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Median expected returns

PortfolioOptimisers.MedianExpectedReturns — Type
struct MedianExpectedReturns{__T_w} <: AbstractExpectedReturnsEstimator

Computes the expected returns as the per-asset median of the asset returns.

w carries optional observation weights. If w is nothing, the median is unweighted. The median resists an outlier that would move the sample mean.

Fields

  • w: Optional observation weights vector observations × 1, or a concrete subtype of DynamicAbstractWeights. If nothing, the computation is unweighted.

Constructors

MedianExpectedReturns(;    w::Option{<:ObsWeights} = nothing) -> MedianExpectedReturns

Keywords correspond to the struct's fields.

Validation

  • If w is not nothing, !isempty(w).

Propagated parameters

When factory is called on this type, the following @fprop-tagged fields are automatically propagated:

Observation weight parameters

When obs_weights_view is called on this type, the following fields are automatically indexed to the selected observations:

Examples

julia> me = MedianExpectedReturns()MedianExpectedReturns  w ┴ nothingjulia> factory(me, StatsBase.Weights([0.1, 0.2, 0.7]))MedianExpectedReturns  w ┴ StatsBase.Weights{Float64, Float64, Vector{Float64}}: [0.1, 0.2, 0.7]

Related

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Statistics.mean — Method
Statistics.mean(me::MedianExpectedReturns, X::MatNum;
                dims::Int = 1, kwargs...)

Compute expected returns as the median of each asset.

This method returns the median of each asset across observations in X. If me.w is nothing, the median is computed directly with Statistics.median(X; dims = dims). Otherwise, the method computes a weighted median for each asset using the observation weights w.

Mathematical definition

Unweighted:

\[\begin{align} \hat{\mu}_j &= \mathrm{median}(r_{1j}, r_{2j}, \ldots, r_{Tj})\,. \end{align}\]

Where:

  • $\hat{\mu}_j$: Median expected return of asset $j$.
  • $r_{tj}$: Return of asset $j$ at time $t$.
  • $T$: Number of observations.

Weighted. The weighted median is the StatsBase weighted quantile at probability $1/2$, which interpolates between two order statistics. Order the returns of asset $j$ so that $r_{(1)j} \leq \ldots \leq r_{(T)j}$, and let $w_{(t)}$ be the weight that travels with each one:

\[\begin{align} S_m &= \sum_{t=1}^{m} w_{(t)}\,, \\ h &= \frac{1}{2} \left( \sum_{t=1}^{T} w_t - w_{(1)} \right) + w_{(1)}\,, \\ k &= \max \left\lbrace m : S_m \leq h \right\rbrace\,, \\ \hat{\mu}_j &= r_{(k)j} + \frac{h - S_k}{S_{k+1} - S_k} \left( r_{(k+1)j} - r_{(k)j} \right)\,. \end{align}\]

Where:

  • $w_t$: Observation weight at time $t$.
  • $w_{(t)}$: Weight of the $t$-th smallest return, so the weights are permuted with the returns.
  • $r_{(t)j}$: $t$-th smallest return of asset $j$.
  • $S_m$: Cumulative weight of the $m$ smallest returns.
  • $h$: Cumulative weight that the probability $1/2$ corresponds to.

The result is therefore not in general one of the observed returns. Under equal weights it reduces to the ordinary median.

Arguments

  • me: Median expected returns estimator.
  • X: Data matrix of asset returns (observations × assets).
  • dims: Dimension along which to perform the computation.
  • kwargs: Additional keyword arguments (ignored).

Validation

  • dims in (1, 2).

Returns

  • mu::Matrix{<:Number}: Median vector, shaped as (1, N) if dims == 1 or (N, 1) if dims == 2.

Related

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