"""Mean Risk Optimization estimator."""
import warnings
# Copyright (c) 2023
# Author: Hugo Delatte <delatte.hugo@gmail.com>
# License: BSD 3 clause
# The optimization features are derived
# from Riskfolio-Lib, Copyright (c) 2020-2023, Dany Cajas, Licensed under BSD 3 clause.
import cvxpy as cp
import numpy as np
import numpy.typing as npt
import pandas as pd
import sklearn as sk
import sklearn.utils.metadata_routing as skm
import skfolio.typing as skt
from skfolio.measures import RiskMeasure
from skfolio.optimization.convex._base import ConvexOptimization, ObjectiveFunction
from skfolio.prior import BasePrior, EmpiricalPrior
from skfolio.uncertainty_set import BaseCovarianceUncertaintySet, BaseMuUncertaintySet
from skfolio.utils.tools import args_names, check_estimator
# noinspection PyUnresolvedReferences
_NON_ANNUALIZED_RISK_MEASURES = [rm for rm in RiskMeasure if not rm.is_annualized]
[docs]
class MeanRisk(ConvexOptimization):
r"""Mean-Risk Optimization estimator.
The below 4 objective functions can be optimized:
* Minimize Risk:
.. math:: \begin{cases}
\begin{aligned}
&\min_{w} & & risk_{i}(w) \\
&\text{s.t.} & & w^T \cdot \mu \ge min\_return \\
& & & A \cdot w \ge b \\
& & & risk_{j}(w) \le max\_risk_{j} \quad \forall \; j \ne i
\end{aligned}
\end{cases}
* Maximize Expected Return:
.. math:: \begin{cases}
\begin{aligned}
&\max_{w} & & w^T \cdot \mu \\
&\text{s.t.} & & risk_{i}(w) \le max\_risk_{i} \\
& & & A \cdot w \ge b \\
& & & risk_{j}(w) \le max\_risk_{j} \quad \forall \; j \ne i
\end{aligned}
\end{cases}
* Maximize Utility:
.. math:: \begin{cases}
\begin{aligned}
&\max_{w} & & w^T \cdot \mu - \lambda \times risk_{i}(w)\\
&\text{s.t.} & & risk_{i}(w) \le max\_risk_{i} \\
& & & w^T \cdot \mu \ge min\_return \\
& & & A \cdot w \ge b \\
& & & risk_{j}(w) \le max\_risk_{j} \quad \forall \; j \ne i
\end{aligned}
\end{cases}
* Maximize Ratio:
.. math:: \begin{cases}
\begin{aligned}
&\max_{w} & & \frac{w^T \cdot \mu - r_{f}}{risk_{i}(w)}\\
&\text{s.t.} & & risk_{i}(w) \le max\_risk_{i} \\
& & & w^T \cdot \mu \ge min\_return \\
& & & A \cdot w \ge b \\
& & & risk_{j}(w) \le max\_risk_{j} \quad \forall \; j \ne i
\end{aligned}
\end{cases}
With :math:`risk_{i}` 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, regularization, uncertainty set, and additional constraints can also be added
to the optimization problem (see the parameters description).
The assets expected returns, covariance matrix and returns are estimated from the
:ref:`prior estimator <prior>`.
Parameters
----------
objective_function : ObjectiveFunction, default=ObjectiveFunction.MINIMIZE_RISK
:class:`~skfolio.optimization.ObjectiveFunction` of the optimization.
Can be any of:
* MINIMIZE_RISK
* MAXIMIZE_RETURN
* MAXIMIZE_UTILITY
* MAXIMIZE_RATIO
The default is `ObjectiveFunction.MINIMIZE_RISK`.
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_RATIO
The default is `RiskMeasure.VARIANCE`.
risk_aversion : float, default=1.0
Risk aversion factor :math:`\lambda` of the utility function. Only used for
`objective_function=ObjectiveFunction.MAXIMIZE_UTILITY`.
The default value is `1.0`.
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,
returns and Cholesky decomposition of the covariance.
The default (`None`) is to use :class:`~skfolio.prior.EmpiricalPrior`.
efficient_frontier_size : int, optional
If provided, it represents the number of Pareto-optimal portfolios along the
efficient frontier to be computed. This parameter can only be used with
`objective_function = ObjectiveFunction.MINIMIZE_RISK`.
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 minium 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).
Examples:
* `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_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.
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.
cardinality : int, optional
Specifies the cardinality constraint to limit the number of invested assets
(non-zero weights). This feature requires a mixed-integer solver. For an
open-source option, we recommend using SCIP by setting `solver="SCIP"`.
