Manual#

Here we explain some concepts that apply throughout the library and are useful to know. The code blocks in this document assume the following imports

import cvxportfolio as cvx
import numpy as np
import pandas as pd

Note

This document is a work in progress. We are prioritizing topics which are more important to understand when using Cvxportfolio.

Quickstart#

Cvxportfolio is designed for ease of use and extension. With it, you can quickly experiment with financial trading strategies and develop new ones.

For example, a classic Markowitz optimization strategy

\[\begin{split}\begin{array}{ll} \text{maximize} & w^T \mathbf{E}[r] - \frac{1}{2} w^T \mathbf{Var}[r] w \\ \text{subject to} & w \geq 0, w^T \mathbf{1} <= 1 \end{array}\end{split}\]

where the expected returns and covariances are the simple sample ones, is specified as follows

objective = cvx.ReturnsForecast() - 0.5 * cvx.FullCovariance()
constraints = [cvx.LongOnly(), cvx.LeverageLimit(1)]

strategy = cvx.SinglePeriodOptimization(objective, constraints)

Here we instantiated an optimization-based trading policy. Cvxportfolio defines two (the other is a simple extension of this). Optimization-based policies are defined by an objective function, which is maximized, and a list of constraints, that are imposed on the solution. The objective function is specified as a linear combination of simple terms, we provide many, and it’s easy to define new ones. We provide as well many constraints to choose from, and it’s even easier to define new ones.

Where are the assets’ names?#

Cvxportfolio policies are symbolic, they only define the logic of a trading strategy. They work with any selection of assets. The two objective terms defined above, however, need to know the values of the expected returns and the covariance matrix. You can pass Pandas dataframes for those, and in that case your dataframes should contain the assets’ names you want to trade. Or, if you want, or you can rely on forecasters to compute those (iteratively, in back-testing) using past data. That is what happens in the code shown above, the default parameters of ReturnsForecast and FullCovariance are forecasters that compute historical means and historical covariances respectively, at each point in time if running in a back-test or once, if running live, by using the policy’s execute() method.

Passing Data#

Most Cvxportfolio objects, such as policies, constraints, and objective terms, accept user-provided data. These data can either have values that are constant in time, varying in time, constant for all assets, or specific for each asset. They are specified as follows.

  • Python scalars. These represents values that are constant for all assets (if the object requires a value per each asset) and for all times. For example

    cvx.LeverageLimit(3)

    is a constraint that requires the leverage of the portfolio to be less or equal than three, at all times. Or,

    cvx.HoldingCost(short_fees=5.25)

    is a cost object that models 5.25% annual fees on short positions, for all assets and at all times.

  • Pandas series. These represent either values that are constant in time and vary for each asset, or values that vary in time and are constant for all assets. For example

    my_forecast = pd.Series([0.001, 0.0005], index=['AAPL', 'GOOG'])
cvx.ReturnsForecast(r_hat=my_forecast)

    is an objective term that models market returns forecasts of 0.1% and 0.05% for the two stocks that are specified, over the trading period used. The forecasts are constant in time.

    Note

    During a back-test the trading universe may change. Cvxportfolio objects are aware of the current trading universe at each point of a back-test. If you pass data that vary for each asset, Cvxportfolio objects will try to slice it using the current universe. If they fail, they throw an error. So, you should always provide data for all assets that ever appear in a back-test.

    If instead the pandas series has datetime index it is assumed to contain values that are varying in time. For example

    datetime_index_2020 = pd.date_range('2020-01-01', '2020-12-31')
short_fees_2020 = pd.Series(5.0, index=datetime_index_2020)

datetime_index_2021 = pd.date_range('2021-01-01', '2021-12-31')
short_fees_2021 = pd.Series(5.25, index=datetime_index_2021)

historical_short_fees = pd.concat([short_fees_2020, short_fees_2021])

cvx.HoldingCost(short_fees=historical_short_fees)

    is a cost object that models annual fees on short positions, for all assets, of 5% in 2020 and 5.25% in 2021.

    Note

    You should be careful and make sure that the timestamps used match the timestamps used by the market data server: for example they must have the same timezone. To find the correct timestamps you can call the trading_calendar() method of a market data object.

