{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Riskfolio-Lib Tutorial: \n", "
\n", "
\n", "\n", "
\n", "\n", "\n", "
Buy Me a Coffee at ko-fi.com \n", "
\n", "
__[Financionerioncios](https://financioneroncios.wordpress.com)__\n", "
__[Orenji](https://www.linkedin.com/company/orenj-i)__\n", "
__[Riskfolio-Lib](https://riskfolio-lib.readthedocs.io/en/latest/)__\n", "
__[Dany Cajas](https://www.linkedin.com/in/dany-cajas/)__\n", "\n", "## Tutorial 20: Black Litterman with Factors Models Mean Risk Optimization\n", "\n", "## 1. Downloading the data:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 09018036072018003600720010800
Date         
2017-11-16 00:00:000.0000%0.0059%0.0108%0.0178%0.0246%0.0213%0.0075%-0.0048%-0.0093%
2017-11-15 00:00:000.0180%0.0247%0.0303%0.0391%0.0495%0.0558%0.0512%0.0450%0.0417%
2017-11-14 00:00:00-0.1800%-0.1710%-0.1624%-0.1460%-0.1167%-0.0506%0.0140%0.0676%0.0861%
2017-11-13 00:00:000.0000%0.0013%0.0025%0.0048%0.0088%0.0174%0.0258%0.0334%0.0364%
2017-11-10 00:00:000.0000%0.0026%0.0043%0.0054%0.0017%-0.0248%-0.0615%-0.0936%-0.1054%
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 APACMCSACNPHPQPSASEEZIONPEP11900D031PEP13000D012PEP13000M088PEP23900M103PEP70101M530PEP70101M571PEP70310M156
Date              
2017-11-16 00:00:00-1.3161%-0.2958%-1.0903%0.9831%1.7234%1.4016%-0.8387%-0.0411%-0.0380%-0.0597%-0.0737%-0.0116%0.0076%-0.0633%
2017-11-15 00:00:00-2.0296%0.8682%-1.1202%0.0000%-1.3479%-0.3326%0.0215%-0.1626%-0.3076%-0.3041%-0.2286%-0.4459%-0.4651%-0.2146%
2017-11-14 00:00:00-3.7020%-1.0470%1.0800%0.8975%-0.1548%0.2668%2.6950%0.2320%0.0236%0.1040%0.2373%-0.2741%-0.3932%0.2151%
2017-11-13 00:00:00-1.4503%1.0855%0.7480%-0.2826%0.8179%1.4205%3.4145%0.0906%0.0064%-0.0767%0.0354%-0.0835%-0.1114%-0.0081%
2017-11-10 00:00:00-2.4536%0.7932%-1.3418%-0.5155%0.0710%-0.9381%-0.0910%0.1194%0.3792%0.3047%0.1355%0.7118%0.8207%-0.0090%
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Uploading Data\n", "########################################################################\n", "\n", "import pandas as pd\n", "import numpy as np\n", "import warnings\n", "\n", "warnings.filterwarnings(\"ignore\")\n", "\n", "# Interest Rates Data\n", "kr = pd.read_excel('KeyRates.xlsx', engine='openpyxl', index_col=0, header=0)/100\n", "\n", "# Prices Data\n", "assets = pd.read_excel('Assets.xlsx', engine='openpyxl', index_col=0, header=0)\n", "\n", "# Find common dates\n", "a = pd.merge(left=assets, right=kr, how='inner', on='Date')\n", "dates = a.index\n", "\n", "# Calculate interest rates returns\n", "kr_returns = kr.loc[dates,:].sort_index().diff().dropna()\n", "kr_returns.sort_index(ascending=False, inplace=True)\n", "\n", "# List of instruments\n", "equity = ['APA','CMCSA','CNP','HPQ','PSA','SEE','ZION']\n", "bonds = ['PEP11900D031', 'PEP13000D012', 'PEP13000M088',\n", " 'PEP23900M103','PEP70101M530','PEP70101M571',\n", " 'PEP70310M156']\n", "factors = ['MTUM','QUAL','SIZE','USMV','VLUE']\n", "\n", "# Calculate assets returns\n", "assets_returns = assets.loc[dates, equity + bonds]\n", "assets_returns = assets_returns.sort_index().pct_change().dropna()\n", "assets_returns.sort_index(ascending=False, inplace=True)\n", "\n", "# Calculate factors returns\n", "factors_returns = assets.loc[dates, factors]\n", "factors_returns = factors_returns.sort_index().pct_change().dropna()\n", "factors_returns.sort_index(ascending=False, inplace=True)\n", "\n", "# Show tables\n", "display(kr_returns.head().style.format(\"{:.4%}\"))\n", "display(assets_returns.head().style.format(\"{:.4%}\"))" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Durations Matrix\n" ] }, { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 R 0R 90R 180R 360R 720R 1800R 3600R 7200R 10800
PEP11900D0310.00120.00570.01920.07300.36853.04160.00300.00000.0000
PEP13000D0120.00000.00780.01420.06170.33271.09024.80550.20740.0000
PEP13000M0880.00130.00040.01470.05010.27702.46263.07640.00000.0000
PEP23900M1030.00000.00050.01170.04050.22743.97260.03810.00000.0000
PEP70101M5300.00000.00520.01010.04420.24880.88264.91473.55370.0000
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "Convexities Matrix\n" ] }, { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 R^2 0R^2 90R^2 180R^2 360R^2 720R^2 1800R^2 3600R^2 7200R^2 10800
PEP11900D0310.00040.00320.01670.09280.774115.56170.00000.00000.0000
PEP13000D0120.00000.00570.00700.07560.72104.498445.21590.11050.0000
PEP13000M0880.00100.00010.01920.07360.61618.847916.28800.00000.0000
PEP23900M1030.00000.00000.01560.06440.516122.12720.00220.00000.0000
PEP70101M5300.00000.00380.00520.05610.55303.737338.231526.14640.0000
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Uploading Duration and Convexity Matrixes\n", "########################################################################\n", "\n", "durations = pd.read_excel('durations.xlsx', index_col=0, header=0)\n", "convexity = pd.read_excel('convexity.xlsx', index_col=0, header=0)\n", "\n", "print('Durations Matrix')\n", "display(durations.head().style.format(\"{:.4f}\").background_gradient(cmap='YlGn'))\n", "print('')\n", "print('Convexities Matrix')\n", "display(convexity.head().style.format(\"{:.4f}\").background_gradient(cmap='YlGn'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. Estimating Black Litterman with Factors for Fixed Income Portfolios\n", "\n", "### 2.1 Building the loadings matrix and risk factors returns\n", "\n", "This part shows how to build a personalized loadings matrix that will be used by Riskfolio-Lib to calculate the expected returns and covariance matrix." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 R 0R 90R 180R 360R 720R 1800R 3600R 7200R 10800R^2 0R^2 90R^2 180R^2 360R^2 720R^2 1800R^2 3600R^2 7200R^2 10800
