The microstructure of covered interest arbitrage in a market with a dominant market maker

Int. Fin. Markets, Inst. and Money 24 (2013) 25–41

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Journal of International Financial Markets, Institutions & Money j o ur na l ho me pa ge : w w w . e l s e v i e r . c o m / l o c a t e / i n t f i n

The microstructure of covered interest arbitrage in a market with a dominant market maker Hao-Chen Liu 1, Mark David Witte ∗ Department of Economics and Finance, 5 Liberty Street, College of Charleston, Charleston, SC 29401, United States

a r t i c l e

i n f o

Article history: Received 18 January 2012 Accepted 29 November 2012 Available online 8 December 2012 JEL classification: F31 F41 Keywords: Exchange rates Covered interest Foreign exchange microstructure Forward swap Hedging

a b s t r a c t Measured by transaction volume, foreign exchange swaps are the largest market in the world. However, there are very few empirical studies of swap rates. Theoretically, covered interest parity is commonly assumed. But what factors determine arbitrage opportunities? We create a unique microstructure model of exchange rate activity to identify theoretical predictions regarding covered interest arbitrage in a market with a dominant market maker. Using a unique data set of actual, recorded swap transactions, not price quotes, the model is verified as we find economically significant returns that depend in part on market volatility, contract irregularity and trader identity. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Swap transactions consist of both a spot exchange and forward exchange. According to the Bank for International Settlements, in 2010 the spot market was responsible for 37% of all foreign exchange transactions, the outright forward market accounted for 18% and the swap market was approximately 45%. Despite the large contribution from swap transactions to the world’s largest financial market, the swap market has received very little attention in the literature. In this study we develop a theoretical model on covered interest return in a market with a dominant market maker and examine empirically the model predictions with actual swap transactions in the New Taiwanese Dollar/US Dollar market.

∗ Corresponding author. Tel.: +1 843 953 3986. E-mail addresses: [email protected] (H.-C. Liu), [email protected] (M.D. Witte). 1 Tel.: +1 843 953 1988. 1042-4431/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.intfin.2012.11.012

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Based on the theoretical model of Burnside et al. (2009) we construct a unique theoretical representation specifically designed to best represent the NTD/USD market. The theoretical model herein adds to the literature by including not only arbitrage traders seeking profit but also hedgers who merely wish to reduce exchange rate risk. The model is designed to represent smaller countries where the swap exchange is dominated by a market maker, not necessarily larger, more liquid markets. Akram et al. (2008) show that covered interest returns occasionally appear in large countries. Using tick quotes the authors find that positive returns are more likely to occur in periods of high volatility or in less active/liquid markets. Taylor (1989) finds more profitable deviations from covered interest parity during periods of greater volatility or when the contract duration is longer. Hui et al. (2009) and Baba and Packer (2009) examine covered interest returns during the recent credit crisis. These works are limited to using quoted forward spot rates to construct potential covered interest arbitrage opportunities. Because the data set herein is composed of actual spot and swap rates we are uniquely able to examine what the implied covered interest returns are for each individual swap transaction. As to our knowledge, this is the first study able to calculate actual, not potential, returns to covered interest arbitrage. The unique data set allows us to test theories regarding the standardization of swap contracts, which to our knowledge has never been empirically examined. The theoretical model makes three predictions that are verified in the data. One, economically significant returns are available from the swap market. Around 42% of swap transactions have an implied covered interest return over 0.1%. Given that the average swap contract is a little more than $10 million USD, a 0.1% return for an averaged sized transaction is equal to $10,000 USD. Approximately 4.4% of transactions have a return greater than 1%; equivalent to a $100,000 USD return on average. Because these are covered interest returns they are free from exchange rate risk and are in addition to the available returns from prevailing interest rates. Two, the difference between the spot rate and swap rate (defined as the swap points by traders) will increase for more irregular contracts. While some contract durations are more standardized, other contract durations are quite irregular as they are witnessed only a few times in the data set. Results show that as the contract duration becomes more irregular the returns to covered interest arbitrage increase. Three, greater market volatility will decrease the ex-ante covered interest returns for irregular contracts but may not decrease the ex-ante returns for more standardized contracts. This result directly conflicts with Akram et al. (2008) who suggest that higher returns are more common when market volatility is high. However, the data set in Akram et al. (2008) is limited to quoted forward and spot rates of the most standardized contract durations. Using actual swap transactions we find that irregular contracts experience decreased returns under heightened market volatility. Yet, we confirm that with the most standardized contracts an increase in market volatility may increase covered interest returns. The paper is organized as follows. In Section 2, we construct a model of the microstructure of the spot and forward NTD/USD markets to create predictions regarding the swap market. In Section 3, we describe the swap transaction data set in detail and define the return to covered interest arbitrage. In Section 4, we estimate the swap returns in a seemingly unrelated regression (SUR) to account for the potential cross-correlation in the error terms for the three components negotiated in a swap transaction: the swap returns, the contract duration and the amount in millions of USD for the contract. Section 5 concludes. 2. Theory Our model for the equilibrium in the swap market follows that of the spot and forward exchange rate microstructure model in Burnside et al. (2009). We elect to use this model as a point of reference because it creates a stable equilibrium with an ability to mimic empirical outcomes from the literature. We will adapt the model to the specific USD/NTD market we empirically examine in Section 4. In the USD/NTD swap market participants negotiate the swap points (the difference between the forward rate and the spot rate). The market maker, the swap desk in the USD/NTD market, quotes reference prices around which the participants negotiate the swap transaction. When the participants agree to the swap points the swap desk retrieves the current spot rate from the spot desk and merely adds the swap points to the current spot rate to record the forward rate. While our model below asserts greater

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price-setting power to the swap desk than the real USD/NTD market would suggest, the model’s results should be robust because of the swap desk’s role as a common reference for traders. Traders could be willing to negotiate around the swap desk’s reported rate to achieve a more favorable contract size or duration. 2.1. Spot rate determination The spot exchange rate follows an exogenous process detailed in Eq. (1). et+1 − et = ϕt + εt+1 + ωt+1 et

