25a_mutual_fund_perfeval.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Performance evaluation\n",
    "... or let's be Morningstar for a minute\n",
    "\n",
    "We will evaluate just under 2,000 active mutual funds relative to a market model.  The returns data are already in excess of the risk-free rate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import statsmodels.api as sm\n",
    "import plotly.graph_objects as go"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# mf = pd.read_csv(\"../data/mf_rets.csv\")\n",
    "mf = pd.read_csv(\"https://www.dropbox.com/s/e1cuahqqdn56l0y/mf_rets.csv?dl=1\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['Date', 'MKT-RF', 'RF', '1', '2', '3', '4', '5', '6', '7',\n",
       "       ...\n",
       "       '1904', '1905', '1906', '1907', '1908', '1909', '1910', '1911', '1912',\n",
       "       '1913'],\n",
       "      dtype='object', length=1916)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mf.columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "cols = mf.columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "cols = cols[3:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['1', '2', '3', '4', '5', '6', '7', '8', '9', '10',\n",
       "       ...\n",
       "       '1904', '1905', '1906', '1907', '1908', '1909', '1910', '1911', '1912',\n",
       "       '1913'],\n",
       "      dtype='object', length=1913)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cols"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fund Performance: Market Model Alpha"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                      1   R-squared:                       0.926\n",
      "Model:                            OLS   Adj. R-squared:                  0.926\n",
      "Method:                 Least Squares   F-statistic:                     2804.\n",
      "Date:                Mon, 10 Apr 2023   Prob (F-statistic):          2.84e-128\n",
      "Time:                        11:09:30   Log-Likelihood:                 676.10\n",
      "No. Observations:                 225   AIC:                            -1348.\n",
      "Df Residuals:                     223   BIC:                            -1341.\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const         -0.0009      0.001     -1.151      0.251      -0.003       0.001\n",
      "MKT-RF         0.9245      0.017     52.949      0.000       0.890       0.959\n",
      "==============================================================================\n",
      "Omnibus:                       36.057   Durbin-Watson:                   1.952\n",
      "Prob(Omnibus):                  0.000   Jarque-Bera (JB):              240.860\n",
      "Skew:                           0.295   Prob(JB):                     4.99e-53\n",
      "Kurtosis:                       8.034   Cond. No.                         21.7\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "y = mf['1']\n",
    "X = sm.add_constant(mf['MKT-RF'])\n",
    "results = sm.OLS(y,X, missing='drop').fit()\n",
    "print(results.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "const    -0.000933\n",
       "MKT-RF    0.924485\n",
       "dtype: float64"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = pd.DataFrame(dtype=float,columns=['alpha','beta'], index=cols)\n",
    "X = sm.add_constant(mf['MKT-RF'])\n",
    "for c in cols:\n",
    "    y = mf[c]\n",
    "    results = sm.OLS(y,X, missing='drop').fit()\n",
    "    df.loc[c,'alpha'] = results.params[0]\n",
    "    df.loc[c,'beta'] = results.params[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>alpha</th>\n",
       "      <th>beta</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.000933</td>\n",
       "      <td>0.924485</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.000034</td>\n",
       "      <td>1.066703</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.001191</td>\n",
       "      <td>1.005108</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.001104</td>\n",
       "      <td>0.729851</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.002545</td>\n",
       "      <td>1.264373</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1909</th>\n",
       "      <td>-0.001335</td>\n",
       "      <td>0.997558</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1910</th>\n",
       "      <td>0.000863</td>\n",
       "      <td>1.099683</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1911</th>\n",
       "      <td>-0.000391</td>\n",
       "      <td>0.509993</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1912</th>\n",
       "      <td>0.000417</td>\n",
       "      <td>1.000028</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1913</th>\n",
       "      <td>-0.002054</td>\n",
       "      <td>1.241786</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>1913 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "         alpha      beta\n",
       "1    -0.000933  0.924485\n",
       "2     0.000034  1.066703\n",
       "3    -0.001191  1.005108\n",
       "4    -0.001104  0.729851\n",
       "5    -0.002545  1.264373\n",
       "...        ...       ...\n",
       "1909 -0.001335  0.997558\n",
       "1910  0.000863  1.099683\n",
       "1911 -0.000391  0.509993\n",
       "1912  0.000417  1.000028\n",
       "1913 -0.002054  1.241786\n",
       "\n",
       "[1913 rows x 2 columns]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.41766858337689494\n"
     ]
    }
   ],
   "source": [
    "frac_pos = (df.alpha >0).mean()\n",
