随机利率下的寿险精算与实务研究

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3.0 赵德峰 2024-11-19 4 4 2.61MB 58 页 15积分
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I
摘 要
在现代保险业经营与发展的过程中，科学的理论与方法，特别是精算学原理
起着至关重要的作用，几乎保险业经营的所有的重要环节，如新险种设计、保险
费率和责任准备金计算等，都必须经过精算分析处理．在传统的精算理论中，利
率都是假定为常数，但，由于寿险是长期性的经济形为，保险期间内，政府政策、
经济周期等因素都会造成利率的不确定性，如近年来，我国多次下调利率且降幅
较大，对寿险公司产生了严重冲击，最终导致辞了利率风险，从而随机利率下的
寿险精算理论与方法的研究成为近年来研究的重点与热点问题．
为了研究保险公司在承保后所面临的风险，本文在随机利率的假设方法的基
础上，从理论上研究随机利率对寿险公司的影响并进行相关的实证分析．本文主
要是利用布朗运动的一些性质建立了随机利率的寿险精算模型，并利用该模型对
寿险精算中所涉变量的性质进行了分析，如推导了寿险及生存年金的递推公式，
以及寿险变量与生存年金变量的一阶矩、二阶矩对应变量的关系式，并在此基础
上计算出了生存年金的方差；另外，计算了随机利率下终身寿险的均衡纯保费和
寿险责任准备金，并对保险人在承保时以及在任一时刻的未来损失现值随机变量
的方差进行了分析，最后在所建模型进行了实证分析，得出了一些有实际指导价
值的结果．
关键词：随机利率 布朗运动 保险精算 精算模型 准备金 递推关系
II
ABSTRACT
In the management and development of insurance industry, the scientific theories
and methods are very important, specially the science of actuary. Almost all of the
critical sections of insurance are analyzed and solved by the actuary, such as the design
of the new product, the computation of the premium and of the reserve. Usually, the
traditional actuarial theory is based on a fix interest rate. However, since the life
insurance is a long-term economic action, the factors of government policy and
economic cycles may cause the interest rate to be uncertain during the time of insurance.
For example, in the recent years, the fluctuation of interest rate had been played an
important influence on Life Insurance Company and led to great interest risk on china.
So, the study on actuarial theory and method with random interest rate has become an
important and popular topic.
In this paper, with the random interest rate based on the Brown process, the net
premium and net premium reserves are calculated, meanwhile, the variance of the future
loss present value are computed by using the relations between mortality insurance and
survival annuity with random interest rate, and the results are simple in structure and
may be easily used for practical calculations. And then, two examples are analyzed on
the results of the model, and some suggests are given.
Key word: stochastic interest rate, Brown process, actuary,
actuarial model, reserve, recurrence relation
I
目 录
中文摘要
ABSTRACT
第一章 绪论 ·······················································································1
§1.1 人寿保险概述 ··········································································1
§1.2 课题的来源及意义 ····································································2
§1.3 目前国内外的研究现状 ······························································4
§1.4 本文的主要工作 ·······································································5
第二章 常利率下的寿险精算原理 ··························································· 6
§2.1 利息的度量与年金 ·····································································6
§2.2 生存分布与生命表 ····································································7
§2.3 寿险与生存年金 ····································································· 10
§2.4 均衡纯保费与责任准备金 ························································· 12
第三章 随机利率下的寿险与年金 ··························································16
§3.1 布朗运动 ·············································································· 16
§3.2 随机利率下的寿险 ·································································· 17
§3.2.1 随机利率下连续型寿险的现值和方差 ··································· 17
§3.2.2 随机利率下离散型寿险的现值和方差 ··································· 18
§3.2.3 随机利率下离散型寿险与连续型寿险的变换关系 ···················· 19
§3.2.4 随机利率下离散型终身寿险趸缴保费的递推公式 ···················· 21
§3.2.5 随机利率下连续型终身寿险趸缴保费的微分方程式 ················· 22
§3.3 随机利率下的生存年金 ···························································· 25
§3.3.1 随机利率下离散型生存年金 ··············································· 25
§3.3.2 随机利率下连续型生存年金 ··············································· 25
§3.3.3 随机利率下寿险与生存年金的关系 ······································ 26
§3.3.4 随机利率下连续型生存年金现值的方差 ································ 28
