Fundamentals of Actuarial Mathematics
暫譯: 精算數學基礎

S. David Promislow

Fundamentals of Actuarial Mathematics
  • 出版商: Wiley
  • 出版日期: 2006-02-10
  • 售價: $1,078
  • 語言: 英文
  • 頁數: 392
  • 裝訂: Hardcover
  • ISBN: 0470016892
  • ISBN-13: 9780470016893

    • 下單後立即進貨 (約5~7天)

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Description

Actuarial work is the application of mathematics and statistics to the analysis of financial problems in life insurance, pensions, general insurance and investments. This unique introduction to the topic employs both a deterministic and stochastic treatment of the subject. It combines interest theory and life contingencies in a unified manner as well as covering basic risk theory. Fundamentals of Actuarial Mathematics presents the concepts in an original, accessible style, assuming a minimal formal background.

  • Provides a complete review of necessary probability theory.
  • Covers the Society of Actuaries’ syllabus on Actuarial Models.
  • Orders the topics specifically to facilitate learning, beginning with the simplest case of the deterministic discrete model, and then moving to the more complicated stochastic, continuous models.
  • Employs modern calculation and computing techniques, such as spreadsheets.
  • Contains a variety of exercises, both computational and theoretical.
  • Supported by a website featuring exercises and further examples.
  • Written by a highly respected academic with over 35 years teaching experience.

 

Table of Contents

Preface.

Notation index.

PART I: THE DETERMINISTIC MODEL.

1. Introduction and motivation.

1.1 Risk and insurance.

1.2 Deterministic versus stochastic models.

1.3 Finance and investments.

1.4 Adequacy and equity.

1.5 Reassessment.

1.6 Conclusion.

2. The basic deterministic model.

2.1 Cashflows.

2.2 An analogy with currencies.

2.3 Discount functions.

2.4 Calculating the discount function.

2.5 Interest and discount rates.

2.6 The constant interest case.

2.7 Values and actuarial equivalence.

2.8 The case of equal cashflows.

2.9 Balances and reserves.

2.10 Time shifting and the splitting identity.

*2.11 Change of discount function.

*2.12 Internal rate of return.

2.13 Standard notation and terminology.

2.14 Spreadsheet calculations.

2.15 Notes and references.

Exercises.

3. The life table.

3.1 Basic definitions.

3.2 Probabilities.

3.3 Constructing the life table from the values of qx.

3.4 Life expectancy.

3.5 Choice of life tables.

3.6 Standard notation and terminology.

3.7 A sample table.

3.8 Notes and references.

Exercises.

4. Life annuities.

4.1 Introduction.

4.2 Calculating annuity premiums.

4.3 The interest and survivorship discount function.

4.4 Guaranteed payments.

4.5 Deferred annuities with annual premiums.

4.6 Some practical considerations.

4.7 Standard notation and terminology.

4.8 Spreadsheet calculations.

Exercises.

5. Life insurance.

5.1 Introduction.

5.2 Calculating life insurance premiums.

5.3 Types of life insurance.

5.4 Combined benefits.

5.5 Insurances viewed as annuities.

5.6 Summary of formulas.

5.7 A general insurance–annuity identity.

5.8 Standard notation and terminology.

5.9 Spreadsheet applications.

Exercises.

6. Insurance and annuity reserves.

6.1 Introduction to reserves.

6.2 The general pattern of reserves.

6.3 Recursion.

6.4 Detailed analysis of an insurance or annuity contract.

6.5 Bases for reserves.

6.6 Nonforfeiture values.

6.7 Policies involving a return of the reserve.

6.8 Premium difference and paid-up formulas.

6.9 Standard notation and terminology.

6.10 Spreadsheet applications.

Exercises.

7. Fractional durations.

7.1 Introduction.

7.2 Cashflows discounted with interest only.

7.3 Life annuities paid mthly.

7.4 Immediate annuities.

7.5 Approximation and computation.

7.6 Fractional period premiums and reserves.

7.7 Reserves at fractional durations.

7.8 Notes and references.

Exercises.

