On the limit of conditional Spearman’s rho under the common factor model

Under the common factor structural model of credit risk, we study a limit of a conditional Spearman’s rho coefficient. The conditioning event is that the common factor stays below a threshold and the limit is taken as the threshold tends to − ∞ . The main result is established through a relation with the classical theory of regular variation. We identify the relationship between the limiting Spearman’s rho and the tail thickness of the distribution function of the common factor. In particular, a necessary condition for the limiting Spearman’s rho to be strictly less than 1, is that the inverse cumulative distribution function of the common random variable is slowly varying at zero. As an illustration, the calculation of Stress Value-at-Risk for portfolio credit losses is discussed.