The Limits of Diversification when Losses May be Large

Abstract

Recent results in value at risk analysis show that, for extremely heavy-tailed risks with unbounded distribution support, diversification may increase value at risk, and that generally it is difficult to construct an appropriate risk measure for such distributions. We further analyze the limitations of diversification for heavy-tailed risks. We provide additional insight in two ways. First, we show that similar nondiversification results are valid for a large class of risks with bounded support, as long as the risks are concentrated on a sufficiently large interval. The required length of the support depends on the number of risks available and on the degree of heavy-tailedness. Second, we relate the value at risk approach to more general risk frameworks. We argue that in markets for risky assets where the number of assets is limited compared with the (bounded) distribution support of the risks, unbounded heavy-tailed risks may provide a reasonable approximation. We suggest that this type of analysis may have a role in explaining various types of market failures in markets for assets with possibly large negative outcomes. JEL classification: G11Keywords: value at risk, coherent measures of risk, heavy-tailed risks, portfolios, riskiness, diversification, catastrophe insurance, risk bounds * An extended working paper version of this paper is available as . The authors are grateful to Gary Chamberlain and Oliver Hart for helpful comments and for posing questions about the stylized facts on diversification under boundedness assumptions that are answered in this paper. We also thank Donald Andrews, Donald Brown, Xavier Gabaix, Tilmann Gneiting, Dwight Jaffee, Peter Phillips, Jacob Sagi, Herbert Scarf, Martin Weitzman and the participants at seminars at the Departments of Economics at Yale University, University of British Columbia, the University of California at San Diego, Harvard University, the London School of Economics and Political Science, Massachusetts Institute of Technology, the Université de Montréal, McGill University and New York University, the Division of the Humanities and Social Sciences at California Institute of Technology, Nuffield College, University of Oxford, and the Department of Statistics at Columbia University for helpful discussions. We are grateful to Jonathan Ingersoll and David Pollard for useful discussions on extremal problems for expectations of functions of random variables with fixed moments. Finally, we thank a referee for helpful comments and suggestions.1 Tel

Value at risk and efficiency under dependence and heavy-tailedness: models with common shocks

“…For instance, using in the proof the extensions of the results in Propositions 3.1 and 3.2 to the case of dependence discussed in Ibragimov (2005Ibragimov ( , 2009b and Ibragimov and Walden (2007) one obtains that all the results in the paper also hold in settings where the risks R i , C j and U i j in (1.2) and (1.3) are dependent among themselves or are bounded. These generalizations include models (1.2) and (1.3) in which the vectors of common shocks (R 1 , .…”

“…As discussed in Ibragimov (2005Ibragimov ( , 2009b, and Ibragimov and Walden (2007), convolutions of α−symmetric distributions exhibit both heavy-tailedness in marginals and dependence among them. For instance, convolutions of models (7.4) with α < 1 have extremely heavy-tailed marginal distributions with infinite means.…”

“…. , U rc ) have distributions which are convolutions of α− symmetric distributions (see Fang et al 1990, and the review in Ibragimov 2005, 2009b, and Ibragimov and Walden 2007.…”

“…3). Recently, Ibragimov and Walden (2007) showed that, with a value at risk approach with bounded risks concentrated on a sufficiently large interval, diversification may be suboptimal up to a certain number of risks and then become optimal. Ibragimov et al (2009) demonstrate how this analysis can be used to explain low levels of reinsurance among insurance providers in markets for catastrophe reinsurance.…”

“…(see Ibragimov 2005, 2009band Ibragimov and Walden 2007, models (1.2) and (1.3) provide a natural framework for modeling risks subject to (additive) common shocks R i and C j (such as political or macroeconomic ones, see the discussion in Andrews 2003Andrews , 2005. The common shocks R i affect all risks Y i j , j = 1, .…”