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).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The functions ϕ p have the same form as measures of diversification considered in Bouchaud and Potters (2004), Chap. 12, p. 205.
- 2.
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).
- 3.
Here, as usual, \(\Gamma (r)\) and B(a, b) denote the Gamma and Beta functions.
- 4.
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.
- 5.
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.
- 6.
- 7.
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.
- 8.
- 9.
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.”
- 10.
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).
- 11.
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.
- 12.
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.
- 13.
That is, in the case of an absolutely continuous risk Z, P(Z ≤ VaR q (Z)) = q.
- 14.
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.
- 15.
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).
- 16.
These data are from the Insurance Information Institute; see http://iii.org/media/industry/.
- 17.
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.
- 18.
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.
- 19.
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.
- 20.
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.
- 21.
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.
- 22.
This is the moment magnitude version of the celebrated Gutenberg–Richter exponential law for the Richter magnitude.
- 23.
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.
- 24.
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).
- 25.
1. 10 ≈ 1. 84 × 0. 6.
- 26.
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).
- 27.
0. 3 ≈ 1. 10∕3. 7, 0. 76 ≈ 1. 10∕1. 44.
- 28.
0. 6 ≈ 1. 10∕(3. 7∕2), 1. 5 ≈ 1. 10∕(1. 44∕2).
- 29.
Our analysis applies to banks, but more broadly to general financial intermediaries, like pension funds, insurance companies, and hedge funds.
- 30.
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.
- 31.
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.
- 32.
- 33.
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.
- 34.
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.
- 35.
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.
- 36.
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.
- 37.
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.
- 38.
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.
- 39.
It is easy to show that λ ∈ [0, T], and that λ is increasing in δ and T.
- 40.
This follows trivially, since 1 − q M = P(∑ i −ξ i > MK) ≥ P(∩ i { −ξ i > K}) = (1 − q)M.
- 41.
This also follows trivially, since 1 − q M ≥ (1 − q)M and therefore, if q ≥ q M , then \(\frac{1+\delta q} {1+\lambda (1-q)^{M}} \geq \frac{1+\delta q} {1+\lambda (1-q_{M})} \geq \frac{1+\delta q_{M}} {1+\lambda (1-q)^{M}}\).
- 42.
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.
- 43.
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.
- 44.
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.
- 45.
The terms “reservation prices” and “valuations” are used as synonyms in this section, in accordance with the well-established tradition in the bundling literature.
- 46.
- 47.
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.
- 48.
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).
- 49.
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.
- 50.
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 +.
- 51.
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).
- 52.
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.
Bibliography
Acharya, V. (2009). A theory of systemic risk and design of prudent bank regulation. Journal of Financial Stability, 5, 224–255.
Adams, W. J., & Yellen, J. L. (1976). Commodity bundling and the burden of monopoly. Quarterly Journal of Economics, 90, 475–498.
Adrian, T., & Brunnermeier, M. (2011). CoVar. Federal Reserve Bank of New York Staff Report no. 348. Available at http://www.newyorkfed.org/research/staff_reports/sr348.pdf.
Agarwal, R., & Gort, M. (1996). The evolution of markets and entry, exit and survival of firms. Review of Economics and Statistics, 78, 489–498.
Allen, F., & Gale, D. (2000). Financial contagion. Journal of Political Economy, 108, 1–33.
An, M. Y. (1998). Logconcavity versus logconvexity: A complete characterization. Journal of Economic Theory 80, 350–369.
Andrews, D. W. K. (1993). Tests for parameter instability and structural change with unknown change point. Econometrica, 61(4), 821–856. http://dx.doi.org/10.2307/2951764.
Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.
Audretsch, D. B. (1991). New firm survival and the technological regime. Review of Economics and Statistics, 73, 441–450.
Azariadis, C., & Stachurski, J. J. (2006). Poverty traps. In P. Aghion & S. Durlauf (Eds.), Handbook of Economic Growth. Amsterdam: Elsevier.
