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Equilibrium expected returns: private API

PortfolioOptimisers.equilibrium_mu — Function
equilibrium_mu(l::Number, sigma::MatNum, w::Option{<:VecNum})

Compute equilibrium expected returns from a risk aversion parameter, a covariance block, and equilibrium weights.

equilibrium_mu is the single owner of the $\lambda \mathbf{\Sigma} \boldsymbol{w}$ expression and of its equal-weight fallback. EquilibriumExpectedReturns, FactorBlackLittermanPrior and AugmentedBlackLittermanPrior all reach it, so the fallback and the length check are stated once.

The result is an excess return. Reverse optimisation implies a risk premium, so no risk-free rate is in it. This is why the Black-Litterman members apply apply_rf to the result of this function, and only on the branch where they call it: a mean taken from a wrapped prior estimator is a total return already and needs no conversion.

sigma is a covariance block, not necessarily a square covariance matrix. Its columns are the assets the weights are written over, so size(sigma, 2) is the length w must have. A square covariance gives the plain equilibrium returns. A rectangular block gives the equilibrium returns of the rows it spans, which is how the factor Black-Litterman members build a prior mean over factors from asset weights.

Mathematical definition

\[\begin{align} \hat{\boldsymbol{\mu}}_{\text{eq}} &= \lambda \, \mathbf{\Sigma} \, \boldsymbol{w}\,. \end{align}\]

Where:

  • $\lambda$: Risk aversion parameter.
  • $\mathbf{\Sigma}$: $M \times N$ covariance block.
  • $\boldsymbol{w}$: $N \times 1$ equilibrium portfolio weights.

Algorithm

  1. When w is nothing, read the asset count N from size(sigma, 2), and build the equal-weight vector fill(inv(N), N).
  2. When w is a vector, check that length(w) equals size(sigma, 2).
  3. Apply the expression above to l, sigma and the weights of step 1 or step 2, giving mu.

Arguments

  • l: Risk aversion parameter.
  • sigma: Covariance block whose columns are the assets.
  • w: Equilibrium weights, or nothing for equal weights.

Validation

  • If w is a vector, length(w) == size(sigma, 2).

Returns

  • mu::VecNum: Equilibrium expected returns vector of length size(sigma, 1).

Related

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