Financial and Insurance Formulas (eBook)

Contents 6
List of Symbols 12
1 Introduction 15
Part I Financial Formulas 17
2 Simple Interest and Discount 18
2.1 Simple Interest 18
2.2 Calendar Conventions 20
2.3 Simple Interest with Principals Credited mthly 22
2.4 Simple Discount 22
Further Reading 23
3 Compound Interest and Discount 24
3.1 Compound Interest 24
3.2 Compound Discount 25
3.3 Compound Interest and Discount Convertible mthly 26
3.4 Combination of Simple and Compound Interest 28
Further Reading 29
4 Continuous Interest and Discount 30
Further Reading 32
5 Classical Analysis of Interest Rates 33
5.1 Risk-Free Interest Rate and Real Interest Rate 33
5.2 Term Structure of Interest Rates 35
Further Reading 36
6 Systems of Cash Flows 37
6.1 Present and Future Value 37
6.2 Internal Rate of Return 40
6.3 Payback Period 42
6.4 Duration 42
6.5 Convexity 45
Further Reading 46
7 Annuities 47
7.1 Annuity Calculus 48
7.2 Dynamic Annuities 52
7.3 Annuities Payable mthly 56
7.4 Continuously Payable Annuities 57
7.5 Amortization of Debt 58
Further Reading 60
8 Depreciation 62
Further Reading 64
9 Financial Instruments 65
9.1 Discount Securities 65
9.2 Bonds 67
9.3 Stocks 75
9.4 Currencies 81
Further Reading 81
10 Derivative Securities 83
10.1 General Classification 83
10.2 Forwards 84
10.3 Futures 88
10.4 Swaps 90
10.5 Options 91
Further Reading 99
11 Utility Theory 100
Further Reading 102
12 Rate of Return and Financial Risk 103
12.1 Rate of Return 103
12.2 Financial Risk 106
12.3 Value at Risk VaR 110
12.4 Credit at Risk CaR 113
Further Reading 115
13 Portfolio Analysis and CAPM Model 116
13.1 Construction of Portfolio 116
13.2 Portfolio with a Risk-Free Asset 121
13.3 CAPM Model 123
Further Reading 125
14 Arbitrage Theory 126
Further Reading 129
15 Financial Stochastic Analysis 130
15.1 Wiener Process in Finance 130
15.2 Poisson Process in Finance 132
15.3 Ito Stochastic Integral 133
15.4 Stochastic Differential Equations SDE 135
15.5 Itos Lemma 136
15.6 Girsanov Theorem on Equivalent Martingale Probability 137
15.7 Theorem on Martingale Representation 139
15.8 Derivatives Pricing by Means of Equivalent Martingale Probabilities 139
15.9 Derivatives Pricing by Means of Partial Differential Equations PDE 141
15.10 Term Structure Modeling 142
Further Reading 146
Part II Insurance Formulas 147
16 Insurance Classification 148
Further Reading 153
17 Actuarial Demography 154
17.1 Selected Population Indicators 154
17.2 Life Tables 158
17.3 Mortality and Survival Modeling 162
17.4 Multiple Decrement Models 166
17.5 Multiple Life Functions 167
17.6 Commutation Functions 167
Further Reading 169
18 Classical Life Insurance 170
18.1 Basic Concepts of Life Insurance 170
18.2 Symbols and Calculation Principles of Life Insurance 172
18.3 Technical Provisions in Life Insurance 175
18.4 Pure Endowments 179
18.5 Whole Life and Term Insurance 180
18.6 Further Products of Capital Life Insurance 185
18.7 Life Annuities 188
18.8 Multiple Life Insurance 194
18.9 Premium Reserve and Its Implications 195
18.10 Medical Underwriting 197
Further Reading 198
19 Modern Approaches to Life Insurance 200
19.1 Critical Illness Insurance 200
19.2 Flexible Products of Life Insurance 202
19.3 Unit Linked 203
19.4 Profit Testing 206
19.5 Embedded Value 208
19.6 Fair Value 210
Further Reading 212
20 Pension Insurance 213
20.1 Basic Concepts of Pension Insurance 213
20.2 Defined Contribution Plan 215
20.3 Defined Benefit Plan 217
Further Reading 221
