Non-Optimisation Risk Measures

julia

struct MeanReturn{__T_w, __T_flag} <: NonOptimisationRiskMeasure

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

MeanReturn computes the arithmetic (or geometric, when flag = true) mean of portfolio returns. It is used as the numerator in risk-adjusted performance ratios such as MeanReturnRiskRatio.

Mathematical definition

For flag = false (arithmetic mean):

\begin{aligned} \bar{x} & = \frac{1}{T} \sum_{t = 1}^{T} x_{t} . \end{aligned}

Where:

$\bar{x}$ : Arithmetic mean portfolio return.
$x$ : Portfolio returns vector $T \times 1$ .
$T$ : Number of observations.

For flag = true (log-return mean):

\begin{aligned} {\bar{x}}_{\log} & = \frac{1}{T} \sum_{t = 1}^{T} \log (1 + x_{t}) . \end{aligned}

Where:

${\bar{x}}_{\log}$ : Log-return mean portfolio return.
$x$ : Portfolio returns vector $T \times 1$ .
$T$ : Number of observations.

For observation-weighted samples, the weighted mean is used instead.

Fields

w: Optional portfolio weights.
flag: Algorithm selection flag.

Constructors

julia

MeanReturn(;
    w::Option{<:ObsWeights} = nothing,
    flag::Bool = false
) -> MeanReturn

Keywords correspond to the struct's fields.

Validation

If w is not nothing: !isempty(w).

Functor

julia

(r::MeanReturn)(x::VecNum)

Computes the mean return of a portfolio returns vector x.

Arguments

x::VecNum: Portfolio returns vector.

Examples

julia

julia> MeanReturn()
MeanReturn
     w ┼ nothing
  flag ┴ Bool: false

Related

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PortfolioOptimisers.factory Method

julia

factory(
    r::MeanReturn,
    pr::AbstractPriorResult,
    args...
) -> MeanReturn{_A, Bool} where _A

Create an instance of MeanReturn by selecting observation weights from the risk-measure instance or falling back to the prior result.

Related

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PortfolioOptimisers.port_opt_view Method

julia

port_opt_view(r::MeanReturn, _, args...) -> MeanReturn

Return the MeanReturn risk measure r unchanged.

MeanReturn does not have asset-level parameters to slice.

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PortfolioOptimisers.MeanReturnRiskRatio Type

julia

struct MeanReturnRiskRatio{__T_rt, __T_rk, __T_rf} <: NonOptimisationRiskMeasure

Represents a mean return to risk ratio measure.

MeanReturnRiskRatio computes the ratio of the mean portfolio return (minus a risk-free rate) to a risk measure, used for performance analysis and comparison. It generalises the Sharpe ratio by allowing any risk measure in the denominator.

Mathematical definition

\begin{aligned} MRRR (x) & = \frac{\bar{x} - r_{f}}{ρ (x)} . \end{aligned}

Where:

$MRRR (x)$ : Mean return to risk ratio.
$x$ : Portfolio returns vector $T \times 1$ .
$\bar{x}$ : Mean portfolio return (computed by rt).
$r_{f}$ : Risk-free rate.
$ρ$ : Base risk measure (computed by rk).

Fields

rt: Mean return estimator.
rk: Risk measure for ratio computation.
rf: Risk-free rate.

Constructors

julia

MeanReturnRiskRatio(;
    rt::MeanReturn = MeanReturn(),
    rk::AbstractBaseRiskMeasure = ConditionalValueatRisk(),
    rf::Number = 0.0
) -> MeanReturnRiskRatio

Keywords correspond to the struct's fields.

Related

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PortfolioOptimisers.needs_previous_weights Method

julia

needs_previous_weights(r::MeanReturnRiskRatio) -> Any

Return whether MeanReturnRiskRatio r requires previous portfolio weights.

Delegates to the inner risk measure r.rk.

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PortfolioOptimisers.factory Method

julia

factory(
    r::MeanReturnRiskRatio,
    args...;
    kwargs...
) -> MeanReturnRiskRatio{MeanReturn{__T_w, __T_flag}, <:AbstractBaseRiskMeasure, <:Number} where {__T_w, __T_flag}

Create an instance of MeanReturnRiskRatio by updating both the return measure and risk measure from the optimisation context.

Related

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PortfolioOptimisers.factory Method

julia

factory(
    r::MeanReturnRiskRatio,
    w::AbstractVector{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}}
) -> MeanReturnRiskRatio{MeanReturn{__T_w, __T_flag}, <:AbstractBaseRiskMeasure, <:Number} where {__T_w, __T_flag}

Create an instance of MeanReturnRiskRatio updating the risk measure from new portfolio weights w.

The return measure rt is preserved unchanged.

Related

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PortfolioOptimisers.ThirdCentralMoment Type

julia

struct ThirdCentralMoment{__T_w, __T_mu} <: NonOptimisationRiskMeasure

Represents the Third Central Moment risk measure.

ThirdCentralMoment computes the third central moment of portfolio returns about a specified centre. It is used as a measure of the asymmetry (skewness) of the return distribution in higher-order portfolio optimisation.

