SHORT-RUN EXCHANGE-RATE DYNAMICS:

EVIDENCE, THEORY, AND EVIDENCE

John A. Carlson, Purdue University

Carol L. Osler, Brandeis International Business School

Abstract

Research in currency market microstructure has by now revealed a wealth of information about the relationship between currency trading and exchange rates. This paper develops a model of short-run exchange-rate dynamics consistent with these observations, and applies it to the forward premium puzzle. Despite the newness of the evidence, the underlying structure of the model is not new. Indeed, over the past two decades a number of authors have independently developed models based on the same underlying structure (e.g., Black 1985, Driskill and McCafferty 1980a, 1980b, 1982, 1992, Driskill, Mark, and Sheffrin 1987, Osler 1995, 1998, Carlson and Osler 2000, Hau and Rey 2004, Sager and Taylor 2005a). Using calibrated simulations we show that the model's predicted short-run exchange-rate behavior fits all the stylized facts associated with the forward premium puzzle. The model implies a linear equation governing short-run exchange-rate dynamics. Regressions based on this linear equation, using quarterly data for five currency-pairs, strongly support the model. In addition to their broad support for this microstructure-based model, these regression results suggest that microstructure may be relevant to exchange-rate dynamics at macro horizons. [JEL classifications: F31, G12, G15]

March 2005

Corresponding author: Carol Osler, Brandeis International Business School, Mailstop 032, Brandeis University, Waltham, MA 02454. Tel.: 781-736-4826. Fax: 781-736-2269. Email: cosler@brandeis.edu. The authors thank Gijoon Hong and Filippo Rotollo for excellent research assistance. We take responsibility for all errors.

SHORT-RUN EXCHANGE-RATE DYNAMICS:

EVIDENCE, THEORY, AND EVIDENCE

Research in currency market microstructure has by now revealed a wealth of information about

the relationship between currency trading and exchange rates. It has taught us that exchange-rates are

strongly influenced by interdealer order flow (Evans and Lyons 2002). It has highlighted the significance

of heterogeneity by showing that exchange-rates are positively cointegrated with order-flow initiated by

financial institutions and negatively cointegrated with order-flow initiated by importers and exporters

(Bjønnes et al. 2004, Mende et al. 2005). The importance of heterogeneity is also revealed by the fact that

importers and exporters’ net demand for currency is strongly correlated with the current account while

financial institutions’ net demand is strongly related to bond and stock returns, but not to the current

account (Bjønnes et al. 2004). Microstructure analysis has also shown that currencies are traded in a

wholesale market where virtually all trading agents work for institutions, and that speculative traders are

motivated by profit-determined bonuses and constrained in their risk-taking by loss-limits (see below).

This paper develops a model of short-run exchange-rate dynamics consistent with these

observations, and applies it to the forward premium puzzle. Thus the model’s structure is based on new

but existing microstructure evidence, and its implications are consistent with further evidence we provide

here. Despite the newness of the microstructure evidence, the underlying structure of the model is

anything but new. Indeed, over the past two decades a number of authors have independently developed

models of short-run exchange-rate dynamics based on the same underlying structure (e.g., Black 1985,

Driskill and McCafferty 1980a, 1980b, 1982, 1992, Driskill, Mark, and Sheffrin 1987, Osler 1995, 1998,

Carlson and Osler 2000, Hau and Rey 2004, Sager and Taylor 2005a). The model has not become widely

familiar, however, presumably because some of its key elements are not widely accepted despite their

strong empirical support. We suggest that, in light of the microstructure evidence as well as some new

theoretical arguments, renewed consideration of this model is appropriate.

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The underlying structure of the model has three key elements: 1. Risk-averse speculators

concerned with profits; 2. Non-speculative agents that respond linearly to exchange rate levels; 3. An

equilibrium condition requiring equality of flow currency demand and supply. For a long time the strong

correspondence between these elements and reality could not be formally documented, given the paucity

of data on currency trading. That constraint has now been eased.

Elements (1) and (2) imply that the model’s agents are heterogeneous. The heterogeneity of

currency market participants is stressed in Sager and Taylor’s (2005a) detailed description of the market.

They divide participants into two groups that correspond fairly closely to the model’s speculative and

non-speculative agents.

The two key properties of the model’s speculators – their focus on profits, rather than

consumption, and their limited willingness to take on risk – are both consistent with the reality we strive

to reflect in our model. Currencies are traded in wholesale markets where virtually all participants work

for larger institutions. The institutions, which determine the traders’ incentives, consciously focus the

traders on profits by compensating them with profit-determined bonuses. Meanwhile, they limit the

traders’ ability to take on risk via explicit loss limits and position limits.

The responsiveness of non-speculative agents to exchange-rate levels, rather than to speculative

considerations, is consistent with the fact that the order flow of importers and exporters, as a group, is

negatively related to a currency’s value (Bjønnes et al. 2004, Mende et al. 2005, Sager and Taylor 2005b).

The approximate identification of non-speculative agents with current-account traders such as importers

and exporters is further supported by evidence showing that non-commercial deal flow in SEK/EUR is

related to current account dynamics while financial-customer deal flow is not (Bjønnes et al. 2004).

The third key element of the model’s structure, its equilibrium condition, fits well with the

evidence that order flow – defined as the difference between trades initiated by buyers and those initiated

by sellers – is a key influence on short-run exchange-rate returns (see Lyons (2001), and Evans and Lyons

(2002), inter alia). Our equilibrium condition implies that flow currency demand and supply determine

exchange rates, thus capturing the essence of that key lesson.

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Beyond the microstructure evidence, we support the use of an equilibrium condition based on

flow, instead of a more traditional stock equilibrium condition, with two additional arguments, one old

and one new. Both arguments attempt to explain why exchange-rate models based on the traditional

assumption have not been more empirically successful (Meese and Rogoff 1983, 1997). The old argument

points out that the poor empirical record could reflect, in part, the models’ typical assumption of short-run

purchasing power parity, the empirical failure of which is well-documented.

The new argument suggests a theoretical failing in the traditional condition that aggregate money

demand equals aggregate money supply. A straightforward application of the Baumol-Tobin model of

money demand (Baumol 1952, Tobin 1956) shows that money demand is unlikely to be uniquely

determined at short horizons. Similarly, standard institutional aspects of the money supply process imply

that money supplies may also not be uniquely determined at short horizons. As a result, money demand

and supply are unlikely to constrain each other at short horizons and thus exchange rates cannot be

uniquely determined in the short run by the traditional money-market stock-equilibrium condition. This

argument is explored more closely in Section I.

The model we analyze is in some ways an updated version of the familiar portfolio balance model

(Branson 1975). It is updated in the sense that, unlike speculators in the original models, our speculators

are fully rational, and their portfolio choices represent closed-form solutions to an explicit utility-

maximization problem. Indeed, our version of the model adopts many off-the-shelf structures now

standard in finance, such as the interaction of speculative and non-speculative agents. In the international

context such non-speculative agents, known as “liquidity traders” in finance, can be easily identified with

real-world counterparts such as importers and exporters.

We demonstrate the model’s empirical relevance by applying it to the familiar forward premium

puzzle and related empirical regularities. We focus on the forward premium puzzle because our model is

explicitly intended to explain short-run exchange-rate dynamics and the forward premium puzzle has

primarily been documented with respect to short-run returns (meaning returns under one year).1 Research

1 Surveys of this literature can be found in Hodrick 1987, Froot and Thaler 1990, Lewis 1995, and Engel 1996).

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shows that high-interest rate currencies tend to appreciate and that currency risk premiums appear to be

highly variable and strongly related to interest differentials. In addition, the volatility of exchange rates is

high, exceeding both the estimated volatility of risk premiums and the volatility of interest differentials

(Bekaert ‘s (1996) “volatility puzzle”), and the autocorrelation of exchange-rate changes is close to zero

while that of interest differentials is fairly high (Bekaert’s “persistence puzzle”).

Using calibrated simulations and regressions we show that our model is consistent with all these

puzzles. The simulations, which require the model to fit reality in a number of key dimensions, show that

it can account for all the empirical regularities associated with the forward premium puzzle. Our

regression equation, which comes directly from the model, suggests that traditional risk premium

regressions are mis-specified due to the exclusion of two important terms. The regression results, based

on quarterly data for five currency-pairs, strongly support the model's specific predictions. The

coefficients are generally statistically indistinguishable from their predicted values, and the regressions’

predictive power far exceeds that of traditional risk premium regressions.

Our results suggest that this model is broadly relevant for understanding short-run exchange-rate

dynamics. It also suggests that currency microstructure, or the study of currency trading per se, may have

macroeconomic relevance. It was currency microstructure research that demonstrated empirically the

importance of order-flow for high-frequency exchange-rate dynamics. At horizons of a few minutes,

hours, days, and even weeks the importance of order flow is demonstrable, but at horizons of a quarter or

longer its relevance remains questioned. This paper joins other recent works in highlighting the potential

relevance of microstructure for exchange-rate dynamics at macroeconomic horizons (e.g., Bjønnes et al.

2004, Evans and Lyons 2005, and Evans and Lyons 2004)

Our empirical support for this model is not the first it has gathered. Most notably, Driskill, Mark,

and Sheffrin (1987) shows that their version of the model “is broadly consistent with recent Swiss/U.S.

data.” The model explains the empirical record well in-sample and it also out-performs the random walk

out-of-sample. Driskill and McCafferty (1992), which analyzes an extended version of the model

including explicit labor and goods markets and a more clearly elaborated money market, show that the

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model ‘can … account for the ‘stylized facts’ of the open economy. …Our model gives rise to persistent

deviations of relative prices from purchasing power parity, … [and] is consistent with higher variability of

the exchange rate relative to the price level” (p. 260). Nonetheless, the model has not gained wide

familiarity within the international economics community, in part because some of its assumptions are

non-standard. Given the empirical support now available for each element of the model’s underlying

structure, and given the model’s empirical success at matching the observed properties of risk premiums,

renewed consideration of this model as a baseline for analyzing short-run exchange-rate dynamics may be

appropriate.

The forward premium puzzle has been examined by many other researchers; we are thus forced to

be selective in our review of the relevant literature. Research that considers the forward premium in a

portfolio-balance context has not been considered successful by the profession, in part because the U.S.

overall net asset position does not change sign with sufficient frequency to explain the variable behavior

of risk premiums (Lewis 1995). Despite its short-run focus, our model shares many properties with

portfolio-balance models, including the importance of net international asset positions for risk premiums.

However, the net asset positions of interest in our model are those associated with short-run trading,

rather than a country’s overall net asset position. We propose that short-run assets are the most relevant to

the forward premium puzzle, since the puzzle applies only to short-run forward premiums (Chinn and

Meredith 2002). Net positions in short-run assets could well change sign more frequently than net

positions of all assets. While appropriate data do not exist to test this hypothesis, we note that net futures

positions of large currency speculators changed sign between 4 and 9 times per year during the period

January 1993 through May 2003.

Other researchers have examined the forward premium puzzle in models where trading decisions

are based on consumption, and exchange rates are determined by stock equilibrium in the money market

(e.g., Bekaert 1996, Moore and Roche 2002, Backus et al. 1993). Exchange rates in these studies match

some but not all of the empirical regularities associated with the forward premium puzzle.

