"""Distributionally Robust CVaR Optimization estimator."""
# Copyright (c) 2023-2026
# Author: Hugo Delatte <hugo.delatte@skfoliolabs.com>
# SPDX-License-Identifier: BSD-3-Clause
from __future__ import annotations
import cvxpy as cp
import numpy as np
import sklearn.utils.metadata_routing as skm
import sklearn.utils.validation as skv
import skfolio.typing as skt
from skfolio.measures import RiskMeasure
from skfolio.optimization.convex._base import ConvexOptimization
from skfolio.prior import BasePrior, EmpiricalPrior
from skfolio.typing import ArrayLike
from skfolio.utils.tools import check_estimator
[docs]
class DistributionallyRobustCVaR(ConvexOptimization):
r"""Distributionally Robust CVaR.
The Distributionally Robust CVaR model constructs a Wasserstein ball in the space of
multivariate and non-discrete probability distributions centered at the uniform
distribution on the training samples and finds the allocation that minimizes the
CVaR of the worst-case distribution within this Wasserstein ball.
Esfahani and Kuhn [1]_ proved that for piecewise linear objective functions,
which is the case of CVaR [2]_, the distributionally robust optimization problem
over a Wasserstein ball can be reformulated as finite convex programs.
Only piecewise linear functions are supported, which means that transaction costs
and regularization are not permitted.
A solver like `Mosek` that can handle a high number of constraints is preferred.
Parameters
----------
cvar_beta : float, default=0.95
CVaR (Conditional Value at Risk) confidence level.
risk_aversion : float, default=1.0
Risk aversion factor of the utility function: return - risk_aversion * cvar.
wasserstein_ball_radius: float, default=0.02
Radius of the Wasserstein ball.
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,
returns and Cholesky decomposition of the covariance.
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).
If a float is provided, it is applied to each asset.
`None` is equivalent to `-np.Inf` (no lower bound).
If a dictionary is provided, its (key/value) pair must be the
(asset name/asset minimum 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 `0.0`.
The default value 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 = -2` --> each weight must be above -200%.
* `min_weights = {"SX5E": 0, "SPX": -2}`
* `min_weights = [0, -2]`
max_weights : float | dict[str, float] | array-like of shape (n_assets, ) | None, default=1.0
Maximum assets weights (weights upper bounds).
If a float is provided, it is applied to each asset.
`None` is equivalent to `+np.Inf` (no upper bound).
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 value is `1.0` (each asset is below 100%).
Example:
* `max_weights = 0` --> no long position (short only portfolio).
* `max_weights = None` --> no upper bound.
* `max_weights = 2` --> each weight must be below 200%.
* `max_weights = {"SX5E": 1, "SPX": 2}`
* `max_weights = [1, 2]`
budget : float | None, default=1.0
Investment budget. It is the sum of long positions and short positions (sum of
all weights). `None` means no budget constraints.
The default value is `1.0` (fully invested portfolio).
For example:
* `budget = 1` --> fully invested portfolio.
* `budget = 0` --> market neutral portfolio.
* `budget = None` --> no constraints on the sum of weights.
min_budget : float, optional
Minimum budget. It is the lower bound of the sum of long and short positions
(sum of all weights). If provided, you must set `budget=None`.
The default (`None`) means no minimum budget constraint.
max_short : float, optional
Maximum short position. The short position is defined as the sum of negative
weights (in absolute term).
The default (`None`) means no maximum short position.
max_long : float, optional
Maximum long position. The long position is defined as the sum of positive
weights.
The default (`None`) means no maximum long position.
max_budget : float, optional
Maximum budget. It is the upper bound of the sum of long and short positions
(sum of all weights). If provided, you must set `budget=None`.
The default (`None`) means no maximum budget constraint.
linear_constraints : array-like of shape (n_constraints,), optional
Linear constraints on portfolio weights or factor exposures.
