# Non-US investors and ETF currencies

Among the considerations for non-US investors investing through ETFs is the effect of currency risk on returns. The issue can be particularly difficult to untangle where an ETF might have different denomination and trading currencies, perhaps neither of which is the same as either the investor's home currency or the currency of the assets held by the ETF.

This page demonstrates why it is that for investors in these ETFs, the only currency exchange rate that impacts long term returns is the one between the investor's home currency and the currency of the ETF's assets. During the holding period, the investor owns only assets valued in the currency of the assets. Other currencies besides the asset's and investor's own are irrelevant.

## Introduction

Non-US investors are frequently faced with a choice of investing through ETFs that have a bewildering set of currencies associated with them.

An ETF will typically have a denomination currency, and then one or more trading currencies. The denomination currency is the currency in which the ETF's performance is calculated, and is sometimes also the currency in which dividends are paid. The trading currencies are those with which the investor can buy and sell ETF shares, and usually align with the currency or currencies associated with the exchange(s) on which the ETF trades.

Alongside this, the investor will themselves have a home currency which may differ from the ETF's denomination and trading currencies, and the ETF will hold assets in a country which uses yet another different currency.

All of this means that there may be as many as four different currencies involved in a non-US investor's use of an ETF. The question then becomes, how much additional currency risk appears when all of these different currencies combine in this way?

## Conceptual currency flows in ETF trades

The following diagram shows how funds might flow between currencies in an ETF purchase, and then later on in a sale.

In this example the ETF's denomination currency is the US Dollar (USD), and its trading currency is UK Pound Sterling (GBP). It tracks the Nikkei index and so holds Japanese stocks whose pricing currency is the Japanese Yen (JPY). We can consider two different potential investors, one whose home currency is GBP, the other from Europe and whose currency is the Euro (EUR).

While this diagram conceptually shows the ETF provider exchanging currencies and buying or selling Japanese stocks to create or destroy ETF shares, in practice they may not -- and in fact, probably will not -- do all of these things.

## Short explanation of currency effects

The short explanation of why all the intermediate currencies have no effect whatsoever on returns to the investor who buys an ETF holding Japanese stocks which are priced in JPY is that on purchase, the investor exchanges some of their local currency for an equivalent value of stocks priced in JPY, and on sale the reverse happens. In between purchase and sale, the investor holds only JPY assets, so other currencies besides the investor's own are irrelevant.

However, some people find this explanation insufficient or unconvincing. If this is you, read on. Even finance writers and investment professionals regularly get confused over this, so it is nothing to be embarrassed about.

## Longer explanation of currency effects

There are two key realisations to understanding why the intermediate currencies in this ETF purchase and sale have no effect on the long term return to the investor:

 Currency markets ensure that the result of exchanging currency A into currency B and then immediately into currency C is the exact same as directly exchanging currency A into currency C. If this were not the case, a trader could make money by simply cycling cash around the three currencies. If such an opportunity does arise, markets will immediately arbitrage it away. During the ETF holding period, the investor owns only assets valued in the currency of the assets that the ETF holds.

### Sterling investor and Nikkei tracker ETF example

Suppose a GBP investor buys a share in this example ETF. The provider conceptually exchanges GBP to obtain an amount in the ETF's denomination currency of USD, converts this amount to JPY, and then buys Japanese stocks. However, because currency markets are liquid and highly efficient, this GBP to USD to JPY transition produces the exact same result as a direct GBP to JPY exchange. For this investor then, the GBP to USD exchange rate plays no part in the outcome.

Similarly, when this investor sells their share, the provider conceptually exchanges currencies in the reverse direction. They sell Japanese stocks to realise JPY, convert this into a value expressed in the ETF's denomination currency of USD, and then return that to the investor as GBP. The GBP to USD exchange rate plays no part in the outcome here either, for the same reason as when buying.

In between buying and selling, this investor holds only assets valued in JPY, so their investment outcome results from combining the performance of the Nikkei index with the GBP to JPY exchange rate. No other currencies are involved.

### Euro investor and Nikkei tracker ETF example

What about the Euro investor, who has to exchange from EUR to GBP to buy this ETF, and then back on sale?

Here again, the EUR to GBP exchange rate plays no part in the long term return to this investor. The EUR to GBP conversions on buy and sell will produce the same result as if directly between EUR and JPY for the same reasons as with USD above. Currency markets will ensure that on buying, the EUR to GBP conversion followed by GBP to JPY -- and the converse on sale -- will produce the same result as directly converting EUR to JPY.

Currency exchanges have a cost. If the investor has the opportunity to buy an ETF in their currency then they can avoid this cost.

### Dividends paid in different currencies

Finally, what about ETF dividends paid out in a currency other than that of the investor?

