Minimum-Variance Currency Hedge Ratios

You own a German stock worth €1,000,000 and you're a dollar investor. The obvious move is to sell €1,000,000 forward against dollars, cancelling the currency exposure exactly. But that's only the right hedge if the stock's euro value never moves when the euro itself moves. In reality, a euro-area stock often rises when the euro weakens, because a cheaper currency helps that company's exporters. Hedge the full €1,000,000 and you've over-corrected — removed currency risk but introduced a new risk, one that's negatively correlated with your unhedged exposure and would have partly cancelled it out.

Think of it like insulating a house: you don't seal every gap to zero, because some gaps let out heat that would otherwise build up dangerously behind a wall. The right hedge ratio is not "1," it's whatever ratio empirically minimises the swings in your total portfolio.

Building the ratio

Let $R_A$ be the return on your foreign asset measured in the foreign currency, and $R_{FX}$ the return on that currency versus your home currency. Your total dollar return, roughly, is the sum of the asset's local return plus the currency's move (ignoring cross terms, which are small). If you hedge a fraction $h$ of the currency exposure with a forward, your hedged dollar return becomes:

$$R_{\text{hedged}} = R_A + (1-h)\,R_{FX}.$$

In words: the unhedged part of the currency move, $(1-h)$ of it, still passes through into your return; the hedge cancels the rest. The minimum-variance hedge ratio is the $h$ that minimises the variance of $R_{\text{hedged}}$, and calculus turns this into a familiar regression coefficient:

$$h^{*} = \frac{\text{Cov}(R_A, R_{FX})}{\text{Var}(R_{FX})}.$$

In words: run a regression of the asset's local-currency return on the currency's return; the slope of that regression is the hedge ratio that minimises total portfolio variance. If the covariance is zero, $h^* = 0$ — don't hedge at all, since currency moves don't correlate with the asset. If the asset value falls exactly as much as the currency falls (dollar terms are currency-driven with no offset), $h^*$ lands near 1, the naive full hedge.

Two worked examples

Example 1 — a case for full hedging. A US investor holds a UK government bond. Historically, monthly local-currency bond returns and sterling's return against the dollar have shown covariance of 0.00018 and sterling's return variance is 0.00020. Then $h^* = 0.00018 / 0.00020 = 0.90$. Hedge 90% of the currency notional — close to full hedging, sensible for a bond, since bond prices are largely insensitive to the currency's own strength, so there's little natural offset to preserve.

Example 2 — a case for partial hedging. A US investor holds Japanese exporter equities. Local-currency equity returns have historically had covariance of $-0.00006$ with yen returns (yen weakness helps exporters, so local stock returns and yen strength move opposite ways) and yen return variance is 0.00015. Then $h^* = -0.00006 / 0.00015 = -0.40$. A negative hedge ratio means the minimum-variance position isn't to sell yen forward at all — it's to buy yen forward, adding to currency exposure rather than cancelling it, because the natural negative correlation between the stock and the yen already does some of the hedging work, and adding a normal short-yen hedge would fight that natural offset rather than help it.

exporter equity: negative slope h* ≈ −0.40

Where this matters

Global equity and bond desks re-estimate hedge ratios periodically rather than trusting a single historical number forever, because the local-return/currency-return relationship drifts with the business cycle and with which sectors dominate a market's index. A hedge ratio estimated on ten-year-old data can be actively harmful if the underlying correlation has flipped sign since.