A Pocket Guide to Risk Mathematics
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Matthew Leitch (Epsom, UK) is an author on a mission to make risk control easier, more natural, and much more valuable. His insightful, readable books are at the leading edge of thinking and practice in internal control and risk management. He frequently carries out original research on topical questions, such as how our use of words affects the way we think about uncertainty, and what expertise auditors need. He is a qualified chartered accountant and holds a BSc in psychology from University College London. He is author of Intelligent Internal Control and Risk Management, and runs the website, www.internalcontrolsdesign.co.uk. He speaks at numerous risk and audit conferences for organizations including the IIA and IIR.
Content
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
