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Two-stage mean-risk stochastic mixed integer optimization model for location-allocation problems under uncertain environment
General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes
1. | School of Mathematical Sciences, Xiamen University, Fujian 361005, China |
2. | Department of Statistics and Actuarial Science, Xi'an Jiaotong-Liverpool University, Suzhou, 215123, China |
For spectrally negative Lévy risk processes we consider a generalized version of the De Finetti's optimal dividend problem with fixed transaction costs, where the ruin time is replaced by a general drawdown time in the framework. We identify a condition under which a band–type impulse dividend strategy is optimal among all admissible impulse strategies. As a consequence, we are able to extend the previous results on ruin time based impulse dividend optimization problem to those on drawdown time based impulse dividend optimization problems. A new type of drawdown function is proposed at end, and various numerical examples are presented to illustrate the existence of those optimal impulse dividend strategies under different assumptions.
References:
[1] |
S. Asmussen, F. Avram and M. R. Pistorius,
Russian and american put options under exponential phase-type lévy models, Stochastic Processes and their Applications, 109 (2004), 79-111.
doi: 10.1016/j.spa.2003.07.005. |
[2] |
S. Asmussen and M. Taksar,
Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.
doi: 10.1016/S0167-6687(96)00017-0. |
[3] |
F. Avram, A. Kyprianouy and M. Pistoriusz,
Exit problems for spectrally negative l evy processes and applications to russian, american and canadized options, Ann. Appl. Probab, 14 (2004), 215-238.
doi: 10.1214/aoap/1075828052. |
[4] |
F. Avram, Z. Palmowski and M. R. Pistorius,
On the optimal dividend problem for a spectrally negative lévy process, The Annals of Applied Probability, 17 (2007), 156-180.
doi: 10.1214/105051606000000709. |
[5] |
F. Avram, Z. Palmowski and M. R. Pistorius,
On gerber–shiu functions and optimal dividend distribution for a lévy risk process in the presence of a penalty function, The Annals of Applied Probability, 25 (2015), 1868-1935.
doi: 10.1214/14-AAP1038. |
[6] |
F. Avram, N. L. Vu and X. Zhou,
On taxed spectrally negative lévy processes with draw-down stopping, Insurance: Mathematics and Economics, 76 (2017), 69-74.
doi: 10.1016/j.insmatheco.2017.06.005. |
[7] |
J. Azéma and M. Yor, Une solution simple au probleme de skorokhod, in Séminaire de Probabilités XIII, Springer, 721 (1979), 90–115. |
[8] |
E. J. Baurdoux, Z. Palmowski and M. R. Pistorius,
On future drawdowns of lévy processes, Stochastic Processes and their Applications, 127 (2017), 2679-2698.
doi: 10.1016/j.spa.2016.12.008. |
[9] |
E. Bayraktar, A. E. Kyprianou and K. Yamazaki,
On optimal dividends in the dual model, ASTIN Bulletin: The Journal of the IAA, 43 (2013), 359-372.
doi: 10.1017/asb.2013.17. |
[10] |
J. Bertoin, Lévy Processes, vol. 121, Cambridge university press Cambridge, 1996. |
[11] |
G. Burghardt and R. Duncan, Deciphering drawdown, Risk management for investors, September, S16–S20. Google Scholar |
[12] |
P. Carr, First-order calculus and option pricing, Journal of Financial Engineering, 1 (2014), 1450009.
doi: 10.1142/s2345768614500093. |
[13] |
B. De Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, in Transactions of the XVth international congress of Actuaries, vol. 2, New York, 1957,433–443. Google Scholar |
[14] |
D. C. Dickson and H. R. Waters,
Some optimal dividends problems, ASTIN Bulletin: The Journal of the IAA, 34 (2004), 49-74.
doi: 10.1017/S0515036100013878. |
[15] |
B. HøJgaard and M. Taksar,
Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.
doi: 10.1111/1467-9965.00066. |
[16] |
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Elsevier, 2014. Google Scholar |
[17] |
J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288, Springer-Verlag, Berlin, 1987.
doi: 10.1007/978-3-662-02514-7. |
[18] |
A. E. Kyprianou, R. Loeffen and J.-L. Pérez,
Optimal control with absolutely continuous strategies for spectrally negative lévy processes, Journal of Applied Probability, 49 (2012), 150-166.