PEP11900D031-0.0012-0.0057-0.0192-0.0730-0.3685-3.0416-0.0030-0.0000-0.00000.00020.00160.00830.04640.38717.78090.00000.00000.0000
PEP13000D012-0.0000-0.0078-0.0142-0.0617-0.3327-1.0902-4.8055-0.2074-0.00000.00000.00290.00350.03780.36052.249222.60800.05530.0000
PEP13000M088-0.0013-0.0004-0.0147-0.0501-0.2770-2.4626-3.0764-0.0000-0.00000.00050.00000.00960.03680.30814.42408.14400.00000.0000
PEP23900M103-0.0000-0.0005-0.0117-0.0405-0.2274-3.9726-0.0381-0.0000-0.00000.00000.00000.00780.03220.258111.06360.00110.00000.0000
PEP70101M530-0.0000-0.0052-0.0101-0.0442-0.2488-0.8826-4.9147-3.5537-0.00000.00000.00190.00260.02800.27651.868619.115713.07320.0000
PEP70101M571-0.0015-0.0039-0.0126-0.0501-0.2829-1.0108-2.5878-6.0312-0.45010.00020.00160.00640.03190.31232.133610.163249.90210.4523
PEP70310M156-0.0000-0.0039-0.0097-0.0403-0.2614-3.8920-0.0000-0.0000-0.00000.00000.00100.00300.02680.250810.68130.00000.00000.0000
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Building The Loadings Matrix\n", "########################################################################\n", "\n", "loadings = pd.concat([-1.0 * durations, 0.5 * convexity], axis = 1)\n", "\n", "display(loadings.style.format(\"{:.4f}\").background_gradient(cmap='YlGn'))" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 R 0R 90R 180R 360R 720R 1800R 3600R 7200R 10800R^2 0R^2 90R^2 180R^2 360R^2 720R^2 1800R^2 3600R^2 7200R^2 10800
Date                  
2017-11-16 00:00:000.0000%0.0059%0.0108%0.0178%0.0246%0.0213%0.0075%-0.0048%-0.0093%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%
2017-11-15 00:00:000.0180%0.0247%0.0303%0.0391%0.0495%0.0558%0.0512%0.0450%0.0417%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%
2017-11-14 00:00:00-0.1800%-0.1710%-0.1624%-0.1460%-0.1167%-0.0506%0.0140%0.0676%0.0861%0.0003%0.0003%0.0003%0.0002%0.0001%0.0000%0.0000%0.0000%0.0001%
2017-11-13 00:00:000.0000%0.0013%0.0025%0.0048%0.0088%0.0174%0.0258%0.0334%0.0364%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%
2017-11-10 00:00:000.0000%0.0026%0.0043%0.0054%0.0017%-0.0248%-0.0615%-0.0936%-0.1054%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0000%0.0001%0.0001%
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Building the risk factors returns matrix\n", "########################################################################\n", "\n", "kr_returns_2 = kr_returns ** 2\n", "cols = loadings.columns\n", "\n", "X = pd.concat([kr_returns, kr_returns_2], axis=1)\n", "X.columns = cols\n", "\n", "display(X.head().style.format(\"{:.4%}\"))" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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PEP11900D031PEP13000D012PEP13000M088PEP23900M103PEP70101M530PEP70101M571PEP70310M156
Date
2017-11-16-0.000411-0.000380-0.000597-0.000737-0.0001160.000076-0.000633
2017-11-15-0.001626-0.003076-0.003041-0.002286-0.004459-0.004651-0.002146
2017-11-140.0023200.0002360.0010400.002373-0.002741-0.0039320.002151
2017-11-130.0009060.000064-0.0007670.000354-0.000835-0.001114-0.000081
2017-11-100.0011940.0037920.0030470.0013550.0071180.008207-0.000090
\n", "
" ], "text/plain": [ " PEP11900D031 PEP13000D012 PEP13000M088 PEP23900M103 \\\n", "Date \n", "2017-11-16 -0.000411 -0.000380 -0.000597 -0.000737 \n", "2017-11-15 -0.001626 -0.003076 -0.003041 -0.002286 \n", "2017-11-14 0.002320 0.000236 0.001040 0.002373 \n", "2017-11-13 0.000906 0.000064 -0.000767 0.000354 \n", "2017-11-10 0.001194 0.003792 0.003047 0.001355 \n", "\n", " PEP70101M530 PEP70101M571 PEP70310M156 \n", "Date \n", "2017-11-16 -0.000116 0.000076 -0.000633 \n", "2017-11-15 -0.004459 -0.004651 -0.002146 \n", "2017-11-14 -0.002741 -0.003932 0.002151 \n", "2017-11-13 -0.000835 -0.001114 -0.000081 \n", "2017-11-10 0.007118 0.008207 -0.000090 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Building the asset returns matrix\n", "########################################################################\n", "\n", "Y = assets_returns[bonds]\n", "\n", "display(Y.head())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2.2 Building views on risk factors" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "R 0 -0.002028\n", "R 90 -0.001955\n", "R 180 -0.001819\n", "R 360 -0.001463\n", "R 720 -0.000791\n", "R 1800 -0.000530\n", "R 3600 -0.002041\n", "R 7200 -0.003312\n", "R 10800 -0.003533\n", "R^2 0 0.000173\n", "R^2 90 0.000080\n", "R^2 180 0.000049\n", "R^2 360 0.000038\n", "R^2 720 0.000046\n", "R^2 1800 0.000050\n", "R^2 3600 0.000057\n", "R^2 7200 0.000069\n", "R^2 10800 0.000091\n", "dtype: float64" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Showing annualized returns of Fixed Income Risk Factors\n", "########################################################################\n", "\n", "display(X.mean()*252)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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DisabledFactorSignValueRelative Factor
0FalseR 10800>=0.001R 7200
1FalseR 1800<=-0.001
2FalseR 3600<=-0.003