(1)

et is the spot exchange rate defined as the number of NTD for one USD. The spot exchange rate is impacted by predictable trends, ϕt , based on publicly available information (an example would be monetary policy). The information is available to all market participants. We assume that ϕt is independently and identically distributed with a binominal distribution as arbitrage traders are concerned mostly with the direction of the exchange rate. Pr(ϕt = ϕ > 0) = Pr(ϕt = −ϕ < 0) =

1 2

(2)

The directional change of the exchange rate is also determined by unknown but somewhat predictable trends, εt+1 , as well as unknown and unpredictable trends, εt+1 . While εt+1 is not observed at time t it is however predicted accurately by a group of informed traders. The informed traders could be able to predict this value either by receiving an advance signal or simply by being better able to process known events. This variable is also independently and identically distributed according to a binominal distribution outlined below: Pr(εt+1 = ε > 0) = Pr(εt+1 = −ε < 0) =

1 2

(3)

None of the traders know, or have any ability to forecast, ωt+1 . The variable is independently and identically distributed with mean zero and a known but unstable variance  ω,t . While traders may know the value of  ω,t the variance can fluctuate over time. We include this characteristic in our model because spot exchange rates typically have different levels of volatility at different times. Because the exchange rate generation process in Eq. (1) is built from the values of ϕt , εt+1 (which both have constant variance) and εt+1 then in order to introduce the possibility of a fluctuating variance in the exchange rate process we must allow  ω,t to be unstable. 2.2. Market participants There are four main groups involved in this model. Three groups are potential buyers and sellers of forward contracts: uninformed arbitrage traders, informed arbitrage traders and hedgers. The fourth market participant is the market maker who receives a commission on every trade they broker and as such wish to maximize the volume of activity. The arbitrage traders seek profit by purchasing NTD forward contracts (or selling NTD forward contracts) when they expect the NTD to appreciate (depreciate). For simplicity the arbitrage traders are risk neutral with finite limits on the volume of their trades. Because of asymmetric information there are two groups of arbitrage traders: uninformed traders that cannot predict εt+1 and informed traders that can. The proportion of informed traders is denoted by ˛1 and the proportion of uninformed traders is˛1 . If the public signal (ϕt ) is positive then the uninformed traders will purchase the NTD forward with a probability of ( ω,t ). When the volatility of the spot rate increases the uninformed traders use less discretion and are more likely to follow the simple behavioral rule of buying the NTD forward when they expect appreciation of the NTD. If the public signal is negative then the uninformed traders purchase the NTD forward with probability 1 − ( ω,t ). Informed traders also follow the simple behavioral rule of buying the NTD forward when they expect appreciation of the NTD. The informed traders receive a signal at time t about the value of εt+1

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with a probability of q. When εt+1 = ε > 0 the informed traders will purchase the NTD forward with a probability equal to the likelihood of receiving the signal of εt+1 . The third group consists of hedgers where the proportion of hedgers is given by ˛3 = 1 − ˛1 − ˛2 . Hedgers assume the exchange rate market follows a random walk but they are risk averse. These are firms or banks with commercial interests who wish to reduce exchange rate risks and have no desire to attempt to profit from exchange rate arbitrage. Hedgers are more likely to purchase a forward contract when the asking price of the forward contract is equal to the current spot rate (i.e., the swap points are equal to zero). The hedgers will tend to purchase contracts with more standard time periods as they may lack the ability (or desire) to negotiate on the floor of the exchange. Thus, if the contract duration is more standard (such as 7 days, 30 days, 90 days, 365 days) then the hedgers will be a larger portion of the market’s volume compared to the arbitrage traders. If  ω,t is large then the market is more volatile and hedgers are more likely to engage in forward contracts regardless of the swap points. H( ω,t ) represents the price elasticity of demand for the hedgers so that H < 0 and the lim H(ω,t ) = 0. The probability of a hedger buying the NTD forward is denoted ω,t →∞

by h( ω,t ). We define h( ω,t ) so that h > 0 while the

lim h(ω,t ) = 1 and the lim h(ω,t ) = 0.

ω,t →∞

ω,t →0

The market maker receives a commission based on the volume of contracts that it brokers. The market maker maximizes profit by reducing the number of contracts lost as a result of poorly quoted pricing. If the quoted swap points are either too large or too small then there will be either excess supply or excess demand for NTD forward contracts among the arbitrage traders. However, the hedgers wish to buy forward contracts when the swap points are zero but can be induced to buy forward contracts if volatility in the spot rate is high. The market maker wishes to minimize the following loss function.

1

LossVolumet = −

2

2

(fta + ftb ) − E(et+1 )

− DH(ω,t )(fta − et )

2

(4)

The forward ask rate and the forward bid rate at time t are denoted by fta and ftb , respectively. The first portion of the loss function represents the volume of activity lost because the arbitrage traders have either excess supply or excess demand while the second portion of the loss function represents the lost volume from hedgers who would like to pay an asking price equal to the current spot rate. However, the hedgers are more likely to purchase forward contracts at any given forward rate as the volatility of the spot rate increases. D denotes how standard or irregular the forward contract duration is. When the contract duration is more irregular, hedgers make up a smaller proportion of the market’s volume and D becomes smaller; more or less, D measures the volume of the market driven by hedgers. For simplicity, we allow the optimal difference between the bid and ask forward rate to be denominated as  such that  = ftb /fta . Then by optimizing with respect to fta the optimal forward asking rate can be defined as follows: fta =

E(et+1 )(1 + ) + 2DH(ω,t )et (1 + ) + 2DH(ω,t )

(5)

Given that  ∼ = 1 we can simplify Eq. (5). E(et+1 ) + DH(ω,t )et fta ∼ = 1 + DH(ω,t )

(6)

This is equivalent to the setting of the forward ask rate in Burnside et al. (2009) when H( ω,t ) = 0 or D = 0. Given that the market maker’s information set includes the value of ϕt we can rewrite Eq. (6) as follows: fta (ϕt ) ∼ =

et (1 + ϕt + E(εt+1 |buy, ϕt )) + DH(ω,t )et 1 + DH(ω,t )

(7)

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2.3. Equilibrium The forward ask rate is still impacted by the expected value of the future spot rate. Suppose that ϕt = ϕ > 0 and εt+1 = ε > 0, then the probability of retrieving a buy order from the continuum of arbitrage traders and hedgers is as follows: Pr(buy|εt+1 = ε, ϕ) = ˛2 v(ω,t ) + ˛1 q + ˛3 Dh(ω,t )

