10_leverage.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Leverage"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import yfinance as yf\n",
    "pd.options.display.float_format = '{:.2f}'.format\n",
    "import plotly.graph_objects as go\n",
    "from scipy.stats import norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "d:\\bbcx-investments-quarto\\quarto\\venv\\lib\\site-packages\\pandas\\core\\arrays\\datetimes.py:1162: UserWarning: Converting to PeriodArray/Index representation will drop timezone information.\n",
      "  warnings.warn(\n"
     ]
    }
   ],
   "source": [
    "# Pull S&P 500 returns\n",
    "ticker = 'SPY'\n",
    "ret = yf.download(ticker, start='2010-01-01', progress=False)\n",
    "ret.index = ret.index.to_period('D')\n",
    "ret = ret[\"Adj Close\"].resample(\"M\").last()\n",
    "ret = ret.pct_change()\n",
    "ret.name = \"ret\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Leverage\n",
    "leverage = 0.50  # 50% leverage\n",
    "# leverage = 1.0  # 100% leverage\n",
    "\n",
    "# Borrowing rate\n",
    "r_b = 0.02 / 12\n",
    "levered_ret = ret + leverage * (ret - r_b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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   "source": [
    "# Levered returns are riskier\n",
    "fig = go.Figure()\n",
    "trace0= go.Scatter(x=ret.index.to_timestamp(\"M\"), y=ret, mode=\"lines\", name=ticker)\n",
    "trace1= go.Scatter(x=levered_ret.index.to_timestamp(\"M\"), y=levered_ret, mode=\"lines\", name=f'Levered: {leverage:0.1%}')\n",
    "fig.add_trace(trace0)\n",
    "fig.add_trace(trace1)\n",
    "fig.layout.yaxis[\"title\"] = \"Monthly Return\"\n",
    "fig.update_yaxes(tickformat=\".1%\")\n",
    "fig.update_layout(legend=dict(yanchor=\"top\", y =0.99, xanchor=\"left\", x=0.01))\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Unlevered return SD is:\t 0.0427\n",
      "Levered return SD is:\t 0.0641\n"
     ]
    }
   ],
   "source": [
    "print(f'Unlevered return SD is:\\t {ret.std():.4f}')\n",
    "print(f'Levered return SD is:\\t {levered_ret.std():.4f}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Levered return SD:\n",
      "  at leverage of 0.00:\t 0.0427\n",
      "  at leverage of 0.50:\t 0.0641\n",
      "  at leverage of 1.00:\t 0.0854\n",
      "  at leverage of 1.50:\t 0.1068\n",
      "  at leverage of 2.00:\t 0.1281\n"
     ]
    }
   ],
   "source": [
    "# Risk at different leverage\n",
    "print(f'Levered return SD:')\n",
    "for leverage in [0.0, 0.5, 1.0, 1.5, 2.0]:\n",
    "    levered_ret = ret + leverage * (ret - r_b)\n",
    "    print(f'  at leverage of {leverage:.2f}:\\t {levered_ret.std():.4f}')"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Margin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial percent margin is:  66.7%\n"
     ]
    }
   ],
   "source": [
    "# Initial margin\n",
    "MARGIN = 50000\n",
    "EQUITY = 100000\n",
    "INIT_VALUE = MARGIN + EQUITY\n",
    "\n",
    "\n",
    "pct_margin = EQUITY / INIT_VALUE\n",
    "print(f'Initial percent margin is: {pct_margin: .1%}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "New percent margin is:  63.0%\n"
     ]
    }
   ],
   "source": [
    "# Percent margin after price move (ignoring interest exp accumulation)\n",
    "RETURN = -0.10\n",
    "\n",
    "new_asset_value = INIT_VALUE*(1+RETURN)\n",
    "pct_margin = (new_asset_value-MARGIN)/new_asset_value\n",
    "print(f'New percent margin is: {pct_margin: .1%}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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       "      <th></th>\n",
