14_portfolios_borrowing_frictions.ipynb

{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Optimal portfolios with frictions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import plotly.graph_objects as go\n",
    "from scipy.optimize import minimize\n",
    "from cvxopt import matrix\n",
    "from cvxopt.solvers import qp as Solver, options as SolverOptions"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Inputs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.0225    -0.0002625]\n",
      " [-0.0002625  0.001225 ]]\n"
     ]
    }
   ],
   "source": [
    "##### Inputs\n",
    "# Risk-free rate\n",
    "RF = 0.02\n",
    "RF_BORROW = 0.0325\n",
    "\n",
    "# Expected returns\n",
    "MNS = np.array([0.06, 0.035])\n",
    "\n",
    "# Standard deviations\n",
    "SDS = np.array([0.15, 0.035])\n",
    "\n",
    "# Correlations\n",
    "C  = np.identity(2)\n",
    "C[0, 1] = C[1, 0] = -0.05\n",
    "\n",
    "# Covariance matrix\n",
    "COV = np.diag(SDS) @ C @ np.diag(SDS)\n",
    "print(COV)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tangency function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def tangency(means, cov, rf):\n",
    "    n = len(means)\n",
    "    def f(w):\n",
    "        mn = w @ means\n",
    "        sd = np.sqrt(w @ cov @ w)\n",
    "        return -(mn - rf) / sd\n",
    "    # Initial guess (equal-weighted)\n",
    "    w0 = (1/n)*np.ones(n)\n",
    "    # Constraint: fully-invested portfolio\n",
    "    A = np.ones(n)\n",
    "    b = 1\n",
    "    cons = [{\"type\": \"eq\", \"fun\": lambda x: A @ x - b}]\n",
    "    # No short-sale constraint\n",
    "    bnds = [(None, None) for i in range(n)] \n",
    "    # Optimization\n",
    "    TOL = 10**(-10)\n",
    "    wgts = minimize(f, w0, bounds=bnds, constraints=cons, options={'ftol':TOL}).x\n",
    "    return wgts"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tangency portfolio relative to savings rate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangency (relative to savings rate) portfolio weights: [0.13203413 0.86796587]\n"
     ]
    }
   ],
   "source": [
    "wgts_tang_save = tangency(MNS, COV, RF)\n",
    "print(f'Tangency (relative to savings rate) portfolio weights: {wgts_tang_save}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Portfolio Expected Return:\t\t\t  0.0383\n",
      "Portfolio Standard Deviation:\t\t\t  0.0354\n",
      "Portfolio Sharpe Ratio (using savings rate):\t  0.5166\n"
     ]
    }
   ],
   "source": [
    "# Portfolio expected return\n",
    "port_expret = wgts_tang_save @ MNS\n",
    "print(f'Portfolio Expected Return:\\t\\t\\t {port_expret: ,.4f}')\n",
    "\n",
    "# Portfolio standard deviation\n",
    "port_sd = np.sqrt(wgts_tang_save @ COV @ wgts_tang_save)\n",
    "print(f'Portfolio Standard Deviation:\\t\\t\\t {port_sd: ,.4f}')\n",
    "\n",
    "# Portfolio sharpe ratio\n",
    "port_sr = (port_expret - RF)/port_sd\n",
    "print(f'Portfolio Sharpe Ratio (using savings rate):\\t {port_sr: ,.4f}')"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tangency portfolio relative to borrowing rate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangency (relative to borrowing rate) portfolio weights: [0.35111804 0.64888196]\n"
     ]
    }
   ],
   "source": [
    "wgts_tang_borrow = tangency(MNS, COV, RF_BORROW)\n",
    "print(f'Tangency (relative to borrowing rate) portfolio weights: {wgts_tang_borrow}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Portfolio Expected Return:\t\t\t  0.0438\n",
      "Portfolio Standard Deviation:\t\t\t  0.0563\n",
      "Portfolio Sharpe Ratio (using savings rate):\t  0.2003\n"
     ]
    }
   ],
   "source": [
    "# Portfolio expected return\n",
