July  2018, 14(3): 1179-1201. 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  July 2018 Early access  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 and Management Optimization, 2018, 14 (3) : 1179-1201. 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. AsmussenF. 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. ChengH. 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. ChesneyM. 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. CossetteD. 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. CzarnaZ. 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. KimH.-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. LandriaultJ.-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. LkabousI. 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. LoeffenI. 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. 

show all references

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. AsmussenF. 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. ChengH. 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. ChesneyM. 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. CossetteD. 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. CzarnaZ. 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. KimH.-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. LandriaultJ.-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. LkabousI. 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. LoeffenI. 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. 

Figure 1.  Illustration of the threshold-based dividend strategy: Standard waiting period
Figure 2.  Illustration of the threshold-based dividend strategy: Cumulative waiting period
Figure 3.  Layering of the recursive algorithm for $V_i(u, m)$
Figure 4.  Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 1
Figure 5.  Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 2
Figure 6.  Plots of $V(10, 80)$ and $\psi(10, 80)$ under Distribution 3
Figure 7.  Plot of $V(10, 80)$ against $(h, b)$ under Distribution 2
Figure 8.  Plot of $V(10, 80)$ against $\psi(10, 80)$ under Distribution 2
Figure 9.  Convergence of $V(10, m)$ under Distribution 2
Figure 10.  Illustration of the generalized threshold-based dividend strategy
Figure 11.  Plot of $V(10,100)$ against $(h, b)$ under the generalized threshold-based dividend strategy
Figure 12.  Plot of $\psi(10,100)$ against $(h, b)$ under the generalized threshold-based dividend strategy
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