1. 1
Financial Econometrics Concerning Carry Trade: Empirical Analysis
Francis Dixon

2. 2
Part 1: Purpose, Notation, and an Introduction into International Currency Exchange Markets
The purpose of this paper is to examine--from an econometric approach--the metrics at play in
international currencymarkets, the potential drivers of the frictionbetweenthese marketswhich cause
deviation from Purchase Power Parity in the short run, theorized statistical methods through which
economists generate exchange rate forecasts and investors exploit arbitrage opportunities, and an
analysisof the empirical resultsof these statistical instruments.We will conclude withanexplanationof
our findingsfroma strictlyempirical perspective. Letusfirstlay the foundation forthis study by defining
the keyterms andgeneral notations thatwill be usedthroughoutthispaper,and byreviewingtheof short
andlongrunequilibrium conditionof domesticmoneymarketsaswellastheirrelationshiptoequilibrium
in foreign exchange markets.
The notation to specify the time interval associated with a “holding period” will be represented
by subscripts. The beginningof anasset’s holdingperiodwill be denotedbysubscript(t),withthe endof
that the holding period denoted by subscript (t+1).
The notationtospecifywhetheravariable isa characteristicof the domesticeconomyoraforeign
economywillbe representedbysuperscripts.Domesticvariablesof interestwillbedenotedbysuperscript
(*), withforeignvariablesof interest will be denotedbylackof a subscript. The domesticeconomy willbe
the economyinwhichanassetisfunded,andthe foreigneconomywill be the economyinwhichanasset
is borrowed. For simplicity strictlywhen generating formulas or expressions the domestic economy will
be the United States, and the foreign economy will be Europe, unless noted otherwise.
The nominal interestrate,denoted asi,represents the interestearnedon financial assetsheldin
an economy.The real interestrate,denotedasr, representsthe interestearnedonfinancial assetsheld
in an economy less that economy’s rate of inflation, where the inflation rate is denoted as π. The
mathematical representationof nominal andrealinterestrates are then,i =(1+i) andr= i –π respectively.
The nominal exchange rate,denotedas ϵ,isthe price of oneUSdollarintermsof foreigncurrency.
For Example,if itcostsone andahalf unitsof foreigncurrencytopurchase one unitof domesticcurre ncy,
the nominal exchange rate would equal 1.5€/$ or ϵ = 1.5. The real exchange rate, denoted as q, is the
price of one unitof US goodsin termsof foreigngoods,wherethe domesticprice levelwill be denotedas
P*, and the foreign price level will be denoted as P. The mathematical expression for the domestic real
exchange rate is then, q = ϵ(P*/P).For Example, If a US shirt costs 50 dollars, the nominal exchange rate
is 6, and a European shirt costs 100 euros, then q = ϵ(P*/P) would be calculated as 3 = 6*(50/100).

3. 3
These terms are essential in understanding the dynamics between the short run supply of and
demand for domestic and foreign currency, as well as how short run equilibrium is determined in the
nominal exchange market.1
Review Figure 1, which plots the nominal exchange rate on the vertical axis
and the quantityof currency exchanged onthe horizontal axis. The upwardslopingcurve representsthe
supply of domestic currency to foreign investors. To understand why it slopes upward, notice when the
nominal exchange rate rises the purchasing power of domestic currency rises relative to that of foreign
currency(makingdomesticimportsof foreigngoodsandservicescheaper).The downwardslopingcurve
represents the demand for domestic currency by foreign investors. To understand why it slopes
downward,notice whenthe nominalexchange rate falls the purchasingpowerof domesticcurrencyfalls
relative to that of foreign currency (making foreign imports of domestic goods and services cheaper).
Short run equilibrium in the nominal exchange market is the point where the supply of and demand for
the domestic currency intersect. This equilibrium, shown in Figure 1, relates the quantity of domestic
currencyexchangedforforeigncurrencyatanygivennominal exchangerate. Note thatinthismodel,the
nominal exchange rate is measured as €/$.
Supply in the nominal exchange market is essentially a combination of domestic demand for
foreigngoodsandservices,anddomesticdemandforforeignfinancial assets (suchasreal estate,mines,
factories,bankaccounts,stocks,bondsand treasurybills).There are twocausesof shiftsinthe supplyof
domesticcurrency:achange inthe domesticlevelof income,andchange inforeignnominal interestrate.
Anincrease todomesticincome shiftsthe supplyof domesticcurrencytothe right,asdomesticinvestors
1 For more information on international financesee“International Economics”

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will have the ability to buy more foreign goods and services. Rising foreign interest rates also shiftsthe
supplyof domesticcurrency rightward.Thisis because domesticinvestorswill wanttoholdmore assets
inforeigncurrency. A rightwardsupplyshiftresultsinadecreasethe domesticnominal exchangerate and
an increase indomesticoutput.The resultingdecreaseinthe domesticnominalexchangerate represents
a domestic currency depreciation.
Similarly,demandinthe nominalexchange marketisessentiallyacombinationof foreigndemand
for domestic goods and services, and foreign demand for domestic financial assets (such as real estate,
mines, factories, bank accounts, stocks, bonds and treasury bills). There are two causes of shifts in the
demandfor domesticcurrency:a change in the foreignlevel of income andchange in domesticnominal
interest rate. An increase to foreign income shifts the demand for domestic currency to the right, as
foreign investors will have the ability to buy more domestic goods and services. Demand for domestic
currencyalsoshiftsrightif domesticinterestratesrise.Thisis because foreigninvestorswillwanttohold
more assetsindomesticcurrency. A rightwarddemandshiftresultsinanincreasesthe domesticnominal
exchange rate and domestic output. The resulting increase in the domestic nominal exchange rate
represents a domestic currency appreciation.
Using Figure 1 which relates the nominal exchange rate to the short run equilibriumquantityof
currencyexchange atthe intersectionof the supplyof anddemandforcurrency,we are able toconstruct
a graphicrepresentationthatrelatesthe nominalexchangerates tothelevel ofdomesticeconomicoutput
inthe short run inFigure 2. Here the DD schedule isderivedfromall potential combinationsof the supply
of and demandforcurrency inthe shortrunequilibrium conditionwe discussedinFigure 1.Ourobjective
isnowto relate the domesticmoneymarkettothe foreignexchange market.InFigure 3, the AA schedule
represents all possible combinationsof real exchange rates and output levels that keep the domestic
money market and foreignexchange market in equilibrium. Figure 4 combines these schedules, relating
these two markets. Creating this graphic representation, we are able to better understand how foreign
exchange rates relate to interest rates, the level of income, the price level, and the money supply.

