Details

Essential Mathematics for Market Risk Management


Essential Mathematics for Market Risk Management


The Wiley Finance Series, Band 642 1. Aufl.

von: Simon Hubbert

41,99 €

Verlag: Wiley
Format: EPUB
Veröffentl.: 12.12.2011
ISBN/EAN: 9781119953029
Sprache: englisch
Anzahl Seiten: 352

DRM-geschütztes eBook, Sie benötigen z.B. Adobe Digital Editions und eine Adobe ID zum Lesen.

Beschreibungen

<b>Everything you need to know in order to manage risk effectively within your organization</b> <p>You cannot afford to ignore the explosion in mathematical finance in your quest to remain competitive. This exciting branch of mathematics has very direct practical implications: when a new model is tested and implemented it can have an immediate impact on the financial environment.</p> <p>With risk management top of the agenda for many organizations, this book is essential reading for getting to grips with the mathematical story behind the subject of financial risk management. It will take you on a journey—from the early ideas of risk quantification up to today's sophisticated models and approaches to business risk management.</p> <p>To help you investigate the most up-to-date, pioneering developments in modern risk management, the book presents statistical theories and shows you how to put statistical tools into action to investigate areas such as the design of mathematical models for financial volatility or calculating the value at risk for an investment portfolio.</p> <ul> <li>Respected academic author Simon Hubbert is the youngest director of a financial engineering program in the U.K. He brings his industry experience to his practical approach to risk analysis</li> <li>Captures the essential mathematical tools needed to explore many common risk management problems</li> <li>Website with model simulations and source code enables you to put models of risk management into practice</li> <li>Plunges into the world of high-risk finance and examines the crucial relationship between the risk and the potential reward of holding a portfolio of risky financial assets</li> </ul> <p>This book is your one-stop-shop for effective risk management.</p>
<b>Preface xiii</b> <p><b>1 Introduction 1</b></p> <p>1.1 Basic Challenges in Risk Management 1</p> <p>1.2 Value at Risk 3</p> <p>1.3 Further Challenges in Risk Management 6</p> <p><b>2 Applied Linear Algebra for Risk Managers 11</b></p> <p>2.1 Vectors and Matrices 11</p> <p>2.2 Matrix Algebra in Practice 17</p> <p>2.3 Eigenvectors and Eigenvalues 21</p> <p>2.4 Positive Definite Matrices 24</p> <p><b>3 Probability Theory for Risk Managers 27</b></p> <p>3.1 Univariate Theory 27</p> <p>3.1.1 Random variables 27</p> <p>3.1.2 Expectation 31</p> <p>3.1.3 Variance 32</p> <p>3.2 Multivariate Theory 33</p> <p>3.2.1 The joint distribution function 33</p> <p>3.2.2 The joint and marginal density functions 34</p> <p>3.2.3 The notion of independence 34</p> <p>3.2.4 The notion of conditional dependence 35</p> <p>3.2.5 Covariance and correlation 35</p> <p>3.2.6 The mean vector and covariance matrix 37</p> <p>3.2.7 Linear combinations of random variables 38</p> <p>3.3 The Normal Distribution 39</p> <p><b>4 Optimization Tools 43</b></p> <p>4.1 Background Calculus 43</p> <p>4.1.1 Single-variable functions 43</p> <p>4.1.2 Multivariable functions 44</p> <p>4.2 Optimizing Functions 47</p> <p>4.2.1 Unconstrained quadratic functions 48</p> <p>4.2.2 Constrained quadratic functions 50</p> <p>4.3 Over-determined Linear Systems 52</p> <p>4.4 Linear Regression 54</p> <p><b>5 Portfolio Theory I 63</b></p> <p>5.1 