Source code for skfolio.optimization.cluster.hierarchical._herc

"""Hierarchical Equal Risk Contribution estimator."""

# Copyright (c) 2023
# Author: Hugo Delatte <delatte.hugo@gmail.com>
# License: BSD 3 clause
# Weight constraints is a novel implementation, see docstring for more details.

import numpy as np
import numpy.typing as npt
import pandas as pd
import scipy.cluster.hierarchy as sch
import sklearn.utils.metadata_routing as skm

import skfolio.typing as skt
from skfolio.cluster import HierarchicalClustering
from skfolio.distance import BaseDistance, PearsonDistance
from skfolio.measures import ExtraRiskMeasure, RiskMeasure
from skfolio.optimization.cluster.hierarchical._base import (
    BaseHierarchicalOptimization,
)
from skfolio.prior import BasePrior, EmpiricalPrior
from skfolio.utils.stats import minimize_relative_weight_deviation
from skfolio.utils.tools import check_estimator


[docs] class HierarchicalEqualRiskContribution(BaseHierarchicalOptimization): r"""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: .. math:: \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} 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_measure : RiskMeasure or ExtraRiskMeasure, default=RiskMeasure.VARIANCE :class:`~skfolio.meta.RiskMeasure` or :class:`~skfolio.meta.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_estimator : BasePrior, optional :ref:`Prior estimator <prior>`. The prior estimator is used to estimate the :class:`~skfolio.prior.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 :class:`~skfolio.prior.EmpiricalPrior`. distance_estimator : BaseDistance, optional :ref:`Distance estimator <distance>`. 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 :class:`~skfolio.distance.PearsonDistance`. hierarchical_clustering_estimator : HierarchicalClustering, optional :ref:`Hierarchical Clustering estimator <hierarchical_clustering>`. 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 :class:`~skfolio.cluster.HierarchicalClustering`. min_weights : float | 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_weights : float | 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_costs : float | 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: .. math:: total\_cost = \sum_{i=1}^{N} c_{i} \times |w_{i} - w\_prev_{i}| with :math:`c_{i}` the transaction cost of asset i, :math:`w_{i}` its weight and :math:`w\_prev_{i}` its previous weight (defined in `previous_weights`). The float :math:`total\_cost` is impacting the portfolio expected return in the optimization: .. math:: expected\_return = \mu^{T} \cdot w - total\_cost with :math:`\mu` the vector af assets' expected returns and :math:`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 :math:`\mu`. For example, if the input `X` is composed of **daily** returns, the `transaction_costs` need to be expressed as **daily** costs. (See :ref:`sphx_glr_auto_examples_1_mean_risk_plot_6_transaction_costs.py`) management_fees : float | 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: .. math:: total\_fee = \sum_{i=1}^{N} f_{i} \times w_{i} with :math:`f_{i}` the management fee of asset i and :math:`w_{i}` its weight. The float :math:`total\_fee` is impacting the portfolio expected return in the optimization: .. math:: expected\_return = \mu^{T} \cdot w - total\_fee with :math:`\mu` the vector af assets expected returns and :math:`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 :math:`\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 :math:`\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_weights : float | 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_params : dict, 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. solver : str, 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_params : dict, 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). """ def __init__( self, risk_measure: RiskMeasure | ExtraRiskMeasure = RiskMeasure.VARIANCE, prior_estimator: BasePrior | None = None, distance_estimator: BaseDistance | None = None, hierarchical_clustering_estimator: HierarchicalClustering | None = None, min_weights: skt.MultiInput | None = 0.0, max_weights: skt.MultiInput | None = 1.0, solver: str = "CLARABEL", solver_params: dict | None = None, transaction_costs: skt.MultiInput = 0.0, management_fees: skt.MultiInput = 0.0, previous_weights: skt.MultiInput | None = None, portfolio_params: dict | None = None, ): super().__init__( risk_measure=risk_measure, prior_estimator=prior_estimator, distance_estimator=distance_estimator, hierarchical_clustering_estimator=hierarchical_clustering_estimator, min_weights=min_weights, max_weights=max_weights, transaction_costs=transaction_costs, management_fees=management_fees, previous_weights=previous_weights, portfolio_params=portfolio_params, ) self.solver = solver self.solver_params = solver_params
