Parametric & Volatility-managed portfolio.ipynb

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    "## Data\n",
    "\n",
    "We apply the parametric portfolio policies developed by Brandt, Santa-Clara, and Valkanov (2009). The data file contains monthly returns for nine firm-specific characteristics: Market, SMB, HML, RMW, CMA, UMD, ROE, IA, BAB.\n",
    "- Assume that these returns were generated by Nt stocks and that the number of stocks is constant over time.\n",
    "- The first five characteristics (Market, SMB, HML, RMW, CMA) are from Fama and French (2015), the sixth (UMD) is from Carhart (1997), the profitability (ROE) and investment (IA) factors are from Hou, Xue, and Zhang (2015), and the betting-against-beta (BAB) factor is from Frazzini and Pedersen (2014).\n",
    "- All factors are returns in excess of the risk-free rate.\n",
    "- In particular, every factor (besides MKT and BAB) is the return of a long-short portfolio of stocks with 1usd in the long leg and 1usd in the short leg, and thus, their returns equal their excess returns.\n",
    "- The MKT and BAB factors are also long-short portfolios because they are returns in excess of the risk-free rate."
   ]
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   "source": [
    "## Parametric portfolio"
   ]
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   "source": [
    "Parametric portfolio policies show us how to reduce the dimensionality of the portfolio problem when investing in factors.\n",
    "Instead of modeling asset returns in terms of factors, and then constructing the optimal portfolio, parametric portfolio policies directly specify portfolio weights in terms of factors.\n",
    "\n",
    "$$w_{t}(\\theta) = w_{b,t} + (F_{1,t} \\theta_{1} + F_{2,t} \\theta_{2} + ... + F_{K,t} \\theta_{K})/N_{t}$$\n",
    "\n",
    "where $w_{b,t}$ is the weight of the benchmark portfolio, $F_{k,t}$ is the factor k's return at time t and $N_{t}$ is the number of assets."
   ]
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   "id": "bf490574-a3fe-464e-b38d-2df9deed08ff",
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   "source": [
    "**Finding the optimal $\\theta$ vector for a mean-variance investor with risk aversion of $\\gamma=5$, assuming the investor can invest in only these nine factors. (No out-of-sample analysis).**"
   ]
  },
  {
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   "id": "482e6ef0-55db-40fe-aeb6-3a8438c2b831",
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   "source": [
    "First, let's import the libraries we need for this problem"
   ]
  },
  {
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    "lastSuccessfullyExecutedCode": "import pandas as pd\nimport numpy as np\nfrom scipy.optimize import minimize"
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   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "from scipy.optimize import minimize"
   ]
  },
  {
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   "id": "04df0ce0-6f3e-4049-8df9-d2efe592a22d",
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   "source": [
    "Then we load the given data set. Since we consider the market factor as the benchmark weights, we will allocate the returns of this factor into the excess return benchmark, and the other 8 factors into the excess returns Parametric Portfolio."
   ]
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Market</th>\n",
       "      <th>SMB</th>\n",
       "      <th>HML</th>\n",
       "      <th>RMW</th>\n",
       "      <th>CMA</th>\n",
       "      <th>UMD</th>\n",
       "      <th>ROE</th>\n",
       "      <th>IA</th>\n",
       "      <th>BAB</th>\n",
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       "    <tr>\n",
       "      <th>Dates</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
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       "    <tr>\n",
       "      <th>196702</th>\n",
       "      <td>0.0078</td>\n",
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       "      <td>-0.0151</td>\n",
       "      <td>0.0142</td>\n",
       "      <td>0.018876</td>\n",
       "      <td>-0.016933</td>\n",
       "      <td>0.0081</td>\n",
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       "    <tr>\n",
       "      <th>196704</th>\n",
       "      <td>0.0389</td>\n",
       "      <td>0.0062</td>\n",
       "      <td>-0.0264</td>\n",
       "      <td>0.0243</td>\n",
       "      <td>-0.0375</td>\n",
       "      <td>0.0064</td>\n",
       "      <td>0.010983</td>\n",
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       "      <td>0.0171</td>\n",
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       "      <th>196705</th>\n",
       "      <td>-0.0433</td>\n",
       "      <td>0.0198</td>\n",
       "      <td>0.0080</td>\n",
       "      <td>-0.0175</td>\n",
       "      <td>0.0161</td>\n",
       "      <td>0.0067</td>\n",
       "      <td>0.005234</td>\n",
       "      <td>0.024686</td>\n",
       "      <td>0.0201</td>\n",
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       "      <th>196706</th>\n",
       "      <td>0.0241</td>\n",
       "      <td>0.0596</td>\n",
       "      <td>0.0096</td>\n",
       "      <td>-0.0064</td>\n",
       "      <td>-0.0239</td>\n",
       "      <td>0.0603</td>\n",
       "      <td>0.002945</td>\n",
       "      <td>-0.021700</td>\n",
       "      <td>-0.0163</td>\n",
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       "      <th>...</th>\n",
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       "      <td>...</td>\n",
       "      <td>...</td>\n",
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       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202008</th>\n",
       "      <td>0.0763</td>\n",
       "      <td>-0.0094</td>\n",
       "      <td>-0.0294</td>\n",
       "      <td>0.0427</td>\n",
       "      <td>-0.0144</td>\n",
       "      <td>0.0051</td>\n",
       "      <td>-0.008682</td>\n",
       "      <td>-0.028534</td>\n",
       "      <td>-0.0399</td>\n",
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       "    <tr>\n",
       "      <th>202009</th>\n",
       "      <td>-0.0363</td>\n",
       "      <td>0.0007</td>\n",
       "      <td>-0.0251</td>\n",
       "      <td>-0.0115</td>\n",
       "      <td>-0.0177</td>\n",
       "      <td>0.0305</td>\n",
