Riskfolio-Lib Tutorial:¶

__[Financionerioncios](https://financioneroncios.wordpress.com)__
__[Orenji](https://www.linkedin.com/company/orenj-i)__
__[Riskfolio-Lib](https://riskfolio-lib.readthedocs.io/en/latest/)__
__[Dany Cajas](https://www.linkedin.com/in/dany-cajas/)__

Tutorial 20: Black Litterman with Factors Models Mean Risk Optimization¶

1. Downloading the data:¶

In [2]:

########################################################################
# Uploading Data
########################################################################

import pandas as pd
import numpy as np
import warnings

warnings.filterwarnings("ignore")

# Interest Rates Data
kr = pd.read_excel('KeyRates.xlsx', engine='openpyxl', index_col=0, header=0)/100

# Prices  Data
assets = pd.read_excel('Assets.xlsx', engine='openpyxl', index_col=0, header=0)

# Find common dates
a = pd.merge(left=assets, right=kr, how='inner', on='Date')
dates = a.index

# Calculate interest rates returns
kr_returns = kr.loc[dates,:].sort_index().diff().dropna()
kr_returns.sort_index(ascending=False, inplace=True)

# List of instruments
equity = ['APA','CMCSA','CNP','HPQ','PSA','SEE','ZION']
bonds = ['PEP11900D031', 'PEP13000D012', 'PEP13000M088',
         'PEP23900M103','PEP70101M530','PEP70101M571',
         'PEP70310M156']
factors = ['MTUM','QUAL','SIZE','USMV','VLUE']

# Calculate assets returns
assets_returns = assets.loc[dates, equity + bonds]
assets_returns = assets_returns.sort_index().pct_change().dropna()
assets_returns.sort_index(ascending=False, inplace=True)

# Calculate factors returns
factors_returns = assets.loc[dates, factors]
factors_returns = factors_returns.sort_index().pct_change().dropna()
factors_returns.sort_index(ascending=False, inplace=True)

# Show tables
display(kr_returns.head().style.format("{:.4%}"))
display(assets_returns.head().style.format("{:.4%}"))

	0	90	180	360	720	1800	3600	7200	10800
Date
2017-11-16 00:00:00	0.0000%	0.0059%	0.0108%	0.0178%	0.0246%	0.0213%	0.0075%	-0.0048%	-0.0093%
2017-11-15 00:00:00	0.0180%	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 00:00:00	0.0000%	0.0013%	0.0025%	0.0048%	0.0088%	0.0174%	0.0258%	0.0334%	0.0364%
2017-11-10 00:00:00	0.0000%	0.0026%	0.0043%	0.0054%	0.0017%	-0.0248%	-0.0615%	-0.0936%	-0.1054%

	APA	CMCSA	CNP	HPQ	PSA	SEE	ZION	PEP11900D031	PEP13000D012	PEP13000M088	PEP23900M103	PEP70101M530	PEP70101M571	PEP70310M156
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%

In [3]:

########################################################################
# Uploading Duration and Convexity Matrixes
########################################################################

durations = pd.read_excel('durations.xlsx', index_col=0, header=0)
convexity = pd.read_excel('convexity.xlsx', index_col=0, header=0)

print('Durations Matrix')
display(durations.head().style.format("{:.4f}").background_gradient(cmap='YlGn'))
print('')
print('Convexities Matrix')
display(convexity.head().style.format("{:.4f}").background_gradient(cmap='YlGn'))

Durations Matrix

	R 0	R 90	R 180	R 360	R 720	R 1800	R 3600	R 7200
PEP11900D031	0.0012	0.0057	0.0192	0.0730	0.3685	3.0416	0.0030	0.0000
PEP13000D012	0.0000	0.0078	0.0142	0.0617	0.3327	1.0902	4.8055	0.2074
PEP13000M088	0.0013	0.0004	0.0147	0.0501	0.2770	2.4626	3.0764	0.0000
PEP23900M103	0.0000	0.0005	0.0117	0.0405	0.2274	3.9726	0.0381	0.0000
PEP70101M530	0.0000	0.0052	0.0101	0.0442	0.2488	0.8826	4.9147	3.5537

Convexities Matrix

	R^2 0	R^2 90	R^2 180	R^2 360	R^2 720	R^2 1800	R^2 3600	R^2 7200
PEP11900D031	0.0004	0.0032	0.0167	0.0928	0.7741	15.5617	0.0000	0.0000
PEP13000D012	0.0000	0.0057	0.0070	0.0756	0.7210	4.4984	45.2159	0.1105
PEP13000M088	0.0010	0.0001	0.0192	0.0736	0.6161	8.8479	16.2880	0.0000
PEP23900M103	0.0000	0.0000	0.0156	0.0644	0.5161	22.1272	0.0022	0.0000
PEP70101M530	0.0000	0.0038	0.0052	0.0561	0.5530	3.7373	38.2315	26.1464

2. Estimating Black Litterman with Factors for Fixed Income Portfolios¶

2.1 Building the loadings matrix and risk factors returns¶

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.