doi: 10.1239/jap/1331216839. |
[19] |
A. E. Kyprianou and Z. Palmowski,
Distributional study of de finetti's dividend problem for a general lévy insurance risk process, Journal of Applied Probability, 44 (2007), 428-443.
doi: 10.1239/jap/1183667412. |
[20] |
D. Landriault, B. Li and S. Li,
Analysis of a drawdown-based regime-switching lévy insurance model, Insurance: Mathematics and Economics, 60 (2015), 98-107.
doi: 10.1016/j.insmatheco.2014.11.005. |
[21] |
D. Landriault, B. Li and S. Li,
Drawdown analysis for the renewal insurance risk process, Scandinavian Actuarial Journal, 2017 (2017), 267-285.
doi: 10.1080/03461238.2015.1123174. |
[22] |
J. P. Lehoczky,
Formulas for stopped diffusion processes with stopping times based on the maximum, The Annals of Probability, 5 (1977), 601-607.
doi: 10.1214/aop/1176995770. |
[23] |
B. Li, N. L. Vu and X. Zhou,
Exit problems for general draw-down times of spectrally negative lévy processes, Journal of Applied Probability, 56 (2019), 441-457.
doi: 10.1017/jpr.2019.31. |
[24] |
R. L. Loeffen,
On optimality of the barrier strategy in de finetti's dividend problem for spectrally negative lévy processes, The Annals of Applied Probability, 18 (2008), 1669-1680.
doi: 10.1214/07-AAP504. |
[25] |
R. L. Loeffen,
An optimal dividends problem with a terminal value for spectrally negative lévy processes with a completely monotone jump density, Journal of Applied Probability, 46 (2009), 85-98.
doi: 10.1017/S0021900200005246. |
[26] |
R. L. Loeffen,
An optimal dividends problem with transaction costs for spectrally negative lévy processes, Insurance: Mathematics and Economics, 45 (2009), 41-48.
doi: 10.1016/j.insmatheco.2009.03.002. |
[27] |
R. L. Loeffen and J.-F. Renaud,
De finetti's optimal dividends problem with an affine penalty function at ruin, Insurance: Mathematics and Economics, 46 (2010), 98-108.
doi: 10.1016/j.insmatheco.2009.09.006. |
[28] |
D. B. Madan and M. Yor,
Making markov martingales meet marginals: With explicit constructions, Bernoulli, 8 (2002), 509-536.
|
[29] |
X. Peng, M. Chen and J. Guo,
Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.
doi: 10.1016/j.insmatheco.2012.08.004. |
[30] |
G. Peskir,
Designing options given the risk: The optimal skorokhod-embedding problem, Stochastic Processes and Their Applications, 81 (1999), 25-38.
doi: 10.1016/S0304-4149(98)00097-0. |
[31] |
J.-F. Renaud and X. Zhou,
Distribution of the present value of dividend payments in a lévy risk model, Journal of Applied Probability, 44 (2007), 420-427.
doi: 10.1239/jap/1183667411. |
[32] |
N. Scheer and H. Schmidli,
Optimal dividend strategies in a cramer–lundberg model with capital injections and administration costs, European Actuarial Journal, 1 (2011), 57-92.
doi: 10.1007/s13385-011-0007-3. |
[33] |
F. Schuhmacher and M. Eling, Sufficient conditions for expected utility to imply drawdown-based performance rankings, Journal of Banking & Finance, 35 (2011), 2311-2318. Google Scholar |
[34] |
L. Shepp and A. N. Shiryaev,
The russian option: Reduced regret, The Annals of Applied Probability, 3 (1993), 631-640.
doi: 10.1214/aoap/1177005355. |
[35] |
H. M. Taylor,
A stopped brownian motion formula, The Annals of Probability, 3 (1975), 234-246.
doi: 10.1214/aop/1176996395. |
[36] |
S. Thonhauser and H. Albrecher,
Dividend maximization under consideration of the time value of ruin, Insurance: Mathematics and Economics, 41 (2007), 163-184.
doi: 10.1016/j.insmatheco.2006.10.013. |
[37] |
M. Vierkötter and H. Schmidli,
On optimal dividends with exponential and linear penalty payments, Insurance: Mathematics and Economics, 72 (2017), 265-270.
doi: 10.1016/j.insmatheco.2016.12.001. |
[38] |
W. Wang, Y. Wang and X. Wu, Dividend and capital injection optimization with transaction cost for spectrally negative lévy risk processes, arXiv preprint, arXiv: 1807.11171. Google Scholar |
[39] |
W. Wang and Z. Zhang,
Optimal loss-carry-forward taxation for lévy risk processes stopped at general draw-down time, Advances in Applied Probability, 51 (2019), 865-897.
