"""Hierarchical Risk Parity Optimization estimator."""
# Copyright (c) 2023-2026
# Author: Hugo Delatte <hugo.delatte@skfoliolabs.com>
# SPDX-License-Identifier: BSD-3-Clause
# The risk measure generalization and constraint features are derived
# from Riskfolio-Lib, Copyright (c) 2020-2023, Dany Cajas, Licensed under BSD 3 clause.
from __future__ import annotations
import numpy as np
import pandas as pd
import scipy.cluster.hierarchy as sch
import sklearn.utils.metadata_routing as skm
import sklearn.utils.validation as skv
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.typing import ArrayLike, FloatArray
from skfolio.utils.tools import bisection, check_estimator
[docs]
class HierarchicalRiskParity(BaseHierarchicalOptimization):
r"""Hierarchical Risk Parity estimator.
Hierarchical Risk Parity is a portfolio optimization method developed by Marcos
Lopez de Prado [1]_.
This algorithm uses a distance matrix to compute hierarchical clusters using the
Hierarchical Tree Clustering algorithm. It then employs seriation to rearrange the
assets in the dendrogram, minimizing the distance between leaves.
The final step is the recursive bisection where each cluster is split between two
sub-clusters by starting with the topmost cluster and traversing in a top-down
manner. For each sub-cluster, we compute the total cluster risk of an inverse-risk
allocation. A weighting factor is then computed from these two sub-cluster risks,
which is used to update the cluster weight.
.. note ::
The original paper uses the variance as the risk measure and the single-linkage
method for the Hierarchical Tree Clustering algorithm. Here we generalize it to
multiple risk measures and linkage methods.
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 [2]_.
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.ReturnDistribution`
containing estimates of expected asset 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). The default is 0.0 (no short
selling). Negative weights are not allowed. If a float is provided, it is
applied to each asset. `None` is equivalent to the default `0.0`. If a
dictionary is provided, its (key/value) pair must be the (asset name/asset
minimum weight) and the input `X` of the `fit` methods must be a DataFrame with
the asset names in columns. When using a dictionary, assets values that are not
provided are assigned the default minimum weight of `0.0`.
Example:
* `min_weights = 0.0` --> long only portfolio (default).
* `min_weights = {"SX5E": 0.1, "SPX": 0.2}`
* `min_weights = [0.1, 0.2]`
max_weights : float | dict[str, float] | array-like of shape (n_assets, ), default=1.0
Maximum assets weights (weights upper bounds). The default is 1.0 (each asset
is below 100%). Weights above 1.0 are not allowed. If a float is provided, it is
applied to each asset. `None` is equivalent to the default `1.0`. 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 asset names in columns. When using a dictionary, assets values that are not
provided are assigned the default maximum weight of `1.0`.
Example:
* `max_weights = 1.0` --> each weight must be below 100% (default).
* `max_weights = 0.5` --> each weight must be below 50%.
* `max_weights = {"SX5E": 0.8, "SPX": 0.9}`
* `max_weights = [0.8, 0.9]`
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 of 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 asset names in columns.
The default value is `0.0`.
.. warning::
Based on the above formula, the periodicity of the transaction costs
must match 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. A transaction cost is paid once per
rebalancing while a position earns its expected return on every period it
is held, so the one-off cost is converted by dividing it by the expected
investment duration (e.g. `0.001 / 21` for a 10 bps cost with daily
returns and a one-month expected holding period).
(See :ref:`Periodicity Convention <periodicity_convention>`)
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 of 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 asset names in columns.
The default value is `0.0`.
.. warning::
Based on the above formula, the periodicity of the management fees
must match 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. Unlike transaction costs, management fees
accrue with holding time, so a stated annual fee converts directly to the
return periodicity (e.g. `0.02 / 252` for a 2% annual fee on daily
returns).
.. 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 asset names in columns.
The default (`None`) means no previous weights.
Additionally, when `fallback="previous_weights"`, failures will fall back to
these weights if provided.
portfolio_params : dict, optional
Portfolio parameters forwarded to the resulting `Portfolio` in `predict`.
If not provided and if available on the estimator, the following
attributes are propagated to the portfolio by default: `name`,
`transaction_costs`, `management_fees`, `previous_weights` and `risk_free_rate`.
fallback : BaseOptimization | "previous_weights" | list[BaseOptimization | "previous_weights"], optional
Fallback estimator or a list of estimators to try, in order, when the primary
optimization raises during `fit`. Alternatively, use `"previous_weights"`
(alone or in a list) to fall back to the estimator's `previous_weights`.
When a fallback succeeds, its fitted `weights_` are copied back to the primary
estimator so that `fit` still returns the original instance. For traceability,
`fallback_` stores the successful estimator (or the string `"previous_weights"`)
and `fallback_chain_` stores each attempt with the associated outcome.
raise_on_failure : bool, default=True
Controls error handling when fitting fails.
If True, any failure during `fit` is raised immediately, no `weights_` are
set and subsequent calls to `predict` will raise a `NotFittedError`.
If False, errors are not raised; instead, a warning is emitted, `weights_`
is set to `None` and subsequent calls to `predict` will return a
`FailedPortfolio`. When fallbacks are specified, this behavior applies only
after all fallbacks have been exhausted.
