"""Benchmark Tracker estimator."""
# Copyright (c) 2023-2026
# Author: Hugo Delatte <hugo.delatte@skfoliolabs.com>
# SPDX-License-Identifier: BSD-3-Clause
from __future__ import annotations
import numpy as np
import pandas as pd
import sklearn.utils.validation as skv
import skfolio.typing as skt
from skfolio.measures import RiskMeasure
from skfolio.optimization.convex._base import ObjectiveFunction
from skfolio.optimization.convex._mean_risk import MeanRisk
from skfolio.prior import BasePrior
from skfolio.typing import ArrayLike
[docs]
class BenchmarkTracker(MeanRisk):
r"""Benchmark Tracker Optimization estimator.
Optimize a portfolio to track a benchmark by minimizing the risk of
benchmark-relative (excess) returns.
This estimator minimizes the tracking risk between portfolio returns and a
benchmark's returns by optimizing directly on excess (active) returns, defined as
portfolio returns minus benchmark returns.
Tracking risk can be defined using any of the available risk measures. Typical
choices include standard deviation (tracking-error volatility), semi-deviation and
mean absolute deviation.
.. seealso::
:ref:`Tracking Error Optimization <tracking_error_optimization>`
Notes
-----
It is implemented as a special case of :class:`~skfolio.optimization.MeanRisk` where
the input asset returns `X` and benchmark returns `y` are first transformed to
excess returns `X_excess = X - y` before optimization. A full-investment constraint
(`budget = 1.0`) is always enforced because:
.. math::
r_t^{excess}(w) = \sum_i w_i(r_{t,i} - r_{b,t}) = r_{p,t} - (\sum_i w_i)r_{b,t}
where :math:`r_{p,t}` is the portfolio return and :math:`r_{b,t}` the benchmark
return, coincides with the active (benchmark-relative) return
.. math::
r_{p,t} - r_{b,t}
when :math:`\sum_i w_i = 1`
Parameters
----------
risk_measure : RiskMeasure, default=RiskMeasure.STANDARD_DEVIATION
:class:`~skfolio.measures.RiskMeasure` to minimize on excess returns.
The default is `RiskMeasure.STANDARD_DEVIATION`.
prior_estimator : BasePrior, optional
:ref:`Prior estimator <prior>`.
The prior estimator is used to estimate the :class:`~skfolio.prior.ReturnDistribution`
of excess returns (portfolio returns - benchmark returns).
The default (`None`) is to use :class:`~skfolio.prior.EmpiricalPrior`.
min_weights : float | dict[str, float] | array-like of shape (n_assets, ) | None, default=0.0
Minimum assets weights (weights lower bounds).
See :class:`~skfolio.optimization.MeanRisk` for details.
max_weights : float | dict[str, float] | array-like of shape (n_assets, ) | None, default=1.0
Maximum assets weights (weights upper bounds).
See :class:`~skfolio.optimization.MeanRisk` for details.
max_short : float, optional
Maximum short position.
See :class:`~skfolio.optimization.MeanRisk` for details.
max_long : float, optional
Maximum long position.
See :class:`~skfolio.optimization.MeanRisk` for details.
cardinality : int, optional
Cardinality constraint to limit the number of invested assets.
See :class:`~skfolio.optimization.MeanRisk` for details.
group_cardinalities : dict[str, int], optional
Cardinality constraints for specific groups of assets.
See :class:`~skfolio.optimization.MeanRisk` for details.
threshold_long : float | dict[str, float] | array-like of shape (n_assets, ), optional
Minimum weight threshold for long positions.
See :class:`~skfolio.optimization.MeanRisk` for details.
threshold_short : float | dict[str, float] | array-like of shape (n_assets, ), optional
Maximum weight threshold for short positions.
See :class:`~skfolio.optimization.MeanRisk` for details.
transaction_costs : float | dict[str, float] | array-like of shape (n_assets, ), default=0.0
Transaction costs of the assets.
See :class:`~skfolio.optimization.MeanRisk` for details.
management_fees : float | dict[str, float] | array-like of shape (n_assets, ), default=0.0
Management fees of the assets.
See :class:`~skfolio.optimization.MeanRisk` for details.
previous_weights : float | dict[str, float] | array-like of shape (n_assets, ), optional
Previous weights of the assets.
See :class:`~skfolio.optimization.MeanRisk` for details.
l1_coef : float, default=0.0
L1 regularization coefficient.
See :class:`~skfolio.optimization.MeanRisk` for details.
l2_coef : float, default=0.0
L2 regularization coefficient.
See :class:`~skfolio.optimization.MeanRisk` for details.
groups : dict[str, list[str]] or array-like of shape (n_groups, n_assets), optional
The assets groups.
See :class:`~skfolio.optimization.MeanRisk` for details.
linear_constraints : array-like of shape (n_constraints,), optional
Linear constraints.
See :class:`~skfolio.optimization.MeanRisk` for details.
left_inequality : array-like of shape (n_constraints, n_assets), optional
Left inequality matrix.
See :class:`~skfolio.optimization.MeanRisk` for details.
right_inequality : array-like of shape (n_constraints, ), optional
Right inequality vector.
See :class:`~skfolio.optimization.MeanRisk` for details.
risk_free_rate : float, default=0.0
Risk-free interest rate.
See :class:`~skfolio.optimization.MeanRisk` for details.
solver : str, default="CLARABEL"
The solver to use.
See :class:`~skfolio.optimization.MeanRisk` for details.
solver_params : dict, optional
Solver parameters.
See :class:`~skfolio.optimization.MeanRisk` for details.
scale_objective : float, optional
Scale each objective element by this value.
See :class:`~skfolio.optimization.MeanRisk` for details.
scale_constraints : float, optional
Scale each constraint element by this value.
