Financial and Actuarial Statistics: An Introduction

398 Pages·2013·4.06 MB·English

by  Dale S. Borowiak

SECOND SECOND EDITION
Statistics
EDITION
Financial and Actuarial Statistics: An Introduction, Second Edition enables
you to obtain the mathematical and statistical background required in the current
financial and actuarial industries. It also advances the application and theory of A
statistics in modern financial and actuarial modeling. Like its predecessor, this C FINANCIAL
second edition considers financial and actuarial modeling from a statistical point T
of view while adding a substantial amount of new material. UF
AI
New to the Second Edition N
R
• Nomenclature and notations standard to the actuarial field A
I ACTUARIAL
• Excel™ exercises with solutions that demonstrate how to use Excel functions AN
for statistical and actuarial computations LC
AND
• Problems dealing with standard probability and statistics theory, along with SI
A
detailed equation links
T
L
• A chapter on Markov chains and actuarial applications A

• Expanded discussions of simulation techniques and applications, such as TA STATISTICS
IN
investment pricing
S
D
• Sections on the maximum likelihood approach to parameter estimation as T

well as asymptotic applications I
C
• Discussions of diagnostic procedures for nonnegative random variables and
S
Pareto, lognormal, Weibull, and left truncated distributions
AN INTRODUCTION
• Expanded material on surplus models and ruin computations
• Discussions of nonparametric prediction intervals, option pricing diagnostics,
variance of the loss function associated with standard actuarial models, and
Gompertz and Makeham distributions
• Sections on the concept of actuarial statistics for a collection of stochastic
status models
The book presents a unified approach to both financial and actuarial modeling
through the use of general status structures. The authors define future time-
B DALE S. BOROWIAK
dependent financial actions in terms of a status structure that may be either S
O
H
deterministic or stochastic. They show how deterministic status structures lead to R
A
classical interest and annuity models, investment pricing models, and aggregate P O
claim models. They also employ stochastic status structures to develop financial IR W ARNOLD F. SHAPIRO
I
and actuarial models, such as surplus models, life insurance, and life annuity O A
K
models.
C8508
C8508_Cover.indd 1 10/8/13 8:53 AM
FINANCIAL
ACTUARIAL
AND
STATISTICS
AN INTRODUCTION
SECOND EDITION
DALE S. BOROWIAK
University of Akron
Ohio, USA
ARNOLD F. SHAPIRO
Pennsylvania State University
USA
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2014 by Taylor & Francis Group, LLC
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Version Date: 20130923
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Contents
Preface ......................................................................................................................ix
1 Statistical Concepts ........................................................................................1
1.1 Probability ..............................................................................................1
1.2 Random Variables .................................................................................7
1.2.1 Discrete Random Variables ....................................................8
1.2.2 Continuous Random Variables ............................................10
1.2.3 Mixed Random Variables .....................................................13
1.3 Expectations .........................................................................................14
1.4 Moment Generating Function ...........................................................20
1.5 Survival Functions ..............................................................................22
1.6 Nonnegative Random Variables .......................................................25
1.6.1 Pareto Distribution ................................................................25
1.6.2 Lognormal Distribution ........................................................26
1.6.3 Weibull Distribution ..............................................................26
1.6.4 Gompertz Distribution..........................................................27
1.6.5 Makeham Distribution ..........................................................28
1.7 Conditional Distributions ..................................................................29
1.8 Joint Distributions ...............................................................................31
Problems ..........................................................................................................36
Excel Problems ...............................................................................................38
Solutions ..........................................................................................................38
2 Statistical Techniques ..................................................................................41
2.1 Sampling Distributions and Estimation ..........................................41
2.1.1 Point Estimation .....................................................................42
2.1.2 Confidence Intervals ..............................................................44
2.1.3 Percentiles and Prediction Intervals ...................................45
2.1.4 Confidence and Prediction Sets ...........................................46
2.2 Sums of Independent Variables ........................................................49
2.3 Order Statistics and Empirical Prediction Intervals ......................54
2.4 Approximating Aggregate Distributions ........................................57
2.4.1 Central Limit Theorem .........................................................57
2.4.2 Haldane Type A Approximation .........................................61
2.4.3 Saddlepoint Approximation .................................................62
2.5 Compound Aggregate Variables .......................................................65
2.5.1 Expectations of Compound Aggregate Variables .............65
2.5.2 Limiting Distributions for Compound Aggregate
Variables ..................................................................................66
