Currency Hedged Return Calculations

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Currency Hedged Return Calculations

  1. 1. Research Note Andreas Steiner Consulting GmbH February 2011Currency Hedged ReturnCalculationsIntroductionAs of first quarter 2011 it is probably not necessary to elaborate on the importance ofcurrency risk. Significant movements in major exchange rates like the USD or EUR tookplace just recently and have had a major impact on the performance and riskcharacteristics of international portfolios. Nevertheless, we see a contrast between theimportance of the currency risk factor in modern investment management and itstreatment in portfolio analytics like performance attribution and risk budgeting. Part ofthis can be explained by conceptual complexities: currencies are not just another assetclass, but a risk exposure embedded in any assets and therefore affecting the overallportfolio in non-trivial ways. This research note addresses one particular aspect ofcurrency risk analytics, namely the calculation of hedged asset currency returns. Suchcalculations are used in “paper portfolios” like benchmarks and generally ex anteperformance and risk analytics.Unhedged Foreign Asset ReturnsIn order to prepare a framework for the discussion of currency-hedged asset returns, wewill first analyze unhedged asset returns.Let us consider an investor with the base currency CHF that invests a certain amount Xin 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© 2011, Andreas Steiner Consulting GmbH. All rights reserved 1/8
  2. 2. The situation can be illustrated graphically as follows… Start of investment period t0 End of investment period t1Asset Currency USD X0, USD X1, USDBase Currency CHF X0, CHF X1, CHF rA, CHFWhat we are interested in is to calculate the CHF return of the USD asset rA, CHF. Thecalculation 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, CHFThe 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. 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, CHFThe 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 · S1It 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 anadditional 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 + rSFrom this approximation, we can clearly see the nature of currency risk: the return of aforeign 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.© 2011, Andreas Steiner Consulting GmbH. All rights reserved 3/8
  4. 4. Note that the approximation works if rA, USD · rS is “small”, which is the case when bothcurrency spot and asset return are small. This is the case in “normal markets” only, notin turbulent times. As many calculations in spreadsheets and commercial performanceand risk systems are based on this approximation, we end up with the paradoxicalsituation that approximate analytics fail when we need them the most.The notation and illustration developed above can now be used to derive the formulasfor hedged returns. We will analyze three different types of hedge implementations. In allof them, we assume that the goal is to “fully hedge” currency risk. The results can beeasily extended to partially hedged assets, with is simply a portfolio consisting of theunhedged and fully hedged assets, with the weight of the fully hedged being the hedgeratio.Case I: The Perfect HedgeThe “perfect hedge” is a situation which we define as follows: We eliminate uncertaintyabout the future spot rate (and therefore currency risk) by entering into a currencyforward contract at the beginning of the investment period. A currency forward contractis an agreement to exchange a certain amount in a certain currency into a certaindifferent currency at a certain exchange rate. This contractually agreed exchange rate isthe “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, CHFThe only change to the unhedged situation is that F1 replaces S1. The formula for theperfectly 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 / S0The 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)© 2011, Andreas Steiner Consulting GmbH. All rights reserved 4/8
  5. 5. As before, the above expression can be approximated… RA, CHF ≈ rA, USD + rFNote that in the above calculations, we assume that we can exchange the amount X1, USDat the rate F1 agreed in t0. This is unrealistic: X1, USD is only known in t1 due to uncertaintyabout the asset’s gains and losses during the investment period. The amount to beexchanged in forward rate agreements, on the other hand, has to be specified in t0already. The “perfect hedge” therefore implies “perfect foresight” regarding the futureasset value.The forward exchange rate cannot be set at arbitrary values, as this would createarbitrage 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 ratemultiplied 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 – rUSDThis can be read as follows: the return of the perfectly hedged foreign asset equals itsreturn in local currency plus the difference in riskfree rates between the base currencyand asset currency.Note that currency hedging completely removes the uncertainty regarding the futurespot exchange rate (it can be shown that the volatility of the perfectly hedged foreignasset in base currency equals its volatility in asset currency). The perfect hedge doesnot result in the investor earning the local return of foreign assets; he earns the localreturn plus an interest rate differential. Generally speaking, the local return of foreignassets is not an investable asset; investable are only hedged, partially hedged orunhedged returns. The contribution of a foreign asset in an international portfolio cannotbe altered without changing the contribution of the interest rate differential. These are© 2011, Andreas Steiner Consulting GmbH. All rights reserved 5/8
  6. 6. the reasons why performance attribution models based on the Karnovsky/Singerapproach have not become popular among practitioners: attribution effects need to beindependent and tied to investable assets (we will elaborate on this point in a futureresearch note).Case II: Real-World HedgingAs we have discussed, “perfect hedging” is only feasible if the ending market value ofrisk assets is known. This is generally not the case in real-world portfolios. Real-worldhedging is typically performed by hedging the beginning market value. Depending onwhether 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, CHFThe 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 – rUSDWe can now “approximate the approximation”… RA, CHF ≈ rA, USD + rCHF – rUSDWe 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 ratedifferentials and small currency and local asset returns. We can expect the second orderapproximation to be less accurate that the “perfect hedge” approximation.© 2011, Andreas Steiner Consulting GmbH. All rights reserved 6/8
  7. 7. Case III: Realistic Money Market HedgingThe uncovered interest rate parity also tells us that a forward contract can be interpretedas a derivative instrument. Its total return can be statically replicated with a long positionin the domestic riskfree asset and a short position in the foreign riskfree asset. Asriskfree rates are hypothetical constructs that do not exist on real-world financialmarkets, one can use money market instruments as proxies.Such a strategy would involve buying the foreign asset plus a long position in a domesticmoney market instrument and a short position in a money market instrument in the assetcurrency. 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, CHFNote that we assume “realistic money market hedging”, i.e. we assume that we onlyhedge beginning market values. The exact and approximation formulas can be derivedas 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 – rUSDThe first order approximation of “money market hedging” is equal to the first orderapproximation of a “perfect hedge” and the second order approximation of a “real-world”hedge.Numerical ExamplesLet 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© 2011, Andreas Steiner Consulting GmbH. All rights reserved 7/8
  8. 8. Based on these values, we can compare exact result and first and second order variousapproximations for the asset return… NORMAL MARKETS Exact Approximation I Approximation IIUnhedged 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 constellationtypical 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 IIUnhedged -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 157.1bpConclusions  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.© 2011, Andreas Steiner Consulting GmbH. All rights reserved 8/8

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