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This chapter provides an analysis of the risk aggregation process in banks in both a regulatory and a non-regulatory context. It starts with a brief introduction defining risk aggregation and explains that understanding total risk can lead to improved decision-making. Risk aggregation is then explained at an individual asset risk level involving credit and market risk. Because risk aggregation involves a choice for banks in terms of the risk metric used such as value-at-risk and expected shortfall, the chapter compares these metrics and discusses their relative strengths and limitations. The risk aggregation process offers different ways to aggregate risks, which are broadly classified into the top-down approach and the bottom-up approach. The chapter also reviews the literature on risk aggregation approaches and their pros and cons. Finally, it provides a perspective into new stress testing regulations such as the Dodd-Frank Act that require banks to aggregate risk (capital) and returns (profit/losses) over a longer period of time.

(1) Why does it seem that we don’t know we will lose a lottery, while it seems we do know other things with respect about which we are more likely to be wrong? (2) And do we really fail to know that we will lose a lottery, while we do know those other things? This chapter defends an insensitivity answer to (1): You seem not to know that you will lose the lottery because you would have believed you would lose even if you were the winner. As for (2), a solution to this lottery puzzle is defended on which you do know that you will lose the lottery, according to ordinary standards for knowledge (unless you are the winner, in which case you are rational to think you know that you will lose). Key to this solution is defending a certain understanding of the closure principle for knowledge.

According to the principle of multiple premise closure, if one has justification for believing each of a series of propositions, one has justification for believing their joint deductive consequences. A powerful objection to this principle arises from the phenomenon of risk aggregation, made vivid by the preface paradox. In this chapter two theories of justification are outlined—a pure normic theory that respects multiple premise closure and a hybrid theory that violates it. It is shown that multiple premise closure is derivable from a range of further formal principles for justification and, as such, the formal consequences of rejecting multiple premise closure are difficult to anticipate. A case is made for the possibility of justifiably believing propositions that run a high risk of falsity, and some consequences for the psychology of human reasoning are explored.

A definition of fallibility shows that agents are fallible about necessary truths. It is shown that fallibility of agents implies a denial of parity reasoning. Moorean paradoxes appear to undercut fallibility, but they are due entirely to the factivity of “know.” Kripke’s dogmatism paradox is explained: the key is recognizing that knowledge fallibility applies to the knowledge that all evidence against something one knows is misleading. That we do not know we will lose a lottery is denied. Fallibility shows this. And that people argue over this also indicates this. Knowledge closure fails because of fallibility; so does aggregation of assumptions. Vagueness shows why debates about whether we know outcomes of lotteries before winning tickets are drawn are irresolvable. Irrational penny reasoning is analyzed; it applies to nonfactive attitudes such as being really really sure. Preface paradoxes are explained. That it is sometimes rational to believe contradictory propositions is explained.