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Financial Instruments, Markets and Conventions / Forwards, Futures, Swaps, and Contracts for Difference

Derivation — The Minimum-Variance Hedge Ratio: Beta Analogy and OLS Interpretation

The formula h∗=ρ σS/σFh^* = \rho \, \sigma_S / \sigma_F is stated and used in the hedging lesson. This note derives it from first principles: a single step of calculus on the variance of a hedged portfolio. Three connections make the result important beyond the formula itself:
  1. OLS regression. h∗h^* is the slope of the least-squares regression of ΔS\Delta S on ΔF\Delta F, which is the foundation for estimating it empirically.
  2. CAPM beta. For a diversified equity portfolio hedged with index futures, h∗=βh^* = \beta — the hedge ratio is identical to the portfolio's systematic risk coefficient.
  3. Hedge effectiveness. The proportion of variance eliminated is ρ2\rho^2, the coefficient of determination from the regression. This is the most interpretable measure of how well a futures contract hedges the underlying exposure.

In this lesson

  1. Setup
  2. Derivation
  3. The OLS regression interpretation
  4. The beta analogy
  5. Worked example: verifying the formula

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