skfolio.optimization.HierarchicalEqualRiskContribution#

class skfolio.optimization.HierarchicalEqualRiskContribution(risk_measure=Variance, prior_estimator=None, distance_estimator=None, hierarchical_clustering_estimator=None, min_weights=0.0, max_weights=1.0, solver='CLARABEL', solver_params=None, transaction_costs=0.0, management_fees=0.0, previous_weights=None, portfolio_params=None)[source]#

Hierarchical Equal Risk Contribution estimator.

The Hierarchical Equal Risk Contribution is a portfolio optimization method developed by Thomas Raffinot [2].

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 [4].

Also, the initial paper does not provide an algorithm for handling weight constraints, and no standard solution currently exists. In contrast to HRP (Hierarchical Risk Parity), where weight constraints can be applied to the split factor at each bisection step, HERC (Hierarchical Equal Risk Contribution) cannot incorporate weight constraints during the intermediate steps of the allocation. Therefore, in HERC, the weight constraints must be enforced after the top-down allocation has been completed. In skfolio, we minimize the relative deviation of the final weights from the initial weights. This is formulated as a convex optimization problem:

\[\begin{split}\begin{cases} \begin{aligned} &\min_{w} & & \Vert \frac{w - w_{init}}{w_{init}} \Vert_{2}^{2} \\ &\text{s.t.} & & \sum_{i=1}^{N} w_{i} = 1 \\ & & & w_{min} \leq w_i \leq w_{max}, \quad \forall i \end{aligned} \end{cases}\end{split}\]

The reason for minimizing the relative deviation (as opposed to the absolute deviation) is that we want to limit the impact on the risk contribution of each asset. Since HERC allocates inversely to risk, adjusting the weights based on relative deviation ensures that the assets’ risk contributions remain proportionally consistent with the initial allocation.

Parameters:
risk_measureRiskMeasure or ExtraRiskMeasure, default=RiskMeasure.VARIANCE

RiskMeasure or ExtraRiskMeasure of the optimization. Can be any of:

  • MEAN_ABSOLUTE_DEVIATION

  • FIRST_LOWER_PARTIAL_MOMENT

  • VARIANCE

  • SEMI_VARIANCE

  • CVAR

  • EVAR

  • WORST_REALIZATION

  • CDAR

  • MAX_DRAWDOWN

  • AVERAGE_DRAWDOWN

  • EDAR

  • ULCER_INDEX

  • GINI_MEAN_DIFFERENCE_RATIO

  • VALUE_AT_RISK

  • DRAWDOWN_AT_RISK

  • ENTROPIC_RISK_MEASURE

  • FOURTH_CENTRAL_MOMENT

  • FOURTH_LOWER_PARTIAL_MOMENT

The default is RiskMeasure.VARIANCE.

prior_estimatorBasePrior, optional

Prior estimator. The prior estimator is used to estimate the PriorModel containing the estimation of assets expected returns, covariance matrix and returns. The moments and returns estimations are used for the risk computation and the returns estimation are used by the distance matrix estimator. The default (None) is to use EmpiricalPrior.

distance_estimatorBaseDistance, optional

Distance estimator. The distance estimator is used to estimate the codependence and the distance matrix needed for the computation of the linkage matrix. The default (None) is to use PearsonDistance.

hierarchical_clustering_estimatorHierarchicalClustering, optional

Hierarchical Clustering estimator. The hierarchical clustering estimator is used to compute the linkage matrix and the hierarchical clustering of the assets based on the distance matrix. The default (None) is to use HierarchicalClustering.

min_weightsfloat | dict[str, float] | array-like of shape (n_assets, ), default=0.0

Minimum assets weights (weights lower bounds). Negative weights are not allowed. If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset minium weight) and the input X of the fit methods must be a DataFrame with the assets names in columns. When using a dictionary, assets values that are not provided are assigned a minimum weight of 0.0. The default is 0.0 (no short selling).

