Capability catalogue

Everything PortfolioOptimisers.jl can do, grouped by the job it does rather than by the file it lives in. Each entry links to its docstring.

This page is generated (see docs/generate_capability_catalogue.jl): the grouping is curated in docs/capability_catalogue.jl, and every description is the first sentence of the corresponding docstring, so the two can never disagree. A test asserts that every estimator and algorithm in the package appears here, so the page cannot fall behind the code.

For the same types arranged by subtyping rather than by capability, see the type hierarchy.

Core abstractions

Every component is an Estimator (a configuration encoding a method and its hyperparameters), an Algorithm (a behaviour selector consumed through an Estimator), or a Result (computed output). Estimators and Algorithms are what you choose; Results are what you get back.

Because every struct is immutable, runtime values are propagated down a composed estimator tree by rebuilding it.

• No-op factory function for constructing objects with a uniform interface. factory

Preprocessing

• Convert TimeSeries.TimeArray price data to returns. prices_to_returns and ReturnsResult

• A container for aligned, time-indexed price-level data. PricesResult

• Preprocessing estimator converting price-level data into returns-level data. PricesToReturns, fit_preprocessing, and apply_preprocessing

  • Preprocessing estimator dropping assets and observations with excessive missing data from price-level data. MissingDataFilter and MissingDataFilterResult

  • Preprocessing estimator imputing missing price observations from per-asset statistics fitted on the training window. Imputer and ImputerResult

• Asset selection

  • Asset selector that scores every asset with a risk measure and keeps the assets a rule admits. ScoreSelector, ZeroVarianceFilter, and CompleteAssetSelector

  • Asset selector that discards assets which duplicate information already carried by others. RedundancySelector

  • Fitted result of any AbstractAssetSelector. AssetSelectorResult

  • Selection rules

    • RankRule with the tail sizes given as fractions of the asset universe. QuantileRule

    • Take best and/or worst assets from the tails of the score ordering, then keep or drop them. RankRule

    • Keep assets whose score falls strictly inside the band (lo, hi). ThresholdRule

• Cut price- or returns-level data into a training window (the head) and a held-out test window (the tail). train_test_split, TrainTestSplit, and TrainTestSplitResult

• Return a ReturnsResult appropriate for benchmark-tracking optimisations. returns_result_picker

Matrix processing

• Projects a matrix to the nearest positive definite matrix, typically used for co-moment matrices. Posdef, posdef!, and posdef

• Configures and applies denoising algorithms to covariance or correlation matrices. Denoise, denoise!, and denoise

  • Denoises by setting the smallest num_factors eigenvalues to zero. SpectralDenoise

  • Denoises by replacing the smallest num_factors eigenvalues with their average. FixedDenoise

  • Denoises by shrinking the smallest num_factors eigenvalues towards the diagonal. ShrunkDenoise

• Removes the largest n principal components (market modes) from a covariance or correlation matrix. Detone, detone!, and detone

• Configures and applies matrix processing routines. MatrixProcessing, matrix_processing!, matrix_processing_step!, and matrix_processing

Regression models

Factor prior models and implied volatility use regression in their estimation, which return a Regression object.

Regression targets

• Regression target type for standard linear models. LinearModel

• Regression target type for generalised linear models (GLMs). GeneralisedLinearModel

Regression types

• Estimator for stepwise regression-based moment estimation. StepwiseRegression

  • Algorithms

    • Stepwise regression algorithm: forward selection. ForwardSelection

    • Stepwise regression algorithm: backward elimination. BackwardElimination

  • Selection criteria

    • Stepwise regression criterion based on p-value thresholding. PValue

    • Akaike Information Criterion (AIC) for stepwise regression. AIC

    • Corrected Akaike Information Criterion (AICC) for stepwise regression. AICC

    • Bayesian Information Criterion (BIC) for stepwise regression. BIC

    • Coefficient of determination (R²) for stepwise regression. RSquared

    • Adjusted coefficient of determination (Adjusted R²) for stepwise regression. AdjustedRSquared

• Estimator for dimension reduction regression-based moment estimation. DimensionReductionRegression

  • Principal Component Analysis (PCA) dimension reduction target. PCA

  • Probabilistic Principal Component Analysis (PPCA) dimension reduction target. PPCA

Moment estimation

Expected returns

Overloads Statistics.mean.

• A simple expected returns estimator for PortfolioOptimisers.jl, representing the sample mean with optional observation weights. SimpleExpectedReturns

• Container type for equilibrium expected returns estimators. EquilibriumExpectedReturns

• Container type for excess expected returns estimators. ExcessExpectedReturns

• Container type for shrinkage-based expected returns estimators. ShrunkExpectedReturns

  • Algorithms

    • James-Stein JamesStein

    • Bayes-Stein BayesStein

    • Bodnar-Okhrin-Parolya BodnarOkhrinParolya

  • Targets: all algorithms can have any of the following targets

    • Grand Mean GrandMean

    • Volatility Weighted VolatilityWeighted

    • Mean Squared Error MeanSquaredError

• Expected returns estimator that returns the asset standard deviations. StandardDeviationExpectedReturns

• Expected returns estimator that returns the asset variances. VarianceExpectedReturns

• Expected returns estimator that returns the optionally weighted asset medians. MedianExpectedReturns

• Expected returns estimator that returns custom values for each asset. CustomValueExpectedReturns

• Expected returns estimator that restricts computation to a rolling or indexed observation window. WindowedExpectedReturns

Variance and standard deviation

Overloads Statistics.var and Statistics.std.

• A flexible variance estimator for PortfolioOptimisers.jl supporting optional expected returns estimators, observation weights, and bias correction. SimpleVariance

• Variance estimator that restricts computation to a rolling or indexed observation window. WindowedVariance

Covariance and correlation

Overloads Statistics.cov and Statistics.cor.

• A simple wrapper around a StatsBase.CovarianceEstimator, optional StatsBase.AbstractWeights, and an optional index. GeneralCovariance

• Estimates the covariance matrix of asset returns from a centring estimator, a covariance estimator, and a moment algorithm. Covariance

  • FullMoment is used to indicate that all deviations are included in the moment estimation process. FullMoment

  • SemiMoment is used for semi-moment estimators, where only observations below a target are considered. SemiMoment

• Configures and applies Gerber covariance estimators. GerberCovariance

  • Implements the original Gerber covariance algorithm. Gerber0

  • Implements the first variant of the Gerber covariance algorithm. Gerber1

  • Implements the second variant of the Gerber covariance algorithm. Gerber2

• Configures and applies Smyth-Broby covariance estimators. SmythBrobyCovariance

  • Implements the original Smyth-Broby covariance algorithm. SmythBroby0

  • Implements the first variant of the Smyth-Broby covariance algorithm. SmythBroby1

  • Implements the second variant of the Smyth-Broby covariance algorithm. SmythBroby2

  • Implements the original Smyth-Broby covariance algorithm scaled by vote counts. SmythBrobyGerber0

  • Implements the first variant of the Smyth-Broby covariance algorithm scaled by vote counts. SmythBrobyGerber1

  • Implements the second variant of the Smyth-Broby covariance algorithm scaled by vote counts. SmythBrobyGerber2

  • Implements the original Smyth-Broby covariance algorithm using vote counts only. SmythBrobyCount0

  • Implements the first variant of the Smyth-Broby covariance algorithm using vote counts only. SmythBrobyCount1

  • Implements the second variant of the Smyth-Broby covariance algorithm using vote counts only. SmythBrobyCount2

• Gerber Information Quality GerberIQCovariance with custom variance, demeaning, temporal decay and numerator + denominator estimators

  • Implements the basic Gerber IQ covariance template. BasicGerberIQ

  • Gerber Information Quality template with asymmetric thresholds. PartialGerberIQ

  • Gerber Information Quality template with fine-grained asymmetric thresholds. FullGerberIQ

  • Exponential Gerber IQ temporal decay. ExpGerberIQDecay

  • Scales the threshold parameters using the individual asset volatilities. AssetVolatilityGerberIQScaler

• Configures and applies distance-based covariance estimators. DistanceCovariance

• Lower tail dependence covariance estimator. LowerTailDependenceCovariance

• Rank covariances

  • Robust covariance estimator based on Kendall's tau rank correlation. KendallCovariance