    "print(frac_pos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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          "text": "Monthly Alpha"
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         }
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       }
      }
     },
     "metadata": {},
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   ],
   "source": [
    "# Plot the distribution of alphas\n",
    "fig = go.Figure()\n",
    "trace= go.Histogram(x=df.alpha, histnorm='percent',hovertemplate=\"<br>%{y:.2}% of funds<br><extra></extra>\")\n",
    "fig.add_trace(trace)\n",
    "# some formatting\n",
    "fig.update_traces(marker_line_width=1, marker_line_color='black')\n",
    "fig.layout.xaxis[\"title\"] = \"Monthly Alpha\"\n",
    "fig.layout.yaxis[\"title\"] = \"Percent of Funds\"\n",
    "fig.add_vline(x=0, line_width=4, line_dash=\"dash\", line_color=\"black\")\n",
    "fig.add_annotation(x=df.alpha.max()*0.7, y=8,\n",
    "            text=\"Fraction of Funds with Positive Alpha <br>\"+f'{frac_pos:.1%} of funds', showarrow=False)\n",
    "fig.show()"
   ]
  },
  {
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Predicting Survival"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "last = mf.iloc[-1]\n",
    "last_obs = pd.DataFrame(last[cols])\n",
    "last_obs.columns=['last']\n",
    "df = df.merge(last_obs,left_index=True, right_index=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "df['died'] = np.isnan(df['last'])*1.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>alpha</th>\n",
       "      <th>beta</th>\n",
       "      <th>last</th>\n",
       "      <th>died</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.000933</td>\n",
       "      <td>0.924485</td>\n",
       "      <td>0.0276</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.000034</td>\n",
       "      <td>1.066703</td>\n",
       "      <td>0.0585</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.001191</td>\n",
       "      <td>1.005108</td>\n",
       "      <td>0.0512</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.001104</td>\n",
       "      <td>0.729851</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
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       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.002545</td>\n",
       "      <td>1.264373</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
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       "      <td>...</td>\n",
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       "    <tr>\n",
       "      <th>1909</th>\n",
       "      <td>-0.001335</td>\n",
       "      <td>0.997558</td>\n",
       "      <td>0.0283</td>\n",
       "      <td>0.0</td>\n",
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       "    <tr>\n",
       "      <th>1910</th>\n",
       "      <td>0.000863</td>\n",
       "      <td>1.099683</td>\n",
       "      <td>0.0662</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1911</th>\n",
       "      <td>-0.000391</td>\n",
       "      <td>0.509993</td>\n",
       "      <td>0.0195</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1912</th>\n",
       "      <td>0.000417</td>\n",
       "      <td>1.000028</td>\n",
       "      <td>0.0434</td>\n",
       "      <td>0.0</td>\n",
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       "    <tr>\n",
       "      <th>1913</th>\n",
       "      <td>-0.002054</td>\n",
       "      <td>1.241786</td>\n",
       "      <td>0.0543</td>\n",
       "      <td>0.0</td>\n",
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       "  </tbody>\n",
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       "<p>1913 rows × 4 columns</p>\n",
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      "text/plain": [
       "         alpha      beta    last  died\n",
       "1    -0.000933  0.924485  0.0276   0.0\n",
       "2     0.000034  1.066703  0.0585   0.0\n",
       "3    -0.001191  1.005108  0.0512   0.0\n",
       "4    -0.001104  0.729851     NaN   1.0\n",
       "5    -0.002545  1.264373     NaN   1.0\n",
       "...        ...       ...     ...   ...\n",
       "1909 -0.001335  0.997558  0.0283   0.0\n",
       "1910  0.000863  1.099683  0.0662   0.0\n",
       "1911 -0.000391  0.509993  0.0195   0.0\n",
       "1912  0.000417  1.000028  0.0434   0.0\n",
       "1913 -0.002054  1.241786  0.0543   0.0\n",
       "\n",
       "[1913 rows x 4 columns]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
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   ],
   "source": [
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                   died   R-squared:                       0.099\n",
      "Model:                            OLS   Adj. R-squared:                  0.099\n",
      "Method:                 Least Squares   F-statistic:                     210.9\n",
      "Date:                Mon, 10 Apr 2023   Prob (F-statistic):           2.10e-45\n",
      "Time:                        11:12:35   Log-Likelihood:                -1273.5\n",
      "No. Observations:                1913   AIC:                             2551.\n",
      "Df Residuals:                    1911   BIC:                             2562.\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const          0.4143      0.011     38.031      0.000       0.393       0.436\n",
      "alpha        -54.9354      3.783    -14.522      0.000     -62.355     -47.516\n",
      "==============================================================================\n",
      "Omnibus:                    11703.735   Durbin-Watson:                   1.711\n",
