Applying simulation optimization to dynamic financial analysis for the asset–liability management of a property–casualty insurer

Abstract

The Dynamic Financial Analysis (DFA) system is a useful decision-support system for the insurer, but it lacks optimization capability. This article applies a simulation optimization technique to a DFA system and use the enhanced system to search an Asset–Liability Management (ALM) solution for a Property–Casualty (P&C) insurance company. The simulation optimization technique used herein is a Genetic Algorithm (GA), and the optimization problem is a constrained, multi-period asset allocation problem that takes account of insurance liability dynamics. We find that coupling a DFA system with simulation optimization results in significant improvements over the search method currently available to the DFA system. The results were robust across random number sets. Furthermore, the resulting asset allocations changes with the asset–liability setting in a way that is consistent with the differences in the settings. Applying simulation optimization to a DFA system is therefore promising.

Acknowledgements

The authors are grateful to Jia-Le Lin for his competent programming assistance, to the National Center for High-Performance Computing of Taiwan for use of its facility, and to the National Science Council of Taiwan for its financial support (project number NSC 94-2416-H-004-041 and NSC 96-2416-H-004-026-MY3). T.-Y. Yu would like to thank Georgia State University and Santa Clara University for their kind support during his visits in writing this article.

Notes

¹ The feasible region is defined by the practical limits on the ranges of the controllable inputs. Examples of practical limits include short-sale constraints and upper bounds on portfolio weights faced by most financial institutions.

² The impossibility is due to three reasons. First, the system contains several types of stochastic processes. The function represented by the simulation model is thus unknown. Second, the variations of the outcomes generated by the system are significant, heterogeneous over the feasible region, and not normally distributed. Third, the system has 12 controllable variables over real intervals. The feasible region is rather large.

³ The articles using scenario trees usually considered several periods with dozens of scenarios only, while those utilizing simulations allow for more periods with thousands of scenarios.

⁴ We are aware that simulation optimization is a heuristic search method. The existence of the optimal solution is not proven, and there is no verification theorem to show that the resulted solution from simulation optimization is at least as good as all other solutions. The word ‘optimal’ is used loosely in this article to mean ‘the best known solution’.

⁵ The normal distribution might be reasonable for aggregate losses arising from a population with a known mean and independent individual losses (Bustic, 1994). Eling and Holzmüller (2008) also modelled the sum of small claims using a normal distribution.

⁶ The correlation coefficients are estimated using the historical data on S&P 500 index, indexes of All Publicly Traded Real Estate Investment Trusts (REITs), Treasury-Bill rates and the loss ratios published in Best's Aggregates and Averages. The sampling period is from 1972 to 1999. dW_{LR ( S )} is not in the matrix because we assume that the loss ratio of short-tail businesses is independent of other processes. We use REITs, a securitization investment tool, as a subjective proxy to the insurer's short-term investments.

⁷ We assume that the return on cash is r(t).

⁸ The multiple, also called the adjustment factor in this article, is to account for the effect of growth and time value of money. For an insurer that does not have growth in premiums written, calendar-year loss ratios are equal to the ratio of the ultimate losses to the premiums written. For a growing insurer, however, calendar-year loss ratios will be less than the ratio of the ultimate losses to the premiums written because the denominators of loss ratios grow with time. Furthermore, time value of money should be considered.

⁹ Reserves might be smaller than loss payments in extreme cases. We set reserves as zero in these cases and deduct the deficit from surplus.

¹⁰ Notice that IP(t + 1) = IP(t) * B(t) * (1 + G(L)) + IP(t) * (1 − B(t)) * (1 + G(S)).

¹¹ After a simple spreadsheet work, we obtain an adjustment factor of 0.9449 for the long-tail businesses given a growth rate of 5%, a discount rate of 7% and the specified D_L (dy). The adjustment factor for the short-tail businesses is 0.9905 given a growth rate of 4%, a discount rate of 7% and the specified D_S (dy).

¹² The last period starting from the fourth asset allocation is therefore 7 years.

¹³ In a discrete space, decision variables take a discrete set of values such as the number of machines in a system. The feasible region in a continuous space, on the other hand, consists of real-valued decision variables such as the release time of factory orders.

¹⁴ The following descriptions are drawn from Chen et al. (1995, 1996).

¹⁵ Remember that the return from cash is the 1-year short rate. Since the simulated yield curve is usually upward-sloping, the 1-year bond is smaller than longer-maturity bonds. The insolvency probability of the runner-up is the same as the number-one choice implies that the risk of the bond portfolio is small.

¹⁶ The alternative parameter sets are described in the Appendix. They are specified rather arbitrarily. We intentionally make them ‘unreasonable’ to see whether our GA can still find solutions under odd settings. The random number set used for these two alternative parameter sets is the same as the one used in the benchmark case.