Statistical Tools for Finance and Insurance (eBook)

Pavel Cizek, Wolfgang Karl Härdle, Rafał Weron (Herausgeber)

eBook Download: PDF

2005 | 2005
IV, 518 Seiten
Springer Berlin (Verlag)
978-3-540-27395-0 (ISBN)

Infos

Written in an accessible and engaging style, this self-instructional book makes a good use of extensive examples and full explanations.

The electronic edition, allowing the reader to run, modify, and enhance all quantlets on the spot, can be downloaded at no cost via the attached license registration card.

Inhalt

Contents 5
Contributors 17
Preface 19
Part I Finance 23
1 Stable Distributions 25
1.1 Introduction 25
1.2 Definitions and Basic Characteristics 26
1.2.1 Characteristic Function Representation 28
1.2.2 Stable Density and Distribution Functions 30
1.3 Simulation of stable Variables 32
1.4 Estimation of Parameters 34
1.4.1 Tail Exponent Estimation 35
1.4.2 Quantile Estimation 37
1.4.3 Characteristic Function Approaches 38
1.4.4 Maximum Likelihood Method 39
1.5 Financial Applications of Stable Laws 40
2 Extreme Value Analysis and Copulas 49
2.1 Introduction 49
2.1.1 Analysis of Distribution of the Extremum 50
2.1.2 Analysis of Conditional Excess Distribution 51
2.1.3 Examples 52
2.2 Multivariate Time Series 57
2.2.1 Copula Approach 57
2.2.2 Examples 60
2.2.3 Multivariate Extreme Value Approach 61
2.2.4 Examples 64
2.2.5 Copula Analysis for Multivariate Time Series 65
2.2.6 Examples 66
3 Tail Dependence 69
3.1 Introduction 69
3.2 What is Tail Dependence? 70
3.3 Calculation of the Tail-dependence Coefficient 73
3.3.1 Archimedean Copulae 73
3.3.2 Elliptically-contoured Distributions 74
3.3.3 Other Copulae 78
3.4 Estimating the Tail-dependence Coefficient 79
3.5 Comparison of TDC Estimators 82
3.6 Tail Dependence of Asset and FX Returns 85
3.7 Value at Risk – a Simulation Study 88
4 Pricing of Catastrophe Bonds 97
4.1 Introduction 97
4.1.1 The Emergence of CAT Bonds 98
4.1.2 Insurance Securitization 100
4.1.3 CAT Bond Pricing Methodology 101
4.2 Compound Doubly Stochastic Poisson Pricing Model 103
4.3 Calibration of the Pricing Model 104
4.4 Dynamics of the CAT Bond Price 108
5 Common Functional Implied Volatility Analysis 119
5.1 Introduction 119
5.2 Implied Volatility Surface 120
5.3 Functional Data Analysis 122
5.4 Functional Principal Components 125
5.4.1 Basis Expansion 127
5.5 Smoothed Principal Components Analysis 129
5.5.1 Basis Expansion 130
5.6 Common Principal Components Model 131
6 Implied Trinomial Trees 139
6.1 Option Pricing 140
6.2 Trees and Implied Trees 142
6.3 Implied Trinomial Trees 144
6.3.1 Basic Insight 144
6.3.2 State Space 146
6.3.3 Transition Probabilities 148
6.3.4 Possible Pitfalls 149
6.4 Examples 151
6.4.1 Pre-speci.ed Implied Volatility 151
6.4.2 German Stock Index 156
7 Heston’s Model and the Smile 165
7.1 Introduction 165
7.2 Heston’s Model 167
7.3 Option Pricing 170
7.3.1 Greeks 172
7.4 Calibration 173
7.4.1 Qualitative E.ects of Changing Parameters 175
7.4.2 Calibration Results 177
8 FFT-based Option Pricing 187
8.1 Introduction 187
8.2 Modern Pricing Models 187
8.2.1 Merton Model 188
8.2.2 Heston Model 189
8.2.3 Bates Model 191
8.3 Option Pricing with FFT 192
8.4 Applications 196
9 Valuation of Mortgage Backed Securities: from Optimality to Reality 205
9.1 Introduction 205
9.2 Optimally Prepaid Mortgage 208
9.2.1 Financial Characteristics and Cash Flow Analysis 208
9.2.2 Optimal Behavior and Price 208
9.3 Valuation of Mortgage Backed Securities 216
9.3.1 Generic Framework 217
9.3.2 A Parametric Speci.cation of the Prepayment Rate 219
9.3.3 Sensitivity Analysis 222
10 Predicting Bankruptcy with Support Vector Machines 229
10.1 Bankruptcy Analysis Methodology 230
10.2 Importance of Risk Classification in Practice 234
10.3 Lagrangian Formulation of the SVM 237
10.4 Description of Data 240
10.5 Computational Results 241
11 Econometric and Fuzzy Modelling of Indonesian Money Demand 253
11.1 Speci.cation of Money Demand Functions 254
11.2 The EconometricApproach to Money Demand 256
11.2.1 Econometric Estimation of Money Demand Functions 256
11.2.2 Modelling Indonesian Money Demand with Econometric Techniques 258
11.3 The Fuzzy Approach to Money Demand 264
11.3.1 Fuzzy Clustering 264
11.3.2 The Takagi-Sugeno Approach 265
11.3.3 Model Identi.cation 266
11.3.4 Modelling Indonesian Money Demand with Fuzzy Techniques 267
11.4 Conclusions 270
12 Nonparametric Productivity Analysis 275
12.1 The Basic Concepts 276
12.2 Nonparametric Hull Methods 280
12.2.1 Data Envelopment Analysis 281
12.2.2 Free Disposal Hull 282
12.3 DEA in Practice: Insurance Agencies 283
12.4 FDH in Practice: Manufacturing Industry 285
Part II 292
13 Loss Distributions 293
13.1 Introduction 293
13.2 Empirical Distribution Function 294
13.3 Analytical Methods 296
13.3.1 Log-normal Distribution 296
13.3.2 Exponential Distribution 297
13.3.3 Pareto Distribution 299
13.3.4 Burr Distribution 302
13.3.5 Weibull Distribution 302
13.3.6 Gamma Distribution 304
13.3.7 Mixture of Exponential Distributions 306
13.4 Statistical Validation Techniques 307
13.4.1 Mean Excess Function 307
13.4.2 Tests Based on the Empirical Distribution Function 309
13.4.3 Limited Expected Value Function 313
13.5 Applications 315
14 Modeling of the Risk Process 323
14.1 Introduction 323
14.2 Claim Arrival Processes 325
