Actuarial Mahematics

Actuarial Mathematics
and Life-Table Statistics
Eric V. Slud
Mathematics Department
University of Maryland, College Park
March 22, 2009

c 2009
Eric V. Slud
Statistics Program
Mathematics Department
University of Maryland
College Park, MD 20742

Contents
0.1

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Basics of Probability & Interest
1.1

iv1
1

1.1.1

Random Variables and Expectations . . . . . . . . . .

6

Theory of Interest . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2.1

Interest Rates and Compounding . . . . . . . . . . . .

9

1.2.2

Present Values and Payment Streams . . . . . . . . . . 14

1.2.3

Principal and Interest, and Discount Rates . . . . . . . 17

1.2.4

Variable InterestRates . . . . . . . . . . . . . . . . . . 20

1.2.5

1.2

Probabilities about Lifetimes . . . . . . . . . . . . . . . . . . .

Continuous-time Payment Streams . . . . . . . . . . . 24

1.3

Exercise Set 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.4

Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5

Useful Formulas . . . . . . . . . . . .. . . . . . . . . . . . . . 31

2 Interest & Force of Mortality
2.1

33

More on Theory of Interest . . . . . . . . . . . . . . . . . . . . 33
2.1.1

Annuities & Actuarial Notation . . . . . . . . . . . . . 34

2.1.2

Loan Repayment: Mortgage, Bond, Sinking Fund . . . 39
i

ii

CONTENTS
2.1.3
2.1.4

Illustration on Mortgage Refinancing . . . . . . . . . . 42

2.1.5
2.2Loan Amortization & Mortgage Refinancing . . . . . . 41

Computational illustration in R . . . . . . . . . . . . . 44

Force of Mortality & Analytical Models . . . . . . . . . . . . . 48
2.2.1

Comparison of Forces of Mortality . . . . . . . . . . . . 55

2.3

Exercise Set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.4

Worked Examples . . . . . . . . . . . . . . .. . . . . . . . . . 65

2.5

Useful Formulas from Chapter 2 . . . . . . . . . . . . . . . . . 69

3 Probability & Life Tables
3.1

73

Binomial Variables & Law of Large Numbers . . . . . . . . . . 74
3.1.1

Probability Bounds & Approximations . . . . . . . . . 77

3.2

Simulation of Discrete Lifetimes . . . . . . . . . . . . . . . . . 80

3.3

Expectation of Discrete RandomVariables . . . . . . . . . . . 84
3.3.1

Rules for Manipulating Expectations . . . . . . . . . . 87

3.3.2

Curtate Expectation of Life . . . . . . . . . . . . . . . 90

3.4

Interpreting Force of Mortality . . . . . . . . . . . . . . . . . . 91

3.5

Interpolation Between Integer Ages . . . . . . . . . . . . . . . 92
3.5.1

Life Expectancy – Definition and Approximation . . . 963.6

Some Special Integrals . . . . . . . . . . . . . . . . . . . . . . 97

3.7

Exercise Set 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.8

Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.9

Appendix on Large Deviation Probabilities . . . . . . . . . . . 109

3.10 Useful Formulas from Chapter 3 . . . . . . . . . . . . . . . . . 112
4Expected Present Values of Payments

115

CONTENTS

iii

4.1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.2

Types of Contracts . . . . . . . . . . . . . . . . . . . . . . . . 118
4.2.1

Formal Relations, m = 1 . . . . . . . . . . . . . . . . . 121

4.2.2

Formulas for Net Single Premiums . . . . . . . . . . . 123

4.3

Extension toMultiple Payments per Year . . . . . . . . . . . . 125

4.4

Interpolation Formulas in Risk Premiums . . . . . . . . . . . . 130

4.5

Continuous Risk Premium Formulas . . . . . . . . . . . . . . . 132
4.5.1

Continuous Contracts & Residual Life . . . . . . . . . 133

4.5.2

Integral Formulas . . . . . . . . . . . . . . . . . . . . . 134

4.5.3

Risk Premiums under Theoretical Models...