Example:

  • min_weights = 0 –> long only portfolio (no short selling).

  • min_weights = None –> no lower bound (same as -np.Inf).

  • min_weights = {“SX5E”: 0, “SPX”: 0.1}

  • min_weights = [0, 0.1]

max_weightsfloat | dict[str, float] | array-like of shape (n_assets, ), default=1.0

Maximum assets weights (weights upper bounds). Weights above 1.0 are not allowed. If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset maximum weight) and the input X of the fit method must be a DataFrame with the assets names in columns. When using a dictionary, assets values that are not provided are assigned a minimum weight of 1.0. The default is 1.0 (each asset is below 100%).

Example:

  • max_weights = 0 –> no long position (short only portfolio).

  • max_weights = 0.5 –> each weight must be below 50%.

  • max_weights = {“SX5E”: 1, “SPX”: 0.25}

  • max_weights = [1, 0.25]

transaction_costsfloat | dict[str, float] | array-like of shape (n_assets, ), default=0.0

Transaction costs of the assets. It is used to add linear transaction costs to the optimization problem:

\[total\_cost = \sum_{i=1}^{N} c_{i} \times |w_{i} - w\_prev_{i}|\]

with \(c_{i}\) the transaction cost of asset i, \(w_{i}\) its weight and \(w\_prev_{i}\) its previous weight (defined in previous_weights). The float \(total\_cost\) is impacting the portfolio expected return in the optimization:

\[expected\_return = \mu^{T} \cdot w - total\_cost\]

with \(\mu\) the vector af assets’ expected returns and \(w\) the vector of assets weights.

If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset cost) and the input X of the fit method must be a DataFrame with the assets names in columns. The default value is 0.0.

Warning

Based on the above formula, the periodicity of the transaction costs needs to be homogenous to the periodicity of \(\mu\). For example, if the input X is composed of daily returns, the transaction_costs need to be expressed as daily costs. (See Transaction Costs)

management_feesfloat | dict[str, float] | array-like of shape (n_assets, ), default=0.0

Management fees of the assets. It is used to add linear management fees to the optimization problem:

\[total\_fee = \sum_{i=1}^{N} f_{i} \times w_{i}\]

with \(f_{i}\) the management fee of asset i and \(w_{i}\) its weight. The float \(total\_fee\) is impacting the portfolio expected return in the optimization:

\[expected\_return = \mu^{T} \cdot w - total\_fee\]

with \(\mu\) the vector af assets expected returns and \(w\) the vector of assets weights.

If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset fee) and the input X of the fit method must be a DataFrame with the assets names in columns. The default value is 0.0.

Warning

Based on the above formula, the periodicity of the management fees needs to be homogenous to the periodicity of \(\mu\). For example, if the input X is composed of daily returns, the management_fees need to be expressed in daily fees.

Note

Another approach is to directly impact the management fees to the input X in order to express the returns net of fees. However, when estimating the \(\mu\) parameter using for example Shrinkage estimators, this approach would mix a deterministic value with an uncertain one leading to unwanted bias in the management fees.

previous_weightsfloat | dict[str, float] | array-like of shape (n_assets, ), optional

Previous weights of the assets. Previous weights are used to compute the portfolio total cost. If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset previous weight) and the input X of the fit method must be a DataFrame with the assets names in columns. The default (None) means no previous weights.

portfolio_paramsdict, optional

Portfolio parameters passed to the portfolio evaluated by the predict and score methods. If not provided, the name, transaction_costs, management_fees, previous_weights and risk_free_rate are copied from the optimization model and passed to the portfolio.

solverstr, default=”CLARABEL”