Constraint names can reference:
* Asset names: individual asset weights (e.g. `"SPX"`, `"AAPL"`)
* Group names: sums of weights in groups defined by `groups`
* Factor names: portfolio factor exposure (requires factor model prior)
* Factor families: sum of portfolio exposures to all factors in one family
Supported equation patterns include:
* `"name <= value"` or `"name >= value"`
* `"name == value"`
* `"a * name1 + b * name2 <= c * name3 + d"`
For example:
* `"SPX >= 0.10"` --> SPX weight >= 10%
* `"SX5E + SPX >= 0.2"` --> sum of SX5E and SPX weights >= 20%
* `"US == 0.7"` --> sum of weights in US group == 70%
* `"Equity == 3 * Bond"` --> sum of weights in Equity group == 3x sum of weights in Bond group
* `"Momentum <= 0.30"` --> portfolio Momentum exposure <= 30%
* `"style <= 0.50"` --> sum of all style factor exposures (Momentum, Value, Size, etc.) <= 50%
Factor constraints require a prior estimator (e.g.
:class:`~skfolio.prior.TimeSeriesFactorModel`,
:class:`~skfolio.prior.CharacteristicsFactorModel`)
that provides `loading_matrix`, `factor_names` and optionally `factor_families`
in its :class:`~skfolio.prior.FactorModel`.
Asset, group, factor, and factor family names must be unique.
groups : dict[str, list[str]] or array-like of shape (n_groups, n_assets), optional
The assets groups referenced in `linear_constraints`.
If a dictionary is provided, its (key/value) pair must be the
(asset name/asset groups) and the input `X` of the `fit` method must be a
DataFrame with the assets names in columns.
For example:
* `groups = {"SX5E": ["Equity", "Europe"], "SPX": ["Equity", "US"], "TLT": ["Bond", "US"]}`
* `groups = [["Equity", "Equity", "Bond"], ["Europe", "US", "US"]]`
left_inequality : array-like of shape (n_constraints, n_assets), optional
Left inequality matrix :math:`A` of the linear
constraint :math:`A \cdot w \leq b`.
right_inequality : array-like of shape (n_constraints, ), optional
Right inequality vector :math:`b` of the linear
constraint :math:`A \cdot w \leq b`.
risk_free_rate : float, default=0.0
Risk-free interest rate.
The default value is `0.0`.
add_constraints : Callable[[cp.Variable], cp.Expression | list[cp.Expression]], optional
Add a custom constraint or a list of constraints to the existing constraints.
It must be a function taking the CVXPY weight variable `w` as its first
positional argument and, optionally, the estimator instance as its second.
It must return a CVXPY expression or a list of CVXPY expressions, evaluated
when `fit` is called.
For example, to require an effective number of assets of at least 20:
>>> import cvxpy as cp
>>> from skfolio.optimization import DistributionallyRobustCVaR
>>> model = DistributionallyRobustCVaR(
... add_constraints=lambda w: cp.sum_squares(w) <= 1 / 20
... )
The optional second argument gives access to the estimator's attributes,
including quantities estimated during `fit`. For example, to cap each
position size in risk units at 20 bps, using the volatilities estimated
by the prior:
>>> import numpy as np
>>> def position_risk_cap(w, model):
... covariance = model.prior_estimator_.return_distribution_.covariance
... vols = np.sqrt(np.diag(covariance))
... return cp.multiply(vols, w) <= 0.002
>>> model = DistributionallyRobustCVaR(add_constraints=position_risk_cap)
overwrite_expected_return : Callable[[cp.Variable], cp.Expression], optional
Overwrite the expected return :math:`\mu \cdot w` with a custom CVXPY
expression. It must be a function taking the CVXPY weight variable `w` as
its first positional argument and, optionally, the estimator instance as
its second. It must return a concave CVXPY expression, evaluated when
`fit` is called. The custom expression replaces the expected return in the
objective function and in the constraints where the expected return is
used.
