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2015 | OriginalPaper | Buchkapitel

2. Implications of Heavy-Tailedness

verfasst von : Marat Ibragimov, Rustam Ibragimov, Johan Walden

Erschienen in: Heavy-Tailed Distributions and Robustness in Economics and Finance

Verlag: Springer International Publishing

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Abstract

This chapter demonstrates how majorization theory provides a powerful tool for the study of robustness of many important models in economics, finance, econometrics, statistics, risk management, and insurance to heavy-tailedness assumptions. The majorization relation is a formalization of the concept of diversity in the components of vectors. Over the past decades, majorization theory, which focuses on the study of this relation and functions that preserve it, has found applications in disciplines ranging from statistics, probability theory, and economics to mathematical genetics, linear algebra, and geometry (see Marshall et al. 2011, and the references therein).

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Vorheriges Kapitel Introduction

Nächstes Kapitel Inference and Empirical Examples

The functions ϕ _p have the same form as measures of diversification considered in Bouchaud and Potters (2004), Chap. 12, p. 205.

Throughout the chapter, we interpret the negative values of risks X as a risk holder’s losses. This interpretation is similar to that in Artzner et al. (1999) and Christoffersen (2012) and is in contrast to Chap. 2 in McNeil et al. (2005) who interpret positive values of risks X as losses. All the results presented and discussed in the chapter can be easily reformulated in terms of interpretation of positive values of the risks as losses (see Ibragimov 2009a,b, for details).

Here, as usual, \(\Gamma (r)\) and B(a, b) denote the Gamma and Beta functions.

This section draws upon material from the following articles: Ibragimov (2009b) “Portfolio diversification and VaR under thick-tailedness”, Quantitative Finance, Vol. 9, No. 5, 565–580, and Ibragimov (2009a) “Heavy-tailed densities,” “The New Palgrave Dictionary of Economics,” Eds. Steven N. Durlauf and Lawrence E. Blume, Palgrave Macmillan, reproduced with permission of Palgrave Macmillan. The full published version of this publication is available from: http://www.dictionaryofeconomics.com/article?id=pde2009_H000191.

In particular, the results Theorems 2.1.1 and 2.1.2 and their analogues under dependence provided by Theorems 5.1 and 5.2 in Ibragimov (2009b) substantially generalize the riskiness analysis for uniform (equal weights) portfolios of independent stable risks considered, among others, in the papers by Fama (1965b), Samuelson (1967a) and Ross (1976): 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 2.1.1 and 2.1.2 and this property it follows that the theorems continue to hold for convolutions of distributions from the classes \(\overline{\mathcal{C}\mathcal{S}\mathcal{L}\mathcal{C}}\) and \(\underline{\mathcal{C}\mathcal{S}}\) with Cauchy distributions S ₁(σ).

See also Ibragimov et al. (2014) for the analysis of the interplay of dependence modeled using different copula structures, the degree of heavy-tailedness and the values of loss probabilities and disaster levels in problems of portfolio diversification in VaR frameworks.

Proschan’s (1965) results are an example of many results and inequalities in probability whose formulation or proofs are based on majorization theory (for further examples of such inequalities, see Chap. 3 in Marshall et al. 2011).

The main results in Proschan (1965) are reviewed in Sect. 12.J in Marshall et al. (2011). The work by Proschan (1965) is also presented, in a rearranged form, in Sect. 11 of Chap. 7 in Karlin (1968). Peakedness results in Proschan (1965) and Karlin (1968) are formulated for “PF2 densities,” which is the same as “log-concave densities.”

The analysis of tail probabilities of linear combinations of r.v.’s is related to the field of probability and moment inequalities in probability and statistics (see, among others, Sect. 12 in de la Peña and Giné 1999; de la Peña et al. 2003; Hansen 2015; Ibragimov and Ibragimov 2008; Ibragimov and Sharakhmetov 1997, 2002; Marshall et al. 2011; Nze and Doukhan 2004; Utev 1985, and the references therein for a number of results in the field and their statistical and econometric applications).

