Multihomogeneous resultant formulae by means of complexes

We provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants arising from Weyman complexes associated to line bundles in products of projective spaces. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. We characterize the systems that admit a purely B\'ezout-type matrix and show a bijection of such matrices with the permutations of the variable groups. We conclude with examples showing the hybrid matrices that may be encountered, and illustrations of our Maple implementation. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing results by Sturmfels-Zelevinsky, and Weyman-Zelevinsky.Comment: 30 pages. To appear: Journal of Symbolic Computatio

Sectoral vs. country diversification benefits and downside risk

Recently, the advantage of country diversification relative to sector diversification has been questioned especially against the background of the European monetary and financial integration. Correct estimates of the correlation matrix are central for the evaluation of the relative diversification gains. These estimates should take into account the time-varying and asymmetric behaviour of the correlation process particularly in the context of major changes in volatility and market trends. In this paper, the ADCC (Asymmetric Dynamic Conditional Correlation) model developed by Cappiello et al. (2003) is used to estimate the conditional correlation and volatility of weekly country, sector and country/sector returns indexes over 1990-2003. This model offers a relatively flexible specification for the conditional correlation process that is still computationally feasible for estimation on larger portfolios. The estimation results point to an increase in the average correlation between country indexes during the last five years, but at the same time there is an important decline in the correlation between sector indexes. This trend is observed in both the euro area and the word-wide portfolios and is therefore not specific to the European integration process. At the same time, the volatility in the sector indexes has increased remarkably and in a relatively stronger way compared to the volatility in the country indexes. Both trends tend to cancel out in the calculations of optimal portfolio variance: lower sector correlation is offset by higher sector volatility and higher country correlation is neutralised by the relative lower volatility in country indexes. Therefore no clear trend appears from comparing the relative conditional variances of sector and country portfolios. After taking into account the effect of average returns, it turns out that country diversification is still the dominant strategy for world portfolios, whereas sector diversification is more interesting for euro area portfolios.portfolio diversification, correlation, international finance

A software framework for computing Newton polytopes of resultants and (reduced) discriminants

We present a new software for computing Newton polytopes of resultant and discriminant polynomials. We illustrate its use with a number of examples