optimalportfolios 5.3.1

Implementation of optimisation analytics for constructing and backtesting optimal portfolios in Python

🚀 Optimal Portfolios Construction and Backtesting: optimalportfolios

Production-grade multi-asset portfolio construction and backtesting in Python — from covariance estimation to rolling optimisation to factsheet reporting, in a single pipeline that handles real-world data

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Why optimalportfolios

Most Python portfolio optimisation packages (PyPortfolioOpt, Riskfolio-Lib, skfolio) solve single-period allocation problems: given a covariance matrix and expected returns, find the optimal weights. This is useful for textbook exercises but insufficient for running a real multi-asset portfolio.

optimalportfolios solves the production problem end-to-end: estimate covariance → compute alpha signals → optimise with constraints → rebalance on schedule → backtest with transaction costs — all in a single roll-forward pipeline that handles incomplete data, mixed-frequency assets, and illiquid positions.

Key differentiators

Production multi-asset portfolio construction. The package implements the full pipeline from the ROSAA framework: factor model covariance estimation (via factorlasso) → risk-budgeted SAA → alpha signal computation → TE-constrained TAA → rolling backtest. No other open-source package handles universes where equities rebalance monthly, alternatives rebalance quarterly, and private equity enters the allocation set only when sufficient return history is available. The constraint system (weight bounds, group allocation limits, tracking error budgets, turnover controls, rebalancing indicators for frozen positions) matches what real institutional PM teams need.

HCGL factor covariance estimation. The Hierarchical Clustering Group LASSO factor model (published in JPM, 2026) produces sparse, structured covariance matrices for heterogeneous multi-asset universes. The LASSO/Group LASSO/HCGL solver is implemented in the standalone factorlasso package — a general-purpose sparse factor model estimator with sign constraints, prior-centered regularisation, and scikit-learn compatible API. optimalportfolios builds on top of factorlasso with finance-specific functionality: FactorCovarEstimator handles multi-frequency asset returns, rolling estimation schedules, factor covariance assembly (Σ_y = β Σ_x β' + D), and integration with qis for performance attribution. The separation means the LASSO solver can be used independently for any multi-output regression problem (genomics, macro-econometrics), while the portfolio-specific rolling pipeline stays in optimalportfolios.

Drift-aware rolling backtests (new in v5.3.1). Turnover constraints and transaction-cost penalties act on the realised current holdings, not the previous target. This eliminates the "phantom turnover budget" issue where the optimiser thinks it's trading X but the NAV simulator actually trades X·(1 + drift fraction). Controlled by OptimiserConfig.use_drifted_weights_0 (default True); set to False to reproduce pre-v5.3.1 behaviour for legacy comparisons.

NaN-aware rolling backtesting. The three-layer architecture (solver / wrapper / rolling) automatically handles real-world data: assets with missing prices receive zero weight, assets entering the universe mid-sample are included when sufficient history is available, and the rebalancing indicator system freezes illiquid positions at their current weight while re-optimising the liquid portion. When the freeze produces group-constraint overshoots due to drift, the constraint is relaxed for that rebalance with a logged warning rather than aborting. No data cleaning or pre-filtering required.

Research-backed methodology. The package is the reference implementation for the ROSAA framework published in The Journal of Portfolio Management (Sepp, Ossa, Kastenholz, 2026). The optimisation solvers, covariance estimators, and alpha signals are battle-tested on live multi-asset portfolios.

Quick-start: rolling backtest in 10 lines

import qis as qis
from optimalportfolios import (EwmaCovarEstimator, Constraints,
                               PortfolioObjective, compute_rolling_optimal_weights)

prices = ...  # pd.DataFrame of asset prices (may have NaNs, different start dates)
time_period = qis.TimePeriod('31Dec2004', '15Mar2026')

# estimate covariance → optimise → get rolling weights
estimator = EwmaCovarEstimator(returns_freq='W-WED', span=52, rebalancing_freq='QE')
covar_dict = estimator.fit_rolling_covars(prices=prices, time_period=time_period)
weights = compute_rolling_optimal_weights(prices=prices,
                                          portfolio_objective=PortfolioObjective.MAX_DIVERSIFICATION,
                                          constraints=Constraints(is_long_only=True),
                                          time_period=time_period,
                                          covar_dict=covar_dict)

# backtest with transaction costs
portfolio = qis.backtest_model_portfolio(prices=prices, weights=weights,
                                         rebalancing_costs=0.001, ticker='MaxDiv')

That's it — from prices to backtested portfolio in 10 lines, with automatic NaN handling, roll-forward estimation (no hindsight bias), drift-aware turnover accounting, and any optimisation objective.