8. Continuous payments.

8.1 Introduction to continuous annuities.

8.2 The force of discount.

8.3 The constant interest case.

8.4 Continuous life annuities.

8.5 The force of mortality.

8.6 Insurances payable at the moment of death.

8.7 Premiums and reserves.

8.8 The general insurance–annuity identity in the continuous case.

8.9 Differential equations for reserves.

8.10 Some examples of exact calculation.

8.11 Standard notation and terminology.

8.12 Notes and references.

Exercises.

9. Select mortality.

9.1 Introduction.

9.2 Select and ultimate tables.

9.3 Changes in formulas.

9.4 Further remarks.

Exercises.

10. Multiple-life contracts.

10.1 Introduction.

10.2 The joint-life status.

10.3 Joint-life annuities and insurances.

10.4 Last-survivor annuities and insurances.

10.5 Moment of death insurances.

10.6 The general two-life annuity contract.

10.7 The general two-life insurance contract.

10.8 Contingent insurances.

10.9 Standard notation and terminology.

10.10 Spreadsheet applications.

10.11 Notes and references.

Exercises.

11. Multiple-decrement theory.

11.1 Introduction.

11.2 The basic model.

11.3 Insurances.

11.4 Determining the model from the forces of decrement.

11.5 The analogy with joint-life statuses.

11.6 A machine analogy.

11.7 Associated single-decrement tables.

11.8 Notes and references.

Exercises.

12. Expenses.

12.1 Introduction.

12.2 Effect on reserves.

12.3 Realistic reserve and balance calculations.

12.4 Notes and references.

Exercises.

PART II: THE STOCHASTIC MODEL.

13. Survival distributions and failure times.

13.1 Introduction to survival distributions.

13.2 The discrete case.

13.3 The continuous case.

13.4 Examples.

13.5 Shifted distributions.

13.6 The standard approximation.

13.7 The stochastic life table.

13.8 Life expectancy in the stochastic model.

13.9 Notes and references.

Exercises.

14. The stochastic approach to insurance and annuities.

14.1 Introduction.

14.2 The stochastic approach to insurance benefits.

14.3 The stochastic approach to annuity benefits.

14.4 Deferred contracts.

14.5 The stochastic approach to reserves.

14.6 The stochastic approach to premiums.

14.7 The variance of rL.

14.8 Standard notation and terminology.

14.9 Notes and references.

Exercises.

15. Simplifications under constant benefit contracts.

15.1 Introduction.

15.2 Variance calculations in the continuous case.

15.3 Variance calculations in the discrete case.

15.4 Exact distributions.

15.5 Some nonconstant benefit examples.

Exercises.

16. The minimum failure time.

16.1 Introduction.

16.2 Joint distributions.

16.3 The distribution of T.

16.4 The joint distribution of (T J).

16.5 Approximations.

16.6 Other problems.

16.7 The common shock model.

16.8 Copulas.

16.9 Notes and references.

Exercises.

PART III: RISK THEORY.

17. Compound distributions.

17.1 Introduction.

17.2 The mean and variance of S.

17.3 Generating functions.

17.4 Exact distribution of S.

17.5 Choosing a frequency distribution.

17.6 Choosing a severity distribution.

17.7 Handling the point mass at 0.

17.8 Counting claims of a particular type.

17.9 The sum of two compound Poisson distributions.

17.10 Deductibles and other modifications.

17.11 A recursion formula for S.

17.12 Notes and references.

Exercises.

18. An introduction to stochastic processes.

18.1 Introduction.

18.2 Markov chains.

18.3 Examples.

18.4 Martingales.

18.5 Finite-state Markov chains.

18.6 Multi-state insurances and annuities.

18.7 Notes and references.

Exercises.

19. Poisson processes.

19.1 Introduction.

19.2 Definition of a Poisson process.

19.3 Waiting times.

19.4 Some properties of the Poisson process.

19.5 Nonhomogeneous Poisson processes.

19.6 Compound Poisson processes.

19.7 Notes and references.

Exercises.