Bagnoli, M., & Bergstrom, T. (2005). Log-concave probability and its applications. Economic theory, 26, 445–469.
Bakos, Y., & Brynjolfsson, E. (1999). Bundling information goods: Pricing, profits and efficiency. Management Science, 45, 1613–1630.
Bakos, Y., & Brynjolfsson, E. (2000a). Aggregation and disaggregation of information goods: Implications for bundling, site licensing and micropayment systems. In D. Hurley, B. Kahin & H. Varian (Eds.), Proceedings of Internet Publishing and Beyond: The Economics of Digital Information and Intellectual Property. Cambridge, MA: MIT.
Bakos, Y., & Brynjolfsson, E. (2000b). Bundling and competition on the internet. Marketing Science, 19, 63–82.
Bakun, W. H., Johnston, A. C., & Hopper, M. G. (2003). Estimating locations and magnitudes of earthquakes in eastern North American from Modified Mercalli intensities. Bulletin of the Seismological Society of America, 93, 190–202.
Barro, R. J., & Sala-i-Martin, X. (2004). Economic growth. Cambridge: MIT.
Berger, J. O. (1985). Statistical decision theory and Bayesian analysis. New York: Springer.
Bernanke, B. S. (1983). Nonmonetary effects of the financial crisis in the propagation of the great depression. American Economic Review, 73, 257–276.
Bickel, P. J., & Lehmann, E. L. (1975a). Descriptive statistics for nonparametric models. I. Introduction. Annals of Statistics, 3, 1038–1044.
Bickel, P. J., & Lehmann, E. L. (1975b). Descriptive statistics for nonparametric models. II. Location. Annals of Statistics, 3, 1045–1069.
Bouchaud, J.-P. and Potters, M. (2004). Theory of financial risk and derivative pricing: From statistical physics to risk management (2nd edn.). Cambridge: Cambridge University Press.
Box, G. E. P., & Tiao, G. C. (1973). Bayesian inference in statistical analysis. Reading/London/Don Mills: Addison-Wesley.
Brunnermeier, M. (2009). Deciphering the liquidity and credit crunch 2007–2008. Journal of Economic Perspectives, 23, 77–100.
Brunnermeier, M., & Pedersen, L. H. (2009). Market liquidity and funding liquidity. Review of Financial Studies, 22, 2201–2238.
Caballero, R., & Krishnamurthy, A. (2008). Collective risk management in a flight to quality episode. Journal of Finance, 63, 2195–2230.
Carlin, B. P., & Louis, T. A. (2000). Bayes and empirical Bayes methods for data analysis. London: Chapman & Hall.
Carroll, S., LaTourrette, T., Chow, B. G., Jones, G. S., & Martin, C. (2005). Distribution of losses from large terrorist attacks under the Terrorism Risk Insurance Act. Santa Monica, Calif: Rand Corporation.
Chakraborty, I. (1999). Bundling decisions for selling multiple objects. Economic Theory, 13, 723–733.
Chan, W., Park, D. H., & Proschan, F. (1989). Peakedness of weighted averages of jointly distributed random variables. In L. J. Gleser, M. D. Perlman, S. J. Press & A. R. Sampson (Eds.), Contributions to Probability and Statistics (pp. 58–62). New York: Springer.
Christoffersen, P. F. (2012). Elements of financial risk management (2nd edn.). New York: Elsevier.
Chu, C. S., Leslie, P., & Sorensen, A. (2011). Bundle-size pricing as an approximation to mixed bundling. American Economic Review, 101, 263–303.
Conley, T. G. (1999). GMM estimation with cross sectional dependence. Journal of Econometrics, 92, 1–45.
Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance, 1, 223–236.
Cummins, J. D. (2006). Should the government provide insurance for catastrophes? Federal Reserve Bank of St. Louis Review, 88, 337–379.
Cummins, J. D., Doherty, N., & Lo, A. (2002). Can insurers pay for the ‘big one’? Measuring the capacity of the insurance market to respond to catastrophic losses. Journal of Banking and Finance, 26, 557–583.