Design scope

The optimisation solvers use quadratic and conic objective functions (variance, tracking error, Sharpe ratio, diversification ratio, CARA utility). The package does not implement non-quadratic risk measures (CVaR, MAD, drawdown constraints). For these, use Riskfolio-Lib or skfolio. The solver architecture (three-layer: mathematical / wrapper / rolling) makes it straightforward to add new solvers — each solver lives in its own module in optimization/solvers/ and plugs into the rolling backtester via a single dispatch function.

Package overview

optimalportfolios/
├── alphas/                        # Alpha signal computation
│   ├── signals/
│   │   ├── momentum.py            # compute_momentum_alpha()
│   │   ├── low_beta.py            # compute_low_beta_alpha()
│   │   └── managers_alpha.py      # compute_managers_alpha()
│   ├── alpha_data.py              # AlphasData container
│   ├── backtest_alphas.py         # Signal backtesting tool
│   └── tests/
│       └── signals_test.py
├── covar_estimation/              # Covariance matrix estimation
│   ├── covar_estimator.py         # CovarEstimator ABC
│   ├── ewma_covar_estimator.py    # EwmaCovarEstimator
│   ├── factor_covar_estimator.py  # FactorCovarEstimator (uses factorlasso)
│   ├── factor_covar_data.py       # CurrentFactorCovarData, RollingFactorCovarData
│   └── covar_reporting.py         # Rolling covariance diagnostics
├── optimization/                  # Portfolio optimisation
│   ├── constraints.py             # Constraints, GroupLowerUpperConstraints
│   ├── config.py                  # OptimiserConfig (incl. use_drifted_weights_0)
│   ├── wrapper_rolling_portfolios.py  # compute_rolling_optimal_weights()
│   └── solvers/
│       ├── quadratic.py           # min variance, max quadratic utility
│       ├── risk_budgeting.py      # constrained risk budgeting (pyrb)
│       ├── max_diversification.py # maximum diversification ratio
│       ├── max_sharpe.py          # maximum Sharpe ratio
│       ├── tracking_error.py      # alpha-over-tracking-error
│       ├── target_return.py       # alpha with target return constraint
│       └── carra_mixure.py        # CARA utility under Gaussian mixture
├── utils/                         # Auxiliary analytics
│   ├── filter_nans.py             # NaN-aware covariance/vector filtering
│   ├── portfolio_funcs.py         # Risk contributions, diversification ratio
│   ├── weights_drift.py           # apply_drift_to_weights_0 (new in v5.3.1)
│   ├── gaussian_mixture.py        # Gaussian mixture fitting (pure numpy/scipy EM)
│   └── returns_unsmoother.py      # AR(1) return unsmoothing for PE/PD
├── reports/                       # Performance reporting
│   └── marginal_backtest.py       # Marginal asset contribution analysis
└── examples/                      # Worked examples — see examples/README.md
    ├── data/                      # Shared universe fixtures
    ├── solvers/                   # One demo per single-objective solver
    ├── backtests/                 # End-to-end rolling workflows
    ├── comparisons/               # A-vs-B sweeps (incl. drift_policy)
    └── covar_estimation/          # Covariance estimator demos

# External dependency:
# factorlasso (pip install factorlasso)
#   └── LassoModel, solve_lasso_cvx_problem, solve_group_lasso_cvx_problem
#       Sign-constrained LASSO/Group LASSO/HCGL solver (domain-agnostic)
#       https://github.com/ArturSepp/factorlasso

Architecture: factorlasso vs optimalportfolios

factorlasso is the domain-agnostic LASSO solver — it estimates sparse factor loadings β in Y_t = α + β X_t + ε_t with sign constraints, prior-centered regularisation, and HCGL clustering. It provides LassoModel (scikit-learn compatible estimator), CurrentFactorCovarData (single-date covariance decomposition Σ_y = β Σ_x β' + D), and RollingFactorCovarData (time-indexed collection). It knows nothing about finance, asset returns, frequencies, or rebalancing schedules.

optimalportfolios adds two finance-specific layers on top:

estimate_lasso_factor_covar_data() — the core estimation function in covar_estimation/factor_covar_estimator.py. It handles everything between raw market data and the factorlasso solver:

• Computes factor returns from prices at the specified frequency
• Estimates annualised factor covariance Σ_x via EWMA
• Calls factorlasso.LassoModel.fit() separately per frequency for mixed-frequency universes (e.g., monthly equities + quarterly alternatives)
• Annualises residual variances, R², and alphas across frequencies
• Merges multi-frequency betas into a single (N × M) loading matrix
• Returns a factorlasso.CurrentFactorCovarData with the full decomposition

