Step3_FactorCombination_BarraKalmanFilter.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step Three: Barra's Risk Model\n",
    "\n",
    "Firstly, we define the dynamics of the price process with respect to factors(determined by step1 and step2). \n",
    "\n",
    "Secondly, using three different schemes to calibrate the coefficients.\n",
    "\n",
    "* Scheme 1: Cross-sectional regression and weighted average\n",
    "* Scheme 2: Optimization problem: minimize the exponential weighted average of squared error\n",
    "* Scheme 3: Dynamic linear model using Kalman filter\n",
    "\n",
    "## 1 Dynamics of Price\n",
    "Based on the celebrated APT model and Berra's risk model, I proposed the stock forecasting model as the following expression.\n",
    "\n",
    "$$r_{i}(t) = \\alpha(t) + \\sum_{k=1}^K \\beta_{ik}(t) F_{ik}(t)$$\n",
    "\n",
    "or more consistently,\n",
    "\n",
    "$$r_{t,i} = \\alpha_t + \\sum_{k=1}^K \\beta_{t,i,k} F_{t,i,k}$$\n",
    "\n",
    "The factor exposures $\\beta$ of the stock i to the factor k in the time t is assumed to be known in this model (a postulate), and valuated as the rescaled (normalized) factor values. \n",
    "\n",
    "For model calibration, there are three ways to handle it: a) cross-sectional regression, b) gradient descend scheme for the loss function, and c) kalman filter."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.1 Model Definition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import statsmodels.api as sm\n",
    "from statsmodels import regression,stats\n",
    "import itertools\n",
    "from pykalman import KalmanFilter\n",
    "idx = pd.IndexSlice\n",
    "\n",
    "class MultiFactorModel:\n",
    "    def __init__(self, price_df, equity_df, benchmark_df, factor_list, universe):\n",
    "        self.price_df = price_df\n",
    "        self.equity_df = equity_df\n",
    "        self.benchmark_df = benchmark_df\n",
    "        self.factor_list = factor_list\n",
    "        self.universe = universe\n",
    "        self.valid_universe = Nones\n",
    "        self.subset = None\n",
    "        \n",
    "        # z-score in universe\n",
    "        tmp_equity_df = self.calculate_subset_df(self.equity_df,self.universe)\n",
    "        tmp_equity_df = tmp_equity_df[factor_list].fillna(0) # TODO: fill or drop??\n",
    "        self.factor_zscore = (tmp_equity_df - tmp_equity_df.groupby(level='date').mean())/tmp_equity_df.groupby(level='date').std()\n",
    "        \n",
    "        # valide_universe\n",
    "        self.valid_universe = self.calculate_valid_universe()\n",
    "        self.valid_factor_zscore = self.factor_zscore.loc[idx[:,self.valid_universe],:]\n",
    "\n",
    "\n",
    "    def calculate_valid_universe(self):\n",
    "        valid_universe = set(self.universe)\n",
    "\n",
    "        for date,group in self.factor_zscore.groupby(level=0):\n",
    "            size = group.shape[0]\n",
    "            new_set = set(group.loc[idx[:,valid_universe],:].index.get_level_values(1).values)\n",
    "            if(new_set<valid_universe):\n",
    "                valid_universe = new_set\n",
    "        # self.valid_universe = valid_universe        \n",
    "        return valid_universe\n",
    "\n",
    "    def calculate_subset_df(self,df,subset):\n",
    "        return df.loc[idx[:,subset],:]\n",
    "        \n",
    "    def calculate_portfolio_risk(self,hp):\n",
    "        if(~hasattr(self,'V') or self.V==None):\n",
    "            self.cross_section_regression()\n",