    "port_expret = wgts_tang_borrow @ MNS\n",
    "print(f'Portfolio Expected Return:\\t\\t\\t {port_expret: ,.4f}')\n",
    "\n",
    "# Portfolio standard deviation\n",
    "port_sd = np.sqrt(wgts_tang_borrow @ COV @ wgts_tang_borrow)\n",
    "print(f'Portfolio Standard Deviation:\\t\\t\\t {port_sd: ,.4f}')\n",
    "\n",
    "# Portfolio sharpe ratio\n",
    "port_sr = (port_expret - RF_BORROW)/port_sd\n",
    "print(f'Portfolio Sharpe Ratio (using savings rate):\\t {port_sr: ,.4f}')"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Capital allocation and risk aversion"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Method #1: Find risk-aversions that represent each tangency portfolio\n",
    "- Method #2: Directly maximize MV utility using all assets"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Method 1\n",
    "Set $w^*=1$ in the expression $w^* = \\frac{E[r_p-r_f]}{A\\cdot \\text{var}(r_p)}$ and solve for $A$:\n",
    "\n",
    "$$A = \\frac{E[r_p-r_f]}{\\text{var}(r_p)}.$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Risk aversion upper threshold:\t  14.6\n",
      "Risk aversions above this level will do some saving\n"
     ]
    }
   ],
   "source": [
    "lo_expret = wgts_tang_save @ MNS\n",
    "lo_var    = wgts_tang_save @ COV @ wgts_tang_save\n",
    "raver_upper_threshold = (lo_expret - RF) / lo_var\n",
    "print(f'Risk aversion upper threshold:\\t {raver_upper_threshold: ,.1f}')\n",
    "print(f'Risk aversions above this level will do some saving')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Risk aversion lower threshold:\t  3.6\n",
      "Risk aversions below this level will invest in levered positions\n",
      "Risk aversions between  3.6 and  14.6 will invest along the risky asset frontier\n"
     ]
    }
   ],
   "source": [
    "hi_expret = wgts_tang_borrow @ MNS\n",
    "hi_var    = wgts_tang_borrow @ COV @ wgts_tang_borrow\n",
    "raver_lower_threshold = (hi_expret - RF_BORROW) / hi_var\n",
    "print(f'Risk aversion lower threshold:\\t {raver_lower_threshold: ,.1f}')\n",
    "print(f'Risk aversions below this level will invest in levered positions')\n",
    "print(f'Risk aversions between {raver_lower_threshold: ,.1f} and {raver_upper_threshold: ,.1f} will invest along the risky asset frontier')\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For risk aversions between the thresholds, we solve for the frontier portfolio that maximizes mean-variance utility at the given risk aversion.  We make use of the fact that any frontier portfolio can be expressed as a portfolio of two frontier portfolios.  So we consider the portfolio that invests $a$ in the efficient low-risk portfolio and $1-a$ in the efficient high-risk portfolio and maximizes utility:\n",
    "$$ \\underset{a}\\max E[r_p] - 0.5\\cdot A \\cdot \\text{var}[r_p]\\,. $$  \n",
    "with $r_p = a \\cdot r_{\\text{LO}} + (1-a) \\cdot r_{\\text{HI}}$.  In order to calculate the portfolio variance, we need to calculate the covariance between the two frontier portfolios.  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# What is the covariance between the efficient low and high risk portfolios?\n",
    "cov_hilo = wgts_tang_save @ COV @ wgts_tang_borrow"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def opt_allocation(means, cov, rf_save, rf_borrow, risk_aversion):\n",
    "    # Calculate two tangencies\n",
    "    wgts_tang_save = tangency(means, cov, rf_save)\n",
    "    wgts_tang_borrow = tangency(means, cov, rf_borrow)\n",
    "\n",
    "    # Find risk aversion thresholds\n",
    "    lo_expret = wgts_tang_save @ means\n",
    "    lo_var    = wgts_tang_save @ cov @ wgts_tang_save\n",
