CVaR2 Risk (cvar2_risk)
Contents
CVaR2 Risk, which is an element of CVaR (Superquantile) quadrangle (see [1])
Syntax
cvar2_risk(α, matrix) |
short call |
cvar2_risk_name(α, matrix) |
call with optional name |
Parameters
is a confidence level.
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.
Mathematical Definition
CVaR2 Risk function is calculated as follows:
,
where integral of CVaR is equal:
.
is CVaR function,
is VaR function.
For discrete distributions considered in PSG, when models are based on scenarios and finite sampling, calculation of the integral of CVaR includes the following steps:
1. Calculate values of the Loss function for all scenarios:
.
2. Sort scenarios
,
is an order.
3. If 
, then 
.
4. If 
then determine 
, and an index 
such that 
and 
. Then
,
where 
are coefficients depending on 
and an order 
(in case of uniform distribution 
does not depend on an order)
,
,
where 
.
.
Example
References
[1] Rockafellar,R. T. , Royset, J. O., and S. I. Miranda (2014): Superquantile Regression with Applications to Buffered Reliability, Uncertainty Quantification and Conditional Value-at-Risk. European J. Operations Research 234 (2014), 140-154.
See also