What a foreign sleeve adds
A US client in European shares makes two bets: the shares and the euro. Hedging removes the second, at a price set by interest rates, not by anyone's view.
- Compound a local return and a currency move into a dollar return
- Compute a hedged return from the interest-rate gap
- Find the currency move at which hedging and not hedging tie
- Say why hedging can raise volatility, and why foreign bonds are the stronger case
The intuition
A US client buys a European equity fund. The fund rises 8% in euros; the euro falls 6% against the dollar. Most of the gain disappears on the way home: not 2%, but 1.52%, because the currency move applies to the gain as well as the original money. The client did not mean to bet on the euro. They did.
Hedging sells euros forward for the year, and the price of doing so is set by interest rates: when dollar rates are higher than euro rates, the forward pays the hedger the gap, and the hedged return is roughly the local return plus that gap. On risk the answer is subtler. The currency adds a moderate amount of volatility to equities and can swamp bonds entirely, which is why most advisors hedge foreign bonds and argue about foreign equities. And when the currency tends to move against the shares, hedging can make the sleeve more volatile, not less.
Unhedged $ return = (1 + local return)(1 + currency move) − 1. Hedged $ return ≈ local return + (US rate − euro rate). They tie when the currency move equals (1 + local + rate gap) ÷ (1 + local) − 1, which is close to the rate gap. Unhedged volatility² = local² + currency² + 2 × correlation × local × currency; hedged ≈ local.
Why it works
- The conventions here: a US-dollar investor holds euro-area equities for one year. The hedge is a one-year forward on the starting value; its return is the local return plus the US-minus-euro rate gap, and the currency's effect on the year's gain or loss is ignored. Unhedged compounds the two returns.
- Compound, do not add. The currency move applies to the whole position at year end, including the gain. The cross term, local return × currency move, is small in a normal year and matters in a big one.
- The hedge's cost is carry, not a forecast. Covered interest parity prices the euro forward below spot when dollar rates are higher, so selling euros forward locks in the rate gap as a gain. When euro rates are higher the same logic makes the hedge cost the gap.
- Not hedging is a bet that the currency beats the forward. The break-even currency move is almost exactly the rate gap: the unhedged sleeve wins only if the euro does better than the forward already priced in.
- Variances add, so a currency layered on equities adds less than its own volatility, and that is all a hedge can take away. With a negative correlation between the shares and the euro, the currency was partly offsetting the equity risk, and hedging removes the offset: volatility rises.
- Bonds are the stronger case. Currency volatility is similar to or larger than high-grade bond volatility, so an unhedged foreign bond is mostly a currency bet.
| Unhedged: 1.08 × 0.94 − 1 | 1.52% (the simple sum says 2.00%) |
| Rate gap: 5% − 2% | 3%, which the hedge earns |
| Hedged: 8% + 3% | 11.00% |
| Break-even euro move: 1.11 ÷ 1.08 − 1 | +2.78% |
| Unhedged volatility: √(18² + 10²) | 20.59%, against 18% hedged |
The euro fell 6% when the break-even needed it to rise 2.78%, so hedging won by nearly nine and a half points this year. The volatility saving is 2.6 points, because variances add.
The formulas
Compound the two; do not add them.
The forward replaces the currency move with the rate gap.
Where hedged and unhedged tie; close to the rate gap.
Variances add, with a correlation term.
Only the shares' own risk remains.
Worked example
The rate gap first: that is what the hedge earns or costs before the currency moves. Add it to the local return, compare with the unhedged return, and read whether hedging helped.
See it move
Same client and the same fund. Change the local return, the euro's move, the two interest rates, the two volatilities and the correlation between the shares and the euro.
- Raise the euro's move. The unhedged return rises; the hedged return does not move, because the hedge sold the euro forward.
- Raise the dollar rate. The hedged return rises; the unhedged return does not move: the rate gap is what the forward pays.
- Raise the correlation between the shares and the euro. Unhedged volatility rises; hedged volatility does not move. Take the correlation below zero and watch the unhedged bar fall toward, or under, the hedged one.
- Slide the euro's move until the curve crosses the hedged line. That is the break-even in the readout, and it sits close to the rate gap.
Run it backwards
Hedged and unhedged are known as formulas; set them equal and solve for the euro move. The follow-up compares the answer with the rate gap itself.
Set (1 + local)(1 + x) − 1 equal to local + rate gap. Then 1 + x = (1 + local + gap) ÷ (1 + local), and x is that less one.
It lands close to the rate gap because the forward has already priced that gap in. Choosing not to hedge is a bet that the currency beats the forward, not a bet that it rises.
Traps
Say it in the interview
“Should this client hedge the currency on their international sleeve?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Unhedged = (1 + local)(1 + currency) − 1; compound, do not add.
- Hedged ≈ local + (US rate − foreign rate); the cost is carry, not a forecast.
- Break-even currency move ≈ the rate gap.
- Variances add; hedging can raise volatility when the correlation is negative. Hedge the bonds first.