    "        sigmap = np.sqrt(np.dot(np.dot(hp.T,self.V),hp))\n",
    "        MCTR = np.dot(self.V,hp)/sigmap\n",
    "        return self.V, sigmap, MCTR\n",
    "        \n",
    "    def time_series_regression(self):\n",
    "        ts_factor_zscore = sm.add_constant(self.factor_zscore)\n",
    "        Y = [self.equity_df.xs(asset,level=1)['return'] for asset in self.valid_universe]\n",
    "        X = [ts_factor_zscore.xs(asset,level=1)[factor_list+['const']] for asset in self.valid_universe]\n",
    "        reg_results = [regression.linear_model.OLS(y,x).fit().params for y,x in zip(Y,X) if not(x.empty or y.empty)]\n",
    "        indices = [asset for y, x, asset in zip(Y, X, self.valid_universe) if not(x.empty or y.empty)]\n",
    "        ts_result_df = pd.DataFrame(reg_results, index=indices)\n",
    "        return ts_result_df\n",
    "    \n",
    "#     def regression(self,tau):\n",
    "#         Xtik = self.factor_zscore.loc[idx[:,self.valid_universe],:]\n",
    "#         n_factors = len(self.factor_list)\n",
    "#         starting_point = np.zeros(n_factors)\n",
    "#         Fk = starting_point\n",
    "#         at = self.benchmark_df['return']\n",
    "#         rti = self.equity_df.loc[idx[:,self.valid_universe],'return']\n",
    "        \n",
    "#         def grad_loss(X,F,r,a,tau):\n",
    "#             n_factor = len(F)\n",
    "#             n_time = len(self.equity_df.index.get_level_values(level=0).unique())\n",
    "#             n_stock = len(self.valid_universe)\n",
    "#             grad = np.zeros(n_factor)\n",
    "#             for k in range(n_factor):\n",
    "#                 gradk = 0\n",
    "#                 for t in range(n_time):\n",
    "#                     for i in range(n_stock):\n",
    "#                         gradk += 2*tau**(n_time-t)*(a[t]+np.dot(X[t,i,:],F[:]) - r[t,i])*X[t,i,k]\n",
    "#                 grad[k] = gradk\n",
    "#             return grad\n",
    "                \n",
    "        \n",
    "    \n",
    "    \n",
    "#TODO: exponential decay regression, alpha contribution\n",
    "    def cross_section_regression(self, ndays=22, latest=True):\n",
    "        valid_factor_zscore = self.factor_zscore.loc[idx[:,self.valid_universe],:]\n",
    "        if(latest):\n",
    "            dates = valid_factor_zscore.index.get_level_values(level=0).unique()[-ndays:]\n",
    "        else:\n",
    "            dates = valid_factor_zscore.index.get_level_values(level=0).unique()[:ndays]\n",
    "        valid_factor_zscore = valid_factor_zscore.loc[idx[dates,:],:]\n",
    "        \n",
    "        result_list = []\n",
    "        date_list = []\n",
    "        for date,group in valid_factor_zscore.groupby(level=0):\n",
    "            X = sm.add_constant(group.loc[:,self.factor_list])\n",
    "            y = list(equity_df.loc[(date,list(self.valid_universe)),'return'])          \n",
    "            results = sm.regression.linear_model.OLS(y,X).fit()\n",
    "            result_list.append(results.params)\n",
    "            date_list.append(date)\n",
    "\n",
    "        cs_result_df = pd.DataFrame(result_list,index=date_list)\n",
    "        F = np.cov(cs_result_df.iloc[:,1:].T)\n",
    "        X = self.valid_factor_zscore.groupby(level=1).mean()\n",
    "        V = np.inner(np.inner(X,F),X)\n",
    "        self.F = F\n",
    "        self.X = X\n",
    "        self.V = V\n",
    "        \n",
    "        return cs_result_df\n",
    "    \n",