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It is important to understand the factors that cause the AA and DD schedule to shift. The DD
schedule will shift rightward in reaction to an increase in domestic government spending,a decrease in
domestictaxes,anincrease ininvestmentdemand,adecrease inthe domesticprice level relative tothe
foreign price level, a decrease in domestic savings, and an increase in demand for domestic goods and
services.The AA schedulewillshiftsrightwardinreactiontoanincrease inthe domesticmoneysupply,a
decrease in the domestic price level,an increase in the expectedexchange rate at the end of an asset’s
holdingperiod,anincrease inthe foreigninterestrate,anda decrease indomesticreal money demand.
Itisobviousatthispointof the natural linkbetweenthe moneymarketsandthe foreignexchange
markets.We are able to showgraphicallythislinkthroughFigures5and6, where we take acloserlookat
the twocauses of marketfluctuationwe discussedinFigure 1. Figure 5 displayshow a change in foreign
interest rates (due to an increase in the foreign economy’s money supply) affects the equilibrium
condition in both the domestic money market and the foreign exchange market. Similarly, Figure 6

6. 6
displays how an increase in the domestic level of income (outward shift in the money demand curve)
affects the equilibrium condition in the domestic money market and foreign exchange market.
Understandinghowthese marketsrelate toone anotherisessential inforecastingtheequilibrium
conditionof these markets inthe longrun.Now we willlook atthe longruneffectof apermanentchange
in the moneysupply,which as statedpreviously drivesfluctuationin bothmarketsby changingthe level
of inflation (therefore changing the expected exchange rate) at the end of an asset’s holding period.
Figure 7 showsthe relationship betweenthe foreignexchange marketandthe moneymarketin
longrun equilibrium where the purchasingpowerparityconditionholds.Figure 8displaysmovementto
short run equilibrium inresponse toapermanentincrease inthe domesticmoneysupply onthe leftside
of the graph, as well as movement from this short run equilibrium to long run equilibrium on the right
side of the graph.
Let us look more closely at Figure 8. Note that in this model, the exchange rate is measuredas
$/€. Initially the markets are in long run equilibriumat point 1. If the domestic central bank were to
permanently increase the money supply by raising the rate of monetary growth (engaging in easy
monetarypolicies),there wouldbe animmediate increaseinthesupplyof domesticcurrency,whichshifts
the real domestic money supply downward. Observe how the increase in the domestic money supply

7. 7
shiftsdomesticMSdownward fromMS1
US toMS2
US, and causesthe domesticrate of returnshiftleftward,
a decrease fromR1
$ to R2
$ in the domesticmoneymarket.To understandwhy,rememberour definition
of the real interest rate is the nominal interest rate less the inflation rate. Permanently increasing the
money supply would then permanently increase the expected rate of inflation, resulting in the lower
return to domestic assets. Because the return to domestic currency has decreased, the expected return
to foreigncurrencyincreases,whichisrepresentedinthe rightwardshiftinthe expectedeuroreturn.The
resulting short run equilibrium is now at point 2, where the equilibrium exchange rate has increased in
the foreign exchange market.The increase in the exchange rate associate with point 2 is a depreciation
of domestic currency against the foreign currency.
In the long run,pricesbecome flexible sothatthe domesticprice level adjuststothe permanent
change in the money supply. As this process takes place, the real domestic money supply shifts back
upward to its initial level do to the increase in P1
US to P2
US, which increases the return to assets held
domesticallyfromR2
$ back to itsinitial level atR1
$ as well.The curve that representsthe expectedreturn
to foreign currency, however, is not affected by the increase in the domestic price level. The long run
equilibrium therefore decreases to point 4. Therefore, in the long run the permanent increase in the
money supply has no effect on the domestic money market, but results in a permanent increase in the
exchange rate in the foreign exchange market: the domestic currency has permanently depreciated
against the foreign currency.

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Part 2: Spot Transactions, Futures Contracts, and the Three Purchase Interest Parity Conditions
Now that we have an understanding of what factors drive the fluctuation of exchange
rates,we can beginanalyzinghowarbitrage opportunitiesarise.The “Law of One Price”statesthatwhen
convertedintoacommoncurrency,the pricesof identical goodsandservicessoldindifferent economies
shouldhave the same value.2
Recall fromPart1 thatthe real exchange rate equalsthe nominal exchange
rate adjusted for difference in the price levels of two economies, q = ϵ(P*/P). To convert the different
price levelsintoone commoncurrency.Byrearrangingthe termsof this formulatosolveforthe domestic
price level. This conditioncreates what economist and currency investors refer to as “Purchasing Power
Parity”(PPP). Figure9exhibitshowthe conceptof purchasepriceparityrelatestothe longrunequilibrium
of money markets and foreign exchange markets:
2 For details on the link between interest rates, exchange rates, and the parity conditions see“A note on Parity
Conditions (CIRP,“FP”, UIRP, PPP) and Carry Trades”