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" ], "text/plain": [ " Disabled Factor Sign Value Relative Factor\n", "0 False R 10800 >= 0.001 R 7200\n", "1 False R 1800 <= -0.001 \n", "2 False R 3600 <= -0.003 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Building views on some Risk Factors\n", "########################################################################\n", "\n", "views = {'Disabled': [False, False, False],\n", " 'Factor': ['R 10800','R 1800','R 3600'],\n", " 'Sign': ['>=', '<=', '<='],\n", " 'Value': [0.001, -0.001, -0.003],\n", " 'Relative Factor': ['R 7200', '', '']}\n", "\n", "views = pd.DataFrame(views)\n", "\n", "display(views)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Matrix of factors views P_f\n", "[[ 0. 0. 0. 0. 0. 0. 0. -1. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]]\n", "\n", "Matrix of returns of factors views Q_f\n", "[[0.001]\n", " [0.001]\n", " [0.003]]\n" ] } ], "source": [ "########################################################################\n", "# Building views matrixes P_f and Q_f\n", "########################################################################\n", "\n", "import riskfolio as rp\n", "\n", "P_f, Q_f = rp.factors_views(views, loadings, const=False)\n", "\n", "print('Matrix of factors views P_f')\n", "print(P_f)\n", "print('\\nMatrix of returns of factors views Q_f')\n", "print(Q_f)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2.3 Building Portfolios with mean vector and covariance matrix from Black Litterman with Factors." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "########################################################################\n", "# Building the Portfolio Object\n", "########################################################################\n", "\n", "# Building the portfolio object\n", "port = rp.Portfolio(returns=Y)\n", "\n", "# Select method and estimate input parameters:\n", "method_mu='hist' # Method to estimate expected returns based on historical data.\n", "method_cov='hist' # Method to estimate covariance matrix based on historical data.\n", "\n", "port.assets_stats(method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "port.factors = X\n", "port.factors_stats(method_mu=method_mu,\n", " method_cov=method_cov,\n", " B=loadings,\n", " const=False)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "You must convert self.cov_bl_fm to a positive definite matrix\n", "You must convert self.cov_bl_fm to a positive definite matrix\n" ] }, { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 Pure FactorsBayesian BLAugmented BL
PEP11900D0310.0000%0.0000%0.0000%
PEP13000D0126.6680%33.1060%85.2406%
PEP13000M0880.0000%0.0000%0.0000%
PEP23900M1030.0000%0.0000%0.0000%
PEP70101M53032.3833%0.0001%14.7594%
PEP70101M57160.9486%66.8939%0.0000%
PEP70310M1560.0000%0.0000%0.0000%
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Calculating optimum portfolios using Mean Vector and\n", "# Covariance Matrix of Black Litterman with Factors\n", "########################################################################\n", "\n", "port.alpha = 0.05\n", "rm = 'MV' # Risk measure used, this time will be variance\n", "obj = 'Sharpe' # Objective function, could be MinRisk, MaxRet, Utility or Sharpe\n", "hist = False # False: BL covariance and risk factors scenarios\n", " # True: historical covariance and scenarios\n", " # 2: risk factors covariance and scenarios\n", "rf = 0 # Risk free rate\n", "l = 0 # Risk aversion factor, only useful when obj is 'Utility'\n", "\n", "w_fm = port.optimization(model='FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist) \n", "\n", "# Estimate Portfolio weights using Black Litterman Bayesian Model:\n", "port.blfactors_stats(flavor='BLB',\n", " B=loadings,\n", " P_f=P_f,\n", " Q_f=Q_f/252,\n", " rf=0,\n", " delta=None,\n", " eq=True,\n", " const=False,\n", " diag=False,\n", " method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "w_blb = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)\n", "\n", "# Estimate Portfolio weights using Augmented Black Litterman Model:\n", "port.blfactors_stats(flavor='ABL',\n", " B=loadings,\n", " P_f=P_f,\n", " Q_f=Q_f/252,\n", " rf=0,\n", " delta=None,\n", " eq=True,\n", " const=False,\n", " diag=False,\n", " method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "w_abl = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)\n", "\n", "ws = pd.concat([w_fm, w_blb, w_abl], axis=1)\n", "ws.columns = ['Pure Factors', 'Bayesian BL', 'Augmented BL']\n", "\n", "display(ws.style.format(\"{:.4%}\").background_gradient(cmap='YlGn'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can see that the we got a messsage that the covariance matrix is not a positive definite matrix, this is common when we work with views on assets or risk factors. In this case, Riskfolio-Lib replace the negative eigenvalues of covariance matrix with zeros and reconstruct the covariance matrix. The problem with this approach is common that the weights that we will get are highly concetrated in few assets.\n", "\n", "Other approach is using the mean vector estimated with Black Litterman with Factors and the covariance matrix that we get from historical returns or a factor model. An example with this approach follows:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2.4 Building Portfolios with mean vector from Black Litterman with Factors." ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "You must convert self.cov_bl_fm to a positive definite matrix\n", "You must convert self.cov_bl_fm to a positive definite matrix\n" ] }, { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 Pure FactorsBayesian BLAugmented BL