This section draws upon material from the following articles: Ibragimov (2007) “Efficiency of linear estimators under heavy-tailedness: Convolutions of α-symmetric distributions,” Econometric Theory, Volume 23(3), pp. 501–517 (2010) \(\copyright\) Cambridge University Press, reproduced with permission, and Ibragimov (2009a) “Heavy-tailed densities,” “The New Palgrave Dictionary of Economics,” Eds. Steven N. Durlauf and Lawrence E. Blume, Palgrave Macmillan, reproduced with permission of Palgrave Macmillan. The full published version of this publication is available from: http://www.dictionaryofeconomics.com/article?id=pde2009_H000191.

This section is based on the article Ibragimov and Walden (2007), which was published in the Journal of Banking and Finance, Vol. 31, Issue 8, pp. 2551–2569, Copyright Elsevier (2007).

This definition of truncation moves probability mass to the edges of the distributions. The results in this section continue to hold for the more commonly used truncations XI( | X | ≤ a) which move probability mass to the center.

That is, in the case of an absolutely continuous risk Z, P(Z ≤ VaR_q(Z)) = q.

This section is based on the article Ibragimov et al. (2009), which was published in the Review of Financial Studies, Vol. 22, Issue 3, pp. 959–993.

See Cummins (2006), Jaffee (2006a), and Jaffee and Russell (2006) for recent discussions and references to the literature. OECD (2005a) and OECD (2005b) discuss government interventions around the world to reactivate terrorism insurance. Kunreuther and Michel-Kerjan (2006) discuss the specific issue of terrorism insurance in the United States.

There is generally open access to scientific forecasts of natural disasters, much of it provided by governments. Terrorists may be more strategic in their choice of targets, but this does not create a moral hazard on the part of those purchasing terrorism insurance (unless the terrorists particularly target insured properties).

These data are from the Insurance Information Institute; see http://iii.org/media/industry/.

Insurance is unique among U.S. financial services in that it is regulated in the United States only at the state level. The structure of a catastrophe insurance market is well illustrated by California’s earthquake risk market. As of 2005, 70 % of the coverage was provided by the California Earthquake Authority, an entity created by the State of California following the 1994 Northridge quake. With no major quakes since then, private insurers have slowly reentered the market, now representing about 30 % of the market. However, still only 35 private insurance groups are offering California earthquake coverage (based on annual written premiums of USD 1 million or more). Furthermore, the top 5, 10, and 20 firms represent 46, 66, and 89 % of the total private market, respectively.

We assume that a third party, perhaps the government, covers the excess losses to policy holders. This avoids the complications of any impact on policyholder demand.

Here, in line with the previous discussion on reduced strategy space, \(\tilde{q}_{m}\) does not need to be conditioned on the participation choices \(q_{m^{{\prime}}}\) of agents m ^′ = 1, …, m − 1. This is the case as the equilibrium mapping \(q = \mathcal{E}(p) \in \{ 0, 1\}^{M}\) is known, so \(q_{m^{{\prime}}}\) is uniquely implied by p in equilibrium.

For example, in a calibration to earthquake insurance (see Ibragimov et al. 2009), we arrive at nondiversification trap arising for annual insurance premiums, λ, between USD 1,840 and USD 2,300 per household. Below USD 1,840, the only equilibrium is the nondiversification equilibrium, and above USD 2,300, the only equilibrium is the full-diversification equilibrium. The range of λ for which a nondiversification trap arises is thus about 20 % of the premium—i.e., (2,300–1,840)/2,340. With other parameter values, we have derived ranges from a few percent up to an order of magnitude.

The moment magnitude is almost the same as the Richter magnitude, M _R, for M ≤ 6. 5, but provides a more accurate measure for earthquakes of larger magnitudes.

This is the moment magnitude version of the celebrated Gutenberg–Richter exponential law for the Richter magnitude.

Although for very high levels, physical arguments imply that there has to be an upper bound on the energy released; see Knopoff and Kagan (1977), and Kagan and Knopoff (1984). However, even if there is an upper bound, say at M = 10 to 11, this still leads to an approximate Pareto law for over 15 magnitudes of energy release. The upper bound is well beyond the limited liability threshold of most insurance markets and is therefore not crucial for our trap argument.

This estimate may be somewhat outdated, as building structures nowadays may be stronger. However, this does not change our general conclusions, only the constants in the formulae (personal communication with William L. Ellsworth, Chief Scientist, Western Region Earthquake Hazards Team, United States Geological Survey).

1. 10 ≈ 1. 84 × 0. 6.

