Risk Parity - Variance

This tutorial uses the RiskBudgeting optimization to find the risk parity portfolio with variance as the risk measure.

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.optimization import InverseVolatility, RiskBudgeting
from skfolio.preprocessing import prices_to_returns

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 the risk parity model and then fit it on the training set:

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

array([0.04135261, 0.03210859, 0.03372588, 0.03785038, 0.06105344,
       0.04432799, 0.04252186, 0.06593408, 0.03451951, 0.06469301,
       0.05418934, 0.05209466, 0.04535479, 0.06568174, 0.05104143,
       0.06894357, 0.0404652 , 0.04667615, 0.05627036, 0.06119539])

To compare this model, we use an inverse volatility benchmark using the InverseVolatility estimator.

bench = InverseVolatility(portfolio_params=dict(name="Inverse Vol"))
bench.fit(X_train)
bench.weights_

array([0.03306735, 0.02548697, 0.03551377, 0.0296872 , 0.06358463,
       0.05434705, 0.04742354, 0.07049715, 0.03882539, 0.06697905,
       0.05570808, 0.05576851, 0.04723274, 0.06351213, 0.05581397,
       0.0676481 , 0.02564642, 0.03970752, 0.05744543, 0.06610498])

Risk Contribution Analysis

Let’s analyze the risk contribution of both models on the training set. As expected, the risk parity model has the same variance contribution for each asset:

ptf_model_train = model.predict(X_train)
ptf_model_train.plot_contribution(measure=RiskMeasure.ANNUALIZED_VARIANCE)

And the inverse volatility model has non-equal variance contribution. This is because the correlation is not taken into account in an inverse volatility model:

ptf_bench_train = bench.predict(X_train)
ptf_bench_train.plot_contribution(measure=RiskMeasure.ANNUALIZED_VARIANCE)

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)

The predict method returns a Portfolio object.

Analysis

For improved analysis, we load both predicted portfolios into a Population:

population = Population([ptf_model_test, ptf_bench_test])

Let’s plot each portfolio composition:

population.plot_composition()

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 - Variance	Inverse Vol
Mean	0.065%	0.064%
Annualized Mean	16.40%	16.06%
Variance	0.010%	0.010%
Annualized Variance	2.63%	2.56%
Semi-Variance	0.0054%	0.0053%
Annualized Semi-Variance	1.36%	1.33%
Standard Deviation	1.02%	1.01%
Annualized Standard Deviation	16.22%	16.00%
Semi-Deviation	0.73%	0.73%
Annualized Semi-Deviation	11.66%	11.54%
Mean Absolute Deviation	0.66%	0.65%
CVaR at 95%	2.37%	2.35%
EVaR at 95%	5.23%	5.29%
Worst Realization	10.34%	10.49%
CDaR at 95%	12.39%	12.22%
MAX Drawdown	34.30%	34.83%
Average Drawdown	2.40%	2.34%
EDaR at 95%	19.86%	20.05%
First Lower Partial Moment	0.33%	0.32%
Ulcer Index	0.041	0.040
Gini Mean Difference	1.00%	0.98%
Value at Risk at 95%	1.46%	1.45%
Drawdown at Risk at 95%	8.97%	8.79%
Entropic Risk Measure at 95%	3.00	3.00
Fourth Central Moment	0.000023%	0.000022%
Fourth Lower Partial Moment	0.000011%	0.000011%
Skew	-5.68%	-14.94%
Kurtosis	2065.52%	2117.02%
Sharpe Ratio	0.064	0.063
Annualized Sharpe Ratio	1.01	1.00
Sortino Ratio	0.089	0.088
Annualized Sortino Ratio	1.41	1.39
Mean Absolute Deviation Ratio	0.098	0.098
First Lower Partial Moment Ratio	0.20	0.20
Value at Risk Ratio at 95%	0.045	0.044
CVaR Ratio at 95%	0.027	0.027
Entropic Risk Measure Ratio at 95%	0.00022	0.00021
EVaR Ratio at 95%	0.012	0.012
Worst Realization Ratio	0.0063	0.0061
Drawdown at Risk Ratio at 95%	0.0073	0.0073
CDaR Ratio at 95%	0.0053	0.0052
Calmar Ratio	0.0019	0.0018
Average Drawdown Ratio	0.027	0.027
EDaR Ratio at 95%	0.0033	0.0032
Ulcer Index Ratio	0.016	0.016
Gini Mean Difference Ratio	0.065	0.065
Effective Number of Assets	18.99586745481971	18.460872007821077
Assets Number	20	20