Risk Functions





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This module has functions that calculates several risk measures that are widely used by the asset management industry and academics.

Module Functions

RiskFunctions.MAD(X)[source]

Calculate the Mean Absolute Deviation (MAD) of a returns series.

\[\text{MAD}(X) = \frac{1}{T}\sum_{t=1}^{T} | X_{t} - \mathbb{E}(X_{t}) |\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Returns:

value – MAD of a returns series.

Return type:

float

RiskFunctions.SemiDeviation(X)[source]

Calculate the Semi Deviation of a returns series.

\[\text{SemiDev}(X) = \left [ \frac{1}{T-1}\sum_{t=1}^{T} \min (X_{t} - \mathbb{E}(X_{t}), 0)^2 \right ]^{1/2}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Semi Deviation of a returns series.

Return type:

float

RiskFunctions.Kurtosis(X)[source]

Calculate the Square Root Kurtosis of a returns series.

\[\text{Kurt}(X) = \left [ \frac{1}{T}\sum_{t=1}^{T} (X_{t} - \mathbb{E}(X_{t}))^{4} \right ]^{1/2}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Square Root Kurtosis of a returns series.

Return type:

float

RiskFunctions.SemiKurtosis(X)[source]

Calculate the Semi Square Root Kurtosis of a returns series.

\[\text{SemiKurt}(X) = \left [ \frac{1}{T}\sum_{t=1}^{T} \min (X_{t} - \mathbb{E}(X_{t}), 0)^{4} \right ]^{1/2}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Semi Square Root Kurtosis of a returns series.

Return type:

float

RiskFunctions.EvenMoment(X, p: int = 2)[source]

Calculate the p-th Root of Even Moment of order 2 * p of a returns series.

\[\text{EM}_{p}(X) = \left [ \frac{1}{T}\sum_{t=1}^{T} (X_{t} - \mathbb{E}(X_{t}))^{2p} \right ]^{1/p}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

p : int, optional

Order of the Even Moment. It must be an integer greater than or equal to 2. The default is 2.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – p-th Root Even Moment of order 2 * p of a returns series.

Return type:

float

RiskFunctions.EvenSemiMoment(X, p: int = 2)[source]

Calculate the p-th Root of Semi Even Moment of order 2 * p of a returns series.

\[\text{ESM}_{p}(X) = \left [ \frac{1}{T}\sum_{t=1}^{T} \min (X_{t} - \mathbb{E}(X_{t}), 0)^{2p} \right ]^{1/p}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

p : int, optional

Order of the Semi Even Moment. It must be an integer greater than or equal to 2. The default is 2.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – p-th Root Semi Even Moment of order 2 * p of a returns series.

Return type:

float

RiskFunctions.VaR_Hist(X, alpha=0.05)[source]

Calculate the Value at Risk (VaR) of a returns series.

\[\text{VaR}_{\alpha}(X) = -\inf_{t \in (0,T)} \left \{ X_{t} \in \mathbb{R}: F_{X}(X_{t})>\alpha \right \}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of VaR. The default is 0.05.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – VaR of a returns series.

Return type:

float

RiskFunctions.CVaR_Hist(X, alpha=0.05)[source]

Calculate the Conditional Value at Risk (CVaR) of a returns series.

\[\text{CVaR}_{\alpha}(X) = \text{VaR}_{\alpha}(X) + \frac{1}{\alpha T} \sum_{t=1}^{T} \max(-X_{t} - \text{VaR}_{\alpha}(X), 0)\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of CVaR. The default is 0.05.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – CVaR of a returns series.

Return type:

float

RiskFunctions.WR(X)[source]

Calculate the Worst Realization (WR) or Worst Scenario of a returns series.

\[\text{WR}(X) = \max(-X)\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – WR of a returns series.

Return type:

float

RiskFunctions.LPM(X, MAR=0, p=1)[source]

Calculate the First or Second Lower Partial Moment of a returns series.

\[\begin{split}\text{LPM}(X, \text{MAR}, 1) &= \frac{1}{T}\sum_{t=1}^{T} \max(\text{MAR} - X_{t}, 0) \\ \text{LPM}(X, \text{MAR}, 2) &= \left [ \frac{1}{T-1}\sum_{t=1}^{T} \max(\text{MAR} - X_{t}, 0)^{2} \right ]^{\frac{1}{2}} \\\end{split}\]

Where:

\(\text{MAR}\) is the minimum acceptable return. \(p\) is the order of the \(\text{LPM}\).

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

MAR : float, optional

Minimum acceptable return. The default is 0.

p : int or float, optional

Order of the \(\text{LPM}\). Only 1 and 2 are supported. The default is 1.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – p-th Lower Partial Moment of a returns series.

Return type:

float

RiskFunctions.Entropic_RM(X, z=1, alpha=0.05)[source]

Calculate the Entropic Risk Measure (ERM) of a returns series.

\[\text{ERM}_{\alpha}(X) = z\ln \left (\frac{M_X(z^{-1})}{\alpha} \right )\]

Where:

\(M_X(z)\) is the moment generating function of X.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

z : float, optional

Risk aversion parameter, must be greater than zero. The default is 1.

alpha : float, optional

Significance level of EVaR. The default is 0.05.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – ERM of a returns series.

Return type:

float

RiskFunctions.EVaR_Hist(X, alpha=0.05, solver='CLARABEL')[source]

Calculate the Entropic Value at Risk (EVaR) of a returns series.

\[\text{EVaR}_{\alpha}(X) = \inf_{z>0} \left \{ z \ln \left (\frac{M_X(z^{-1})}{\alpha} \right ) \right \}\]

Where:

\(M_X(t)\) is the moment generating function of X.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of EVaR. The default is 0.05.

solver : str, optional

Solver available for CVXPY that supports exponential cone programming. Used to calculate EVaR, EVRG and EDaR. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

(value, z) – EVaR of a returns series and value of z that minimize EVaR.

