Common mistake 1 with Loss severity fitting

Choosing a distribution by how it looks in the middle. The tail drives the cost of a limit or reinsurance layer.

Common mistake 2 with Loss severity fitting

Fitting to claims already reduced by a deductible as if they were ground-up.

Common mistake 3 with Loss severity fitting

Including zero or negative amounts, which every candidate here rules out.

Common mistake 4 with Loss severity fitting

Pricing a high layer from a fit with few large claims; the uncertainty there is large.

EconStatsCraft › Loss severity fitting

Loss severity fitting

Fits five candidate severity distributions to a column of claim amounts by maximum likelihood, ranks them by AIC, then prices a deductible and a limit with the best fit: limited expected values, the loss elimination ratio, and payment per loss and per payment.

What it needs

Claim amounts

The intuition

Insurers need to know not just how often claims arrive but how big they are. Claim sizes are positive and skewed, with many small claims and a few very large ones, and the large ones drive the cost. Fitting a distribution summarises the data and, more importantly, lets you price policy features such as a deductible or a limit that change which part of the distribution the insurer pays for.

When to use it

Pricing a deductible or a limit, building the severity half of an aggregate loss model, reinsurance layer pricing, and the loss models chapters of SOA ASTAM and IFoA CS1 and CS2.

Where the estimator comes from

For each candidate density f(x;\theta), maximum likelihood chooses \hat\theta to maximise \sum_i\log f(x_i;\theta). AIC =2k-2\log L trades fit against the number of parameters k. With the chosen fit, the limited expected value is E[X\wedge m]=\int_0^m S(x)\,dx, where S=1-F is the survival function. With deductible d and limit u, the insurer pays \min(X,u)-\min(X,d), whose mean is E[X\wedge u]-E[X\wedge d]; dividing by S(d) gives the mean per payment.

The model

E[X\wedge d] = \int_0^{d} S(x)\,dx,\qquad \text{payment per loss} = E[X\wedge u]-E[X\wedge d],\qquad \text{LER} = \frac{E[X\wedge d]}{E[X]}

What it assumes

Claims are independent and come from one distribution. Mixed lines of business need separate fits.
Claim amounts are ground-up, not already net of a deductible or capped by a limit; truncated data needs a truncated likelihood.
Inflation and trend have been removed, so old and new claims are comparable.

Statistical properties

Exponential is the one-parameter baseline; gamma and Weibull add shape; lognormal and Pareto have heavier tails.
Of these, only the Pareto has moments that can fail to exist, which matters for high layers.
Payment per payment exceeds payment per loss, because small claims below the deductible are removed.
Inflation raises the payment under a deductible by more than the inflation rate (leveraging).

How to read the output

Choose the distribution by AIC or BIC, which penalise extra parameters, and check the KS distance and the histogram, especially in the right tail.
The loss elimination ratio is the share of expected loss a deductible removes, the basis of the premium discount for taking one.
Payment per payment is larger than payment per loss, because it only counts claims that exceed the deductible.
Pareto and lognormal have heavier tails than gamma and Weibull. When they fit almost equally, the choice matters for large-loss layers.

A worked reading

The actuarial sample's class_loss column holds 205 claims with a mean of 534 and a coefficient of variation of 0.87, generated from gamma distributions. Gamma wins on AIC with Weibull essentially tied, and lognormal is clearly worst: these claims have a lighter tail than lognormal. With a 100 deductible and a 2,000 limit, the deductible removes E[X ∧ 100] = 95.16, a loss elimination ratio of 17.8%. The expected payment is 433.86 per loss and 487.40 per payment, since 89% of claims exceed the deductible.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Two fits almost tie on AIC	The data cannot tell them apart	Look at the right tail and the purpose of the fit
Lognormal fits poorly	Data lighter-tailed than lognormal	Prefer gamma or Weibull
Fit ignores the largest claims	Body dominates the likelihood	Fit the tail separately, e.g. with EVT
Many amounts dropped	Zeros or negatives in the column	Severity uses positive claims only

Common mistakes

Choosing a distribution by how it looks in the middle. The tail drives the cost of a limit or reinsurance layer.
Fitting to claims already reduced by a deductible as if they were ground-up.
Including zero or negative amounts, which every candidate here rules out.
Pricing a high layer from a fit with few large claims; the uncertainty there is large.

Where to read this properly

Klugman, Panjer and Willmot, Loss Models: From Data to Decisions (2019), ch. 3–5 and 13.
IFoA, CS1 and CS2, on loss distributions and reinsurance.
SOA, ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

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Loss severity fitting

What it needs

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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