To install it, use: `pip install cvxpy[SCIP]`. For commercial solvers,
supported options include MOSEK, GUROBI, or CPLEX.
group_cardinalities : dict[str, int], optional
A dictionary specifying cardinality constraints for specific groups of assets.
The keys represent group names (strings), and the values specify the maximum
number of assets allowed in each group. You must provide the groups using the
`groups` parameter. This requires a mixed-integer solver (see `cardinality`
for more details).
threshold_long : float | dict[str, float] | array-like of shape (n_assets, ), optional
Specifies the minimum weight threshold for assets in the portfolio to be
considered as a long position. Assets with weights below this threshold
will not be included as part of the portfolio's long positions. This
constraint can help eliminate insignificant allocations.
This requires a mixed-integer solver (see `cardinality` for more details).
It follows the same format as `min_weights` and `max_weights`.
threshold_short : float | dict[str, float] | array-like of shape (n_assets, ), optional
Specifies the maximum weight threshold for assets in the portfolio to be
considered as a short position. Assets with weights above this threshold
will not be included as part of the portfolio's short positions. This
constraint can help control the magnitude of short positions.
This requires a mixed-integer solver (see `cardinality` for more details).
It follows the same format as `min_weights` and `max_weights`.
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 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.
l1_coef : float, default=0.0
L1 regularization coefficient.
It is used to penalize the objective function by the L1 norm:
.. math:: l1\_coef \times \Vert w \Vert_{1} = l1\_coef \times \sum_{i=1}^{N} |w_{i}|
Increasing this coefficient will reduce the number of non-zero weights
(cardinality). It tends to increase robustness (out-of-sample stability) but
reduces diversification.
The default value is `0.0`.
l2_coef : float, default=0.0
L2 regularization coefficient.
It is used to penalize the objective function by the L2 norm:
.. math:: l2\_coef \times \Vert w \Vert_{2}^{2} = l2\_coef \times \sum_{i=1}^{N} w_{i}^2
It tends to increase robustness (out-of-sample stability).
The default value is `0.0`.
mu_uncertainty_set_estimator : BaseMuUncertaintySet, optional
:ref:`Mu Uncertainty set estimator <uncertainty_set_estimator>`.
If provided, the assets expected returns are modelled with an ellipsoidal
uncertainty set. It is called worst-case optimization and is a class of robust
optimization. It reduces the instability that arises from the estimation errors
of the expected returns.
The worst case portfolio expect return is:
.. math:: w^T \cdot \hat{\mu} - \kappa_{\mu} \lVert S_{\mu}^\frac{1}{2} \cdot w \rVert_{2}
with :math:`\kappa` the size of the ellipsoid (confidence region) and
:math:`S` its shape.
The default (`None`) means that no uncertainty set is used.
covariance_uncertainty_set_estimator : BaseCovarianceUncertaintySet, optional
:ref:`Covariance Uncertainty set estimator <uncertainty_set_estimator>`.
If provided, the assets covariance matrix is modelled with an ellipsoidal
uncertainty set. It is called worst-case optimization and is a class of robust
optimization. It reduces the instability that arises from the estimation errors
of the covariance matrix.
The default (`None`) means that no uncertainty set is used.
linear_constraints : array-like of shape (n_constraints,), optional
Linear constraints.
The linear constraints must match any of following patterns:
* "2.5 * ref1 + 0.10 * ref2 + 0.0013 <= 2.5 * ref3"
* "ref1 >= 2.9 * ref2"
* "ref1 == ref2"
* "ref1 >= ref1"
With "ref1", "ref2" ... the assets names or the groups names provided
in the parameter `groups`. Assets names can be referenced without the need of
`groups` if the input `X` of the `fit` method is a DataFrame with these
assets names in columns.
Examples:
* "SPX >= 0.10" --> SPX weight must be greater than 10% (note that you can also use `min_weights`)
* "SX5E + TLT >= 0.2" --> the sum of SX5E and TLT weights must be greater than 20%
* "US == 0.7" --> the sum of all US weights must be equal to 70%
* "Equity == 3 * Bond" --> the sum of all Equity weights must be equal to 3 times the sum of all Bond weights.
* "2*SPX + 3*Europe <= Bond + 0.05" --> mixing assets and group constraints
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.
Examples:
* 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`.
max_tracking_error : float, optional
Upper bound constraint on the tracking error.
