"""Risk Budgeting Optimization estimator."""
# Copyright (c) 2023-2026
# Author: Hugo Delatte <hugo.delatte@skfoliolabs.com>
# SPDX-License-Identifier: BSD-3-Clause
# Some optimization features are derived
# from Riskfolio-Lib, Copyright (c) 2020-2023, Dany Cajas, Licensed under BSD 3 clause.
# Credits: Daniel P. Palomar (improvements)
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, FloatArray
from skfolio.utils.tools import args_names, check_estimator
[docs]
class RiskBudgeting(ConvexOptimization):
r"""Risk Budgeting Optimization estimator.
The Risk Budgeting estimator solves the below convex problem:
.. math:: \begin{cases}
\begin{aligned}
& \min_{w,s} && \mathrm{Risk}(w) \\
& \text{s.t.} && budget^{\top}\log(w) \ge 0 \\
& && \mathbf{1}^{\top} w = s \\
& && expected\_return(w) \ge s\, min\_return \\
& && A w \le s\, b \\
& && w \ge 0
\end{aligned}
\end{cases}
with :math:`budget` the risk budget vector and :math:`min\_return` the minimum
expected return constraint.
And :math:`Risk` a risk measure among:
* Mean Absolute Deviation
* First Lower Partial Moment
* Variance
* Semi-Variance
* CVaR (Conditional Value at Risk)
* EVaR (Entropic Value at Risk)
* Worst Realization (worst return)
* CDaR (Conditional Drawdown at Risk)
* Maximum Drawdown
* Average Drawdown
* EDaR (Entropic Drawdown at Risk)
* Ulcer Index
* Gini Mean Difference
Cost and additional constraints can also be added to the optimization problem (see
the parameters description).
Limitations are imposed on some constraints including long only weights to ensure
convexity.
The expected asset returns, covariance matrix and returns are estimated from the
:ref:`prior estimator <prior>`.
Parameters
----------
risk_measure : RiskMeasure, default=RiskMeasure.VARIANCE
:class:`~skfolio.meta.RiskMeasure` of the optimization.
Can be any of:
* VARIANCE
* SEMI_VARIANCE
* STANDARD_DEVIATION
* SEMI_DEVIATION
* MEAN_ABSOLUTE_DEVIATION
* FIRST_LOWER_PARTIAL_MOMENT
* CVAR
* EVAR
* WORST_REALIZATION
* CDAR
* MAX_DRAWDOWN
* AVERAGE_DRAWDOWN
* EDAR
* ULCER_INDEX
* GINI_MEAN_DIFFERENCE
The default is `RiskMeasure.VARIANCE`.
risk_budget : dict[str, float] | array-like of shape (n_assets,), optional
Risk budget allocated to each asset.
If a dictionary is provided, its (key/value) pair must be the
(asset name/asset risk budget) and the input `X` of the `fit` method must be a
DataFrame with the assets names in columns.
The default (`None`) is to use the identity vector, reducing the risk
budgeting to a risk-parity (each asset contributing equally to the total risk).
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]`
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 assets 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 assets 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 cost and the portfolio turnover.
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.
Additionally, when `fallback="previous_weights"`, failures will fall back to
these weights if provided.
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`.
min_return : float | array-like of shape (n_optimization), optional
Lower bound constraint on the expected return.
min_acceptable_return : float, optional
The minimum acceptable return used to distinguish "downside" and "upside"
returns for the computation of lower partial moments:
* First Lower Partial Moment
* Semi-Variance
* Semi-Deviation
The default (`None`) is to use the mean.
cvar_beta : float, default=0.95
CVaR (Conditional Value at Risk) confidence level.
The default value is `0.95`.
evar_beta : float, default=0.95
EVaR (Entropic Value at Risk) confidence level.
The default value is `0.95`.
cdar_beta : float, default=0.95
CDaR (Conditional Drawdown at Risk) confidence level.
The default value is `0.95`.
edar_beta : float, default=0.95
EDaR (Entropic Drawdown at Risk) confidence level.
The default value is `0.95`.
add_objective : Callable[[cp.Variable], cp.Expression], optional
Add a custom objective to the existing objective expression.
It is a function that must take as argument the weights `w` and returns a
CVXPY expression.
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 RiskBudgeting
>>> model = RiskBudgeting(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 = RiskBudgeting(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 RiskBudgeting
>>> 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 = RiskBudgeting(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 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.
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 complete tutorials on risk budgeting optimization, see the
:ref:`risk_budgeting_examples` gallery.
