Riskfolio-Lib Tutorial:¶

Financionerioncios
Orenji
Riskfolio-Lib
Dany Cajas

Tutorial 2: Portfolio Optimization with Risk Factors using Stepwise Regression¶

1. Downloading the data:¶

In [1]:

import numpy as np
import pandas as pd
import yfinance as yf
import warnings

warnings.filterwarnings("ignore")
pd.options.display.float_format = '{:.4%}'.format

# Date range
start = '2016-01-01'
end = '2019-12-30'

# Tickers of assets
assets = ['JCI', 'TGT', 'CMCSA', 'CPB', 'MO', 'APA', 'MMC', 'JPM',
          'ZION', 'PSA', 'BAX', 'BMY', 'LUV', 'PCAR', 'TXT', 'TMO',
          'DE', 'MSFT', 'HPQ', 'SEE', 'VZ', 'CNP', 'NI', 'T', 'BA']

assets.sort()

# Tickers of factors

factors = ['MTUM', 'QUAL', 'VLUE', 'SIZE', 'USMV']
factors.sort()

tickers = assets + factors
tickers.sort()

# Downloading data
data = yf.download(tickers, start = start, end = end)
data = data.loc[:,('Adj Close', slice(None))]
data.columns = tickers

[*********************100%***********************]  30 of 30 completed

In [2]:

# Calculating returns

X = data[factors].pct_change().dropna()
Y = data[assets].pct_change().dropna()

display(X.head())

	MTUM	QUAL	SIZE	USMV	VLUE
Date
2016-01-05 00:00:00-05:00	0.4735%	0.2672%	0.0000%	0.6780%	0.1635%
2016-01-06 00:00:00-05:00	-0.5267%	-1.1914%	-0.5380%	-0.6253%	-1.8277%
2016-01-07 00:00:00-05:00	-2.2293%	-2.3798%	-1.7181%	-1.6215%	-2.1609%
2016-01-08 00:00:00-05:00	-0.9549%	-1.1376%	-1.1978%	-1.0086%	-1.0873%
2016-01-11 00:00:00-05:00	0.6043%	0.1479%	-0.5898%	0.1491%	-0.6183%

2. Estimating Mean Variance Portfolios¶

2.1 Estimating the loadings matrix.¶

This part is just to visualize how Riskfolio-Lib calculates a loadings matrix.

In [3]:

import riskfolio as rp

step = 'Forward' # Could be Forward or Backward stepwise regression
loadings = rp.loadings_matrix(X=X, Y=Y, stepwise=step)

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

Out[3]:

	const	MTUM	QUAL	SIZE	USMV	VLUE
APA	-0.0006	-0.6551	0.0000	0.9406	-0.7883	1.7237
BA	0.0005	0.0000	1.1744	0.3616	-0.4322	0.0000
BAX	0.0003	0.3146	0.0000	0.0000	0.7717	0.0000
BMY	-0.0003	0.0000	0.8123	0.0000	0.0000	0.0000
CMCSA	0.0001	0.0000	0.4958	0.0000	0.4962	0.0000
CNP	0.0001	-0.5595	-0.2157	0.0000	1.8341	0.0000
CPB	-0.0003	-0.4782	-0.5993	0.0000	2.0793	0.0000
DE	0.0004	0.0000	0.0000	0.3631	0.0000	0.8090
HPQ	0.0002	0.0000	0.0000	0.0000	0.0000	1.2514
JCI	0.0001	0.0000	0.0000	0.3412	0.0000	0.5797
JPM	0.0005	0.0000	0.3076	0.0000	-0.5485	1.1512
LUV	-0.0001	0.0000	0.0000	0.3642	0.0000	0.6767
MMC	0.0003	-0.2693	0.6808	0.0000	0.6066	0.0000
MO	-0.0004	-0.4572	0.0000	0.0000	1.4359	0.0000
MSFT	0.0005	1.0302	0.6530	-0.2640	-0.2595	0.0000
NI	-0.0000	-0.4649	-0.4611	0.0000	2.3257	-0.3532
PCAR	0.0003	-0.4420	1.0319	0.2217	-0.4831	0.7536
PSA	-0.0004	-0.3311	0.0000	0.0000	1.7073	-0.4915
SEE	-0.0005	-0.3267	0.0000	0.2594	0.6973	0.4801
T	0.0000	-0.4414	-0.3693	-0.2898	1.2928	0.7563
TGT	0.0005	0.0000	0.0000	0.0000	0.0000	0.7562
TMO	0.0002	0.2867	0.5400	0.0000	0.3864	0.0000
TXT	-0.0003	0.0000	0.4485	0.4429	-0.6363	0.8108
VZ	0.0000	-0.5099	-0.5271	-0.2886	1.8941	0.4321
ZION	0.0004	-0.2371	0.0000	0.3554	-0.8203	1.6431

