风险函数

该模块具有计算几个风险指标的功能，这些指标被资产管理行业和学术界广泛使用。

模块函数

RiskFunctions.MAD(X)

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

$$\text{MAD}(X)=\frac{1}{T}\sum _{t=1}^T|X_t-\mathbb{E}(X_t)|$$

参数:

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

返回:

value – MAD of a returns series.

返回类型:

float

RiskFunctions.SemiDeviation(X)

Calculate the Semi Deviation of a returns series.

$$\text{SemiDev}(X)={[\frac{1}{T-1}\sum _{t=1}^{T}min(X_t-\mathbb{E}(X_t),0{)}^2]}^{1/2}$$

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Semi Deviation of a returns series.

返回类型:

float

RiskFunctions.Kurtosis(X)

Calculate the Square Root Kurtosis of a returns series.

$$\text{Kurt}(X)={[\frac{1}{T}\sum _{t=1}^T(X_t-\mathbb{E}(X_t){)}^4]}^{1/2}$$

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Square Root Kurtosis of a returns series.

返回类型:

float

RiskFunctions.SemiKurtosis(X)

Calculate the Semi Square Root Kurtosis of a returns series.

$$\text{SemiKurt}(X)={[\frac{1}{T}\sum _{t=1}^{T}min(X_t-\mathbb{E}(X_t),0{)}^4]}^{1/2}$$

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Semi Square Root Kurtosis of a returns series.

返回类型:

float

RiskFunctions.VaR_Hist(X, alpha=0.05)

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

$${\text{VaR}}_{\alpha }(X)=-\underset{t\in (0,T)}{inf}\{X_t\in \mathbb{R}:F_X(X_t)>\alpha \}$$

参数:

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

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – VaR of a returns series.

返回类型:

float

RiskFunctions.CVaR_Hist(X, alpha=0.05)

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

$${\text{CVaR}}_{\alpha }(X)={\text{VaR}}_{\alpha }(X)+\frac{1}{\alpha T}\sum _{t=1}^{T}max(-X_t-{\text{VaR}}_{\alpha }(X),0)$$

参数:

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

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – CVaR of a returns series.

返回类型:

float

RiskFunctions.WR(X)

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

$$\text{WR}(X)=max(-X)$$

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – WR of a returns series.

返回类型:

float

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

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

$$\begin{array}{rl}\text{LPM}(X,\text{MAR},1)& =\frac{1}{T}\sum _{t=1}^{T}max(\text{MAR}-X_t,0)\\ \text{LPM}(X,\text{MAR},2)& ={[\frac{1}{T-1}\sum _{t=1}^{T}max(\text{MAR}-X_t,0{)}^2]}^{\frac{1}{2}}\end{array}$$

Where:

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

参数:

• 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.

抛出:

ValueError – When the value cannot be calculated.

返回:

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

返回类型:

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.

参数:

• 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.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – ERM of a returns series.

返回类型:

float

RiskFunctions.EVaR_Hist(X, alpha=0.05)

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

$${\text{EVaR}}_{\alpha }(X)=\underset{z>0}{inf}\{z\ln \left(\frac{M_X(z^{-1})}{\alpha }\right)\}$$

Where:

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

参数:

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

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

抛出:

ValueError – When the value cannot be calculated.

返回:

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

返回类型:

tuple

RiskFunctions.RLVaR_Hist(X, alpha=0.05, kappa=0.01, solver=None)

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

$$\text{begin{array} ended with end{split}}$$

Where:

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

$\kappa$ is the deformation parameter.

参数:

• 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 0.01.

• solver (str, optional) – Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR and RLDaR. The default value is None.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – RLVaR of a returns series.

返回类型:

tuple

RiskFunctions.MDD_Abs(X)

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

$$\text{MDD}(X)=\underset{j\in (0,T)}{max}[\underset{t\in (0,j)}{max}\left(\sum _{i=0}^t{X}_i\right)-\sum _{i=0}^j{X}_i]$$

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – MDD of an uncompounded cumulative returns.

