Optimal Portfolios Backtester

optimalportfolios package implements analytics for backtesting of optimal portfolios including:

1. computing of inputs (covariance matrices, returns) for roll forward computations (to avoid hindsight bias) and for generation of rolling optimal portfolios
2. implementation of core optimisation solvers:
   1. Minimum variance
   2. Maximum quadratic utility
   3. Equal risk contribution
   4. Maximum diversification
   5. Maximum Sharpe ratio
   6. Maximum Cara utility under Gaussian mixture model
   7. Tracking error minimisation
3. the rolling backtests are compatible with incomplete time series for roll forward analysis so that the portfolio universe can include instruments with different price histories
4. computing performances of simulated portfolios
5. reporting of backtested portfolios and cross-sectional analysis

OptimalPortfolios package is split into 5 main modules with the dependecy path increasing sequentially as follows.

1. optimisation with sub-package solvers contains implementation of various quadratic and nonlinear solvers. Each solver is implemented in a module independently from other solvers.

2. reports is module for computing performance statistics and performance attribution including returns, volatilities, etc.

3. examples.solvers provides example of running all implemented solvers.

4. examples.crypto_allocation is module for computations and visualisations for paper "Optimal Allocation to Cryptocurrencies in Diversified Portfolios" https://ssrn.com/abstract=4217841 (see paper for description of the rolling-forward methodology and estimation of inputs)

Table of contents

1. Analytics
2. Installation
3. Portfolio Optimisers
   1. Implementation structure
   2. Example of implementation for Maximum Diversification Solver
   3. Constraints
   4. Wrapper for implemented rolling portfolios
   5. Adding an optimiser
   6. Default parameters
   7. Price time series data
4. Examples
   1. Optimal Portfolio Backtest
   2. Customised reporting
   3. Parameters sensitivity backtest
   4. Multi optimisers cross backtest
   5. Optimal allocation to cryptocurrencies
5. Contributions
6. Updates
7. Disclaimer

Installation

install using

pip install optimalportfolios

upgrade using

pip install --upgrade optimalportfolios

close using

git clone https://github.com/ArturSepp/OptimalPortfolios.git

Core dependencies: python = ">=3.8,<3.11", numba = ">=0.56.4", numpy = ">=1.22.4", scipy = ">=1.9.0", pandas = ">=2.2.2", matplotlib = ">=3.2.2", seaborn = ">=0.12.2", scikit_learn = ">=1.3.0", cvxpy = ">=1.3.2", qis = ">=2.1.1",

Optional dependencies: yfinance ">=0.2.3" (for getting test price data), pybloqs ">=1.2.13" (for producing html and pdf factsheets)

To use pybloqs for pandas > 2.x, locate file "...\Lib\site-packages\pybloqs\jinja\table.html" and change line 44 from:

{% for col_name, cell in row.iteritems() %}

to:

{% for col_name, cell in row.items() %}

Portfolio optimisers

1. Implementation structure

The implementation of each solver is split into the 3 layers:

1. Mathematical layer which takes clean inputs, formulates the optimisation problem and solves the optimisation problem using Scipy or CVXPY solvers. The logic of this layer is to solve the problem algorithmically by taking clean inputs.

2. Wrapper level which takes inputs potentially containing nans, filters them out and calls the solver in layer 1). The output weights of filtered out assets are set to zero.

3. Rolling level function with takes price time series as inputs and implements the estimation of covariance matrix and other inputs on roll-forward basis. For each update date the rolling layer call the wrapper layer 2) with estimated inputs as of the update date.

The recommended usage is as follows.

Layer 2) is used for live portfolios or for backtests which are implemented using data augmentation

Layer 3) is applied for roll forward backtests where all available data is processed using roll forward analysis

For implementation of different estimation methods for covariance matrices and other inputs, we recommend to implement specific layers 2) or 3) with the implementation of the estimation logic.

Layer 1 works with provided covariance matrices and inputs.

2. Example of implementation for Maximum Diversification Solver

Using example of optimization.solvers.max_diversification.py

1. Scipy solver opt_maximise_diversification() which takes "clean" inputs of the covariance matrix of type np.ndarray without nans and Constraints dataclass which implements constraints for the solver.

The lowest level of each optimisation method is opt_... or cvx_... function taking clean inputs and producing the optimal weights.

The logic of this layer is to implement pure quant logic for the optimiser with cvx solver.

2. Wrapper function wrapper_maximise_diversification() which takes inputs covariance matrix of type pd.Dataframe potentially containing nans or assets with zero variance (when their time series are missing in the estimation period) and filters out non-nan "clean" inputs and updated constraints for OPT/CVX solver in layer 1.

