Reference 0002

Portfolio Risk as a Quadratic Form

The compressed facts behind the risk term w' Sigma w.

If asset returns are r and weights are w, portfolio return is r_p = w' r.

If Sigma is the covariance matrix of asset returns, then Var(r_p) = w' Sigma w.

Sigma_ij measures how asset i and asset j move together. Diagonal entries are individual variances.

Diversification comes from imperfect comovement. Low or negative covariance can make combined risk lower than a weighted average of standalone risks.

A valid covariance matrix is positive semidefinite, so w' Sigma w >= 0 for every w.

With Sigma positive semidefinite, minimizing w' Sigma w with linear constraints is a convex quadratic program.

Two-asset expansion

Var(r_p) =
  w_A^2 sigma_A^2
+ w_B^2 sigma_B^2
+ 2 w_A w_B rho_AB sigma_A sigma_B

The last term is the diversification term. When correlation rho_AB is low, it contributes less risk; when it is negative, it subtracts risk.

Use this page while reading Lesson 0002.