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Our studyof the manyvariantsof purchase powerparity beginsbydefiningabsolute andrelative
PPP. If absolute PPP holds, the real exchange rate should equal one. In practice, absolute PPP at many
times will fail to hold true for some goods and most services. Relative PPP, a less restrictive variation,
claimsthat the differentinthe twoeconomies inflationrateswill measure the degree of appreciationor
depreciationin acurrencyalongthe holdingperiod.Relative PPP,iscomputed mathematicallyas Δϵ / ϵ =
π – π*, where Δϵ represents the change in the nominal exchange rate, π represents the rate of foreign
inflationand π*representstherate of domesticinflation. WhereasinabsolutePPP,if the law failstohold
the real exchange rate fluctuatesaround1, with relative PPPthe real exchange rate remainsconstantas
movementsinpricelevelsare offsetbymovementsinthe nominalexchange rate.Inpractice,relative PPP
has proven empirically to the more accurate measure in explaining exchange rate fluctuation in the
medium to long run.
It is also important to understand two types of currency transactions: spotsand futures. A spot
transactionwill be denotedas ϵ for simplicity,because the transactionsprice of aspot transactioninthe
currenttime period issame asthe nominalexchange rate.Withfuturescontracts,denotedasF,the buyer
and seller of the currency instrument agree to reverse the transaction at a future date, but hold the
repaymentprice paidatthe endof the holdingperiod atthe currentnominal exchangerate.Thisholding
period could be a week, month, years,etc. It is important to understand in a futures contract, investors
have potential tospeculate and hedge againstrisk,because the future nominal exchange rate may have
increasedordecreased.The difference inthese transactions fostertwoPPPconditions we willfindtobe
major elements in our study: covered and uncovered interest parity.
Covered Interest Parity takes place in the presence of futures contracts. It states that a trader
investingatthe domesticnominal interestrate (i*) mustearnthe same return as an investorexchanging
domestic currency for foreign currency in the current time period, earning the foreign interest rate (i)
during the holding period, and converting the investment back to domestic currency at the end of the
holding period at the nominal interest rate locked in by the futures contract. There are two ways to
express covered interest parity mathematically: CIP exact and CIP approximate.
CIP exactis measuredas CIPexact = 1 + i = (ϵ*
t/F)(1 + i). It is typicallymore accurate to use the CIP
exact when forecasting against developing economies due to the significantly high interest rates
associated with emerging markets. A more commonly used expression known as CIP approximate is
measuredas CIPapproximate = (F – ε*
t/ ε*
t) = i – i*. WhenCIPholds,as itshould,there shouldbe noexistence
of arbitrage opportunities.If conditionsprovoke CIPtofail,traderstendtoborrow assetsatlow domestic

10. 10
interestratesandfund assetsat the higherinterest ratesof foreigneconomies. The exploitationof these
arbitrage opportunities exerts pressure on price levels in the two economies until CIP is restored.
Uncovered Interest Parity takes place when there is no presence of a futures contract. It states
thatonaverage,tradersinvestingindomesticcurrencyshouldmakethesame returnontheirinvestments
as investors buying foreign currency, earning the foreign nominal interest rate, and converting their
investmentsback to domesticcurrencyat the domesticnominal interestrate at the time of maturity,or
(ε*
t+1). The mathematical representationof uncoveredinterestparity is then, UIPapproximate = (E(ε*
t+1) – ε*
t/
ε*
t) = i – i*, where E(ε*
t+1) denotesthe expectedvalue of the domesticnominal exchangerate at the end
of the holding period.
Part 3: The “Forward Premium Puzzle” and the Theories that Seek Explain its Existence
The formof currencytradingof we will focusonforthe durationof thispaperiscarrytrade.Carry
traders concentrate their investments in high-yielding currencies in an attempt to profit from the yield
spread. One of the most basic principles of financial econometrics states that on average a zero-cost
investment, suchascarry trade, shouldgenerate zeroexpectedreturn. However,historicallythisformof
trading generates small positive returns the majority of the time it is executed. Our study in Part 2
explained twovariantsof PPP, coveredand uncoveredinterestparity.The UIP conditionstatedthat the
degree of appreciationof the fundedcurrencyneededtoeliminate arbitrage isdirectlyrelatedtothe yield
spreadbetweenthe twoeconomies,where the yieldspreadisconsideredtobe an unbiasedpredictorof
fluctuation in the spot exchange rate. The CIP condition stated that due to forward contracts, there is a
natural link between the spot and forward exchange rates, where the forward exchange rate acts an
unbiased predictor as well. As we stated, these conditions are compared to the PPP condition, the
observationthatnationalprice levelsshouldequaloneanotherwhenconvertedtoone commoncurrency.
Most economists agree that the UIP and CIP conditions should hold, meaning a trader investing
inforwardcontractswithdevelopingcountriesthathave relativelyhigherinterestratesshouldexpectthe
funded currency to depreciate along the holding period (eliminating arbitrage opportunities). However,
empirical studies have indicated the exact opposite: high interest rate countries tend to appreciate
creating the positive average returns captured in carry trade. This is the basis of one of the largest
international financial economic anomalies today, known as the “Forward Premium Puzzle.” Over the
recent decades, many economists have sought to solve this puzzle. While some economists have been
able to partially explain potential causes of this anomaly, none have succeeded in deriving a complete

