Correlation versus co-fractality: Evidence from foreign-exchange-rate variances

The concept of correlation appears to be the cornerstone of modern finance as it is applied in almost all finance-related research studies. However, Fama (1963) argued that "if the [population] variance is infinite, other sta-tistical tools (e.g., least-squares regression) which are based on the assumption of finite variance will, at best, be considerably weakened and may in fact give very misleading answers" (p. 421). This study shows variances of foreign exchange rates to be governed by power laws with a tail exponent of alpha < 3, suggesting infinite second moments. We derive a new concept to measure dependencies between power-law processes with this tail exponent, which we term co-fractality. We show that risk diversification based on the concept of correlation indeed gives misleading results. Notably, foreign-exchange-rate variances lacking co-fractality in our earlier subsample do not show evidence for co-fractality in our later subsample. We argue that co-fractality, as opposed to correlation, should be used to measure the dependency between processes governed by power laws.