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CVaR Group

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CVaR Group of functions defined on Loss and Gain includes the following functions:

Full Name	Brief Name	Short Description
CVaR	cvar_risk	Conditional Value-at-Risk for Linear Loss scenarios (also called Expected Shortfall and Tail VaR), i.e., the average of largest (1-α)% of Losses.
CVaR for Gain	cvar_risk_g	Conditional Value-at-Risk for -(Linear Loss ) scenarios (also called Expected Shortfall and Tail VaR), i.e., the average of largest (1-α)% of -(Losses).
CVaR Normal Independent	cvar_risk_ni	Special case of the CVaR when all coefficients in Linear Loss function are independent normally distributed random values.
CVaR for Gain Normal Independent	cvar_risk_ni_g	Special case of the CVaR for Gain when all coefficients in -(Linear Loss ) function are independent normally distributed random values.
CVaR Normal Dependent	cvar_risk_nd	Special case of the CVaR when all coefficients in Linear Loss function are mutually dependent normally distributed random values.
CVaR for Gain Normal Dependent	cvar_risk_nd_g	Special case of the CVaR for Gain when all coefficients in -(Linear Loss ) function are mutually dependent normally distributed random values
CVaR Deviation	cvar_dev	Conditional Value-at-Risk for (Linear Loss) - (Average over Linear Loss scenarios) , i.e., the average of largest (1-α)% of (Linear Loss) - (Average over Linear Loss scenarios) scenarios.
CVaR Deviation for Gain	cvar_dev_g	Conditional Value-at-Risk for -(Linear Loss ) + (Average over scenarios Linear Loss) , i.e., the average of largest (1-α)% of - (Linear Loss) + (Average over scenarios Linear Loss) scenarios.
CVaR Deviation Normal Independent	cvar_ni_dev	Special case of the CVaR Deviation when all coefficients in Linear Loss function are independent normally distributed random values.
CVaR Deviation for Gain Normal Independent	cvar_ni_dev_g	Special case of the CVaR Deviation for Gain when all coefficients in -(Linear Loss ) function are independent normally distributed random values.
CVaR Deviation Normal Dependent	cvar_nd_dev	Special case of the CVaR Deviation when all coefficients in Linear Loss function are mutually dependent normally distributed random values
CVaR Deviation for Gain Normal Dependent	cvar_nd_dev_g	Special case of the CVaR Deviation for Gain when all coefficients in -(Linear Loss ) function are mutually dependent normally distributed random values
CVaR for Mixture of Normal Independent	avg_cvar_risk_ni	Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all Linear Loss functions are independent normally distributed random values. Avg_cvar_risk_ni is the CVaR of the mixture of Normally Independent random values.
CVaR for Gain for Mixture of Normal Independent	avg_cvar_risk_ni_g	Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all Linear Loss functions are independent normally distributed random values. avg_cvar_risk_ni_g is the CVaR of the mixture of Normally Independent random values.
CVaR Deviation for Mixture of Normal Independent	avg_cvar_ni_dev	Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all Linear Loss functions are independent normally distributed random values. avg_cvar_ni_dev is the CVaR of the mixture of Normally Independent random values.
CVaR Deviation for Gain for Mixture of Normal Independent	avg_cvar_ni_dev_g	Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all Linear Loss functions are independent normally distributed random values. avg_cvar_ni_dev_g is the CVaR of the mixture of Normally Independent random values.
CVaR Max	cvar_max_risk	There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Loss scenarios function is calculated by maximizing losses over Linear Loss functions (over M functions for every scenario). CVaR Max is calculated by taking CVaR of the Maximum Loss scenarios.
CVaR Max for Gain	cvar_max_risk_g	There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Gain scenarios function is calculated by maximizing losses over -(Linear Loss) functions (over M functions for every scenario). CVaR Max for Gain is calculated by taking CVaR of the Maximum Gain scenarios.
CVaR Max Deviation	cvar_max_dev	There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Loss scenarios function is calculated by maximizing losses over (Linear Loss)-(Expected Linear Loss) functions (over M functions for every scenario). CVaR Max Deviation is calculated by taking CVaR of the Maximum Loss scenarios.
CVaR Max Deviation for Gain	cvar_max_dev_g	There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Gain scenarios function is calculated by maximizing losses over -(Linear Loss)+(Expected Linear Loss) functions (over M functions for every scenario). CVaR Max Deviation for Gain is calculated by taking CVaR of the Maximum Gain scenarios.
CVaR for Discrete Distribution as Function of Atom Probabilities	pcvar	This function is similar to the standard CVaR function, but decision variables are probabilities of scenarios.
CVaR for Mixture of Normal Distributions as Function of Mixture Weights	wcvar_ni	This function calculates CVaR for a mixture of normal distributions as a function of variable weights in this mixture
CVaR Recourse	cvar_risk(recourse(.))	CVaR of Recourse scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.
CVaR for Gain Recourse	cvar_risk_g(recourse(.)	CVaR of -(Recourse) scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.
CVaR Deviation Recourse	cvar_dev(recourse(.))	CVaR of (Deviation Recourse) = (Recourse-(Expected Recourse)) scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.
CVaR Deviation for Gain Recourse	cvar_dev_g(recourse(.))	CVaR of -(Deviation Recourse) = (-Recourse+(Expected Recourse)) scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Remarks

1.	The confidence level, α, satisfies the following condition: .

2.	Functions from the CVaR group are calculated with double precision.

3.	Any function from this group may be called by its "brief name" or by "brief name" with "optional name"

•	The optional name of any function from this group may contain up to 128 symbols.

•	Optional names of these functions may include only alphabetic characters, numbers, and the underscore sign, "_".

•	Optional names of these functions are "insensitive" to the case, i.e. there is no difference between low case and upper case in these names.