PEP11900D0310.0000%0.0000%6.1662%
PEP13000D0126.6680%16.6281%16.4519%
PEP13000M0880.0000%2.2781%5.2878%
PEP23900M1030.0000%0.0000%11.2163%
PEP70101M53032.3833%35.1625%26.2773%
PEP70101M57160.9486%45.9313%23.8393%
PEP70310M1560.0000%0.0000%10.7612%
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Calculating optimum portfolios using only Mean Vector\n", "# of Black Litterman with Factors and Factor Covariance Matrix\n", "########################################################################\n", "\n", "hist = 2 # False: BL covariance and risk factors scenarios\n", " # True: historical covariance and scenarios\n", " # 2: risk factors covariance and scenarios (Only in BL_FM)\n", "\n", "# Estimate Portfolio weights using Black Litterman Bayesian Model:\n", "port.blfactors_stats(flavor='BLB',\n", " B=loadings,\n", " P_f=P_f,\n", " Q_f=Q_f/252,\n", " rf=0,\n", " delta=None,\n", " eq=True,\n", " const=False,\n", " diag=False,\n", " method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "w_blb = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)\n", "\n", "# Estimate Portfolio weights using Augmented Black Litterman Model:\n", "port.blfactors_stats(flavor='ABL',\n", " B=loadings,\n", " P_f=P_f,\n", " Q_f=Q_f/252,\n", " rf=0,\n", " delta=None,\n", " eq=True,\n", " const=False,\n", " diag=False,\n", " method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "w_abl = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)\n", "\n", "ws = pd.concat([w_fm, w_blb, w_abl], axis=1)\n", "ws.columns = ['Pure Factors', 'Bayesian BL', 'Augmented BL']\n", "\n", "display(ws.style.format(\"{:.4%}\").background_gradient(cmap='YlGn'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can see that when we only use the mean vector of Black Litterman with Factors, the weights that we get are more diversified. Also, we can see that the Augmented Black Litterman creates more diversified portfolios than the Bayesian Black Litterman." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3. Estimating Mean Variance Portfolio for Equity and Fixed Income Portfolio\n", "\n", "### 3.1 Building the loadings matrix and risk factors returns.\n", "\n", "This part shows how to build a personalized loadings matrix that will be used by Riskfolio-Lib to calculate the expected returns and covariance matrix." ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 constMTUMQUALSIZEUSMVVLUE
APA-0.0008-0.89991.26740.0000-0.57501.4874
CMCSA-0.00000.24260.0000-0.15690.54830.3623
CNP-0.0001-0.3827-0.35790.00001.95560.0000
HPQ0.00030.00000.51640.00000.00000.7550
PSA-0.00000.0000-0.46230.00001.8071-0.3115
SEE-0.00010.00000.49150.00000.50430.2535
ZION0.00020.00001.15890.0000-1.32031.2202
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "B = rp.loadings_matrix(factors_returns, assets_returns[equity])\n", "\n", "display(B.style.format(\"{:.4f}\").background_gradient(cmap='YlGn'))" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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APACMCSACNPHPQPSASEEZIONPEP11900D031PEP13000D012PEP13000M088PEP23900M103PEP70101M530PEP70101M571PEP70310M156
Date
2017-11-16-0.013161-0.002958-0.0109030.0098310.0172340.014016-0.008387-0.000411-0.000380-0.000597-0.000737-0.0001160.000076-0.000633
2017-11-15-0.0202960.008682-0.0112020.000000-0.013479-0.0033260.000215-0.001626-0.003076-0.003041-0.002286-0.004459-0.004651-0.002146
2017-11-14-0.037020-0.0104700.0108000.008975-0.0015480.0026680.0269500.0023200.0002360.0010400.002373-0.002741-0.0039320.002151
2017-11-13-0.0145030.0108550.007480-0.0028260.0081790.0142050.0341450.0009060.000064-0.0007670.000354-0.000835-0.001114-0.000081
2017-11-10-0.0245360.007932-0.013418-0.0051550.000710-0.009381-0.0009100.0011940.0037920.0030470.0013550.0071180.008207-0.000090
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" ], "text/plain": [ " APA CMCSA CNP HPQ PSA SEE \\\n", "Date \n", "2017-11-16 -0.013161 -0.002958 -0.010903 0.009831 0.017234 0.014016 \n", "2017-11-15 -0.020296 0.008682 -0.011202 0.000000 -0.013479 -0.003326 \n", "2017-11-14 -0.037020 -0.010470 0.010800 0.008975 -0.001548 0.002668 \n", "2017-11-13 -0.014503 0.010855 0.007480 -0.002826 0.008179 0.014205 \n", "2017-11-10 -0.024536 0.007932 -0.013418 -0.005155 0.000710 -0.009381 \n", "\n", " ZION PEP11900D031 PEP13000D012 PEP13000M088 PEP23900M103 \\\n", "Date \n", "2017-11-16 -0.008387 -0.000411 -0.000380 -0.000597 -0.000737 \n", "2017-11-15 0.000215 -0.001626 -0.003076 -0.003041 -0.002286 \n", "2017-11-14 0.026950 0.002320 0.000236 0.001040 0.002373 \n", "2017-11-13 0.034145 0.000906 0.000064 -0.000767 0.000354 \n", "2017-11-10 -0.000910 0.001194 0.003792 0.003047 0.001355 \n", "\n", " PEP70101M530 PEP70101M571 PEP70310M156 \n", "Date \n", "2017-11-16 -0.000116 0.000076 -0.000633 \n", "2017-11-15 -0.004459 -0.004651 -0.002146 \n", "2017-11-14 -0.002741 -0.003932 0.002151 \n", "2017-11-13 -0.000835 -0.001114 -0.000081 \n", "2017-11-10 0.007118 0.008207 -0.000090 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Building the asset returns matrix\n", "########################################################################\n", "\n", "Y = pd.concat([assets_returns[equity], Y], axis=1)\n", "\n", "display(Y.head())" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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MTUMQUALSIZEUSMVVLUER 0R 90R 180R 360R 720...R 10800R^2 0R^2 90R^2 180R^2 360R^2 720R^2 1800R^2 3600R^2 7200R^2 10800
