Projects/1-Trading-with-momentum/project_1_starter.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Project 1: Trading with Momentum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window.Plotly) {{require(['plotly'],function(plotly) {window.Plotly=plotly;});}}</script>"
      ],
      "text/vnd.plotly.v1+html": [
       "<script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window.Plotly) {{require(['plotly'],function(plotly) {window.Plotly=plotly;});}}</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import helper\n",
    "import project_helper\n",
    "import project_tests"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Market Data\n",
    "### Load Data\n",
    "The data we use for most of the projects is end of day data. This contains data for many stocks, but we'll be looking at stocks in the S&P 500. We also made things a little easier to run by narrowing down our range of time period instead of using all of the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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       "<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>ticker</th>\n",
       "      <th>A</th>\n",
       "      <th>AAL</th>\n",
       "      <th>AAP</th>\n",
       "      <th>AAPL</th>\n",
       "      <th>ABBV</th>\n",
       "      <th>ABC</th>\n",
       "      <th>ABT</th>\n",
       "      <th>ACN</th>\n",
       "      <th>ADBE</th>\n",
       "      <th>ADI</th>\n",
       "      <th>...</th>\n",
       "      <th>XL</th>\n",
       "      <th>XLNX</th>\n",
       "      <th>XOM</th>\n",
       "      <th>XRAY</th>\n",
       "      <th>XRX</th>\n",
       "      <th>XYL</th>\n",
       "      <th>YUM</th>\n",
       "      <th>ZBH</th>\n",
       "      <th>ZION</th>\n",
       "      <th>ZTS</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2013-07-01</th>\n",
       "      <td>29.99418563</td>\n",
       "      <td>16.17609308</td>\n",
       "      <td>81.13821681</td>\n",
       "      <td>53.10917319</td>\n",
       "      <td>34.92447839</td>\n",
       "      <td>50.86319750</td>\n",
       "      <td>31.42538772</td>\n",
       "      <td>64.69409505</td>\n",
       "      <td>46.23500000</td>\n",
       "      <td>39.91336014</td>\n",
       "      <td>...</td>\n",
       "      <td>27.66879066</td>\n",
       "      <td>35.28892781</td>\n",
       "      <td>76.32080247</td>\n",
       "      <td>40.02387348</td>\n",
       "      <td>22.10666494</td>\n",
       "      <td>25.75338607</td>\n",
       "      <td>45.48038323</td>\n",
       "      <td>71.89882693</td>\n",
       "      <td>27.85858718</td>\n",
       "      <td>29.44789315</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-07-02</th>\n",
       "      <td>29.65013670</td>\n",
       "      <td>15.81983388</td>\n",
       "      <td>80.72207258</td>\n",
       "      <td>54.31224742</td>\n",
       "      <td>35.42807578</td>\n",
       "      <td>50.69676639</td>\n",
       "      <td>31.27288084</td>\n",
       "      <td>64.71204071</td>\n",
       "      <td>46.03000000</td>\n",
       "      <td>39.86057632</td>\n",
       "      <td>...</td>\n",
       "      <td>27.54228410</td>\n",
       "      <td>35.05903252</td>\n",
       "      <td>76.60816761</td>\n",
       "      <td>39.96552964</td>\n",
       "      <td>22.08273998</td>\n",
       "      <td>25.61367511</td>\n",
       "      <td>45.40266113</td>\n",
       "      <td>72.93417195</td>\n",
       "      <td>28.03893238</td>\n",
       "      <td>28.57244125</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-07-03</th>\n",
       "      <td>29.70518453</td>\n",
       "      <td>16.12794994</td>\n",
       "      <td>81.23729877</td>\n",
       "      <td>54.61204262</td>\n",
       "      <td>35.44486235</td>\n",
       "      <td>50.93716689</td>\n",
       "      <td>30.72565028</td>\n",
       "      <td>65.21451912</td>\n",
       "      <td>46.42000000</td>\n",
       "      <td>40.18607651</td>\n",
       "      <td>...</td>\n",
       "      <td>27.33445191</td>\n",
       "      <td>35.28008569</td>\n",
       "      <td>76.65042719</td>\n",
       "      <td>40.00442554</td>\n",
       "      <td>22.20236479</td>\n",
       "      <td>25.73475794</td>\n",
       "      <td>46.06329899</td>\n",
       "      <td>72.30145844</td>\n",
       "      <td>28.18131017</td>\n",
       "      <td>28.16838652</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-07-05</th>\n",
       "      <td>30.43456826</td>\n",
       "      <td>16.21460758</td>\n",
       "      <td>81.82188233</td>\n",
       "      <td>54.17338125</td>\n",
       "      <td>35.85613355</td>\n",
       "      <td>51.37173702</td>\n",
       "      <td>31.32670680</td>\n",
       "      <td>66.07591068</td>\n",
       "      <td>47.00000000</td>\n",
       "      <td>40.65233352</td>\n",
       "      <td>...</td>\n",
       "      <td>27.69589920</td>\n",
       "      <td>35.80177117</td>\n",
       "      <td>77.39419581</td>\n",
       "      <td>40.67537968</td>\n",
       "      <td>22.58516418</td>\n",
       "      <td>26.06075017</td>\n",
       "      <td>46.41304845</td>\n",
       "      <td>73.16424628</td>\n",
       "      <td>29.39626730</td>\n",
       "      <td>29.02459772</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-07-08</th>\n",