返回类型:

float

RiskFunctions.ADD_Abs(X)

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

$$\text{ADD}(X)=\frac{1}{T}\sum _{j=0}^T[\underset{t\in (0,j)}{max}\left(\sum _{i=0}^t{X}_i\right)-\sum _{i=0}^j{X}_i]$$

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – ADD of an uncompounded cumulative returns.

返回类型:

float

RiskFunctions.DaR_Abs(X, alpha=0.05)

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

$$\begin{array}{rl}{\text{DaR}}_{\alpha }(X)& =\underset{j\in (0,T)}{max}\{\text{DD}(X,j)\in \mathbb{R}:F_{\text{DD}}(\text{DD}(X,j))<1-\alpha \}\\ \text{DD}(X,j)& =\underset{t\in (0,j)}{max}\left(\sum _{i=0}^t{X}_i\right)-\sum _{i=0}^j{X}_i\end{array}$$

参数:

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

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – DaR of an uncompounded cumulative returns series.

返回类型:

float

RiskFunctions.CDaR_Abs(X, alpha=0.05)

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

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

Where:

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

参数:

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

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – CDaR of an uncompounded cumulative returns series.

返回类型:

float

RiskFunctions.EDaR_Abs(X, alpha=0.05)

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

$$\begin{array}{rl}{\text{EDaR}}_{\alpha }(X)& =\underset{z>0}{inf}\{z\ln \left(\frac{M_{\text{DD}(X)}(z^{-1})}{\alpha }\right)\}\\ \text{DD}(X,j)& =\underset{t\in (0,j)}{max}\left(\sum _{i=0}^t{X}_i\right)-\sum _{i=0}^j{X}_i\end{array}$$

参数:

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

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

抛出:

ValueError – When the value cannot be calculated.

返回:

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

返回类型:

tuple

RiskFunctions.RLDaR_Abs(X, alpha=0.05, kappa=0.01, solver=None)

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

$$\begin{array}{rl}{\text{RLDaR}}_{\alpha }^{\kappa }(X)& ={\text{RLVaR}}_{\alpha }^{\kappa }(DD(X))\\ \text{DD}(X,j)& =\underset{t\in (0,j)}{max}\left(\sum _{i=0}^t{X}_i\right)-\sum _{i=0}^j{X}_i\end{array}$$

参数:

• 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 0.01.

• solver (str, optional) – Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR and RLDaR. The default value is None.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – RLDaR of an uncompounded cumulative returns series.

返回类型:

tuple

RiskFunctions.UCI_Abs(X)

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

$$\text{UCI}(X)=\sqrt{\frac{1}{T}\sum _{j=0}^T{[\underset{t\in (0,j)}{max}\left(\sum _{i=0}^t{X}_i\right)-\sum _{i=0}^j{X}_i]}^2}$$

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Ulcer Index of an uncompounded cumulative returns.

返回类型:

float

RiskFunctions.MDD_Rel(X)

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

$$\text{MDD}(X)=\underset{j\in (0,T)}{max}[\underset{t\in (0,j)}{max}(\prod _{i=0}^t(1+X_i))-\prod _{i=0}^j(1+X_i)]$$

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – MDD of a cumpounded cumulative returns.

返回类型:

float

RiskFunctions.ADD_Rel(X)

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

$$\text{ADD}(X)=\frac{1}{T}\sum _{j=0}^T[\underset{t\in (0,j)}{max}(\prod _{i=0}^t(1+X_i))-\prod _{i=0}^j(1+X_i)]$$

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – ADD of a cumpounded cumulative returns.

返回类型:

float

RiskFunctions.DaR_Rel(X, alpha=0.05)

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

$$\begin{array}{rl}{\text{DaR}}_{\alpha }(X)& =\underset{j\in (0,T)}{max}\{\text{DD}(X,j)\in \mathbb{R}:F_{\text{DD}}(\text{DD}(X,j))<1-\alpha \}\\ \text{DD}(X,j)& =\underset{t\in (0,j)}{max}(\prod _{i=0}^t(1+X_i))-\prod _{i=0}^j(1+X_i)\end{array}$$

参数:

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

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – DaR of a cumpounded cumulative returns series.

