This document discusses currency hedged return calculations. It presents formulas for calculating unhedged, perfectly hedged, real-world hedged, and money market hedged foreign asset returns. The perfect hedge eliminates currency risk by entering a forward contract. Real-world and money market hedges only hedge the initial value, so can be over or under hedged. Numerical examples show the approximations are very close under normal markets but diverge more in turbulent times.

1. Research Note
Andreas Steiner Consulting GmbH
February 2011
Currency Hedged Return
Calculations
Introduction
As of first quarter 2011 it is probably not necessary to elaborate on the importance of
currency risk. Significant movements in major exchange rates like the USD or EUR took
place just recently and have had a major impact on the performance and risk
characteristics of international portfolios. Nevertheless, we see a contrast between the
importance of the currency risk factor in modern investment management and its
treatment in portfolio analytics like performance attribution and risk budgeting. Part of
this can be explained by conceptual complexities: currencies are not just another asset
class, but a risk exposure embedded in any assets and therefore affecting the overall
portfolio in non-trivial ways. This research note addresses one particular aspect of
currency risk analytics, namely the calculation of hedged asset currency returns. Such
calculations are used in “paper portfolios” like benchmarks and generally ex ante
performance and risk analytics.
Unhedged Foreign Asset Returns
In order to prepare a framework for the discussion of currency-hedged asset returns, we
will first analyze unhedged asset returns.
Let us consider an investor with the base currency CHF that invests a certain amount X
in an asset denominated in USD for a certain period of time.
We use the following notation…
X0, CHF Capital in base currency at the beginning of the investment period
X0, USD Capital in asset currency at the beginning of the investment period
X1, CHF Capital in base currency at the end of the investment period
X1, USD Capital in asset currency at the end of the investment period
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2. The situation can be illustrated graphically as follows…
Start of investment period t0 End of investment period t1
Asset Currency USD X0, USD X1, USD
Base Currency CHF X0, CHF X1, CHF
rA, CHF
What we are interested in is to calculate the CHF return of the USD asset rA, CHF. The
calculation is basically a three step procedure…
1. Convert the amount to be invested from base currency to asset currency and
buy the asset.
2. The asset incurs capital gains and losses in its currency over the investment
period.
3. Sell the asset at the end of the investment period and convert the proceeds
back into base currency.
Graphically, the three steps are…
t0 t1
USD X0, USD X1, USD

 
CHF X0, CHF X1, CHF
The formulas involved in the three steps are…
S0 Beginning spot exchange rate of the asset currency relative to
base currency
S1 Ending spot exchange rate of the asset currency relative to base
currency
rA, USD Asset return over investment period in asset currency
© 2011, Andreas Steiner Consulting GmbH. All rights reserved 2/8

3. t0 t1
X1, USD = X0, USD · (1 + rA, USD)
USD X0, USD X1, USD

X0, CHF / S0   X1, USD · S1
CHF X0, CHF X1, CHF
The calculations in the three steps are…
1. X0, USD = X0, CHF / S0
2. X1, USD = X0, USD · (1 + rA, USD)
3. X1, CHF = X1, USD · S1
It follows that rA, CHF can be expressed as…
1 + rA, CHF = X1, CHF / X0, CHF
= X1, USD · S1 / X0, CHF
= X0, USD · (1 + rA, USD) · S1 / X0, CHF
= ( X0, CHF / S0 ) · (1 + rA, USD) · S1 / X0, CHF
= (1 + rA, USD) · S1 / S0
…and finally…
1 + rA, CHF = (1 + rA, USD) · (1 + rS)
…with rS as the spot currency return over the investment period. Note that since S1,
USD/CHF is not known at the beginning of the investment period, currency risk as an
additional source of investment risk enters the equation.
From this exact formula, we can derive a well-known approximation…
1 + rA, CHF = 1 + rA, USD + rS + rA, USD · rS
rA, CHF ≈ rA, USD + rS
From this approximation, we can clearly see the nature of currency risk: the return of a
foreign asset in based currency is nothing other than the return of a leveraged portfolio,
i.e. a portfolio invested 100% in the foreign asset and 100% in the asset’s currency.
Total risk exposure is 200%; currency risk “doubles” the risk exposure.
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4. Note that the approximation works if rA, USD · rS is “small”, which is the case when both
currency spot and asset return are small. This is the case in “normal markets” only, not
in turbulent times. As many calculations in spreadsheets and commercial performance
and risk systems are based on this approximation, we end up with the paradoxical
situation that approximate analytics fail when we need them the most.
The notation and illustration developed above can now be used to derive the formulas
for hedged returns. We will analyze three different types of hedge implementations. In all
of them, we assume that the goal is to “fully hedge” currency risk. The results can be
easily extended to partially hedged assets, with is simply a portfolio consisting of the
unhedged and fully hedged assets, with the weight of the fully hedged being the hedge
ratio.
Case I: The Perfect Hedge
The “perfect hedge” is a situation which we define as follows: We eliminate uncertainty
about the future spot rate (and therefore currency risk) by entering into a currency
forward contract at the beginning of the investment period. A currency forward contract
is an agreement to exchange a certain amount in a certain currency into a certain
different currency at a certain exchange rate. This contractually agreed exchange rate is
the “forward rate” F1.
The perfect hedge can be illustrated as follows…
t0 t1
X1, USD = X0, USD · (1 + rA, USD)
USD X0, USD X1, USD