       "      <td>0.012255</td>\n",
       "      <td>-0.021992</td>\n",
       "      <td>0.0129</td>\n",
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       "    <tr>\n",
       "      <th>202010</th>\n",
       "      <td>-0.0210</td>\n",
       "      <td>0.0476</td>\n",
       "      <td>0.0403</td>\n",
       "      <td>-0.0060</td>\n",
       "      <td>-0.0053</td>\n",
       "      <td>-0.0303</td>\n",
       "      <td>-0.024671</td>\n",
       "      <td>-0.007387</td>\n",
       "      <td>-0.0201</td>\n",
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       "    <tr>\n",
       "      <th>202011</th>\n",
       "      <td>0.1247</td>\n",
       "      <td>0.0675</td>\n",
       "      <td>0.0211</td>\n",
       "      <td>-0.0278</td>\n",
       "      <td>0.0105</td>\n",
       "      <td>-0.1225</td>\n",
       "      <td>-0.144600</td>\n",
       "      <td>0.030672</td>\n",
       "      <td>-0.0509</td>\n",
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       "      <th>202012</th>\n",
       "      <td>0.0463</td>\n",
       "      <td>0.0467</td>\n",
       "      <td>-0.0136</td>\n",
       "      <td>-0.0215</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0242</td>\n",
       "      <td>-0.035415</td>\n",
       "      <td>-0.027001</td>\n",
       "      <td>0.0534</td>\n",
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       "<p>647 rows × 9 columns</p>\n",
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       "        Market     SMB     HML     RMW     CMA     UMD       ROE        IA  \\\n",
       "Dates                                                                        \n",
       "196702  0.0078  0.0334 -0.0217  0.0194 -0.0094  0.0356  0.035317 -0.002064   \n",
       "196703  0.0399  0.0163  0.0031  0.0090 -0.0151  0.0142  0.018876 -0.016933   \n",
       "196704  0.0389  0.0062 -0.0264  0.0243 -0.0375  0.0064  0.010983 -0.029519   \n",
       "196705 -0.0433  0.0198  0.0080 -0.0175  0.0161  0.0067  0.005234  0.024686   \n",
       "196706  0.0241  0.0596  0.0096 -0.0064 -0.0239  0.0603  0.002945 -0.021700   \n",
       "...        ...     ...     ...     ...     ...     ...       ...       ...   \n",
       "202008  0.0763 -0.0094 -0.0294  0.0427 -0.0144  0.0051 -0.008682 -0.028534   \n",
       "202009 -0.0363  0.0007 -0.0251 -0.0115 -0.0177  0.0305  0.012255 -0.021992   \n",
       "202010 -0.0210  0.0476  0.0403 -0.0060 -0.0053 -0.0303 -0.024671 -0.007387   \n",
       "202011  0.1247  0.0675  0.0211 -0.0278  0.0105 -0.1225 -0.144600  0.030672   \n",
       "202012  0.0463  0.0467 -0.0136 -0.0215  0.0000 -0.0242 -0.035415 -0.027001   \n",
       "\n",
       "           BAB  \n",
       "Dates           \n",
       "196702  0.0262  \n",
       "196703  0.0081  \n",
       "196704  0.0171  \n",
       "196705  0.0201  \n",
       "196706 -0.0163  \n",
       "...        ...  \n",
       "202008 -0.0399  \n",
       "202009  0.0129  \n",
       "202010 -0.0201  \n",
       "202011 -0.0509  \n",
       "202012  0.0534  \n",
       "\n",
       "[647 rows x 9 columns]"
      ]
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     "execution_count": 2,
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   ],
   "source": [
    "excess_returns = pd.read_excel('QPM-FactorsData.xlsx', index_col=[0])\n",
    "excess_returns"
   ]
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       "      <th>HML</th>\n",
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       "      <th>196702</th>\n",
       "      <td>0.0334</td>\n",
       "      <td>-0.0217</td>\n",
       "      <td>0.0194</td>\n",
       "      <td>-0.0094</td>\n",
       "      <td>0.0356</td>\n",
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       "      <th>196703</th>\n",
       "      <td>0.0163</td>\n",
       "      <td>0.0031</td>\n",
       "      <td>0.0090</td>\n",
       "      <td>-0.0151</td>\n",
       "      <td>0.0142</td>\n",
       "      <td>0.018876</td>\n",
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       "      <th>196704</th>\n",
       "      <td>0.0062</td>\n",
       "      <td>-0.0264</td>\n",
       "      <td>0.0243</td>\n",
       "      <td>-0.0375</td>\n",
       "      <td>0.0064</td>\n",
       "      <td>0.010983</td>\n",
       "      <td>-0.029519</td>\n",
       "      <td>0.0171</td>\n",
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       "    <tr>\n",
       "      <th>196705</th>\n",
       "      <td>0.0198</td>\n",
       "      <td>0.0080</td>\n",
       "      <td>-0.0175</td>\n",
       "      <td>0.0161</td>\n",
       "      <td>0.0067</td>\n",
       "      <td>0.005234</td>\n",
       "      <td>0.024686</td>\n",
       "      <td>0.0201</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196706</th>\n",
       "      <td>0.0596</td>\n",
       "      <td>0.0096</td>\n",
       "      <td>-0.0064</td>\n",
       "      <td>-0.0239</td>\n",
       "      <td>0.0603</td>\n",
       "      <td>0.002945</td>\n",
       "      <td>-0.021700</td>\n",
       "      <td>-0.0163</td>\n",
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       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202008</th>\n",
       "      <td>-0.0094</td>\n",
       "      <td>-0.0294</td>\n",
       "      <td>0.0427</td>\n",
       "      <td>-0.0144</td>\n",
       "      <td>0.0051</td>\n",
       "      <td>-0.008682</td>\n",
       "      <td>-0.028534</td>\n",
       "      <td>-0.0399</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202009</th>\n",
       "      <td>0.0007</td>\n",
       "      <td>-0.0251</td>\n",
       "      <td>-0.0115</td>\n",
       "      <td>-0.0177</td>\n",
       "      <td>0.0305</td>\n",
       "      <td>0.012255</td>\n",
       "      <td>-0.021992</td>\n",
       "      <td>0.0129</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202010</th>\n",
       "      <td>0.0476</td>\n",
       "      <td>0.0403</td>\n",
       "      <td>-0.0060</td>\n",
       "      <td>-0.0053</td>\n",
       "      <td>-0.0303</td>\n",
       "      <td>-0.024671</td>\n",
       "      <td>-0.007387</td>\n",
       "      <td>-0.0201</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202011</th>\n",
       "      <td>0.0675</td>\n",
       "      <td>0.0211</td>\n",
       "      <td>-0.0278</td>\n",
       "      <td>0.0105</td>\n",