       "      <td>30.52402098</td>\n",
       "      <td>16.31089385</td>\n",
       "      <td>82.95141667</td>\n",
       "      <td>53.86579916</td>\n",
       "      <td>36.66188936</td>\n",
       "      <td>52.03746147</td>\n",
       "      <td>31.76628544</td>\n",
       "      <td>66.82065546</td>\n",
       "      <td>46.62500000</td>\n",
       "      <td>40.25645492</td>\n",
       "      <td>...</td>\n",
       "      <td>27.98505704</td>\n",
       "      <td>35.20050655</td>\n",
       "      <td>77.96892611</td>\n",
       "      <td>40.64620776</td>\n",
       "      <td>22.48946433</td>\n",
       "      <td>26.22840332</td>\n",
       "      <td>46.95062632</td>\n",
       "      <td>73.89282298</td>\n",
       "      <td>29.57661249</td>\n",
       "      <td>29.76536472</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 495 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "ticker               A         AAL         AAP        AAPL        ABBV  \\\n",
       "date                                                                     \n",
       "2013-07-01 29.99418563 16.17609308 81.13821681 53.10917319 34.92447839   \n",
       "2013-07-02 29.65013670 15.81983388 80.72207258 54.31224742 35.42807578   \n",
       "2013-07-03 29.70518453 16.12794994 81.23729877 54.61204262 35.44486235   \n",
       "2013-07-05 30.43456826 16.21460758 81.82188233 54.17338125 35.85613355   \n",
       "2013-07-08 30.52402098 16.31089385 82.95141667 53.86579916 36.66188936   \n",
       "\n",
       "ticker             ABC         ABT         ACN        ADBE         ADI  \\\n",
       "date                                                                     \n",
       "2013-07-01 50.86319750 31.42538772 64.69409505 46.23500000 39.91336014   \n",
       "2013-07-02 50.69676639 31.27288084 64.71204071 46.03000000 39.86057632   \n",
       "2013-07-03 50.93716689 30.72565028 65.21451912 46.42000000 40.18607651   \n",
       "2013-07-05 51.37173702 31.32670680 66.07591068 47.00000000 40.65233352   \n",
       "2013-07-08 52.03746147 31.76628544 66.82065546 46.62500000 40.25645492   \n",
       "\n",
       "ticker         ...              XL        XLNX         XOM        XRAY  \\\n",
       "date           ...                                                       \n",
       "2013-07-01     ...     27.66879066 35.28892781 76.32080247 40.02387348   \n",
       "2013-07-02     ...     27.54228410 35.05903252 76.60816761 39.96552964   \n",
       "2013-07-03     ...     27.33445191 35.28008569 76.65042719 40.00442554   \n",
       "2013-07-05     ...     27.69589920 35.80177117 77.39419581 40.67537968   \n",
       "2013-07-08     ...     27.98505704 35.20050655 77.96892611 40.64620776   \n",
       "\n",
       "ticker             XRX         XYL         YUM         ZBH        ZION  \\\n",
       "date                                                                     \n",
       "2013-07-01 22.10666494 25.75338607 45.48038323 71.89882693 27.85858718   \n",
       "2013-07-02 22.08273998 25.61367511 45.40266113 72.93417195 28.03893238   \n",
       "2013-07-03 22.20236479 25.73475794 46.06329899 72.30145844 28.18131017   \n",
       "2013-07-05 22.58516418 26.06075017 46.41304845 73.16424628 29.39626730   \n",
       "2013-07-08 22.48946433 26.22840332 46.95062632 73.89282298 29.57661249   \n",
       "\n",
       "ticker             ZTS  \n",
       "date                    \n",
       "2013-07-01 29.44789315  \n",
       "2013-07-02 28.57244125  \n",
       "2013-07-03 28.16838652  \n",
       "2013-07-05 29.02459772  \n",
       "2013-07-08 29.76536472  \n",
       "\n",
       "[5 rows x 495 columns]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv('eod-quotemedia.csv', parse_dates=['date'], index_col=False)\n",
    "\n",
    "close = df.reset_index().pivot(index='date', columns='ticker', values='adj_close')\n",
    "\n",
    "close.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Stock Example\n",
    "Let's see what a single stock looks like from the closing prices. For this example and future display examples in this project, we'll use Apple's stock (AAPL). If we tried to graph all the stocks, it would be too much information."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "apple_ticker = 'AAPL'\n",
    "project_helper.plot_stock(close[apple_ticker], '{} Stock'.format(apple_ticker))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Resample Adjusted Prices\n",
    "\n",
    "The trading signal you'll develop in this project does not need to be based on daily prices, for instance, you can use month-end prices to perform trading once a month. To do this, you must first resample the daily adjusted closing prices into monthly buckets, and select the last observation of each month.\n",
    "\n",
    "Implement the `resample_prices` to resample `close_prices` at the sampling frequency of `freq`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tests Passed\n"
     ]
    }
   ],
   "source": [
    "def resample_prices(close_prices, freq='M'):\n",
    "    \"\"\"\n",
    "    Resample close prices for each ticker at specified frequency.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    close_prices : DataFrame\n",
    "        Close prices for each ticker and date\n",
    "    freq : str\n",