    "    raver_upper_threshold = (lo_expret - rf_save) / lo_var  \n",
    "\n",
    "    hi_expret = wgts_tang_borrow @ means\n",
    "    hi_var    = wgts_tang_borrow @ cov @ wgts_tang_borrow\n",
    "    raver_lower_threshold = (hi_expret - rf_borrow) / hi_var    \n",
    "    \n",
    "    # Find optimal allocation based on risk aversion\n",
    "    wgts = np.zeros(4)  # 1st element is risk-free saving, 2nd element is risk-free borrowing, 3nd is low-risk tangency, 4th\n",
    "    if risk_aversion >= raver_upper_threshold:\n",
    "        # Savings CAL\n",
    "        wgts[2] = (lo_expret - rf_save) / (risk_aversion * lo_var)\n",
    "        wgts[0] = 1-wgts[2]\n",
    "        wgts[1] = 0.0\n",
    "        wgts[3] = 0.0\n",
    "    elif risk_aversion >= raver_lower_threshold:\n",
    "        # Risky asset frontier\n",
    "        cov_hilo = wgts_tang_save @ cov @ wgts_tang_borrow\n",
    "        wgts[0] = 0.0\n",
    "        wgts[1] = 0.0\n",
    "        wgts[2] = (lo_expret-hi_expret - risk_aversion * (cov_hilo  - hi_var))/(risk_aversion*(lo_var + hi_var - 2*cov_hilo))\n",
    "        wgts[3] = 1-wgts[2]\n",
    "    else:\n",
    "        # Borrowing CAL\n",
    "        wgts[0] = 0.0  \n",
    "        wgts[2] = 0.0  \n",
    "        wgts[3] = (hi_expret - rf_borrow) / (risk_aversion * hi_var)\n",
    "        wgts[1] = 1-wgts[3]\n",
    "    return wgts\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For risk aversion of  2.0, invest:\n",
      "\tRisk-free saving\t 0.0%\n",
      "\tRisk-free borrowing\t-77.9%\n",
      "\tEfficient low-risk\t 0.0%\n",
      "\tEfficient high-risk\t 177.9%\n"
     ]
    }
   ],
   "source": [
    "RAVER = 2\n",
    "wgts = opt_allocation(MNS, COV, RF, RF_BORROW, RAVER)\n",
    "print(f'For risk aversion of {RAVER: .1f}, invest:')\n",
    "print(f'\\tRisk-free saving\\t{wgts[0]: ,.1%}')\n",
    "print(f'\\tRisk-free borrowing\\t{wgts[1]: ,.1%}')\n",
    "print(f'\\tEfficient low-risk\\t{wgts[2]: ,.1%}')\n",
    "print(f'\\tEfficient high-risk\\t{wgts[3]: ,.1%}')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.02      , 0.0325    , 0.03830085, 0.04377795])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Expected returns for risk-free savings + borrowing + low-risk efficient + high-risk efficient\n",
    "new_mns = np.array([RF, RF_BORROW, lo_expret, hi_expret])\n",
    "new_mns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.        , 0.        , 0.        , 0.        ],\n",
       "       [0.        , 0.        , 0.        , 0.        ],\n",
       "       [0.        , 0.        , 0.00125495, 0.00163053],\n",
       "       [0.        , 0.        , 0.00163053, 0.00317006]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# COV for risk-free savings + borrowing + low-risk efficient + high-risk efficient\n",
    "new_cov = np.zeros((4,4))\n",
    "new_cov[2,2] = lo_var\n",
    "new_cov[3,3] = hi_var\n",
    "new_cov[2,3] = new_cov[3,2] = cov_hilo\n",
    "new_cov"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For risk aversion of  2.0, Portfolio Expected Return:\t\t  0.0526\n",
      "For risk aversion of  2.0, Portfolio Standard Deviation:\t  0.1002\n"
     ]
    }
   ],
   "source": [
    "# Portfolio expected return\n",
    "optport_expret = wgts @ new_mns\n",
    "print(f'For risk aversion of {RAVER: .1f}, Portfolio Expected Return:\\t\\t {optport_expret: ,.4f}')\n",
    "\n",
    "# Portfolio standard deviation\n",
    "optport_sd     = np.sqrt(wgts @ new_cov @ wgts)\n",
    "print(f'For risk aversion of {RAVER: .1f}, Portfolio Standard Deviation:\\t {optport_sd: ,.4f}')"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Method #2: Directly maximize MV utility using all assets\n",