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 asset names that are all strings.
fallback_ : BaseOptimization | "previous_weights" | None
The fallback estimator instance, or the string `"previous_weights"`, that
produced the final result. `None` if no fallback was used.
fallback_chain_ : list[tuple[str, str]] | None
Sequence describing the optimization fallback attempts. Each element is a
pair `(estimator_repr, outcome)` where `estimator_repr` is the string
representation of the primary estimator or a fallback (e.g. `"EqualWeighted()"`,
`"previous_weights"`), and `outcome` is `"success"` if that step produced
a valid solution, otherwise the stringified error message. For successful
fits without any fallback, this is `None`.
error_ : str | list[str] | None
Captured error message(s) when `fit` fails. For multi-portfolio outputs
(`weights_` is 2D), this is a list aligned with portfolios.
Notes
-----
All estimators should specify all parameters as explicit keyword arguments in
`__init__` (no `*args` or `**kwargs`), following scikit-learn conventions.
References
----------
.. [1] "A robust estimator of the efficient frontier",
SSRN Electronic Journal,
Marcos López de Prado (2019).
.. [2] "A review of two decades of correlations, hierarchies, networks and
clustering in financial markets",
Gautier Marti, Frank Nielsen, Mikołaj Bińkowski, Philippe Donnat (2020).
.. [3] "Portfolio Optimization: Theory and Application", Chapter 12,
Daniel P. Palomar (2025)
.. [4] "Building diversified portfolios that outperform out of sample",
The Journal of Portfolio Management,
Marcos López de Prado (2016).
.. [5] "Machine Learning for Asset Managers",
Elements in Quantitative Finance. Cambridge University Press,
Marcos López de Prado (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,
transaction_costs: skt.MultiInput = 0.0,
management_fees: skt.MultiInput = 0.0,
previous_weights: skt.MultiInput | None = None,
portfolio_params: dict | None = None,
fallback: skt.Fallback = None,
raise_on_failure: bool = True,
):
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,
fallback=fallback,
raise_on_failure=raise_on_failure,
)
[docs]
def fit(self, X: ArrayLike, y: None = None, **fit_params) -> HierarchicalRiskParity:
"""Fit the Hierarchical Risk Parity Optimization 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.
Returns
-------
self : HierarchicalRiskParity
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 in HRP"
)
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)
return_distribution = self.prior_estimator_.return_distribution_
returns = return_distribution.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
)
X = skv.validate_data(self, X)
n_assets = X.shape[1]
min_weights, max_weights = self._convert_weights_bounds(n_assets=n_assets)
assets_risks = self._unitary_risks(return_distribution=return_distribution)
ordered_linkage_matrix = sch.optimal_leaf_ordering(
self.hierarchical_clustering_estimator_.linkage_matrix_,
self.hierarchical_clustering_estimator_.condensed_distance_,
)
sorted_assets = sch.leaves_list(ordered_linkage_matrix)
weights = np.ones(n_assets)
items = [sorted_assets]
while len(items) > 0:
new_items = []
for clusters_ids in bisection(items):
new_items += clusters_ids
risks = []
for ids in clusters_ids:
inv_risk_w = np.zeros(n_assets)
inv_risk_w[ids] = 1 / assets_risks[ids]
inv_risk_w /= inv_risk_w.sum()
risks.append(
self._risk(
weights=inv_risk_w, return_distribution=return_distribution
)
)
left_risk, right_risk = risks
left_cluster, right_cluster = clusters_ids
alpha = 1 - left_risk / (left_risk + right_risk)
# Weights constraints
alpha = _apply_weight_constraints_to_split_factor(
alpha=alpha,
weights=weights,
max_weights=max_weights,
min_weights=min_weights,
left_cluster=left_cluster,
right_cluster=right_cluster,
)
weights[left_cluster] *= alpha
weights[right_cluster] *= 1 - alpha
items = new_items
self.weights_ = weights
return self
def _apply_weight_constraints_to_split_factor(
alpha: float,
max_weights: FloatArray,
min_weights: FloatArray,
weights: FloatArray,
left_cluster: FloatArray,
right_cluster: FloatArray,
) -> float:
"""
Apply weight constraints to the split factor alpha of the ,Hierarchical Tree
Clustering algorithm.
Parameters
----------
alpha : float
The split factor alpha of the Hierarchical Tree Clustering algorithm.
min_weights : ndarray of shape (n_assets,)
The weight lower bound 1D array.
max_weights : ndarray of shape (n_assets,)
The weight upper bound 1D array.
weights : FloatArray of shape (n_assets,)
The assets weights.
left_cluster : ndarray of shape (n_left_cluster,)
Indices of the left cluster weights.
right_cluster : ndarray of shape (n_right_cluster,)
Indices of the right cluster weights.
Returns
-------
value : float
The transformed split factor alpha incorporating the weight constraints.
"""
alpha = min(
np.sum(max_weights[left_cluster]) / weights[left_cluster[0]],
max(np.sum(min_weights[left_cluster]) / weights[left_cluster[0]], alpha),
)
alpha = 1 - min(
np.sum(max_weights[right_cluster]) / weights[right_cluster[0]],
max(
np.sum(min_weights[right_cluster]) / weights[right_cluster[0]],
1 - alpha,
),
)
return alpha
