<a id="skfolio-optimization-hierarchicalequalriskcontribution"></a>

# skfolio.optimization.HierarchicalEqualRiskContribution

<a id="skfolio.optimization.HierarchicalEqualRiskContribution"></a>

### *class* skfolio.optimization.HierarchicalEqualRiskContribution(risk_measure=Variance, prior_estimator=None, distance_estimator=None, hierarchical_clustering_estimator=None, min_weights=0.0, max_weights=1.0, solver='CLARABEL', solver_params=None, transaction_costs=0.0, management_fees=0.0, previous_weights=None, portfolio_params=None, fallback=None, raise_on_failure=True)

Hierarchical Equal Risk Contribution estimator.

The Hierarchical Equal Risk Contribution is a portfolio optimization method
developed by Thomas Raffinot [[2]](#rc628ffe0b6ca-2).

This algorithm uses a distance matrix to compute hierarchical clusters using the
Hierarchical Tree Clustering algorithm. It then computes, for each cluster, the
total cluster risk of an inverse-risk allocation.

The final step is the top-down recursive division of the dendrogram, where the
assets weights are updated using a naive risk parity within clusters.

It differs from the Hierarchical Risk Parity by exploiting the dendrogram shape
during the top-down recursive division instead of bisecting it.

#### NOTE
The default linkage method is set to the Ward variance minimization algorithm,
which is more stable and has better properties than the single-linkage
method [[4]](#rc628ffe0b6ca-4).

Also, the initial paper does not provide an algorithm for handling weight
constraints, and no standard solution currently exists.
In contrast to HRP (Hierarchical Risk Parity), where weight constraints
can be applied to the split factor at each bisection step, HERC
(Hierarchical Equal Risk Contribution) cannot incorporate weight constraints
during the intermediate steps of the allocation. Therefore, in HERC, the
weight constraints must be enforced after the top-down allocation has been
completed.
In skfolio, we minimize the relative deviation of the final weights from
the initial weights. This is formulated as a convex optimization problem:

$$
\begin{cases}
\begin{aligned}
&\min_{w} & & \Vert \frac{w - w_{init}}{w_{init}} \Vert_{2}^{2} \\
&\text{s.t.} & & \sum_{i=1}^{N} w_{i} = 1 \\
& & & w_{min} \leq w_i \leq w_{max}, \quad \forall i
\end{aligned}
\end{cases}

$$

The reason for minimizing the relative deviation (as opposed to the absolute
deviation) is that we want to limit the impact on the risk contribution of
each asset. Since HERC allocates inversely to risk, adjusting the weights
based on relative deviation ensures that the assets’ risk contributions
remain proportionally consistent with the initial allocation.

* **Parameters:**
  **risk_measure** *RiskMeasure or ExtraRiskMeasure, default=RiskMeasure.VARIANCE*
  : `RiskMeasure` or `ExtraRiskMeasure`
    of the optimization.
    Can be any of:
    > * MEAN_ABSOLUTE_DEVIATION
    > * FIRST_LOWER_PARTIAL_MOMENT
    > * VARIANCE
    > * SEMI_VARIANCE
    > * CVAR
    > * EVAR
    > * WORST_REALIZATION
    > * CDAR
    > * MAX_DRAWDOWN
    > * AVERAGE_DRAWDOWN
    > * EDAR
    > * ULCER_INDEX
    > * GINI_MEAN_DIFFERENCE_RATIO
    > * VALUE_AT_RISK
    > * DRAWDOWN_AT_RISK
    > * ENTROPIC_RISK_MEASURE
    > * FOURTH_CENTRAL_MOMENT
    > * FOURTH_LOWER_PARTIAL_MOMENT
    <br/>
    The default is `RiskMeasure.VARIANCE`.

