Abstract
In this paper the solutions to several variants of the so-called dividend-distribution problem in a multi-dimensional, diffusion setting are studied. In a nutshell, the manager of a firm must balance the retention of earnings (so as to ward off bankruptcy and earn interest) and the distribution of dividends (so as to please the shareholders). A dynamic-programming approach is used, where the state variables are the current levels of cash reserves and of the stochastic short-rate, as well as time. This results in a family of Hamilton–Jacobi–Bellman variational inequalities whose solutions must be approximated numerically. To do so, a finite element approximation and a time-marching scheme are employed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Considering stochastic discounting instead of a stochastic interest rate in our setting could be done with minor modifications to our numerical schemes.
A notable exception is Hoejgaard and Taksar (2004), where the same problem is tackled within an Itô-diffusion setting.
In the sequel we equate dividend distribution to the consumption of the latter and use the terms indistinctly.
References
Achdou, Y., & Pironneau, O. (2005). Computational methods for option pricing, Frontiers in applied mathematics. Philadelphia: Society for Industrial and Applied Mathematics.
Akyildirim, E., Güney, I. E., Rochet, J.-C., & Soner, H. M. (2014). Optimal dividend policy with random interest rates. Journal of Mathematical Economics, 51, 93–101.
Azcue, P., & Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model. Mathematical Finance, 15(2), 261–308.
Azcue, P., & Muler, N. (2010). Optimal investment policy and dividend payment strategy in an insurance company. The Annals of Applied Probability, 29(4), 1253–1302.
Barth, A., & Moreno-Bromberg, S. (2014). Optimal risk and liquidity management with costly refinancing opportunities. Insurance: Mathematics and Economics, 57, 31–44.
Bass, R. F., & Hsu, P. (1990). The semimartingale structure of reflecting Brownian motion. Proceedings of the American Mathematical Society, 108(4), 1007–1010.
Bolton, P., Chen, H., & Wang, N. (2014). Debt, taxes and liquidity, Technical Report, Columbia University and MIT Sloan.
Brennan, M., & Schwartz, E. (1977). The valuation of American put options. Journal of Finance, 32, 449–462.
Daskalopoulos, P., & Feehan, P. (2012) \({C}^{1,1}\) regularity for degenerate elliptic obstacle problems in mathematical finance. Technical Report, arvix, http://arxiv.org/abs/1206.0831
De Finetti, B. (1957). Su un impostazione alternativa dell teoria collettiva del rischio. Transactions of the XVth International Congress of Actuaries, 2, 433–443.
Dumas, B. (1991). Super contact and related optimality conditions. Journal of Economic Dynamics and Control, 15(4), 675–685.
Ekström, E., & Tysk, J. (2010). The Black–Scholes equation in stochastic volatility models. Journal of Mathematical Analysis and Applications, 368(2), 498–507.
Elworthy, K. D., Truman, A., & Zhao, H. Z. (2007). Generalized Itô formulae and space-time Lebesgue–Stieltjes integrals of local times., Lecture Notes in Mathematics 1899 Berlin: Springer.
Evans, L. C. (1998). Partial differential equations. Providence: American Mathematical Society, AMS.
Gerber, H. U., & Shiu, E. S. W. (2004). Optimal dividends: Analysis with Brownian motion. North American Actuarial Journal, 8(1), 1–20.
Hilber, N., Reichmann, O., Schwab, Ch., & Winter, Ch. (2013). Computational methods for quantitative finance: Finite element methods for derivative pricing. Berlin: Springer Finance, Springer, London, Limited.
Hipp, C., & Plum, M. (2000). Optimal investment for insurers. Insurance: Mathematics and Economics, 27, 215–228.
Hoejgaard, B., & Taksar, M. (1998). Controlling risk exposure and dividends pay-out schemes: Insurance company example. Mathematical Finance, 9, 153–182.
Hoejgaard, B., & Taksar, M. (2004). Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Quantitative Finance, 4, 315–327.
Hugonnier, J., Malamud, S., & Morellec, E. (2015). Capital supply uncertainty, cash holdings and investment. Review of Financial Studies, 2(28), 391–445.
Jeanblanc-Picqué, M., & Shiryaev, A. N. (1995). Optimization of the flow of dividends. Uspekhi Matematicheskikh Nauk, 502(203), 25–46.
Jiang, Z., & Pistorius, M. (2012). Optimal dividend distribution under Markov regime switching. Finance and Stochastics, 16(3), 449–476.
Johnson, C., Nävert, U., & Pitkäranta, J. (1984). Finite element methods for linear hyperbolic problems. Computer Methods in Applied Mechanics and Engineering, 45(1–3), 285–312.
Kruk, L., Lehoczky, J., Ramanan, K., & Shreve, S. E. (2007). An explicit formula for the Skorokhod map on \([0, a]\). The Annals of Probability, 35(5), 1740–1768.
Laurence, P., & Salsa, S. (2009). Regularity of the free boundary of an American option on several assets. Communications on Pure and Applied Mathematics, 62, 969–994.
Lokka, A., & Zervos, M. (2008). Optimal dividend and issuance of equity policies in the presence of proportional costs. Insurance: Mathematics and Economics, 42(3), 954–961.
Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. The Review of Economics and Statistics, 51(3), 247–257.
Pham, H. (2010). Continuous-time stochastic control and optimization with financial applications, stochastic modelling and applied probability. Berlin: Springer.
Prokaj, V. (2009). Unfolding the Skorokhod reflection of a semimartingale. Statistics and Probability Letters, 79, 534–536.
Radner, R., & Shepp, L. (1996). Risk versus profit potential: A model for corporate strategy. Journal of Economic Dynamics and Control, 20(8), 1373–13793.
Revouz, D., & Yor, M. (1999). Continuous martingales and Brownian motion. Berlin: Springer.
Rochet, J.-C., & Villeneuve, S. (2005). Corporate portfolio management. Annals of Finance, 1(3), 225–243.
Schmidli, H. (2002). On minimizing the ruin probability by investment and reinsurance. The Annals of Applied Probability, 12, 890–907.
Scott, L. O. (1987). Option pricing when the variance changes randomly: theory, estimation, and an application. Journal of Financial and Quantitative Analysis, 22, 419–438.
Stein, E. M., & Stein, J. C. (1991). Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies, 4(4), 727–752.
Acknowledgments
We would like to thank the editor and an anonymous referee for their comments and suggestions, which allowed us to improve our original manuscript. It goes without saying that we assume full responsibility for any remaining mistakes. The research leading to these results has received funding form the ERC (Grant agreements AdG 247277 and 249415-RMAC), from NCCR FinRisk (Project “Banking and Regulation”), from the Swiss Finance Institute (Project “Systemic Risk and Dynamic Contract Theory”), from the SNF (Grant 144130) and from the German Research Foundation (DFG) as part of the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart, and it is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barth, A., Moreno–Bromberg, S. & Reichmann, O. A Non-stationary Model of Dividend Distribution in a Stochastic Interest-Rate Setting. Comput Econ 47, 447–472 (2016). https://doi.org/10.1007/s10614-015-9502-y
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10614-015-9502-y
Keywords
- Dividend distribution
- Singular stochastic control
- Numerical methods for partial differential equations
- Finite element method