  • Pandas dataframes. The same conventions used for Pandas series apply, so you should read the above, including the two notes. With dataframes you can specify data that varies both for each asset and in time, or multi-dimensional data that varies in time, or for each assets, or both. If you provide data that varies in time, the datetime index should always be the index (not the columns) and in case of a multi-index it should be the first level. For example

    my_forecast = pd.DataFrame(
    [[0.1, 0.05], [0.15, 0.06]],
    index=[pd.Timestamp('2020-01-01'), pd.Timestamp('2021-01-01')],
    columns=['AAPL', 'GOOG'])
cvx.ReturnsForecast(r_hat=my_forecast)

    is an objective term that models returns forecasts of 10% and 5% for the first period, and 15% and 6% for the second period, for the two assets specified. In this case the two periods are one year each (you get that, for example, by setting the trading_frequency attribute of a market data server as 'annual'). Remember again that the timestamps must match those provided by the trading_calendar() method of the market data server used.

    Multi-dimensional data constant in time is modeled as follows

    exposures = pd.DataFrame(
    [[1, -.5], [-.25, .75]],
    index=['AAPL', 'GOOG'],
    columns=['factor_1', 'factor_2'])
cvx.FactorNeutral(factor_exposure=exposures)

    so the resulting constraint requires neutrality of the portfolio with respect to those two factors. The index must contain all assets that appear in a back-test, and it will be sliced if at some point in time of a back-test only a subset of those assets is traded (see the note above).

    Multi-dimensional data can also vary in time. It is modeled as a Pandas multi-indexed dataframe. If the data is time-varying, the first level of the multi-index should be a Pandas datetime index.

    multi_index = pd.MultiIndex.from_product(
    [[pd.Timestamp('2020-01-01'), pd.Timestamp('2021-01-01')],
    ['AAPL', 'GOOG']])

exposures = pd.DataFrame(
    [[1, -.5], [-.25, .75], [.9, -.3], [-.1, .9]],
    index=multi_index,
    columns=['factor_1', 'factor_2'])
cvx.FactorNeutral(factor_exposure=exposures)

    All the conventions above apply (timestamps should match the ones provided by the trading_calendar() method of the market data server, assets’ names should include all the ones that are traded, …).

    Another example are factor covariances that appear in low-rank factor model covariances. These are specified as follows

    multi_index = pd.MultiIndex.from_product(
    [[pd.Timestamp('2020-01-01'), pd.Timestamp('2021-01-01')],
    ['factor_1', 'factor_2']])

factor_covariances = pd.DataFrame(
     # factor covariance at '2020-01-01'
    [[1, 0.25],
     [0.25, 1],
     # factor covariance at '2021-01-01'
     [1, .1],
     [.1, 1]],
    index=multi_index,
    columns=['factor_1', 'factor_2'])

cvx.FactorModelCovariance(
    F=exposures, Sigma_F=factor_covariances, d=0.01)

  • Numpy arrays. These are not recommended but can be used in simple cases. One use-case is to model data that is constant in time and vary for the assets. If the trading universe varies through a back-test these can’t be used, an error is thrown whenever the sizes of the trading universe and of the array don’t match. For example

    my_forecast = np.array([0.001, 0.0005])
cvx.ReturnsForecast(r_hat=my_forecast)

    models returns’ forecasts of 0.1% for the first asset in the universe and 0.05% for the second asset in the universe. The ordering is the one of the data provided by the serve() method of the market data server.

    Another usecase, less problematic, is to model data that varies across other dimensions, such as risk factors. For example a constant factor covariance can be provided as follows

    factor_covariance = np.array( # constant in time
    [[1, 0.25],
     [0.25, 1]])

cvx.FactorModelCovariance(
    F=exposures, Sigma_F=factor_covariance, d=0.01)

    the ordering used here is that of the columns of the provided exposures dataframe.

Missing values#

When Cvxportfolio objects access user-provided data, after they locate the right time and slice with the current trading universe (if applicable), they check that the resulting data does not contain any np.nan missing value. If any is found, they throw an error. Thus, you should make sure that no np.nan values are contained in any data passed that will be accessed. It is fine to have np.nan values for assets that are not traded at a certain time (for example, because they didn’t exist) because that data won’t be accessed.