FactorCovarEstimator — a CovarEstimator subclass that wraps estimate_lasso_factor_covar_data() in a rolling estimation schedule using qis.TimePeriod and qis.generate_dates_schedule. It provides two APIs:

• fit_rolling_covars() → Dict[Timestamp, DataFrame] (plain covariance matrices, plug into any solver)
• fit_rolling_factor_covars() → RollingFactorCovarData (full decomposition with betas, R², clusters, residuals over time)

Alpha signals module

New in v4.1.1. The alphas module provides standalone alpha signal computation functions with a consistent interface. Each function handles single-frequency and mixed-frequency universes, supports within-group cross-sectional scoring, and returns both a dimensionless score and the raw signal for diagnostics.

Naming convention

Stage	What it is	Example
Raw signal	Observable quantity with units	Cumulative return, EWMA beta, regression residual
Score	Cross-sectional z-score, dimensionless	Momentum rank, negated beta rank
Alpha	Portfolio-ready signal after CDF mapping	Combined score mapped to [-1, 1]

Pipeline: raw signal → score → alpha.

Available signals

Momentum (compute_momentum_alpha) — EWMA-filtered risk-adjusted excess returns relative to a benchmark, converted to cross-sectional scores.

from optimalportfolios.alphas import compute_momentum_alpha

score, raw_momentum = compute_momentum_alpha(
    prices=prices, benchmark_price=benchmark, returns_freq='ME',
    group_data=asset_class_groups, long_span=12)

Low Beta (compute_low_beta_alpha) — EWMA regression beta to benchmark, negated and cross-sectionally scored ("betting against beta").

from optimalportfolios.alphas import compute_low_beta_alpha

score, raw_beta = compute_low_beta_alpha(
    prices=prices, benchmark_price=benchmark, returns_freq='ME',
    group_data=asset_class_groups, beta_span=12)

Managers Alpha (compute_managers_alpha) — factor model regression residuals using pre-estimated betas from FactorCovarEstimator, EWMA-smoothed and cross-sectionally scored.

from optimalportfolios.alphas import compute_managers_alpha

score, raw_alpha = compute_managers_alpha(
    prices=asset_prices, risk_factor_prices=factor_prices,
    estimated_betas=rolling_data.get_y_betas(),
    returns_freq='ME', alpha_span=12)

Mixed-frequency support

All signal functions accept returns_freq as a string (uniform) or a pd.Series (per-asset frequency). When mixed, the function groups by frequency, computes per group, and merges.

# equities monthly, alternatives quarterly
returns_freq = pd.Series({'SPY': 'ME', 'EZU': 'ME', 'HF_Macro': 'QE', 'PE': 'QE'})
score, raw = compute_momentum_alpha(prices, returns_freq=returns_freq, ...)

AlphasData container

AlphasData holds the combined alpha scores and all intermediate components:

from optimalportfolios.alphas import AlphasData

data = AlphasData(alpha_scores=combined, momentum_score=mom, beta_score=beta, ...)
snapshot = data.get_alphas_snapshot(date=pd.Timestamp('2024-12-31'))

See the alphas module README for full documentation.

Table of contents

1. Why optimalportfolios
2. Package overview
3. Alpha signals module
4. Installation
5. 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
   8. Drift-aware rolling backtests (v5.3.1)
6. Examples
7. Updates
8. Disclaimer

Installation

install using

pip install optimalportfolios

upgrade using

pip install --upgrade optimalportfolios

clone using

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

Core dependencies: python = ">=3.9", numba = ">=0.60.0", numpy = ">=2.0", scipy = ">=1.12.0", pandas = ">=2.2.0", matplotlib = ">=3.8.0", seaborn = ">=0.13.0", cvxpy = ">=1.3.0", quadprog = ">=0.1.11", qis = ">=3.5.7", factorlasso = ">=0.1.0"

Optional dependencies: yfinance ">=0.2.40" (for getting test price data), pybloqs ">=1.2.13" (for producing html and pdf factsheets)

Portfolio optimisers

1. Implementation structure

The implementation of each solver is split into 3 layers:

1. Mathematical layer which takes clean inputs, formulates the optimisation problem and solves it using Scipy or CVXPY solvers. The logic of this layer is to solve the problem algorithmically by taking clean inputs.
2. Wrapper layer 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. Includes rebalancing indicator support for freezing specific assets at their previous weights, and (as of v5.3.1) automatic relaxation of group bounds when frozen-position drift causes overshoot.
3. Rolling layer which takes price time series as inputs and implements the estimation of covariance matrix and other inputs on a roll-forward basis. For each update date the rolling layer calls the wrapper layer 2) with estimated inputs as of the update date. As of v5.3.1, the rolling layer also drifts weights_0 between rebalances using realised price returns, so that turnover constraints and transaction-cost penalties measure actual trades rather than notional trades against a stale baseline.

