# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018005

## 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

Received  April 2016 Revised  August 2017 Published  January 2018

Fund Project: 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

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.

Citation: Steve Drekic, Jae-Kyung Woo, Ran Xu. A threshold-based risk process with a waiting period to pay dividends. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018005
##### References:
 [1] J. Akahori, Some formulae for a new type of path-dependent option, Annals of Appled Probability, 5 (1995), 383-388. doi: 10.1214/aoap/1177004769. [2] S. Asmussen, F. Avram and M. Usabel, Erlangian approximations for finite-horizon ruin probabilities, ASTIN Bulletin, 32 (2002), 267-281. doi: 10.2143/AST.32.2.1029. [3] A. S. Alfa and S. Drekic, Algorithmic analysis of the Sparre Andersen model in discrete time, ASTIN Bulletin, 37 (2007), 293-317. doi: 10.1017/S0515036100014872. [4] B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251. doi: 10.1080/10920277.2009.10597549. [5] B. Bao, A note on the compound binomial model with randomized dividend strategy, Applied Mathematics and Computation, 194 (2007), 276-286. doi: 10.1016/j.amc.2007.04.023. [6] S. Cheng, H. U. Gerber and E. S. W. Shiu, Discounted probabilities and ruin theory in the compound binomial model, Insurance: Mathematics and Economics, 26 (2000), 239-250. doi: 10.1016/S0167-6687(99)00053-0. [7] M. Chesney, M. Jeanblanc-Picqué and M. Yor, Brownian excursions and Parisian barrier options, Advances in Applied Probability, 29 (1997), 165-184. doi: 10.1017/S000186780002783X. [8] E. C. K. Cheung and J. T. Y. Wong, On the dual risk model with Parisian implementation delays in dividend payments, European Journal of Operational Research, 257 (2017), 159-173. doi: 10.1016/j.ejor.2016.09.018. [9] H. Cossette, D. Landriault and E. Marceau, Ruin probabilities in the discrete time renewal risk model, Insurance: Mathematics and Economics, 38 (2006), 309-323. doi: 10.1016/j.insmatheco.2005.09.005. [10] I. Czarna and Z. Palmowski, Ruin probability with Parisian delay for a spectrally negative Lévy process, Journal of Applied Probability, 48 (2011), 984-1002. doi: 10.1017/S0021900200008573. [11] I. Czarna and Z. Palmowski, Dividend problem with Parisian delay for a spectrally negative Lévy process, Journal of Optimization Theory and Applications, 161 (2014), 239-256. doi: 10.1007/s10957-013-0283-y. [12] I. Czarna, Z. Palmowski and P. Świątek, Discrete time ruin probability with Parisian delay, Scandinavian Actuarial Journal, 2017 (2017), 854-869. doi: 10.1080/03461238.2016.1261734. [13] A. Dassios, The distribution of the quantile of a Brownian motion with drift and the pricing of related path-dependent options, Annals of Appled Probability, 5 (1995), 389-398. doi: 10.1214/aoap/1177004770. [14] A. Dassios and S. Wu, On barrier strategy dividends with Parisian implementation delay for classical surplus processes, Insurance: Mathematics and Economics, 45 (2009), 195-202. doi: 10.1016/j.insmatheco.2009.05.013. [15] B. de Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. [16] D. C. M. Dickson, Some comments on the compound binomial model, ASTIN Bulletin, 24 (1994), 33-45. doi: 10.2143/AST.24.1.2005079. [17] D. C. M. Dickson and H. R. Water, Some optimal dividends problems, ASTIN Bulletin, 34 (2004), 49-74. doi: 10.1017/S0515036100013878. [18] S. Drekic and A. M. Mera, Ruin