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Financial Econometrics Concerning Carry Trade: Empirical Analysis

Francis Dixon

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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 currency markets, the potential drivers of the friction between these markets which 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

analysis of the empirical results of these statistical instruments. We will conclude with an explanation of

our findings from a strictly empirical perspective. Let us first lay the foundation for this study by defining

the key terms and general notations that will be used throughout this paper, and by reviewing the of short

and long run equilibrium condition of domestic money markets as well as their relationship to equilibrium

in foreign exchange markets.

The notation to specify the time interval associated with a “holding period” will be represented

by subscripts. The beginning of an asset’s holding period will be denoted by subscript (t), with the end of

that the holding period denoted by subscript (t+1).

The notation to specify whether a variable is a characteristic of the domestic economy or a foreign

economy will be represented by superscripts. Domestic variables of interest will be denoted by superscript

(*), with foreign variables of interest will be denoted by lack of a subscript. The domestic economy will be

the economy in which an asset is funded, and the foreign economy will be the economy in which an asset

is borrowed. For simplicity strictly when generating formulas or expressions the domestic economy will

be the United States, and the foreign economy will be Europe, unless noted otherwise.

The nominal interest rate, denoted as i, represents the interest earned on financial assets held in

an economy. The real interest rate, denoted as r, represents the interest earned on financial assets held

in an economy less that economy’s rate of inflation, where the inflation rate is denoted as π. The

mathematical representation of nominal and real interest rates are then, i = (1 + i) and r = i – π respectively.

The nominal exchange rate, denoted as ϵ, is the price of one US dollar in terms of foreign currency.

For Example, if it costs one and a half units of foreign currency to purchase one unit of domestic curre 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 unit of US goods in terms of foreign goods, where the domestic price level will be denoted as

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).

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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 quantity of currency exchanged on the horizontal axis. The upward sloping curve represents the

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 (making domestic imports of foreign goods and services cheaper). The downward sloping curve

represents the demand for domestic currency by foreign investors. To understand why it slopes

downward, notice when the nominal exchange rate falls the purchasing power of domestic currency falls

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

currency exchanged for foreign currency at any given nominal exchange rate. Note that in this model, the

nominal exchange rate is measured as €/$.

Supply in the nominal exchange market is essentially a combination of domestic demand for

foreign goods and services, and domestic demand for foreign financial assets (such as real estate, mines,

factories, bank accounts, stocks, bonds and treasury bills). There are two causes of shifts in the supply of

domestic currency: a change in the domestic level of income, and change in foreign nominal interest rate.

An increase to domestic income shifts the supply of domestic currency to the right, as domestic investors

1 For more information on international finance see “International Economics”

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will have the ability to buy more foreign goods and services. Rising foreign interest rates also shifts the

supply of domestic currency rightward. This is because domestic investors will want to hold more assets

in foreign currency. A rightward supply shift results in a decrease the domestic nominal exchange rate and

an increase in domestic output. The resulting decrease in the domestic nominal exchange rate represents

a domestic currency depreciation.

Similarly, demand in the nominal exchange market is essentially a combination of foreign demand

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

demand for domestic currency: a change in the foreign level of income and change in domestic nominal

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

currency also shifts right if domestic interest rates rise. This is because foreign investors will want to hold

more assets in domestic currency. A rightward demand shift results in an increases the domestic nominal

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 equilibrium quantity of

currency exchange at the intersection of the supply of and demand for currency, we are able to construct

a graphic representation that relates the nominal exchange rates to the level of domestic economic output

in the short run in Figure 2. Here the DD schedule is derived from all potential combinations of the supply

of and demand for currency in the short run equilibrium condition we discussed in Figure 1. Our objective

is now to relate the domestic money market to the foreign exchange market. In Figure 3, the AA schedule

represents all possible combinations of real exchange rates and output levels that keep the domestic

money market and foreign exchange 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

domestic taxes, an increase in investment demand, a decrease in the domestic price level relative to the

foreign price level, a decrease in domestic savings, and an increase in demand for domestic goods and

services. The AA schedule will shifts rightward in reaction to an increase in the domestic money supply, a

decrease in the domestic price level, an increase in the expected exchange rate at the end of an asset’s

holding period, an increase in the foreign interest rate, and a decrease in domestic real money demand.