    "        What frequency to sample at\n",
    "        For valid freq choices, see http://pandas.pydata.org/pandas-docs/stable/timeseries.html#offset-aliases\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    prices_resampled : DataFrame\n",
    "        Resampled prices for each ticker and date\n",
    "    \"\"\"\n",
    "    \n",
    "    return close_prices.resample(freq).last()\n",
    "\n",
    "project_tests.test_resample_prices(resample_prices)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### View Data\n",
    "Let's apply this function to `close` and view the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>ABC</th>\n",
       "      <th>ABT</th>\n",
       "      <th>ACN</th>\n",
       "      <th>ADBE</th>\n",
       "      <th>ADI</th>\n",
       "      <th>...</th>\n",
       "      <th>XL</th>\n",
       "      <th>XLNX</th>\n",
       "      <th>XOM</th>\n",
       "      <th>XRAY</th>\n",
       "      <th>XRX</th>\n",
       "      <th>XYL</th>\n",
       "      <th>YUM</th>\n",
       "      <th>ZBH</th>\n",
       "      <th>ZION</th>\n",
       "      <th>ZTS</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2013-07-31</th>\n",
       "      <td>30.77861719</td>\n",
       "      <td>18.63139292</td>\n",
       "      <td>81.73270857</td>\n",
       "      <td>58.73000866</td>\n",
       "      <td>38.52144972</td>\n",
       "      <td>53.87744989</td>\n",
       "      <td>32.99081455</td>\n",
       "      <td>66.22844876</td>\n",
       "      <td>47.28000000</td>\n",
       "      <td>43.44107832</td>\n",
       "      <td>...</td>\n",
       "      <td>28.32843198</td>\n",
       "      <td>41.28388974</td>\n",
       "      <td>79.23671352</td>\n",
       "      <td>41.69639686</td>\n",
       "      <td>23.20721320</td>\n",
       "      <td>23.21996075</td>\n",
       "      <td>47.44778390</td>\n",
       "      <td>80.02819050</td>\n",
       "      <td>28.13385091</td>\n",
       "      <td>28.74031861</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-08-31</th>\n",
       "      <td>32.09288410</td>\n",
       "      <td>15.55986096</td>\n",
       "      <td>79.33492514</td>\n",
       "      <td>63.64994327</td>\n",
       "      <td>36.09056668</td>\n",
       "      <td>53.01732111</td>\n",
       "      <td>30.01866909</td>\n",
       "      <td>64.82868748</td>\n",
       "      <td>45.75000000</td>\n",
       "      <td>41.01373554</td>\n",
       "      <td>...</td>\n",
       "      <td>26.71095532</td>\n",
       "      <td>38.60058125</td>\n",
       "      <td>74.17847651</td>\n",
       "      <td>40.83096325</td>\n",
       "      <td>23.87711213</td>\n",
       "      <td>23.18866549</td>\n",
       "      <td>45.56080402</td>\n",
       "      <td>75.81970579</td>\n",
       "      <td>26.58466104</td>\n",
       "      <td>28.10400159</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-09-30</th>\n",
       "      <td>35.34697923</td>\n",
       "      <td>18.25587647</td>\n",
       "      <td>81.98212977</td>\n",
       "      <td>62.28266407</td>\n",
       "      <td>37.88620154</td>\n",
       "      <td>56.91072241</td>\n",
       "      <td>29.89257807</td>\n",
       "      <td>66.07591068</td>\n",
       "      <td>51.94000000</td>\n",
       "      <td>41.69611619</td>\n",
       "      <td>...</td>\n",
       "      <td>27.97701372</td>\n",
       "      <td>41.65075616</td>\n",
       "      <td>73.22528819</td>\n",
       "      <td>42.27738327</td>\n",
       "      <td>24.75516170</td>\n",
       "      <td>26.13637721</td>\n",
       "      <td>46.45223934</td>\n",
       "      <td>78.93635707</td>\n",
       "      <td>26.06190225</td>\n",
       "      <td>30.00331147</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-10-31</th>\n",
       "      <td>35.00902763</td>\n",
       "      <td>21.15409315</td>\n",
       "      <td>98.34285959</td>\n",
       "      <td>68.28583759</td>\n",
       "      <td>41.39637606</td>\n",
       "      <td>60.85069550</td>\n",
       "      <td>33.05576095</td>\n",
       "      <td>66.82299697</td>\n",
       "      <td>54.22000000</td>\n",
       "      <td>43.69008561</td>\n",
       "      <td>...</td>\n",
       "      <td>27.75007493</td>\n",
       "      <td>40.38659157</td>\n",
       "      <td>76.27208656</td>\n",
       "      <td>45.87069305</td>\n",
       "      <td>23.91314940</td>\n",
       "      <td>32.41710942</td>\n",
       "      <td>44.24404652</td>\n",
       "      <td>84.05847719</td>\n",
       "      <td>26.98861102</td>\n",
       "      <td>30.58582019</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-11-30</th>\n",
       "      <td>36.94707663</td>\n",
       "      <td>22.60801580</td>\n",
       "      <td>100.15741326</td>\n",
       "      <td>73.07037475</td>\n",
       "      <td>41.39637606</td>\n",
       "      <td>65.91719111</td>\n",
       "      <td>34.53897430</td>\n",
       "      <td>70.43234797</td>\n",
       "      <td>56.78000000</td>\n",
       "      <td>42.73298029</td>\n",
       "      <td>...</td>\n",
       "      <td>29.03908724</td>\n",
       "      <td>39.71607441</td>\n",
       "      <td>80.10221273</td>\n",
       "      <td>46.31377055</td>\n",
       "      <td>27.37742858</td>\n",
       "      <td>32.47348700</td>\n",
