Which of the following is closest to the description of a ‘risk functional’?

Which of the following is closest to the description of a ‘risk functional’?
A . A risk functional is the distribution that models the severity of a risk
B . A risk functional is a model distribution that is an approximation of the true loss distribution of a risk
C . Risk functional refers to the Kolmogorov-Smirnov distance
D . A risk functional assigns a penalty value for the difference between a model distribution and a risk’s severity distribution

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Answer: D

Explanation:

For operational risk modeling, both frequency and severity distributions need to be modeled. Modeling severity involves finding an analytical distribution, such as log-normal or other that approximates the distribution best represented by known data – whether from the internal loss database, the external loss database or scenario data. A ‘risk functional’ is a measure of the deviation of the model distribution from the risk’s actual severity distribution. It assigns a penalty value for the deviation, using a statistical measure, such as the KS distance (Kolmogorov-Smirnov distance).

The problem of finding the right distribution then becomes the problem of optimizing the risk functional. For example, if F is the model distribution, and G is the actual, or empirical severity distribution, and we are using the KS test, then the Risk Functional R is defined as follows:

Note that supx stands for ‘supremum’, which is a more technical way of saying ‘maximum’. In other words, we are calculating the maximum absolute KS distance between the two distributions. (Note that the KS distance is the max of the distance between identical percentiles of the two distributions using the CDFs of the two.)

Once the risk functional is identified, we can minimize it to determine the best fitting distribution for severity.