Working Out A Hedge Ratio In Practice

You're long $10m of a stock and want to hedge out its market exposure with an index future. How many dollars of the future do you sell? The textbook answer is the position's beta to the index, estimated by regressing the stock's returns on the index's returns. In practice, that number moves depending on choices the textbook doesn't make for you — how much history you use, what frequency you sample at, and whether you override the regression when it clashes with what you already know.

The regression, and its knobs

$$\beta = \frac{\text{Cov}(r_{\text{stock}}, r_{\text{index}})}{\text{Var}(r_{\text{index}})}$$

In words: beta is how much the stock tends to move for each unit the index moves, measured from how the two have co-varied historically. The hedge notional is then $\beta \times$ position size.

The knobs that change the answer: lookback window (60 trading days versus 250 trading days can give noticeably different betas, especially if the stock's relationship to the market has shifted); sampling frequency (daily versus weekly returns can differ if there's staleness or thin trading distorting daily correlation); and outlier handling (a single earnings-day gap can drag a short-window beta estimate around by a lot).

Worked example

You regress the stock's daily returns against the index over the trailing 60 days and get $\beta = 1.35$. Over the trailing 250 days, the same regression gives $\beta = 1.15$. Which do you use?

Check what's driving the difference: the 60-day window includes a stretch where the stock had two large earnings-related gaps that happened to move in the same direction as a market swing — those few days are doing a lot of work in a small sample. Excluding just those two days from the 60-day window brings the beta down to 1.18, much closer to the longer-window estimate. That tells you the 250-day beta of 1.15 is the more stable, trustworthy number here, and the short-window spike was mostly noise from two idiosyncratic days, not a genuine shift in the stock's market sensitivity.

Using $\beta = 1.15$ on the $10m position: hedge notional $= 1.15 \times 10{,}000{,}000 = 11{,}500{,}000$, so you sell $11.5m of index exposure, not $10m and not the $13.5m the noisier short-window estimate would have suggested.

What to actually do

Run the regression on more than one window and frequency before trusting a single number, and look at what's driving any disagreement between them — a few outlier days versus a genuine, gradual shift in the relationship are different problems with different fixes. If the estimates are stable across windows, use the longer one for a stable hedge that won't need constant adjustment. If they disagree because of a real regime shift (a merger, a change in the company's business mix), the shorter window is probably closer to the current truth and the hedge ratio needs to be revisited more often going forward, not just this once.