For rolling level function, the estimated covariance matrix can be passed as Dict[pd.Timestamp, pd.DataFrame] with DataFrames containing covariance matrices for the universe and with keys being rebalancing times.

Covariance can be estimated using EwmaCovarEstimator (simple EWMA) or FactorCovarEstimator (HCGL factor model using factorlasso.LassoModel for sparse beta estimation, with finance-specific annualisation, multi-frequency returns, and rolling schedule management).

Important design principle (v4.1.1): covariance estimation is separated from portfolio optimisation. The recommended workflow is to estimate covariance matrices first, then pass them as covar_dict to any solver:

from optimalportfolios import EwmaCovarEstimator, FactorCovarEstimator

# estimate once
estimator = EwmaCovarEstimator(returns_freq='W-WED', span=52, rebalancing_freq='QE')
covar_dict = estimator.fit_rolling_covars(prices=prices, time_period=time_period)

# reuse across multiple solvers
weights_rb = rolling_risk_budgeting(prices=prices, covar_dict=covar_dict, ...)
weights_md = rolling_maximise_diversification(prices=prices, covar_dict=covar_dict, ...)
weights_te = rolling_maximise_alpha_over_tre(prices=prices, covar_dict=covar_dict, ...)

This separation provides three benefits: (1) the same covariance matrices can be reused across multiple solvers without re-estimation, (2) covariance diagnostics and reporting can be inspected independently of the optimiser, and (3) different covariance estimators can be swapped in without modifying the solver code. For the HCGL factor model, use FactorCovarEstimator with asset_returns_dict for mixed-frequency universes (e.g., monthly equities + quarterly alternatives).

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.

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 updates 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 slices these accordingly and at each rebalancing step calls the wrapper in layer 2. In the end, the function outputs the time series of optimal weights 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 which produces the time series of optimal portfolio weights.

The logic of this layer is to facilitate 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.solvers 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

Frozen-position relaxation (new in v5.3.1). When rebalancing_indicators freeze illiquid positions for a given rebalance date, update_with_valid_tickers pins their min_weights and max_weights to the current (drifted) weights_0. If the resulting group-loading sum exceeds group_max_allocation (or falls below group_min_allocation), the group bound is automatically relaxed for that rebalance and a UserWarning is emitted. This prevents ValueError: Infeasible constraints detected errors that would otherwise occur when illiquid sleeves drift over their group cap between low-frequency rebalances. The relaxation is audit-trailable: each event surfaces in logs with the group name, the original bound, and the relaxed bound.

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 in the examples folder and the examples/README.md for the full demo index.

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 as hedge 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 recommend 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 and ending at different times.

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

8. Drift-aware rolling backtests (v5.3.1)

Every rolling optimiser carries weights_0 forward from one rebalance date to the next so that turnover constraints (Constraints.turnover_constraint, turnover_costs, turnover_utility_weight, group_turnover_constraint) act on a sensible baseline. The choice of baseline matters.

Legacy behaviour (pre-v5.3.1, also use_drifted_weights_0=False). weights_0 at each rebalance equals the previous-period target weights, with no adjustment for realised drift over the holding period. The optimiser's L1 turnover budget constrains ||w_new − w_prev_target||_1, but the simulator actually trades ||w_new − w_drift||_1. The two differ by the realised one-period drift, which is typically 1–3% of NAV for diversified portfolios at quarterly frequency. Cumulative effect: realised turnover exceeds the optimiser's budget by roughly the same fraction.

New default (v5.3.1, use_drifted_weights_0=True). Before each rebalance, the helper apply_drift_to_weights_0 (in utils/weights_drift.py) drifts the previous-period target weights to the current date using realised price returns under the self-financing identity

w_drift_i = w_i · (1 + r_i) / (1 + Σ_j w_j · r_j)

The denominator is portfolio NAV growth. For long-only fully-invested portfolios this reduces to the conventional gross / sum(gross) form, but the formula remains correct for long-short and variable-exposure mandates. The helper is constraint-agnostic and silently falls back to passing weights_0 unchanged whenever any input is missing or pathological (NaN prices, NAV collapse, zero weights_0, first rebalance, etc.).