The tracking error is defined as the RMSE (root-mean-square error) of the
portfolio returns compared to a target returns. If `max_tracking_error` is
provided, the target returns `y` must be provided in the `fit` method.
max_turnover : float, optional
Upper bound constraint of the turnover.
The turnover is defined as the absolute difference between the portfolio weights
and the `previous_weights`. Note that another way to control for turnover is by
using the `transaction_costs` parameter.
max_mean_absolute_deviation : float | array-like of shape (n_optimization), optional
Upper bound constraint on the Mean Absolute Deviation.
max_first_lower_partial_moment : float | array-like of shape (n_optimization), optional
Upper bound constraint on the First Lower Partial Moment.
max_variance : float | array-like of shape (n_optimization), optional
Upper bound constraint on the Variance.
max_standard_deviation : float | array-like of shape (n_optimization), optional
Upper bound constraint on the Standard deviation.
max_semi_variance : float | array-like of shape (n_optimization), optional
Upper bound constraint on the Semi-Variance (Second Lower Partial Moment or
Downside Variance).
max_semi_deviation : float | array-like of shape (n_optimization), optional
Upper bound constraint on the Semi-Standard deviation.
max_worst_realization : float | array-like of shape (n_optimization), optional
Upper bound constraint on the Worst Realization (Worst Return).
max_cvar : float | array-like of shape (n_optimization), optional
Upper bound constraint on the CVaR (Conditional Value-at-Risk or Expected
Shortfall).
max_evar : float | array-like of shape (n_optimization), optional
Upper bound constraint on the EVaR (Entropic Value at Risk).
max_max_drawdown : float | array-like of shape (n_optimization), optional
Upper bound constraint on the Maximum Drawdown.
max_average_drawdown : float | array-like of shape (n_optimization), optional
Upper bound constraint on the Average Drawdown.
max_cdar : float | array-like of shape (n_optimization), optional
Upper bound constraint on the CDaR (Conditional Drawdown at Risk).
max_edar : float | array-like of shape (n_optimization), optional
Upper bound constraint on the EDaR (Entropic Drawdown at Risk).
max_ulcer_index : float | array-like of shape (n_optimization), optional
Upper bound constraint on the Ulcer Index.
max_gini_mean_difference : float | array-like of shape (n_optimization), optional
Upper bound constraint on the Gini Mean Difference.
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
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 is a function that must take as argument the weights `w` and returns a
CVPXY expression or a list of CVPXY expressions.
overwrite_expected_return : Callable[[cp.Variable], cp.Expression], optional
Overwrite the expected return :math:`\mu \cdot w` with a custom expression.
It is a function that must take as argument the weights `w` and returns a
CVPXY expression.
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 "CLARABEL", `{"numerics/feastol": 1e-8, "limits/gap": 1e-8}` for SCIP
and the solver default otherwise.
For more details about solver arguments, check the CVXPY documentation:
https://www.cvxpy.org/tutorial/solvers
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`.
raise_on_failure : bool, default=True
If this is set to True, an error is raised when the optimization fail otherwise
it passes with a warning.
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.
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`.
mu_uncertainty_set_estimator_ : BaseMuUncertaintySet
Fitted `mu_uncertainty_set_estimator` if provided.
covariance_uncertainty_set_estimator_ : BaseCovarianceUncertaintySet
Fitted `covariance_uncertainty_set_estimator` if provided.
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.