       "      <td>-0.1225</td>\n",
       "      <td>-0.144600</td>\n",
       "      <td>0.030672</td>\n",
       "      <td>-0.0509</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202012</th>\n",
       "      <td>0.0467</td>\n",
       "      <td>-0.0136</td>\n",
       "      <td>-0.0215</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0242</td>\n",
       "      <td>-0.035415</td>\n",
       "      <td>-0.027001</td>\n",
       "      <td>0.0534</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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       "<p>647 rows × 8 columns</p>\n",
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      "text/plain": [
       "           SMB     HML     RMW     CMA     UMD       ROE        IA     BAB\n",
       "Dates                                                                     \n",
       "196702  0.0334 -0.0217  0.0194 -0.0094  0.0356  0.035317 -0.002064  0.0262\n",
       "196703  0.0163  0.0031  0.0090 -0.0151  0.0142  0.018876 -0.016933  0.0081\n",
       "196704  0.0062 -0.0264  0.0243 -0.0375  0.0064  0.010983 -0.029519  0.0171\n",
       "196705  0.0198  0.0080 -0.0175  0.0161  0.0067  0.005234  0.024686  0.0201\n",
       "196706  0.0596  0.0096 -0.0064 -0.0239  0.0603  0.002945 -0.021700 -0.0163\n",
       "...        ...     ...     ...     ...     ...       ...       ...     ...\n",
       "202008 -0.0094 -0.0294  0.0427 -0.0144  0.0051 -0.008682 -0.028534 -0.0399\n",
       "202009  0.0007 -0.0251 -0.0115 -0.0177  0.0305  0.012255 -0.021992  0.0129\n",
       "202010  0.0476  0.0403 -0.0060 -0.0053 -0.0303 -0.024671 -0.007387 -0.0201\n",
       "202011  0.0675  0.0211 -0.0278  0.0105 -0.1225 -0.144600  0.030672 -0.0509\n",
       "202012  0.0467 -0.0136 -0.0215  0.0000 -0.0242 -0.035415 -0.027001  0.0534\n",
       "\n",
       "[647 rows x 8 columns]"
      ]
     },
     "execution_count": 3,
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   ],
   "source": [
    "excess_returns_bm = excess_returns['Market']\n",
    "excess_returns_PP = excess_returns.iloc[:,1:]\n",
    "excess_returns_PP"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "431b030c-2d7e-483e-8fda-e2f374a2716b",
   "metadata": {},
   "source": [
    "Next, let's define a function ```MVP_weights``` to return objective function of the optimization problem and optimal weight for mean-variance portfolio."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "0b9e1183-d633-465c-a3eb-4c50c7043784",
   "metadata": {
    "executionCancelledAt": null,
    "executionTime": 17,
    "lastExecutedAt": 1700780360520,
    "lastScheduledRunId": null,
    "lastSuccessfullyExecutedCode": "# Define the objective function to be minimized (portfolio variance)\ndef obj_MVP(weights,bm_return,char_return):\n    portfolio_return = bm_return + char_return @ weights\n    mean_return = np.mean(portfolio_return)\n    portfolio_risk = np.var(portfolio_return)\n    return (gamma/2) * portfolio_risk - mean_return\n\n# Design a function to calculate the optimal weight of GMV-C portfolio:\ndef MVP_C_w(bm_return,char_return):\n    #mu = returns.mean()\n    #V = returns.cov()\n    # Define the equality constraint (weights sum to 0)\n    constraints = ({'type': 'eq', 'fun': lambda weights: np.sum(weights)})\n\n    # Define the initial guess for weights\n    initial_weights = np.ones(len(char_return.columns)) / (len(char_return.columns))\n\n    # Use scipy.optimize.minimize to find the optimal weights\n    result = minimize(obj_MVP, initial_weights, args=(bm_return,char_return,), method='SLSQP', constraints=constraints)\n    # Extract the optimal weights\n    optimal_weights = result.x\n    \n    return optimal_weights"
   },
   "outputs": [],
   "source": [
    "# Define the objective function to be minimized (portfolio variance)\n",
    "def obj_MVP(weights,bm_return,char_return):\n",
    "    portfolio_return = bm_return + char_return @ weights\n",
    "    mean_return = np.mean(portfolio_return)\n",
    "    portfolio_risk = np.var(portfolio_return)\n",
    "    return (gamma/2) * portfolio_risk - mean_return\n",
    "\n",
    "# Design a function to calculate the optimal weight of GMV-C portfolio:\n",
    "def MVP_C_w(bm_return,char_return):\n",
    "\n",
    "    # Define the initial guess for weights\n",
    "    initial_weights = np.ones(len(char_return.columns)) / (len(char_return.columns))\n",
    "\n",
    "    # Use scipy.optimize.minimize to find the optimal weights\n",
    "    result = minimize(obj_MVP, initial_weights, args=(bm_return,char_return,), method='SLSQP')\n",
    "    \n",
    "    # Extract the optimal weights\n",
    "    optimal_weights = result.x\n",
    "    \n",
    "    return optimal_weights"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1932693f-0c9c-4d45-98b1-f188eef664d5",
   "metadata": {},
   "source": [
    "Now let's calculate our theta for 8 factors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c03081c1-95aa-4364-abdb-2ce542a009f0",
   "metadata": {
    "executionCancelledAt": null,
    "executionTime": 88,
    "lastExecutedAt": 1700780362910,
    "lastScheduledRunId": null,
    "lastSuccessfullyExecutedCode": "# Given value of risk tolerance\ngamma = 5\n\n# Calculate theta vector\ntheta = MVP_C_w(excess_returns_bm,excess_returns_PP)\ntheta"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.01658799, -0.136824  ,  0.54525987,  1.07803223,  0.20377353,\n",
       "        1.71734256,  1.61456511,  0.72318278])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Given value of risk tolerance\n",
    "gamma = 5\n",
    "\n",
    "# Calculate theta vector\n",
    "theta = MVP_C_w(excess_returns_bm,excess_returns_PP)\n",
    "theta"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cdae0460-3092-45b0-bb6b-4edf1da21867",
   "metadata": {},
   "source": [
    "**Calculation of the Sharpe ratio for each of the nine factors and comparison to that of the parametric portfolio.**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c7015f2-4079-430d-a5cf-8bf8702043b5",
   "metadata": {},
   "source": [
    "First, let's calculate the Sharpe Ratio for each factor's returns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "5aa66b26-c83f-4789-bdc9-567121e6f5ca",
   "metadata": {
    "executionCancelledAt": null,
    "executionTime": 16,
    "lastExecutedAt": 1700780393148,
    "lastScheduledRunId": null,