The solver used for the weights constraints optimization. The default is “CLARABEL” which is written in Rust and has better numerical stability and performance than ECOS and SCS. For more details about available solvers, check the CVXPY documentation: https://www.cvxpy.org/tutorial/advanced/index.html#choosing-a-solver

solver_paramsdict, optional

Solver parameters. For example, solver_params=dict(verbose=True). The default (None) is to use the CVXPY default. For more details about solver arguments, check the CVXPY documentation: https://www.cvxpy.org/tutorial/advanced/index.html#setting-solver-options

Attributes:
weights_ndarray of shape (n_assets,)

Weights of the assets.

distance_estimator_BaseDistance

Fitted distance_estimator.

hierarchical_clustering_estimator_HierarchicalClustering

Fitted hierarchical_clustering_estimator.

n_features_in_int

Number of assets seen during fit.

feature_names_in_ndarray of shape (n_features_in_,)

Names of assets seen during fit. Defined only when X has assets names that are all strings.

References

[1]

“Hierarchical clustering-based asset allocation”, The Journal of Portfolio Management, Thomas Raffinot (2017).

[2]

“The hierarchical equal risk contribution portfolio”, Thomas Raffinot (2018).

[3]

“Application of two-order difference to gap statistic”. Yue, Wang & Wei (2009).

[4]

“A review of two decades of correlations, hierarchies, networks and clustering in financial markets”, Gautier Marti, Frank Nielsen, Mikołaj Bińkowski, Philippe Donnat (2020).

Methods

fit(X[, y])

Fit the Hierarchical Equal Risk Contribution estimator.

fit_predict(X)

Perform fit on X and returns the predicted Portfolio or Population of Portfolio on X based on the fitted weights.

get_metadata_routing()

Get metadata routing of this object.

get_params([deep])

Get parameters for this estimator.

predict(X)

Predict the Portfolio or Population of Portfolio on X based on the fitted weights.

score(X[, y])

Prediction score.

set_params(**params)

Set the parameters of this estimator.
fit(X, y=None, **fit_params)[source]#

Fit the Hierarchical Equal Risk Contribution estimator.

Parameters:
Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

yIgnored

Not used, present for API consistency by convention.

**fit_paramsdict

Parameters to pass to the underlying estimators. Only available if enable_metadata_routing=True, which can be set by using sklearn.set_config(enable_metadata_routing=True). See Metadata Routing User Guide for more details.

Returns:
selfHierarchicalEqualRiskContribution

Fitted estimator.

fit_predict(X)#

Perform fit on X and returns the predicted Portfolio or Population of Portfolio on X based on the fitted weights. For factor models, use fit(X, y) then predict(X) separately.

Parameters:
Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

Returns:
predictionPortfolio | Population

Portfolio or Population of Portfolio estimated on X based on the fitted weights.

get_metadata_routing()#

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:
routingMetadataRequest

A MetadataRequest encapsulating routing information.

get_params(deep=True)#

Get parameters for this estimator.

Parameters:
deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:
paramsdict

Parameter names mapped to their values.

predict(X)#

Predict the Portfolio or Population of Portfolio on X based on the fitted weights.

Optimization estimators can return a 1D or a 2D array of weights. For a 1D array, the prediction returns a Portfolio. For a 2D array, the prediction returns a Population of Portfolio.

If name is not provided in the portfolio arguments, we use the first 500 characters of the estimator name.

Parameters:
Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

Returns:
predictionPortfolio | Population

Portfolio or Population of Portfolio estimated on X based on the fitted weights.

score(X, y=None)#

Prediction score. If the prediction is a single Portfolio, the score is the Sharpe Ratio. If the prediction is a Population of Portfolio, the score is the mean of all the portfolios Sharpe Ratios in the population.

Parameters:
Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

yIgnored

Not used, present here for API consistency by convention.

Returns:
scorefloat

The Sharpe Ratio of the portfolio if the prediction is a single Portfolio or the mean of all the portfolios Sharpe Ratios if the prediction is a Population of Portfolio.

set_params(**params)#

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:
**paramsdict

Estimator parameters.

Returns:
selfestimator instance

Estimator instance.