Why Currency Hedge Ratios Can Include a One-for-One Base

Summary

The document reconciles two minimum-variance hedge ratio expressions: the textbook covariance-based ratio and a currency-hedging version that adds one. It explains the difference through an example of a European investor holding US Treasury bonds. The investor has direct USD exposure, which motivates a one-for-one short forward hedge when asset returns and exchange-rate changes are uncorrelated.

The added adjustment reflects the relationship between bond prices and exchange rates. Because both can respond to interest rates, the bond’s return may be correlated with currency movements. The proposed decomposition treats the foreign bond exposure as a USD currency position plus a bond-yield-related component; the currency is hedged one-for-one, while the correlated component is adjusted using the textbook covariance approach. This is an intuitive reconciliation, not a full derivation. The hedge ratio depends on the chosen return definitions, instruments, and assumptions, and the example does not establish that the added-one form applies universally to all currency hedges.

Key ideas

• A direct foreign-currency exposure can motivate a one-for-one baseline forward hedge.
• The textbook minimum-variance ratio adjusts hedge size according to correlation and relative volatility.
• The added-one expression can be interpreted as a base currency hedge plus an adjustment for correlated asset returns.
• Foreign bond prices and exchange rates may share interest-rate sensitivities, making a zero-correlation assumption questionable.
• The explanation is an intuition and does not establish a universal formula for every hedge setup.

Tags

# Minimum variance hedge ratio for currency hedging


# Minimum variance hedge ratio for currency hedging












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 by nbbo2 (score 1)

https://quant.stackexchange.com/a/79089

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).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.