Actuarial and Quantitative Risk
The Global Risks Alliance (GRA)
A loss distribution describes how often losses occur and how large they are, for one policy, a portfolio or a whole line of business. Actuaries combine frequency and severity models into aggregate loss distributions that drive premiums, reserves, reinsurance and capital. Choices about data, distribution family and the tail change results substantially. This page sets out how loss distributions are built, where they mislead, the data behind them and who signs off.
GRA · In development · service level R0
Loss distributions are statistical descriptions of how often losses occur and how large they are, combined into aggregate distributions for pricing, reserving, reinsurance and capital. Data adjustment, distribution choice and tail treatment decide results. Pricing and reserving actuaries, chief actuaries, reinsurance buyers and supervisors hold the decisions.
Which places drive loss in a severe year, and how far it reaches into reinsurance. Illustrative design values.
Frequency, severity and aggregate loss, and how tails are estimated. The dossier below sets out how the risk develops, what is measured, where the data come from, how it is modelled and who decides.
Claim counts and claim sizes are modelled separately and combined. Each depends on exposure, risk factors and trends that must be projected to the period being priced.
Most claims are small, but capital and reinsurance depend on rare large losses. A distribution that fits the body well can understate the tail badly.
Past losses must be adjusted for inflation, exposure growth, policy changes and claims practice. Errors in trending carry straight into prices.
Deductibles hide small losses and policy limits cap large ones. Fitting without accounting for this biases both frequency and severity.
Each term as this page uses it. Where a standard-setting body defines the term, the wording follows that body.
| Term | Definition |
|---|---|
| Frequency distribution | A probability distribution of the number of claims in a period. |
| Severity distribution | A probability distribution of the size of individual claims, given that a claim occurs. |
| Censoring | Partial observation of losses, such as when policy limits cap the recorded amount. |
Each indicator carries its unit, its cadence, the kind of source behind it and the evidence label a value would carry. No values are shown: a programme supplies them under its own authority.
| Indicator | Unit | Cadence | Source type | Label |
|---|---|---|---|---|
| Claim frequency | claims per unit of exposure | annual | Claims and exposure records | OBSERVED |
| Mean claim severity | currency per claim | annual | Claims records | OBSERVED |
| Coefficient of variation of aggregate loss | dimensionless | per analysis | Fitted distributions | MODELLED |
| Large-loss share of total | percent of incurred losses | annual | Claims records | OBSERVED |
| Severity trend | annual change in mean severity, percent | annual | Claims analysis | ESTIMATED |
| Goodness-of-fit statistics | test statistic by distribution | per analysis | Model fitting | ESTIMATED |
The topic as a computational graph: named sources feed the methods, an evidence review gates the outputs, and the decisions stay with the authorities that hold them.
Fig. 02Nodes carry their type, two metric rows and a state; connectors show what feeds what; dashed frames name the owners; the timeline is the cadence. Design of the pipeline, not a record of a run.
Named public datasets and institutional records, each with its licence class. Sources by agreement are used only under their terms and never republished.
| Source | What it gives | Licence class |
|---|---|---|
| Casualty Actuarial Society monographs and study notes | Methods for loss distributions and ratemaking | Free to read, CAS terms |
| Insurance supervisory statistical returns | Aggregate claims and premiums by line | Public records, terms per supervisor |
| R actuar and fitdistrplus packages | Open-source tools for fitting loss distributions | Open source, GPL |
| Society of Actuaries research and data | Experience studies and methods | Free to read, SOA terms |
| OECD Global Insurance Statistics | Market claims and premium aggregates | Free, OECD terms |
Actuarial and Quantitative Risk prepares evidence. The decisions below are taken by the authorities named, under their own rules, and never by the platform.
| Who decides | The question | Evidence this page serves |
|---|---|---|
| Pricing and reserving actuaries | Which distributions and trends to use | Data, fit diagnostics and professional standards |
| Chief actuaries and risk committees | Whether loss models are fit for pricing and capital | Validation and sensitivity evidence |
| Reinsurance buyers | How much tail protection to buy | Aggregate loss distributions with uncertainty |
| Supervisors | Whether loss modelling supports reserves and capital | Actuarial reports and reviews |
The classical collective risk model separates claim frequency, often modelled with Poisson or negative binomial distributions, from claim severity, modelled with distributions such as lognormal, gamma, Pareto or Weibull. Aggregate loss is the sum of a random number of random severities, computed through recursive methods, fast Fourier transforms or simulation. For catastrophe risk, event-based models simulate many possible years of events and their losses to build the aggregate distribution.
The tail is where choices matter most. Heavy-tailed severity distributions, common in property catastrophe, liability, cyber and operational risk, mean that the largest losses dominate the total, and that sample averages can be unstable. Small changes in the assumed tail shape can change estimated capital needs substantially. Parameter uncertainty, the fact that distribution parameters are themselves estimated from limited data, adds a further layer that is often neglected.
Inflation, changes in exposure and changes in the legal or physical environment mean that historical losses must be adjusted before they can inform the future, and each adjustment carries its own assumptions.
Maximum likelihood and Bayesian fitting, with goodness-of-fit tests that are weak in the tail.
Simulating aggregate losses; tail precision needs many simulations and careful convergence checks.
Bayesian and bootstrap methods to carry parameter uncertainty into results.
Limits: the tail is estimated from few observations, so different defensible models can give very different answers. Results should be reported with their sensitivity to model choice.
The current service level is R0: information and enquiries. An enquiry receives a reference and names the next step; applications, payments, compute allocations and registrations open only when their team, terms and review capacity exist.