Statistical Tools for Finance and Insurance

STATISTICAL TOOLS FOR FINANCE AND INSURANCE......Page 1
Springerlink......Page 0
Half-title......Page 2
Title Page......Page 3
Copyright Page......Page 4
Contents......Page 5
Contributors......Page 17
Preface......Page 19
Part I: Finance......Page 23
1.1 Introduction......Page 25
1.2 Definitions and Basic Characteristics......Page 26
1.2.1 Characteristic Function Representation......Page 28
1.2.2 Stable Density and Distribution Functions......Page 30
1.3 Simulation of α-stable Variables......Page 32
1.4 Estimation of Parameters......Page 34
1.4.1 Tail Exponent Estimation......Page 35
1.4.2 Quantile Estimation......Page 37
1.4.3 Characteristic Function Approaches......Page 38
1.4.4 Maximum Likelihood Method......Page 39
1.5 Financial Applications of Stable Laws......Page 40
Bibliography......Page 46
2.1 Introduction......Page 49
2.1.1 Analysis of Distribution of the Extremum......Page 50
2.1.2 Analysis of Conditional Excess Distribution......Page 51
2.1.3 Examples......Page 52
2.2.1 Copula Approach......Page 57
2.2.2 Examples......Page 60
2.2.3 Multivariate Extreme Value Approach......Page 61
2.2.4 Examples......Page 64
2.2.5 Copula Analysis for Multivariate Time Series......Page 65
2.2.6 Examples......Page 66
Bibliography......Page 68
3.1 Introduction......Page 69
3.2 What is Tail Dependence?......Page 70
3.3.1 Archimedean Copulae......Page 73
3.3.2 Elliptically-contoured Distributions......Page 74
3.3.3 Other Copulae......Page 78
3.4 Estimating the Tail-dependence Coefficient......Page 79
3.5 Comparison of TDC Estimators......Page 82
3.6 Tail Dependence of Asset and FX Returns......Page 85
3.7 Value at Risk – a Simulation Study......Page 88
Bibliography......Page 93
4.1 Introduction......Page 97
4.1.1 The Emergence of CAT Bonds......Page 98
4.1.2 Insurance Securitization......Page 100
4.1.3 CAT Bond Pricing Methodology......Page 101
4.2 Compound Doubly Stochastic Poisson Pricing Model......Page 103
4.3 Calibration of the Pricing Model......Page 104
4.4 Dynamics of the CAT Bond Price......Page 108
Bibliography......Page 116
5.1 Introduction......Page 119
5.2 Implied Volatility Surface......Page 120
5.3 Functional Data Analysis......Page 122
5.4 Functional Principal Components......Page 125
5.4.1 Basis Expansion......Page 127
5.5 Smoothed Principal Components Analysis......Page 129
5.5.1 Basis Expansion......Page 130
5.6 Common Principal Components Model......Page 131
Bibliography......Page 137
Pavel Čížek and Karel Komorád......Page 139
6.1 Option Pricing......Page 140
6.2 Trees and Implied Trees......Page 142
6.3.1 Basic Insight......Page 144
6.3.2 State Space......Page 146
6.3.3 Transition Probabilities......Page 148
6.3.4 Possible Pitfalls......Page 149
6.4.1 Pre-specified Implied Volatility......Page 151
6.4.2 German Stock Index......Page 156
Bibliography......Page 162
7.1 Introduction......Page 165
7.2 Heston’s Model......Page 167
7.3 Option Pricing......Page 170
7.3.1 Greeks......Page 172
7.4 Calibration......Page 173
7.4.1 Qualitative Effects of Changing Parameters......Page 175
7.4.2 Calibration Results......Page 177
Bibliography......Page 183
8.2 Modern Pricing Models......Page 187
8.2.1 Merton Model......Page 188
8.2.2 Heston Model......Page 189
8.2.3 Bates Model......Page 191
8.3 Option Pricing with FFT......Page 192
8.4 Applications......Page 196
Bibliography......Page 203
9.1 Introduction......Page 205
9.2.2 Optimal Behavior and Price......Page 208
9.3 Valuation of Mortgage Backed Securities......Page 216
9.3.1 Generic Framework......Page 217
9.3.2 A Parametric Specification of the Prepayment Rate......Page 219
9.3.3 Sensitivity Analysis......Page 222
