>=2.8.2->pandas) (1.17.0)\r\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Note: you may need to restart the kernel to use updated packages.\n"
]
}
],
"source": [
"# Install dependencies\n",
"%pip install numpy scipy gurobipy pandas matplotlib"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "6cea9bcb",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-03T11:04:07.077288Z",
"iopub.status.busy": "2026-07-03T11:04:07.077081Z",
"iopub.status.idle": "2026-07-03T11:04:07.672256Z",
"shell.execute_reply": "2026-07-03T11:04:07.670917Z"
}
},
"outputs": [],
"source": [
"import gurobipy as gp\n",
"import pandas as pd\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "6ebd068e",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-03T11:04:07.674259Z",
"iopub.status.busy": "2026-07-03T11:04:07.673977Z",
"iopub.status.idle": "2026-07-03T11:04:07.683794Z",
"shell.execute_reply": "2026-07-03T11:04:07.682608Z"
},
"nbsphinx": "hidden"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Set parameter WLSAccessID\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Set parameter WLSSecret\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Set parameter LicenseID to value 2443533\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"WLS license 2443533 - registered to Gurobi GmbH\n"
]
}
],
"source": [
"# Hidden cell to avoid licensing messages\n",
"# when docs are generated.\n",
"with gp.Model():\n",
" pass"
]
},
{
"cell_type": "markdown",
"id": "2785c8a5",
"metadata": {},
"source": [
"## Input Data\n",
"\n",
"The following input data is used within the model:\n",
"\n",
"- $S$: set of stocks\n",
"- $\\mu$: vector of expected returns\n",
"- $\\Sigma$: PSD variance-covariance matrix\n",
" - $\\sigma_{ij}$ covariance between returns of assets $i$ and $j$\n",
" - $\\sigma_{ii}$ variance of return of asset $i$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "e32d9d5c",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-03T11:04:07.685547Z",
"iopub.status.busy": "2026-07-03T11:04:07.685369Z",
"iopub.status.idle": "2026-07-03T11:04:07.691126Z",
"shell.execute_reply": "2026-07-03T11:04:07.690235Z"
}
},
"outputs": [],
"source": [
"# Import some example data set\n",
"Sigma = pd.read_pickle(\"sigma.pkl\")\n",
"mu = pd.read_pickle(\"mu.pkl\")"
]
},
{
"cell_type": "markdown",
"id": "4a9d3c5b",
"metadata": {},
"source": [
"## Formulation\n",
"Mathematically, this results in a convex quadratically constrained optimization problem.\n",
"\n",
"### Model Parameters\n",
"\n",
"The following parameters are used within the model:\n",
"\n",
"- $\\bar\\sigma^2$: maximal admissible variance for the portfolio return\n",
"- $c_\\text{rf}$: interest on the risk-free asset. For simplicity, we assume the same interest rate for lending and borrowing.\n",
"- $\\ell_\\text{rf}$: maximal short on risk-free asset\n",
"- $u_\\text{rf}$: maximal investment in risk-free asset"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "13f2f98d",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-03T11:04:07.692747Z",
"iopub.status.busy": "2026-07-03T11:04:07.692593Z",
"iopub.status.idle": "2026-07-03T11:04:07.696153Z",
"shell.execute_reply": "2026-07-03T11:04:07.695384Z"
}
},
"outputs": [],
"source": [
"# Values for the model parameters:\n",
"V = 4.0 # Maximal admissible variance (sigma^2)\n",
"c_rf = 2 / 52 # interest rate on risk-free asset\n",
"l_rf = -0.3 # maximal borrowing of risk-free asset\n",
"u_rf = 1 # maximal investment in risk-free asset"
]
},
{
"cell_type": "markdown",
"id": "e3a880eb",
"metadata": {},
"source": [
"### Decision Variables\n",
"We require two types of decision variables:\n",
"\n",
"1. The proportions of capital invested among the considered stocks. The corresponding vector of positions is denoted by $x$ with its component $x_i$ denoting the proportion of capital invested in stock $i$.\n",
"\n",
"2. The proportion $x_\\text{rf}$ invested in the risk-free asset. This may be positive or negative. If positive, we gain a risk-free return; if negative, we pay interest on the borrowed amount.\n",
"\n",
"\n",
"### Variable Bounds\n",
"\n",
"Each position must be nonnegative:\n",
"\n",
"$$ x_i\\geq 0 \\;, \\, i \\in S$$\n",
"\n",
"The risk-free position must be within its bounds:\n",