2.2 Calculating the portfolio that maximizes Sharpe ratio.¶

In [4]:

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

# Calculating optimal 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, d=0.94)

port.factors = X
port.factors_stats(method_mu=method_mu, method_cov=method_cov, d=0.94)

# Estimate optimal portfolio:

port.alpha = 0.05
model='FM' # Factor Model
rm = 'MV' # Risk measure used, this time will be variance
obj = 'Sharpe' # Objective function, could be MinRisk, MaxRet, Utility or Sharpe
hist = False # Use historical scenarios for risk measures that depend on scenarios
rf = 0 # Risk free rate
l = 0 # Risk aversion factor, only useful when obj is 'Utility'

w = port.optimization(model=model, rm=rm, obj=obj, rf=rf, l=l, hist=hist)

display(w.T)

(1004, 25)

	APA	BA	BAX	BMY	CMCSA	CNP	CPB	DE	HPQ	JCI	...	NI	PCAR	PSA	SEE	T	TGT	TMO	TXT	VZ	ZION
weights	0.0000%	5.9508%	11.8691%	0.0000%	0.0000%	9.7091%	0.0000%	4.3627%	0.0000%	0.0000%	...	10.4798%	0.0000%	0.0000%	0.0000%	0.0000%	5.5106%	1.3023%	0.0000%	4.0977%	0.0000%

1 rows × 25 columns

2.3 Plotting portfolio composition¶

In [5]:

# Plotting the composition of the portfolio

ax = rp.plot_pie(w=w, title='Sharpe FM Mean Variance', others=0.05, nrow=25, cmap = "tab20",
                 height=6, width=10, ax=None)

2.3 Calculate efficient frontier¶

In [6]:

points = 50 # Number of points of the frontier

frontier = port.efficient_frontier(model=model, rm=rm, points=points, rf=rf, hist=hist)

display(frontier.T.head())

	APA	BA	BAX	BMY	CMCSA	CNP	CPB	DE	HPQ	JCI	...	NI	PCAR	PSA	SEE	T	TGT	TMO	TXT	VZ	ZION
0	0.0000%	0.0000%	1.9915%	4.4203%	3.5982%	8.4947%	5.0090%	0.0000%	0.0000%	4.4340%	...	12.0935%	0.0000%	13.3196%	1.0089%	8.3014%	4.0842%	0.0000%	0.0000%	10.3131%	1.6678%
1	0.0000%	1.6589%	6.2599%	1.2538%	3.4251%	10.4170%	3.1168%	0.3986%	0.0000%	2.5870%	...	13.7067%	0.0000%	7.7369%	0.0000%	6.7067%	4.9334%	0.0000%	0.0000%	10.5792%	1.6507%
2	0.0000%	2.3022%	7.3158%	0.0000%	3.0350%	11.1001%	2.4078%	0.9815%	0.0000%	1.6503%	...	14.3023%	0.0000%	5.5454%	0.0000%	6.0226%	5.1519%	0.0000%	0.0000%	10.6635%	1.4140%
3	0.0000%	2.8523%	8.2124%	0.0000%	2.5287%	11.5853%	1.6264%	1.4954%	0.0000%	0.6300%	...	14.6708%	0.0000%	3.1895%	0.0000%	5.2024%	5.3156%	0.0757%	0.0000%	10.5638%	1.1082%
4	0.0000%	3.3061%	8.9457%	0.0000%	2.0093%	11.9269%	0.8759%	1.9173%	0.0000%	0.0000%	...	14.8951%	0.0000%	0.9663%	0.0000%	4.3810%	5.4380%	0.4388%	0.0000%	10.3772%	0.7801%