Date
2017-11-160.0094780.0075560.0066750.0064080.0145090.000000.0000590.0001080.0001780.000246...-0.0000930.000000e+003.527647e-091.163831e-083.181621e-086.048648e-084.552420e-085.573518e-092.314283e-098.695936e-09
2017-11-15-0.003381-0.005884-0.001974-0.006750-0.0037710.000180.0002470.0003030.0003910.000495...0.0004173.225005e-086.082093e-089.197391e-081.529780e-072.448864e-073.108575e-072.620815e-072.025117e-071.736147e-07
2017-11-14-0.0016870.000375-0.0003700.001932-0.004380-0.00180-0.001710-0.001624-0.001460-0.001167...0.0008613.241235e-062.925701e-062.636999e-062.131988e-061.362113e-062.555747e-071.966922e-084.572289e-077.417774e-07
2017-11-130.0024880.0008770.0035890.0032960.0007510.000000.0000130.0000250.0000480.000088...0.0003640.000000e+001.681950e-106.400900e-102.324301e-097.748577e-093.033345e-086.657174e-081.116368e-071.328391e-07
2017-11-100.0028940.001255-0.001730-0.0009680.0012540.000000.0000260.0000430.0000540.000017...-0.0010540.000000e+006.770924e-101.820644e-092.881542e-092.791573e-106.131368e-083.779975e-078.759856e-071.110697e-06
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5 rows × 23 columns

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" ], "text/plain": [ " MTUM QUAL SIZE USMV VLUE R 0 \\\n", "Date \n", "2017-11-16 0.009478 0.007556 0.006675 0.006408 0.014509 0.00000 \n", "2017-11-15 -0.003381 -0.005884 -0.001974 -0.006750 -0.003771 0.00018 \n", "2017-11-14 -0.001687 0.000375 -0.000370 0.001932 -0.004380 -0.00180 \n", "2017-11-13 0.002488 0.000877 0.003589 0.003296 0.000751 0.00000 \n", "2017-11-10 0.002894 0.001255 -0.001730 -0.000968 0.001254 0.00000 \n", "\n", " R 90 R 180 R 360 R 720 ... R 10800 \\\n", "Date ... \n", "2017-11-16 0.000059 0.000108 0.000178 0.000246 ... -0.000093 \n", "2017-11-15 0.000247 0.000303 0.000391 0.000495 ... 0.000417 \n", "2017-11-14 -0.001710 -0.001624 -0.001460 -0.001167 ... 0.000861 \n", "2017-11-13 0.000013 0.000025 0.000048 0.000088 ... 0.000364 \n", "2017-11-10 0.000026 0.000043 0.000054 0.000017 ... -0.001054 \n", "\n", " R^2 0 R^2 90 R^2 180 R^2 360 \\\n", "Date \n", "2017-11-16 0.000000e+00 3.527647e-09 1.163831e-08 3.181621e-08 \n", "2017-11-15 3.225005e-08 6.082093e-08 9.197391e-08 1.529780e-07 \n", "2017-11-14 3.241235e-06 2.925701e-06 2.636999e-06 2.131988e-06 \n", "2017-11-13 0.000000e+00 1.681950e-10 6.400900e-10 2.324301e-09 \n", "2017-11-10 0.000000e+00 6.770924e-10 1.820644e-09 2.881542e-09 \n", "\n", " R^2 720 R^2 1800 R^2 3600 R^2 7200 \\\n", "Date \n", "2017-11-16 6.048648e-08 4.552420e-08 5.573518e-09 2.314283e-09 \n", "2017-11-15 2.448864e-07 3.108575e-07 2.620815e-07 2.025117e-07 \n", "2017-11-14 1.362113e-06 2.555747e-07 1.966922e-08 4.572289e-07 \n", "2017-11-13 7.748577e-09 3.033345e-08 6.657174e-08 1.116368e-07 \n", "2017-11-10 2.791573e-10 6.131368e-08 3.779975e-07 8.759856e-07 \n", "\n", " R^2 10800 \n", "Date \n", "2017-11-16 8.695936e-09 \n", "2017-11-15 1.736147e-07 \n", "2017-11-14 7.417774e-07 \n", "2017-11-13 1.328391e-07 \n", "2017-11-10 1.110697e-06 \n", "\n", "[5 rows x 23 columns]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Building the asset returns matrix\n", "########################################################################\n", "\n", "X = pd.concat([factors_returns, X], axis=1)\n", "\n", "display(X.head())" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 constMTUMQUALSIZEUSMVVLUER 0R 90R 180R 360R 720R 1800R 3600R 7200R 10800R^2 0R^2 90R^2 180R^2 360R^2 720R^2 1800R^2 3600R^2 7200R^2 10800
APA-0.0008-0.89991.26740.0000-0.57501.48740.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
CMCSA-0.00000.24260.0000-0.15690.54830.36230.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
CNP-0.0001-0.3827-0.35790.00001.95560.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
HPQ0.00030.00000.51640.00000.00000.75500.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
PSA-0.00000.0000-0.46230.00001.8071-0.31150.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
SEE-0.00010.00000.49150.00000.50430.25350.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
ZION0.00020.00001.15890.0000-1.32031.22020.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
PEP11900D0310.00000.00000.00000.00000.00000.0000-0.0012-0.0057-0.0192-0.0730-0.3685-3.0416-0.0030-0.0000-0.00000.00020.00160.00830.04640.38717.78090.00000.00000.0000
PEP13000D0120.00000.00000.00000.00000.00000.0000-0.0000-0.0078-0.0142-0.0617-0.3327-1.0902-4.8055-0.2074-0.00000.00000.00290.00350.03780.36052.249222.60800.05530.0000
PEP13000M0880.00000.00000.00000.00000.00000.0000-0.0013-0.0004-0.0147-0.0501-0.2770-2.4626-3.0764-0.0000-0.00000.00050.00000.00960.03680.30814.42408.14400.00000.0000
PEP23900M1030.00000.00000.00000.00000.00000.0000-0.0000-0.0005-0.0117-0.0405-0.2274-3.9726-0.0381-0.0000-0.00000.00000.00000.00780.03220.258111.06360.00110.00000.0000
PEP70101M5300.00000.00000.00000.00000.00000.0000-0.0000-0.0052-0.0101-0.0442-0.2488-0.8826-4.9147-3.5537-0.00000.00000.00190.00260.02800.27651.868619.115713.07320.0000
PEP70101M5710.00000.00000.00000.00000.00000.0000-0.0015-0.0039-0.0126-0.0501-0.2829-1.0108-2.5878-6.0312-0.45010.00020.00160.00640.03190.31232.133610.163249.90210.4523
PEP70310M1560.00000.00000.00000.00000.00000.0000-0.0000-0.0039-0.0097-0.0403-0.2614-3.8920-0.0000-0.0000-0.00000.00000.00100.00300.02680.250810.68130.00000.00000.0000