Return type:

tuple

RiskFunctions.RLVaR_Hist(X, alpha=0.05, kappa=0.3, solver='CLARABEL')[source]

Calculate the Relativistic Value at Risk (RLVaR) of a returns series. I recommend only use this function with MOSEK solver.

\[\begin{split}\text{RLVaR}^{\kappa}_{\alpha}(X) & = \left \{ \begin{array}{ll} \underset{z, t, \psi, \theta, \varepsilon, \omega}{\text{inf}} & t + z \ln_{\kappa} \left ( \frac{1}{\alpha T} \right ) + \sum^T_{i=1} \left ( \psi_{i} + \theta_{i} \right ) \\ \text{s.t.} & -X - t + \varepsilon + \omega \leq 0\\ & z \geq 0 \\ & \left ( z \left ( \frac{1+\kappa}{2\kappa} \right ), \psi_{i} \left ( \frac{1+\kappa}{\kappa} \right ), \varepsilon_{i} \right) \in \mathcal{P}_3^{1/(1+\kappa),\, \kappa/(1+\kappa)} \\ & \left ( \omega_{i}\left ( \frac{1}{1-\kappa} \right ), \theta_{i}\left ( \frac{1}{\kappa} \right), -z \left ( \frac{1}{2\kappa} \right ) \right ) \in \mathcal{P}_3^{1-\kappa,\, \kappa} \\ \end{array} \right .\end{split}\]

Where:

\(\mathcal{P}_3^{\alpha,\, 1-\alpha}\) is the power cone 3D.

\(\kappa\) is the deformation parameter.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of EVaR. The default is 0.05.

kappa : float, optional

Deformation parameter of RLVaR, must be between 0 and 1. The default is 0.3.

solver : str, optional

Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR and RLDaR. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – RLVaR of a returns series.

Return type:

tuple

RiskFunctions.MDD_Abs(X)[source]

Calculate the Maximum Drawdown (MDD) of a returns series using uncompounded cumulative returns.

\[\text{MDD}(X) = \max_{j \in (0,T)} \left [\max_{t \in (0,j)} \left ( \sum_{i=0}^{t}X_{i} \right ) - \sum_{i=0}^{j}X_{i} \right ]\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – MDD of an uncompounded cumulative returns.

Return type:

float

RiskFunctions.ADD_Abs(X)[source]

Calculate the Average Drawdown (ADD) of a returns series using uncompounded cumulative returns.

\[\text{ADD}(X) = \frac{1}{T}\sum_{j=0}^{T}\left [ \max_{t \in (0,j)} \left ( \sum_{i=0}^{t}X_{i} \right ) - \sum_{i=0}^{j}X_{i} \right ]\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – ADD of an uncompounded cumulative returns.

Return type:

float

RiskFunctions.DaR_Abs(X, alpha=0.05)[source]

Calculate the Drawdown at Risk (DaR) of a returns series using uncompounded cumulative returns.

\[\begin{split}\text{DaR}_{\alpha}(X) & = \max_{j \in (0,T)} \left \{ \text{DD}(X,j) \in \mathbb{R}: F_{\text{DD}} \left ( \text{DD}(X,j) \right )< 1-\alpha \right \} \\ \text{DD}(X,j) & = \max_{t \in (0,j)} \left ( \sum_{i=0}^{t}X_{i} \right )- \sum_{i=0}^{j}X_{i}\end{split}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of DaR. The default is 0.05.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – DaR of an uncompounded cumulative returns series.

Return type:

float

RiskFunctions.CDaR_Abs(X, alpha=0.05)[source]

Calculate the Conditional Drawdown at Risk (CDaR) of a returns series using uncompounded cumulative returns.

\[\text{CDaR}_{\alpha}(X) = \text{DaR}_{\alpha}(X) + \frac{1}{\alpha T} \sum_{j=0}^{T} \max \left [ \max_{t \in (0,j)} \left ( \sum_{i=0}^{t}X_{i} \right ) - \sum_{i=0}^{j}X_{i} - \text{DaR}_{\alpha}(X), 0 \right ]\]

Where:

\(\text{DaR}_{\alpha}\) is the Drawdown at Risk of an uncompounded cumulated return series \(X\).

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of CDaR. The default is 0.05.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – CDaR of an uncompounded cumulative returns series.

Return type:

float

RiskFunctions.EDaR_Abs(X, alpha=0.05, solver='CLARABEL')[source]

Calculate the Entropic Drawdown at Risk (EDaR) of a returns series using uncompounded cumulative returns.

\[\begin{split}\text{EDaR}_{\alpha}(X) & = \inf_{z>0} \left \{ z \ln \left (\frac{M_{\text{DD}(X)}(z^{-1})}{\alpha} \right ) \right \} \\ \text{DD}(X,j) & = \max_{t \in (0,j)} \left ( \sum_{i=0}^{t}X_{i} \right )- \sum_{i=0}^{j}X_{i} \\\end{split}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of EDaR. The default is 0.05.

Raises:

ValueError – When the value cannot be calculated.

Returns:

(value, z) – EDaR of an uncompounded cumulative returns series and value of z that minimize EDaR.

Return type:

tuple

RiskFunctions.RLDaR_Abs(X, alpha=0.05, kappa=0.3, solver='CLARABEL')[source]

Calculate the Relativistic Drawdown at Risk (RLDaR) of a returns series using uncompounded cumulative returns. I recommend only use this function with MOSEK solver.

\[\begin{split}\text{RLDaR}^{\kappa}_{\alpha}(X) & = \text{RLVaR}^{\kappa}_{\alpha}(\text{DD}(X)) \\ \text{DD}(X,j) & = \max_{t \in (0,j)} \left ( \sum_{i=0}^{t}X_{i} \right )- \sum_{i=0}^{j}X_{i} \\\end{split}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of EVaR. The default is 0.05.

kappa : float, optional

Deformation parameter of RLDaR, must be between 0 and 1. The default is 0.3.

solver : str, optional

Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR, RVRG and RLDaR. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – RLDaR of an uncompounded cumulative returns series.

Return type:

tuple

RiskFunctions.UCI_Abs(X)[source]

Calculate the Ulcer Index (UCI) of a returns series using uncompounded cumulative returns.

\[\text{UCI}(X) =\sqrt{\frac{1}{T}\sum_{j=0}^{T} \left [ \max_{t \in (0,j)} \left ( \sum_{i=0}^{t}X_{i} \right ) - \sum_{i=0}^{j}X_{i} \right ] ^2}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of an uncompounded cumulative returns.

