Hedgefundie_Value_at_risk.ipynb

hedgefundie_value_at_risk.ipynb

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    "colab": {
      "name": "Hedgefundie_Value_at_risk.ipynb",
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      "name": "python3",
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    {
      "cell_type": "markdown",
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        "id": "view-in-github",
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      "source": [
        "<a href=\"https://colab.research.google.com/gist/spiyer99/e5c62362e564ce95437060ebfe15c06c/hedgefundie_value_at_risk.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BSfacWOFGUS_"
      },
      "source": [
        "# Attempt 2: Get daily returns from backtrader"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "9DBffc0GGqsN",
        "outputId": "952f7f1a-fbca-4e12-cd49-2d71a07eeaf8"
      },
      "source": [
        "!pip install backtrader yfinance arch"
      ],
      "execution_count": 25,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Requirement already satisfied: backtrader in /usr/local/lib/python3.7/dist-packages (1.9.76.123)\n",
            "Requirement already satisfied: yfinance in /usr/local/lib/python3.7/dist-packages (0.1.63)\n",
            "Requirement already satisfied: arch in /usr/local/lib/python3.7/dist-packages (5.0.1)\n",
            "Requirement already satisfied: numpy>=1.15 in /usr/local/lib/python3.7/dist-packages (from yfinance) (1.19.5)\n",
            "Requirement already satisfied: lxml>=4.5.1 in /usr/local/lib/python3.7/dist-packages (from yfinance) (4.6.3)\n",
            "Requirement already satisfied: multitasking>=0.0.7 in /usr/local/lib/python3.7/dist-packages (from yfinance) (0.0.9)\n",
            "Requirement already satisfied: requests>=2.20 in /usr/local/lib/python3.7/dist-packages (from yfinance) (2.23.0)\n",
            "Requirement already satisfied: pandas>=0.24 in /usr/local/lib/python3.7/dist-packages (from yfinance) (1.1.5)\n",
            "Requirement already satisfied: pytz>=2017.2 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.24->yfinance) (2018.9)\n",
            "Requirement already satisfied: python-dateutil>=2.7.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.24->yfinance) (2.8.1)\n",
            "Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.7.3->pandas>=0.24->yfinance) (1.15.0)\n",
            "Requirement already satisfied: idna<3,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests>=2.20->yfinance) (2.10)\n",
            "Requirement already satisfied: chardet<4,>=3.0.2 in /usr/local/lib/python3.7/dist-packages (from requests>=2.20->yfinance) (3.0.4)\n",
            "Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests>=2.20->yfinance) (2021.5.30)\n",
            "Requirement already satisfied: urllib3!=1.25.0,!=1.25.1,<1.26,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests>=2.20->yfinance) (1.24.3)\n",
            "Requirement already satisfied: statsmodels>=0.11 in /usr/local/lib/python3.7/dist-packages (from arch) (0.12.2)\n",
            "Requirement already satisfied: scipy>=1.3 in /usr/local/lib/python3.7/dist-packages (from arch) (1.4.1)\n",
            "Requirement already satisfied: property-cached>=1.6.4 in /usr/local/lib/python3.7/dist-packages (from arch) (1.6.4)\n",
            "Requirement already satisfied: patsy>=0.5 in /usr/local/lib/python3.7/dist-packages (from statsmodels>=0.11->arch) (0.5.1)\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hiXXnfV8GmFd"
      },
      "source": [
        "import backtrader as bt\n",
        "from pandas_datareader import data as pdr\n",
        "from pathlib import Path\n",
        "import os\n",
        "from arch import arch_model\n",
        "import numpy as np\n",
        "from random import gauss, seed\n",
        "from scipy.stats import binom\n",
        "import pandas as pd\n",
        "import datetime\n",
        "import yfinance as yf\n",
        "\n",
        "yf.pdr_override()"
      ],
      "execution_count": 26,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "e5HKTM_sGZah"
      },
      "source": [
        "def sim_leverage(proxy, leverage=1, expense_ratio = 0.0, initial_value=1.0):\n",
        "    pct_change = proxy.pct_change(1)\n",
        "    pct_change = (pct_change - expense_ratio / 252) * leverage\n",
        "    sim = (1 + pct_change).cumprod() * initial_value\n",
        "    sim[0] = initial_value\n",
        "    return sim"
      ],
      "execution_count": 27,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "eeACbOfeGbtK",
        "outputId": "6e0fb02f-ff72-4439-a861-d6693e480f58"
      },
      "source": [
        "start = datetime.datetime(1986, 5, 19)\n",
        "end = datetime.datetime(2021, 6, 12)\n",
        "\n",
        "vfinx = pdr.get_data_yahoo(\"VFINX\", start, end)[\"Adj Close\"]\n",
        "qqq = pdr.get_data_yahoo(\"QQQ\", datetime.datetime(2000, 3, 1), end)[\"Adj Close\"]\n",
        "vustx = pdr.get_data_yahoo(\"VUSTX\", start, end)[\"Adj Close\"]\n",
        "psq = pdr.get_data_yahoo(\"PSQ\", start, end)[\"Adj Close\"]\n",
        "\n",
        "upro_sim = sim_leverage(vfinx, leverage=3.0, expense_ratio=0.0092).to_frame(\"close\")\n",
        "tqqq_sim = sim_leverage(qqq, leverage=3.0, expense_ratio=0.0092).to_frame(\"close\")\n",
        "tmf_sim = sim_leverage(vustx, leverage=3.0, expense_ratio=0.0109).to_frame(\"close\")"
      ],
      "execution_count": 28,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[*********************100%***********************]  1 of 1 completed\n",
            "[*********************100%***********************]  1 of 1 completed\n",
            "[*********************100%***********************]  1 of 1 completed\n",
            "[*********************100%***********************]  1 of 1 completed\n"
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          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5lK-GYF1GfaL"
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      "source": [
        "for column in [\"open\", \"high\", \"low\"]:\n",
        "    upro_sim[column] = upro_sim[\"close\"]\n",
        "    tmf_sim[column] = tmf_sim[\"close\"]\n",
        "    tqqq_sim[column] = tqqq_sim[\"close\"]\n",
        "    \n",
        "upro_sim[\"volume\"] = 0\n",
        "tqqq_sim[\"volume\"] = 0\n",
        "tmf_sim[\"volume\"] = 0\n",
        "\n",
        "upro_sim = bt.feeds.PandasData(dataname=upro_sim)\n",
        "tmf_sim = bt.feeds.PandasData(dataname=tmf_sim)\n",
        "tqqq_sim = bt.feeds.PandasData(dataname=tqqq_sim) "
      ],
      "execution_count": 29,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Jz8J-1YBGhaz"
      },
      "source": [
        "def backtest(datas, strategy, plot=False, **kwargs):\n",
        "    cerebro = bt.Cerebro()\n",
        "    for data in datas:\n",
        "        cerebro.adddata(data)\n",
        "    cerebro.addanalyzer(bt.analyzers.SharpeRatio, riskfreerate=0.0)\n",
        "    cerebro.addanalyzer(bt.analyzers.Returns)\n",
        "    cerebro.addanalyzer(bt.analyzers.DrawDown)\n",
        "    if Path('hedgefundie_strategy_daily_returns.csv').exists(): os.remove('hedgefundie_strategy_daily_returns.csv')\n",
        "    cerebro.addwriter(bt.WriterFile, csv = True, out='hedgefundie_strategy_daily_returns.csv')\n",
        "    cerebro.addstrategy(strategy, **kwargs)\n",
        "    results = cerebro.run()\n",
        "    if plot:\n",
        "        cerebro.plot()\n",
        "    return (results[0].analyzers.drawdown.get_analysis()['max']['drawdown'],\n",
        "            results[0].analyzers.returns.get_analysis()['rnorm100'],\n",
        "            results[0].analyzers.sharperatio.get_analysis()['sharperatio'])"
      ],
      "execution_count": 30,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "b_6h1ydhGovR"
      },
      "source": [
        "class AssetAllocation(bt.Strategy):\n",
        "    params = (\n",
        "        ('equity',0.6),\n",
        "    )\n",
        "    def __init__(self):\n",
        "        self.UPRO = self.datas[0]\n",
        "        self.TMF = self.datas[1]\n",
        "        self.counter = 0\n",
        "        \n",
        "    def next(self):\n",
        "        if  self.counter % 252 == 0: # annual rebalance\n",
        "            self.order_target_percent(self.UPRO, target=self.params.equity)\n",
        "            self.order_target_percent(self.TMF, target=(1 - self.params.equity))\n",
        "        self.counter += 1"
      ],
      "execution_count": 31,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "YB7Ge_xrGzVb",
        "outputId": "634358bc-2723-4c2f-ad6b-7fc30edc7b32"
      },
      "source": [
        "dd, cagr, sharpe = backtest([upro_sim, tmf_sim], AssetAllocation, plot=True, equity=0.55)\n",
        "print(f\"Max Drawdown: {dd:.2f}%\\nCAGR: {cagr:.2f}%\\nSharpe: {sharpe:.3f}\")"
      ],
      "execution_count": 32,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Max Drawdown: 54.66%\n",
            "CAGR: 16.27%\n",
            "Sharpe: 0.592\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "04u598UWslkR"
      },
      "source": [
        "def fix_csv(name = 'hedgefundie_strategy_daily_returns.csv'):\n",
        "  # fix the fix exported by cerebro\n",
        "  with open(name,'r') as f:\n",
        "    new_name = Path(name).stem + '_fixed.csv'\n",
        "    if Path(new_name).exists(): os.remove(new_name)\n",
        "    with open(Path(name).stem + '_fixed.csv','w') as f1:\n",
        "        next(f) # skip header line\n",
        "        for line in f:\n",
        "            f1.write(line)\n",
        "fix_csv()"
      ],
      "execution_count": 33,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 0
        },
        "id": "hF935hNrqkN1",
        "outputId": "5b97ab15-26bf-4104-b1f9-22eabcc0b298"
      },
      "source": [
        "backtest = pd.read_csv('hedgefundie_strategy_daily_returns_fixed.csv', parse_dates = True)\n",
        "cols = ['datetime', 'open', 'high', 'low', 'close', 'volume', 'value']\n",
        "backtest = backtest[backtest['datetime'].notnull()][cols].reset_index(drop=True)\n",
        "backtest.head()"
      ],
      "execution_count": 34,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
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              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>datetime</th>\n",
              "      <th>open</th>\n",
              "      <th>high</th>\n",
              "      <th>low</th>\n",
              "      <th>close</th>\n",
              "      <th>volume</th>\n",
              "      <th>value</th>\n",
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              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>1986-05-19 00:00:00</td>\n",
              "      <td>1.000000</td>\n",
              "      <td>1.000000</td>\n",
              "      <td>1.000000</td>\n",
              "      <td>1.000000</td>\n",
              "      <td>0.0</td>\n",
              "      <td>10000.000000</td>\n",
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              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>1986-05-20 00:00:00</td>\n",
              "      <td>1.037538</td>\n",
              "      <td>1.037538</td>\n",
              "      <td>1.037538</td>\n",
              "      <td>1.037538</td>\n",
              "      <td>0.0</td>\n",
              "      <td>10000.000000</td>\n",
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              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>1986-05-21 00:00:00</td>\n",
              "      <td>1.030191</td>\n",
              "      <td>1.030191</td>\n",
              "      <td>1.030191</td>\n",
              "      <td>1.030191</td>\n",
              "      <td>0.0</td>\n",
              "      <td>9959.593644</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>1986-05-22 00:00:00</td>\n",
              "      <td>1.091267</td>\n",
              "      <td>1.091267</td>\n",
              "      <td>1.091267</td>\n",
              "      <td>1.091267</td>\n",
              "      <td>0.0</td>\n",
              "      <td>10295.507652</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>1986-05-23 00:00:00</td>\n",
              "      <td>1.107348</td>\n",
              "      <td>1.107348</td>\n",
              "      <td>1.107348</td>\n",
              "      <td>1.107348</td>\n",
              "      <td>0.0</td>\n",
              "      <td>10383.958368</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "              datetime      open      high  ...     close  volume         value\n",
