随机利率下年金的精算模型

文档编号:XXLW109 文档字数:16580,页数:42

摘要
 年金在日常生活中被广泛应用，但以往大多研究的是固定年金以及随机利率下的确定年
金．本文考虑了利率随机波动对生命年金的影响，运用随机利率模型，得出年金精算现值较为简单的递推关系式，并举例说明利率的随机波动对年金精算现值的影响程度，结果表明利率的波动对年金的定价影响非常大，不容忽视。
 年金是经济、金融、年金领域中的重要概念,随机利率下关于年金问题的研究近年来引起了应用概率统计界的极大关注,并且取得了一系列深刻的研究成果.由于随机利率的复杂性和重要性,关于这一问题的研究方兴未艾.文中试图在随机利率条件下研究年金现值的数字特征.基于随机利率下的年金思想方法以及鞅、反射Brownian运动和随机序的理论,得到了控制随机利率下连续年金现值的期望,并给出了离散随机年金现值函数在凸序意义下的上界,讨论了上界的分布函数和停止损失保费,进而得到了离散的随机年金现值的期望。
 
 关键字:年金 随机利率 精算模型 期望 精算

Abstract
 
 Annuity is widely used in daily life, but past experience most of the fixed annuity, as well as stochastic interest rate determined under the annuity. This paper considers the random fluctuations in interest rates impact on life annuities, using stochastic interest rate model, obtained Annuity relatively simple recurrence relation, and illustrates the interest rate of the random fluctuations of the Annuity's impact, the results show that the interest rate The impact of fluctuations on the pricing of annuities is very large, can not be overlooked. random interest rate on the annuity issue of applied probability and statistics in recent years has aroused great concern to industry, and has made a series of profound research.
 Annuity is the economic, financial, insurance, important concepts in the field, As the complexity of stochastic interest rates and importance of research on this issue in full swing. This paper studied under stochastic interest rate annuity present value of the digital features. annuities under random rates of interest-based way of thinking and martingale, reflecting Brownian motion and random order of the theory, control stochastic interest rate the present value of expected continuous annuity, annuity given the present value of discrete random function in the sense of convex order upper bound on the community to discuss the distribution function and the stop-loss premiums, and then get the discrete random annuity is value expectations.
 Keywords: annuities   stochastic interest rates   actuarial model   expectations   actuarial

目录
中文摘要…………………………………………………………………………………………………I
英文摘要……………………………………………………………..…………………………………II
目录……………………………………………………………………………………………………III
第一章  年金问题的综述……………………………………………………………………………....1
 1.1年金的基本概念………………………………………………………………………….……1
 1.2年金建立的好处…………………………………………………………………………….…2
 1.3年金计算方式…………………………………………………………………………….……3
 1.4精算学及其发展………………………………………………………………………….……3
 1.5年金中的数学原理………………………….…………………………………………………4
 1.6随机利率情况下目前研究的状况……………………………………………….……………5
 1.7本文研究的主要内容…………………………………………………………….……………6
第二章  利息论基础……………………………………………………………………………………7
 2.1  利息的度量…………………………………………………………..………………………7
     2.1.1积累函数与金额函数……………………………………………………….…………7
     2.1.2现值…………………………………………………………………………….………9
 2.1.3离散时间下的利息度量：利息率与贴现率……………………………….…………9
 2.1.4连续时间下的利息度量：利息力……………………………………………………10
 2.1.5单利与复利……………………………………………………………………………11
 2.2  年金…………………………………………………………………………………….……12
     2.2.1年金的定义……………………………………………………………………………12
     2.2.2年金的分类……………………………………………………………………………12
     2.2.3常见的确定年金………………………………………………………………………12
第三章  随机利率相关知识 ……………………………………………………………….…………16
 3.1  离散时间下的随机利息………………………………………………………….…………16
     3.1.1独立同分布的情形……………………………………………………………………16
 3.1.2时间序列模型…………………………………………………………………………17
 3.2  连续时间下的随机利息………………………………………………………….…………20
 3.3  其它在本文中用到的知识(举例分析------生存年金精算模型)…………….…………22
 3.3.1 岁生命的剩余寿命[8]…………………………………………………….…………22
 3.3.2死亡力…………………………………………………………………………………23
 3.3.3Wiener过程……………………………………………………………………………24
第四章  随机利率下的综合年金模型…………………………………………………...……………25
 4.1  利率对年金的影响…………………………………………….……………………………25
 4.2  模型的建立………………………………………….………………………………………25
     4.2.1年金责任………………………………………………………………………………25
 4.2.2保费计算………………………………………………………………………………25
 4.3  参数的取值范围………………………………………………………….…………………34
 4.4  结论…………………………………………………………………….……………………34
第五章  展望…………………………………………………………………………...………………35
结束语…………………………………………………………………...………………………………36
致谢………………………………………………………………………...……………………………37
参考文献…………………………………………………...……………………………………………38

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