analysis of a threshold strategy in a discrete-time Sparre Andersen Model, Methodology and Computing in Applied Probability, 13 (2011), 723-747. doi: 10.1007/s11009-010-9184-9. [19] H. U. Gerber, Mathematical fun with compound binomial process, ASTIN Bulletin, 18 (1988), 161-168. doi: 10.2143/AST.18.2.2014949. [20] H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78. doi: 10.1080/10920277.1998.10595671. [21] S.S. Kim and S. Drekic, Ruin analysis of a discrete-time dependent Sparre Andersen model with external financial activities and randomized dividends, Risks, 4 (2016), p2. doi: 10.3390/risks4010002. [22] B. Kim, H.-S. Kim and J. Kim, A risk model with paying dividends and random environment, Insurance: Mathematics and Economics, 42 (2008), 717-726. doi: 10.1016/j.insmatheco.2007.08.001. [23] D. Landriault, Randomized dividends in the compound binomial model with a general premium rate, Scandinavian Actuarial Journal, 2008 (2008), 1-15. [24] D. Landriault, J.-F. Renaud and X. Zhou, An insurance risk model with Parisian implementation delays, Methodology and Computing in Applied Probability, 16 (2014), 583-607. doi: 10.1007/s11009-012-9317-4. [25] M. A. Lkabous, I. Czarna and J.-F. Renaud, Parisian ruin for a refracted Lévy process, Insurance: Mathematics and Economics, 74 (2017), 153-163. doi: 10.1016/j.insmatheco.2017.03.005. [26] S. Li, On a class of discrete time renewal risk models, Scandinavian Actuarial Journal, 2005 (2005), 241-260. [27] S. Li, Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time renewal risk models, Scandinavian Actuarial Journal, 2005 (2005), 271-284. [28] R. Loeffen, I. Czarna and Z. Palmowski, Parisian ruin probability for spectrally negative Lévy process, Bernoulli, 19 (2013), 599-609. doi: 10.3150/11-BEJ404. [29] K. P. Pavlova and G. E. Willmot, The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function, Insurance: Mathematics and Economics, 35 (2004), 267-277. doi: 10.2143/AST.32.2.1029. [30] A. Pechtl, Some applications of occupation times of Brownian motion with drift in mathematical finance, Journal of Applied Mathematics and Decision Sciences, 3 (1999), 63-73. doi: 10.1155/S1173912699000048. [31] E. S. W. Shiu, The probability of eventual ruin in the compound binomial model, ASTIN Bulletin, 19 (1989), 179-190. doi: 10.2143/AST.19.2.2014907. [32] D. W. Sommer, The impact of firm risk on property-liability insurance prices, Journal of Risk and Insurance, 63 (1996), 501-514. doi: 10.2307/253623. [33] J. Tan and X. Yang, The compound binomial model with randomized decisions on paying dividends, Insurance: Mathematics and Economics, 39 (2006), 1-18. doi: 10.1016/j.insmatheco.2006.01.001. [34] G. Venter and A. Underwood, Value of risk reduction, Casualty Actuary Society E-Forum, 2 (2012), 1-19. [35] G. E. Willmot, Ruin probabilities in the compound binomial model, Insurance: Mathematics and Economics, 12 (1993), 133-142. doi: 10.1016/0167-6687(93)90823-8. [36] J. T. Y. Wong and E. C. K. Cheung, On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps, Insurance: Mathematics and Economics, 65 (2015), 280-290. doi: 10.1016/j.insmatheco.2015.10.001. [37] J.-K. Woo, A generalized penalty function for a class of discrete renewal processes, Scandinavian Actuarial Journal, 2012 (2012), 130-152. [38] X. Wu and S. Li, On the discounted penalty function in a discrete time renewal risk model with general interclaim times, Scandinavian Actuarial Journal, 2009 (2009), 281-294.