  • Robust covariance estimator based on Spearman's rho rank correlation. SpearmanCovariance

• Covariance estimator based on mutual information. MutualInfoCovariance

  • Abstract supertype for all histogram binning algorithms based on a bin width selection rule. BinWidthBins

    • Knuth's optimal bin width Knuth

    • Freedman Diaconis bin width FreedmanDiaconis

    • Scott's bin width Scott

  • Histogram binning algorithm using the Hacine-Gharbi–Ravier rule. HacineGharbiRavier

  • Predefined number of bins

• Convenience constructor. DenoiseCovariance

• Convenience constructor. DetoneCovariance

• Convenience constructor. ProcessedCovariance

• Covariance estimator based on implied volatility scaling. ImpliedVolatility

  • Implied volatility algorithm that scales implied volatility by a user-supplied premium factor. ImpliedVolatilityPremium

  • Implied volatility algorithm that predicts realised volatility via regression on implied volatility. ImpliedVolatilityRegression

• Composite covariance estimator with post-processing. PortfolioOptimisersCovariance

• A covariance estimator that returns the correlation matrix as both the covariance and correlation. CorrelationCovariance

• Covariance estimator that restricts computation to a rolling or indexed observation window. WindowedCovariance

• Regime-adjusted covariance and variance

  • Online exponentially weighted covariance estimator with regime-state adjustment. RegimeAdjustedExpWeightedCovariance

  • Online exponentially weighted variance estimator with regime-state adjustment. RegimeAdjustedExpWeightedVariance

  • Regime adjustment methods

    • Regime adjustment method that scales variance by the ratio of the mean absolute deviation of standardised returns to the first-moment normalisation constant x. FirstMomentRegimeAdjusted

    • Regime adjustment method that scales variance exponentially with the smoothed log-deviation of standardised squared returns from its expected value under stationarity. LogRegimeAdjusted

    • Regime adjustment method that scales variance by the square root of the mean of the standardised squared returns. RootMeanSquaredAdjusted

  • Shrinkage targets

    • Regime-adjustment target that uses a diagonal baseline covariance structure. DiagonalTarget

    • Regime-adjustment target that uses a Mahalanobis-distance-based baseline covariance structure. MahalanobisTarget

    • Regime-adjustment target that uses a portfolio-weighted baseline covariance structure. PortfolioTarget

  • Demeaning

    • Centres the returns series using the (weighted) mean before computing the Median Absolute Deviation. MeanCentering

    • Centres the returns series using the (weighted) median before computing the Median Absolute Deviation. MedianCentering

  • Correlation smoothing

    • Greedy pairwise correlation pruning: drop assets until no surviving pair exceeds t. PairwiseCorrelation

    • Group assets by connected component of the over-threshold correlation graph, and keep the best-scoring member of each. CorrelationComponents

Coskewness

Implements coskewness.

• Container type for coskewness estimators. Coskewness

  • FullMoment is used to indicate that all deviations are included in the moment estimation process. FullMoment

  • SemiMoment is used for semi-moment estimators, where only observations below a target are considered. SemiMoment

• Coskewness estimator that restricts computation to a rolling or indexed observation window. WindowedCoskewness

Cokurtosis

Implements cokurtosis.

• Container type for cokurtosis estimators. Cokurtosis

  • FullMoment is used to indicate that all deviations are included in the moment estimation process. FullMoment

  • SemiMoment is used for semi-moment estimators, where only observations below a target are considered. SemiMoment

• Cokurtosis estimator that restricts computation to a rolling or indexed observation window. WindowedCokurtosis

Windowed moments

Every windowed estimator wraps a base moment estimator and recomputes it over a trailing window, so a moment can vary across the folds of a cross-validation scheme. The window is set by a fixed length or by a WindowSizeEstimator.

• Abstract supertype for estimators that determine the rolling window size. WindowSizeEstimator

Distance matrices

Implements distance and cor_and_dist.

• If power is not nothing, computes the generalised distance estimator. Distance

• Distance-of-distances estimator for portfolio optimization. DistanceDistance

The distance estimators are used together with various distance matrix algorithms.

• Simple distance algorithm for portfolio optimization. SimpleDistance

• Simple absolute distance algorithm for portfolio optimization. SimpleAbsoluteDistance

• Logarithmic distance algorithm for portfolio optimization. LogDistance

• Correlation distance algorithm for portfolio optimization. CorrelationDistance

• Variation of Information (VI) distance algorithm for portfolio optimization. VariationInfoDistance

  • Abstract supertype for all histogram binning algorithms based on a bin width selection rule. BinWidthBins

    • Knuth's optimal bin width Knuth

    • Freedman Diaconis bin width FreedmanDiaconis

    • Scott's bin width Scott

  • Histogram binning algorithm using the Hacine-Gharbi–Ravier rule. HacineGharbiRavier

  • Predefined number of bins

• Canonical distance algorithm for portfolio optimization. CanonicalDistance

Phylogeny

PortfolioOptimisers.jl can make use of asset relationships to perform optimisations, define constraints, and compute relatedness characteristics of portfolios.

Clustering

Phylogeny constraints and clustering optimisations make use of clustering algorithms via ClustersEstimator, Clusters, and clusterise. Most clustering algorithms come from Clustering.jl.

• Estimator type for selecting the optimal number of clusters. OptimalNumberClusters and VectorToScalarMeasure

  • Algorithm type for estimating the optimal number of clusters using the second-order difference method. SecondOrderDifference

  • Algorithm type for estimating the optimal number of clusters using the standardised silhouette score. SilhouetteScore

  • Predefined number of clusters.

  • Select the optimal number of clusters for a hierarchical clustering tree. optimal_number_clusters

• Get the vector of cluster indices for each point. assignments

Hierarchical

• Algorithm type for hierarchical clustering. HClustAlgorithm

• Direct Bubble Hierarchical Trees DBHT and Local Global sparsification of the covariance matrix LoGo, logo!, and logo

  • Root selection

    • A DBHT root selection method that enforces a unique root in the hierarchy. UniqueRoot

    • A DBHT root selection method that creates a root from the adjacency tree of all root candidates. EqualRoot

Non-hierarchical

Non-hierarchical clustering algorithms are incompatible with hierarchical clustering optimisations, but they can be used for phylogeny constraints and NestedClustered optimisations.

• K-means clustering algorithm configuration for non-hierarchical clustering. KMeansAlgorithm

Networks

Adjacency matrices

Adjacency matrices encode asset relationships either with clustering or graph theory via phylogeny_matrix and PhylogenyResult.

• Network adjacency NetworkEstimator with custom tree algorithms, covariance, and distance estimators

  • Algorithm type for Kruskal's minimum spanning tree (MST). KruskalTree, BoruvkaTree, and PrimTree

  • Triangulated Maximally Filtered Graph with various similarity matrix estimators

    • Maximum distance similarity MaximumDistanceSimilarity

    • Exponential similarity ExponentialSimilarity

    • General exponential similarity GeneralExponentialSimilarity

• Estimator type for clustering. ClustersEstimator and Clusters

• Estimator type for network-based phylogeny analysis. NetworkClustersEstimator

• Group assets by clustering them, and keep the best-scoring member of each cluster. ClusterGroups

Centrality and phylogeny measures

• Centrality estimator CentralityEstimator with custom adjacency matrix estimators (clustering and network) and centrality measures

  • Betweenness BetweennessCentrality

  • Closeness ClosenessCentrality

  • Degree DegreeCentrality

  • Eigenvector EigenvectorCentrality

  • Katz KatzCentrality

  • Pagerank Pagerank

  • Radiality RadialityCentrality

  • Stress StressCentrality

• Fallback no-op for returning a validated centrality vector result as-is. centrality_vector

• Compute the weighted average centrality for a network and centrality algorithm. average_centrality

• Compute the asset phylogeny score for a set of weights and a phylogeny matrix. asset_phylogeny

Cluster trees

Hierarchical clustering produces a tree of ClusterNodes, walked by to_tree, pre_order, and is_leaf.