Cummins, J. D., & Mahul, O. (2003). Optimal insurance with divergent beliefs about insurer total default risk. The Journal of Risk and Uncertainty, 27, 121–138.
Dansby, R. E., & Conrad, C. (1984). Commodity bundling. American Economic Review, 74, 377–381.
Davis, R. A., & Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Annals of Statistics, 26, 2049–2080.
de la Peña, V. H., & Giné, E. (1999). Decoupling: From dependence to independence. Probability and Its Applications. New York: Springer.
de la Peña, V. H., Ibragimov, R., & Sharakhmetov, S. (2003). On extremal distributions and sharp L p -bounds for sums of multilinear forms. Annals of Probability 31, 630–675.
Diamond, D., & Dybvig, P. (1983). Bank runs, deposit insurance, and liquidity. Journal of Political Economy, 91, 401–419.
Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. New York: Springer.
Embrechts, P., McNeil, A., & Straumann, D. (2002). Correlation and dependence in risk management: Properties and pitfalls. In M. A. H. Dempster (Ed.), Risk Management: Value at Risk and Beyond (pp. 176–223). Cambridge: Cambridge University Press.
Embrechts, P., Neslehova, J., & Wüthrich, M. (2009). Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness. Insurance: Mathematics and Economics, 44, 164–169.
Fama, E. (1965b). Portfolio analysis in a stable Paretian market. Management Science, 11, 404–419.
Fang, H., & Norman, P. (2006). To bundle or not to bundle. Marketing Science, 37, 946–963.
Flannery, M. (2005). No pain, no gain: Effecting market discipline via reverse convertible debentures. In H. Scott (Ed.),Capital Adequacy beyond Basel: Banking, Securities, and Insurance (Chap. 5, pp. 171–197.). Oxford: Oxford University Press.
Fölmer, H., & Schied, A. (2002). Convex measures of risk and trading constraints. Finance and Stochastics, 6, 429–447.
Frittelli, M., & Gianin, E. R. (2002). Putting order in risk measures. Journal of Banking & Finance, 26, 1473–1486.
Froot, K. A. (2001). The market for catastrophe risk: A clinical examination. Journal of Financial Economics, 60, 529–571.
Froot, K. A., & Posner, S. (2002). The pricing of event risks with parameter uncertainty. The Geneva Papers on Risk and Insurance, 27(2), 153–165.
Froot, K. A., Scharfstein, D., & Stein, J. (1993). Risk management: Coordinating corporate investment and financing policies. Journal of Finance, 48, 1629–1658.
Froot, K. A., & Stein, J. (1998). Risk management, capital budgeting, and capital structure policy for financial institutions: An integrated approach. Journal of Financial Economics, 47, 55–82.
Gordy, M. B. (2000). Credit VaR and risk-bucket capital rules: A reconciliation. Federal Reserve Bank of Chicago (May) (pp. 406–417).
Gordy, M. B. (2003). A risk-factor based model foundation for ratings-based bank capital rules. Journal of Financial Intermediation, 12, 199–232.
Granger, C. W. J., & Orr, D. (1972). Infinite variance and research strategy in time series analysis. Journal of the American Statistical Association, 67, 275–285.
Hanks, T., & Johnston, A. (1992). Common features of the excitation and propagation of strong ground motion for north american earthquakes. Bulletin of the Seismological Society of America, 82, 1–23.
Hanks, T., & Kanamori, H. (1979). A moment magnitude scale. Journal of Geophysical Research, 84, 2348–2350.
Hansen, B. E. (2015). The integrated mean squared error of series regression and a Rosenthal hilbert-space inequality. Econometric Theory, 31, 337–361.
Hennessy, D. A., & Lapan, H. E. (2009). Harmonic symmetries of imperfect competition on circular city. Journal of Mathematical Economics, 45, 124–146.
Hitt, L. M., & Chen, P.-y. (2005). Bundling with customer self-selection: A simple approach to bundling low-marginal-cost goods. Management Science, 51, 1481–1493.