    "\n",
    "The optimal portfolio for investor with risk aversion $A$:\n",
    "$$ \\underset{w_{\\text{saving}},w_{\\text{borrow}},w_1,w_2,\\dots,w_N}{\\text{max}} E[r_p] - 0.5 \\cdot A \\cdot \\text{var}[r_p] $$ \n",
    "subject to the constraints $w_{\\text{saving}} + w_{\\text{borrow}} + \\sum_i w_i=1$, $w_{\\text{saving}} \\ge 0$, and $w_{\\text{borrow}} \\le 0$. \n",
    "\n",
    "We will map this problem into the `cvxopt.solvers.qp` function's general form:\n",
    "\\begin{align*}\n",
    "    \\underset{w}{\\text{min  }}& \\frac{1}{2} w' Q w + p'w \\\\\n",
    "     \\text{subject to  } & Gw \\le h \\\\\n",
    "                        & Aw = b \\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def opt_allocation2(means, cov, rf_save, rf_borrow, risk_aversion):\n",
    "    n=len(means)\n",
    "    Q = np.zeros((n + 2, n + 2))\n",
    "    Q[2:, 2:] = risk_aversion * cov\n",
    "    Q = matrix(Q, tc=\"d\")\n",
    "    p = np.array([-rf_save, -rf_borrow] + list(-means))\n",
    "    p = matrix(p, (n + 2, 1), tc=\"d\")\n",
    "    # Constraint: saving weight positive, borrowing weight negative\n",
    "    G = np.zeros((2, n + 2))\n",
    "    G[0, 0] = -1\n",
    "    G[1, 1] = 1\n",
    "    G = matrix(G, (2, n+2), tc=\"d\")\n",
    "    h = matrix([0, 0], (2, 1), tc=\"d\")\n",
    "    # Constraint: fully-invested portfolio\n",
    "    A = matrix(np.ones(n+2), (1, n+2), tc=\"d\")\n",
    "    b = matrix([1], (1, 1), tc=\"d\")\n",
    "    sol = Solver(Q, p, G, h, A, b)\n",
    "    if sol[\"status\"] == \"optimal\":\n",
    "        wgts_optimal = np.array(sol[\"x\"]).flatten()\n",
    "    else:\n",
    "        wgts_optimal = None\n",
    "    return wgts_optimal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     pcost       dcost       gap    pres   dres\n",
      " 0: -4.2049e-02 -2.9322e-02  2e+00  1e+00  1e+00\n",
      " 1: -2.9913e-02 -6.4796e-02  4e-02  2e-02  2e-02\n",
      " 2: -3.9174e-02 -4.2633e-02  3e-03  6e-05  6e-05\n",
      " 3: -4.2497e-02 -4.2538e-02  4e-05  7e-07  7e-07\n",
      " 4: -4.2530e-02 -4.2531e-02  4e-07  7e-09  7e-09\n",
      " 5: -4.2531e-02 -4.2531e-02  4e-09  7e-11  7e-11\n",
      "Optimal solution found.\n",
      "For risk aversion of  2.0, invest:\n",
      "\tRisk-free saving\t\t 0.0%\n",
      "\tRisk-free borrowing\t\t-77.9%\n",
      "\tTotal weight in risky assets:\t 177.9%\n",
      "Weights in each risky asset of [0.6246 1.1542]\n"
     ]
    }
   ],
   "source": [
    "wgts = opt_allocation2(MNS, COV, RF, RF_BORROW, RAVER)\n",
    "print(f'For risk aversion of {RAVER: .1f}, invest:')\n",
    "print(f'\\tRisk-free saving\\t\\t{wgts[0]: ,.1%}')\n",
    "print(f'\\tRisk-free borrowing\\t\\t{wgts[1]: ,.1%}')\n",
    "print(f'\\tTotal weight in risky assets:\\t{np.sum(wgts[2:]): .1%}')\n",
    "print(f'Weights in each risky asset of {wgts[2:].round(4)}')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For risk aversion of  2.0, Portfolio Expected Return:\t\t  0.0526\n",
      "For risk aversion of  2.0, Portfolio Standard Deviation:\t  0.1002\n"
     ]
    }
   ],
   "source": [
    "# Augment undelying asset means and covariances\n",
    "augmented_mns = np.append([RF, RF_BORROW], MNS)\n",
    "n = len(MNS)\n",
    "augmented_cov = np.zeros((n + 2, n + 2))\n",
    "augmented_cov[2:,2:] = COV\n",
    "\n",
    "# Portfolio expected return\n",
    "optport_expret = wgts @ augmented_mns\n",
    "print(f'For risk aversion of {RAVER: .1f}, Portfolio Expected Return:\\t\\t {optport_expret: ,.4f}')\n",
    "\n",
    "# Portfolio standard deviation\n",
    "optport_sd     = np.sqrt(wgts @ augmented_cov @ wgts)\n",