  **prior_estimator** *BasePrior, optional*
  : [Prior estimator](https://skfolio.org/user_guide/prior.html.md#prior).
    The prior estimator is used to estimate the [`ReturnDistribution`](https://skfolio.org/generated/skfolio.prior.ReturnDistribution.html.md#skfolio.prior.ReturnDistribution)
    containing estimates of expected asset returns, covariance matrix and
    returns. The moments and returns estimations are used for the risk computation
    and the returns estimation are used by the distance matrix estimator.
    The default (`None`) is to use [`EmpiricalPrior`](https://skfolio.org/generated/skfolio.prior.EmpiricalPrior.html.md#skfolio.prior.EmpiricalPrior).

  **distance_estimator** *BaseDistance, optional*
  : [Distance estimator](https://skfolio.org/user_guide/distance.html.md#distance).
    The distance estimator is used to estimate the codependence and the distance
    matrix needed for the computation of the linkage matrix.
    The default (`None`) is to use [`PearsonDistance`](https://skfolio.org/generated/skfolio.distance.PearsonDistance.html.md#skfolio.distance.PearsonDistance).

  **hierarchical_clustering_estimator** *HierarchicalClustering, optional*
  : [Hierarchical Clustering estimator](https://skfolio.org/user_guide/cluster.html.md#hierarchical-clustering).
    The hierarchical clustering estimator is used to compute the linkage matrix
    and the hierarchical clustering of the assets based on the distance matrix.
    The default (`None`) is to use
    [`HierarchicalClustering`](https://skfolio.org/generated/skfolio.cluster.HierarchicalClustering.html.md#skfolio.cluster.HierarchicalClustering).

  **min_weights** *float | dict[str, float] | array-like of shape (n_assets, ), default=0.0*
  : Minimum assets weights (weights lower bounds). The default is 0.0 (no short
    selling). Negative weights are not allowed. If a float is provided, it is
    applied to each asset. `None` is equivalent to the default `0.0`. If a
    dictionary is provided, its (key/value) pair must be the (asset name/asset
    minimum weight) and the input `X` of the `fit` methods must be a DataFrame with
    the asset names in columns. When using a dictionary, assets values that are not
    provided are assigned the default  minimum weight of `0.0`.
    <br/>
    Example:
    > * `min_weights = 0.0` –> long only portfolio (default).
    > * `min_weights = {"SX5E": 0.1, "SPX": 0.2}`
    > * `min_weights = [0.1, 0.2]`

  **max_weights** *float | dict[str, float] | array-like of shape (n_assets, ), default=1.0*
  : Maximum assets weights (weights upper bounds). The default is 1.0 (each asset
    is below 100%). Weights above 1.0 are not allowed. If a float is provided, it is
    applied to each asset. `None` is equivalent to the default `1.0`. If a
    dictionary is provided, its (key/value) pair must be the (asset name/asset
    maximum weight) and the input `X` of the `fit` method must be a DataFrame with
    the asset names in columns. When using a dictionary, assets values that are not
    provided are assigned the default maximum weight of `1.0`.
    <br/>
    Example:
    > * `max_weights = 1.0` –> each weight  must be below 100% (default).
    > * `max_weights = 0.5` –> each weight must be below 50%.
    > * `max_weights = {"SX5E": 0.8, "SPX": 0.9}`
    > * `max_weights = [0.8, 0.9]`