              "0  1986-05-19 00:00:00  1.000000  1.000000  ...  1.000000     0.0  10000.000000\n",
              "1  1986-05-20 00:00:00  1.037538  1.037538  ...  1.037538     0.0  10000.000000\n",
              "2  1986-05-21 00:00:00  1.030191  1.030191  ...  1.030191     0.0   9959.593644\n",
              "3  1986-05-22 00:00:00  1.091267  1.091267  ...  1.091267     0.0  10295.507652\n",
              "4  1986-05-23 00:00:00  1.107348  1.107348  ...  1.107348     0.0  10383.958368\n",
              "\n",
              "[5 rows x 7 columns]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 34
        }
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        "outputId": "bfcda8aa-eace-4e22-9939-9d393b65f56f"
      },
      "source": [
        "backtest.plot(figsize=(10,10))"
      ],
      "execution_count": 35,
      "outputs": [
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              "\tthis.context.msBackingStorePixelRatio ||\n",
              "\tthis.context.oBackingStorePixelRatio ||\n",
              "\tthis.context.backingStorePixelRatio || 1;\n",
              "\n",
              "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
              "\n",
              "    var rubberband = $('<canvas/>');\n",
              "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
              "\n",
              "    var pass_mouse_events = true;\n",
              "\n",
              "    canvas_div.resizable({\n",
              "        start: function(event, ui) {\n",
              "            pass_mouse_events = false;\n",
              "        },\n",
              "        resize: function(event, ui) {\n",
              "            fig.request_resize(ui.size.width, ui.size.height);\n",
              "        },\n",
              "        stop: function(event, ui) {\n",
              "            pass_mouse_events = true;\n",
              "            fig.request_resize(ui.size.width, ui.size.height);\n",
              "        },\n",
              "    });\n",
              "\n",
              "    function mouse_event_fn(event) {\n",
              "        if (pass_mouse_events)\n",
              "            return fig.mouse_event(event, event['data']);\n",
              "    }\n",
              "\n",
              "    rubberband.mousedown('button_press', mouse_event_fn);\n",
              "    rubberband.mouseup('button_release', mouse_event_fn);\n",
              "    // Throttle sequential mouse events to 1 every 20ms.\n",
              "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
              "\n",
              "    rubberband.mouseenter('figure_enter', mouse_event_fn);\n",
              "    rubberband.mouseleave('figure_leave', mouse_event_fn);\n",
              "\n",
              "    canvas_div.on(\"wheel\", function (event) {\n",
              "        event = event.originalEvent;\n",
              "        event['data'] = 'scroll'\n",
              "        if (event.deltaY < 0) {\n",
              "            event.step = 1;\n",
              "        } else {\n",
              "            event.step = -1;\n",
              "        }\n",
              "        mouse_event_fn(event);\n",
              "    });\n",
              "\n",
              "    canvas_div.append(canvas);\n",
              "    canvas_div.append(rubberband);\n",
              "\n",
              "    this.rubberband = rubberband;\n",
              "    this.rubberband_canvas = rubberband[0];\n",
              "    this.rubberband_context = rubberband[0].getContext(\"2d\");\n",
              "    this.rubberband_context.strokeStyle = \"#000000\";\n",
              "\n",
              "    this._resize_canvas = function(width, height) {\n",
              "        // Keep the size of the canvas, canvas container, and rubber band\n",
              "        // canvas in synch.\n",
              "        canvas_div.css('width', width)\n",
              "        canvas_div.css('height', height)\n",
              "\n",
              "        canvas.attr('width', width * mpl.ratio);\n",
              "        canvas.attr('height', height * mpl.ratio);\n",
              "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
              "\n",
              "        rubberband.attr('width', width);\n",
              "        rubberband.attr('height', height);\n",
              "    }\n",
              "\n",
              "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
              "    // upon first draw.\n",
              "    this._resize_canvas(600, 600);\n",
              "\n",
              "    // Disable right mouse context menu.\n",
              "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
              "        return false;\n",
              "    });\n",
              "\n",
              "    function set_focus () {\n",
              "        canvas.focus();\n",
              "        canvas_div.focus();\n",
              "    }\n",
              "\n",
              "    window.setTimeout(set_focus, 100);\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._init_toolbar = function() {\n",
              "    var fig = this;\n",
              "\n",
              "    var nav_element = $('<div/>');\n",
              "    nav_element.attr('style', 'width: 100%');\n",
              "    this.root.append(nav_element);\n",
              "\n",
              "    // Define a callback function for later on.\n",
              "    function toolbar_event(event) {\n",
              "        return fig.toolbar_button_onclick(event['data']);\n",
              "    }\n",
              "    function toolbar_mouse_event(event) {\n",
              "        return fig.toolbar_button_onmouseover(event['data']);\n",
              "    }\n",
              "\n",
              "    for(var toolbar_ind in mpl.toolbar_items) {\n",
              "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
              "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
              "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
              "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
              "\n",
              "        if (!name) {\n",
              "            // put a spacer in here.\n",
              "            continue;\n",
              "        }\n",
              "        var button = $('<button/>');\n",
              "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
              "                        'ui-button-icon-only');\n",
              "        button.attr('role', 'button');\n",
              "        button.attr('aria-disabled', 'false');\n",
              "        button.click(method_name, toolbar_event);\n",
              "        button.mouseover(tooltip, toolbar_mouse_event);\n",
              "\n",
              "        var icon_img = $('<span/>');\n",
              "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
              "        icon_img.addClass(image);\n",
              "        icon_img.addClass('ui-corner-all');\n",
              "\n",
              "        var tooltip_span = $('<span/>');\n",
              "        tooltip_span.addClass('ui-button-text');\n",
              "        tooltip_span.html(tooltip);\n",
              "\n",
              "        button.append(icon_img);\n",
              "        button.append(tooltip_span);\n",
              "\n",
              "        nav_element.append(button);\n",
              "    }\n",
              "\n",
              "    var fmt_picker_span = $('<span/>');\n",
              "\n",
              "    var fmt_picker = $('<select/>');\n",
              "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
              "    fmt_picker_span.append(fmt_picker);\n",
              "    nav_element.append(fmt_picker_span);\n",
              "    this.format_dropdown = fmt_picker[0];\n",
              "\n",
              "    for (var ind in mpl.extensions) {\n",
              "        var fmt = mpl.extensions[ind];\n",
              "        var option = $(\n",
              "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
              "        fmt_picker.append(option);\n",
              "    }\n",
              "\n",
              "    // Add hover states to the ui-buttons\n",
              "    $( \".ui-button\" ).hover(\n",
              "        function() { $(this).addClass(\"ui-state-hover\");},\n",
              "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
              "    );\n",
              "\n",
              "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
              "    nav_element.append(status_bar);\n",
              "    this.message = status_bar[0];\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
              "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
              "    // which will in turn request a refresh of the image.\n",
              "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.send_message = function(type, properties) {\n",
              "    properties['type'] = type;\n",
              "    properties['figure_id'] = this.id;\n",
              "    this.ws.send(JSON.stringify(properties));\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.send_draw_message = function() {\n",
              "    if (!this.waiting) {\n",
              "        this.waiting = true;\n",
              "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
              "    }\n",
              "}\n",
              "\n",
              "\n",
              "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
              "    var format_dropdown = fig.format_dropdown;\n",
              "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
              "    fig.ondownload(fig, format);\n",
              "}\n",
              "\n",
              "\n",
              "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
              "    var size = msg['size'];\n",
              "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
              "        fig._resize_canvas(size[0], size[1]);\n",
              "        fig.send_message(\"refresh\", {});\n",
              "    };\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
              "    var x0 = msg['x0'] / mpl.ratio;\n",
              "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
              "    var x1 = msg['x1'] / mpl.ratio;\n",
              "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
              "    x0 = Math.floor(x0) + 0.5;\n",
              "    y0 = Math.floor(y0) + 0.5;\n",
              "    x1 = Math.floor(x1) + 0.5;\n",
              "    y1 = Math.floor(y1) + 0.5;\n",
              "    var min_x = Math.min(x0, x1);\n",
              "    var min_y = Math.min(y0, y1);\n",
              "    var width = Math.abs(x1 - x0);\n",
              "    var height = Math.abs(y1 - y0);\n",
              "\n",
              "    fig.rubberband_context.clearRect(\n",
              "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
              "\n",
              "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
              "    // Updates the figure title.\n",
              "    fig.header.textContent = msg['label'];\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
              "    var cursor = msg['cursor'];\n",
              "    switch(cursor)\n",
              "    {\n",
              "    case 0:\n",
              "        cursor = 'pointer';\n",
              "        break;\n",
              "    case 1:\n",
              "        cursor = 'default';\n",
              "        break;\n",
              "    case 2:\n",
              "        cursor = 'crosshair';\n",
              "        break;\n",
              "    case 3:\n",
              "        cursor = 'move';\n",
              "        break;\n",
              "    }\n",
              "    fig.rubberband_canvas.style.cursor = cursor;\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
              "    fig.message.textContent = msg['message'];\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
              "    // Request the server to send over a new figure.\n",
              "    fig.send_draw_message();\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
              "    fig.image_mode = msg['mode'];\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.updated_canvas_event = function() {\n",
              "    // Called whenever the canvas gets updated.\n",
              "    this.send_message(\"ack\", {});\n",
              "}\n",
              "\n",