Other estimates for the relation are available—e.g., in Bakun et al. (2003). However, as with the strength of building structures, they are qualitatively similar and will not change our main conclusions (personal communication with William L. Ellsworth, Chief Scientist, Western Region Earthquake Hazards Team, United States Geological Survey).

0. 3 ≈ 1. 10∕3. 7, 0. 76 ≈ 1. 10∕1. 44.

0. 6 ≈ 1. 10∕(3. 7∕2), 1. 5 ≈ 1. 10∕(1. 44∕2).

This section is based on the article Ibragimov et al. (2011), which was published in the Journal of Financial Economics, Vol. 99, Issue 2, pp. 333–348, Copyright Elsevier (2010).

Our analysis applies to banks, but more broadly to general financial intermediaries, like pension funds, insurance companies, and hedge funds.

Circulant topological structures have been used in the economics literature to provide a simple spatial “distance” metric without discontinuities, see, e.g., the discussion in Hennessy and Lapan (2009) and the references therein.

We focus on multivariate normal risks, for tractability. Similar results arise with other, thin-tailed, individual risks, e.g., Bernoulli distributions, although the analysis becomes more complex, because other distribution classes are not closed under portfolio formation so the central limit theorem needs to be incorporated into the analysis.

For review and discussion of models with common shocks and modeling approaches for spatially dependent economic and financial data, see, among others, Conley (1999), Andrews (1993), Ibragimov and Walden (2007), and Ibragimov (2009b).

A Toeplitz matrix A = Toeplitz_N[a _−N+1, a _−N+2, …, a ₋₁, a ₀, a ₁, …, a _N−2, a _N−1], is an N × N matrix with the elements given by (A)_ij = a _j−i, 1 ≤ i ≤ N, 1 ≤ j ≤ N. A Toeplitz matrix is banded if (A)_ij = 0 for large | j − i | , corresponding to a _i = 0 for indices i that are large by absolute value. When a _i = 0 if i < −k or i > m, for k < N − 1 or m < N − 1, we use the notation \(a_{-k},a_{-k+1},\ldots,\underline{a_{0}},\ldots,a_{m-1},a_{m}\) to represent the whole sequence generating the Toeplitz matrix. For example, the notation \(\mathbf{A} = \mbox{ Toeplitz}_{N}[a_{-1},\underline{a_{0}}]\) then means that A _ii = a ₀, A _i, i−1 = a ₋₁, and that all other elements of A are zero. For an N × N Toeplitz matrix, if a _N−j = a _−j, then the matrix is, in addition, circulant. See Horn and Johnson (1990) for more on the definition and properties of Toeplitz and circulant matrices.

As we shall see, our approach leads to very tractable formulas. An alternative approach for introducing more complex correlation structures than the standard multivariate normal one is to use copulas, which may also lead to heavy-tailed portfolio distributions.

We allow for short-selling. In the proof, we show that the optimal portfolio does not involve short-selling, so we could equivalently have permitted only nonnegative portfolios, c ∈ R ₊ ^N.

In Ibragimov et al. (2010), equilibrium premiums are derived in a model with multiple risk factors and risk-averse agents. The general model, however, is quite intractable, and the simplifying assumptions in this section allow us to carry through a more complete study of the role of risk distributions.

It may be optimal, if possible, for the owners of the intermediary to infuse more capital even if losses exceed k, to keep the option of generating future profits alive. Equivalently, they may be able to borrow against future profits. Taking such possibilities into account would increase the point of default to a value higher than k, but would qualitatively not change the results, since there would always be some realized loss level beyond which the intermediary would be shut down even with such possibilities.

This is not a critical assumption. Alternatively, we could have assumed that the regulator anticipates whether the individual intermediaries will trade risks, and adjusts the VaR requirements accordingly.

It is easy to show that λ ∈ [0, T], and that λ is increasing in δ and T.

This follows trivially, since 1 − q _M = P(∑ _i −ξ _i > MK) ≥ P(∩_i{ −ξ _i > K}) = (1 − q)^M.

This also follows trivially, since 1 − q _M ≥ (1 − q)^M and therefore, if q ≥ q _M, then \(\frac{1+\delta q} {1+\lambda (1-q)^{M}} \geq \frac{1+\delta q} {1+\lambda (1-q_{M})} \geq \frac{1+\delta q_{M}} {1+\lambda (1-q)^{M}}\).