Ho, P., Sussman, J., & Veneziano, D. (2001). Estimating the direct and indirect losses from a Midwest earthquake. Report SE-10, Mid-America Earthquake Center.
Hogg, R., & Klugman, S. (1983). On the estimation of long tailed skewed distributions with actuarial applications. Journal of Econometrics, 23, 91–102.
Horn, R. A., & Johnson, C. R. (1990) Matrix Analysis. Cambridge: Cambridge University Press; Corrected reprint of the 1985 original.
Hsieh, P. (1999). Robustness of tail index estimation. Journal of Computational Graphical Statistics, 8, 318–332.
Hubbard, G., Deal, B., & Hess, P. (2005). The economic effect of federal participation in terrorism risk. Risk Management & Insurance Review, 8, 177–303.
Ibragimov, M., & Ibragimov, R. (2008). Optimal constants in the rosenthal in- equality for random variables with zero odd moments. Statistics and Probability Letters, 78, 186–189.
Ibragimov, R. (2005). New majorization theory in economics and martingale convergence results in econometrics. Ph.D. Dissertation, Yale University.
Ibragimov, R. (2007). Efficiency of linear estimators under heavy-tailedness: Convolutions of α-symmetric distributions. Econometric Theory, 23, 501–517.
Ibragimov, R. (2009a). Heavy-tailed densities, In S. N. Durlauf & L. E. Blume (Eds.), The New Palgrave Dictionary of Economics Online. Palgrave Macmillan. http://www.dictionaryofeconomics.com/article?id=pde2008_H000191.
Ibragimov, R. (2009b). Portfolio diversification and value at risk under thick-tailedness. Quantitative Finance, 9, 565–580.
Ibragimov, R. (2014). On the robustness of location estimators in models of firm growth under heavy-tailedness. Journal of Econometrics. In press, http://dx.doi.org/10.1016/j.jeconom.2014.02.005. Also available as Harvard Institute of Economic Research Discussion Paper No. 2087. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=774865.
Ibragimov, R., Jaffee, D., & Walden, J. (2009). Nondiversification traps in catastrophe insurance markets. Review of Financial Studies, 22, 959–993.
Ibragimov, R., Jaffee, D., & Walden, J. (2010). Pricing and capital allocation for multiline insurance firms. Journal of Risk and Insurance, 77, 551–578.
Ibragimov, R., Jaffee, D., & Walden, J. (2011). Diversification disasters. Journal of Financial Economics, 99, 333–348.
Ibragimov, R., Jaffee, D., & Walden, J. (2012). Insurance equilibrium with monoline and multiline insurers. Working paper, Imperial College Business School and the University of California at Berkeley.
Ibragimov, R., Mo, J., & Prokhorov, A. (2014). Fat tails and copulas: Limits of diversification revisited. Working paper, Imperial College Business School and the University of Sydney Business School.
Ibragimov, R., & Sharakhmetov, S. (1997). On an exact constant for the Rosenthal inequality. Theory of Probability and Its Applications, 42, 294–302.
Ibragimov, R., & Sharakhmetov, S. (2002). Bounds on moments of symmetric statistics. Studia Scientiarum Mathematicarum Hungarica, 39, 251–275 (Date of submission: 1996).
Ibragimov, R., & Walden, J. (2007). The limits of diversification when losses may be large. Journal of Banking and Finance, 31, 2551–2569.
Ibragimov, R., & Walden, J. (2010). Optimal bundling strategies under heavy-tailed valuations. Management Science, 56, 1963–1976.
Jaffee, D. (2006a). Commentary on Should the government provide insurance for catastrophes? Federal Reserve Bank of St. Louis Review, 88, 381–385.
Jaffee, D. (2006b). Monoline restrictions, with applications to mortgage insurance and title insurance. Review of Industrial Organization, 28, 88–108.
Jaffee, D. (2009). Monoline regulations to control the systemic risk created by investment banks and GSEs. The B.E. Journal of Economic Analysis & Policy, 9(3). Available at http://www.bepress.com/bejeap/vol9/iss3/art17.
