Simple expected returns

The most basic moment is the simple expected return. These types and functions implement it.

PortfolioOptimisers.SimpleExpectedReturns Type

struct SimpleExpectedReturns{__T_w} <: AbstractExpectedReturnsEstimator

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

SimpleExpectedReturns is the standard estimator for computing expected returns as the possibly weighted mean of asset returns.

Fields

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

Constructors

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

Keywords correspond to the struct's fields.

Validation

• If w is not nothing, !isempty(w).

Examples

julia> SimpleExpectedReturns()
SimpleExpectedReturns
  w ┴ nothing

julia> SimpleExpectedReturns(; w = StatsBase.Weights([0.5, 0.5]))
SimpleExpectedReturns
  w ┴ StatsBase.Weights{Float64, Float64, Vector{Float64}}: [0.5, 0.5]

Related

• AbstractExpectedReturnsEstimator

• Option

• StatsBase.AbstractWeights

• mean(me::SimpleExpectedReturns, X::MatNum; dims::Int = 1, kwargs...)

source

Statistics.mean Method

Statistics.mean(
    me::SimpleExpectedReturns,
    X::MatNum;
    dims::Int = 1,
    kwargs...
) -> ArrNum

Compute the mean of asset returns using a SimpleExpectedReturns estimator.

This method computes the expected returns as the sample mean of the input data X according to ce.

Mathematical definition

Unweighted:

$$\begin{array}{rl}{\hat{\mu }}_j& =\frac{1}{T}\sum _{t=1}^T{r}_{tj}.\end{array}$$

Where:

• $\hat{\mathit{\mu }}$: $N\times 1$ vector of estimated expected returns.

• $r_{tj}$: Return of asset $j$ at time $t$.

• $T$: Number of observations.

Weighted:

$$\begin{array}{rl}{\hat{\mu }}_j& =\frac{\sum _{t=1}^T{w}_t{r}_{tj}}{\sum _{t=1}^T{w}_t}.\end{array}$$

Where:

• $\hat{\mathit{\mu }}$: $N\times 1$ vector of estimated expected returns.

• $r_{tj}$: Return of asset $j$ at time $t$.

• $T$: Number of observations.

• $w_t$: Observation weight at time $t$.

Arguments

• me: Expected returns estimator.

• X: Data matrix observations × features if the dims keyword does not exist or dims = 1, features × observations when dims = 2.

• dims: Dimension along which to perform the computation.

• kwargs...: Additional keyword arguments passed to Statistics.mean.

Returns

• mu::ArrNum: Expected returns vector features x 1 if the dims keyword does not exist or dims = 2, 1 x features if dims = 1.

Examples

julia> X = [0.01 0.02; 0.03 0.04];

julia> ser = SimpleExpectedReturns()
SimpleExpectedReturns
  w ┴ nothing

julia> mean(ser, X)
1×2 Matrix{Float64}:
 0.02  0.03

julia> serw = SimpleExpectedReturns(; w = StatsBase.Weights([0.2, 0.8]))
SimpleExpectedReturns
  w ┴ StatsBase.Weights{Float64, Float64, Vector{Float64}}: [0.2, 0.8]

julia> mean(serw, X)
1×2 Matrix{Float64}:
 0.026  0.036

Related

• SimpleExpectedReturns

• MatNum

• VecNum

• Statistics.mean

source