KyperXCapital
  1. Home
  2. Tools
  3. Currency hedging

Tools · Hedged vs unhedged

A hedged return is not the local return.

Two things are almost always dropped when a foreign return is explained. The local return and the currency move compound rather than add, so there is a cross term. And a currency hedge does not return you to the local figure — it earns the interest rate differential between the two currencies, which can be a gain or a cost. Paste a local return series and a currency series and see both effects separated out.

Calculator

This page is showing an example. Over 12 months the local asset returned +7.8% and the currency moved +3.7%. Adding those gives +11.52%; the actual unhedged return is +11.8%, and the −0.46% difference is the cross term. Hedged, at an assumed 1.5% a year differential, the same asset returned +9.4%. Enable JavaScript to paste your own; the table, chart and method on this page do not need it.

Positive means the foreign currency rose against yours, which adds to an unhedged return. If you have the exchange rate the other way round, the sign is inverted.

Your own short rate minus the foreign short rate. Positive when your currency pays more, which makes hedging a source of return rather than a cost.

Local return
+7.8%
compounded, own currency
Currency
+3.7%
compounded
Cross term
−0.46%
what adding the two misses
Unhedged
+11.8%
compounded together
Hedged
+9.4%
local plus the differential
Volatility, unhedged
4.3%
annualised
Volatility, hedged
8.6%
annualised
Volatility, currency
7.0%
the currency alone
Growth of 100, hedged and unhedged12 months
Growth of 100 hedged and unhedged over the same 12 months, ending at 109.4 hedged and 111.8 unhedged.100105110115024681012Growth of 100Months
Growth of 100 hedged and unhedged over the same 12 months, ending at 109.4 hedged and 111.8 unhedged.1001051101150612Growth of 100Months

Solid is hedged, dashed unhedged. Which ends higher is a fact about the period, not a property of hedging: the currency either moved in your favour or it did not. What hedging changes more reliably is the volatility — and not always downwards. In this example the unhedged series is the calmer of the two, at 4.3% against 8.6%, because the currency tended to move against the asset. For an Australian investor holding global assets that is the ordinary case rather than the exception, and it is the reason removing the currency can add risk instead of removing it.

Decomposition of the example unhedged return into the local asset return, the currency movement and the cross term
ComponentExampleWhat it is
Local asset return, compounded+7.8%What the holding did in its own currency.
Currency movement, compounded+3.7%The foreign currency against the Australian dollar over the same months.
Cross term−0.46%The product of the two, summed. Dropped by every additive decomposition.
Unhedged return, compounded+11.8%The three above, compounded together rather than added.
Hedged return, compounded+9.4%Local return plus the 1.5% a year interest differential, compounded. Not the local return.
Volatility, local8.6%Annualised from the monthly series.
Volatility, unhedged4.3%Higher or lower than local depending on the correlation between the asset and the currency.
Volatility, hedged8.6%Close to the local figure: the hedge removes the currency’s variance, not the asset’s.
Volatility of the currency alone7.0%Annualised standard deviation of the monthly currency movement.

Method

The decomposition.

Two identities, both exact, and the approximation that gets used instead of them.

The unhedged return

ru = (1 + rl)(1 + rc) − 1

Which expands to rl + rc + rlrc. The third term is the cross term and it is what makes an additive decomposition wrong. It is small in any one month and it compounds: over the example 12 months, adding the local return and the currency movement gives +11.52% against an actual +11.8%.

The hedged return

rh = (1 + rl)(1 + d ÷ n) − 1

Where d is the annual interest differential and n the periods a year. A forward contract prices at the interest differential between the two currencies — covered interest parity — so a hedge does not remove the currency’s effect and leave the local return. It swaps the currency’s movement for the differential. The differential is known in advance; the movement is not.

Why the cross term is always dropped

Because “asset return plus currency return” is close enough for a single month and reads better in a commentary. It goes wrong in two situations that matter: over long periods, where it compounds, and in the months where both moves are large, which are exactly the months anyone is trying to explain.

What hedging does to volatility

The hedged series has close to the local asset’s own volatility, 8.6% in the example against 8.6% hedged. The unhedged series has 4.3%, which is not the sum of the asset’s and the currency’s 7.0% — it depends on the correlation between them. Where a currency tends to fall as the asset falls, the unhedged position is more volatile than either component; where it tends to rise, the currency is a partial hedge on its own and removing it can increase volatility.