• Preorder traversal strategy that visits nodes by their ID. PreorderTreeByID

Optimisation constraints

Non clustering optimisers support a wide range of constraints, while naive and clustering optimisers only support weight bounds. Furthermore, entropy pooling prior supports a variety of views constraints. It is therefore important to provide users with the ability to generate constraints manually and/or programmatically. We therefore provide a wide, robust, and extensible range of types such as AbstractEstimatorValueAlgorithm and UniformValues, and functions that make this easy, fast, and safe.

Constraints can be defined via their estimators or directly by their result types. Some using estimators need to map key-value pairs to the asset universe, this is done by defining the assets and asset groups in AssetSets. Internally, PortfolioOptimisers.jl uses all the information and calls group_to_val!, and replace_group_by_assets to produce the appropriate arrays.

• Equation parsing parse_equation and ParsingResult

• No-op fallback for returning an existing LinearConstraint object or nothing. linear_constraints, LinearConstraintEstimator, PartialLinearConstraint, and LinearConstraint

• No-op fallback for risk budget constraint generation. risk_budget_constraints, RiskBudgetEstimator, and RiskBudget

• Generate phylogeny-based portfolio constraints from an estimator or result. phylogeny_constraints, centrality_constraints, SemiDefinitePhylogenyEstimator, SemiDefinitePhylogeny, IntegerPhylogenyEstimator, IntegerPhylogeny, and CentralityConstraint

• Generate portfolio weight bounds constraints from a WeightBoundsEstimator and asset set. weight_bounds_constraints, WeightBoundsEstimator, and WeightBounds

• Container for asset set and group information used in constraint generation. AssetSets

• Budget constraints BudgetEstimator and BudgetRange

  • Budget constraint that accounts for linear transaction costs. BudgetCosts

  • Budget constraint that accounts for non-linear (power-law) market impact costs. BudgetMarketImpact

• Constraint values AbstractEstimatorValueAlgorithm

  Where a constraint takes one value per asset or group, these algorithms say how to derive it from data rather than stating it outright.

  • Custom weight bounds constraint for uniformly distributing asset weights, 1/N for lower bounds and 1 for upper bounds, where N is the number of assets. UniformValues

  • Return value for assets or groups, based on a mapping and asset sets. estimator_to_val

  • Algorithm for reducing a vector of real values to its minimum. MinValue

  • Algorithm for reducing a vector of real values to its maximum. MaxValue

  • Algorithm for reducing a vector of real values to its optionally weighted mean. MeanValue

  • Algorithm for reducing a vector of real values to its optionally weighted median. MedianValue

  • Algorithm for reducing a vector of real values to its mode. ModeValue

  • Algorithm for reducing a vector of real values to its sum. SumValue

  • Algorithm for reducing a vector of real values to its product. ProdValue

  • Algorithm for reducing a vector of real values to its optionally weighted standard deviation. StdValue

  • Algorithm for reducing a vector of real values to its optionally weighted variance. VarValue

  • Algorithm for reducing a vector of real values to its optionally weighted mean divided by its optionally weighted standard deviation. StandardisedValue

  • Marker for "no default here", used in the two places a fold-less value may be missing. NoDefault

• Time-dependent constraint: an optimiser input whose value changes across the folds of a cross-validation scheme. TimeDependent

  A time-dependent input takes a different value in each fold of a cross-validation scheme, and is inert outside one.

  • Abstract supertype for callable structs used as time-dependent constraint values. TimeDependentCallable

  • Abstract supertype for callable structs whose per-fold value is an optimiser. TimeDependentOptimiserCallable

  • Per-fold context handed to time-dependent constraints when they are resolved. TimeDependentContext

  • Wrapper marking a callable time-dependent constraint entry as requiring the previous optimisation's weights. PreviousWeightsFunction

• Construct a binary asset-group membership matrix from asset set groupings. asset_sets_matrix and AssetSetsMatrixEstimator

• Propagate or pass through buy-in threshold portfolio constraints. threshold_constraints, ThresholdEstimator, and Threshold

Prior statistics

Many optimisations and constraints use prior statistics computed via prior.

• Container type for low order prior results. LowOrderPrior

  • Empirical prior estimator for asset returns. EmpiricalPrior

  • Factor-based prior estimator for asset returns. FactorPrior

  • Black-Litterman

    • Unified interface for constructing or passing through Black-Litterman investor views. black_litterman_views

    • Black-Litterman prior estimator for asset returns. BlackLittermanPrior

    • Bayesian Black-Litterman prior estimator for asset returns. BayesianBlackLittermanPrior

    • Factor Black-Litterman prior estimator for asset returns. FactorBlackLittermanPrior

    • Augmented Black-Litterman prior estimator for asset returns. AugmentedBlackLittermanPrior

  • Entropy pooling prior estimator for asset returns. EntropyPoolingPrior

    Entropy pooling reweights the observations so that the posterior satisfies the stated views while staying as close as possible to the prior.

    • Container for Black-Litterman investor views in canonical matrix form. BlackLittermanViews

    • View constraint algorithms

      • One-shot entropy pooling. H0_EntropyPooling

      • Uses the initial probabilities to optimise the posterior probabilities at every step. H1_EntropyPooling

      • Uses the previous step's probabilities to optimise the next step's probabilities. H2_EntropyPooling

      • Conditional Value-at-Risk (CVaR) entropy pooling optimiser. CVaREntropyPooling

    • Divergence formulations

      • Exponential entropy pooling optimisation algorithm. ExpEntropyPooling

      • Logarithmic entropy pooling optimisation algorithm. LogEntropyPooling

    • Optimisers

      • Optim.jl-based entropy pooling optimiser. OptimEntropyPooling

      • JuMP.jl-based entropy pooling optimiser. JuMPEntropyPooling

  • Opinion pooling prior estimator for asset returns. OpinionPoolingPrior

    • Linear opinion pooling algorithm for consensus prior estimation. LinearOpinionPooling

    • Logarithmic opinion pooling algorithm for consensus prior estimation. LogarithmicOpinionPooling

• Container type for high order prior results. HighOrderPrior

  • High order prior estimator for asset returns. HighOrderPriorEstimator

  • Represents the High Order Factor Prior Estimator. HighOrderFactorPriorEstimator

Uncertainty sets

In order to make optimisations more robust to noise and measurement error, it is possible to define uncertainty sets on the expected returns and covariance. These can be used in optimisations which use either of these two quantities. These are implemented via ucs, mu_ucs, and sigma_ucs.

PortfolioOptimisers.jl implements two types of uncertainty sets.

• Represents a box uncertainty set for risk or prior statistics in portfolio optimisation. BoxUncertaintySet and BoxUncertaintySetAlgorithm

• EllipsoidalUncertaintySet and EllipsoidalUncertaintySetAlgorithm with various algorithms for computing the scaling parameter via k_ucs

  • Algorithm for computing the scaling parameter k for ellipsoidal uncertainty sets under the assumption of normally distributed returns in portfolio optimisation. NormalKUncertaintyAlgorithm

  • Computes the ellipsoidal uncertainty set scaling parameter k as sqrt((1 - q) / q). GeneralKUncertaintyAlgorithm

  • Algorithm for computing the scaling parameter k for ellipsoidal uncertainty sets using the chi-squared distribution in portfolio optimisation. ChiSqKUncertaintyAlgorithm

  • Predefined scaling parameter

It also implements various estimators for the uncertainty sets, the following two can generate box and ellipsoidal sets.

• Estimator for box or ellipsoidal uncertainty sets under the assumption of normally distributed returns in portfolio optimisation. NormalUncertaintySet

• Bootstrapping via Autoregressive Conditional Heteroscedasticity ARCHUncertaintySet via arch

  • Circular CircularBootstrap

  • Moving MovingBootstrap

  • Stationary StationaryBootstrap

The following estimator can only generate box sets.

• Estimator for box uncertainty sets using delta bounds on mean and covariance statistics in portfolio optimisation. DeltaUncertaintySet

Quintile portfolios are expressed as an uncertainty set on the characteristic vector rather than as an optimiser of their own (ADR 0032).

