Charta is on Google Play: practise on the go. Get the app
Charta

Judge screens, stratification, linear programming and quadratic programming as ways of building portfolios, noting what each does well and badly.

FRM® Part II · Risk Management and Investment Management · Portfolio Construction · FRM2.IM.4.h

The idea

Four construction methods trade simplicity against rigor. Screens rank assets on alpha and hold the top group, usually equal-weighted. They are simple and transparent, but they ignore correlation, the size of alpha and benchmark exposures, so bets can cluster. Stratification splits the universe into cells, such as sector by size, matches the benchmark's cell weights and picks within each cell. It controls unintended exposures but discards information about alpha size. Linear programming maximizes alpha under linear constraints on weights, turnover or exposures, but has no variance penalty for risk. Quadratic programming maximizes alpha less a risk penalty using the full covariance matrix, and it handles costs and constraints. It is the most complete, but it exploits input errors, so inputs must be refined.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which method uses the full covariance matrix to trade alpha against risk directly?

  1. ScreensWhy not: Screens rank on alpha and use no covariance matrix.
  2. StratificationWhy not: Stratification controls exposures by cell weights and does not model covariance.
  3. Linear programmingWhy not: The objective and constraints are linear, so there is no variance term.
  4. Quadratic programming - correct

Quadratic programming has a quadratic objective: alpha less a penalty on portfolio variance, which requires the covariance matrix.

A harder one

What is the main weakness of stratification?

  1. Little alpha information is used, since within a cell only rank matters, and overall risk is not controlled directly. - correct
  2. It cannot control sector or size exposures, because the cells are not linked to any benchmark weights.Why not: Controlling these exposures is the main strength of stratification.
  3. It needs a full covariance matrix, which makes the resulting portfolios unstable and sensitive to errors.Why not: Stratification does not use a covariance matrix.
  4. It requires continuous rebalancing to maintain the cell weights, so that turnover and costs are high.Why not: Frequency of trading is not a feature of stratification as such.

Stratification keeps sector and size exposures near the benchmark by design but discards the magnitude of alpha, and cell choices are arbitrary.

The hardest tier

A quadratic programming portfolio holds large offsetting positions in two highly correlated stocks whose alphas differ by only a small amount. What is the most likely diagnosis?

  1. A software bug, since optimizers never hold large offsetting positions in assets that are highly correlated.Why not: Offsetting positions are a known result of quadratic optimization when correlations are high.
  2. A vertex solution typical of linear programming, which shows the method has been mis-specified for the task.Why not: The vertex behavior belongs to linear programming; this is a quadratic problem.
  3. The optimizer treats the pair as a low-risk spread and levers a small alpha gap, amplifying input errors. - correct
  4. Transaction costs have been set too high, which pushes the optimizer into holding large and offsetting positions.Why not: Higher costs reduce trading, not increase positions.

When two assets are highly correlated, their spread has very low modeled risk, so a small alpha gap justifies a large bet. Refining alphas, constraints and robust risk estimates mitigate this.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Portfolio Construction