Portfolio diversification and value at risk under thick-tailedness†

†The results in this paper constitute a part of the author's dissertation ‘New majorization theory in economics and martingale convergence results in econometrics’ presented to the faculty of the Graduate School of Yale University in candidacy for the degree of Doctor of Philosophy in Economics in March, 2005. The results were originally contained in the work circulated in 2003–2005 under the titles ‘Shifting paradigms: On the robustness of economic models to heavy-tailedness assumptions’ and ‘On the robustness of economic models to heavy-tailedness assumptions’.

Abstract

This paper focuses on the study of portfolio diversification and value at risk analysis under heavy-tailedness. We use a notion of diversification based on majorization theory that will be explained in the text. The paper shows that the stylized fact that portfolio diversification is preferable is reversed for extremely heavy-tailed risks or returns. However, the stylized facts on diversification are robust to heavy-tailedness of risks or returns as long as their distributions are moderately heavy-tailed. Extensions of the results to the case of dependence, including convolutions of α-symmetric distributions and models with common shocks are provided.

†The results in this paper constitute a part of the author's dissertation ‘New majorization theory in economics and martingale convergence results in econometrics’ presented to the faculty of the Graduate School of Yale University in candidacy for the degree of Doctor of Philosophy in Economics in March, 2005. The results were originally contained in the work circulated in 2003–2005 under the titles ‘Shifting paradigms: On the robustness of economic models to heavy-tailedness assumptions’ and ‘On the robustness of economic models to heavy-tailedness assumptions’.

Keywords:

• Value at risk
• Heavy-tailed risks
• Portfolios
• Riskiness
• Diversification
• Risk bounds
• Coherent measures of risk

Acknowledgements

I am indebted to my advisors, Donald Andrews, Peter Phillips and Herbert Scarf, for all their support and guidance in all stages of the current project. I also thank three anonymous referees, Donald Brown, John Campbell, Aydin Cecen, Gary Chamberlain, Brian Dineen, Darrell Duffie, Paul Embrechts, Frank Fabozzi, Xavier Gabaix, Tilmann Gneiting, Philip Haile, Dwight Jaffee, Samuel Karlin, Benoît Mandelbrot, Alex Maynard, Ingram Olkin, Ben Polak, Stephan Ross, Gustavo Soares, Kevin Song, Johan Walden 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 as well as the participants at the 18th New England Statistics Symposium at Harvard University, April 2004, the International Conference on Stochastic Finance, Lisbon, Portugal, September 2004, and Deutsche Bundesbank Conference on Heavy Tails and Stable Paretian Distributions in Finance and Macroeconomics, Eltville, Germany, November 2005, for helpful comments and discussions.

Notes

†The results in this paper constitute a part of the author's dissertation ‘New majorization theory in economics and martingale convergence results in econometrics’ presented to the faculty of the Graduate School of Yale University in candidacy for the degree of Doctor of Philosophy in Economics in March, 2005. The results were originally contained in the work circulated in 2003–2005 under the titles ‘Shifting paradigms: On the robustness of economic models to heavy-tailedness assumptions’ and ‘On the robustness of economic models to heavy-tailedness assumptions’.

‡In addition, as follows from the results in Ibragimov and Walden (2007), the value at risk and expected utility comparisons are closely linked: in particular, non-diversification results for truncations of extremely heavy-tailed distributions implied by the results in this paper also continue to hold in the expected utility framework if investors' utility function becomes convex at any point in the domain of large losses (convexity of utility functions in the loss domain is one of the key foundations of Prospect theory, see Kahneman and Tversky 1979; it also effectively arises if there is limited liability).

§Several recent papers (see among others, Acerbi and Tasche 2002, Tasche 2002) recommend to use the expected shortfall as a coherent alternative to the value at risk (see Artzner et al. 1999, Embrechts et al. 2002 and section 3 in this paper for the definition of coherency for measures of risk). However, the expected shortfall, which is defined as the average of the worst losses of a portfolio, requires existence of first moments of risks to be finite. It is not difficult to see that existence of means of the risks in considerations is also required for finiteness of coherent spectral measures of risk (see Acerbi 2002, Cotter and Dowd 2006) that generalize the expected shortfall.

†Here and throughout the paper, f(x) ≈ g(x) means that 0 < c ≤ f(x)/g(x) ≤ C < ∞ for large x, for constants c and C.

