Futures Hedge Ratio: How Many Contracts Actually Reduce Risk?

A $10 million stock portfolio does not necessarily need $10 million of short index futures. If its fluctuations are larger than those of the futures contract, the lowest-variance position may require more futures notional than the portfolio is worth. But that same position can become an additional source of risk when the relationship changes.

The useful question is not whether a hedge looks balanced in dollars. It is how much of the portfolio’s movement the chosen contract can offset, over the period that matters, without creating a cash obligation the account cannot fund. A complete answer needs a hedge ratio, a contract count and a failure test.

Start with the exposure, not the margin deposit

Consider an existing long portfolio. Let V be its value at the start of the hedge and let h be short futures notional divided by V. Here a positive h means a short futures position. The portfolio return is rP; rF is the change in the futures quote divided by its starting quote. It is not the return on the margin posted.

rH = rP − h × rF

This is combined profit or loss divided by the initial portfolio value, for fixed holdings and fixed contract numbers over one measurement period. The simple model sets aside trading costs, financing, taxes and interim cash flows; those belong in the implementation assessment, not in a different denominator disguised as the same return.

The variance of that combined position is:

σH² = σP² + h²σF² − 2hρσPσF

Here σP and σF are standard deviations for the same horizon, and ρ is the correlation between the two synchronized return series. Minimizing this expression with respect to h gives:

h* = Cov(rP, rF) / Var(rF) = ρ × σP / σF

The ratio is a fitted sensitivity, closely related to beta and residual risk, not a promise about the next market move. Regress portfolio returns on futures returns, with an intercept; reversing the regression does not generally give the reciprocal hedge. A coefficient estimated against a cash index is not automatically the same as one estimated against the actual futures maturity.

A 48-contract hedge, with every input visible

Use hypothetical monthly inputs: portfolio value $10 million, portfolio volatility 6%, futures volatility 4%, and correlation 0.80. Suppose the futures quote is 5,000 index points and the contract multiplier is $50 per point. These are teaching assumptions, not current prices or a claim about a specific portfolio.

h* = 0.80 × 6% / 4% = 1.20

Contract notional = 5,000 × $50 = $250,000
N = h* × V / contract notional = 1.20 × $10,000,000 / $250,000 = 48

The model therefore calls for 48 short contracts. Equal-dollar matching would use 40. That difference comes from the relationship between the two exposures, not from how much margin the broker collects. The table holds the 6% and 4% volatilities fixed and changes only the correlation and hedge size.

At correlation 0.80, the fitted hedge leaves 36% of the original variance: 1 − 0.80² = 0.36. Variance falls by 64%, while standard deviation falls from 6% to 3.60%, a 40% reduction. None of those figures means an 80% chance of protection, a maximum monthly loss of 3.60%, or elimination of company-specific risk.

The improvement over equal-dollar matching is modest here: 3.69% versus 3.60% monthly volatility. That is important economically. A small model advantage may not justify extra turnover, wider spreads or additional funding demands. “Optimal” means optimal for the stated variance objective and inputs, not superior under every practical constraint.

When the old hedge becomes worse than no hedge

Now change correlation to 0.20 while leaving both volatility assumptions unchanged. Keeping 48 short contracts means h remains 1.20, but modeled monthly volatility rises to 6.89%—above the portfolio’s unhedged 6%. The newly fitted ratio is only 0.20 × 6% / 4% = 0.30, or 12 contracts, leaving volatility of 5.88%.

There is a useful diagnostic behind that reversal. For a fixed positive h, subtract unhedged variance from hedged variance. The hedge lowers variance only when:

ρ > hσF / (2σP)

For h = 1.20 and the assumed volatilities, the threshold is 0.40. At 0.40 there is no variance benefit; below it the old position adds variance. This is a conditional result of this model, not a universal correlation stop or an instruction to trade every time a rolling estimate crosses a line. If the volatility ratio changes, so does the threshold.

Nor does a low estimated correlation necessarily justify a smaller live hedge immediately. A short sample, stale equity prices or mismatched closing times can depress the estimate without changing the underlying economics. First determine whether the relationship has changed or the measurement has deteriorated. With negative covariance, the unconstrained solution can even call for long futures; blindly shorting a superficially similar contract is then the wrong sign.

Build a ratio that can survive outside the estimation window

Use the actual portfolio, valuation currency and hedge horizon. Align timestamps and adjust the portfolio series for external deposits, withdrawals and corporate actions. A daily estimate is not automatically valid for a one-month holding period. Longer horizons introduce serial dependence, changing holdings and different basis behavior.

The futures series must represent the contract and roll policy that could actually be traded. Do not feed an artificial roll jump from an unadjusted continuous chart into the covariance calculation as though it were a one-day holding return. Record maturity changes and realized roll costs separately. For commodity hedges estimated from price changes rather than percentage returns, the coefficient has a different unit: use physical exposure and contract units, not this article’s market-value scaling.

Fit the ratio in a training window, freeze it, and compare the resulting residual profit-and-loss variation in the next window with no hedge and equal-dollar matching. Inspect several economically defensible windows without simply selecting the best hindsight result. Sector concentration, acquisitions, currency exposure or a changed contract basis can invalidate an otherwise neat historical regression.

If the calculation produces a fractional number of contracts, evaluate the two neighboring integers with their actual h values. Include bid–ask spreads, commissions, roll costs, funding and any position constraints. Variance reduction and expected return are different outputs; a futures short gives up some upside as well as cushioning correlated declines. This statistical calculation also does not establish eligibility for hedge accounting.

A profitable combined position can still need cash

Keep the 48-contract short and suppose the futures quote rises from 5,000 to 5,400. Cumulative futures settlement losses are 48 × 400 × $50 = $960,000. In a separate illustrative scenario, the stock portfolio rises 12%, producing a $1.2 million unrealized gain. Combined profit is $240,000 before costs, yet the futures account has still had to fund the settlement losses.

The unrealized equity gain is not automatically available as settlement cash. The $960,000 excludes initial margin, possible increases in required collateral and financing costs; an adverse move followed by a reversal can also create a larger interim cash need than the final net loss suggests. A low terminal variance is therefore not a liquidity plan.

Before implementation, a defensible review should produce five things:

1. An exposure statement: what is owned, in which currency, and until when.
2. A matched data set and documented estimation window, with the sign and units of h explicit.
3. A contract count and the residual risk after integer rounding, compared with simpler alternatives.
4. A stress result for weaker correlation, different volatilities and a portfolio-specific shock.
5. A cash schedule covering settlement losses, collateral changes and the feasible way to close or resize the position.

The hedge is ready for a decision only when all five are consistent. A precise contract count cannot compensate for an unsuitable benchmark or an unfunded margin call.