Measuring Returns 63</p> <p>5.1.1 A comparison of the standard and log returns 64</p> <p>5.2 Setting Up the Optimal Portfolio Problem 67</p> <p>5.3 Solving the Optimal Portfolio Problem 70</p> <p><b>6 Portfolio Theory II 77</b></p> <p>6.1 The Two-Fund Investment Service 77</p> <p>6.2 A Mathematical Investigation of the Optimal Frontier 78</p> <p>6.2.1 The minimum variance portfolio 78</p> <p>6.2.2 Covariance of frontier portfolios 78</p> <p>6.2.3 Correlation with the minimum variance portfolio 79</p> <p>6.2.4 The zero-covariance portfolio 79</p> <p>6.3 A Geometrical Investigation of the Optimal Frontier 80</p> <p>6.3.1 Equation of a tangent to an efficient portfolio 80</p> <p>6.3.2 Locating the zero-covariance portfolio 82</p> <p>6.4 A Further Investigation of Covariance 83</p> <p>6.5 The Optimal Portfolio Problem Revisited 86</p> <p><b>7 The Capital Asset Pricing Model (CAPM) 91</b></p> <p>7.1 Connecting the Portfolio Frontiers 91</p> <p>7.2 The Tangent Portfolio 94</p> <p>7.2.1 The market’s supply of risky assets 94</p> <p>7.3 The CAPM 95</p> <p>7.4 Applications of CAPM 96</p> <p>7.4.1 Decomposing risk 97</p> <p><b>8 Risk Factor Modelling 101</b></p> <p>8.1 General Factor Modelling 101</p> <p>8.2 Theoretical Properties of the Factor Model 102</p> <p>8.3 Models Based on Principal Component Analysis (PCA) 105</p> <p>8.3.1 PCA in two dimensions 106</p> <p>8.3.2 PCA in higher dimensions 112</p> <p><b>9 The Value at Risk Concept 117</b></p> <p>9.1 A Framework for Value at Risk 117</p> <p>9.1.1 A motivating example 120</p> <p>9.1.2 Defining value at risk 121</p> <p>9.2 Investigating Value at Risk 122</p> <p>9.2.1 The suitability of value at risk to capital allocation 124</p> <p>9.3 Tail Value at Risk 126</p> <p>9.4 Spectral Risk Measures 127</p> <p><b>10 Value at Risk under a Normal Distribution 131</b></p> <p>10.1 Calculation of Value at Risk 131</p> <p>10.2 Calculation of Marginal Value at Risk 132</p> <p>10.3 Calculation of Tail Value at Risk 134</p> <p>10.4 Sub-additivity of Normal Value at Risk 135</p> <p><b>11 Advanced Probability Theory for Risk Managers 137</b></p> <p>11.1 Moments of a Random Variable 137</p> <p>11.2 The Characteristic Function 140</p> <p>11.2.1 Dealing with the sum of several random variables 142</p> <p>11.2.2 Dealing with a scaling of a random variable 143</p> <p>11.2.3 Normally distributed random variables 143</p> <p>11.3 The Central Limit Theorem 145</p> <p>11.4 The Moment-Generating Function 147</p> <p>11.5 The Log-normal Distribution 148</p> <p><b>12 A Survey of Useful Distribution Functions 151</b></p> <p>12.1 The Gamma Distribution 151</p> <p>12.2 The Chi-Squared Distribution 154</p> <p>12.3 The Non-central Chi-Squared Distribution 157</p> <p>12.4 The F-Distribution 161</p> <p>12.5 The <i>t</i>-Distribution 164</p> <p><b>13 A Crash Course on Financial Derivatives 169</b></p> <p>13.1 The Black–Scholes Pricing Formula 169</p> <p>13.1.1 A model for asset returns 170</p> <p>13.1.2 A second-order approximation 172</p> <p>13.1.3 The Black–Scholes formula 174</p> <p>13.2 Risk-Neutral Pricing 176</p> <p>13.3 A Sensitivity Analysis 179</p> <p>13.3.1 Asset price sensitivity: The delta and gamma measures 179</p> <p>13.3.2 Time decay sensitivity: The theta measure 182</p> <p>13.3.3 The remaining sensitivity measures 183</p> <p><b>14 Non-linear Value at Risk 185</b></p> <p>14.1 Linear Value at Risk Revisited 185</p> <p>14.2 Approximations for Non-linear Portfolios 186</p> <p>14.2.1 Delta approximation for the portfolio 188</p> <p>14.2.2 Gamma approximation for the portfolio 189</p> <p>14.3 Value