advanced-long-term-actuarial-mathematics

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Original

English

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Chinese

Advanced Long-Term Actuarial Mathematics

高级长期精算数学

When to Use

适用场景

Build and interpret survival models: (l_x), (q_x), (\mu_x), select vs ultimate mortality
Value life insurance and annuity benefits (term, whole life, endowment, temporary/whole life annuities)
Derive premiums and reserves via equivalence principle, Thiele, prospective vs retrospective views
Model multiple decrements and Markov/multi-state paths (disability, pension, active/retired)
Apply yield curves and discounting for long-dated liability cash flows
Frame mortality improvement and longevity risk at a technical level
Outline profit testing and emerging cost for life products (conceptual)
Estimate and graduate mortality tables; document data and smoothing choices
Connect math to pricing, reserving, and pension liability measurement; hand execution to
```
actuarial-analyst
```

构建并解读生存模型：(l_x)、(q_x)、(\mu_x)、选择死亡率与终极死亡率的对比
对人寿保险和年金福利进行估值（定期寿险、终身寿险、两全保险、临时/终身年金）
通过等价原理、Thiele方法、前瞻法与回溯法推导保费与准备金
建模多减因与马尔可夫/多状态路径（伤残、养老金、在职/退休状态）
为长期负债现金流应用收益率曲线与贴现方法
从技术层面阐述死亡率改善与长寿风险
概述人寿产品的利润测试与显成本（概念层面）
估算并修匀死亡率表；记录数据与平滑选择
将数学方法与定价、准备金计提和养老金负债计量关联；将执行工作转交
```
actuarial-analyst
```

When NOT to Use

不适用场景

P&C frequency-severity, aggregate loss, credibility, or short-tail reserving math →
```
advanced-short-term-actuarial-mathematics
```
Triangle workbooks, exhibits, statutory tie-outs, or model run packs only →
```
actuarial-analyst
```
Appointed actuary opinions, regulatory sign-off, or enterprise capital policy →
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actuary
```
,
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appointed-chief-actuary
```
Enterprise assumption governance, assumption papers, and change control →
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assumption-setting
```
ALM duration matching, LDI hedge design, and investment policy depth →
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asset-liability-management
```
Life/health product mechanics, underwriting, or claims without contingency math focus →
```
life-health-insurance
```
Pension ERISA, PBGC, fiduciary, or plan design narrative without contingency formulas →
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pension-retirement-funds
```
Exam cram or past-exam solutions as the sole deliverable (support professional application; exam study is secondary)
General data science, ML pipelines, or quant research without life-contingency framing →
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data-scientist
```
,
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quantitative-researcher
```
Credential pathway and exam strategy only →
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associate-actuary
```

财产险的频率-severity模型、总损失模型、可信度模型或短尾准备金数学 →
```
advanced-short-term-actuarial-mathematics
```
仅涉及三角工作簿、图表、法定核对或模型运行包的场景 →
```
actuarial-analyst
```
指定精算师意见、监管签字或企业资本政策相关场景 →
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actuary
```
、
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appointed-chief-actuary
```
企业假设治理、假设文档与变更控制相关场景 →
```
assumption-setting
```
资产负债管理久期匹配、负债驱动投资（LDI）对冲设计与投资政策细节相关场景 →
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asset-liability-management
```
未聚焦或有事项数学的人寿/健康产品机制、核保或理赔相关场景 →
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life-health-insurance
```
未包含或有事项公式的养老金ERISA、PBGC、受托责任或计划设计说明相关场景 →
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pension-retirement-funds
```
仅提供考试突击复习或过往考试答案的场景（支持专业应用；考试学习为次要目的）
未结合人寿或有事项框架的通用数据科学、机器学习管道或量化研究相关场景 →
```
data-scientist
```
、
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quantitative-researcher
```
仅涉及认证路径与考试策略的场景 →
```
associate-actuary
```

Related skills

Need	Skill
Workpapers, model I/O, exhibits, analyst QA	`actuarial-analyst`
Sign-off, capital overview, governance memos	`actuary`
Appointed actuary / chief actuary regulatory framing	`appointed-chief-actuary`
ASA/FSO exam pathways and professional standards	`associate-actuary`
Assumption governance and enterprise change control	`assumption-setting`
Life/health products, benefits, and distribution context	`life-health-insurance`
DB/DC pensions, funding policy, liability overview	`pension-retirement-funds`
ALM, duration, LDI, and asset–liability strategy	`asset-liability-management`
Short-term loss models (ASTAM)	`advanced-short-term-actuarial-mathematics`
Statistical/ML modeling beyond standard actuarial methods	`quantitative-researcher`
General ML and predictive pipelines	`data-scientist`

