Estimating Aggregate Loss Moments and Fitting a Distribution
Summary
The document asks how to obtain the first three moments of simulated aggregate losses and whether to use sample averages or formulas from the collective risk model. The answer distinguishes two routes: if the frequency and severity distributions are known, formulas can provide aggregate-loss moments directly; if simulations are available, sample-based statistical estimates can be used instead.
For a simulated sample, the response recommends working with the observed values to estimate quantities of interest, including quantiles and expected shortfall, or applying the method of moments when fitting a distribution such as a translated gamma. If the fitted distribution’s density is known, maximum likelihood is another option and does not require the moments to exist. The post gives general guidance rather than derivations or worked calculations, and it does not provide the referenced moment formulas.
Key ideas
- Known frequency and severity distributions allow aggregate-loss moments to be calculated from collective risk formulas.
- Monte Carlo output can be used directly to estimate sample moments and risk measures such as quantiles.
- The method of moments can fit a distribution to simulated aggregate losses.
- Maximum likelihood is an alternative when a candidate density is available and does not require existing moments.
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Full text
# Compute moments of aggregate loss using Monte Carlo # Compute moments of aggregate loss using Monte Carlo Spin-off from here. Richard referred to me an article that tells me how to get parameters of a translated gamma distribution to which I should consider fitting simulated aggregated loss values. The parameters depend on moments of S (or. in Richard's terms, L): How do I compute the $E(S), E(S^2) and E(S^3)$ given simulations of S? Do I estimate them with mean(S), mean(S^2) and mean(S^3), or do I use the formulas given in the article? I wouldn't know how to compute the $E(S^3)$ ... Cross-posted: https://stats.stackexchange.com/questions/136830/compute-moments-of-aggregate-loss-using-monte-carlo ## Answer by Richi Wa (score 1, accepted) https://quant.stackexchange.com/a/16503 as you post 3 questions on this topic and after reading them: this is homerwork/study material- right? So for comparing Fast Fourier, MC and Panjer there are tons of publications out there. For the formulas for the momemts of $S$ look here or google "moments in the collective risk model". You should notice that: - If you know the distribution of $N$ and $X$ then you know the moments and using those formulas you can calculate the moments of $S$ without MC. Just plug in. - If you do MC then you can work on the sample directly and calculate quantiles (e.g. VaR) or an empirial estimate of expected shortfall. Apply statistics (method of moments) to the sample. - If you fit a distribution and you know its density then use maximum-likelihood - it does not need the moments to exist.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.