The intermediary level of each optimisation method is wrapper_... function taking "dirty" inputs, filtering inputs, and producing the optimal weights. This wrapper can be called either by rolling backtest simulations or by live portfolios for rebalancing.

The logic of this layer is to filter out data and to be an interface for portfolio implementations.

3. Rolling optimiser function rolling_maximise_diversification() takes the time series of data and sliced these accordingly and at each rebalancing step call the wrapper in layer 2. In the end, the function outputs the time series of optiomal weight of assets in the universe. Price data of assets may have gaps and nans which is taken care of in the wrapper level.

The backtesting of each optimisation method is implemented with rolling_... method with produces the time series of optimal portfolio weights.

The logic of this layer is to faciliate the backtest of portfolio optimisation method and to produce time series of portfolio weights using a Markovian setup. These weights are applied for the backtest of the optimal portfolio and the underlying strategy

Each module in optimization.solver implements specific optimisers and estimators for their inputs.

3. Constraints

Dataclass Constraints in optimization.constraints implements optimisation constraints in solver independent way.

The following inputs for various constraints are implemented.

@dataclass
class Constraints:
    is_long_only: bool = True  # for positive allocation weights
    min_weights: pd.Series = None  # instrument min weights  
    max_weights: pd.Series = None  # instrument max weights
    max_exposure: float = 1.0  # for long short portfolios: for long_portfolios = 1
    min_exposure: float = 1.0  # for long short portfolios: for long_portfolios = 1
    benchmark_weights: pd.Series = None  # for minimisation of tracking error 
    tracking_err_vol_constraint: float = None  # annualised sqrt tracking error
    weights_0: pd.Series = None  # for turnover constraints
    turnover_constraint: float = None  # for turnover constraints
    target_return: float = None  # for optimisation with target return
    asset_returns: pd.Series = None  # for optimisation with target return
    max_target_portfolio_vol_an: float = None  # for optimisation with maximum portfolio volatility target
    min_target_portfolio_vol_an: float = None  # for optimisation with maximum portfolio volatility target
    group_lower_upper_constraints: GroupLowerUpperConstraints = None  # for group allocations constraints

Dataclass GroupLowerUpperConstraints implements asset class loading and min and max allocations

@dataclass
class GroupLowerUpperConstraints:
    """
    add constraints that each asset group is group_min_allocation <= sum group weights <= group_max_allocation
    """
    group_loadings: pd.DataFrame  # columns=instruments, index=groups, data=1 if instrument in indexed group else 0
    group_min_allocation: pd.Series  # index=groups, data=group min allocation 
    group_max_allocation: pd.Series  # index=groups, data=group max allocation 

Constraints are updated on the wrapper level to include the valid tickers

    def update_with_valid_tickers(self,  valid_tickers: List[str]) -> Constraints:

On the solver layer, the constants for the solvers are requested as follows.

For Scipy: set_scipy_constraints(self, covar: np.ndarray = None) -> List

For CVXPY: set_cvx_constraints(self, w: cvx.Variable, covar: np.ndarray = None) -> List

4. Wrapper for implemented rolling portfolios

Module optimisation.wrapper_rolling_portfolios.py wraps implementation of of the following solvers enumerated in config.py

Using the wrapper function allows for cross-sectional analysis of different backtest methods and for sensitivity analysis to parameters of estimation and solver methods.

class PortfolioObjective(Enum):
    """
    implemented portfolios in rolling_engine
    """
    # risk-based:
    MAX_DIVERSIFICATION = 1  # maximum diversification measure
    EQUAL_RISK_CONTRIBUTION = 2  # implementation in risk_parity
    MIN_VARIANCE = 3  # min w^t @ covar @ w
    # return-risk based
    QUADRATIC_UTILITY = 4  # max means^t*w- 0.5*gamma*w^t*covar*w
    MAXIMUM_SHARPE_RATIO = 5  # max means^t*w / sqrt(*w^t*covar*w)
    # return-skeweness based
    MAX_CARA_MIXTURE = 6  # carra for mixture distributions

See examples for Parameters sensitivity backtest and Multi optimisers cross backtest

5. Adding an optimiser

1. Add analytics for computing rolling weights using a new estimator in subpackage optimization.solvers. Any third-party packages can be used

2. For cross-sectional analysis, add new optimiser type to config.py and link implemented optimiser in wrapper function compute_rolling_optimal_weights() in optimisation.wrapper_rolling_portfolios.py

6. Default parameters

Key parameters include the specification of the estimation sample.