Empirical comparison. On a min-variance rolling backtest of the 15-ETF benchmark universe with a binding L1 turnover budget of 0.08/quarter:

	Policy A (legacy, drift off)	Policy B (new default, drift on)
Apparent turnover (ann.)	0.2767	0.2766
Realised turnover (ann.)	0.3403	0.2814
Realised / apparent	1.23	1.02
Cumulative TC drag (bps)	19.4	16.0

Under (A), the optimiser believes it's hitting the 0.08/quarter cap but is actually trading 0.085/quarter — the budget is leaky. Under (B), realised trading sits at 0.070/quarter, comfortably under the cap. Cost drag drops by 17.5% relative for the same nominal constraint. See examples/comparisons/drift_policy.py for the reproducible demonstration.

Toggling for legacy comparisons. To reproduce pre-v5.3.1 behaviour exactly (e.g. to validate against published backtest numbers from earlier papers or reports), set:

from optimalportfolios import OptimiserConfig

cfg = OptimiserConfig(use_drifted_weights_0=False)
weights = rolling_quadratic_optimisation(prices=prices, covar_dict=covar_dict,
                                          constraints=constraints,
                                          optimiser_config=cfg)

Examples

The examples/ folder is organised into five purpose-folders. The examples README maps every demo to its role; the headlines are:

examples/
├── data/                  Universe fixtures (fetch_benchmark_universe_data, fetch_minimal_universe_data)
├── solvers/               One demo per single-objective solver
├── backtests/             End-to-end rolling backtest workflows
├── comparisons/           A-vs-B sweeps (covar / optimiser / parameter / drift policy)
├── covar_estimation/      Covariance estimator demos
└── sp500_universe.py      S&P 500 universe loader (top level)

Recommended reading order for newcomers

1. examples/data/universe.py — understand the shared fixture.
2. examples/backtests/minimal_backtest.py — see one full workflow end-to-end.
3. examples/solvers/min_variance.py — minimal solver demo with both single-date and rolling forms.
4. examples/solvers/tracking_error.py — the production TAA pattern (alpha + benchmark + TE constraint).
5. examples/comparisons/optimisers.py — see how objectives differ on the same universe.

Highlighted demos

Optimal portfolio backtest

See script optimalportfolios/examples/backtests/minimal_backtest.py.

import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import qis as qis

from optimalportfolios import (compute_rolling_optimal_weights, PortfolioObjective,
                               Constraints, EwmaCovarEstimator)
from optimalportfolios.examples.data.universe import fetch_minimal_universe_data


# 1. fetch universe (8 ETFs across 6 asset-class groups)
prices, benchmark_prices, group_data = fetch_minimal_universe_data()
time_period = qis.TimePeriod('31Dec2004', '15Mar2026')

# 2. define optimisation setup
portfolio_objective = PortfolioObjective.MAX_DIVERSIFICATION
returns_freq = 'W-WED'
rebalancing_freq = 'QE'
span = 52
constraints = Constraints(is_long_only=True,
                          min_weights=pd.Series(0.0, index=prices.columns),
                          max_weights=pd.Series(0.5, index=prices.columns))

# 3. estimate covariance, then optimise
ewma_estimator = EwmaCovarEstimator(returns_freq=returns_freq, span=span,
                                     rebalancing_freq=rebalancing_freq)
covar_dict = ewma_estimator.fit_rolling_covars(prices=prices, time_period=time_period)
weights = compute_rolling_optimal_weights(prices=prices,
                                          portfolio_objective=portfolio_objective,
                                          constraints=constraints,
                                          time_period=time_period,
                                          rebalancing_freq=rebalancing_freq,
                                          covar_dict=covar_dict)

# 4. backtest with transaction costs (drift-aware under v5.3.1 defaults)
portfolio_data = qis.backtest_model_portfolio(prices=prices.loc[weights.index[0]:, :],
                                              weights=weights,
                                              ticker='MaxDiversification',
                                              weight_implementation_lag=1,
                                              rebalancing_costs=0.0010)

# 5. generate factsheet
portfolio_data.set_group_data(group_data=group_data,
                              group_order=list(group_data.unique()))
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))
qis.save_figs_to_pdf(figs=figs, file_name=f"{portfolio_data.nav.name}_portfolio_factsheet",
                     orientation='landscape', local_path="output/")

Customised reporting

Portfolio data class PortfolioData is implemented in QIS package.

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

Parameter sensitivity backtest

Cross-sectional backtests test the sensitivity of an optimisation method to estimation or solver parameters.