The debt overhang problem arises in recapitalizing a bank because the existing shareholder ownership is diluted while some of the cash inflow benefit accrues as a credit upgrade for the existing bondholders and other bank creditors. The agency problems arise because larger capital ratios provide management greater incentive to carry out risky investments that raise the expected value of compensation but may reduce expected equity returns; for further discussion, see Kashyap et al. (2008). While the tax shield benefit of debt is valuable for the banking industry, it is not necessarily welfare-enhancing for society.

The efficiency costs of capital requirements can be mitigated by setting the requirements in terms of contingent capital in lieu of balance sheet capital. One mechanism is based on bonds that convert to capital if bankruptcy is threatened (Flannery 2005), but that instrument is not particularly directed to systemic risk. Kashyap et al. (2008) take the contingent capital idea a step further by requiring banks to purchase “capital insurance” that provides cash to the bank if industry losses, or some comparable aggregate trigger, hits a specified threshold. This mechanism may reduce or eliminate the costs that are otherwise created by bankruptcy, but it does not eliminate the negative externality that creates the systemic risk.

It is important to note that AIG wrote its CDS contracts from its Financial Products subsidiary, which was chartered as a savings and loan association and not as an insurance firm. Indeed, AIG also owns a monoline mortgage insurer, United Guaranty, but this subsidiary was not the source of the losses that forced the government bailout.

This section has drawn upon material from the article Ibragimov and Walden (2010), which was published in the Management Science, Vol. 56, Issue 11, pp. 1963–1976.

The terms “reservation prices” and “valuations” are used as synonyms in this section, in accordance with the well-established tradition in the bundling literature.

The arguments for the results in this section are based on peakedness, majorization and VaR comparison results for heavy-tailed distributions discussed in Sects. 2.1.3 and 2.1.4.

From Theorems 2.1.1 and 2.1.2 it further follows that the regularity condition is also satisfied for moderately heavy-tailed valuations, but it does not hold for extremely heavy-tailed valuations. Therefore, Chakraborty’s analysis cannot be applied if consumers’ valuations are extremely heavy-tailed.

In particular, the assumptions are satisfied for valuations with a finite support \([\underline{v},\overline{v}]\) distributed as the truncation XI( | X −μ | < h), h > 0, of an arbitrary random variable X with a log-concave density symmetric about \(\mu = (\underline{v} + \overline{v})/2\), where \(h = (\overline{v} -\underline{v})/2\) and I(⋅ ) is the indicator function (see also Remark 2 in An 1998).

Clearly, in the case of discretely distributed valuations X _i, i ∈ M, consumers unanimously prefer \(\mathcal{B}_{1}\) to \(\mathcal{B}_{2}\) ex ante if each of them prefers \(\mathcal{B}_{1}\) to \(\mathcal{B}_{2}\) for all but a finite number of realizations of their stand-alone valuations.

This section has drawn upon material from the article Ibragimov (2014) “On the robustness of location estimators in models of firm growth under heavy-tailedness,” in press in the Journal of Econometrics, reproduced with permission from Elsevier, Copyright Elsevier (2014).

From the proof of the results in this section it follows that they continue to hold in the case of arbitrary loss functions ρ(x, y) = ψ( | x − y | ), where ψ is nonnegative and increasing on R ₊.

By Proposition 4 in JR, in the model of demand-driven innovation and spatial competition over time involving the choice of informational gathering effort z in addition to the choice of output y, larger firms always invest more in information if the function G is convex (G″ > 0). Thus, under this condition, investment z into gathering information in JR is secondary with respect to persistence results comparing to y. One should note that, according to empirical studies, there is a positive relationship between R&D expenditures and firm size, that suggests that G is indeed convex (see Kamien and Schwartz 1982, and the discussion following Proposition 4 in JR).

In JR, the proposition is formulated for the Poisson distribution π ₀. The argument for the distributions π _j, j = 1, 2, is completely similar to that case.

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Titel: Implications of Heavy-Tailedness
verfasst von: Marat Ibragimov
Rustam Ibragimov
Johan Walden
Verlag: Springer International Publishing
Buch: Heavy-Tailed Distributions and Robustness in Economics and Finance
Print ISBN: 978-3-319-16876-0

Electronic ISBN: 978-3-319-16877-7

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-16877-7_2

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