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explanation for deviations from the law of one price. Before we begin discussing how to exploit the
forward premium puzzle, it is important to cover a few of these economist’s findings. Generally, the
researchof economiststhatsucceedsin explainingdeviationinthe short run, fail to explaindeviationin
the long run. Likewise, research that succeeds in explaining deviation in the long run, fails to explain
deviation in the short run.3
First,letuslookat the shortrun. One explanationisthe existence of transportationcosts,tariffs,
andnontariff barrierscreate frictioninthe international markets.Transportationcostshave the potential
to create differences in domestic and foreign price levels.Over the last decade the level of worldtariffs
have beendramaticallyreduced,yetsome remaininplace andwill inevitablydiscourage assetallocation
relative to a hypothetical tariff free environment. With nontariff barriers, exporters operating with a
limited supply must charge a premium price in order to offset the costs of initially surmounting the
barriersaswell asgainingsomeof the rentsassociatedwithinternationaltrade whengovernmentforeign
exchange policy fails to prohibit rent seeking among private entities. Many economists succeeded in
proving empirically that nontariff barriers can partially explain some of the deviation from PPP.
Another way to explain the failure of short run PPP is through price stickiness. Price stickiness,
however,failstocompletelyexplaindeviationsfromPPPinthe longrun.Afteraperiodof abouttwoyears
prices typically become flexible, but convergence to long run PPP usually takes much longer. One of the
most popular economists today, Paul Krugman, explainssome of this puzzle through a concept called
“pricing the market.” Pricing the market takes place with monopolistic firms that refuse to provide
warranty services for goods to consumers in one country who have purchased the goods in another
country. By limiting arbitrage this way, producers then have the ability to discriminate prices across
internationalmarkets. While manyattemptstolimitmonopoliesabilitytoprice the marketthroughfiscal
policies, pricing the market does occur and continues to create some deviation in the short term.
Anotherhypothesisisknownasthe Balassa-SamuelsonHypothesis focusesmore onthe longrun.
This theoryarguesthat whenall price levelsare convertedintodollarsusingthe nominal exchange rate,
there are higher price levelsassociated with rich countries relative to poor countries because wealthier
countriestendto be more productive inthe traded goodssectorof theireconomies.Insmall developing
countries,the price levelof the tradedgoodssectoristiedtothe worldprice level.Therefore,anincrease
inproductivityinthesecountriesresultsinanincrease inwagesearnedbylaborersof tradedgoods.If the
3 For more information on the theories discussed in this section,see“The PurchasingPower Parity Puzzle”

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non-tradedgoodssectorsfail to increase inproductivityatthe same rate as the tradedgoodssector,the
only way that wages will remain competitive across sectors would be to increase prices. Under the
assumption that there is one constant component and one increasing component of the CPI, thiswould
inevitably cause the CPI of the developing country to increase. If both components were to increase in
productivitysimultaneously,there wouldbe norelative price effect,andtherefore nochange tothe real
exchange rate. Compared to the theory of stick priceswhich was a short term explanations in deviation
from PPP, the Balassa-Samuelson theory partially explains long term deviations. Through shifts in the
terms of trade, we are able to account for significant movements in real exchange rates.
A similarlongrun theorypredictsthatrichcountrieswill havehigherexchangerate adjustedprice
levelsduetothe factthattheircapital-laborratiosare higherrelative topoorcountries,ratherthanbeing
dependentonthe assumptionof higherproductivity.Since highercapital-laborratiosimplieshigherwage
rates, labor is cheaper in poor countries. Through the study of international trade there is evidence of
poor countries being labor intensive relative to wealthy capital intensive countries. Coupled together,
these twofactslead us again to the conclusion that wealthy countries should have higher price levels.
Another long run theory takes account deficits into consideration.The argument here is that
sustaineddeficitsare directlyrelatedwithreal exchangerate depreciation.While these twovariablesdo
exhibitempirical correlation,there ismuchdebate as to whetherthiscorrelationisa resultof causation
due to the fact that there are manycausesof currentaccount deficitsthatare not directlyrelatedtoreal
exchange rates.Proponentsof thistheoryare quickto emphasize correlationandcausationbecause the
borrowingandlendingbetweencountriesthat isassociatedwithsustainedcurrentaccountdeficitsleads
to a transfer of wealth across countries. Government spending has also been considered as a potential
cause for fluctuation in the real exchange rate. Because government spending tends to fall more on the
non-traded goods sector, a rise in government spending at many times leads to increases in the real
exchange rate. Proponents argue when taxes are used to finance government spending programs, it is
possible for fiscal policy to cause fluctuation in real exchange rates.
All of these theories can partially explain deviation from PPP, but all fail to provide a complete
explanationof the forwardpremiumpuzzle inthe short,medium, andlongrun. We conclude that inthe
short run, PPP does not hold, and convergence to PPP in the long run can take many years. Most
explanationsthattendtofocuson monetaryand fiscal policiesare able toexplainthe puzzleinthe short
run, but fail to provide evidence for the long run. The theories that focus on shocks to productivity, and
consumer preferences are able to explain some of the long run convergence back to PPP, but fail to

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provide evidence for exchange rate volatility in the short run. In reality a complete explanation of
deviationof PPPwillbe amultivariatemodelcontainingmanyof these theoriessimultaneously,assuming
some variablestobe purelytemporaryandotherspermanent.Everydayinternationalgoodsmarketsare
continuing to become more integrated, but until they become as integrated as domestic goods market
segments these markets will continue to generate various sources of friction among international
economies.
Part 4: The Exploitation of Arbitrage Opportunities and the “Trader’s Decision Problem”
Now that we have created a solid understanding of the dynamics of international currency
marketsas well as howarbitrage opportunitiesarise inthem, the questionbecomes“whatmethodscan
be used bythe carry traderto capture returninthe market.Let usbeginbyrecallingone of the principles
of econometric analysis: a zero cost investments, such as carry trade, should generate zero expected
return on average (assuming the absence of arbitrage opportunities). Throughout our discussion, the
econometricmodelswe will builduponrepresent stochasticprocesses.The modelswe buildwillthenbe
used in generating forecasted outcomes of financial variables of interest.4
Let Et denote the expected
value of returnatthe currenttime period,andletxt+1 denote thereturntoazerocostinvestmentstrategy
for a risk-neutral investor in the absence of arbitrage. We can then express this econometric principle
mathematically by:
What this expressionimpliesisthaton average,one wouldexpecta return of zero to theircarry
portfolio.Nowwe addastochasticdiscountfactorwhichwillsummarize the interactionof outcomesand
consumerpreferences.Thisfactor representsthe potential combinationof pseudo-probabilitiesimplied
bythe investor’schoice of consumption,andisdeterminedbythe outcomesthesepreferences associated
withthe conditionof worldmarketsandothersourcesof risksothatthe value of the trader’sinvestment
is dependent on its ability to generate return at the current condition of the world markets. This
expression, where mt+1 represents the stochastic discount factor, then becomes:
4 For more information on the expressions derived in this section as well as the characteristicsof stochastic
processes see “Carry Trade”