Jaffee, D., & Russell, T. (2006). Should governments provide catastrophe insurance? The Economists’ Voice, 3(5). Available at http://www.bepress.com/ev/vol3/iss5/art6.
Jensen, D. R. (1997). Peakedness of linear forms in ensembles and mixtures. Statistics and Probability Letters 35, 277–282.
Jovanovic, B., & Rob, R. R. (1987). Demand-driven innovation and spatial competition over time. Review of Economic Studies, 54, 63–72.
Jurečková, J., & Sen, P. K. (1996). Robust statistical procedures: Asymptotics and interrelations, New York: Wiley.
Kagan, Y., & Knopoff, L. (1984). A stochastic model of earthquake occurrence. In Proceedings of the 8th International Conference on Earthquake Engineering (Vol. 1, pp. 295–302). Englewood Cliffs, NJ: Prentice-Hall.
Kamien, M., & Schwartz, N. (1982). Market structure and innovation. New York: Cambridge University Press.
Karlin, S. (1968). Total positivity (Vol. I). Stanford, CA: Stanford University Press.
Kashyap, A., Rajan, R., & Stein, J. (2008). Rethinking capital regulation. In Federal Reserve Bank of Kansas City Symposium on Maintaining Stability in a Changing Financial System (pp. 431–471).
Knopoff, L., & Kagan, Y. (1977). Analysis of the theory of extremes as applied to earthquake problems. Journal of Geophysical Research, 82, 5647–5657.
Kodes, L., & Pritsker, M. (2002). A rational expectations model of financial contagion. Journal of Finance, 57, 769–799.
Krishna, V. (2002). Auction theory. San Diego, CA: Academic.
Kunreuther, H., & Michel-Kerjan, E. (2006). Looking beyond TRIA: A clinical examination of potential terrorism loss sharing. NBER Working paper No. 12069. http://www.nber.org/papers/w12069.
Kyle, A., & Xiong, W. (2001). Contagion as a wealth effect. Journal of Finance, 56, 1401–1440.
Lewbel, A. (1985). Bundling of substitutes or complements. International Journal of Industrial Organization, 3, 101–107.
Ma, C. (1998). On peakedness of distributions of convex combinations. Journal of Statistical Planning and Inference, 70, 51–56.
Mahul, O., & Wright, B. (2004). Efficient risk sharing within a catastrophe insurance pool. Paper presented at the 2003 NBER Insurance Project Workshop. Available at http://www.arec.umd.edu/Department/Seminars/mahulwright0204.pdf.
Marshall, A. W., Olkin, I., & Barro, A. (2011). Inequalities: Theory of majorization and its applications (2nd edn.). New York: Springer.
McAfee, R., M. J., & Whinston, M. (1989). Multiproduct monopoly, commodity bundling and correlation of values. Quarterly Journal of Economics, 114, 371–384.
McGuire, R. (2004). Seismic Hazard and Risk Analysis. Oakland, CA: Earthquake Engineering Research Institute.
McNeil, A. J., Frey, R., & Embrechts, P. (2005). Quantitative risk management: Concepts, techniques, and tools. Princeton: Princeton University Press.
Mikosch, T., & Stărică, C. (2000). Limit theory for the sample autocorrelations and extremes of a GARCH (1, 1) process. Annals of Statistics, 28, 1427–1451.
Milgrom, P., & Weber, R. (1982). A theory of auctions and competitive bidding. Econometrica, 50, 1089–1122.
Nelson, C. R., & Plosser, C. I. (1982). Trends and random walks in macroeconomic time series. Journal of Monetary Economics, 10, 139–162.
Nze, P. A., & Doukhan, P. (2004). Weak dependence: Models and applications to econometrics. Econometric Theory, 20, 995–1045.
OECD (2005a). Policy issues in insurance no. 08: Catastrophic risks and insurance. OECD Publishing, August.
OECD (2005b). Policy issues in insurance no. 09: Terrorism risk insurance in OECD countries. OECD Publishing, July.
