Hedging with Stock-Index Futures

A fund manager holding a diversified equity portfolio doesn't want to sell any of the individual stocks — maybe there are tax reasons, maybe the picks are good long-term bets and she just wants to sit out a rough few months, maybe the trading desk is closed for a holiday. What she wants is to temporarily dial the portfolio's exposure to the overall market up or down, without touching a single underlying holding. Stock-index futures do exactly that, and the sizing formula is a direct corollary of the minimum-variance hedge ratio from last lesson.

From h^{*} to beta

A stock's (or a portfolio's) beta, \beta, measures how much its return moves for a given move in the overall market — it's the slope of a regression of portfolio returns on market returns. That's precisely \rho\,\sigma_S/\sigma_F with the market index playing the role of the futures price F. So the minimum-variance hedge ratio for an equity portfolio, hedged with index futures, is its beta — no new theory required.

N^{*} = \beta\,\frac{P}{A},

where P is the current value of the portfolio and A is the value of one index futures contract — the futures index level times the contract's dollar multiplier. Going short N^{*} contracts drives the portfolio's effective beta toward zero: it neutralizes market risk while leaving stock-specific ("idiosyncratic") risk untouched, since that part was never correlated with the index to begin with.

Worked example: dialing risk down before an earnings season

A portfolio manager holds equities worth P = \$5{,}050{,}000 with a portfolio beta of \beta = 1.5 — an aggressive, high-beta book. With a volatile earnings season approaching, she wants to cut that beta to \beta^{*} = 0.75 for six weeks without selling any stock. The relevant index future is at 1,010, with a $250 multiplier, so one contract is worth

A = 1{,}010 \times \$250 = \$252{,}500.

Step 1 — contracts needed.

N = (\beta^{*} - \beta)\,\frac{P}{A} = (0.75 - 1.5)\times\frac{5{,}050{,}000}{252{,}500} = -0.75 \times 20 = -15.

A negative 15 means sell (short) 15 index futures contracts. If the market subsequently drops 4% during earnings season, the equity portfolio (beta 1.5) would have lost about 1.5 \times 4\% \times 5{,}050{,}000 \approx \$303{,}000 unhedged; with the 15-contract short in place, the futures gain roughly offsets 45% of that exposure, leaving a net loss close to what a beta-0.75 portfolio would have suffered on its own — 0.75 \times 4\% \times 5{,}050{,}000 \approx \$151{,}500.

Six weeks later, once the results are out and she wants full exposure back, she simply buys back the 15 contracts — no stock ever traded hands.

Seeing it: hedging flattens the slope

Plot the portfolio's return against the market's return and beta is just the slope of the line. An unhedged high-beta portfolio has a steep line — it amplifies market moves. Overlay the hedged line at the target beta and you can see exactly what the futures position buys: the same portfolio, on a shallower line, moving less for every percent the market moves. Setting the target to 0 flattens the line completely — a fully market-neutral book.

You could — selling enough stock to cut beta-weighted exposure by the same amount achieves a similar market-risk reduction. But futures have three practical advantages that make them the default tool on most trading desks: they're fast (one phone call or a few clicks moves the whole hedge, versus unwinding dozens or hundreds of individual positions); they're cheap (a fraction of a percent in transaction costs, versus commissions and market-impact costs on every line item, not to mention the bid-ask spread on illiquid names); and they're reversible without re-establishing cost basis or triggering tax events on the underlying stock, which selling and rebuying does. A futures overlay is the portfolio-management equivalent of a dimmer switch on the market-risk exposure the manager already has, rather than physically rewiring the lamp each time.

The beta plugged into N^{*} = \beta P/A is almost always an estimated beta from historical returns — typically a regression over the last one to five years of monthly or weekly data. That estimate carries real uncertainty (a standard error, just like any other regression coefficient), and betas genuinely drift over time as a company's business mix, leverage, or the composition of a portfolio changes. A hedge sized off a stale or noisy beta estimate will systematically over- or under-hedge — the market risk doesn't fully go away just because the arithmetic says N^{*} contracts were traded. Desks that run this hedge continuously re-estimate beta periodically and rebalance the futures position, exactly as the next lesson describes for rolling a hedge forward in time.