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Building The Loadings Matrix\n", "########################################################################\n", "\n", "loadings = pd.concat([B, loadings], axis = 1)\n", "loadings.fillna(0, inplace=True)\n", "\n", "display(loadings.style.format(\"{:.4f}\").background_gradient(cmap='YlGn'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 3.2 Building views on risk factors" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "MTUM 0.154379\n", "QUAL 0.113842\n", "SIZE 0.115915\n", "USMV 0.124555\n", "VLUE 0.106612\n", "dtype: float64" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Showing annualized returns of Equity Risk Factors\n", "########################################################################\n", "\n", "display(factors_returns.mean()*252)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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DisabledFactorSignValueRelative Factor
0FalseMTUM>=0.020VLUE
1FalseUSMV>=0.090
2FalseSIZE>=0.120
3FalseR 10800>=0.001R 90
4FalseR 1800<=-0.001
5FalseR 3600<=-0.003
\n", "
" ], "text/plain": [ " Disabled Factor Sign Value Relative Factor\n", "0 False MTUM >= 0.020 VLUE\n", "1 False USMV >= 0.090 \n", "2 False SIZE >= 0.120 \n", "3 False R 10800 >= 0.001 R 90\n", "4 False R 1800 <= -0.001 \n", "5 False R 3600 <= -0.003 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Building views on some Risk Factors\n", "########################################################################\n", "\n", "views = {'Disabled': [False, False, False, False, False, False],\n", " 'Factor': ['MTUM','USMV','SIZE','R 10800','R 1800','R 3600'],\n", " 'Sign': ['>=', '>=', '>=', '>=', '<=', '<='],\n", " 'Value': [0.02, 0.09, 0.12, 0.001, -0.001, -0.003],\n", " 'Relative Factor': ['VLUE', '', '','R 90', '', '']}\n", "views = pd.DataFrame(views)\n", "\n", "display(views)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Matrix of factors views P_f\n", "[[ 1. 0. 0. 0. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0.]\n", " [ 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. -1. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1. 0. 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1. 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0.]]\n", "\n", "Matrix of returns of factors views Q_f\n", "[[0.02 ]\n", " [0.09 ]\n", " [0.12 ]\n", " [0.001]\n", " [0.001]\n", " [0.003]]\n" ] } ], "source": [ "########################################################################\n", "# Building views matrixes P_f and Q_f\n", "########################################################################\n", "\n", "P_f, Q_f = rp.factors_views(views, loadings, const=True)\n", "\n", "print('Matrix of factors views P_f')\n", "print(P_f)\n", "print('\\nMatrix of returns of factors views Q_f')\n", "print(Q_f)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 3.3 Building Portfolios with mean vector and covariance matrix from Black Litterman with Factors." ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [], "source": [ "########################################################################\n", "# Building Portfolio Object\n", "########################################################################\n", "\n", "# Building the portfolio object\n", "port = rp.Portfolio(returns=Y)\n", "\n", "# Calculating optimum portfolio\n", "\n", "# Select method and estimate input parameters:\n", "\n", "method_mu='hist' # Method to estimate expected returns based on historical data.\n", "method_cov='hist' # Method to estimate covariance matrix based on historical data.\n", "\n", "port.assets_stats(method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "port.factors = X\n", "port.factors_stats(method_mu=method_mu,\n", " method_cov=method_cov,\n", " B=loadings,\n", " const=True)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "You must convert self.cov_bl_fm to a positive definite matrix\n" ] }, { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 Pure FactorsBayesian BLAugmented BL
APA0.0000%0.0000%0.0000%
CMCSA10.6669%4.4927%0.0000%
CNP12.2572%7.6298%0.0000%
HPQ16.3690%17.0336%72.8372%
PSA26.0444%19.6223%0.0000%
SEE0.0758%0.0000%0.0000%
ZION7.9950%9.8279%0.0000%
PEP11900D0310.0000%0.0000%0.0000%
PEP13000D0120.0001%6.9747%27.1628%
PEP13000M0880.0000%0.4876%0.0000%
PEP23900M1030.0000%0.0000%0.0000%
PEP70101M5307.8241%14.5620%0.0000%
PEP70101M57118.7674%19.3695%0.0000%
PEP70310M1560.0000%0.0000%0.0000%
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Calculating optimum portfolios\n", "########################################################################\n", "\n", "port.alpha = 0.05\n", "rm = 'MV' # Risk measure used, this time will be variance\n", "obj = 'Sharpe' # Objective function, could be MinRisk, MaxRet, Utility or Sharpe\n", "hist = False # False: BL covariance and risk factors scenarios\n", " # True: historical covariance and scenarios\n", " # 2: risk factors covariance and scenarios\n", "rf = 0 # Risk free rate\n", "l = 0 # Risk aversion factor, only useful when obj is 'Utility'\n", "\n", "w_fm = port.optimization(model='FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist) \n", "\n", "# Estimate Portfolio weights using Black Litterman Bayesian Model:\n", "port.blfactors_stats(flavor='BLB',\n", " B=loadings,\n", " P_f=P_f,\n", " Q_f=Q_f/252,\n", " rf=0,\n", " delta=None,\n", " eq=True,\n", " const=True,\n", " diag=True,\n", " method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "w_blb = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)\n", "\n", "# Estimate