       "      <td>50.82634626</td>\n",
       "      <td>87.84481134</td>\n",
       "      <td>27.91674534</td>\n",
       "      <td>30.09312378</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 495 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "ticker               A         AAL          AAP        AAPL        ABBV  \\\n",
       "date                                                                      \n",
       "2013-07-31 30.77861719 18.63139292  81.73270857 58.73000866 38.52144972   \n",
       "2013-08-31 32.09288410 15.55986096  79.33492514 63.64994327 36.09056668   \n",
       "2013-09-30 35.34697923 18.25587647  81.98212977 62.28266407 37.88620154   \n",
       "2013-10-31 35.00902763 21.15409315  98.34285959 68.28583759 41.39637606   \n",
       "2013-11-30 36.94707663 22.60801580 100.15741326 73.07037475 41.39637606   \n",
       "\n",
       "ticker             ABC         ABT         ACN        ADBE         ADI  \\\n",
       "date                                                                     \n",
       "2013-07-31 53.87744989 32.99081455 66.22844876 47.28000000 43.44107832   \n",
       "2013-08-31 53.01732111 30.01866909 64.82868748 45.75000000 41.01373554   \n",
       "2013-09-30 56.91072241 29.89257807 66.07591068 51.94000000 41.69611619   \n",
       "2013-10-31 60.85069550 33.05576095 66.82299697 54.22000000 43.69008561   \n",
       "2013-11-30 65.91719111 34.53897430 70.43234797 56.78000000 42.73298029   \n",
       "\n",
       "ticker         ...              XL        XLNX         XOM        XRAY  \\\n",
       "date           ...                                                       \n",
       "2013-07-31     ...     28.32843198 41.28388974 79.23671352 41.69639686   \n",
       "2013-08-31     ...     26.71095532 38.60058125 74.17847651 40.83096325   \n",
       "2013-09-30     ...     27.97701372 41.65075616 73.22528819 42.27738327   \n",
       "2013-10-31     ...     27.75007493 40.38659157 76.27208656 45.87069305   \n",
       "2013-11-30     ...     29.03908724 39.71607441 80.10221273 46.31377055   \n",
       "\n",
       "ticker             XRX         XYL         YUM         ZBH        ZION  \\\n",
       "date                                                                     \n",
       "2013-07-31 23.20721320 23.21996075 47.44778390 80.02819050 28.13385091   \n",
       "2013-08-31 23.87711213 23.18866549 45.56080402 75.81970579 26.58466104   \n",
       "2013-09-30 24.75516170 26.13637721 46.45223934 78.93635707 26.06190225   \n",
       "2013-10-31 23.91314940 32.41710942 44.24404652 84.05847719 26.98861102   \n",
       "2013-11-30 27.37742858 32.47348700 50.82634626 87.84481134 27.91674534   \n",
       "\n",
       "ticker             ZTS  \n",
       "date                    \n",
       "2013-07-31 28.74031861  \n",
       "2013-08-31 28.10400159  \n",
       "2013-09-30 30.00331147  \n",
       "2013-10-31 30.58582019  \n",
       "2013-11-30 30.09312378  \n",
       "\n",
       "[5 rows x 495 columns]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "monthly_close = resample_prices(close)\n",
    "monthly_close.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Compute Log Returns\n",
    "\n",
    "Compute log returns ($R_t$) from prices ($P_t$) as your primary momentum indicator:\n",
    "\n",
    "$$R_t = log_e(P_t) - log_e(P_{t-1})$$\n",
    "\n",
    "Implement the `compute_log_returns` function below, such that it accepts a dataframe (like one returned by `resample_prices`), and produces a similar dataframe of log returns. Use Numpy's [log function](https://docs.scipy.org/doc/numpy/reference/generated/numpy.log.html) to help you calculate the log returns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tests Passed\n"
     ]
    }
   ],
   "source": [
    "def compute_log_returns(prices):\n",
    "    \"\"\"\n",
    "    Compute log returns for each ticker.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    prices : DataFrame\n",
    "        Prices for each ticker and date\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    log_returns : DataFrame\n",
    "        Log returns for each ticker and date\n",
    "    \"\"\"\n",
    "    previous_prices = prices.shift(periods = 1)\n",
    "    \n",
    "    return np.log(prices / previous_prices)\n",
    "\n",
    "project_tests.test_compute_log_returns(compute_log_returns)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### View Data\n",
    "Using the same data returned from `resample_prices`, we'll generate the log returns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<style scoped>\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>ticker</th>\n",
       "      <th>A</th>\n",
       "      <th>AAL</th>\n",
       "      <th>AAP</th>\n",
       "      <th>AAPL</th>\n",
       "      <th>ABBV</th>\n",
       "      <th>ABC</th>\n",
       "      <th>ABT</th>\n",
       "      <th>ACN</th>\n",
       "      <th>ADBE</th>\n",
       "      <th>ADI</th>\n",
       "      <th>...</th>\n",
       "      <th>XL</th>\n",
       "      <th>XLNX</th>\n",
       "      <th>XOM</th>\n",
       "      <th>XRAY</th>\n",
       "      <th>XRX</th>\n",
       "      <th>XYL</th>\n",
       "      <th>YUM</th>\n",
       "      <th>ZBH</th>\n",
       "      <th>ZION</th>\n",
