PRMIA风险管理考试 的教材<<The Professional Risk Managers' Handbook>>第二册
书名: The Professional Risk Managers’ Handbook
A Comprehensive Guide to Current Theory and Best Practices

Author:　Carol Alexander　（ＰＲＭＩＡ现任主席) and Elizabeth Sheedy.

Publisher: PRMIA Publications

Volume II: Mathematical Foundations of
Risk Measurement

Contents
Introduction ................................................................................................................................................8
Preface to Volume II: Mathematical Foundations of Risk Measurement..............................10
II.A Foundations ........................................................................................................................................13
II.A.1 Symbols and Rules ..............................................................................................................13
II.A.1.1 Expressions, Functions, Graphs, Equations and Greek......................................13
II.A.1.2 The Algebra of Number............................................................................................15
II.A.1.3 The Order of Operations..........................................................................................17
II.A.2 Sequences and Series...........................................................................................................18
II.A.2.1 Sequences ....................................................................................................................18
II.A.2.2 Series............................................................................................................................19
II.A.3 Exponents and Logarithms ...............................................................................................20
II.A.3.1 Exponents ...................................................................................................................20
II.A.3.2 Logarithms ..................................................................................................................21
II.A.3.3 The Exponential Function and Natural Logarithms............................................22
II.A.4 Equations and Inequalities.................................................................................................25
II.A.4.1 Linear Equations in One Unknown........................................................................25
II.A.4.2 Inequalities...................................................................................................................26
II.A.4.3 Systems of Linear Equations in More Than One Unknown..............................26
II.A.4.4 Quadratic Equations..................................................................................................28
II.A.5 Functions and Graphs ........................................................................................................28
II.A.5.1 Functions .....................................................................................................................28
II.A.5.2 Graphs.........................................................................................................................29
II.A.5.3 The Graphs of Some Functions ..............................................................................30
II.A.6 Case Study Continuous Compounding .......................................................................31
II.A.6.1 Repeated Compounding ...........................................................................................31
II.A.6.2 Discrete versus Continuous Compounding...........................................................34
II.A.7 Summary ...............................................................................................................................35
II.B Descriptive Statistics..........................................................................................................................37
II.B.1 Introduction........................................................................................................................37
II.B.2 Data ......................................................................................................................................38
II.B.2.1 Continuous and Discrete Data.................................................................................38
II.B.2.2 Grouped Data.............................................................................................................39
II.B.2.3 Graphical Representation of Data...........................................................................41
II.B.3 The Moments of a Distribution........................................................................................45
II.B.4 Measures of Location or Central Tendency – Averages...............................................45
II.B.4.1 The Arithmetic Mean ................................................................................................45

II.B.4.2 The Geometric Mean ................................................................................................46
II.B.4.3 The Median and the Mode .......................................................................................48
II.B.5 Measures of Dispersion......................................................................................................48
II.B.5.1 Variance .......................................................................................................................49
II.B.5.2 Standard Deviation ....................................................................................................50
II.B.5.3 Case Study: Calculating Historical Volatility from Returns Data.......................51
II.B.5.4 The Negative Semi-variance and Negative Semi-deviation.................................52
II.B.5.5 Skewness......................................................................................................................53
II.B.5.6 Kurtosis.......................................................................................................................56
II.B.6 Bivariate Data.......................................................................................................................57
II.B.6.1 Covariance...................................................................................................................58
II.B.6.2 The Covariance Matrix..............................................................................................60
II.B.6.3 The Correlation Coefficient......................................................................................61
II.B.6.4 The Correlation Matrix..............................................................................................62
II.B.6.5 Case Study: Calculating the Volatility of a Portfolio ............................................62
II.C Calculus................................................................................................................................................67
II.C.1 Differential Calculus ...........................................................................................................68
II.C.1.1 Functions .....................................................................................................................68
II.C.1.2 The First Derivative....................................................................................................69
II.C.1.3 Notation.......................................................................................................................71
II.C.1.4 Simple Rules................................................................................................................72
II.C.2 Case Study: Modified Duration of a Bond......................................................................75
II.C.3 Higher-Order Derivatives..................................................................................................78
II.C.3.1 Second Derivatives.....................................................................................................78
II.C.3.2 Further Derivatives ....................................................................................................79
II.C.3.3 Taylor Approximations .............................................................................................79
II.C.4 Financial Applications of Second Derivatives ................................................................81
II.C.4.1 Convexity.....................................................................................................................81
II.C.4.2 Convexity in Action...................................................................................................82
II.C.4.3 The Delta and Gamma of an Option .....................................................................83
II.C.5 Differentiating a Function of More than One Variable................................................83
II.C.5.1 Partial Differentiation................................................................................................83
II.C.5.2 Total Differentiation..................................................................................................85
II.C.6 Integral Calculus ..................................................................................................................86
II.C.6.1 Indefinite and Definite Integrals ............................................................................86
II.C.6.2 Rules for Integration...................................................................................................90
II.C.6.3 Guessing ......................................................................................................................90

