A Pocket Guide to Risk Mathematics - Key Concepts Every Auditor Should Know

Start here 1 Good choice! 1 This book 2 How this book works 3 The myth of mathematical clarity 5 The myths of quantification 7 The auditor s mission 8 Auditing simple risk assessments 11 1 Probabilities 12 2 Probabilistic forecaster 13 3 Calibration (also known as reliability) 13 4 Resolution 14 5 Proper score function 15 6 Audit point: Judging probabilities 17 7 Probability interpretations 17 8 Degree of belief 18 9 Situation (also known as an experiment) 19 10 Long run relative frequency 20 11 Degree of belief about long run relative frequency 21 12 Degree of belief about an outcome 22 13 Audit point: Mismatched interpretations of probability 24 14 Audit point: Ignoring uncertainty about probabilities 25 15 Audit point: Not using data to illuminate probabilities 25 16 Outcome space (also known as sample space, or possibility space) 26 17 Audit point: Unspecified situations 27 18 Outcomes represented without numbers 28 19 Outcomes represented with numbers 29 20 Random variable 29 21 Event 30 22 Audit point: Events with unspecified boundaries 31 23 Audit point: Missing ranges 32 24 Audit point: Top 10 risk reporting 32 25 Probability of an outcome 33 26 Probability of an event 34 27 Probability measure (also known as probability distribution, probability function, or even probability distribution function) 34 28 Conditional probabilities 36 29 Discrete random variables 37 30 Continuous random variables 38 31 Mixed random variables (also known as mixed discrete-continuous random variables) 39 32 Audit point: Ignoring mixed random variables 40 33 Cumulative probability distribution function 41 34 Audit point: Ignoring impact spread 43 35 Audit point: Confusing money and utility 44 36 Probability mass function 44 37 Probability density function 45 38 Sharpness 47 39 Risk 49 40 Mean value of a probability distribution (also known as the expected value) 50 41 Audit point: Excessive focus on expected values 51 42 Audit point: Misunderstanding expected 51 43 Audit point: Avoiding impossible provisions 52 44 Audit point: Probability impact matrix numbers 53 45 Variance 54 46 Standard deviation 55 47 Semi-variance 55 48 Downside probability 55 49 Lower partial moment 56 50 Value at risk (VaR) 56 51 Audit point: Probability times impact 58 Some types of probability distribution 61 52 Discrete uniform distribution 62 53 Zipf distribution 62 54 Audit point: Benford s law 64 55 Non-parametric distributions 65 56 Analytical expression 65 57 Closed form (also known as a closed formula or explicit formula) 66 58 Categorical distribution 67 59 Bernoulli distribution 67 60 Binomial distribution 68 61 Poisson distribution 69 62 Multinomial distribution 70 63 Continuous uniform distribution 70 64 Pareto distribution and power law distribution 71 65 Triangular distribution 73 66 Normal distribution (also known as the Gaussian distribution) 74 67 Audit point: Normality tests 77 68 Non-parametric continuous distributions 78 69 Audit point: Multi-modal distributions 78 70 Lognormal distribution 79 71 Audit point: Thin tails 80 72 Joint distribution 80 73 Joint normal distribution 81 74 Beta distribution 82 Auditing the design of business prediction models 83 75 Process (also known as a system) 84 76 Population 84 77 Mathematical model 85 78 Audit point: Mixing models and registers 86 79 Probabilistic models (also known as stochastic models or statistical models) 86 80 Model structure 88 81 Audit point: Lost assumptions 89 82 Prediction formulae 89 83 Simulations 90 84 Optimization 90 85 Model inputs 90 86 Prediction formula structure 91 87 Numerical equation solving 93 88 Prediction algorithm 94 89 Prediction errors 94 90 Model uncertainty 94 91 Audit point: Ignoring model uncertainty 95 92 Measurement uncertainty 96 93 Audit point: Ignoring measurement uncertainty 96 94 Audit point: Best guess forecasts 97 95 Prediction intervals 97 96 Propagating uncertainty 98 97 Audit point: The flaw of averages 99 98 Random 100 99 Theoretically random 101 100 Real life random 102 101 Audit point: Fooled by randomness (1) 102 102 Audit point: Fooled by randomness (2) 104 103 Pseudo random number generation 104 104 Monte Carlo simulation 105 105 Audit point: Ignoring real options 109 106 Tornado diagram 109 107 Audit point: Guessing impact 111 108 Conditional dependence and independence 112 109 Correlation (also known as linear correlation) 113 110 Copulas 113 111 Resampling 114 112 Causal modelling 114 113 Latin hypercube 114 114 Regression 115 115 Dynamic models 116 116 Moving average 116 Auditing model fitting and validation 117 117 Exhaustive, mutually exclusive hypotheses 118 118 Probabilities applied to alternative hypotheses 119 119 Combining evidence 120 120 Prior probabilities 120 121 Posterior probabilities 120 122 Bayes s theorem 121 123 Model fitting 123 124 Hyperparameters 126 125 Conjugate distributions 126 126 Bayesian model averaging 128 127 Audit point: Best versus true explanation 128 128 Hypothesis testing 129 129 Audit point: Hypothesis testing in business 130 130 Maximum a posteriori estimation (MAP) 131 131 Mean a posteriori estimation 131 132 Median a posteriori estimation 132 133 Maximum likelihood estimation (MLE) 132 134 Audit point: Best estimates of parameters 135 135 Estimators 135 136 Sampling distribution 138 137 Least squares fitting 138 138 Robust estimators 140 139 Over-fitting 140 140 Data mining 141 141 Audit point: Searching for significance 142 142 Exploratory data analysis 143 143 Confirmatory data analysis 143 144 Interpolation and extrapolation 143 145 Audit Point: Silly extrapolation 144 146 Cross validation 145 147 R2 (the coefficient of determination) 145 148 Audit point: Happy history 147 149 Audit point: Spurious regression results 147 150 Information graphics 148 151 Audit point: Definition of measurements 148 152 Causation 149 Auditing and samples 151 153 Sample 152 154 Audit point: Mixed populations 152 155 Accessible population 152 156 Sampling frame 153 157 Sampling method 153 158 Probability sample (also known as a random sample) 154 159 Equal probability sampling (also known as simple random sampling) 155 160 Stratified sampling 155 161 Systematic sampling 156 162 Probability proportional to size sampling 156 163 Cluster sampling 156 164 Sequential sampling 157 165 Audit point: Prejudging sample sizes 158 166 Dropouts 159 167 Audit point: Small populations 160 Auditing in the world of high finance 163 168 Extreme values 164 169 Stress testing 165 170 Portfolio models 166 171 Historical simulation 168 172 Heteroskedasticity 169 173 RiskMetrics variance model 169 174 Parametric portfolio model 170 175 Back-testing 170 176 Audit point: Risk and reward 171 177 Portfolio effect 172 178 Hedge 172 179 Black Scholes 173 180 The Greeks 175 181 Loss distributions 176 182 Audit point: Operational loss data 178 183 Generalized linear models 179 Congratulations 181 Useful websites 183 Index 185