What is a corner portfolio, and how does it relate to constructing the efficient frontier with multiple assets?

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AcadiFi · Accepted Answer

Corner portfolios are the building blocks of the efficient frontier when short-selling constraints are imposed. They mark the critical points where an asset either **enters or exits** the optimal portfolio. **Definition** A corner portfolio is a portfolio on the efficient frontier at which the **composition of included assets changes** — specifically, where one asset's weight goes to zero (exits) or moves from zero to positive (enters). Between any two adjacent corner portfolios, the set of included assets remains the same, and efficient portfolios are linear combinations of the two corner portfolios. **Why Corner Portfolios Matter** Without short-selling constraints, the efficient frontier is a smooth hyperbola. With no-short-sale constraints (w_i >= 0), the frontier becomes a **piecewise curve**, and corner portfolios are the kink points. **Worked Example — Prescott Four-Asset Universe:** Assets: Belmont Equity, Courtland Bonds, Dunmore REIT, Eastfield Commodities. | Corner Portfolio | Included Assets | Expected Return | Sigma | |---|---|---|---| | CP1 (GMV) | Belmont, Courtland, Dunmore | 5.8% | 7.2% | | CP2 | Belmont, Courtland, Dunmore, Eastfield | 7.4% | 8.9% | | CP3 | Belmont, Dunmore, Eastfield | 9.1% | 11.3% | | CP4 | Belmont, Eastfield | 11.5% | 15.8% | | CP5 (Max return) | Belmont only | 14.0% | 22.0% | At CP1 (GMV), Eastfield has zero weight and is about to enter. At CP3, Courtland has been driven to zero weight and exits. At CP5, only the highest-return asset remains. **Constructing Portfolios Between Corner Portfolios:** Any efficient portfolio between two adjacent corner portfolios (say CP2 and CP3) can be formed as: **w = lambda * w_CP2 + (1 - lambda) * w_CP3** where 0 <= lambda <= 1 This means you only need to know the corner portfolios to construct **any** portfolio on the efficient frontier. \`\`\`mermaid flowchart LR A["CP1: GMV
σ=7.2%, R=5.8%"] --> B["CP2
σ=8.9%, R=7.4%"] B --> C["CP3
σ=11.3%, R=9.1%"] C --> D["CP4
σ=15.8%, R=11.5%"] D --> E["CP5: Max Return
σ=22.0%, R=14.0%"] A -.->|"Linear combo
of adjacent CPs"| B B -.->|"Asset enters
or exits"| C \`\`\` **Algorithm Summary:** 1. Start at the maximum-return portfolio (100% in highest-return asset). 2. Gradually reduce the target return. 3. At each step, solve the constrained optimization. When a new asset enters or an existing asset exits, record a new corner portfolio. 4. Continue until reaching the GMV portfolio. **Exam Tips:** 1. The number of corner portfolios is at most **2n - 1** for n assets, but typically fewer. 2. Corner portfolios are a **discrete set** — efficient portfolios between them are continuous combinations. 3. The concept is closely related to the **critical line algorithm** developed by Markowitz. 4. In practice, corner portfolios simplify the computational problem of tracing the entire frontier. See our CFA Level II Portfolio Management module for optimization examples.

What is a corner portfolio, and how does it relate to constructing the efficient frontier with multiple assets?

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