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**Risk-neutral valuation of participating life insurance contracts in a stochastic interest rate environment.**
*(English)*
Zbl 1140.91431

Summary: Over the last years, the valuation of life insurance contracts using concepts from financial mathematics has become a popular research area for actuaries as well as financial economists. In particular, several methods have been proposed of how to model and price participating policies, which are characterized by an annual interest rate guarantee and some bonus distribution rules. However, despite the long terms of life insurance products, most valuation models allowing for sophisticated bonus distribution rules and the inclusion of frequently offered options assume a simple Black-Scholes setup and, more specifically, deterministic or even constant interest rates.

We present a framework in which participating life insurance contracts including predominant kinds of guarantees and options can be valuated and analyzed in a stochastic interest rate environment. In particular, the different option elements can be priced and analyzed separately. We use Monte Carlo and discretization methods to derive the respective values.

The sensitivity of the contract and guarantee values with respect to multiple parameters is studied using the bonus distribution schemes as introduced in [D. Bauer et al., Insur. Math. Econ. 39, No. 2, 171–183 (2006; Zbl 1098.91067)]. Surprisingly, even though the value of the contract as a whole is only moderately affected by the stochasticity of the short rate of interest, the value of the different embedded options is altered considerably in comparison to the value under constant interest rates. Furthermore, using a simplified asset portfolio and empirical parameter estimations, we show that the proportion of stock within the insurer’s asset portfolio substantially affects the value of the contract.

We present a framework in which participating life insurance contracts including predominant kinds of guarantees and options can be valuated and analyzed in a stochastic interest rate environment. In particular, the different option elements can be priced and analyzed separately. We use Monte Carlo and discretization methods to derive the respective values.

The sensitivity of the contract and guarantee values with respect to multiple parameters is studied using the bonus distribution schemes as introduced in [D. Bauer et al., Insur. Math. Econ. 39, No. 2, 171–183 (2006; Zbl 1098.91067)]. Surprisingly, even though the value of the contract as a whole is only moderately affected by the stochasticity of the short rate of interest, the value of the different embedded options is altered considerably in comparison to the value under constant interest rates. Furthermore, using a simplified asset portfolio and empirical parameter estimations, we show that the proportion of stock within the insurer’s asset portfolio substantially affects the value of the contract.

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

### Keywords:

participating life insurance contracts; risk-neutral valuation; stochastic interest rates; embedded options### Citations:

Zbl 1098.91067
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\textit{K. Zaglauer} and \textit{D. Bauer}, Insur. Math. Econ. 43, No. 1, 29--40 (2008; Zbl 1140.91431)

### References:

[1] | Ballotta, L.; Haberman, S., The fair valuation problem of guaranteed annuity options: the stochastic mortality environment case, Insurance: mathematics and economics, 38, 195-214, (2006) · Zbl 1101.60045 |

[2] | Barbarin, J.; Devolder, P., Risk measure and fair valuation of an investment guarantee in life insurance, Insurance: mathematics and economics, 37, 297-323, (2005) · Zbl 1125.91061 |

[3] | Bauer, D.; Kiesel, R.; Kling, A.; Ruß, J., Risk-neutral valuation of participating life insurance contracts, Insurance: mathematics and economics, 39, 171-183, (2006) · Zbl 1098.91067 |

[4] | Bernard, C.; Le Courtois, O.; Quittard-Pinon, F., Market value of life insurance contracts under stochastic interest rates and default risk, Insurance: mathematics and economics, 36, 499-516, (2005) · Zbl 1242.60068 |

[5] | Bingham, N.H.; Kiesel, R., Risk-neutral valuation, (2004), Springer Finance, Springer London · Zbl 1058.91029 |

[6] | Briys, E.; de Varenne, F., On the risk of insurance liabilities: debunking some common pitfalls, The journal of risk and insurance, 64, 673-694, (1997) |

[7] | Cox, J.C.; Ingersoll, J.E.; Ross, S.A., A theory of the term structure of interest rates, Econometrica, 53, 385-407, (1985) · Zbl 1274.91447 |

[8] | De Felice, M.; Moriconi, F., Market based tools for managing the life insurance company, ASTIN bulletin, 35, 1, 79-111, (2005) · Zbl 1137.62384 |

[9] | Grosen, A.; Jørgensen, P.L., Fair valuation of life insurance liabilities: the impact of interest guarantees, surrender options, and bonus policies, Insurance: mathematics and economics, 26, 37-57, (2000) · Zbl 0977.62108 |

[10] | Hull, J.C., Options, futures, and other derivatives, (2006), Prentice-Hall Englewood Cliffs, NJ · Zbl 1087.91025 |

[11] | Jensen, B.; Jørgensen, P.L.; Grosen, A., A finite difference approach to the valuation of path dependent life insurance liabilities, Geneva papers on risk and insurance theory, 26, 57-84, (2001) |

[12] | Karatzas, I.; Shreve, S.E., () |

[13] | Kling, A.; Richter, A.; Ruß, J., The interaction of guarantees, surplus distribution, and asset allocation in with profit life insurance policies, Insurance: mathematics and economics, 40, 164-178, (2007) · Zbl 1273.91238 |

[14] | Longstaff, F.A.; Schwartz, E.S., Valuing American options by simulation: A simple least-squares approach, Review of financial studies, 14, 113-147, (2001) |

[15] | Mallier, R.; Deakin, A.S., A green’s function for a convertible bond using the vasicek model, Journal of applied mathematics, 5, 219-232, (2002) · Zbl 1025.91012 |

[16] | Miltersen, K.R.; Persson, S.A., Pricing rate of return guarantees in a heath-jarrow-morton framework, Insurance: mathematics and economics, 25, 307-325, (1999) · Zbl 1028.91566 |

[17] | Nielsen, J.A.; Sandmann, K., Equity-linked life insurance: A model with stochastic interest rates, Insurance: mathematics and economics, 16, 225-253, (1995) · Zbl 0872.62094 |

[18] | Tanskanen, A.J.; Lukkarinen, J., Fair valuation of path-dependent participating life insurance contracts, Insurance: mathematics and economics, 33, 595-609, (2003) · Zbl 1103.91373 |

[19] | Vasic˘ek, O., An equilibrium characterization of the term structure, Journal of financial economics, 5, 177-188, (1977) · Zbl 1372.91113 |

[20] | Walter, U., Die bewertung von zinsoptionen, (1996), Gabler Verlag Wiesbaden, Germany |

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