A taxonomy is a classification system. In this chapter, we present a risk taxonomy, by which we mean that we shall categorize and describe all the major risks that may be faced by a firm or institution. We will describe risks that arise from outside the organization (external risks) and those that come from within the organization (internal risks). External risks are further categorized into economic, political, and environmental categories, while internal risks include operational and strategic risks. Reputational risk may be internally or externally generated. We describe some examples of how risks have arisen in several high-profile cases, showing the intersectionality of the different risk categories – that is, how the different risk types can all be driven by a single risk event.

In this chapter, we discuss the ways that credit risk arises, and how it can be modelled and mitigated. First, we consider the various types of contractual forms for loans and other obligations. We then discuss credit derivatives, which are contracts with payoffs that are contingent on credit events. We consider credit risk models based on the three fundamental components: probability of default, proportionate loss given default, and exposure at default. We consider models of default for individual firms, including the role of credit rating agencies, structural models, which are based on the underlying processes causing default, and reduced form models which are more based on the empirical information, with less emphasis on the underlying story. This is followed by a description of portfolio credit risk models, where the joint credit risk of multiple entities is the modelling objective.

This chapter discusses censored time-to-event data. We review and define right-censored and interval-censored data and common assumptions associated with them, focusing on standard cases when the independent censoring assumption holds. We define the Kaplan–Meier estimator, the nonparametric maximum likelihood estimator (NPMLE) for the survival distribution for right censored data. We describe the beta product confidence procedure which gives pointwise confidence intervals for it, with better coverage than the standard Greenwood intervals. We describe the NPMLE for the survival distribution for interval censored data using the E-M algorithm. We compare the proportional hazards or proportional odds models. For both right- and interval-censored data, we describe the score tests from the proportional hazards or odds models, and show they are different forms of weighted logrank tests. We cover testing the difference in survival distributions at a specific point in time. We discuss issues with interpreting the proportional hazards model causally, showing that a model with individual proportional hazards does not equal the usual population proportional hazards model.

The chapter focuses on two-sample studies with binary responses, mostly on the case where each sample has an independent binomial response. We discuss three parameters of interest based on the functions of the two binomial parameters: the difference, ratio, or odds ratio of the two parameters. The difference and odds ratio have symmetry equivariance, but the ratio does not. The odds ratio is useful for case-control studies. We compare two versions of the two-sided Fisher’s exact test, and recommend the central one. We describe compatible confidence intervals with the Fisher’s exact test using any of the three parameters of interest. Unconditional exact tests generally have more power than conditional ones, such as Fisher’s exact test, but are computationally more complicated. We recommend a modified Boschloo unconditional exact test with associated confidence intervals to have good power. We discuss the Berger–Boos adjustment, and mid-p methods. We compare several methods with respect to confidence interval coverage. We end with a different study design used with COVID-19 vaccines, where the number of total events is fixed in advance.

In this chapter, we discuss some of the common psychological or behavioural factors that influence risk analysis and risk management. We give examples of cases where behavioural biases created a risk management failure, and some ways in which the negative impact of biases can be mitigated. Biases are categorized, loosely, as relating to (i) self-deception, (ii) information processing (both forms of cognitive bias), and (iii) social bias, relating to the pressures created by social norms and expectations. We give examples of a range of common behavioural biases in risk management, and we briefly describe some strategies for overcoming the distortions created by behavioural factors in decision-making. Next, we present the foundational concepts of Cumulative Prospect Theory, which provides a mathematical framework for decision making that reflects some universal cognitive biases.

In this chapter, we review some of the risk management implications of the regulation of banks and insurance companies. Banks are largely regulated through local implementation of the Basel II and Basel III Accords. Insurance regulation is more varied, but the development of the Solvency II framework in the European Union has influenced regulation more widely, and so we focus on Solvency II as an example of a modern insurance regulatory system.

In this chapter we begin with definitions of standard missing data assumptions: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). Under MNAR, the probability that a response is missing may depend on the missing data values. For example, if the response is death, if individuals drop out of the study if they are very sick, and if we do not or cannot measure the variables that indicate which individuals are very sick, then that is MNAR. In the MNAR case, we consider several sensitivity analysis methods: worse case imputation, opposite arm imputation, and tipping point analysis. The tipping point analysis changes the imputed missing data systematically until the inferential results change (e.g., from significant to not significant). In the MAR case, we consider in a very simple case models such as regression imputation and inverse probability weighted estimators. We simulate two scenarios (1) when the MAR model is correctly specified, and (2) when the MAR model is misspecified. Finally, we briefly describe multiple imputation for missing data in a simple MAR scenario.

This chapter addresses multiplicity in testing, the problem that if many hypotheses are tested then unless some multiplicity adjustment is made, the probability of falsely rejecting at least one hypothesis can be unduly high. We define familywise error rate (FWER), the probability that at least one null hypothesis in the family of hypotheses is rejected. We discuss which sets of hypotheses should be grouped into families. We define the false discovery rate (FDR). We describe simple adjustments based only on p-values of the hypotheses in the family, such the Bonferroni, Holm, and Hochberg procedures for FWER control and the Benjamini–Hochberg adjustment for FDR control. We discuss max-t type inferences for controlling FWER in linear models, or other models with asymptotically normal estimators. We describe resampling-based multiplicity adjustments. We demonstrate graphical methods, showing, for example, gatekeeping and fallback methods, and allowing for more complicated methods. We briefly present logical constraints for hypotheses and the theoretically important closure method.

