The source files can be found in examples/.

Advanced Black–Litterman variants

The base BlackLittermanPrior (previous page) works entirely in asset space: views are statements about individual assets or asset groups. But views are often more naturally expressed about factors — "momentum will earn a premium", "quality minus low-volatility will be negative" — and sometimes you hold asset and factor views at once. PortfolioOptimisers ships three variants of the model for exactly these cases:

• BayesianBlackLittermanPrior — the Bayesian formulation that sits on top of a FactorPrior. Views are expressed on the factors, and the factor structure propagates the posterior back to the assets.
• FactorBlackLittermanPrior — expresses views directly on factor premia and pushes them to the assets through a factor regression. The rsd flag controls whether idiosyncratic residual variance is retained, and l is the risk-aversion of the implied factor equilibrium.
• AugmentedBlackLittermanPrior — carries both asset views (a_views) and factor views (f_views) simultaneously, each with its own asset set and confidence.

All three return an asset-space posterior (mu, sigma) you can feed to any optimiser, exactly like the base model.

using PortfolioOptimisers, PrettyTables, DataFramesmmtfmt = (v, i, j) -> begin    if j == 1        return v    else        return isa(v, Number) ? "$(round(v * 100, digits = 4)) %" : v    endend;resfmt = (v, i, j) -> begin    if j == 1        return v    else        return isa(v, Number) ? "$(round(v * 100, digits = 3)) %" : v    endend;

1. Data, sets, and the equilibrium baseline

We load the S&P 500 slice with its factor block, then declare one UniverseSets that all three variants read. It names both axes — assets under xkey, factors under fkey, each in the column order of rd.X and rd.F — plus a couple of asset groups for the augmented asset views. Every factor-view estimator here resolves its names on the declared factor axis, so one object covers a factor-only mandate, an asset-only mandate, and the augmented model that writes both. The EquilibriumExpectedReturns prior is the neutral anchor every Black–Litterman posterior tilts away from.

using CSV, TimeSeriesX = TimeArray(CSV.File(joinpath(@__DIR__, "..", "SP500.csv.gz")); timestamp = :Date)[(end - 252):end]F = TimeArray(CSV.File(joinpath(@__DIR__, "..", "Factors.csv.gz")); timestamp = :Date)[(end - 252):end]rd = prices_to_returns(X, F)universe_sets = UniverseSets(;                             dict = Dict("nx" => rd.nx, "nf" => rd.nf,                                         "tech" => ["AAPL", "AMD", "MSFT"],                                         "energy" => ["CVX"]))tau = 1 / size(rd.X, 1)pr_eq = prior(EmpiricalPrior(; me = EquilibriumExpectedReturns()), rd)

LowOrderPrior
      X ┼ 252×20 Matrix{Float64}
    o_X ┼ nothing
     mu ┼ 20-element Vector{Float64}
  sigma ┼ 20×20 Matrix{Float64}
   chol ┼ nothing
      w ┼ nothing
    ens ┼ nothing
    kld ┼ nothing
     ow ┼ nothing
     rr ┼ nothing
    fpr ┼ nothing
      Z ┴ nothing

The factors available in this dataset:

pretty_table(DataFrame(; factor = rd.nf); title = "Factor names (rd.nf)")

Factor names (rd.nf)
┌────────┐
│ factor │
│ String │
├────────┤
│   MTUM │
│   QUAL │
│   SIZE │
│   USMV │
│   VLUE │
└────────┘

2. Bayesian Black–Litterman: factor views on a factor prior

BayesianBlackLittermanPrior takes a FactorPrior as its base estimator and accepts views written in factor space, resolved against the declared factor axis sets.dict[sets.fkey]. It is the Bayesian formulation: the factor prior supplies the structure, the views update the factor means, and the result is mapped back to an asset-space posterior.