6

Only a few other treatments of the forward premium puzzle have connected currency risk

premiums with interest differentials as we do here: these include Obstfeld and Rogoff (1998), Hagiwara

and Hierce (1999), Mark and Wu (1998), and Meredith and Ma (2002). The conclusions of these papers

are complementary to ours, since they are principally relevant to fairly long time horizons: The

purchasing power assumption of Obstfeld and Rogoff (1998) is widely known to be relevant only at long

horizons. The global portfolio-balance asset-market equilibrium assumption of Hagiwara and Hierce

(1999) also does not seem to be empirically relevant at short horizons. Mark and Wu’s (1999)

overlapping generations model would only be relevant to quarterly data if a trader’s lifetime spanned six

months. The analysis in Meredith and Ma (2002) relies on the endogenous response of monetary policy to

the real effects of currency changes; since real exchange rate changes affect trade and output with

substantial lags, this process that would generally take at least a year.

Section I of this paper introduces our model of exchange-rate determination, provides new

theoretical perspectives on that model's relevance, and shows how the model is consistent with recent

empirical evidence on currency market microstructure. Section II discusses the forward premium puzzle

and some previously proposed explanations. Section III derives the properties of risk premiums in the

basic model. It also demonstrates the model’s flexibility by introducing two modifications: interest-

sensitive non-speculative demand and permanent as well as transitory variations in non-speculative

demand. Section IV uses calibrated simulations to demonstrate that the model can account for all the

well-documented empirical regularities associated with the forward premium puzzle, including the

“volatility” and “persistence” puzzles. Section V presents regression evidence using quarterly data for

five currency pairs that supports the model. Section VI concludes.

I. A MODEL OF SHORT-RUN EXCHANGE-RATE DYNAMICS

This section presents our model of short-run exchange-rate dynamics. As noted in the

introduction, the underlying structure of this model is not new. To the contrary, models sharing the same

structure have been developed independently over the past two decades by numerous researchers,

including Black (1985), Driskill and McCafferty (1980a, 1980b, 1982, 1992), Driskill, Mark, and Sheffrin

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(1987), Osler (1995, 1998), Carlson and Osler (2000), and Hau and Rey (2004). In addition, Sager and

Taylor (2005a) provide a "thumb-nail sketch" of an exchange-rate model that closely fits the model

developed here in many (though not all) respects.

Though the model is not new, our theoretical arguments in support of the model are new, and

much of the empirical evidence we cite has not been applied to evaluate exchange-rate models. Before

presenting the model formally we review this evidence and discuss why the standard money-market

equilibrium condition is unlikely to constrain exchange rates at short horizons.

A. The Model and the Microstructure Evidence

The underlying structure of the model has three key elements. 1. Rational speculators with

constant absolute risk aversion that maximize the expected utility of profits; 2. Non-speculative agents

that respond to exchange rate levels; 3. Equilibrium characterized by equality of flow currency demand

and supply.

The division of agents into speculative and non-speculative types is supported by Sager and

Taylor (2005a), which describes in detail the market’s varied agent types. The paper ultimately suggests

that these agents can be divided into two classes, "push" and "pull" agents. Push agents, who are

essentially active informed speculators, correspond to our rational speculators. Pull customers, who are

"attracted or pulled into the market by price movements; effectively they exercise an option to trade once

the price crosses their implicit 'strike price'" (p. 19), correspond to our non-speculative agents.

The decision to model speculators as motivated by profits, rather than consumption, is driven by

our commitment to reality-based modeling. Currencies are traded in a wholesale market in which the vast

majority of currency trades are initiated by institutions such as banks, corporations, and asset managers.2

Thus the incentives that matter directly to exchange-rate determination are those faced by the currency

traders at these institutions. These traders might be motivated by consumption dynamics if their

incentives were perfectly aligned with those of consumers/shareholders. However, there are many layers

2 Sources at the Bank for International Settlements and in the private sector estimate informally that retail trades account for less than one percent of total currency trading. Exact statistics do not seem to exist.

8

of agency relationships between currency traders and the ultimate consumers/shareholders of their

institutions. At a bank, for example, the line of responsibility begins with the Board of Directors and runs

through the CEO, the Treasurer, the head of trading, the chief dealer, and finally the traders themselves.

Asymmetric information and associated agency problems are present at every link in this chain, and

incentive schemes must be used to minimize the misalignment between the interests of these employees

and those of the institutions’ shareholders. In practice “the interests of shareholders” is taken to be

maximum share value, so the incentive schemes for traders focus on their personal profitability.3 Half or

more of a trader’s compensation typically comes from an annual bonus heavily influenced by his/her

profits (the rest is salary).4 It is for this reason that traders in our model focus on profits rather than

consumption.5

The decision to model currency traders as risk averse might seem inconsistent with our

commitment to reality-based modeling, since currency traders will not survive as such without a high

tolerance for risk. Nonetheless, currency traders face institutional incentives to avoid risk, so they are

likely to exhibit risk-averse behavior whatever their personal level of risk tolerance. Most significantly,

they face the gambler’s ruin problem: if they run into a long series of losses, they will shortly be out of a

job, even if their profitability would ultimately have been outstanding had they been permitted to continue

trading. Such traders will behave as if they are risk averse (Carlson 1998). Traders will also behave as if

they are risk averse because they face explicit position limits and loss limits (these limits, which reflect

the agency problems outlined above, are now standard across all trading institutions (Oberlechner 2005)).

For some short-term traders, risk also directly affects their annual bonus.

3 One of the authors is married to a fifteen-year currency market veteran and has ongoing professional relationships with traders and management at one of the ten biggest currency trading banks. 4 Even though agency problems in currency markets are not yet the subject of widespread research they seem likely to be an important influence on reality. Bensaid and DeBandt (2000) have already explained the use of stop-loss limits for currency traders using agency theory. Agency problems more generally have been a major theme in corporate finance research since Jensen and Meckling (1976), and the real-world importance of such issues was recently highlighted anew by a wave of major corporate scandals. 5 In stock markets, for which these general-equilibrium models were originally developed, individual consumers generally undertake a substantial fraction of all trading, so consumption could more realistically be relevant to equity trading and stock-price dynamics.

9

The third key feature of the model, its equilibrium condition, reflects a key lesson from recent

empirical research on currency microstructure, that exchange-rates are strongly determined by order flow.

Evidence has long existed that order flow is an important determinant of U.S. equity returns (e.g., Shleifer

1986, Holthausen et al. 1990). Nonetheless, the idea that order flow matters for currencies was not widely

accepted among international economists until the late 1990s, when rigorous supporting evidence

appeared based on new high-frequency currency data (Lyons 2001; Evans and Lyons 2002). Since then,

confirming evidence has appeared from many quarters (e.g., Bjønnes and Rime 2001).

Theory and evidence both indicate that the responsiveness of exchange rates to order flow

reflects, at least in part, the fact that order flow contains substantial information about exchange-rate

fundamentals (Evans and Lyons 2004), and may also include important information about transitory

forces (Mende et al. 2005). Exchange rates may also respond to order flow for inventory reasons (Ho and

Stoll 1981), though the exact process through which inventories would affect exchange rates is still

unclear (Osler 2005). Whatever the underlying microtheoretic reasons, flow demand and supply for

currencies is clearly a key determinant of short-run exchange rate dynamics, an insight this model

incorporates by requiring exchange rates to equate flow currency demand to flow currency supply.

The decision to model short-run exchange rates as determined by order flow is also motivated by

a second line of reasoning, which begins with the failure of models based on the more traditional

assumption that aggregate money demand equals aggregate money supply (Meese and Rogoff 1983,

1997). This failure is presumably related in part to the models’ typical assumption of purchasing power

parity, the empirical failure of which at short horizons is well documented.

Just as importantly, however, the models’ failure could reflect the irrelevance of its central

equilibrium condition for short-run exchange-rate dynamics. We next highlight, using long-established

theoretical insights and standard institutional features of banking systems, that money demand and supply

are unlikely to be uniquely determined at short horizons. If either one is not uniquely determined short-

run exchange rates cannot be uniquely determined by the money-market stock-equilibrium condition.

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Non-uniqueness of short-run money demand: In their widely respected inventory-theoretic

models of money demand Baumol (1952) and Tobin (1956) show that, on a given day, individuals have a

range of acceptable levels for their money balances, rather than a single point value. Only when an

individual’s balance reaches the boundaries of his/her range does s/he adjust his money holdings. As a

result, money demand at any point in time is not uniquely determined, and the equilibrium money supply

can fall anywhere within a range (determined by the aggregate of lower and upper bounds for each

individual’s acceptable money inventory) without generating any change in individual behavior, interest

rates, or exchange rates.

Non-uniqueness of short-run money supply: Standard reserve accounting procedures imply that

money supply is also not uniquely determined at short horizons, even if the central bank directly controls

reserves. In the U.S., for example, deposits and reserves are both taken as averages over two-week

periods, so there is inherently some intra-period flexibility in the aggregate supply of deposits. In

addition, the period over which average deposits are calculated begins and ends earlier than the period

over which average reserves are calculated. With neither money demand nor money supply uniquely

determined at short horizons, they seem unlikely to constrain each other sufficiently to have an important

influence on short-run exchange-rate dynamics.6

We now turn to a formal exposition of the model. We discuss the two agent types in turn, and

then present the model’s equilibrium. To close this section we show that evidence supports the model's

implied relationships between exchange rates and the order flow of each agent type, as well as the

model’s implications about the short-run and long-run determinants of exchange rates.

B. Rational Speculators

The model’s rational speculators exploit expected short-run exchange-rate changes to make

profits. Given the model’s short-term focus, these agents are intended to correspond to real-world

currency traders with short horizons, including foreign-exchange dealers, currency-fund managers, and

6 We stress the limited scope of this discussion. For the money market at long horizons, and for other asset markets at any horizon, we assume that stock supply and demand constrain each other sufficiently to affect prices.

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managers of other actively traded portfolios such as mutual funds, pension funds, and insurance funds. As

discussed above, reality dictates two important properties of our speculators: (1) they maximize the utility

of profits, rather than consumption; and (2) they behave as if they are risk averse.

Formally, we assume that speculators choose positions to maximize the expected utility of profits:

(1) Wt = Et(πt+1) - (θ/2)Vart(πt+1) Here, Wt represents welfare expected conditional on information at time t, θ is a measure of risk aversion,

Et(πt+1) denotes expected t+1 profits, and Vart(πt+1) denotes the conditional variance of profits with

information as of time t. This specification of utility, which is standard in asset-pricing models, is

equivalent to maximizing the expected value of a constant-absolute-risk-aversion utility function when

the exchange rate has a conditional normal distribution.

Every period a speculator takes a position of size bt, measured in units of foreign currency. The

profits on this position are proportional to the change in (the log of) the exchange rate, st+1 - st, minus the

short-term interest differential across countries, dt = rt – rt*:

(2) πt+1 = bt [st+1 - st - dt]. The speculator’s optimal bet is proportional to expected profits and inversely proportional to risk aversion

and risk itself:

(3) bt = [Et(st+1) - st - dt]/θ Var(st+1)

Var(st+1) is the expected variance of the exchange rate conditional on information at time t. As shown by

Carlson and Osler (2000), Var(st+1) is constant if exogenous influences on the conditional variance of the

exchange rate are constant. This is assumed here. The risk premium is defined as the rationally expected

excess return to foreign currency, rpt ≡ Et(st+1) - st - dt . When the expected return on foreign assets

exceeds that on domestic assets, the risk premium is positive, and speculators take a long position in

foreign currency. Conversely, speculators take a short position when the risk premium is negative. For

convenience we define q ≡ 1/[θ Var(st+1)] .