In [5]:

########################################################################
# Building The Loadings Matrix
########################################################################

loadings = pd.concat([-1.0 * durations, 0.5 * convexity], axis = 1)

display(loadings.style.format("{:.4f}").background_gradient(cmap='YlGn'))

	R 0	R 90	R 180	R 360	R 720	R 1800	R 3600	R 7200	R 10800	R^2 0	R^2 90	R^2 180	R^2 360	R^2 720	R^2 1800	R^2 3600	R^2 7200	R^2 10800
PEP11900D031	-0.0012	-0.0057	-0.0192	-0.0730	-0.3685	-3.0416	-0.0030	-0.0000	-0.0000	0.0002	0.0016	0.0083	0.0464	0.3871	7.7809	0.0000	0.0000	0.0000
PEP13000D012	-0.0000	-0.0078	-0.0142	-0.0617	-0.3327	-1.0902	-4.8055	-0.2074	-0.0000	0.0000	0.0029	0.0035	0.0378	0.3605	2.2492	22.6080	0.0553	0.0000
PEP13000M088	-0.0013	-0.0004	-0.0147	-0.0501	-0.2770	-2.4626	-3.0764	-0.0000	-0.0000	0.0005	0.0000	0.0096	0.0368	0.3081	4.4240	8.1440	0.0000	0.0000
PEP23900M103	-0.0000	-0.0005	-0.0117	-0.0405	-0.2274	-3.9726	-0.0381	-0.0000	-0.0000	0.0000	0.0000	0.0078	0.0322	0.2581	11.0636	0.0011	0.0000	0.0000
PEP70101M530	-0.0000	-0.0052	-0.0101	-0.0442	-0.2488	-0.8826	-4.9147	-3.5537	-0.0000	0.0000	0.0019	0.0026	0.0280	0.2765	1.8686	19.1157	13.0732	0.0000
PEP70101M571	-0.0015	-0.0039	-0.0126	-0.0501	-0.2829	-1.0108	-2.5878	-6.0312	-0.4501	0.0002	0.0016	0.0064	0.0319	0.3123	2.1336	10.1632	49.9021	0.4523
PEP70310M156	-0.0000	-0.0039	-0.0097	-0.0403	-0.2614	-3.8920	-0.0000	-0.0000	-0.0000	0.0000	0.0010	0.0030	0.0268	0.2508	10.6813	0.0000	0.0000	0.0000

In [6]:

########################################################################
# Building the risk factors returns matrix
########################################################################

kr_returns_2 =  kr_returns ** 2
cols = loadings.columns

X = pd.concat([kr_returns, kr_returns_2], axis=1)
X.columns = cols

display(X.head().style.format("{:.4%}"))

	R 0	R 90	R 180	R 360	R 720	R 1800	R 3600	R 7200	R 10800	R^2 0	R^2 90	R^2 180	R^2 360	R^2 720	R^2 1800	R^2 3600	R^2 7200	R^2 10800
Date
2017-11-16 00:00:00	0.0000%	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 00:00:00	0.0180%	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 00:00:00	0.0000%	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 00:00:00	0.0000%	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%

In [7]:

########################################################################
# Building the asset returns matrix
########################################################################

Y = assets_returns[bonds]

display(Y.head())

	PEP11900D031	PEP13000D012	PEP13000M088	PEP23900M103	PEP70101M530	PEP70101M571	PEP70310M156
Date
2017-11-16	-0.000411	-0.000380	-0.000597	-0.000737	-0.000116	0.000076	-0.000633
2017-11-15	-0.001626	-0.003076	-0.003041	-0.002286	-0.004459	-0.004651	-0.002146
2017-11-14	0.002320	0.000236	0.001040	0.002373	-0.002741	-0.003932	0.002151
2017-11-13	0.000906	0.000064	-0.000767	0.000354	-0.000835	-0.001114	-0.000081
2017-11-10	0.001194	0.003792	0.003047	0.001355	0.007118	0.008207	-0.000090

2.2 Building views on risk factors¶

In [9]:

########################################################################
# Showing annualized returns of Fixed Income Risk Factors
########################################################################

display(X.mean()*252)

R 0         -0.002028
R 90        -0.001955
R 180       -0.001819
R 360       -0.001463
R 720       -0.000791
R 1800      -0.000530
R 3600      -0.002041
R 7200      -0.003312
R 10800     -0.003533
R^2 0        0.000173
R^2 90       0.000080
R^2 180      0.000049
R^2 360      0.000038
R^2 720      0.000046
R^2 1800     0.000050
R^2 3600     0.000057
R^2 7200     0.000069
R^2 10800    0.000091
dtype: float64

In [10]:

########################################################################
# Building views on some Risk Factors
########################################################################

views = {'Disabled': [False, False, False],
        'Factor': ['R 10800','R 1800','R 3600'],
        'Sign': ['>=', '<=', '<='],
        'Value': [0.001, -0.001, -0.003],
        'Relative Factor': ['R 7200', '', '']}

views = pd.DataFrame(views)

display(views)