       "      <th>ZTS</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2013-07-31</th>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>...</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "      <td>nan</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-08-31</th>\n",
       "      <td>0.04181412</td>\n",
       "      <td>-0.18015337</td>\n",
       "      <td>-0.02977582</td>\n",
       "      <td>0.08044762</td>\n",
       "      <td>-0.06518370</td>\n",
       "      <td>-0.01609335</td>\n",
       "      <td>-0.09440968</td>\n",
       "      <td>-0.02136190</td>\n",
       "      <td>-0.03289558</td>\n",
       "      <td>-0.05749847</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.05879217</td>\n",
       "      <td>-0.06720501</td>\n",
       "      <td>-0.06596571</td>\n",
       "      <td>-0.02097402</td>\n",
       "      <td>0.02845720</td>\n",
       "      <td>-0.00134868</td>\n",
       "      <td>-0.04058203</td>\n",
       "      <td>-0.05402072</td>\n",
       "      <td>-0.05663911</td>\n",
       "      <td>-0.02238899</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-09-30</th>\n",
       "      <td>0.09657861</td>\n",
       "      <td>0.15979244</td>\n",
       "      <td>0.03282284</td>\n",
       "      <td>-0.02171531</td>\n",
       "      <td>0.04855545</td>\n",
       "      <td>0.07086509</td>\n",
       "      <td>-0.00420927</td>\n",
       "      <td>0.01905603</td>\n",
       "      <td>0.12689741</td>\n",
       "      <td>0.01650096</td>\n",
       "      <td>...</td>\n",
       "      <td>0.04630944</td>\n",
       "      <td>0.07605219</td>\n",
       "      <td>-0.01293321</td>\n",
       "      <td>0.03481157</td>\n",
       "      <td>0.03611367</td>\n",
       "      <td>0.11966450</td>\n",
       "      <td>0.01937689</td>\n",
       "      <td>0.04028369</td>\n",
       "      <td>-0.01985983</td>\n",
       "      <td>0.06539579</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-10-31</th>\n",
       "      <td>-0.00960698</td>\n",
       "      <td>0.14734639</td>\n",
       "      <td>0.18195865</td>\n",
       "      <td>0.09201927</td>\n",
       "      <td>0.08860637</td>\n",
       "      <td>0.06693948</td>\n",
       "      <td>0.10058564</td>\n",
       "      <td>0.01124304</td>\n",
       "      <td>0.04296064</td>\n",
       "      <td>0.04671322</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.00814469</td>\n",
       "      <td>-0.03082169</td>\n",
       "      <td>0.04076620</td>\n",
       "      <td>0.08157415</td>\n",
       "      <td>-0.03460553</td>\n",
       "      <td>0.21535825</td>\n",
       "      <td>-0.04870385</td>\n",
       "      <td>0.06287079</td>\n",
       "      <td>0.03494040</td>\n",
       "      <td>0.01922875</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013-11-30</th>\n",
       "      <td>0.05388057</td>\n",
       "      <td>0.06647111</td>\n",
       "      <td>0.01828314</td>\n",
       "      <td>0.06772063</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.07997603</td>\n",
       "      <td>0.04389252</td>\n",
       "      <td>0.05260536</td>\n",
       "      <td>0.04613431</td>\n",
       "      <td>-0.02215021</td>\n",
       "      <td>...</td>\n",
       "      <td>0.04540422</td>\n",
       "      <td>-0.01674184</td>\n",
       "      <td>0.04899644</td>\n",
       "      <td>0.00961292</td>\n",
       "      <td>0.13529041</td>\n",
       "      <td>0.00173762</td>\n",
       "      <td>0.13869403</td>\n",
       "      <td>0.04405904</td>\n",
       "      <td>0.03381174</td>\n",
       "      <td>-0.01623981</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 495 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "ticker               A         AAL         AAP        AAPL        ABBV  \\\n",
       "date                                                                     \n",
       "2013-07-31         nan         nan         nan         nan         nan   \n",
       "2013-08-31  0.04181412 -0.18015337 -0.02977582  0.08044762 -0.06518370   \n",
       "2013-09-30  0.09657861  0.15979244  0.03282284 -0.02171531  0.04855545   \n",
       "2013-10-31 -0.00960698  0.14734639  0.18195865  0.09201927  0.08860637   \n",
       "2013-11-30  0.05388057  0.06647111  0.01828314  0.06772063  0.00000000   \n",
       "\n",
       "ticker             ABC         ABT         ACN        ADBE         ADI  \\\n",
       "date                                                                     \n",
       "2013-07-31         nan         nan         nan         nan         nan   \n",
       "2013-08-31 -0.01609335 -0.09440968 -0.02136190 -0.03289558 -0.05749847   \n",
       "2013-09-30  0.07086509 -0.00420927  0.01905603  0.12689741  0.01650096   \n",
       "2013-10-31  0.06693948  0.10058564  0.01124304  0.04296064  0.04671322   \n",
       "2013-11-30  0.07997603  0.04389252  0.05260536  0.04613431 -0.02215021   \n",
       "\n",
       "ticker         ...              XL        XLNX         XOM        XRAY  \\\n",
       "date           ...                                                       \n",
       "2013-07-31     ...             nan         nan         nan         nan   \n",
       "2013-08-31     ...     -0.05879217 -0.06720501 -0.06596571 -0.02097402   \n",
       "2013-09-30     ...      0.04630944  0.07605219 -0.01293321  0.03481157   \n",