5 rows × 25 columns

In [7]:

# Plotting the efficient frontier

label = 'Max Risk Adjusted Return Portfolio' # Title of point
mu = port.mu_fm # Expected returns
cov = port.cov_fm # Covariance matrix
returns = port.returns_fm # Returns of the assets

ax = rp.plot_frontier(w_frontier=frontier, mu=mu, cov=cov, returns=returns, rm=rm,
                      rf=rf, alpha=0.05, cmap='viridis', w=w, label=label,
                      marker='*', s=16, c='r', height=6, width=10, ax=None)

In [8]:

# Plotting efficient frontier composition

ax = rp.plot_frontier_area(w_frontier=frontier, cmap="tab20", height=6, width=10, ax=None)

3. Optimization with Constraints on Risk Factors¶

3.1 Statistics of Risk Factors¶

In [9]:

# Displaying factors statistics

display(loadings.min())
display(loadings.max())
display(X.corr())

const    -0.0583%
MTUM    -65.5125%
QUAL    -59.9324%
SIZE    -28.9796%
USMV    -82.0310%
VLUE    -49.1495%
dtype: float64

const     0.0504%
MTUM    103.0208%
QUAL    117.4359%
SIZE     94.0633%
USMV    232.5675%
VLUE    172.3667%
dtype: float64

	MTUM	QUAL	SIZE	USMV	VLUE
MTUM	100.0000%	90.4264%	79.1173%	87.2320%	78.5394%
QUAL	90.4264%	100.0000%	89.8169%	89.9554%	91.6580%
SIZE	79.1173%	89.8169%	100.0000%	82.5080%	87.9112%
USMV	87.2320%	89.9554%	82.5080%	100.0000%	76.9678%
VLUE	78.5394%	91.6580%	87.9112%	76.9678%	100.0000%

3.2 Creating Constraints on Risk Factors¶

In [10]:

# Creating risk factors constraints

constraints = {'Disabled': [False, False, False, False, False],
               'Factor': ['MTUM', 'QUAL', 'SIZE', 'USMV', 'VLUE'],
               'Sign': ['<=', '<=', '<=', '>=', '<='],
               'Value': [-0.3, 0.8, 0.4, 0.8 , 0.9],
               'Relative Factor': ['', 'USMV', '', '', '']}

constraints = pd.DataFrame(constraints)

display(constraints)

	Disabled	Factor	Sign	Value	Relative Factor
0	False	MTUM	<=	-30.0000%
1	False	QUAL	<=	80.0000%	USMV
2	False	SIZE	<=	40.0000%
3	False	USMV	>=	80.0000%
4	False	VLUE	<=	90.0000%

In [11]:

C, D = rp.factors_constraints(constraints, loadings)

In [12]:

port.ainequality = C
port.binequality = D

w = port.optimization(model=model, rm=rm, obj=obj, rf=rf, l=l, hist=hist)

display(w.T)

	APA	BA	BAX	BMY	CMCSA	CNP	CPB	DE	HPQ	JCI	...	NI	PCAR	PSA	SEE	T	TGT	TMO	TXT	VZ	ZION
weights	0.0000%	6.5843%	1.4896%	0.0000%	0.0000%	17.4848%	0.0000%	4.0876%	0.0000%	0.0000%	...	13.9652%	1.3789%	0.0000%	0.0000%	0.3064%	4.4149%	0.0000%	0.0000%	10.5778%	2.0946%