    "lastSuccessfullyExecutedCode": "SR_factor = (excess_returns.mean() * 12) / (excess_returns.std() * np.sqrt(12))\nSR_factor"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Market    0.432904\n",
       "SMB       0.197861\n",
       "HML       0.277910\n",
       "RMW       0.420687\n",
       "CMA       0.477857\n",
       "UMD       0.507284\n",
       "ROE       0.684304\n",
       "IA        0.626422\n",
       "BAB       0.899567\n",
       "dtype: float64"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SR_factor = (excess_returns.mean() * 12) / (excess_returns.std() * np.sqrt(12))\n",
    "SR_factor"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f1695bf1-dd7f-46b8-953f-44cd78bb46cf",
   "metadata": {},
   "source": [
    "Now we calculate the Sharpe Ratio for the Parametric portfolio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "8e5f6ea4-97fd-42b5-a37b-a3652ff30248",
   "metadata": {
    "executionCancelledAt": null,
    "executionTime": 20,
    "lastExecutedAt": 1700780395030,
    "lastScheduledRunId": null,
    "lastSuccessfullyExecutedCode": "# Vector of weights of 9 characteristics in the Parametric Portfolio\nPP_weight = np.insert(theta, 0 ,1)\n\n# Calculate Sharpe Ratio for Parametric Portfolio\nPP_returns = excess_returns @ PP_weight\nSR_PP = (np.mean(PP_returns) * 12) / (np.std(PP_returns) * np.sqrt(12))\nSR_PP"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.4388440549546873"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Vector of weights of 9 characteristics in the Parametric Portfolio\n",
    "PP_weight = np.insert(theta, 0 ,1)\n",
    "\n",
    "# Calculate Sharpe Ratio for Parametric Portfolio\n",
    "PP_returns = excess_returns @ PP_weight\n",
    "SR_PP = (np.mean(PP_returns) * 12) / (np.std(PP_returns) * np.sqrt(12))\n",
    "SR_PP"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2b2ad58c-013c-43cc-b496-936421241a9d",
   "metadata": {},
   "source": [
    "We can see that the Sharpe Ratio of the parametric portfolio outperforms all the Sharpe Ratios of individual factor portfolios."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "63e9fecd",
   "metadata": {},
   "source": [
    "## Volatility-managed Portfolio"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c79e6c8",
   "metadata": {},
   "source": [
    "Now we build on parametric portfolio policies, by modeling the portfolio weight on each factor, $θ_{k}$, to depend on factor or market volatility"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dfff4125",
   "metadata": {},
   "source": [
    "We use an estimation window of 120 months. Therefore, to facilitate comparison, the in-sample and out-of-sample performance will be evaluated from January 1977 to December 2020 (monthly daily instead of daily data as in the original papers).\n",
    "\n",
    "Define $f_{t+1}$ to be an excess return.\n",
    "\n",
    "Construct a new volatility-managed factor, whose return is:\n",
    "\n",
    "$$f_{t+1}^{\\sigma} = \\frac{c}{\\sigma_t^2 (f)} * f_{t+1}$$ \n",
    "\n",
    "where:\n",
    "- $\\sigma_t (f)$ is the previous 12 month's realized volatility, estimated using monthly data\n",
    "- choose $c$ so $f^{\\sigma}$ has the same unconditional volatility as $f$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "7130ca13",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
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       "    }\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>Market</th>\n",
       "      <th>SMB</th>\n",
       "      <th>HML</th>\n",
       "      <th>RMW</th>\n",
       "      <th>CMA</th>\n",
       "      <th>UMD</th>\n",
       "      <th>ROE</th>\n",
       "      <th>IA</th>\n",
       "      <th>BAB</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Dates</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>196702</th>\n",
       "      <td>0.0078</td>\n",
       "      <td>0.0334</td>\n",
       "      <td>-0.0217</td>\n",
       "      <td>0.0194</td>\n",
       "      <td>-0.0094</td>\n",
       "      <td>0.0356</td>\n",
       "      <td>0.035317</td>\n",
       "      <td>-0.002064</td>\n",
       "      <td>0.0262</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196703</th>\n",
       "      <td>0.0399</td>\n",
       "      <td>0.0163</td>\n",
       "      <td>0.0031</td>\n",
       "      <td>0.0090</td>\n",
       "      <td>-0.0151</td>\n",
       "      <td>0.0142</td>\n",
       "      <td>0.018876</td>\n",
       "      <td>-0.016933</td>\n",
       "      <td>0.0081</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196704</th>\n",
       "      <td>0.0389</td>\n",
       "      <td>0.0062</td>\n",
       "      <td>-0.0264</td>\n",
       "      <td>0.0243</td>\n",
       "      <td>-0.0375</td>\n",
       "      <td>0.0064</td>\n",
       "      <td>0.010983</td>\n",
       "      <td>-0.029519</td>\n",
       "      <td>0.0171</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196705</th>\n",
       "      <td>-0.0433</td>\n",
       "      <td>0.0198</td>\n",
       "      <td>0.0080</td>\n",
       "      <td>-0.0175</td>\n",
       "      <td>0.0161</td>\n",
       "      <td>0.0067</td>\n",
       "      <td>0.005234</td>\n",
       "      <td>0.024686</td>\n",
       "      <td>0.0201</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196706</th>\n",
       "      <td>0.0241</td>\n",
       "      <td>0.0596</td>\n",
       "      <td>0.0096</td>\n",
       "      <td>-0.0064</td>\n",
       "      <td>-0.0239</td>\n",
       "      <td>0.0603</td>\n",
       "      <td>0.002945</td>\n",
       "      <td>-0.021700</td>\n",
       "      <td>-0.0163</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202008</th>\n",
       "      <td>0.0763</td>\n",
       "      <td>-0.0094</td>\n",
       "      <td>-0.0294</td>\n",
       "      <td>0.0427</td>\n",
       "      <td>-0.0144</td>\n",
       "      <td>0.0051</td>\n",
       "      <td>-0.008682</td>\n",
       "      <td>-0.028534</td>\n",
       "      <td>-0.0399</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202009</th>\n",
       "      <td>-0.0363</td>\n",
       "      <td>0.0007</td>\n",
       "      <td>-0.0251</td>\n",
       "      <td>-0.0115</td>\n",
       "      <td>-0.0177</td>\n",
       "      <td>0.0305</td>\n",
       "      <td>0.012255</td>\n",
       "      <td>-0.021992</td>\n",