	Disabled	Factor	Sign	Value	Relative Factor
0	False	R 10800	>=	0.001	R 7200
1	False	R 1800	<=	-0.001
2	False	R 3600	<=	-0.003

In [11]:

########################################################################
# Building views matrixes P_f and Q_f
########################################################################

import riskfolio as rp

P_f, Q_f = rp.factors_views(views, loadings, const=False)

print('Matrix of factors views P_f')
print(P_f)
print('\nMatrix of returns of factors views Q_f')
print(Q_f)

Matrix of factors views P_f
[[ 0.  0.  0.  0.  0.  0.  0. -1.  1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0. -1.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0. -1.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]]

Matrix of returns of factors views Q_f
[[0.001]
 [0.001]
 [0.003]]

2.3 Building Portfolios with mean vector and covariance matrix from Black Litterman with Factors.¶

In [13]:

########################################################################
# Building the Portfolio Object
########################################################################

# Building the portfolio object
port = rp.Portfolio(returns=Y)

# Select method and estimate input parameters:
method_mu='hist' # Method to estimate expected returns based on historical data.
method_cov='hist' # Method to estimate covariance matrix based on historical data.

port.assets_stats(method_mu=method_mu,
                  method_cov=method_cov)

port.factors = X
port.factors_stats(method_mu=method_mu,
                   method_cov=method_cov,
                   B=loadings,
                   const=False)

In [14]:

########################################################################
# Calculating optimum portfolios using Mean Vector and
# Covariance Matrix of Black Litterman with Factors
########################################################################

port.alpha = 0.05
rm = 'MV' # Risk measure used, this time will be variance
obj = 'Sharpe' # Objective function, could be MinRisk, MaxRet, Utility or Sharpe
hist = False # False: BL covariance and risk factors scenarios
         # True: historical covariance and scenarios
         # 2: risk factors covariance and scenarios
rf = 0 # Risk free rate
l = 0 # Risk aversion factor, only useful when obj is 'Utility'

w_fm = port.optimization(model='FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist) 

# Estimate Portfolio weights using Black Litterman Bayesian Model:
port.blfactors_stats(flavor='BLB',
                     B=loadings,
                     P_f=P_f,
                     Q_f=Q_f/252,
                     rf=0,
                     delta=None,
                     eq=True,
                     const=False,
                     diag=False,
                     method_mu=method_mu,
                     method_cov=method_cov)

w_blb = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)

# Estimate Portfolio weights using Augmented Black Litterman Model:
port.blfactors_stats(flavor='ABL',
                     B=loadings,
                     P_f=P_f,
                     Q_f=Q_f/252,
                     rf=0,
                     delta=None,
                     eq=True,
                     const=False,
                     diag=False,
                     method_mu=method_mu,
                     method_cov=method_cov)

w_abl = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)

ws = pd.concat([w_fm, w_blb, w_abl], axis=1)
ws.columns = ['Pure Factors', 'Bayesian BL', 'Augmented BL']

display(ws.style.format("{:.4%}").background_gradient(cmap='YlGn'))

You must convert self.cov_bl_fm to a positive definite matrix
You must convert self.cov_bl_fm to a positive definite matrix

	Pure Factors	Bayesian BL	Augmented BL
PEP11900D031	0.0000%	0.0000%	0.0000%
PEP13000D012	6.6680%	33.1060%	85.2406%
PEP13000M088	0.0000%	0.0000%	0.0000%
PEP23900M103	0.0000%	0.0000%	0.0000%
PEP70101M530	32.3833%	0.0001%	14.7594%
PEP70101M571	60.9486%	66.8939%	0.0000%
PEP70310M156	0.0000%	0.0000%	0.0000%

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.

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:

2.4 Building Portfolios with mean vector from Black Litterman with Factors.¶

In [17]:

########################################################################
# Calculating optimum portfolios using only Mean Vector
# of Black Litterman with Factors and Factor Covariance Matrix
########################################################################

hist = 2 # False: BL covariance and risk factors scenarios
             # True: historical covariance and scenarios
             # 2: risk factors covariance and scenarios (Only in BL_FM)

# Estimate Portfolio weights using Black Litterman Bayesian Model:
port.blfactors_stats(flavor='BLB',
                     B=loadings,
                     P_f=P_f,
                     Q_f=Q_f/252,
                     rf=0,
                     delta=None,
                     eq=True,
                     const=False,
                     diag=False,
                     method_mu=method_mu,
                     method_cov=method_cov)

w_blb = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)

# Estimate Portfolio weights using Augmented Black Litterman Model:
port.blfactors_stats(flavor='ABL',
                     B=loadings,
                     P_f=P_f,
                     Q_f=Q_f/252,
                     rf=0,
                     delta=None,
                     eq=True,
                     const=False,
                     diag=False,
                     method_mu=method_mu,
                     method_cov=method_cov)

w_abl = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)

ws = pd.concat([w_fm, w_blb, w_abl], axis=1)
ws.columns = ['Pure Factors', 'Bayesian BL', 'Augmented BL']

display(ws.style.format("{:.4%}").background_gradient(cmap='YlGn'))

