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Portfolio Theory and Quantitative Investment / Index Models and Portfolio Computation

Computing the Efficient Frontier: Corner Portfolios and Constant Correlation

The single-index model reduces the covariance matrix from N(N−1)/2N(N-1)/2 estimates down to NN betas. But a smaller matrix still requires a quadratic programme (QP) to find the optimal portfolio. For an institution following 200 stocks, that means numerical optimisation with 200 variables and a 200×200200 \times 200 matrix inversion — feasible with computers, but opaque. Every perturbation of an expected return or beta produces a new set of weights with no analytical insight into why.
The payoff of the single-index model goes further. When returns follow the single-index structure, the mean-variance optimisation problem admits a closed-form solution: rank all stocks by their excess-return-to-beta ratio (Rˉi−RF)/βi(\bar{R}_i - R_F)/\beta_i, compute a scalar cutoff rate C∗C^*, and include every stock above the cutoff. Weights are then determined by a single formula for each stock individually. No matrix inversion, no joint optimisation, no solver. The result is identical to what the QP would have produced.

This is important for practitioners for two reasons. First, it makes the construction logic legible: a stock analyst can determine whether any stock belongs in the optimal portfolio by comparing a single number — the excess-return-to-beta ratio — against C∗C^*, without knowledge of any other stock in the universe. Second, it shows that the single-index model's structural assumption (residuals uncorrelated) is what makes portfolio construction decomposable into stock-by-stock decisions — a property that completely fails in the full-covariance case.

Learning objectives. By the end of this note you should be able to:
  • state and interpret the excess-return-to-beta ranking rule;
  • compute the cutoff rate C∗C^* from the iterative formula (9.1) using a worked example;
  • determine portfolio membership and compute portfolio weights from equation (9.3);
  • adapt the procedure to the short-sales-allowed case;
  • apply the analogous constant-correlation selection rule.

In this lesson

  1. The excess-return-to-beta ranking rule
  2. Computing the cutoff rate C∗C^*
  3. Worked example
  4. Portfolio weights
  5. Short sales allowed
  6. The constant-correlation model selection rule
  7. Connection to the fundamental law

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

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