• Estimator for ${\ell }_1$ uncertainty sets on the characteristic vector. CharacteristicUncertaintySet

  • $${\ell }_1$$

    (cross-polytope) uncertainty set on the characteristic vector. L1UncertaintySet and L1UncertaintySetAlgorithm

  • Signed ${\ell }_1$ uncertainty set on the characteristic vector, with a separate error budget per sign. SignedL1UncertaintySet and SignedL1UncertaintySetAlgorithm

  • Radius algorithm that calibrates the ${\ell }_1$ uncertainty radius to a target number of active assets. ActiveAssetsUncertaintyAlgorithm

• Ellipsoidal set classes

  • Represents the class identifier for mean ellipsoidal uncertainty sets in portfolio optimisation. MuEllipsoidalUncertaintySet

  • Represents the class identifier for covariance ellipsoidal uncertainty sets in portfolio optimisation. SigmaEllipsoidalUncertaintySet

Turnover

The turnover is defined as the element-wise absolute difference between the vector of current weights and a vector of benchmark weights. It can be used as a constraint, method for fee calculation, and risk measure. These are all implemented using turnover_constraints, TurnoverEstimator, and Turnover.

Fees

Fees are a non-negligible aspect of active investing. As such PortfolioOptimiser.jl has the ability to account for them in all optimisations but the naive ones. They can also be used to adjust expected returns calculations via calc_fees and calc_asset_fees.

• Generate portfolio transaction fee constraints from a FeesEstimator and asset set. fees_constraints

• Compute the fixed portfolio fees for assets that have been allocated. calc_fixed_fees and calc_asset_fixed_fees

• Estimator for portfolio transaction fees constraints. FeesEstimator and Fees

  • Proportional long

  • Proportional short

  • Fixed long

  • Fixed short

  • Turnover

Portfolio returns and drawdowns

Various risk measures and analyses require the computation of simple and cumulative portfolio returns and drawdowns both in aggregate and per-asset. These are computed by calc_net_returns, calc_net_asset_returns, cumulative_returns, drawdowns.

Tracking

It is often useful to create portfolios that track the performance of an index, indicator, or another portfolio.

• Compute the benchmark portfolio returns for a weights-based tracking algorithm. tracking_benchmark and TrackingError

  • Returns-based tracking algorithm. ReturnsTracking

  • Asset weights-based tracking algorithm. WeightsTracking

The error can be computed using different algorithms using norm_error.

• Norm tracking algorithms

  • Norm-one (NOC) error formulation. L1Norm

  • Second-order cone (SOC) norm-based error formulation. L2Norm

  • Second-order cone (SOC) squared norm-based error formulation. SquaredL2Norm

  • L-p norm error estimator. LpNorm

  • L-infinity norm (maximum absolute deviation) error estimator. LInfNorm

It is also possible to track the error in with risk measures RiskTrackingError using WeightsTracking, which allows for two approaches.

• Dependent variable-based tracking formulation. DependentVariableTracking

• Independent variable-based tracking formulation. IndependentVariableTracking

Risk measures

PortfolioOptimisers.jl provides a wide range of risk measures. These are broadly categorised into two types based on the type of optimisations that support them.

Risk measures for traditional optimisation

These are all subtypes of RiskMeasure, and are supported by all optimisation estimators.

• Represents the portfolio variance using a covariance matrix. Variance

  • Traditional optimisations also support:

    • Risk contribution

    • Formulations

      • Direct quadratic risk expression optimisation formulation for variance-like risk measures. QuadRiskExpr

      • Squared second-order cone risk expression optimisation formulation for applicable risk measures. SquaredSOCRiskExpr

• Represents the portfolio standard deviation using a covariance matrix. StandardDeviation

• Uncertainty set variance UncertaintySetVariance (same as variance when used in non-traditional optimisation)

• Represents a low-order moment risk measure. LowOrderMoment

  • Represents the first lower moment risk measure algorithm. FirstLowerMoment

  • Represents the mean absolute deviation risk measure algorithm. MeanAbsoluteDeviation

  • Represents a second moment (variance or standard deviation) risk measure algorithm. SecondMoment

    • Second squared moments

      • FullMoment is used to indicate that all deviations are included in the moment estimation process. FullMoment

      • SemiMoment is used for semi-moment estimators, where only observations below a target are considered. SemiMoment

      • Traditional optimisation formulations

        • Direct quadratic risk expression optimisation formulation for variance-like risk measures. QuadRiskExpr

        • Squared second-order cone risk expression optimisation formulation for applicable risk measures. SquaredSOCRiskExpr

        • Rotated second-order cone risk expression optimisation formulation for applicable risk measures. RSOCRiskExpr

    • Second-order cone risk expression optimisation formulation for applicable risk measures. SOCRiskExpr

      • FullMoment is used to indicate that all deviations are included in the moment estimation process. FullMoment

      • SemiMoment is used for semi-moment estimators, where only observations below a target are considered. SemiMoment

• Represents the square root kurtosis risk measure. Kurtosis

  • Actual kurtosis

    • FullMoment and semi-kurtosis are supported in traditional optimisers via the kt field. Risk calculation uses

      • FullMoment is used to indicate that all deviations are included in the moment estimation process. FullMoment

      • SemiMoment is used for semi-moment estimators, where only observations below a target are considered. SemiMoment

    • Traditional optimisation formulations

      • Direct quadratic risk expression optimisation formulation for variance-like risk measures. QuadRiskExpr

      • Squared second-order cone risk expression optimisation formulation for applicable risk measures. SquaredSOCRiskExpr

      • Rotated second-order cone risk expression optimisation formulation for applicable risk measures. RSOCRiskExpr

  • Second-order cone risk expression optimisation formulation for applicable risk measures. SOCRiskExpr

    • FullMoment is used to indicate that all deviations are included in the moment estimation process. FullMoment

    • SemiMoment is used for semi-moment estimators, where only observations below a target are considered. SemiMoment

• Represents the Negative Skewness risk measure. NegativeSkewness

  • Squared negative skewness

    • FullMoment and semi-skewness are supported in traditional optimisers via the sk and V fields. Risk calculation uses

      • FullMoment is used to indicate that all deviations are included in the moment estimation process. FullMoment

      • SemiMoment is used for semi-moment estimators, where only observations below a target are considered. SemiMoment

    • Traditional optimisation formulations

      • Direct quadratic risk expression optimisation formulation for variance-like risk measures. QuadRiskExpr

      • Squared second-order cone risk expression optimisation formulation for applicable risk measures. SquaredSOCRiskExpr

    • Second-order cone risk expression optimisation formulation for applicable risk measures. SOCRiskExpr

• Represents the Value-at-Risk (VaR) risk measure. ValueatRisk

  • Traditional optimisation formulations

    • Mixed-integer programming (MIP) formulation for Value-at-Risk. MIPValueatRisk

    • Distribution-based formulation for Value-at-Risk. DistributionValueatRisk

• Represents the Value-at-Risk Range risk measure. ValueatRiskRange

  • Traditional optimisation formulations

    • Mixed-integer programming (MIP) formulation for Value-at-Risk. MIPValueatRisk

    • Distribution-based formulation for Value-at-Risk. DistributionValueatRisk

• Represents the Drawdown-at-Risk (DaR) risk measure. DrawdownatRisk

• Represents the Conditional Value-at-Risk (CVaR) risk measure, also known as Expected Shortfall (ES). ConditionalValueatRisk

• Distributionally Robust Conditional Value at Risk DistributionallyRobustConditionalValueatRisk (same as conditional value at risk when used in non-traditional optimisation)

• Represents the Conditional Value-at-Risk Range (CVaR Range) risk measure. ConditionalValueatRiskRange

• Distributionally Robust Conditional Value at Risk Range DistributionallyRobustConditionalValueatRiskRange (same as conditional value at risk range when used in non-traditional optimisation)

• Represents the Conditional Drawdown-at-Risk (CDaR) risk measure, also known as Expected Maximum Drawdown. ConditionalDrawdownatRisk

• Distributionally Robust Conditional Drawdown at Risk DistributionallyRobustConditionalDrawdownatRisk(same as conditional drawdown at risk when used in non-traditional optimisation)

• Represents the Entropic Value-at-Risk (EVaR) risk measure. EntropicValueatRisk

• Represents the Entropic Value-at-Risk Range (EVaR Range) risk measure. EntropicValueatRiskRange

• Represents the Entropic Drawdown-at-Risk (EDaR) risk measure. EntropicDrawdownatRisk