Return type:

float

RiskFunctions.MDD_Rel(X)[source]

Calculate the Maximum Drawdown (MDD) of a returns series using cumpounded cumulative returns.

\[\text{MDD}(X) = \max_{j \in (0,T)}\left[\max_{t \in (0,j)} \left ( \prod_{i=0}^{t}(1+X_{i}) \right ) - \prod_{i=0}^{j}(1+X_{i}) \right]\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – MDD of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.ADD_Rel(X)[source]

Calculate the Average Drawdown (ADD) of a returns series using cumpounded cumulative returns.

\[\text{ADD}(X) = \frac{1}{T}\sum_{j=0}^{T} \left [ \max_{t \in (0,j)} \left ( \prod_{i=0}^{t}(1+X_{i}) \right )- \prod_{i=0}^{j}(1+X_{i}) \right ]\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – ADD of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.DaR_Rel(X, alpha=0.05)[source]

Calculate the Drawdown at Risk (DaR) of a returns series using cumpounded cumulative returns.

\[\begin{split}\text{DaR}_{\alpha}(X) & = \max_{j \in (0,T)} \left \{ \text{DD}(X,j) \in \mathbb{R}: F_{\text{DD}} \left ( \text{DD}(X,j) \right )< 1 - \alpha \right \} \\ \text{DD}(X,j) & = \max_{t \in (0,j)} \left ( \prod_{i=0}^{t}(1+X_{i}) \right )- \prod_{i=0}^{j}(1+X_{i})\end{split}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of DaR. The default is 0.05.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – DaR of a cumpounded cumulative returns series.

Return type:

float

RiskFunctions.CDaR_Rel(X, alpha=0.05)[source]

Calculate the Conditional Drawdown at Risk (CDaR) of a returns series using cumpounded cumulative returns.

\[\text{CDaR}_{\alpha}(X) = \text{DaR}_{\alpha}(X) + \frac{1}{\alpha T} \sum_{i=0}^{T} \max \left [ \max_{t \in (0,T)} \left ( \prod_{i=0}^{t}(1+X_{i}) \right )- \prod_{i=0}^{j}(1+X_{i}) - \text{DaR}_{\alpha}(X), 0 \right ]\]

Where:

\(\text{DaR}_{\alpha}\) is the Drawdown at Risk of a cumpound cumulated return series \(X\).

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of CDaR. The default is 0.05.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – CDaR of a cumpounded cumulative returns series.

Return type:

float

RiskFunctions.EDaR_Rel(X, alpha=0.05, solver='CLARABEL')[source]

Calculate the Entropic Drawdown at Risk (EDaR) of a returns series using cumpounded cumulative returns.

\[\begin{split}\text{EDaR}_{\alpha}(X) & = \inf_{z>0} \left \{ z \ln \left (\frac{M_{\text{DD}(X)}(z^{-1})}{\alpha} \right ) \right \} \\ \text{DD}(X,j) & = \max_{t \in (0,j)} \left ( \prod_{i=0}^{t}(1+X_{i}) \right )- \prod_{i=0}^{j}(1+X_{i})\end{split}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of EDaR. The default is 0.05.

Raises:

ValueError – When the value cannot be calculated.

Returns:

(value, z) – EDaR of a cumpounded cumulative returns series and value of z that minimize EDaR.

Return type:

tuple

RiskFunctions.RLDaR_Rel(X, alpha=0.05, kappa=0.3, solver='CLARABEL')[source]

Calculate the Relativistic Drawdown at Risk (RLDaR) of a returns series using compounded cumulative returns. I recommend only use this function with MOSEK solver.

\[\begin{split}\text{RLDaR}^{\kappa}_{\alpha}(X) & = \text{RLVaR}^{\kappa}_{\alpha}(\text{DD}(X)) \\ \text{DD}(X,j) & = \max_{t \in (0,j)} \left ( \prod_{i=0}^{t}(1+X_{i}) \right )- \prod_{i=0}^{j}(1+X_{i}) \\\end{split}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of RLDaR. The default is 0.05.

kappa : float, optional

Deformation parameter of RLDaR, must be between 0 and 1. The default is 0.3.

solver : str, optional

Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR, RVRG and RLDaR. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – RLDaR of a compounded cumulative returns series.

Return type:

tuple

RiskFunctions.UCI_Rel(X)[source]

Calculate the Ulcer Index (UCI) of a returns series using cumpounded cumulative returns.

\[\text{UCI}(X) =\sqrt{\frac{1}{T}\sum_{j=0}^{T} \left [ \max_{t \in (0,j)} \left ( \prod_{i=0}^{t}(1+X_{i}) \right )- \prod_{i=0}^{j} (1+X_{i}) \right ] ^2}\]
Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.GMD(X)[source]

Calculate the Gini Mean Difference (GMD) of a returns series.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Gini Mean Difference of a returns series.

Return type:

float

RiskFunctions.TG(X, alpha=0.05, a_sim=100)[source]

Calculate the Tail Gini of a returns series.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of Tail Gini. The default is 0.05.

a_sim : int, optional

Integer number of CVaRs used to approximate Tail Gini. The default is 100.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.RG(X)[source]

Calculate the range of a returns series.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.VRG(X, alpha=0.05, beta=None)[source]

Calculate the VaR range of a returns series.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of VaR of losses. The default is 0.05.

beta : float or None, optional

Significance level of VaR of gains. If None it duplicates alpha value. The default is None.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.CVRG(X, alpha=0.05, beta=None)[source]

Calculate the CVaR range of a returns series.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of CVaR of losses. The default is 0.05.

beta : float or None, optional

Significance level of CVaR of gains. If None it duplicates alpha value. The default is None.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.TGRG(X, alpha=0.05, a_sim=100, beta=None, b_sim=None)[source]

Calculate the Tail Gini range of a returns series.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of Tail Gini of losses. The default is 0.05.

a_sim : int, optional

Integer number of CVaRs used to approximate Tail Gini of losses. The default is 100.

beta : float or None, optional

Significance level of Tail Gini of gains. If None it duplicates alpha value. The default is None.

