The mathematics of banking and finance

Throughout banking, mathematical techniques are used. Some of these are within software products or models; mathematicians use others to analyse data. The current literature on the subject is either very basic or very advanced.

The Mathematics of Banking offers an intermediate guide to the various techniques used in the industry, and a consideration of how each one should be approached. Written in a practical style, it will enable readers to quickly appreciate the purpose of the techniques and, through illustrations, see how they can be applied in practice. Coverage is extensive and includes techniques such as VaR analysis, Monte Carlo simulation, extreme value theory, variance and many others.

A practical review of mathematical techniques needed in banking which does not expect a high level of mathematical competence from the reader

Author Notes

Michael Cox was born on August 30 1948 in Northamptonshire, England. In 1989 he started work at the Oxford University Press. In 1983, Cox published his first book, a biography M. R. James, a Victorian ghost story writer. Between 1983 and 1997 he compiled and edited several anthologies of Victorian short stories for Oxford University Press. His first novel, The Meaning of Night, was published in 2006. Michael Cox died of cancer on March 31, 2009.

(Bowker Author Biography)

Introduction	p. xiii
1 Introduction to How to Display Data and the Scatter Plot	p. 1
1.1 Introduction	p. 1
1.2 Scatter Plots	p. 2
1.3 Data Identification	p. 2
1.3.1 An example of salary against age	p. 2
1.4 Why Draw a Scatter Plot?	p. 3
1.5 Matrix Plots	p. 4
1.5.1 An example of salary against age: Revisited	p. 5
2 Bar Charts	p. 7
2.1 Introduction	p. 7
2.2 Discrete Data	p. 7
2.3 Relative Frequencies	p. 8
2.4 Pie Charts	p. 12
3 Histograms	p. 13
3.1 Continuous Variables	p. 13
3.2 Cumulative Frequency Polygon	p. 14
3.3 Sturges' Formula	p. 20
4 Probability Theory	p. 21
4.1 Introduction	p. 21
4.2 Basic Probability Concepts	p. 21
4.3 Estimation of Probabilities	p. 22
4.4 Exclusive Events	p. 22
4.5 Independent Events	p. 22
4.6 Comparison of Exclusivity and Independence	p. 23
4.7 Venn Diagrams	p. 23
4.8 The Addition Rule for Probabilities	p. 24
4.8.1 A simple probability example using a Venn diagram	p. 25
4.9 Conditional Probability	p. 25
4.9.1 An example of conditional probability	p. 26
4.10 The Multiplication Rule for Probabilities	p. 26
4.10.1 A classical example of conditional probability	p. 27
4.11 Bayes' Theorem	p. 27
4.11.1 An example of Bayes' theorem	p. 28
4.11.2 Bayes' theorem in action for more groups	p. 29
4.11.3 Bayes' theorem applied to insurance	p. 29
4.12 Tree Diagram	p. 30
4.12.1 An example of prediction of success	p. 30
4.12.2 An example from an American game show: The Monty Hall Problem	p. 34
4.13 Conclusion	p. 35
5 Standard Terms in Statistics	p. 37
5.1 Introduction	p. 37
5.2 Maximum and Minimum	p. 37
5.2.1 Mean	p. 37
5.2.2 Median	p. 38
5.2.3 Mode	p. 39
5.3 Upper and Lower Quartile	p. 39
5.4 MQMQM Plot	p. 40
5.5 Skewness	p. 41
5.6 Variance and Standard Deviation	p. 41
5.7 Measures for Continuous Data	p. 44
6 Sampling	p. 47
6.1 Introduction	p. 47
6.2 Planning Data Collection	p. 47
6.3 Methods for Survey Analysis	p. 48
6.3.1 Random samples	p. 49
6.3.2 Systematic sampling	p. 49
6.3.3 Stratified sampling	p. 49
6.3.4 Multistage sampling	p. 50
6.3.5 Quota sampling	p. 50
6.3.6 Cluster sampling	p. 50
6.4 How It Can Go Wrong	p. 50
6.5 What Might Be In a Survey?	p. 51
6.6 Cautionary Notes	p. 51
7 Probability Distribution Functions	p. 53
7.1 Introduction	p. 53
7.2 Discrete Uniform Distribution	p. 53
7.2.1 Counting techniques	p. 54
7.2.2 Combination	p. 54
7.2.3 Permutation	p. 55
7.3 Binomial Distribution	p. 55
7.3.1 Example of a binomial distribution	p. 56
7.3.2 Pascal's triangle	p. 56
7.3.3 The use of the binomial distribution	p. 57
7.4 The Poisson Distribution	p. 58
7.4.1 An example of the Poisson distribution	p. 59