Again, the only exchange rate that matters is that between JPY and the investor's currency, GBP or EUR in this example. The stocks in the ETF pay their dividends in JPY, and if the ETF provider chooses (perhaps for their own convenience) to distribute these to investors in USD, then again because of currency market efficiencies the conversion from JPY to USD and then on to GBP or EUR produces the same outcome as directly converting JPY to GBP or EUR.

### Extension to the general case

The example shown uses the currencies GBP, EUR, USD and JPY to illustrate what might happen for one particular ETF, but the currencies selected here are not important, and the result is completely general. An investor might find an ETF whose trading and denomination currencies match their own, or one where only the denomination currency is different from their own, or one where both the trading and the denomination currency are different from their own.

In all three of these cases though, only the exchange rate between the investor's currency and the currency (or currencies) of the assets held by the ETF will impact long term performance. Any other currencies are simply accounting or trading conveniences.

An investor looking at their ETF's performance reported in a currency other than their own -- that is, where the ETF's denomination currency is different from the investor's currency -- must always convert performance or return figures into their own currency before interpreting the results. For example, a sudden gain of 100% in the value as shown in the denomination currency is less likely to mean that the assets have doubled in value than that the denomination currency has devalued by 50% relative to both the asset's currency and the investor's currency (imagine an ETF denominated in Zimbabwean dollars or Venezuelan bolivars!).

## Worked example showing this in practice

To make this more concrete, here are some numbers from dates picked entirely at random (honestly!). Suppose an investor buys shares of this example Nikkei tracker ETF on 15 Jan 2010 and sells on 18 May 2018.

 On 15 Jan 2010, the Nikkei stood at 10,982.10, the GBP to USD rate was 1.623972, the GBP to JPY rate 147.387674, and the USD to JPY rate 90.757521. A GBP investor buying £100 of this ETF would see this £100 exchanged to $162.40 and then into ‎¥14,738.77 worth of Japanese stocks. Notice however that £100 converted directly into JPY on this date is also exactly ¥14,738.77. The transit through USD had no effect. ### Sale example  On 18 May 2018, the Nikkei stood at 22,930.36, up 108.79% from purchase, the USD to GBP rate was 0.742075, the JPY to GBP rate 0.006702, and the JPY to USD rate 0.009031. The investor's original ¥14,738.77 has risen with the Nikkei to ¥30,744.19. This equates to$277.92 in the ETF's denomination currency, and this converts to £206.24.

Again though, notice that ¥30,744.19 converted directly to GBP is also £206.24. So again, the transit through USD had no effect.

### Long term (non-)effects

 During the holding period, the GBP to JPY rate changed from 147.387674 to 149.211265, indicating a currency loss of 1.23%. This investor's return in GBP is the 108.79% Nikkei gain less this small currency effect; that is, 98.77% of £208.79, so £206.24.

During the holding period the USD 'strengthened' so that GBP to USD rate changed from 1.623972 to 1.347573. At 17.01% this is a far larger currency effect than the 1.23% difference in GBP to JPY over the same period. However, this sizeable change in GBP to USD does not show up at all in the investor's long term returns here. It is nowhere in the numbers above.

Finally, to drive the point home, notice that if you multiply together the two example exchange rates that happen when transiting through USD, you get the third to within five decimal places. At purchase, 1.623972 × 90.757521 = 147.38767(2893), and at sale 0.742075 × 0.009031 = 0.00670(1679325).

### Euro investor's outcome

What of the Euro investor who trades this ETF in GBP?

 On 15 Jan 2010 the EUR to GBP rate was 0.885379 and the EUR to JPY rate 130.493893, and on 18 May 2018 the GBP to EUR rate was 1.144340 and the JYP to EUR rate 0.007669. To obtain £100 to buy this ETF, the Euro investor would need to transfer €112.95. On sale, they would receive €236.01, converted from £206.24. Their gain measured in Euros is 108.95%.

Had this same investor been able to convert directly from EUR to JPY their gain would have been the 108.79% Nikkei gain plus the minuscule gain from the EUR to JPY rate changing from 130.493893 to 130.390705. This is the same outcome as going via both GBP and USD.

## Mathematical treatment

### Simple three currency case

Suppose three currencies ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$. Let the exchange rates be such that ${\displaystyle R_{AB}}$ converts from ${\displaystyle A}$ to ${\displaystyle B}$, ${\displaystyle R_{BC}}$ from ${\displaystyle B}$ to ${\displaystyle C}$, and ${\displaystyle R_{AC}}$ from ${\displaystyle A}$ to ${\displaystyle C}$. Assume currency markets are fully liquid and perfectly efficient.

Exchanging an amount ${\displaystyle V_{A}}$ from currency ${\displaystyle A}$ to ${\displaystyle V_{C}}$ in currency ${\displaystyle C}$ can be accomplished in two ways:

• directly as ${\displaystyle V_{C}=R_{AC}\times V_{A}}$, or
• indirectly via currency ${\displaystyle B}$ as ${\displaystyle V_{C}=(R_{AB}\times V_{A})\times R_{BC}}$.

