Riskfolio-Lib Tutorial:¶

Financionerioncios
Orenji
Riskfolio-Lib
Dany Cajas

Tutorial 39: Mean Semi 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	1.5230%	-0.3778%	0.0475%	-0.8599%	-1.4332%	-3.0011%	0.1113%	1.2968%	3.4360%	-0.8763%	...	1.6740%	0.1475%	0.6339%	-0.1823%	-0.7728%	1.1385%	-1.1145%	-1.2872%	0.4575%	-0.8025%
2018-10-22	0.6110%	1.4325%	-1.9240%	-0.0786%	-0.6335%	-6.2983%	-0.6393%	-1.1024%	0.0527%	-0.3221%	...	-1.0331%	0.8927%	-0.8661%	0.4483%	-2.8972%	-0.6084%	-0.6075%	-0.8634%	0.1457%	-3.4490%
2018-10-23	0.9427%	-1.1513%	-3.6571%	-1.6658%	-0.4202%	-0.4520%	-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	-3.4302%	-5.9083%	-4.5500%	1.3141%	-1.8042%	-3.5933%	-4.2917%	0.8674%	0.9995%	-4.1109%	...	0.7437%	-5.3469%	3.5178%	-4.2683%	-1.3479%	-8.0557%	-1.2403%	-4.2187%	0.3671%	-3.3065%
2018-10-25	2.1898%	7.0887%	0.4741%	2.5716%	0.5186%	0.7782%	5.0411%	-0.5732%	-1.1719%	2.1585%	...	1.3642%	5.8444%	-1.0309%	0.4914%	0.9109%	-1.2516%	4.3662%	1.3799%	-1.7241%	3.3538%

5 rows × 25 columns

2. Estimating Mean Semi Kurtosis Portfolios¶

The Semi Kurtosis portfolio model proposed by Cajas (2022) shows how to optimize the fourth lower moment of portfolio returns using an lower semi cokurtosis matrix as an heuristic in a similar way than lower semi covariance matrix.

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/semi 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,
                  )

# Estimate optimal portfolio:

port.solvers = ['MOSEK'] # It is recommended to use mosek when optimizing GMD
port.sol_params = {'MOSEK': {'mosek_params': {'MSK_IPAR_NUM_THREADS': 2}}}

model ='Classic' # Could be Classic (historical), BL (Black Litterman) or FM (Factor Model)
rm = 'SKT' # 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%	2.2933%	1.5069%	0.0000%	10.1992%	0.0000%	...	0.0000%	0.0000%	0.0000%	4.7230%	5.3222%	0.0000%	5.1388%	0.0000%	27.6922%	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())

The problem doesn't have a solution with actual input parameters

	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%	1.4624%	6.3368%	12.0102%	16.6038%	0.0000%	0.0000%	...	10.2422%	0.0000%	5.6465%	0.0000%	8.8630%	0.0000%	0.0000%	0.0000%	28.2496%	0.0000%
1	0.0000%	0.0000%	0.0000%	0.0000%	1.0674%	5.9072%	11.2322%	11.7571%	0.0000%	0.0000%	...	5.6054%	0.0000%	5.9885%	0.0000%	9.2228%	0.0000%	0.0000%	0.0000%	30.2239%	0.0000%
2	0.0000%	0.0000%	0.0000%	0.0000%	0.5698%	5.4444%	8.8890%	4.5476%	0.0802%	0.0000%	...	0.0000%	0.0000%	7.4434%	0.0000%	9.7238%	0.0000%	0.0000%	0.0000%	33.1218%	0.0000%
3	0.0000%	0.0000%	0.0000%	0.0000%	0.1879%	5.4249%	6.5311%	0.0001%	4.9114%	0.0000%	...	0.0000%	0.0000%	4.7880%	0.0000%	8.6779%	0.0000%	0.0005%	0.0000%	33.0473%	0.0000%
4	0.0000%	0.0000%	0.0000%	0.0000%	0.0001%	4.4636%	4.2342%	0.0000%	8.1199%	0.0000%	...	0.0000%	0.0000%	0.0001%	0.0000%	6.8826%	0.0000%	2.0224%	0.0000%	32.0412%	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 Semi Kurtosis¶

3.1 Calculating the risk parity portfolio for Square Root Semi 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	2.5334%	2.6996%	2.4343%	3.6101%	4.2748%	4.0051%	4.0332%	6.4981%	4.6227%	2.5405%	...	5.6576%	2.9168%	5.8968%	3.4460%	4.6167%	3.9412%	3.5702%	3.2022%	7.6373%	3.4864%

1 rows × 25 columns

3.2 Plotting portfolio composition¶

In [10]:

ax = rp.plot_pie(w=w_rp,
                 title='Risk Parity Square Root Semi 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)

Due to semi kurtosis use an heuristic based on lower semi cokurtosis matrix to approach portfolio lower semi kurtosis, the solution is only an approximation.

In [12]:

# Plotting the efficient frontier
ws = pd.concat([w, w_rp],axis=1)
ws.columns = ["Max Return/ Semi Kurtosis", "Risk Parity Semi 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, c='r', height=6, width=10, ax=None)