Pricing general insurance in a reactive and competitive market. (English) Zbl 1228.91033
Summary: A simple parameterisation is introduced which represents the insurance market’s response to an insurer adopting a pricing strategy determined via optimal control theory. Claims are modelled using a lognormally distributed mean claim size rate, and the market average premium is determined via the expected value principle. If the insurer maximises its expected wealth then the resulting Bellman equation has a moving boundary in state space that determines when it is optimal to stop selling insurance. This stochastic optimisation problem is simplified by the introduction of a stopping time that prevents an insurer leaving and then re-entering the insurance market. Three finite difference schemes are used to verify the existence of a solution to the resulting Bellman equation when there is market reaction. All of the schemes use a front-fixing transformation. If the market reacts, then it is found that the optimal strategy is altered, in that premiums are raised if the strategy is of loss-leading type and lowered if it is optimal for the insurer to set a relatively high premium and sell little insurance.
Keywords:
competitive demand model; reactive insurance market; dynamic programming; moving boundary problemReferences:
| [1] | Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic Processes for Insurance and Finance (1999), Wiley · Zbl 0940.60005 |
| [2] | Emms, P.; Haberman, S., Pricing general insurance using optimal control theory, Astin Bulletin, 35, 2, 427-453 (2005) · Zbl 1155.91401 |
| [3] | Taylor, G. C., Underwriting strategy in a competitive insurance environment, Insurance: Mathematics and Economics, 5, 1, 59-77 (1986) · Zbl 0585.62175 |
| [4] | Emms, P., Dynamic pricing of general insurance in a competitive market, Astin Bulletin, 37, 1, 1-34 (2007) · Zbl 1162.91409 |
| [5] | Emms, P., Pricing general insurance with constraints, Insurance: Mathematics and Economics, 40, 335-355 (2007) · Zbl 1141.91504 |
| [6] | Emms, P.; Haberman, S., Optimal management of an insurer’s exposure in a competitive general insurance market, North American Actuarial Journal, 13, 1, 77-105 (2009) · Zbl 1483.91192 |
| [7] | Merton, R. C., Optimum consumption and portfolio rules in a continuous time model, Journal of Economic Theory, 3, 373-413 (1971) · Zbl 1011.91502 |
| [8] | Dockner, E. J.; Jørgensen, S.; Long, N. V.; Sorger, G., Differential Games in Economics and Management Science (2000), Cambridge University Press · Zbl 0996.91001 |
| [9] | Daykin, C. D.; Pentikäinen, T.; Pesonen, M., Practical Risk Theory for Actuaries (1994), Chapman and Hall · Zbl 1140.62345 |
| [10] | Fleming, W.; Rishel, R., Deterministic and Stochastic Optimal Control (1975), Springer Verlag: Springer Verlag New York · Zbl 0323.49001 |
| [11] | Fleming, W. H.; Soner, H. M., Controlled Markov Processes and Viscosity Solutions (1993), Springer-Verlag · Zbl 0773.60070 |
| [12] | Crank, J., Free and Moving Boundary Problems (1984), Oxford University Press · Zbl 0547.35001 |
| [13] | Javierre, E.; Vuik, C.; Vermolen, F. J.; van der Zwaag, S., A comparison of numerical models for one-dimensional Stefan problems, Journal of Computational and Applied Mathematics, 192, 2, 445-459 (2006) · Zbl 1092.65072 |
| [14] | Nielsen, B. F.; Skavhaug, O.; Tveito, A., Penalty and front-fixing methods for the numerical solution of American option problems, Journal of Computational Finance, 5, 4, 69-97 (2002) |
| [15] | Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods (1985), Oxford University Press · Zbl 0576.65089 |
| [16] | Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes in C++ (2002), Cambridge University Press · Zbl 1078.65500 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
