From VaR to CVaR

Let be the loss function associated with a decision vector chosen from and a random vector of uncertainties . For each vector , can be seen as a random variable in induced by . Let be the probability density function of , which can be derived analytically or constructed by Monte Carlo simulation.

We have the probability that does not exceeding a threshold given by For a fixed , is the cumulative distribution function of at a given , and thus, it completely determines the behaviour of and is fundamental in defining VaR and CVaR measurements.

For a specified probability level , let be the -VaR value for . We then have the definition of as the lowest level of such that, with probability , the loss value does not exceed , Given the definition of -VaR, -CVaR is then defined as the expectatio of the loss values above -VaR, i.e.,

Axioms to assess risk

A function is said to be a coherent risk measure if it satisfies the following properties, for :

• Monotonicity: if , then

• Sub-additivity:

• Translation invariance: , for

• Positive homogeneity: , for

• Convexity: , for

Although VaR is a very popular risk measure, it is not a coherent risk measure due to laking of sub-additivity and convexity. Moreover, VaR is very difficult to optimize when it is calculated from scenarios. On the other hand, CVaR is an important and prominent risk measure that satisfies all the above axioms. Additionally, optimizing CVaR is closely related to optimizing VaR, as shown in their definition in the previous section.

Optimizing CVaR

Let define a function where The advantages of CVaR measure is based on the properties of function as given below.

The above theorem is powerful since functions with such properties like convexity and continuous differentiability are very easy to minimize numerically. Moreover, the theorem also reveals that CVaR can be calculate without calculating VaR, which would be complicated.

In practice, can be calculated by simulations. Particularly, let be samples of the uncertainty following the density function . The function can be approximated by which can then be used to optimize the decision vector .