20. Ruin models.

20.1 Introduction.

20.2 A functional equation approach.

20.3 The martingale approach to ruin theory.

20.4 Distribution of the deficit at ruin.

20.5 Recursion formulas.

20.6 The compound Poisson surplus process.

20.7 The maximal aggregate loss.

20.8 Notes and references.

Exercises.

Appendix: A review of probability theory.

A.1 Introduction.

A.2 Sample spaces and probability measures.

A.3 Conditioning and independence.

A.4 Random variables.

A.5 Distributions.

A.6 Expectations and moments.

A.7 Expectation in terms of the distribution function.

A.8 Joint distributions.

A.9 Conditioning and independence for random variables.

A.10 Convolution.

A.11 Moment generating functions.

A.12 Probability generating functions.

A.13 Mixtures.

Answers to exercises.

References.

Index.

 

商品描述(中文翻譯)

**描述**

精算工作是將數學和統計應用於人壽保險、退休金、一般保險和投資的財務問題分析。這本獨特的入門書籍以確定性和隨機性兩種方式探討主題。它以統一的方式結合了利息理論和生命事件，並涵蓋了基本的風險理論。《精算數學基礎》以原創且易於理解的風格呈現概念，假設讀者具備最少的正式背景。

- 提供必要的概率理論的完整回顧。
- 涵蓋精算學會的精算模型課程大綱。
- 特別安排主題以促進學習，從最簡單的確定性離散模型開始，然後轉向更複雜的隨機連續模型。
- 採用現代計算和計算技術，例如電子表格。
- 包含各種計算和理論的練習。
- 支持一個網站，提供練習和進一步的例子。
- 由一位擁有超過35年教學經驗的受人尊敬的學者撰寫。

**目錄**

**前言**

**符號索引**

**第一部分：確定性模型**

**1. 介紹與動機**
1.1 風險與保險
1.2 確定性模型與隨機模型
1.3 財務與投資
1.4 充足性與公平性
1.5 重新評估
1.6 結論

**2. 基本確定性模型**
2.1 現金流
2.2 與貨幣的類比
2.3 折現函數
2.4 計算折現函數
2.5 利率與折現率
2.6 常數利率情況
2.7 價值與精算等價
2.8 相等現金流的情況
2.9 餘額與準備金
2.10 時間轉移與分割恒等式
*2.11 折現函數的變更
*2.12 內部收益率
2.13 標準符號與術語
2.14 電子表格計算
2.15 註解與參考
練習

**3. 生命表**
3.1 基本定義
3.2 機率
3.3 從qx值構建生命表
3.4 生命期望
3.5 生命表的選擇
3.6 標準符號與術語
3.7 一個示例表
3.8 註解與參考
練習

**4. 生命年金**
4.1 介紹
4.2 計算年金保費
4.3 利息與生存折現函數
4.4 保證支付
4.5 具有年度保費的遞延年金
4.6 一些實際考量
4.7 標準符號與術語
4.8 電子表格計算
練習

**5. 人壽保險**
5.1 介紹
5.2 計算人壽保險保費
5.3 人壽保險的類型
5.4 組合利益
5.5 將保險視為年金
5.6 公式摘要
5.7 一般保險–年金恒等式
5.8 標準符號與術語
5.9 電子表格應用
練習

**6. 保險與年金準備金**
6.1 準備金介紹
6.2 準備金的一般模式
6.3 遞歸
6.4 對保險或年金合約的詳細分析
6.5 準備金的基礎
6.6 不可喪失的價值
6.7 涉及準備金回報的保單
6.8 保費差額與已繳保費公式
6.9 標準符號與術語
6.10 電子表格應用
練習

**7. 分數期間**
7.1 介紹
7.2 僅以利息折現的現金流
7.3 每月支付的生命年金
7.4 即時年金
7.5 近似與計算
7.6 分數期間的保費與準備金
7.7 分數期間的準備金
7.8 註解與參考
練習

**8. 持續支付**
8.1 持續年金介紹
8.2 折現力
8.3 常數利率情況
8.4 持續生命年金
8.5 死亡率
8.6 在死亡時支付的保險
8.7 保費與準備金
8.8 持續情況下的一般保險–年金恒等式
8.9 準備金的微分方程
8.10 一些精確計算的例子
8.11 標準符號與術語
8.12 註解與參考
練習