(8)

However, if εt+1 = −ε < 0 then, Pr(buy|εt+1 = −ε, ϕ) = ˛2 v(ω,t ) + ˛1 (1 − q) + ˛3 Dh(ω,t )

(9)

Thus the probability of retrieving the buy order is as follows: Pr(buy|ϕ) = ˛2 v(ω,t ) +

˛1 + ˛3 Dh(ω,t ) 2

(10)

Bayes’ rule implies: Pr(εt+1 = ε|buy, ϕ) =

Pr(buy|εt+1 = ε, ϕ)Pr(εt+1 = ε) Pr(buy|ϕ)

(11)

Substituting Eqs. (8) and (10) into Eq. (11): ˛2 v(ω,t ) + ˛1 q + ˛3 Dh(ω,t ) 2˛2 v(ω,t ) + ˛1 + 2˛3 Dh(ω,t )

Pr(εt+1 = ε|buy, ϕ) =

(12)

And if εt+1 = −ε < 0: Pr(εt+1 = −ε|buy, ϕ) =

˛2 v(ω,t ) + ˛1 (1 − q) + ˛3 Dh(ω,t ) 2˛2 v(ω,t ) + ˛1 + 2˛3 Dh(ω,t )

(13)

Using Bayes’ rule we can show the market maker’s expectation of εt+1 given his information set. E(εt+1 = ε|buy, ϕ) = (ε)(Pr(εt+1 = ε|buy, ϕ)) + (−ε)(Pr(εt+1 = −ε|buy, ϕ))

(14)

Substituting Eqs. (12) and (13) into Eq. (14): E(εt+1 = ε|buy, ϕ) = Eq. (15) into Eq. (7):



fta (ϕ) ∼ =

ε˛1 (2q − 1) 2˛2 v(ω,t ) + ˛1 + 2˛3 Dh(ω,t )



(15)



et 1 + ϕ + [ε˛1 (2q − 1)]/[2˛2 v(ω,t ) + ˛1 + 2˛3 Dh(ω,t )]

+ DH(ω,t )et

1 + DH(ω,t )

(16)

If these steps are repeated under the assumption that ϕt = ϕ < 0 then the forward ask rate is as follows:



fta (−ϕ)

∼ =





et 1 − ϕ + [ε˛1 (2q − 1)]/[2˛2 (1 − v(ω,t )) + ˛1 + 2˛3 Dh(ω,t )]

+ DH(ω,t )et

1 + DH(ω,t ) (17)

2.4. Predictions from the theory There are three main predictions from the model. First, there are potentially sizable returns to covered interest arbitrage. Because the forward rate is set in part by the actions of hedgers and arbitrage traders, there’s little concern for interest rates. As seen in Eqs. (16) and (17), the equilibrium forward rate is not a function of the interest rates. If NTD interest rates are equal to the USD interest rates then covered interest arbitrage profits are available simply by engaging in a forward contract in which the agent is paid back more in the future than they pay in the

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spot market. Burnside et al. (2009) assume that the agents and market makers act in the interest rate market to achieve covered interest parity in equilibrium; their interest rates are function of the exchange rates. We make no such assumption as the arbitrage traders and hedgers may have limited ability (and limited volume) to impact the relevant interest rates in a meaningful way. Two, |ft − et | increases when the contract duration becomes more irregular. Eq. (6) shows that the forward rate is partially determined by the expected future spot rate and the current spot rate. When the contract duration is most irregular the hedgers are not part of the market’s volume; D = 0 and the forward rate moves farther away from the current spot rate. However, for more standardized contract durations the hedgers make up a larger portion of the market’s volume and the market maker is more likely to reduce the swap points (defined as the difference between the forward rate and the current spot rate). Three, heightened spot rate volatility is much more likely to decrease the difference between the forward rate and the current spot rate for irregular contract durations. Hedgers are more likely to engage in the forward market under standard contract durations as they may have limited ability or desire to negotiate on the trading room floor. Eq. (6) suggests that more standardized contract durations (with a larger value of D) will see an increase in |ft − et | when the spot rate volatility, determined by  ω,t , increases. However, the more irregular contracts (with smaller values of D) will see |ft − et | decrease according to Eqs. (16) and (17). If D = 0, and if  ω,t is very large then the value of v(ω,t ) approaches 1 and the average value of the forward ask rate, fta (ϕt ) approaches et regardless of the value of ϕ. As shown in Eqs. (16) and (17), when the spot rate volatility increases the uninformed traders are more likely to follow a behavioral rule (and avoid discretion) and the market maker’s expectation of εt+1 proceeds to change in the opposite direction of ϕt ; the forward ask rate approaches the current spot rate. Simply, the absolute value of the swap points, |ft − et |, is non-monotonic with the value of  ω,t and depends on the value of D. The more irregular the duration of the contract the more likely that increases in the spot rate volatility will decrease the absolute value of the swap points. Greater details about the impact of  ω,t and D on the value of |ft − et | can be found in Appendix A. 3. Data description A swap transaction combines a spot exchange rate transaction and a forward rate transaction. The swap transaction data used in this study comes from a single forward swap trading desk of a leading foreign exchange brokerage firm in Taiwan.2 This data set is unique because it consists of transactions, not quotes, in the swap market. The data set covers 3692 trade-to-trade swap transactions between August 1st, 2001 and July 31st, 2003. While this data set is limited to only one currency pair, which is far from being one of the most actively traded, it is as far as we know the only study of covered interest arbitrage to make use of actual transactions. The natural log of the difference between the forward rate and spot rate for the transactions is given in Fig. 1. Swaps are a type of forward trade in which two banks simultaneously buy and sell a currency against each other. This transaction combines a spot transaction at the spot date and a forward transaction at the forward date. In a NTD/USD swap transaction, the NTD/USD market uses indirect quotation, i.e., the price of USD in terms of NTD. For example, a swap buyer sells USD (i.e., receives NTD) at the spot date and receives USD (i.e., sells NTD) at the forward date while a swap seller receives USD (i.e., sells NTD) at the spot date and sells USD (i.e., receives NTD) at the forward date. The market practice is to use the prevailing market spot rate at the completion of the trade while the swap rate at the forward date is determined by adding swap points to the spot rate.