It is obvious at this point of the natural link between the money markets and the foreign exchange

markets. We are able to show graphically this link through Figures 5 and 6, where we take a closer look at

the two causes of market fluctuation we discussed in Figure 1. Figure 5 displays how 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

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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.

Understanding how these markets relate to one another is essential in forecasting the equilibrium

condition of these markets in the long run. Now we will look at the long run effect of a permanent change

in the money supply, which as stated previously drives fluctuation in both markets by changing the level

of inflation (therefore changing the expected exchange rate) at the end of an asset’s holding period.

Figure 7 shows the relationship between the foreign exchange market and the money market in

long run equilibrium where the purchasing power parity condition holds. Figure 8 displays movement to

short run equilibrium in response to a permanent increase in the domestic money supply on the left side

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 measured as

$/€. Initially the markets are in long run equilibrium at point 1. If the domestic central bank were to

permanently increase the money supply by raising the rate of monetary growth (engaging in easy

monetary policies), there would be an immediate increase in the supply of domestic currency, which shifts

the real domestic money supply downward. Observe how the increase in the domestic money supply

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shifts domestic MS downward from MS1US to MS2

US, and causes the domestic rate of return shift leftward,

a decrease from R1$ to R2

$ in the domestic money market. To understand why, remember our 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 foreign currency increases, which is represented in the rightward shift in the expected euro return. 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, prices become flexible so that the domestic price level adjusts to the 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 P1US to P2

US, which increases the return to assets held

domestically from R2$ back to its initial level at R1

$ as well. The curve that represents the expected return

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 begin analyzing how arbitrage opportunities arise. The “Law of One Price” states that when

converted into a common currency, the prices of identical goods and services sold in different economies

should have the same value.2 Recall from Part 1 that the real exchange rate equals the nominal exchange

rate adjusted for difference in the price levels of two economies, q = ϵ(P*/P). To convert the different

price levels into one common currency. By rearranging the terms of this formula to solve for the domestic

price level. This condition creates what economist and currency investors refer to as “Purchasing Power

Parity” (PPP). Figure 9 exhibits how the concept of purchase price parity relates to the long run equilibrium

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 study of the many variants of purchase power parity begins by defining absolute and relative

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,

claims that the different in the two economies inflation rates will measure the degree of appreciation or

depreciation in a currency along the holding period. Relative PPP, is computed mathematically as Δϵ / ϵ =

π – π*, where Δϵ represents the change in the nominal exchange rate, π represents the rate of foreign

inflation and π* represents the rate of domestic inflation. Whereas in absolute PPP, if the law fails to hold

the real exchange rate fluctuates around 1, with relative PPP the real exchange rate remains constant as

movements in price levels are offset by movements in the nominal exchange rate. In practice, 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: spots and futures. A spot

transaction will be denoted as ϵ for simplicity, because the transactions price of a spot transaction in the

current time period is same as the nominal exchange rate. With futures contracts, denoted as F, the buyer

and seller of the currency instrument agree to reverse the transaction at a future date, but hold the

repayment price paid at the end of the holding period at the current nominal exchange rate. This holding

period could be a week, month, years, etc. It is important to understand in a futures contract, investors

have potential to speculate and hedge against risk, because the future nominal exchange rate may have

increased or decreased. The difference in these transactions foster two PPP conditions we will find to be

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

investing at the domestic nominal interest rate (i*) must earn the same return as an investor exchanging

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 exact is measured as CIPexact = 1 + i = (ϵ*t/F)(1 + i). It is typically more 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

measured as CIPapproximate = (F – ε*t/ ε*

t) = i – i*. When CIP holds, as it should, there should be no existence

of arbitrage opportunities. If conditions provoke CIP to fail, traders tend to borrow assets at low domestic

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interest rates and fund assets at the higher interest rates of foreign economies. The exploitation of 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

that on average, traders investing in domestic currency should make the same return on their investments

as investors buying foreign currency, earning the foreign nominal interest rate, and converting their

investments back to domestic currency at the domestic nominal interest rate at the time of maturity, or

(ε*t+1). The mathematical representation of uncovered interest parity is then, UIPapproximate = (E(ε*

t+1) – ε*t/

ε*t) = i – i*, where E(ε*

t+1) denotes the expected value of the domestic nominal exchange rate at the end

of the holding period.