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##### References:
 [1] J. Akahori, Some formulae for a new type of path-dependent option, Annals of Appled Probability, 5 (1995), 383-388. doi: 10.1214/aoap/1177004769. [2] S. Asmussen, F. Avram and M. Usabel, Erlangian approximations for finite-horizon ruin probabilities, ASTIN Bulletin, 32 (2002), 267-281. doi: 10.2143/AST.32.2.1029. [3] A. S. Alfa and S. Drekic, Algorithmic analysis of the Sparre Andersen model in discrete time, ASTIN Bulletin, 37 (2007), 293-317. doi: 10.1017/S0515036100014872. [4] B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251. doi: 10.1080/10920277.2009.10597549. [5] B. Bao, A note on the compound binomial model with randomized dividend strategy, Applied Mathematics and Computation, 194 (2007), 276-286. doi: 10.1016/j.amc.2007.04.023. [6] S. Cheng, H. U. Gerber and E. S. W. Shiu, Discounted probabilities and ruin theory in the compound binomial model, Insurance: Mathematics and Economics, 26 (2000), 239-250. doi: 10.1016/S0167-6687(99)00053-0. [7] M. Chesney, M. Jeanblanc-Picqué and M. Yor, Brownian excursions and Parisian barrier options, Advances in Applied Probability, 29 (1997), 165-184. doi: 10.1017/S000186780002783X. [8] E. C. K. Cheung and J. T. Y. Wong, On the dual risk model with Parisian implementation delays in dividend payments, European Journal of Operational Research, 257 (2017), 159-173. doi: 10.1016/j.ejor.2016.09.018. [9] H. Cossette, D. Landriault and E. Marceau, Ruin probabilities in the discrete time renewal risk model, Insurance: Mathematics and Economics, 38 (2006), 309-323. doi: 10.1016/j.insmatheco.2005.09.005. [10] I. Czarna and Z. Palmowski, Ruin probability with Parisian delay for a spectrally negative Lévy process, Journal of Applied Probability, 48 (2011), 984-1002. doi: 10.1017/S0021900200008573. [11] I. Czarna and Z. Palmowski, Dividend problem with Parisian delay for a spectrally negative Lévy process, Journal of Optimization Theory and Applications, 161 (2014), 239-256. doi: 10.1007/s10957-013-0283-y. [12] I. Czarna, Z. Palmowski and P. Świątek, Discrete time ruin probability with Parisian delay, Scandinavian Actuarial Journal, 2017 (2017), 854-869. doi: 10.1080/03461238.2016.1261734. [13] A. Dassios, The distribution of the quantile of a Brownian motion with drift and the pricing of related path-dependent options, Annals of Appled Probability, 5 (1995), 389-398. doi: 10.1214/aoap/1177004770. [14] A. Dassios and S. Wu, On barrier strategy dividends with Parisian implementation delay for classical surplus processes, Insurance: Mathematics and Economics, 45 (2009), 195-202. doi: 10.1016/j.insmatheco.2009.05.013. [15] B. de Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. [16] D. C. M. Dickson, Some comments on the compound binomial model, ASTIN Bulletin, 24 (1994), 33-45. doi: 10.2143/AST.24.1.2005079. [17] D. C. M. Dickson and H. R. Water, Some optimal dividends problems, ASTIN Bulletin, 34 (2004), 49-74. doi: 10.1017/S0515036100013878. [18] S. Drekic and A. M. Mera, Ruin analysis of a threshold strategy in a discrete-time Sparre Andersen Model, Methodology and Computing in Applied Probability, 13 (2011), 723-747. doi: 10.1007/s11009-010-9184-9. [19] H. U. Gerber, Mathematical fun with compound binomial process, ASTIN Bulletin, 18 (1988), 161-168. doi: 10.2143/AST.18.2.2014949. [20] H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78. doi: 10.1080/10920277.1998.10595671. [21] S.S. Kim and S. Drekic, Ruin analysis of a discrete-time dependent Sparre Andersen model with external financial activities and randomized dividends, Risks, 4 (2016), p2. doi: 10.3390/risks4010002. [22] B. Kim, H.