• Represents the Relativistic Value-at-Risk (RVaR) risk measure. RelativisticValueatRisk

• Represents the Relativistic Value-at-Risk Range (RVaR Range) risk measure. RelativisticValueatRiskRange

• Represents the Relativistic Drawdown-at-Risk (RLDaR) risk measure. RelativisticDrawdownatRisk

• Ordered Weights Array

  • Risk measures

    • Ordered Weights Array (OWA) risk measure. OrderedWeightsArray

    • Ordered Weights Array Range (OWA Range) risk measure. OrderedWeightsArrayRange

  • Traditional optimisation formulations

    • OWA formulation that computes exact OWA weights by solving a linear programme. ExactOrderedWeightsArray

    • OWA formulation that approximates OWA weights using a set of p-norm parameters. ApproxOrderedWeightsArray

    • Estimator type for OWA weights using JuMP-based optimization. OWAJuMP

  • One-call OWA measures

    • Callable OWA weight estimator for the Conditional Value at Risk (CVaR) risk measure. OrderedWeightsArrayConditionalValueatRisk

    • Callable OWA weight estimator for the Conditional Value at Risk Range risk measure. OrderedWeightsArrayConditionalValueatRiskRange

    • Callable OWA weight estimator for the tail Gini risk measure. OrderedWeightsArrayTailGini

    • Callable OWA weight estimator for the tail Gini range risk measure. OrderedWeightsArrayTailGiniRange

  • Array functions

    • Gini Mean Difference owa_gmd

    • Worst Realisation owa_wr

    • Range owa_rg

    • Conditional Value at Risk owa_cvar

    • Weighted Conditional Value at Risk owa_wcvar

    • Conditional Value at Risk Range owa_cvarrg

    • Weighted Conditional Value at Risk Range owa_wcvarrg

    • Tail Gini owa_tg

    • Tail Gini Range owa_tgrg

    • Linear moments (L-moments)

      • Compute the linear moment weights for the linear moments convex risk measure (CRM). owa_l_moment

      • Compute Ordered Weights Array (OWA) linear moment convex risk measure (CRM) weights using various estimation methods. owa_l_moment_crm

        • L-moment combination formulations

          • Represents the Maximum Entropy algorithm for Ordered Weights Array (OWA) estimation. MaximumEntropy

            • Entropy formulation for MaximumEntropy OWA that uses the exponential cone entropy constraint in JuMP. ExponentialConeEntropy

            • Entropy formulation for MaximumEntropy OWA that uses the relative entropy cone constraint in JuMP. RelativeEntropy

          • Represents the Minimum Squared Distance algorithm for Ordered Weights Array (OWA) estimation. MinimumSquaredDistance

          • Represents the Minimum Sum of Squares algorithm for Ordered Weights Array (OWA) estimation. MinimumSumSquares

• Represents the Average Drawdown risk measure. AverageDrawdown

• Represents the Ulcer Index risk measure. UlcerIndex

• Represents the Maximum Drawdown risk measure. MaximumDrawdown

• Represents the Brownian Distance Variance (BDVar) risk measure. BrownianDistanceVariance

  • Traditional optimisation formulations

    • Distance matrix constraint formulations

      • Norm-one cone formulation for the Brownian Distance Variance optimisation constraint. NormOneConeBrownianDistanceVariance

      • Inequality formulation for the Brownian Distance Variance optimisation constraint. IneqBrownianDistanceVariance

    • Risk formulation

      • Direct quadratic risk expression optimisation formulation for variance-like risk measures. QuadRiskExpr

      • Rotated second-order cone risk expression optimisation formulation for applicable risk measures. RSOCRiskExpr

• Represents the Worst Realisation risk measure. WorstRealisation

• Represents the Range risk measure. Range

• Represents the Turnover risk measure. TurnoverRiskMeasure

• Represents the Tracking Error risk measure. TrackingRiskMeasure

  • Norm-one (NOC) error formulation. L1Norm

  • Second-order cone (SOC) norm-based error formulation. L2Norm

  • Second-order cone (SOC) squared norm-based error formulation. SquaredL2Norm

  • L-p norm error estimator. LpNorm

  • L-infinity norm (maximum absolute deviation) error estimator. LInfNorm

• Risk Tracking Risk Measure

  • Dependent variable-based tracking formulation. DependentVariableTracking

  • Independent variable-based tracking formulation. IndependentVariableTracking

• Represents the Power Norm Value-at-Risk (PNVaR) risk measure. PowerNormValueatRisk

• Represents the Power Norm Value-at-Risk Range (PNVaRRange) risk measure. PowerNormValueatRiskRange

• Represents the Power Norm Drawdown-at-Risk (PNDDaR) risk measure. PowerNormDrawdownatRisk

• Represents a generic Value-at-Risk range risk measure that combines any pair of XatRisk-type measures applied to the loss and gain sides of the return distribution. GenericValueatRiskRange

• Represents the Risk Tracking risk measure. RiskTrackingRiskMeasure

• Risk measure that contributes no risk. NoRisk

• Risk measure settings

  Every risk measure carries a settings object saying how it enters the problem: as the objective, as a constraint with an upper bound, and with what scale.

  • Settings type for configuring risk measure estimators. RiskMeasureSettings

  • Settings type for configuring hierarchical risk measure estimators. HierarchicalRiskMeasureSettings

  • Settings type for configuring risk measures that expose a lower bound (maximisation direction). MaxRiskMeasureSettings

Risk measures for hierarchical optimisation

These are all subtypes of HierarchicalRiskMeasure, and are only supported by hierarchical optimisation estimators.

• Represents a high-order moment risk measure. HighOrderMoment

  • Represents the unstandardised semi-skewness risk measure algorithm. ThirdLowerMoment

  • Represents a standardised high-order moment risk measure algorithm. StandardisedHighOrderMoment and ThirdLowerMoment

  • Represents the unstandardised fourth moment (kurtosis or semi-kurtosis) risk measure algorithm. FourthMoment

    • FullMoment is used to indicate that all deviations are included in the moment estimation process. FullMoment

    • SemiMoment is used for semi-moment estimators, where only observations below a target are considered. SemiMoment

  • Represents a standardised high-order moment risk measure algorithm. StandardisedHighOrderMoment and FourthMoment

    • FullMoment is used to indicate that all deviations are included in the moment estimation process. FullMoment

    • SemiMoment is used for semi-moment estimators, where only observations below a target are considered. SemiMoment

• Represents the Relative Drawdown-at-Risk risk measure for hierarchical optimisation. RelativeDrawdownatRisk

• Represents the Relative Conditional Drawdown-at-Risk risk measure for hierarchical optimisation. RelativeConditionalDrawdownatRisk

• Represents the Relative Entropic Drawdown-at-Risk (Relative EDaR) risk measure for hierarchical optimisation. RelativeEntropicDrawdownatRisk

• Represents the Relative Relativistic Drawdown-at-Risk (Relative RLDaR) risk measure for hierarchical optimisation. RelativeRelativisticDrawdownatRisk

• Represents the Relative Average Drawdown risk measure for hierarchical optimisation. RelativeAverageDrawdown

• Represents the Relative Ulcer Index risk measure for hierarchical optimisation. RelativeUlcerIndex

• Represents the Relative Maximum Drawdown risk measure for hierarchical optimisation. RelativeMaximumDrawdown

• Represents the Relative Power Norm Drawdown-at-Risk (Relative PNDDaR) risk measure for hierarchical optimisation. RelativePowerNormDrawdownatRisk

• Represents a risk ratio risk measure for hierarchical portfolio optimisation. RiskRatio

• Represents the Equal Risk Measure for hierarchical portfolio optimisation. EqualRisk

• Represents the Median Absolute Deviation (MAD) risk measure for hierarchical portfolio optimisation. MedianAbsoluteDeviation

• Composite risk measure combining variance, skewness, and kurtosis into a single expression. VarianceSkewKurtosis

• Represents an even-order moment risk measure algorithm. EvenMoment

• Callable estimator that generates OWA linear moment convex risk measure (CRM) weights for a given number of observations. LinearMoment

Non-optimisation risk measures

These risk measures are unsuitable for optimisation because they can return negative values. However, they can be used for performance metrics.

