.. _mv_portfolios:

Mean-Variance Portfolio Optimization
------------------------------------

This section provides several self-contained Jupyter notebooks which discuss
the modeling of typical features in mean-variance (M-V) portfolio
optimization.


* `Data Preparation <modeling_notebooks/mean_covariance.ipynb>`_.  This
  notebook prepares the example data that is used throughout the other
  modeling notebooks.  The data is based on historical returns for 462 stocks
  from the S&P 500 and incorporates shrinkage for the first moment.

Basic Markowitz model
^^^^^^^^^^^^^^^^^^^^^

There exist three different basic kinds of M-V models, depending on which
quantity is to be optimized, and which quantity is constrained:

* `Maximizing Return <modeling_notebooks/basic_model_maxmu.ipynb>`_. We show a
  basic M-V problem where we maximize the expected return subject to a
  prescribed maximum variance.

* `Minimizing Variance <modeling_notebooks/basic_model_minsigma.ipynb>`_. We
  show a basic M-V problem where we minimize the expected variance subject to a
  prescribed minimum expected return.

* `Maximizing Utility <modeling_notebooks/basic_model_maxutility.ipynb>`_. We
  show a basic M-V problem where we optimize a combination of risk and
  return, parametrized by a risk-aversion coefficient.

Factor models
^^^^^^^^^^^^^

Another set of tools for representing risk are *factor models*, which
comprise a combination of market risks and idiosyncratic risk information on
the assets.  Such representations can be used either as part of the objective
function, or to formulate constraints on the admissible variance:

* `Factor model in the objective <modeling_notebooks/factor_models_objective.ipynb>`_.

* `Factor model in the constraints <modeling_notebooks/factor_models_constraint.ipynb>`_.

Portfolio constraints
^^^^^^^^^^^^^^^^^^^^^

The following notebooks explain how to model certain portfolio features on top
of the above base formulations:

* `Minimum Buy-In <modeling_notebooks/minimum_buy_in.ipynb>`_.  Tiny
  transaction sizes can be avoided by enforcing a minimum amount whenever a
  position is opened.

* `Cardinality Constraints <modeling_notebooks/concentration.ipynb>`_.  To
  avoid scattering the investment across too many assets, one can impose
  *cardinality constraints* on the portfolio, that is, bounding the number of
  open positions.

* `Enforcing Diversification <modeling_notebooks/concentration.ipynb>`_.  To
  increase diversification, one can impose bounds on the individual positions
  and enforce that a given minimum number of positions is traded.

* `Leverage by Short-Selling
  <modeling_notebooks/leverage_short_selling.ipynb>`_. We show how to
  incorporate short-selling into the base model and how to add limits on the
  total short-sell (e.g., 130/30 strategy).

* `Leverage by Borrowing Cash <modeling_notebooks/leverage_borrow.ipynb>`_.  We
  include a risk-free asset (e.g., money market) that can be used for
  leveraging long positions within prescribed bounds.

* `Limiting Turnover <modeling_notebooks/turnover.ipynb>`_.  If a given
  portfolio is to be rebalanced, we show how to limit the turnover, that is,
  the total change to the starting portfolio.

* `Round Lot Constraints <modeling_notebooks/round_lots.ipynb>`_.  Securities
  are often traded in *lots* (i.e., a fixed number of units). To avoid small
  positions, one might also require a minimum number of units if a position is opened.

* `Sector Allocations <modeling_notebooks/sector_allocation.ipynb>`_.  The stock market comprises different sectors (Technology, Financial Services, Healthcare, etc.), industry groups, industries, and sub-industries. We demonstrate how to limit exposure to individual sectors.


Transaction costs and slippage
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

For trading assets on the market, it is possible to incorporate further pricing
mechanisms into the optimization model:

* `Investing with Transaction Costs
  <modeling_notebooks/transaction_costs.ipynb>`_.  We show how to include
  fixed-charges and proportional transaction costs.

* `Rebalancing with Transaction Costs
  <modeling_notebooks/transaction_costs.ipynb>`_.  Instead of starting from an
  all-cash position, we show how to rebalance an existing portfolio subject to
  fixed-charges and proportional costs.

* `Market Impact Costs <modeling_notebooks/market_impact.ipynb>`_. Huge
  transactions can affect the market price by itself, and in this notebook we
  show one standard approach for modeling such slippage.

.. toctree::
   :maxdepth: 1
   :caption: Portfolio features
   :hidden:

   modeling_notebooks/mean_covariance
   modeling_notebooks/basic_model_maxmu
   modeling_notebooks/basic_model_minsigma
   modeling_notebooks/basic_model_maxutility
   modeling_notebooks/factor_models_objective
   modeling_notebooks/factor_models_constraint
   modeling_notebooks/minimum_buy_in
   modeling_notebooks/concentration
   modeling_notebooks/diversification
   modeling_notebooks/leverage_short_selling
   modeling_notebooks/leverage_borrow
   modeling_notebooks/turnover
   modeling_notebooks/round_lots
   modeling_notebooks/sector_allocation
   modeling_notebooks/transaction_costs
   modeling_notebooks/rebalancing
   modeling_notebooks/market_impact