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A plan of capital injections based on the claims frequency

Published online by Cambridge University Press:  20 June 2018

Ran Xu ,
Jae-Kyung Woo ,
Xixuan Han  and
Hailiang Yang
Ran Xu
Affiliation:
Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong
Jae-Kyung Woo*
Affiliation:
School of Risk and Actuarial Studies, University of New South Wales, Sydney, NSW, Australia
Xixuan Han
Affiliation:
Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong
Hailiang Yang
Affiliation:
Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong
*
*Correspondence to: Jae-Kyung Woo, School of Risk and Actuarial Studies, University of New South Wales, Room 574, Level 5, East Wing, UNSW Business School building, Kensington Campus, UNSW, Sydney NSW 2052, Australia. Tel: +61 2 9385 7559. E-mail: [email protected]
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Abstract

In this work, we propose a capital injection strategy which is periodically implemented based on the number of claims in the classical Poisson risk model. Especially, capital injection decisions are made at a predetermined accumulated number of claim instants, if the surplus is lower than a minimum required level. There appears to be a similar problem found in reliability theory such that preventive maintenance policies are performed at certain shock numbers. Assuming a combination of exponentials for the claim severities, we first derive an explicit expression for the discounted density of the surplus level after a certain number of claims if ruin has not yet occurred. Utilising this result, we study the expected total discounted capital injection until the first ruin time. To solve the differential equation associated with this quantity, we analyse an extended Lundberg’s fundamental equation. Similarly, an expression for the Laplace transform of the time to ruin is also explicitly found. Finally, we illustrate the applicability of the present capital injection strategy and methodologies through various numerical examples. In particular, for exponential claim severities, some optimal capital injection strategy which minimises the expected capital spending per unit time is numerically studied.

Keywords

Preventive maintenanceNumber of claimsClassical Poisson risk modelExpected total discounted capital injectionLaplace transform of time to ruin
Type
Paper
Information
Annals of Actuarial Science , Volume 12 , Issue 2 , September 2018 , pp. 296 - 325
Copyright
© Institute and Faculty of Actuaries 2018 

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