[docs] def fit( self, X: npt.ArrayLike, y: None = None, **fit_params ) -> "HierarchicalEqualRiskContribution": """Fit the Hierarchical Equal Risk Contribution estimator. Parameters ---------- X : array-like of shape (n_observations, n_assets) Price returns of the assets. y : Ignored Not used, present for API consistency by convention. **fit_params : dict 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 :ref:`Metadata Routing User Guide <metadata_routing>` for more details. Returns ------- self : HierarchicalEqualRiskContribution Fitted estimator. """ routed_params = skm.process_routing(self, "fit", **fit_params) # Validate if not isinstance(self.risk_measure, RiskMeasure | ExtraRiskMeasure): raise TypeError( "`risk_measure` must be of type `RiskMeasure` or `ExtraRiskMeasure`" ) if self.risk_measure in [ExtraRiskMeasure.SKEW, ExtraRiskMeasure.KURTOSIS]: # Because Skew and Kurtosis can take negative values raise ValueError( f"risk_measure {self.risk_measure} currently not supported" f"in HERC" ) self.prior_estimator_ = check_estimator( self.prior_estimator, default=EmpiricalPrior(), check_type=BasePrior, ) self.distance_estimator_ = check_estimator( self.distance_estimator, default=PearsonDistance(), check_type=BaseDistance, ) self.hierarchical_clustering_estimator_ = check_estimator( self.hierarchical_clustering_estimator, default=HierarchicalClustering(), check_type=HierarchicalClustering, ) # Fit the estimators self.prior_estimator_.fit(X, y, **routed_params.prior_estimator.fit) prior_model = self.prior_estimator_.prior_model_ returns = prior_model.returns # To keep the asset_names if isinstance(X, pd.DataFrame): returns = pd.DataFrame(returns, columns=X.columns) # noinspection PyArgumentList self.distance_estimator_.fit(returns, y, **routed_params.distance_estimator.fit) distance = self.distance_estimator_.distance_ # To keep the asset_names if isinstance(X, pd.DataFrame): distance = pd.DataFrame(distance, columns=X.columns) # noinspection PyArgumentList self.hierarchical_clustering_estimator_.fit( X=distance, y=None, **routed_params.hierarchical_clustering_estimator.fit ) n_clusters = self.hierarchical_clustering_estimator_.n_clusters_ labels = self.hierarchical_clustering_estimator_.labels_ linkage_matrix = self.hierarchical_clustering_estimator_.linkage_matrix_ X = self._validate_data(X) n_assets = X.shape[1] min_weights, max_weights = self._convert_weights_bounds(n_assets=n_assets) assets_risks = self._unitary_risks(prior_model=prior_model) weights = np.ones(n_assets) clusters_weights = np.ones(n_clusters) clusters = [np.argwhere(labels == i).flatten() for i in range(n_clusters)] clusters_sets = [set(cluster_ids) for cluster_ids in clusters] # Compute cluster total risk based on inverse-risk allocation cluster_risks = [] for cluster_ids in clusters: inv_risk_w = np.zeros(n_assets) inv_risk_w[cluster_ids] = 1 / assets_risks[cluster_ids] inv_risk_w /= inv_risk_w.sum() cluster_risks.append( self._risk(weights=inv_risk_w, prior_model=prior_model) ) weights[cluster_ids] = inv_risk_w[cluster_ids] cluster_risks = np.array(cluster_risks) # Compute the cluster weights using the dendrogram structure. # Recurse from the root until each of the defined cluster is reached and # update the weights using the naive risk parity. def _recurse(node): # Stop when the cluster is reached if set(node.pre_order()) in clusters_sets: return left_node = node.get_left() right_node = node.get_right() left_cluster_tree = set(left_node.pre_order()) right_cluster_tree = set(right_node.pre_order()) left_cluster = [] right_cluster = [] for i, cluster_ids in enumerate(clusters_sets): if cluster_ids.issubset(left_cluster_tree): left_cluster.append(i) elif cluster_ids.issubset(right_cluster_tree): right_cluster.append(i) if not left_cluster or not right_cluster: raise ValueError("Corrupted") left_cluster = np.array(left_cluster) right_cluster = np.array(right_cluster) left_risk = np.sum(cluster_risks[left_cluster]) right_risk = np.sum(cluster_risks[right_cluster]) alpha = 1 - left_risk / (left_risk + right_risk) clusters_weights[left_cluster] *= alpha clusters_weights[right_cluster] *= 1 - alpha _recurse(left_node) _recurse(right_node) root = sch.to_tree(linkage_matrix) _recurse(root) # Combine intra-cluster weights with inter-cluster weights for i, cluster_ids in enumerate(clusters): weights[cluster_ids] *= clusters_weights[i] # Apply weights constraints weights = minimize_relative_weight_deviation( weights=weights, min_weights=min_weights, max_weights=max_weights, solver=self.solver, solver_params=self.solver_params, ) self.weights_ = weights return self