The Statistical Distribution of Incurred Losses and Its Evolution Over Time Ill: Dynamic Models

Casualty Actuarial Society, 2000

38 Pages Posted: 14 Nov 2015

See all articles by Greg Taylor

Greg Taylor

UNSW Australia Business School, School of Risk & Actuarial Studies

Date Written: November 1, 2000

Abstract

This paper is written at the request of, and is partly funded by, the Casualty Actuarial Society's Committee on Theory of Risk. It is the third of a trio of papers whose purpose is to answer the following question, posed by the Committee: Assume you know the aggregate loss distribution at policy inception and you have expected patterns of claims reporting, losses emerging and losses paid and other pertinent information, how do you modify the distribution as the policy matures and more information becomes available? Actuaries have historically dealt with the problem of modifying the expectation conditional on emerged information. This expands the problem to continuously modifying the whole distribution from inception until it decays to a point. One might expect that there are at least two separate states that are important. There is the exposure state. It is during this period that claims can attach to the policy. Once this period is over no new claims can attach. The second state is the discovery or development state. In this state claims that already attached to the policy can become known and their value can begin developing. These two states may have to be treated separately. In general terms, this brief requires the extension of conventional point estimation of incurred losses to their companion distributions. Specifically, the evolution of this distribution over time is required as the relevant period of origin matures. Expressed in this way, the problem takes on a natural Bayesian form. For any particular year of origin (the generic name for an accident year, underwriting year, etc), one begins with a prior distribution of incurred losses, which applies in advance of data collection. As the period of origin develops, loss data accumulate, and may be used for progressive Bayesian revision of the prior. When the period of origin is fully mature, the amount of incurred losses is known with certainty. The Bayesian revision of the prior is then a single point distribution. The present paper addresses the question of how the Bayesian revision of the prior evolves over time from the prior itself to the final degenerate distribution. This evolution can take two distinct forms. On the one hand, one may impose no restrictions on the posterior distributions arising from the Bayesian revisions. These posterior distributions will depend on the empirical distributions of certain observations. Such models are non-parametric. Alternatively, the posterior distributions may be assumed to come from some defined family. For example, it may be assumed that the posterior-to-data distribution of incurred losses, as assessed at a particular point of development of the period of origin, is log normal. Any estimation questions must relate to the parameters which define the distribution within the chosen family. These are parametric models. They are, in certain respects, more flexible than non-parametric, but lead to quite different estimation procedures. The first paper (Taylor 1999a) dealt with non-parametric models only. The second (Taylor 1999b) dealt with parametric models. The present paper addresses the case of dynamic models, in which parameters are allowed to evolve from one period of origin to the next. Familiarity with the earlier papers will be assumed here. In particular, the Bayesian background introduced and described there will be assumed. As far as possible, the notation used here will be common with the earlier papers.

JEL Classification: C11, C14, C61, D81

Suggested Citation

Taylor, Greg, The Statistical Distribution of Incurred Losses and Its Evolution Over Time Ill: Dynamic Models (November 1, 2000). Casualty Actuarial Society, 2000, Available at SSRN: https://ssrn.com/abstract=2661197

Greg Taylor (Contact Author)

UNSW Australia Business School, School of Risk & Actuarial Studies ( email )

Level 6, East Lobby
UNSW Business School Building, UNSW
Sydney, NSW 2052
Australia
+61 421 338 448 (Phone)

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