L11 sets as an exercise the one MV problem it does not solve in lecture: the fully-invested optimiser that combines a return objective with the budget constraint. The solution is the most informative formula in portfolio theory — it splits any optimal portfolio into a risk-anchored part and a return-seeking part — and its derivation rehearses every Lagrangian step used elsewhere in the course. We derive it completely, verify its structure, and extract the active-management solution as a corollary.
In this lesson
- Step 0 — The problem
- Step 1 — Lagrangian and first-order conditions
- Step 2 — Solve the stationarity condition for
- Step 3 — Determine from the budget constraint
- Step 4 — Substitute back and rearrange into anchor + tilt
- Step 5 — Verification and structure
- Marking checklist (for the exam write-up)
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