b_sim : int or None, optional

Number of CVaRs used to approximate Tail Gini of gains. If None, it defaults to a_sim. The default is None.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.EVRG(X, alpha=0.05, beta=None, solver='CLARABEL')[source]

Calculate the CVaR range of a returns series.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of EVaR of losses. The default is 0.05.

beta : float or None, optional

Significance level of EVaR of gains. If None it duplicates alpha value. The default is None.

solver : str, optional

Solver available for CVXPY that supports exponential cone programming. Used to calculate EVaR, EVRG and EDaR. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.RVRG(X, alpha=0.05, beta=None, kappa=0.3, kappa_g=None, solver='CLARABEL')[source]

Calculate the CVaR range of a returns series.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

alpha : float, optional

Significance level of RLVaR of losses. The default is 0.05.

beta : float or None, optional

Significance level of RLVaR of gains. If None it duplicates alpha value. The default is None.

kappa : float, optional

Deformation parameter of RLVaR for losses, must be between 0 and 1. The default is 0.3.

kappa_g : float or None, optional

Deformation parameter of RLVaR for gains, must be between 0 and 1. The default is None.

solver : str, optional

Solver available for CVXPY that supports power cone programming. Used to calculate EVaR, EVRG and EDaR. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.L_Moment(X, k=2)[source]

Calculate the kth l-moment of a returns series.

Where $y_{[i]}$ is the ith-ordered statistic.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

k : int, optional

Order of the l-moment. Must be an integer greater than or equal to 1. The default is 2.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Kth l-moment of a returns series.

Return type:

float

RiskFunctions.L_Moment_CRM(X, k=4, method='MSD', g=0.5, max_phi=0.5, solver='CLARABEL')[source]

Calculate a custom convex risk measure that is a weighted average of first k-th l-moments.

Parameters:
X : array_like

Finite numeric returns series as a one-dimensional array-like or a row/column vector.

k : int, optional

Order of the l-moment. Must be an integer greater than or equal to 2. The default is 4.

method : str, optional

Method to calculate the weights used to combine the l-moments with order higher than 2. The default value is ‘MSD’. Possible values are:

  • ’CRRA’: Normalized Constant Relative Risk Aversion coefficients.

  • ’ME’: Maximum Entropy.

  • ’MSS’: Minimum Sum Squares.

  • ’MSD’: Minimum Square Distance.

g : float, optional

Risk aversion coefficient of CRRA utility function. The default is 0.5.

max_phi : float, optional

Maximum weight constraint of L-moments. The default is 0.5.

solver : str, optional

Solver available for CVXPY. Used to calculate ‘ME’, ‘MSS’ and ‘MSD’ weights. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Custom convex risk measure that is a weighted average of first k-th l-moments of a returns series.

Return type:

float

RiskFunctions.NEA(w)[source]

Calculate the number of effective assets (NEA) that is the inverse of the Herfindahl Hirschman index (HHI).

Parameters:
w : array_like

Portfolio weights as a one-dimensional sequence or row/column vector. NumPy arrays, pandas Series, and one-column or one-row DataFrames are accepted.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – The NEA of the portfolio.

Return type:

float

RiskFunctions.Sharpe_Risk(returns, w=None, cov=None, rm='MV', rf=0, alpha=0.05, a_sim=100, beta=None, b_sim=None, kappa=0.3, kappa_g=None, p_em=2, p_esm=2, solver='CLARABEL')[source]

Calculate the risk measure available on the Sharpe function.

Parameters:
w : array_like, optional

Portfolio weights as a one-dimensional sequence or row/column vector. Required when returns has more than one column; otherwise it defaults to a single weight of 1.

cov : array_like, optional

Covariance matrix of shape (n_assets, n_assets). If None, it is estimated from returns. The default is None.

returns : DataFrame, Series, or array_like of shape (n_samples, n_features)

Numeric returns for the assets. A pandas Series is treated as one asset; a one-dimensional array-like input is treated as one observation across assets.

rm : str, optional

Risk measure used in the denominator of the ratio. The default is ‘MV’. Possible values are:

  • ’MV’: Standard Deviation.

  • ’KT’: Square Root Kurtosis.

  • ’EM’: Even Moment of order 2 * p_em.

  • ’MAD’: Mean Absolute Deviation.

  • ’GMD’: Gini Mean Difference.

  • ’MSV’: Semi Standard Deviation.

  • ’SKT’: Square Root Semi Kurtosis.

  • ’ESM’: Even Semi Moment of order 2 * p_em.

  • ’FLPM’: First Lower Partial Moment (Omega Ratio).

  • ’SLPM’: Second Lower Partial Moment (Sortino Ratio).

  • ’VaR’: Value at Risk.

  • ’CVaR’: Conditional Value at Risk.

  • ’TG’: Tail Gini.

  • ’EVaR’: Entropic Value at Risk.

  • ’RLVaR’: Relativistic Value at Risk. I recommend only use this function with MOSEK solver.

  • ’WR’: Worst Realization (Minimax).

  • ’RG’: Range of returns.

  • ’VRG’ VaR range of returns.

  • ’CVRG’: CVaR range of returns.

  • ’TGRG’: Tail Gini range of returns.

  • ’EVRG’: EVaR range of returns.

  • ’RVRG’: RLVaR range of returns. I recommend only use this function with MOSEK solver.

  • ’MDD’: Maximum Drawdown of uncompounded cumulative returns (Calmar Ratio).

  • ’ADD’: Average Drawdown of uncompounded cumulative returns.

  • ’DaR’: Drawdown at Risk of uncompounded cumulative returns.

  • ’CDaR’: Conditional Drawdown at Risk of uncompounded cumulative returns.

  • ’EDaR’: Entropic Drawdown at Risk of uncompounded cumulative returns.

  • ’RLDaR’: Relativistic Drawdown at Risk of uncompounded cumulative returns. I recommend only use this risk measure with MOSEK solver.

  • ’UCI’: Ulcer Index of uncompounded cumulative returns.

  • ’MDD_Rel’: Maximum Drawdown of compounded cumulative returns (Calmar Ratio).

  • ’ADD_Rel’: Average Drawdown of compounded cumulative returns.

