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2721 results in General statistics and probability

Page 42 of 137

Lectures on the Poisson Process
  • Günter Last, Mathew Penrose
  • Published online:
    26 October 2017
    Print publication:
    26 October 2017
  • The Poisson process, a core object in modern probability, enjoys a richer theory than is sometimes appreciated. This volume develops the theory in the setting of a general abstract measure space, establishing basic results and properties as well as certain advanced topics in the stochastic analysis of the Poisson process. Also discussed are applications and related topics in stochastic geometry, including stationary point processes, the Boolean model, the Gilbert graph, stable allocations, and hyperplane processes. Comprehensive, rigorous, and self-contained, this text is ideal for graduate courses or for self-study, with a substantial number of exercises for each chapter. Mathematical prerequisites, mainly a sound knowledge of measure-theoretic probability, are kept in the background, but are reviewed comprehensively in the appendix. The authors are well-known researchers in probability theory; especially stochastic geometry. Their approach is informed both by their research and by their extensive experience in teaching at undergraduate and graduate levels.

  • 6 - Selected Additional Reserving Topics

  • David Hindley
  • Book:
    Claims Reserving in General Insurance
    Published online:
    27 October 2017
    Print publication:
    26 October 2017, pp 346-409

  • Summary

    Introduction

    This chapter contains a range of additional reserving topics that are not covered elsewhere in the other chapters, including:

  • A section covering the estimation of outwards reinsurance.

  • A section describing the use of roll-forward and Actual vs Expected procedures.

  • A number of sections covering the approaches used to estimate reserves for certain specific types or category of reserve, specifically Unallocated Loss Adjustment Expenses, reinsurance bad debt and latent claims.

  • A section that covers discounting of reserves for the time value of money.

  • A section that summarises reserving under Solvency II.

  • A final section covering a number of miscellaneous topics (e.g. using external data sources and deriving Unearned Premium reserves).

    • Outwards Reinsurance

    This section considers the approaches that can be used for estimation of outwards reinsurance in a reserving exercise, where the purpose of the exercise is assumed to be the derivation of a point estimate of the reserves, rather than a distribution of future claims. Outwards reinsurance may also be referred to as “ceded” reinsurance in some contexts.

    Allowing for outwards reinsurance across a full distribution is considered in the stochastic methods chapter. In particular, some of the approaches that can be used are discussed in the context of the bootstrap method in the outwards reinsurance subsection of Section 4.4.5, and these are also applicable to a number of other stochastic methods. In some cases, even where the principal aim of the exercise is to establish the allowance for outwards reinsurance in respect of a single point estimate, stochastic methods can be helpful, because they can be used to make allowance across the full range of possible future gross claims outgo. Depending on the nature of the outwards reinsurance, this may produce a more accurate estimate of the outwards reinsurance reserve than, for example, netting down a single point estimate of the reserve.

    The reserving of inwards reinsurance business (i.e. reinsurance business written by a company) is a different topic. However, the methods that can be used are generally similar to those for insurance business, with some specific approaches being used, where appropriate. Some of the specific reserving contexts covered in Chapter 7 make reference to such approaches.

  • 4 - Stochastic Reserving Methods

  • David Hindley
  • Book:
    Claims Reserving in General Insurance
    Published online:
    27 October 2017
    Print publication:
    26 October 2017, pp 146-319

  • Summary

    Introduction

    Stochastic reserving methods provide both a point estimate (e.g. a best estimate) of the future claims and an estimate of the variability surrounding that point estimate. Some of these methods also provide an estimate of the full distribution of future claims and their associated cash flows. In most practical reserving situations, if a point estimate for the reserves is required (e.g. as part of a regular reserving exercise for financial reporting purposes) this will often be determined using a combination of deterministic, as opposed to stochastic, reserving methods. Although there is no reason why stochastic methods cannot be used to derive the point estimate, in practice this is not usually the case, at least at present. Despite this, the majority of insurers will be making some use of stochastic reserving methods, in one form or another, and hence an understanding of the different methods available and their limitations is vital for most insurers.

    This chapter provides a description of several stochastic reserving methods, including those that are commonly used in practice (at least in the UK in 2017), along with worked examples of these methods. There is also a brief description of other less commonly used methods and to some more recently developed approaches. Although some details of the mathematical formulation of a number of methods is included, this chapter is not intended to represent a thorough analysis of the underlying theoretical basis for the methods that are described here, but rather an overview of some of the commonly used methods. Readers who require more details of the underlying theory will find many sources in the extensive body of material that is available on stochastic reserving. Several of these sources are referred to in this chapter.

    Most of the stochastic claims reserving techniques that have been developed are designed to be used where the data triangle structure is an appropriate one to use in order to estimate reserves. As discussed in Section 2.5, there are some instances where the triangle format is not the most appropriate format to use.

Page 42 of 137