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Net foreign-currency demand from speculators corresponds to the change in their aggregate

desired foreign-currency position. If there are N speculators, aggregate net speculative demand can be

written: N(bt - bt-1) = Nq(rpt - rpt-1) .

C. Non-Speculative Currency Demand

We model net foreign-currency demand from non-speculative agents, FXt, as a simple linear

function of the exchange rate:

(4) FXt = Ct - S st , S > 0.

Ct summarizes the influence of all factors other than the exchange rate that might affect non-speculative

demand, such as goods and services prices, real incomes, and barriers to trade. We assume that S is

positive, which is equivalent to assuming that net foreign-exchange demand from non-speculative agents

satisfies the Marshall-Lerner-Robinson condition for foreign-exchange market stability familiar from

international trade theory, i.e., that demand for foreign exchange falls as its value rises. At present we

assume for convenience that interest rates do not influence non-speculative demand.

We think of “non-speculative” currency demand as including all traditional current-account

activities, including trade in goods and services, transfers of investment income, and both public and

private unilateral transfers. We also take it to include foreign direct investment, because empirical

evidence suggests that exchange rate levels, rather than their changes, are a major influence on direct

investment (Ray 1989, Froot and Stein 1991, Blonigen 1997). Many other items in the capital account

that are not influenced by exchange-rate expectations, such as official reserve holdings and official aid

flows, may also be included in this category of demand.

Some readers might suggest that our current account traders should be utility-maximizers. It

certainly would be possible to add structures to the model so that these agents become utility-maximizing

hedgers. However, such structures add complexity while contributing only marginally to the model’s

overall relevance. Non-utility-maximizing agents like our non-speculating traders have become a

common feature in asset-pricing models, where they are called “liquidity” or “noise” traders (e.g., Dow

13

and Gorton 1993). In the exchange-rate literature, this type of modeling structure has been used by other

authors including Hau and Rey (2003) and Black (1985).

In the international context, “liquidity” traders have the additional advantage of being readily

recognizable as representing international goods and services traders. By making these agents behave in a

partially random fashion, we capture the orthogonality of these agents’ concerns relative to those of

speculators. As importers and exporters attempt to maximize the profits associated with their primary

activities, they focus on factors that are not directly relevant to short-term speculators, such as overall

economic activity, relative price levels, and recent inventory cycles.7 We capture the influence of such

factors in the non-speculative demand component Ct.

We also assume that these agents do not speculate because, in reality, current-account traders

almost never do speculate. In extensive discussions, currency traders consistently assert that their

“corporate customers,” meaning importing and exporting firms, rarely choose to speculate.8 This choice is

based on a perceived lack of expertise and a conviction that they best serve shareholders by focusing on

"core competencies." Indeed, many of their corporate customers are forbidden in their bylaws from

undertaking purely speculative trades (e.g., Sony). The potential consistency of this decision with full

rationality is highlighted by two observations: first, economists themselves have difficulty forecasting

exchange rates; second, the value of focusing on core competencies is supported by the empirical record

in corporate finance, especially the lackluster performance of diversified firms relative to more focused

firms (Lang and Stulz 1994, Berger and Ofek 1995). In a model intended to match the observed behavior

of exchange rates it is, of course, critically important to ensure that agent behavior is consistent with

reality, within the limits imposed by the requirement of tractability. Thus it is, in our view, a virtue of the

model that its current account traders do not speculate. However, it is important to recognize that this

modeling choice is without loss of generality: the model as structured is consistent with the existence of

agents with both speculative and non-speculative motives. One can simply reinterpret “non-speculative

7 Because they affect current and future exchange rates through the non-speculative agents, these factors are indirectly relevant to speculators, and are captured in the model as such. 8 Sources within the foreign exchange community tell us that the number of corporations willing to speculate actively, always close to zero, has declined in recent years.

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agents” as representing the non-speculative component of all agents’ behavior, and “speculators” as

representing the speculative component of all agents’ behavior.

The responsiveness of our non-speculative agents to exchange-rate levels makes sense

economically. At macroeconomically meaningful horizons (say, a quarter or longer), this responsiveness

captures the effects of real exchange-rate changes (which closely parallel nominal exchange-rate changes)

on trade flows. At higher frequencies, this responsiveness captures the fact that corporate traders often

place orders to buy a certain amount of currency if its value falls to a pre-specified level, or to sell a

certain amount if its value rises to a pre-specified level.9 These agents often have liquidity needs that are

not immediate, in which case they may place orders with their dealing bank hoping to improve on the

instantaneous prevailing rate. The placement of such orders, called “take-profit orders,” is standard

operating procedure for such firms. Conditional trading practices are also adopted by other current-

account agents, such as the “Japanese exporters” who loom large in the conversations of market

participants because they must dispose of large quantities of dollars. These agents monitor the market

closely and enter the market in large volume if the rate hits their intraday trigger. (Costs and benefits of

placing orders rather than dealing immediately are discussed in Handa and Schwartz 1996, Foucault 1999,

and Hollifield et al. 2002, inter alia.)

In aggregate, the take-profit orders on the books of individual banks and other conditional trading

effectively create a demand curve for currency of the sort assumed in our model. To illustrate, Figure 1

shows all outstanding take-profit orders at the Royal Bank of Scotland (formerly NatWest Markets), a

large FX dealing bank, at 20:53 G.M.T. on January 26, 2000. The negative relationship between the

exchange-rate level and this component of currency demand is readily apparent.

9 These orders, commonly known as “take profit orders,” are discussed at greater length in Osler (2003) and Osler (2005).

15

D. Equilibrium

So far we have described the model’s two types of agents, rational speculators who attempt to

anticipate exchange-rate changes and non-speculative agents who respond to current exchange-rate levels.

How do these agents interact?

As discussed above, extensive evidence confirms the critical role of flow currency demand and

supply in the determination of exchange rates (Lyons 1995; Goodhart and Payne 1996; Evans 1998;

Evans and Lyons 2002). We incorporate this by assuming that equilibrium in the currency market occurs

when total net foreign-currency demand equals zero:

(5) FXt + N(bt - bt-1) = 0.

Note that the model takes the importance of order flow as given, without taking a stand on why

order flow affects exchange rates. Reasons suggested in the literature include information, inventories (or,

more broadly, “price pressures”), and the possibility that the demand for financial assets may be

downward sloping.10 All of these explanations are consistent with our model.11

Earlier “flow” models of exchange-rate determination did not emphasize that expected returns

determine asset holdings rather than changes in holdings, (e.g., Mundell 1963, Fleming 1962). In our

model, by contrast, rational utility-maximizing speculators must be satisfied with their currency holdings

each period, consistent with portfolio-balance and other now-standard models.

Under the assumption of rational expectations, condition (5) becomes:

(6) Etst+1 - (1 + S/Nq)st - Et-1st + st-1 = - Ct/Nq + ∆t ,

10 The information connection is highlighted in Lyons (2001) and Evans and Lyons (2002); inventory considerations are highlighted in (Garman (1976); Amihud and Mendelson (1980); Ho and Stoll (1981); the possibility of a downward sloping demand for financial assets is discussed in Shleifer (1986). 11 The model captures the central insight of the order-flow results, that flow currency demand and supply dominate short-run exchange-rate movements. If there were an excess of buyer (seller) initiated trades, the price would rise (fall) until the excess demand were eliminated. We do not have “initiated” trades explicitly in our model, because we abstract from the dealers per se in the interest of achieving a broader focus. Nonetheless, a flow equilibrium in which excess demand is always brought to zero by exchange-rate movements is consistent with a balance between buyer- and seller-initiated trades.

16

where ∆t ≡ dt - dt-1 represents the change in the interest differential.12 The bubble-free solution, derived in

the appendix, is:

(7) st = λ st-1 + (1-λ) - λ λj

jt t j t t jE C E C S

=

∞

+ − +∑ −0

1( ) / )(1 1

0jttjtt

j

j EE +−+

∞

=

∆−∆− ∑ λλ

λλ

.

The term λ is the smaller root of the associated characteristic equation and λ rises monotonically with

speculative activity from a lower bound of zero to an upper bound of unity (see Appendix).

Equation (7) states that the current exchange rate depends on its own lagged value, on expected

future values of Ct (representing external influences on non-speculative demand), and on expected future

values of ∆t, the change in the interest differential. To derive a closed-form solution, we must be more

specific about the behavior of Ct and ∆t, which are the system’s two sources of randomness. With regard

to Ct, suppose for now that this component of non-speculative currency demand is subject to i.i.d. mean-

zero shocks denoted by εt: Ct = C + εt. In this case, EtCt+j = C for all j > 0. Later we permit Ct to be

disturbed by permanent shocks, as well as transitory shocks. The exchange rate’s equilibrium in the

absence of speculators is Ct /S, and we define ≡ _s C /S. We note that this long-run level is determined

independently of speculative activity, a point to which we return later.

We assume interest differentials are mean-reverting, consistent with evidence provided by

McCallum (1994) and others. As in Mark and Wu (1998), we also assume that interest differentials are

exogenous; this seems like a reasonable representation of reality, given that a country’s monetary policy

is the main determinant of its short-run interest rates, and that monetary policy is exogenous from a short-

run perspective.13 However, the assumption of exogenous interest rates is not necessary. Interest rates are

endogenous in many earlier interpretations of this model, including Black (1985), Driskill and

McCafferty (1980a, 1980b, 1982, 1992), and Driskill, Mark, and Sheffrin (1987).

12 This expression is derived by substituting from (3) times N and from (4) into (5), letting q = 1/θvar(s), and collecting terms. 13 The assumption that interest rates are strictly exogenous is not critical to the results, which are unchanged so long as interest rates are subject to at least one influence exogenous to the rest of the model, such as national monetary policies.

17

We also assume that interest differentials are stationary, and more specifically that they are

AR(1): dt = ρ dt-1 + ηt, where 0 < ρ < 1 and ηt represents a mean zero i.i.d. shock. The strong observed

autocorrelation of interest differentials would be represented by a value of ρ close to unity. With these

assumptions, the solution for the exchange rate becomes (details in Appendix):

(8) st+1 = + λ(ss_

t - ) + (1-λ) ε_s t+1 -

ρλλ

−1 ηt+1 +

ρλρλ

−−

1)1(

dt

The first term on the right-hand side of (8) shows that , the long-run exchange rate in the absence of

speculators, is still an important determinant of s

_s

t when speculators are active. The second term shows

that, in the absence of other influences, the exchange rate will converge to monotonically, eliminating

the fraction 1-λ of any discrepancy between and s

_s

_s t each period. Since the remaining three exchange-

rate determinants (dt, εt+1, and ηt+1) all have a central tendency of zero, the long-run exchange rate remains

= _s C /S in the presence of speculators.

