Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T21:39:08.957Z Has data issue: false hasContentIssue false

Optimal dividend strategies for two collaborating insurance companies

Published online by Cambridge University Press:  26 June 2017

Hansjörg Albrecher*
Affiliation:
University of Lausanne and Swiss Finance Institute
Pablo Azcue*
Affiliation:
Universidad Torcuato Di Tella
Nora Muler*
Affiliation:
Universidad Torcuato Di Tella
*
* Postal address: Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, CH-1015 Lausanne, Switzerland. Email address: [email protected]
** Postal address: Departamento de Matematicas, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350, C1428BCW Buenos Aires, Argentina.
** Postal address: Departamento de Matematicas, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350, C1428BCW Buenos Aires, Argentina.

Abstract

We consider a two-dimensional optimal dividend problem in the context of two insurance companies with compound Poisson surplus processes, who collaborate by paying each other's deficit when possible. We study the stochastic control problem of maximizing the weighted sum of expected discounted dividend payments (among all admissible dividend strategies) until ruin of both companies, by extending results of univariate optimal control theory. In the case that the dividends paid by the two companies are equally weighted, the value function of this problem compares favorably with the one of merging the two companies completely. We identify the optimal value function as the smallest viscosity supersolution of the respective Hamilton–Jacobi–Bellman equation and provide an iterative approach to approximate it numerically. Curve strategies are identified as the natural analogue of barrier strategies in this two-dimensional context. A numerical example is given for which such a curve strategy is indeed optimal among all admissible dividend strategies, and for which this collaboration mechanism also outperforms the suitably weighted optimal dividend strategies of the two stand-alone companies.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Albrecher, H. and Lautscham, V. (2015). Dividends and the time of ruin under barrier strategies with a capital-exchange agreement. Anal. Inst. Actuarios Espaoles 3, 130. Google Scholar
[2] Albrecher, H. and Thonhauser, S. (2009). Optimality results for dividend problems in insurance. Rev. R. Acad. Cien. Ser. A Math. 103, 295320. Google Scholar
[3] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 14), 2nd edn. World Scientific, Hackensack, NJ. Google Scholar
[4] Avanzi, B. (2009). Strategies for dividend distribution: a review. N. Amer. Actuarial J. 13, 217251. CrossRefGoogle Scholar
[5] Avram, F., Palmowski, Z. and Pistorius, M. (2008). A two-dimensional ruin problem on the positive quadrant. Insurance Math. Econom. 42, 227234. CrossRefGoogle Scholar
[6] Avram, F., Palmowski, Z. and Pistorius, M. R. (2008). Exit problem of a two-dimensional risk process from the quadrant: exact and asymptotic results. Ann. Appl. Prob. 18, 24212449. Google Scholar
[7] Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15, 261308. Google Scholar
[8] Azcue, P. and Muler, N. (2013). Minimizing the ruin probability allowing investments in two assets: a two-dimensional problem. Math. Meth. Operat. Res. 77, 177206. Google Scholar
[9] Azcue, P. and Muler, N. (2014). Stochastic Optimization in Insurance. Springer, New York. Google Scholar
[10] Badescu, A., Gong, L. and Lin, S. (2015). Optimal capital allocations for a bivariate risk process under a risk sharing strategy. Preprint, University of Toronto. Google Scholar
[11] Badila, E. S., Boxma, O. J. and Resing, J. A. C. (2015). Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times. Insurance Math. Econom. 61, 4861. Google Scholar
[12] Crandall, M. G. and Lions, P.-L. (1983). Viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc. 277, 142. CrossRefGoogle Scholar
[13] Czarna, I. and Palmowski, Z. (2011). De Finetti's dividend problem and impulse control for a two-dimensional insurance risk process. Stoch. Models 27, 220250. Google Scholar
[14] De Finetti, B. (1957). Su un'impostazione alternativa della teoria collettiva del rischio. In Transactions XVth Congress of Actuaries, Vol. 2, Mallon, New York, pp. 433443. Google Scholar
[15] Gerber, H. U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-prozess. Schweiz. Aktuarver. Mitt. 1, 185227. Google Scholar
[16] Gerber, H. U. and Shiu, S. W. (2006). On the merger of two companies. N. Amer. Actuarial J. 10, 6067. Google Scholar
[17] Ivanovs, J. and Boxma, O. (2015). A bivariate risk model with mutual deficit coverage. Insurance Math. Econom. 64, 126134. Google Scholar
[18] Kulenko, N. and Schmidli, H. (2008). Optimal dividend strategies in a Cramér–Lundberg model with capital injections. Insurance Math. Econom. 43, 270278. CrossRefGoogle Scholar
[19] Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes. Ann. Appl. Prob. 18, 16691680. Google Scholar
[20] Loeffen, R. L. and Renaud, J.-F. (2010). De Finetti's optimal dividends problem with an affine penalty function at ruin. Insurance Math. Econom. 46, 98108. CrossRefGoogle Scholar
[21] Radner, R. and Shepp, L. (1996). Risk vs. profit potential: a model for corporate strategy. J. Econom. Dynam. Control 20, 13731393. CrossRefGoogle Scholar
[22] Schmidli, H. (2008). Stochastic Control in Insurance. Springer, London. Google Scholar
[23] Soner, H. M. (1986). Optimal control with state-space constraint. I. SIAM J. Control Optimization 24, 552561. Google Scholar
[24] Thonhauser, S. and Albrecher, H. (2007). Dividend maximization under consideration of the time value of ruin. Insurance Math. Econom. 41, 163184. Google Scholar