1. returns_freq defines the frequency of returns for covariance matrix estimation. This parameter affects all methods.

The default (assuming daily price data) is weekly Wednesday returns returns_freq = 'W-WED'.

For price data with monthly observations (such us hedged funds), monthly returns should be used returns_freq = 'ME'.

2. span defines the estimation span for ewma covariance matrix. This parameter affects all methods which use EWMA covariance matrix:

PortfolioObjective in [MAX_DIVERSIFICATION, EQUAL_RISK_CONTRIBUTION, MIN_VARIANCE]

and

PortfolioObjective in [QUADRATIC_UTILITY, MAXIMUM_SHARPE_RATIO]

The span is defined as the number of returns for the half-life of EWMA filter: ewma_lambda = 1 - 2 / (span+1). span=52 with weekly returns means that last 52 weekly returns (one year of data) contribute 50% of weight to estimated covariance matrix

The default (assuming weekly returns) is 52: span=52.

For monthly returns, I recommend to use span=12 or span=24.

3. rebalancing_freq defines the frequency of weights update. This parameter affects all methods.

The default value is quarterly rebalancing rebalancing_freq='QE'.

For the following methods

PortfolioObjective in [QUADRATIC_UTILITY, MAXIMUM_SHARPE_RATIO, MAX_CARA_MIXTURE]

Rebalancing frequency is also the rolling sample update frequency when mean returns and mixture distributions are estimated.

4. roll_window defines the number of past returns applied for estimation of rolling mean returns and mixture distributions.

This parameter affects the following optimisers

PortfolioObjective in [QUADRATIC_UTILITY, MAXIMUM_SHARPE_RATIO, MAX_CARA_MIXTURE]

and it is linked to rebalancing_freq.

Default value is roll_window=20 which means that data for past 20 (quarters) are used in the sample with rebalancing_freq='QE'

For monthly rebalancing, I recomend to use roll_window=60 which corresponds to using past 5 years of data

7. Price time series data

The input to all optimisers is dataframe prices which contains dividend and split adjusted prices.

The price data can include assets with prices starting an ending at different times.

All optimisers will set maximum weight to zero for assets with missing prices in the estimation sample period.

Examples

1. Optimal Portfolio Backtest

See script in optimalportfolios.examples.optimal_portfolio_backtest.py

import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import yfinance as yf
from typing import Tuple
import qis as qis

# package
from optimalportfolios import compute_rolling_optimal_weights, PortfolioObjective, Constraints

def fetch_universe_data() -> Tuple[pd.DataFrame, pd.DataFrame, pd.Series]:
    """
    fetch universe data for the portfolio construction:
    1. dividend and split adjusted end of day prices: price data may start / end at different dates
    2. benchmark prices which is used for portfolio reporting and benchmarking
    3. universe group data for portfolio reporting and risk attribution for large universes
    this function is using yfinance to fetch the price data
    """
    universe_data = dict(SPY='Equities',
                         QQQ='Equities',
                         EEM='Equities',
                         TLT='Bonds',
                         IEF='Bonds',
                         LQD='Credit',
                         HYG='HighYield',
                         GLD='Gold')
    tickers = list(universe_data.keys())
    group_data = pd.Series(universe_data)
    prices = yf.download(tickers, start=None, end=None, ignore_tz=True)['Adj Close']
    prices = prices[tickers]  # arrange as given
    prices = prices.asfreq('B', method='ffill')  # refill at B frequency
    benchmark_prices = prices[['SPY', 'TLT']]
    return prices, benchmark_prices, group_data


# 2. get universe data
prices, benchmark_prices, group_data = fetch_universe_data()
time_period = qis.TimePeriod('31Dec2004', '16Aug2024')   # period for computing weights backtest

# 3.a. define optimisation setup
portfolio_objective = PortfolioObjective.MAX_DIVERSIFICATION  # define portfolio objective
returns_freq = 'W-WED'  # use weekly returns
rebalancing_freq = 'QE'  # weights rebalancing frequency: rebalancing is quarterly on WED
span = 52  # span of number of returns_freq-returns for covariance estimation = 12y
constraints0 = Constraints(is_long_only=True,
                           min_weights=pd.Series(0.0, index=prices.columns),
                           max_weights=pd.Series(0.5, index=prices.columns))

