According to the fundamental law of active management: IR = IC × √BR × TC

Where:

• IR = Information Ratio
• IC = Information Coefficient
• BR = Breadth
• TC = Transfer Coefficient

Expected active return = IR × Active Risk

Given:

• Breadth increases by factor of 4 → √BR increases by factor of 2 (since √4 = 2)
• Active risk decreases by factor of 4
• All else equal (IC and TC unchanged)

IR_new = IC × (2 × √BR_old) × TC = 2 × IR_old

Expected active return_new = IR_new × (Active Risk_old / 4) = (2 × IR_old) × (Active Risk_old / 4) = 0.5 × (IR_old × Active Risk_old)

Therefore, the expected active return would remain unchanged.