Actuarial Mahematics
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...
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