# Get distribution for aggregate loss using Monte Carlo

I am given two data sets containing dates and losses (in some currency).

Given a distribution for the amount of losses and an (a,b,0) distribution for frequency of losses, how can I use Monte Carlo simulations to get a distribution for aggregate losses?

The papers and books I see online seem to state how to simulate aggregate losses (by simulating # of losses and losses given such #), but how do I come up with a distribution given all that data?

There's this book I found "Operational Risk with Excel and VBA". It describes the procedure and ends with the mean, standard deviation and other moment stuff. Is that sufficient to describe the distribution of aggregate losses?

• Your question is not clear to me, what do you want to know: 1. Do you want to know how to do a Monte-Carlo simulation given a frequency and severity distribution? 2. Do you want to know how to calibrate a specific parametric distribution to your data at hand? Or do you want to know how to choose such a parametric distribution in the first place? – g g Feb 6 '15 at 11:05
• And by the way, given that losses from operational risk often are very heavy tailed, I would consider carefully whether a distribution for such losses should have finite variance – g g Feb 6 '15 at 11:12
• @gg I want to know how to fit a distribution of aggregate losses after simulating them. What's weird is that the book doesn't mention how to after simply stating the moments – BCLC Feb 6 '15 at 18:37
• I am still not sure I understand because what you seem to want sounds somewhat unusual. Normally people fit frequency and severity and then simulate exactly to avoid doing a fit to the aggregate distribution. Why do you need a parametric representation of the aggregate losses, if you can simulate them? – g g Feb 7 '15 at 11:44
• @gg I have no idea with my professor. :| – BCLC Feb 7 '15 at 11:50

You can do the following:

• For each $i$ in $1$ to number of Mont-Carlo runs $K$
• simulate the number of losses $N_i$
• simulate $N_i$ many loss-sizes $X_{i,1},\ldots,X_{i,N_i}$
• calculate $L_i = \sum_{j=1}^{N_i} X_{i,j}$

Doing this you get a sample of losses $L_1,\ldots,L_K$ and you can do all sorts of hisograms, density fits, VaR, ES calculations on it.

EDIT: on this sample you could try to fit a loss distribution (e.g. Gamma or translated Gamma see here) by maximum likelihood or method of moments. But you can apply the method of moments even without MC becauase if you assume that the number of losses and the loss sizes are independent then $$E[L] = E[N]E[X] \text{ and } V[L] = E[N]V[X] +E[X]^2 V[N]$$ for these fromulas and fitting distributions see e.g. again here.

• "density fits" How? That is precisely what I want to know how to do given all those Li's – BCLC Feb 6 '15 at 8:58
• See the EDIT and the link for the moment and the Gamma distribution. – Ric Feb 6 '15 at 9:15
• THANK YOU, @Richard...I think. I'll check it out and see it if helps. – BCLC Feb 7 '15 at 9:01
• Yes you can average $L^3$ from the sample or you look here on page 8 for the formla of the skewness in the collective risk model. If you like my answer then don't forger to accept ;) – Ric Feb 9 '15 at 8:04
• You try to fit this $x_0$ by the method of moments (as I have seen in the other question). Maybe this estimte has too much variance. Try maximum-likeliohood instead. It should work too. – Ric Feb 9 '15 at 9:14