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AbstractWe study how the presence of dependencies between risks in a population of prospective insurance customers translates into risk exposure for an insurance company, depending on the company's market share on the various risks. It turns out that the dependency structure in the insurer's portfolio may differ significantly from the dependency structure of those risks in the general population, even when policyholders for different risks are selected independently. We obtain an upper bound for the difference between the ruin probability and its estimate based on the company's portfolio marginal distributions. Under certain conditions, dependencies between risks in the portfolio of a company with small market shares are mild. We characterize the optimal loadings and market shares, assuming generic demand functions for the different risks.

This is an accepted manuscript of an article published by Taylor & Francis in Emerging Markets Finance and Trade on 2014/12/7, available online: http://dx.doi.org/10.2753/REE1540-496X490504. ; We analyse the impact of default probability in four leading Latin American stock markets (Argentina, Brazil, Chile and Mexico). We find no positive default risk premium except in the case of Brazil, and in fact find a negative risk premium for Argentina and Mexico. The latter effect tends to fade when the analysis accounts for size and BTM market variables. Although we find no size effect in any of the markets considered, the BTM effect is very strong in all of them, and our results reveal a consistent relationship, analogous to that found in more developed markets, between default probability and the size and book-to-market variables. ; This paper has received financial supp ort from the Spanish Ministry of Science and Innovation (ECO2009-12819) and the Ministry of Economy and Competitiveness (ECO2012-35946-C02-01). Isabel Abinzano particularly acknowledges the financial support of the Andalusian Regional Government (P09-SEJ-4467).

How much should we pay to remove the interdependence of biased information sources? This question is relevant in both statistics and political economy. When there are many information sources or variables, their dependence may be unknown, which creates multivariate ambiguity. One approach to answer our leading question involves use of decoupling inequalities from probability theory. We present a new inequality, designed to cope with this question, which holds for any type of dependence across information sources. We apply our method to a simple formalization of a political echo chamber. For a given set of marginal information, this bound is the sup over all possible joint distributions connecting the marginals. Our method highlights a price to pay for facing summed dependent (multivariate) data, similar to the probability premium required for univariate data. We show that a conservative decisionmaker will pay approximately 50% more than if the data were independent, in order to freely neglect the correlations.

The seminal Barro (2006) closed-economy model of the equity risk premium in the presence of extreme events (disasters) allowed for leverage in the form of risky corporate debt which defaulted only in states when the Government defaulted on its debt. The probability of default was therefore exogenous and independent of the degree of leverage. In this paper, we take the model a step closer reality by assuming that, on the one hand, the Government never defaults, and on the the other hand, that the corporate sector in the form of the Lucas tree owner pays its debts in full if and only if its asset value is suffcient, which is always the case in non-crisis states. Otherwise, in exceptionally severe crises, it defaults and hands over the whole firm to its creditors. The probability of default by the tree owner is thus endogenous, dependent both on the volume of debt issued (taken as exogenous) and on the uncertain value of output. We show, using data from both Barro (2006) and Barro and Ursua (2008), that the model can generate values of the riskless rate, equity risk premium and credit risk spread broadly consistent with those typically observed in the data.

Game-theoretic probability and finance come of age Glenn Shafer and Vladimir Vovk Probability and Finance, published in 2001, showed that perfect-information games can be used to define mathematical probability. Based on fifteen years of further research, Game-Theoretic Foundations for Probability and Finance presents a mature view of the foundational role game theory can play. Its account of probability theory opens the way to new methods of prediction and testing and makes many statistical methods more transparent and widely usable. Its contributions to finance theory include purely game-theoretic accounts of Ito stochastic calculus, the capital asset pricing model, the equity premium, and portfolio theory. Game-Theoretic Foundations for Probability and Finance is a book of research. It is also a teaching resource. Each chapter is supplemented with carefully designed exercises and notes relating the new theory to its historical context. Praise from early readers ver since Kolmogorov's Grundbegriffe, the standard mathematical treatment of probability theory has been measure-theoretic. In this ground-breaking work, Shafer and Vovk give a game-theoretic foundation instead. While being just as rigorous, the game-theoretic approach allows for vast and useful generalizations of classical measure-theoretic results, while also giving rise to new, radical ideas for prediction, statistics and mathematical finance without stochastic assumptions. The authors set out their theory in great detail, resulting in what is definitely one of the most important books on the foundations of probability to have appeared in the last few decades. Peter Gr130 0Wiley series in probability and statisticsnwald, CWI and University of Leiden hafer and Vovk have thoroughly re-written their 2001 book on the game-theoretic foundations for probability and for finance. They have included an account of the tremendous growth that has occurred since, in the game-theoretic and pathwise approaches to stochastic analysis and in their applications to continuous-time finance. This new book will undoubtedly spur a better understanding of the foundations of these very important fields, and we should all be grateful to its authors. Ioannis Karatzas, Columbia University

We extend the Barro (2006) closed-economy model of the equity risk premium in the presence of extreme events (disasters) to a two-country world. In this more general setting, both the output risk of rare disasters and the associated risk of a default on Government debt, can be diversiÖed. The extent to which agents in one country can diversify away the risk of extreme events depends on the relative size of the two countries, and critically on the probability of a disaster in one country conditional on a disaster in the other. We show that, using Barroís own calibration in combination with a broad range of plausible values for the additional parameters, the model implies levels of the equity risk premium far lower than those typically observed in the data. We conclude that the model is unlikely to explain the equity risk premium.

We document the return to investing in U.S. nonpublicly traded equity. Entrepreneurial investment is extremely concentrated, yet despite its poor diversification, we find that the returns to private equity are no higher than the returns to public equity. Given the large public equity premium, it is puzzling why households willingly invest substantial amounts in a single privately held firm with a seemingly far worse risk-return trade-off. We briefly discuss how large nonpecuniary benefits, a preference for skewness, or overestimates of the probability of survival could potentially explain investment in private equity despite these findings.

In this study, the tail probability of a class of distributions commonly used in assessing the severity of insurance losses was examined. Without specifying any particular distribution, the use of an algebraic functional form Cx−α to approximate the tail behavior of the distributions in the class was demonstrated. Norwegian fire insurance data were examined, and the algebraic functional form was applied to derive the expected loss of a reinsurance treaty that covers all losses exceeding a retention limit. It was shown that (1) the expected loss is insensitive to the parameter α for a high retention limit (e.g., a catastrophe treaty), and (2) with a low retention limit (e.g., a largest claim treaty), a reliable estimate of the parameter α and a sound judgment on the maximum potential loss of the treaty could provide useful and defensible summary statistics for pricing the treaty. Thus, when dealing with the losses of certain reinsurance treaties, it was concluded that knowledge of a specific probability distribution is not critical, and the summary statistics derived from the model are robust with respect to a large class of loss distributions.