"\n",
"$$ \\ell_\\text{rf} \\leq x_\\text{rf} \\leq u_\\text{rf} $$\n",
"\n",
"Setting the upper bound $u_\\text{rf}=1$ means the portfolio is allowed to be fully invested in the risk-free asset."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "8b819c98",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-03T11:04:07.697883Z",
"iopub.status.busy": "2026-07-03T11:04:07.697706Z",
"iopub.status.idle": "2026-07-03T11:04:07.702697Z",
"shell.execute_reply": "2026-07-03T11:04:07.701782Z"
}
},
"outputs": [],
"source": [
"%%capture\n",
"# Create an empty optimization model\n",
"m = gp.Model()\n",
"\n",
"# Add variables: x[i] denotes the proportion invested in stock i\n",
"x = m.addMVar(len(mu), name=\"x\")\n",
"\n",
"# Risk-free allocation\n",
"x_rf = m.addVar(lb=l_rf, ub=u_rf, name=\"x_rf\")"
]
},
{
"cell_type": "markdown",
"id": "26503978",
"metadata": {},
"source": [
"### Constraints\n",
"\n",
"The budget constraint ensures that all capital (both initial and borrowed) is invested:\n",
"\n",
"$$\\sum_{i \\in S} x_i + x_\\text{rf} = 1$$\n",
"\n",
"The estimated risk must not exceed a prespecified maximal admissible level of variance $\\bar\\sigma^2$:\n",
"\n",
"$$x^\\top \\Sigma x \\leq \\bar\\sigma^2$$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "aa4ac569",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-03T11:04:07.704456Z",
"iopub.status.busy": "2026-07-03T11:04:07.704288Z",
"iopub.status.idle": "2026-07-03T11:04:07.831521Z",
"shell.execute_reply": "2026-07-03T11:04:07.830566Z"
}
},
"outputs": [],
"source": [
"%%capture\n",
"# Budget constraint: all investments sum up to 1\n",
"m.addConstr(x.sum() + x_rf == 1, name=\"Budget_Constraint\")\n",
"\n",
"# Upper bound on variance\n",
"risk_constr = m.addConstr(x @ Sigma.to_numpy() @ x <= V, name=\"Variance\")"
]
},
{
"cell_type": "markdown",
"id": "1df9520a",
"metadata": {},
"source": [
"### Objective Function\n",
"\n",
"The objective is to maximize the expected return of the portfolio. We need to account for risk-free returns and costs for borrowing cash:\n",
"\\begin{equation*}\n",
"\\max_x \\underbrace{c_\\text{rf} x_\\text{rf}}_{\\substack{\\text{cost for borrowing}\\\\\\text{or risk-free return}}} + \\underbrace{\\mu^\\top x}_\\text{expected return from stocks}\n",
"\\end{equation*}"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "c71fbd74",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-03T11:04:07.833813Z",
"iopub.status.busy": "2026-07-03T11:04:07.833564Z",
"iopub.status.idle": "2026-07-03T11:04:07.837904Z",
"shell.execute_reply": "2026-07-03T11:04:07.837057Z"
}
},
"outputs": [],
"source": [
"m.setObjective(c_rf * x_rf + mu.to_numpy() @ x, gp.GRB.MAXIMIZE)"
]
},
{
"cell_type": "markdown",
"id": "5eb02213",
"metadata": {},
"source": [
"We now solve the optimization problem:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "9d88095d",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-03T11:04:07.839594Z",
"iopub.status.busy": "2026-07-03T11:04:07.839423Z",
"iopub.status.idle": "2026-07-03T11:04:08.101793Z",
"shell.execute_reply": "2026-07-03T11:04:08.100858Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Gurobi Optimizer version 13.0.2 build v13.0.2rc1 (linux64 - \"Ubuntu 24.04.4 LTS\")\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU model: AMD EPYC 7763 64-Core Processor, instruction set [SSE2|AVX|AVX2]\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Thread count: 1 physical cores, 2 logical processors, using up to 2 threads\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"WLS license 2443533 - registered to Gurobi GmbH\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Optimize a model with 1 rows, 463 columns and 463 nonzeros (Max)\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Model fingerprint: 0xc5359fbf\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Model has 463 linear objective coefficients\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Model has 1 quadratic constraint\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Coefficient statistics:\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" Matrix range [1e+00, 1e+00]\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" QMatrix range [3e-03, 