doi: 10.1017/apr.2019.33. |
[40] |
W. Wang and X. Zhou,
General drawdown-based de finetti optimization for spectrally negative lévy risk processes, Journal of Applied Probability, 55 (2018), 513-542.
doi: 10.1017/jpr.2018.33. |
[41] |
R. Xu and J.-K. Woo,
Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments, Insurance: Mathematics and Economics, 92 (2020), 1-16.
doi: 10.1016/j.insmatheco.2020.02.008. |
[42] |
C. Yin and Y. Wen,
Optimal dividend problem with a terminal value for spectrally positive levy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773.
doi: 10.1016/j.insmatheco.2013.09.019. |
[43] |
Y. Zhao, P. Chen and H. Yang,
Optimal periodic dividend and capital injection problem for spectrally positive lévy processes, Insurance: Mathematics and Economics, 74 (2017), 135-146.
doi: 10.1016/j.insmatheco.2017.03.006. |
show all references
References:
[1] |
S. Asmussen, F. Avram and M. R. Pistorius,
Russian and american put options under exponential phase-type lévy models, Stochastic Processes and their Applications, 109 (2004), 79-111.
doi: 10.1016/j.spa.2003.07.005. |
[2] |
S. Asmussen and M. Taksar,
Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.
doi: 10.1016/S0167-6687(96)00017-0. |
[3] |
F. Avram, A. Kyprianouy and M. Pistoriusz,
Exit problems for spectrally negative l evy processes and applications to russian, american and canadized options, Ann. Appl. Probab, 14 (2004), 215-238.
doi: 10.1214/aoap/1075828052. |
[4] |
F. Avram, Z. Palmowski and M. R. Pistorius,
On the optimal dividend problem for a spectrally negative lévy process, The Annals of Applied Probability, 17 (2007), 156-180.
doi: 10.1214/105051606000000709. |
[5] |
F. Avram, Z. Palmowski and M. R. Pistorius,
On gerber–shiu functions and optimal dividend distribution for a lévy risk process in the presence of a penalty function, The Annals of Applied Probability, 25 (2015), 1868-1935.
doi: 10.1214/14-AAP1038. |
[6] |
F. Avram, N. L. Vu and X. Zhou,
On taxed spectrally negative lévy processes with draw-down stopping, Insurance: Mathematics and Economics, 76 (2017), 69-74.
doi: 10.1016/j.insmatheco.2017.06.005. |
[7] |
J. Azéma and M. Yor, Une solution simple au probleme de skorokhod, in Séminaire de Probabilités XIII, Springer, 721 (1979), 90–115. |
[8] |
E. J. Baurdoux, Z. Palmowski and M. R. Pistorius,
On future drawdowns of lévy processes, Stochastic Processes and their Applications, 127 (2017), 2679-2698.
doi: 10.1016/j.spa.2016.12.008. |
[9] |
E. Bayraktar, A. E. Kyprianou and K. Yamazaki,
On optimal dividends in the dual model, ASTIN Bulletin: The Journal of the IAA, 43 (2013), 359-372.
doi: 10.1017/asb.2013.17. |
[10] |
J. Bertoin, Lévy Processes, vol. 121, Cambridge university press Cambridge, 1996. |
[11] |
G. Burghardt and R. Duncan, Deciphering drawdown, Risk management for investors, September, S16–S20. Google Scholar |
[12] |
P. Carr, First-order calculus and option pricing, Journal of Financial Engineering, 1 (2014), 1450009.
doi: 10.1142/s2345768614500093. |
[13] |
B. De Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, in Transactions of the XVth international congress of Actuaries, vol. 2, New York, 1957,433–443. Google Scholar |
[14] |
D. C. Dickson and H. R. Waters,
Some optimal dividends problems, ASTIN Bulletin: The Journal of the IAA, 34 (2004), 49-74.
doi: 10.1017/S0515036100013878. |
[15] |
B. HøJgaard and M. Taksar,
Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.
doi: 10.1111/1467-9965.00066. |
[16] |
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Elsevier, 2014. Google Scholar |
[17] |
J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288, Springer-Verlag, Berlin, 1987.
doi: 10.1007/978-3-662-02514-7. |
[18] |
A. E. Kyprianou, R. Loeffen and J.-L. Pérez,
Optimal control with absolutely continuous strategies for spectrally negative lévy processes, Journal of Applied Probability, 49 (2012), 150-166.