    "    def kalman_filter_calibration(self):\n",
    "        asset_size = len(self.valid_universe)\n",
    "        factor_size = len(self.factor_list)\n",
    "        observation_transition_matrix = []\n",
    "        for date,group in self.valid_factor_zscore.groupby(level=0):\n",
    "            exposure_matrix = group.values\n",
    "#             print(np.shape(exposure_matrix))\n",
    "#             print(asset_size)\n",
    "            observation_transition_matrix.append(\n",
    "                np.concatenate((np.eye(asset_size),exposure_matrix),axis=1).tolist()\n",
    "            )\n",
    "        observations = self.equity_df.loc[idx[:,self.valid_universe],'return']\n",
    "        observation_list = []\n",
    "        for date,group in observations.groupby(level=0):\n",
    "            observation_list.append(group.values.tolist())\n",
    "            \n",
    "        state_transition_matrix = np.eye(factor_size+asset_size)\n",
    "        state_covariance_matrix = np.eye(factor_size+asset_size)*0.01\n",
    "        observation_matrices = observation_transition_matrix\n",
    "        observation_covariance_matrix = np.eye(asset_size)*0.0\n",
    "        initial_state_mean = np.zeros(factor_size+asset_size)\n",
    "        initial_state_covariance = np.eye(factor_size+asset_size)*0.01\n",
    "\n",
    "\n",
    "        nstate = factor_size+asset_size\n",
    "        nobs = asset_size\n",
    "#         print(state_transition_matrix.shape)\n",
    "#         print(state_covariance_matrix.shape)\n",
    "#         print(np.shape(observation_matrices))\n",
    "#         print(observation_covariance_matrix.shape==(nobs,nobs))\n",
    "#         print(initial_state_mean.shape==(nstate,))\n",
    "#         print(initial_state_covariance.shape==(nstate,nstate))\n",
    "\n",
    "\n",
    "        kf = KalmanFilter(transition_matrices=state_transition_matrix,\n",
    "                 transition_covariance=state_covariance_matrix,\n",
    "                 observation_matrices=observation_matrices,\n",
    "                 observation_covariance=observation_covariance_matrix,\n",
    "                 initial_state_mean=initial_state_mean,\n",
    "                 initial_state_covariance=initial_state_covariance,\n",
    "                 n_dim_state=nstate,\n",
    "                 n_dim_obs=nobs)\n",
    "        returns = kf.filter(observation_list)\n",
    "        filtered_state_means = returns[0]\n",
    "        # a and F\n",
    "        a_list = []\n",
    "        F_list = []\n",
    "        for state in filtered_state_means:\n",
    "            a = state[:asset_size]\n",
    "            F = state[asset_size:]\n",
    "            a_list.append(a)\n",
    "            F_list.append(F)\n",
    "            \n",
    "        a_df = pd.Series(list(itertools.chain(*a_list)),index = self.valid_factor_zscore.index,name='alpha')\n",
    "        \n",
    "        F_list_temp = [[F,]*asset_size for F in F_list]\n",
    "        # subset_factor_zscore.index.get_level_values(0).unique()\n",
    "        F_list_temp = list(itertools.chain(*F_list_temp))\n",
    "        F_df = pd.DataFrame(F_list_temp,columns=[\"{}_F\".format(f) for f in factor_list],index=self.valid_factor_zscore.index)\n",
    "        \n",
    "        return_df = self.equity_df.loc[idx[:,self.valid_universe],'return']\n",
    "        close_df = self.equity_df.loc[idx[:,self.valid_universe],'close']\n",
    "        kf_df = pd.concat([self.valid_factor_zscore,F_df,a_df,return_df,close_df],axis=1)\n",