       "2013-10-31     ...     -0.00814469 -0.03082169  0.04076620  0.08157415   \n",
       "2013-11-30     ...      0.04540422 -0.01674184  0.04899644  0.00961292   \n",
       "\n",
       "ticker             XRX         XYL         YUM         ZBH        ZION  \\\n",
       "date                                                                     \n",
       "2013-07-31         nan         nan         nan         nan         nan   \n",
       "2013-08-31  0.02845720 -0.00134868 -0.04058203 -0.05402072 -0.05663911   \n",
       "2013-09-30  0.03611367  0.11966450  0.01937689  0.04028369 -0.01985983   \n",
       "2013-10-31 -0.03460553  0.21535825 -0.04870385  0.06287079  0.03494040   \n",
       "2013-11-30  0.13529041  0.00173762  0.13869403  0.04405904  0.03381174   \n",
       "\n",
       "ticker             ZTS  \n",
       "date                    \n",
       "2013-07-31         nan  \n",
       "2013-08-31 -0.02238899  \n",
       "2013-09-30  0.06539579  \n",
       "2013-10-31  0.01922875  \n",
       "2013-11-30 -0.01623981  \n",
       "\n",
       "[5 rows x 495 columns]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "monthly_close_returns = compute_log_returns(monthly_close)\n",
    "monthly_close_returns.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Shift Returns\n",
    "Implement the `shift_returns` function to shift the log returns to the previous or future returns in the time series. For example, the parameter `shift_n` is 2 and `returns` is the following:\n",
    "\n",
    "```\n",
    "                           Returns\n",
    "               A         B         C         D\n",
    "2013-07-08     0.015     0.082     0.096     0.020     ...\n",
    "2013-07-09     0.037     0.095     0.027     0.063     ...\n",
    "2013-07-10     0.094     0.001     0.093     0.019     ...\n",
    "2013-07-11     0.092     0.057     0.069     0.087     ...\n",
    "...            ...       ...       ...       ...\n",
    "```\n",
    "\n",
    "the output of the `shift_returns` function would be:\n",
    "```\n",
    "                        Shift Returns\n",
    "               A         B         C         D\n",
    "2013-07-08     NaN       NaN       NaN       NaN       ...\n",
    "2013-07-09     NaN       NaN       NaN       NaN       ...\n",
    "2013-07-10     0.015     0.082     0.096     0.020     ...\n",
    "2013-07-11     0.037     0.095     0.027     0.063     ...\n",
    "...            ...       ...       ...       ...\n",
    "```\n",
    "Using the same `returns` data as above, the `shift_returns` function should generate the following with `shift_n` as -2:\n",
    "```\n",
    "                        Shift Returns\n",
    "               A         B         C         D\n",
    "2013-07-08     0.094     0.001     0.093     0.019     ...\n",
    "2013-07-09     0.092     0.057     0.069     0.087     ...\n",
    "...            ...       ...       ...       ...       ...\n",
    "...            ...       ...       ...       ...       ...\n",
    "...            NaN       NaN       NaN       NaN       ...\n",
    "...            NaN       NaN       NaN       NaN       ...\n",
    "```\n",
    "_Note: The \"...\" represents data points we're not showing._"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tests Passed\n"
     ]
    }
   ],
   "source": [
    "def shift_returns(returns, shift_n):\n",
    "    \"\"\"\n",
    "    Generate shifted returns\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    returns : DataFrame\n",
    "        Returns for each ticker and date\n",
    "    shift_n : int\n",
    "        Number of periods to move, can be positive or negative\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    shifted_returns : DataFrame\n",
    "        Shifted returns for each ticker and date\n",
    "    \"\"\"\n",
    "    \n",
    "    return returns.shift(periods = shift_n)\n",
    "\n",
    "project_tests.test_shift_returns(shift_returns)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### View Data\n",
    "Let's get the previous month's and next month's returns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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   "source": [
    "prev_returns = shift_returns(monthly_close_returns, 1)\n",
    "lookahead_returns = shift_returns(monthly_close_returns, -1)\n",
    "\n",
    "project_helper.plot_shifted_returns(\n",
    "    prev_returns.loc[:, apple_ticker],\n",
    "    monthly_close_returns.loc[:, apple_ticker],\n",
    "    'Previous Returns of {} Stock'.format(apple_ticker))\n",
    "project_helper.plot_shifted_returns(\n",
    "    lookahead_returns.loc[:, apple_ticker],\n",
    "    monthly_close_returns.loc[:, apple_ticker],\n",
    "    'Lookahead Returns of {} Stock'.format(apple_ticker))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generate Trading Signal\n",
    "\n",
    "A trading signal is a sequence of trading actions, or results that can be used to take trading actions. A common form is to produce a \"long\" and \"short\" portfolio of stocks on each date (e.g. end of each month, or whatever frequency you desire to trade at). This signal can be interpreted as rebalancing your portfolio on each of those dates, entering long (\"buy\") and short (\"sell\") positions as indicated.\n",
