Lesson 2

Suppose the total budget is \(B\), and the target weights are \(w_{1},\dotsc,w_{n}\), where \(\sum_{i} w_{i}=1\). The target amount for stock \(i\) is \[t_{i}=Bw_{i}.\]

For stock \(i\), let \(p_{i}\) be the price of one share, let \(t_{i}\) be the desired amount of money invested in this stock, and let \(m_{i}\) be the maximum number of shares we are allowed to buy. If we buy \(a_{i}\) shares, then the actual amount invested in this stock is \(p_{i} a_{i}\). The tracking error of stock \(i\) is therefore \(|p_{i} a_{i}-t_{i}|\). If shares were divisible, the answer would be boring: buy \(t_{i}/p_{i}\) shares of each stock. But shares are indivisible, so each \(a_{i}\) must be an integer.