This chapter is a cheat sheet describing the important R functions in the ggm, piecewiseSEM, pwSEM, and lavaan packages of R.

This chapter revisits the distinction between implicit and explicit latent variables and concentrates on explicit latents and how to model them in covariance-based SEM via a ‘measurement’ model or else a ‘composite’ latent. It explains how to model these using covariance-based SEM via the lavaan package.

This chapter looks at how risk appetite can be defined and how risk can be measured and used in further analysis.

This chapter describes the concept of economic capital and the construction of an economic capital model, as well as principles of capital allocation.

This chapter describes various definitions of risk, including financial and operational, both from a regulatory and a non-regulatory standpoint.

A dsep test uses the notion of d-separation to produce an inferential statistical test comparing observed data to the hypothesized mechanism (piecewise SEM). This involves obtaining a subset of d-separation claims that logically imply all others (the union basis set), obtaining the null probability of each of the (conditional) independence claims implied by the d-separation claims in this basis set, and combining them. Two ways of obtaining the null probabilities of each of these d-separation claims are explained: using regression slopes and using the generalized covariance statistic. These are implemented in the ‘piecewiseSEM’ and ‘pwSEM’ packages of R respectively. The rules for interpreting and manipulating the resulting path coefficients are explained.

This chapter describes the ways in which institutions can respond to each of the various risks they face.

This chapter describes how the various risks can be quantified, exploring the range of models that can be used.

This chapter describes a number of case studies relating to both financial and non-financial institutions.

This chapter describes the internal environment of a financial institution.

This chapter describes a number of important univariate and multivariate statistical distributions and their uses, as well as discussing various copulas.

This chapter describes the main approaches to identifying risk, in terms of tools and techniques.

A1.1 Basic matrix operations

A1.1.1 Definition (matrix)

A matrix is an ordered rectangular array of numbers (or functions). The numbers (or functions) are called the elements or the entries of the matrix.

A1.1.2 Dimension/size of a matrix

A matrix having m rows and a columns is a matrix of dimension m x n.

CHANCE PERMEATES OUR physical and mental universe. While the role of chance in human lives has had a longer history, starting with the more authoritative influence of the nobility, the more rationally sound theory of probability and statistics has come into practice in diverse areas of science and engineering starting from the early to mid-twentieth century. Practical applications of statistical theories proliferated to such an extent in the previous century that the American government-sponsored RAND corporation published a 600-page book that wholly consisted of a random number table and a table of standard normal deviates. One of the primary objectives of this book was to enable a computer-simulated approximate solution of an exact but unsolvable problem by a procedure known as the Monte Carlo method devised by Fermi, von Neumann, and Ulam in the 1930s–40s.

Statistical methods are the mainstay of conducting modern scientific experiments. One such experimental paradigm is known as a randomized control trial, which is widely used in a variety of fields such as psychology, drug verification, testing the efficacy of vaccines, agricultural sciences, and demography. These statistical experiments require sophisticated sampling techniques in order to nullify experimental biases. With the explosion of information in the modern era, the need to develop advanced and accurate predictive capabilities has grown manifold. This has led to the emergence of modern artificial intelligence (AI) technologies. Further, climate change has become a reality of modern civilization. Accurate prediction of weather and climatic patterns relies on sophisticated AI and statistical techniques. It is impossible to think of a modern economy and social life without the influence and role of chance, and hence without the influence of technological interventions based on statistical principles. We must begin this journey by learning the foundational tenets of probability and statistics.