Stable-½ Bridges and Insurance

To appear in: Advances in Mathematics of Finance (A. Palczewski and L. Stettner, editors.), Banach Center Publications, Polish Academy of Science, Institute of Mathematics.

27 Pages Posted: 11 Apr 2014

See all articles by Edward Hoyle

Edward Hoyle

Man AHL

Lane Hughston

Goldsmiths University of London

Andrea Macrina

University College London; University of Cape Town (UCT)

Date Written: April 9, 2014

Abstract

We develop a class of non-life reserving models using a stable-½ random bridge to simulate the accumulation of paid claims, allowing for an essentially arbitrary choice of a priori distribution for the ultimate loss. Taking an information-based approach to the reserving problem, we derive the process of the conditional distribution of the ultimate loss. The “best-estimate ultimate loss process” is given by the conditional expectation of the ultimate loss. We derive explicit expressions for the best-estimate ultimate loss process, and for expected recoveries arising from aggregate excess-of loss reinsurance treaties. Use of a deterministic time change allows for the matching of any initial (increasing) development pattern for the paid claims. We show that these methods are well-suited to the modelling of claims where there is a non-trivial probability of catastrophic loss. The generalized inverse-Gaussian (GIG) distribution is shown to be a natural choice for the a priori ultimate loss distribution. For particular GIG parameter choices, the best-estimate ultimate loss process can be written as a rational function of the paid-claims process. We extend the model to include a second paid-claims process, and allow the two processes to be dependent. The results obtained can be applied to the modelling of multiple lines of business or multiple origin years. The multi-dimensional model has the property that the dimensionality of calculations remains low, regardless of the number of paid-claims processes. An algorithm is provided for the simulation of the paid-claims processes.

Keywords: non-life reserving, claims development, reinsurance, best estimate of ultimate loss, information-based asset pricing, Levy processes, stable processes

Suggested Citation

Hoyle, Edward and Hughston, Lane and Macrina, Andrea, Stable-½ Bridges and Insurance (April 9, 2014). To appear in: Advances in Mathematics of Finance (A. Palczewski and L. Stettner, editors.), Banach Center Publications, Polish Academy of Science, Institute of Mathematics. , Available at SSRN: https://ssrn.com/abstract=2422919

Edward Hoyle

Man AHL ( email )

Riverbank House
2 Swan Lane
London, EC4R 3AD
United Kingdom

Lane Hughston

Goldsmiths University of London ( email )

London, England SE14 6NW
United Kingdom

Andrea Macrina (Contact Author)

University College London ( email )

Gower Street
London, WC1E 6BT
United Kingdom

University of Cape Town (UCT) ( email )

Private Bag X3
Rondebosch, Western Cape 7701
South Africa

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