doi: 10.1239/jap/1331216839. |
[19] |
A. E. Kyprianou and Z. Palmowski,
Distributional study of de finetti's dividend problem for a general lévy insurance risk process, Journal of Applied Probability, 44 (2007), 428-443.
doi: 10.1239/jap/1183667412. |
[20] |
D. Landriault, B. Li and S. Li,
Analysis of a drawdown-based regime-switching lévy insurance model, Insurance: Mathematics and Economics, 60 (2015), 98-107.
doi: 10.1016/j.insmatheco.2014.11.005. |
[21] |
D. Landriault, B. Li and S. Li,
Drawdown analysis for the renewal insurance risk process, Scandinavian Actuarial Journal, 2017 (2017), 267-285.
doi: 10.1080/03461238.2015.1123174. |
[22] |
J. P. Lehoczky,
Formulas for stopped diffusion processes with stopping times based on the maximum, The Annals of Probability, 5 (1977), 601-607.
doi: 10.1214/aop/1176995770. |
[23] |
B. Li, N. L. Vu and X. Zhou,
Exit problems for general draw-down times of spectrally negative lévy processes, Journal of Applied Probability, 56 (2019), 441-457.
doi: 10.1017/jpr.2019.31. |
[24] |
R. L. Loeffen,
On optimality of the barrier strategy in de finetti's dividend problem for spectrally negative lévy processes, The Annals of Applied Probability, 18 (2008), 1669-1680.
doi: 10.1214/07-AAP504. |
[25] |
R. L. Loeffen,
An optimal dividends problem with a terminal value for spectrally negative lévy processes with a completely monotone jump density, Journal of Applied Probability, 46 (2009), 85-98.
doi: 10.1017/S0021900200005246. |
[26] |
R. L. Loeffen,
An optimal dividends problem with transaction costs for spectrally negative lévy processes, Insurance: Mathematics and Economics, 45 (2009), 41-48.
doi: 10.1016/j.insmatheco.2009.03.002. |
[27] |
R. L. Loeffen and J.-F. Renaud,
De finetti's optimal dividends problem with an affine penalty function at ruin, Insurance: Mathematics and Economics, 46 (2010), 98-108.
doi: 10.1016/j.insmatheco.2009.09.006. |
[28] |
D. B. Madan and M. Yor,
Making markov martingales meet marginals: With explicit constructions, Bernoulli, 8 (2002), 509-536.
|
[29] |
X. Peng, M. Chen and J. Guo,
Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.
doi: 10.1016/j.insmatheco.2012.08.004. |
[30] |
G. Peskir,
Designing options given the risk: The optimal skorokhod-embedding problem, Stochastic Processes and Their Applications, 81 (1999), 25-38.
doi: 10.1016/S0304-4149(98)00097-0. |
[31] |
J.-F. Renaud and X. Zhou,
Distribution of the present value of dividend payments in a lévy risk model, Journal of Applied Probability, 44 (2007), 420-427.
doi: 10.1239/jap/1183667411. |
[32] |
N. Scheer and H. Schmidli,
Optimal dividend strategies in a cramer–lundberg model with capital injections and administration costs, European Actuarial Journal, 1 (2011), 57-92.
doi: 10.1007/s13385-011-0007-3. |
[33] |
F. Schuhmacher and M. Eling, Sufficient conditions for expected utility to imply drawdown-based performance rankings, Journal of Banking & Finance, 35 (2011), 2311-2318. Google Scholar |
[34] |
L. Shepp and A. N. Shiryaev,
The russian option: Reduced regret, The Annals of Applied Probability, 3 (1993), 631-640.
doi: 10.1214/aoap/1177005355. |
[35] |
H. M. Taylor,
A stopped brownian motion formula, The Annals of Probability, 3 (1975), 234-246.
doi: 10.1214/aop/1176996395. |
[36] |
S. Thonhauser and H. Albrecher,
Dividend maximization under consideration of the time value of ruin, Insurance: Mathematics and Economics, 41 (2007), 163-184.
doi: 10.1016/j.insmatheco.2006.10.013. |
[37] |
M. Vierkötter and H. Schmidli,
On optimal dividends with exponential and linear penalty payments, Insurance: Mathematics and Economics, 72 (2017), 265-270.