You must convert self.cov_bl_fm to a positive definite matrix
You must convert self.cov_bl_fm to a positive definite matrix

	Pure Factors	Bayesian BL	Augmented BL
PEP11900D031	0.0000%	0.0000%	6.1662%
PEP13000D012	6.6680%	16.6281%	16.4519%
PEP13000M088	0.0000%	2.2781%	5.2878%
PEP23900M103	0.0000%	0.0000%	11.2163%
PEP70101M530	32.3833%	35.1625%	26.2773%
PEP70101M571	60.9486%	45.9313%	23.8393%
PEP70310M156	0.0000%	0.0000%	10.7612%

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.

3. Estimating Mean Variance Portfolio for Equity and Fixed Income Portfolio¶

3.1 Building the loadings matrix and risk factors returns.¶

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.

In [20]:

B = rp.loadings_matrix(factors_returns, assets_returns[equity])

display(B.style.format("{:.4f}").background_gradient(cmap='YlGn'))

	const	MTUM	QUAL	SIZE	USMV	VLUE
APA	-0.0008	-0.8999	1.2674	0.0000	-0.5750	1.4874
CMCSA	-0.0000	0.2426	0.0000	-0.1569	0.5483	0.3623
CNP	-0.0001	-0.3827	-0.3579	0.0000	1.9556	0.0000
HPQ	0.0003	0.0000	0.5164	0.0000	0.0000	0.7550
PSA	-0.0000	0.0000	-0.4623	0.0000	1.8071	-0.3115
SEE	-0.0001	0.0000	0.4915	0.0000	0.5043	0.2535
ZION	0.0002	0.0000	1.1589	0.0000	-1.3203	1.2202

In [21]:

########################################################################
# Building the asset returns matrix
########################################################################

Y = pd.concat([assets_returns[equity], Y], axis=1)

display(Y.head())

	APA	CMCSA	CNP	HPQ	PSA	SEE	ZION	PEP11900D031	PEP13000D012	PEP13000M088	PEP23900M103	PEP70101M530	PEP70101M571	PEP70310M156
Date
2017-11-16	-0.013161	-0.002958	-0.010903	0.009831	0.017234	0.014016	-0.008387	-0.000411	-0.000380	-0.000597	-0.000737	-0.000116	0.000076	-0.000633
2017-11-15	-0.020296	0.008682	-0.011202	0.000000	-0.013479	-0.003326	0.000215	-0.001626	-0.003076	-0.003041	-0.002286	-0.004459	-0.004651	-0.002146
2017-11-14	-0.037020	-0.010470	0.010800	0.008975	-0.001548	0.002668	0.026950	0.002320	0.000236	0.001040	0.002373	-0.002741	-0.003932	0.002151
2017-11-13	-0.014503	0.010855	0.007480	-0.002826	0.008179	0.014205	0.034145	0.000906	0.000064	-0.000767	0.000354	-0.000835	-0.001114	-0.000081
2017-11-10	-0.024536	0.007932	-0.013418	-0.005155	0.000710	-0.009381	-0.000910	0.001194	0.003792	0.003047	0.001355	0.007118	0.008207	-0.000090

In [22]:

########################################################################
# Building the asset returns matrix
########################################################################

X = pd.concat([factors_returns, X], axis=1)

display(X.head())

	MTUM	QUAL	SIZE	USMV	VLUE	R 0	R 90	R 180	R 360	R 720	...	R 10800	R^2 0	R^2 90	R^2 180	R^2 360	R^2 720	R^2 1800	R^2 3600	R^2 7200	R^2 10800
Date
2017-11-16	0.009478	0.007556	0.006675	0.006408	0.014509	0.00000	0.000059	0.000108	0.000178	0.000246	...	-0.000093	0.000000e+00	3.527647e-09	1.163831e-08	3.181621e-08	6.048648e-08	4.552420e-08	5.573518e-09	2.314283e-09	8.695936e-09
2017-11-15	-0.003381	-0.005884	-0.001974	-0.006750	-0.003771	0.00018	0.000247	0.000303	0.000391	0.000495	...	0.000417	3.225005e-08	6.082093e-08	9.197391e-08	1.529780e-07	2.448864e-07	3.108575e-07	2.620815e-07	2.025117e-07	1.736147e-07
2017-11-14	-0.001687	0.000375	-0.000370	0.001932	-0.004380	-0.00180	-0.001710	-0.001624	-0.001460	-0.001167	...	0.000861	3.241235e-06	2.925701e-06	2.636999e-06	2.131988e-06	1.362113e-06	2.555747e-07	1.966922e-08	4.572289e-07	7.417774e-07
2017-11-13	0.002488	0.000877	0.003589	0.003296	0.000751	0.00000	0.000013	0.000025	0.000048	0.000088	...	0.000364	0.000000e+00	1.681950e-10	6.400900e-10	2.324301e-09	7.748577e-09	3.033345e-08	6.657174e-08	1.116368e-07	1.328391e-07
2017-11-10	0.002894	0.001255	-0.001730	-0.000968	0.001254	0.00000	0.000026	0.000043	0.000054	0.000017	...	-0.001054	0.000000e+00	6.770924e-10	1.820644e-09	2.881542e-09	2.791573e-10	6.131368e-08	3.779975e-07	8.759856e-07	1.110697e-06