Part 3: The “Forward Premium Puzzle” and the Theories that Seek Explain its Existence

The form of currency trading of we will focus on for the duration of this paper is carry trade. 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, such as carry trade, should generate zero expected return. However, historically this form of

trading generates small positive returns the majority of the time it is executed. Our study in Part 2

explained two variants of PPP, covered and uncovered interest parity. The UIP condition stated that the

degree of appreciation of the funded currency needed to eliminate arbitrage is directly related to the yield

spread between the two economies, where the yield spread is considered to be an unbiased predictor of

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

observation that national price levels should equal one another when converted to one common currency.

Most economists agree that the UIP and CIP conditions should hold, meaning a trader investing

in forward contracts with developing countries that have relatively higher interest rates should expect the

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

research of economists that succeeds in explaining deviation in the short run, fail to explain deviation in

the long run. Likewise, research that succeeds in explaining deviation in the long run, fails to explain

deviation in the short run.3

First, let us look at the short run. One explanation is the existence of transportation costs, tariffs,

and nontariff barriers create friction in the international markets. Transportation costs have the potential

to create differences in domestic and foreign price levels. Over the last decade the level of world tariffs

have been dramatically reduced, yet some remain in place and will inevitably discourage asset allocation

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

barriers as well as gaining some of the rents associated with international trade when government foreign

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, fails to completely explain deviations from PPP in the long run. After a period of about two years

prices typically become flexible, but convergence to long run PPP usually takes much longer. One of the

most popular economists today, Paul Krugman, explains some 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

international markets. While many attempts to limit monopolies ability to price the market through fiscal

policies, pricing the market does occur and continues to create some deviation in the short term.

Another hypothesis is known as the Balassa-Samuelson Hypothesis focuses more on the long run.

This theory argues that when all price levels are converted into dollars using the nominal exchange rate,

there are higher price levels associated with rich countries relative to poor countries because wealthier

countries tend to be more productive in the traded goods sector of their economies. In small developing

countries, the price level of the traded goods sector is tied to the world price level. Therefore, an increase

in productivity in these countries results in an increase in wages earned by laborers of traded goods. If the

3 For more information on the theories discussed in this section, see “The Purchasing Power Parity Puzzle”

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non-traded goods sectors fail to increase in productivity at the same rate as the traded goods sector, 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, this would

inevitably cause the CPI of the developing country to increase. If both components were to increase in

productivity simultaneously, there would be no relative price effect, and therefore no change to the real

exchange rate. Compared to the theory of stick prices which 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 similar long run theory predicts that rich countries will have higher exchange rate adjusted price

levels due to the fact that their capital-labor ratios are higher relative to poor countries, rather than being

dependent on the assumption of higher productivity. Since higher capital -labor ratios implies higher wage

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 two facts lead 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

sustained deficits are directly related with real exchange rate depreciation. While these two variables do

exhibit empirical correlation, there is much debate as to whether this correlation is a result of causation

due to the fact that there are many causes of current account deficits that are not directly related to real

exchange rates. Proponents of this theory are quick to emphasize correlation and causation because the

borrowing and lending between countries that is associated with sustained current account deficits leads

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

explanation of the forward premium puzzle in the short, medium, and long run. We conclude that in the

short run, PPP does not hold, and convergence to PPP in the long run can take many years. Most

explanations that tend to focus on monetary and fiscal policies are able to explain the puzzle in the 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 volatili ty in the short run. In reality a complete explanation of

deviation of PPP will be a multivariate model containing many of these theories simultaneously, assuming

some variables to be purely temporary and others permanent. Every day international goods markets are

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

markets as well as how arbitrage opportunities arise in them, the question becomes “what methods can

be used by the carry trader to capture return in the market. Let us begin by recalling one 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

econometric models we will build upon represent stochastic processes. The models we build will then be

used in generating forecasted outcomes of financial variables of interest.4 Let Et denote the expected

value of return at the current time period, and let xt+1 denote the return to a zero cost investment strategy

for a risk-neutral investor in the absence of arbitrage. We can then express this econometric principle

mathematically by:

What this expression implies is that on average, one would expect a return of zero to their carry

portfolio. Now we add a stochastic discount factor which will summarize the interaction of outcomes and

consumer preferences. This factor represents the potential combination of pseudo-probabilities implied

by the investor’s choice of consumption, and is determined by the outcomes these preferences associated

with the condition of world markets and other sources of risk so that the value of the trader’s investment

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 characteristics of stochastic

processes see “Carry Trade”

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To get a better idea of what this formula implies, assume we have borrowed one unit of domestic

currency at the domestic interest rate by selling the domestic security short, then we purchase the

equivalent unit of foreign currency with the same date of maturity that yields the foreign interest rate. 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

owned by an investor, and therefore must return the same security to the lender at an agreed upon date.