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To geta betterideaof whatthisformulaimplies,assume wehave borrowedone unitof domestic
currency at the domestic interest rate by selling the domestic security short, then we purchase the
equivalentunitof foreigncurrencywiththe same date of maturitythatyieldsthe foreigninterestrate.At
the time of maturity, the transaction is reversed.
For those not familiar with Investment jargon, selling a currency short means it is not currently
ownedbyaninvestor,andthereforemustreturnthe same securitytothe lenderatanagreedupondate.
This strategyis usedwhenanticipatinga decline invalue,resultinginrepaymentof the securitylessthe
depreciated value which leaves the difference as return to the investor who sold short. By the same
measure, the investment of foreign currency would be a process known as going long.
At the time of maturity,the fundedforeigncurrencyistransformed backtodomesticcurrencyat
the spot exchange rate,denotedas ε*
t+1. Alongthe term, the returngeneratedbythe foreignsecurity is
equal to the foreign interest rate, so that return is represented mathematically by (1 + it). Proceeds will
then be used to repay the borrowed principle as well as the interest associated with it. We will denote
the value of this interest payment as it+1. Remembering this form of investment should generate zero
expectedreturn onaverage, the expressionwithwhich we can measure the expectedcarrytrade return
of a spot transaction is then:
With respect to the stochastic process, we must then take the natural logarithms, using the
approximation ln(1 + it) ≈ it, denoting ln(ε*
t) = et, and Δet+1 = et+1 – et. The stochastic expression for the
expected carry trade return of a spot transaction is then:
Assuming the trade is operating under risk-neutrality and assuming the absence of arbitrage implies:
This measure also expresses uncovered interest parity in that:
Reverting back to and modifying expression (3) for the expected carry trade return of a spot
transaction, we can express the expected carry trade return of an investor who instead purchased a

15. 15
futurescontract,denotedasFt.Holdingthe assumptionthatthe investorhasriskneutralpreferences,this
measure expresses covered interest parity where the forward rate is again an unbiasedpredictor of the
spot exchange rate at the time to maturity. The expected carry trade return of a futures transaction is
then:
To complete the trinityof parity conditions,wemodify versionof ourexpressionforthe expected
carry trade returnof aspottransaction once more toreflectthe valueof the expectedreturn inrealrather
than nominal terms.Let the real interestrate,denotedrt, equal the nominal interestrate in the current
time period less the inflation over the course of the holding period, so that r*t = i*t – π*t+1 where π*t+1 =
Δp*t+1 an letp*t = ln(P*t).Let the same notation methodapplyfor foreign economies so that rt
= it
– πt+1
where πt+1 = Δpt+1 and by letting pt = ln(Pt). With respect to the stochastic process, the logarithm of the
real exchange rate, denoted as qt+1, would be measured by qt+1 = et+1 + (pt+1 – pt+1).
The stochasticconditionknownas “first orderweaklystationary”statesthat for a processto be
firstorder weakly,all randomvariablesmusthave the same meanor expectedvalue.Forthisexpression
tobecome aweaklystationaryprocess,wedenotethe fundamentalequilibriumexchange rate (FEER) that
the natural logarithmof the real exchange rate (qt+1) revertsto inthe longrun as, ɋ. The expression then
becomes qt+1 = ɋ + Ø(pt+1 – p*t+1).
Now expression(4) which weobtainedbytakingthenatural logarithmof theexpected carrytrade
return of a spot transaction can be rewritten as:
Again,assumingthe absenceof arbitrage,the degreeof expectedreal exchangerate appreciationequals
the expected value of the spread between the real interest rates, that is, Et(qt+1) = Et(rt – r*t).
We can nowuse interactionof interestrates, exchangerates,andlongrunequilibrium condition
under purchasing power parity as a stochastic process regressed against the nominal exchange rate to
generate forecasts for future time intervals. Review the stationary random vector for Δyt+1 below:

16. 16
If the stochasticprocessof thissystemislinear,we use the VectorErrorCorrectionModel (VECM). VECM
is one of the most common models used among economists and currency traders. There are four
characteristics of VECM models that differentiate them from other statistical models. First, they are
bivariate systems,wherewe modeltwostochasticprocessesjointly.Second,eachequationhasthe same
regressorsaswell asthe same numberof lags,builtuponan autoregressive structure.Third,all variables
onthe leftandrighthandside of the equationare stationaryduetothe cointegrationthatexistsbetween
them. Fourth, there must be at least one adjustment coefficient different from zero (otherwise there
wouldn’tbe cointegration). The first order of the vector error correction model for the system is then:
Comprised within this expression are three “signals” that can be analyzed individually by carry
traders:the carry signal,the value signal,andthe momentumsignal.Thesesignalsare sub-strategiesthat
can be used individually based on the carry trader’s preferences. Collectively, CMV signals are the
foundation from which entire VCM portfolios have been built. Let us look at each of the three sub-
strategies individually before covering the VECMmodel in more detail.
The carry strategy (C) can be expressed mathematically as Δět+1 = 0. This strategy focuses solely
on the expectedvalue of the real domesticinterestrate lessthe real foreigninterestrate indetermining
which currency should be sold short, and which to go long in. The momentum strategy (M) can be
expressedmathematicallyasΔět+1 =βeΔet. Assumingβe to be equal tozero,thisstrategyusesthe value of
the nominal exchange rate inthe current periodto be the best forecastof the exchange rate at the end
of the term. The value strategy (V) can be expressed mathematically as Δět+1 = ϒ(qt - ɋ), and is used to
forecast the degree of currency appreciation or depreciation via the value of the PPP signal. The VECM
model is used in forecasting deviations of the real exchange rate from the fundamental equilibrium
exchange rate,andisoneof themostaccurate methodswithwhichwe canexpresstheuncoveredinterest
parity.
The currencies which an investor should borrow and which an investor should fundcan then be
taken by inputting the results from any of these strategies into the expression (10). This expression
represents the direction of a carry trade for it to be profitable at the end of the holding period. In this
expression, xt+1 is the VECMstrategy result:

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The ex-post realized return of the investment is then:
Expressions(10) and (11) reveal a veryimportantcharacteristicof carry trade.The probabilityof
generatinga positive investment ismore dependent onthe investor’sabilitytoforecastthe directionof
the trade correctly than the investor’s ability to accurately forecast the expected change in the real
exchange rates (Δet+1) at the end of the holding period.
All of these trading strategiesare generatedthrough statistical modelsusingthe propertiesof an
asset’s first and second moments. The “Trader’s Decision Problem” is that an investor must, however,
develop aninvestmentstrategythatreliesmore heavilyongenerating performance criteriathatappeals
to aportfolio’s requiredreturnwhilerespectingthe degreeof riskassociatedwithindividualperformance
preferences. We beginhow to solve this problem by modifying the expressions we coveredin Part 4 to
determine the direction of a carry trade as well ascalculate ex-postrealizedreturns.We will use amore
general notation knownas a “genericscoringclassifier,”whichenables the analysisof multiple classifiers
to generate forecasts,ratherthanjustusingthe conditional momentsof ourinformationset.The implied
increase in independent variables is simply the transformation of our information set to a multi-variant
econometric model. Incorporate this scoring classifier, the trade directionexpressionthen will become:
dt+1
= sign(δt+1
– c), where δt+1
represents the classifier and c is a scalar that can take any value in the
interval cЄ {-∞,∞}. Incorporatingthisnotation, the total numberof observations thatencompassacarry
portfoliocanbe expressedbyeachindividual forecast(decision) andtheirassociatedoutcomes.Observe
Figure 10:
In thistable,TN(c) refersto the true classificationratesof negative outcomes,while TP(c) refers
to the true classificationratesof positive outcomes.Likewise,FN(c) referstothe false classificationrates

18. 18
of negative outcomes, while FP(c) refers to the false classification rates of positive outcomes. Negative
rates andpositive ratesare representative of currencysoldshortandcurrency fundedlong,respectfully.
Aswe wouldexpect mathematically,the sumof TN(c) andFP(c) is equal toone,andthe sumof FN(c) and
TP(c) is equal toone. If we letthe total numberof observationsinaportfolio(N),equalsthe sumof total
shorts (S) and the total numberof longs(L),then N = S + L. Throughthese denotations,we cancompute
the empirical values of TN(c) and TP(c) as:
All possiblecombinationsof true outcomescanbe representedgraphicallyinthe same mannera
production possibilities frontier is constructed. This PPF is shown in Figure 11, where the Correct
Classification Frontier (CC Frontier) represents all possible combinations of true outcomes. The Perfect
ClassifierFrontierextendshorizontallyfromthe Y-axisat the value of 1 and verticallyas well asX-axisat
the value of 1, and represents the points along the frontier in which all realized outcomes matchedan
investor’s trade directionpredictions.The UninformativeClassifierFrontier,representsaclassifierwhere
TP(c) = FP(c) =1 – TN(c) foranyscalarvalue.Thislineisoftenreferredtoasthe “coin-toss”diagonal. Utility
optimization, as we would expect, occurs at the point of tangency along the CC frontier. In equilibrium
(pointof tangency) the marginal rate of substitutionbetweenprofitable longsand shortswouldbe equal
to -1.

19. 19
We are then able to define the utility of classification using the expression:
With the point of tangency along the utility classification function calculated as:
Another useful statistic is the Kolmogorov-Smirnoff statistic (KS) which compares the average
correct classificationabilityof aclassifieragainstacoin-tosser.The KSstatisticis measuredbythe vertical
distance between the CC frontier and the coin-toss diagonal. The formula used in measuring the KS
statistic is calculated:
The area underthe CC frontier,the AUC,is an alternative tothe KS statisticthat summarizesthe
characteristicsof the CCfrontierwithbetterdefinition.Inthe graph,the AUCforaperfectclassifierequals
1, and for a coin-tosserequals.5.Inpractice,mostportfoliosresultinanAUC value somewhere between
perfectclassificationandcoin-toss.The KSstatisticandAUCare bothmeasuresof aninvestor’ssuccessin
forecasting the direction of the trades that makeup a carry portfolio.
In practice, the trades that make up a carry portfolio are not identical in term length, quantity
exchanged,orwhentrades are made with differingamountsof leverage.Therefore,the weightedvalue
of each trade must be adjusted to generate an accurate measure of a currency portfolio’s profitability.
Using modifiedstatistics toadjustforthese differences,we definethe total portfolioreturnequal to the
sum of total returnto short trades (BS) and total returnto long trades(BL).These totalsare measuredby
the following two expressions:
These measurements are used to adjust for the difference in weights of each positive and negative
outcome so that the return to BS and BL after being adjusted for their respective weight is calculated by
the following two expressions:

20. 20
We these statisticswe can measure the valuesof TN(c) and TP(c) adjustedfor weight,which are
needed in generating a KS statistic and AUC that also reflect return-weighted directional performance.
Thus the adjusted measurements of TN(c) and TP(c) are expressed by:
By normalizing net portfolio profit by total potential portfolio profit we can measure portfolio
performance in terms of total portfolio gainsand losses. Let gains(G) equal the product of total returns
to shorts and the return-weightedstatisticfortotal numberof true negative classificationratesplusthe
product of total returns to longs and the return weighted statistic for total number of true positive
classificationrates. Likewise,let losses(L) equal the product of total returns to shorts and the return-
weightedstatisticfortotal numberof false positive classificationratesplusthe productof total returnsto
longs and the return weighted statistic for total number of true negative classification rates. Net profit
would then be the value of gains less losses where:
Using the expressions for a portfolio’s gains and losses, we can then normalize portfolio profit by total
potential portfolio profit to express a portfolio’s utility. This measure of utility would then be:
Because thisfunctionforportfolioutilitycanbe expressedintermsof the portfolio’sgains-losses
ratio, maximizingthe ratioof gains to lossesisthe same as maximizingaportfolio’sutility.The portfolio
utility function in terms of portfolio gains and losses is expressed by:

21. 21
Using this adjusted functionfor portfolio utility, we define the point at which optimal portfolio utilityis
being achieved. At this point, the slope of the CC frontier, represented by –BS/BL, again equals -1. The
followingderivativerepresentsthe pointalongthe portfolioutilityfunction where utility is maximized:
In practice, portfolio returnsare rarely symmetric even in the presence of risk neutrality.This is
due to a number of external factors such as leverage limits and transaction costs. Assuming that an
investor’s portfolio return is symmetrically distributed, then the gain-loss ratio that maximizes utility
coincides with the point at which the KS statistic adjusted for return-weight is calculated so that:
Inthe absence of arbitrage opportunitiesaportfolio’sexpectedreturniszero,where the valueof
total portfolio gains is the same as the value of the total portfolio losses.This implies the gain-lossratio
wouldbe equal to one.In the presence of arbitrage,thisratio can take any approximationinan interval
from negative infinityto positive infinity where a gain-loss ratio of positive infinity would imply perfect
classification.
Part 5: Empirical Analysis Assuming the Existence of Arbitrage and Concluding Research
Historically, the empirical analysis of the results generated by these models exhibits positive
average returns to carry trade portfolios. One theory as to how this is possible argues that carry trade
returns are, in fact, compensation for risk. Proponents of this theory argue that shocks to supply and
demandinthe currency marketsencourage investorstoreduce theirholdingsof riskyassets,resultingin
potential losses in the short run via the order flow effect which impacts the price of trades.
Orderflowis a standardpractice inthe brokerage industry where brokeragefirmsreceive asmall
per-share rebate on orders routed to certain market makers for the execution of their transaction. In
addition, compensation may be rewarded that is not directly related to specific per-share values from
marketcenters,butbased onotherfactorssuchasquantity,quality,and type of the orderflow presented
tothe market. Statistical analysisindicates,however,thanorderflow doesnotcompletelyaccountforthe
positive average returns.

22. 22
Another potential explanation is in the context of a certain class of consumption-based asset
pricing models. The argument here is that brokers make infrequent adjustments to their portfolio
decisions because the losses incurred are insignificant relative to portfolio management fees. The very
structure of our financial system provokes infrequent portfolio adjustments by investors because many
of the majorfinancial instrumentsthatmake upour marketscan be boughtand soldat any pointintime
while others cannot.
Part 3 of this paper elaborated upon the existence of deviation from the PPP condition, which
createsfrictionbetweeninternational economiesresultinginarbitrageopportunities.Understandingthat
the arbitrage opportunities do in fact exist in these markets fosters an environment where generating
positive average returns becomes a possibility.
The potential positiveexpectedreturnstocarrytrade strategiescanbe improvedfurtherthrough
strategiesthatallowaninvestortoremaininacashpositionif theyexpectreturnstobe small oruncertain,
such as an optimally designed portfolio. While carry trade is inarguably a risky form of financial
investment, returns to carry portfolios are hardly justified on the basis of trader’s preferences of risk
exposure alone.
To validate this study we will build a model that focusesexclusivelyon the carry sub-strategyof
the VECM model.5
In this model we regress excess return to three different currencies on the current
differences in interest rates between their respective economies. Because of the short memory
characteristicof financial forecasts,wewillbe forecastingatamonthlyfrequencyaswell asataquarterly
frequency. The trade direction will be an investment in the Brazilian Real, the Canadian Dollar, and the
Chinese Yuan from the United States Dollar. The Regression will then be:
In the calculationof excessreturn,the variable∆st+1 representsthe change inthe logexchange rate from
one periodto the next.Make note that inthismodel the measure of exchange rate isin termsof foreign
currency per United States Dollar and that i* denotes the “funded” economy (The U.S. Dollar nominal
interest rate is denoted by i) that the carry trade will invest in. After running the regression, we will use
the descriptive statisticsgeneratedintestingforstatistical significance. Uncoveredinterestparityimplies
5 For details on this model see “Infrequent Portfolio Decisions:A Solution to the Forward DiscountPuzzle”