II.C.7 Optimisation.........................................................................................................................90
II.C.7.1 Finding the Minimum or Maximum of a Function of One Variable ................91
II.C.7.2 Maxima and Minima of Functions of More than One Variable..........................94
II.C.7.3 Optimization Subject to Constraints: Lagrange Multipliers...............................95
II.C.7.4 Applications ................................................................................................................97
References ...............................................................................................................................................97
II.D Linear Mathematics and Matrix Algebra.........................................................................................99
II.D.1 Matrix Algebra ................................................................................................................... 100
II.D.1.1 Matrices..................................................................................................................... 100
II.D.1.2 Vectors and Transposes .......................................................................................... 100
II.D.1.3 Manipulation of Matrices ....................................................................................... 101
II.D.1.4 Matrix Multiplication.............................................................................................. 103
II.D.1.5 Inverting a Matrix.................................................................................................... 105
II.D.2 Application of Matrix Algebra to Portfolio Construction.......................................... 106
II.D.2.1 Calculating the Risk of an Existing Portfolio ...................................................... 107
II.D.2.2 Deriving Asset Weights for the Minimum Risk Portfolio................................. 110
II.D.2.3 Hedging a Vanilla Option Position ....................................................................... 113
II.D.3 Quadratic Forms............................................................................................................... 117
II.D.3.1 The Variance of Portfolio Returns as a Quadratic Form................................. 119
II.D.3.2 Definition of Positive Definiteness....................................................................... 119
II.D.4 Cholesky Decomposition................................................................................................ 120
II.D.4.1 The Cholesky Arithmetic ........................................................................................ 121
II.D.4.2 Simulation in Excel .................................................................................................. 121
II.D.5 Eigenvalues and Eigenvectors........................................................................................ 125
II.D.5.1 Matrices as Transformations ................................................................................. 125
II.D.5.2 Definition of Eigenvector and Eigenvalue........................................................... 126
II.D.5.3 Determinants............................................................................................................. 128
II.D.5.4 The Characteristic Equation................................................................................... 129
II.D.5.5 Principal Components ............................................................................................. 131
References ............................................................................................................................................ 132
II.E Probability Theory in Finance........................................................................................................ 135
II.E.1 Definitions and Rules ...................................................................................................... 136
II.E.1.1 Definitions................................................................................................................ 136
II.E.1.2 Rules for Probability............................................................................................... 137
II.E.2 Probability Distributions................................................................................................. 139
II.E.2.1 Random Variables ................................................................................................... 139
II.E.2.2 Probability Density Functions and Histograms ................................................. 141