>>> from skfolio import RiskMeasure
>>> from skfolio.datasets import load_sp500_dataset
>>> from skfolio.optimization import RiskBudgeting
>>> from skfolio.preprocessing import prices_to_returns
>>>
>>> # Load historical prices and convert them to returns
>>> prices = load_sp500_dataset()
>>> X = prices_to_returns(prices)
>>>
>>> # Variance risk parity optimization
>>> model = RiskBudgeting(risk_measure=RiskMeasure.VARIANCE)
>>> model.fit(X)
RiskBudgeting()
>>> print(model.weights_)
[0.0422 0.0314 0.0343 ... 0.0473 0.0603 0.0565]
>>>
>>> # CVaR risk budgeting with custom asset budgets
>>> risk_budget = {asset: 1.0 for asset in X.columns}
>>> risk_budget["AAPL"] = 1.5
>>> risk_budget["GE"] = 0.2
>>> risk_budget["JPM"] = 0.2
>>> model = RiskBudgeting(
... risk_measure=RiskMeasure.CVAR,
... risk_budget=risk_budget,
... )
>>> model.fit(X)
RiskBudgeting(...)
>>> print(model.weights_)
[0.0623 0.0319 0.0347 ... 0.0502 0.0659 0.0595]
>>>
>>> portfolio = model.predict(X)
>>> print(portfolio.cvar)
0.0251...
References
----------
.. [1] "Constrained Risk Budgeting Portfolios: Theory, Algorithms, Applications",
Journal of Portfolio Management, Richard, J.-C., & Roncalli, T. (2019)
.. [2] "Portfolio Optimization: Theory and Application", Chapter 11,
Daniel P. Palomar (2025)
"""
def __init__(
self,
risk_measure: RiskMeasure = RiskMeasure.VARIANCE,
risk_budget: FloatArray | None = None,
prior_estimator: BasePrior | 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,
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,
min_return: skt.Target | None = None,
min_acceptable_return: skt.Target | None = None,
cvar_beta: float = 0.95,
evar_beta: float = 0.95,
cdar_beta: float = 0.95,
edar_beta: float = 0.95,
solver: str = "CLARABEL",
solver_params: dict | None = None,
scale_objective: float | None = None,
scale_constraints: float | None = None,
save_problem: bool = False,
raise_on_failure: bool = True,
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,
):
super().__init__(
risk_measure=risk_measure,
prior_estimator=prior_estimator,
min_weights=min_weights,
max_weights=max_weights,
budget=1,
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,
risk_free_rate=risk_free_rate,
min_acceptable_return=min_acceptable_return,
cvar_beta=cvar_beta,
evar_beta=evar_beta,
cdar_beta=cdar_beta,
edar_beta=edar_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,
raise_on_failure=raise_on_failure,
)
self.min_return = min_return
self.risk_budget = risk_budget
[docs]
def fit(self, X: ArrayLike, y=None, **fit_params) -> RiskBudgeting:
"""Fit the Risk Budgeting 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`.
Returns
-------
self : RiskBudgeting
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"
)
if not isinstance(self.risk_measure, RiskMeasure):
raise TypeError("risk_measure must be of type `RiskMeasure`")
# 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_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)
# Risk budget
risk_budget = self.risk_budget
if risk_budget is None:
risk_budget = np.ones(n_assets)
else:
risk_budget = self._clean_input(
self.risk_budget,
n_assets=n_assets,
fill_value=1e-10,
name="risk_budget",
)
risk_budget[risk_budget == 0] = 1e-10
# Variables
w = cp.Variable(n_assets)
factor = cp.Variable()
# Expected returns
expected_return = (
self._cvx_expected_return(return_distribution=return_distribution, w=w)
- self._cvx_transaction_cost(
return_distribution=return_distribution, w=w, factor=factor
)
- self._cvx_management_fee(return_distribution=return_distribution, w=w)
)
# risk budgeting constraint
constraints = [risk_budget @ cp.log(w) * self._scale_constraints >= 0]
# weight constraints
constraints += self._get_weight_constraints(
n_assets=n_assets,
w=w,
factor=factor,
allow_negative_weights=False,
return_distribution=return_distribution,
)
parameters_values = []
# min_return constraint
if self.min_return is not None:
parameter = cp.Parameter(nonneg=False)
constraints += [
expected_return * self._scale_constraints
>= parameter * factor * self._scale_constraints
]
parameters_values.append((parameter, self.min_return))
# risk and risk constraints
risk_func = getattr(self, f"_{self.risk_measure.value}_risk")
args = {}
for arg_name in args_names(risk_func):
if arg_name == "return_distribution":
args[arg_name] = return_distribution
elif arg_name == "w":
args[arg_name] = w
elif arg_name == "factor":
if self.risk_measure in [RiskMeasure.FIRST_LOWER_PARTIAL_MOMENT]:
args[arg_name] = factor
else:
args[arg_name] = cp.Constant(1)
else:
args[arg_name] = getattr(self, arg_name)
risk, constraints_i = risk_func(**args)
constraints += constraints_i
# custom objectives and constraints
custom_objective = self._get_custom_objective(w=w)
constraints += self._get_custom_constraints(w=w)
objective = cp.Minimize(
risk * self._scale_objective + custom_objective * self._scale_objective
)
# problem
# noinspection PyTypeChecker
problem = cp.Problem(objective, constraints)
# results
self._solve_problem(
problem=problem,
w=w,
factor=factor,
parameters_values=parameters_values,
expressions={
"expected_return": expected_return,
"risk": risk,
"factor": factor,
},
)
return self