# 3.b. compute solvers portfolio weights rebalanced every quarter
weights = compute_rolling_optimal_weights(prices=prices,
                                          portfolio_objective=portfolio_objective,
                                          constraints0=constraints0,
                                          time_period=time_period,
                                          rebalancing_freq=rebalancing_freq,
                                          span=span)

# 4. given portfolio weights, construct the performance of the portfolio
funding_rate = None  # on positive / negative cash balances
rebalancing_costs = 0.0010  # rebalancing costs per volume = 10bp
weight_implementation_lag = 1  # portfolio is implemented next day after weights are computed
portfolio_data = qis.backtest_model_portfolio(prices=prices.loc[weights.index[0]:, :],
                                              weights=weights,
                                              ticker='MaxDiversification',
                                              funding_rate=funding_rate,
                                              weight_implementation_lag=weight_implementation_lag,
                                              rebalancing_costs=rebalancing_costs)

# 5. using portfolio_data run the reporting with strategy factsheet
# for group-based reporting set_group_data
portfolio_data.set_group_data(group_data=group_data, group_order=list(group_data.unique()))
# set time period for portfolio reporting
figs = qis.generate_strategy_factsheet(portfolio_data=portfolio_data,
                                       benchmark_prices=benchmark_prices,
                                       time_period=time_period,
                                       **qis.fetch_default_report_kwargs(time_period=time_period))
# save report to pdf and png
qis.save_figs_to_pdf(figs=figs,
                     file_name=f"{portfolio_data.nav.name}_portfolio_factsheet",
                     orientation='landscape',
                     local_path="C://Users//Artur//OneDrive//analytics//outputs")
qis.save_fig(fig=figs[0], file_name=f"example_portfolio_factsheet1", local_path=f"figures/")
qis.save_fig(fig=figs[1], file_name=f"example_portfolio_factsheet2", local_path=f"figures/")

2. Customised reporting

Portfolio data class PortfolioData is implemented in QIS package

# 6. can create customised reporting using portfolio_data custom reporting
def run_customised_reporting(portfolio_data) -> plt.Figure:
    with sns.axes_style("darkgrid"):
        fig, axs = plt.subplots(3, 1, figsize=(12, 12), tight_layout=True)
    perf_params = qis.PerfParams(freq='W-WED', freq_reg='ME')
    kwargs = dict(x_date_freq='YE', framealpha=0.8, perf_params=perf_params)
    portfolio_data.plot_nav(ax=axs[0], **kwargs)
    portfolio_data.plot_weights(ncol=len(prices.columns)//3,
                                legend_stats=qis.LegendStats.AVG_LAST,
                                title='Portfolio weights',
                                freq='QE',
                                ax=axs[1],
                                **kwargs)
    portfolio_data.plot_returns_scatter(benchmark_price=benchmark_prices.iloc[:, 0],
                                        ax=axs[2],
                                        **kwargs)
    return fig


# run customised report
fig = run_customised_reporting(portfolio_data)
# save png
qis.save_fig(fig=fig, file_name=f"example_customised_report", local_path=f"figures/")

3. Parameters sensitivity backtest

Cross-sectional backtests are applied to test the sensitivity of optimisation method to a parameter of estimation or solver methods.

See script in optimalportfolios.examples.parameter_sensitivity_backtest.py

4. Multi optimisers cross backtest

Multiple optimisation methods can be analysed using the wrapper function compute_rolling_optimal_weights()

See example script in optimalportfolios.examples.multi_optimisers_backtest.py

5. Optimal allocation to cryptocurrencies

Computations and visualisations for paper "Optimal Allocation to Cryptocurrencies in Diversified Portfolios" https://ssrn.com/abstract=4217841 are implemented in module optimalportfolios.examples.crypto_allocation, see README in this module

Updates

8 July 2023, Version 1.0.1 released

Implementation of optimisation methods and data considered in "Optimal Allocation to Cryptocurrencies in Diversified Portfolios" by A. Sepp published in Risk Magazine, October 2023, 1-6. The draft is available at SSRN: https://ssrn.com/abstract=4217841

2 September 2023, Version 1.0.8 released

Added subpackage optimisation.rolling_engine with optimisers grouped by the type of inputs and data they require.

18 August 2024, Version 2.1.1 released

Refactor the implementation of solvers with the 3 layers.

Add new solvers for tracking error and target return optimisations.

Add exmples of running all solvers

Disclaimer

OptimalPortfolios package is distributed FREE & WITHOUT ANY WARRANTY under the GNU GENERAL PUBLIC LICENSE.

See the LICENSE.txt in the release for details.

Please report any bugs or suggestions by opening an issue.