       "      <td>0.0129</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202010</th>\n",
       "      <td>-0.0210</td>\n",
       "      <td>0.0476</td>\n",
       "      <td>0.0403</td>\n",
       "      <td>-0.0060</td>\n",
       "      <td>-0.0053</td>\n",
       "      <td>-0.0303</td>\n",
       "      <td>-0.024671</td>\n",
       "      <td>-0.007387</td>\n",
       "      <td>-0.0201</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202011</th>\n",
       "      <td>0.1247</td>\n",
       "      <td>0.0675</td>\n",
       "      <td>0.0211</td>\n",
       "      <td>-0.0278</td>\n",
       "      <td>0.0105</td>\n",
       "      <td>-0.1225</td>\n",
       "      <td>-0.144600</td>\n",
       "      <td>0.030672</td>\n",
       "      <td>-0.0509</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202012</th>\n",
       "      <td>0.0463</td>\n",
       "      <td>0.0467</td>\n",
       "      <td>-0.0136</td>\n",
       "      <td>-0.0215</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0242</td>\n",
       "      <td>-0.035415</td>\n",
       "      <td>-0.027001</td>\n",
       "      <td>0.0534</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>647 rows × 9 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "        Market     SMB     HML     RMW     CMA     UMD       ROE        IA  \\\n",
       "Dates                                                                        \n",
       "196702  0.0078  0.0334 -0.0217  0.0194 -0.0094  0.0356  0.035317 -0.002064   \n",
       "196703  0.0399  0.0163  0.0031  0.0090 -0.0151  0.0142  0.018876 -0.016933   \n",
       "196704  0.0389  0.0062 -0.0264  0.0243 -0.0375  0.0064  0.010983 -0.029519   \n",
       "196705 -0.0433  0.0198  0.0080 -0.0175  0.0161  0.0067  0.005234  0.024686   \n",
       "196706  0.0241  0.0596  0.0096 -0.0064 -0.0239  0.0603  0.002945 -0.021700   \n",
       "...        ...     ...     ...     ...     ...     ...       ...       ...   \n",
       "202008  0.0763 -0.0094 -0.0294  0.0427 -0.0144  0.0051 -0.008682 -0.028534   \n",
       "202009 -0.0363  0.0007 -0.0251 -0.0115 -0.0177  0.0305  0.012255 -0.021992   \n",
       "202010 -0.0210  0.0476  0.0403 -0.0060 -0.0053 -0.0303 -0.024671 -0.007387   \n",
       "202011  0.1247  0.0675  0.0211 -0.0278  0.0105 -0.1225 -0.144600  0.030672   \n",
       "202012  0.0463  0.0467 -0.0136 -0.0215  0.0000 -0.0242 -0.035415 -0.027001   \n",
       "\n",
       "           BAB  \n",
       "Dates           \n",
       "196702  0.0262  \n",
       "196703  0.0081  \n",
       "196704  0.0171  \n",
       "196705  0.0201  \n",
       "196706 -0.0163  \n",
       "...        ...  \n",
       "202008 -0.0399  \n",
       "202009  0.0129  \n",
       "202010 -0.0201  \n",
       "202011 -0.0509  \n",
       "202012  0.0534  \n",
       "\n",
       "[647 rows x 9 columns]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "original_factors = pd.read_excel('QPM-FactorsData.xlsx', index_col=[0])\n",
    "original_factors"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "075b6b47",
   "metadata": {},
   "source": [
    "Next we calculate the previous 12-months realized variance of each factor $\\sigma_{t}^2 (f)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "16ed1c7a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Market</th>\n",
       "      <th>SMB</th>\n",
       "      <th>HML</th>\n",
       "      <th>RMW</th>\n",
       "      <th>CMA</th>\n",
       "      <th>UMD</th>\n",
       "      <th>ROE</th>\n",
       "      <th>IA</th>\n",
       "      <th>BAB</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Dates</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>196801</th>\n",
       "      <td>0.001046</td>\n",
       "      <td>0.000371</td>\n",
       "      <td>0.000595</td>\n",
       "      <td>0.000346</td>\n",
       "      <td>0.000807</td>\n",
       "      <td>0.000795</td>\n",
       "      <td>0.000298</td>\n",
       "      <td>0.000611</td>\n",
       "      <td>0.000490</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196802</th>\n",
       "      <td>0.001221</td>\n",
       "      <td>0.000618</td>\n",
       "      <td>0.000565</td>\n",
       "      <td>0.000312</td>\n",
       "      <td>0.000858</td>\n",
       "      <td>0.000917</td>\n",
       "      <td>0.000228</td>\n",
       "      <td>0.000677</td>\n",
       "      <td>0.000483</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196803</th>\n",
       "      <td>0.001096</td>\n",
       "      <td>0.000713</td>\n",
       "      <td>0.000569</td>\n",
       "      <td>0.000317</td>\n",
       "      <td>0.000848</td>\n",
       "      <td>0.000965</td>\n",
       "      <td>0.000222</td>\n",
       "      <td>0.000661</td>\n",
       "      <td>0.000567</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196804</th>\n",
       "      <td>0.001671</td>\n",
       "      <td>0.000816</td>\n",
       "      <td>0.000512</td>\n",
       "      <td>0.000335</td>\n",
       "      <td>0.000842</td>\n",
       "      <td>0.001113</td>\n",
       "      <td>0.000315</td>\n",
       "      <td>0.000642</td>\n",
       "      <td>0.000579</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196805</th>\n",
       "      <td>0.001448</td>\n",
       "      <td>0.000957</td>\n",
       "      <td>0.000512</td>\n",
       "      <td>0.000306</td>\n",
       "      <td>0.000849</td>\n",
       "      <td>0.001158</td>\n",
       "      <td>0.000330</td>\n",
       "      <td>0.000639</td>\n",
       "      <td>0.000577</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202007</th>\n",
       "      <td>0.004751</td>\n",
       "      <td>0.001022</td>\n",
       "      <td>0.002410</td>\n",
       "      <td>0.000181</td>\n",
       "      <td>0.000382</td>\n",
       "      <td>0.002688</td>\n",
       "      <td>0.000506</td>\n",
       "      <td>0.000393</td>\n",
       "      <td>0.002099</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202008</th>\n",
       "      <td>0.004933</td>\n",
       "      <td>0.000999</td>\n",
       "      <td>0.002367</td>\n",
       "      <td>0.000328</td>\n",
       "      <td>0.000389</td>\n",
       "      <td>0.002256</td>\n",
       "      <td>0.000513</td>\n",
       "      <td>0.000442</td>\n",
       "      <td>0.002191</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202009</th>\n",
