Lesson 5

But rounding ignores the budget, and that can be costly. Keep stock A with price \(1\) and target \(1\), and let stock B have price \(100\) and target \(100\), each allowed at most one share, with budget \(100\). Both ideal holdings equal exactly one share, so rounding says “buy one of each.” That costs \(1+100=101\), one dollar over budget, so something has to give. Keeping the cheap, perfectly tracked stock A and dropping the expensive B looks sensible and leaves error \(|1-1|+|0-100|=100\). Yet that single dollar spent on A is exactly what made B unaffordable: the budget couples the two stocks, so they cannot be rounded one at a time.

Problem. For this second two-stock market with budget \(100\), what is the minimum possible tracking error?