Abstract. Maximum drawdown, the largest cumulative loss from peak to trough, is one of the most widely used indicators of risk in the fund management industry, but one of the least developed in the context of measures of risk. We formalize drawdown risk as Conditional Expected Drawdown (CED), which is the tail mean of maximum drawdown distributions. We show that CED is a degree one positive homogenous risk measure, so that it can be linearly attributed to factors; and convex, so that it can be used in quantitative optimization. We empirically explore the differences in risk attributions based on CED, Expected Shortfall (ES) and volatility. An important feature of CED is its sensitivity to serial correlation. In an empirical study that fits AR(1) models to US Equity and US Bonds, we find substantially higher correlation between the autoregressive parameter and CED than with ES or with volatility. We are grateful to Robert Anderson for insightful comments on the material discussed in this article; to Alexei Chekhlov, Stan Uryasev, and Michael Zabarankin for their feedback on a previous draft of this work; to Vladislav Dubikovsky, Michael Hayes, and Márk Horváth for their contributions to an earlier version of this article; to Carlo Acerbi for providing detailed comments on a previous draft; and to the referees and editors of Mathematics and Financial Economics for their valuable feedback.
SummarySecond-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This dissertation completes the algebraic foundations of second-order languages from the viewpoint of categorical algebra.In particular, the dissertation introduces the notion of second-order algebraic theory. A main role in the definition is played by the second-order theory of equality , representing the most elementary operators and equations present in every second-order language. We show that can be described abstractly via the universal property of being the free cartesian category on an exponentiable object.Thereby, in the tradition of categorical algebra, a second-order algebraic theory consists of a cartesian category and a strict cartesian identity-on-objects functor M : → that preserves the universal exponentiable object of .At the syntactic level, we establish the correctness of our definition by showing a categorical equivalence between second-order equational presentations and second-order algebraic theories. This equivalence, referred to as the Second-Order Syntactic Categorical Type Theory Correspondence, involves distilling a notion of syntactic translation between second-order equational presentations that corresponds to the canonical notion of morphism between second-order algebraic theories. Syntactic translations provide a mathematical formalisation of notions such as encodings and transforms for second-order languages.On top of the aforementioned syntactic correspondence, we furthermore establish the Second-Order Semantic Categorical Type Theory Correspondence. This involves generalising Lawvere's notion of functorial model of algebraic theories to the second-order setting. By this semantic correspondence, second-order functorial semantics is shown to correspond to the model theory of second-order universal algebra.We finally show that the core of the theory surrounding Lawvere theories generalises to the second order as well. Instances of this development are the existence of algebraic functors and monad morphisms in the second-order universe. Moreover, we define a notion of translation homomorphism that allows us to establish a 2-categorical type theory correspondence. Es nimmt der Augenblick, was Jahre geben.-Johann Wolfgang von Goethe 10
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.