2.6 Regression Modeling ..........................................................................70
iii
iv Contents
2.6.1 Least Squares Estimation ......................................................71
2.6.2 Regression Model-Based Inference .....................................74
2.7 Autoregressive Systems .....................................................................75
2.8 Model Diagnostics ..............................................................................78
2.8.1 Probability Plotting ...............................................................79
2.8.2 Generalized Least Squares Diagnostic ...............................83
2.8.3 Interval Data Diagnostic .......................................................84
Problems ..........................................................................................................87
Excel Problems ...............................................................................................88
Solutions ..........................................................................................................90
3 Financial Computational Models ..............................................................93
3.1 Fixed Financial Rate Models .............................................................94
3.1.1 Financial Rate-Based Calculations ......................................94
3.1.2 General Period Discrete Rate Models .................................99
3.1.3 Continuous-Rate Models ....................................................100
3.2 Fixed-Rate Annuities ........................................................................101
3.2.1 Discrete Annuity Models ...................................................101
3.2.2 Continuous Annuity Models .............................................104
3.3 Stochastic Rate Models .....................................................................106
3.3.1 Discrete Stochastic Rate Model ..........................................106
3.3.2 Continuous Stochastic Rate Models ..................................112
3.3.3 Discrete Stochastic Annuity Models .................................114
3.3.4 Continuous Stochastic Annuity Models...........................116
Problems ........................................................................................................117
Excel Problems .............................................................................................119
Solutions ........................................................................................................120
4 Deterministic Status Models ...................................................................123
4.1 Basic Loss Model ...............................................................................123
4.1.1 Deterministic Loss Models .................................................124
4.1.2 Stochastic Rate Models ........................................................126
4.2 Stochastic Loss Criterion ..................................................................128
4.2.1 Risk Criteria ..........................................................................129
4.2.2 Percentile Criteria ................................................................130
4.3 Single-Risk Models ...........................................................................131
4.3.1 Insurance Pricing .................................................................131
4.3.2 Investment Pricing ...............................................................135
4.3.3 Options Pricing ....................................................................136
4.3.4 Option Pricing Diagnostics ................................................139
4.4 Collective Aggregate Models ..........................................................140
4.4.1 Fixed Number of Variables .................................................141
4.4.2 Stochastic Number of Variables .........................................143
4.4.3 Aggregate Stop-Loss Reinsurance and Dividends .........145
4.5 Stochastic Surplus Model .................................................................148
Contents v
4.5.1 Discrete Surplus Model.......................................................148
4.5.2 Continuous Surplus Model ................................................152
Problems ........................................................................................................155
Excel Problems .............................................................................................158
Solutions ........................................................................................................159
5 Future Lifetime Random Variables and Life Tables ...........................163
5.1 Continuous Future Lifetime ............................................................164
5.2 Discrete Future Lifetime ..................................................................167
5.3 Force of Mortality ..............................................................................169
5.4 Fractional Ages ..................................................................................175
5.5 Select Future Lifetimes .....................................................................177
5.6 Survivorship Groups ........................................................................179
5.7 Life Models and Life Tables .............................................................182
5.8 Life Table Confidence Sets and Prediction Intervals ...................185
5.9 Life Models and Life Table Parameters .........................................187
5.9.1 Population Parameters ........................................................188
5.9.2 Aggregate Parameters .........................................................191
5.9.3 Fractional Age Adjustments ...............................................193
5.10 Select and Ultimate Life Tables .......................................................194
Problems ........................................................................................................198
Excel Problems .............................................................................................200
Solutions ........................................................................................................200
