Module 3 — Statistics for Markets

Correlation and Covariance

Returns, volatility, covariance — and the tail behaviour that breaks naïve models.

Module lessons

Simple vs Log ReturnsVolatility — Measuring UncertaintyCorrelation and Covariance

Learning objectives

  • ▸Compute the covariance matrix.
  • ▸Distinguish correlation, beta, and copulas.
  • ▸Understand why correlations jump to 1 in crises.

FORMULA

Pearson correlation

ρ_{X,Y} = Cov(X, Y) / (σ_X · σ_Y),   ρ ∈ [-1, 1]

TEXT

Correlation is for diversification, beta is for hedging

Two assets with ρ = 0.2 can both have β to SPX of 1.5 — beta is about sensitivity to a benchmark, correlation is about co-movement. Portfolio construction uses the full covariance matrix Σ; risk management often boils that down to beta against systematic factors.

CODE

Covariance matrix in pandas

returns = log_returns_df  # DataFrame of asset returns
Σ = returns.cov() * 252      # annualised
ρ = returns.corr()           # correlation
print(ρ.round(2))

TEXT

Crises break diversification

In normal markets a stock/bond portfolio shows ρ ≈ -0.2. In March 2020 it briefly hit +0.4 as everything sold off together. This 'tail dependence' is exactly why copulas (which model the joint distribution beyond just the linear correlation) are used for credit and crisis modeling.

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