See optimalportfolios/examples/comparisons/parameter_sensitivity.py.

Multi-optimiser cross-backtest

Multiple optimisation methods can be analysed using compute_rolling_optimal_weights().

See optimalportfolios/examples/comparisons/optimisers.py.

Multi-covariance-estimator backtest

Multiple covariance estimators can be backtested for the same optimisation method.

See optimalportfolios/examples/comparisons/covar_estimators.py.

Drift-policy comparison (new in v5.3.1)

Compares OptimiserConfig.use_drifted_weights_0 = True (production default) vs False (legacy) using rolling_quadratic_optimisation with a binding L1 turnover budget. Shows that under the legacy convention the realised turnover exceeds the optimiser's apparent turnover by ~23%; under the new default the two agree.

See optimalportfolios/examples/comparisons/drift_policy.py.

Optimal allocation to cryptocurrencies

Computations and visualisations for the paper "Optimal Allocation to Cryptocurrencies in Diversified Portfolios" are implemented in optimalportfolios.examples.crypto_allocation. See the README in that module.

Published reference: Sepp A. (2023), "Optimal Allocation to Cryptocurrencies in Diversified Portfolios", Risk Magazine, October 2023, 1-6. Available at SSRN.

Robust optimisation of strategic and tactical asset allocation

Computations and visualisations for the paper "Robust Optimization of Strategic and Tactical Asset Allocation for Multi-Asset Portfolios" are implemented in optimalportfolios.examples.robust_optimisation_saa_taa. See the README in that module.

The paper presents the ROSAA framework — a unified approach to strategic and tactical asset allocation for multi-asset portfolios. Key contributions: the HCGL (Hierarchical Clustering Group LASSO) factor covariance estimator for heterogeneous multi-asset universes, constrained risk budgeting for SAA with group allocation limits, and alpha-over-tracking-error optimisation for TAA. The framework handles real-world challenges including mixed-frequency assets, incomplete return histories, and illiquid positions requiring rebalancing indicators. The optimalportfolios package is the reference implementation of the full ROSAA pipeline.

Published reference: Sepp A., Ossa I., and Kastenholz M. (2026), "Robust Optimization of Strategic and Tactical Asset Allocation for Multi-Asset Portfolios", The Journal of Portfolio Management, 52(4), 86-120. Paper link.

Updates

May 2026, Version 5.3.1 released

Drift-aware weights_0 in rolling backtests (default behaviour change).

Every rolling optimiser now drifts the previous-period weights to the current rebalance date using realised price returns before passing them as weights_0 to the next single-date optimisation. The new helper apply_drift_to_weights_0 in utils/weights_drift.py implements the self-financing identity w_drift_i = w_i · (1 + r_i) / (1 + Σ_j w_j · r_j), which is correct for long-only, long-short, and variable-exposure mandates.

Controlled by OptimiserConfig.use_drifted_weights_0 — default True. Set to False to reproduce pre-v5.3.1 behaviour for legacy comparisons. The change affects all nine rolling optimisers: rolling_risk_budgeting, rolling_maximize_portfolio_sharpe, rolling_maximize_cara_mixture, rolling_maximise_diversification, rolling_quadratic_optimisation, rolling_max_return_target_vol, rolling_min_variance_target_return, rolling_maximise_alpha_with_target_return, rolling_maximise_alpha_over_tre.

Impact: for backtests with a binding turnover constraint or non-zero transaction-cost penalty, realised turnover and TC drag will differ from pre-v5.3.1 numbers. The optimiser now constrains ||w_new − w_drift||_1 rather than ||w_new − w_prev_target||_1, which matches what the NAV simulator actually trades. On a min-variance / L1 0.08-per-quarter backtest: legacy realised/apparent turnover ratio 1.23, new default 1.02; cumulative TC drag drops from 19.4 bps to 16.0 bps (17.5% relative reduction).

For backtests without a turnover-related constraint or penalty, weights_0 only affects the CVXPY warm-start and the SciPy convergence path. Numerical differences may exist but are typically below 1 bp/year on Sharpe.

Frozen-position overshoot relaxation in Constraints.update_with_valid_tickers.

When rebalancing_indicators freeze illiquid positions (PE, HF, CAT bonds, private credit) over multiple TAA rebalance dates, the frozen positions can drift above their group's group_max_allocation. Previously Constraints.__post_init__ raised ValueError: Infeasible constraints detected. The new behaviour automatically relaxes the offending bound by the overshoot amount with an audit-trail UserWarning, treating the rebalance as a one-period compliance waiver — the optimiser can no longer trade frozen assets, and the relaxed cap prevents tradable members from adding more on top of the inherited overhang.