1e+02]\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" Objective range [4e-02, 6e-01]\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" Bounds range [3e-01, 1e+00]\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" RHS range [1e+00, 1e+00]\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" QRHS range [4e+00, 4e+00]\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Presolve time: 0.03s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Presolved: 463 rows, 926 columns, 107878 nonzeros\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Presolved model has 1 second-order cone constraint\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Ordering time: 0.01s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Barrier statistics:\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" AA' NZ : 1.070e+05\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" Factor NZ : 1.074e+05 (roughly 1 MB of memory)\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" Factor Ops : 3.319e+07 (less than 1 second per iteration)\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" Threads : 1\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" Objective Residual\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Iter Primal Dual Primal Dual Compl Time\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0 1.35263322e+01 1.38517017e+00 5.74e+01 5.93e-02 4.06e-02 0s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 1 1.72294179e+00 1.45034312e+00 5.96e+00 6.53e-08 5.06e-03 0s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 2 5.65980685e-01 7.13794601e-01 1.23e+00 5.20e-09 1.14e-03 0s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 3 2.76492884e-01 4.52651953e-01 1.35e-06 9.08e-10 1.90e-04 0s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 4 3.04811140e-01 3.88375383e-01 1.49e-12 2.76e-10 9.00e-05 0s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 5 3.32848350e-01 3.72022345e-01 1.44e-15 4.16e-16 4.22e-05 0s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 6 3.60987566e-01 3.62373983e-01 3.11e-15 2.78e-17 1.49e-06 0s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 7 3.61633548e-01 3.61707202e-01 2.89e-15 1.11e-16 7.94e-08 0s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 8 3.61665207e-01 3.61667012e-01 8.88e-16 1.58e-15 1.94e-09 0s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 9 3.61665648e-01 3.61666258e-01 6.48e-14 1.36e-13 6.56e-10 0s\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Barrier solved model in 9 iterations and 0.25 seconds (0.59 work units)\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Optimal objective 3.61665648e-01\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n"
]
}
],
"source": [
"m.optimize()"
]
},
{
"cell_type": "markdown",
"id": "76a91575",
"metadata": {},
"source": [
"Display basic solution data for all non-negligible positions; for clarity we've rounded all solution quantities to five digits."
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "7fbbcd56",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-03T11:04:08.103598Z",
"iopub.status.busy": "2026-07-03T11:04:08.103425Z",
"iopub.status.idle": "2026-07-03T11:04:08.120231Z",
"shell.execute_reply": "2026-07-03T11:04:08.119459Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Expected return: 0.361666\n",
"Variance: 3.999995\n",
"Solution time: 0.25 seconds\n",
"\n",
"Total investment: 1.180835\n",
"Risk-free allocation: -0.180836\n",
"Number of positions: 31\n"
]
},
{
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" \n",
" | MKTX | \n",
" 0.018729 | \n",
"
\n",
" \n",
" | AZO | \n",
" 0.018145 | \n",
"
\n",
" \n",
" | MNST | \n",
" 0.015005 | \n",
"
\n",
" \n",
" | CLX | \n",
" 0.014785 | \n",
"
\n",
" \n",
" | META | \n",
" 0.013694 | \n",
"
\n",
" \n",
" | HRL | \n",
" 0.010083 | \n",
"
\n",
" \n",
" | NFLX | \n",
" 0.009696 | \n",
"
\n",
" \n",
" | WMT | \n",
" 0.008265 | \n",
"
\n",
" \n",
" | UNH | \n",
" 0.007742 | \n",
"
\n",
" \n",
" | XEL | \n",
" 0.004691 | \n",
"
\n",
" \n",
" | DXCM | \n",
" 0.004483 | \n",
"
\n",
" \n",
" | CBOE | \n",
" 0.003388 | \n",
"
\n",
" \n",
" | MOH | \n",
" 0.000747 | \n",
"
\n",
" \n",
" | WEC | \n",
" 0.000449 | \n",
"
\n",
" \n",
" | CME | \n",
" 0.000096 | \n",
"
\n",
" \n",
"
\n",
"