Lesson 1

A portfolio gives a point \((b,e)\), where \(b\) is the spent amount and \(e\) is the tracking error. Some points are plainly worse than others. We say that \((b,e)\) is dominated by \((b',e')\) if \(b'\le b\), \(e'\le e\), and at least one of these inequalities is strict. A dominated point is never useful: another portfolio spends no more and tracks at least as well.

The nondominated points form the frontier. The frontier is a compact picture of the search space. It does not show every portfolio, but it shows the tradeoff between spending more money and reducing error. In the language of the earlier units, the frontier is the useful part of the generating function \(F(z,w)\): among all its terms \(z^{b} w^{e}\), it keeps only those that no cheaper-or-equal term beats on error.