       "      <th>assets</th>\n",
       "      <th>margin</th>\n",
       "      <th>equity</th>\n",
       "      <th>ret</th>\n",
       "      <th>pct_margin</th>\n",
       "      <th>margin call?</th>\n",
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       "      <th>0</th>\n",
       "      <td>150000.00</td>\n",
       "      <td>50000</td>\n",
       "      <td>100000.00</td>\n",
       "      <td>-0.04</td>\n",
       "      <td>0.67</td>\n",
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       "      <td>0.68</td>\n",
       "      <td>0</td>\n",
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       "      <td>50000</td>\n",
       "      <td>44556.34</td>\n",
       "      <td>-0.39</td>\n",
       "      <td>0.47</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>129358.19</td>\n",
       "      <td>50000</td>\n",
       "      <td>79358.19</td>\n",
       "      <td>0.37</td>\n",
       "      <td>0.61</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>88133.40</td>\n",
       "      <td>50000</td>\n",
       "      <td>38133.40</td>\n",
       "      <td>-0.32</td>\n",
       "      <td>0.43</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>76821.53</td>\n",
       "      <td>50000</td>\n",
       "      <td>26821.53</td>\n",
       "      <td>-0.13</td>\n",
       "      <td>0.35</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>87620.81</td>\n",
       "      <td>50000</td>\n",
       "      <td>37620.81</td>\n",
       "      <td>0.14</td>\n",
       "      <td>0.43</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>69303.01</td>\n",
       "      <td>50000</td>\n",
       "      <td>19303.01</td>\n",
       "      <td>-0.21</td>\n",
       "      <td>0.28</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>57411.28</td>\n",
       "      <td>50000</td>\n",
       "      <td>7411.28</td>\n",
       "      <td>-0.17</td>\n",
       "      <td>0.13</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>49270.27</td>\n",
       "      <td>50000</td>\n",
       "      <td>-729.73</td>\n",
       "      <td>-0.14</td>\n",
       "      <td>-0.01</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>56675.14</td>\n",
       "      <td>50000</td>\n",
       "      <td>6675.14</td>\n",
       "      <td>0.15</td>\n",
       "      <td>0.12</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>84924.39</td>\n",
       "      <td>50000</td>\n",
       "      <td>34924.39</td>\n",
       "      <td>0.50</td>\n",
       "      <td>0.41</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>89026.91</td>\n",
       "      <td>50000</td>\n",
       "      <td>39026.91</td>\n",
       "      <td>0.05</td>\n",
       "      <td>0.44</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>72682.89</td>\n",
       "      <td>50000</td>\n",
       "      <td>22682.89</td>\n",
       "      <td>-0.18</td>\n",
       "      <td>0.31</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>83426.27</td>\n",
       "      <td>50000</td>\n",
       "      <td>33426.27</td>\n",
       "      <td>0.15</td>\n",
       "      <td>0.40</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>76816.25</td>\n",
       "      <td>50000</td>\n",
       "      <td>26816.25</td>\n",
       "      <td>-0.08</td>\n",
       "      <td>0.35</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>79594.99</td>\n",
       "      <td>50000</td>\n",
       "      <td>29594.99</td>\n",
       "      <td>0.04</td>\n",
       "      <td>0.37</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>101483.63</td>\n",