1 rows × 25 columns

To check if the constraints are verified, I will make a regression among the portfolio returns and risk factors:

In [13]:

import statsmodels.api as sm

X1 = sm.add_constant(X)
y = np.matrix(returns) * np.matrix(w)
results = sm.OLS(y, X1).fit()
coefs = results.params

print(coefs)

const     0.0229%
MTUM    -30.0000%
QUAL     15.3036%
SIZE      1.7736%
USMV     92.6900%
VLUE     21.9835%
dtype: float64

3.3 Plotting portfolio composition¶

In [14]:

ax = rp.plot_pie(w=w, title='Sharpe FM Mean Variance', others=0.05, nrow=25, cmap = "tab20",
                 height=6, width=10, ax=None)

3.4 Calculate efficient frontier¶

In [15]:

points = 50 # Number of points of the frontier

frontier = port.efficient_frontier(model=model, rm=rm, points=points, rf=rf, hist=hist)

display(frontier.T.head())

	APA	BA	BAX	BMY	CMCSA	CNP	CPB	DE	HPQ	JCI	...	NI	PCAR	PSA	SEE	T	TGT	TMO	TXT	VZ	ZION
0	0.0000%	0.0000%	1.9926%	4.4202%	3.5980%	8.4936%	5.0087%	0.0000%	0.0000%	4.4337%	...	12.0932%	0.0000%	13.3197%	1.0130%	8.3001%	4.0840%	0.0000%	0.0000%	10.3121%	1.6674%
1	0.0000%	1.0475%	4.5030%	1.9695%	3.3300%	10.3898%	3.6703%	0.0001%	0.0000%	2.9604%	...	13.4551%	0.0000%	9.0989%	0.0000%	7.4172%	4.6595%	0.0000%	0.0000%	10.8691%	1.9813%
2	0.0000%	1.7533%	4.8601%	0.7459%	2.9456%	11.4506%	3.1270%	0.3774%	0.0000%	2.0699%	...	14.1330%	0.0000%	7.2013%	0.0000%	7.1319%	4.7870%	0.0000%	0.0000%	11.2907%	2.0815%
3	0.0000%	2.2850%	5.0718%	0.0000%	2.6007%	12.3054%	2.6939%	0.7947%	0.0000%	1.3267%	...	14.6720%	0.0000%	5.6643%	0.0000%	6.8992%	4.8726%	0.0000%	0.0000%	11.6306%	2.1410%
4	0.0000%	2.7534%	5.1132%	0.0000%	2.1862%	13.1588%	2.2596%	1.1574%	0.0000%	0.5350%	...	15.1825%	0.0000%	4.1051%	0.0000%	6.6679%	4.9234%	0.0000%	0.0000%	11.9726%	2.1890%

5 rows × 25 columns

In [16]:

# Plotting efficient frontier composition

ax = rp.plot_frontier(w_frontier=frontier, mu=mu, cov=cov, returns=returns, rm=rm,
                      rf=rf, alpha=0.05, cmap='viridis', w=w, label=label,
                      marker='*', s=16, c='r', height=6, width=10, ax=None)

In [17]:

# Plotting efficient frontier composition

ax = rp.plot_frontier_area(w_frontier=frontier, cmap="tab20", height=6, width=10, ax=None)

In [18]:

display(returns)

