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 38: Mean Kurtosis Optimization¶

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', 'AMZN', 'CMCSA', 'CPB', 'MO', 'APA', 'MMC', 'JPM',
          'ZION', 'AAPL', 'BAX', 'BMY', 'LUV', 'PCAR', 'TXT', 'TMO',
          'DE', 'MSFT', 'HPQ', 'SEE', 'VZ', 'CNP', 'NI', 'T', 'BA']
assets.sort()

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

[*********************100%***********************]  25 of 25 completed

In [2]:

# Calculating returns

Y = data[assets].iloc[-300:,:].pct_change().dropna()

display(Y.head())

	AAPL	AMZN	APA	BA	BAX	BMY	CMCSA	CNP	CPB	DE	...	MO	MSFT	NI	PCAR	SEE	T	TMO	TXT	VZ	ZION
Date
2018-10-19 00:00:00+00:00	1.5230%	-0.3778%	0.0475%	-0.8599%	-1.4333%	-3.0011%	0.1113%	1.2968%	3.4360%	-0.8764%	...	1.6741%	0.1475%	0.6339%	-0.1823%	-0.7728%	1.1384%	-1.1145%	-1.2872%	0.4574%	-0.8025%
2018-10-22 00:00:00+00:00	0.6110%	1.4325%	-1.9240%	-0.0786%	-0.6334%	-6.2983%	-0.6392%	-1.1025%	0.0528%	-0.3221%	...	-1.0331%	0.8927%	-0.8661%	0.4483%	-2.8972%	-0.6084%	-0.6075%	-0.8635%	0.1457%	-3.4490%
2018-10-23 00:00:00+00:00	0.9427%	-1.1513%	-3.6571%	-1.6658%	-0.4202%	-0.4521%	-0.2797%	-0.5034%	0.1844%	-3.9948%	...	0.8808%	-1.3956%	0.4766%	-5.1240%	-0.0321%	1.0713%	-1.0807%	-1.8308%	4.0560%	4.0353%
2018-10-24 00:00:00+00:00	-3.4302%	-5.9083%	-4.5501%	1.3141%	-1.8042%	-3.5933%	-4.2917%	0.8674%	0.9995%	-4.1108%	...	0.7437%	-5.3469%	3.5178%	-4.2683%	-1.3479%	-8.0558%	-1.2403%	-4.2187%	0.3671%	-3.3065%
2018-10-25 00:00:00+00:00	2.1898%	7.0887%	0.4741%	2.5716%	0.5186%	0.7782%	5.0410%	-0.5733%	-1.1718%	2.1585%	...	1.3642%	5.8444%	-1.0310%	0.4914%	0.9109%	-1.2516%	4.3662%	1.3800%	-1.7241%	3.3538%

5 rows × 25 columns

2. Estimating Mean Kurtosis Portfolios¶

The Kurtosis portfolio model proposed by Cajas (2022) shows how to optimize the fourth moment of portfolio returns in a similar way than portfolio variance.

It is recommended to use MOSEK to optimize Kurtosis for a large number of assets due the model use semidefinite programming. Also, for a large number of assets is recommended to use the relaxed version of this model based only on second order cone programming. To use the relaxed version we have to use a number of assets higher than the property n_max_kurt, so for example if number of assets is 30 and we set port.n_max_kurt = 25, riskfolio-lib is going to use the relaxed version.

Instructions to install MOSEK are in this link, is better to install using Anaconda. Also you will need a license, I recommend you that ask for an academic license here.

2.1 Calculating the portfolio that optimize return/kurtosis ratio.¶

In [3]:

import riskfolio as rp
import mosek

# 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.
method_kurt='hist' # Method to estimate cokurtosis square matrix based on historical data.

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

port.solvers = ['MOSEK'] # It is recommended to use mosek when optimizing Kurtosis

# Estimate optimal portfolio:
model ='Classic' # Could be Classic (historical), BL (Black Litterman) or FM (Factor Model)
rm = 'KT' # Risk measure used, this time will be Tail Gini Range
obj = 'Sharpe' # Objective function, could be MinRisk, MaxRet, Utility or Sharpe
hist = True # 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)

You must convert self.kurt to a positive definite matrix
You must convert self.skurt to a positive definite matrix