For example, to adjust the expected return for volatility drag,
approximating the portfolio geometric mean return:
>>> import cvxpy as cp
>>> from skfolio.optimization import DistributionallyRobustCVaR
>>> def geometric_expected_return(w, model):
... dist = model.prior_estimator_.return_distribution_
... return dist.mu @ w - 0.5 * cp.quad_form(w, dist.covariance)
>>> model = DistributionallyRobustCVaR(
... overwrite_expected_return=geometric_expected_return
... )
solver : str, default="CLARABEL"
The solver to use. The default is "CLARABEL" which is written in Rust and has
better numerical stability and performance than ECOS and SCS. Cvxpy will replace
its default solver "ECOS" by "CLARABEL" in future releases.
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 `{"tol_gap_abs": 1e-9, "tol_gap_rel": 1e-9}`
for the solver "CLARABEL" and the CVXPY default otherwise.
For more details about solver arguments, check the CVXPY documentation:
https://www.cvxpy.org/tutorial/advanced/index.html#setting-solver-options
scale_objective : float, optional
Scale each objective element by this value.
It can be used to increase the optimization accuracies in specific cases.
The default (`None`) is set depending on the problem.
scale_constraints : float, optional
Scale each constraint element by this value.
It can be used to increase the optimization accuracies in specific cases.
The default (`None`) is set depending on the problem.
save_problem : bool, default=False
If this is set to True, the CVXPY Problem is saved in `problem_`.
The default is `False`.
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.
previous_weights : float | dict[str, float] | array-like of shape (n_assets,), optional
When `fallback="previous_weights"`, failures will fall back to these weights if
provided.
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,) or (n_optimizations, n_assets)
Weights of the assets.
problem_values_ : dict[str, float] | list[dict[str, float]] of size n_optimizations
Expression values retrieved from the CVXPY problem.
prior_estimator_ : BasePrior
Fitted `prior_estimator`.
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`. Defined only when `X`
has assets 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.
Examples
--------
For a complete tutorial on distributionally robust CVaR optimization, see the
:ref:`distributionally_robust_examples` gallery.
>>> from skfolio.datasets import load_sp500_dataset
>>> from skfolio.optimization import DistributionallyRobustCVaR
>>> from skfolio.preprocessing import prices_to_returns
>>>
>>> # Load recent historical prices and convert them to returns
>>> prices = load_sp500_dataset()["2020":]
>>> X = prices_to_returns(prices)
>>>
>>> # Distributionally robust CVaR optimization
>>> model = DistributionallyRobustCVaR(wasserstein_ball_radius=0.01)
>>> model.fit(X)
>>> print(model.weights_)
>>>
>>> # Increasing the radius increases the uncertainty around the distribution,
>>> # which brings the weights closer to equal weighting
>>> model = DistributionallyRobustCVaR(wasserstein_ball_radius=0.10)
>>> model.fit(X)
>>> print(model.weights_)
References
----------
.. [1] "Data-driven distributionally robust optimization using the Wasserstein
metric: performance guarantees and tractable reformulations".
Esfahani and Kuhn (2018).
.. [2] "Optimization of conditional value-at-risk".
Rockafellar and Uryasev (2000).
"""
def __init__(
self,
risk_aversion: float = 1.0,
cvar_beta: float = 0.95,
wasserstein_ball_radius: float = 0.02,
prior_estimator: BasePrior | None = None,
min_weights: skt.MultiInput | None = 0.0,
max_weights: skt.MultiInput | None = 1.0,
budget: float | None = 1,
min_budget: float | None = None,
max_budget: float | None = None,
max_short: float | None = None,
max_long: float | 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,
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,
overwrite_expected_return: skt.ExpressionFunction | None = None,
portfolio_params: dict | None = None,
fallback: skt.Fallback = None,
previous_weights: skt.MultiInput | None = None,
raise_on_failure: bool = True,
):
super().__init__(
risk_measure=RiskMeasure.CVAR,
prior_estimator=prior_estimator,
min_weights=min_weights,
max_weights=max_weights,
budget=budget,
min_budget=min_budget,
max_budget=max_budget,
max_short=max_short,
max_long=max_long,
groups=groups,
linear_constraints=linear_constraints,
left_inequality=left_inequality,
right_inequality=right_inequality,
risk_free_rate=risk_free_rate,
cvar_beta=cvar_beta,
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,
overwrite_expected_return=overwrite_expected_return,
portfolio_params=portfolio_params,
fallback=fallback,
previous_weights=previous_weights,
raise_on_failure=raise_on_failure,
)
self.risk_aversion = risk_aversion
self.wasserstein_ball_radius = wasserstein_ball_radius
[docs]
def fit(
self, X: ArrayLike, y: ArrayLike | None = None, **fit_params
) -> DistributionallyRobustCVaR:
"""Fit the Distributionally Robust CVaR Optimization estimator.