2 The authors have permission to use this data but, having signed a confidentiality agreement, are not permitted to disclose the data or use it for non-academic purposes.

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Fig. 1. Natural log (spot/forward) – NTD/USD. Source: Author’s calculations.

For each transaction we can identify the specific traders between 59 separate financial institutions. Because of the sensitive nature of this information, we will not disclose the actual identity of the financial institutions involved though we have classified the firms into three categories: non-Taiwanese institutions, old Taiwanese institutions and new Taiwanese institutions. Eun and Sabherwal (2002) suggest a degree of information asymmetry between local and foreign banks. They find home-country advantages for home-banks in forecasting a country’s currency rate. In another study, Deutsche Bank was identified as a price leader in the Deutsche mark and US Dollar market (see Peiers, 1997). Furfine (2004) also shows the effects of the bank identity on lending activities in the U.S. Fed Funds market. To control for possible information asymmetry we categorize the banks according to the following method. A Taiwanese institution is considered “new” when the institution was established after the banking deregulation in the 1990s. In Taiwan, before the banking industry was deregulated, there were only a few state-owned banks. We label these banks “old”. Because the government owned a large portion of the old banks before the industry was deregulated these old banks may have better networks with the central bank and may possess a different information set compared with new and foreign banks. However, prior to being deregulated these old banks were the only banks available in the market; with little competition over market share it is possible that the old banks are less innovative and less competitive in foreign exchange. The old banks were more involved in traditional banking business, not financial market speculation, and attained a greater share of their profits from deposits and lending. Conversely, the new and foreign banks depend more on product innovation and profit from trading. In the sample, there are 59 financial institutions, including 27 foreign banks, 13 old banks, and 19 new banks (see Panel A, Table 1). With regard to trading frequency, among the 3692 transactions, on the swap-buyer side, foreign banks account for 74.7% of transactions, old banks account for 12.5%, and new banks account for 12.8%. On the swap-seller side, foreign banks account for 58.4%, old banks account for 25%, and new banks account for 16.6% (see Panel B, Table 1). Panel C of Table 1 further breaks down each transaction by bank type and transaction side (i.e., swap-buyer vs. swap-seller). For example, there are 283 trades in which the swap-buyer is an old bank and the swap-seller is a foreign bank and 612 trades in which a foreign bank is the swap-buyer and a new bank is the swap-seller.

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Table 1 Transactions by bank identity. Foreign (New, Old) equals 1 if the arbitrage taker is a foreign (new Taiwanese, old Taiwanese) bank. Panel A shows the quantity of banks by each type found within the data. Panel B shows how often in the data that the different types of banks are either swaps buyer or swaps sellers. Panel C shows how often in the data that different pairwise combinations are found. Panel A: Bank types Bank type

Obs.

Percentage

Foreign Old New Total

27 13 19 59

45.8% 22.0% 32.2% 100%

Panel B: Breakdown of bank type by transaction side Bank type

Swaps-buyer

Foreign Old New Total

Swaps-seller

Obs.

Percentage

Obs.

Percentage

2758 461 473 3692

74.7% 12.5% 12.8% 100%

2157 924 611 3692

58.4% 25.0% 16.5% 100%

Panel C: Breakdown of transactions by the counter-parties based on bank type Transaction type

Swaps-seller

Total

Swaps-buyer

Foreign

Old

New

Foreign Old New Total

1656 283 218 2157

490 52 69 611

612 126 186 924

2758 461 473 3692

Following Akram et al. (2008), we use deposit rates for interest rates.3 As the authors note, this can be limiting because it implies that potential arbitrageurs have credit agreements with financial institutions. Luckily in this data set, the 59 observed participants in the market are all financial institutions which suggest that they already possess credit agreements with other institutions. Fig. 2 shows the difference between interest rates on Taiwanese Dollar deposits and US Dollar deposits. We follow Akram et al. (2008) by including deposit rates for durations of one month, three month, six month and one year. For the most part, interest rates in Taiwan tend to be higher than interest rates in the United States and the differential is larger for deposits with longer durations. Covered interest parity for the Taiwanese/US market could be expressed by the following equation: (1 + rT ) =

FT /U ST /U

(1 + rU )

(18)

The interest rates in Taiwan and the United States are represented by rT and rU , respectively. The forward (F) and spot (S) exchange rates are defined as Taiwanese Dollars per US Dollar (NTD/USD). Given the daily interest rate differential and the actual swap transaction data, it is possible to construct the implied covered interest arbitrage returns for each transaction. In previous research where the forward and spot rates are taken from indicative quoted data it is necessary to construct a window of no arbitrage set by the bid-ask spread (Akram et al., 2008; Szilagyi and Batten, 2006; Taylor, 1987,

3 Daily data on deposit rates come from Thomson Reuter’s Datastream. Due to a low volume market the Taiwanese interest rates on deposits do not always change on daily basis. While we would prefer to use market based bond rates, the Taiwanese bond market is highly illiquid and does not have bond durations of the varieties available for U.S. Treasuries. Regardless, we really want to measure the difference between Taiwanese and U.S. interest rates which deposit rates will provide. Lastly, as we will show below the main results of this paper are unchanged if we subtract the interest rate differential and examine only swap points as opposed to covered interest arbitrage.

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Fig. 2. Deposit interest rate differential (Taiwan rate minus U.S. rate). Source: Thomson Reuter’s Datastream and author’s calculations.

