Lesson 4

A tempting first idea is to treat the shares as if they were divisible. The ideal continuous holding of stock \(i\) is \(t_{i}/p_{i}\) shares, so we round each one to the nearest whole number, \(a_{i}\approx t_{i}/p_{i}\), and then repair the budget if we overspent.

Sometimes this is exactly right. Take stock A with price \(3\) and target \(8\), and stock B with price \(5\) and target \(12\), each allowed up to three shares, with budget \(20\). The ideal holdings are \(8/3\approx 2.7\) and \(12/5=2.4\), which round to \(a_{A}=3\) and \(a_{B}=2\). This spends \(9+10=19\le 20\), so it is feasible, and its error \(|9-8|+|10-12|=1+2=3\) cannot be beaten: each stock already sits at its own best share count.