       "      <td>0.005196</td>\n",
       "      <td>0.001005</td>\n",
       "      <td>0.001494</td>\n",
       "      <td>0.000320</td>\n",
       "      <td>0.000238</td>\n",
       "      <td>0.001766</td>\n",
       "      <td>0.000505</td>\n",
       "      <td>0.000328</td>\n",
       "      <td>0.002202</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202010</th>\n",
       "      <td>0.005301</td>\n",
       "      <td>0.001251</td>\n",
       "      <td>0.001961</td>\n",
       "      <td>0.000324</td>\n",
       "      <td>0.000239</td>\n",
       "      <td>0.001914</td>\n",
       "      <td>0.000576</td>\n",
       "      <td>0.000327</td>\n",
       "      <td>0.002233</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202011</th>\n",
       "      <td>0.006336</td>\n",
       "      <td>0.001666</td>\n",
       "      <td>0.002174</td>\n",
       "      <td>0.000374</td>\n",
       "      <td>0.000266</td>\n",
       "      <td>0.003314</td>\n",
       "      <td>0.002379</td>\n",
       "      <td>0.000467</td>\n",
       "      <td>0.002383</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>635 rows × 9 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "          Market       SMB       HML       RMW       CMA       UMD       ROE  \\\n",
       "Dates                                                                          \n",
       "196801  0.001046  0.000371  0.000595  0.000346  0.000807  0.000795  0.000298   \n",
       "196802  0.001221  0.000618  0.000565  0.000312  0.000858  0.000917  0.000228   \n",
       "196803  0.001096  0.000713  0.000569  0.000317  0.000848  0.000965  0.000222   \n",
       "196804  0.001671  0.000816  0.000512  0.000335  0.000842  0.001113  0.000315   \n",
       "196805  0.001448  0.000957  0.000512  0.000306  0.000849  0.001158  0.000330   \n",
       "...          ...       ...       ...       ...       ...       ...       ...   \n",
       "202007  0.004751  0.001022  0.002410  0.000181  0.000382  0.002688  0.000506   \n",
       "202008  0.004933  0.000999  0.002367  0.000328  0.000389  0.002256  0.000513   \n",
       "202009  0.005196  0.001005  0.001494  0.000320  0.000238  0.001766  0.000505   \n",
       "202010  0.005301  0.001251  0.001961  0.000324  0.000239  0.001914  0.000576   \n",
       "202011  0.006336  0.001666  0.002174  0.000374  0.000266  0.003314  0.002379   \n",
       "\n",
       "              IA       BAB  \n",
       "Dates                       \n",
       "196801  0.000611  0.000490  \n",
       "196802  0.000677  0.000483  \n",
       "196803  0.000661  0.000567  \n",
       "196804  0.000642  0.000579  \n",
       "196805  0.000639  0.000577  \n",
       "...          ...       ...  \n",
       "202007  0.000393  0.002099  \n",
       "202008  0.000442  0.002191  \n",
       "202009  0.000328  0.002202  \n",
       "202010  0.000327  0.002233  \n",
       "202011  0.000467  0.002383  \n",
       "\n",
       "[635 rows x 9 columns]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "realized_variance = original_factors.rolling(12).var()\n",
    "realized_variance = realized_variance[11:-1]\n",
    "realized_variance"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a67a733d",
   "metadata": {},
   "source": [
    "Finally we construct the volatility-managed factor by multiplying $\\frac{c}{\\sigma_{t}^2 (f)}$ by the monthly excess return of sed factor $f_{t+1}$. \n",
    "\n",
    "We set `c` as an average of the realized variance of the 9 original factors. Changing `c` won't impact the Sharpe Ratio but the leverage used in the portfolio."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "b7ab9651",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Market_vol</th>\n",
       "      <th>SMB_vol</th>\n",
       "      <th>HML_vol</th>\n",
       "      <th>RMW_vol</th>\n",
       "      <th>CMA_vol</th>\n",
       "      <th>UMD_vol</th>\n",
       "      <th>ROE_vol</th>\n",
       "      <th>IA_vol</th>\n",
       "      <th>BAB_vol</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Dates</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>196802</th>\n",
       "      <td>-0.033684</td>\n",
       "      <td>-0.074830</td>\n",
       "      <td>0.018486</td>\n",
       "      <td>-0.003802</td>\n",
       "      <td>0.028999</td>\n",
       "      <td>-0.040310</td>\n",
       "      <td>-0.011930</td>\n",
       "      <td>0.042310</td>\n",
       "      <td>0.040892</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196803</th>\n",
       "      <td>0.001540</td>\n",
       "      <td>-0.019471</td>\n",
       "      <td>-0.009808</td>\n",
       "      <td>0.035286</td>\n",
       "      <td>-0.012602</td>\n",
       "      <td>0.032589</td>\n",
       "      <td>0.069197</td>\n",
       "      <td>-0.015741</td>\n",
       "      <td>-0.030732</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196804</th>\n",
       "      <td>0.077641</td>\n",
       "      <td>0.075525</td>\n",
       "      <td>-0.017011</td>\n",
       "      <td>0.082682</td>\n",
       "      <td>-0.040557</td>\n",
       "      <td>0.050188</td>\n",
       "      <td>0.159124</td>\n",
       "      <td>-0.037051</td>\n",
       "      <td>0.002818</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196805</th>\n",
       "      <td>0.012827</td>\n",
       "      <td>0.074055</td>\n",
       "      <td>0.015422</td>\n",
       "      <td>0.010954</td>\n",
       "      <td>-0.020318</td>\n",
       "      <td>0.031496</td>\n",
       "      <td>0.058408</td>\n",
       "      <td>-0.033911</td>\n",
       "      <td>0.029373</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196806</th>\n",
       "      <td>0.004479</td>\n",
       "      <td>-0.001670</td>\n",
       "      <td>0.012289</td>\n",
       "      <td>-0.040791</td>\n",
       "      <td>0.029688</td>\n",
       "      <td>-0.015587</td>\n",
       "      <td>-0.013588</td>\n",
       "      <td>0.035958</td>\n",
       "      <td>0.112239</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202008</th>\n",
       "      <td>0.015096</td>\n",
       "      <td>-0.008646</td>\n",
       "      <td>-0.011468</td>\n",
       "      <td>0.222098</td>\n",
       "      <td>-0.035416</td>\n",
       "      <td>0.001784</td>\n",
       "      <td>-0.016131</td>\n",
       "      <td>-0.068272</td>\n",