Portfolio weights using Augmented Black Litterman Model:\n", "port.blfactors_stats(flavor='ABL',\n", " B=loadings,\n", " P_f=P_f,\n", " Q_f=Q_f/252,\n", " rf=0,\n", " delta=None,\n", " eq=True,\n", " const=True,\n", " diag=True,\n", " method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "w_abl = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)\n", "\n", "ws = pd.concat([w_fm, w_blb, w_abl], axis=1)\n", "ws.columns = ['Pure Factors', 'Bayesian BL', 'Augmented BL']\n", "\n", "display(ws.style.format(\"{:.4%}\").background_gradient(cmap='YlGn'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can see that the we got a messsage that the covariance matrix is not a positive definite matrix, this is common when we work with views on assets or risk factors. In this case, Riskfolio-Lib replace the negative eigenvalues of covariance matrix with zeros and reconstruct the covariance matrix. The problem with this approach is common that the weights that we will get are highly concetrated in few assets. \n", "\n", "We can see that in this case the Black Litterman Bayesian model creates a more diversified portfolio than the Augmented Black Litterman model.\n", "\n", "Other approach is using the mean vector estimated with Black Litterman with Factors and the covariance matrix that we get from historical returns or a factor model." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 3.4 Building Portfolios with mean vector from Black Litterman with Factors." ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "You must convert self.cov_bl_fm to a positive definite matrix\n" ] }, { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 Pure FactorsBayesian BLAugmented BL
APA0.0000%0.0000%0.0000%
CMCSA10.6669%5.2672%7.3134%
CNP12.2572%5.9762%1.6711%
HPQ16.3690%17.3220%21.2692%
PSA26.0444%18.0273%12.0783%
SEE0.0758%0.0000%0.0000%
ZION7.9950%11.3383%20.9051%
PEP11900D0310.0000%0.0000%2.3440%
PEP13000D0120.0001%6.2308%6.3664%
PEP13000M0880.0000%0.2338%2.0322%
PEP23900M1030.0000%0.0000%4.1502%
PEP70101M5307.8241%14.6896%9.6157%
PEP70101M57118.7674%20.9148%8.2701%
PEP70310M1560.0000%0.0000%3.9845%
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########################################################################\n", "# Calculating optimum portfolios\n", "########################################################################\n", "\n", "hist = 2 # False: BL covariance and risk factors scenarios\n", " # True: historical covariance and scenarios\n", " # 2: risk factors covariance and scenarios (Only in BL_FM)\n", "\n", "# Estimate Portfolio weights using Black Litterman Bayesian Model:\n", "port.blfactors_stats(flavor='BLB',\n", " B=loadings,\n", " P_f=P_f,\n", " Q_f=Q_f/252,\n", " rf=0,\n", " delta=None,\n", " eq=True,\n", " const=True,\n", " diag=True,\n", " method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "w_blb = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)\n", "\n", "# Estimate Portfolio weights using Augmented Black Litterman Model:\n", "port.blfactors_stats(flavor='ABL',\n", " B=loadings,\n", " P_f=P_f,\n", " Q_f=Q_f/252,\n", " rf=0,\n", " delta=None,\n", " eq=True,\n", " const=True,\n", " diag=True,\n", " method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "w_abl = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)\n", "\n", "ws = pd.concat([w_fm, w_blb, w_abl], axis=1)\n", "ws.columns = ['Pure Factors', 'Bayesian BL', 'Augmented BL']\n", "\n", "display(ws.style.format(\"{:.4%}\").background_gradient(cmap='YlGn'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can see that when we only use the mean vector of Black Litterman with Factors, the weights that we get are more diversified. Also, we can see that the Augmented Black Litterman creates a more diversified portfolio than the Bayesian Black Litterman." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4. Estimating Black Litterman with Factors Mean Risk Portfolios\n", "\n", "When we use risk measures different than Standard Deviation, Riskfolio-Lib only considers the vector of expected returns, and use historical returns to calculate risk measures.\n", "\n", "### 4.1 Calculate Black Litterman Bayesian Portfolios for Several Risk Measures" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [], "source": [ "# Risk Measures available:\n", "#\n", "# 'MV': Standard Deviation.\n", "# 'MAD': Mean Absolute Deviation.\n", "# 'MSV': Semi Standard Deviation.\n", "# 'FLPM': First Lower Partial Moment (Omega Ratio).\n", "# 'SLPM': Second Lower Partial Moment (Sortino Ratio).\n", "# 'CVaR': Conditional Value at Risk.\n", "# 'EVaR': Entropic Value at Risk.\n", "# 'WR': Worst Realization (Minimax)\n", "# 'MDD': Maximum Drawdown of uncompounded cumulative returns (Calmar Ratio).\n", "# 'ADD': Average Drawdown of uncompounded cumulative returns.\n", "# 'CDaR': Conditional Drawdown at Risk of uncompounded cumulative returns.\n", "# 'EDaR': Entropic Drawdown at Risk of uncompounded cumulative returns.\n", "# 'UCI': Ulcer Index of uncompounded cumulative returns.\n", "\n", "rms = ['MV', 'MAD', 'MSV', 'FLPM', 'SLPM', 'CVaR',\n", " 'EVaR', 'WR', 'MDD', 'ADD', 'CDaR', 'UCI', 'EDaR']\n", "\n", "w_s = pd.DataFrame([])\n", "port.alpha = 0.05\n", "\n", "port.blfactors_stats(flavor='BLB',\n", " B=loadings,\n", " P_f=P_f,\n", " Q_f=Q_f/252,\n", " rf=0,\n", " delta=None,\n", " eq=True,\n", " const=True,\n", " diag=True,\n", " method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "model = 'BL_FM'\n", "obj = 'Sharpe'\n", "\n", "for i in rms:\n", " if i == 'MV':\n", " hist = 2\n", " else:\n", " hist = True\n", " w = port.optimization(model=model, rm=i, obj=obj, rf=rf, l=l, hist=hist)\n", " w_s = pd.concat([w_s, w], axis=1)\n", " \n", "w_s.columns = rms" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