5 rows × 23 columns

In [23]:

########################################################################
# Building The Loadings Matrix
########################################################################

loadings = pd.concat([B, loadings], axis = 1)
loadings.fillna(0, inplace=True)

display(loadings.style.format("{:.4f}").background_gradient(cmap='YlGn'))

	const	MTUM	QUAL	SIZE	USMV	VLUE	R 0	R 90	R 180	R 360	R 720	R 1800	R 3600	R 7200	R 10800	R^2 0	R^2 90	R^2 180	R^2 360	R^2 720	R^2 1800	R^2 3600	R^2 7200	R^2 10800
APA	-0.0008	-0.8999	1.2674	0.0000	-0.5750	1.4874	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
CMCSA	-0.0000	0.2426	0.0000	-0.1569	0.5483	0.3623	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
CNP	-0.0001	-0.3827	-0.3579	0.0000	1.9556	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
HPQ	0.0003	0.0000	0.5164	0.0000	0.0000	0.7550	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
PSA	-0.0000	0.0000	-0.4623	0.0000	1.8071	-0.3115	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
SEE	-0.0001	0.0000	0.4915	0.0000	0.5043	0.2535	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
ZION	0.0002	0.0000	1.1589	0.0000	-1.3203	1.2202	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
PEP11900D031	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	-0.0012	-0.0057	-0.0192	-0.0730	-0.3685	-3.0416	-0.0030	-0.0000	-0.0000	0.0002	0.0016	0.0083	0.0464	0.3871	7.7809	0.0000	0.0000	0.0000
PEP13000D012	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	-0.0000	-0.0078	-0.0142	-0.0617	-0.3327	-1.0902	-4.8055	-0.2074	-0.0000	0.0000	0.0029	0.0035	0.0378	0.3605	2.2492	22.6080	0.0553	0.0000
PEP13000M088	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	-0.0013	-0.0004	-0.0147	-0.0501	-0.2770	-2.4626	-3.0764	-0.0000	-0.0000	0.0005	0.0000	0.0096	0.0368	0.3081	4.4240	8.1440	0.0000	0.0000
PEP23900M103	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	-0.0000	-0.0005	-0.0117	-0.0405	-0.2274	-3.9726	-0.0381	-0.0000	-0.0000	0.0000	0.0000	0.0078	0.0322	0.2581	11.0636	0.0011	0.0000	0.0000
PEP70101M530	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	-0.0000	-0.0052	-0.0101	-0.0442	-0.2488	-0.8826	-4.9147	-3.5537	-0.0000	0.0000	0.0019	0.0026	0.0280	0.2765	1.8686	19.1157	13.0732	0.0000
PEP70101M571	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	-0.0015	-0.0039	-0.0126	-0.0501	-0.2829	-1.0108	-2.5878	-6.0312	-0.4501	0.0002	0.0016	0.0064	0.0319	0.3123	2.1336	10.1632	49.9021	0.4523
PEP70310M156	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	-0.0000	-0.0039	-0.0097	-0.0403	-0.2614	-3.8920	-0.0000	-0.0000	-0.0000	0.0000	0.0010	0.0030	0.0268	0.2508	10.6813	0.0000	0.0000	0.0000

3.2 Building views on risk factors¶

In [25]:

########################################################################
# Showing annualized returns of Equity Risk Factors
########################################################################

display(factors_returns.mean()*252)

MTUM    0.154379
QUAL    0.113842
SIZE    0.115915
USMV    0.124555
VLUE    0.106612
dtype: float64

In [26]:

########################################################################
# Building views on some Risk Factors
########################################################################

views = {'Disabled': [False, False, False, False, False, False],
        'Factor': ['MTUM','USMV','SIZE','R 10800','R 1800','R 3600'],
        'Sign': ['>=', '>=', '>=', '>=', '<=', '<='],
        'Value': [0.02, 0.09, 0.12, 0.001, -0.001, -0.003],
        'Relative Factor': ['VLUE', '', '','R 90', '', '']}
views = pd.DataFrame(views)

display(views)

	Disabled	Factor	Sign	Value	Relative Factor
0	False	MTUM	>=	0.020	VLUE
1	False	USMV	>=	0.090
2	False	SIZE	>=	0.120
3	False	R 10800	>=	0.001	R 90
4	False	R 1800	<=	-0.001
5	False	R 3600	<=	-0.003

In [27]:

########################################################################
# Building views matrixes P_f and Q_f
########################################################################

P_f, Q_f = rp.factors_views(views, loadings, const=True)

print('Matrix of factors views P_f')
print(P_f)
print('\nMatrix of returns of factors views Q_f')
print(Q_f)

Matrix of factors views P_f
[[ 1.  0.  0.  0. -1.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.
   0.  0.  0.  0.  0.]
 [ 0.  0.  0.  1.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.
   0.  0.  0.  0.  0.]
 [ 0.  0.  1.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.
   0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0. -1.  0.  0.  0.  0.  0.  0.  1.  0.  0.  0.  0.
   0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0. -1.  0.  0.  0.  0.  0.  0.  0.
   0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0. -1.  0.  0.  0.  0.  0.  0.
   0.  0.  0.  0.  0.]]