    "print(f'For risk aversion of {RAVER: .1f}, Portfolio Standard Deviation:\\t {optport_sd: ,.4f}')"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Optimal allocation dataframe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>risk_aversion</th>\n",
       "      <th>saving</th>\n",
       "      <th>borrowing</th>\n",
       "      <th>low_risk</th>\n",
       "      <th>high_risk</th>\n",
       "      <th>port_expret</th>\n",
       "      <th>port_sd</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.25</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-13.230593</td>\n",
       "      <td>0.0</td>\n",
       "      <td>14.230593</td>\n",
       "      <td>0.192992</td>\n",
       "      <td>0.801229</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.50</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-6.115297</td>\n",
       "      <td>0.0</td>\n",
       "      <td>7.115297</td>\n",
       "      <td>0.112746</td>\n",
       "      <td>0.400614</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.75</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-3.743531</td>\n",
       "      <td>0.0</td>\n",
       "      <td>4.743531</td>\n",
       "      <td>0.085997</td>\n",
       "      <td>0.267076</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1.00</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-2.557648</td>\n",
       "      <td>0.0</td>\n",
       "      <td>3.557648</td>\n",
       "      <td>0.072623</td>\n",
       "      <td>0.200307</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1.25</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-1.846119</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.846119</td>\n",
       "      <td>0.064598</td>\n",
       "      <td>0.160246</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   risk_aversion  saving  borrowing  low_risk  high_risk  port_expret  \\\n",
       "0           0.25     0.0 -13.230593       0.0  14.230593     0.192992   \n",
       "1           0.50     0.0  -6.115297       0.0   7.115297     0.112746   \n",
       "2           0.75     0.0  -3.743531       0.0   4.743531     0.085997   \n",
       "3           1.00     0.0  -2.557648       0.0   3.557648     0.072623   \n",
       "4           1.25     0.0  -1.846119       0.0   2.846119     0.064598   \n",
       "\n",
       "    port_sd  \n",
       "0  0.801229  \n",
       "1  0.400614  \n",
       "2  0.267076  \n",
       "3  0.200307  \n",
       "4  0.160246  "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RAVER_LEVELS = np.arange(0.25,30.25,0.25)\n",
    "\n",
    "# Dataframe\n",
    "df_opt = pd.DataFrame(dtype='float',columns=['risk_aversion','saving','borrowing','low_risk','high_risk','port_expret', 'port_sd'], index=np.arange(len(RAVER_LEVELS)))\n",
    "df_opt.risk_aversion = RAVER_LEVELS\n",
    "for i in df_opt.index:\n",
    "    wgts = opt_allocation(MNS, COV, RF, RF_BORROW, df_opt.loc[i,'risk_aversion'])\n",
    "    df_opt.loc[i,['saving','borrowing','low_risk','high_risk']]=wgts\n",
    "    df_opt.loc[i,'port_expret']= wgts @ new_mns\n",
    "    df_opt.loc[i,'port_sd']    = np.sqrt(wgts @ new_cov @ wgts)\n",
    "df_opt.head()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The full picture\n",
    "Note we will now also calculate the sharpe ratio for each frontier portfolio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Calculate frontier portfolios (from last time)\n",
    "def frontier(means, cov, target):\n",
    "    n = len(means)\n",
    "    Q = matrix(cov, tc=\"d\")\n",
    "    p = matrix(np.zeros(n), (n, 1), tc=\"d\")\n",
    "    # Constraint: short-sales allowed\n",
    "    G = matrix(np.zeros((n,n)), tc=\"d\")\n",
    "    h = matrix(np.zeros(n), (n, 1), tc=\"d\")\n",