  **transaction_costs** *float | dict[str, float] | array-like of shape (n_assets, ), default=0.0*
  : Transaction costs of the assets. It is used to add linear transaction costs to
    the optimization problem:
    $$
    total\_cost = \sum_{i=1}^{N} c_{i} \times |w_{i} - w\_prev_{i}|
    <br/>
    $$
    <br/>
    with $c_{i}$ the transaction cost of asset i, $w_{i}$ its weight
    and $w\_prev_{i}$ its previous weight (defined in `previous_weights`).
    The float $total\_cost$ is impacting the portfolio expected return in the
    optimization:
    $$
    expected\_return = \mu^{T} \cdot w - total\_cost
    <br/>
    $$
    <br/>
    with $\mu$ the vector of assets’ expected returns and $w$ the
    vector of assets weights.
    <br/>
    If a float is provided, it is applied to each asset.
    If a dictionary is provided, its (key/value) pair must be the
    (asset name/asset cost) and the input `X` of the `fit` method must be a
    DataFrame with the asset names in columns.
    The default value is `0.0`.
    <br/>
    #### WARNING
    Based on the above formula, the periodicity of the transaction costs
    must match the periodicity of $\mu$. 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 [Periodicity Convention](https://skfolio.org/user_guide/data_preparation.html.md#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:
    $$
    total\_fee = \sum_{i=1}^{N} f_{i} \times w_{i}
    <br/>
    $$
    <br/>
    with $f_{i}$ the management fee of asset i and $w_{i}$ its weight.
    The float $total\_fee$ is impacting the portfolio expected return in the
    optimization:
    $$
    expected\_return = \mu^{T} \cdot w - total\_fee
    <br/>
    $$
    <br/>
    with $\mu$ the vector of assets’ expected returns and $w$ the vector
    of assets weights.
    <br/>
    If a float is provided, it is applied to each asset.
    If a dictionary is provided, its (key/value) pair must be the
    (asset name/asset fee) and the input `X` of the `fit` method must be a
    DataFrame with the asset names in columns.
    The default value is `0.0`.
    <br/>
    #### WARNING
    Based on the above formula, the periodicity of the management fees
    must match the periodicity of $\mu$. 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).
    <br/>
    #### 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
    $\mu$ parameter using for example Shrinkage estimators, this approach
    would mix a deterministic value with an uncertain one leading to unwanted
    bias in the management fees.

  **previous_weights** *float | dict[str, float] | array-like of shape (n_assets, ), optional*
  : Previous weights of the assets. Previous weights are used to compute the
    portfolio total cost. If a float is provided, it is applied to each asset.
    If a dictionary is provided, its (key/value) pair must be the
    (asset name/asset previous weight) and the input `X` of the `fit` method must
    be a DataFrame with the asset names in columns.
    The default (`None`) means no previous weights.
    Additionally, when `fallback="previous_weights"`, failures will fall back to
    these weights if provided.

  **solver** *str, default=”CLARABEL”*
  : The solver used for the weights constraints optimization. The default is
    “CLARABEL” which is written in Rust and has better numerical stability and
    performance than ECOS and SCS.
    For more details about available solvers, check the CVXPY documentation:
    [https://www.cvxpy.org/tutorial/advanced/index.html#choosing-a-solver](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 the CVXPY default.
    For more details about solver arguments, check the CVXPY documentation:
    [https://www.cvxpy.org/tutorial/advanced/index.html#setting-solver-options](https://www.cvxpy.org/tutorial/advanced/index.html#setting-solver-options)

  **portfolio_params** *dict, optional*
  : Portfolio parameters forwarded to the resulting `Portfolio` in `predict`.
    If not provided and if available on the estimator, the following
    attributes are propagated to the portfolio by default: `name`,
    `transaction_costs`, `management_fees`, `previous_weights` and `risk_free_rate`.

  **fallback** *BaseOptimization | “previous_weights” | list[BaseOptimization | “previous_weights”], optional*
  : Fallback estimator or a list of estimators to try, in order, when the primary
    optimization raises during `fit`. Alternatively, use `"previous_weights"` 
    (alone or in a list) to fall back to the estimator’s `previous_weights`.
    When a fallback succeeds, its fitted `weights_` are copied back to the primary 
    estimator so that `fit` still returns the original instance. For traceability, 
    `fallback_` stores the successful estimator (or the string `"previous_weights"`)
    and `fallback_chain_` stores each attempt with the associated outcome.

  **raise_on_failure** *bool, default=True*
  : Controls error handling when fitting fails.
    If True, any failure during `fit` is raised immediately, no `weights_` are
    set and subsequent calls to `predict` will raise a `NotFittedError`.
    If False, errors are not raised; instead, a warning is emitted, `weights_`
    is set to `None` and subsequent calls to `predict` will return a
    `FailedPortfolio`. When fallbacks are specified, this behavior applies only
    after all fallbacks have been exhausted.
* **Attributes:**
  **weights_** *ndarray of shape (n_assets,)*
  : Weights of the assets.