Cash account#

Many Cvxportfolio internal variables, such as the weights and holdings vectors that you can access in a result.BacktestResult object, include the cash account as their last element. In most cases used-provided data is not concerned with the cash account (such as all examples above) and so it can be ignored. Exceptions are noted in the documentation of each object.

Multi-Period Optimization#

Multi-period optimization is the signature portfolio allocation model defined by Cvxportfolio (although it is not the only one). It is discussed in section 5 of the paper, and it is based on model-predictive control, the industrial engineering standard for dynamic control. The relevant Cvxportfolio object we deal with here is the MultiPeriodOptimization policy.

We note that the simpler single-period optimization model, implemented by SinglePeriodOptimization and defined in section 4 of the paper, is in fact a special case of multi-period optimization with the planning horizon equal to 1. So, understanding the content of this section also helps you use the single-period optimization model more effectively (and, in fact, there are various situations in which single-period is preferable).

A general formulation of the model is given by the following optimization problem, which is solved at time \(t\)

\[\begin{split}\begin{array}{ll} \text{maximize} & \sum_{\tau = t}^{t + H - 1} \left( \hat {r}_{\tau | t}^T w_\tau^T - \gamma^\text{risk} \psi_\tau(w_\tau) - \gamma^\text{hold} \hat{\phi}^\text{hold}_{\tau|t}(w_\tau) - \gamma^\text{trade} \hat{\phi}^\text{trade}_{\tau|t}(w_\tau - w_{\tau - 1}) \right) \\ \text{subject to} & \mathbf{1}^T w = 1, \ w_\tau - w_{\tau -1} \in \mathcal{Z}_\tau, \ w_\tau \in \mathcal{W}_\tau, \ \text{for } \tau = t+1, \ldots, t+H \end{array}\end{split}\]

This is copied from the equations in the paper, and uses all the definitions made there. We summarize them here for clarity:

  • The time index \(\tau\) spans the planning horizon \(H \geq 1\), and if \(H = 1\) we have single-period optimization. The policy plans for allocations over a few time-steps in the future, and only the first time-step, \(\tau = t\), is used. (This is, in a nutshell, model-predictive control.) The time indexes used here are natural numbers.

  • The weight vector \(w_\tau\) is the allocation of wealth among assets, at each step in the planning horizon. Its last element is used by the cash account, which is always there. The vector always sums to one, because of the self-financing condition defined in the paper.

  • The vector \(\hat {r}_{\tau | t}\) are the forecasts of the returns, for all assets (and cash), made at (execution) time \(t\) for (prediction) time \(\tau\). The return \(r_t\) for a single asset is simply the ratio \(\frac{p_{t+1} - p_t}{p_t}\) between consecutive prices. (Note that often returns by data vendors are defined differently, shifted by one.)

  • The \(\gamma\)’s are hyper-parameters, as discussed in section 4.8 of the paper. These are positive numbers.

  • The objective term \(\psi_{\tau|t}\) is a risk model, like a factor model covariance. We can have different risk models at different planning steps.

  • The terms \(\hat{\phi}^\text{hold}_{\tau|t}\) and \(\hat{\phi}^\text{trade}_{\tau|t}\) model the forecasts, made at time \(t\), for the holding and trading costs respectively that will be incurred at time \(\tau\).

  • The sets \(\mathcal{Z}_\tau\) and \(\mathcal{W}_\tau\) represent the allowed space of trade allocation vectors at each planning step, and are defined by a selection of imposed constraints.

    A close translation of the above equation in Cvxportfolio code looks like this. Here we use \(H = 2\).

    same_period_returns_forecast = pd.DataFrame(...)
next_period_returns_forecast = pd.DataFrame(...) # indexed by the time of execution, not of the forecast!