14.2.1 Homogeneous Poisson Process 325
14.2.2 Non-homogeneous Poisson Process 327
14.2.3 Mixed Poisson Process 330
14.2.4 Cox Process 331
14.2.5 Renewal Process 332
14.3 Simulation of Risk Processes 333
14.3.1 Catastrophic Losses 333
14.3.2 Danish Fire Losses 338
15 Ruin Probabilities in Finite and Infinite Time 345
15.1 Introduction 345
15.1.1 Light- and Heavy-tailed Distributions 347
15.2 Exact Ruin Probabilities in Infinite Time 350
15.2.1 No Initial Capital 351
15.2.2 Exponential Claim Amounts 351
15.2.3 Gamma Claim Amounts 351
15.2.4 Mixture of Two Exponentials Claim Amounts 353
15.3 Approximations of the Ruin Probability in Infinite Time 354
15.3.1 Cram´ er–Lundberg Approximation 355
15.3.2 Exponential Approximation 356
15.3.3 Lundberg Approximation 356
15.3.4 Beekman–Bowers Approximation 357
15.3.5 Renyi Approximation 358
15.3.6 De Vylder Approximation 359
15.3.7 4-moment Gamma De Vylder Approximation 360
15.3.8 Heavy Tra.c Approximation 362
15.3.9 Light Tra.c Approximation 363
15.3.10 Heavy-light Tra.c Approximation 364
15.3.11 Subexponential Approximation 364
15.3.12 Computer Approximation via the Pollaczek-Khinchin Formula 365
15.3.13 Summary of the Approximations 366
15.4 Numerical Comparison of the Infinite Time Approximations 367
15.5 Exact Ruin Probabilities in Finite Time 371
15.5.1 Exponential Claim Amounts 372
15.6 Approximations of the Ruin Probability in Finite Time 372
15.6.1 Monte Carlo Method 373
15.6.2 Segerdahl Normal Approximation 373
15.6.3 Diffusion Approximation 375
15.6.4 Corrected Di.usion Approximation 376
15.6.5 Finite Time De Vylder Approximation 377
15.6.6 Summary of the Approximations 378
15.7 Numerical Comparison of the Finite Time Approximations 378
16 Stable Difiusion Approximation of the Risk Process 385
16.1 Introduction 385
16.2 Brownian Motion and the Risk Model for Small Claims 386
16.2.1 Weak Convergence of Risk Processes to Brownian Motion 387
16.2.2 Ruin Probability for the Limit Process 387
16.2.3 Examples 388
16.3 Stable Levy Motion and the Risk Model for Large Claims 390
16.3.1 Weak Convergence of Risk Processes to a-stable Levy Motion 391
16.3.2 Ruin Probability in the Limit Risk Model for Large Claims 392
16.3.3 Examples 394
17 Risk Model of Good and Bad 399
17.1 Introduction 399
17.2 Fractional Brownian Motion and the Risk Model of Good and Bad Periods 400
17.3 Ruin Probability in the Limit Risk Model of Good and Bad Periods 403
17.4 Examples 406
18 Premiums in the Individual and Collective Risk Models 411
18.1 Premium Calculation Principles 412
18.2 Individual Risk Model 414
18.2.1 General Premium Formulae 415
18.2.2 Premiums in the Case of the Normal Approximation 416
18.2.3 Examples 417
18.3 Collective Risk Model 420
18.3.1 General Premium Formulae 421
18.3.2 Premiums in the Case of the Normal and Translated Gamma Approximations 422
18.3.3 Compound Poisson Distribution 424
18.3.4 Compound Negative Binomial Distribution 425
18.3.5 Examples 427
19 Pure Risk Premiums under Deductibles 431
19.1 Introduction 431
19.2 General Formulae for Premiums Under Deductibles 432
19.2.1 Franchise Deductible 433
19.2.2 Fixed Amount Deductible 435
19.2.3 Proportional Deductible 436
19.2.4 Limited Proportional Deductible 436
19.2.5 Disappearing Deductible 438
19.3 Premiums Under Deductibles for Given Loss Distributions 440
19.3.1 Log-normal Loss Distribution 441
19.3.2 Pareto Loss Distribution 442
19.3.3 Burr Loss Distribution 445
19.3.4 Weibull Loss Distribution 449
19.3.5 Gamma Loss Distribution 451
19.3.6 Mixture of Two Exponentials Loss Distribution 453
19.4 Final Remarks 454
20 Premiums, Investments, and Reinsurance 457
20.1 Introduction 457
20.2 Single-period Criterion and the Rate of Return on Capital 460
20.2.1 Risk Based Capital Concept 460
20.2.2 How to Choose Parameter Values? 461
20.3 The Top-down Approach to Individual Risks Pricing 463
20.3.1 Approximations of Quantiles 463
20.3.2 Marginal Cost Basis for Individual Risk Pricing 464
20.3.3 Balancing Problem 465
20.3.4 A Solution for the Balancing Problem 466
20.3.5 Applications 466
20.4 Rate of Return and Reinsurance Under the Short Term Criterion 467
20.4.1 General Considerations 468
20.4.2 Illustrative Example 469
20.4.3 Interpretation of Numerical Calculations in Example 2 471
20.5 Ruin Probability Criterion when the Initial Capital is Given 473
20.5.1 Approximation Based on Lundberg Inequality 473
20.5.2 Zero” Approximation 475
20.5.3 Cram´ er–Lundberg Approximation 475
20.5.4 Beekman–Bowers Approximation 476
20.5.5 Di.usion Approximation 477
20.5.6 De Vylder Approximation 478
20.5.7 Subexponential Approximation 479
20.5.8 Panjer Approximation 479
20.6 Ruin Probability Criterion and the Rate of Return 481
20.6.1 Fixed Dividends 481
20.6.2 Flexible Dividends 483
20.7 Ruin Probability, Rate of Return and Reinsurance 485
20.7.1 Fixed Dividends 485
20.7.2 Interpretation of Solutions Obtained in Example 5 486
20.7.3 Flexible Dividends 488
20.7.4 Interpretation of Solutions Obtained in Example 6 489
20.8 Final Remarks 491
Part III 494
21 Working with the XQC 495
21.1 Introduction 495
21.2 The XploRe Quantlet Client 496
21.2.1 Con.guration 496
21.2.2 Getting Connected 497
21.3 Desktop 498
21.3.1 XploRe Quantlet Editor 499
21.3.2 Data Editor 500
21.3.3 Method Tree 505
21.3.4 Graphical Output 507
Index 511