1989). The bid-ask spread represents the transaction costs to engage in the exchange rate market. However, the data set herein is actual transaction prices which include the transaction cost involved in each trade, not quoted data. Covered interest arbitrage research using indicative quoted data with bid-ask spreads assumes that a trader sells at the bid price and buys at the ask price for both spot and forward transactions. However, in practice, swap prices are quoted by using swap points. When a swap transaction is completed, the swap points are added to the last traded spot price. Therefore, the spot price is not necessarily the bid or ask rate. Our data set includes not only the swap price but also the spot price for each trade. Because we possess actual transaction prices we do not necessarily have a window of non-existent covered interest returns created by the bid-ask spread. Therefore, we can more accurately capture any covered interest arbitrage. We ignore brokerage costs as being too small to impact potential returns; evidence suggest that they are merely 0.001% of the order volume. For these reasons we can describe the implied covered interest return for each transaction as follows:

  

Implied covered interest return = (1 + rT ) −

FT /U ST /U

 

(1 + rU )

(19)

The implied covered interest return (ICIR) is an absolute value because for each transaction; there is one financial firm that is achieving a positive covered interest return while the other institution is accepting a negative covered interest return. Contrast the implied covered interest returns with the percentage value of the swap points given below:

   FT /U − ST /U    ST /U 

SwapPoint% = 

(20)

Fig. 3 denotes the level of similarity between the implied covered interest return and the swap point percentage. The two measurements are highly correlated with a correlation coefficient of 0.925 suggesting that most of the covered interest return in this market is derived from the swap point percentage and not the differences in interest rates; the gap between the two measurements of return becomes more pronounced over values of 0.1% denoting that a large swap point percentage corresponds to a large difference in interest rates. While these returns may not seem particularly high, the average swap contract is just over $10 million USD. Thus, a 0.1% return is a risk-free $10,000 USD. Over 42% of transactions have a return equal to or greater than 0.1%. Approximately 4.4% of transactions have an implied covered interest return of over 1% indicating a risk-free return over $100,000 USD per

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Fig. 3. Returns from swap market (percentage). Source: Author’s calculations.

transaction. Additionally, as suggested by the ICIR from Eq. (19); the returns from the swap market are in excess of the prevailing interest rates. Fig. 3 shows that the first prediction from the theoretical section appears to be accurate. Risk-free economically significant returns can be achieved by traders in the swap market. Just how quickly are these risk free returns realized? The duration of the swap contract varies dramatically in the data set. Fig. 4 shows the distribution of swap contract duration for the 3692 transactions observed. Durations corresponding to one week, one month, two months, three months,

Fig. 4. Duration of swap contracts by frequency of occurrence. Source: Author’s calculations.

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six months, nine months and one year tend to occur more often than the irregular terms. In fact, there are 26 swap contracts where the duration only occurs once in the data set and 60 swap contracts with durations that occur ten times or less. Given the non-standard duration of many of these transactions we use different deposit rates for the interest rates depending on the duration of the swap contract. For durations less than 61 days we use one month interest rates, three month interest rates are used for durations between 61 and 137 days, six month interest rates are used for durations between 137 and 274 days and one year interest rates are used for durations greater than 274 days. The windows are roughly halfway between any two deposit rate durations. Because the swap contract term may not perfectly match the term of the interest rate it is possible that the timing mismatch may bias the results. However, each bank likely has their own financing constraints for each currency and has their own internal return of holding each currency. Because we cannot possibly know each bank’s internal interest rates at the time of each transaction for every different term, we instead approximate the internal interest rates with market interest rates. As the calculation of the swap transaction return includes only the difference between the USD and NTD rates, even if there are inconsistencies in the recorded interest rates, the difference between the two rates should be consistently approximated. Lastly, and perhaps most importantly, for a robustness check we examine both the covered interest return and the swap point percentage and find similar results. The calculation of the swap point percentage occurs without the interest rates and, as shown in Fig. 3, is highly correlated with deviations from covered interest parity.

4. Results To analyze covered interest returns we make use of a Seemingly Unrelated Regression (SUR) approach. When traders agree to a swap transaction there are three separate elements negotiated. The first element is the swap return, which is included as either the swap point percentage given in Eq. (20) (the effective focus of the theoretical section) or the implied covered interest return from Eq. (19) (which is highly correlated with the swap point percentage as shown in Fig. 3). The second element is the size of the contract in millions of USD. The third element is the duration of the contract in days. Because these three separate elements are presumably negotiated as a package, we wish to control for potential cross-equation correlation in the error terms. Additionally, because we know that timing effects are of vital importance in financial markets we use maximum likelihood estimation to optimize the SUR so as to allow for clustering and robust standard errors. We cluster the observations by the date of the transaction to control for any news events that may have been impacting the market on that given day. Because we cluster the observations in this methodology this implies that the standard errors are calculated as Huber–White robust standard errors. The regression formation is given below: SwapReturni,t = ˇi,t Xi,t + ε1,i,t TradeSizei,t = ˇi,t Xi,t + ε2,i,t

(21)

ContractDaysi,t = ˇi,t Xi,t + ε3,i,t Again, the SwapReturn could be either the swap point percentage in Eq. (20), SwapPoint%, or the implied covered interest return from Eq. (19), ICIR. Descriptive statistics for the dependent variables are given in Table 2. To analyze the determinants of the factors listed in Table 2 we use a variety of potential explanatory variables listed in Table 3. The second prediction from the theoretical section says that more irregular durations will be associated with a greater swap points. High, Medium and Low are indicator variables that denote the regularity of the swap contract within the data set. High is equal to 1 when the number of days of the contract is found in more than 5% of the data set (7 days, 31 days, 92 days and 365 days). Medium suggests a mostly standard contract duration appearing in the data set in 1–5% of the observations while Low is for irregular contract durations when the number of days comprises less than 1% of observations in the database (and occasionally represented only once in the data set).

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Table 2 Descriptive statistics. Variable

Obs.

Mean

Std. Dev.

Min

Max

Swap points Covered interest return Trade size Contract days

3692 3692 3692 3692

−0.03994 0.002068 10.07765 122.539

0.068141 0.003045 5.742084 122.1984

−0.39 2.78E−07 1 3

0.072998 0.016607 50 547

Table provides summary statistics of the following variables, swap points, covered interest return, trade size, and contract days. Swap points is defined as the difference in the forward rate and spot rate. Covered interest return is defined as |(1 + rT ) − (FT/U /ST/U ) × (1 + rU )|, where rT (rU ) is the NTD (USD) interest rate, and FT/U (ST/U ) is forward (spot) rate. Trade size is the size of each transaction in terms of million USD. Contract days is the number of days in each transaction.

