Methodologies of fitting distributions
If you remember we had started fitting distributions using scenario analysis and Interval approach, where experts mention the frequency of losses estimated within specific loss intervals.
This time we discuss the percentile approach – data is collected for specific percentiles/quantiles of loss severity from experts. In this tutorial we would discuss the Interval Approach. In the following illustrations, we will fit continuous distributions to scenario data collected for loss severities.
Assume that the output of a scenario workshop is that the median loss severity and 90th percentile loss severity are USD 30000 and USD 160000 respectively.
Using Quantile Matching for fitting severity distribution to data collected through Quantile approach
Step-1: Decide on a distribution to be fitted to data.
One of the decision criteria could be fatness of the tail.
- Usually, thin tailed distributions (exponential, Weibull with shape parameter>1, gamma) should be fitted to HFLS cells (High Frequency, Low Severity)
- Medium and fat tailed distribution (such as lognormal, Weibull with shape<1, extreme value) should be fitted to LFHS cells.
- Whether the cell is HFLS or LFHS may be based on expert judgement and empirical studies.
- One of the problems with Quantile approach is that ex-ante. The modeller has to decide which distribution would best describe operational losses in a cell.
- This is in contrast with Loss Data Analysis where the modeller can fit various distributions to data and the check which distribution best fits the data (using information criteria and other goodness of fit tests).
For this illustration, let us fit a lognormal distribution to scenario data.
Step-2: Decide on seed values of distribution parameters to calculate theoretical Quantile
- For lognormal distribution, both parameters need to be positive.
Step-3: Calculate the theoretical Quantile
- Theoretical Quantile = INVCDF(probability)
- In our illustration, theoretical median would be = LOGINV(.5)
- And theoretical 90th Quantile would be = LOGINV(.9)
Step-4: Compare theoretical quantiles with empirical/scenario quantiles.
- Calculate the sum of squared differences between theoretical and scenario quantiles.
- Squaring up magnifies deviances and will help us in penalising large deviations.
Step-5: Use an optimization algorithm (like Excel Solver) to minimise sum of squared differences between theoretical and empirical quantiles by changing parameter values.
In our illustration, change in parameter-1 (log_mean) to 10.31 and parameter-2 (log_stdev) to 1.31 reduces the squared deviation between theoretical quantiles and expert opinion to zero.
Therefore, lognormal (10.31, 1.31) may be used for severity modeling in OpVaR estimation.
One of the common issues is how many quantiles should be elicited from the experts. For fitting a two-parameter continuous distribution, atleast two quantiles should be elicited.
For practical considerations, it may be difficult to ask experts about more than three-four quantiles, lest they will be confused. BCBS in its July-2009 paper on ‘Results from the 2008 Loss Data Collection Exercise for Operational Risk’ observes that ‘the median number of severity percentiles for banks using the percentile approach was four, with a narrow inter-quartile range indicating that at least three quarters of these banks used four or fewer percentiles’.
Both the MLE approach and Quantile approach to fit continuous distributions to scenario analysis data are also discussed in BCBS paper on ‘Results from the 2008 Loss Data Collection Exercise for Operational Risk’, Annexure-C.
Templates to download
I have created a template for you, where the subheadings are given and you have to link the model to get the cash numbers! You can download the same from here. You can go through the case and fill in the yellow boxes. I also recommend that you try to create this structure on your own (so that you get a hang of what information is to be recorded).
Also you can download this filled template and check, if the information you recorded, matches mine or not!