Converting a covariance matrix from local currencies to a common currency

This short paper demonstrates how a covariance matrix estimated using log returns of multiple assets in their respective base currencies can be converted directly into a covariance matrix in a single common currency by using basic matrix multiplication. This approach eliminates the need to convert returns into a common currency, simplifying the estimation process. In addition to describing the conversion process, this note also addresses the conversion of covariances between two currencies. By applying the proposed methodology, asset managers can efficiently analyze the covariance between assets denominated in diverse currencies, saving time and resources. It is thus a valuable tool for asset managers seeking to optimize portfolio allocation across different currencies.