需求	技能
工作底稿、模型输入输出、图表、分析师质量保证	`actuarial-analyst`
签字确认、资本概览、治理备忘录	`actuary`
指定精算师/首席精算师的监管框架	`appointed-chief-actuary`
ASA/FSO考试路径与专业标准	`associate-actuary`
假设治理与企业变更控制	`assumption-setting`
人寿/健康产品、福利与分销背景	`life-health-insurance`
确定给付/确定缴费型养老金、筹资政策、负债概览	`pension-retirement-funds`
资产负债管理、久期、负债驱动投资（LDI）与资产负债策略	`asset-liability-management`
短期损失模型（ASTAM）	`advanced-short-term-actuarial-mathematics`
超出标准精算方法的统计/机器学习建模	`quantitative-researcher`
通用机器学习与预测管道	`data-scientist`

Core Workflows

核心工作流程

1. Problem framing (ALTAM-aligned)

1. 问题框架（符合ALTAM范畴）

Before deriving formulas:

Horizon — Long-duration; valuation vs pricing vs funding purpose
State space — Single life, joint life, or multi-state (active/disabled/retired/dead)
Benefit/premium basis — Net vs gross; continuous vs discrete; payment timing
Mortality basis — Ultimate vs select; table vintage; improvement scale (if any)
Interest/discount — Flat rate vs term structure; real vs nominal
Deliverable — Formula spec, APV tables, reserve recursion, sensitivity—not filing sign-off
Peer execution — Route spreadsheet builds and filing exhibits to
```
actuarial-analyst
```

See
references/altam_scope_and_principles.md
.

推导公式前：

时间范围 — 长期；估值、定价或筹资目的
状态空间 — 单生命、联合生命或多状态（在职/伤残/退休/身故）
福利/保费基础 — 净保费 vs 总保费；连续型 vs 离散型；支付时点
死亡率基础 — 终极死亡率 vs 选择死亡率；表格年代；改善尺度（如有）
利率/贴现 — 固定利率 vs 期限结构；实际利率 vs 名义利率
交付成果 — 公式规范、精算现值表、准备金递推、敏感性分析——而非申报签字文件
Peer执行 — 将电子表格构建与申报图表转交
```
actuarial-analyst
```

参见
references/altam_scope_and_principles.md
。

2. Survival models and mortality

2. 生存模型与死亡率

Define life table functions: (l_x), (d_x), (q_x), (p_x), (\mu_x)
Distinguish ultimate vs select mortality and duration since selection
State independence assumptions for joint lives unless modeling dependence
Document radix, age basis, and table source (regulatory, industry, experience)
Link force of mortality to discrete (q_x) under stated assumptions

See
references/survival_models_and_mortality.md
.

定义生命表函数：(l_x)、(d_x)、(q_x)、(p_x)、(\mu_x)
区分终极与选择死亡率及选择后的持续期
阐述联合生命的独立性假设（除非建模依赖性）
记录基数、年龄基准与表格来源（监管、行业、经验数据）
在既定假设下将死亡率与离散型(q_x)关联

参见
references/survival_models_and_mortality.md
。

3. Life insurance and annuities

3. 人寿保险与年金

Map benefit types: level term, whole life, endowment, deferred benefits
Write actuarial present value (APV) with correct survival and payment timing
Cover annuities: temporary, whole life, due vs immediate, varying payments
Address joint-life and last-survivor structures at formula level
Flag riders and options for product context →
```
life-health-insurance
```

See
references/life_insurance_and_annuities.md
.

梳理福利类型：定额定期寿险、终身寿险、两全保险、递延福利
结合正确的生存概率与支付时点撰写精算现值（APV）
覆盖年金：临时年金、终身年金、期初付 vs 期末付、变额支付
在公式层面处理联合生命与最后生存者结构
标注附加条款与选择权以提供产品背景 →
```
life-health-insurance
```

参见
references/life_insurance_and_annuities.md
。

4. Premiums and reserves

4. 保费与准备金

Apply equivalence principle for net/gross premiums; define expense and profit loadings
Build prospective and retrospective reserves; reconcile where standard
Use Thiele's equation for continuous reserves; discrete analogs where needed
Distinguish net level premium vs gross premium reserves
Summarize deficiency and emerging cost implications (conceptual)

See
references/premiums_and_reserves.md
.

应用等价原理计算净/总保费；定义费用与利润加载
构建前瞻法与回溯法准备金；在标准场景下进行核对
使用Thiele方程计算连续型准备金；必要时使用离散型类比
区分均衡净保费与总保费准备金
概述不足准备金与显成本的影响（概念层面）

参见
references/premiums_and_reserves.md
。

5. Multiple decrement and state models

5. 多减因与状态模型

Set up multiple decrement table: dependent vs independent decrements
Compute probabilities of single and combined decrements; associated single decrement rates
Extend to Markov multi-state models (transition intensities, Kolmogorov forward equations)
Apply to disability, pension, and long-term care state spaces (math only)
Hand pension plan law and governance narrative to
```
pension-retirement-funds
```

See
references/multiple_decrement_and_state_models.md
.