X0, CHF / S0, USD/CHF   X1, USD · F1
CHF X0, CHF X1, CHF
The only change to the unhedged situation is that F1 replaces S1. The formula for the
perfectly hedged foreign asset in base currency RA, CHF is…
1 + RA, CHF = ( X0, CHF / S0 ) · (1 + rA, USD) · F1 / X0, CHF
= (1 + rA, USD) · F1 / S0
The expression F1 / S0 - 1 is called the “forward premium” or “forward discount” rF,
depending on whether the forward rate is above or below the current spot rate…
1 + RA, CHF = (1 + rA, USD) · (1 + rF)
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5. As before, the above expression can be approximated…
RA, CHF ≈ rA, USD + rF
Note that in the above calculations, we assume that we can exchange the amount X1, USD
at the rate F1 agreed in t0. This is unrealistic: X1, USD is only known in t1 due to uncertainty
about the asset’s gains and losses during the investment period. The amount to be
exchanged in forward rate agreements, on the other hand, has to be specified in t0
already. The “perfect hedge” therefore implies “perfect foresight” regarding the future
asset value.
The forward exchange rate cannot be set at arbitrary values, as this would create
arbitrage opportunities. The forward rate is determined by what is known as the
“covered interest rate parity”, which states that the forward rate must equal the spot rate
multiplied by the relative ratio of foreign and domestic riskfree rates rCHF and rUSD…
F1 = S0 · (1 + rA, CHF) / (1 + rA, USD)
Therefore, the forward premium is…
rF = F1 / S0 = (1 + rCHF) / (1 + rUSD)
The exact formula for the return of the perfectly hedged foreign asset is…
1 + RA, CHF = (1 + rA, USD) · (1 + rCHF) / (1 + rUSD)
And the approximation formula becomes…
RA, CHF ≈ rA, USD + rCHF – rUSD
This can be read as follows: the return of the perfectly hedged foreign asset equals its
return in local currency plus the difference in riskfree rates between the base currency
and asset currency.
Note that currency hedging completely removes the uncertainty regarding the future
spot exchange rate (it can be shown that the volatility of the perfectly hedged foreign
asset in base currency equals its volatility in asset currency). The perfect hedge does
not result in the investor earning the local return of foreign assets; he earns the local
return plus an interest rate differential. Generally speaking, the local return of foreign
assets is not an investable asset; investable are only hedged, partially hedged or
unhedged returns. The contribution of a foreign asset in an international portfolio cannot
be altered without changing the contribution of the interest rate differential. These are
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6. the reasons why performance attribution models based on the Karnovsky/Singer
approach have not become popular among practitioners: attribution effects need to be
independent and tied to investable assets (we will elaborate on this point in a future
research note).
Case II: Real-World Hedging
As we have discussed, “perfect hedging” is only feasible if the ending market value of
risk assets is known. This is generally not the case in real-world portfolios. Real-world
hedging is typically performed by hedging the beginning market value. Depending on
whether the asset loses or gains in value, the asset will be “over-“ or “under-hedged”.
Graphically, this case can be illustrated as follows…
t0 t1
X1, USD = X0, USD · (1 + rA, USD)
USD X0, USD X1, USD