7.4.2 Uses of the Poisson distribution	p. 60
7.5 Uses of the Binomial and Poisson Distributions	p. 60
7.5.1 Is suicide a Poisson process?	p. 62
7.6 Continuous Uniform Distribution	p. 64
7.7 Exponential Distribution	p. 66
8 Normal Distribution	p. 67
8.1 Introduction	p. 67
8.2 Normal Distribution	p. 67
8.2.1 A simple example of normal probabilities	p. 69
8.2.2 A second example of normal probabilities	p. 69
8.3 Addition of Normal Variables	p. 70
8.4 Central Limit Theorem	p. 70
8.4.1 An example of the Central Limit Theorem	p. 70
8.5 Confidence Intervals for the Population Mean	p. 71
8.5.1 An example of confidence intervals for the population mean	p. 71
8.6 Normal Approximation to the Binomial Distribution	p. 72
8.6.1 An example of the normal approximation to the binomial distribution	p. 72
8.7 Normal Approximation to the Poisson Distribution	p. 72
8.7.1 An example of fitting a normal curve to the Poisson distribution	p. 73
9 Comparison of the Means, Sample Sizes and Hypothesis Testing	p. 75
9.1 Introduction	p. 75
9.2 Estimation of the Mean	p. 75
9.2.1 An example of estimating a confidence interval for an experimental mean	p. 76
9.3 Choice of the Sample Size	p. 77
9.3.1 An example of selecting sample size	p. 77
9.4 Hypothesis Testing	p. 77
9.4.1 An example of hypothesis testing	p. 78
9.5 Comparison of Two Sample Means	p. 79
9.5.1 An example of a two-sample t test	p. 79
9.6 Type I and Type II Errors	p. 80
9.6.1 An example of type I and type II errors	p. 80
10 Comparison of Variances	p. 83
10.1 Introduction	p. 83
10.2 Chi-Squared Test	p. 83
10.2.1 An example of the chi-squared test	p. 83
10.3 F Test	p. 85
10.3.1 An example of the F test	p. 85
10.3.2 An example considering the normal distribution	p. 85
11 Chi-squared Goodness of Fit Test	p. 91
11.1 Introduction	p. 91
11.2 Contingency Tables	p. 92
11.3 Multiway Tables	p. 94
11.3.1 An example of a four by four table	p. 94
12 Analysis of Paired Data	p. 97
12.1 Introduction	p. 97
12.2 t Test	p. 97
12.3 Sign Test	p. 98
12.4 The U Test	p. 99
12.4.1 An example of the use of the U test	p. 101
13 Linear Regression	p. 103
13.1 Introduction	p. 103
13.2 Linear Regression	p. 103
13.3 Correlation Coefficient	p. 104
13.3.1 An example of examining correlation	p. 105
13.4 Estimation of the Uncertainties	p. 109
13.5 Statistical Analysis and Interpretation of Linear Regression	p. 110
13.6 ANOVA for Linear Regression	p. 110
13.7 Equations for the Variance of a and b	p. 112
13.8 Significance Test for the Slope	p. 112
13.8.1 An example of slope analysis	p. 113
13.8.2 A further example of correlation and linear regression	p. 115
14 Analysis of Variance	p. 121
14.1 Introduction	p. 121
14.2 Formal Background to the ANOVA Table	p. 121
14.3 Analysis of the ANOVA Table	p. 122
14.4 Comparison of Two Causal Means	p. 123
14.4.1 An example of extinguisher discharge times	p. 123
14.4.2 An example of the lifetime of lamps	p. 125
15 Design and Approach to the Analysis of Data	p. 129
15.1 Introduction	p. 129
15.2 Randomised Block Design	p. 129
15.2.1 An example of outsourcing	p. 130
15.3 Latin Squares	p. 131
15.4 Analysis of a Randomised Block Design	p. 132
15.5 Analysis of a Two-way Classification	p. 135
15.5.1 An example of two-way analysis	p. 137
15.5.2 An example of a randomised block	p. 140
15.5.3 An example of the use of the Latin square	p. 143
16 Linear Programming: Graphical Method	p. 149
16.1 Introduction	p. 149
16.2 Practical Examples	p. 149
16.2.1 An example of an optimum investment strategy	p. 149
16.2.2 An example of the optimal allocation of advertising	p. 154
17 Linear Programming: Simplex Method	p. 159
17.1 Introduction	p. 159
17.2 Most Profitable Loans	p. 159
17.2.1 An example of finance selection	p. 164
17.3 General Rules	p. 167
17.3.1 Standardisation	p. 167
17.3.2 Introduction of additional variables	p. 167
17.3.3 Initial solution	p. 167
17.3.4 An example to demonstrate the application of the general rules for linear programming	p. 167
17.4 The Concerns with the Approach	p. 170