6 Stochastic Status Models ..........................................................................203
6.1 Stochastic Present Value Functions ................................................204
6.2 Risk Evaluations ................................................................................205
6.2.1 Continuous-Risk Calculations ...........................................205
6.2.2 Discrete Risk Calculations ..................................................206
6.2.3 Mixed Risk Calculations .....................................................207
6.3 Percentile Evaluations ......................................................................208
6.4 Life Insurance ....................................................................................210
6.4.1 Types of Unit Benefit Life Insurance .................................212
6.5 Life Annuities ....................................................................................215
6.5.1 Types of Unit Payment Life Annuities..............................217
6.5.2 Apportionable Annuities ....................................................220
6.6 Relating Risk Calculations ...............................................................223
6.6.1 Relations among Insurance Expectations ........................223
6.6.2 Relations among Insurance and Annuity Expectations ...225
6.6.3 Relations among Annuity Expectations ...........................226
6.7 Actuarial Life Tables .........................................................................227
6.8 Loss Models and Insurance Premiums .........................................230
6.8.1 Unit Benefit Premium Notation .........................................232
6.8.2 Variance of the Loss Function ............................................235
6.9 Reserves ..............................................................................................237
vi Contents
6.9.1 Unit Benefit Reserves Notations ........................................240
6.9.2 Relations among Reserve Calculations ............................241
6.9.3 Survivorship Group Approach to Reserve Calculations ....243
6.10 General Time Period Models ...........................................................244
6.10.1 General Period Expectation ................................................245
6.10.2 Relations among General Period Expectations ...............246
6.11 Expense Models and Computations ..............................................249
Problems ........................................................................................................252
Excel Problems .............................................................................................254
Solutions ........................................................................................................254
7 Advanced Stochastic Status Models .......................................................257
7.1 Multiple Future Lifetimes ................................................................257
7.1.1 Joint Life Status ....................................................................258
7.1.2 Last Survivor Status ............................................................260
7.1.3 General Contingent Status ..................................................263
7.2 Multiple-Decrement Models ............................................................264
7.2.1 Continuous Multiple Decrements .....................................264
7.2.2 Forces of Decrement ............................................................266
7.2.3 Discrete Multiple Decrements ...........................................268
7.2.4 Single-Decrement Probabilities .........................................269
7.2.5 Uniformly Distributed Single-Decrement Rates .............271
7.2.6 Single-Decrement Probability Bounds .............................273
7.2.7 Multiple-Decrement Life Tables.........................................275
7.2.8 Single-Decrement Life Tables .............................................278
7.2.9 Multiple-Decrement Computations ..................................279
7.3 Pension Plans .....................................................................................280
7.3.1 Multiple-Decrement Benefits..............................................281
7.3.2 Pension Contributions .........................................................285
7.3.3 Future Salary-Based Benefits and Contributions ............287
7.3.4 Yearly Based Retirement Benefits ......................................288
Problems ........................................................................................................290
Excel Problems .............................................................................................291
Solutions ........................................................................................................292
8 Markov Chain Methods ............................................................................295
8.1 Introduction to Markov Chains ......................................................296
8.2 Nonhomogeneous Stochastic Status Chains.................................297
8.2.1 Single-Decrement Chains ...................................................298
8.2.2 Actuarial Chains ..................................................................299
8.2.3 Multiple-Decrement Chains ...............................................300
8.2.4 Multirisk Strata Chains .......................................................303
8.3 Homogeneous Stochastic Status Chains .......................................307
Contents vii
8.3.1 Expected Curtate Future Lifetime .....................................309
8.3.2 Actuarial Chains ..................................................................310
8.4 Survivorship Chains .........................................................................312
8.4.1 Single-Decrement Models...................................................313
8.4.2 Multiple-Decrement Models ..............................................314
8.4.3 Multirisk Strata Models ......................................................315
Problems ........................................................................................................316
Excel Problems .............................................................................................317