This change is independent of the drift policy: it also helps when weights_0 comes from a live PMS that is slightly out of compliance due to intra-period flows, corporate actions, or settlement gaps. Each relaxation event surfaces in logs with the group name, original bound, and relaxed bound.

Examples folder reorganised.

The flat examples/ layout has been replaced with five purpose-folders:

Old path	New path
examples.universe	examples.data.universe
examples.optimal_portfolio_backtest	examples.backtests.minimal_backtest
examples.solve_risk_budgets_balanced_portfolio	examples.backtests.balanced_risk_budgets
examples.computation_of_tracking_error	examples.backtests.tracking_error_decomposition
examples.multi_optimisers_backtest	examples.comparisons.optimisers
examples.multi_covar_estimation_backtest	examples.comparisons.covar_estimators
examples.parameter_sensitivity_backtest	examples.comparisons.parameter_sensitivity
examples.risk_budgeting_pyrb_vs_scipy	examples.comparisons.pyrb_vs_scipy
examples.sp500_minvar	examples.comparisons.sp500_minvar_spans
examples.long_short_optimisation	examples.solvers.long_short
examples.sp500_universe	unchanged (kept at top level)

The new layout adds an examples/README.md indexing every demo. Six wrong docstrings in solvers/ corrected (carra_mixture, max_diversification, max_sharpe, min_variance, risk_budgeting, tracking_error — all were boilerplate copies of "example of minimization of tracking error" regardless of the file's contents). Two helpers in data/universe.py: fetch_benchmark_universe_data() (15-ETF universe, 6-tuple return) and fetch_minimal_universe_data() (8-ETF universe, 3-tuple return) replace the inline loaders previously duplicated across minimal_backtest.py and long_short.py.

Migration from v5.0.x:

• If you have notebooks or scripts referencing the old examples.* paths, see the table above. The package public API (everything under from optimalportfolios import ...) is unchanged.
• If your backtests rely on the legacy weights_0 behaviour for reproducibility (e.g. validating against published numbers), pass OptimiserConfig(use_drifted_weights_0=False).
• If you previously caught ValueError: Infeasible constraints detected from a long-running backtest of illiquid universes, those backtests will now run to completion with UserWarning messages instead. Consider capturing the warnings at the runner level and emitting a summary line rather than per-event logs.

March 2026, Version 5.0.4 released

Removed scikit-learn dependency. The Gaussian mixture model in utils/gaussian_mixture.py previously used sklearn.mixture.GaussianMixture. This has been replaced with a pure numpy/scipy EM implementation (fit_gmm) using scipy.stats.multivariate_normal for the E-step and scipy.cluster.vq.kmeans2 for K-means initialisation. The public API (fit_gaussian_mixture, Params, plot_mixure1, plot_mixure2, estimate_rolling_mixture) is unchanged.

This removes the last scikit-learn import from optimalportfolios, eliminating the transitive dependency on joblib, threadpoolctl, and the scikit-learn binary itself — a meaningful reduction in install footprint.

March 2026, Version 5.0.0 released

LASSO estimator extracted to factorlasso package. The lasso/ module has been removed from optimalportfolios. The LASSO/Group LASSO/HCGL solver is now in the standalone factorlasso package — a domain-agnostic sparse factor model estimator with sign constraints, prior-centered regularisation, NaN-aware estimation, and scikit-learn compatible API (fit / predict / score / coef_ / intercept_). factorlasso is a required dependency of optimalportfolios v5.0.0. All existing imports (from optimalportfolios import LassoModel) continue to work via re-exports.

License changed from GPL-3.0 to MIT.

Dependencies cleaned:

• Removed easydev, pyarrow, fsspec, statsmodels, ecos (unused)
• yfinance, pandas-datareader moved to [data] optional
• numpy unpinned from ==2.2.6 to >=2.0
• Build system simplified (removed unused poetry-core, hatchling)
• Dev tooling: black/flake8/isort/mypy replaced with ruff

CI added: GitHub Actions test pipeline across Python 3.10–3.12.

Migration from v4.x: No code changes required. All existing imports (from optimalportfolios import LassoModel, LassoModelType) continue to work via re-exports from factorlasso. The only exception: if your code imports directly from the deleted module path (from optimalportfolios.lasso.lasso_estimator import ...), change to from optimalportfolios import ....

