Risk Functions

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)

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 (1d-array) – Returns series, must have Tx1 size.

Returns:

value – MAD of a returns series.

Return type:

float

RiskFunctions.SemiDeviation(X)

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 (1d-array) – Returns series, must have Tx1 size.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Semi Deviation of a returns series.

Return type:

float

RiskFunctions.Kurtosis(X)

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 (1d-array) – Returns series, must have Tx1 size.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Square Root Kurtosis of a returns series.

Return type:

float

RiskFunctions.SemiKurtosis(X)

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 (1d-array) – Returns series, must have Tx1 size.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Semi Square Root Kurtosis of a returns series.

Return type:

float

RiskFunctions.VaR_Hist(X, alpha=0.05)

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 (1d-array) – Returns series, must have Tx1 size.

• 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)

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 (1d-array) – Returns series, must have Tx1 size.

• 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)

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

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

Parameters:

X (1d-array) – Returns series, must have Tx1 size.

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)

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 (1d-array) – Returns series, must have Tx1 size.

• MAR (float, optional) – Minimum acceptable return. The default is 0.

• p (float, optional can be {1,2}) – order of the \(\text{LPM}\). 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)

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 (1d-array) – Returns series, must have Tx1 size.

• theta (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')

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 (1d-array) – Returns series, must have Tx1 size.

• 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 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')

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

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

Where:

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

\(\kappa\) is the deformation parameter.

Parameters:

• X (1d-array) – Returns series, must have Tx1 size.

• 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)

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 (1d-array) – Returns series, must have Tx1 size.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – MDD of an uncompounded cumulative returns.

Return type:

float

RiskFunctions.ADD_Abs(X)

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 (1d-array) – Returns series, must have Tx1 size.

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)

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 (1d-array) – Returns series, must have Tx1 size..

• 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)

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 (1d-array) – Returns series, must have Tx1 size..

• 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)

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 (1d-array) – Returns series, must have Tx1 size..

• 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')

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

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

Parameters:

• X (1d-array) – Returns series, must have Tx1 size.

• 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 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)

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 (1d-array) – Returns series, must have Tx1 size.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of an uncompounded cumulative returns.

Return type:

float

RiskFunctions.MDD_Rel(X)

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 (1d-array) – Returns series, must have Tx1 size.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – MDD of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.ADD_Rel(X)

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 (1d-array) – Returns series, must have Tx1 size.

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)

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 (1d-array) – Returns series, must have Tx1 size..

• 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)

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 (1d-array) – Returns series, must have Tx1 size..

• 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)

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 (1d-array) – Returns series, must have Tx1 size..

• 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')

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

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

Parameters:

• X (1d-array) – Returns series, must have Tx1 size.

• 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 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)

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 (1d-array) – Returns series, must have Tx1 size.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Ulcer Index of a cumpounded cumulative returns.

Return type:

float

RiskFunctions.GMD(X)

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

Parameters:

X (1d-array) – Returns series, must have Tx1 size.

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)

Calculate the Tail Gini of a returns series.

Parameters:

• X (1d-array) – Returns series, must have Tx1 size.

• alpha (float, optional) – Significance level of Tail Gini. The default is 0.05.

• a_sim (float, optional) – 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)

Calculate the range of a returns series.

Parameters:

X (1d-array) – Returns series, must have Tx1 size.

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)

Calculate the CVaR range of a returns series.

Parameters:

• X (1d-array) – Returns series, must have Tx1 size.

• alpha (float, optional) – Significance level of CVaR of losses. The default is 0.05.

• beta (float, 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)

Calculate the Tail Gini range of a returns series.

Parameters:

• X (1d-array) – Returns series, must have Tx1 size.

• alpha (float, optional) – Significance level of Tail Gini of losses. The default is 0.05.

• a_sim (float, optional) – Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

• beta (float, optional) – Significance level of Tail Gini of gains. If None it duplicates alpha value. The default is None.

• b_sim (float, optional) – Number of CVaRs used to approximate Tail Gini of gains. If None it duplicates a_sim 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.L_Moment(X, k=2)

Calculate the kth l-moment of a returns series.

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

Parameters:

• X (1d-array) – Returns series, must have Tx1 size.

• k (int) – Order of the l-moment. Must be an integer higher or equal than 1.

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')

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

Parameters:

• X (1d-array) – Returns series, must have Tx1 size.

• k (int) – Order of the l-moment. Must be an integer higher or equal than 2.

• 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 None.

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.Sharpe_Risk(w, cov=None, returns=None, rm='MV', rf=0, alpha=0.05, a_sim=100, beta=None, b_sim=None, kappa=0.3, solver='CLARABEL')

Calculate the risk measure available on the Sharpe function.

Parameters:

• w (DataFrame or 1d-array of shape (n_assets, 1)) – Weights matrix, where n_assets is the number of assets.

• cov (DataFrame or nd-array of shape (n_features, n_features)) – Covariance matrix, where n_features is the number of features.

• returns (DataFrame or nd-array of shape (n_samples, n_features)) – Features matrix, where n_samples is the number of samples and n_features is the number of features.

• 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.

  • ’MAD’: Mean Absolute Deviation.

  • ’GMD’: Gini Mean Difference.

  • ’MSV’: Semi Standard Deviation.

  • ’SKT’: Square Root Semi Kurtosis.

  • ’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.

  • ’CVRG’: CVaR range of returns.

  • ’TGRG’: Tail Gini 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 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.

  • ’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 (float, optional) – Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

• beta (float, optional) – Significance level of CVaR and Tail Gini of gains. If None it duplicates alpha value. The default is None.

• b_sim (float, 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, 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 – Risk measure of the portfolio.

