Ulcer Index

PortfolioOptimisers.UlcerIndex Type

struct UlcerIndex{__T_settings} <: RiskMeasure

Represents the Ulcer Index risk measure.

UlcerIndex measures the depth and duration of drawdowns in portfolio returns. It is defined as the root-mean-square of the absolute drawdown series, capturing both the magnitude and persistence of losses.

Mathematical definition

Define the absolute drawdown series:

$$\begin{array}{rl}{c}_t& =\sum _{s=1}^t{x}_s,\\ d_t& =c_t-\underset{0\le s\le t}{max}{c}_s\le 0.\end{array}$$

Where:

• $\mathit{x}$: Portfolio returns vector $T\times 1$.

• $c_t$: Cumulative simple portfolio return at period $t$.

• $d_t\le 0$: Absolute drawdown at period $t$.

The Ulcer Index is:

$$\begin{array}{rl}\mathrm{UI}(\mathit{x})& =\sqrt{\frac{1}{T}\sum _{t=1}^T{d}_t^2}=\frac{\Vert \mathit{d}{\Vert }_2}{\sqrt{T}}.\end{array}$$

Where:

• $\mathrm{UI}(\mathit{x})$: Ulcer Index (root-mean-square drawdown).

• $T$: Number of observations.

• $d_t\le 0$: Absolute drawdown at period $t$.

• $\mathit{d}$: Absolute drawdown series vector $T\times 1$.

Fields

• settings: Risk measure settings.

Constructors

UlcerIndex(;
    settings::RiskMeasureSettings = RiskMeasureSettings()
) -> UlcerIndex

Keywords correspond to the struct's fields.

Functor

(r::UlcerIndex)(x::VecNum)

Computes the Ulcer Index of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> UlcerIndex()
UlcerIndex
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true

Related

• RiskMeasure

• RiskMeasureSettings

• AverageDrawdown

• MaximumDrawdown

• RelativeUlcerIndex

source

PortfolioOptimisers.RelativeUlcerIndex Type

struct RelativeUlcerIndex{__T_settings} <: HierarchicalRiskMeasure

Represents the Relative Ulcer Index risk measure for hierarchical optimisation.

RelativeUlcerIndex applies the Ulcer Index framework to the relative (compounded) drawdown series.

Mathematical definition

Define the relative drawdown series:

$$\begin{array}{rl}{C}_t& =\prod _{s=1}^t(1+x_s),\\ r{d}_t& =\frac{C_t}{\underset{0\le s\le t}{max}{C}_s}-1\le 0.\end{array}$$

Where:

• $\mathit{x}$: Portfolio returns vector $T\times 1$.

• $C_t$: Compound wealth process at period $t$.

• $r{d}_t\le 0$: Relative drawdown at period $t$.

The Relative Ulcer Index is:

$$\begin{array}{rl}\mathrm{RUI}(\mathit{x})& =\frac{\Vert \mathit{r}\mathit{d}{\Vert }_2}{\sqrt{T}}.\end{array}$$

Where:

• $\mathrm{RUI}(\mathit{x})$: Relative Ulcer Index (root-mean-square relative drawdown).

• $T$: Number of observations.

• $r{d}_t\le 0$: Relative drawdown at period $t$.

• $\mathit{r}\mathit{d}$: Relative drawdown series vector $T\times 1$.

Fields

• settings: Risk measure settings.

Constructors

RelativeUlcerIndex(;
    settings::HierarchicalRiskMeasureSettings = HierarchicalRiskMeasureSettings()
) -> RelativeUlcerIndex

Keywords correspond to the struct's fields.

Functor

(r::RelativeUlcerIndex)(x::VecNum)

Computes the Relative Ulcer Index of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativeUlcerIndex()
RelativeUlcerIndex
  settings ┼ HierarchicalRiskMeasureSettings
           │   scale ┴ Float64: 1.0

Related

• HierarchicalRiskMeasure

• HierarchicalRiskMeasureSettings

• UlcerIndex

• RelativeAverageDrawdown

source