Textauszug

18 Premiums in the Individual and Collective Risk Models (p.407)

Jan Iwanik and Joanna Nowicka-Zagrajek

The premium is the price for the good "insurance" sold by an insurance company. The right pricing is vital since too low a price level results in a loss, while with too high prices a company can price itself out of the market. It is the actuary’s task to and methods of premium calculation (also called premium calculation principles), i.e. rules saying what premium should be assigned to a given risk.

We present the most important types of premiums in Section 18.1; for more premium calculation principles, that are not considered here, see Straub (1988) and Young (2004). We focus on the monetary payout made by the insurer in connection with insurable losses and we ignore premium loading for expenses and profit.

The goal of insurance modeling is to develop a probability distribution for the total amount paid in bene.ts. This allows the insurance company to manage its capital account and honor its commitments. Therefore, we describe two standard models: the individual risk model in Section 18.2 and the collective risk model in Section 18.3. In both cases, we determine the expectation and variance of the portfolio, consider the approximation of the distribution of the aggregate claims, and present formulae for the considered premiums. It is worth mentioning here that the collective risk model can also be applied to quantifying regulatory capital for operational risk, for example to model a yearly operational risk variable (Embrechts, Furrer, and Kaufmann, 2003).

Produkt-Details

• Erscheint lt. Verlag: 6.12.2005
• Zusatzinfo: IV, 518 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Allgemeines / Lexika
• Schlagworte: algorithms • Calculus • Catastrophe Bonds • Compound Risk Model • extreme value theory • Fuzzy Identification Model • Loss distributions • Modeling • Option pricing • Quantitative Finance • reinsurance • Ruin Probability • Stable Distributions • Support Vector Machine • Tail Dependence • VOL • VOLA Surfaces
• ISBN-10: 3-540-27395-6 / 3540273956
• ISBN-13: 978-3-540-27395-0 / 9783540273950