Matrix of returns of factors views Q_f
[[0.02 ]
 [0.09 ]
 [0.12 ]
 [0.001]
 [0.001]
 [0.003]]

3.3 Building Portfolios with mean vector and covariance matrix from Black Litterman with Factors.¶

In [29]:

########################################################################
# Building Portfolio Object
########################################################################

# Building the portfolio object
port = rp.Portfolio(returns=Y)

# Calculating optimum portfolio

# Select method and estimate input parameters:

method_mu='hist' # Method to estimate expected returns based on historical data.
method_cov='hist' # Method to estimate covariance matrix based on historical data.

port.assets_stats(method_mu=method_mu,
                  method_cov=method_cov)

port.factors = X
port.factors_stats(method_mu=method_mu,
                   method_cov=method_cov,
                   B=loadings,
                   const=True)

In [30]:

########################################################################
# Calculating optimum portfolios
########################################################################

port.alpha = 0.05
rm = 'MV' # Risk measure used, this time will be variance
obj = 'Sharpe' # Objective function, could be MinRisk, MaxRet, Utility or Sharpe
hist = False # False: BL covariance and risk factors scenarios
             # True: historical covariance and scenarios
             # 2: risk factors covariance and scenarios
rf = 0 # Risk free rate
l = 0 # Risk aversion factor, only useful when obj is 'Utility'

w_fm = port.optimization(model='FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist) 

# Estimate Portfolio weights using Black Litterman Bayesian Model:
port.blfactors_stats(flavor='BLB',
                     B=loadings,
                     P_f=P_f,
                     Q_f=Q_f/252,
                     rf=0,
                     delta=None,
                     eq=True,
                     const=True,
                     diag=True,
                     method_mu=method_mu,
                     method_cov=method_cov)

w_blb = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)

# Estimate Portfolio weights using Augmented Black Litterman Model:
port.blfactors_stats(flavor='ABL',
                     B=loadings,
                     P_f=P_f,
                     Q_f=Q_f/252,
                     rf=0,
                     delta=None,
                     eq=True,
                     const=True,
                     diag=True,
                     method_mu=method_mu,
                     method_cov=method_cov)

w_abl = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)

ws = pd.concat([w_fm, w_blb, w_abl], axis=1)
ws.columns = ['Pure Factors', 'Bayesian BL', 'Augmented BL']

display(ws.style.format("{:.4%}").background_gradient(cmap='YlGn'))

You must convert self.cov_bl_fm to a positive definite matrix

	Pure Factors	Bayesian BL	Augmented BL
APA	0.0000%	0.0000%	0.0000%
CMCSA	10.6669%	4.4927%	0.0000%
CNP	12.2572%	7.6298%	0.0000%
HPQ	16.3690%	17.0336%	72.8372%
PSA	26.0444%	19.6223%	0.0000%
SEE	0.0758%	0.0000%	0.0000%
ZION	7.9950%	9.8279%	0.0000%
PEP11900D031	0.0000%	0.0000%	0.0000%
PEP13000D012	0.0001%	6.9747%	27.1628%
PEP13000M088	0.0000%	0.4876%	0.0000%
PEP23900M103	0.0000%	0.0000%	0.0000%
PEP70101M530	7.8241%	14.5620%	0.0000%
PEP70101M571	18.7674%	19.3695%	0.0000%
PEP70310M156	0.0000%	0.0000%	0.0000%

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.

We can see that in this case the Black Litterman Bayesian model creates a more diversified portfolio than the Augmented Black Litterman model.

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.

3.4 Building Portfolios with mean vector from Black Litterman with Factors.¶

In [33]:

########################################################################
# Calculating optimum portfolios
########################################################################

hist = 2 # False: BL covariance and risk factors scenarios
         # True: historical covariance and scenarios
         # 2: risk factors covariance and scenarios (Only in BL_FM)

# Estimate Portfolio weights using Black Litterman Bayesian Model:
port.blfactors_stats(flavor='BLB',
                     B=loadings,
                     P_f=P_f,
                     Q_f=Q_f/252,
                     rf=0,
                     delta=None,
                     eq=True,
                     const=True,
                     diag=True,
                     method_mu=method_mu,
                     method_cov=method_cov)

w_blb = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)