    "        self.kf_df = kf_df\n",
    "        return kf_df\n",
    "\n",
    "        "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.2 Pull Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from rqdata_utils import *\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import alphalens as al\n",
    "from pykalman import KalmanFilter\n",
    "\n",
    "\n",
    "price_df,instrument_df,equity_df = get_price_instrument_equity(\"cn_stock_price_2012_2018.csv\",\"cn_instrument_info_2012_2018.csv\",\"cn_equity_daily_2012_2018.csv\",\"sectorCode\")\n",
    "healthcare_universe = instrument_df.index[instrument_df.sectorCode=='HealthCare'].values\n",
    "benchmark_df = benchmark_reader(\"cn_SH_healthcare_index_2012_2018.csv\")\n",
    "factor_list = ['market_cap', 'pb_ratio', 'ps_ratio']\n",
    "universe_list = instrument_df.index[instrument_df.sectorCode=='HealthCare'].values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.3 Inject Data into the Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "myModel = MultiFactorModel(price_df, equity_df, benchmark_df, factor_list, universe_list)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model Calibration\n",
    "\n",
    "* Scheme 1: Cross-sectional regression and weighted average\n",
    "* Scheme 2: Optimization problem: minimize the exponential weighted average of squared error\n",
    "* Scheme 3: Dynamic linear model using Kalman filter\n",
    "\n",
    "The schemes one, two, three are slightly evolutional related. Scheme one is just the original BARRA's risk model defined, scheme two further defined a objective function to obtain the coefficients, and scheme three introduce Kalman filter to fit the dynamic linear model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "### 2.1 Calibration Scheme 1: Cross-Sectional Regression and Weighted Average"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "            market_cap  pb_ratio  ps_ratio\n",
      "2017-10-16    0.004067  0.001988 -0.002809\n",
      "2017-10-17    0.001545 -0.003527  0.003828\n",
      "2017-10-18    0.003857  0.001502 -0.004133\n",
      "2017-10-19    0.006112 -0.000609  0.001779\n",
      "2017-10-20   -0.002669 -0.001985  0.005158\n",
      "V:\n",
      "[[ 3.19844460e-05 -7.42713103e-06  5.18342510e-06 ...  2.25855232e-06\n",
      "  -1.12620455e-07 -2.46779972e-05]\n",
      " [-7.42713103e-06  1.60948004e-05 -1.76671909e-05 ... -9.67456183e-06\n",
      "  -8.91835400e-08  1.85188600e-05]\n",
      " [ 5.18342510e-06 -1.76671909e-05  1.98030463e-05 ...  1.16950009e-05\n",
      "  -4.77268788e-07 -1.67809236e-05]\n",
      " ...\n",
      " [ 2.25855232e-06 -9.67456183e-06  1.16950009e-05 ...  1.30630599e-05\n",
      "  -4.88037744e-06  5.75765299e-06]\n",
      " [-1.12620455e-07 -8.91835400e-08 -4.77268788e-07 ... -4.88037744e-06\n",
      "   3.45766356e-06 -1.09477448e-05]\n",
      " [-2.46779972e-05  1.85188600e-05 -1.67809236e-05 ...  5.75765299e-06\n",
      "  -1.09477448e-05  6.49979346e-05]]\n",
      "sigma\n",
      "0.03377233630871653\n",
      "MCTR:\n",
      "[ 0.00078128  0.00349245 -0.00401757  0.00233878  0.00041072  0.0004083\n",
      "  0.00222235 -0.00621658  0.00352282 -0.00051361 -0.00095923  0.0025695\n",
      " -0.0020911   0.00181597  0.00121508  0.00254621  0.00259181 -0.00777772\n",
      "  0.0022341   0.0001535  -0.00194782 -0.00624887  0.00143136  0.00212528\n",
      "  0.00095299  0.000577    0.00218747  0.00055355  0.00315648  0.00213663\n",