       "      <td>50000</td>\n",
       "      <td>51483.63</td>\n",
       "      <td>0.28</td>\n",
       "      <td>0.51</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>90363.65</td>\n",
       "      <td>50000</td>\n",
       "      <td>40363.65</td>\n",
       "      <td>-0.11</td>\n",
       "      <td>0.45</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>94141.30</td>\n",
       "      <td>50000</td>\n",
       "      <td>44141.30</td>\n",
       "      <td>0.04</td>\n",
       "      <td>0.47</td>\n",
       "      <td>0</td>\n",
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       "      assets  margin    equity   ret  pct_margin  margin call?\n",
       "0  150000.00   50000 100000.00 -0.04        0.67             0\n",
       "1  154312.00   50000 104312.00  0.03        0.68             0\n",
       "2   94556.34   50000  44556.34 -0.39        0.47             0\n",
       "3  129358.19   50000  79358.19  0.37        0.61             0\n",
       "4   88133.40   50000  38133.40 -0.32        0.43             0\n",
       "5   76821.53   50000  26821.53 -0.13        0.35             1\n",
       "6   87620.81   50000  37620.81  0.14        0.43             0\n",
       "7   69303.01   50000  19303.01 -0.21        0.28             1\n",
       "8   57411.28   50000   7411.28 -0.17        0.13             1\n",
       "9   49270.27   50000   -729.73 -0.14       -0.01             1\n",
       "10  56675.14   50000   6675.14  0.15        0.12             1\n",
       "11  84924.39   50000  34924.39  0.50        0.41             0\n",
       "12  89026.91   50000  39026.91  0.05        0.44             0\n",
       "13  72682.89   50000  22682.89 -0.18        0.31             1\n",
       "14  83426.27   50000  33426.27  0.15        0.40             0\n",
       "15  76816.25   50000  26816.25 -0.08        0.35             1\n",
       "16  79594.99   50000  29594.99  0.04        0.37             0\n",
       "17 101483.63   50000  51483.63  0.28        0.51             0\n",
       "18  90363.65   50000  40363.65 -0.11        0.45             0\n",
       "19  94141.30   50000  44141.30  0.04        0.47             0"
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     "execution_count": 9,
     "metadata": {},
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   ],
   "source": [
    "# Simulate levered account\n",
    "MAINTENANCE_MARGIN = 0.35\n",
    "T = 20\n",
    "\n",
    "# set up account\n",
    "cols = ['assets', 'margin', 'equity', 'ret', 'pct_margin', 'margin call?']\n",
    "indx = np.arange(0,T)\n",
    "acct = pd.DataFrame(dtype=float,columns=cols, index=indx)\n",
    "\n",
    "# Initialize accounts\n",
    "\n",
    "acct.loc[0,'assets'] = INIT_VALUE\n",
    "acct['margin']   = MARGIN  # assume interest expense will not accumulate in our time horizon\n",
    "acct.loc[0,'equity'] = EQUITY\n",
    "\n",
    "# simulate T returns (NOTE: remove random_state to make this truly random)\n",
    "acct.ret = norm.rvs(loc=0.04, scale = 0.2, size=T, random_state=2)  \n",
    "# acct.ret = norm.rvs(loc=0.04, scale = 0.2, size=T)  \n",
    "\n",
    "# Calculate asset and equity values + percent margin each period\n",
    "for t in acct.index[1:]:\n",
    "    acct.loc[t,'assets'] = acct.loc[t-1,'assets'] * (1+acct.loc[t,'ret'])\n",
    "acct.equity = acct.assets - acct.margin\n",
    "acct.pct_margin = acct.equity/acct.assets\n",
    "\n",
    "# Margin calls occur when pct_margin drops below maintenance margin\n",
    "acct['margin call?'] = np.where(acct.pct_margin < MAINTENANCE_MARGIN, 1, 0)\n",