      "Prob(Omnibus):                  0.000   Jarque-Bera (JB):              221.317\n",
      "Skew:                           0.333   Prob(JB):                     8.74e-49\n",
      "Kurtosis:                       1.472   Cond. No.                         351.\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "# Use alpha to predict leaving sample\n",
    "y = df['died']\n",
    "X = sm.add_constant(df['alpha'])\n",
    "results = sm.OLS(y,X, missing='drop').fit()\n",
    "print(results.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                   died   R-squared:                       0.099\n",
      "Model:                            OLS   Adj. R-squared:                  0.099\n",
      "Method:                 Least Squares   F-statistic:                     190.5\n",
      "Date:                Mon, 10 Apr 2023   Prob (F-statistic):           2.23e-41\n",
      "Time:                        11:12:58   Log-Likelihood:                -1273.5\n",
      "No. Observations:                1913   AIC:                             2551.\n",
      "Df Residuals:                    1911   BIC:                             2562.\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:                  HC3                                         \n",
      "==============================================================================\n",
      "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const          0.4381      0.011     40.647      0.000       0.417       0.459\n",
      "alpha         -0.1565      0.011    -13.802      0.000      -0.179      -0.134\n",
      "==============================================================================\n",
      "Omnibus:                    11703.735   Durbin-Watson:                   1.711\n",
      "Prob(Omnibus):                  0.000   Jarque-Bera (JB):              221.317\n",
      "Skew:                           0.333   Prob(JB):                     8.74e-49\n",
      "Kurtosis:                       1.472   Cond. No.                         1.00\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors are heteroscedasticity robust (HC3)\n"
     ]
    }
   ],
   "source": [
    "# Let's standardize the alpha first\n",
    "y = df['died']\n",
    "X = sm.add_constant((df.alpha - df.alpha.mean())/df.alpha.std())\n",
    "results = sm.OLS(y,X, missing='drop').fit(cov_type='HC3')\n",
    "print(results.summary())"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### How should we interpret the above result?"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Survivorship bias"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>alpha</th>\n",
       "      <th>beta</th>\n",
       "      <th>last</th>\n",
       "      <th>died</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>1075.000000</td>\n",
       "      <td>1075.000000</td>\n",
       "      <td>1075.000000</td>\n",
       "      <td>1075.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>0.000360</td>\n",
       "      <td>1.038410</td>\n",
       "      <td>0.046399</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>0.002086</td>\n",
       "      <td>0.156203</td>\n",
       "      <td>0.015075</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>-0.010812</td>\n",
       "      <td>-0.022087</td>\n",
       "      <td>-0.012500</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>-0.000973</td>\n",
       "      <td>0.946906</td>\n",
       "      <td>0.034750</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>0.000064</td>\n",
       "      <td>1.025742</td>\n",
       "      <td>0.046400</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>0.001559</td>\n",
       "      <td>1.125275</td>\n",
       "      <td>0.057550</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>0.010883</td>\n",
       "      <td>2.157380</td>\n",
       "      <td>0.098600</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             alpha         beta         last    died\n",
       "count  1075.000000  1075.000000  1075.000000  1075.0\n",
       "mean      0.000360     1.038410     0.046399     0.0\n",
       "std       0.002086     0.156203     0.015075     0.0\n",
       "min      -0.010812    -0.022087    -0.012500     0.0\n",
       "25%      -0.000973     0.946906     0.034750     0.0\n",
       "50%       0.000064     1.025742     0.046400     0.0\n",
       "75%       0.001559     1.125275     0.057550     0.0\n",
       "max       0.010883     2.157380     0.098600     0.0"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = df[df['died']==0]\n",
    "df.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.5237209302325582\n"
     ]
    }
   ],
   "source": [
    "frac_pos = (df.alpha >0).mean()\n",
    "print(frac_pos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
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    "# Plot the distribution of alphas WITH SURVIVORSHIP BIAS\n",
    "fig = go.Figure()\n",
    "trace= go.Histogram(x=df.alpha, histnorm='percent',hovertemplate=\"<br>%{y:.2}% of funds<br><extra></extra>\")\n",
    "fig.add_trace(trace)\n",
    "# some formatting\n",
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    "fig.layout.xaxis[\"title\"] = \"Monthly Alpha\"\n",
    "fig.layout.yaxis[\"title\"] = \"Percent of Funds\"\n",
    "fig.add_vline(x=0, line_width=4, line_dash=\"dash\", line_color=\"black\")\n",
    "fig.add_annotation(x=df.alpha.max()*0.7, y=8,\n",
    "            text=\"Fraction of Funds with Positive Alpha <br>\"+f'{frac_pos:.1%} of funds', showarrow=False)\n",
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