              "// A function to construct a web socket function for onmessage handling.\n",
              "// Called in the figure constructor.\n",
              "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
              "    return function socket_on_message(evt) {\n",
              "        if (evt.data instanceof Blob) {\n",
              "            /* FIXME: We get \"Resource interpreted as Image but\n",
              "             * transferred with MIME type text/plain:\" errors on\n",
              "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
              "             * to be part of the websocket stream */\n",
              "            evt.data.type = \"image/png\";\n",
              "\n",
              "            /* Free the memory for the previous frames */\n",
              "            if (fig.imageObj.src) {\n",
              "                (window.URL || window.webkitURL).revokeObjectURL(\n",
              "                    fig.imageObj.src);\n",
              "            }\n",
              "\n",
              "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
              "                evt.data);\n",
              "            fig.updated_canvas_event();\n",
              "            fig.waiting = false;\n",
              "            return;\n",
              "        }\n",
              "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
              "            fig.imageObj.src = evt.data;\n",
              "            fig.updated_canvas_event();\n",
              "            fig.waiting = false;\n",
              "            return;\n",
              "        }\n",
              "\n",
              "        var msg = JSON.parse(evt.data);\n",
              "        var msg_type = msg['type'];\n",
              "\n",
              "        // Call the  \"handle_{type}\" callback, which takes\n",
              "        // the figure and JSON message as its only arguments.\n",
              "        try {\n",
              "            var callback = fig[\"handle_\" + msg_type];\n",
              "        } catch (e) {\n",
              "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
              "            return;\n",
              "        }\n",
              "\n",
              "        if (callback) {\n",
              "            try {\n",
              "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
              "                callback(fig, msg);\n",
              "            } catch (e) {\n",
              "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
              "            }\n",
              "        }\n",
              "    };\n",
              "}\n",
              "\n",
              "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
              "mpl.findpos = function(e) {\n",
              "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
              "    var targ;\n",
              "    if (!e)\n",
              "        e = window.event;\n",
              "    if (e.target)\n",
              "        targ = e.target;\n",
              "    else if (e.srcElement)\n",
              "        targ = e.srcElement;\n",
              "    if (targ.nodeType == 3) // defeat Safari bug\n",
              "        targ = targ.parentNode;\n",
              "\n",
              "    // jQuery normalizes the pageX and pageY\n",
              "    // pageX,Y are the mouse positions relative to the document\n",
              "    // offset() returns the position of the element relative to the document\n",
              "    var x = e.pageX - $(targ).offset().left;\n",
              "    var y = e.pageY - $(targ).offset().top;\n",
              "\n",
              "    return {\"x\": x, \"y\": y};\n",
              "};\n",
              "\n",
              "/*\n",
              " * return a copy of an object with only non-object keys\n",
              " * we need this to avoid circular references\n",
              " * http://stackoverflow.com/a/24161582/3208463\n",
              " */\n",
              "function simpleKeys (original) {\n",
              "  return Object.keys(original).reduce(function (obj, key) {\n",
              "    if (typeof original[key] !== 'object')\n",
              "        obj[key] = original[key]\n",
              "    return obj;\n",
              "  }, {});\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.mouse_event = function(event, name) {\n",
              "    var canvas_pos = mpl.findpos(event)\n",
              "\n",
              "    if (name === 'button_press')\n",
              "    {\n",
              "        this.canvas.focus();\n",
              "        this.canvas_div.focus();\n",
              "    }\n",
              "\n",
              "    var x = canvas_pos.x * mpl.ratio;\n",
              "    var y = canvas_pos.y * mpl.ratio;\n",
              "\n",
              "    this.send_message(name, {x: x, y: y, button: event.button,\n",
              "                             step: event.step,\n",
              "                             guiEvent: simpleKeys(event)});\n",
              "\n",
              "    /* This prevents the web browser from automatically changing to\n",
              "     * the text insertion cursor when the button is pressed.  We want\n",
              "     * to control all of the cursor setting manually through the\n",
              "     * 'cursor' event from matplotlib */\n",
              "    event.preventDefault();\n",
              "    return false;\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
              "    // Handle any extra behaviour associated with a key event\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.key_event = function(event, name) {\n",
              "\n",
              "    // Prevent repeat events\n",
              "    if (name == 'key_press')\n",
              "    {\n",
              "        if (event.which === this._key)\n",
              "            return;\n",
              "        else\n",
              "            this._key = event.which;\n",
              "    }\n",
              "    if (name == 'key_release')\n",
              "        this._key = null;\n",
              "\n",
              "    var value = '';\n",
              "    if (event.ctrlKey && event.which != 17)\n",
              "        value += \"ctrl+\";\n",
              "    if (event.altKey && event.which != 18)\n",
              "        value += \"alt+\";\n",
              "    if (event.shiftKey && event.which != 16)\n",
              "        value += \"shift+\";\n",
              "\n",
              "    value += 'k';\n",
              "    value += event.which.toString();\n",
              "\n",
              "    this._key_event_extra(event, name);\n",
              "\n",
              "    this.send_message(name, {key: value,\n",
              "                             guiEvent: simpleKeys(event)});\n",
              "    return false;\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
              "    if (name == 'download') {\n",
              "        this.handle_save(this, null);\n",
              "    } else {\n",
              "        this.send_message(\"toolbar_button\", {name: name});\n",
              "    }\n",
              "};\n",
              "\n",
              "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
              "    this.message.textContent = tooltip;\n",
              "};\n",
              "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
              "\n",
              "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
              "\n",
              "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
              "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
              "    // object with the appropriate methods. Currently this is a non binary\n",
              "    // socket, so there is still some room for performance tuning.\n",
              "    var ws = {};\n",
              "\n",
              "    ws.close = function() {\n",
              "        comm.close()\n",
              "    };\n",
              "    ws.send = function(m) {\n",
              "        //console.log('sending', m);\n",
              "        comm.send(m);\n",
              "    };\n",
              "    // Register the callback with on_msg.\n",
              "    comm.on_msg(function(msg) {\n",
              "        //console.log('receiving', msg['content']['data'], msg);\n",
              "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
              "        ws.onmessage(msg['content']['data'])\n",
              "    });\n",
              "    return ws;\n",
              "}\n",
              "\n",
              "mpl.mpl_figure_comm = function(comm, msg) {\n",
              "    // This is the function which gets called when the mpl process\n",
              "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
              "\n",
              "    var id = msg.content.data.id;\n",
              "    // Get hold of the div created by the display call when the Comm\n",
              "    // socket was opened in Python.\n",
              "    var element = $(\"#\" + id);\n",
              "    var ws_proxy = comm_websocket_adapter(comm)\n",
              "\n",
              "    function ondownload(figure, format) {\n",
              "        window.open(figure.imageObj.src);\n",
              "    }\n",
              "\n",
              "    var fig = new mpl.figure(id, ws_proxy,\n",
              "                           ondownload,\n",
              "                           element.get(0));\n",
              "\n",
              "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
              "    // web socket which is closed, not our websocket->open comm proxy.\n",
              "    ws_proxy.onopen();\n",
              "\n",
              "    fig.parent_element = element.get(0);\n",
              "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
              "    if (!fig.cell_info) {\n",
              "        console.error(\"Failed to find cell for figure\", id, fig);\n",
              "        return;\n",
              "    }\n",
              "\n",
              "    var output_index = fig.cell_info[2]\n",
              "    var cell = fig.cell_info[0];\n",
              "\n",
              "};\n",
              "\n",
              "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
              "    var width = fig.canvas.width/mpl.ratio\n",
              "    fig.root.unbind('remove')\n",
              "\n",
              "    // Update the output cell to use the data from the current canvas.\n",
              "    fig.push_to_output();\n",
              "    var dataURL = fig.canvas.toDataURL();\n",
              "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
              "    // the notebook keyboard shortcuts fail.\n",
              "    IPython.keyboard_manager.enable()\n",
              "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
              "    fig.close_ws(fig, msg);\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.close_ws = function(fig, msg){\n",
              "    fig.send_message('closing', msg);\n",
              "    // fig.ws.close()\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
              "    // Turn the data on the canvas into data in the output cell.\n",
              "    var width = this.canvas.width/mpl.ratio\n",
              "    var dataURL = this.canvas.toDataURL();\n",
              "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.updated_canvas_event = function() {\n",
              "    // Tell IPython that the notebook contents must change.\n",
              "    IPython.notebook.set_dirty(true);\n",
              "    this.send_message(\"ack\", {});\n",
              "    var fig = this;\n",
              "    // Wait a second, then push the new image to the DOM so\n",
              "    // that it is saved nicely (might be nice to debounce this).\n",
              "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._init_toolbar = function() {\n",
              "    var fig = this;\n",
              "\n",
              "    var nav_element = $('<div/>');\n",
              "    nav_element.attr('style', 'width: 100%');\n",
              "    this.root.append(nav_element);\n",
              "\n",
              "    // Define a callback function for later on.\n",
              "    function toolbar_event(event) {\n",
              "        return fig.toolbar_button_onclick(event['data']);\n",
              "    }\n",
