  Measures of risk concentration and their asymptotic behavior for portfolios
with heavy-tailed risk factors is of interest in risk management. Second order
regular variation is a structural assumption often imposed on such risk factors
to study their convergence rates. In this paper, we provide the asymptotic rate
of convergence of the measure of risk concentration for a portfolio of
heavy-tailed risk factors, when the portfolio admits the so-called second order
regular variation property. Moreover, we explore the relationship between
multivariate second order regular variation for a vector (e.g., risk factors)
and the second order regular variation property for the sum of its components
(e.g., the portfolio of risk factors). Results are illustrated with a variety
of examples.