      "  0.00238647  0.00040797  0.00277732  0.00337277  0.00037938  0.00223035\n",
      "  0.00230273 -0.001752    0.0020087   0.00203213  0.00160285 -0.00023433\n",
      "  0.00242781  0.00174764  0.00070743  0.00312676  0.00245787 -0.00103323\n",
      " -0.00720428  0.00291931  0.00133933  0.00127603  0.0015967  -0.00119847\n",
      " -0.00927106  0.00151585  0.00106341  0.0023      0.0027298  -0.00160792\n",
      " -0.00174121 -0.00041166  0.00386674]\n"
     ]
    }
   ],
   "source": [
    "factorReturn_df = myModel.cross_section_regression(ndays=132).iloc[:,(1,2,3)]\n",
    "print(factorReturn_df.head())\n",
    "hp = np.ones(len(myModel.valid_universe))\n",
    "V,sigma, MCTR = myModel.calculate_portfolio_risk(hp)\n",
    "print(\"V:\\n{}\\nsigma\\n{}\\nMCTR:\\n{}\".format(V,sigma,MCTR))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "market_cap    0.000550\n",
       "pb_ratio      0.000032\n",
       "ps_ratio      0.000151\n",
       "dtype: float64"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def weight_avg(array_like, tau):\n",
    "    n_element = len(array_like)\n",
    "    weights = [tau**(n_element - i-1) for i in range(n_element)]\n",
    "    return np.average(array_like,weights=weights)\n",
    "factorReturn_df.apply(lambda x: weight_avg(x,1),axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From this result, we can see the factor returns are under-estimated. Moreover, inconsistant with the market size effect. Here factor return for market cap should not be positive. An alternative choice would be the long-short portfolio based on each factor, which should be more practical as it can be more straight-forward to represent the factor returns. However, before doing so, let's try another optimization algorithm, gradient descend."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2 Calibration Scheme 2: Optimization Problem and Gradient Descend\n",
    "\n",
    "The loss function:\n",
    "$$obj = \\sum_{t=1}^T \\tau^{T-t} \\sum_{i=1}^N (a_t + \\sum_{k=1}^K X_{t,i,k} f_k - r_{t,i})^2$$\n",
    "Corresponding gradient function:\n",
    "$$\\frac{\\partial obj}{\\partial f_{k^\\prime}} = \\sum_{t=1}^T \\tau^{T-t} \\sum_{i=1}^N 2 (a_t + \\sum_{k=1}^K X_{t,i,k} f_k - r_{t,i}) X_{t,i,k^{\\prime}}$$\n",
    "\n",
    "Here $\\tau$ represents the half-life period. I used a dampened sum of squared error as the loss function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def grad_loss(X,F,r,a,tau,n_factor,n_time,n_stock,n_days=None):\n",
    "    grad = np.zeros(n_factor)\n",
    "    for k in range(n_factor):\n",
    "        gradk = 0\n",
    "        for t in range(n_time):\n",
    "            if(n_days!=None and n_time-t>n_days):\n",
    "                continue\n",
    "            else:\n",
    "                damper = tau**(n_time-t)\n",
    "                for i in range(n_stock):\n",
    "                    gradk += 2*damper*(a[t]+np.dot(X[t,i,:],F[:]) - r[t,i])*X[t,i,k]\n",
    "        grad[k] = gradk\n",
    "    return grad\n",
    "        \n",
    "def loss(X,F,r,a,tau,n_factor,n_time,n_stock):\n",
    "    loss = 0\n",
    "    for t in range(n_time):\n",
    "        for i in range(n_stock):\n",
    "            loss += a[t] -r[t,i]\n",
    "            loss += np.dot(X[t,i,:],F[:])\n",
    "            loss = loss**2 * tau**(n_time-t)\n",
    "    return loss\n",
    "                "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X_df = myModel.factor_zscore.loc[idx[:,myModel.valid_universe],:]\n",