建立多减因表：相依减因 vs 独立减因
计算单一减因与组合减因的概率；关联单一减因率
扩展至马尔可夫多状态模型（转移强度、柯尔莫哥洛夫向前方程）
将模型应用于伤残、养老金与长期护理状态空间（仅数学层面）
将养老金计划法律与治理说明转交
```
pension-retirement-funds
```

参见
references/multiple_decrement_and_state_models.md
。

6. Longevity, improvement, estimation, and discounting

6. 长寿风险、死亡率改善、估算与贴现

Frame mortality improvement scales and longevity risk (framework-level)
Describe estimation from experience; graduation and smoothing methods
Apply yield curves to long cash flows; note ALM interaction at high level →
```
asset-liability-management
```
Outline profit testing cash flows and sensitivity to mortality and interest
Package assumptions for actuary review; governance →
```
assumption-setting
```

See
references/longevity_improvement_and_estimation.md
.

构建死亡率改善尺度与长寿风险的框架（框架层面）
描述基于经验数据的估算方法；修匀与平滑方法
为长期现金流应用收益率曲线；提及与资产负债管理的高层交互 →
```
asset-liability-management
```
概述利润测试现金流及对死亡率与利率的敏感性
整理假设供精算师审核；治理相关工作转交
```
assumption-setting
```

参见
references/longevity_improvement_and_estimation.md
。

Deliverable standards

交付成果标准

Deliverable	Minimum content
Model specification	States, benefits, premium type, mortality/interest basis, payment timing
Formula sheet	APV, premium, reserve recursions with defined notation
Numerical illustration	Small worked example or table with stated assumptions
Sensitivity	Mortality, interest, improvement, or lapse drivers (as relevant)
Limitations	Table vintage, extrapolation, select period, data volume

Label output as technical modeling support, not actuarial opinion, legal advice, or filed regulatory submission.

交付成果	最低内容要求
模型规范	状态、福利、保费类型、死亡率/利率基础、支付时点
公式表	精算现值、保费、准备金递推公式及定义符号
数值示例	小型演算示例或带既定假设的表格
敏感性分析	死亡率、利率、改善率或退保率驱动因素（如适用）
局限性说明	表格年代、外推法、选择期、数据量

将输出标注为技术建模支持，而非精算意见、法律建议或已提交的监管申报文件。

Assignment type matrix

任务类型矩阵

Trigger phrase	Primary workflow	Lead reference
survival model / force of mortality	Life table and (\mu_x)	`survival_models_and_mortality.md`
life insurance APV / endowment	Benefit APVs	`life_insurance_and_annuities.md`
annuity valuation	Annuity due/immediate	`life_insurance_and_annuities.md`
equivalence principle / net premium	Premium derivation	`premiums_and_reserves.md`
Thiele equation / reserve	Reserve recursion	`premiums_and_reserves.md`
multiple decrement	Decrement table	`multiple_decrement_and_state_models.md`
Markov / disability / pension states	State models	`multiple_decrement_and_state_models.md`
mortality improvement / longevity risk	Improvement frameworks	`longevity_improvement_and_estimation.md`
graduation / mortality table	Estimation and smooth	`longevity_improvement_and_estimation.md`
yield curve / discount long cash flows	Discounting + ALM cross-ref	`longevity_improvement_and_estimation.md`

触发短语	主要工作流程	主导参考文档
survival model / force of mortality	生命表与(\mu_x)	`survival_models_and_mortality.md`
life insurance APV / endowment	福利精算现值	`life_insurance_and_annuities.md`
annuity valuation	期初付/期末付年金	`life_insurance_and_annuities.md`
equivalence principle / net premium	保费推导	`premiums_and_reserves.md`
Thiele equation / reserve	准备金递推	`premiums_and_reserves.md`
multiple decrement	减因表	`multiple_decrement_and_state_models.md`
Markov / disability / pension states	状态模型	`multiple_decrement_and_state_models.md`
mortality improvement / longevity risk	改善框架	`longevity_improvement_and_estimation.md`
graduation / mortality table	估算与平滑	`longevity_improvement_and_estimation.md`
yield curve / discount long cash flows	贴现 + 资产负债管理交叉参考	`longevity_improvement_and_estimation.md`

When to load references

何时加载参考文档

Scope, ALTAM alignment, principles →

references/altam_scope_and_principles.md

Survival models and mortality →

references/survival_models_and_mortality.md

Life insurance and annuities →

references/life_insurance_and_annuities.md

Premiums and reserves →
```
references/premiums_and_reserves.md
```

Multiple decrement and state models →

references/multiple_decrement_and_state_models.md

Longevity, improvement, estimation, discounting →

references/longevity_improvement_and_estimation.md

范围、ALTAM对齐、原则 →

references/altam_scope_and_principles.md

生存模型与死亡率 →

references/survival_models_and_mortality.md

人寿保险与年金 →

references/life_insurance_and_annuities.md

保费与准备金 →
```
references/premiums_and_reserves.md
```

多减因与状态模型 →

references/multiple_decrement_and_state_models.md

长寿风险、死亡率改善、估算、贴现 →
```
references/longevity_improvement_and_estimation.md
```