What is not modelled

The cost of running the hedge itself: dealing spreads, the operational cost of rolling contracts, and the cash flow of settling them. Nor is the mismatch that arises because the hedge is placed on a notional amount that the asset’s own movement immediately makes wrong, which in practice requires rebalancing and produces a tracking difference against the figures here.

Annualisation

Returns are compounded. The differential is divided by the number of periods a year rather than compounded down, which is the market convention for a short-dated forward and differs from the exact figure by a negligible amount at these rates. Volatilities are annualised by √12 from the monthly series.

Limits

What this does not settle.

  1. It describes one period that has already happened. Whether hedging helped over that period is a fact about how the currency moved. It is not evidence about the next period, and a decomposition of the past is not an argument for a policy.
  2. The differential is treated as constant. In reality it moves with both central banks, and a hedge rolled monthly earns whatever the differential is at each roll. Over a period in which rate differentials reversed, a single assumed figure will be materially wrong.
  3. A real hedge is never exact. Hedge ratios drift as the asset moves, contracts are rolled on dates that do not match the asset’s cash flows, and the cash settlement of a hedge has to be funded. All of this produces a difference from the figures here, and it is larger in volatile periods.
  4. The volatility comparison depends on the correlation between the asset and the currency. That correlation is unstable, and it has historically changed sign around stress. A hedge sized on a calm-period relationship is not sized for the period it is wanted in.
  5. Nothing here is a tax calculation. Currency gains and hedge settlements are treated differently from asset returns in many circumstances, and that treatment can dominate the arithmetic above.
  6. The two series have to line up. Same months, same order, and the currency quoted as the foreign currency against your own. An inverted quote gives an answer with the currency contribution exactly the wrong way round.
General information

This calculator is provided for general information only and is directed to wholesale and professional investors. It is not personal advice: it does not take into account the objectives, financial situation or needs of any person, and it is not an offer, invitation or recommendation to acquire any financial product. It is not tax advice. Investing involves risk, including the possible loss of capital. Past performance is not a reliable indicator of future performance.

Definitions

Terms on this page.

Each links to the glossary entry, which states the convention the term assumes as well as what it means.

  • Hedged return — local return plus the hedge differential, not the local return.
  • Unhedged return — local return and currency movement compounded together.
  • Volatility — annualised standard deviation of the monthly series.
  • Correlation — what decides whether hedging raises or lowers volatility.

Questions

Common questions.

Should I hedge my international holdings?

That is a portfolio decision that depends on what the holdings are for, what else is in the portfolio, and what liabilities it has to meet — none of which this page knows. What this page can settle is the arithmetic: hedging exchanges an unknown currency movement for a known interest differential, it moves volatility much more reliably than it moves return, and it is not free of operational cost. This is general information and not personal advice.

Is a hedged return the same as the local return?

No, and this is the most common misunderstanding. A hedge is priced at the interest rate differential between the two currencies, so the hedged return is the local return plus that differential. Where your own currency pays a higher short rate, hedging adds return; where it pays less, hedging costs return. Over the example period at 1.5% a year the hedged figure is +9.4% against a local +7.8%.

Why does adding the asset return and the currency return give the wrong answer?

Because they compound rather than add. The exact identity is (1 + local)(1 + currency) − 1, which expands to local + currency + local × currency. The third term is small monthly and accumulates: +11.52% added against +11.8% actual over the example 12 months, a −0.46% difference.

Does hedging always reduce volatility?

Usually but not always. It removes the currency’s own variance, which is 7.0% in the example, but the effect on the total depends on the correlation between the currency and the asset. Where a currency has tended to rise when the asset falls, it has been acting as a partial hedge already, and removing it can leave the position more volatile rather than less.

What differential should I use?

Approximately your own currency’s short-term rate minus the foreign currency’s, over the period being measured. For a precise historical figure, use the actual forward points paid at each roll rather than a single assumed rate, because the differential moves and a constant assumption misstates any period in which it changed.

Is anything I paste sent to KyperX Capital?

No. Both series are parsed and every figure computed in your browser. Nothing is transmitted, recorded or stored, and this website receives none of it.