High, Medium and Low are also very important to the third prediction from the theoretical section: heightened spot volatility will likely decrease the swap points (and implied covered interest returns) when the contract duration is irregular. We will analyze the impact of the interaction between Spot.volatility and the contract duration indicator variables to test this prediction. Spot.volatility is the variance of daily spot rate changes in the 30 days prior to the swap contract date. Because we know both the swap seller and the swap buyer, we can examine whether certain categories of traders have significantly higher returns than others. Foreign is an indicator variable equal to 1 when the profiting trader is a foreign bank. New denotes a “new” Taiwanese bank established after the banking deregulation in the 1990s while Old is a Taiwanese bank established before the wave of banking regulation. We also examine whether certain types of bank are consistently profiting when trading with another category of bank. Foreign.Foreign (New.New, Old.Old) equals 1 if both swap buyer and swap seller are foreign (new Taiwanese, old Taiwanese) banks. For cross-category swaps we use New.Foreign1, New.Old1 and Foreign.Old1. New.Foreign1 equals 1 (−1) if the profiting trader is a new (foreign) bank. New.Old1 equals 1 (−1) if the profiting trader is a new (old) bank. Foreign.Old1 equals 1 (−1) if the profiting trader is a foreign (old) bank. For control we include Total.volume, the total trading volume during the day, as market liquidity could impact returns. Forward.spread is the prevailing bid-ask spread for forward contracts of a similar duration during the trading day as transaction costs could impact trading behavior and returns. Following Moulton (2005) Terr911 is an indicator variable for those trading days immediately following Sept. 11, 2001. Q.end and Q.beg are indicator variables for the trades conducted in the last week of the Table 3 Descriptive statistics. Variable

Obs

Mean

Std. Dev.

Min

Max

Total.volume Spot.volatility Forward.spread Foreign New Old Terr911 Medium Low High Q.end Q.beg Buyer.init

3692 3692 3692 3692 3692 3692 3692 3692 3692 3692 3692 3692 3692

103.7926 5.75E−07 0.086926 0.680932 0.188787 0.130282 0.006771 0.493229 0.089924 0.416847 0.091549 0.07584 0.783044

54.51414 5.84E−07 0.044469 0.466179 0.391392 0.336659 0.082021 0.500022 0.286112 0.493104 0.288428 0.264777 0.412228

7 1.28E−08 0.009 0 0 0 0 0 0 0 0 0 0

275 2.90E−06 0.25 1 1 1 1 1 1 1 1 1 1

Table provides summary statistics of the explanatory variables. Total.volume is the total daily trading volume. Spot.volatility is the standard deviation of 30-day spot rate prior to the transaction day. Forward.spread is the forward bid and ask spread. Foreign (New, Old) equals 1 if the arbitrage taker is a foreign (new, old) bank. Medium equals 1 if the duration of a contract occurs between 1% and 5% in the sample. Low equals 1 if the duration of a contract occurs less than 1% in the sample. High equals 1 if the duration of a contract occurs more than 5% in the sample. Terr911 equals 1 if a trade occurs within 3 days following September 11th, 2001. Q.end equals 1 if a trade occurs within 1 week at a quarter’s end. Q.beg equals 1 if a trade occurs within 1 week at a quarter’s beginning. Buyer.init equals 1 if the arbitrage taker is a swap buyer.

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quarter and the first week of the quarter, respectively, included to control for potential market timing effects on the part of traders. Lastly, Buyer.init is an indicator variable equal to 1 when the profiting trader is the swap buyer (buys NTD at spot rate and sells NTD in the future – forfeits short-term USD liquidity). Buyer.init is included because there could be a premium to hold USD liquidity in Taiwan. Results for the SUR with the swap point percentage are given in Table 4 and the results with the implied covered interest returns are given in Table 5. A cursory glance at Tables 4 and 5 shows similar results; significant determinants of the swap point percentage are also significant determinants of the implied covered interest return. Because the results are so similar using either SwapPoint% or ICIR there’s little reason to believe that mismatches in interest rate terms drive the results simply because the SwapPoint% is calculated without the interest rates and is highly correlated with ICIR as shown in Fig. 3. But what of the predictions made in the theoretical section? In Reg. 1 through Reg. 6 of both tables we find that contracts with a more irregular duration have a greater swap point percentage and a larger implied covered interest return. Low, which denotes the most irregular contract duration, is positive and significant. Medium, is also positive and though not always statistically significant. Obviously, these coefficients are relative to the omitted variable, High, denoting the most standardized contract duration. More irregular contracts have higher swap points confirming the second prediction from the theoretical model. As contract durations become more standardized it is likely that more market activity is driven by hedgers who would prefer to merely lock in the current spot rate. The third prediction is also confirmed. Market volatility impacts certain types of swap contracts differently. The most common contracts (High) experience increased returns when the market volatility increases (Reg. 4). As shown in the theoretical section, hedgers may be less price-sensitive when market volatility is high and as such will be willing to pay more swap points. However, the most irregular contract durations (Low) will have little activity from hedgers and high market volatility decreases the swap points (Reg. 2). This result directly conflicts with Akram et al. (2008) who suggest that higher returns are more common when market volatility is high. Akram et al. examine quoted forward and spot rates, not actual transactions. Therefore, their data set includes only the most standardized contract duration. We confirm that heightened volatility increases the potential swap returns for standardized contract duration but contradict this finding as indicative of all swap market activity. The last prediction from the theory suggests that some traders should be able to make higher returns than others. In the simplest formation Foreign and New are significant and positive suggesting that foreign banks and new Taiwanese banks consistently achieve higher returns when they profit from a swap contract (Reg. 1–4). In Reg. 5 we compare the returns when the swap contract includes swap buyers and swap sellers of the same type – the implied omitted category being swap contracts conducted across categories. Foreign.Foreign is a positive and significant determinant of implied covered interest return but is not a significant determinant of swap points. Both New.New and Old.Old are significantly negative for both the SwapPoint% and ICIR. When Taiwanese banks trade with one another they are less likely to earn larger profits – this could be because they are herders who are attracted to smaller swap points or because they have similar information sets or abilities to find profits from the swap market. Reg. 6 examines the cross-category swap transactions, omitting swap transaction between firms of the same type. Results from New.Foreign1 and New.Old1 imply that new Taiwanese banks, on average, achieve higher returns when trading with foreign and old Taiwanese banks. Recall that when New.Foreign1 (New.Old1) is equal to −1, the foreign firm (old Taiwanese firm) is profiting from a new Taiwanese bank. But the coefficients on New.Foreign1 and New.Old1 suggest that even when foreign banks or old Taiwanese banks are able to profit from a swap transaction with a new Taiwanese bank, the foreign banks and old Taiwanese banks earn consistently smaller returns. Likewise, Foreign.Old1 is positive and significant; when foreign banks profit from swap transactions with old Taiwanese banks they achieve higher returns. But if old Taiwanese firms profit over foreign banks then Foreign.Old1 is equal to −1: suggesting that old Taiwanese banks do not make very high profits from the swap market even when they successfully negotiate a profitable deal. The simple explanation, which we cannot necessarily prove, for these results from the theoretical section is that old Taiwanese banks are more concerned about hedging or they are the least knowledgeable of active traders in the swap market.