    "n_factors = len(myModel.factor_list)\n",
    "n_time = len(myModel.factor_zscore.index.get_level_values(0).unique())\n",
    "n_stocks = len(myModel.valid_universe)\n",
    "starting_point = np.zeros(n_factors)\n",
    "F = starting_point\n",
    "a_df = myModel.benchmark_df['return']\n",
    "r_df = myModel.equity_df.loc[idx[:,myModel.valid_universe],'return']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X = X_df.values.reshape(n_time,n_stocks,-1)\n",
    "a = a_df.values\n",
    "r = r_df.values.reshape(n_time,n_stocks)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0010854417361952566 [1.00971943e-04 5.04011777e-05 2.01246325e-05]\n",
      "0.0010768884651663005 [1.78955730e-04 8.62188223e-05 3.40693426e-05]\n",
      "0.0010700715099318495 [2.39302675e-04 1.11486567e-04 4.36077670e-05]\n",
      "0.0010646276874447063 [2.86094112e-04 1.29152446e-04 5.00238621e-05]\n",
      "0.0010602726560927807 [3.22447687e-04 1.41365992e-04 5.42438643e-05]\n",
      "0.0010567830237503634 [3.50748807e-04 1.49690293e-04 5.69332518e-05]\n",
      "0.0010539827255753424 [3.72825728e-04 1.55258458e-04 5.85678911e-05]\n",
      "0.0010517325971302425 [3.90082151e-04 1.58888940e-04 5.94861575e-05]\n",
      "0.0010499223528246523 [4.03597774e-04 1.61170436e-04 5.99270497e-05]\n",
      "0.001048464385585839 [4.14204622e-04 1.62524290e-04 6.00580054e-05]\n",
      "0.001047288953295496 [4.22545091e-04 1.63250299e-04 5.99951560e-05]\n",
      "0.0010463404268252507 [4.29116128e-04 1.63560271e-04 5.98180350e-05]\n",
      "0.0010455743548112811 [4.34302927e-04 1.63602562e-04 5.95802290e-05]\n",
      "0.00104495515969231 [4.38404655e-04 1.63479976e-04 5.93170627e-05]\n",
      "0.0010444543237308852 [4.41654135e-04 1.63262786e-04 5.90511222e-05]\n",
      "0.0010440489568392809 [4.44232925e-04 1.62998183e-04 5.87962063e-05]\n",
      "0.0010437206629753278 [4.46282893e-04 1.62717104e-04 5.85601374e-05]\n",
      "0.001043454640779563 [4.47915113e-04 1.62439147e-04 5.83467480e-05]\n",
      "0.0010432389685325883 [4.49216730e-04 1.62176091e-04 5.81572732e-05]\n",
      "0.0010430640345463738 [4.50256243e-04 1.61934409e-04 5.79913175e-05]\n",
      "0.0010429220825942866 [4.51087602e-04 1.61717040e-04 5.78475180e-05]\n",
      "0.001042806848546405 [4.51753382e-04 1.61524635e-04 5.77239915e-05]\n",
      "0.0010427132694675084 [4.52287237e-04 1.61356425e-04 5.76186301e-05]\n",
      "0.0010426372503991334 [4.52715822e-04 1.61210806e-04 5.75292905e-05]\n",
      "0.0010425754771439687 [4.53060287e-04 1.61085749e-04 5.74539108e-05]\n",
      "0.0010425252657977324 [4.53337436e-04 1.60979055e-04 5.73905764e-05]\n",
      "0.0010424844416809442 [4.53560649e-04 1.60888528e-04 5.73375537e-05]\n",
      "0.0010424512418257842 [4.53740590e-04 1.60812075e-04 5.72933014e-05]\n",
      "0.001042424236360253 [4.53885775e-04 1.60747763e-04 5.72564680e-05]\n",
      "0.0010424022650715478 [4.54003014e-04 1.60693850e-04 5.72258815e-05]\n",
      "0.0010423843861760186 [4.54097758e-04 1.60648787e-04 5.72005347e-05]\n",
      "0.0010423698349156275 [4.54174378e-04 1.60611218e-04 5.71795677e-05]\n",
      "0.00104235799007269 [4.54236381e-04 1.60579968e-04 5.71622514e-05]\n",
      "0.00104234834687104 [4.54286588e-04 1.60554025e-04 5.71479703e-05]\n",
      "0.0010423404950324745 [4.54327265e-04 1.60532526e-04 5.71362071e-05]\n",
      "0.0010423341009979277 [4.54360239e-04 1.60514737e-04 5.71265286e-05]\n",
      "0.0010423288935155822 [4.54386981e-04 1.60500039e-04 5.71185732e-05]\n",