    "acct\n"
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   "source": [
    "# Plot return time-series\n",
    "fig = go.Figure()\n",
    "trace= go.Scatter(x=acct.index, y=acct.pct_margin, hovertemplate=\"<br>Percent Margin: %{y:.1%}<br><extra></extra>\")\n",
    "fig.add_trace(trace)\n",
    "# some formatting\n",
    "fig.update_traces(marker_line_width=1, marker_line_color='black')\n",
    "fig.layout.xaxis[\"title\"] = \"Time\"\n",
    "fig.layout.yaxis[\"title\"] = \"Percent Margin\"\n",
    "fig.add_hline(y=MAINTENANCE_MARGIN, line_width=4, line_dash=\"dash\", line_color=\"black\")\n",
    "fig.add_annotation(x=3, y=MAINTENANCE_MARGIN*1.1,\n",
    "            text=\"Maintenance margin: \"+f'{MAINTENANCE_MARGIN:.1%}', showarrow=False)\n",
    "fig.show()"
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   "source": [
    "### Repo cash flows"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial cash is: $19,380,266.38\n",
      "Day 1 cash flow is: $19,383,496.43\n",
      "Interest on cash loan is: $3,230.04\n"
     ]
    }
   ],
   "source": [
    "# Example 1\n",
    "MV = 19576026.65\n",
    "REPO_RATE = 0.06\n",
    "HAIRCUT = 0.01\n",
    "\n",
    "# Day 0 cash flow\n",
    "cf0 = MV*(1-HAIRCUT)\n",
    "print(f'Initial cash is: ${cf0:,.2f}')\n",
    "\n",
    "# Day 1 cash flow\n",
    "cf1 = cf0*(1+REPO_RATE/360)\n",
    "print(f'Day 1 cash flow is: ${cf1:,.2f}')\n",
    "print(f'Interest on cash loan is: ${cf1-cf0:,.2f}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Haircut is:\t\t 2.00%\n",
      "Repo rate is:\t\t 3.67%\n",
      "Initial equity:\t\t $20.00\n",
      "Initial leverage:\t 4900%\n",
      "Initial pct margin:\t 2.0%\n"
     ]
    }
   ],
   "source": [
    "# Example 2\n",
    "\n",
    "MV          = 1000\n",
    "LOAN        = 980\n",
    "REPO_PRICE  = 981\n",
    "TERM        = 10   # in days\n",
    "\n",
    "haircut = 1 - LOAN/MV\n",
    "print(f'Haircut is:\\t\\t {haircut:.2%}')\n",
    "\n",
    "repo_rate = (REPO_PRICE/LOAN - 1)*360/TERM\n",
    "print(f'Repo rate is:\\t\\t {repo_rate:.2%}')\n",
    "\n",
    "equity = MV - LOAN\n",
    "print(f'Initial equity:\\t\\t ${equity:.2f}')\n",
    "\n",
    "leverage = LOAN/equity\n",
    "print(f'Initial leverage:\\t {leverage:.0%}')\n",
    "\n",
    "pct_margin = equity/MV\n",
    "print(f'Initial pct margin:\\t {pct_margin:.1%}')"
   ]
  },
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   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "New equity:\t\t $24.00\n",
      "New leverage:\t\t 4088%\n",
      "New pct margin:\t\t 2.4%\n",
      "Levered return:\t\t 20.0%\n",
      "Levered return (check):\t 20.0%\n"
     ]
    }
   ],
   "source": [
    "# Example 2 (cont'd): Suppose some total return on the bond\n",
    "RETURN = 0.005\n",
    "\n",
    "new_mv   = MV*(1+RETURN)\n",
    "new_loan = REPO_PRICE\n",
    "\n",
    "new_equity = new_mv - new_loan\n",
    "print(f'New equity:\\t\\t ${new_equity:.2f}')\n",
    "\n",
    "new_leverage = new_loan/new_equity\n",
    "print(f'New leverage:\\t\\t {new_leverage:.0%}')\n",
    "\n",
    "pct_margin = new_equity/new_mv\n",
    "print(f'New pct margin:\\t\\t {pct_margin:.1%}')\n",
    "\n",
    "levered_ret = new_equity/equity - 1\n",
    "print(f'Levered return:\\t\\t {levered_ret:.1%}')\n",
    "\n",
    "# Levered return using formula\n",
    "levered_ret = RETURN + leverage * (RETURN - repo_rate*TERM/360)\n",
    "print(f'Levered return (check):\\t {levered_ret:.1%}')\n"
   ]
  }
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