  • ’DaR_Rel’: Drawdown at Risk of compounded cumulative returns.

  • ’CDaR_Rel’: Conditional Drawdown at Risk of compounded cumulative returns.

  • ’EDaR_Rel’: Entropic Drawdown at Risk of compounded cumulative returns.

  • ’RLDaR_Rel’: Relativistic Drawdown at Risk of compounded cumulative returns. I recommend only use this risk measure with MOSEK solver.

  • ’UCI_Rel’: Ulcer Index of compounded cumulative returns.

rf : float, optional

Risk free rate. The default is 0.

alpha : float, optional

Significance level of VaR, CVaR, EVaR, RLVaR, DaR, CDaR, EDaR, RLDaR and Tail Gini of losses. The default is 0.05.

a_sim : int, optional

Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

beta : float or None, optional

Significance level of CVaR and Tail Gini of gains. If None it duplicates alpha value. The default is None.

b_sim : int or None, optional

Number of CVaRs used to approximate Tail Gini of gains. If None it duplicates a_sim value. The default is None.

kappa : float, optional

Deformation parameter of RLVaR and RLDaR for losses, must be between 0 and 1. The default is 0.3.

kappa_g : float or None, optional

Deformation parameter of RLVaR and RLDaR for gains, must be between 0 and 1. The default is None.

p_em : int, optional

Order of the Even Moment of order 2 * p_em. It must be an integer higher equal than 2. The default value is 2.

p_esm : int, optional

Order of the Even Semi Moment of order 2 * p_esm. It must be an integer higher equal than 2. The default value is 2.

solver : str, optional

Solver available for CVXPY that supports exponential and power cone programming. Used to calculate RLVaR and RLDaR. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Risk measure of the portfolio.

Return type:

float

RiskFunctions.Sharpe(returns, w=None, mu=None, cov=None, rm='MV', rf=0, alpha=0.05, a_sim=100, beta=None, b_sim=None, kappa=0.3, kappa_g=None, p_em=2, p_esm=2, solver='CLARABEL')[source]

Calculate the Risk Adjusted Return Ratio from a portfolio returns series.

\[\text{Sharpe}(X) = \frac{\mathbb{E}(X) - r_{f}}{\phi(X)}\]

Where:

\(X\) is the vector of portfolio returns.

\(r_{f}\) is the risk free rate, when the risk measure is

\(\text{LPM}\) uses instead of \(r_{f}\) the \(\text{MAR}\).

\(\phi(X)\) is a convex risk measure. The risk measures availabe are:

Parameters:
returns : DataFrame, Series, or array_like of shape (n_samples, n_features)

Numeric returns for the assets. A pandas Series is treated as one asset; a one-dimensional array-like input is treated as one observation across assets.

w : array_like, optional

Portfolio weights as a one-dimensional sequence or row/column vector. Required when returns has more than one column; otherwise it defaults to a single weight of 1.

mu : array_like, optional

Expected returns as a one-dimensional vector or a row vector of shape (1, n_assets). If None, the sample means are used. The default is None.

cov : array_like, optional

Covariance matrix of shape (n_assets, n_assets). If None, it is estimated from returns. The default is None.

rm : str, optional

Risk measure used in the denominator of the ratio. The default is ‘MV’. Possible values are:

  • ’MV’: Standard Deviation.

  • ’KT’: Square Root Kurtosis.

  • ’EM’: Even Moment of order 2 * p_em.

  • ’MAD’: Mean Absolute Deviation.

  • ’GMD’: Gini Mean Difference.

  • ’MSV’: Semi Standard Deviation.

  • ’SKT’: Square Root Semi Kurtosis.

  • ’ESM’: Even Semi Moment of order 2 * p_em.

  • ’FLPM’: First Lower Partial Moment (Omega Ratio).

  • ’SLPM’: Second Lower Partial Moment (Sortino Ratio).

  • ’VaR’: Value at Risk.

  • ’CVaR’: Conditional Value at Risk.

  • ’TG’: Tail Gini.

  • ’EVaR’: Entropic Value at Risk.

  • ’RLVaR’: Relativistic Value at Risk. I recommend only use this function with MOSEK solver.

  • ’WR’: Worst Realization (Minimax).

  • ’RG’: Range of returns.

  • ’VRG’ VaR range of returns.

  • ’CVRG’: CVaR range of returns.

  • ’TGRG’: Tail Gini range of returns.

  • ’EVRG’: EVaR range of returns.

  • ’RVRG’: RLVaR range of returns. I recommend only use this function with MOSEK solver.

  • ’MDD’: Maximum Drawdown of uncompounded cumulative returns (Calmar Ratio).

  • ’ADD’: Average Drawdown of uncompounded cumulative returns.

  • ’DaR’: Drawdown at Risk of uncompounded cumulative returns.

  • ’CDaR’: Conditional Drawdown at Risk of uncompounded cumulative returns.

  • ’EDaR’: Entropic Drawdown at Risk of uncompounded cumulative returns.

  • ’RLDaR’: Relativistic Drawdown at Risk of uncompounded cumulative returns. I recommend only use this function with MOSEK solver.

  • ’UCI’: Ulcer Index of uncompounded cumulative returns.

  • ’MDD_Rel’: Maximum Drawdown of compounded cumulative returns (Calmar Ratio).

  • ’ADD_Rel’: Average Drawdown of compounded cumulative returns.

  • ’DaR_Rel’: Drawdown at Risk of compounded cumulative returns.

  • ’CDaR_Rel’: Conditional Drawdown at Risk of compounded cumulative returns.

  • ’EDaR_Rel’: Entropic Drawdown at Risk of compounded cumulative returns.

  • ’RLDaR_Rel’: Relativistic Drawdown at Risk of compounded cumulative returns. I recommend only use this function with MOSEK solver.