Our factor views: momentum earns 5 bps/day, and quality underperforms low-volatility by 3 bps/day.

factor_views = LinearConstraintEstimator(;                                         val = ["MTUM == 0.0005", "QUAL - USMV == -0.0003"])pr_bayes = prior(BayesianBlackLittermanPrior(; pe = FactorPrior(; pe = EmpiricalPrior()),                                             sets = universe_sets, tau = tau,                                             views = factor_views), rd)

LowOrderPrior
      X ┼ 252×20 Matrix{Float64}
    o_X ┼ 252×20 Matrix{Float64}
     mu ┼ 20-element Vector{Float64}
  sigma ┼ 20×20 Matrix{Float64}
   chol ┼ nothing
      w ┼ nothing
    ens ┼ nothing
    kld ┼ nothing
     ow ┼ nothing
     rr ┼ Regression
        │   M ┼ 20×5 SubArray{Float64, 2, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, UnitRange{Int64}}, true}
        │   L ┼ 20×5 SubArray{Float64, 2, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, UnitRange{Int64}}, true}
        │   b ┴ 20-element SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}
    fpr ┼ LowOrderPrior
        │       X ┼ 252×5 Matrix{Float64}
        │     o_X ┼ nothing
        │      mu ┼ Vector{Float64}: [0.0004935513857205137, 0.00013981052864866136, 0.0002941301399949491, 0.0004387517589914853, 0.0003055527028387581]
        │   sigma ┼ 5×5 Matrix{Float64}
        │    chol ┼ nothing
        │       w ┼ nothing
        │     ens ┼ nothing
        │     kld ┼ nothing
        │      ow ┼ nothing
        │      rr ┼ nothing
        │     fpr ┼ nothing
        │       Z ┴ nothing
      Z ┴ nothing

3. Factor Black–Litterman: views on factor premia

FactorBlackLittermanPrior also takes factor views, but propagates them to the assets through the factor regression rather than a factor prior. Its views resolve against the declared factor axis, sets.dict[sets.fkey]: the names must be looked up on the axis they were written in, and the asset axis is what a view over a subset of assets slices. Two knobs matter:

• rsd — keep the idiosyncratic residual variance (true) or drop it (false), i.e. whether the posterior covariance is the full asset covariance or only its factor-explained part.
• l — the risk-aversion of the implied factor equilibrium.

We build both rsd settings and a higher-risk-aversion variant to show each knob moves the posterior.

pr_fbl_rsd = prior(FactorBlackLittermanPrior(; pe = EmpiricalPrior(), rsd = true,                                             sets = universe_sets, tau = tau,                                             views = factor_views), rd)pr_fbl_nors = prior(FactorBlackLittermanPrior(; pe = EmpiricalPrior(), rsd = false,                                              sets = universe_sets, tau = tau,                                              views = factor_views), rd)pr_fbl_l = prior(FactorBlackLittermanPrior(; pe = EmpiricalPrior(), rsd = true, l = 5.0,                                           sets = universe_sets, tau = tau,                                           views = factor_views), rd)i_aapl = findfirst(==("AAPL"), rd.nx)pretty_table(DataFrame(; quantity = ["AAPL posterior mean", "AAPL posterior variance"],                       rsd_true = [pr_fbl_rsd.mu[i_aapl], pr_fbl_rsd.sigma[i_aapl, i_aapl]],                       rsd_false = [pr_fbl_nors.mu[i_aapl],                                    pr_fbl_nors.sigma[i_aapl, i_aapl]]);             formatters = [mmtfmt], title = "Factor BL: residual variance on vs off")

      Factor BL: residual variance on vs off
┌─────────────────────────┬──────────┬───────────┐
│                quantity │ rsd_true │ rsd_false │
│                  String │  Float64 │   Float64 │
├─────────────────────────┼──────────┼───────────┤
│     AAPL posterior mean │ -0.044 % │  -0.044 % │
│ AAPL posterior variance │ 0.0501 % │  0.0384 % │
└─────────────────────────┴──────────┴───────────┘

Dropping the residual variance (rsd = false) leaves only the factor-explained part of the covariance, so the posterior variance is smaller — the model trusts the factor structure to explain all risk.

4. Augmented Black–Litterman: asset and factor views together

AugmentedBlackLittermanPrior is the most general: it takes asset views (a_views) and factor views (f_views) in the same posterior. Use it when you have a stock-specific call and a factor call you do not want to choose between.