    "\n",
    "Here's a strategy that we will try:\n",
    "> For each month-end observation period, rank the stocks by _previous_ returns, from the highest to the lowest. Select the top performing stocks for the long portfolio, and the bottom performing stocks for the short portfolio.\n",
    "\n",
    "Implement the `get_top_n` function to get the top performing stock for each month. Get the top performing stocks from `prev_returns` by assigning them a value of 1. For all other stocks, give them a value of 0. For example, using the following `prev_returns`:\n",
    "\n",
    "```\n",
    "                                     Previous Returns\n",
    "               A         B         C         D         E         F         G\n",
    "2013-07-08     0.015     0.082     0.096     0.020     0.075     0.043     0.074\n",
    "2013-07-09     0.037     0.095     0.027     0.063     0.024     0.086     0.025\n",
    "...            ...       ...       ...       ...       ...       ...       ...\n",
    "```\n",
    "\n",
    "The function `get_top_n` with `top_n` set to 3 should return the following:\n",
    "```\n",
    "                                     Previous Returns\n",
    "               A         B         C         D         E         F         G\n",
    "2013-07-08     0         1         1         0         1         0         0\n",
    "2013-07-09     0         1         0         1         0         1         0\n",
    "...            ...       ...       ...       ...       ...       ...       ...\n",
    "```\n",
    "*Note: You may have to use Panda's [`DataFrame.iterrows`](https://pandas.pydata.org/pandas-docs/version/0.21/generated/pandas.DataFrame.iterrows.html) with [`Series.nlargest`](https://pandas.pydata.org/pandas-docs/version/0.21/generated/pandas.Series.nlargest.html) in order to implement the function. This is one of those cases where creating a vecorization solution is too difficult.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tests Passed\n"
     ]
    }
   ],
   "source": [
    "def get_top_n(prev_returns, top_n):\n",
    "    \"\"\"\n",
    "    Select the top performing stocks\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    prev_returns : DataFrame\n",
    "        Previous shifted returns for each ticker and date\n",
    "    top_n : int\n",
    "        The number of top performing stocks to get\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    top_stocks : DataFrame\n",
    "        Top stocks for each ticker and date marked with a 1\n",
    "    \"\"\"\n",
    "    # TODO: Implement Function\n",
    "    top_stocks = pd.DataFrame(index = prev_returns.index, columns = prev_returns.columns)\n",
    "    for index, row in prev_returns.iterrows():\n",
    "        top_stocks.loc[index] = row.isin(row.nlargest(top_n)).astype(np.int)\n",
    "        \n",
    "    return top_stocks\n",
    "\n",
    "project_tests.test_get_top_n(get_top_n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### View Data\n",
    "We want to get the best performing and worst performing stocks. To get the best performing stocks, we'll use the `get_top_n` function. To get the worst performing stocks, we'll also use the `get_top_n` function. However, we pass in `-1*prev_returns` instead of just `prev_returns`. Multiplying by negative one will flip all the positive returns to negative and negative returns to positive. Thus, it will return the worst performing stocks."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10 Most Longed Stocks:\n",
      "INCY, AMD, AVGO, NFX, SWKS, NFLX, ILMN, UAL, NVDA, MU\n",
      "10 Most Shorted Stocks:\n",
      "RRC, FCX, CHK, MRO, GPS, WYNN, DVN, FTI, SPLS, TRIP\n"
     ]
    }
   ],
   "source": [
    "top_bottom_n = 50\n",
    "df_long = get_top_n(prev_returns, top_bottom_n)\n",
    "df_short = get_top_n(-1*prev_returns, top_bottom_n)\n",
    "project_helper.print_top(df_long, 'Longed Stocks')\n",
    "project_helper.print_top(df_short, 'Shorted Stocks')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Projected Returns\n",
    "It's now time to check if your trading signal has the potential to become profitable!\n",
    "\n",
    "We'll start by computing the net returns this portfolio would return. For simplicity, we'll assume every stock gets an equal dollar amount of investment. This makes it easier to compute a portfolio's returns as the simple arithmetic average of the individual stock returns.\n",
    "\n",
    "Implement the `portfolio_returns` function to compute the expected portfolio returns. Using `df_long` to indicate which stocks to long and `df_short` to indicate which stocks to short, calculate the returns using `lookahead_returns`. To help with calculation, we've provided you with `n_stocks` as the number of stocks we're investing in a single period."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tests Passed\n"
     ]
    }
   ],
   "source": [
    "def portfolio_returns(df_long, df_short, lookahead_returns, n_stocks):\n",
    "    \"\"\"\n",
    "    Compute expected returns for the portfolio, assuming equal investment in each long/short stock.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    df_long : DataFrame\n",
    "        Top stocks for each ticker and date marked with a 1\n",
    "    df_short : DataFrame\n",