-S. Kim and J. Kim, A risk model with paying dividends and random environment, Insurance: Mathematics and Economics, 42 (2008), 717-726. doi: 10.1016/j.insmatheco.2007.08.001. [23] D. Landriault, Randomized dividends in the compound binomial model with a general premium rate, Scandinavian Actuarial Journal, 2008 (2008), 1-15. [24] D. Landriault, J.-F. Renaud and X. Zhou, An insurance risk model with Parisian implementation delays, Methodology and Computing in Applied Probability, 16 (2014), 583-607. doi: 10.1007/s11009-012-9317-4. [25] M. A. Lkabous, I. Czarna and J.-F. Renaud, Parisian ruin for a refracted Lévy process, Insurance: Mathematics and Economics, 74 (2017), 153-163. doi: 10.1016/j.insmatheco.2017.03.005. [26] S. Li, On a class of discrete time renewal risk models, Scandinavian Actuarial Journal, 2005 (2005), 241-260. [27] S. Li, Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time renewal risk models, Scandinavian Actuarial Journal, 2005 (2005), 271-284. [28] R. Loeffen, I. Czarna and Z. Palmowski, Parisian ruin probability for spectrally negative Lévy process, Bernoulli, 19 (2013), 599-609. doi: 10.3150/11-BEJ404. [29] K. P. Pavlova and G. E. Willmot, The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function, Insurance: Mathematics and Economics, 35 (2004), 267-277. doi: 10.2143/AST.32.2.1029. [30] A. Pechtl, Some applications of occupation times of Brownian motion with drift in mathematical finance, Journal of Applied Mathematics and Decision Sciences, 3 (1999), 63-73. doi: 10.1155/S1173912699000048. [31] E. S. W. Shiu, The probability of eventual ruin in the compound binomial model, ASTIN Bulletin, 19 (1989), 179-190. doi: 10.2143/AST.19.2.2014907. [32] D. W. Sommer, The impact of firm risk on property-liability insurance prices, Journal of Risk and Insurance, 63 (1996), 501-514. doi: 10.2307/253623. [33] J. Tan and X. Yang, The compound binomial model with randomized decisions on paying dividends, Insurance: Mathematics and Economics, 39 (2006), 1-18. doi: 10.1016/j.insmatheco.2006.01.001. [34] G. Venter and A. Underwood, Value of risk reduction, Casualty Actuary Society E-Forum, 2 (2012), 1-19. [35] G. E. Willmot, Ruin probabilities in the compound binomial model, Insurance: Mathematics and Economics, 12 (1993), 133-142. doi: 10.1016/0167-6687(93)90823-8. [36] J. T. Y. Wong and E. C. K. Cheung, On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps, Insurance: Mathematics and Economics, 65 (2015), 280-290. doi: 10.1016/j.insmatheco.2015.10.001. [37] J.-K. Woo, A generalized penalty function for a class of discrete renewal processes, Scandinavian Actuarial Journal, 2012 (2012), 130-152. [38] X. Wu and S. Li, On the discounted penalty function in a discrete time renewal risk model with general interclaim times, Scandinavian Actuarial Journal, 2009 (2009), 281-294.
Illustration of the threshold-based dividend strategy: Standard waiting period
Illustration of the threshold-based dividend strategy: Cumulative waiting period
Layering of the recursive algorithm for $V_i(u, m)$
Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 1
Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 2
Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 3
Plot of $V(10, 80)$ against $(h, b)$ under Distribution 2
Plot of $V(10, 80)$ against $\psi(10, 80)$ under Distribution 2
Convergence of $V(10, m)$ under Distribution 2
Illustration of the generalized threshold-based dividend strategy
Plot of $V(10,100)$ against $(h, b)$ under the generalized threshold-based dividend strategy
Plot of $\psi(10,100)$ against $(h, b)$ under the generalized threshold-based dividend strategy
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