This strategy is used when anticipating a decline in value, resulting in repayment of the security less the

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 funded foreign currency is transformed back to domestic currency at

the spot exchange rate, denoted as ε*t+1. Along the term, the return generated by the foreign security 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

expected return on average, the expression with which we can measure the expected carry trade 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

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futures contract, denoted as Ft. Holding the assumption that the investor has risk neutral preferences, this

measure expresses covered interest parity where the forward rate is again an unbiased predictor of the

spot exchange rate at the time to maturity. The expected carry trade return of a futures transaction is

then:

To complete the trinity of parity conditions, we modify version of our expression for the expected

carry trade return of a spot transaction once more to reflect the value of the expected return in real rather

than nominal terms. Let the real interest rate, denoted rt, equal the nominal interest rate 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 let p*t = ln(P*t). Let the same notation method apply for 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 stochastic condition known as “first order weakly stationary” states that for a process to be

first order weakly, all random variables must have the same mean or expected value. For this expression

to become a weakly stationary process, we denote the fundamental equilibrium exchange rate (FEER) that

the natural logarithm of the real exchange rate (qt+1) reverts to in the long run as, ɋ. The expression then

becomes qt+1 = ɋ + Ø(pt+1 – p*t+1).

Now expression (4) which we obtained by taking the natural logarithm of the expected carry trade

return of a spot transaction can be rewritten as:

Again, assuming the absence of arbitrage, the degree of expected real exchange rate appreciation equals

the expected value of the spread between the real interest rates, that is, Et(qt+1) = Et(rt – r*t).

We can now use interaction of interest rates, exchange rates, and long run equilibrium 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:

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If the stochastic process of this system is linear, we use the Vector Error Correction Model (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, where we model two stochastic processes jointly. Second, each equation has the same

regressors as well as the same number of lags, built upon an autoregressive structure. Third, all variables

on the left and right hand side of the equation are stationary due to the cointegration that exists between

them. Fourth, there must be at least one adjustment coefficient different from zero (otherwise there

wouldn’t be 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, and the momentum signal. These signals are sub-strategies that

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 VECM model in more detail.

The carry strategy (C) can be expressed mathematically as Δět+1 = 0. This strategy focuses solely

on the expected value of the real domestic interest rate less the real foreign interest rate in determining

which currency should be sold short, and which to go long in. The momentum strategy (M) can be

expressed mathematically as Δě t+1 = βeΔet. Assuming βe to be equal to zero, this strategy uses the value of

the nominal exchange rate in the current period to be the best forecast of the exchange rate at the end

of the term. The value strategy (V) can be expressed mathematically as Δě t+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, and is one of the most accurate methods with which we can express the uncovered interest

parity.

The currencies which an investor should borrow and which an investor should fund can 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 VECM strategy result:

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The ex-post realized return of the investment is then:

Expressions (10) and (11) reveal a very important characteristic of carry trade . The probability of

generating a positive investment is more dependent on the investor’s ability to forecast the direction of

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 strategies are generated through statistical models using the properties of an

asset’s first and second moments. The “Trader’s Decision Problem” is that an investor must, however,

develop an investment strategy that relies more heavily on generating performance criteria that appeals

to a portfolio’s required return while respecting the degree of risk associated with individual performance

preferences. We begin how to solve this problem by modifying the expressions we covered in Part 4 to

determine the direction of a carry trade as well as calculate ex-post realized returns. We will use a more

general notation known as a “generic scoring classifier,” which enables the analysis of multiple classifiers

to generate forecasts, rather than just using the conditional moments of our information set. 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 direction expression then 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 c Є {-∞,∞}. Incorporating this notation, the total number of observations that encompass a carry

portfolio can be expressed by each individual forecast (decision) and their associated outcomes. Observe

Figure 10:

In this table, TN(c) refers to the true classification rates of negative outcomes, while TP(c) refers

to the true classification rates of positive outcomes. Likewise, FN(c) refers to the false cl assification rates

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of negative outcomes, while FP(c) refers to the false classification rates of positive outcomes. Negative

rates and positive rates are representative of currency sold short and currency funded long, respectfully.