Solutions ........................................................................................................320
9 Scenario and Simulation Testing ............................................................323
9.1 Scenario Testing ................................................................................323
9.1.1 Deterministic Status Scenarios ..........................................324
9.1.2 Stochastic Status Scenarios .................................................325
9.1.3 Stochastic Rate Scenarios ....................................................328
9.2 Simulation Techniques .....................................................................330
9.2.1 Bootstrap Sampling .............................................................331
9.2.2 Simulation Sampling ...........................................................332
9.2.3 Simulation Probabilities ......................................................335
9.2.4 Simulation Prediction Intervals .........................................337
9.3 Investment Pricing Applications ....................................................340
9.4 Stochastic Surplus Application .......................................................343
9.5 Future Directions in Simulation Analysis .....................................344
Problems ........................................................................................................346
Excel Problems .............................................................................................348
Solutions ........................................................................................................350
10 Further Statistical Considerations ..........................................................353
10.1 Mortality Adjustment Models .........................................................354
10.1.1 Linear Mortality Acceleration Models ..............................355
10.1.2 Mean Mortality Acceleration Models ...............................357
10.1.3 Survival-Based Mortality Acceleration Models ..............360
10.2 Mortality Trend Modeling ...............................................................361
10.3 Actuarial Statistics ............................................................................364
10.3.1 Normality-Based Prediction Intervals ..............................365
10.3.2 Prediction Set-Based Prediction Intervals ........................366
10.3.3 Simulation-Based Prediction Intervals .............................368
10.4 Data Set Simplifications ...................................................................370
Problems ........................................................................................................371
Excel Problems .............................................................................................371
Solutions ........................................................................................................373
Appendix A: Excel Statistical Functions, Basic Mathematical
Functions, and Add-Ins ....................................................................................375
viii Contents
Appendix B: Acronyms and Principal Sections ..........................................377
References ...........................................................................................................379
Preface
Financial and actuarial modeling is an ever-changing field with an increased
reliance on statistical techniques. This is seen in the changing of competency
exams, especially at the upper levels, where topics include more statistical
concepts and techniques. In the years since the first edition was published
statistical techniques such as reliability measurement, simulation, regres-
sion, and Markov chain modeling have become more prominent. This influx
in statistics has put an increased pressure on students to secure both strong
mathematical and statistical backgrounds and the knowledge of statistical
techniques in order to have successful careers.
As in the first edition, this text approaches financial and actuarial model-
ing from a statistical point of view. The goal of this text is twofold. The first
is to provide students and practitioners a source for required mathemati-
cal and statistical background. The second is to advance the application and
theory of statistics in financial and actuarial modeling.
This text presents a unified approach to both financial and actuarial
modeling through the utilization of general status structures. Future time-
dependent financial actions are defined in terms of a status structure that
may be either deterministic or stochastic. Deterministic status structures
lead to classical interest and annuity models, investment pricing models,
and aggregate claim models. Stochastic status structures are used to develop
financial and actuarial models, such as surplus models, life insurance, and
life annuity models.
This edition is updated with the addition of nomenclature and notations
standard to the actuarial field. This is essential to the interchange of concepts
and applications between actuarial, financial, and statistical practitioners.
Throughout this edition exercise problems have been added along with solu-
tions listing detailed equation links. After each chapter a series of applica-
tion problems listed as “Excel Problems,” along with solutions listing useful
library functions, are newly included. Specific changes in this edition, listed
by chapter, are now discussed.
Chapter 1 from the first edition is now split into two new chapters. Chapter 1
gives basic statistical theory and applications. Additional examples to help
prepare students for the initial actuarial exams are also given along with a
new section on nonnegative variables, namely, the Pareto, lognormal, and
Weibull. Chapter 2 consists of statistical models and techniques includ-
ing a new section on model diagnostics. Probability plotting, least squares,
and interval data diagnostics are explored. In Chapter 4 the discussions of
option pricing and stochastic surplus models are expanded. New discus-
sions include option pricing diagnostics and upper and lower bounds on the
probability of ruin for standard surplus models. Further, ruin computations
ix