Ledoit-Wolf Shrinkage

Problem

In high-dimensional portfolio management, the sample covariance matrix is noisy and can be singular. Ledoit-Wolf shrinkage stabilizes it by blending with a structured target:

$\hat{\Sigma}_{\text{shrunk}} = \alpha F + (1 - \alpha) \hat{\Sigma}$

Given a 2x2 sample covariance matrix $\hat{\Sigma}$, a shrinkage target $F$, and the optimal shrinkage intensity $\alpha$, compute the shrunk covariance matrix.

Input Format

Three lines:

• Line 1: four floats s11 s12 s21 s22 (sample covariance, row-major)
• Line 2: four floats f11 f12 f21 f22 (shrinkage target, row-major)
• Line 3: one float alpha (shrinkage intensity)

Output Format

Three space-separated values: c11 c12 c22 (the 3 unique elements of the symmetric shrunk matrix), each to 4 decimal places.