Mathematics and statistics for financial risk management (Q2892692)

Today's economics faces the financial crisis all over the world. Because of this crisis the importance of financial risk management has arisen. Moreover, financial products and strategies are becoming more complex. In this light, it becomes more and more important to know and to understand the tools of mathematics and statistics.NEWLINENEWLINEThe book under review is a guide to modern financial risk management for both practitioners and academics. Many mathematical and statistical tools, used in risk management, have been adapted from other fields, but risk managers have developed their own vocabulary for characterizing risk. These tools and vocabulary becomes more standard and this book serve to fill the gap between theory and practice.NEWLINENEWLINEAll the chapters of the book focus on different topics of the mathematics and statistics. Different techniques are introduced and various applications to actual risk management problems are presented. Each chapter is accompanied with the exercises which solutions are given at the end of the book. These exercises the reader to practise the techniques and to follow the progress. Many exercises are with the special icon indicating that Excel examples can be found at the accompanying John Wiley \& Sons' web site.NEWLINENEWLINEThe first chapter ``Some basic mathematics'' begins with presenting the background for logarithms and it's application in finance for computing the log returns. Further, the author of the book focuses on the compounding return and limited liability. Another topic related to the idea of log returns is continuously compound returns. The next topic is elementary combinatorics and its applications to the discount factor. The chapter ends with an introduction on geometric series which play a role in many financial scenarios.NEWLINENEWLINEIn the second chapter ``Probabilities'', the reader is introduced to discrete and continuous random variables. There are basics on probability density function, cumulative and inverse cumulative distribution functions. Further, the author of the book talks about mutually exclusive events and independent events. Also, the reader gets to know about probability matrices, conditional probability and Bayes' theorem. All the topics are followed by applications in finance.NEWLINENEWLINEThe third chapter is devoted to descriptive statistics. The author introduces averages for discrete and continuous random variables. Next he explains the properties of expectations and turns to the variance and standard deviation. In addition, in this chapter we can find introduction to standardized variables, covariance and correlation. The author of the book shows how we can apply the presented tools for estimation of portfolio variance and hedging. The chapter ends with the introduction to higher moments and best linear unbiased estimator (BLUE).NEWLINENEWLINEAfter descriptive statistics, the author introduce probability distributions most widely used in risk management: uniform, Bernoulli, binomial, Poisson, normal, lognormal, chi squared and student's distributions. In addition, the author of the book describes F-distributions and mixture of distinct distributions. Furthermore, we find in this chapter an illustration of the central limit theorem and a description of Monte Carlo simulations for creating normal random variables.NEWLINENEWLINEIn the fifth chapter, one can find two closely related topics: hypothesis testing and confidence intervals. At first the author goes back to sample mean and sample variance by constructing there confidence intervals. Then he shows the methods of hypothesis testing and explains in which way of testing one should choose. In addition, the author discusses Chebyshev's inequality and gives its applications to Value at Risk (VaR).NEWLINENEWLINEElements of matrix algebra are presented in the sixth chapter. Here, the author briefly introduces matrix notations and explains the main operations with the matrices. Then he gives some applications of this theory to transition matrices and to Monte Carlo simulations of the Cholesky decomposition.NEWLINENEWLINEThe seventh chapter ``Vector spaces'' is devoted to the introduction to vectors and their main properties (orthogonality, rotation). The central section of this chapter is devoted to principal component analysis. The material of this chapter can be applied to the analysis of the dynamic term structure of interest rates and the structure of global equity markets.NEWLINENEWLINEIn the eighth chapter, the author turns to linear regression models. He introduces bivariate and multivariate linear regressions, least squares estimates for the coefficients and regression adequacy testing. He also discusses multicollinearity as a common problem. The examples of the regression applications are given for factor analysis and stress testing.NEWLINENEWLINEAnother important tool used in risk management is the time series model. To this, the author devotes the ninth chapter. This chapter starts from the description of random walks and drift-diffusion. Then the author talks about autoregression, its variance and autocorrelation. Further, in this chapter, the stationarity and moving average of time series models are discussed. In addition, we can find a section about continuous time models. The chapter ends with applications to GARCH, jump-diffusion and interest rate models.NEWLINENEWLINEIn the last chapter ``Decay factors'', we can find the class of estimators that are very popular in finance and risk management for historical data analysis. The main discussed estimators are: mean estimators with the different weights (decay factor), variance estimators with the decay factor, weighted least squares estimator. The application section of this chapter focuses on hybrid VaR.NEWLINENEWLINEThe book ends with the ``Appendices''. Here we can find brief mathematical background for binary numbers, Taylor expansions and vector spaces. Also, the author gives the Greek alphabet and common abbreviations.