Bibliography......Page 227
Wolfgang Härdle, Rouslan Moro, and Dorothea Schäfer......Page 229
10.1 Bankruptcy Analysis Methodology......Page 230
10.2 Importance of Risk Classification in Practice......Page 234
10.3 Lagrangian Formulation of the SVM......Page 237
10.4 Description of Data......Page 240
10.5 Computational Results......Page 241
10.6 Conclusions......Page 247
Bibliography......Page 250
Noer Azam Achsani, Oliver Holtemöller, and Hizir Sofyan......Page 253
11.1 Specification of Money Demand Functions......Page 254
11.2.1 Econometric Estimation of Money Demand Functions......Page 256
11.2.2 Modelling Indonesian Money Demand with Econometric Techniques......Page 258
11.3.1 Fuzzy Clustering......Page 264
11.3.2 The Takagi–Sugeno Approach......Page 265
11.3.3 Model Identification......Page 266
11.3.4 Modelling Indonesian Money Demand with Fuzzy Techniques......Page 267
11.4 Conclusions......Page 270
Bibliography......Page 271
Wolfgang Härdle and Seok-Oh Jeong......Page 275
12.1 The Basic Concepts......Page 276
12.2 Nonparametric Hull Methods......Page 280
12.2.1 Data Envelopment Analysis......Page 281
12.2.2 Free Disposal Hull......Page 282
12.3 DEA in Practice: Insurance Agencies......Page 283
12.4 FDH in Practice: Manufacturing Industry......Page 285
Bibliography......Page 289
Part II: Insurance......Page 291
13.1 Introduction......Page 293
13.2 Empirical Distribution Function......Page 294
13.3.1 Log-normal Distribution......Page 296
13.3.2 Exponential Distribution......Page 297
13.3.3 Pareto Distribution......Page 299
13.3.5 Weibull Distribution......Page 302
13.3.6 Gamma Distribution......Page 304
13.3.7 Mixture of Exponential Distributions......Page 306
13.4.1 Mean Excess Function......Page 307
13.4.2 Tests Based on the Empirical Distribution Function......Page 309
13.4.3 Limited Expected Value Function......Page 313
13.5 Applications......Page 315
Bibliography......Page 320
14.1 Introduction......Page 323
14.2.1 Homogeneous Poisson Process......Page 325
14.2.2 Non-homogeneous Poisson Process......Page 327
14.2.3 Mixed Poisson Process......Page 330
14.2.4 Cox Process......Page 331
14.2.5 Renewal Process......Page 332
14.3.1 Catastrophic Losses......Page 333
14.3.2 Danish Fire Losses......Page 338
Bibliography......Page 342
15.1 Introduction......Page 345
15.1.1 Light- and Heavy-tailed Distributions......Page 347
15.2 Exact Ruin Probabilities in Infinite Time......Page 350
15.2.3 Gamma Claim Amounts......Page 351
15.2.4 Mixture of Two Exponentials Claim Amounts......Page 353
15.3 Approximations of the Ruin Probability in Infinite Time......Page 354
15.3.1 Cramér–Lundberg Approximation......Page 355
15.3.3 Lundberg Approximation......Page 356
15.3.4 Beekman–Bowers Approximation......Page 357
15.3.5 Renyi Approximation......Page 358
15.3.6 De Vylder Approximation......Page 359
15.3.7 4-moment Gamma De Vylder Approximation......Page 360
15.3.8 Heavy Traffic Approximation......Page 362
15.3.9 Light Traffic Approximation......Page 363
15.3.11 Subexponential Approximation......Page 364
15.3.12 Computer Approximation via the Pollaczek–Khinchin Formula......Page 365
15.3.13 Summary of the Approximations......Page 366
15.4 Numerical Comparison of the Infinite Time Approximations......Page 367
15.5 Exact Ruin Probabilities in Finite Time......Page 371
15.6 Approximations of the Ruin Probability in Finite Time......Page 372
15.6.2 Segerdahl Normal Approximation......Page 373
15.6.3 Diffusion Approximation......Page 375
15.6.4 Corrected Diffusion Approximation......Page 376
15.6.5 Finite Time De Vylder Approximation......Page 377
15.7 Numerical Comparison of the Finite Time Approximations......Page 378