‡In addition, as discussed in Ibragimov and Walden (2007) and Ibragimov et al. (2009), truncation arguments imply that (non-) diversification results in value at risk models for stable distributions with unbounded support continue to hold in the framework of bounded risks.

†One should note here that commonly used approaches to inference on the tail indices, such as OLS log-log rank-size regression estimators and Hill's estimator, are strongly biased in small samples and are very sensitive to deviations from power laws (1) in the form of regularly varying tails (see, among others, the discussion in Embrechts et al. 1997, Weron 2001, Borak et al. 2005 and Gabaix and Ibragimov 2006). In particular, these procedures tend to overestimate the tail index for observations from infinite variance stable distributions with α < 2 and sample sizes typical in applications (see McCulloch 1997, Borak et al. 2005). Therefore, point estimates of the tail index greater than 1 do not necessarily exclude heavy-tailedness with infinite means and true values α < 1 in the same way as point estimates of the tail exponent greater than 2 do not necessarily exclude stable regimes with infinite variances, as discussed in McCulloch (1997) and Weron (2001).

‡An n-dimensional distribution is called α-symmetric if its characteristic function can be written as , where φ is a continuous function and α > 0. Such distributions should not be confused with multivariate spherically symmetric stable distributions, which have characteristic functions , 0 < β ≤ 2. Obviously, spherically symmetric stable distributions are particular examples of α-symmetric distributions with α = 2 (that is, of spherical distributions) and φ(x) = exp(−x ^β).

§As recently demonstrated in Ibragimov and Walden (2007), the conclusions in this paper also continue to hold for a wide class of bounded r.v.s concentrated on a sufficiently large interval with distributions given by truncations of stable and α-symmetric ones.

†This concept of (univariate) symmetry is not to be confused with joint α-symmetric or spherical distributions discussed in sections 1.3 and 5, which provide models of dependence among the components of random vectors.

†Remark 2 and appendix A discuss the properties that explain, in particular, why one can pool the risks from the classes ℒ𝒞 and in the results obtained in the paper.

†Heterogeneity in distributions of risks may require altering of formalizations of portfolio diversification using majorization. Appendix C contains some suggestions on these extensions motivated by VaR analysis for skewed and non-identically distributed risks.

†The functions φ_α have the same form as measures of diversification considered in Bouchaud and Potters (2004, Ch. 12, p. 205).

‡In particular, the results Theorems 4.1 and 4.2 and their analogues under dependence provided by Theorems 5.1 and 5.2 substantially generalize the riskiness analysis for uniform (equal weights) portfolios of independent stable risks considered, among others, in Fama (1965a), Ross (1976), and Samuelson (1967): these theorems demonstrate that the formalization of portfolio diversification on the basis of majorization pre-ordering allows one to obtain comparisons of riskiness for portfolios of heavy-tailed and possibly dependent risks with arbitrary, rather than equal, weights.

§From the proof of Theorems 4.1 and 4.2 and this property it follows that the theorems continue to hold for convolutions of distributions from the classes and with Cauchy distributions S ₁(σ, 0, 0).

†Taking the absolute values here and inside the function U in the comparisons that follow is needed because, for Z _v and Z _w with symmetric distributions, the value at risk comparisons VaR _q (Z _v ) ≤ VaR _q (Z _w ), q ∈ (0, 1/2), imply the opposite inequalities for the tail probabilities with negative x : P(Z _v > x) ≥ P(Z _w > x) for x < 0.

†The main results in Proschan (1965) are reviewed in section 12. J in Marshall and Olkin (1979). The work by Proschan (1965) is also presented, in a rearranged form, in section 11 of Chapter 7 in Karlin (1968). Peakedness results in Karlin (1968), and Proschan (1965) are formulated for ‘PF2 densities’, which is the same as ‘log-concave densities’.

†This is true because if one assumes that r.v.s X ₁, …, X _n , n ≥ 2, have an α-symmetric distribution with α < 1 and that E|X _i | < ∞, i = 1, …, n, then, by the triangle inequality, E|X ₁ + ··· + X _n | ≤ E|X ₁| + ··· + E|X _n | = nE|X ₁|. This inequality, however, cannot hold since, according to (3), (X ₁ + ··· + X _n ) ∼ n ^1/α X ₁ and thus, under the above assumptions, E|X ₁ + ··· + X _n | > nE|X ₁|. Similarly, one can show that α-symmetric distributions with α < r have infinite marginal moments of order r.