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