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

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