X0, CHF / S0, USD/CHF   X0, USD · F1 + X0, USD ·rA, USD · S1
CHF X0, CHF X1, CHF
The exact and approximate formulas are derived as follows…
X1,CHF = X0,CHF · (1 + rCHF) / (1 + rUSD) + ( X0,CHF / S0 ) · rA,USD · S1
1 + RA, CHF = (1 + rCHF) / (1 + rUSD) + (1 + rS) · rA, USD
RA, CHF ≈ rA, USD · (1 + rS) + rCHF – rUSD
We can now “approximate the approximation”…
RA, CHF ≈ rA, USD + rCHF – rUSD
We can see that in a second order approximation, the “real-world” result is equal to the
“perfect hedge”. Note that this result is only valid in the case of small interest rate
differentials and small currency and local asset returns. We can expect the second order
approximation to be less accurate that the “perfect hedge” approximation.
© 2011, Andreas Steiner Consulting GmbH. All rights reserved 6/8

7. Case III: Realistic Money Market Hedging
The uncovered interest rate parity also tells us that a forward contract can be interpreted
as a derivative instrument. Its total return can be statically replicated with a long position
in the domestic riskfree asset and a short position in the foreign riskfree asset. As
riskfree rates are hypothetical constructs that do not exist on real-world financial
markets, one can use money market instruments as proxies.
Such a strategy would involve buying the foreign asset plus a long position in a domestic
money market instrument and a short position in a money market instrument in the asset
currency. We can illustrate this situation as follows…
X1, USD =
X0, USD · (1 + rA, USD)
t0 - X0, USD · (1 + rUSD) t1
+ X0, USD · S1 ·(1 + rCHF)
USD X0, USD X1, USD

X0, CHF / S0   X1, USD · S1
CHF X0, CHF X1, CHF
Note that we assume “realistic money market hedging”, i.e. we assume that we only
hedge beginning market values. The exact and approximation formulas can be derived
as follows…
X1,CHF = (X0,CHF /S0 )·(1+rA, USD)·S1-( X0,CHF / S0 )·(1+rUSD)·S1+X0,CHF ·(1+rCHF)
1 + RA, CHF = (1 + rS) · (1+rA, USD)- (1 + rS) · (1+rUSD) +(1+rCHF)
RA, CHF ≈ rS + rA, USD - rS - rUSD + rCHF ≈ rA, USD + rCHF – rUSD
The first order approximation of “money market hedging” is equal to the first order
approximation of a “perfect hedge” and the second order approximation of a “real-world”
hedge.
Numerical Examples
Let us feed the formulas derived above with some figures…
S0 = 1.45 S1 = 1.435
X0,CHF = 100 rA, USD = 5.5%
rUSD = 1.5% rCHF = 2% F1 = 1.4571
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8. Based on these values, we can compare exact result and first and second order various
approximations for the asset return…
NORMAL MARKETS
Exact Approximation I Approximation II
Unhedged 4.4086% 4.4655% 4.4655%
Perfect Hedge 6.0197% 6.0000% 6.0000%
Real-World Hedge 5.9357% 5.9431% 6.0000%
Money Market Hedge 5.9586% 6.0000% 6.0000%
The range of the various possible calculations is 8.4bp for a parameter constellation
typical for “normal market conditions”.
In order to see what happens in turbulent market conditions, let us set S1 = 1.25 and rA,
USD = 5.5%...
TURBULENT MARKETS
Exact Approximation I Approximation II
Unhedged -21.9828% -23.2931% -23.2931%
Perfect Hedge -9.0542% -9.0000% -9.0000%
Real-World Hedge -7.6970% -7.6897% -9.0000%
Money Market Hedge -7.4828% -9.0000% -9.0000%
The range of the various possible calculations is now 157.1bp
Conclusions
 There are several “correct” formulas to calculate hedged returns, reflecting
various ways of implementing a currency hedge. When taking into account
approximations, the number of available formulas explodes, as various degrees
of approximations can be performed.
 Certain “correct” formulas are based on unrealistic assumptions and cannot be
implemented in real-world portfolios. Such formulas should not be used in
calculating benchmarks for performance analysis purposes.
 When implementing hedged return formulas, further realistic features should be
considered, like variable hedge horizon and costs.
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