• Represents a simple mean return measure for use in non-optimisation contexts. MeanReturn

• Represents the Third Central Moment risk measure. ThirdCentralMoment

• Represents the standardised Skewness risk measure. Skewness

• Return-based risk measure. ExpectedReturn

• Ratio-based risk measure. ExpectedReturnRiskRatio

• Represents a mean return to risk ratio measure. MeanReturnRiskRatio

• Represents a non-optimisation risk ratio measure. NonOptimisationRiskRatio

Performance metrics

• Compute the expected value of a risk measure. expected_risk

• Compute the effective number of assets (Herfindahl-Hirschman inverse index). number_effective_assets

• Risk contribution

  • Compute the risk contribution of each asset to the total portfolio risk using numerical differentiation. risk_contribution

  • Compute the risk contribution of each factor (and the idiosyncratic component) to the total portfolio risk using a factor regression. factor_risk_contribution

• Compute the expected portfolio return using the specified return estimator. expected_return

  • Arithmetic ArithmeticReturn

  • Logarithmic LogarithmicReturn

• Compute the expected risk of a measure from a precomputed net-return series. expected_risk_from_returns

• Compute the expected risk of a risk measure over rolling windows of the returns data. rolling_window_measure

• Sort the successful paths in a PopulationPredictionResult by their expected risk under r. sort_by_measure

• Compute the expected risk-adjusted return ratio for a portfolio. expected_ratio and expected_risk_ret_ratio

• Compute the risk-adjusted ratio information criterion (SRIC) for a portfolio. expected_sric and expected_risk_ret_sric

• Compute Brinson performance attribution aggregated per asset class [1]. brinson_attribution

Portfolio optimisation

Optimisations are implemented via optimise. Optimisations consume an estimator and return a result.

Naive

These return a NaiveOptimisationResult.

• Inverse Volatility portfolio optimiser. InverseVolatility

• Equal-weighted portfolio optimiser. EqualWeighted

• Random-weighted portfolio optimiser. RandomWeighted

Naive optimisation features

• Estimator for portfolio weight bounds constraints. WeightBoundsEstimator, UniformValues, and WeightBounds

• Weight finalisers

  • Iteratively projects weights into the feasible region defined by weight bounds. IterativeWeightFinaliser

  • Uses a JuMP optimisation model to enforce weight bounds. JuMPWeightFinaliser

    • Minimises the L1 norm of relative weight deviations when enforcing weight bounds. RelativeErrorWeightFinaliser

    • Minimises the L2 norm (squared) of relative weight deviations when enforcing weight bounds. SquaredRelativeErrorWeightFinaliser

    • Minimises the L1 norm of absolute weight deviations when enforcing weight bounds. AbsoluteErrorWeightFinaliser

    • Minimises the L2 norm (squared) of absolute weight deviations when enforcing weight bounds. SquaredAbsoluteErrorWeightFinaliser

Traditional

These optimisations are implemented as JuMP problems and make use of JuMPOptimiser, which encodes all supported constraints.

Objective function optimisations

These optimisations support a variety of objective functions.

• Container for configuring a JuMP solver and its settings. Solver

• Main JuMP-based portfolio optimiser configuration. JuMPOptimiser

• Objective functions

  • Minimum risk MinimumRisk

  • Maximum utility MaximumUtility

  • Maximum return over risk ratio MaximumRatio

  • Maximum return MaximumReturn

• Mean-Risk portfolio optimiser. MeanRisk and NearOptimalCentering

  • Defines the number of points on the efficient frontier (Pareto Front). Frontier

    • Return based

    • Risk based

    • Bound spacing FrontierBoundEstimator

      • Passes bound values through unchanged (identity transformation). LinearBound

      • Applies a square-root transformation to bound values before enforcing them. SquareRootBound

      • Applies a squaring transformation to bound values before enforcing them. SquaredBound

• Optimisation estimators

  • Mean-Risk MeanRisk returns a MeanRiskResult

  • Near Optimal Centering NearOptimalCentering returns a NearOptimalCenteringResult

  • Factor Risk Contribution FactorRiskContribution returns a FactorRiskContributionResult

• Near Optimal Centering formulations NearOptimalCentering

  • Constrained Near Optimal Centering algorithm. ConstrainedNearOptimalCentering

  • Unconstrained Near Optimal Centering algorithm. UnconstrainedNearOptimalCentering

  • Intermediate result type storing the setup data for Near Optimal Centering. NearOptimalSetup

Risk budgeting optimisations

These optimisations attempt to achieve weight values according to a risk budget vector. This vector can be provided on a per asset or per factor basis.

• Budget targets

  • Asset-level Risk Budgeting algorithm. AssetRiskBudgeting

  • Fromulations

    • Log-barrier formulation for Risk Budgeting. LogRiskBudgeting

    • Mixed-integer formulation for Risk Budgeting. MixedIntegerRiskBudgeting

  • Factor-level Risk Budgeting algorithm. FactorRiskBudgeting

• Optimisation estimators

  • Risk Budgeting RiskBudgeting returns a RiskBudgetingResult

  • Relaxed Risk Budgeting RelaxedRiskBudgeting returns a RiskBudgetingResult

    • Basic Relaxed Risk Budgeting formulation. BasicRelaxedRiskBudgeting

    • Regularised Relaxed Risk Budgeting formulation. RegularisedRelaxedRiskBudgeting

    • Regularised and penalised Relaxed Risk Budgeting formulation. RegularisedPenalisedRelaxedRiskBudgeting

Traditional optimisation features

• Abstract supertype for custom JuMP objective implementations. CustomJuMPObjective

• Abstract supertype for custom JuMP constraint implementations. CustomJuMPConstraint

• Estimator for portfolio weight bounds constraints. WeightBoundsEstimator, UniformValues, and WeightBounds

• Budget

  • Directionality

    • Long

    • Short

  • Type

    • Exact

    • Specifies the portfolio budget constraint as a closed interval $[\mathrm{lb},\mathrm{ub}]$ on the sum of weights. BudgetRange

• Estimator for buy-in threshold portfolio constraints. ThresholdEstimator and Threshold

  • Directionality

    • Long

    • Short

  • Type

    • Asset

    • Estimator for constructing asset set membership matrices from asset groupings. AssetSetsMatrixEstimator

• Container for one or more linear constraint equations to be parsed and converted into constraint matrices. LinearConstraintEstimator and LinearConstraint

• Estimator type for centrality-based analysis. CentralityEstimator

• Cardinality

  • Asset

  • Container for one or more linear constraint equations to be parsed and converted into constraint matrices. LinearConstraintEstimator and LinearConstraint

  • Set(s)

  • Container for one or more linear constraint equations to be parsed and converted into constraint matrices. LinearConstraintEstimator and LinearConstraint

• Estimator for turnover portfolio constraints. TurnoverEstimator and Turnover

• Estimator for portfolio transaction fees constraints. FeesEstimator and Fees

• Tracking error result type. TrackingError

• Estimator for generating integer phylogeny-based constraints. IntegerPhylogenyEstimator and SemiDefinitePhylogenyEstimator

• Portfolio returns

  • Arithmetic ArithmeticReturn

    • Represents a box uncertainty set for risk or prior statistics in portfolio optimisation. BoxUncertaintySet, BoxUncertaintySetAlgorithm, EllipsoidalUncertaintySet, and EllipsoidalUncertaintySetAlgorithm

    • Custom expected returns vector

  • Logarithmic LogarithmicReturn

• Risk vector scalarisation

  • Scalariser that combines multiple risk measures using a weighted sum. SumScalariser

  • Scalariser that selects the risk expression whose scaled value is the largest. MaxScalariser

  • Scalariser that aggregates multiple risk measures using the log-sum-exp function. LogSumExpScalariser

• Custom constraint

• Number of effective assets

• Regularisation penalty

  • L1

  • L2-norm regularisation term added to the optimisation objective. L2Regularisation

  • Lp-norm regularisation term added to the optimisation objective. LpRegularisation

  • L-Inf

• Weight-norm constraints

  Where a regularisation penalty prices a norm in the objective, these bound it instead.