返回类型:

float

RiskFunctions.CDaR_Rel(X, alpha=0.05)

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

$${\text{CDaR}}_{\alpha }(X)={\text{DaR}}_{\alpha }(X)+\frac{1}{\alpha T}\sum _{i=0}^{T}max[\underset{t\in (0,T)}{max}(\prod _{i=0}^t(1+X_i))-\prod _{i=0}^j(1+X_i)-{\text{DaR}}_{\alpha }(X),0]$$

Where:

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

参数:

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

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – CDaR of a cumpounded cumulative returns series.

返回类型:

float

RiskFunctions.EDaR_Rel(X, alpha=0.05)

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

$$\begin{array}{rl}{\text{EDaR}}_{\alpha }(X)& =\underset{z>0}{inf}\{z\ln \left(\frac{M_{\text{DD}(X)}(z^{-1})}{\alpha }\right)\}\\ \text{DD}(X,j)& =\underset{t\in (0,j)}{max}(\prod _{i=0}^t(1+X_i))-\prod _{i=0}^j(1+X_i)\end{array}$$

参数:

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

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

抛出:

ValueError – When the value cannot be calculated.

返回:

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

返回类型:

tuple

RiskFunctions.RLDaR_Rel(X, alpha=0.05, kappa=0.01, solver=None)

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

$$\begin{array}{rl}{\text{RLDaR}}_{\alpha }^{\kappa }(X)& ={\text{RLVaR}}_{\alpha }^{\kappa }(DD(X))\\ \text{DD}(X,j)& =\underset{t\in (0,j)}{max}(\prod _{i=0}^t(1+X_i))-\prod _{i=0}^j(1+X_i)\end{array}$$

参数:

• 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 0.01.

• solver (str, optional) – Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR and RLDaR. The default value is None.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – RLDaR of a compounded cumulative returns series.

返回类型:

tuple

RiskFunctions.UCI_Rel(X)

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

$$\text{UCI}(X)=\sqrt{\frac{1}{T}\sum _{j=0}^T{[\underset{t\in (0,j)}{max}(\prod _{i=0}^t(1+X_i))-\prod _{i=0}^j(1+X_i)]}^2}$$

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Ulcer Index of a cumpounded cumulative returns.

返回类型:

float

RiskFunctions.GMD(X)

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

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Gini Mean Difference of a returns series.

返回类型:

float

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

Calculate the Tail Gini of a returns series.

参数:

• 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.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Ulcer Index of a cumpounded cumulative returns.

返回类型:

float

RiskFunctions.RG(X)

Calculate the range of a returns series.

参数:

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

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Ulcer Index of a cumpounded cumulative returns.

返回类型:

float

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

Calculate the CVaR range of a returns series.

参数:

• 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.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Ulcer Index of a cumpounded cumulative returns.

返回类型:

float

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

Calculate the Tail Gini range of a returns series.

参数:

• 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.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Ulcer Index of a cumpounded cumulative returns.

返回类型:

float

RiskFunctions.L_Moment(X, k=2)

Calculate the kth l-moment of a returns series.

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

参数:

• 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.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Kth l-moment of a returns series.

返回类型:

float

RiskFunctions.L_Moment_CRM(X, k=4, method='MSD', g=0.5, max_phi=0.5, solver=None)

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

参数:

• 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.

抛出:

ValueError – When the value cannot be calculated.

返回:

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

返回类型:

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=0.01, solver=None)

Calculate the risk measure available on the Sharpe function.

参数:

• 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 0.01.

• solver (str, optional) – Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR and RLDaR. The default value is None.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Risk measure of the portfolio.

返回类型:

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=0.01, solver=None)

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

$$\text{Sharpe}(X)=\frac{\mathbb{E}(X)-r_f}{\varphi (X)}$$

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 $\text{MAR}$.

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

参数:

• 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 0.01.

• solver (str, optional) – Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR and RLDaR. The default value is None.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Risk adjusted return ratio of $X$.

返回类型:

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=0.01, solver=None)

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

参数:

• 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 0.01.

• solver (str, optional) – Solver available for CVXPY that supports power cone programming. Used to calculate RLVaR and RLDaR. The default value is None.

抛出:

ValueError – When the value cannot be calculated.

返回:

value – Risk measure of the portfolio.

返回类型:

float