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Table 4 Swap points. Swap.lag Trade.size.lag Days.lag Total.volume Forward.spread Spot.vol

(1)

(2)

(3)

(4)

(5)

(6)

0.5445 (18.99)** −4.20E−06 (0.99) −3.78E−06 (11.18)** 2.87E−06 (3.59)** 0.0047 (5.43)** −60.71 (1.41)

0.5411 (19.16)** −3.90E−06 (0.92) −3.74E−06 (11.14)** 2.80E−06 (3.46)** 0.0047 (5.39)** 6.76 (0.13) −822.05 (3.38)**

0.544 (18.91)** −4.20E−06 (0.99) −3.77E−06 (11.16)** 2.85E−06 (3.58)** 0.0047 (5.43)** −2.91 (0.04)

0.5414 (18.87)** −4.00E−06 (0.95) −3.74E−06 (11.14)** 2.77E−06 (3.48)** 0.0047 (5.44)** −238.37 (3.44)**

0.5459 (19.17)** −5.00E−06 (1.18) −3.82E−06 (11.32)** 2.86E−06 (3.57)** 0.0045 (5.20)** −64.21 (1.50)

0.5441 (18.93)** −4.50E−06 (1.05) −3.77E−06 (11.15)** 2.91E−06 (3.61)** 0.0045 (5.22)** −60.97 (1.41)

Spot.vol × Low Spot.vol × Medium

−127.77 (1.10)

Spot.vol × High Foreign New

0.0002 (2.85)** 0.0003 (3.06)**

0.0002 (3.05)** 0.0003 (3.27)**

0.0002 (2.87)** 0.0003 (3.08)**

381.64 (3.03)** 0.0002 (3.03)** 0.0003 (3.23)**

Foreign.Foreign

0.0001 (2.23)* −0.0002 (2.11)* −0.0004 (2.37)*

New.New Old.Old New.Foreign1 New.Old1 Foreign.Old1 Low Medium Terr911 Q.end Q.beg Buyer.init Constant Observations

0.0006 (3.34)** 0.0002 (2.55)* 0.0002 (1.26) −0.0001 (0.69) −0.0002 (1.92)+ 0.0014 (15.12)** −0.001 (5.69)** 3691

0.0011 (4.24)** 0.0002 (2.59)** 0.0002 (1.25) −0.0001 (0.80) −0.0002 (1.91)+ 0.0014 (15.31)** −0.0010 (5.94)** 3691

0.0006 (3.33)** 0.0003 (2.36)* 0.0002 (1.25) −0.0001 (0.68) −0.0002 (1.91)+ 0.0014 (15.20)** −0.001 (5.54)** 3691

0.0009 (4.13)** 0.0004 (3.53)** 0.0002 (1.24) −0.0001 (0.68) −0.0002 (1.89)+ 0.0014 (15.34)** −0.0011 (6.00)** 3691

0.0006 (3.29)** 0.0002 (2.48)* 0.0002 (1.16) −0.0001 (0.63) −0.0002 (1.87)+ 0.0013 (15.47)** −0.0007 (5.22)** 3691

0.0003 (4.64)** 0.0003 (2.82)** 0.0001 (1.11) 0.0006 (3.26)** 0.0002 (2.43)* 0.0002 (1.06) −0.0001 (0.68) −0.0002 (1.85)+ 0.0014 (16.22)** −0.0008 (5.70)** 3691

Table reports regression estimation on swap points. Three models (Swap points, Trade size, and Contract days) are estimated simultaneously using seemingly unrelated regression optimized by maximum likelihood. Observations are clustered by the contract date and Huber–White robust standard errors are used to calculate statistical significance. Only results of Swap points are reported to save space. Results of Trade size and Contract days are available upon request. Swap points is defined as |(Fi,t − Si,t )/Si,t |, where F is swap rate and S is spot rate for transaction i on day t. Swap.lag, Trade.size.lag, and Days.lag are the lag variables for Swap points, Trade size, and Contract days, respectively. Total.volume is the total daily trading volume. Forward.spread is the forward bid and ask spread. Spot.vol is the standard deviation of 30-day spot rate prior to the transaction day. Low equals 1 if the duration of a contract occurs less than 1% in the sample. Medium equals 1 if the duration of a contract occurs between 1% and 5% in the sample. Spot.vol × Low is an interaction variable between Spot.vol and Low. Spot.vo × Medium is an interaction variable between Spot.vol and Medium. Spot.vol × High is an interaction variable between Spot.vol and High, where High equals 1 if the duration of a contract occurs more than 5% in the sample. Foreign (New, Old) equals 1 if the arbitrage taker is a foreign (new Taiwanese, old Taiwanese) bank. Foreign.Foreign (New.New, Old.Old) equals 1 if both buyer and seller are foreign (new Taiwanese, old Taiwanese) banks. New.Foreign1 equals 1 (−1) if the arbitrage taker is a new Taiwanese (foreign) bank. New.Old1 equals 1 (−1) if the arbitrage taker is a new Taiwanese (old Taiwanese) bank. Foreign.Old1 equals 1 (−1) if the arbitrage taker is a foreign (old Taiwanese) bank. Terr911 equals 1 if a trade occurs within 3 days following September 11th, 2001. Q.end equals 1 if a trade occurs within 1 week at a quarter’s end. Q.beg equals 1 if a trade occurs within 1 week at a quarter’s beginning. Buyer.init equals 1 if the arbitrage taker is a swap buyer. Robust z statistics in parenthesis. + * **

Significant at 10%. Significant at 5%. Significant at 1%.

