We consider models for analyzing the surplus of an insurance portfolio. Suppose an insurance business begins with a start-up capital, called the initial surplus. The insurance company receives premium payments and pays claim losses. The premium payments are assumed to be coming in at a constant rate. When there are claims, losses are paid out to policyholders. Unlike the constant premium payments, losses are random and uncertain, in both timing and amount. The net surplus through time is the excess of the initial capital and aggregate premiums received over the losses paid out. The insurance business is in ruin if the surplus falls to or below zero. The main purpose of this chapter is to consider the probability of ruin as a function of time, the initial surplus and the claim distribution. Ultimate ruin refers to the situation where in ruin occurs at finite time, irrespective of the time of occurrence.

Chapter 12: In the preceding chapter, we found that each square complex matrix A is similar to a direct sum of upper triangular unispectral matrices. We now show that A is similar to a direct sum of Jordan blocks (unispectral upper bidiagonal matrices with 1s in the superdiagonal) that is unique up to permutation of its direct summands.

This chapter offers a primer of the literature on the origins of proportional representation. Subsequent chapters use this literature to illustrate the elements of CHA. It therefore provides the background information to make those illustration easier tofollow.

Short-term insurance policies often take multiple years before the final settlement of all losses incurred is completed. This is especially true for liability insurance policies, which may drag on for a long time due to legal proceedings. To set the premiums of insurance policies appropriately so that the premiums are competitive and yet sufficient to cover the losses and expenses with a reasonable profit margin, accurate projection of losses cannot be overemphasized. Loss reserving refers to the techniques to project future payments of insurance losses based on policies in the past.

We have discussed the limited-fluctuation credibility method, the Bühlmann and Bühlmann–Straub credibility methods, as well as the Bayesian method for future loss prediction. The implementation of these methods requires the knowledge or assumptions of some unknown parameters of the model. For the limited-fluctuation credibility method, Poisson distribution is usually assumed for claim frequency.

Longue durée analysis employs descriptive statistics and data visualization of explore univariate, bivariate, and multivariate trends. These trends tracks historical historical changes in a thinner manner than eventful analysis, but in return they explore such trends over longer time periods and larger numbers of cases. It introduces four graph types: serial scatterplots, contextualized line graphs, developmental typologies, and serial bar graphs that are used to visualize trends. Longue durée analysis frequently redescribes mi-described trends by paying close attention to three sources of misdescription: inadequate baselining, unfreezing of time scales, and mismeasurements. It discusses how using stock instead of flow indicators increases the ability of time series to capture historical changes.

Chapter 19: In this chapter, we introduce new examples of norms, with special attention to submultiplicative norms on matrices. These norms are well-adapted to applications involving power series of matrices and iterative numerical algorithms. We use them to prove a formula for the spectral radius that is the key to a fundamental theorem on positive matrices in the next chapter.

Chapter 17: In this chapter, we investigate applications and consequences of the singular value decomposition. For example, it provides a systematic way to approximate a matrix by a matrix of lower rank. It also permits us to define a generalized inverse for matrices that are not invertible (and need not even be square). The singular value decomposition has a pleasant special form for complex symmetric matrices. The largest singular value is especially important; it turns out to be a norm (the spectral norm) on matrices. We use the spectral norm to study how the solution of a linear system changes if the system is perturbed, and how the eigenvalues of a matrix can change if it is perturbed.

Chapter 5: Many abstract concepts that make linear algebra a powerful mathematical tool have their roots in plane geometry, so we begin the study of inner product spaces with a review of basic properties of lengths and angles in the real two-dimensional plane. Guided by these geometrical properties, we formulate axioms for inner products and norms, which provide generalized notions of length (norm) and perpendicularity (orthogonality) in abstract vector spaces.

Various CHA scholars have contributed to causal process tracing and helped establish it as principal alternative to VBA. A recent, Bayesian-informed version is particularly relevant to CHA because it shares the same historiographical sensibilities of seeing knowledge evolve through a close dialogue between new findings and the existing foreknowledge. It makes causal inferences conditional on a pre-testing articulation of testworthy hypotheses and juxtaposes new test results onto older ones at the post-testing stage. Process tracing defines the testworthiness of hypotheses according to the number and diversity of empirical implications a theory has before the testing starts. It also defines test strength in terms of the specificity of the hypotheses that are tested and of their uniqueness vis-à-vis the alternative explanations against whichthey are paired. The chapter introduces a new tool, the theory ledger, to help evaluate test strength and to update confidence in causal inferencing.

While the classical credibility theory addresses the important problem of combining claim experience and prior information to update the prediction for loss, it does not provide a very satisfactory solution. The method is based on arbitrary selection of the coverage probability and the accuracy parameter. Furthermore, for tractability some restrictive assumptions about the loss distribution have to be imposed.

Historical time is a plural entity, not a singular one. Methodologicals rest on difficult ontological assumptions, which translate into four potential notions of history: cyclical, bounded, serial, and eventful. Cyclical history freezes history the most, by assuming that the present merely repeats the past, and thus makes history de facto reversible. It in effect freezes history, and CHA regards this notion of history as ahistorical. Bounded history freezes certain time intervals during which it assumes that history stands still and the past in effect repeats itself, at least for a limited period of time. It is not interested in how the bounded period is qualitatively different from the periods preceeding or following it. Serial history partially unfreezes history and uses time series data to track secular trends through time. Eventful history is the most unfrozen treatment of the past and tries to identify continuities and discontinuities, or distinct periods. Historical tourism refers to notions of history so static and so frozen that they cease to be historical in any meaningful sense of the word. It identifies variants of historical tourism in history proper and in CHA.