	APA	BA	BAX	BMY	CMCSA	CNP	CPB	DE	HPQ	JCI	...	NI	PCAR	PSA	SEE	T	TGT	TMO	TXT	VZ	ZION
2016-01-05 00:00:00-05:00	-0.6212%	0.0684%	0.7015%	0.1909%	0.4772%	0.9283%	0.9912%	0.1761%	0.2252%	0.1002%	...	1.1712%	-0.1122%	0.8782%	0.3502%	0.6946%	0.1709%	0.5662%	-0.2137%	0.9753%	-0.3551%
2016-01-06 00:00:00-05:00	-2.8766%	-1.2758%	-0.6189%	-0.9939%	-0.8928%	-0.5879%	-0.3663%	-1.6301%	-2.2664%	-1.2376%	...	-0.0189%	-2.1655%	-0.0371%	-1.3275%	-1.3599%	-1.3348%	-1.0117%	-1.8914%	-0.9195%	-2.5117%
2016-01-07 00:00:00-05:00	-2.6603%	-2.6676%	-1.9233%	-1.9591%	-1.9763%	-1.2059%	-0.9113%	-2.3282%	-2.6834%	-1.8334%	...	-0.8785%	-2.6708%	-1.0103%	-1.9321%	-1.3673%	-1.5868%	-2.5264%	-2.5833%	-1.1151%	-2.2578%
2016-01-08 00:00:00-05:00	-1.6385%	-1.2855%	-1.0494%	-0.9502%	-1.0563%	-1.0629%	-0.9909%	-1.2707%	-1.3400%	-1.0335%	...	-0.9977%	-1.3239%	-0.9137%	-1.2706%	-0.9353%	-0.7750%	-1.2535%	-1.3153%	-0.9453%	-1.1138%
2016-01-11 00:00:00-05:00	-2.1923%	-0.0564%	0.3345%	0.0940%	0.1556%	-0.0892%	-0.0996%	-0.6705%	-0.7531%	-0.5542%	...	0.2116%	-0.7575%	0.3162%	-0.5897%	-0.4231%	-0.4203%	0.3350%	-0.8259%	-0.1979%	-1.4465%
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
2019-12-23 00:00:00-05:00	0.6972%	0.2476%	-0.4231%	-0.0181%	-0.1988%	-0.5600%	-0.7397%	0.1196%	0.1321%	0.0606%	...	-0.8527%	0.4842%	-0.6862%	-0.1714%	-0.3171%	0.1146%	-0.2473%	0.3183%	-0.5772%	0.6376%
2019-12-24 00:00:00-05:00	-0.1965%	0.0028%	0.1506%	-0.0503%	0.0391%	0.0929%	0.1005%	0.0557%	-0.0072%	0.0206%	...	0.1566%	-0.1185%	0.0728%	-0.0238%	0.0208%	0.0304%	0.0896%	-0.0882%	0.0777%	-0.0760%
2019-12-26 00:00:00-05:00	0.0839%	0.3934%	0.3568%	0.2717%	0.3344%	0.2787%	0.1966%	0.3099%	0.3689%	0.2053%	...	0.2526%	0.3537%	0.2094%	0.2125%	0.2761%	0.2577%	0.4284%	0.2202%	0.2812%	0.2258%
2019-12-27 00:00:00-05:00	-0.7229%	-0.0384%	0.2654%	-0.0101%	0.1392%	0.3665%	0.3917%	-0.1305%	-0.2432%	-0.1203%	...	0.5586%	-0.2999%	0.4284%	-0.0293%	0.0874%	-0.1122%	0.1726%	-0.3568%	0.2897%	-0.5416%
2019-12-30 00:00:00-05:00	-0.0583%	0.0476%	0.0293%	-0.0261%	0.0082%	0.0074%	-0.0320%	0.0439%	0.0207%	0.0054%	...	-0.0045%	0.0257%	-0.0422%	-0.0464%	0.0022%	0.0473%	0.0243%	-0.0347%	0.0028%	0.0447%

1004 rows × 25 columns

4. Estimating Portfolios Using Risk Factors with Other Risk Measures¶

In this part I will calculate optimal portfolios for several risk measures. I will find the portfolios that maximize the risk adjusted return for all available risk measures.

4.1 Calculate Optimal Portfolios for Several Risk Measures.¶

I will mantain the constraints on risk factors.