The third term on the right-hand side of (8) shows that a positive shock to non-speculative

foreign-currency demand, εt > 0, tends to appreciate the foreign currency. The fourth and fifth terms show

that the exchange rate is influenced by the level and the change in interest differentials: not surprisingly, a

rise in domestic interest rates (a positive ηt+1) immediately depreciates the foreign currency. To

understand why the coefficient on the current interest differential is positive, keep in mind that, with mean

reversion, a high current interest-rate differential means declining differentials over the future. This, in

turn, implies that speculators will be planning concurrent decreases in their holdings of foreign exchange.

The effect is stronger when mean reversion occurs more rapidly (when ρ is smaller).

The introduction of speculators transforms exchange-rate determination. In the absence of

speculation the exchange rate always satisfies st = Ct/S = (C + εt)/S, and interest differentials have no

effect whatsoever on exchange rates. In the presence of speculators, both the level and the change in

interest differentials affect current exchange rates. In the absence of speculators, any nonzero value for

the shock to non-speculative demand, εt, is immediately and fully reflected in the current exchange rate

18

and has no impact thereafter. By contrast, when speculators are present, the exchange rate’s immediate

response to a shock to non-speculative demand is reduced, but such shocks also affect all future exchange

rates.

The influence of speculators can be summarized by the variable λ. Since λ is monotonically

related to Nq = N/θVar(s), which in turn can be viewed as a measure of average speculative activity, we

can take λ as a measure of average speculative activity so long as other exogenous variables, such as risk

aversion and the statistical distributions of the shocks, remain constant. This implies that increasing the

activity of speculators reduces the initial effect of a shock to non-speculative demand (ε) and lengthens

the exchange-rate’s convergence towards its long-run equilibrium.14 Increasing the activity of speculators

also intensifies the exchange rate’s response to the change in interest differentials and to past differentials.

E. The Model and The Microstructure Evidence, Continued

We earlier showed that the model’s specific structural elements each, individually, have strong

empirical support. Having presented the model, it becomes possible to discuss microstructure evidence

supporting some of the model’s implications for equilibrium behavior. For example, in Sager and

Taylor’s (2005a) examination of market practices the authors conclude that net demand from push

customers/active speculators should be inversely related to net demand from pull customers/current-

account traders. Bjønnes et al. (2004) show that this theoretical negative relationship is actually observed

in reality, using data covering virtually all trading in SEK/EUR during 1993 to 2002. They show that

flows from financial institutions, the paradigmatic push customers, are negatively related to non-financial

flows, and that this relationship holds at all horizons. Our equilibrium condition (5) implies the same

negative relationship between net demand from speculative and non-speculative agents.

Sager and Taylor (2005a) further asserts that net push-customer demand should be positively

related to exchange rates at short horizons, which implies that net pull-customer demand should be

negatively related to exchange rates. This conclusion is also supported empirically. Evans and Lyons

(2004) examine net demand from various types of customers at the largest foreign exchange dealing bank,

14 This point has been made by Osler (1998) in a version of this model that excludes interest-rate differentials.

19

Citibank, and how it is related to exchange rates. Defining the short run as one day or one week, they

show that the relationship between exchange rates and excess demand from short-term speculative agents

is positive, while the corresponding relationship for importers and exporters is negative. Consistent results

are provided in Mende et al. (2005) based on higher frequency data for a relatively small bank. Further

confirming evidence is presented in Sager and Taylor (2005b) and Bjønnes et al. (2004). A positive

relationship between speculative flows and the value of the commodity currency is implied by our

theoretical model when interest-rate shocks dominate exchange-rate behavior at short horizons, a

condition that seems plausible.15

Our interpretation of non-speculative agents as, in part, representing current-account traders,

should imply that net demand from these agents should be strongly related to the current account. This is

supported by evidence presented in Bjønnes et al. (2004) showing that net demand for SEK from non-

financial agents is strongly positively related to the Swedish current account and trade balance; in

contrast, the paper finds essentially no relationship between the current account and net demand from

financial agents.

Our review of empirical evidence consistent with the model’s implications ends with long-run

implications. Our model implies that non-speculative agents dominate exchange rates in the long run, as

noted earlier. It is not entirely accidental that this is consistent with the views of currency traders, whom

one must respect if one takes rational expectations and market efficiency seriously.16 Importantly, the

long-run dominance of non-speculative agents is also consistent with the empirical record of purchasing

power parity (PPP), which captures one of the dominant forces driving non-speculative agents, relative

price levels across countries. As is well known, PPP generally fails to hold in the short run but seems to

hold approximately in the long run (Rogoff 1996).

15 In a more general model, with anticipated monetary policy shocks, speculators’ order flow will be positively correlated with the exchange rate if current and anticipated monetary policy drives exchange rates. This seems even more plausible. 16 If currency dealers don't embody the rational expectations assumption, given the heavy reliance of their compensation on huge profit-based bonuses, it may not be likely that other agents will act rationally, either. For an efficient rational expectations equilibrium to hold the dominant agents must be rational.

20

II. CURRENCY RISK PREMIUMS

The previous section sketched out our model, the underlying structure of which is shared with

models developed independently by a number of other researchers, and provided theoretical and empirical

support not only for that general structure but also for some of the model’s implications. In the rest of the

paper we show that the model can be used to improve our understanding of the forward bias puzzle.

Before plunging into that analysis, however, we review the puzzle, its implications for currency risk

premiums, and a few existing theoretical responses to the puzzle.

A. The Puzzling Behavior of Realized Risk Premiums

A foreign-exchange risk premium, as noted earlier, represents the market’s anticipated excess

return to holding foreign currency relative to holding domestic currency:

(9) rpt = Et(st+1) - st + r*t - rt

Here, rpt represents the risk premium anticipated at time t, st represents the (log) spot exchange rate at

time t, measured as domestic currency units per foreign currency unit; rt* and rt represent foreign and

domestic interest rates, respectively, Et indicates that anticipations are formed at time t.

The strictest version of uncovered interest parity states that risk premiums should be identically

zero, implying that a currency’s expected appreciation should equal the current interest differential. Thus,

“a” should be zero and “b” should be unity in the following equation:

(10) Et(st+1) - s t = a+ b(rt - r*t) .

A less restrictive version of this condition permits the constant term, a, to represent a constant risk

premium. Under rational expectations, one can estimate the following version of (10):

(11) st+k - st = α + β(rt – r*t) + µt+k ,

where the expected appreciation is replaced by the actual appreciation, and µt+k represents a mean-zero

random disturbance. Under the null hypothesis, rpt = 0 and the true value of β is unity.

21

The joint hypothesis of (i) uncovered interest parity and (ii) rational expectations has been widely

tested and rejected. In fact, estimates of β are typically closer to minus one than to plus one (Hodrick

1987, Froot and Thaler 1990, Lewis 1995, Engel 1996).17 This is the “forward premium puzzle.”

B. Time-Varying Risk Premiums

Empirical evidence shows that realized excess returns to currencies vary substantially over time

and are strongly related to interest-rate differentials. The high variability of excess returns is fully

consistent with standard equilibrium asset-pricing models. In the Capital Asset Pricing Model, for

example, the risk premium depends on variables such as the risk-free rate prevailing at time t, the return

on an appropriate market portfolio from period t through t+1 expected at time t, and the asset’s market

beta (Hodrick 1987). Since these determinants of risk premiums vary considerably over time, it is natural

that the risk premiums should also vary.

If these are the only determinants of risk premiums, then it may be difficult to explain their

behavior during certain periods. In the early 1980s, for example, when U.S. interest rates were high, this

risk premium interpretation would require that “dollar-denominated assets were perceived to be much

riskier than assets denominated in other currencies” (Froot and Thaler, 1990). However, this was “exactly

the opposite of the ‘safe haven’ hypothesis which was frequently offered at that time as an explanation for

the dollar’s strength.” Thus, Froot and Thaler conclude that it “is hard to see how one could rely on the

risk-premium interpretation alone to explain the dollar of the 1980’s.”

This perspective on currency risk premiums omits another potentially important factor

highlighted by the portfolio-balance models of the 1970s and early 1980s: the magnitude of international

asset holdings. From a portfolio-balance perspective, any open position in a foreign currency, whether

long or short, creates exchange-rate risk; and the larger the open position, the greater the variability of a

speculator’s overall portfolio. At some point, a risk-averse individual speculator will decide that the

17 Positive coefficients arise when the interest differentials are very large, as shown by Flood and Taylor (1996), or there are long horizons as shown by Chinn and Meredith (2004).

22

marginal increase in portfolio variability is not worth the marginal expected excess return from increasing

an open position.

This intuition is reflected in a portfolio-balance model reviewed by Lewis (1995), who comments

that “the sign of the risk premium would [also] depend on the difference between … domestic holdings of

foreign bonds and foreign holdings of domestic bonds. When domestic residents are net creditors … then

the overall effect on the risk premium is to compensate domestic investors for net holdings of foreign

deposits” (pp. 1926-1927). Lewis is critical of that model, stressing that countries’ net asset positions

change sign infrequently. “[I]nfrequent shifts between net debtor to creditor positions … suggest that this

model cannot explain the changes in sign in predictable returns.” Implicitly Lewis assumes that the

international asset holdings relevant to currency risk premiums encompass the entire range of capital-

account items, including assets intended to be held for many years, such as foreign direct investment,

official holdings, and loans.

However, the insights from the portfolio-balance model may yet be relevant. By definition, the

short-term risk premiums studied in the literature should reward international investors with horizons of

corresponding short length. As shown in Table 1, the U.S.’s overall credit position is dominated by assets

that are generally held for long periods and are, therefore, insensitive to short-term expected excess

returns, such as direct investment, foreign official assets, and international loans (which represent a large

share of “Claims reported by banks, not included elsewhere”). By contrast, the international asset

positions relevant to short-term risk premiums should presumably be restricted to items such as short-

term time deposits, trading accounts, and a subset of high-turnover stock and bond holdings.

Though data are not available on holdings of such short-term assets, informal observation suggest

that net holdings of short-term assets are quite variable and could well change sign frequently. For

example, early in the fall of 1998 it was widely reported that traders were heavily long dollars and short

yen. After the crash of the yen in October, this net position was reported to have narrowed quickly. More

formally, note that net open positions of large speculative futures traders change sign a few times per

year. Weekly data from the International Money Market over the period January 1993 through May 2003

23

show that these traders’ net open position changed sign 75 times for U.K. Pound contracts, or on average

7.2 times per year on average. Comparable figures for other currencies are: 5.1 times per year for Swiss

Franc contracts, 3.7 times per year for yen contracts, 9.1 times per year for DEM contracts, and 4.3 times

per year for Euro contracts.18

These observations re-open the possibility that a model including portfolio-balance factors could

be helpful in explaining the behavior of short-term risk premiums. The model developed above shares

many of the key attributes of portfolio balance models, and it also assumes the international assets

relevant to short-term risk premiums are those associated with short-term speculation. Thus it is natural to

suspect that the model might do a reasonable job explaining the empirical behavior of risk premiums. In

the rest of this paper we examine whether risk premiums in this model behave consistently with empirical

evidence on currency risk premiums.

III. RISK PREMIUMS AND LONG-RUN EQUILIBRIUM IN THE MODEL

This section demonstrates the empirical relevance of the model developed in Section I, focusing

on the behavior of currency risk premiums. We first show that risk premiums in the model are determined

endogenously, that they vary across time, and that they are strongly influenced by interest differentials.