    "        Bottom stocks for each ticker and date marked with a 1\n",
    "    lookahead_returns : DataFrame\n",
    "        Lookahead returns for each ticker and date\n",
    "    n_stocks: int\n",
    "        The number of stocks chosen for each month\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    portfolio_returns : DataFrame\n",
    "        Expected portfolio returns for each ticker and date\n",
    "    \"\"\"\n",
    "    # TODO: Implement Function\n",
    "    \n",
    "    return (df_long - df_short) * lookahead_returns / n_stocks\n",
    "\n",
    "project_tests.test_portfolio_returns(portfolio_returns)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### View Data\n",
    "Time to see how the portfolio did."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
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    "expected_portfolio_returns = portfolio_returns(df_long, df_short, lookahead_returns, 2*top_bottom_n)\n",
    "project_helper.plot_returns(expected_portfolio_returns.T.sum(), 'Portfolio Returns')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Statistical Tests\n",
    "### Annualized Rate of Return"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Mean:                       0.003185\n",
      "Standard Error:             0.002158\n",
      "Annualized Rate of Return:  3.90%\n",
      "\n"
     ]
    }
   ],
   "source": [
    "expected_portfolio_returns_by_date = expected_portfolio_returns.T.sum().dropna()\n",
    "portfolio_ret_mean = expected_portfolio_returns_by_date.mean()\n",
    "portfolio_ret_ste = expected_portfolio_returns_by_date.sem()\n",
    "portfolio_ret_annual_rate = (np.exp(portfolio_ret_mean * 12) - 1) * 100\n",
    "\n",
    "print(\"\"\"\n",
    "Mean:                       {:.6f}\n",
    "Standard Error:             {:.6f}\n",
    "Annualized Rate of Return:  {:.2f}%\n",
    "\"\"\".format(portfolio_ret_mean, portfolio_ret_ste, portfolio_ret_annual_rate))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The annualized rate of return allows you to compare the rate of return from this strategy to other quoted rates of return, which are usually quoted on an annual basis. \n",
    "\n",
    "### T-Test\n",
    "Our null hypothesis ($H_0$) is that the actual mean return from the signal is zero. We'll perform a one-sample, one-sided t-test on the observed mean return, to see if we can reject $H_0$.\n",
    "\n",
    "We'll need to first compute the t-statistic, and then find its corresponding p-value. The p-value will indicate the probability of observing a t-statistic equally or more extreme than the one we observed if the null hypothesis were true. A small p-value means that the chance of observing the t-statistic we observed under the null hypothesis is small, and thus casts doubt on the null hypothesis. It's good practice to set a desired level of significance or alpha ($\\alpha$) _before_ computing the p-value, and then reject the null hypothesis if $p < \\alpha$.\n",
    "\n",
    "For this project, we'll use $\\alpha = 0.05$, since it's a common value to use.\n",
    "\n",
    "Implement the `analyze_alpha` function to perform a t-test on the sample of portfolio returns. We've imported the `scipy.stats` module for you to perform the t-test.\n",
    "\n",
    "Note: [`scipy.stats.ttest_1samp`](https://docs.scipy.org/doc/scipy-1.0.0/reference/generated/scipy.stats.ttest_1samp.html) performs a two-sided test, so divide the p-value by 2 to get 1-sided p-value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tests Passed\n"
     ]
    }
   ],
   "source": [
    "from scipy import stats\n",
    "\n",
    "def analyze_alpha(expected_portfolio_returns_by_date):\n",
    "    \"\"\"\n",
    "    Perform a t-test with the null hypothesis being that the expected mean return is zero.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    expected_portfolio_returns_by_date : Pandas Series\n",
    "        Expected portfolio returns for each date\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    t_value\n",
    "        T-statistic from t-test\n",
    "    p_value\n",
    "        Corresponding p-value\n",
    "    \"\"\"\n",
    "    t_stastic, p_value = stats.ttest_1samp(expected_portfolio_returns_by_date, 0)\n",
    "    return t_stastic, p_value / 2\n",
    "\n",
    "project_tests.test_analyze_alpha(analyze_alpha)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### View Data\n",
    "Let's see what values we get with our portfolio. After you run this, make sure to answer the question below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Alpha analysis:\n",
      " t-value:        1.476\n",
      " p-value:        0.073339\n",
      "\n"
     ]
    }
   ],
   "source": [
    "t_value, p_value = analyze_alpha(expected_portfolio_returns_by_date)\n",
    "print(\"\"\"\n",
    "Alpha analysis:\n",
    " t-value:        {:.3f}\n",
    " p-value:        {:.6f}\n",
    "\"\"\".format(t_value, p_value))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question: What p-value did you observe? And what does that indicate about your signal?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "p-value = 0.073359. p-value > 0.05 and we cannot reject the null hypothesis which is the population mean is zero."
   ]
  }
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