  **distance_estimator_** *BaseDistance*
  : Fitted `distance_estimator`.

  **hierarchical_clustering_estimator_** *HierarchicalClustering*
  : Fitted `hierarchical_clustering_estimator`.

  **n_features_in_** *int*
  : Number of assets seen during `fit`.

  **feature_names_in_** *ndarray of shape (`n_features_in_`,)*
  : Names of assets seen during `fit`. Defined only when `X`
    has asset names that are all strings.

  **fallback_** *BaseOptimization | “previous_weights” | None*
  : The fallback estimator instance, or the string `"previous_weights"`, that
    produced the final result. `None` if no fallback was used.

  **fallback_chain_** *list[tuple[str, str]] | None*
  : Sequence describing the optimization fallback attempts. Each element is a
    pair `(estimator_repr, outcome)` where `estimator_repr` is the string
    representation of the primary estimator or a fallback (e.g. `"EqualWeighted()"`,
    `"previous_weights"`), and `outcome` is `"success"` if that step produced
    a valid solution, otherwise the stringified error message. For successful
    fits without any fallback, this is `None`.

  **error_** *str | list[str] | None*
  : Captured error message(s) when `fit` fails. For multi-portfolio outputs
    (`weights_` is 2D), this is a list aligned with portfolios.

### Methods

| [`fit`](#skfolio.optimization.HierarchicalEqualRiskContribution.fit)(X[, y])            | Fit the Hierarchical Equal Risk Contribution estimator.                                                                         |
|-------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------|
| [`fit_predict`](#skfolio.optimization.HierarchicalEqualRiskContribution.fit_predict)(X)         | Perform `fit` on `X` and returns the predicted `Portfolio` or `Population` of `Portfolio` on `X` based on the fitted `weights`. |
| [`get_metadata_routing`](#skfolio.optimization.HierarchicalEqualRiskContribution.get_metadata_routing)() | Get metadata routing of this object.                                                                                            |
| [`get_params`](#skfolio.optimization.HierarchicalEqualRiskContribution.get_params)([deep])     | Get parameters for this estimator.                                                                                              |
| [`predict`](#skfolio.optimization.HierarchicalEqualRiskContribution.predict)(X)             | Predict the `Portfolio` or a `Population` of portfolios on `X`.                                                                 |
| [`score`](#skfolio.optimization.HierarchicalEqualRiskContribution.score)(X[, y])          | Prediction score using the Sharpe Ratio.                                                                                        |
| [`set_params`](#skfolio.optimization.HierarchicalEqualRiskContribution.set_params)(\*\*params) | Set the parameters of this estimator.                                                                                           |

### Notes

All estimators should specify all parameters as explicit keyword arguments in
`__init__` (no `*args` or `**kwargs`), following scikit-learn conventions.

### References

* <a id='rc628ffe0b6ca-1'>**[1]**</a> “Hierarchical clustering-based asset allocation”, The Journal of Portfolio Management, Thomas Raffinot  (2017).
* <a id='rc628ffe0b6ca-2'>**[2]**</a> “The hierarchical equal risk contribution portfolio”, Thomas Raffinot (2018).
* <a id='rc628ffe0b6ca-3'>**[3]**</a> “Application of two-order difference to gap statistic”. Yue, Wang & Wei (2009).
* <a id='rc628ffe0b6ca-4'>**[4]**</a> “A review of two decades of correlations, hierarchies, networks and clustering in financial markets”, Gautier Marti, Frank Nielsen, Mikołaj Bińkowski, Philippe Donnat (2020).

<a id="skfolio.optimization.HierarchicalEqualRiskContribution.fit"></a>

#### fit(X, y=None, \*\*fit_params)

Fit the Hierarchical Equal Risk Contribution estimator.

* **Parameters:**
  **X** *array-like of shape (n_observations, n_assets)*
  : Price returns of the assets.

  **y** *Ignored*
  : Not used, present for API consistency by convention.

  **\*\*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 [Metadata Routing User Guide](https://skfolio.org/user_guide/metadata_routing.html.md#metadata-routing) for
    more details.
* **Returns:**
  **self** *HierarchicalEqualRiskContribution*
  : Fitted estimator.

