A threshold-based risk process with a waiting period to pay dividends

Steve Drekic; Jae-Kyung Woo; Ran Xu

doi:10.3934/jimo.2018005

Article Contents

2018, Volume 14, Issue 3: 1179-1201. Doi: 10.3934/jimo.2018005

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A threshold-based risk process with a waiting period to pay dividends

1.
Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada
2.
School of Risk and Actuarial Studies, Australian School of Business, University of New South Wales, Australia
3.
Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong, China

* Corresponding author: Jae-Kyung Woo
* Corresponding author: Jae-Kyung Woo

Received: April 2016

Revised: August 2017

Early access: January 15, 2018

Published online: January 15, 2018

This work has been supported by the Natural Sciences and Engineering Research Council of Canada, through the Discovery grant (#238675-2010-RGPIN) of Dr. Drekic.

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Abstract

In this paper, a modified dividend strategy is proposed by delaying dividend payments until the insurer's surplus level remains at or above a threshold level b for a predetermined period of time h. We consider two cases depending on whether the period of time sustained at or above level b is counted either consecutively or accumulatively (referred to as standard or cumulative waiting period). In both cases, we develop a recursive computational procedure to calculate the expected total discounted dividend payments made prior to ruin for a discrete-time Sparre Andersen renewal risk process. By varying the values of b and h, a numerical study of the trade-off effects between finite-time ruin probabilities and expected total discounted dividend payments is investigated under a variety of scenarios. Finally, a generalized threshold-based strategy with a delayed dividend payment rule is studied under the compound binomial model.

Keywords:

Discrete-time Sparre Andersen renewal risk process,
threshold strategy,
waiting period,
dividend payments,
ruin probabilities,
Parisian-type model,
compound binomial model.

Mathematics Subject Classification: Primary: 60G50, 60K05; Secondary: 91B30, 62P05.

Citation:

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Figure 1. Illustration of the threshold-based dividend strategy: Standard waiting period

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Figure 2. Illustration of the threshold-based dividend strategy: Cumulative waiting period

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Figure 3. Layering of the recursive algorithm for $V_i(u, m)$

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Figure 4. Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 1

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Figure 5. Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 2

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Figure 6. Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 3

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Figure 7. Plot of $V(10, 80)$ against $(h, b)$ under Distribution 2

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Figure 8. Plot of $V(10, 80)$ against $\psi(10, 80)$ under Distribution 2

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Figure 9. Convergence of $V(10, m)$ under Distribution 2

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Figure 10. Illustration of the generalized threshold-based dividend strategy

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Figure 11. Plot of $V(10,100)$ against $(h, b)$ under the generalized threshold-based dividend strategy

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Figure 12. Plot of $\psi(10,100)$ against $(h, b)$ under the generalized threshold-based dividend strategy

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