Under the assumption that currency markets are perfectly efficient, these will always produce the same value of ${\displaystyle V_{C}}$. Therefore

${\displaystyle (R_{AB}\times V_{A})\times R_{BC}=R_{AC}\times V_{A}}$, factoring out ${\displaystyle V_{A}}$ produces ${\displaystyle R_{AB}\times R_{BC}=R_{AC}}$.

Currency ${\displaystyle B}$ does not participate in ${\displaystyle R_{AC}}$, so currency ${\displaystyle B}$ and its relationship to any other currencies can be ignored in any situation where ${\displaystyle R_{AB}\times R_{BC}}$ is applied. Generalised, this becomes:

${\displaystyle R_{ik}\times R_{kj}=R_{ij}}$, for all currency triples formed by ${\displaystyle i}$, ${\displaystyle j}$, and ${\displaystyle k}$.

For inverse conversions, because currency markets are perfectly efficient, ${\displaystyle R_{ij}=1/R_{ji}}$ for all currency pairs formed by ${\displaystyle i}$ and ${\displaystyle j}$. Thus

${\displaystyle (1/R_{AB})\times (1/R_{BC})=1/R_{AC}}$, equivalent to ${\displaystyle R_{CB}\times R_{BA}=R_{CA}}$.

An ETF may trade in currency ${\displaystyle A}$, be denominated in currency ${\displaystyle B}$, and hold assets in currency ${\displaystyle C}$. Because of the above three currency pattern, the ETF's denomination currency can be ignored since it plays no part in the ETF's return. Neither the conversion from ${\displaystyle A}$ to ${\displaystyle C}$ on purchase or the conversion from ${\displaystyle C}$ to ${\displaystyle A}$ on sale are affected by ${\displaystyle B}$.

### Extension to more than three currencies

An ETF may hold assets valued in multiple currencies. Currency ${\displaystyle C}$ and exchange rates ${\displaystyle R_{BC}}$ and ${\displaystyle R_{AC}}$ become multiple element sets, and let currency set ${\displaystyle C}$ be the currencies of the assets held by the ETF. For each set element indexed by ${\displaystyle i}$ and forming the currency triple ${\displaystyle \{A,B,C_{i}\}}$, by applying the three currency pattern the relationship ${\displaystyle R_{AB}\times R_{BC_{i}}=R_{AC_{i}}}$ holds, such that currency ${\displaystyle B}$ can be ignored for all ${\displaystyle i}$. And symmetrically, ${\displaystyle R_{C_{i}B}\times R_{BA}=R_{C_{i}A}}$.

An investor may need to convert their currency to the ETF's trading currency, producing a longer chain of exchanges. Suppose currencies ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$ ... ${\displaystyle Z}$ and rates ${\displaystyle R_{AB}}$, ${\displaystyle R_{BC}}$, ... ${\displaystyle R_{YZ}}$, and ${\displaystyle R_{AZ}}$ such that ${\displaystyle R_{AB}}$ converts ${\displaystyle A}$ to ${\displaystyle B}$, ${\displaystyle R_{BC}}$ converts ${\displaystyle B}$ to ${\displaystyle C}$ ..., and ${\displaystyle R_{AZ}}$ converts ${\displaystyle A}$ directly to ${\displaystyle Z}$. And suppose an arbitrary length chain of conversions such that

${\displaystyle V_{Z}=R_{AB}\times R_{BC}\times {\text{...}}\times R_{YZ}\times V_{A}}$.

Repeated application of the three currency pattern reduces the chain of conversions to ${\displaystyle V_{Z}=R_{AZ}\times V_{A}}$, such that currencies ${\displaystyle B}$ to ${\displaystyle Y}$ can be ignored.

## Caveats and details glossed over

All of the above makes the assumption that the ETF is not currency-hedged. If it is, this will significantly affect outcomes, either positively or negatively. In general, investors should probably avoid currency-hedged ETFs where the currency being hedged by the ETF is not the investor's currency, although one possible exception would be where the investor's currency is pegged to the currency hedged by the ETF.

It also ignores any costs or losses involved in actual currency exchanges. For conceptual currency exchanges that happen internally to the ETF, there either should be no actual costs involved, or if there are then the ETF provider will absorb them into the costs of running the ETF.

For real currency exchanges though, where the investor may not hold any of the currencies in which an ETF trades, there is an obvious loss to foreign currency exchange on both purchase and sale. Investors should probably prefer to buy and sell ETFs that trade in the currency they hold, at least as far as is possible. For other cases, it is likely to be best to exchange first into whichever of the available ETF trading currencies has the lowest conversion losses for the currency held.