March 2026, Version 4.1.1 released

Alpha signals module (optimalportfolios.alphas):

• New alphas/ package with three standalone signal functions: compute_momentum_alpha, compute_low_beta_alpha, compute_managers_alpha
• Each function handles single-frequency and mixed-frequency universes via returns_freq (string or per-asset pd.Series)
• Within-group cross-sectional scoring via group_data parameter
• AlphasData container moved from utils/manager_alphas.py to alphas/alpha_data.py
• backtest_alphas.py moved from reports/ to alphas/ with fixed function names (typo corrections: backtest_alpha_signas → backtest_alpha_signals, etc.)
• Comprehensive test suite in alphas/tests/signals_test.py

Deprecated and removed:

• utils/factor_alphas.py — all functions migrated to alphas/signals/. The 9-function variant explosion (3 signal types × 3 frequency variants) is replaced by 3 functions, each handling all dispatch modes internally
• utils/manager_alphas.py — AlphasData moved to alphas/alpha_data.py. compute_joint_alphas() is replaced by external aggregation (see migration guide below)
• reports/backtest_alphas.py — moved to alphas/backtest_alphas.py

Risk budgeting fixes:

• Fixed total_to_good_ratio computation in wrapper_risk_budgeting: previously used len(pd_covar.columns) / len(clean_covar.columns) which over-inflated budgets when zero-budget and NaN assets coexisted. Now uses n_eligible / n_valid where n_eligible counts assets with positive risk budget
• Replaced all print() fallback messages with warnings.warn() for proper logging
• Removed unused FactorCovarEstimator import

Solver docstrings:

• Full docstrings added to all optimisation solvers (quadratic, risk_budgeting, max_diversification, max_sharpe, tracking_error, target_return, cara_mixture)
• Full docstrings for the rolling portfolio dispatcher

Covariance estimation separation:

• Covariance estimation is now clearly separated from portfolio optimisation. The recommended workflow is to estimate covariance matrices upfront using EwmaCovarEstimator or FactorCovarEstimator, then pass the resulting covar_dict to any solver. This enables reusing the same covariance across multiple solvers, inspecting covariance diagnostics independently, and swapping estimators without modifying solver code.

05 January 2025, Version 3.1.1 released

Added Lasso estimator and Group Lasso estimator using cvxpy quadratic problems.

Added covariance estimator using factor model with Lasso betas.

Estimated covariance matrices can be passed to rolling solvers, CovarEstimator type is added for different covariance estimators.

Risk budgeting is implemented using pyrb package with pyrb forked for optimalportfolios package.

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 examples of running all solvers.

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.

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

Disclaimer

OptimalPortfolios package is distributed FREE & WITHOUT ANY WARRANTY under the MIT License.

See the LICENSE.txt in the release for details.

Please report any bugs or suggestions by opening an issue.

References

Sepp A. (2023), "Optimal Allocation to Cryptocurrencies in Diversified Portfolios", Risk Magazine, October 2023, 1-6. Available at https://ssrn.com/abstract=4217841

Sepp A., Ossa I., and Kastenholz M. (2026), "Robust Optimization of Strategic and Tactical Asset Allocation for Multi-Asset Portfolios", The Journal of Portfolio Management, 52(4), 86-120. Paper link

Sepp A., Hansen E., and Kastenholz M. (2026), "Capital Market Assumptions and Strategic Asset Allocation Using Multi-Asset Tradable Factors", Under revision at the Journal of Portfolio Management.

BibTeX Citations for optimalportfolios Package

If you use optimalportfolios in your research, please cite it as:

@software{sepp2024optimalportfolios,
  author={Sepp, Artur},
  title={OptimalPortfolios: Implementation of optimisation analytics for constructing and backtesting optimal portfolios in Python},
  year={2024},
  url={https://github.com/ArturSepp/OptimalPortfolios}
}

@article{sepp2023,
  title={Optimal allocation to cryptocurrencies in diversified portfolios},
  author={Sepp, Artur},
  journal={Risk Magazine},
  pages={1--6},
  month={October},
  year={2023},
  url={https://ssrn.com/abstract=4217841}
}

@article{sepp2026rosaa,
  author={Sepp, Artur and Ossa, Ivan and Kastenholz, Mika},
  title={Robust Optimization of Strategic and Tactical Asset Allocation for Multi-Asset Portfolios},
  journal={The Journal of Portfolio Management},
  volume={52},
  number={4},
  pages={86--120},
  year={2026}
}

@article{sepphansenkastenholz2026,
  title={Capital Market Assumptions and Strategic Asset Allocation Using Multi-Asset Tradable Factors},
  author={Sepp, Artur and Hansen, Emilie H. and Kastenholz, Mika},
  journal={Working Paper},
  year={2026}
}