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IntermediateOptions, Derivatives & FinancePython

Run this module

cd "Finance - Covariance Estimation"
python "covariance_estimation.py"

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Robust Covariance Estimation

Sample covariance is noisy and often poorly conditioned with many assets. Shrinkage estimators blend sample covariance with a structured target for more stable portfolio optimization.

Functions

Function Description
sample_covariance(returns) Standard MLE covariance
ledoit_wolf_shrinkage(returns) Analytical LW shrinkage toward scaled identity
constant_correlation_shrinkage(returns) Shrink toward equal-correlation matrix
ewma_covariance(returns, lambda_) RiskMetrics exponentially weighted covariance
condition_number(cov) Ratio of max/min eigenvalue (lower = better)

Key Concepts

  • Curse of dimensionality: With N assets and T observations, sample covariance has N(N+1)/2 parameters. When T < N, it's singular.
  • Ledoit-Wolf: Finds optimal alpha blending S toward mu*I. Minimizes expected Frobenius distance to true covariance.
  • Constant Correlation: Preserves sample variances but equalizes all correlations to the cross-sectional mean. Works well for equity portfolios.
  • EWMA: Downweights old data — lambda=0.94 (daily) makes the half-life ~11 days. Captures volatility clustering.

Example

import numpy as np
from covariance_estimation import ledoit_wolf_shrinkage, condition_number

T, N = 252, 50
returns = np.random.randn(T, N) * 0.01
S = np.cov(returns.T)

lw = ledoit_wolf_shrinkage(returns)
print(f"Sample cov condition number: {condition_number(S):.0f}")
print(f"LW shrunk condition number:  {condition_number(lw['shrunk_cov']):.0f}")
print(f"Shrinkage intensity alpha:   {lw['alpha']:.3f}")

Rule of Thumb

  • T > 10N: Sample covariance is fine
  • T ~ 3N: Use Ledoit-Wolf or constant correlation
  • T < 2N: Consider factor-based covariance models

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