On multivariate higher order von Mises expansions

In this paper we review the von Mises calculations of higher order for statistical functionals of one variable. For the functionals of several variables, the higher order Gateaux differentials are defined leading to the corresponding multivariate higher order von Mises expansions. These expansions are used to analyze the bias of the corresponding statistical functionals. The second and third order derivatives are computed for M-estimates. Applications of these expansions to study the bias of M-estimates of location and to simultaneous M-estimates of location and scale are also given. Received: July 2000     

Diversity and relative arbitrage in equity markets

An equity market is called diverse if no single stock is ever allowed to dominate the entire market in terms of relative capitalization. In the context of the standard Itô-process model initiated by Samuelson (1965) we formulate this property (and the allied, successively weaker notions of weak diversity and asymptotic weak diversity) in precise terms. We show that diversity is possible to achieve, but delicate. Several examples are provided which illustrate these notions and show that weakly-diverse markets contain relative arbitrage opportunities: it is possible to outperform or underperform such markets over any given time-horizon. The existence of this type of relative arbitrage does not interfere with the development of contingent claim valuation, and has consequences for the pricing of long-term warrants and for put-call parity. Several open questions are suggested for further study.Received: January 2004, Mathematics Subject Classification (2000): 60H10, 91B28; 60J55JEL Classification: G10We are grateful for the helpful remarks offered by seminar audiences at Columbia, Yale, Princeton, the Sloan School of MIT, Boston University, the Mathematical Institute in Oberwolfach, and the Universities of Athens, Connecticut/Storrs and Texas/Austin. Special thanks go to Professors Jérôme Detemple, Julien Hugonnier, Ralf Korn, Andrew Lo, Mark Lowenstein and Steven Shreve. We are also indebted to Dr. Adrian Banner for a number of discussions that helped sharpen our thinking about these problems, and to the referees and editors for suggestions that improved the exposition. A significant part of this work was completed in the spring semester of 2002, while the second author was on sabbatical leave at the Cowles Foundation for Research in Economics, Yale University. He is grateful to the Foundation for its hospitality.     

A second-order stock market model

A first-order model for a stock market assigns to each stock a return parameter and a variance parameter that depend only on the rank of the stock. A second-order model assigns these parameters based on both the rank and the name of the stock. First- and second-order models exhibit stability properties that make them appropriate as a backdrop for the analysis of the idiosyncratic behavior of individual stocks. Methods for the estimation of the parameters of second-order models are developed in this paper.     

Terrorism and the Invisible Hook

This article investigates and compares the root causes of transnational terrorism and piracy. In order to accomplish this, we construct a novel data set that catalogues terrorist activity and piracy over the years 1992–2008, paying particular attention to the Arabian Sea and Indian Ocean since 1991. These data are then merged with other information detailing the economic, political, and security posture of each organization during the same period so that we can examine the relationship between economic prosperity, security, terrorism, and piracy. The main conclusion is that terrorism is mostly unrelated to economic conditions, while piracy responds to both economic payoffs and military deterrents.     

EXCHANGE RATE MANIPULATION AND CONSTRUCTIVE AMBIGUITY

I consider dynamic models in which investors are heterogeneously informed about both foreign exchange interventions and exchange rate fundamentals and show that transparency sometimes exacerbates misalignment between the exchange rate and fundamentals. Although transparency reveals information about fundamentals, it also increases the precision of the exchange rate as a signal of those fundamentals that remain unknown (the signal‐precision effect of transparency). If a central bank announcement reveals little information about fundamentals, then this effect dominates and transparency magnifies misalignment. One implication is that ambiguity can increase the effectiveness of intervention to support a declining currency during times of crisis.     

Robustness issues regarding content-corrected tolerance limits

This article reviews the content-corrected method for tolerance limits proposed by Fernholz and Gillespie (2001) and addresses some robustness issues that affect the length of the corresponding interval as well as the corrected content value. The content-corrected method for k-factor tolerance limits consists of obtaining a bootstrap corrected value p ^* that is robust in the sense of preserving the confidence coefficient for a variety of distributions. We propose several location/scale robust alternatives to obtain robust corrected-content k-factor tolerance limits that produce shorter intervals when outlying observations are present. We analyze the Hadamard differentiability to insure bootstrap consistency for large samples. We define the breakdown point for the particular statistic to be bootstrapped, and we obtain the influence function and the value of the breakdown point for the traditional and the robust statistics. Two examples showing the advantage of using these robust alternatives are also included.     

The distributional effects of progressive capital taxes

Rising inequality since the 1980s has spurred much research examining the underlying causes and potential policy responses. Among the more controversial, One of the more controversial policy proposals is a progressive capital tax in response to rising top wealth shares around the world proposes a progressive capital tax in response to rising top wealth shares around the world. This paper introduces rank-based econometric methods for dynamic power laws as a tool for estimating the effect of progressive capital taxes on the distribution of wealth under different assumptions about the impact of these taxes on household behavior. In most scenarios, we find that a small tax levied on 1% of households would substantially reshape the US wealth distribution and reduce inequality.     

Relative arbitrage in volatility-stabilized markets

Summary. We provide simple, easy-to-test criteria for the existence of relative arbitrage in equity markets. These criteria postulate essentially that the excess growth rate of the market portfolio, a positive quantity that can be estimated or even computed from a given market structure, be sufficiently large. We show that conditions which satisfy these criteria are manifestly present in the U.S. equity market. We then construct examples of abstract markets in which the criteria hold. These abstract markets allow us to isolate conditions similar to those prevalent in actual markets, and to construct explicit portfolios under these conditions. We study in some detail a specific example of an abstract market which is volatility-stabilized, in that the return from the market portfolio has constant drift and variance rates while the smallest stocks are assigned the largest volatilities. A rather interesting probabilistic structure emerges, in which time changes and the asymptotic theory for planar Brownian motion play crucial roles. The largest stock and the overall market grow at the same, constant rate, though individual stocks fluctuate widely.We are indebted to Adrian Banner and Constantinos Kardaras for several helpful discussions; in particular, the computation of (4.27) is due to Adrian Banner. We are also grateful to Professors Chris Rogers and Marc Yor for their expert advice on the subject of planar Brownian motion, and to Professor Peter Bank for an observation that helped us correct an error in an earlier version of the paper.This revised version was published online in January 2005 with corrections to the Cover date.     

Short-term relative arbitrage in volatility-stabilized markets

We answer in the affirmative the following open question posed in Fernholz and Karatzas (Ann Finance 1:149–177, 2005): do there exist relative arbitrage opportunities over arbitrarily short time horizons in the context of certain volatility-stabilized market models?     

The implied liquidity premium for equities

Over the long term, the returns on smaller stocks are likely to be higher than the returns on larger stocks. This phenomenon has been called size effect, and a number of explanations have been proposed to account for it. Here we show that the difference in return between the larger and the smaller stocks can be accounted for by a liquidity premium for the smaller stocks, and we estimate the value of this premium using structural parameters for the capital distribution of the U.S. stock market during the 1990s The authors wish to express their gratitude to an anonymous referee for a very thorough and incisive reading, as well as for many constructive suggestions that have significantly improved this paper. The authors wish to express their gratitude to an anonymous referee for a very thorough and incisive reading, as well as for many constructive suggestions that have significantly improved this paper.