	AAPL	AMZN	APA	BA	BAX	BMY	CMCSA	CNP	CPB	DE	...	MO	MSFT	NI	PCAR	SEE	T	TMO	TXT	VZ	ZION
weights	0.0000%	0.0000%	0.0000%	0.0000%	0.0000%	4.7028%	0.0000%	0.0000%	11.1102%	0.0000%	...	0.0000%	0.0000%	2.9795%	5.4946%	8.0196%	4.8792%	1.2582%	0.0000%	23.3527%	0.0000%

1 rows × 25 columns

2.2 Plotting portfolio composition¶

In [4]:

# Plotting the composition of the portfolio

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

2.3 Plotting risk measures¶

In [5]:

ax = rp.plot_hist(returns=Y,
                  w=w,
                  alpha=0.05,
                  bins=50,
                  height=6,
                  width=10,
                  ax=None)

2.4 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())

	AAPL	AMZN	APA	BA	BAX	BMY	CMCSA	CNP	CPB	DE	...	MO	MSFT	NI	PCAR	SEE	T	TMO	TXT	VZ	ZION
0	0.0000%	0.0000%	0.0000%	0.0000%	0.6894%	6.6516%	4.7593%	17.4514%	0.0000%	0.0000%	...	11.7019%	0.0000%	10.5489%	0.0000%	8.5918%	0.5055%	0.0000%	0.0000%	26.1992%	3.3163%
1	0.0000%	0.0000%	0.0000%	0.0000%	0.0000%	6.6966%	1.6972%	6.4671%	3.6874%	0.0000%	...	0.9022%	0.0000%	10.1653%	0.0000%	10.1175%	4.9220%	0.0000%	0.0000%	27.3730%	0.0000%
2	0.0000%	0.0000%	0.0000%	0.0000%	0.0000%	6.1761%	0.0000%	0.0000%	7.7140%	0.0000%	...	0.0000%	0.0000%	8.0389%	1.2117%	9.4790%	5.8509%	0.0000%	0.0000%	26.0669%	0.0000%
3	0.0000%	0.0000%	0.0000%	0.0000%	0.0000%	4.9226%	0.0000%	0.0000%	10.8113%	0.0000%	...	0.0000%	0.0000%	3.4604%	5.0189%	8.1531%	4.9600%	0.8984%	0.0000%	23.6468%	0.0000%
4	0.0000%	0.0000%	0.0000%	0.0000%	0.0000%	3.4452%	0.0000%	0.0000%	12.6920%	0.0000%	...	0.0000%	0.0000%	0.5352%	8.2394%	7.3098%	4.4609%	3.2102%	0.0000%	21.7050%	0.0000%

5 rows × 25 columns

In [7]:

# Plotting the efficient frontier

label = 'Max Risk Adjusted Return Portfolio' # Title of point
mu = port.mu # Expected returns
cov = port.cov # Covariance matrix
returns = port.returns # 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. Estimating Risk Parity Portfolios for Square Root Kurtosis¶

3.1 Calculating the risk parity portfolio for Square Root Kurtosis.¶

In [9]:

b = None # Risk contribution constraints vector

w_rp = port.rp_optimization(model=model, rm=rm, rf=rf, b=b, hist=hist)

display(w_rp.T)

	AAPL	AMZN	APA	BA	BAX	BMY	CMCSA	CNP	CPB	DE	...	MO	MSFT	NI	PCAR	SEE	T	TMO	TXT	VZ	ZION
weights	2.4551%	2.3390%	2.3561%	2.9710%	3.8989%	4.2350%	4.0183%	6.7929%	5.1240%	2.6398%	...	5.7290%	2.7138%	7.0910%	3.2300%	4.6015%	4.4500%	3.2760%	3.3235%	7.4288%	3.4346%

1 rows × 25 columns

3.2 Plotting portfolio composition¶

In [10]:

ax = rp.plot_pie(w=w_rp,
                 title='Risk Parity Square Root Kurtosis',
                 others=0.05,
                 nrow=25,
                 cmap="tab20",
                 height=6,
                 width=10,
                 ax=None)

3.3 Plotting Risk Composition¶

In [11]:

ax = rp.plot_risk_con(w_rp, cov=port.cov, returns=port.returns, rm=rm, rf=0, alpha=0.05,
                      color="tab:blue", height=6, width=10, ax=None)

In [12]:

# Plotting the efficient frontier
ws = pd.concat([w, w_rp],axis=1)
ws.columns = ["Max Return/Kurtosis", "Risk Parity Kurtosis"]

mu = port.mu # Expected returns
cov = port.cov # Covariance matrix
returns = port.returns # 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=ws,
                      marker='*', s=16, height=6, width=10, ax=None)