23. 23
the value of excess return to be equal to one. We are testing if the difference in interest rates has
predictive powerinthe excessreturninthe nextperiod.There willbe predictivepowerif βisstatistically
differentfromzero.The testwillthenbe onthe null hypothesisthatβ=0, andthe alternative will be that
β < 0. The model estimation results at monthly and quarterly frequencies are as follows:
Monthly Predictable Excess Returns
qt+1 = α + β(it – it*) + Ɛt+1
Currency β σ(β) R2
T-Statistic P-Value
CanadianDollar -1.071980 .063570 .356183 -16.86309 .00000
BrazilianReal -.967640 .067705 .551669 -14.29203 .00000
Chinese Yuan -.986021 .009943 .983402 -99.17174 .00000
Average -1.01 0.05 0.63 -43.44 0.00
Let us focus on the resultsof our model at the monthlyfrequency. All three tradesresultedina
negative expectedvaluesof excess return.Consideringthe large t-statisticsaswell asthe low p-valuesof
all three processes,the resultsindicate the predictabilitycoefficientsare statisticallydifferentfromzero.
All three processesrejectthe null hypothesisinfavorof the alternative,thusthe modelsexhibitpredictive
power. The R2
, which measures the “goodness of fit” of the model indicates how much the regression
capturesthe variation of the dependentvariable.We cansayat a monthlyfrequency,the model we have
builtexplains35.62%of the returnto the CanadianDollar,55.17% of the returnto the BrazilianReal,and

24. 24
98.34% of the returnto the Chinese Yuan. Consideringourmodelsonlycapture interestrate differentials
inexplaininginterestrate fluctuation,one mightinquire astohow the variationof the Chinese Yuanisso
high (a near perfect fit), but the reason is actually quite simple. Currencies with fixed exchange rates
againstthe dollar(suchas the Yuan) generate a value of close to zero in the coefficient∆st+1.This causes
the equation to become –(i – i*) = α + β(it – it*) + Ɛt+1.
Quarterly Predictable Excess Returns
qt+1 = α + β(it – it*) + Ɛt+1
Currency β σ(β) R2
T-Statistic P-Value
CanadianDollar -.998531 .118277 .295403 -8.442313 .00000
BrazilianReal -.902532 .245386 .200329 -3.678014 .00005
Chinese Yuan -.976701 .042434 .907501 -23.01711 .00000
Average -0.96 0.14 0.47 -11.71 0.00
Now let us examine the results of our model at a three month frequency. Again, all three
processesresultedinanegativeexpectedvaluesof excessreturn,large t-statistics,andlowp-values.With
the t-statistics statistically different from zero, we once again reject the null hypothesis in favor of the
alternative inall three cases.Thisiswhat we wouldexpect,asthe change in frequencyshouldhave little
effect on the statistical significance of the regression parameters. The R2
, measures have all decreased
from their monthly values, which reflects the additional movement in the variables over the additional
twomonthsof marketactivitybetweentrades.Forecastingatquarterlyhorizons, the modelwe havebuilt
explains 29.54% of the return to the Canadian Dollar, 20.03% of the return to the Brazilian Real, and
90.75% of the return to the Chinese Yuan.
Using the descriptive statisticsof these processeswe are able to graphthe fittedline againstthe
realized returns to generate the residual series for each process:

25. 25
All six time series seem to exhibit the stochastic properties we are looking for. To inquire as to
whether the residuals are white noise processes, we compute their autocorrelation and partial
autocorrelation functions as follows:
The resultsindicate,however,thatall of our processesexhibitsome lineardependence.Inall six
series,there isone to three spikesinthe PACFand a slow decay towardzeroin the ACF.This impliesthe
dependent variable would be better explained through an autoregressive structure of higher order
(depending on the series) that includes lagged values of the regressors. This is very much expected
consideringwe’ve alreadydiscussedmany sourcesof deviationfromUIP thatcan cause wedgesbetween
price levelsforyearsatatime. Manyfinancial time seriesare improvedbyincluding laggedvaluesof their
regressive variables due tononsynchronoustrading,andinfrequentportfoliodecisions.Aswe nowknow,
the equilibrium expected returnsare positive and time varying, with portfolios optimally designed,one
can hedge against changes to future expected returns.
To conclude our study, we observe the β coefficients of our empirical analysis.In each of our
models, the β coefficients were all statistically different from zero. This proves not only that UIP fails to
hold in the short run, but also that there are variables such as interest rates that yield predictive power
over the deviation in UIP. Executing carry trades based on models such as this carry strategy will, on
average,be successful ingeneratingpositive portfolioreturns.Includingothervariables(assuggestedin
the VECM) will only increase the accuracy of the models used to forecast carry trade.

26. 26
Footnote References
1
International Economics:TheoryandPolicy
Paul R. Krugman; Maurice Obstfeld;Marc J.Melitz
Textbook, Part3
Stable URL:
http://www.acsu.buffalo.edu/~twang28/Krugman-%20International%20Economics%209ed%202011.pdf
2
A note on ParityConditions(CIRP,“FP”,UIRP,PPP) andCarry Trades
Michel A.Robe
WorkingPaper, Kogod School of Business,AmericanUniversity,Washington,DC
Stable URL:
http://www1.american.edu/academic.depts/ksb/finance_realestate/mrobe/302/Handouts/IRP_note.pd
f
3
Exchange Rate DynamicsRedux
Maurice Obstfeld;KennethRogoff
The Journal of Political Economy,Vol.103, No.3. (Jun.,1995), pp.624-660.
Stable URL:
http://links.jstor.org/sici?sici=0022-3808%28199506%29103%3A3%3C624%3AERDR%3E2.0.CO%3B2-6
4
Carry Trade
Òscar Jordà
WorkingPapers,Universityof California,Departmentof Economics,No.10, 18. (2010)
Stable URL:
http://hdl.handle.net/10419/58381
5
InfrequentPortfolioDecisions:A Solutiontothe ForwardDiscountPuzzle
Bacchetta,P. and E. van Wincoop
WorkingPapers,CEPRand NBER, (2007)
Stable URL:
http://www.hec.unil.ch/pbacchetta/PDF/forexnew51-1.pdf