As we would expect mathematically, the sum of TN(c) and FP(c) is equal to one, and the sum of FN(c) and

TP(c) is equal to one. If we let the total number of observations in a portfolio (N), equals the sum of total

shorts (S) and the total number of longs (L), then N = S + L. Through these denotations, we can compute

the empirical values of TN(c) and TP(c) as:

All possible combinations of true outcomes can be represented graphically in the same manner a

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

Classifier Frontier extends horizontally from the Y-axis at the value of 1 and vertically as well as X-axis at

the value of 1, and represents the points along the frontier in which all realized outcomes matched an

investor’s trade direction predictions. The Uninformative Classifier Frontier, represents a classifier where

TP(c) = FP(c) = 1 – TN(c) for any scalar value. This line is often referred to as the “coin-toss” diagonal. Utility

optimization, as we would expect, occurs at the point of tangency along the CC frontier. In equilibrium

(point of tangency) the marginal rate of substitution between profitable longs and shorts would be equal

to -1.

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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 classification ability of a classifier against a coin-tosser. The KS statistic is measured by the vertical

distance between the CC frontier and the coin-toss diagonal. The formula used in measuring the KS

statistic is calculated:

The area under the CC frontier, the AUC, is an alternative to the KS statistic that summarizes the

characteristics of the CC frontier with better definition. In the graph, the AUC for a perfect classifier equals

1, and for a coin-tosser equals .5. In practice, most portfolios result in an AUC value somewhere between

perfect classification and coin-toss. The KS statistic and AUC are both measures of an investor’s success in

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, or when trades are made with differing amounts of leverage . Therefore, the weighted value

of each trade must be adjusted to generate an accurate measure of a currency portfolio’s profitability.

Using modified statistics to adjust for these differences, we define the total portfolio return equal to the

sum of total return to short trades (BS) and total return to long trades (BL). These totals are measured by

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:

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We these statistics we can measure the values of TN(c) and TP(c) adjusted for 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 gains and losses. Let gains (G) equal the product of total returns

to shorts and the return-weighted statistic for total number of true negative classification rates plus the

product of total returns to longs and the return weighted statistic for total number of true positive

classification rates. Likewise, let losses(L) equal the product of total returns to shorts and the return-

weighted statistic for total number of false positive classification rates plus the product of total returns to

longs and the return weighted statistic for total number of true negative cl assification 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 this function for portfolio utility can be expressed in terms of the portfolio’s gains -losses

ratio, maximizing the ratio of gains to losses is the same as maximizing a portfolio’s utility. The portfolio

utility function in terms of portfolio gains and losses is expressed by:

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Using this adjusted function for portfolio utility, we define the point at which optimal portfolio utility is

being achieved. At this point, the slope of the CC frontier, represented by –BS/BL, again equals -1. The

following derivative represents the point along the portfolio utility function where utility is maximized:

In practice, portfolio returns are 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:

In the absence of arbitrage opportunities a portfolio’s expected return is zero, where the value of

total portfolio gains is the same as the value of the total portfolio losses. This implies the gain-loss ratio

would be equal to one. In the presence of arbitrage, this ratio can take any approximation in an interval

from negative infinity to 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

demand in the currency markets encourage investors to reduce their holdings of risky assets , resulting in

potential losses in the short run via the order flow effect which impacts the price of trades.

Order flow is a standard practice in the brokerage industry where brokerage firms receive a small

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

market centers, but based on other factors such as quantity, quality, and type of the order flow presented

to the market. Statistical analysis indicates, however, than order flow does not completely account for the

positive average returns.

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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 major financial instruments that make up our markets can be bought and sold at any point in time

while others cannot.

Part 3 of this paper elaborated upon the existence of deviation from the PPP condition, which

creates friction between international economies resulting in arbitrage opportunities. Understanding that

the arbitrage opportunities do in fact exist in these markets fosters an environment where generating

positive average returns becomes a possibility.