       "      <td>-0.017867</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202009</th>\n",
       "      <td>-0.006916</td>\n",
       "      <td>0.000658</td>\n",
       "      <td>-0.009969</td>\n",
       "      <td>-0.032935</td>\n",
       "      <td>-0.042777</td>\n",
       "      <td>0.012708</td>\n",
       "      <td>0.022433</td>\n",
       "      <td>-0.046807</td>\n",
       "      <td>0.005534</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202010</th>\n",
       "      <td>-0.003799</td>\n",
       "      <td>0.044524</td>\n",
       "      <td>0.025357</td>\n",
       "      <td>-0.017602</td>\n",
       "      <td>-0.020915</td>\n",
       "      <td>-0.016130</td>\n",
       "      <td>-0.045884</td>\n",
       "      <td>-0.021148</td>\n",
       "      <td>-0.008581</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202011</th>\n",
       "      <td>0.022109</td>\n",
       "      <td>0.050706</td>\n",
       "      <td>0.010111</td>\n",
       "      <td>-0.080685</td>\n",
       "      <td>0.041262</td>\n",
       "      <td>-0.060158</td>\n",
       "      <td>-0.236057</td>\n",
       "      <td>0.088150</td>\n",
       "      <td>-0.021426</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202012</th>\n",
       "      <td>0.006869</td>\n",
       "      <td>0.026353</td>\n",
       "      <td>-0.005881</td>\n",
       "      <td>-0.054072</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>-0.006863</td>\n",
       "      <td>-0.013991</td>\n",
       "      <td>-0.054382</td>\n",
       "      <td>0.021059</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>635 rows × 9 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "        Market_vol   SMB_vol   HML_vol   RMW_vol   CMA_vol   UMD_vol  \\\n",
       "Dates                                                                  \n",
       "196802   -0.033684 -0.074830  0.018486 -0.003802  0.028999 -0.040310   \n",
       "196803    0.001540 -0.019471 -0.009808  0.035286 -0.012602  0.032589   \n",
       "196804    0.077641  0.075525 -0.017011  0.082682 -0.040557  0.050188   \n",
       "196805    0.012827  0.074055  0.015422  0.010954 -0.020318  0.031496   \n",
       "196806    0.004479 -0.001670  0.012289 -0.040791  0.029688 -0.015587   \n",
       "...            ...       ...       ...       ...       ...       ...   \n",
       "202008    0.015096 -0.008646 -0.011468  0.222098 -0.035416  0.001784   \n",
       "202009   -0.006916  0.000658 -0.009969 -0.032935 -0.042777  0.012708   \n",
       "202010   -0.003799  0.044524  0.025357 -0.017602 -0.020915 -0.016130   \n",
       "202011    0.022109  0.050706  0.010111 -0.080685  0.041262 -0.060158   \n",
       "202012    0.006869  0.026353 -0.005881 -0.054072  0.000000 -0.006863   \n",
       "\n",
       "         ROE_vol    IA_vol   BAB_vol  \n",
       "Dates                                 \n",
       "196802 -0.011930  0.042310  0.040892  \n",
       "196803  0.069197 -0.015741 -0.030732  \n",
       "196804  0.159124 -0.037051  0.002818  \n",
       "196805  0.058408 -0.033911  0.029373  \n",
       "196806 -0.013588  0.035958  0.112239  \n",
       "...          ...       ...       ...  \n",
       "202008 -0.016131 -0.068272 -0.017867  \n",
       "202009  0.022433 -0.046807  0.005534  \n",
       "202010 -0.045884 -0.021148 -0.008581  \n",
       "202011 -0.236057  0.088150 -0.021426  \n",
       "202012 -0.013991 -0.054382  0.021059  \n",
       "\n",
       "[635 rows x 9 columns]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = realized_variance.mean().mean()\n",
    "vol_managed = (c/realized_variance.values) * original_factors[12:]\n",
    "vol_managed = vol_managed.rename(columns=lambda x: x + \"_vol\")\n",
    "vol_managed"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a3e4274",
   "metadata": {},
   "source": [
    "Now we use mean-variance optimization to combine:\n",
    "- The original (without timing) factor, $f_{t+1}$;\n",
    "- The volatility-managed version of this factor, $f_{t+1}^{\\sigma}$\n",
    "\n",
    "First let's merge $f_{t+1}$ and $f_{t+1}^{\\sigma}$ for each factor onto a separate dataframe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "de356f14",
   "metadata": {},
   "outputs": [],
   "source": [
    "factor_names = [\"Market\", \"SMB\", \"HML\", \"RMW\", \"CMA\", \"UMD\", \"ROE\", \"IA\", \"BAB\"]\n",
    "factor_dfs = []\n",
    "\n",
    "for i, factor_name in enumerate(factor_names):\n",
    "    factor_df = pd.concat([original_factors.iloc[12:, i], vol_managed.iloc[:, i]], axis=1)\n",
    "    factor_dfs.append(factor_df)\n",
    "\n",
    "Market, SMB, HML, RMW, CMA, UMD, ROE, IA, BAB = factor_dfs"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aab2b476",
   "metadata": {},
   "source": [
    "Let's create a function `mvo` to perform mean-variance optimization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "bd51170c",
   "metadata": {},
   "outputs": [],
   "source": [
    "def mvo(dataframe):\n",
    "    \n",
    "    mu_sample = np.mean(dataframe, axis=0).values\n",
    "    \n",
    "    # Calculate inverse of covariance matrix\n",
    "    sample_cov = dataframe.cov().values\n",
    "    sample_Vinverse = np.linalg.inv(sample_cov)\n",
    "\n",
    "    # Calculate the w_MVU\n",
    "    w_MVU = (1/gamma) * sample_Vinverse @ mu_sample\n",
    "    \n",
    "    return w_MVU"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d0ecdd23",
   "metadata": {},
   "source": [
    "Now we loop over each dataframe using the `mvo` function and a 120-month estimation window, to compute the weights for each window.\n",
    "\n",
    "We then loop over again, this time to compute the out-of-sample $t+1$ monthly returns using the weights we just found. The returns are stored in a dictionary (with the key being the factor)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "f179817c",
   "metadata": {},
   "outputs": [],
   "source": [
    "factor_returns = {} # Create an empty dictionary to store the returns for each factor\n",
    "\n",
    "gamma = 5\n",
    "\n",
    "for factor in factor_names:\n",
    "    weights = []\n",
    "    \n",
    "    # Loop to calculate weights\n",
    "    for i in range(120, len(eval(factor))):\n",
    "        w_MVU = mvo(eval(factor)[i-120:i])\n",
    "        weights.append(w_MVU)\n",
    "\n",
    "    weights = np.array(weights)\n",
