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Kathryn A. Watts MSc ASA Debbie J. Dupuis PhD Bruce L. Jones PhD FSA FCIA 《North American actuarial journal : NAAJ》2013,17(4):162-178
Abstract Extreme value theory describes the behavior of random variables at extremely high or low levels. The application of extreme value theory to statistics allows us to fit models to data from the upper tail of a distribution. This paper presents a statistical analysis of advanced age mortality data, using extreme value models to quantify the upper tail of the distribution of human life spans. Our analysis focuses on mortality data from two sources. Statistics Canada publishes the annual number of deaths in Canada, broken down by angender and age. We use the deaths data from 1949 to 1997 in our analysis. The Japanese Ministry of Health, Labor, and Welfare also publishes detailed annual mortality data, including the 10 oldest reported ages at death in each year. We analyze the Japanese data over the period from 1980 to 2000. Using the r-largest and peaks-over-threshold approaches to extreme value modeling, we fit generalized extreme value and generalized Pareto distributions to the life span data. Changes in distribution by birth cohort or over time are modeled through the use of covariates. We then evaluate the appropriateness of the fitted models and discuss reasons for their shortcomings. Finally, we use our findings to address the existence of a finite upper bound on the life span distribution and the behavior of the force of mortality at advanced ages. 相似文献
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Tak Kuen Siu PhD 《North American actuarial journal : NAAJ》2013,17(2):62-75
Abstract The autoregressive random variance (ARV) model introduced by Taylor (1980, 1982, 1986) is a popular version of stochastic volatility (SV) models and a discrete-time simplification of the continuous-time diffusion SV models. This paper introduces a valuation model for options under a discrete-time ARV model with general stock and volatility innovations. It employs the discretetime version of the Esscher transform to determine an equivalent martingale measure under an incomplete market. Various parametric cases of the ARV models, are considered, namely, the log-normal ARV models, the jump-type Poisson ARV models, and the gamma ARV models, and more explicit pricing formulas of a European call option under these parametric cases are provided. A Monte Carlo experiment for some parametric cases is also conducted. 相似文献
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Abstract This paper derives a class of efficient factor models that bridge a gap between factor models and Heath-Jarrow-Morton models. These efficient factor models provide arbitrage-free dynamics for the yield curve, can be readily extended to fit the current yield curve, and have closed-form formulas for pricing default-free zero-coupon bonds. The short rate is a state variable in these efficient factor models. There are no restrictions imposed on the functional form of the volatility of the short rate except for certain technical conditions to ensure the solvability of the associated stochastic differential equations. The stochastic volatility of the short rate can be one of the state variables. The paper also presents a closed-form solution for default-free discount bond prices in the Malkiel model and provides a new method to derive the Ritchken-Sankarasubramanian model. 相似文献
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Abstract This paper gives analytic approximations for the distribution of a stochastic life annuity. It is assumed that returns follow a geometric Brownian motion. The distribution of the stochastic annuity may be used to answer questions such as “What is the probability that an amount F is sufficient to fund a pension with annual amount y to a pensioner aged x?” The main idea is to approximate the future lifetime distribution with a combination of exponentials, and then apply a known formula (due to Marc Yor) related to the integral of geometric Brownian motion. The approximations are very accurate in the cases studied. 相似文献
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Abstract In this paper we study the Gerber-Shiu discounted penalty function for the ordinary renewal risk model modified by the constant interest on the surplus. Explicit answers are expressed by an infinite series, and a relational formula for some important joint density functions is derived. Applications of the results to the compound Poisson model are given. Finally, a lower bound and an upper bound for the ultimate ruin probability are derived. 相似文献