<a id="skfolio.optimization.HierarchicalEqualRiskContribution.fit_predict"></a>

#### fit_predict(X)

Perform `fit` on `X` and returns the predicted `Portfolio` or
`Population` of `Portfolio` on `X` based on the fitted `weights`.
For factor models, use `fit(X, factors=...)` then `predict(X)` separately.

If fitting fails and `raise_on_failure=False`, this returns a
`FailedPortfolio`.

* **Parameters:**
  **X** *array-like of shape (n_observations, n_assets)*
  : Price returns of the assets.
* **Returns:**
  Portfolio | Population
  : The predicted `Portfolio` or `Population` based on the fitted `weights`.

<a id="skfolio.optimization.HierarchicalEqualRiskContribution.get_metadata_routing"></a>

#### get_metadata_routing()

Get metadata routing of this object.

Please check [User Guide](https://skfolio.org/user_guide/metadata_routing.html.md#metadata-routing) on how the routing
mechanism works.

* **Returns:**
  **routing** *MetadataRequest*
  : A `MetadataRequest` encapsulating
    routing information.

<a id="skfolio.optimization.HierarchicalEqualRiskContribution.get_params"></a>

#### get_params(deep=True)

Get parameters for this estimator.

* **Parameters:**
  **deep** *bool, default=True*
  : If True, will return the parameters for this estimator and
    contained subobjects that are estimators.
* **Returns:**
  **params** *dict*
  : Parameter names mapped to their values.

<a id="skfolio.optimization.HierarchicalEqualRiskContribution.needs_previous_weights"></a>

#### *property* needs_previous_weights

Whether `previous_weights` must be propagated between folds/rebalances.

Used by `cross_val_predict` and `online_predict` to decide whether to run
sequentially and pass the weights from the previous rebalancing to the next.
This is `True` when `portfolio_params` sets `weight_drift=True`, or when
transaction costs, a maximum turnover, or a fallback depending on
`previous_weights` are present.

<a id="skfolio.optimization.HierarchicalEqualRiskContribution.predict"></a>

#### predict(X)

Predict the `Portfolio` or a `Population` of portfolios on `X`.

Optimization estimators can return a 1D or a 2D array of `weights`.
For a 1D array, the prediction is a single `Portfolio`.
For a 2D array, the prediction is a `Population` of `Portfolio`.

If `name` is not provided in the portfolio parameters, the estimator
class name is used.

* **Parameters:**
  **X** *array-like of shape (n_observations, n_assets) | ReturnDistribution*
  : Asset returns or a `ReturnDistribution` carrying returns and optional
    sample weights.
* **Returns:**
  Portfolio | Population
  : The predicted `Portfolio` or `Population` based on the fitted `weights`.

<a id="skfolio.optimization.HierarchicalEqualRiskContribution.score"></a>

#### score(X, y=None)

Prediction score using the Sharpe Ratio.
If the prediction is a single `Portfolio`, the score is its Sharpe Ratio.
If the prediction is a `Population`, the score is the mean Sharpe Ratio
across portfolios.

* **Parameters:**
  **X** *array-like of shape (n_observations, n_assets)*
  : Price returns of the assets.

  **y** *Ignored*
  : Not used, present here for API consistency by convention.
* **Returns:**
  **score** *float*
  : The Sharpe Ratio of the portfolio if the prediction is a single `Portfolio`
    or the mean of all the portfolio Sharpe Ratios if the prediction is a
    `Population` of `Portfolio`.

<a id="skfolio.optimization.HierarchicalEqualRiskContribution.set_params"></a>

#### set_params(\*\*params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects
(such as `Pipeline`). The latter have
parameters of the form `<component>__<parameter>` so that it’s
possible to update each component of a nested object.

* **Parameters:**
  **\*\*params** *dict*
  : Estimator parameters.
* **Returns:**
  **self** *estimator instance*
  : Estimator instance.