Bibliography......Page 383
16.1 Introduction......Page 385
16.2 Brownian Motion and the Risk Model for Small Claims......Page 386
16.2.2 Ruin Probability for the Limit Process......Page 387
16.2.3 Examples......Page 388
16.3 Stable Lévy Motion and the Risk Model for Large Claims......Page 390
16.3.1 Weak Convergence of Risk Processes to α-stable Lévy Motion......Page 391
16.3.2 Ruin Probability in the Limit Risk Model for Large Claims......Page 392
16.3.3 Examples......Page 394
Bibliography......Page 397
17.1 Introduction......Page 399
17.2 Fractional Brownian Motion and the Risk Model of Good and Bad Periods......Page 400
17.3 Ruin Probability in the Limit Risk Model of Good and Bad Periods......Page 403
17.4 Examples......Page 406
Bibliography......Page 409
Jan Iwanik and Joanna Nowicka-Zagrajek......Page 411
18.1 Premium Calculation Principles......Page 412
18.2 Individual Risk Model......Page 414
18.2.1 General Premium Formulae......Page 415
18.2.2 Premiums in the Case of the Normal Approximation......Page 416
18.2.3 Examples......Page 417
18.3 Collective Risk Model......Page 420
18.3.1 General Premium Formulae......Page 421
18.3.2 Premiums in the Case of the Normal and Translated Gamma Approximations......Page 422
18.3.3 Compound Poisson Distribution......Page 424
18.3.4 Compound Negative Binomial Distribution......Page 425
18.3.5 Examples......Page 427
Bibliography......Page 430
19.1 Introduction......Page 431
19.2 General Formulae for Premiums Under Deductibles......Page 432
19.2.1 Franchise Deductible......Page 433
19.2.2 Fixed Amount Deductible......Page 435
19.2.4 Limited Proportional Deductible......Page 436
19.2.5 Disappearing Deductible......Page 438
19.3 Premiums Under Deductibles for Given Loss Distributions......Page 440
19.3.1 Log-normal Loss Distribution......Page 441
19.3.2 Pareto Loss Distribution......Page 442
19.3.3 Burr Loss Distribution......Page 445
19.3.4 Weibull Loss Distribution......Page 449
19.3.5 Gamma Loss Distribution......Page 451
19.3.6 Mixture of Two Exponentials Loss Distribution......Page 453
19.4 Final Remarks......Page 454
Bibliography......Page 456
20.1 Introduction......Page 457
20.2.1 Risk Based Capital Concept......Page 460
20.2.2 How to Choose Parameter Values?......Page 461
20.3.1 Approximations of Quantiles......Page 463
20.3.2 Marginal Cost Basis for Individual Risk Pricing......Page 464
20.3.3 Balancing Problem......Page 465
20.3.5 Applications......Page 466
20.4 Rate of Return and Reinsurance Under the Short Term Criterion......Page 467
20.4.1 General Considerations......Page 468
20.4.2 Illustrative Example......Page 469
20.4.3 Interpretation of Numerical Calculations in Example 2......Page 471
20.5.1 Approximation Based on Lundberg Inequality......Page 473
20.5.3 Cramér–Lundberg Approximation......Page 475
20.5.4 Beekman–Bowers Approximation......Page 476
20.5.5 Diffusion Approximation......Page 477
20.5.6 De Vylder Approximation......Page 478
20.5.8 Panjer Approximation......Page 479
20.6.1 Fixed Dividends......Page 481
20.6.2 Flexible Dividends......Page 483
20.7.1 Fixed Dividends......Page 485
20.7.2 Interpretation of Solutions Obtained in Example 5......Page 486
20.7.3 Flexible Dividends......Page 488
20.7.4 Interpretation of Solutions Obtained in Example 6......Page 489
20.8 Final Remarks......Page 491
Bibliography......Page 492
Part III: General......Page 493
21.1 Introduction......Page 495
21.2.1 Configuration......Page 496
21.2.2 Getting Connected......Page 497
21.3 Desktop......Page 498
21.3.1 XploRe Quantlet Editor......Page 499
21.3.2 Data Editor......Page 500
21.3.3 Method Tree......Page 505
21.3.4 Graphical Output......Page 507
Index......Page 511