at Risk for Derivative Portfolios 190</p> <p>14.3.1 Multi-factor delta approximation 190</p> <p>14.3.2 Single-factor gamma approximation 191</p> <p>14.3.3 Multi-factor gamma approximation 192</p> <p><b>15 Time Series Analysis 197</b></p> <p>15.1 Stationary Processes 197</p> <p>15.1.1 Purely random processes 198</p> <p>15.1.2 White noise processes 198</p> <p>15.1.3 Random walk processes 199</p> <p>15.2 Moving Average Processes 199</p> <p>15.3 Auto-regressive Processes 201</p> <p>15.4 Auto-regressive Moving Average Processes 203</p> <p><b>16 Maximum Likelihood Estimation 207</b></p> <p>16.1 Sample Mean and Variance 209</p> <p>16.2 On the Accuracy of Statistical Estimators 211</p> <p>16.2.1 Sample mean example 211</p> <p>16.2.2 Sample variance example 212</p> <p>16.3 The Appeal of the Maximum Likelihood Method 215</p> <p><b>17 The Delta Method for Statistical Estimates 217</b></p> <p>17.1 Theoretical Framework 217</p> <p>17.2 Sample Variance 219</p> <p>17.3 Sample Skewness and Kurtosis 221</p> <p>17.3.1 Analysis of skewness 222</p> <p>17.3.2 Analysis of kurtosis 223</p> <p><b>18 Hypothesis Testing 227</b></p> <p>18.1 The Testing Framework 227</p> <p>18.1.1 The null and alternative hypotheses 227</p> <p>18.1.2 Hypotheses: simple vs compound 228</p> <p>18.1.3 The acceptance and rejection regions 228</p> <p>18.1.4 Potential errors 229</p> <p>18.1.5 Controlling the testing errors/defining the acceptance region 229</p> <p>18.2 Testing Simple Hypotheses 230</p> <p>18.2.1 Testing the mean when the variance is known 231</p> <p>18.3 The Test Statistic 233</p> <p>18.3.1 Example: Testing the mean when the variance is unknown 234</p> <p>18.3.2 The <i>p</i>-value of a test statistic 236</p> <p>18.4 Testing Compound Hypotheses 237</p> <p><b>19 Statistical Properties of Financial Losses 241</b></p> <p>19.1 Analysis of Sample Statistics 244</p> <p>19.2 The Empirical Density and Q–Q Plots 247</p> <p>19.3 The Auto-correlation Function 247</p> <p>19.4 The Volatility Plot 252</p> <p>19.5 The Stylized Facts 253</p> <p><b>20 Modelling Volatility 255</b></p> <p>20.1 The RiskMetrics Model 256</p> <p>20.2 ARCH Models 258</p> <p>20.2.1 The ARCH(1) volatility model 260</p> <p>20.3 GARCH Models 264</p> <p>20.3.1 The GARCH(1, 1) volatility model 265</p> <p>20.3.2 The RiskMetrics model revisited 268</p> <p>20.3.3 Summary 269</p> <p>20.4 Exponential GARCH 269</p> <p><b>21 Extreme Value Theory 271</b></p> <p>21.1 The Mathematics of Extreme Events 271</p> <p>21.1.1 A naive attempt 273</p> <p>21.1.2 Example 1: Exponentially distributed losses 273</p> <p>21.1.3 Example 2: Normally distributed losses 274</p> <p>21.1.4 Example 3: Pareto distributed losses 275</p> <p>21.1.5 Example 4: Uniformly distributed losses 275</p> <p>21.1.6 Example 5: Cauchy distributed losses 276</p> <p>21.1.7 The extreme value theorem 277</p> <p>21.2 Domains of Attraction 278</p> <p>21.2.1 The Fr´echet domain of attraction 280</p> <p>21.3 Extreme Value at Risk 283</p> <p>21.4 Practical Issues 286</p> <p>21.4.1 Parameter estimation 286</p> <p>21.4.2 The choice of threshold 287</p> <p><b>22 Simulation Models 291</b></p> <p>22.1 Estimating the Quantile of a Distribution 291</p> <p>22.1.1 Asymptotic behaviour 293</p> <p>22.2 Historical Simulation 296</p> <p>22.3 Monte Carlo Simulation 299</p> <p>22.3.1 The Choleski algorithm 300</p> <p>22.3.2 Generating random numbers 302</p> <p><b>23 Alternative Approaches to VaR 309</b></p> <p>23.1 The <i>t</i>-Distributed Assumption 309</p> <p>23.2 Corrections to the Normal Assumption 313</p> <p><b>24 Backtesting 319</b></p> <p>24.1 Quantifying the Performance of VaR 319</p> <p>24.2 Testing the Proportion of VaR Exceptions 320</p> <p>24.3 Testing the Independence of VaR Exceptions 323</p> <p><b>References 327</b></p> <p><b>Index 331</b></p>