This chapter briefly covers many general methods for calculating p-values and confidence intervals. We discuss likelihood ratios for inferences with a one-dimensional parameter. Pivot functions are defined (e.g., the probability integral transformation). Basic results for normal and asymptotic normal inferences are given, such as some central limit theorems and the delta method. Three important likelihood-based asymptotic methods (the Wald, score, and likelihood ratio test) are defined and compared. We describe the sandwich method for estimating variance, which requires fewer assumptions than the likelihood-based methods. General permutation tests are presented, along with implementation descriptions including equivalent forms, permutational central limit theorem, and Monte Carlo methods. The nonparametric bootstrap is described, as well as some bootstrap confidence interval methods such as the BCa methods. The melding method of combining two confidence intervals is described, which gives an automatically efficient method to calculate confidence intervals for the differences or ratios of two parameters. Finally, we discuss within-cluster resampling.

The chapter covers inferences with ordinal or numeric responses, with the focus on medians or means. We discuss choosing between the mean and median for describing the central tendency. We give an exact test and associated exact central confidence interval for the median that is applicable without making assumptions on the distribution. For the mean, we show the need for making some restrictive assumptions on the class of distributions for testing the mean, otherwise tests on the mean are not possible. We discuss the one-sample t-test, and how with the normality assumption it is uniformly most powerful unbiased test. We show through some asymptotic results and simulations that with less restrictive assumptions the t-test can still be approximately valid. By simulation, we compare the t-test to some bootstrap inferential methods on the mean, suggesting that the bootstrap-t interval is slightly better for skewed data. We discuss making inferences on rate or count data after making either Poisson or overdispersed Poisson assumptions on the counts. Finally, we discuss testing the variance, standard deviation, or coefficient of variation under certain normality assumptions.

In this chapter, we distinguish funding liquidity from market liquidity, and idiosyncratic liquidity from systemic liquidity. We discuss the nature of highly liquid assets, and methods by which a firm might acquire liquid assets to cover short-term cash flow problems, either in normal operations, or in more extreme crises. As liquidity risk is a problem of cash flow management, we explain how cash flow scenario tests can be used to identify and mitigate risks. We describe liquidity adjusted risk measures used in banking. Finally, we describe how firms might create emergency plans for managing extreme and unexpected liquidity shocks.

This chapter first focuses on goodness-of-fit tests. A simple case is testing for normality (e.g., the Shapiro–Wilks test). We generally recommend against this because large sample sizes can find statistically significant differences even if those differences are not important, and vice versa. We show Q-Q plots to graphically check for the largeness of departures from normality. We discuss the Kolmogorov–Smirnoff test for any difference between two distributions. We review goodness-of-fit tests for contingency tables (Pearson’s chi-squared test and Fisher’s exact test) and for logistic regression (the Hosmer–Lemeshow test). The rest of the chapter is devoted to equivalence or noninferiority tests. The margin of equivalence or noninferiority must be prespecified, and for noninferiority tests of a new drug against a standard, the margin should be larger than the difference between the placebo and the standard. We discuss the constancy assumption and biocreep. We note that while poor design (poor compliance, poor study population choice, poor measurement) generally decreases power in superiority design, these can lead to high Type I error rates in noninferiority designs.

This chapter handles power and sample size estimation used for study design. A very flexible method to estimate power is to simulate. Binomial confidence intervals can be used on the simulated power estimates to determine the number of simulations needed for the precision desired. To determine sample size by simulation, we introduce an algorithm based on methods for dose finding studies, that does not need a large number of replications at each different sample size tried. We present a general normal theory approximation, to give approximate sample sizes for designs that use simple tests such as the two-sample t-test, and the two-sample difference in binomial proportions test. We generalize to cases with unequal allocation between arms, or more complicated tests such as the logrank test. We discuss modifications to sample sizes for nonadherence or missing data.

In this chapter, we describe some numerical methods used for calculating VaR and Expected Shortfall for losses related to investment portfolios, measured over short time horizons – typically 10 days or less. These are techniques commonly used for regulatory capital calculations under Basel III. We start with simple portfolios investments, and then add derivatives. We review the covariance approach, the delta-normal approach, and the delta-gamma-normal approach to portfolio risk measures. Each of these approaches ultimately uses a normal approximation to the distribution of the portfolio value. We also consider the use of historical simulation, based on the empirical distribution of asset prices over the recent past. Finally, we discuss backtesting the risk measure distributions. Backtesting is required under the Basel regulations.

The chapter begins with a cautionary story of a study on hydroxychloroquine treatment for COVID-19. We detail weaknesses in the study that led to misinterpretation of its results, emphasizing that there is more to proper application of hypothesis tests than calculating a p-value. We then discuss reproducibility in science and the p-value controversy. Some argue that since p-values are often misunderstood and misused that they should be replaced with other statistics. We counter that point of view, arguing that frequentist hypothesis tests, when properly applied, are well suited to address reproducibility issues. This motivates the book which provides guiding principles and tools for designing studies and properly applying hypothesis testing for many different scientific applications. We agree with many of the concerns about overreliance on p-values, hence the approach of the book is to present not just methods for hypothesis tests, but also methods for compatible confidence intervals on parameters that can accompany them. The chapter ends with an overview of the book, describing its level and the intended audience.