All these properties are consistent with the empirical results just discussed. In the second subsection we

examine the model’s long-run equilibrium, in which risk premiums are unlikely to be driven to zero by

competition among market participants. In the third subsection we enhance the model’s realism in two

ways, first by making non-speculative trading interest-sensitive and second by adding a permanent

component to current account shocks. The first modification permits exchange rate changes and interest

differentials to be negatively related, consistent with the forward-premium puzzle.

A. Risk Premiums

To begin, note that the expected change in the exchange rate at time t can be derived from

equation (8) as:

18 We are grateful to Thomas Klitgaard, Federal Reserve Bank of New York, for providing these data.

24

(12) Etst+1 - st = (1-λ) ( - s_s t) + β dt

where 0 < β = λ ρ

ρλ( )1

1−

− < 1. The expected excess return to foreign currency, or risk premium, then

takes the form:

(13) rpt = (1-λ)( - s_s t) + (β-1) dt

So long as λ < 1, the model predicts that the risk premium varies over time and is determined by

the gap between current and long-run exchange rates, which we call the “exchange-rate gap,” and by the

interest differential. Other factors affect the risk premium through λ and β. These include the extent of

non-speculative activity and the number of speculators, speculators' risk aversion, the volatility of non-

speculative demand and of interest differentials, and the autocorrelation of interest differentials.

The connection between risk premiums and the exchange-rate gap can be explained through an

example. Suppose that the interest differential is fixed at zero, and that the exchange rate is below its

long-run value. Speculators would choose to hold a long position in foreign currency so long as they

could expect some compensation for the associated risk. In equilibrium, a larger exchange-rate gap is

associated with a larger open currency position and higher expected compensation for risk.

The importance of portfolio allocations for risk also explains the relationship between risk

premiums and interest differentials. Once again, an example may clarify the intuition. Suppose dollar

interest rates rise relative to interest rates on assets denominated in other currencies, as they did in the

early 1980s. Other things equal, foreign speculators will choose to own more dollar assets, and thus

increase their exposure to currency risk. In equilibrium, a larger expected excess return (risk premium) on

dollars will be required to compensate speculators for their increased exposure.

In our model the risk relevant for risk premiums does not arise exclusively from exchange-rate

volatility. Risk also depends on the size of short-term speculative positions. As the example suggests, this

interpretation of risk could be useful in understanding the dollar’s behavior in the early 1980s. The risk

premiums of that era may not have been the associated with the general economic risk relevant to the

25

“safe haven” hypothesis cited by Froot and Thaler (1990); instead, it could have been position risk, as

foreign agents accumulated ever-larger exposure to dollar-denominated assets in response to relatively

attractive returns in the U.S. That is, the causation may have run from interest differentials to speculative

positions to risk premiums, rather than the other way. (See Goodhart 1988 and Carlson 1998 for further

discussion of this view.)

B. Long-Run Equilibrium

If speculators were extremely active, λ would be close to unity, risk premiums would be

approximately constant at zero, and the model would conform approximately to uncovered interest parity.

However, foreign exchange speculation is naturally limited by competition from other markets, so λ will

almost certainly fall short of unity in long-run equilibrium. We model this point explicitly below,

following Osler (1995).

We endogenize speculative activity by assuming that speculators (or their employers) compare

unconditional expected welfare from trading in the market, E(Wt), with an appropriate exogenous

benchmark, denoted W*. W* can simply summarize the large fixed costs associated with speculators:

training them, providing them space and real-time information, and paying their base salaries. W* can,

instead, represent expected welfare from alternative activities, presumably speculating in other markets.19

If E(Wt) is greater than W*, then people in other realms of activity have an incentive to become foreign

exchange speculators; conversely, if E(Wt) is less than W* some speculators will drop out of this market.

Thus in the long run, the number of speculators, N, is endogenous.

We model the participation choice with respect to unconditional expected welfare in order to

enhance the correspondence of our model with the reality of foreign-exchange markets, where short-term

trading is dominated by interbank traders. Chief dealers hire and train their traders and usually keep them

for a matter of years. For these reasons, decisions about the extent of participation in foreign-exchange

19 If W* is taken to be an opportunity cost of speculating, we assume that it is bounded away from arbitrarily low levels. If one restricts the relevant “alternative activities” to asset market speculation, then there must be some finite limit to the total possible amount of speculation; this seems reasonable.

26

markets are made on a low-frequency basis, while a single period in the model corresponds to medium- or

high-frequency exchange-rate dynamics.

Unconditional expected welfare depends on the extent of speculative activity, λ, risk aversion, θ,

and parameters governing the behavior of shocks: Var(ε), Var(η), and ρ:

(14) E(Wt) = )1(2

)1( 3

λθλ+

− [

)var()var(

νε

+ )1()1(

)1(23 ρρλ

ρλ−−

+)var()var(

νη

]

(see Appendix for details).

The implications of (14) for the long-run level of speculative activity are not immediately obvious

because the relationship between unconditional welfare and λ, or equivalently between unconditional

welfare and the amount of speculative activity, is not necessarily monotonic. However, as evident in

equation (14), unconditional expected welfare becomes arbitrarily small for large values of λ. By way of

illustration, Figure 2 depicts E(Wt) as a function of λ.20

A stable long-run equilibrium occurs when E(W) cuts W * from above. If E(W) is greater than

W*, there is an incentive for additional speculators to enter the market, so that N and hence λ increases.

When E(W) is less than W*, there is an incentive for speculators to drop out so that N decreases. To

illustrate that the equilibrium number of speculators will be finite, Figure 3 depicts a mapping from values

of W * to the number of speculators N (parameters for these solutions are as for Figure 2 plus S = 100).

For lower values of W*, there will be more speculators, but the number approaches infinity only when W*

approaches zero. In summary, in long-run equilibrium, i.e., when E(Wt) = W* > 0 there is no incentive for

additional speculators to enter the foreign exchange market. Long-run equilibrium speculation is finite,

and equilibrium values of λ are bounded away from unity. In short, in this model there is no presumption

that uncovered interest parity will hold even approximately.

C. Modifications of the Model

20 These parameters are consistent with those used in simulations reported below. If the coefficient of risk aversion θ were larger, E(W) would be lower in Figure 2, and the equilibrium value for λ would also be lower.

27

We now increase the model’s realism in two ways. First, we allow non-speculative activity to

depend on interest-rate levels. This permits exchange-rate changes to be negatively related to interest

differentials, consistent with the forward premium puzzle. Second, we incorporate permanent shocks to

the current account. In addition to bringing in greater realism, this exercise permits us to demonstrate the

model’s flexibility.

1. Interest Sensitivity of Current-Account Trading

Import demand seems likely to be negatively affected by higher domestic interest rates and export

demand negatively affected by higher foreign interest rates, with corresponding effects on currency

demands. More specifically, we modify the expression for net non-speculative foreign-currency demand

by adding a term related to the interest differential, as follows:

(15) FXt = Ct - S st - I dt where I > 0 represents the sensitivity of these flows to interest-rate differentials. When foreign interest

rates rise, for example, foreign importers presumably reduce their demand for domestic goods, and net

demand for foreign currency rises. Empirical evidence for such a negative relationship is provided in

Bjønnes et al. (2004), which shows that net commercial-customer demand for SEK was negatively related

to the difference between Swedish and German bond yields during 1993-2002.

As shown in the appendix, the solution for the exchange rate now takes this form:

st+1 = + λ(ss_

t - ) + (1-λ) ε_s t+1

(16) - λ λ

ρλ+ −

−( )( /11

I S) ηt+1 +

λ ρ ρ λρλ

( ) ( / ) ( )1 11

2− − −−I S

dt.

The foreign-exchange risk premium becomes:

(17) rpt = (1-λ)( - s_s t) + [

λ ρ ρ λρλ

( ) ( / ) ( )1 11

2− − −−I S

- 1] dt .

In this version of the model, the coefficient on the interest differential dt can be negative in the

exchange-rate equation (16) and less than minus one in the risk-premium equation (17). That is, this

version of the model is consistent with a negative relation between interest differentials and risk

28

premiums, as found often in the empirical literature. To fulfill this condition, the interest sensitivity of

non-speculative net demand must be relatively high; more specifically, we must have IS

>λ ρρ λ

( )( )

11 2

−−

.

Some recent research suggests that the relationship between interest differentials and exchange-rate

changes became positive and potentially much above unity in the 1990s (Baillie and Bollerslev 2000 and

Flood and Rose 2002). This is consistent with our model, so long as IS

<λ ρρ λ

( )( )

11 2

−−

.

Interest sensitivity of non-speculative currency demand is presumably not a sine qua non for a

negative relationship between interest differentials and risk premiums. The model is flexible, and

numerous other modification might produce the same results. In particular, we note that incorporating the

well-documented deviation of exchange-rate expectations from the paradigm of full rationality might

generate the same negative relationship. Gruen and Gizycky (1993) show that such a relationship arises

when some agents anchor their exchange-rate expectations to current exchange rates and interest

differentials, a condition that seems plausible given the well-documented importance of anchoring in

normal human cognitive functioning (Yates 1988) and the well-known absence of alternative short-run

forecasts based on a clear and reliable understanding of exchange rates.

2. Permanent Shifts in Fundamentals

In reality, non-speculative currency demand is subject to permanent as well as transitory shifts.

Among the potential sources of permanent shifts, relative prices are potentially most important. As shown

in the appendix, if the ε-shocks have permanent and transitory components, the solution for the change in

the exchange rate has a new term:

(18) st+1 - st = (Et+1 s t+2 - Et s t+1) + (1-λ)( Et s t+1 - st) + β dt + (1-λ) εt+1 - ρλ

λ−1

ηt+1 .

Here, β < 1, 0 < λ < 1, 0 < ρ < 1. Et s t+1 is an expectation, based on information available at time t, of the

exchange rate that would be reached in the long run if fundamental variables remained constant at their

time-t values. For brevity we still refer to s t as "the long-run equilibrium exchange-rate."

29

This new expression given by (18) shows that exchange-rate changes are still affected by the

exchange-rate gap and the interest differential. However, they are now also influenced by the perceived

change in the long-run equilibrium rate. The risk premium becomes:

(19) rpt = (E t s t+2 - Et s t+1) + (1-λ)( Et s t+1 - st) + (β-1) dt ,

assuming E tEt+1 s t+2 = E t s t+2 by the law of iterative expectations.

IV. SIMULATION ANALYSIS

In this section we use simulations calibrated to observed properties of exchange rates to show that

our model can replicate all the significant stylized facts of exchange-rate dynamics associated with the

forward premium puzzle. We have already shown theoretically that the model can generate a strongly

negative relationship between realized excess returns and interest differentials. The simulations show that

exchange rates in the model satisfy the conditions noted by Bekaert (1996). First, the volatility of

exchange-rate changes is high. Second, the volatility of exchange-rate changes exceeds the volatility of

risk premiums, Var(∆st) > Var(rpt). Third, the volatility of exchange-rate changes exceeds the volatility of

forward premiums: Var(∆st) > Var(dt). Finally, forward premiums are strongly persistent while the

autocorrelation of exchange-rate changes is close to zero: Corr(dt, dt -1) >> Corr(∆st, ∆st-1) ≈ 0.