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APA0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%
CMCSA5.27%6.48%8.28%6.69%8.34%11.76%4.02%0.00%8.15%10.28%2.03%10.10%5.55%
CNP5.98%6.22%2.27%7.88%2.13%0.00%0.00%0.00%7.08%5.60%23.19%11.26%18.31%
HPQ17.32%19.10%15.45%18.15%15.33%15.79%15.73%15.78%8.45%0.00%1.99%0.00%6.33%
PSA18.03%19.43%18.60%18.69%18.55%18.59%12.37%3.43%43.61%34.45%39.36%34.46%38.65%
SEE0.00%0.00%0.00%0.00%0.00%0.00%0.00%3.03%4.16%10.15%6.27%9.26%7.64%
ZION11.34%10.70%10.67%10.02%10.67%9.50%12.81%14.88%7.76%11.92%13.28%12.52%9.74%
PEP11900D0310.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%
PEP13000D0126.23%12.34%14.33%12.51%14.52%23.34%16.79%5.21%0.00%0.00%0.00%0.00%0.00%
PEP13000M0880.23%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%
PEP23900M1030.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%
PEP70101M53014.69%0.00%5.41%0.00%5.61%4.35%3.99%10.37%0.00%0.00%0.00%0.00%0.00%
PEP70101M57120.91%25.72%24.98%26.06%24.84%16.66%34.29%47.30%20.80%27.60%13.88%22.40%13.78%
PEP70310M1560.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%
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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "\n", "# Plotting a comparison of assets weights for each portfolio\n", "\n", "fig = plt.gcf()\n", "fig.set_figwidth(14)\n", "fig.set_figheight(6)\n", "ax = fig.subplots(nrows=1, ncols=1)\n", "\n", "w_s.plot.bar(ax=ax)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.2 Calculate Augmented Black Litterman Portfolios for Several Risk Measures" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "You must convert self.cov_bl_fm to a positive definite matrix\n" ] } ], "source": [ "# Risk Measures available:\n", "#\n", "# 'MV': Standard Deviation.\n", "# 'MAD': Mean Absolute Deviation.\n", "# 'MSV': Semi Standard Deviation.\n", "# 'FLPM': First Lower Partial Moment (Omega Ratio).\n", "# 'SLPM': Second Lower Partial Moment (Sortino Ratio).\n", "# 'CVaR': Conditional Value at Risk.\n", "# 'EVaR': Entropic Value at Risk.\n", "# 'WR': Worst Realization (Minimax)\n", "# 'MDD': Maximum Drawdown of uncompounded cumulative returns (Calmar Ratio).\n", "# 'ADD': Average Drawdown of uncompounded cumulative returns.\n", "# 'CDaR': Conditional Drawdown at Risk of uncompounded cumulative returns.\n", "# 'EDaR': Entropic Drawdown at Risk of uncompounded cumulative returns.\n", "# 'UCI': Ulcer Index of uncompounded cumulative returns.\n", "\n", "rms = ['MV', 'MAD', 'MSV', 'FLPM', 'SLPM', 'CVaR',\n", " 'EVaR', 'WR', 'MDD', 'ADD', 'CDaR', 'UCI', 'EDaR']\n", "\n", "w_s = pd.DataFrame([])\n", "port.alpha = 0.05\n", "\n", "port.blfactors_stats(flavor='ABL',\n", " B=loadings,\n", " P_f=P_f,\n", " Q_f=Q_f/252,\n", " rf=0,\n", " delta=None,\n", " eq=True,\n", " const=True,\n", " diag=True,\n", " method_mu=method_mu,\n", " method_cov=method_cov)\n", "\n", "model = 'BL_FM'\n", "obj = 'Sharpe'\n", "\n", "for i in rms:\n", " if i == 'MV':\n", " hist = 2\n", " else:\n", " hist = True\n", " w = port.optimization(model=model, rm=i, obj=obj, rf=rf, l=l, hist=hist)\n", " w_s = pd.concat([w_s, w], axis=1)\n", " \n", "w_s.columns = rms" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
 MVMADMSVFLPMSLPMCVaREVaRWRMDDADDCDaRUCIEDaR
APA0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%
CMCSA7.31%7.34%8.79%6.38%8.88%8.36%8.79%5.47%6.11%15.07%5.27%11.85%7.29%
CNP1.67%3.53%0.00%2.67%0.00%0.00%0.00%0.00%18.42%4.20%25.33%10.06%20.55%
HPQ21.27%22.38%19.00%24.95%18.78%16.58%15.88%14.88%13.48%3.41%6.93%1.57%10.91%
PSA12.08%15.68%15.84%15.57%15.77%15.63%7.81%0.00%38.43%31.22%41.50%34.67%41.70%
SEE0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%7.49%12.88%4.78%11.08%7.65%
ZION20.91%17.88%18.28%18.92%18.22%17.27%15.12%14.63%6.48%13.10%13.29%14.57%10.48%
PEP11900D0312.34%6.29%0.00%3.83%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%
PEP13000D0126.37%15.62%19.18%15.37%19.39%25.39%24.55%18.52%0.00%0.00%0.00%0.00%0.00%
PEP13000M0882.03%0.00%0.00%0.00%0.00%0.00%0.00%0.00%9.59%0.00%0.00%0.00%0.00%
PEP23900M1034.15%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%2.27%0.00%0.00%0.00%
PEP70101M5309.62%5.86%4.04%4.75%4.03%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%
PEP70101M5718.27%5.43%14.87%7.56%14.93%16.77%27.86%46.51%0.00%17.84%2.90%16.20%1.41%
PEP70310M1563.98%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%0.00%
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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "\n", "# Plotting a comparison of assets weights for each portfolio\n", "\n", "fig = plt.gcf()\n", "fig.set_figwidth(14)\n", "fig.set_figheight(6)\n", "ax = fig.subplots(nrows=1, ncols=1)\n", "\n", "w_s.plot.bar(ax=ax)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can see that in the examples the Augmented Black Litterman model and the Bayesian Black Litterman model increase the diversification when we only consider the mean vector obtained through these methods and we use the covariance and scenarios from the factor model." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.11.7" } }, "nbformat": 4, "nbformat_minor": 4 }