              "    function toolbar_mouse_event(event) {\n",
              "        return fig.toolbar_button_onmouseover(event['data']);\n",
              "    }\n",
              "\n",
              "    for(var toolbar_ind in mpl.toolbar_items){\n",
              "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
              "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
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              "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
              "\n",
              "        if (!name) { continue; };\n",
              "\n",
              "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
              "        button.click(method_name, toolbar_event);\n",
              "        button.mouseover(tooltip, toolbar_mouse_event);\n",
              "        nav_element.append(button);\n",
              "    }\n",
              "\n",
              "    // Add the status bar.\n",
              "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
              "    nav_element.append(status_bar);\n",
              "    this.message = status_bar[0];\n",
              "\n",
              "    // Add the close button to the window.\n",
              "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
              "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
              "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
              "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
              "    buttongrp.append(button);\n",
              "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
              "    titlebar.prepend(buttongrp);\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._root_extra_style = function(el){\n",
              "    var fig = this\n",
              "    el.on(\"remove\", function(){\n",
              "\tfig.close_ws(fig, {});\n",
              "    });\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._canvas_extra_style = function(el){\n",
              "    // this is important to make the div 'focusable\n",
              "    el.attr('tabindex', 0)\n",
              "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
              "    // off when our div gets focus\n",
              "\n",
              "    // location in version 3\n",
              "    if (IPython.notebook.keyboard_manager) {\n",
              "        IPython.notebook.keyboard_manager.register_events(el);\n",
              "    }\n",
              "    else {\n",
              "        // location in version 2\n",
              "        IPython.keyboard_manager.register_events(el);\n",
              "    }\n",
              "\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
              "    var manager = IPython.notebook.keyboard_manager;\n",
              "    if (!manager)\n",
              "        manager = IPython.keyboard_manager;\n",
              "\n",
              "    // Check for shift+enter\n",
              "    if (event.shiftKey && event.which == 13) {\n",
              "        this.canvas_div.blur();\n",
              "        // select the cell after this one\n",
              "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
              "        IPython.notebook.select(index + 1);\n",
              "    }\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
              "    fig.ondownload(fig, null);\n",
              "}\n",
              "\n",
              "\n",
              "mpl.find_output_cell = function(html_output) {\n",
              "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
              "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
              "    // IPython event is triggered only after the cells have been serialised, which for\n",
              "    // our purposes (turning an active figure into a static one), is too late.\n",
              "    var cells = IPython.notebook.get_cells();\n",
              "    var ncells = cells.length;\n",
              "    for (var i=0; i<ncells; i++) {\n",
              "        var cell = cells[i];\n",
              "        if (cell.cell_type === 'code'){\n",
              "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
              "                var data = cell.output_area.outputs[j];\n",
              "                if (data.data) {\n",
              "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
              "                    data = data.data;\n",
              "                }\n",
              "                if (data['text/html'] == html_output) {\n",
              "                    return [cell, data, j];\n",
              "                }\n",
              "            }\n",
              "        }\n",
              "    }\n",
              "}\n",
              "\n",
              "// Register the function which deals with the matplotlib target/channel.\n",
              "// The kernel may be null if the page has been refreshed.\n",
              "if (IPython.notebook.kernel != null) {\n",
              "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
              "}\n"
            ],
            "text/plain": [
              "<IPython.core.display.Javascript object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "<div id='af41151d-0c6a-4abb-ba73-74abbeaa4080'></div>"
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              "<IPython.core.display.HTML object>"
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          "metadata": {
            "tags": []
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        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.axes._subplots.AxesSubplot at 0x7fe4e8012b90>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 35
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Y-H7z0GCvwWh"
      },
      "source": [
        "backtest['return'] = backtest['value'].pct_change()"
      ],
      "execution_count": 36,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "2LPrjSOwvtQR"
      },
      "source": [
        "return_list=backtest['return'].tolist()[1:]"
      ],
      "execution_count": 37,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "d2NjCsJLvtQS"
      },
      "source": [
        "GARCH(1,1) model is used to determined the volatility."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "mOtX5ZA4vtQS",
        "outputId": "c43a2680-8969-41b5-91b2-0b4f406e60d0"
      },
      "source": [
        "# Using GARCH(1,1) to determine the volatility\n",
        "model = arch_model(return_list, mean='Zero', vol='GARCH', p=1, q=1)\n",
        "model_fit = model.fit()\n",
        "volatility = model_fit.conditional_volatility\n",
        "print(model_fit.summary())"
      ],
      "execution_count": 38,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/arch/univariate/base.py:310: DataScaleWarning: y is poorly scaled, which may affect convergence of the optimizer when\n",
            "estimating the model parameters. The scale of y is 0.0003138. Parameter\n",
            "estimation work better when this value is between 1 and 1000. The recommended\n",
            "rescaling is 100 * y.\n",
            "\n",
            "This warning can be disabled by either rescaling y before initializing the\n",
            "model or by setting rescale=False.\n",
            "\n",
            "  data_scale_warning.format(orig_scale, rescale), DataScaleWarning\n"
          ],
          "name": "stderr"
        },
        {
          "output_type": "stream",
          "text": [
            "Iteration:      1,   Func. Count:      5,   Neg. LLF: -23438.325824725303\n",
            "Optimization terminated successfully.    (Exit mode 0)\n",
            "            Current function value: -23438.325854447183\n",
            "            Iterations: 5\n",
            "            Function evaluations: 5\n",
            "            Gradient evaluations: 1\n",
            "                       Zero Mean - GARCH Model Results                        \n",
            "==============================================================================\n",
            "Dep. Variable:                      y   R-squared:                       0.000\n",
            "Mean Model:                 Zero Mean   Adj. R-squared:                  0.000\n",
            "Vol Model:                      GARCH   Log-Likelihood:                23438.3\n",
            "Distribution:                  Normal   AIC:                          -46870.7\n",
            "Method:            Maximum Likelihood   BIC:                          -46849.4\n",
            "                                        No. Observations:                 8693\n",
            "Date:                Fri, Jul 30 2021   Df Residuals:                     8693\n",
            "Time:                        01:05:55   Df Model:                            0\n",
            "                              Volatility Model                              \n",
            "============================================================================\n",
            "                 coef    std err          t      P>|t|      95.0% Conf. Int.\n",
            "----------------------------------------------------------------------------\n",
            "omega      6.2879e-06  6.998e-13  8.985e+06      0.000 [6.288e-06,6.288e-06]\n",
            "alpha[1]       0.0500  1.334e-04    374.781      0.000 [4.974e-02,5.026e-02]\n",
            "beta[1]        0.9300  1.715e-03    542.304      0.000     [  0.927,  0.933]\n",
            "============================================================================\n",
            "\n",
            "Covariance estimator: robust\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "V6yQunngvtQX"
      },
      "source": [
        "Estimate the Volatility in the next date with the GARCH(1,1) model"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "eHMM0akDvtQX",
        "outputId": "3fab0049-0add-4214-81bc-93a6a55438df"
      },
      "source": [
        "# Estimate the volatility in the next date\n",
        "omega = model_fit.params['omega'] \n",
        "alpha = model_fit.params['alpha[1]']\n",
        "beta = model_fit.params['beta[1]']\n",
        "vola_estimate = omega + alpha * np.power(return_list[-1] , 2)  + beta * np.power( volatility[-1] , 2 )\n",
        "vola_estimate = np.sqrt(vola_estimate)\n",
        "print('The volatility in the next date=', vola_estimate )\n",
        "#return_scienarios = []\n",
        "#for i in range(0,return_list)"
      ],
      "execution_count": 39,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "The volatility in the next date= 0.016502061111247215\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mamGxP2zvtQY"
      },
      "source": [
        "Value at Risk Calculation: historical simulation with volatility updating"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "FdBXb1-VvtQY"
      },
      "source": [
        "# calculate the predicted value in the next date under different scenario and inporporate volatility updating\n",
        "Market_scienarios = []\n",
        "N_history = len(return_list)\n",
        "Value_current = backtest.loc[backtest.index[-1], 'value']\n",
        "backtest.loc[backtest.index[-1], 'value']\n",
        "for i in range(0, N_history ):\n",
        "  Market_value = Value_current *  ( 1 + return_list[i] * vola_estimate/volatility[i] )\n",
        "  Market_scienarios.append(Market_value)"
      ],
      "execution_count": 40,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "NqvRVtjNvtQZ",
        "outputId": "c3190e4d-b281-4a93-ad69-ad06d43754cf"
      },
      "source": [
        "# Value at Risk (VaR) determination\n",
        "# CL_Var: condifence level for VaR\n",
        "CL_VaR = 95\n",
        "Value_current = backtest.loc[backtest.index[-1], 'value']\n",
        "Value_at_Risk = np.percentile(Market_scienarios,100-CL_VaR) -  Value_current\n",
        "print(\"The 1 day\", CL_VaR,\"% VaR:\", -Value_at_Risk)"
      ],
      "execution_count": 41,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "The 1 day 95 % VaR: 47264.910781804705\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "icgRS2X2vtQZ"
      },
      "source": [
        "# Make the histogram of loss for the scenaros considered between current date and the next date\n",
        "import matplotlib.pyplot as plt\n",
        "mss = np.asarray(Value_current - Market_scienarios )\n",
        "\n",
        "plt.xlim([min(mss), max(mss)])\n",
        "\n",
        "plt.hist(mss, bins='auto', alpha=0.5)\n",
        "plt.title('Plot of loss for the scenaros')\n",
        "plt.xlabel('Loss')\n",