Following the literature, we use simulations of exchange rates at quarterly horizons (e.g., Bekaert

1996; Moore and Roche 2002). For our simulations, all parameters are permitted a low, a medium, and a

high value. We first identify the equilibrium when all variables are at their medium level, which we refer

to as the base case. We then allow each variable to change up or down independently of the others,

creating a total of eleven simulations. Each run uses the same 10,000 draws of the random shocks.

We calibrate the underlying parameters of the model so that the model fits reality in numerous

dimensions. The autocorrelation of interest differentials and the variance of interest-differential shocks

are set to be roughly comparable to corresponding figures for quarterly differentials between U.S. three-

month euro rates and those of Germany, Japan, Switzerland, the U.K., and Canada, which average 0.89

(differentials for Germany, Switzerland, and the U.K. begin in 1970; otherwise they begin in 1978).

30

Likewise, the quarterly standard deviation of exchange rate returns is set at or near 5.4 percent, which is

the average for exchange rates between the five country pairs listed above. Finally, β was set close to –2

to conform to traditional regression estimates, as well as estimates reported below. The parameter values

that satisfied these criteria are shown in Table 2.21

Results reported in Table 3 show that, when the model fits reality as described above, it can

conform to all the well-documented empirical regularities associated with the forward premium puzzle:

a) For the base case and all ten parameter variations, risk premiums vary inversely with interest

differentials, consistent with reality. Simulated values of β vary from –1.45 to –2.57.

b) For the base case and all ten parameter variations, the standard deviation of exchange-rate changes is

high, since it uniformly comes close to the 5.4 percent average listed above.

c) For the base case and all ten parameter variations, the standard deviation of exchange-rate changes

exceeds, by a large margin, the standard deviation of both risk premiums and interest differentials.22

(c) For the base case and all ten parameter variations, the autocorrelations of risk premiums and of interest

differentials (ρ) are substantial while that of exchange-rate changes is close to zero. The consistently

small negative signs for these simulated exchange-rate autocorrelations conform to the negative-but-

small quarterly autocorrelations for the Canadian Dollar/U.S. dollar and U.K. Pound/U.S. dollar

exchange rates over recent decades (-0.09 and -0.13, respectively).

In short, our order-flow-based model of short-run exchange-rate determination can, when

calibrated to match observed properties of actual currency markets, generate exchange-rate behavior that

conforms to all the well-documented empirical regularities associated with the forward premium puzzle.

As discussed in the introduction, other models do not seem to have achieved this level of success in fitting

21 Though λ is not a parameter, it is closely related to the parameter W,* and it is easier to solve for equilibrium when varying λ than W.* 22 Note that our results are also consistent with Bakaert’s condition that his estimated lower bound for the volatility of risk premiums σ(dt)|β-1|, exceeds the volatility of interest differentials. This condition is satisfied whenever whenever |β−1|>1, which is true in all our simulations. However, his estimation approach implicitly assumes that changes in interest differentials have only one, direct connection to risk premiums, an assumption that is not correct according to our model. The indirect connection works through the unconditional negative covariance between the interest-rate differential and the exchange-rate gap (see equation (A.29) in the appendix).

31

the stylized facts. The alternative models of which we are aware are all unable to match one or more of

the properties examined above.

V. REGRESSION ANALYSIS

The clean analytical solutions to our model are linear in three critical variables: the interest

differential, the gap between the current and long-run exchange rate, and the most recent change in the

expected long-run exchange rate. Thus, the model lends itself naturally to regression tests, which we

undertake in this section. Using data for five major currency-pairs⎯DEM/EURO, JPY, CHF, GBP, and

CAD⎯we find that the model fits well. From this we infer that this microstructure-based model can be

used to analyze exchange rates at macro horizons.

A. Traditional Risk Premium Regressions

First, however, we replicate the standard regression in the literature, though we place excess

returns on the left hand side rather than exchange-rate changes:

(20) xrt+1 = α + γ dt + ξ t+1 .

We estimate this expression using quarterly data on exchange rates and three-month eurocurrency rates

for the five currencies listed above over January, 1973, through June, 2003.

As shown in Table 4, when equation (20) is estimated using OLS the estimated values of γ are

consistently negative, greater than unity in absolute value, and highly statistically significant. Consistent

with earlier studies, the explanatory power is fairly low: adjusted R2s average only 0.10.

B. Regressions of the Model

In our model, the excess return to currency can be expressed as follows:

(21) xrt+1 = (Et+1 s t+2 - Et s t+1) + (1-λ)( Et s t+1 - st) + (β-1) dt + νt+1

where νt+1 ≡ (1-λ) εt+1 - ρλ

λ−1

ηt+1 is a mean-zero random variable. Thus, our model suggests that the

standard regression is mis-specified due to the exclusion of two variables, the change in the expected

long-run equilibrium exchange rate and the exchange-rate gap.

32

Estimating our model requires a measure of the expected long-run equilibrium exchange rate. In

the absence of a stronger fundamentals-based model we rely on PPP. We assume that the long-run

equilibrium real exchange rate is constant at some value M, in which case the long-run-consistent

nominal exchange rate, s t, satisfies M = s t - pt + pt*, or s t = M + pt - pt* (we use log producer price

indexes for pt and pt*). To calculate the expected long-run-consistent nominal exchange rate, we replace

actual price levels with fitted values from an autoregression with 12 lags of monthly prices, beginning

three months prior to the observation date. Only the constant term is affected by our choice of M, which

we set at the mean observed (log) real exchange rate.

23

Our estimating equation, based on (21), is:

(22) xrt+1 = α + δ(Et+1 s t+2 - Et s t+1) + (1−λ)( Et s t+1 – st) + (β−1) dt + υt+1 According to the model, the constant term, α, should be zero, the coefficient on the change-in-expected-

long-run exchange rate, δ , should be unity, (1−λ) should be between zero and one, and (β−1) should not

exceed zero.

OLS estimates of (22), reported in Table 5, are quite favorable to the model. The explanatory

power of the model-based regressions is substantially higher than that of the standard regressions:

adjusted R2s now average 0.28, almost three times their average with the traditional model. The negative

estimates of β-1 are similar to the γ estimates reported for the standard excess-return regressions.

As required by the model, the constant term is not significantly different from zero in three of the

five regressions, and only marginally significant in a fourth. This term could differ from zero for two

reasons; first, our model could be mis-specified; second, we might not be accurate in assuming that the

equilibrium long-run real exchange-rate is constant at the average realized real exchange rate. Since this

assumption is really quite crude, the fact that the constant terms are generally close to zero is almost

surprising, and certainly supports the model.

23 This permits expectations to be based on all available price data, rather than simply quarterly observations of prices, while still ensuring that information used to create forecasts in period t is truly available in period t.

33

The coefficients on the change-in-long-run-exchange-rate are generally in the vicinity of one, as

required by the model, and three out of the five do not differ significantly from unity. Again, given the

simplicity of our model of long-run exchange rates, this is quite supportive of the model. These results

certainly suggest that, with a more refined model of the long-run real exchange-rate, our results could

improve further.

The coefficients (1-λ) on the exchange-rate gap all have the theoretically expected positive sign,

and their magnitudes are economically sensible since they imply that between 7 and 10 percent of any

deviation from PPP is eliminated per quarter. Our mean real exchange half-life is only 8 quarters,

noticeably below Rogoff’s (1996) “consensus view” of three to five years and closer to the relatively

accurate 14-month half-life found by Imbs, et al. (2002) using disaggregated price data.24 Three of our

exchange-rate gap coefficients are statistically significant at the five percent level, and the remaining two

are significant at the 10 percent level. This contrasts with the difficulty economists usually encounter, in

datasets of this length, detecting statistically significant convergence to PPP.

Overall, the empirical analysis supports the model’s implications that risk premiums are

endogenously determined by interest rate differentials and other forces. They also support a number of

other conclusions. First, they support our contention that explaining short-run exchange rate dynamics

requires a model that incorporates the microstructure lesson that order flow matters.

In addition, these results imply that this microstructure-based model can be used to analyze

exchange rates at macro horizons. Like the model, this conclusion is not new. It was already demonstrated

in Driskill, Mark, and Sheffrin (1987), which shows that their relatively elaborate version of the model

broadly fits data for the economies of Switzerland and the United States and the associated exchange rate.

Their model not only explains the empirical record well in-sample, but it forecasts better than the random

walk in out-of-sample tests. Further evidence supporting the relevance of microstructure analysis at macro

horizons is provided in Bjønnes et al. (2004), which shows that order flow and returns are significantly

related at quarterly horizons; Evans and Lyons (2005), which shows that order flow can forecast exchange 24 Since we do not have disaggretated data, average half-life will naturally be higher than 14 months due to the aggregation bias highlighted by Imbs, et al. (2002).

34

rates better than the random walk model for at least one month; and Evans and Lyons (2004), which

shows that order flow forecasts macro fundamentals at quarterly horizons.

VI. CONCLUSION

This paper analyzes a model of short-run exchange-rate dynamics based on evidence from

currency microstructure. The structure of the model is not new. To the contrary, models with a similar

underlying structure have been independently developed by numerous researchers over the past two

decades (Black 1985, Driskill and McCafferty 1982, Osler 1995, 1998, Carlson and Osler 2000, Hau and

Rey 2004). This structure has three key elements: 1. Rational speculators with constant absolute risk

aversion that maximize the expected utility of profits; 2. Non-speculative agents that respond linearly to

exchange rate levels; 3. Equilibrium characterized by equality of flow currency demand and supply.

We provide strong empirical and theoretical justification for each of these elements. For example,

we note that virtually all currency traders work for larger institutions, and thus won’t be motivated by

consumption or consumption risk. Instead, they will be motivated by profit-driven annual bonuses and

will behave in a risk-averse fashion because they face institutional constraints on risk-taking. We

highlight that the model implies a structure of heterogeneity among agents consistent with that observed

in reality, as described in Sager and Taylor (2005a). The specific form of heterogeneity we assume is

consistent with evidence in Bjønnes et al. (2004) and Mende et al. (2005) showing that net demand of

short-term speculators is positively related to exchange rates while net demand of importers and exporters

is negatively related to exchange rates. It is also consistent with the strong observed relationship between

non-speculative net demand and current account behavior in the market for SEK/EUR (Bjønnes et al.

2004).

The model’s equilibrium condition distills recent evidence highlighting that order flow is a key

determinant of high-frequency exchange-rate movements (e.g., Evans and Lyons 2002). In addition, we

provide new theoretical insights showing that the equilibrium condition common to traditional exchange-

rate models, in which stock money demand must equal stock money supply, is unlikely to be relevant to

35

short-run exchange-rate determination. Though new, the insights are based on well-established frames of

reference like the Baumol-Tobin model of money demand (Baumol 1952, Tobin 1956).

To demonstrate the model’s empirical relevance we apply it to the forward premium puzzle. This

is appropriate given that most existing empirical analyses of risk premiums focus on horizons of a year or

less. In the model’s equilibrium we find that exchange-rate risk premiums are determined endogenously

by interest differentials, the gap between current and long-run equilibrium exchange rates, and the

expected change in long-run exchange rates. The relationship between risk premiums and interest

differentials can be strongly negative, consistent with existing empirical evidence.