18 Transport Problems	p. 171
18.1 Introduction	p. 171
18.2 Transport Problem	p. 171
19 Dynamic Programming	p. 179
19.1 Introduction	p. 179
19.2 Principle of Optimality	p. 179
19.3 Examples of Dynamic Programming	p. 180
19.3.1 An example of forward and backward recursion	p. 180
19.3.2 A practical example of recursion in use	p. 182
19.3.3 A more complex example of dynamic programming	p. 184
19.3.4 The 'Travelling Salesman' problem	p. 185
20 Decision Theory	p. 189
20.1 Introduction	p. 189
20.2 Project Analysis Guidelines	p. 190
20.3 Minimax Regret Rule	p. 192
21 Inventory and Stock Control	p. 195
21.1 Introduction	p. 195
21.2 The Economic Order Quantity Model	p. 195
21.2.1 An example of the use of the economic order quantity model	p. 196
21.3 Non-zero Lead Time	p. 199
21.3.1 An example of Poisson and continuous approximation	p. 200
22 Simulation: Monte Carlo Methods	p. 203
22.1 Introduction	p. 203
22.2 What is Monte Carlo Simulation?	p. 203
22.2.1 An example of the use of Monte Carlo simulation: Theory of the inventory problem	p. 203
22.3 Monte Carlo Simulation of the Inventory Problem	p. 205
22.4 Queuing Problem	p. 208
22.5 The Bank Cashier Problem	p. 209
22.6 Monte Carlo for the Monty Hall Problem	p. 212
22.7 Conclusion	p. 214
23 Reliability: Obsolescence	p. 215
23.1 Introduction	p. 215
23.2 Replacement at a Fixed Age	p. 215
23.3 Replacement at Fixed Times	p. 217
24 Project Evaluation	p. 219
24.1 Introduction	p. 219
24.2 Net Present Value	p. 219
24.2.1 An example of net present value	p. 219
24.3 Internal Rate of Return	p. 220
24.3.1 An example of the internal rate of return	p. 220
24.4 Price/Earnings Ratio	p. 222
24.5 Payback Period	p. 222
24.5.1 Mathematical background to the payback period	p. 222
24.5.2 Mathematical background to producing the tables	p. 223
25 Risk and Uncertainty	p. 227
25.1 Introduction	p. 227
25.2 Risk	p. 227
25.3 Uncertainty	p. 227
25.4 Adjusting the Discount Rate	p. 228
25.5 Adjusting the Cash Flows of a Project	p. 228
25.5.1 An example of expected cash flows	p. 228
25.6 Assessing the Margin of Error	p. 229
25.6.1 An example of break-even analysis	p. 229
25.7 The Expected Value of the Net Present Value	p. 231
25.7.1 An example of the use of the distribution approach to the evaluation of net present value	p. 231
25.8 Measuring Risk	p. 232
25.8.1 An example of normal approximation	p. 234
26 Time Series Analysis	p. 235
26.1 Introduction	p. 235
26.2 Trend Analysis	p. 236
26.3 Seasonal Variations	p. 236
26.4 Cyclical Variations	p. 240
26.5 Mathematical Analysis	p. 240
26.6 Identification of Trend	p. 241
26.7 Moving Average	p. 241
26.8 Trend and Seasonal Variations	p. 242
26.9 Moving Averages of Even Numbers of Observations	p. 244
26.10 Graphical Methods	p. 247
27 Reliability	p. 249
27.1 Introduction	p. 249
27.2 Illustrating Reliability	p. 249
27.3 The Bathtub Curve	p. 249
27.4 The Continuous Case	p. 251
27.5 Exponential Distribution	p. 252
27.5.1 An example of exponential distribution	p. 252
27.5.2 An example of maximum of an exponential distribution	p. 254
27.6 Weibull Distribution	p. 255
27.6.1 An example of a Weibull distribution	p. 256
27.7 Log-Normal Distribution	p. 257
27.8 Truncated Normal Distribution	p. 260
28 Value at Risk	p. 261
28.1 Introduction	p. 261
28.2 Extreme Value Distributions	p. 262
28.2.1 A worked example of value at risk	p. 262
28.3 Calculating Value at Risk	p. 264
29 Sensitivity Analysis	p. 267
29.1 Introduction	p. 267
29.2 The Application of Sensitivity Analysis to Operational Risk	p. 267
30 Scenario Analysis	p. 271
30.1 Introduction to Scenario Analysis	p. 271
30.2 Use of External Loss Data	p. 271
30.3 Scaling of Loss Data	p. 272
30.4 Consideration of Likelihood	p. 272
30.5 Anonimised Loss Data	p. 273
31 An Introduction to Neural Networks	p. 275
31.1 Introduction	p. 275
31.2 Neural Algorithms	p. 275
Appendix Mathematical Symbols and Notation	p. 279
Index	p. 285

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