See :class:`~skfolio.optimization.MeanRisk` for details.
save_problem : bool, default=False
If this is set to True, the CVXPY Problem is saved in `problem_`.
See :class:`~skfolio.optimization.MeanRisk` for details.
add_objective : Callable[[cp.Variable], cp.Expression], optional
Add a custom objective to the existing objective expression.
See :class:`~skfolio.optimization.MeanRisk` for details.
add_constraints : Callable[[cp.Variable], cp.Expression | list[cp.Expression]], optional
Add a custom constraint or a list of constraints to the existing constraints.
See :class:`~skfolio.optimization.MeanRisk` for details.
portfolio_params : dict, optional
Portfolio parameters.
See :class:`~skfolio.optimization.MeanRisk` for details.
fallback : BaseOptimization | "previous_weights" | list[BaseOptimization | "previous_weights"], optional
Fallback estimator or list of estimators.
See :class:`~skfolio.optimization.MeanRisk` for details.
raise_on_failure : bool, default=True
Controls error handling when fitting fails.
See :class:`~skfolio.optimization.MeanRisk` for details.
Attributes
----------
weights_ : ndarray of shape (n_assets,)
Weights of the assets.
problem_values_ : dict[str, float]
Expression values retrieved from the CVXPY problem.
prior_estimator_ : BasePrior
Fitted `prior_estimator` on excess returns.
problem_: cvxpy.Problem
CVXPY problem used for the optimization.
Only when `save_problem` is set to `True`.
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`.
fallback_ : BaseOptimization | "previous_weights" | None
The fallback estimator instance that produced the final result.
fallback_chain_ : list[tuple[str, str]] | None
Sequence describing the optimization fallback attempts.
error_ : str | None
Captured error message when `fit` fails.
References
----------
.. [1] "Portfolio Optimization: Theory and Application", Chapter 13,
Daniel P. Palomar (2025)
"""
def __init__(
self,
risk_measure: RiskMeasure = RiskMeasure.STANDARD_DEVIATION,
prior_estimator: BasePrior | None = None,
min_weights: skt.MultiInput | None = 0.0,
max_weights: skt.MultiInput | None = 1.0,
max_short: float | None = None,
max_long: float | None = None,
cardinality: int | None = None,
group_cardinalities: dict[str, int] | None = None,
threshold_long: skt.MultiInput | None = None,
threshold_short: skt.MultiInput | None = None,
transaction_costs: skt.MultiInput = 0.0,
management_fees: skt.MultiInput = 0.0,
previous_weights: skt.MultiInput | None = None,
groups: skt.Groups | None = None,
linear_constraints: skt.LinearConstraints | None = None,
left_inequality: skt.Inequality | None = None,
right_inequality: skt.Inequality | None = None,
l1_coef: float = 0.0,
l2_coef: float = 0.0,
risk_free_rate: float = 0.0,
solver: str = "CLARABEL",
solver_params: dict | None = None,
scale_objective: float | None = None,
scale_constraints: float | None = None,
save_problem: bool = False,
add_objective: skt.ExpressionFunction | None = None,
add_constraints: skt.ExpressionFunction | None = None,
portfolio_params: dict | None = None,
fallback: skt.Fallback = None,
raise_on_failure: bool = True,
):
super().__init__(
objective_function=ObjectiveFunction.MINIMIZE_RISK,
risk_measure=risk_measure,
prior_estimator=prior_estimator,
min_weights=min_weights,
max_weights=max_weights,
budget=1.0,
min_budget=None,
max_budget=None,
max_short=max_short,
max_long=max_long,
cardinality=cardinality,
group_cardinalities=group_cardinalities,
threshold_long=threshold_long,
threshold_short=threshold_short,
transaction_costs=transaction_costs,
management_fees=management_fees,
previous_weights=previous_weights,
groups=groups,
linear_constraints=linear_constraints,
left_inequality=left_inequality,
right_inequality=right_inequality,
l1_coef=l1_coef,
l2_coef=l2_coef,
risk_free_rate=risk_free_rate,
solver=solver,
solver_params=solver_params,
scale_objective=scale_objective,
scale_constraints=scale_constraints,
save_problem=save_problem,
add_objective=add_objective,
add_constraints=add_constraints,
portfolio_params=portfolio_params,
fallback=fallback,
raise_on_failure=raise_on_failure,
)
[docs]
def fit(self, X: ArrayLike, y: ArrayLike, **fit_params) -> BenchmarkTracker:
"""Fit the Return-Based Tracker estimator.
Parameters
----------
X : array-like of shape (n_observations, n_assets)
Price returns of the assets.
y : array-like of shape (n_observations, 1) or (n_observations,)
Price returns of the benchmark.
**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 : BenchmarkTracker
Fitted estimator.
"""
if y is None:
raise ValueError(
"y (benchmark returns) must be provided for BenchmarkTracker"
)
if self.budget != 1.0:
raise ValueError("Budget must be 1.0 for BenchmarkTracker")
y = np.asarray(y)
if y.ndim == 2 and y.shape[1] == 1:
y = y.ravel()
if y.ndim != 1:
raise ValueError(
"y (benchmark returns) must be 1-dimensional or a single-column "
f"DataFrame/array, got shape {y.shape}."
)
X, y = skv.validate_data(self, X, y, ensure_all_finite="allow-nan")
excess_returns = X - y[:, np.newaxis]
# Wrap as DataFrame to preserve feature_names_in_ through the super().fit() call
if hasattr(self, "feature_names_in_") and self.feature_names_in_ is not None:
excess_returns = pd.DataFrame(
excess_returns, columns=self.feature_names_in_
)
super().fit(excess_returns, y=None, **fit_params)
return self