It is the one estimator that reads both declared axes, and it reads them out of the same universe_sets every variant above used: a_views resolve against sets.dict[sets.xkey] — so the tech group is in scope — and f_views against sets.dict[sets.fkey].

asset_views = LinearConstraintEstimator(; val = ["AAPL == 0.0008", "tech == 0.0006"])pr_aug = prior(AugmentedBlackLittermanPrior(; sets = universe_sets, tau = tau,                                            a_views = asset_views, f_views = factor_views),               rd)

LowOrderPrior
      X ┼ 252×20 Matrix{Float64}
    o_X ┼ 252×20 Matrix{Float64}
     mu ┼ 20-element Vector{Float64}
  sigma ┼ 20×20 Matrix{Float64}
   chol ┼ nothing
      w ┼ nothing
    ens ┼ nothing
    kld ┼ nothing
     ow ┼ nothing
     rr ┼ Regression
        │   M ┼ 20×5 SubArray{Float64, 2, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, UnitRange{Int64}}, true}
        │   L ┼ 20×5 SubArray{Float64, 2, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, UnitRange{Int64}}, true}
        │   b ┴ 20-element SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}
    fpr ┼ LowOrderPrior
        │       X ┼ 252×5 Matrix{Float64}
        │     o_X ┼ nothing
        │      mu ┼ Vector{Float64}: [0.00017381016950241063, 0.00010766089837441542, 0.00023286771481700712, 0.00031498660793923987, 0.00021554387985553804]
        │   sigma ┼ 5×5 Matrix{Float64}
        │    chol ┼ nothing
        │       w ┼ nothing
        │     ens ┼ nothing
        │     kld ┼ nothing
        │      ow ┼ nothing
        │      rr ┼ nothing
        │     fpr ┼ nothing
        │       Z ┴ nothing
      Z ┴ nothing

5. Comparing the posteriors

Each variant produces a different asset-space posterior mean. Below we line them up against the equilibrium anchor for the first few assets — the factor views move the whole cross-section, and the augmented model additionally tilts the assets named in its asset views.

cmp = DataFrame(; Assets = rd.nx, Equilibrium = pr_eq.mu, Bayesian = pr_bayes.mu,                Factor = pr_fbl_rsd.mu, Augmented = pr_aug.mu)pretty_table(first(cmp, 8); formatters = [mmtfmt],             title = "Posterior expected returns (first 8 assets)")using StatsPlots, GraphRecipesplot_mu(pr_aug, rd.nx)

6. From views to portfolios

The payoff is the same as for the base model: each posterior reshapes the optimal portfolio. We solve a maximum-ratio portfolio under the equilibrium prior and under each variant, then compare the compositions.

using Clarabelslv = Solver(; name = :clarabel1, solver = Clarabel.Optimizer,             settings = Dict("verbose" => false),             check_sol = (; allow_local = true, allow_almost = true))rf = 4.2 / 100 / 252priors = ["Equilibrium" => pr_eq, "Bayesian" => pr_bayes, "Factor" => pr_fbl_rsd,          "Augmented" => pr_aug]res = [optimise(MeanRisk(; obj = MaximumRatio(; rf = rf),                         opt = JuMPOptimiser(; pe = p, slv = slv))) for (_, p) in priors]pretty_table(DataFrame(hcat(rd.nx, [r.w for r in res]...),                       [:assets; Symbol.(first.(priors))...]); formatters = [resfmt],             title = "Maximum-ratio weights by prior")plot_stacked_bar_composition(res, rd; xticks = (1:length(priors), first.(priors)))

Summary

The Black–Litterman family extends well past asset-space views:

• BayesianBlackLittermanPrior places factor views on a FactorPrior.
• FactorBlackLittermanPrior propagates factor-premia views through a regression, with rsd controlling residual variance and l the implied equilibrium risk-aversion.
• AugmentedBlackLittermanPrior blends asset views and factor views in one posterior.

All three return a standard asset-space posterior, so they drop into any optimiser exactly like the base model — the difference is only in where the views live.

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