**9. 選擇性死亡率**
9.1 介紹
9.2 選擇性與最終表
9.3 公式的變更
9.4 進一步的說明
練習

**10. 多重生命合約**
10.1 介紹
10.2 聯合生命狀態
10.3 聯合生命年金與保險
10.4 最後生存者年金與保險
10.5 死亡時的保險
10.6 一般雙生命年金合約
10.7 一般雙生命保險合約
10.8 附帶保險
10.9 標準符號與術語
10.10 電子表格應用
10.11 註解與參考
練習

**11. 多重減少理論**
11.1 介紹
11.2 基本模型
11.3 保險
11.4 從減少力確定模型
11.5 與聯合生命狀態的類比
11.6 機器類比
11.7 相關的單減少表
11.8 註解與參考
練習

**12. 費用**
12.1 介紹
12.2 對準備金的影響
12.3 實際的準備金與餘額計算
12.4 註解與參考
練習

**第二部分：隨機模型**

**13. 生存分佈與失敗時間**
13.1 生存分佈介紹
13.2 離散情況
13.3 連續情況
13.4 例子
13.5 移位分佈
13.6 標準近似
13.7 隨機生命表
13.8 隨機模型中的生命期望
13.9 註解與參考
練習

**14. 隨機方法的保險與年金**
14.1 介紹
14.2 隨機方法的保險利益
14.3 隨機方法的年金利益
14.4 遞延合約
14.5 隨機方法的準備金
14.6 隨機方法的保費
14.7 rL的變異數
14.8 標準符號與術語
14.9 註解與參考
練習

**15. 在固定利益合約下的簡化**
15.1 介紹
15.2 連續情況下的變異數計算
15.3 離散情況下的變異數計算
15.4 精確分佈
15.5 一些非固定利益的例子
練習

**16. 最小失敗時間**
16.1 介紹
16.2 聯合分佈
16.3 T的分佈
16.4 (T J)的聯合分佈
16.5 近似
16.6 其他問題
16.7 共同衝擊模型
16.8 Copulas
16.9 註解與參考
練習

**第三部分：風險理論**

**17. 複合分佈**
17.1 介紹
17.2 S的均值與變異數
17.3 生成函數
17.4 S的精確分佈
17.5 選擇頻率分佈
17.6 選擇嚴重性分佈
17.7 處理0的點質量
17.8 計算特定類型的索賠
17.9 兩個複合泊松分佈的總和
17.10 自負額與其他修改
17.11 S的遞歸公式
17.12 註解與參考
練習

**18. 隨機過程簡介**
18.1 介紹
18.2 馬可夫鏈
18.3 例子
18.4 養馬
18.5 有限狀態馬可夫鏈
18.6 多狀態保險與年金
18.7 註解與參考
練習

**19. 泊松過程**
19.1 介紹
19.2 泊松過程的定義
19.3 等待時間
19.4 泊松過程的一些性質
19.5 非均勻泊松過程
19.6 複合泊松過程
19.7 註解與參考
練習

**20. 破產模型**
20.1 介紹
20.2 功能方程方法
20.3 養馬方法的破產理論
20.4 破產時的赤字分佈
20.5 遞歸公式
20.6 複合泊松盈餘過程
20.7 最大總損失
20.8 註解與參考
練習

**附錄：概率理論回顧**
A.1 介紹
A.2 樣本空間與概率測度
A.3 條件與獨立性
A.4 隨機變數
A.5 分佈
A.6 期望與矩
A.7 以分佈函數表示的期望
A.8 聯合分佈
A.9 隨機變數的條件與獨立性
A.10 卷積
A.11 矩生成函數
A.12 概率生成函數
A.13 混合

**練習答案**

**參考文獻**

**索引**

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