# Estimate Portfolio weights using Augmented Black Litterman Model:
port.blfactors_stats(flavor='ABL',
                     B=loadings,
                     P_f=P_f,
                     Q_f=Q_f/252,
                     rf=0,
                     delta=None,
                     eq=True,
                     const=True,
                     diag=True,
                     method_mu=method_mu,
                     method_cov=method_cov)

w_abl = port.optimization(model='BL_FM', rm=rm, obj=obj, rf=rf, l=l, hist=hist)

ws = pd.concat([w_fm, w_blb, w_abl], axis=1)
ws.columns = ['Pure Factors', 'Bayesian BL', 'Augmented BL']

display(ws.style.format("{:.4%}").background_gradient(cmap='YlGn'))

You must convert self.cov_bl_fm to a positive definite matrix

	Pure Factors	Bayesian BL	Augmented BL
APA	0.0000%	0.0000%	0.0000%
CMCSA	10.6669%	5.2672%	7.3134%
CNP	12.2572%	5.9762%	1.6711%
HPQ	16.3690%	17.3220%	21.2692%
PSA	26.0444%	18.0273%	12.0783%
SEE	0.0758%	0.0000%	0.0000%
ZION	7.9950%	11.3383%	20.9051%
PEP11900D031	0.0000%	0.0000%	2.3440%
PEP13000D012	0.0001%	6.2308%	6.3664%
PEP13000M088	0.0000%	0.2338%	2.0322%
PEP23900M103	0.0000%	0.0000%	4.1502%
PEP70101M530	7.8241%	14.6896%	9.6157%
PEP70101M571	18.7674%	20.9148%	8.2701%
PEP70310M156	0.0000%	0.0000%	3.9845%

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.

4. Estimating Black Litterman with Factors Mean Risk Portfolios¶

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.

4.1 Calculate Black Litterman Bayesian Portfolios for Several Risk Measures¶

In [36]:

# Risk Measures available:
#
# 'MV': Standard Deviation.
# 'MAD': Mean Absolute Deviation.
# 'MSV': Semi Standard Deviation.
# 'FLPM': First Lower Partial Moment (Omega Ratio).
# 'SLPM': Second Lower Partial Moment (Sortino Ratio).
# 'CVaR': Conditional Value at Risk.
# 'EVaR': Entropic Value at Risk.
# 'WR': Worst Realization (Minimax)
# 'MDD': Maximum Drawdown of uncompounded cumulative returns (Calmar Ratio).
# 'ADD': Average Drawdown of uncompounded cumulative returns.
# 'CDaR': Conditional Drawdown at Risk of uncompounded cumulative returns.
# 'EDaR': Entropic Drawdown at Risk of uncompounded cumulative returns.
# 'UCI': Ulcer Index of uncompounded cumulative returns.

rms = ['MV', 'MAD', 'MSV', 'FLPM', 'SLPM', 'CVaR',
       'EVaR', 'WR', 'MDD', 'ADD', 'CDaR', 'UCI', 'EDaR']

w_s = pd.DataFrame([])
port.alpha = 0.05

port.blfactors_stats(flavor='BLB',
                     B=loadings,
                     P_f=P_f,
                     Q_f=Q_f/252,
                     rf=0,
                     delta=None,
                     eq=True,
                     const=True,
                     diag=True,
                     method_mu=method_mu,
                     method_cov=method_cov)

model = 'BL_FM'
obj = 'Sharpe'

for i in rms:
    if i == 'MV':
        hist = 2
    else:
        hist = True
    w = port.optimization(model=model, rm=i, obj=obj, rf=rf, l=l, hist=hist)
    w_s = pd.concat([w_s, w], axis=1)
    
w_s.columns = rms

In [37]:

w_s.style.format("{:.2%}").background_gradient(cmap='YlGn')

Out[37]:

	MV	MAD	MSV	FLPM	SLPM	CVaR	EVaR	WR	MDD	ADD	CDaR	UCI	EDaR
APA	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
CMCSA	5.27%	6.48%	8.28%	6.69%	8.34%	11.76%	4.02%	0.00%	8.15%	10.28%	2.03%	10.10%	5.55%
CNP	5.98%	6.22%	2.27%	7.88%	2.13%	0.00%	0.00%	0.00%	7.08%	5.60%	23.19%	11.26%	18.31%
HPQ	17.32%	19.10%	15.45%	18.15%	15.33%	15.79%	15.73%	15.78%	8.45%	0.00%	1.99%	0.00%	6.33%
PSA	18.03%	19.43%	18.60%	18.69%	18.55%	18.59%	12.37%	3.43%	43.61%	34.45%	39.36%	34.46%	38.65%
SEE	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	3.03%	4.16%	10.15%	6.27%	9.26%	7.64%
ZION	11.34%	10.70%	10.67%	10.02%	10.67%	9.50%	12.81%	14.88%	7.76%	11.92%	13.28%	12.52%	9.74%
PEP11900D031	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
PEP13000D012	6.23%	12.34%	14.33%	12.51%	14.52%	23.34%	16.79%	5.21%	0.00%	0.00%	0.00%	0.00%	0.00%
PEP13000M088	0.23%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
PEP23900M103	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
PEP70101M530	14.69%	0.00%	5.41%	0.00%	5.61%	4.35%	3.99%	10.37%	0.00%	0.00%	0.00%	0.00%	0.00%
PEP70101M571	20.91%	25.72%	24.98%	26.06%	24.84%	16.66%	34.29%	47.30%	20.80%	27.60%	13.88%	22.40%	13.78%
PEP70310M156	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%