"""
def __init__(
self,
objective_function: ObjectiveFunction = ObjectiveFunction.MINIMIZE_RISK,
risk_measure: RiskMeasure = RiskMeasure.VARIANCE,
risk_aversion: float = 1.0,
efficient_frontier_size: int | None = None,
prior_estimator: BasePrior | None = None,
min_weights: skt.MultiInput | None = 0.0,
max_weights: skt.MultiInput | None = 1.0,
budget: float | None = 1.0,
min_budget: float | None = None,
max_budget: float | None = None,
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,
mu_uncertainty_set_estimator: BaseMuUncertaintySet | None = None,
covariance_uncertainty_set_estimator: (
BaseCovarianceUncertaintySet | None
) = None,
risk_free_rate: float = 0.0,
min_return: skt.Target | None = None,
max_tracking_error: skt.Target | None = None,
max_turnover: skt.Target | None = None,
max_mean_absolute_deviation: skt.Target | None = None,
max_first_lower_partial_moment: skt.Target | None = None,
max_variance: skt.Target | None = None,
max_standard_deviation: skt.Target | None = None,
max_semi_variance: skt.Target | None = None,
max_semi_deviation: skt.Target | None = None,
max_worst_realization: skt.Target | None = None,
max_cvar: skt.Target | None = None,
max_evar: skt.Target | None = None,
max_max_drawdown: skt.Target | None = None,
max_average_drawdown: skt.Target | None = None,
max_cdar: skt.Target | None = None,
max_edar: skt.Target | None = None,
max_ulcer_index: skt.Target | None = None,
max_gini_mean_difference: 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,
):
super().__init__(
risk_measure=risk_measure,
prior_estimator=prior_estimator,
mu_uncertainty_set_estimator=mu_uncertainty_set_estimator,
covariance_uncertainty_set_estimator=covariance_uncertainty_set_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,
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,
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,
raise_on_failure=raise_on_failure,
add_objective=add_objective,
add_constraints=add_constraints,
overwrite_expected_return=overwrite_expected_return,
portfolio_params=portfolio_params,
)
self.objective_function = objective_function
self.risk_aversion = risk_aversion
self.efficient_frontier_size = efficient_frontier_size
self.min_return = min_return
self.max_tracking_error = max_tracking_error
self.max_turnover = max_turnover
self.max_mean_absolute_deviation = max_mean_absolute_deviation
self.max_first_lower_partial_moment = max_first_lower_partial_moment
self.max_variance = max_variance
self.max_standard_deviation = max_standard_deviation
self.max_semi_variance = max_semi_variance
self.max_semi_deviation = max_semi_deviation
self.max_worst_realization = max_worst_realization
self.max_cvar = max_cvar
self.max_evar = max_evar
self.max_max_drawdown = max_max_drawdown
self.max_average_drawdown = max_average_drawdown
self.max_cdar = max_cdar
self.max_edar = max_edar
self.max_ulcer_index = max_ulcer_index
self.max_gini_mean_difference = max_gini_mean_difference
def _validation(self) -> None:
"""Validate the input parameters"""
if not isinstance(self.risk_measure, RiskMeasure):
raise TypeError("risk_measure must be of type `RiskMeasure`")
if not isinstance(self.objective_function, ObjectiveFunction):
raise TypeError("objective_function must be of type `ObjectiveFunction`")
if self.efficient_frontier_size is not None:
if self.efficient_frontier_size <= 1:
raise ValueError(
"`efficient_frontier_size` must be strictly greater than one"
)
if self.objective_function != ObjectiveFunction.MINIMIZE_RISK:
raise ValueError(
"`efficient_frontier_size` must be used only with "
"`objective_function = ObjectiveFunction.MINIMIZE_RISK`"
)
[docs]
def fit(
self, X: npt.ArrayLike, y: npt.ArrayLike | None = None, **fit_params
) -> "MeanRisk":
"""Fit the Mean-Risk 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_targets), optional
Price returns of factors or a target benchmark.
The default is `None`.
Returns
-------
self : MeanRisk
Fitted estimator.
"""
routed_params = skm.process_routing(self, "fit", **fit_params)
self._check_feature_names(X, reset=True)