doi: 10.1016/j.insmatheco.2016.12.001. |
[38] |
W. Wang, Y. Wang and X. Wu, Dividend and capital injection optimization with transaction cost for spectrally negative lévy risk processes, arXiv preprint, arXiv: 1807.11171. Google Scholar |
[39] |
W. Wang and Z. Zhang,
Optimal loss-carry-forward taxation for lévy risk processes stopped at general draw-down time, Advances in Applied Probability, 51 (2019), 865-897.
doi: 10.1017/apr.2019.33. |
[40] |
W. Wang and X. Zhou,
General drawdown-based de finetti optimization for spectrally negative lévy risk processes, Journal of Applied Probability, 55 (2018), 513-542.
doi: 10.1017/jpr.2018.33. |
[41] |
R. Xu and J.-K. Woo,
Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments, Insurance: Mathematics and Economics, 92 (2020), 1-16.
doi: 10.1016/j.insmatheco.2020.02.008. |
[42] |
C. Yin and Y. Wen,
Optimal dividend problem with a terminal value for spectrally positive levy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773.
doi: 10.1016/j.insmatheco.2013.09.019. |
[43] |
Y. Zhao, P. Chen and H. Yang,
Optimal periodic dividend and capital injection problem for spectrally positive lévy processes, Insurance: Mathematics and Economics, 74 (2017), 135-146.
doi: 10.1016/j.insmatheco.2017.03.006. |








(3.9689, 11.6898) | (3.9689, 11.6902) | (2.0000, 11.2002) | (2.0000, 10.1503) | |
(3.4688, 11.1898) | (3.4688, 11.1893) | (1.5000, 10.0581) | (1.5330, 10.3051) | |
(3.1688, 10.8898) | (3.1680, 10.8917) | (3.1688, 10.8898) | (1.6154, 10.3557) | |
(2.9688, 10.6898) | (2.9687, 10.6901) | (2.9688, 10.6894) | (1.7244, 10.3693) |
(3.9689, 11.6898) | (3.9689, 11.6902) | (2.0000, 11.2002) | (2.0000, 10.1503) | |
(3.4688, 11.1898) | (3.4688, 11.1893) | (1.5000, 10.0581) | (1.5330, 10.3051) | |
(3.1688, 10.8898) | (3.1680, 10.8917) | (3.1688, 10.8898) | (1.6154, 10.3557) | |
(2.9688, 10.6898) | (2.9687, 10.6901) | (2.9688, 10.6894) | (1.7244, 10.3693) |
Ruin Time | ||||
(4.5855, 10.5305) | (4.0855, 10.0305) | (2.6070, 9.3218) | (2.5855, 8.5305) | |
(3.9689, 11.6898) | (3.4688, 11.1893) | (1.7244, 10.3693) | (1.9688, 9.6898) | |
(3.5369, 12.6025) | (3.0369, 12.1025) | (1.0605, 11.1901) | (1.5369, 10.6025) | |
(3.1911, 13.3968) | (1.5000, 12.8618) | (1.0000, 11.9132) | (1.1912, 11.3968) | |
(2.8968, 14.1194) | (1.5000, 13.5114) | (1.0000, 12.6039) | (0.8968, 12.1194) |
Ruin Time | ||||
(4.5855, 10.5305) | (4.0855, 10.0305) | (2.6070, 9.3218) | (2.5855, 8.5305) | |
(3.9689, 11.6898) | (3.4688, 11.1893) | (1.7244, 10.3693) | (1.9688, 9.6898) | |
(3.5369, 12.6025) | (3.0369, 12.1025) | (1.0605, 11.1901) | (1.5369, 10.6025) | |
(3.1911, 13.3968) | (1.5000, 12.8618) | (1.0000, 11.9132) | (1.1912, 11.3968) | |
(2.8968, 14.1194) | (1.5000, 13.5114) | (1.0000, 12.6039) | (0.8968, 12.1194) |
| ||||
| (3.8442, 12.3162) | (3.8441, 12.3176) | (3.0519, 11.7522) | (2.2989, 10.8580) |
| (3.3442, 11.8162) | (3.3441, 11.8162) | (3.0575, 11.6979) | (2.2989, 10.8580) |