In [38]:

import matplotlib.pyplot as plt

# Plotting a comparison of assets weights for each portfolio

fig = plt.gcf()
fig.set_figwidth(14)
fig.set_figheight(6)
ax = fig.subplots(nrows=1, ncols=1)

w_s.plot.bar(ax=ax)

Out[38]:

<Axes: >

4.2 Calculate Augmented Black Litterman Portfolios for Several Risk Measures¶

In [40]:

# Risk Measures available:
#
# 'MV': Standard Deviation.
# 'MAD': Mean Absolute Deviation.
# 'MSV': Semi Standard Deviation.
# 'FLPM': First Lower Partial Moment (Omega Ratio).
# 'SLPM': Second Lower Partial Moment (Sortino Ratio).
# 'CVaR': Conditional Value at Risk.
# 'EVaR': Entropic Value at Risk.
# 'WR': Worst Realization (Minimax)
# 'MDD': Maximum Drawdown of uncompounded cumulative returns (Calmar Ratio).
# 'ADD': Average Drawdown of uncompounded cumulative returns.
# 'CDaR': Conditional Drawdown at Risk of uncompounded cumulative returns.
# 'EDaR': Entropic Drawdown at Risk of uncompounded cumulative returns.
# 'UCI': Ulcer Index of uncompounded cumulative returns.

rms = ['MV', 'MAD', 'MSV', 'FLPM', 'SLPM', 'CVaR',
       'EVaR', 'WR', 'MDD', 'ADD', 'CDaR', 'UCI', 'EDaR']

w_s = pd.DataFrame([])
port.alpha = 0.05

port.blfactors_stats(flavor='ABL',
                     B=loadings,
                     P_f=P_f,
                     Q_f=Q_f/252,
                     rf=0,
                     delta=None,
                     eq=True,
                     const=True,
                     diag=True,
                     method_mu=method_mu,
                     method_cov=method_cov)

model = 'BL_FM'
obj = 'Sharpe'

for i in rms:
    if i == 'MV':
        hist = 2
    else:
        hist = True
    w = port.optimization(model=model, rm=i, obj=obj, rf=rf, l=l, hist=hist)
    w_s = pd.concat([w_s, w], axis=1)
    
w_s.columns = rms

You must convert self.cov_bl_fm to a positive definite matrix

In [41]:

w_s.style.format("{:.2%}").background_gradient(cmap='YlGn')

Out[41]:

	MV	MAD	MSV	FLPM	SLPM	CVaR	EVaR	WR	MDD	ADD	CDaR	UCI	EDaR
APA	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
CMCSA	7.31%	7.34%	8.79%	6.38%	8.88%	8.36%	8.79%	5.47%	6.11%	15.07%	5.27%	11.85%	7.29%
CNP	1.67%	3.53%	0.00%	2.67%	0.00%	0.00%	0.00%	0.00%	18.42%	4.20%	25.33%	10.06%	20.55%
HPQ	21.27%	22.38%	19.00%	24.95%	18.78%	16.58%	15.88%	14.88%	13.48%	3.41%	6.93%	1.57%	10.91%
PSA	12.08%	15.68%	15.84%	15.57%	15.77%	15.63%	7.81%	0.00%	38.43%	31.22%	41.50%	34.67%	41.70%
SEE	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	7.49%	12.88%	4.78%	11.08%	7.65%
ZION	20.91%	17.88%	18.28%	18.92%	18.22%	17.27%	15.12%	14.63%	6.48%	13.10%	13.29%	14.57%	10.48%
PEP11900D031	2.34%	6.29%	0.00%	3.83%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
PEP13000D012	6.37%	15.62%	19.18%	15.37%	19.39%	25.39%	24.55%	18.52%	0.00%	0.00%	0.00%	0.00%	0.00%
PEP13000M088	2.03%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	9.59%	0.00%	0.00%	0.00%	0.00%
PEP23900M103	4.15%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	2.27%	0.00%	0.00%	0.00%
PEP70101M530	9.62%	5.86%	4.04%	4.75%	4.03%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
PEP70101M571	8.27%	5.43%	14.87%	7.56%	14.93%	16.77%	27.86%	46.51%	0.00%	17.84%	2.90%	16.20%	1.41%
PEP70310M156	3.98%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%

In [42]:

import matplotlib.pyplot as plt

# Plotting a comparison of assets weights for each portfolio

fig = plt.gcf()
fig.set_figwidth(14)
fig.set_figheight(6)
ax = fig.subplots(nrows=1, ncols=1)

w_s.plot.bar(ax=ax)

Out[42]:

<Axes: >

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.