II.E.2.3 The Cumulative Distribution Function............................................................... 142
II.E.2.4 The Algebra of Random Variables....................................................................... 143
II.E.2.5 The Expected Value of a Discrete Random Variable ....................................... 144
II.E.2.6 The Variance of a Discrete Random Variable.................................................... 144
II.E.2.7 The Algebra of Continuous Random Variables................................................. 146
II.E.3 Joint Distributions............................................................................................................ 146
II.E.3.1 Bivariate Random Variables .................................................................................. 147
II.E.3.2 Covariance................................................................................................................ 148
II.E.3.3 Correlation................................................................................................................ 149
II.E.3.4 The Expected Value and Variance of a Linear Combination of Random
Variables 151
II.E.4 Specific Probability Distributions.................................................................................. 151
II.E.4.1 The Binomial Distribution..................................................................................... 151
II.E.4.2 The Poisson Distribution....................................................................................... 155
II.E.4.3 The Uniform Continuous Distribution............................................................... 158
II.E.4.4 The Normal Distribution....................................................................................... 159
II.E.4.5 The Lognormal Probability Distribution ............................................................ 163
II.E.4.7 The Bivariate Normal Distribution...................................................................... 168
II.F Regression Analysis in Finance ...................................................................................................... 171
II.F.1 Simple Linear Regression................................................................................................ 172
II.F.1.1 The Model..................................................................................................................... 172
II.F.1.2 The Scatter Plot............................................................................................................ 174
II.F.1.3 Estimating the Parameters.......................................................................................... 175
II.F.2 Multiple Linear Regression............................................................................................. 180
II.F.2.1 The Model..................................................................................................................... 180
II.F.2.2 Estimating the Parameters.......................................................................................... 181
II.F.3 Evaluating the Regression Model .................................................................................. 182
II.F.3.1 Intuitive Interpretation ............................................................................................... 183
II.F.3.2 Adjusted R2 ................................................................................................................... 183
II.F.3.3 Testing for Statistical Significance............................................................................. 184
II.F.4 Confidence Intervals........................................................................................................ 184
II.F.4.1 Confidence Intervals for the Regression Parameters............................................. 184
II.F.5 Hypothesis Testing........................................................................................................... 186
II.F.5.1 Significance Tests for the Regression Parameters .................................................. 187
II.F.5.2 Significance Test for R2 .............................................................................................. 188
II.F.5.3 Type I and Type II Errors.......................................................................................... 188
II.F.6 Prediction.......................................................................................................................... 189

II.F.7 Breakdown of the OLS Assumptions........................................................................... 191
II.F.7.1 Heteroscedasticity........................................................................................................ 191
II.F.7.2 Autocorrelation............................................................................................................ 192
II.F.7.3 Multicollinearity............................................................................................................ 192
II.F.8 Random Walks and Mean Reversion............................................................................ 193
II.F.9 Maximum Likelihood Estimation.................................................................................. 196
II.F.10 Summary ............................................................................................................................. 197
References ............................................................................................................................................ 197
II.G Numerical Methods........................................................................................................................ 199
II.G.1 Solving (Non-differential) Equations............................................................................ 199
II.G.1.1 Three Problems ....................................................................................................... 199
II.G.1.2 Bisection ................................................................................................................... 201
II.G.1.3 Newton–Raphson................................................................................................... 205
II.G.1.4 Goal Seek.................................................................................................................. 207
II.G.2 Numerical Optimisation.................................................................................................. 207
II.G.2.1 The Problem............................................................................................................ 208
II.G.2.2 Unconstrained Numerical Optimisation............................................................. 209
II.G.2.3 Constrained Numerical Optimisation.................................................................. 211
II.G.2.4 Portfolio Optimisation Revisited.......................................................................... 212
II.G.3 Numerical Methods for Valuing Options .................................................................... 213
II.G.3.1 Binomial Lattices..................................................................................................... 214
II.G.3.2 Finite Difference Methods .................................................................................... 220
II.G.3.3 Simulation................................................................................................................. 223
II.G.4 Summary ............................................................................................................................ 226
References ............................................................................................................................................ 226