Palfrey, T. R. (1983). Bundling decisions by a multiproduct monopolist with incomplete information. Econometrica 51, 463–483.
Proschan, F. (1965). Peakedness of distributions of convex combinations. Annals of Mathematical Statistics, 36, 1703–1706.
Rachev, S. T., & Mittnik, S. (2000). Stable Paretian Models in Finance, New York: Wiley.
Rochet, J. C., & Tirole, J. (1996). Interbank lending and systemic risk. Journal of Money, Credit, and Banking, 28, 773–762.
Ross, S. A. (1976). A note on a paradox in portfolio theory. Philadelphia: Mimeo, University of Pennsylvania.
Rousseeuw, P. J., & Leroy, A. M. (1987). Robust regression and outlier detection. New York: Wiley.
Roy, A. D. (1952). Safety first and the holding of assets. Econometrica, 20, 431–449.
Salinger, M. A. (1995). A graphical analysis of bundling. Journal of Business, 68, 85–98.
Samuelson, P. A. (1967a). Efficient portfolio selection for Pareto-Lévy investments. Journal of Financial and Quantitative Analysis, 2, 107–122.
Samuelson, P. A. (1967b). General proof that diversification pays. The Journal of Financial and Quantitative Analysis, 2, 1–13.
Schmalensee, R. L. (1984). Gaussian demand and commodity bundling. Journal of Business, 57, S211–S230.
Schwartz, D., & Coppersmith, K. (1984). Fault behavior and characteristic earthquakes. Journal of Geophysical Research, 89, 5681–5698.
Shaffer, S. (1994). Pooling intensifies joint failure risk. Research in Financial Services, 6, 249–280.
Sornette, D., Knopoff, L., Kagan, Y., & Vanneste, C. (1996). Rank-ordering statistics of extreme events: Application to the distribution of large earthquakes. Journal of Geophysical Research, 101, 13883–13894.
Stremersch, S., & Tellis, G. J. (2002). Strategic bundling of products and prices: A new synthesis. Journal of Marketing, 66, 55–72.
SwissRe (2006). Sigma. Sigma, 2, 267–270.
Tong, Y. L. (1994). Some recent developments on majorization inequalities in probability and statistics. Linear Algebra and its Applications, 199, 69–90.
Uchaikin, V. V., & Zolotarev, V. M. (1999). Chance and stability. Stable distributions and their applications. Utrecht: VSP.
Utev, S. A. (1985). Extremal problems in moment inequalities. Proceedings of the Mathematical Institute of the Siberian Branch of the USSR Academy of Sciences, 5, 56–75. (In Russian).
Venkatesh, R., & Kamakura, W. (2003) Optimal bundling and pricing under a monopoly: Contrasting complements and substitutes from independently valued products. Journal of Business 76, 211–231.
Volcker, P. A. (2010). Statement for the committee on banking, housing, and urban affairs of the United States Senate, February 2, 2010. Available at http://banking.senate.gov/public/index.cfm?FuseAction=Files.View&FileStore_id=ec787c56-dbd2-4498-bbbd-ddd23b58c1c4.
Wagner, W. (2010). Diversification at financial institutions and systemic crises. Journal of Financial Intermediation, 19, 373–386.
Woo, G. (1999). The Mathematics of Natural Catastrophes. London: Imperial College Press.
Wu, S.-y., Hitt, L. M., Chen, P.-y., & Anandalingam, G. (2008). Customized bundle pricing for information goods: A nonlinear mixed-integer programming approach. Management Science, 54, 608–622.
Zolotarev, V. M. (1986). One-dimensional stable distributions. Providence: American Mathematical Society.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Ibragimov, M., Ibragimov, R., Walden, J. (2015). Implications of Heavy-Tailedness. In: Heavy-Tailed Distributions and Robustness in Economics and Finance. Lecture Notes in Statistics, vol 214. Springer, Cham. https://doi.org/10.1007/978-3-319-16877-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-16877-7_2
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16876-0
Online ISBN: 978-3-319-16877-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)