  • ’UCI_Rel’: Ulcer Index of compounded cumulative returns.

rf : float, optional

Risk free rate. The default is 0.

alpha : float, optional

Significance level of VaR, CVaR, EVaR, RLVaR, DaR, CDaR, EDaR, RLDaR and Tail Gini of losses. The default is 0.05.

a_sim : int, optional

Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

beta : float or None, optional

Significance level of CVaR and Tail Gini of gains. If None it duplicates alpha value. The default is None.

b_sim : int or None, optional

Number of CVaRs used to approximate Tail Gini of gains. If None it duplicates a_sim value. The default is None.

kappa : float, optional

Deformation parameter of RLVaR and RLDaR for losses, must be between 0 and 1. The default is 0.3.

kappa_g : float or None, optional

Deformation parameter of RLVaR and RLDaR for gains, must be between 0 and 1. The default is None.

p_em : int, optional

Order of the Even Moment of order 2 * p_em. It must be an integer higher equal than 2. The default value is 2.

p_esm : int, optional

Order of the Even Semi Moment of order 2 * p_esm. It must be an integer higher equal than 2. The default value is 2.

solver : str, optional

Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR and RLDaR. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Risk adjusted return ratio of \(X\).

Return type:

float

RiskFunctions.Risk_Contribution(w, returns, cov=None, rm='MV', rf=0, alpha=0.05, a_sim=100, beta=None, b_sim=None, kappa=0.3, kappa_g=None, p_em=2, p_esm=2, solver='CLARABEL')[source]

Calculate the risk contribution for each asset based on the selected risk measure.

Parameters:
w : array_like

Portfolio weights as a one-dimensional sequence or row/column vector. NumPy arrays, pandas Series, and one-column or one-row DataFrames are accepted.

returns : DataFrame, Series, or array_like of shape (n_samples, n_features)

Numeric returns for the assets. A pandas Series is treated as one asset; a one-dimensional array-like input is treated as one observation across assets.

cov : array_like, optional

Covariance matrix of shape (n_assets, n_assets). If None, it is estimated from returns. The default is None.

rm : str, optional

Risk measure used in the denominator of the ratio. The default is ‘MV’. Possible values are:

  • ’MV’: Standard Deviation.

  • ’KT’: Square Root Kurtosis.

  • ’EM’: Even Moment of order 2 * p_em.

  • ’MAD’: Mean Absolute Deviation.

  • ’GMD’: Gini Mean Difference.

  • ’MSV’: Semi Standard Deviation.

  • ’SKT’: Square Root Semi Kurtosis.

  • ’ESM’: Even Semi Moment of order 2 * p_esm.

  • ’FLPM’: First Lower Partial Moment (Omega Ratio).

  • ’SLPM’: Second Lower Partial Moment (Sortino Ratio).

  • ’VaR’: Value at Risk.

  • ’CVaR’: Conditional Value at Risk.

  • ’TG’: Tail Gini.

  • ’EVaR’: Entropic Value at Risk.

  • ’RLVaR’: Relativistic Value at Risk. I recommend only use this function with MOSEK solver.

  • ’WR’: Worst Realization (Minimax).

  • ’RG’: Range of returns.

  • ’VRG’ VaR range of returns.

  • ’CVRG’: CVaR range of returns.

  • ’TGRG’: Tail Gini range of returns.

  • ’EVRG’: EVaR range of returns.

  • ’RVRG’: RLVaR range of returns. I recommend only use this function with MOSEK solver.

  • ’MDD’: Maximum Drawdown of uncompounded cumulative returns (Calmar Ratio).

  • ’ADD’: Average Drawdown of uncompounded cumulative returns.

  • ’DaR’: Drawdown at Risk of uncompounded cumulative returns.

  • ’CDaR’: Conditional Drawdown at Risk of uncompounded cumulative returns.

  • ’EDaR’: Entropic Drawdown at Risk of uncompounded cumulative returns.

  • ’RLDaR’: Relativistic Drawdown at Risk of uncompounded cumulative returns. I recommend only use this function with MOSEK solver.

  • ’UCI’: Ulcer Index of uncompounded cumulative returns.

  • ’MDD_Rel’: Maximum Drawdown of compounded cumulative returns (Calmar Ratio).

  • ’ADD_Rel’: Average Drawdown of compounded cumulative returns.

  • ’CDaR_Rel’: Conditional Drawdown at Risk of compounded cumulative returns.

  • ’EDaR_Rel’: Entropic Drawdown at Risk of compounded cumulative returns.

  • ’RLDaR_Rel’: Relativistic Drawdown at Risk of compounded cumulative returns. I recommend only use this function with MOSEK solver.

  • ’UCI_Rel’: Ulcer Index of compounded cumulative returns.

rf : float, optional

Risk free rate. The default is 0.

alpha : float, optional

Significance level of VaR, CVaR, EVaR, RLVaR, DaR, CDaR, EDaR, RLDaR and Tail Gini of losses. The default is 0.05.

a_sim : int, optional

Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

beta : float or None, optional

Significance level of CVaR and Tail Gini of gains. If None it duplicates alpha value. The default is None.

b_sim : int or None, optional

Number of CVaRs used to approximate Tail Gini of gains. If None it duplicates a_sim value. The default is None.

kappa : float, optional

Deformation parameter of RLVaR and RLDaR for losses, must be between 0 and 1. The default is 0.3.

kappa_g : float or None, optional

Deformation parameter of RLVaR and RLDaR for gains, must be between 0 and 1. The default is None.

p_em : int, optional

Order of the Even Moment of order 2 * p_em. It must be an integer higher equal than 2. The default value is 2.

p_esm : int, optional

Order of the Even Semi Moment of order 2 * p_esm. It must be an integer higher equal than 2. The default value is 2.

solver : str, optional

Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR and RLDaR. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Risk measure of the portfolio.

Return type:

float

RiskFunctions.Risk_Margin(w, returns, cov=None, rm='MV', rf=0, alpha=0.05, a_sim=100, beta=None, b_sim=None, kappa=0.3, kappa_g=None, p_em=2, p_esm=2, solver='CLARABEL')[source]

Calculate the risk margin for each asset based on the risk measure selected.

Parameters:
w : array_like

Portfolio weights as a one-dimensional sequence or row/column vector. NumPy arrays, pandas Series, and one-column or one-row DataFrames are accepted.

returns : DataFrame, Series, or array_like of shape (n_samples, n_features)

Numeric returns for the assets. A pandas Series is treated as one asset; a one-dimensional array-like input is treated as one observation across assets.

cov : array_like, optional

Covariance matrix of shape (n_assets, n_assets). If None, it is estimated from returns. The default is None.

rm : str, optional

Risk measure used in the denominator of the ratio. The default is ‘MV’. Possible values are:

  • ’MV’: Standard Deviation.