    "    # Fully-invested constraint + E[r] = target\n",
    "    A = matrix(np.vstack((np.ones(n), means)), (2, n), tc=\"d\")\n",
    "    b = matrix([1, target], (2, 1), tc=\"d\")\n",
    "    sol = Solver(Q, p, G, h, A, b)\n",
    "    wgts = np.array(sol[\"x\"]).flatten() if sol[\"status\"] == \"optimal\" else np.array(n * [np.nan])\n",
    "    return wgts\n",
    "SolverOptions['show_progress'] = False\n",
    "\n",
    "NUM_TARGETS = 30\n",
    "df = pd.DataFrame(dtype='float',columns=['target_expret','w1','w2','port_expret','port_sd','sharpe'], index=np.arange(NUM_TARGETS))\n",
    "df.target_expret = np.linspace(MNS.min(), MNS.max(),NUM_TARGETS)\n",
    "for i in df.index:\n",
    "    wgts = frontier(MNS, COV, df.loc[i,'target_expret'])\n",
    "    df.loc[i,['w1','w2']] = wgts\n",
    "    df.loc[i,'port_expret'] = wgts @ MNS\n",
    "    df.loc[i,'port_sd'] = np.sqrt(wgts @ COV @ wgts)\n",
    "df.sharpe = (df.port_expret - RF)/df.port_sd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
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   "source": [
    "fig = go.Figure()\n",
    "\n",
    "# Plot frontier\n",
    "string =  \"Stock: %{customdata[0]:.1%}<br>\"\n",
    "string += \"Bond: %{customdata[1]:.1%}<br>\"\n",
    "string += \"<extra></extra>\"\n",
    "trace= go.Scatter(x=df.port_sd, y=df.port_expret,mode=\"lines\",\n",
    "    customdata=df[['w1','w2']],hovertemplate=string, name='Frontier')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Plot underlying assets\n",
    "trace1= go.Scatter(x=[SDS[0]], y=[MNS[0]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Stock')\n",
    "fig.add_trace(trace1)\n",
    "trace2= go.Scatter(x=[SDS[1]], y=[MNS[1]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Bond')\n",
    "fig.add_trace(trace2)\n",
    "\n",
    "# Plot low-risk tangency\n",
    "trace= go.Scatter(x=[np.sqrt(lo_var)], y=[lo_expret] ,\n",
    "    mode=\"markers\", marker=dict(size=10, color=\"black\"),\n",
    "    customdata=wgts_tang_save.reshape(1,2),hovertemplate=string, name='Efficient Low-Risk Tangency')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Plot high-risk tangency\n",
    "trace= go.Scatter(x=[np.sqrt(hi_var)], y=[hi_expret] ,\n",
    "    mode=\"markers\", marker=dict(size=10, color=\"black\"),\n",
    "    customdata=wgts_tang_borrow.reshape(1,2),hovertemplate=string, name='Efficient High-Risk Tangency')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Plot optimal allocation\n",
    "string =  \"Saving: %{customdata[0]:.1%}<br>\"\n",
    "string += \"Borrowing: %{customdata[1]:.1%}<br>\"\n",
    "string += \"Efficient Low-Risk: %{customdata[2]:.1%}<br>\"\n",
    "string += \"Efficient High-Risk: %{customdata[3]:.1%}<br>\"\n",
    "string += \"Optimal if risk aversion is: %{customdata[4]:.1f}<br>\"\n",
    "string += \"<extra></extra>\"\n",
    "trace= go.Scatter(x=df_opt.port_sd, y=df_opt.port_expret,mode=\"lines\",marker=dict(color=\"black\"),\n",
    "    customdata=df_opt[['saving','borrowing','low_risk','high_risk','risk_aversion']],hovertemplate=string, name='Optimal allocation')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Formatting\n",
    "fig.layout.yaxis[\"title\"] = \"Expected Return\"\n",
    "fig.layout.xaxis[\"title\"] = \"Standard Deviation\"\n",
    "fig.update_yaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(range=[0.5 * df[\"port_sd\"].min(), 1.25 * df[\"port_sd\"].max()])\n",
    "fig.update_yaxes(range=[0.7 * df[\"port_expret\"].min(), 1.25 * df[\"port_expret\"].max()])\n",
    "fig.update_layout(legend=dict(yanchor=\"top\", y=0.99, xanchor=\"left\", x=0.01))\n",
    "fig.show()"
   ]
  }
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