      "0.0010423246519527952 [4.54408679e-04 1.60487909e-04 5.71120399e-05]\n",
      "0.001042321196812921 [4.54426292e-04 1.60477909e-04 5.71066788e-05]\n",
      "0.001042318382037992 [4.54440595e-04 1.60469674e-04 5.71022826e-05]\n",
      "0.0010423160887586495 [4.54452213e-04 1.60462899e-04 5.70986799e-05]\n",
      "0.0010423142202174666 [4.54461653e-04 1.60457328e-04 5.70957291e-05]\n",
      "0.0010423126976440806 [4.54469327e-04 1.60452752e-04 5.70933136e-05]\n",
      "0.0010423114569027092 [4.54475566e-04 1.60448994e-04 5.70913371e-05]\n",
      "0.0010423104457666936 [4.54480640e-04 1.60445911e-04 5.70897205e-05]\n",
      "0.0010423096217022463 [4.54484767e-04 1.60443383e-04 5.70883987e-05]\n",
      "0.00104230895006586 [4.54488125e-04 1.60441310e-04 5.70873184e-05]\n",
      "0.001042308402637866 [4.54490858e-04 1.60439612e-04 5.70864356e-05]\n",
      "0.00104230795642922 [4.54493082e-04 1.60438221e-04 5.70857145e-05]\n",
      "0.0010423075927104399 [4.54494893e-04 1.60437082e-04 5.70851256e-05]\n",
      "0.0010423072962212094 [4.54496367e-04 1.60436150e-04 5.70846447e-05]\n",
      "0.0010423070545269321 [4.54497568e-04 1.60435388e-04 5.70842521e-05]\n",
      "0.0010423068574948486 [4.54498545e-04 1.60434764e-04 5.70839316e-05]\n",
      "0.00104230669686744 [4.54499342e-04 1.60434254e-04 5.70836701e-05]\n",
      "0.0010423065659150156 [4.54499990e-04 1.60433837e-04 5.70834567e-05]\n",
      "0.001042306459152757 [4.54500519e-04 1.60433496e-04 5.70832826e-05]\n",
      "0.0010423063721102283 [4.54500949e-04 1.60433217e-04 5.70831406e-05]\n",
      "0.0010423063011436198 [4.54501300e-04 1.60432989e-04 5.70830247e-05]\n",
      "0.0010423062432827785 [4.54501586e-04 1.60432803e-04 5.70829302e-05]\n",
      "0.0010423061961065791 [4.54501819e-04 1.60432651e-04 5.70828531e-05]\n",
      "0.0010423061576413683 [4.54502009e-04 1.60432527e-04 5.70827902e-05]\n",
      "0.0010423061262782149 [4.54502164e-04 1.60432426e-04 5.70827389e-05]\n",
      "0.0010423061007054737 [4.54502290e-04 1.60432343e-04 5.70826971e-05]\n",
      "0.001042306079853827 [4.54502393e-04 1.60432275e-04 5.70826630e-05]\n",
      "0.0010423060628514935 [4.54502476e-04 1.60432220e-04 5.70826352e-05]\n",
      "0.00104230604898772 [4.54502545e-04 1.60432175e-04 5.70826126e-05]\n",
      "0.001042306037683028 [4.54502600e-04 1.60432138e-04 5.70825941e-05]\n",
      "0.0010423060284649548 [4.54502646e-04 1.60432108e-04 5.70825790e-05]\n",
      "0.0010423060209482882 [4.54502683e-04 1.60432084e-04 5.70825667e-05]\n",
      "0.0010423060148189451 [4.54502713e-04 1.60432064e-04 5.70825567e-05]\n",
      "0.0010423060098208336 [4.54502738e-04 1.60432048e-04 5.70825485e-05]\n",
      "0.001042306005745147 [4.54502758e-04 1.60432034e-04 5.70825419e-05]\n",
      "0.0010423060024216248 [4.54502774e-04 1.60432023e-04 5.70825365e-05]\n",
      "0.0010423059997114394 [4.54502787e-04 1.60432015e-04 5.70825320e-05]\n",
      "0.0010423059975013906 [4.54502798e-04 1.60432007e-04 5.70825284e-05]\n",
      "0.0010423059956991738 [4.54502807e-04 1.60432001e-04 5.70825255e-05]\n",
      "0.0010423059942295228 [4.54502814e-04 1.60431997e-04 5.70825231e-05]\n",
      "0.0010423059930310635 [4.54502820e-04 1.60431993e-04 5.70825211e-05]\n",
      "0.0010423059920537487 [4.54502825e-04 1.60431989e-04 5.70825195e-05]\n",
      "0.0010423059912567697 [4.54502829e-04 1.60431987e-04 5.70825182e-05]\n",
      "0.001042305990606847 [4.54502832e-04 1.60431985e-04 5.70825172e-05]\n",
