.. _brinson_fachler:

Brinson-Fachler
===============

Decomposes a portfolio's active return against a benchmark into **allocation**, **selection**
and **interaction** effects, computed per asset and per date, then aggregated across time.

Formulas
--------

For each asset on each date, with asset return :math:`r`, portfolio weight :math:`w_p` and
benchmark weight :math:`w_b`:

.. math::

   r_p          &= w_p \cdot r &&\text{portfolio contribution} \\
   r_b          &= w_b \cdot r &&\text{benchmark contribution} \\
   \alpha       &= r_p - r_b   &&\text{active return} \\
   \text{allocation}  &= (w_p - w_b) \cdot r_b \\
   \text{selection}   &= \alpha \cdot w_p \\
   \text{interaction} &= \alpha - \text{allocation} - \text{selection}

Per-date totals are the sums across assets, and by construction

.. math::

   \alpha = \text{allocation} + \text{selection} + \text{interaction}

which is also equal to :math:`\text{portfolio\_return} - \text{benchmark\_return}`.

Implementation
---------------

The PyArrow-based implementation lives in
:class:`~kaxanuk.attribution_analysis.interfaces.BrinstonFachlerArrowAttribution`, implementing
:class:`~kaxanuk.attribution_analysis.interfaces.BrinstonFachlerArrowInterface`. A pandas-based
reference implementation is also available as
:class:`~kaxanuk.attribution_analysis.interfaces.BrinstonFachlerAttribution`. See
:ref:`attribution_methodologies_brinson_fachler_ref` for the full API reference.

Enable this methodology by setting ``brinson_fachler_method = yes`` in
``attribution_analysis_parameters.xlsx`` (see :ref:`configuration`).