SHORT-RUN EXCHANGE-RATE DYNAMICS: EVIDENCE, THEORY, AND EVIDENCE

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1 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, 198, 199, Driskill, Mark, and Sheffrin 1987, Osler 1995, 1998, Carlson and Osler 000, Hau and Rey 004, Sager and Taylor 005a). 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, G1, G15] March 005 Corresponding author: Carol Osler, Brandeis International Business School, Mailstop 03, Brandeis University, Waltham, MA Tel.: Fax: cosler@brandeis.edu. The authors thank Gijoon Hong and Filippo Rotollo for excellent research assistance. We take responsibility for all errors.

2 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 00). 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. 004, Mende et al. 005). 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. 004). 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, 198, 199, Driskill, Mark, and Sheffrin 1987, Osler 1995, 1998, Carlson and Osler 000, Hau and Rey 004, Sager and Taylor 005a). 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.

3 The underlying structure of the model has three key elements: 1. Risk-averse speculators concerned with profits;. 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 () imply that the model s agents are heterogeneous. The heterogeneity of currency market participants is stressed in Sager and Taylor s (005a) 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. 004, Mende et al. 005, Sager and Taylor 005b). 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. 004). 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 (001), and Evans and Lyons (00), inter alia). Our equilibrium condition implies that flow currency demand and supply determine exchange rates, thus capturing the essence of that key lesson.

4 3 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 195, 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 utilitymaximization 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).

5 4 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. 004, Evans and Lyons 005, and Evans and Lyons 004) 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 (199), which analyzes an extended version of the model including explicit labor and goods markets and a more clearly elaborated money market, show that the

6 5 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. 60). 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 00). 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 003. 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 00, Backus et al. 1993). Exchange rates in these studies match some but not all of the empirical regularities associated with the forward premium puzzle.

7 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 (00). 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 (00) 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: interestsensitive 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, 198, 199), Driskill, Mark, and Sheffrin

8 7 (1987), Osler (1995, 1998), Carlson and Osler (000), and Hau and Rey (004). In addition, Sager and Taylor (005a) 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;. 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 (005a), 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. 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 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.

9 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 005)). 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 (000) 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.

10 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 001; Evans and Lyons 00). Since then, confirming evidence has appeared from many quarters (e.g., Bjønnes and Rime 001). 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 004), and may also include important information about transitory forces (Mende et al. 005). 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 005). 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 shortrun exchange rates cannot be uniquely determined by the money-market stock-equilibrium condition.

11 10 Non-uniqueness of short-run money demand: In their widely respected inventory-theoretic models of money demand Baumol (195) 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.

12 11 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 () they behave as if they are risk averse. Formally, we assume that speculators choose positions to maximize the expected utility of profits: (1) W t = E t (π t+1 ) - (θ/)var t (π t+1 ) Here, W t represents welfare expected conditional on information at time t, θ is a measure of risk aversion, E t (π t+1 ) denotes expected t+1 profits, and Var t (π t+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 b t, measured in units of foreign currency. The profits on this position are proportional to the change in (the log of) the exchange rate, s t+1 - s t, minus the short-term interest differential across countries, d t = r t r t *: () π t+1 = b t [s t+1 - s t - d t ]. The speculator s optimal bet is proportional to expected profits and inversely proportional to risk aversion and risk itself: (3) b t = [E t (s t+1 ) - s t - d t ]/θ Var(s t+1 ) Var(s t+1 ) is the expected variance of the exchange rate conditional on information at time t. As shown by Carlson and Osler (000), Var(s t+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, rp t E t (s t+1 ) - s t - d t. 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(s t+1 )].

13 1 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(b t - b t-1 ) = Nq(rp t - rp t-1 ). C. Non-Speculative Currency Demand We model net foreign-currency demand from non-speculative agents, FX t, as a simple linear function of the exchange rate: (4) FX t = C t - S s t, S > 0. C t 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

14 13 and Gorton 1993). In the exchange-rate literature, this type of modeling structure has been used by other authors including Hau and Rey (003) 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 C t. 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.

15 14 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 currentaccount 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. 00, 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 0:53 G.M.T. on January 6, 000. 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 (003) and Osler (005).

16 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 00). We incorporate this by assuming that equilibrium in the currency market occurs when total net foreign-currency demand equals zero: (5) FX t + N(b t - b t-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 196). 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) E t s t+1 - (1 + S/Nq)s t - E t-1 s t + s t-1 = - C t /Nq + t, 10 The information connection is highlighted in Lyons (001) and Evans and Lyons (00); 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.

17 16 where t d t - d t-1 represents the change in the interest differential. 1 The bubble-free solution, derived in the appendix, is: j λ (7) s t = λ s t-1 + (1-λ) λ ( EC t t+ j λet 1Ct+ j)/ S - j λ ( E t t+ j λet 1 t+ j ). 1 λ j= 0 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 C t (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 C t and t, which are the system s two sources of randomness. With regard to C t, suppose for now that this component of non-speculative currency demand is subject to i.i.d. meanzero shocks denoted by ε t : C t = C + ε t. In this case, E t C t+j = C for all j > 0. Later we permit C t to be disturbed by permanent shocks, as well as transitory shocks. The exchange rate s equilibrium in the j= 0 absence of speculators is C t /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 shortrun 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, 198, 199), and Driskill, Mark, and Sheffrin (1987). 1 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.

18 17 We also assume that interest differentials are stationary, and more specifically that they are AR(1): d t = ρ d t-1 + η t, where 0 < ρ < 1 and η t 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) s t+1 = s + λ(s t - _ s ) + (1-λ) εt+1-1 The first term on the right-hand side of (8) shows that λ ρλ η t+1 + λ (1 ρ) 1 ρλ d t, the long-run exchange rate in the absence of speculators, is still an important determinant of st when speculators are active. The second term shows _ that, in the absence of other influences, the exchange rate will converge to s monotonically, eliminating _ the fraction 1-λ of any discrepancy between s and st each period. Since the remaining three exchangerate determinants (d t, ε t+1, and η t+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, ε t > 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 η t+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 s t = C t /S = (C + ε t )/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, ε t, is immediately and fully reflected in the current exchange rate _ s

19 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 (005a) 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/currentaccount traders. Bjønnes et al. (004) show that this theoretical negative relationship is actually observed in reality, using data covering virtually all trading in SEK/EUR during 1993 to 00. 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 (005a) 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 (004) 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.

20 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. (005) based on higher frequency data for a relatively small bank. Further confirming evidence is presented in Sager and Taylor (005b) and Bjønnes et al. (004). 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. (004) showing that net demand for SEK from nonfinancial 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.

21 0 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) rp t = E t (s t+1 ) - s t + r* t - r t Here, rp t represents the risk premium anticipated at time t, s t represents the (log) spot exchange rate at time t, measured as domestic currency units per foreign currency unit; r t * and r t represent foreign and domestic interest rates, respectively, E t 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) E t (s t+1 ) - s t = a+ b(r t - 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) s t+k - s t = α + β(r t r* t ) + µ t+k, where the expected appreciation is replaced by the actual appreciation, and µ t+k represents a mean-zero random disturbance. Under the null hypothesis, rp t = 0 and the true value of β is unity.