Return type:

float

RiskFunctions.Sharpe(w, mu, cov=None, returns=None, rm='MV', rf=0, alpha=0.05, a_sim=100, beta=None, b_sim=None, kappa=0.3, solver='CLARABEL')

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:

• w (DataFrame or 1d-array of shape (n_assets, 1)) – Weights matrix, where n_assets is the number of assets.

• mu (DataFrame or nd-array of shape (1, n_assets)) – Vector of expected returns, where n_assets is the number of assets.

• cov (DataFrame or nd-array of shape (n_features, n_features)) – Covariance matrix, where n_features is the number of features.

• returns (DataFrame or nd-array of shape (n_samples, n_features)) – Features matrix, where n_samples is the number of samples and n_features is the number of features.

• 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.

  • ’MAD’: Mean Absolute Deviation.

  • ’GMD’: Gini Mean Difference.

  • ’MSV’: Semi Standard Deviation.

  • ’SKT’: Square Root Semi Kurtosis.

  • ’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.

  • ’CVRG’: CVaR range of returns.

  • ’TGRG’: Tail Gini 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 (float, optional) – Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

• beta (float, optional) – Significance level of CVaR and Tail Gini of gains. If None it duplicates alpha value. The default is None.

• b_sim (float, 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, 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 None.

Raises:

ValueError – When the value cannot be calculated.

Returns:

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

Return type:

float

RiskFunctions.Risk_Contribution(w, cov=None, returns=None, rm='MV', rf=0, alpha=0.05, a_sim=100, beta=None, b_sim=None, kappa=0.3, solver='CLARABEL')

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

Parameters:

• w (DataFrame or 1d-array of shape (n_assets, 1)) – Weights matrix, where n_assets is the number of assets.

• cov (DataFrame or nd-array of shape (n_features, n_features)) – Covariance matrix, where n_features is the number of features.

• returns (DataFrame or nd-array of shape (n_samples, n_features)) – Features matrix, where n_samples is the number of samples and n_features is the number of features.

• 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.

  • ’MAD’: Mean Absolute Deviation.

  • ’GMD’: Gini Mean Difference.

  • ’MSV’: Semi Standard Deviation.

  • ’SKT’: Square Root Semi Kurtosis.

  • ’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.

  • ’CVRG’: CVaR range of returns.

  • ’TGRG’: Tail Gini 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 (float, optional) – Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

• beta (float, optional) – Significance level of CVaR and Tail Gini of gains. If None it duplicates alpha value. The default is None.

• b_sim (float, 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, 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 None.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Risk measure of the portfolio.

Return type:

float

RiskFunctions.Risk_Margin(w, cov=None, returns=None, rm='MV', rf=0, alpha=0.05, a_sim=100, beta=None, b_sim=None, kappa=0.3, solver='CLARABEL')

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

Parameters:

• w (DataFrame or 1d-array of shape (n_assets, 1)) – Weights matrix, where n_assets is the number of assets.

• cov (DataFrame or nd-array of shape (n_features, n_features)) – Covariance matrix, where n_features is the number of features.

• returns (DataFrame or nd-array of shape (n_samples, n_features)) – Features matrix, where n_samples is the number of samples and n_features is the number of features.

• 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.

  • ’MAD’: Mean Absolute Deviation.

  • ’GMD’: Gini Mean Difference.

  • ’MSV’: Semi Standard Deviation.

  • ’SKT’: Square Root Semi Kurtosis.

  • ’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.

  • ’CVRG’: CVaR range of returns.

  • ’TGRG’: Tail Gini 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 (float, optional) – Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

• beta (float, optional) – Significance level of CVaR and Tail Gini of gains. If None it duplicates alpha value. The default is None.

• b_sim (float, 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, 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 None.

Raises:

ValueError – When the value cannot be calculated.

Returns:

value – Risk margin of the portfolio.

Return type:

float

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

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

Parameters:

• w (DataFrame or 1d-array of shape (n_assets, 1)) – Weights matrix, where n_assets is the number of assets.

• cov (DataFrame or nd-array of shape (n_features, n_features)) – Covariance matrix, where n_features is the number of features.

• returns (DataFrame or nd-array of shape (n_samples, n_features)) – Features matrix, where n_samples is the number of samples and n_features is the number of features.

• factors (DataFrame or nd-array of shape (n_samples, n_factors)) – Factors matrix, where n_samples is the number of samples and n_factors is the number of factors.

• B (DataFrame of shape (n_assets, n_features), optional) – Loadings matrix. If is not specified, is estimated using stepwise regression. 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.

  • ’MAD’: Mean Absolute Deviation.

  • ’GMD’: Gini Mean Difference.

  • ’MSV’: Semi Standard Deviation.

  • ’SKT’: Square Root Semi Kurtosis.

  • ’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.

  • ’CVRG’: CVaR range of returns.

  • ’TGRG’: Tail Gini 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 (float, optional) – Number of CVaRs used to approximate Tail Gini of losses. The default is 100.

• beta (float, optional) – Significance level of CVaR and Tail Gini of gains. If None it duplicates alpha value. The default is None.

• b_sim (float, 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, 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 None.

• feature_selection (str 'stepwise' or 'PCR', optional) – Indicate the method used to estimate the loadings matrix. The default is ‘stepwise’.

• stepwise (str 'Forward' or 'Backward', optional) – Indicate the method used for stepwise regression. The default is ‘Forward’.

• 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 (scalar, optional) – Is the maximum p-value for each variable that will be accepted in the model. The default is 0.05.

• n_components (int, float, None or str, optional) – if 1 < n_components (int), it represents the number of components that will be keep. if 0 < n_components < 1 (float), it represents the percentage of variance that the is explained by the components kept. 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