Two standardised signals each have an information coefficient of 0.05 against next month's return, and their cross correlation is 0.6. What is the information coefficient of their equally weighted average, and are equal weights optimal?

Two standardised signals each have an information coefficient of 0.05 against next month's return, and their cross correlation is 0.6. What is the information coefficient of their equally weighted average, and are equal weights optimal?

Approach: The combined signal has to be restandardised, so divide the summed covariance with returns by the standard deviation of the sum.

0.0559. The sum of two standardised signals has standard deviation sqrt(2 + 2*rho) = sqrt(3.2) = 1.7889, while its covariance with the return is the sum of the two individual covariances, so the combined information coefficient is (0.05 + 0.05)/1.7889 = 0.0559. That is 12% better than either signal alone, because a signal correlation of 0.6 means most of the second forecast repeats the first. At rho = 0 the same arithmetic gives 0.1/sqrt(2) = 0.0707, a 41% gain, and at rho = 0.9 it gives 0.0513. Equal weights are optimal here by symmetry, since the optimal combination is proportional to the inverse correlation matrix times the vector of individual coefficients and that vector is flat.

Follow-up: What are the optimal weights when the coefficients are 0.05 and 0.03 and the correlation is 0.6?

Key concepts: information coefficient, signal correlation, signal combination, inverse correlation matrix.