gamma_risk = cvx.Gamma(initial_value = 0.5)
gamma_hold = cvx.Gamma(initial_value = 1.0)
gamma_trade = cvx.Gamma(initial_value = 1.0)

objective_1 = cvx.ReturnsForecast(r_hat = same_period_returns_forecast) \
    - gamma_risk * cvx.FullCovariance() \
    - gamma_hold * cvx.HoldingCost(short_fees = 1.) \
    - gamma_trade * cvx.TransactionCost(a = 2E-4)

objective_2 = cvx.ReturnsForecast(r_hat = next_period_returns_forecast) \
    - gamma_risk * cvx.FullCovariance() \
    - gamma_hold * cvx.HoldingCost(short_fees = 1.) \
    - gamma_trade * cvx.TransactionCost(a = 2E-4)

constraints_1 = [cvx.LongOnly(applies_to_cash = True)]
constraints_2 = [cvx.LongOnly(applies_to_cash = True)]

policy = cvx.MultiPeriodOptimization(
    objective = [objective_1, objective_2],
    constraints = [constraints_1, constraints_2]
)

Here we use a mixture of data provided by the user (for the returns’ forecast and the cost) and data estimated internally by forecasters (for the risk model).

One thing to note: If providing time-indexed data like returns forecast, the time index always refers to the time of execution, not the time of the forecast. When the policy is evaluated at time \(t\), every provided dataframe (if applicable) will be searched for entries with index \(t\), regardless of which step in the planning horizon it is at.

We note also that there’s a simpler way to initialize cvxportfolio.MultiPeriodOptimization, by providing a single objective and a single list of constraints, and specifying the planning_horizon argument to, for example, 2. This simply copies the terms for each step of planning horizon.

Disk Access#

If you only use Cvxportfolio with user-provided data, meaning via the UserProvidedMarketData server, and are careful to specify the cash_key attribute so that the risk-free rates are not downloaded from the FED website, Cvxportfolio will not access on-disk storage.

Objects in the Data Interfaces submodule, to be precise the data.SymbolData classes, may try to access on-disk storage. These are objects that download single-symbol historical data from the Internet and store the time series locally. On subsequent runs, only the most recent data gets updated, and older observations are kept to the values stored beforehand. That is important for reproducibility. The storage format can be controlled with the storage_backend option to data.SymbolData classes, which is also exposed by the DownloadedMarketData constructor. In addition to the default 'pickle', we provide a 'csv' backend which is very useful when manual inspection of the data is needed, and a 'sqlite' one which doubles as a prototype for SQL backends. In all cases the data is stored in a single folder, which is specified by the base_location argument to the data interfaces objects, also taken by the market simulator objects. That is, by default, a folder called 'cvxportfolio_data' in your user home directory. It contains subfolders named after each of the single symbol data interfaces, which in turn contain the relevant files, with intuitive names. You can delete that folder, or any of its subfolders, if you wish. That will simply make Cvxportfolio objects re-download each series from the start.

On-disk storage by the classes that download historical data from the Internet can not be disabled, but if you wish you can simply remove the folder after each Cvxportfolio run.

In addition, internal Cvxportfolio interfaces are used to store back-test cache files, in the subdirectory with the same name of the 'cvxportfolio_data' folder, or whichever you specify as base_location to the market simulator constructor. You can see in the back-test timing example how that is indeed very useful in speeding up the work-flow of re-running a similar back-test multiple times.

Back-tests executed with the UserProvidedMarketData server do not use on-disk caching (because that class does not define the data.MarketData.partial_universe_signature() method).

Lastly, some examples and the test suite use temporary file-system storage, via the temporary directory object from Python’s Standard Library. That is otherwise not used by Cvxportfolio.

Network Access#

The same objects in the data interfaces submodule discussed in the previous section, i.e., the YahooFinance and Fred single-symbol data interfaces, are the only two objects that access the Internet in the Cvxportfolio library code. They make standard HTTPS GET calls through the requests module, which is one of our dependencies.

There are other internet calls in the examples, for example the script that downloads stock indexes components, but the examples are not part of the Cvxportfolio library (they are not included in the pip packages).

If you wish to use Cvxportfolio in an environment without Internet access, or are otherwise concerned about accessing the Internet, you simply need to use Cvxportfolio via the UserProvidedMarketData object (and not DownloadedMarketData), being careful to specify the cash_key attribute so that the risk-free rates are not downloaded by Fred.

The test-suite makes a limited number of calls to the two classes that require internet access. If you run the test suite (by python -m cvxportfolio.tests) on a computer without internet access you should see a few tests, mostly in the test_data.py module, failing, but most of the test suite will run.

