**Relevance:**

Modelling of loss distribution from a banking aspect. For instance, the Basel report suggest an approach of looking at operational risk charge from an Advanced Measurement Approach (AMA) and the Loss Distribution Approach (LDA) constitutes a huge component in AMA.

Modelling of loss distribution from an insurer’s aspect. To model for economic capital for regulatory purposes, it requires the insurer to anticipate losses resulted from claims. In this case, the studies on loss distribution would be useful for the insurer to determine their requirement for economic capital.

**Background:**

Although the examples given above are related to the finance and banking sector, it is important for us to understand that the relevance of loss distribution are not limited to these. One example is in the domain of hydrology where water levels are being modelled as a form of loss distribution.

**Aim:**

The aim of this write up is to introduce the method of Peak over Threshold (POT) in the modelling of loss distribution. Often, modelling of loss distribution involves dealing with two different type of losses, 1) High Frequency and Low Severity 2) Low Frequency and High Severity. If we were to use a single parametric distribution to model for the loss distribution, the goodness of fit would often be unsound. Therefore, this point us to the method of POT.

**Separating the loss distribution into type 1 and 2:**

As mentioned above, loss distribution could often be separated into type 1 and type 2. Here, we shall introduce Pickands-Balkema-de Haan Theorem which suggest a way to identify the threshold for differentiating this two types of losses.

Consider a high threshold value, for the losses and let be a finite or infinite right end point of the overall loss distribution F such that . To model the losses above a threshold we define the distribution of such losses as below:

for .

Then the theorem states that under maximum domain of attraction (MDA) condition, the generalised Pareto distribution is the limiting distribution for the extreme values as the threshold , tends towards . Without going into much detail, the MDA condition can be interpreted as the condition of convergence to the generalised extreme value distribution under the method of sample maxima.

For the above method of tail fitting by using the peak over threshold, intuition might suggest that the selection of threshold is a tricky problem. Indeed, if we choose a very high threshold, there are fewer data points available above the threshold. This will result in volatile estimations of parameters . However, if we choose a threshold that is too low, it might result in a biased estimate arising due to non-convergence towards the limiting distribution of generalised Pareto distribution. In the following, we shall introduce some guiding principles which help us in the selection of threshold. Two criteria will be listed, and they can be accessed via a graphical diagnostic as suggested by Coles (2001).

__Using a graphical diagnostic plot__

If the exceedance of threshold indeed converges to a generalised Pareto distribution, then if we choose a threshold such that , the exceedances of the data from this new threshold should also converge to a generalised Pareto distribution by the theorem of Pickands-Balkema-de Haan such that:

Note that the equality is true if:

These conditions can be checked by the graphical plots.

**Type 1 losses:**

After differentiating the type 1 and type 2 losses, we can now proceed to fit a distribution to explain for the losses. Note the type 1 losses are defined from 0 to as defined by Pickands-Balkema-de Haan Theorem. Hence we are now ready to try out of distribution such as truncated normal, truncated exponential distribution etc. Selection criteria could be made using the goodness of fit test such as ks.test and chi-square goodness of fit etc.

**Type 2 losses:**

As specified by Pickands-Balkema-de Haan Theorem, the type 2 losses will have a generalised Pareto distribution.