In [19]:

# 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.

# port.reset_linear_constraints() # To reset linear constraints (factor constraints)

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

w_s = pd.DataFrame([])

# When we use hist = True the risk measures all calculated
# using historical returns, while when hist = False the
# risk measures are calculated using the expected returns 
#  based on risk factor model: R = a + B * F

hist = False
for i in rms:
    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 [20]:

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

Out[20]:

	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%
BA	6.58%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
BAX	1.49%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
BMY	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	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%
CNP	17.48%	42.32%	53.62%	41.97%	53.62%	41.32%	53.62%	48.75%	0.00%	46.96%	23.97%	46.45%	48.69%
CPB	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%
DE	4.09%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
HPQ	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%
JCI	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%
JPM	9.54%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
LUV	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%
MMC	28.07%	23.48%	0.00%	24.20%	0.00%	0.00%	0.00%	0.00%	0.00%	13.82%	61.60%	14.88%	0.00%
MO	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%
MSFT	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%
NI	13.97%	0.00%	0.00%	0.00%	0.00%	14.80%	0.00%	0.00%	64.54%	0.00%	0.00%	0.00%	5.93%
PCAR	1.38%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
PSA	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%
SEE	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%
T	0.31%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
TGT	4.41%	34.20%	46.38%	33.83%	46.38%	43.88%	46.38%	39.77%	35.46%	39.21%	14.43%	38.66%	45.38%
TMO	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%
TXT	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%
VZ	10.58%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
ZION	2.09%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	11.48%	0.00%	0.00%	0.00%	0.00%	0.00%

In [21]:

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[21]:

<AxesSubplot:>

In [22]:

w_s = pd.DataFrame([])

# When we use hist = True the risk measures all calculated
# using historical returns, while when hist = False the
# risk measures are calculated using the expected returns 
#  based on risk factor model: R = a + B * F

hist = True
for i in rms:
    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 [23]:

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

Out[23]:

	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%
BA	7.45%	6.99%	8.25%	6.48%	8.43%	8.52%	8.49%	0.00%	7.16%	13.63%	16.38%	15.08%	8.71%
BAX	1.03%	1.30%	0.36%	0.95%	0.22%	3.48%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
BMY	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	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%
CNP	17.82%	13.25%	17.50%	12.81%	17.98%	21.30%	31.96%	29.80%	55.32%	12.02%	27.50%	16.60%	44.54%
CPB	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.10%	7.20%	0.00%	0.00%	0.00%	0.00%	0.00%
DE	3.77%	1.99%	2.68%	2.93%	2.82%	3.09%	1.73%	0.00%	5.45%	0.38%	0.00%	1.08%	0.00%
HPQ	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%
JCI	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%
JPM	7.34%	10.09%	6.11%	10.54%	5.68%	0.00%	0.00%	7.70%	0.00%	1.62%	0.00%	3.72%	0.00%
LUV	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%
MMC	32.90%	31.48%	35.58%	30.04%	35.91%	36.88%	44.83%	33.51%	15.77%	36.47%	34.25%	36.34%	24.80%
MO	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%
MSFT	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.96%	8.42%	0.00%	3.35%	0.00%	6.91%
NI	11.40%	14.37%	11.04%	15.88%	10.64%	6.20%	0.00%	0.00%	0.00%	20.22%	9.44%	14.85%	5.47%
PCAR	0.89%	2.93%	0.60%	2.70%	0.34%	0.00%	0.00%	4.18%	7.87%	1.22%	0.80%	0.00%	6.73%
PSA	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%
SEE	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%
T	0.00%	1.59%	0.00%	0.15%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	1.62%	0.00%
TGT	6.02%	4.56%	7.40%	4.97%	7.70%	8.05%	12.90%	16.64%	0.00%	3.79%	0.25%	2.60%	2.84%
TMO	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%
TXT	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%
VZ	11.37%	11.46%	10.48%	12.54%	10.28%	12.48%	0.00%	0.00%	0.00%	3.63%	8.03%	5.08%	0.00%
ZION	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	7.02%	0.00%	3.04%	0.00%

In [24]:

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[24]:

<AxesSubplot:>