The Statistics of The Information Ratio

In a recent paper, Lo (2002) derives the asymptotic distribution of the Sharpe ratio under several sets of assumptions for the return-generating process. In this paper, we extend his work to the information ratio (IR), the ratio of the excess return of a portfolio over his benchmark to its tracking-error volatility. We assume that each return generating process is i.i.d., allowing however for cross-correlation. First, given the cross-dependency between the portfolio and the benchmark returns, we derive the analytic expression of the asymptotic variance of the IR and we show explicitly how the higher order covariance influence the precision of the variance estimation. On the other hand we study the partial derivatives of the asymptotic variance of the IR with respect to the different moments of the returns. Second, we conduct some simulations in order to highlight the behavior of the IR’s asymptotic variance