第四章 随机利率下的纯保费和准备金 ····················································32
§4.1 随机利率下的未来损失及均衡纯保费 ·········································· 32
§4.1.1 随机利率下全离散型的均衡纯保费 ······································ 32
§4.1.2 随机利率下全连续型的均衡纯保费 ······································ 32
§4.1.3 随机利率下未来损失的方差分析 ········································· 33
§4.2 随机利率下的寿险准备金 ························································· 36
II
§4.2.1 随机利率下全离散型的寿险准备金 ······································ 36
§4.2.2 随机利率下全连续型的寿险准备金 ······································ 37
§4.2.3 随机利率下全连续型责任准备金的微分方程 ·························· 39
§4.2.4 全连续型寿命 t时刻的未来损失风险分析 ······························40
第五章 随机利率下寿险实务研究 ··························································43
§5.1 实例分析 ·············································································· 43
§5.1.1 基本假设与设计步骤 ························································ 43
§5.1.2 结果分析 ········································································44
§5.2 实务研究 ············································································· 49
第六章 总结与展望 ··············································································51
参考文献 ··························································································· 53
在读期间公开发表的论文和承担科研项目及取得成果 ································· 55
致 谢 ······························································································ 56
第一章 绪论
1
第一章 绪论
§1.1 人寿保险概述
人寿保险是以人的寿命为保险标的的人身保险．当被保险人死亡或达到保险
合同约定的年龄或期限时，由保险人承担给付保险金的责任．显然保险人对保额
的支付的责任并不是确定的，它依赖于被保人自保单生效之日起的剩余寿命，而
且不仅保险人保险责任的支付时间是随机的，保额本身也是可以变动的．
人寿保险是人身保险中产生得最早的一种，主要被用于处理两类人身风险．一
是被保险人死得过早，未能完成其社会责任，而使依靠其维持生活的人或与其合
作的人陷于困境，二是被保人长寿但又无充分的物质准备，而使自己年老时的生
活失去依靠．此外，购买人寿保险也可作为一种投资手段．
人寿保险具有以下特点．
(i)运用生命表
现代人寿保险的经营离不开生命表．寿险经验生命表的发明，揭示了被保险
人寿命的运动规律，使寿险经营建立于科学的基础之上，也使寿险精算较早地成
为一门科学，并被有效地利用．根据生命表，人寿保险所承保的风险，从整体上
看具有相对稳定性．同时，就每个人而言，人的风险具有变动性．
(ii)均衡保费制
均衡保费制是现代人寿保险的重要理论基础．根据这一理论，投保人每期缴
纳相同的保费，而不随被保险人死亡率的变化而逐年变化．这样做，不致因费用
负担过重而使被保险人在晚年得不到保险保障．在这一制度下，保险前期的均衡
费率往往高于自然费率，而保险后期的均衡费率则低于自然费率．保险人用前期
多收的保费及其利息弥补后期不足的保险费．
(iii)长期性
和意外伤害保险和健康保险不同，人寿保险绝大多数是长期性业务，保险期
限可以是几年、几十年．因而寿险保单一般不必年年更新，手续、环节因此减少．而
且保险人可以从长期稳定的保费收入中取得巨额的、可供长期运用的资金，从而
获得投资收益．这种收益也可以一定方式返还给投保方．因此，寿险精算必须考
虑利息因素．
( iv)储蓄性
由于采用均衡保费制，在一定时期内保险人收取的纯保费超过该时期所支付
的保险金，超过的部分是投保人提前交给保险人的资金，相当于投保人在保险人
随机利率下的寿险精算与实务研究
2
处的一种长期存款，故谓之储蓄保费．保险人将此用于投资，得到收益．只要保
单没有到期，就有责任准备金，它属于保单所有人之权益，保险人要为将来的给
付做准备．人寿保险以外的其他保险，一般没有储蓄性．
§1.2 课题的来源及意义
由于 20 世纪 70 年代以前，各个国家的利率一般都是由国家控制的，并且在
一定时期内是不变的，此时，保险公司在费率厘定及制定准备金时很少考虑利率
带来的影响．但是，70 年代以后，利率管制被冲破，利率向着市场化、自由化的
方向发展，利率的波动更加频繁，并对保险业产生了极大的影响．二十世纪八十
年代末到九十年代初，美国寿险业曾出现过较大规模的偿付能力危机，九十年代
后期日本寿险公司出现了大规模倒闭现象．
表一：近十五年我国利率水平变动情况
调整时间
活期
一年定期
调整时间
一年定期
1990.04.15
2.88
10.08
1998.03.25
5.22
1990.08.21
2.16
8.64
1998.07.01
4.77
1991.04.21
1.8
7.56
1998.12.07
3.78
1993.05.15
2.16
9.18
1999.06.10
2.25
1993.07.11
3.15
10.98
2002.02.21
1.98
1996.05.01
2.97
9.18
2004.10.29
2.25
1996.08.23
1.98
7.47
2006.08.19
2.52
1997.10.23
1.71
5.67
引自中国人民银行官方网站
上个世纪 80 年代末到 90 年代初，我国经济以 8％的速度增长，同期银行存款
利率也水涨船高．寿险公司在这个时期推出的产品由于受银行存款利率的影响，
预定利率也相对较高，最高时达到 13％．以后，央行连续几次下调银行存款利率，
使寿险产品的预定利率也紧跟下降，由十年前的 7%-9%下降到现在的 2.5%．但是，
高预定利率时期发展的寿险业务留下的利差损，则是在今后相当一个时期都难以
弥补的．中国的利差损问题与日本的利差损问题有类似的成因，而且中国近几年
寿险经营环境的改变和产品的创新与 70 年代后期的美国十分相似．
利率的波动对寿险公司主要有以下几个方面的影响：
（1）利率波动对寿险公司资产的影响
寿险公司资产主要来源于自有资本和保费收入，利率波动对这两部分资产的
影响既有直接的，也有间接的．直接影响是指利率波动会引起公司资产的贬值或
增值．比如，利率上升会引起公司持有的股票的贬值．而间接影响包括两个方面：
一是需求效应，利率波动引起寿险需求的变化从而影响实收保费的数量；二是投
摘要：

I摘要在现代保险业经营与发展的过程中，科学的理论与方法，特别是精算学原理起着至关重要的作用，几乎保险业经营的所有的重要环节，如新险种设计、保险费率和责任准备金计算等，都必须经过精算分析处理．在传统的精算理论中，利率都是假定为常数，但，由于寿险是长期性的经济形为，保险期间内，政府政策、经济周期等因素都会造成利率的不确定性，如近年来，我国多次下调利率且降幅较大，对寿险公司产生了严重冲击，最终导致辞了利率风险，从而随机利率下的寿险精算理论与方法的研究成为近年来研究的重点与热点问题．为了研究保险公司在承保后所面临的风险，本文在随机利率的假设方法的基础上，从理论上研究随机利率对寿险公司的影响并进行相关的...

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作者：赵德峰 分类：高等教育资料 价格：15积分 属性：58 页 大小：2.61MB 格式：PDF 时间：2024-11-19

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