  • L2 (l2c)

  • Lp (lpc)

  • L-Inf (linfc)

• Estimator type for normalised constant relative risk aversion (CRRA) OWA weights. NormalisedConstantRelativeRiskAversion

• Scalariser that selects the risk expression whose scaled value is the largest. MinScalariser

Clustering optimisation

Clustering optimisations make use of asset relationships to either minimise the risk exposure by breaking the asset universe into subsets which are hierarchically or individually optimised.

• Base configuration for hierarchical clustering-based portfolio optimisers. HierarchicalOptimiser

Hierarchical clustering optimisation

These optimisations minimise risk by hierarchically splitting the asset universe into subsets, computing the risk of each subset, and combining them according to their hierarchy.

• Hierarchical Risk Parity HierarchicalRiskParity returns a HierarchicalResult

• Hierarchical Equal Risk Contribution HierarchicalEqualRiskContribution returns a HierarchicalResult

Hierarchical clustering optimisation features

• Estimator for portfolio weight bounds constraints. WeightBoundsEstimator, UniformValues, and WeightBounds

• Estimator for portfolio transaction fees constraints. FeesEstimator and Fees

• Risk vector scalarisation

  • Scalariser that combines multiple risk measures using a weighted sum. SumScalariser

  • Scalariser that selects the risk expression whose scaled value is the largest. MaxScalariser

  • Scalariser that aggregates multiple risk measures using the log-sum-exp function. LogSumExpScalariser

• Weight finalisers

  • Iteratively projects weights into the feasible region defined by weight bounds. IterativeWeightFinaliser

  • Uses a JuMP optimisation model to enforce weight bounds. JuMPWeightFinaliser

    • Minimises the L1 norm of relative weight deviations when enforcing weight bounds. RelativeErrorWeightFinaliser

    • Minimises the L2 norm (squared) of relative weight deviations when enforcing weight bounds. SquaredRelativeErrorWeightFinaliser

    • Minimises the L1 norm of absolute weight deviations when enforcing weight bounds. AbsoluteErrorWeightFinaliser

    • Minimises the L2 norm (squared) of absolute weight deviations when enforcing weight bounds. SquaredAbsoluteErrorWeightFinaliser

Schur complementary optimisation

Schur complementary hierarchical risk parity provides a bridge between mean variance optimisation and hierarchical risk parity by using an interpolation parameter. It converges to hierarchical risk parity, and approximates mean variance by adjusting this parameter. It uses the Schur complement to adjust the weights of a portfolio according to how much more useful information is gained by assigning more weight to a group of assets.

• Schur Complementary Hierarchical Risk Parity SchurComplementHierarchicalRiskParity returns a SchurComplementHierarchicalRiskParityResult

• Parameters for the Schur Complement step of SCHRP. SchurComplementParams

  • Monotonic Schur Complement algorithm variant for SCHRP. MonotonicSchurComplement

  • Non-monotonic Schur Complement algorithm variant for SCHRP. NonMonotonicSchurComplement

Schur complementary optimisation features

• Estimator for portfolio weight bounds constraints. WeightBoundsEstimator, UniformValues, and WeightBounds

• Estimator for portfolio transaction fees constraints. FeesEstimator and Fees

• Weight finalisers

  • Iteratively projects weights into the feasible region defined by weight bounds. IterativeWeightFinaliser

  • Uses a JuMP optimisation model to enforce weight bounds. JuMPWeightFinaliser

    • Minimises the L1 norm of relative weight deviations when enforcing weight bounds. RelativeErrorWeightFinaliser

    • Minimises the L2 norm (squared) of relative weight deviations when enforcing weight bounds. SquaredRelativeErrorWeightFinaliser

    • Minimises the L1 norm of absolute weight deviations when enforcing weight bounds. AbsoluteErrorWeightFinaliser

    • Minimises the L2 norm (squared) of absolute weight deviations when enforcing weight bounds. SquaredAbsoluteErrorWeightFinaliser

Nested clusters optimisation

Nested clustered optimisation breaks the asset universe of size N into C smaller subsets and treats every subset as an individual portfolio. The weights assigned to each asset are placed in an N × C matrix. In each column, non-zero values correspond to assets assigned to that subset, this means that assets only contribute to the column (and therefore synthetic asset) corresponding to their assigned subset. In other words, each row of the matrix contains a single non-zero value and each row contains as many non-zero values as there are assets in that subset.

From here there are two options: 2. Compute the returns matrix of the synthetic assets directly by multiplying the original T × N matrix by the N × C matrix of asset weights to produce a T × C matrix of predicted returns, where T is the number of observations.

3. For each subset perform a cross validation prediction, yielding a vector of returns for that subset. These vectors are then horizontally concatenated into a Y × C matrix of cross-validation predicted returns, where Y ≤ T because the cross validation may not use the full history.

This matrix of predicted returns is then used by the outer optimisation estimator to generate an optimisation of the synthetic assets. This produces a C × 1 vector, essentially optimising a portfolio of asset clusters. The final weights are the product of the original N × C matrix of asset weights per cluster by the C × 1 vector of optimal synthetic asset weights to produce the final N × 1 vector of asset weights.

• Nested Clustered NestedClustered returns a NestedClusteredResult

Nested clusters optimisation features

• Any features supported by the inner and outer estimators.

• Estimator for portfolio weight bounds constraints. WeightBoundsEstimator, UniformValues, and WeightBounds

• Estimator for portfolio transaction fees constraints. FeesEstimator and Fees

• Weight finalisers

  • Iteratively projects weights into the feasible region defined by weight bounds. IterativeWeightFinaliser

  • Uses a JuMP optimisation model to enforce weight bounds. JuMPWeightFinaliser

    • Minimises the L1 norm of relative weight deviations when enforcing weight bounds. RelativeErrorWeightFinaliser

    • Minimises the L2 norm (squared) of relative weight deviations when enforcing weight bounds. SquaredRelativeErrorWeightFinaliser

    • Minimises the L1 norm of absolute weight deviations when enforcing weight bounds. AbsoluteErrorWeightFinaliser

    • Minimises the L2 norm (squared) of absolute weight deviations when enforcing weight bounds. SquaredAbsoluteErrorWeightFinaliser

• Cross validation predictor for the outer estimator

Ensemble optimisation

This works similarly to the Nested Clustered estimator, only instead of breaking the asset universe into subsets, a list of inner estimators is provided. The procedure is then exactly the same as the nested clusters optimisation, only instead of an N × C matrix of asset weights where each column corresponds to a subset of assets, each column corresponds to a completely independent and isolated inner estimator, which also means there is no enforced sparsity pattern on this matrix.

• Stacking Stacking returns a StackingResult

Ensemble optimisation features

• Any features supported by the inner and outer estimators.

• Estimator for portfolio transaction fees constraints. FeesEstimator and Fees

• Estimator for portfolio weight bounds constraints. WeightBoundsEstimator, UniformValues, and WeightBounds

• Weight finalisers

  • Iteratively projects weights into the feasible region defined by weight bounds. IterativeWeightFinaliser

  • Uses a JuMP optimisation model to enforce weight bounds. JuMPWeightFinaliser

    • Minimises the L1 norm of relative weight deviations when enforcing weight bounds. RelativeErrorWeightFinaliser

    • Minimises the L2 norm (squared) of relative weight deviations when enforcing weight bounds. SquaredRelativeErrorWeightFinaliser

    • Minimises the L1 norm of absolute weight deviations when enforcing weight bounds. AbsoluteErrorWeightFinaliser

    • Minimises the L2 norm (squared) of absolute weight deviations when enforcing weight bounds. SquaredAbsoluteErrorWeightFinaliser

• Cross validation predictor for the outer estimator

Subset resampling optimisation

This optimiser takes ideas from MultipleRandomised cross validation to randomly sample the asset universe and optimise each sample individually using a given optimiser. The final asset weights are the average weight per asset across all samples, if an asset does not appear in a sample, it is taken to be zero.

• SubsetResampling returns a SubsetResamplingResult

Subset resampling optimisation features

• Any features supported by the inner estimator.

