Risk & Portfolio

A U.S. investor buys 100 euros of European stocks at 1.10. The stocks gain 8% in euros, but the euro falls 10% to 0.99. Unhedged the return in dollars is −2.8%. Selling 100 euros forward at the start — at 1.1214, set by U.S. and euro rates of 5% and 3% — turns it into +9.1%: the stock return plus the rate difference, minus a small mismatch on the 8 euros of gain left unhedged.

• The hedge has a cost or a gain set by the interest-rate difference; hedging a high-rate currency back to a low-rate one costs every year.
• Over long horizons currency moves partly cancel, and some currencies (the dollar, the yen, the Swiss franc) tend to rise in crises, so leaving them unhedged can reduce risk (Campbell, Serfaty-de Medeiros & Viceira 2010).
• Hedging a fixed amount leaves the gains or losses on top of it exposed; hedges need rolling and resizing.

eur, s0, s1 = 100, 1.10, 0.99                             # 100 EUR of stocks; EUR/USD falls 10%
local = 0.08                                               # the stocks return 8% in euros
fwd = s0 * 1.05 / 1.03                                     # 1-year forward from USD and EUR rates of 5% and 3%
cost = eur * s0
unhedged = eur * (1 + local) * s1 / cost - 1
hedged = (eur * (1 + local) * s1 + eur * (fwd - s1)) / cost - 1   # sell the starting 100 EUR forward

Assumes import numpy as np and import pandas as pd. Each snippet carries its own example numbers; the comments say which are made up or simulated.