      "0.0010423059900768448 [4.54502835e-04 1.60431983e-04 5.70825163e-05]\n",
      "0.0010423059896446348 [4.54502837e-04 1.60431982e-04 5.70825156e-05]\n",
      "0.0010423059892921722 [4.54502839e-04 1.60431980e-04 5.70825150e-05]\n",
      "0.0010423059890047417 [4.54502840e-04 1.60431979e-04 5.70825146e-05]\n",
      "0.0010423059887703437 [4.54502841e-04 1.60431979e-04 5.70825142e-05]\n",
      "0.0010423059885791928 [4.54502842e-04 1.60431978e-04 5.70825139e-05]\n",
      "0.0010423059884233088 [4.54502843e-04 1.60431978e-04 5.70825136e-05]\n",
      "0.0010423059882961867 [4.54502843e-04 1.60431977e-04 5.70825134e-05]\n",
      "0.001042305988192518 [4.54502844e-04 1.60431977e-04 5.70825132e-05]\n",
      "0.0010423059881079753 [4.54502844e-04 1.60431977e-04 5.70825131e-05]\n",
      "0.0010423059880390307 [4.54502845e-04 1.60431976e-04 5.70825130e-05]\n",
      "0.0010423059879828057 [4.54502845e-04 1.60431976e-04 5.70825129e-05]\n",
      "0.0010423059879369543 [4.54502845e-04 1.60431976e-04 5.70825128e-05]\n",
      "0.0010423059878995613 [4.54502845e-04 1.60431976e-04 5.70825128e-05]\n",
      "0.0010423059878690672 [4.54502846e-04 1.60431976e-04 5.70825127e-05]\n",
      "0.0010423059878441991 [4.54502846e-04 1.60431976e-04 5.70825127e-05]\n",
      "0.0010423059878239191 [4.54502846e-04 1.60431976e-04 5.70825126e-05]\n",
      "0.00104230598780738 [4.54502846e-04 1.60431976e-04 5.70825126e-05]\n",
      "0.0010423059877938928 [4.54502846e-04 1.60431975e-04 5.70825126e-05]\n"
     ]
    }
   ],
   "source": [
    "F = np.zeros(n_factors)\n",
    "for step in range(100):\n",
    "    sse = loss(X[-100:,:,(0,1,2)],F,r[-100:,:],a[-100:],1,n_factors,100,n_stocks)\n",
    "    F -= grad_loss(X[-100:,:,(0,1,2)],F,r[-100:,:],a[-100:],1,n_factors,n_time=100,n_stock=n_stocks)*0.00001\n",
    "    print(sse,F)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Unfortunately, the gredient descend scheme cannot guarantee to lead us to the global optimum. And it is still not helpful in interpreting market size effect."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.3 Calibration Scheme 3: Kalman Filter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "kf_df = myModel.kalman_filter_calibration()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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Ok3/gBM57dM8lXcdP7iQ29p2pwhnsgdng7vzY3WQCL7k/q/8ufLDL57OKofdLuPUKOXmb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\nd1DrzD6b/868osGC8tPH/e5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9Tj6s+FrZ3hdyGoIm/Snsr/F4omSoTwqJgUMXq5TlWFQT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\nWY9+73pRMKYU9FfamJKWloa7uztLlizBzs6OL7/8EicnJ+rUqVNk2jd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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1187bdb70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "kf_df['market_cap_F'].plot()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Even more unreliable. The factor return for market cap size is highly positive and volatile."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "So, from all the three schemes we have discussed above, we concluded that even though BARRA's risk model has the virtue of simplicity, it is not feasible in interpreting factor returns. Or the regular calibration schemes need amendment. Personally I prefer the scheme 2 as it reserve the simplicity of the objective, but may need a better approach to get the globel optimization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}