We use simulation results to show that the model, when calibrated to observed properties of

exchange rates, conforms to three sets of empirical regularities highlighted in Bekaert (1996): (1)

exchange-rate volatility is high; (2) exchange-rate volatility exceeds the volatility of both risk-premiums

and interest-differentials; and (3) the autocorrelation of interest differentials is substantial, while the

autocorrelation of exchange-rate changes is close to zero. Regressions on quarterly data for five currency-

pairs provide further support for the model. The strong empirical support for this model suggests that

currency microstructure research, which initially documented the importance of order-flow at high

frequencies, may also be relevant for macroeconomic analysis.

The paper’s general line of reasoning comes directly from portfolio-balance models, and the

model similarly highlights the importance of the net foreign asset positions. However, our analysis

suggests that the net asset position relevant to short-term risk premiums will not be a country’s overall net

credit or debt position. Instead, the relevant asset position will include only those assets involved in short-

term trading. The limited available evidence on the positions of short-term speculators suggests that net

positions change sign far more frequently than the U.S. overall net asset position, consistent with the

relatively frequent sign changes of estimated risk premiums.

In future research it would be appropriate to use this model to address other puzzling aspects of

currency markets, such as the rise of exchange-rate volatility that accompanied the transition to floating

exchange rates in the mid-1970s, when there was no commensurate increase in the volatility of

36

fundamentals (Flood and Rose 1995). In this model, exchange-rate volatility need not mirror the volatility

of fundamentals because it is partially influenced by speculative behavior (Carlson and Osler, 2000).

37

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41

Appendix

Derivation of the solution for the model Our model implies the following difference equation: (A.1) Etst+1 - (1 + S/Q)st - Et-1st + st-1 = - Xt with Xt = [Ct -Q(dt - dt-1)]/Q and Q ≡ Nq. To find the solution, first take expectations of (A.1) as of time t-1, and denote by F the forward operator which increases the date on s but not the date on the expectations operator E and by L = F-1 the lag operator that decreases the date on s but does not change the date of the expectations operator. Then collect terms: [F2 - (2 +S/Q)F + 1] L Et-1st = - Et-1XtBy factorization:

(A.2) (F - λ)(F - 1λ

) L Et-1st = - Et-1Xt

where λ is the smaller root of the characteristic equation: λ2 - (2+S/Q)λ + 1 = 0. Multiply (A.2) through by -λ/(1-λF) and expand to get:

(A.3) E s s E X Kt t tj

jt t j

t− −

=

∞

− +−= + +∑1 1

01λ λ λ λ

where K is an arbitrary constant. With the assumption of no explosive bubbles, K = 0. When (A.3), with K = 0, is used to substitute in (A.1) for Et-1st and, with a suitable change in the time index, for Etst+1, the resulting expression after collecting terms is:

(A.4) ( ) ( )1 1 11

01

01− + = − + + −−

+

=

∞

+ +=

∞

− +∑ ∑λ λ λ λ λSQ

s s X E X Et t tj

jt t j

j

jt tX j

From the factorization, the sum of the roots can be written λλ

+ = +1 2 S

Q and so:

(A.5) 1 1− + =

−λ λλ

SQ

( )

and

(A.6) λ

λλ

( )( )

11

−= −

QS

From (A.5) and (A.6)

(A.7) 1

11

( )( )

− += −

λλS

Q

QS

Multiply both sides of (A.4) by (A.7), and note that (1-λ)2Q/S = λ, to get

s s QS

E X E Xt tj

jt t j t t j= + − −−

=

∞

+ −∑λ λ λ λ10

11( ) [ + ]

/

or substituting for Xt = [Ct – Q ∆dt ]/Q, and noting again that Q/S = λ/(1-λ)2 ,

(A.8) st = λ st-1 + (1-λ) λ λj

jt t j t t jE C E C S

=

∞

+ − +∑ −0

1( )

- )(1 1

0jttjtt

j

j dEdE +−+

∞

=

∆−∆− ∑ λλ

λλ

42

Assume that Ct is subject to both permanent and transitory shocks, as postulated by Muth (1960). His model can be written: Ct = At + ut

At = At-1 + vt. Ct is observed but not ut and vt separately. Assume initially that u and v are i.i.d. random variables with zero means and variances σu

2 and σv2, respectively.25 In that case, Muth proved that the optimal forecast

of C is adjusted adaptively: Et Ct+1 = γCt + (1-γ)Et-1 Ct

where Et denotes an expected value at time t and γ (0 < γ < 1) is an increasing function of and σv2/σu

2 . If most of the variability is transitory, then γ is close to 0; if most of the variability is permanent, then γ is close to 1. It is also the case with the zero means of u and v that Et Ct+j = Et Ct+1 for j = 2,3,… The summation involving C terms can then be written

(A.9) (1-λ) = [(1-λ) Cλ λj

jt t j t t jE C E C S

=

∞

+ − +∑ −0

1( ) / t + λ EtCt+1 - λ Et-1Ct]/S

= [(1-λ)( Ct – EtCt+1) + EtCt+1 - λ Et-1Ct]/S = Et s t+1 - λ Et-1 s t + (1-λ) εt

The last line in (A.9) is obtained from the following definitions. Define EtCt+1/S by Et s t+1 and Et-1Ct/S by Et-1 s t . Also let εt = [Ct - EtCt+1]/S denote the perceived transitory shock.

For the interest differential, we assume that dt = ρdt-1 + ηt. In that case dt - dt-1 = (ρ−1)dt-1 + ηt , Et-1 (dt - dt-1) = (ρ−1)dt-1, Et (dt+j - dt-1+j) = (ρ−1)ρ j-1 dt = (ρ−1)(ρ j dt-1 + ρ j-1 ηt) Et-1 (dt+j - dt-1+j) = (ρ−1)ρ j dt-1With these substitutions and using (A9) in (A.8), the solution can be written:

(A.10) st - Et s t+1 = λ(st-1 - Et-1 s t)+ (1-λ) εt - ρλλ

−1 ηt +

ρλρλ

−−

1)1(

dt-1.

(A.10) can also be written alternatively as:

st - st-1 = (Et s t+1 - Et-1 s t) + (1-λ)( Et-1 s t - st-1)+ (1-λ) εt - ρλλ

−1 ηt +

ρλρλ

−−

1)1(

dt-1.

Expected WelfareConditional expected welfare is given as: (A.11) Wt = Et(πt+1) - (θ/2)Vart(πt+1) The profitability of position bt is: (A.12) πt+1 = bt [st+1 - st - dt]. The position itself is given by

(A.13) bt = q rpt where q = 1

θVar(s)

The risk premium rpt is defined by (A.14) rpt = Etst+1 - st - dt with dt = rt - r*t. The unanticipated change in the exchange rate takes the form:

25 An alternative formulation with fully anticipated changes in C occurs when u = 0 and v equals a known value. An intermediate case arises when v has a non-zero expected value but there are also stochastic unanticipated changes in C.

43

(A.15) νt+1 = (1-λ) εt+1 - ρλλ

−1 ηt+1

Therefore (A.16) st+1 = Etst+1 + νt+1 and

(A.17) Var(s) = (1-λ)2 var(ε) + λρλ

2

21( )− var(η) = var(ν)

The payoff to a rational speculative position, after substituting (A.13), (A.14) and (A.16) into (A.12) is:

(A.18) πt+1 = q rpt [rpt + νt+1] From (A.18), we have (A.19) Et(πt+1) = q (rpt)2

(A.20) Vart(πt+1) = q2 (rpt)2 Var(s) Substituting these into (A.11) and simplifying yields

(A.21) Wt = )(2

)( 2

sVarrpt

θ

In the model, since Et s t+1 = s t, the risk premium is given by:

(A.22) rpt = (1-λ)( s t - st) + (β -1) dt where β = ρλ

ρλ−

−1

)1(

Substitute this into (A.21) and take the unconditional expected value:

(A.23) E(Wt) = )(2

})]1/(1[){()1( 2_

2

sVardssE ttt

θλρλ −−−−

The numerator in (A.23) can be written

(A.24) (1-λ)2 { E(st -_s t)2 +

)1(2ρλ−

Edt(st -_s t)+ 2)1(

1ρλ−

E dt2 }

with (A.25) dt = ρ dt-1 + ηt 0 < ρ < 1.

(A.26) st - _s t = λ(st-1 -

_s t-1) + νt +

ρλρλ

−−

1)1(

dt-1

To evaluate E dt2, note from (A.25) that dt can be written in moving average form:

dt = ηt + ρηt-1 + ρ2ηt-2 + … Therefore, assuming independent, mean-zero η‘s:

(A.27) E dt2 = var(d) = 21

1ρ−

var(η)

For the first term in (A.24), the moving average representation for (st - _s t) can be written (with b =

ρλρλ

−−

1)1(

):

st - s_

t = νt + λ νt-1 + λ2 νt-2 + … + bdt-1 + λbdt-2 + λ2 bdt-3 + λ3 bdt-4 … Therefore:

E(st - _s t)2 = (1 + λ2 + λ4 …)var(ν)

44

+ b2 E[dt-12 + λ2dt-2

3 + λ4dt-32 …

+ 2 λ dt-1dt-2 + 2 λ2 dt-2dt-3 + 2λ5 dt-3dt-4 + … + 2 λ2 dt-1dt-3 + 2 λ4 dt-2dt-4 + … ]

+2b E[λνt-1dt-1 + λ2 νt-2dt-1 + λ3 νt-3dt-1 + … + λ3 νt-2dt-2 + λ4 νt-3dt-2 + … ]

= 211λ−

{var(ν) + b2 [1 + ρλ

ρλ−12

]var(d) -2bρλ

λ−1 ρλ

λ−1

var(η) }

= 211λ−

{(1-λ)2 var(ε) + 2

2

)1( ρλλ

− var(η)

+ [ 3

22

)1()1(

ρλρλ

−−

211

ρρλ

−+

- 3

3

)1()1(2

ρλρλ

−−

] var(η) }

= λλ

+−

11

var(ε) +

)1()1)(1(

)]1)(1(2)1()1()1)(1[(232

2222

ρρλλρρλρλρρρλλ

−−−−−−+−+−−

var(η)

= λλ

+−

11

var(ε) +

)1()1)(1(

)]1(2)1)(1()1)(1)[(1(232

22

ρρλλρλρλρρρλρλ

−−−−−+−++−−

var(η)

= λλ

+−

11

var(ε) +)1()1)(1(

)1(2)1(232

2

ρρλλλρλ−−−

−− var(η)

(A.28) E(st - _s t)2 =

λλ

+−

11

var(ε) +)1()1)(1(

23

2

ρρλλλ

+−+ var(η)

The foregoing used E dtdt-j = ρj var(d) and E νt-j dt = -ρλ

λ−1

ρj var(η).

For the middle term in (A.24)

E (st - _s t)dt = E(νt + λ νt-1 + λ2 νt-2 … bdt-1 + λbdt-2 + λ2 bdt-3 …) dt

= - ρλ

λ−1

(1+ ρλ +(ρλ)2 + …)var(η) + bρ(1 + ρλ +(ρλ)2 + …)var(d)

= 2

2

)1()1()1(

ρλρρλρλ

−−+−−

var(d) = 2)1()1(

ρλρλ

−−−

var(d)

(A.29) E (st - _s t)dt =

)1()1( 2 ρρλλ

+−−

var(η)

Note the negative unconditional covariance between the interest rate differential and deviations in the exchange rate from its long-run level. Higher domestic interest rates tend to be associated with an appreciated currency.