• Estimator for portfolio transaction fees constraints. FeesEstimator and Fees

• Estimator for portfolio weight bounds constraints. WeightBoundsEstimator, UniformValues, and WeightBounds

• Weight finalisers

  • Iteratively projects weights into the feasible region defined by weight bounds. IterativeWeightFinaliser

  • Uses a JuMP optimisation model to enforce weight bounds. JuMPWeightFinaliser

    • Minimises the L1 norm of relative weight deviations when enforcing weight bounds. RelativeErrorWeightFinaliser

    • Minimises the L2 norm (squared) of relative weight deviations when enforcing weight bounds. SquaredRelativeErrorWeightFinaliser

    • Minimises the L1 norm of absolute weight deviations when enforcing weight bounds. AbsoluteErrorWeightFinaliser

    • Minimises the L2 norm (squared) of absolute weight deviations when enforcing weight bounds. SquaredAbsoluteErrorWeightFinaliser

Finite allocation optimisation

Unlike all other estimators, finite allocation does not yield an "optimal" value, but rather the optimal attainable solution based on a finite amount of capital. They use the result of other estimations, the latest prices, and a cash amount.

• Discrete Allocation portfolio optimiser. DiscreteAllocation

  • Weight finalisers

    • Iteratively projects weights into the feasible region defined by weight bounds. IterativeWeightFinaliser

    • Uses a JuMP optimisation model to enforce weight bounds. JuMPWeightFinaliser

      • Minimises the L1 norm of relative weight deviations when enforcing weight bounds. RelativeErrorWeightFinaliser

      • Minimises the L2 norm (squared) of relative weight deviations when enforcing weight bounds. SquaredRelativeErrorWeightFinaliser

      • Minimises the L1 norm of absolute weight deviations when enforcing weight bounds. AbsoluteErrorWeightFinaliser

      • Minimises the L2 norm (squared) of absolute weight deviations when enforcing weight bounds. SquaredAbsoluteErrorWeightFinaliser

• Greedy Allocation portfolio optimiser. GreedyAllocation

• Problem data fed to a finite allocation optimiser. FiniteAllocationInput

Cross validation

• Prediction on unseen data PredictionReturnsResult, PredictionResult, MultiPeriodPredictionResult, PopulationPredictionResult via predict(res::NonFiniteAllocationOptimisationResult, rd::ReturnsResult), fit_and_predict

• Union of concrete PredictionScorer subtypes and plain functions that score a PopulationPredictionResult. PredictionCrossValScorer, NearestQuantilePrediction, and quantile_by_measure

• Run cross-validated portfolio optimisation and return predictions over all folds. cross_val_predict

• Fit optimisation estimator opt on returns data rd and immediately produce a PredictionResult for the same data. fit_predict

• Return the number of cross-validation splits (folds) that would be produced by cv for the given returns data rd. n_splits

• Find the optimal (n_folds, n_test_folds) pair for combinatorial cross-validation by minimising a weighted cost that balances the average training size against the number of test paths. optimal_number_folds

• Split str into an array of substrings on occurrences of the delimiter(s) dlm. split and fit_and_predict

  • K-Fold KFold returns a KFoldResult

  • Combinatorial CombinatorialCrossValidation returns a CombinatorialCrossValidationResult

  • Walk forward WalkForwardEstimator return a WalkForwardResult

    • Implements index-based walk-forward cross-validation for time series, supporting purging and flexible train/test windowing. IndexWalkForward and DateWalkForward

  • Multiple randomised MultipleRandomised returns a MultipleRandomisedResult

• Performs grid search cross-validation for portfolio optimisation estimators. search_cross_validation

  • Performs grid search cross-validation for portfolio optimisation estimators. GridSearchCrossValidation

  • Randomised search cross-validation estimator for portfolio optimisation. RandomisedSearchCrossValidation

  • Scoring a parameter set CrossValidationSearchScorer

    • A CrossValidationSearchScorer that selects the parameter set with the highest mean score across cross-validation splits. HighestMeanScore

• Wraps a cross-validation scheme and an optional scorer to form a complete optimisation cross-validation pipeline. OptimisationCrossValidation

• Abstract supertype for estimators that determine the number of random subsets to draw. NumberSubsetsEstimator and SubsetSizeEstimator

Pipeline

A Pipeline reifies an end-to-end workflow as data: an ordered list of steps run left-to-right over a PipelineContext, so preprocessing, priors, and the optimiser travel together as one estimator and can be cross-validated or tuned as a unit.

• A reified end-to-end portfolio workflow: an ordered list of steps executed left-to-right over a PipelineContext. Pipeline and PipelineResult

• Explicit pipeline step wrapper — used when a step's slots or its routing intent must be stated rather than inferred. PipelineStep

• The accumulating blackboard threaded through a pipeline's steps. PipelineContext

• The mu/sigma pair held by the uncertainty slot of a PipelineContext. PipelineUncertaintySets

Plotting

Visualising the results is quite a useful way of summarising the portfolio characteristics or evolution. To this extent we provide a few plotting functions with more to come.

• Simple or compound cumulative returns.

  • Plot the cumulative returns of a portfolio. plot_portfolio_cumulative_returns

  • Plot the cumulative returns of individual assets, selecting the most relevant via N. plot_asset_cumulative_returns

• Portfolio composition.

  • Plot portfolio composition as a bar chart of asset weights. plot_composition

  • Multi portfolio.

    • Plot portfolio composition as a stacked bar chart. plot_stacked_bar_composition

    • Plot portfolio composition as a stacked area chart. plot_stacked_area_composition

• Risk contribution.

  • Plot per-asset risk contribution as a bar chart. plot_risk_contribution

  • Plot per-factor risk contribution as a bar chart, including the constant (idiosyncratic) term. plot_factor_risk_contribution

• Plot a hierarchical clustering dendrogram with coloured cluster regions. plot_dendrogram

• Plot a reordered correlation/covariance heatmap with flanking dendrograms and coloured cluster boxes. plot_clusters

• Plot portfolio drawdown over time. plot_drawdowns

• Line plot of the rolling maximum drawdown over a sliding window. plot_rolling_drawdowns

• Plot a histogram of portfolio returns with vertical risk-measure lines and an optional fitted Normal distribution. plot_histogram

• Scatter plot of risk/return measures across a collection of portfolio weight vectors. plot_measures

• Line plot of a risk or return measure evaluated over a rolling window of portfolio returns. plot_rolling_measure

• Sort a collection of portfolio results by risk (x), connect them with a line to trace the efficient frontier, and optionally annotate the minimum-risk and maximum-score portfolios. plot_efficient_frontier

• Box plot of per-asset weight distributions across cross-validation folds or population members. plot_weight_stability

• Line plot of portfolio turnover (L1 weight change) over time. plot_turnover

• Overlay portfolio cumulative returns against one or more benchmark return series from rd.B. plot_benchmark

• Moments and priors

  • Bar chart of per-asset expected returns (μ vector). plot_mu

  • Bar chart of per-asset volatility (√diag(Σ)). plot_sigma

  • Standalone correlation (or covariance) heatmap without clustering or dendrograms. plot_correlation

  • Heatmap of the coskewness matrix (N × N²) from a HighOrderPrior. plot_coskewness

  • Eigenvalue spectrum of the cokurtosis matrix (N² × N²) from a HighOrderPrior. plot_cokurtosis

  • Bar chart of eigenvalues of the covariance/correlation matrix, sorted in descending order. plot_eigenspectrum

  • Three-panel composite plot summarising a prior result: plot_prior

• Factor models

  • Bar chart of per-factor expected returns (the f_mu vector from a factor model prior). plot_factor_mu

  • Correlation/covariance heatmap of the factor covariance matrix (pr.f_sigma). plot_factor_sigma

  • Heatmap of the factor loadings matrix B (assets × factors) from a prior with a regression model. plot_factor_loadings

• Phylogeny

  • Plot the asset network (MST, PMFG, TMFG, or adjacency) as a graph using GraphRecipes.graphplot. plot_network

  • Bar chart of asset centrality scores, sorted in descending order. plot_centrality

• Cross validation

  • Bar chart of cross-validation scores (one bar per fold or population member). plot_cv_scores

  • Four-panel composite plot for a walk-forward cross-validation result: plot_cv_dashboard

• Dashboards

  • Four-panel composite plot for a single optimisation result: plot_portfolio_dashboard

  • Bar chart of annualised portfolio performance metrics: annualised return, annualised volatility, Sharpe ratio, Sortino ratio, Calmar ratio, maximum drawdown %, and CVaR %. plot_performance_summary