<p><b><i>About the author</i></b></p> <p><b>DR SIMON HUBBERT is a lecturer in Mathematics and Mathematical Finance at Birkbeck College, University of London, where he is currently the programme director for the graduate diploma in Financial Engineering. He has taught masters level courses on Risk Management and Financial Mathematics for many years and also has valuable experience in the financial industry having engaged in consultation work with IBM global business services and as a risk analyst for the debt management office, a branch of HM Treasury.</b></p>
<p><b>Essential mathematics for market risk management</b><p><b>Simon Hubbert</b><p><b> In finance the universally held view is that the more risk we take the more reward we stand to gain but, just as importantly, the greater the chance of loss. The role of the financial risk manager is to be aware of the presence of risk, to understand how it can damage a potential investment and, most of all, be able to reduce the exposure to it in order to avert a potential disaster.</b><p><b><b><i>Essential Mathematics for Market Risk Management</i></b> provides readers with the mathematical tools for managing and controlling the major sources of risk in the financial markets. Unlike most books on investment risk management which tend to be either panoptic in their coverage or narrowly focused on advanced mathematical procedures, this book offers a thorough understanding of the basic mathematical concepts and procedures required to satisfy the two key criteria of financial risk management: to ensure a healthy return on investment for a tolerable amount of risk, and to insulate a portfolio against catastrophic market events.</b><p><b> To this end, Dr Simon Hubbert, has drawn from his previous experience in the financial industry to develop a format which clearly and methodically</b><ul><li><b>Traces the evolution of quantitative risk management – from Markowitz’s landmark solution to the portfolio problem in the 1950s, to the emergence of Value at Risk (VaR) in the mid 1990s and its subsequent impact.</b></li><li><b>Provides the basic mathematical tools needed to understand and solve common risk management problems, including applied linear algebra, probability theory and mathematical optimization.</b></li><li><b>Introduces and explains the statistical theory, tools and techniques behind cutting-edge research into financial risk management taking place in professional and academic institutions globally.</b></li><li><b>Explores a range of advanced topics in quantitative risk management, including derivative pricing, non-linear Value at Risk, volatility modelling and extreme value theory.</b></li></ul><p><b> By focusing on the key issues a typical financial risk manager faces on both a daily and longterm basis – from monitoring portfolio performance to modelling the volatility of specific assets – this book is essential reading for finance professionals and students who recognize the need to be conversant in modern quantitative methods for financial risk management. </b>

Diese Produkte könnten Sie auch interessieren:

Agile Project Management
Agile Project Management
von: Project Management Journal
EPUB ebook
23,99 €
Make Change Work
Make Change Work
von: Randy Pennington
PDF ebook
14,99 €
Nonprofit Law Made Easy
Nonprofit Law Made Easy
von: Bruce R. Hopkins
EPUB ebook
53,99 €