        "plt.ylabel('Frequency')\n",
        "plt.axvline( -Value_at_Risk , color='k', linestyle='dashed', linewidth=1)\n",
        "min_ylim, max_ylim = plt.ylim()\n",
        "plt.text(-Value_at_Risk*1.1, max_ylim*0.9, '{}% VaR: {:.2f}'.format(CL_VaR ,-Value_at_Risk))\n",
        "plt.show()"
      ],
      "execution_count": 42,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1PyJ8mx7vtQa"
      },
      "source": [
        "Value-at-Risk backtesting:\n",
        "here the binomial one-tailed test is used"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "O9rBa7H4vtQa",
        "outputId": "b17d47b8-3b62-4059-b2f5-7c00b6b16e5e"
      },
      "source": [
        "# Confidence level for backtesting\n",
        "CL_backtesting = 95\n",
        "# The probability of the VaR beeing exceeded on given day\n",
        "p = 1 - CL_VaR/100\n",
        "N_confidence = binom.ppf( CL_backtesting*0.01, N_history, p)\n",
        "N_exceeded = sum( v < Value_at_Risk + Value_current for v in Market_scienarios)\n",
        "\n",
        "print(\"We look at total\", N_history,\"days. VaR level is reached at\", N_exceeded , \"days.\")\n",
        "if N_exceeded <= N_confidence:\n",
        "  print (\"This VaR model is accepted at\", CL_backtesting,\"% confidence level.\")\n",
        "else:\n",
        "  print (\"This VaR model is rejected at\", CL_backtesting,\"% confidence level.\")"
      ],
      "execution_count": 43,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "We look at total 8693 days. VaR level is reached at 435 days.\n",
            "This VaR model is accepted at 95 % confidence level.\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "J6dtFBbawKaa"
      },
      "source": [
        ""
      ],
      "execution_count": 43,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BzkzWtjk-3vv"
      },
      "source": [
        "# Attempt 1: Single long VAR"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "av5Mm9F_0egA",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "42b7b99a-9c7f-4a32-ecca-31b0df871dee"
      },
      "source": [
        "!pip install arch"
      ],
      "execution_count": 61,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Requirement already satisfied: arch in /usr/local/lib/python3.7/dist-packages (5.0.1)\n",
            "Requirement already satisfied: scipy>=1.3 in /usr/local/lib/python3.7/dist-packages (from arch) (1.4.1)\n",
            "Requirement already satisfied: pandas>=1.0 in /usr/local/lib/python3.7/dist-packages (from arch) (1.1.5)\n",
            "Requirement already satisfied: numpy>=1.17 in /usr/local/lib/python3.7/dist-packages (from arch) (1.19.5)\n",
            "Requirement already satisfied: statsmodels>=0.11 in /usr/local/lib/python3.7/dist-packages (from arch) (0.12.2)\n",
            "Requirement already satisfied: property-cached>=1.6.4 in /usr/local/lib/python3.7/dist-packages (from arch) (1.6.4)\n",
            "Requirement already satisfied: python-dateutil>=2.7.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=1.0->arch) (2.8.1)\n",
            "Requirement already satisfied: pytz>=2017.2 in /usr/local/lib/python3.7/dist-packages (from pandas>=1.0->arch) (2018.9)\n",
            "Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.7.3->pandas>=1.0->arch) (1.15.0)\n",
            "Requirement already satisfied: patsy>=0.5 in /usr/local/lib/python3.7/dist-packages (from statsmodels>=0.11->arch) (0.5.1)\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XC7Ce1A_zoF_"
      },
      "source": [
        "# example of ARCH model\n",
        "import numpy as np\n",
        "from random import gauss\n",
        "from random import seed\n",
        "from matplotlib import pyplot\n",
        "from arch import arch_model\n",
        "import datetime\n",
        "import pandas as pd\n",
        "# from pandas_datareader import data as pdr\n",
        "# import yfinance as yf\n",
        "from scipy.stats import binom"
      ],
      "execution_count": 62,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TlKwBVLjQqUY"
      },
      "source": [
        "Read and preprocess the data such as calculating the continously compounded return."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "e_-w3EY12eHo",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 204
        },
        "outputId": "08eee641-4f8d-429f-c938-150281d18b69"
      },
      "source": [
        "# hedgefundie monthly returns taken from: \n",
        "# https://www.portfoliovisualizer.com/backtest-portfolio?s=y&timePeriod=2&startYear=1985&firstMonth=8&endYear=2021&lastMonth=12&calendarAligned=true&includeYTD=false&initialAmount=10000&annualOperation=0&annualAdjustment=0&inflationAdjusted=true&annualPercentage=0.0&frequency=4&rebalanceType=3&absoluteDeviation=5.0&relativeDeviation=25.0&reinvestDividends=true&showYield=false&showFactors=false&factorModel=3&benchmark=VFINX&portfolioNames=true&portfolioName1=3x+AWG&portfolioName2=Hedgefundie+55%2F45&portfolioName3=Portfolio+3&symbol1=VFINX&allocation1_1=90&allocation1_2=165&symbol2=VUSTX&allocation2_1=120&allocation2_2=135&symbol3=VFITX&allocation3_1=45&symbol4=%5EGOLD&allocation4_1=30&symbol5=GSG&symbol6=CASHX&allocation6_1=-185&allocation6_2=-200&symbol7=IJH&symbol8=IJR\n",
        "data = pd.read_csv(\"hedgefundie_monthly_returns.csv\", dtype = {'Year': int, 'Month': int, 'Return': 'string', 'Balance': 'string'}) \n",
        "data['Return'] = data['Return'].str.replace('%', '').astype(float)\n",
        "data['Return'] = (data['Return']/100).astype(float)\n",
        "data['Balance'] = data['Balance'].str.replace('$', '')\n",
        "data['Balance'] = data['Balance'].str.replace(',', '').astype(float)\n",
        "data.head()"
      ],
      "execution_count": 118,
      "outputs": [
        {
          "output_type": "execute_result",
          "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>Year</th>\n",
              "      <th>Month</th>\n",
              "      <th>Return</th>\n",
              "      <th>Balance</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>1991</td>\n",
              "      <td>11</td>\n",
              "      <td>-0.0740</td>\n",
              "      <td>9260.0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>1991</td>\n",
              "      <td>12</td>\n",
              "      <td>0.2778</td>\n",
              "      <td>11833.0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>1992</td>\n",
              "      <td>1</td>\n",
              "      <td>-0.0809</td>\n",
              "      <td>10875.0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>1992</td>\n",
              "      <td>2</td>\n",
              "      <td>0.0239</td>\n",
              "      <td>11136.0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>1992</td>\n",
              "      <td>3</td>\n",
              "      <td>-0.0557</td>\n",
              "      <td>10515.0</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   Year  Month  Return  Balance\n",
              "0  1991     11 -0.0740   9260.0\n",
              "1  1991     12  0.2778  11833.0\n",
              "2  1992      1 -0.0809  10875.0\n",
              "3  1992      2  0.0239  11136.0\n",
              "4  1992      3 -0.0557  10515.0"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 118
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "EjaJZtf7Kt7A"
      },
      "source": [
        "return_list=data['Return'].tolist()"
      ],
      "execution_count": 119,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "41E2WxK-P9RO"
      },
      "source": [
        "GARCH(1,1) model is used to determined the volatility."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "E-RHDDEcN7bf",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "85e07ac5-888d-4d7d-9f68-f09eabf87477"
      },
      "source": [
        "# Using GARCH(1,1) to determine the volatility\n",
        "model = arch_model(return_list, mean='Zero', vol='GARCH', p=1, q=1)\n",
        "model_fit = model.fit()\n",
        "volatility = model_fit.conditional_volatility\n",
        "print(model_fit.summary())"
      ],
      "execution_count": 120,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Iteration:      1,   Func. Count:      5,   Neg. LLF: -428.1139665133687\n",
            "Iteration:      2,   Func. Count:     13,   Neg. LLF: -428.1925706963546\n",
            "Iteration:      3,   Func. Count:     20,   Neg. LLF: -428.336400163769\n",
            "Iteration:      4,   Func. Count:     26,   Neg. LLF: -428.4391125510042\n",
            "Iteration:      5,   Func. Count:     32,   Neg. LLF: -428.4751234916761\n",
            "Iteration:      6,   Func. Count:     37,   Neg. LLF: -428.4817581531861\n",
            "Iteration:      7,   Func. Count:     42,   Neg. LLF: -428.48270478855306\n",
            "Iteration:      8,   Func. Count:     47,   Neg. LLF: -428.48271083311005\n",
            "Optimization terminated successfully.    (Exit mode 0)\n",
            "            Current function value: -428.4827108330161\n",
            "            Iterations: 8\n",
            "            Function evaluations: 47\n",
            "            Gradient evaluations: 8\n",
            "                       Zero Mean - GARCH Model Results                        \n",
            "==============================================================================\n",
            "Dep. Variable:                      y   R-squared:                       0.000\n",
            "Mean Model:                 Zero Mean   Adj. R-squared:                  0.003\n",
            "Vol Model:                      GARCH   Log-Likelihood:                428.483\n",
            "Distribution:                  Normal   AIC:                          -850.965\n",
            "Method:            Maximum Likelihood   BIC:                          -839.341\n",
            "                                        No. Observations:                  356\n",
            "Date:                Thu, Jul 29 2021   Df Residuals:                      356\n",
            "Time:                        23:42:20   Df Model:                            0\n",
            "                               Volatility Model                              \n",
            "=============================================================================\n",
            "                 coef    std err          t      P>|t|       95.0% Conf. Int.\n",
            "-----------------------------------------------------------------------------\n",
            "omega      6.8528e-04  3.967e-04      1.728  8.405e-02 [-9.216e-05,1.463e-03]\n",
            "alpha[1]       0.1651  6.942e-02      2.378  1.741e-02    [2.902e-02,  0.301]\n",
            "beta[1]        0.7145      0.109      6.549  5.780e-11      [  0.501,  0.928]\n",
            "=============================================================================\n",
            "\n",
            "Covariance estimator: robust\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/arch/univariate/base.py:310: DataScaleWarning: y is poorly scaled, which may affect convergence of the optimizer when\n",
            "estimating the model parameters. The scale of y is 0.005512. Parameter\n",
            "estimation work better when this value is between 1 and 1000. The recommended\n",
            "rescaling is 10 * y.\n",
            "\n",
            "This warning can be disabled by either rescaling y before initializing the\n",
            "model or by setting rescale=False.\n",
            "\n",
            "  data_scale_warning.format(orig_scale, rescale), DataScaleWarning\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gXxF-vwnUFfU"
      },
      "source": [
        "Estimate the Volatility in the next date with the GARCH(1,1) model"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "e3EuHIKqUFGf",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "6b60c36d-bbe0-4979-9887-c2d4cb306219"
      },
      "source": [
        "# Estimate the volatility in the next date\n",
        "omega = model_fit.params['omega'] \n",
        "alpha = model_fit.params['alpha[1]']\n",
        "beta = model_fit.params['beta[1]']\n",
        "vola_estimate = omega + alpha * np.power(return_list[-1] , 2)  + beta * np.power( volatility[-1] , 2 )\n",
        "vola_estimate = np.sqrt(vola_estimate)\n",
        "print('The volatility in the next date=', vola_estimate )\n",
        "#return_scienarios = []\n",
        "#for i in range(0,return_list)"
      ],
      "execution_count": 121,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "The volatility in the next date= 0.07786391557051343\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4KJXGsPbLYov"
