Minimum variance hedge ratio for currency hedging

Question

The textbook formula for minimum variance hedge ratio (MVHR) is correl (Y,X) * (STDEV Y / STDEV X)

However, I would like to reconcile the textbook formula with the following website https://research-center.amundi.com/article/currency-hedging-policy-institutions#section-title-9076 which adds a 1 + term to derive minimum variance hedge ratio i.e.

MVHR = 1 + correl (underlying assets, FX forward) * [STDEV (underlying assets) / STDEV (FX forward) ]

Would anyone be able to help me with reconciling this?

Answer 1

The simplest case of FX hedging is when there is no correlation between the assets and the FX rate.

Example: a European investor buys 1 million USD in US Treasury bonds. How to hedge the EURUSD risk? A quick and approximate answer is the investor should short 1 Million USD in the forward market, typically at a 1 month or 3 month horizon. In this case (on the assumption of no correlation) the hedge ratio is MVHR = 1.

Bond prices (like exchange rates) are interest rate sensitive, so the assumption of no-correlation seems questionable in the case of bonds (unlike say wheat or soybeans). To take into account the correlation you may want to hedge more (or less) than this and you end up with MVHR = 1 + correl (stdev1 / stdev2) that you mention.

(If you want you can look at it this way: the USTR bonds can be modeled as 1 million of USD currency plus a EUR derivative instrument based on US bond yields, the USD needs to be hedged for sure on a 1:1 basis, the derivative should be hedged to the extent it is correlated to exchange rates, according to the standard textbook formula).