The potential positive expected returns to carry trade strategies can be improved further through

strategies that allow an investor to remain in a cash position if they expect returns to be small or uncertain,

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 focuses exclusively on the carry sub-strategy of

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

characteristic of financial forecasts, we will be forecasting at a monthly frequency as well as at a quarterly

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 calculation of excess return, the variable ∆st+1 represents the change in the log exchange rate from

one period to the next. Make note that in this model the measure of exchange rate is in terms of 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 statistics generated in testing for statistical significance. Uncovered interest parity implies

5 For details on this model see “Infrequent Portfolio Decisions: A Solution to the Forward Discount Puzzle”

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the value of excess return to be equal to one. We are testing if the difference in interest rates has

predictive power in the excess return in the next period. There will be predictive power if β is statistically

different from zero. The test will then be on the null hypothesis that β = 0, and the 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

Canadian Dollar -1.071980 .063570 .356183 -16.86309 .00000 Brazilian Real -.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 results of our model at the monthly frequency. All three trades resulted in a

negative expected values of excess return. Considering the large t-statistics as well as the low p-values of

all three processes, the results indicate the predictability coefficients are statistically different from zero.

All three processes reject the null hypothesis in favor of the alternative, thus the models exhibit predictive

power. The R2, which measures the “goodness of fit” of the model indicates how much the regression

captures the variation of the dependent variable. We can say at a monthly frequency, the model we have

built explains 35.62% of the return to the Canadian Dollar, 55.17% of the return to the Brazilian Real, and

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98.34% of the return to the Chinese Yuan. Considering our models only capture interest rate differentials

in explaining interest rate fluctuation, one might inquire as to how the variation of the Chinese Yuan is so

high (a near perfect fit), but the reason is actually quite simple. Currencies with fixed exchange rates

against the dollar (such as 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

Canadian Dollar -.998531 .118277 .295403 -8.442313 .00000 Brazilian Real -.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

processes resulted in a negative expected values of excess return, large t-statistics, and low p-values. With

the t-statistics statistically different from zero, we once again reject the null hypothesis in favor of the

alternative in all three cases. This is what we would expect, as the change in frequency should have 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

two months of market activity between trades. Forecasting at quarterly horizons, the model we have built

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 statistics of these processes we are able to graph the fitted line against the

realized returns to generate the residual series for each process:

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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 results indicate, however, that all of our processes exhibit some linear dependence. In all six

series, there is one to three spikes in the PACF and a slow decay toward zero in the ACF. This implies the

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

considering we’ve already discussed many sources of deviation from UIP that can cause wedges between

price levels for years at a time. Many financial time series are improved by including lagged values of their

regressive variables due to nonsynchronous trading, and infrequent portfolio decisions. As we now know,

the equilibrium expected returns are 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 predict ive power

over the deviation in UIP. Executing carry trades based on models such as this carry strategy will, on

average, be successful in generating positive portfolio returns. Including other variables (as suggested in

the VECM) will only increase the accuracy of the models used to forecast carry trade.

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Footnote References

1 International Economics: Theory and Policy Paul R. Krugman ; Maurice Obstfeld ; Marc J. Melitz Textbook, Part 3 Stable URL: http://www.acsu.buffalo.edu/~twang28/Krugman-%20International%20Economics%209ed%202011.pdf 2

A note on Parity Conditions (CIRP, “FP”, UIRP, PPP) and Carry Trades Michel A. Robe Working Paper, Kogod School of Business, American University, Washington, DC Stable URL: http://www1.american.edu/academic.depts/ksb/finance_realestate/mrobe/302/Handouts/IRP_note.pdf 3

Exchange Rate Dynamics Redux Maurice Obstfeld; Kenneth Rogoff 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 Working Papers, University of California, Department of Economics, No. 10, 18. (2010) Stable URL: http://hdl.handle.net/10419/58381 5 Infrequent Portfolio Decisions: A Solution to the Forward Discount Puzzle Bacchetta, P. and E. van Wincoop Working Papers, CEPR and NBER, (2007) Stable URL: http://www.hec.unil.ch/pbacchetta/PDF/forexnew51-1.pdf