Hierarchical Equal Risk Contribution - CDaR

This tutorial introduces the HierarchicalEqualRiskContribution optimization.

The Hierarchical Equal Risk Contribution (HERC) is a portfolio optimization method developed by Thomas Raffinot.

This algorithm uses a distance matrix to compute hierarchical clusters using the Hierarchical Tree Clustering algorithm. It then computes, for each cluster, the total cluster risk of an inverse-risk allocation.

The final step is the top-down recursive division of the dendrogram, where the assets weights are updated using a naive risk parity within clusters.

It differs from the Hierarchical Risk Parity by exploiting the dendrogram shape during the top-down recursive division instead of bisecting it.

Note

The default linkage method is set to the Ward variance minimization algorithm, which is more stable and has better properties than the single-linkage method

In this example, we will use the CDaR 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 2020-01-02 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.cluster import HierarchicalClustering, LinkageMethod
from skfolio.datasets import load_sp500_dataset
from skfolio.distance import KendallDistance
from skfolio.optimization import (
    EqualWeighted,
    HierarchicalEqualRiskContribution,
)
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=0.5, shuffle=False)

Model

We create a CVaR Hierarchical Equal Risk Contribution model and then fit it on the training set:

model1 = HierarchicalEqualRiskContribution(
    risk_measure=RiskMeasure.CDAR, portfolio_params=dict(name="HERC-CDaR-Ward-Pearson")
)
model1.fit(X_train)
model1.weights_

array([0.05814153, 0.04927288, 0.09420898, 0.05726203, 0.02409012,
       0.08083036, 0.08070112, 0.02983878, 0.06656721, 0.02730592,
       0.01757816, 0.01140416, 0.0933616 , 0.04726087, 0.02170548,
       0.02428936, 0.00827991, 0.02575021, 0.14895433, 0.03319698])

Risk Contribution

Let’s analyze the risk contribution of the model on the training set:

ptf1 = model1.predict(X_train)
ptf1.plot_contribution(measure=RiskMeasure.CDAR)

Dendrogram

To analyze the clusters structure, we plot the dendrogram. The blue lines represent distinct clusters composed of a single asset. The remaining colors represent clusters of more than one asset:

fig = model1.hierarchical_clustering_estimator_.plot_dendrogram(heatmap=False)
show(fig)

The horizontal axis represents the assets. The links between clusters are represented as upside-down U-shaped lines. The height of the U indicates the distance between the clusters. For example, the link representing the cluster containing Assets HD and WMT has a distance of 0.5 (called cophenetic distance).

When heatmap is set to True, the heatmap of the reordered distance matrix is displayed below the dendrogram and clusters are outlined with yellow squares:

model1.hierarchical_clustering_estimator_.plot_dendrogram()

Linkage Methods

The clustering can be greatly affected by the choice of the linkage method. In the HierarchicalEqualRiskContribution estimator, the default linkage method is set to the Ward variance minimization algorithm which is more stable and has better properties than the single-linkage method, which suffers from the chaining effect.

And because HERC rely on the dendrogram structure as opposed to HRP, the choice of the linkage method will have a greater impact on the allocation.

To show this effect, let’s create a second model with the single-linkage method:

model2 = HierarchicalEqualRiskContribution(
    risk_measure=RiskMeasure.CDAR,
    hierarchical_clustering_estimator=HierarchicalClustering(
        linkage_method=LinkageMethod.SINGLE,
    ),
    portfolio_params=dict(name="HERC-CDaR-Single-Pearson"),
)
model2.fit(X_train)
model2.hierarchical_clustering_estimator_.plot_dendrogram(heatmap=True)

We can see that the clustering has been greatly affected by the change of the linkage method. Let’s analyze the risk contribution of this model on the training set:

ptf2 = model2.predict(X_train)
ptf2.plot_contribution(measure=RiskMeasure.CDAR)

The risk of that second model is very concentrated. We can already conclude that the single-linkage method is not appropriate for this dataset. This will be confirmed below on the test set.

Distance Estimator

The distance metric used has also an important effect on the clustering. The default is to use the distance of the pearson correlation matrix. This can be changed using the distance estimators. For example, let’s create a third model with a distance computed from the absolute value of the Kendal correlation matrix:

model3 = HierarchicalEqualRiskContribution(
    risk_measure=RiskMeasure.CDAR,
    distance_estimator=KendallDistance(absolute=True),
    portfolio_params=dict(name="HERC-CDaR-Ward-Kendal"),
)
model3.fit(X_train)
model3.hierarchical_clustering_estimator_.plot_dendrogram(heatmap=True)

To compare the models, we use an equal weighted benchmark using the EqualWeighted estimator:

bench = EqualWeighted()
bench.fit(X_train)
bench.weights_

array([0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05,
       0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05])

Prediction

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

population_test = Population([])
for model in [model1, model2, model3, bench]:
    population_test.append(model.predict(X_test))

population_test.plot_cumulative_returns()

Composition

From the below composition, we notice that the model with single-linkage method is highly concentrated:

population_test.plot_composition()