| (3.0442, 11.5162) | (3.0441, 11.5162) | (3.0397, 11.5094) | (2.2989, 10.8580) |
| (2.8442, 11.3162) | (2.8439, 11.3162) | (2.8441, 11.3166) | (2.2989, 10.8580) |
| ||||
| (3.8442, 12.3162) | (3.8441, 12.3176) | (3.0519, 11.7522) | (2.2989, 10.8580) |
| (3.3442, 11.8162) | (3.3441, 11.8162) | (3.0575, 11.6979) | (2.2989, 10.8580) |
| (3.0442, 11.5162) | (3.0441, 11.5162) | (3.0397, 11.5094) | (2.2989, 10.8580) |
| (2.8442, 11.3162) | (2.8439, 11.3162) | (2.8441, 11.3166) | (2.2989, 10.8580) |
Ruin Time | ||||
(3.9695, 9.9869) | (3.4695, 9.4869) | (2.4539, 8.5596) | (1.9695, 7.9869) | |
(3.8442, 12.3162) | (3.3441, 11.8162) | (2.2989, 10.8580) | (1.8442, 10.3162) | |
(3.7664, 14.1668) | (3.2664, 13.6668) | (2.2024, 12.6891) | (1.7664, 12.1668) | |
(3.7087, 15.7657) | (3.2087, 15.2657) | (2.1307, 14.2736) | (1.7087, 13.7657) | |
(3.6623, 17.2027) | (3.1623, 16.7027) | (2.0730, 15.6991) | (1.6623, 15.2027) |
Ruin Time | ||||
(3.9695, 9.9869) | (3.4695, 9.4869) | (2.4539, 8.5596) | (1.9695, 7.9869) | |
(3.8442, 12.3162) | (3.3441, 11.8162) | (2.2989, 10.8580) | (1.8442, 10.3162) | |
(3.7664, 14.1668) | (3.2664, 13.6668) | (2.2024, 12.6891) | (1.7664, 12.1668) | |
(3.7087, 15.7657) | (3.2087, 15.2657) | (2.1307, 14.2736) | (1.7087, 13.7657) | |
(3.6623, 17.2027) | (3.1623, 16.7027) | (2.0730, 15.6991) | (1.6623, 15.2027) |
(5.4394, 14.7877) | (5.4394, 14.7879) | (5.4393, 14.7878) | (2.0000, 13.4730) | |
(4.9394, 14.2876) | (4.9394, 14.2876) | (4.9394, 14.2875) | (2.1303, 13.7094) | |
(4.6394, 13.9876) | (4.6394, 13.9876) | (4.6386, 13.9915) | (2.7529, 13.7678) | |
(4.4394, 13.7876) | (4.4394, 13.7865) | (4.4394, 13.7877) | (3.5068, 13.7285) |
(5.4394, 14.7877) | (5.4394, 14.7879) | (5.4393, 14.7878) | (2.0000, 13.4730) | |
(4.9394, 14.2876) | (4.9394, 14.2876) | (4.9394, 14.2875) | (2.1303, 13.7094) | |
(4.6394, 13.9876) | (4.6394, 13.9876) | (4.6386, 13.9915) | (2.7529, 13.7678) | |
(4.4394, 13.7876) | (4.4394, 13.7865) | (4.4394, 13.7877) | (3.5068, 13.7285) |
Ruin Time | ||||
(6.2530, 13.5120) | (5.7530, 13.0120) | (5.2514, 12.5135) | (4.2530, 11.5120) | |
(5.4394, 14.7876) | (4.9394, 14.2876) | (3.5068, 13.7285) | (3.4394, 12.7876) | |
(4.8685, 15.7668) | (4.3685, 15.2668) | (2.0611, 14.5761) | (2.8685, 13.7667) | |
(4.4172, 16.6037) | (3.9172, 16.1037) | (1.2175, 15.3164) | (2.4172, 14.6037) | |
(4.0462, 17.3555) | (3.5462, 16.8555) | (1.0000, 15.9652) | (2.0462, 15.3555) |
Ruin Time | ||||
(6.2530, 13.5120) | (5.7530, 13.0120) | (5.2514, 12.5135) | (4.2530, 11.5120) | |
(5.4394, 14.7876) | (4.9394, 14.2876) | (3.5068, 13.7285) | (3.4394, 12.7876) | |
(4.8685, 15.7668) | (4.3685, 15.2668) | (2.0611, 14.5761) | (2.8685, 13.7667) | |
(4.4172, 16.6037) | (3.9172, 16.1037) | (1.2175, 15.3164) | (2.4172, 14.6037) | |
(4.0462, 17.3555) | (3.5462, 16.8555) | (1.0000, 15.9652) | (2.0462, 15.3555) |
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