Parallel back-testing#

You can run multiple back-tests in parallel with the MarketSimulator.backtest_many() method. It takes a list of policies and returns the corresponding list of cvxportfolio.result.BacktestResult. Also MarketSimulator.optimize_hyperparameters() uses the same approach, to search over the space of hyper-parameters efficiently.

Note

It is not recommended to run multiple Cvxportfolio programs at the same time, unless you are careful to not access on-disk storage. If you want to run many back-tests at the same time, you should run a single program with MarketSimulator.backtest_many().

Note

If your Cvxportfolio program uses custom objects, for example a forecaster, and in that you call complex third party libraries, like machine-learning ones, parallel back-testing can be problematic. You should in those cases make sure to initialize and finalize all resources you use. Alternatively, Cvxportfolio supports the multiprocess parallel execution library, which may help in such cases. Simply install multiprocess in the Python environment to make Cvxportfolio use it.

CVXPY#

CVXPY is an object-oriented Python library that offers a simple user interface for the specification of optimization programs, translates them into the formats required by high-performance optimization solvers, runs a solver chosen by the user (or, heuristically, by CVXPY itself, if the user doesn’t require a specific one), and returns the solution in the original format specified by the user. This is done automatically, so the intricacies of numerical optimization are not exposed to the end user, and the solver can be easily replaced (each solver typically requires a different specification format, but those are handled by CVXPY). CVXPY was born not very long before Cvxportfolio, and is now a very successful library with lots of users and contributors, in both the applied and theoretical optimization communities.

In the days before such high-level libraries were available, users typically had to code their application programs against the APIs offered directly by the solvers, resulting in complex, difficult to debug, and un-maintainable codes. Today, the maintainers of the numerical solvers are themselves involved in developing and maintaining the interfaces from CVXPY to their solvers, ensuring best compatibility. Cvxportfolio users need not be expert or even familiar with CVXPY, since Cvxportfolio offers an even higher level interface, automating the definition and management of optimization objects like variables and constraints.

One area however in which awareness of the underlying CVXPY process might be useful is the choice and configuration of the numerical solver. Cvxportfolio exposes the relevant CVXPY API through the constructor of the optimization-based policies. For example:

policy = cvx.SinglePeriodOptimization(
    objective = cvx.ReturnsForecast(),
    constraints = [
        cvx.FullCovariance() <= target_daily_vol**2,
        cvx.LongOnly(),
        cvx.LeverageLimit(1),
    ]
    # the following **kwargs are passed to cvxpy.Problem.solve
    solver='SCS',
    eps=1e-14,
    verbose=True,
)

This policy object is instructed to solve its optimization program with the CVXPY-interfaced solver called 'SCS', which is a modern first-order conic solver, it also requires a target accuracy at convergence of \(10^{-14}\), and will print verbose output from both CVXPY and SCS.

The full list of solvers available, along with their options, can be found on CVXPY’s documentation website, and it’s always growing. Other options to the solve method can also be useful, like the ignore_dpp=True which is sometimes used in the examples; that disables a form of matrix caching done by CVXPY which can slow down execution in certain instances.

Familiarity with CVXPY syntax (and peculiarities!) is needed only if you wish to extend Cvxportfolio with user-defined optimization terms, like custom costs and constraints. Cvxportfolio is designed to make that process as simple as possible, while maintaning a unified system that provides accurate accounting, manages market and derived data, enables multi-processing for parallel back-tests, …. In the next section we explain how Cvxportfolio policy objects work, which is useful to know when extending them.

Policy execution model#

Cvxportfolio policy objects and their internal components inherit from the cvxportfolio.estimator.Estimator base class. That abstraction provides three methods with both a recursive and a simple version. Then, the cvxportfolio.estimator.CvxpyExpressionEstimator subclass adds one other method that is used to provide the CVXPY specific code for optimization policies and objects.

All objects that inherit from these base classes typically inherit empty base implementations for all methods. Then each object redefines what it needs in order to function. In this way we manage to keep the infrastructural code limited to one module, and each specific object the user interacts with contains the code that is strictly pertaining to it. (That is very much unlike how CVXPY is designed, encapsulation is much less strict in its codebase.)

Here’s a brief discussion of these methods are and what they are or can be used for:

To be continued.