Parameters
----------
X : array-like of shape (n_observations, n_assets)
Price returns of the assets.
y : array-like of shape (n_observations, n_factors), optional
Price returns of factors.
The default is `None`.
**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 : DistributionallyRobustCVaR
Fitted estimator.
"""
routed_params = skm.process_routing(self, "fit", **fit_params)
# `X` is unchanged and only `feature_names_in_` is performed
_ = skv.validate_data(
self, X, skip_check_array=True, ensure_all_finite="allow-nan"
)
# Used to avoid adding multiple times similar constrains linked to identical
# risk models
self.prior_estimator_ = check_estimator(
self.prior_estimator,
default=EmpiricalPrior(),
check_type=BasePrior,
)
self.prior_estimator_.fit(X, y, **routed_params.prior_estimator.fit)
return_distribution = self._prepare_investable_distribution(
self.prior_estimator_.return_distribution_, slim=True
)
n_observations, n_assets = return_distribution.returns.shape
# set solvers params
if self.solver == "CLARABEL":
self._set_solver_params(default={"tol_gap_abs": 1e-9, "tol_gap_rel": 1e-9})
else:
self._set_solver_params(default=None)
# set scale
self._set_scale_objective(default=1)
self._set_scale_constraints(default=1)
a1 = -1
b1 = cp.Constant(self.risk_aversion)
a2 = -1 - self.risk_aversion / (1 - self.cvar_beta)
b2 = cp.Constant(self.risk_aversion * (1 - 1 / (1 - self.cvar_beta)))
ones = np.ones(n_assets)
w = cp.Variable(n_assets)
u = cp.Variable((n_observations, n_assets))
v = cp.Variable((n_observations, n_assets))
lb = cp.Variable()
tau = cp.Variable()
s = cp.Variable(n_observations)
factor = cp.Constant(1)
# constraints
constraints = self._get_weight_constraints(
n_assets=n_assets, w=w, factor=factor
)
constraints += [
u * self._scale_constraints >= cp.Constant(0),
v * self._scale_constraints >= cp.Constant(0),
b1 * tau * self._scale_constraints
+ a1 * (return_distribution.returns @ w) * self._scale_constraints
+ cp.multiply(u, (1 + return_distribution.returns))
@ ones
* self._scale_constraints
<= s * self._scale_constraints,
b2 * tau * self._scale_constraints
+ a2 * (return_distribution.returns @ w) * self._scale_constraints
+ cp.multiply(v, (1 + return_distribution.returns))
@ ones
* self._scale_constraints
<= s * self._scale_constraints,
]
for i in range(n_observations):
# noinspection PyTypeChecker
constraints.append(
cp.norm(-u[i] - a1 * w, np.inf) * self._scale_constraints
<= lb * self._scale_constraints
)
# noinspection PyTypeChecker
constraints.append(
cp.norm(-v[i] - a2 * w, np.inf) * self._scale_constraints
<= lb * self._scale_constraints
)
# custom objectives and constraints
custom_objective = self._get_custom_objective(w=w)
constraints += self._get_custom_constraints(w=w)
if return_distribution.sample_weight is None:
risk = cp.sum(s) / n_observations * self._scale_objective
else:
risk = (
cp.sum(cp.multiply(return_distribution.sample_weight, s))
* self._scale_objective
)
objective = cp.Minimize(
cp.Constant(self.wasserstein_ball_radius) * lb * self._scale_objective
+ risk
+ custom_objective * self._scale_objective
)
# problem
problem = cp.Problem(objective, constraints)
# results
self._solve_problem(problem=problem, w=w, factor=factor)
return self