  • ’KT’: Square Root Kurtosis.

  • ’EM’: Even Moment of order 2 * p_em.

  • ’MAD’: Mean Absolute Deviation.

  • ’GMD’: Gini Mean Difference.

  • ’MSV’: Semi Standard Deviation.

  • ’SKT’: Square Root Semi Kurtosis.

  • ’ESM’: Even Semi Moment of order 2 * p_esm.

  • ’FLPM’: First Lower Partial Moment (Omega Ratio).

  • ’SLPM’: Second Lower Partial Moment (Sortino Ratio).

  • ’VaR’: Value at Risk.

  • ’CVaR’: Conditional Value at Risk.

  • ’TG’: Tail Gini.

  • ’EVaR’: Entropic Value at Risk.

  • ’RLVaR’: Relativistic Value at Risk. I recommend only use this function with MOSEK solver.

  • ’WR’: Worst Realization (Minimax).

  • ’RG’: Range of returns.

  • ’VRG’ VaR range of returns.

  • ’CVRG’: CVaR range of returns.

  • ’TGRG’: Tail Gini range of returns.

  • ’EVRG’: EVaR range of returns.

  • ’RVRG’: RLVaR range of returns.

  • ’MDD’: Maximum Drawdown of uncompounded cumulative returns (Calmar Ratio).

  • ’ADD’: Average Drawdown of uncompounded cumulative returns.

  • ’DaR’: Drawdown at Risk of uncompounded cumulative returns.

  • ’CDaR’: Conditional Drawdown at Risk of uncompounded cumulative returns.

  • ’EDaR’: Entropic Drawdown at Risk of uncompounded cumulative returns.

  • ’RLDaR’: Relativistic Drawdown at Risk of uncompounded cumulative returns. I recommend only use this function with MOSEK solver.

  • ’UCI’: Ulcer Index of uncompounded cumulative returns.

  • ’MDD_Rel’: Maximum Drawdown of compounded cumulative returns (Calmar Ratio).

  • ’ADD_Rel’: Average Drawdown of compounded cumulative returns.

  • ’CDaR_Rel’: Conditional Drawdown at Risk of compounded cumulative returns.

  • ’EDaR_Rel’: Entropic Drawdown at Risk of compounded cumulative returns.

  • ’RLDaR_Rel’: Relativistic Drawdown at Risk of compounded cumulative returns. I recommend only use this function with MOSEK solver.

  • ’UCI_Rel’: Ulcer Index of compounded cumulative returns.

rf : float, optional

Risk free rate. The default is 0.

alpha : float, optional

Significance level of VaR, CVaR, EVaR, RLVaR, DaR, CDaR, EDaR, RLDaR and Tail Gini of losses. The default is 0.05.

a_sim : int, optional

Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

beta : float or None, optional

Significance level of VaR, CVaR, Tail Gini, EVaR and RLVaR of gains. If None it duplicates alpha value. The default is None.

b_sim : int or None, optional

Number of CVaRs used to approximate Tail Gini of gains. If None it duplicates a_sim value. The default is None.

kappa : float, optional

Deformation parameter of RLVaR and RLDaR for losses, must be between 0 and 1. The default is 0.3.

kappa_g : float or None, optional

Deformation parameter of RLVaR and RLDaR for gains, must be between 0 and 1. The default is None.

p_em : int, optional

Order of the Even Moment of order 2 * p_em. It must be an integer higher equal than 2. The default value is 2.

p_esm : int, optional

Order of the Even Semi Moment of order 2 * p_esm. It must be an integer higher equal than 2. The default value is 2.

solver : str, optional

Solver available for CVXPY that supports exponential and power cone programming. Used to calculate EVaR, EVRG, EDaR, RLVaR, RVRG and RLDaR. The default value is ‘CLARABEL’.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Risk margin of the portfolio.

Return type:

float

RiskFunctions.Factors_Risk_Contribution(w, returns, factors, cov=None, B=None, const=False, rm='MV', rf=0, alpha=0.05, a_sim=100, beta=None, b_sim=None, kappa=0.3, kappa_g=None, p_em=2, p_esm=2, solver='CLARABEL', feature_selection='stepwise', stepwise='Forward', criterion='pvalue', threshold=0.05, n_components=0.95)[source]

Calculate the risk contribution for each factor based on the selected risk measure.

Parameters:
w : array_like

Portfolio weights as a one-dimensional sequence or row/column vector. NumPy arrays, pandas Series, and one-column or one-row DataFrames are accepted.

returns : DataFrame of shape (n_samples, n_assets)

Numeric asset returns. This parameter must be a pandas DataFrame.

factors : DataFrame of shape (n_samples, n_factors)

Numeric factor returns. This parameter must be a pandas DataFrame with the same index as returns.

cov : array_like, optional

Covariance matrix of shape (n_assets, n_assets). If None, it is estimated from returns. The default is None.

B : DataFrame, optional

Loadings matrix, where n_assets is the number assets and n_factors is the number of risk factors (with an optional constant column). If not specified, it is estimated using the selected feature method. The default is None.

const : bool, optional

Indicate if the loadings matrix has a constant. The default is False.

rm : str, optional

Risk measure used in the denominator of the ratio. The default is ‘MV’. Possible values are:

  • ’MV’: Standard Deviation.

  • ’KT’: Square Root Kurtosis.

  • ’EM’: Even Moment of order 2 * p_em.

  • ’MAD’: Mean Absolute Deviation.

  • ’GMD’: Gini Mean Difference.

  • ’MSV’: Semi Standard Deviation.

  • ’SKT’: Square Root Semi Kurtosis.

  • ’ESM’: Even Semi Moment of order 2 * p_esm.

  • ’FLPM’: First Lower Partial Moment (Omega Ratio).

  • ’SLPM’: Second Lower Partial Moment (Sortino Ratio).

  • ’VaR’: Value at Risk.

  • ’CVaR’: Conditional Value at Risk.