      },
      "source": [
        "Value at Risk Calculation: historical simulation with volatility updating"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "V_5xkz4Wl1Pz"
      },
      "source": [
        "# calculate the predicted value in the next date under different scenario and inporporate volatility updating\n",
        "Market_scienarios = []\n",
        "N_history = len(return_list)\n",
        "Value_current = df.loc[df.index[-1], 'value']\n",
        "df.loc[df.index[-1], 'value']\n",
        "for i in range(0, N_history ):\n",
        "  Market_value = Value_current *  ( 1 + return_list[i] * vola_estimate/volatility[i] )\n",
        "  Market_scienarios.append(Market_value)"
      ],
      "execution_count": 122,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "jg2kWLUWl1CM",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "5a751bb2-4611-43be-e597-10fb4b2ea2b4"
      },
      "source": [
        "# Value at Risk (VaR) determination\n",
        "# CL_Var: condifence level for VaR\n",
        "CL_VaR = 95\n",
        "Value_current = df.loc[df.index[-1], 'value']\n",
        "Value_at_Risk = np.percentile(Market_scienarios,100-CL_VaR) -  Value_current\n",
        "print(\"The 1 Month\", CL_VaR,\"% VaR:\", -Value_at_Risk)"
      ],
      "execution_count": 123,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "The 1 Month 95 % VaR: 1.3969630967506657\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ZLxs5LYPl04T",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        },
        "outputId": "ba5822a3-eed9-4ef9-8735-3fec5d43f8cd"
      },
      "source": [
        "# Make the histogram of loss for the scenaros considered between current date and the next date\n",
        "import matplotlib.pyplot as plt\n",
        "mss = np.asarray(Value_current - Market_scienarios )\n",
        "\n",
        "plt.xlim([min(mss), max(mss)])\n",
        "\n",
        "plt.hist(mss, bins='auto', alpha=0.5)\n",
        "plt.title('Plot of loss for the scenaros')\n",
        "plt.xlabel('Loss')\n",
        "plt.ylabel('Frequency')\n",
        "plt.axvline( -Value_at_Risk , color='k', linestyle='dashed', linewidth=1)\n",
        "min_ylim, max_ylim = plt.ylim()\n",
        "plt.text(-Value_at_Risk*1.1, max_ylim*0.9, '{}% VaR: {:.2f}'.format(CL_VaR ,-Value_at_Risk))\n",
        "plt.show()"
      ],
      "execution_count": 124,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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              "        return WebSocket;\n",
              "    } else if (typeof(MozWebSocket) !== 'undefined') {\n",
              "        return MozWebSocket;\n",
              "    } else {\n",
              "        alert('Your browser does not have WebSocket support. ' +\n",
              "              'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
              "              'Firefox 4 and 5 are also supported but you ' +\n",
              "              'have to enable WebSockets in about:config.');\n",
              "    };\n",
              "}\n",
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              "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n",
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              "    this.supports_binary = (this.ws.binaryType != undefined);\n",
              "\n",
              "    if (!this.supports_binary) {\n",
              "        var warnings = document.getElementById(\"mpl-warnings\");\n",
              "        if (warnings) {\n",
              "            warnings.style.display = 'block';\n",
              "            warnings.textContent = (\n",
              "                \"This browser does not support binary websocket messages. \" +\n",
              "                    \"Performance may be slow.\");\n",
              "        }\n",
              "    }\n",
              "\n",
              "    this.imageObj = new Image();\n",
              "\n",
              "    this.context = undefined;\n",
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              "    this.rubberband_canvas = undefined;\n",
              "    this.rubberband_context = undefined;\n",
              "    this.format_dropdown = undefined;\n",
              "\n",
              "    this.image_mode = 'full';\n",
              "\n",
              "    this.root = $('<div/>');\n",
              "    this._root_extra_style(this.root)\n",
              "    this.root.attr('style', 'display: inline-block');\n",
              "\n",
              "    $(parent_element).append(this.root);\n",
              "\n",
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              "                fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n",
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              "    this.ws.onmessage = this._make_on_message_function(this);\n",
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              "\n",
              "mpl.figure.prototype._init_header = function() {\n",
              "    var titlebar = $(\n",
              "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
              "        'ui-helper-clearfix\"/>');\n",
              "    var titletext = $(\n",
              "        '<div class=\"ui-dialog-title\" style=\"width: 100%; ' +\n",
              "        'text-align: center; padding: 3px;\"/>');\n",
              "    titlebar.append(titletext)\n",
              "    this.root.append(titlebar);\n",
              "    this.header = titletext[0];\n",
              "}\n",
              "\n",
              "\n",
              "\n",
              "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n",
              "\n",
              "}\n",
              "\n",
              "\n",
              "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n",
              "\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._init_canvas = function() {\n",
              "    var fig = this;\n",
              "\n",
              "    var canvas_div = $('<div/>');\n",
              "\n",
              "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
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              "    function canvas_keyboard_event(event) {\n",
              "        return fig.key_event(event, event['data']);\n",
              "    }\n",
              "\n",
              "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
              "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
              "    this.canvas_div = canvas_div\n",
              "    this._canvas_extra_style(canvas_div)\n",
              "    this.root.append(canvas_div);\n",
              "\n",
              "    var canvas = $('<canvas/>');\n",
              "    canvas.addClass('mpl-canvas');\n",
              "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
              "\n",
              "    this.canvas = canvas[0];\n",
              "    this.context = canvas[0].getContext(\"2d\");\n",
              "\n",
              "    var backingStore = this.context.backingStorePixelRatio ||\n",
              "\tthis.context.webkitBackingStorePixelRatio ||\n",
              "\tthis.context.mozBackingStorePixelRatio ||\n",
              "\tthis.context.msBackingStorePixelRatio ||\n",
              "\tthis.context.oBackingStorePixelRatio ||\n",
              "\tthis.context.backingStorePixelRatio || 1;\n",
              "\n",
              "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
              "\n",
              "    var rubberband = $('<canvas/>');\n",
              "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
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              "        resize: function(event, ui) {\n",
              "            fig.request_resize(ui.size.width, ui.size.height);\n",
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              "        stop: function(event, ui) {\n",
              "            pass_mouse_events = true;\n",
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              "        if (pass_mouse_events)\n",
              "            return fig.mouse_event(event, event['data']);\n",
              "    }\n",
              "\n",
              "    rubberband.mousedown('button_press', mouse_event_fn);\n",
              "    rubberband.mouseup('button_release', mouse_event_fn);\n",
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              "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
              "\n",
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              "    rubberband.mouseleave('figure_leave', mouse_event_fn);\n",
              "\n",
              "    canvas_div.on(\"wheel\", function (event) {\n",
              "        event = event.originalEvent;\n",
              "        event['data'] = 'scroll'\n",
              "        if (event.deltaY < 0) {\n",
              "            event.step = 1;\n",
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              "        mouse_event_fn(event);\n",
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              "\n",
              "    canvas_div.append(canvas);\n",
              "    canvas_div.append(rubberband);\n",
              "\n",
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              "    this.rubberband_canvas = rubberband[0];\n",
              "    this.rubberband_context = rubberband[0].getContext(\"2d\");\n",
              "    this.rubberband_context.strokeStyle = \"#000000\";\n",
              "\n",
              "    this._resize_canvas = function(width, height) {\n",
              "        // Keep the size of the canvas, canvas container, and rubber band\n",
              "        // canvas in synch.\n",
              "        canvas_div.css('width', width)\n",
              "        canvas_div.css('height', height)\n",
              "\n",
              "        canvas.attr('width', width * mpl.ratio);\n",
              "        canvas.attr('height', height * mpl.ratio);\n",
              "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
              "\n",
              "        rubberband.attr('width', width);\n",
              "        rubberband.attr('height', height);\n",
              "    }\n",
              "\n",
              "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
              "    // upon first draw.\n",
              "    this._resize_canvas(600, 600);\n",
              "\n",
              "    // Disable right mouse context menu.\n",
              "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
              "        return false;\n",
              "    });\n",
              "\n",
              "    function set_focus () {\n",
              "        canvas.focus();\n",
              "        canvas_div.focus();\n",
              "    }\n",
              "\n",
              "    window.setTimeout(set_focus, 100);\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._init_toolbar = function() {\n",
              "    var fig = this;\n",
              "\n",
              "    var nav_element = $('<div/>');\n",
              "    nav_element.attr('style', 'width: 100%');\n",
              "    this.root.append(nav_element);\n",
              "\n",
              "    // Define a callback function for later on.\n",
              "    function toolbar_event(event) {\n",
              "        return fig.toolbar_button_onclick(event['data']);\n",
              "    }\n",
              "    function toolbar_mouse_event(event) {\n",
              "        return fig.toolbar_button_onmouseover(event['data']);\n",
              "    }\n",
              "\n",
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              "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
              "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
              "\n",
              "        if (!name) {\n",
              "            // put a spacer in here.\n",
              "            continue;\n",
              "        }\n",
              "        var button = $('<button/>');\n",
              "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
              "                        'ui-button-icon-only');\n",
              "        button.attr('role', 'button');\n",
              "        button.attr('aria-disabled', 'false');\n",
              "        button.click(method_name, toolbar_event);\n",
              "        button.mouseover(tooltip, toolbar_mouse_event);\n",
              "\n",
              "        var icon_img = $('<span/>');\n",