Putting (A.27), (A.28) and (A.29) into (A.24) and the result into (A.23) yields:

2θ E(Wt) = λλ

+−

1)1( 3

var( )var( )

εν

45

+ 2

2

)1()1(

ρλλ

−−

[)1)(1)(1(

2 2

ρρλλλ

+−+-

)1)(1(2

ρρλλ

+− + 21

1ρ−

])var()var(

νη

= λλ

+−

1)1( 3

)var()var(

νε

+(1-λ)2 [)1()1)(1(

)1)(1()1)(1(2)1(223

2

ρρλλρλλρλλρλ

−−+−++−+−−

])var()var(

νη

Therefore

(A.30) E(Wt) = )1(2

)1( 3

λθλ+

− [

)var()var(

νε

+ )1()1(

)1(23 ρρλ

ρλ−−

+)var()var(

νη

]

Interest Rates and Non-Speculating Traders

To include non-speculating traders who respond to interest-rate differences, write the net demand as: FXt = Ct - S st - I dt. In this case Xt has the term -(I/Q)dt and the right side of (A.9) has the added expression:

][)1( 11

jttjttj

j dEdEQI

SQ

+−+

∞

=

−−− ∑ λλλ

= ][)1( 11

1−

+∞

=

−−− ∑ tj

tj

j

j ddSI λρρλλ

= ][)1( 11

11

1−

+−

+∞

=

−+−− ∑ tj

tj

tj

j

j ddSI λρηρρλλ

= ⎥⎦

⎤⎢⎣

⎡−

+−

−−− − ttd

SI η

ρλρλλρλ

11

1)1()1( 1

Add this to the right side of (A.10) to get the equation (19) in the text.

Mapping between λ and N. We can manipulate the characteristic equation, λ2 - (2+S/Nq)λ +1 = 0, to obtain (A.31) N = Sλ/(q(1-λ)2). We also know that q = 1/θvar(ν) and that var(ν) = (1-λ)2var(ε) + (λ/(1-ρλ))2var(η) Therefore, by substitution for q and var(ν) in (A.31): N = Sλθ(var(ε) + (λ/((1-λ)(1-ρλ)))2var(η)). This monotonic relationship between N and λ was used to construct Figure 3 from the data underlying Figure 2.

46

Figure 1: Take-Profit Orders Create a Currency Demand Curve The figure plots open the cumulative value of all dollar-yen take-profit orders at Royal Bank of Scotland on January 26, 2000, at 20:53 GMT. The horizontal axis plots the exchange rate, with the contemporaneous market midrate, ¥105.77/$, shown by the vertical line. The vertical axis represents the cumulative dollar value of orders.

-350

-300

-250

-200

-150

-100

-50

0

50

100

150

95 97 99 101 103 105 107 109 111 113 115

47

Figure 2: Welfare and Speculative Activity The figure shows how unconditional expected welfare E{W}, is related to aggregate speculative activity, as measured by λ. (E{W} has been multiplied by 100 for convenience.) Equilibrium combinations of E{W} and λ are plotted for the following parameter values: SD(ε) = 6.0; SD(η) = 0.4; ρ = 0.85; θ = 5. These parameters are consistent with the calibrated model simulated in Section IV.

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

0.05 0.20 0.35 0.50 0.65 0.80 0.95

λ

Unc

ondi

tiona

l Exp

ecte

d W

elfa

re,

E{W

}

48

Figure 3: Welfare and the Number of Speculators The figure shows the equilibrium number of speculators, N, for various values of welfare in alternative speculative activities, W*. (W*, has been multiplied by 100 for convenience.) As welfare in other markets increases, currency speculators choose to leave the market. The underlying parameters for this simulation are those of Figure 2: SD(ε) = 6.0; SD(η) = 0.4; ρ = 0.85; θ = 5.

0

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

Welfare in Other Markets, W*

Num

ber o

f Spe

cula

tors

, N

49

Table 1

International Investment Position of the United States at Year-End 2001 Type of investment $ Millions Percent Net international investment position of the United States: With direct investment positions at market value -2,309,117 U.S.-owned assets abroad: With direct investment at market value 6,862,943 100.0 U.S. official reserve assets 129,961 1.9 U.S. Government assets, other than official reserve assets 85,650 1.2 U.S. private assets: Direct investment abroad, at market values 2,289,926 33.4 Foreign securities

1 Bonds 545,782 8.02 Corporate stocks 1,564,738 22.8 U.S. claims on unaffiliated foreigners reported by U.S. nonbanking concerns 830,111 12.1 U.S. claims reported by U.S. banks, not included elsewhere 1,416,775 20.6 Foreign-owned assets in the United States: With direct investment at market value 9,172,060 100.0 Foreign official assets in the United States 1,021,738 11.1 Other foreign assets in the United States: Direct investment in the United States, at market value: 2,526,711 27.5

3 U.S. Treasury securities 388,774 4.2 U.S. securities other than U.S. Treasury securities

4 Corporate and other bonds 1,392,620 15.25 Corporate stocks 1,464,034 16.06 U.S. currency 275,569 3.0

U.S. liabilities to unaffiliated foreigners reported by U.S. nonbanking concerns 804,417 8.8 U.S. liabilities reported by U.S. banks, not included elsewhere 1,298,197 14.2 Source: Bureau of Economic Analysis: http://bea.gov/bea/di/intinv1976_2001.xls

50

Table 2: Parameters for Simulations The table shows all the values used for the model's salient properties in simulating it's

behavior. σ(ε) represents the standard deviation of shocks to non-speculative currency demand; σ (η)represents the standard deviation of interest differential shocks; ρ represents the autocorrelation of interest differentials; represents, for non-speculative currency demand, the ratio of its interest-differential sensitivity to its exchange-rate sensitivity; λ summarizes speculative activity. Though l is not an exogenous parameter, it will be uniquely determined by the exogenous parameter W*, welfare associated with speculation in alternative assets.

σ(ε) σ (η) ρ I/S λ 5.0 0.3 0.80 10 0.65 6.0 0.4 0.85 12 0.70 7.0 0.5 0.90 14 0.75

51

Table 3: Simulations of the Model The table shows properties of model equilibria for various sets of parameters. Base case parameters: SD(ε) = 6.0; I/S = 12; SD(η) = 0.4; ρ = 0.85; θ = 5 (these are the midpoints of the ranges shown below). The model’s final exogenous variable is W*, welfare in alternative speculative markets. For computational reasons we choose to vary λ from a base value of 0.70 rather than vary W*, since the two are closely related.

β−1 SD(∆st) SD(rpt) SD(dt) Corr(rpt,rpt-1) Corr(∆st,∆st)Base Case -3.01 5.40 0.848 0.747 0.750 -0.085 .ρ = 0.80 .ρ = 0.90

-2.65 -3.34

5.14 5.87

0.908 0.586

0.655 0.747

0.755 0.745

-0.090 -0.070

SD(ε) = 5.0 SD(ε) = 7.0

-3.01 -3.01

5.11 5.72

0.735 0.966

0.747 0.747

0.766 0.731

-0.081 -0.091

SD(η) = 0.3 SD(η) = 0.5

-3.01 -3.01

4.55 6.32

0.814 0.891

0.561 0.934

0.751 0.770

-0.072 -0.080

I/S = 10 I/S = 14

-2.63 -3.39

4.92 5.90

0.801 0.915

0.747 0.747

0.722 0.781

-0.093 -0.078

.λ = 0.65

.λ = 0.75 -3.57 -2.45

5.68 5.11

1.058 0.660

0.747 0.747

0.695 0.805

-0.097 -0.075

52

Table 4: Econometric Estimates of the Standard Risk Premium Equation

The table shows econometric estimates of the following equation: xrt+1 = α + γdt + ξ t+1

where xrt+1 is the excess return to a currency, dt is the interest differential on three-month eurocurrency deposits relative to the U.S. (maturity-adjusted, continuous-time equivalents of quoted annual rates), and ξ t+1 is a random disturbance. Quarterly data from April 1978 through April 2003. Coefficient Newey-West

Standard Error

Marginal Significance

DEM dt -1.625 0.669 0.017 Adj. R2 0.036 JPY dt -4.105 0.763 0.000 Adj. R2 0.143 CHF dt -2.109 0.596 0.001 Adj. R2 0.078 GBP dt -3.230 1.171 0.007 Adj. R2 0.125 CAD dt -1.961 0.490 0.000 Adj. R2 0.105

53

Table 5: Econometric Estimates of the Model The model predicts that excess returns should be determined as follows:

xrt+1 = (Et+1 s t+2 - Et s t+1) + (1−λ)( Et s t+1 – st) + (β−1) dt + υt+1

where xrt+1 is the excess return, Et+1 s t+2 - Et s t+1 represents the change in the perceived long-run equilibrium nominal exchange rate, Et s t+1 – st represents the gap between this perceived long-run equilibrium real exchange rate and the current real exchange rate, dt is the interest differential, and υt+1 is a random disturbance.

Table shows econometric estimates of the following equation:

xrt+1 = α + δ(Et+1 s t+2 - Et s t+1) + (1−λ)( Et s t+1 – st) + (β−1) dt + υt+1

The constant is included to adjust for errors in our estimate of the expected long-term real exchange rate. Interest differentials are for three-month eurocurrency deposits relative to the U.S. (maturity-adjusted, continuous-time equivalents of quoted annual rates). The expected long-run nominal exchange rate is calculated on the assumption that the expected long-run real exchange rate is constant. Quarterly data from April 1978 through April 2003 include three-month euro-market interest rates and producer prices, as well as exchange rates. Coefficient Newey-West

Standard Error

Marginal significance

vs.(#) DEM Constant -0.002 -0.006 0.766 (0)

Et+1 s t+2 - Et s t+1 1.186 0.256 0.767 (1)

Et s_

t+1 – st 0.077 0.035 0.030 (0)

dt -2.040 0.637 0.002 (0) Adjusted R2 0.171 JPY Constant -0.014 0.007 0.046 (0)

Et+1 s t+2 - Et s t+11.745 0.302 0.008 (1)

Et s_

t+1 – st0.077 0.029 0.010 (0)

dt -3.277 0.627 0.002 (0) Adjusted R2 0.489 CHF Constant -0.011 0.008 0.164 (0)

Et+1 s t+2 - Et s t+11.495 0.301 0.133 (1)

Et s_

t+1 – st 0.098 0.036 0.091 (0)

dt -2.761 0.497 0.000 (0) Adjusted R2 0.239 GBP Constant -0.067 0.047 0.153 (0)

Et+1 s t+2 - Et s t+10.803 0.176 0.052 (1)

Et s_

t+1 – st0.071 0.040 0.083 (0)

dt -3.756 0.997 0.000 (0) Adjusted R2 0.202 CAD Constant 0.004 0.002 0.049 (0)

Et+1 s t+2 - Et s t+10.619 0.147 0.005 (1)

Et s_

t+1 – st 0.082 0.048 0.091 (0)

dt -2.340 0.520 0.000 (0) Adjusted R2 0.284

54