    "\n",
    "    # Loop to calculate returns\n",
    "    factor_returns[factor] = [] # Create an empty list for each factor\n",
    "    for i in range(120, len(eval(factor))-1):\n",
    "        factor_return = np.dot(weights[i-120].T, eval(factor).iloc[i])\n",
    "        factor_returns[factor].append(factor_return)\n",
    "\n",
    "    factor_returns[factor] = np.array(factor_returns[factor])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6fbd6698",
   "metadata": {},
   "source": [
    "Let's compare the Sharpe ratios of:\n",
    "- the portfolio with just the original factor and\n",
    "- the portfolio that includes the volatility-timed factor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "4c7738bb",
   "metadata": {},
   "outputs": [],
   "source": [
    "def sharpe_ratio(returns):\n",
    "    '''\n",
    "    Function to return the Sharpe Ratio of a series of returns\n",
    "    '''\n",
    "\n",
    "    m_mean_return = returns.mean() # Monthly mean return\n",
    "    m_vol = returns.std() # Monthly volatility\n",
    "    \n",
    "    y_mean_return = m_mean_return * 12 # Annualize the mean monthly return\n",
    "    y_vol = m_vol * np.sqrt(12) # Transfer to annual volatility\n",
    "    \n",
    "    # Compute the Sharpe Ratio\n",
    "    SR = y_mean_return / y_vol\n",
    "    \n",
    "    return SR\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "cdc4c4f3",
   "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>Sharpe Ratio Vol-Managed</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Market</th>\n",
       "      <td>0.368052</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>SMB</th>\n",
       "      <td>-0.159306</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>HML</th>\n",
       "      <td>0.564673</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>RMW</th>\n",
       "      <td>0.248971</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CMA</th>\n",
       "      <td>0.237078</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>UMD</th>\n",
       "      <td>0.597577</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>ROE</th>\n",
       "      <td>0.940693</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>IA</th>\n",
       "      <td>0.544935</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>BAB</th>\n",
       "      <td>0.717256</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        Sharpe Ratio Vol-Managed\n",
       "Market                  0.368052\n",
       "SMB                    -0.159306\n",
       "HML                     0.564673\n",
       "RMW                     0.248971\n",
       "CMA                     0.237078\n",
       "UMD                     0.597577\n",
       "ROE                     0.940693\n",
       "IA                      0.544935\n",
       "BAB                     0.717256"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SR = {}\n",
    "\n",
    "for factor in factor_names:\n",
    "    SR[factor] = sharpe_ratio(factor_returns[factor])\n",
    "    \n",
    "SR = pd.DataFrame.from_dict(SR, orient='index', columns=['Sharpe Ratio Vol-Managed'])\n",
    "\n",
    "SR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "f12d12e9",
   "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",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Sharpe Ratio Vol-Managed</th>\n",
       "      <th>Sharpe Ratio Original</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Market</th>\n",
       "      <td>0.368052</td>\n",
       "      <td>0.554375</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>SMB</th>\n",
       "      <td>-0.159306</td>\n",
       "      <td>0.168859</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>HML</th>\n",
       "      <td>0.564673</td>\n",
       "      <td>0.158015</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>RMW</th>\n",
       "      <td>0.248971</td>\n",
       "      <td>0.510641</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CMA</th>\n",
       "      <td>0.237078</td>\n",
       "      <td>0.406058</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>UMD</th>\n",
       "      <td>0.597577</td>\n",
       "      <td>0.461568</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>ROE</th>\n",
       "      <td>0.940693</td>\n",
       "      <td>0.702961</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>IA</th>\n",
       "      <td>0.544935</td>\n",
       "      <td>0.506108</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>BAB</th>\n",
       "      <td>0.717256</td>\n",
       "      <td>0.867589</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        Sharpe Ratio Vol-Managed  Sharpe Ratio Original\n",
       "Market                  0.368052               0.554375\n",
       "SMB                    -0.159306               0.168859\n",
       "HML                     0.564673               0.158015\n",
       "RMW                     0.248971               0.510641\n",
       "CMA                     0.237078               0.406058\n",
       "UMD                     0.597577               0.461568\n",
       "ROE                     0.940693               0.702961\n",
       "IA                      0.544935               0.506108\n",
       "BAB                     0.717256               0.867589"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "original_SR = {}\n",
    "\n",
    "for factor in factor_names:\n",
    "    original_SR[factor] = sharpe_ratio(original_factors[132:][factor])\n",
    "    \n",
    "original_SR = pd.DataFrame.from_dict(original_SR, orient='index', columns=['Sharpe Ratio Original'])\n",
    "\n",
    "# Concatenate the two sharpe ratio dataframes\n",
    "SR_merged = pd.concat([SR, original_SR], axis = 1)\n",
    "\n",
    "SR_merged"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f50dd2a",
   "metadata": {},
   "source": [
    "The portfolios including volatility-timed factors which have **higher** Sharpe Ratios than their original counterparts are: HML, UMD, ROE and IA.\n",
    "\n",
    "The portfolios including volatility-timed factors which have **lower** Sharpe Ratios than their original counterparts are: Market, SMB, RMW, CMA and BAB.\n",
    "\n",
    "Therefore, volatility scaling generates a higher Sharpe ratio for ONLY 4 out of the 9 factors we analyse."
   ]
  }
 ],
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