              "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
              "        icon_img.addClass(image);\n",
              "        icon_img.addClass('ui-corner-all');\n",
              "\n",
              "        var tooltip_span = $('<span/>');\n",
              "        tooltip_span.addClass('ui-button-text');\n",
              "        tooltip_span.html(tooltip);\n",
              "\n",
              "        button.append(icon_img);\n",
              "        button.append(tooltip_span);\n",
              "\n",
              "        nav_element.append(button);\n",
              "    }\n",
              "\n",
              "    var fmt_picker_span = $('<span/>');\n",
              "\n",
              "    var fmt_picker = $('<select/>');\n",
              "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
              "    fmt_picker_span.append(fmt_picker);\n",
              "    nav_element.append(fmt_picker_span);\n",
              "    this.format_dropdown = fmt_picker[0];\n",
              "\n",
              "    for (var ind in mpl.extensions) {\n",
              "        var fmt = mpl.extensions[ind];\n",
              "        var option = $(\n",
              "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
              "        fmt_picker.append(option);\n",
              "    }\n",
              "\n",
              "    // Add hover states to the ui-buttons\n",
              "    $( \".ui-button\" ).hover(\n",
              "        function() { $(this).addClass(\"ui-state-hover\");},\n",
              "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
              "    );\n",
              "\n",
              "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
              "    nav_element.append(status_bar);\n",
              "    this.message = status_bar[0];\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
              "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
              "    // which will in turn request a refresh of the image.\n",
              "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.send_message = function(type, properties) {\n",
              "    properties['type'] = type;\n",
              "    properties['figure_id'] = this.id;\n",
              "    this.ws.send(JSON.stringify(properties));\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.send_draw_message = function() {\n",
              "    if (!this.waiting) {\n",
              "        this.waiting = true;\n",
              "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
              "    }\n",
              "}\n",
              "\n",
              "\n",
              "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
              "    var format_dropdown = fig.format_dropdown;\n",
              "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
              "    fig.ondownload(fig, format);\n",
              "}\n",
              "\n",
              "\n",
              "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
              "    var size = msg['size'];\n",
              "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
              "        fig._resize_canvas(size[0], size[1]);\n",
              "        fig.send_message(\"refresh\", {});\n",
              "    };\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
              "    var x0 = msg['x0'] / mpl.ratio;\n",
              "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
              "    var x1 = msg['x1'] / mpl.ratio;\n",
              "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
              "    x0 = Math.floor(x0) + 0.5;\n",
              "    y0 = Math.floor(y0) + 0.5;\n",
              "    x1 = Math.floor(x1) + 0.5;\n",
              "    y1 = Math.floor(y1) + 0.5;\n",
              "    var min_x = Math.min(x0, x1);\n",
              "    var min_y = Math.min(y0, y1);\n",
              "    var width = Math.abs(x1 - x0);\n",
              "    var height = Math.abs(y1 - y0);\n",
              "\n",
              "    fig.rubberband_context.clearRect(\n",
              "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
              "\n",
              "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
              "    // Updates the figure title.\n",
              "    fig.header.textContent = msg['label'];\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
              "    var cursor = msg['cursor'];\n",
              "    switch(cursor)\n",
              "    {\n",
              "    case 0:\n",
              "        cursor = 'pointer';\n",
              "        break;\n",
              "    case 1:\n",
              "        cursor = 'default';\n",
              "        break;\n",
              "    case 2:\n",
              "        cursor = 'crosshair';\n",
              "        break;\n",
              "    case 3:\n",
              "        cursor = 'move';\n",
              "        break;\n",
              "    }\n",
              "    fig.rubberband_canvas.style.cursor = cursor;\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
              "    fig.message.textContent = msg['message'];\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
              "    // Request the server to send over a new figure.\n",
              "    fig.send_draw_message();\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
              "    fig.image_mode = msg['mode'];\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.updated_canvas_event = function() {\n",
              "    // Called whenever the canvas gets updated.\n",
              "    this.send_message(\"ack\", {});\n",
              "}\n",
              "\n",
              "// A function to construct a web socket function for onmessage handling.\n",
              "// Called in the figure constructor.\n",
              "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
              "    return function socket_on_message(evt) {\n",
              "        if (evt.data instanceof Blob) {\n",
              "            /* FIXME: We get \"Resource interpreted as Image but\n",
              "             * transferred with MIME type text/plain:\" errors on\n",
              "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
              "             * to be part of the websocket stream */\n",
              "            evt.data.type = \"image/png\";\n",
              "\n",
              "            /* Free the memory for the previous frames */\n",
              "            if (fig.imageObj.src) {\n",
              "                (window.URL || window.webkitURL).revokeObjectURL(\n",
              "                    fig.imageObj.src);\n",
              "            }\n",
              "\n",
              "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
              "                evt.data);\n",
              "            fig.updated_canvas_event();\n",
              "            fig.waiting = false;\n",
              "            return;\n",
              "        }\n",
              "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
              "            fig.imageObj.src = evt.data;\n",
              "            fig.updated_canvas_event();\n",
              "            fig.waiting = false;\n",
              "            return;\n",
              "        }\n",
              "\n",
              "        var msg = JSON.parse(evt.data);\n",
              "        var msg_type = msg['type'];\n",
              "\n",
              "        // Call the  \"handle_{type}\" callback, which takes\n",
              "        // the figure and JSON message as its only arguments.\n",
              "        try {\n",
              "            var callback = fig[\"handle_\" + msg_type];\n",
              "        } catch (e) {\n",
              "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
              "            return;\n",
              "        }\n",
              "\n",
              "        if (callback) {\n",
              "            try {\n",
              "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
              "                callback(fig, msg);\n",
              "            } catch (e) {\n",
              "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
              "            }\n",
              "        }\n",
              "    };\n",
              "}\n",
              "\n",
              "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
              "mpl.findpos = function(e) {\n",
              "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
              "    var targ;\n",
              "    if (!e)\n",
              "        e = window.event;\n",
              "    if (e.target)\n",
              "        targ = e.target;\n",
              "    else if (e.srcElement)\n",
              "        targ = e.srcElement;\n",
              "    if (targ.nodeType == 3) // defeat Safari bug\n",
              "        targ = targ.parentNode;\n",
              "\n",
              "    // jQuery normalizes the pageX and pageY\n",
              "    // pageX,Y are the mouse positions relative to the document\n",
              "    // offset() returns the position of the element relative to the document\n",
              "    var x = e.pageX - $(targ).offset().left;\n",
              "    var y = e.pageY - $(targ).offset().top;\n",
              "\n",
              "    return {\"x\": x, \"y\": y};\n",
              "};\n",
              "\n",
              "/*\n",
              " * return a copy of an object with only non-object keys\n",
              " * we need this to avoid circular references\n",
              " * http://stackoverflow.com/a/24161582/3208463\n",
              " */\n",
              "function simpleKeys (original) {\n",
              "  return Object.keys(original).reduce(function (obj, key) {\n",
              "    if (typeof original[key] !== 'object')\n",
              "        obj[key] = original[key]\n",
              "    return obj;\n",
              "  }, {});\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.mouse_event = function(event, name) {\n",
              "    var canvas_pos = mpl.findpos(event)\n",
              "\n",
              "    if (name === 'button_press')\n",
              "    {\n",
              "        this.canvas.focus();\n",
              "        this.canvas_div.focus();\n",
              "    }\n",
              "\n",
              "    var x = canvas_pos.x * mpl.ratio;\n",
              "    var y = canvas_pos.y * mpl.ratio;\n",
              "\n",
              "    this.send_message(name, {x: x, y: y, button: event.button,\n",
              "                             step: event.step,\n",
              "                             guiEvent: simpleKeys(event)});\n",
              "\n",
              "    /* This prevents the web browser from automatically changing to\n",
              "     * the text insertion cursor when the button is pressed.  We want\n",
              "     * to control all of the cursor setting manually through the\n",
              "     * 'cursor' event from matplotlib */\n",
              "    event.preventDefault();\n",
              "    return false;\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
              "    // Handle any extra behaviour associated with a key event\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.key_event = function(event, name) {\n",
              "\n",
              "    // Prevent repeat events\n",
              "    if (name == 'key_press')\n",
              "    {\n",
              "        if (event.which === this._key)\n",
              "            return;\n",
              "        else\n",
              "            this._key = event.which;\n",
              "    }\n",
              "    if (name == 'key_release')\n",
              "        this._key = null;\n",
              "\n",
              "    var value = '';\n",
              "    if (event.ctrlKey && event.which != 17)\n",
              "        value += \"ctrl+\";\n",
              "    if (event.altKey && event.which != 18)\n",
              "        value += \"alt+\";\n",
              "    if (event.shiftKey && event.which != 16)\n",
              "        value += \"shift+\";\n",
              "\n",
              "    value += 'k';\n",
              "    value += event.which.toString();\n",
              "\n",
              "    this._key_event_extra(event, name);\n",
              "\n",
              "    this.send_message(name, {key: value,\n",
              "                             guiEvent: simpleKeys(event)});\n",