  • ’TG’: Tail Gini.

  • ’EVaR’: Entropic Value at Risk.

  • ’RLVaR’: Relativistic Value at Risk. I recommend only use this function with MOSEK solver.

  • ’WR’: Worst Realization (Minimax).

  • ’RG’: Range of returns.

  • ’VRG’ VaR range of returns.

  • ’CVRG’: CVaR range of returns.

  • ’TGRG’: Tail Gini range of returns.

  • ’EVRG’: EVaR range of returns.

  • ’RVRG’: RLVaR range of returns. I recommend only use this function with MOSEK solver.

  • ’MDD’: Maximum Drawdown of uncompounded cumulative returns (Calmar Ratio).

  • ’ADD’: Average Drawdown of uncompounded cumulative returns.

  • ’DaR’: Drawdown at Risk of uncompounded cumulative returns.

  • ’CDaR’: Conditional Drawdown at Risk of uncompounded cumulative returns.

  • ’EDaR’: Entropic Drawdown at Risk of uncompounded cumulative returns.

  • ’RLDaR’: Relativistic Drawdown at Risk of uncompounded cumulative returns. I recommend only use this function with MOSEK solver.

  • ’UCI’: Ulcer Index of uncompounded cumulative returns.

  • ’MDD_Rel’: Maximum Drawdown of compounded cumulative returns (Calmar Ratio).

  • ’ADD_Rel’: Average Drawdown of compounded cumulative returns.

  • ’CDaR_Rel’: Conditional Drawdown at Risk of compounded cumulative returns.

  • ’EDaR_Rel’: Entropic Drawdown at Risk of compounded cumulative returns.

  • ’RLDaR_Rel’: Relativistic Drawdown at Risk of compounded cumulative returns. I recommend only use this function with MOSEK solver.

  • ’UCI_Rel’: Ulcer Index of compounded cumulative returns.

rf : float, optional

Risk free rate. The default is 0.

alpha : float, optional

Significance level of VaR, CVaR, EVaR, RLVaR, DaR, CDaR, EDaR, RLDaR and Tail Gini of losses. The default is 0.05.

a_sim : int, optional

Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

beta : float or None, optional

Significance level of CVaR and Tail Gini of gains. If None it duplicates alpha value. The default is None.

b_sim : int or None, optional

Number of CVaRs used to approximate Tail Gini of gains. If None it duplicates a_sim value. The default is None.

kappa : float, optional

Deformation parameter of RLVaR and RLDaR for losses, must be between 0 and 1. The default is 0.3.

kappa_g : float or None, optional

Deformation parameter of RLVaR and RLDaR for gains, must be between 0 and 1. The default is None.

p_em : int, optional

Order of the Even Moment of order 2 * p_em. It must be an integer higher equal than 2. The default value is 2.

p_esm : int, optional

Order of the Even Semi Moment of order 2 * p_esm. It must be an integer higher equal than 2. The default value is 2.

solver : str, optional

Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR and RLDaR. The default value is ‘CLARABEL’.

feature_selection : str, optional

Indicate the method used to estimate the loadings matrix. The default is ‘stepwise’. Supported values are ‘stepwise’ and ‘PCR’.

stepwise : str, optional

Indicate the method used for stepwise regression. The default is ‘Forward’. Supported values are ‘Forward’ and ‘Backward’.

criterion : str, optional

The default is ‘pvalue’. Possible values of the criterion used to select the best features are:

  • ’pvalue’: select the features based on p-values.

  • ’AIC’: select the features based on lowest Akaike Information Criterion.

  • ’SIC’: select the features based on lowest Schwarz Information Criterion.

  • ’R2’: select the features based on highest R Squared.

  • ’R2_A’: select the features based on highest Adjusted R Squared.

threshold : float, optional

Is the maximum p-value for each variable that will be accepted in the model. The default is 0.05.

n_components : int or float, optional

If greater than or equal to 1, it specifies the number of components to keep (integer values are expected). If between 0 and 1, it specifies the fraction of variance to retain. See PCA for more details. The default is 0.95.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Risk measure of the portfolio.

Return type:

float

RiskFunctions.BrinsonAttribution(prices, w, wb, start, end, asset_classes, classes_col, method='nearest')[source]

Creates a DataFrame with the Brinson Performance Attribution per class and aggregate based on [F1].

Parameters:
prices : DataFrame of shape (n_samples, n_assets)

Assets prices DataFrame, where n_samples is the number of observations and n_assets is the number of assets.

w : DataFrame or Series

Portfolio weights. A Series or a single-row or single-column DataFrame with one weight per asset.

wb : DataFrame or Series

Benchmark weights. A Series or a single-row or single-column DataFrame with one weight per asset.

start : str or datetime-like

Start date, usually in ‘YYYY-MM-DD’ format.

end : str or datetime-like

End date, usually in ‘YYYY-MM-DD’ format.

asset_classes : DataFrame of shape (n_assets, n_cols)

Asset’s classes DataFrame, where n_assets is the number of assets and n_cols is the number of columns of the DataFrame where the first column is the asset list and the next columns are the different asset’s classes sets. This parameter must be a pandas DataFrame.

classes_col : str or int

If value is str, it is the column name of the set of classes from asset_classes dataframe. If value is int, it is the column number of the set of classes from asset_classes DataFrame.

method : str or None, optional

Matching method used to locate the start and end dates when an exact match is unavailable. The default is ‘nearest’; None requires an exact match. See get_indexer for more details.

Raises:

ValueError – When the value cannot be calculated.

Returns:

  • BrinAttr (DataFrame) – A DataFrame with the Brinson Performance Attribution per class and aggregate.

  • (start_, end_) (tuple) – Start and end dates calculated using get_indexer method in string format.

Example

BrinAttr, (start, end) = BrinsonAttribution(
    prices=data,
    w=w,
    wb=wb,
    start='2019-01-07',
    end='2019-12-06',
    asset_classes=asset_classes,
    classes_col='Industry',
    )
../_images/BrinAttr.png

Bibliography

[F1]

Gary P Brinson and Nimrod Fachler. Measuring non-US. equity portfolio performance. J. Portf. Manag., 11(3):73–76, 04 1985.