# Validate
self._validation()
# 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)
prior_model = self.prior_estimator_.prior_model_
n_observations, n_assets = prior_model.returns.shape
# set solvers params
match self.solver:
case "CLARABEL":
self._set_solver_params(
default={"tol_gap_abs": 1e-9, "tol_gap_rel": 1e-9}
)
case "SCIP":
self._set_solver_params(
default={"numerics/feastol": 1e-8, "limits/gap": 1e-8}
)
case _:
self._set_solver_params(default=None)
# set scales and check measure
if self.objective_function == ObjectiveFunction.MAXIMIZE_RATIO:
if self.overwrite_expected_return is not None:
if self.risk_measure == RiskMeasure.VARIANCE:
warnings.warn(
"When selecting 'MAXIMIZE_RATIO' with 'VARIANCE', the "
"optimization will return the maximum Sharpe Ratio portfolio. "
"This is because the mean/variance ratio is not a "
"1-homogeneous function, unlike the mean/std. To suppress this"
"warning, replace 'VARIANCE' by 'STANDARD_DEVIATION'",
stacklevel=2,
)
elif self.risk_measure == RiskMeasure.SEMI_VARIANCE:
warnings.warn(
"When selecting 'MAXIMIZE_RATIO' with 'SEMI_VARIANCE', the "
"optimization will return the maximum Sortino Ratio portfolio. "
"This is because the mean/semi-variance ratio is not a "
"1-homogeneous function, unlike the mean/semi-std ratio. To "
"suppress this warning, replace 'SEMI_VARIANCE' by "
"'SEMI_DEVIATION'",
stacklevel=2,
)
self._set_scale_objective(default=1)
self._set_scale_constraints(default=1)
else:
match self.risk_measure:
case (
RiskMeasure.MEAN_ABSOLUTE_DEVIATION
| RiskMeasure.FIRST_LOWER_PARTIAL_MOMENT
| RiskMeasure.CVAR
| RiskMeasure.WORST_REALIZATION
| RiskMeasure.AVERAGE_DRAWDOWN
| RiskMeasure.MAX_DRAWDOWN
| RiskMeasure.CDAR
| RiskMeasure.ULCER_INDEX
):
self._set_scale_objective(default=1e-1)
self._set_scale_constraints(default=1e2)
case RiskMeasure.EVAR:
self._set_scale_objective(default=1)
self._set_scale_constraints(default=1e-2)
case RiskMeasure.EDAR:
self._set_scale_objective(default=1)
self._set_scale_constraints(default=1e2)
case _:
self._set_scale_objective(default=1)
self._set_scale_constraints(default=1)
# Init weight variable and constraints
w = cp.Variable(n_assets)
constraints = []
if self.objective_function == ObjectiveFunction.MAXIMIZE_RATIO:
factor = cp.Variable()
else:
factor = cp.Constant(1)
# Mu uncertainty set
if self.mu_uncertainty_set_estimator is None:
mu_uncertainty_set = cp.Constant(0)
else:
# noinspection PyTypeChecker
self.mu_uncertainty_set_estimator_ = sk.clone(
self.mu_uncertainty_set_estimator
)
self.mu_uncertainty_set_estimator_.fit(
X, y, **routed_params.mu_uncertainty_set_estimator.fit
)
mu_uncertainty_set = self._cvx_mu_uncertainty_set(
mu_uncertainty_set=self.mu_uncertainty_set_estimator_.uncertainty_set_,
w=w,
)
# Expected returns
expected_return = (
self._cvx_expected_return(prior_model=prior_model, w=w)
- self._cvx_transaction_cost(prior_model=prior_model, w=w, factor=factor)
- self._cvx_management_fee(prior_model=prior_model, w=w)
- mu_uncertainty_set
)
# Regularization
regularization = self._cvx_regularization(w=w)
# Tracking error
if self.max_tracking_error is not None:
if y is None:
raise ValueError(
"If `max_tracking_error` is provided, `y` must also be provided"
)
if isinstance(y, pd.DataFrame):
if y.shape[1] > 1:
raise ValueError(
"If `max_tracking_error` is provided, `y` must be a"
" 1d-array, a single-column DataFrame or a Series"
)
y = y[y.columns[0]]
_, y = self._validate_data(X, y)
tracking_error = self._tracking_error(
prior_model=prior_model, w=w, y=y, factor=factor
)
constraints += [
tracking_error * self._scale_constraints
<= self.max_tracking_error * self._scale_constraints
]
# Turnover
if self.max_turnover is not None:
turnover = self._turnover(n_assets=n_assets, w=w, factor=factor)
constraints += [
turnover * self._scale_constraints
<= self.max_turnover * factor * self._scale_constraints
]
# weight constraints
constraints += self._get_weight_constraints(
n_assets=n_assets, w=w, factor=factor
)
parameters_values = []
# Efficient frontier
if self.efficient_frontier_size is not None:
# We find the lower and upper bounds of the expected returns.
# noinspection PyTypeChecker
model: MeanRisk = sk.clone(self)
# noinspection PyTypeChecker
model.set_params(
objective_function=ObjectiveFunction.MINIMIZE_RISK,
efficient_frontier_size=None,
portfolio_params=dict(annualized_factor=1),
)
model.fit(X, y, **fit_params)
min_return = model.problem_values_["expected_return"]
# noinspection PyTypeChecker
model.set_params(objective_function=ObjectiveFunction.MAXIMIZE_RETURN)
model.fit(X, y, **fit_params)
max_return = model.problem_values_["expected_return"]
if max_return <= 0:
raise ValueError(
"Unable to compute the Efficient Frontier with only negative"
" expected returns"
)
targets = np.linspace(
max(min_return, 1e-10) * 1.01,
max_return,
num=self.efficient_frontier_size,
)
parameter = cp.Parameter(nonneg=False)
constraints += [expected_return >= parameter * factor]
parameters_values.append((parameter, targets))
# 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 = None
for r_m in _NON_ANNUALIZED_RISK_MEASURES:
risk_limit = getattr(self, f"max_{r_m.value}")
if self.risk_measure == r_m or risk_limit is not None:
# Add covariance uncertainty set if provided
if (
r_m == RiskMeasure.VARIANCE
and self.covariance_uncertainty_set_estimator is not None
):
risk_func = self._worst_case_variance_risk
else:
risk_func = getattr(self, f"_{r_m.value}_risk")
args = {}
for arg_name in args_names(risk_func):
if arg_name == "prior_model":
args[arg_name] = prior_model
elif arg_name == "w":
args[arg_name] = w
elif arg_name == "factor":
args[arg_name] = factor
elif arg_name == "covariance_uncertainty_set":
# noinspection PyTypeChecker
self.covariance_uncertainty_set_estimator_ = sk.clone(
self.covariance_uncertainty_set_estimator
)
self.covariance_uncertainty_set_estimator_.fit(
X,
y,
**routed_params.covariance_uncertainty_set_estimator.fit,
)
args[arg_name] = (
self.covariance_uncertainty_set_estimator_.uncertainty_set_
)
else:
args[arg_name] = getattr(self, arg_name)
risk_i, constraints_i = risk_func(**args)
constraints += constraints_i
if risk_limit is not None:
parameter = cp.Parameter(nonneg=True)
constraints += [
risk_i * self._scale_constraints
<= parameter * factor * self._scale_constraints
]
parameters_values.append((parameter, risk_limit))
if self.risk_measure == r_m:
risk = risk_i
# custom objectives and constraints
custom_objective = self._get_custom_objective(w=w)
constraints += self._get_custom_constraints(w=w)
match self.objective_function:
case ObjectiveFunction.MAXIMIZE_RETURN:
objective = cp.Maximize(
expected_return * self._scale_objective
- regularization * self._scale_objective
+ custom_objective * self._scale_objective
)
case ObjectiveFunction.MINIMIZE_RISK:
objective = cp.Minimize(
risk * self._scale_objective
+ regularization * self._scale_objective
+ custom_objective * self._scale_objective
)
case ObjectiveFunction.MAXIMIZE_UTILITY:
objective = cp.Maximize(
expected_return * self._scale_objective
- self.risk_aversion * risk * self._scale_objective
- regularization * self._scale_objective
+ custom_objective * self._scale_objective
)
case ObjectiveFunction.MAXIMIZE_RATIO:
homogenization_factor = _optimal_homogenization_factor(
mu=prior_model.mu
)
if expected_return.is_affine():
# Charnes-Cooper's variable transformation for Fractional
# Programming problem Max(f1/f2) with f2 linear and with
# 1-homogeneous function (homogeneous technique)
constraints += [
expected_return * self._scale_constraints
- cp.Constant(self.risk_free_rate)
* factor
* self._scale_constraints
== cp.Constant(homogenization_factor) * self._scale_constraints
]
else:
# Schaible's generalization of Charnes-Cooper's variable
# transformation for Fractional Programming problem :Max(f1/f2)
# with f1 concave instead of linear and with 1-homogeneous function.
# (homogeneous technique)
# Schaible,"Parameter-free Convex Equivalent and Dual Programs of
# Fractional Programming Problems".
# The condition to work is f1 >= 0, so we need to raise an user
# warning when it's not the case.
# TODO: raise user warning when f1<0
constraints += [
expected_return * self._scale_constraints
- cp.Constant(self.risk_free_rate)
* factor
* self._scale_constraints
>= cp.Constant(homogenization_factor) * self._scale_constraints
]
objective = cp.Minimize(
risk * self._scale_objective
+ regularization * self._scale_objective
+ custom_objective * self._scale_objective
)
case _:
raise ValueError(
f"objective_function {self.objective_function} is not valid"
)
# problem
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,
"mu_uncertainty_set": mu_uncertainty_set,
"regularization": regularization,
"factor": factor,
},
)
return self
def _optimal_homogenization_factor(mu: np.ndarray) -> float:
"""
Compute the optimal homogenization factor for ratio optimization based on expected
returns.
While a default value of 1 is commonly used in textbooks for simplicity,
fine-tuning this factor based on the underlying data can enhance convergence.
Additionally, using a data-driven approach to determine this factor can improve the
robustness of certain constraints, such as the calibration of big M methods.
Parameters
----------
mu : ndarray of shape (n_assets,)
Vector of expected returns.
Returns
-------
value : float
Homogenization factor.
"""
return min(1e3, max(1e-3, np.mean(np.abs(mu))))