Mathematical definition

Let $μ$ be the specified centre and $δ_{t} = x_{t} - μ$ the centred deviations. The third central moment is:

\begin{aligned} m_{3} (x) & = \frac{1}{T} \sum_{t = 1}^{T} δ_{t}^{3} . \end{aligned}

Where:

$m_{3} (x)$ : Third central moment of portfolio returns.
$x$ : Portfolio returns vector $T \times 1$ .
$T$ : Number of observations.
$μ$ : Specified centre of the distribution.
$δ_{t} = x_{t} - μ$ : Centred deviation at period $t$ .

For observation-weighted samples, the weighted mean is used.

Fields

w: Optional portfolio weights.
mu: Optional mean for centering.

Constructors

julia

ThirdCentralMoment(;
    w::Option{<:ObsWeights} = nothing,
    mu::Option{<:Num_VecNum_VecScalar} = nothing
) -> ThirdCentralMoment

Keywords correspond to the struct's fields.

Validation

If mu is a VecNum: !isempty(mu).
If w is not nothing: !isempty(w).

Functor

julia

(r::ThirdCentralMoment)(w::VecNum, X::MatNum, fees = nothing)

Computes the third central moment of the portfolio returns.

Arguments

w::VecNum: Portfolio weights vector.
X::MatNum: Asset returns matrix ( $T \times N$ ).
fees: Optional fee structure.

Examples

julia

julia> ThirdCentralMoment()
ThirdCentralMoment
   w ┼ nothing
  mu ┴ nothing

Related

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PortfolioOptimisers.factory Method

julia

factory(
    r::ThirdCentralMoment,
    pr::AbstractPriorResult,
    args...;
    kwargs...
) -> ThirdCentralMoment

Create an instance of ThirdCentralMoment by selecting observation weights and expected returns from the risk-measure instance or falling back to the prior result.

Related

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PortfolioOptimisers.port_opt_view Method

julia

port_opt_view(
    r::ThirdCentralMoment,
    i,
    args...
) -> ThirdCentralMoment

Return a view of ThirdCentralMoment r sliced to asset indices i.

Slices the expected returns mu for cluster-based optimisation.

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PortfolioOptimisers.TCM_Sk Type

julia

const TCM_Sk{T1, T2} = Union{<:ThirdCentralMoment{T1, T2}, <:Skewness{<:Any, <:Any, T1, T2}}

Parameterised union of ThirdCentralMoment and Skewness sharing the same observation-weight (T1) and target-mean (T2) type parameters.

Used for unified dispatch on moment-target calculation methods.

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PortfolioOptimisers.calc_moment_target Method

julia

calc_moment_target(::TCM_Sk{Nothing, Nothing}, ::Any, x::VecNum)
calc_moment_target(r::TCM_Sk{<:StatsBase.AbstractWeights, Nothing}, ::Any, x::VecNum)
calc_moment_target(r::TCM_Sk{<:Any, <:VecNum}, w::VecNum, ::Any)
calc_moment_target(r::TCM_Sk{<:Any, <:VecScalar}, w::VecNum, ::Any)
calc_moment_target(r::TCM_Sk{<:Any, <:Number}, ::Any, ::Any)

Compute the centering target for ThirdCentralMoment and Skewness risk measures.

Dispatches on the observation-weight type T1 and mean type T2 of TCM_Sk:

No weights, no mu: arithmetic mean of x.
AbstractWeights, no mu: weighted mean of x.
VecNum mu: dot product $w^{⊺} μ$ .
VecScalar mu: $w^{⊺} μ_{v} + μ_{s}$ .
Number mu: the scalar r.mu directly.

Related

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PortfolioOptimisers.calc_deviations_vec Function

julia

calc_deviations_vec(
    r::Union{ThirdCentralMoment{T1, T2}, Skewness{<:Any, <:Any, <:Any, T1, T2}} where {T1, T2},
    w::AbstractVector{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}},
    X::AbstractMatrix{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}}
) -> Any
calc_deviations_vec(
    r::Union{ThirdCentralMoment{T1, T2}, Skewness{<:Any, <:Any, <:Any, T1, T2}} where {T1, T2},
    w::AbstractVector{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}},
    X::AbstractMatrix{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}},
    fees::Union{Nothing, Fees}
) -> Any

Compute the vector of deviations from the centering target for ThirdCentralMoment and Skewness risk measures.

Related

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PortfolioOptimisers.calc_deviations_vec Method

julia

calc_deviations_vec(
    r::Union{ThirdCentralMoment{T1, T2}, Skewness{<:Any, <:Any, <:Any, T1, T2}} where {T1, T2},
    x::AbstractVector{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}}
) -> Any

Compute the vector of deviations from the centering target for a precomputed returns series for ThirdCentralMoment and Skewness risk measures.

Single-argument form used by the precomputed-returns functor r(x::VecNum) (ADR 0007).

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PortfolioOptimisers.supports_precomputed_returns Method

julia

supports_precomputed_returns(r::ThirdCentralMoment) -> Any

Return whether ThirdCentralMoment r supports precomputed-return evaluation.

Delegates to weight_independent_target on r.mu: true iff the target is Nothing, a Number, or a MedianCenteringFunction; false for per-asset targets.

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Non-Optimisation Risk Measures ​

Non-Optimisation Risk Measures