CVaR. Conditional Value-at-Risk (also called Expected Shortfall and Tail VaR) for Linear Loss  scenarios, i.e., the average of largest (1-α)% of Losses.

 

Syntax

 

 

Parameters

matrix        is a Matrix of Scenarios:

       

where the header row contains names of variables (except scenario_probability, and scenario_benchmark). Other rows contain numerical data. The scenario_probability, and scenario_benchmark columns are optional.

       is a confidence level.

 

Mathematical Definition

For continuous distributions, given and any specified probability level in  the -CVaR risk for Loss is

 

where        

and is a probability distribution function of the loss , and is a density of the random vector

.

 

For discrete distributions considered in PSG, when models are based on scenarios and finite sampling, calculation of the CVaR Risk for Loss includes the following steps:

1. Calculate values of the Loss function for all scenarios:

.

2. Sort scenarios

.

 

3. If , then

 

Setting for CVaR is not recommended because PSG contains the Average Loss function (avg) (section Average Group) dedicated for this purpose. This function calculates the same value in a more efficient way.

 

4. If then

 

Setting , for CVaR is not recommended because PSG contains the function Maximum Risk for Loss (max_risk) ( see section Maximum Group) dedicated to calculating the same in a more efficient way.

5. Let .

Determine an index such that and . If then last sum equals to 0.

6. If the index  is such that the confidence level equals ,

 

then the CVaR Risk for Loss equals

 

7. If , then the CVaR Risk for Loss equals the interpolation between CVaR Risks for Loss with confidence levels

 

and ,

 

i.e.,

 .

 

 

Example

 

Case Studies with CVaR

 

See also

CVaR for Gain,

CVaR Normal Independent, CVaR Normal Dependent,

CVaR Deviation, CVaR Deviation Normal Independent, CVaR Deviation Normal Dependent,

CVaR for Mixture of Normal Independent, CVaR Deviation for Mixture of Normal Independent

CVaR Max, CVaR Max Deviation,

CVaR for Discrete Distribution as Function of Atom Probabilities, CVaR for Mixture of Normal Distributions as Function of Mixture Weights