21 Classical Non-Life Insurance 222
21.1 Basic Concepts of Non-Life Insurance 222
21.2 Premium Calculations in Non-Life Insurance 225
21.3 Forms of Non-Life Insurance and Deductibles 228
21.4 Technical Provisions in Non-Life Insurance 231
21.5 Bonus-Malus Systems 236
Further Reading 237
22 Risk Theory in Insurance 238
22.1 Collective Risk Model 238
22.2 Aggregate Claim Distribution 241
22.3 Copula 245
22.4 Credibility Premium 246
22.5 Ruin Probability 249
22.6 Deductible 251
22.7 Calculations for Bonus-Malus Systems 253
Further Reading 255
23 Health Insurance 257
Further Reading 259
24 Reinsurance 260
24.1 Basic Concepts of Reinsurance 260
24.2 Types of Reinsurance 262
24.3 Solvency 268
24.4 Alternative Risk Transfer ART 270
Further Reading 272
Part III Formulas of Related Disciplines 274
25 Mathematical Compendium 275
25.1 Powers with Integral Exponents 275
25.2 Roots of Real Numbers 275
25.3 Powers with Rational Exponents 275
25.4 Powers with Real Exponents 276
25.5 Formulas an ± bn 276
25.6 Logarithms 276
25.7 Factorial and Binomial Coefficients 277
25.8 Binomial Theorem 278
25.9 Sums of Powers of Natural Numbers 278
25.10 Numerical Series 278
25.11 Means 280
25.12 Beta and Gamma Function 280
Further Reading 281
26 Probability Theory 282
26.1 Random Events and Probability 282
26.2 Conditional Probability and Independent Events 283
26.3 Random Variables and Their Basic Characteristics 284
26.4 Important Discrete Distributions 287
26.5 Important Continuous Distributions 288
26.6 Random Vectors and Their Basic Characteristics 290
26.7 Transformation of Random Variables 293
26.8 Conditional Mean Value 294
26.9 Martingales 295
26.10 Generating Function 297
26.11 Convolutions and Sums of Random Variables 299
26.12 Random Sums of Random Variables 300
26.13 Some Inequalities 300
26.14 Limit Theorems of Probability Theory 301
Further Reading 304
27 Descriptive and Mathematical Statistics 305
27.1 Sampling Theory: Simple Random Sample 305
27.2 Sampling Theory: Stratified Random Sample 306
27.3 Elementary Statistical Treatment 307
27.4 Sample Quantiles 308
27.5 Measures of Sample Level 308
27.6 Measures of Sample Variability 310
27.7 Measures of Sample Concentration 311
27.8 Measures of Sample Dependence 312
27.9 Point and Interval Estimators 313
27.10 Hypothesis Testing 316
27.11 Regression Analysis 318
27.12 Analysis of Variance (ANOVA) 325
27.13 Multivariate Statistical Analysis 326
Further Reading 327
28 Econometrics 328
28.1 Multicollinearity 328
28.2 A Priori Restrictions 330
28.3 Qualitative Regressors 330
28.4 Probit and Logit Models 331
28.5 Random Regressors and Instrumental Variable Estimation 332
28.6 Simultaneous Equation Models and 2SLS-Estimator 333
Further Reading 335
29 Index Numbers 336
29.1 Indices as Instruments of Comparison 336
29.2 Indices in Practice 337
29.3 Stock Exchange Indicators 339
Further Reading 340
30 Stochastic Processes 341
30.1 Classification and Basic Characteristics of Stochastic Processes 341
30.2 Markov Chains 343
30.3 Markov Processes 347
30.4 Important Stochastic Processes 349
30.5 Spectral Properties of Stochastic Processes 353
Further Reading 356
31 Statistical Analysis of Time Series 357
31.1 Predictions in Time Series 357
31.2 Decomposition of (Economic) Time Series 358
31.3 Estimation of Correlation and Spectral Characteristics 365
31.4 Linear Time Series 367
31.5 Nonlinear and Financial Time Series 372
31.6 Multivariate Time Series 377
31.7 Kalman Filter 378
Further Reading 381
Index 382