Risk Parity - Covariance shrinkage

This tutorial shows how to incorporate covariance shrinkage in the RiskBudgeting optimization.

Data

We load the S&P 500 dataset composed of the daily prices of 20 assets from the S&P 500 Index composition starting from 1990-01-02 up to 2022-12-28:

from plotly.io import show
from sklearn.model_selection import train_test_split

from skfolio import Population, RiskMeasure
from skfolio.datasets import load_sp500_dataset
from skfolio.moments import ShrunkCovariance
from skfolio.optimization import RiskBudgeting
from skfolio.preprocessing import prices_to_returns
from skfolio.prior import EmpiricalPrior

prices = load_sp500_dataset()

X = prices_to_returns(prices)
X_train, X_test = train_test_split(X, test_size=0.33, shuffle=False)

Model

We create a risk parity model by using ShrunkCovariance as the covariance estimator then fit it on the training set:

model = RiskBudgeting(
    risk_measure=RiskMeasure.VARIANCE,
    prior_estimator=EmpiricalPrior(
        covariance_estimator=ShrunkCovariance(shrinkage=0.9)
    ),
    portfolio_params=dict(name="Risk Parity - Covariance Shrinkage"),
)
model.fit(X_train)
model.weights_

array([0.04774388, 0.04370282, 0.04503242, 0.04647739, 0.05284608,
       0.04907465, 0.0485292 , 0.05373939, 0.04539524, 0.0536071 ,
       0.05178498, 0.0513796 , 0.04927133, 0.05375727, 0.05112687,
       0.05417611, 0.04755307, 0.0498814 , 0.05199177, 0.05292942])

To compare this model, we use a basic risk parity without covariance shrinkage:

bench = RiskBudgeting(
    risk_measure=RiskMeasure.VARIANCE,
    portfolio_params=dict(name="Risk Parity - Basic"),
)
bench.fit(X_train)
bench.weights_

array([0.04135234, 0.03210845, 0.03372587, 0.03785082, 0.06105296,
       0.04432849, 0.04252282, 0.06593432, 0.03451821, 0.06469294,
       0.05418924, 0.05209364, 0.04535247, 0.06568089, 0.05104083,
       0.06894458, 0.04046576, 0.04667751, 0.05627127, 0.0611966 ])

Prediction

We predict the model and the benchmark on the test set:

ptf_model_test = model.predict(X_test)
ptf_bench_test = bench.predict(X_test)

Analysis

For improved analysis, it’s possible to load both predicted portfolios into a Population:

population = Population([ptf_model_test, ptf_bench_test])

Let’s plot each portfolio cumulative returns:

fig = population.plot_cumulative_returns()
show(fig)

Finally, we print a full summary of both strategies evaluated on the test set:

population.summary()

	Risk Parity - Covariance Shrinkage	Risk Parity - Basic
Mean	0.068%	0.065%
Annualized Mean	17.04%	16.40%
Variance	0.011%	0.010%
Annualized Variance	2.84%	2.63%
Semi-Variance	0.0058%	0.0054%
Annualized Semi-Variance	1.47%	1.36%
Standard Deviation	1.06%	1.02%
Annualized Standard Deviation	16.86%	16.22%
Semi-Deviation	0.76%	0.73%
Annualized Semi-Deviation	12.11%	11.66%
Mean Absolute Deviation	0.69%	0.66%
CVaR at 95%	2.46%	2.37%
EVaR at 95%	5.37%	5.23%
Worst Realization	10.66%	10.34%
CDaR at 95%	13.02%	12.39%
MAX Drawdown	34.58%	34.30%
Average Drawdown	2.58%	2.40%
EDaR at 95%	20.30%	19.86%
First Lower Partial Moment	0.35%	0.33%
Ulcer Index	0.043	0.041
Gini Mean Difference	1.04%	1.00%
Value at Risk at 95%	1.51%	1.46%
Drawdown at Risk at 95%	9.30%	8.97%
Entropic Risk Measure at 95%	3.00	3.00
Fourth Central Moment	0.000025%	0.000023%
Fourth Lower Partial Moment	0.000012%	0.000011%
Skew	-3.68%	-5.68%
Kurtosis	1998.27%	2067.02%
Sharpe Ratio	0.064	0.064
Annualized Sharpe Ratio	1.01	1.01
Sortino Ratio	0.089	0.089
Annualized Sortino Ratio	1.41	1.41
Mean Absolute Deviation Ratio	0.097	0.098
First Lower Partial Moment Ratio	0.19	0.20
Value at Risk Ratio at 95%	0.045	0.045
CVaR Ratio at 95%	0.027	0.027
Entropic Risk Measure Ratio at 95%	0.00023	0.00022
EVaR Ratio at 95%	0.013	0.012
Worst Realization Ratio	0.0063	0.0063
Drawdown at Risk Ratio at 95%	0.0073	0.0073
CDaR Ratio at 95%	0.0052	0.0053
Calmar Ratio	0.0020	0.0019
Average Drawdown Ratio	0.026	0.027
EDaR Ratio at 95%	0.0033	0.0033
Ulcer Index Ratio	0.016	0.016
Gini Mean Difference Ratio	0.065	0.065
Effective Number of Assets	19.921054710361332	18.995845702541676
Assets Number	20	20