Common mistake 1 with Aggregate loss model

Assuming Poisson frequency when the data is over-dispersed; it understates the tail and the capital.

Common mistake 2 with Aggregate loss model

Choosing a discretisation span so coarse that the severity shape is lost, or so fine that the recursion is impractically long.

Common mistake 3 with Aggregate loss model

Reading VaR as the worst case - it is a threshold that is exceeded with the stated probability, and the average breach (TVaR) is materially larger for heavy-tailed severities.

Common mistake 4 with Aggregate loss model

Ignoring policy limits and deductibles when fitting severity. A limited loss distribution has a different mean and a very different tail from the ground-up one, and fitting the limited data as though it were unlimited understates the risk.

EconStatsCraft › Aggregate loss model

Aggregate loss model

The collective risk model: a claim-count distribution (Poisson or negative binomial) and a severity distribution (lognormal, gamma, Pareto or exponential). Reports the mean and standard deviation of the aggregate loss, the full distribution by Panjer recursion, VaR, TVaR and the stop-loss premium, checked against a simulation.

The intuition

An insurer's total annual loss has two independent sources of randomness: how many claims arrive, and how large each one is. Model those separately and combine them, and you get the whole distribution of the year's outcome rather than just its average. The average is the easy part and the least useful; what a capital calculation needs is the far right tail, and that is driven almost entirely by severity.

When to use it

Setting the risk margin or capital for a portfolio, pricing aggregate stop-loss reinsurance, choosing a retention, or answering how bad next year's total claims bill could plausibly be.

Where the estimator comes from

The collective risk model sets S=\sum_{i=1}^{N}X_i with claim count N independent of the i.i.d. severities X_i. Conditioning on N and using the law of total variance gives E[S]=E[N]E[X] and \mathrm{Var}(S)=E[N]\mathrm{Var}(X)+\mathrm{Var}(N)E[X]^2. For a Poisson count with mean \lambda, \mathrm{Var}(N)=\lambda and the two terms collapse to the compact \mathrm{Var}(S)=\lambda\,E[X^2]. The full distribution has no closed form, so it is obtained by Panjer recursion (exact for the (a,b,0) count family on a discretised severity), by fast Fourier transform, or by simulation.

The model

E[S] = E[N]\,E[X],\qquad \operatorname{Var}(S) = E[N]\operatorname{Var}(X) + \operatorname{Var}(N)\,E[X]^2,\qquad f_S(s) = \frac{1}{1-a f_X(0)}\sum_{j=1}^{s}\left(a + \frac{b j}{s}\right) f_X(j)\,f_S(s-j)

What it assumes

Claim sizes are independent of each other and of the number of claims.
The frequency distribution is in the (a, b, 0) class - Poisson, binomial or negative binomial - which is what makes the Panjer recursion exact.
The severity distribution is discretised onto a grid; a fine enough span makes the discretisation error negligible, at the cost of a longer recursion.
The frequency and severity distributions were estimated on data relevant to the period being modelled. Inflation between the observation period and the exposure period shifts severity and is not automatically handled.

Statistical properties

For a compound Poisson, the variance of the total is lambda times the SECOND moment of severity, not the variance. A heavy-tailed severity therefore drives the aggregate variance almost by itself.
The coefficient of variation of the total falls as the square root of the expected claim count. Doubling the portfolio does not halve the relative risk; it divides it by root two. This is the whole argument for pooling.
A Normal approximation to the aggregate is adequate near the mean and poor in the tail, which is the only region a capital calculation cares about.
If claim counts are over-dispersed - variance above mean - a negative binomial count is the standard fix. Assuming Poisson when counts are over-dispersed understates capital.
Panjer recursion is exact given a discretised severity, but discretisation itself introduces bias; a finer grid costs quadratic time.
Reinsurance changes the severity distribution, not the count. An excess-of-loss layer truncates each claim, so it cuts the second moment sharply and therefore cuts aggregate variance far more than it cuts the mean.

How to read the output

The variance of S has two sources: the spread of individual claim sizes, and the uncertainty in how many claims there are. A negative binomial frequency (over-dispersed) widens S noticeably versus Poisson.
The mass at S = 0 is the probability of no claims at all - visible as the spike at the left of the density.
VaR is a quantile of S; TVaR (expected shortfall) is the average of S in the cases beyond VaR, and unlike VaR it is coherent - it never penalises diversification.
The stop-loss premium E[(S - d)+] is the pure cost of a reinsurance layer that pays everything above the retention d.

A worked reading

Poisson claim counts with lambda = 3 and lognormal severity with a mean of 1,000 and a coefficient of variation of 1.5. Panjer recursion gives E[S] = 3,000.00 exactly - frequency mean times severity mean, as it must - with a standard deviation of 3,122.50, so the aggregate loss has a CV above 1: the spread is larger than the mean. Simulation agrees to within a percent on both (3,020.61 and 3,201.09), which is the check worth doing, because the two methods share no arithmetic. The tail is where they start to part company. VaR at 99% is 14,350.00 by recursion and 14,777.13 by simulation, and TVaR is 18,868.46 against 20,380.31 - a 8% gap. That gap is not a bug in either: the simulated tail rests on the handful of draws beyond the 99th percentile and is therefore noisy, while the recursion is exact for the discretised severity and carries a discretisation error instead. For a capital number, quote the recursion and use the simulation to confirm the body.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Simulated mean matching but tail too thin	Too few simulations for the quantile you want	A 99.5% quantile needs tens of thousands of paths at least
Variance far above lambda times mean squared	Correct: it uses the second moment	Not an error
Observed counts more variable than Poisson	Over-dispersion	Switch to negative binomial
Normal approximation far below the simulated quantile	Skewness, as always here	Do not use the Normal in the tail
One severity draw dominating the total	Very heavy tail	Check whether the severity model has finite variance at all
Aggregate distribution bimodal	A mixture of claim types	Model the types separately

Common mistakes

Assuming Poisson frequency when the data is over-dispersed; it understates the tail and the capital.
Choosing a discretisation span so coarse that the severity shape is lost, or so fine that the recursion is impractically long.
Reading VaR as the worst case - it is a threshold that is exceeded with the stated probability, and the average breach (TVaR) is materially larger for heavy-tailed severities.
Ignoring policy limits and deductibles when fitting severity. A limited loss distribution has a different mean and a very different tail from the ground-up one, and fitting the limited data as though it were unlimited understates the risk.

Where to read this properly

Klugman, Panjer and Willmot, Loss Models: From Data to Decisions (2019), ch. 9.
Panjer (1981), "Recursive Evaluation of a Family of Compound Distributions", ASTIN Bulletin 12(1).
Embrechts, Kluppelberg and Mikosch, Modelling Extremal Events (1997), for the heavy-tailed case.

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Aggregate loss model

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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