              "    return false;\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
              "    if (name == 'download') {\n",
              "        this.handle_save(this, null);\n",
              "    } else {\n",
              "        this.send_message(\"toolbar_button\", {name: name});\n",
              "    }\n",
              "};\n",
              "\n",
              "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
              "    this.message.textContent = tooltip;\n",
              "};\n",
              "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
              "\n",
              "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
              "\n",
              "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
              "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
              "    // object with the appropriate methods. Currently this is a non binary\n",
              "    // socket, so there is still some room for performance tuning.\n",
              "    var ws = {};\n",
              "\n",
              "    ws.close = function() {\n",
              "        comm.close()\n",
              "    };\n",
              "    ws.send = function(m) {\n",
              "        //console.log('sending', m);\n",
              "        comm.send(m);\n",
              "    };\n",
              "    // Register the callback with on_msg.\n",
              "    comm.on_msg(function(msg) {\n",
              "        //console.log('receiving', msg['content']['data'], msg);\n",
              "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
              "        ws.onmessage(msg['content']['data'])\n",
              "    });\n",
              "    return ws;\n",
              "}\n",
              "\n",
              "mpl.mpl_figure_comm = function(comm, msg) {\n",
              "    // This is the function which gets called when the mpl process\n",
              "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
              "\n",
              "    var id = msg.content.data.id;\n",
              "    // Get hold of the div created by the display call when the Comm\n",
              "    // socket was opened in Python.\n",
              "    var element = $(\"#\" + id);\n",
              "    var ws_proxy = comm_websocket_adapter(comm)\n",
              "\n",
              "    function ondownload(figure, format) {\n",
              "        window.open(figure.imageObj.src);\n",
              "    }\n",
              "\n",
              "    var fig = new mpl.figure(id, ws_proxy,\n",
              "                           ondownload,\n",
              "                           element.get(0));\n",
              "\n",
              "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
              "    // web socket which is closed, not our websocket->open comm proxy.\n",
              "    ws_proxy.onopen();\n",
              "\n",
              "    fig.parent_element = element.get(0);\n",
              "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
              "    if (!fig.cell_info) {\n",
              "        console.error(\"Failed to find cell for figure\", id, fig);\n",
              "        return;\n",
              "    }\n",
              "\n",
              "    var output_index = fig.cell_info[2]\n",
              "    var cell = fig.cell_info[0];\n",
              "\n",
              "};\n",
              "\n",
              "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
              "    var width = fig.canvas.width/mpl.ratio\n",
              "    fig.root.unbind('remove')\n",
              "\n",
              "    // Update the output cell to use the data from the current canvas.\n",
              "    fig.push_to_output();\n",
              "    var dataURL = fig.canvas.toDataURL();\n",
              "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
              "    // the notebook keyboard shortcuts fail.\n",
              "    IPython.keyboard_manager.enable()\n",
              "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
              "    fig.close_ws(fig, msg);\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.close_ws = function(fig, msg){\n",
              "    fig.send_message('closing', msg);\n",
              "    // fig.ws.close()\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
              "    // Turn the data on the canvas into data in the output cell.\n",
              "    var width = this.canvas.width/mpl.ratio\n",
              "    var dataURL = this.canvas.toDataURL();\n",
              "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.updated_canvas_event = function() {\n",
              "    // Tell IPython that the notebook contents must change.\n",
              "    IPython.notebook.set_dirty(true);\n",
              "    this.send_message(\"ack\", {});\n",
              "    var fig = this;\n",
              "    // Wait a second, then push the new image to the DOM so\n",
              "    // that it is saved nicely (might be nice to debounce this).\n",
              "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._init_toolbar = function() {\n",
              "    var fig = this;\n",
              "\n",
              "    var nav_element = $('<div/>');\n",
              "    nav_element.attr('style', 'width: 100%');\n",
              "    this.root.append(nav_element);\n",
              "\n",
              "    // Define a callback function for later on.\n",
              "    function toolbar_event(event) {\n",
              "        return fig.toolbar_button_onclick(event['data']);\n",
              "    }\n",
              "    function toolbar_mouse_event(event) {\n",
              "        return fig.toolbar_button_onmouseover(event['data']);\n",
              "    }\n",
              "\n",
              "    for(var toolbar_ind in mpl.toolbar_items){\n",
              "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
              "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
              "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
              "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
              "\n",
              "        if (!name) { continue; };\n",
              "\n",
              "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
              "        button.click(method_name, toolbar_event);\n",
              "        button.mouseover(tooltip, toolbar_mouse_event);\n",
              "        nav_element.append(button);\n",
              "    }\n",
              "\n",
              "    // Add the status bar.\n",
              "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
              "    nav_element.append(status_bar);\n",
              "    this.message = status_bar[0];\n",
              "\n",
              "    // Add the close button to the window.\n",
              "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
              "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
              "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
              "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
              "    buttongrp.append(button);\n",
              "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
              "    titlebar.prepend(buttongrp);\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._root_extra_style = function(el){\n",
              "    var fig = this\n",
              "    el.on(\"remove\", function(){\n",
              "\tfig.close_ws(fig, {});\n",
              "    });\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._canvas_extra_style = function(el){\n",
              "    // this is important to make the div 'focusable\n",
              "    el.attr('tabindex', 0)\n",
              "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
              "    // off when our div gets focus\n",
              "\n",
              "    // location in version 3\n",
              "    if (IPython.notebook.keyboard_manager) {\n",
              "        IPython.notebook.keyboard_manager.register_events(el);\n",
              "    }\n",
              "    else {\n",
              "        // location in version 2\n",
              "        IPython.keyboard_manager.register_events(el);\n",
              "    }\n",
              "\n",
              "}\n",
              "\n",
              "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
              "    var manager = IPython.notebook.keyboard_manager;\n",
              "    if (!manager)\n",
              "        manager = IPython.keyboard_manager;\n",
              "\n",
              "    // Check for shift+enter\n",
              "    if (event.shiftKey && event.which == 13) {\n",
              "        this.canvas_div.blur();\n",
              "        // select the cell after this one\n",
              "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
              "        IPython.notebook.select(index + 1);\n",
              "    }\n",
              "}\n",
              "\n",
              "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
              "    fig.ondownload(fig, null);\n",
              "}\n",
              "\n",
              "\n",
              "mpl.find_output_cell = function(html_output) {\n",
              "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
              "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
              "    // IPython event is triggered only after the cells have been serialised, which for\n",
              "    // our purposes (turning an active figure into a static one), is too late.\n",
              "    var cells = IPython.notebook.get_cells();\n",
              "    var ncells = cells.length;\n",
              "    for (var i=0; i<ncells; i++) {\n",
              "        var cell = cells[i];\n",
              "        if (cell.cell_type === 'code'){\n",
              "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
              "                var data = cell.output_area.outputs[j];\n",
              "                if (data.data) {\n",
              "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
              "                    data = data.data;\n",
              "                }\n",
              "                if (data['text/html'] == html_output) {\n",
              "                    return [cell, data, j];\n",
              "                }\n",
              "            }\n",
              "        }\n",
              "    }\n",
              "}\n",
              "\n",
              "// Register the function which deals with the matplotlib target/channel.\n",
              "// The kernel may be null if the page has been refreshed.\n",
              "if (IPython.notebook.kernel != null) {\n",
              "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
              "}\n"
            ],
            "text/plain": [
              "<IPython.core.display.Javascript object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "<div id='acde8980-d232-405f-92f9-ee122b5a829b'></div>"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GdR_yJpCELCA"
      },
      "source": [
        "Value-at-Risk backtesting:\n",
        "here the binomial one-tailed test is used"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XCj_nmDdEKAJ",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "a6e1951f-34d7-4f44-c14d-c099408bf032"
      },
      "source": [
        "# Confidence level for backtesting\n",
        "CL_backtesting = 95\n",
        "# The probability of the VaR beeing exceeded on given day\n",
        "p = 1 - CL_VaR/100\n",
        "N_confidence = binom.ppf( CL_backtesting*0.01, N_history, p)\n",
        "N_exceeded = sum( v < Value_at_Risk + Value_current for v in Market_scienarios)\n",
        "\n",
        "print(\"We look at total\", N_history,\"months. VaR level is reached at\", N_exceeded , \"months.\")\n",
        "if N_exceeded <= N_confidence:\n",
        "  print (\"This VaR model is accepted at\", CL_backtesting,\"% confidence level.\")\n",
        "else:\n",
        "  print (\"This VaR model is rejected at\", CL_backtesting,\"% confidence level.\")"
      ],
      "execution_count": 127,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "We look at total 356 months. VaR level is reached at 18 months.\n",
            "This VaR model is accepted at 95 % confidence level.\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ImAjg4ceBJgf"
      },
      "source": [
        ""
      ],
      "execution_count": null,
      "outputs": []
    }
  ]
}