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?

*Elaboration: Aggregate losses is a sum of IID RVs distributed either with loglogistic or k-point mixture with a random number of terms N, with an (a,b,0) distribution. The only kinds of (a,b,0) distributions are Poisson, Binomial and Negative Binomial

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Cross-posted: https://quant.stackexchange.com/questions/16464/get-distribution-for-aggregate-loss-using-monte-carlo

  • $\begingroup$ what is an (a,b,0) distribution? you need to provide more details about the statistics aspects of your question. $\endgroup$
    – Xi'an
    Feb 6, 2015 at 7:37
  • $\begingroup$ @Xi'an it is a probability distribution that satisfies a certain equation. The only distributions that satisfy such are Poisson, binomial and negative binomial $\endgroup$
    – BCLC
    Feb 6, 2015 at 8:36
  • $\begingroup$ BCLC Most people probably won't be familiar with $(a,b,0)$ distributions unless they've seen Panjer recursion (which I expect most people here won't have). For the curious, see Wikipedia's page on the class, here. $\endgroup$
    – Glen_b
    Feb 6, 2015 at 9:42

1 Answer 1


Moments don't pin distributions down very well, so mean, variance, skewness and kurtosis don't necessarily tell you about tail behavior -- a distribution with a different amount of tail riskiness may have very similar first few moments.

In the case of aggregate loss distributions, the upper tail is an aspect of particular interest (hopefully, for obvious reasons).

Moments don't tell you all that much about particular exceedance probabilities.

The first couple of moments are definitely useful, of course, and it's nice to have the skewness and kurtosis, but you really need some sense of how bad things might be. Some people look at VaR or, increasingly TailVaR (conditional tail expectation) (in both cases for the right tail) for one or more extreme tail quantiles. In some applications of this stuff - at least in some countries - you may be required to do so.

These risk measures are not completely informative either, but can be a useful way to compare one aspect of the extremes.

To fully understand the distribution, the survival function of the distribution (or perhaps the cdf) would be the most obvious choice. One could also plot TailVaR or VaR against its corresponding tail quantile across a range of tail quantiles.


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