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I am into risk management and deal with Operational risk. As a part of BASEL II guidelines, we need to arrive at the capital charge the banks must set aside to counter any operational risk, if it happens. As a part of Loss Distribution Approach (LDA), we need to collate past loss events and use these loss amounts. The usual process as being practiced in the industry is as follows -

Using these historical loss amounts and using the various statistical tests like KS test, AD test, PP plot, Q-Q plot etc, we try to identify best statistical (continuous) distribution fitting this historical loss data. Then using these estimated parameters w.r.t. the statistical distribution, we simulate say 1 million loss amounts and then taking appropriate percentile (say 99.9%), we arrive at the capital charge.

However, many a times, loss data is such that fitting of distribution to loss data is not possible. May be loss data is multimodal or has significant variability, making the fitting of distribution impossible. Can someone guide me how to deal with such data and what can be done to simulate losses using this historical loss data in R.

My data is as follows

mydat <- c(829.53,4000,6000,1000,1063904,102400,22000,4000,4200,2000,10000,400, 459006, 7276,4000,100,4000,10000,613803.36, 825,1000,5000,4000,3000,84500,200, 2000,68000,97400,6267.8, 49500,27000,2100,10489.92,2200,2000,2000,1000,1900, 6000,5600,100,4000,14300,100,94100,1200,7000,2000,3000,1100,6900,1000,18500,6000,2000,4000,8400,11200,1000,15100,23300,4000,13100,4500,200,2000,50000,3900,3200,2000,2000,67000,2000,500,2000,1000,1900,10400,1900,2000,3200,6500,10000,2900,1000,14300,1000,2700,1500,12000,40000,25000,2800,5000,15000,4000,1000,21000,15000,16000,54000,1500,19200,2000,2000,1000,39000,5000,1100,18000,10000,3500,1000,10000,5000,14000,1800,4000,1000,300,4000,1000,100,1000,4400,2000,2000,12000,200,100,1000,1000,2000,1600,2000,4000,14000,4000,13500,1000,200,200,1000,18000,23000,41400,60000,500,3000,21000,6900,14600,1900,4000,4500,1000,2000,2000,1000,4100,2000,1000,2000,8000,3000,1500,2000,2000,3500,2000,2000,1000,3800,30000,55000,500,1000,1000,2000,62400,2000,3000,200,2000,3500,2000,2000,500,3000,4500,1000,10000,2000,3000,3600,1000,2000,2000,5000,23000,2000,1900,2000,60000,2000,60000,20000,2000,2000,4600,1000,2000,1000,18000,6000,62000,68000,26800,50000,45900,16900,21500,2000,22700,2000,2000,32000,10000,5000,138000,159700,13000,2000,17619,2000,1000,4000,2000,1500,4000,20000,158900,74100,6000,24900,60000,500,1000,40000,10000,50000,800,4000,4900,6500,5000,400,500,3000,32300,24000,300,11500,2000,5000,1000,500,5000,5500,17450,56800,2000,1000,21400,22000,60000,3000,7500,3000,1000,1000,2000,1500,83700,2000,4000,170005,70000,6700,1500,3500,2000,10563.97,1500,25000,2000,2000,2267.57,1100,3100,2000,3500,10000,2000,6000,1500,200,20000,4000,46400,296900,150000,3700,7500,20000,48500,3500,12000,2500,4000,8500,1000,14500,1000,11000,2000,2000,120000,20000,7600,3000,2000,8000,1600,40000,2000,5000,34187.67,279100,9900,31300,814000,43500,5100,49500,4500,6262.38,100,10400,2400,1500,5000,2500,15000,40000,32500,41100,358600,109600,514300,258200,225900,402700,274300,75000,1000,56000,10000,4100,1000,15000,100,40000,7900,5000,105000,15100,2000,1100,2900,1500,600,500,1300,100,5000,5000,10000,10100,7000,40000,10500,5000,9500,1000,15200,2000,10000,10000,100,7800,3500,189900,58000,345000,151700,11000,6000,7000,15700,6000,3000,5000,10000,2000,1000,36000,1000,500,8000,9000,6000,2000,26500,6000,5000,97200,2000,5100,17000,2500,25500,24000,5400,90000,41500,6200,7500,5000,7000,41000,25000,1500,40000,5000,10000,21500,100,32000,32500,70000,500,66400,21000,5000,5000,12600,3000,6200,38900,10000,1000,60000,41100,1200,31300,2500,58000,4100,58000,42500)

Sorry for the inconvenience. I do understand fitting of distribution to such data is not a full proof method, but this is what is the procedure that has been followed in the risk management risk industry. Please note that my question is not pertaining to operational risk. My question if if distributions are not fitting to a particular data, how do we proceed further to simulate data based on this data.

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Firstly, I would recommend taking the logarithm of your data. This is because you are interested in financial losses that cannot go negative.

As illustrated in the figure below, I do not think a well known distribution will fit nicely to your data. You can try non (or semi) parametric solutions (kernel density estimation is the method generating the figure below).

For your particular case however, I would personally consider using a lognormal distribution (which is equivalent to using a normal distribution for the logarithms). Even if your data will not pass normality tests, this law is easy to simulate and may still provide good results.

enter image description here

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  • $\begingroup$ Thanks a lot for your suggestion. As suggested, I took log of the data and tried to fit some distribution. Unfortunately, it didn't. It's not unfortunately I can select Log normal just like that as client wont be convinced. If I try Kernel density estimation, will I be able to simulate say 50000 losses? Can you guide me please. Regards $\endgroup$ – Amelia Marsh Jul 22 '15 at 16:34
  • $\begingroup$ Then, you can go for the kernel density estimation. You can simulate log-losses by inverting the kernel cumulative distribution function. You can check this answer stats.stackexchange.com/questions/108342/… for more information on these computations. $\endgroup$ – ThePawn Jul 23 '15 at 1:01

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