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Simulating Aggregate Loss Distributions from Frequency and Severity

Article Quant Q&A · Author: BCLC
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Summary

The document describes a Monte Carlo method for constructing an aggregate loss distribution from separate models for loss frequency and loss size. For each simulation run, draw a loss count from the frequency distribution, then draw that many individual severities from the loss-size distribution and sum them. Repeating this process produces simulated aggregate outcomes that can be examined with histograms, density estimates, value at risk, and expected shortfall.

The simulated sample can also be used to fit a parametric aggregate-loss distribution, for example by maximum likelihood or moments. When frequency and severity are assumed independent, the aggregate mean is the product of their means, and the variance combines expected frequency times severity variance with severity mean squared times frequency variance. These moment formulas do not require Monte Carlo, but they rely on the independence assumption. The document gives a procedure and moment relationships, without specifying a best-fit family or assessing how well any model captures extreme losses.

Key ideas

  • Simulate a loss count for each Monte Carlo run from the frequency model.
  • Draw as many individual loss sizes as the simulated count and sum them to form aggregate loss.
  • Use simulated aggregate outcomes to estimate tail measures such as value at risk and expected shortfall.
  • Under frequency-severity independence, aggregate mean and variance follow from component moments.
  • Moment formulas depend on independence and do not by themselves identify the full loss distribution.

Tags

StatisticsRisk managementBacktesting
Full text
# Get distribution for aggregate loss using Monte Carlo


# 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?

Cross-posted: https://stats.stackexchange.com/questions/136541/get-distribution-for-aggregate-loss-using-monte-carlo

## Answer by Richi Wa (score 1, accepted)

https://quant.stackexchange.com/a/16465

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.