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Table 5 Implied covered interest return, ICIR. ICIR.lag Trade.size.lag Days.lag Total.volume Forward.spread Spot.vol

(1)

(2)

(3)

(4)

(5)

(6)

0.5258 (16.33)** −3.51E−06 (0.45) −6.24E−06 (9.34)** 3.96E−06 (3.07)** 0.0087 (6.20)** −54.23 (0.82)

0.521 (16.25)** −3.01E−06 (0.38) −6.14E−06 (9.22)** 3.87E−06 (2.97)** 0.0087 (6.17)** 42.59 (0.56) −1180.90 (3.05)**

0.5257 (16.32)** −3.48E−06 (0.45) −6.23E−06 (9.33)** 3.94E−06 (3.06)** 0.0087 (6.20)** 2.81 (0.02)

0.5235 (16.28)** −3.20E−06 (0.41) −6.19E−06 (9.27)** 3.83E−06 (2.98)** 0.0087 (6.21)** −282.00 (2.54)*

0.5319 (16.39)** −5.27E−06 (0.68) −6.43E−06 (9.52)** 3.90E−06 (3.00)** 0.0084 (5.94)** −60.18 (0.91)

0.5256 (16.17)** −3.94E−06 (0.51) −6.20E−06 (9.21)** 4.02E−06 (3.07)** 0.0085 (6.00)** −50.25 (0.75)

Spot.vol × Low Spot.vol × Medium

−125.86 (0.64)

Spot.vol × High Foreign New

0.0005 (3.97)** 0.0005 (3.22)**

0.0006 (4.14)** 0.0005 (3.39)**

0.0005 (3.97)** 0.0005 (3.23)**

490.05 (2.42)* 0.0006 (4.11)** 0.0005 (3.35)**

Foreign.Foreign

0.0004 (3.42)** −0.0004 (2.29)* −0.0009 (2.99)**

New.New Old.Old New.Foreign1 New.Old1 Foreign.Old1 Low Medium Terr911 Q.end Q.beg Buyer.init Constant Observations

0.001 (3.25)** 0.0001 (0.83) 0.0014 (2.22)* −0.0001 (0.30) −0.0003 (1.17) 0.0022 (16.76)** −0.0017 (5.90)** 3691

0.0017 (4.10)** 0.0001 (0.85) 0.0014 (2.22)* −0.0001 (0.40) −0.0003 (1.16) 0.0022 (16.95)** −0.0018 (6.17)** 3691

0.001 (3.25)** 0.0002 (0.89) 0.0014 (2.21)* −0.0001 (0.29) −0.0003 (1.16) 0.0022 (16.83)** −0.0017 (5.59)** 3691

0.0013 (3.79)** 0.0004 (1.82)+ 0.0014 (2.21)* −0.0001 (0.29) −0.0003 (1.15) 0.0022 (16.99)** −0.0019 (6.01)** 3691

0.0010 (3.20)** 0.0001 (0.75) 0.0013 (2.13)* −0.00004 (0.23) −0.0003 (1.15) 0.0022 (17.46)** −0.0012 (5.17)** 3691

0.0004 (4.27)** 0.0005 (3.18)** 0.0001 (1.07) 0.0010 (3.17)** 0.0001 (0.69) 0.0014 (2.10)* −0.0001 (0.30) −0.0003 (1.09) 0.0024 (18.37)** −0.0013 (5.62)** 3691

Table reports regression estimation on the size of implied covered interest return (ICIR). Three models (ICIR, Trade size, and Contract days) are estimated simultaneously using seemingly unrelated regression optimized by maximum likelihood. Observations are clustered by the contract date and Huber–White robust standard errors are used to calculate statistical significance. Only results of ICIR are reported to save space. Results of Trade size and Contract days USD NTD × [Daysi,t /360]) − (Fi,t /Si,t ) × (1 + ii,t × [Daysi,t /360])|, where F is swap rate and S is spot rate. are available upon request. ICIR is defined as |(1 + ii,t ICIR.lag, Trade.size.lag, and Days.lag are the lag variables for ICIR, Trade size, and Contract days, respectively. Total.volume is the total daily trading volume. Forward.spread is the forward bid and ask spread. Spot.volatility is the standard deviation of 30-day spot rate prior to the transaction day. Low equals 1 if the duration of a contract occurs less than 1% in the sample. Medium equals 1 if the duration of a contract occurs between 1% and 5% in the sample. Spot.volatility × Low is an interaction variable between Spot.volatility and Low. Spot.volatility × Medium is an interaction variable between Spot.volatility and Medium. Spot.volatility × High is an interaction variable between Spot.volatility and High, where High equals 1 if the duration of a contract occurs more than 5% in the sample. Foreign (New, Old) equals 1 if the arbitrage taker is a foreign (new Taiwanese, old Taiwanese) bank. Foreign.Foreign (New.New, Old.Old) equals 1 if both buyer and seller are foreign (new Taiwanese, old Taiwanese) banks. New.Foreign1 equals 1 (−1) if the arbitrage taker is a new Taiwanese (foreign) bank. New.Old1 equals 1 (−1) if the arbitrage taker is a new Taiwanese (old Taiwanese) bank. Foreign.Old1 equals 1 (−1) if the arbitrage taker is a foreign (old Taiwanese) bank. Terr911 equals 1 if a trade occurs within 3 days following September 11th, 2001. Q.end equals 1 if a trade occurs within 1 week at a quarter’s end. Q.beg equals 1 if a trade occurs within 1 week at a quarter’s beginning. Buyer.init equals 1 if the arbitrage taker is a swap buyer. Robust z statistics in parenthesis. + * **

Significant at 10%. Significant at 5%. Significant at 1%.

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5. Conclusion The theoretical model herein adds to the literature by introducing hedgers to the microstructure exchange rate models. Results from the theoretical model are confirmed. One, returns to swap market transactions can be quite sizable. Two, swap returns are larger for irregular contracts. Three, heightened market volatility increases returns in the swap markets for standardized contracts but decreases returns for irregular contracts. The major caveat to these results is that we cannot know whether or not these patterns exist in larger, more liquid markets. The NTD/USD market is dominated by a single market maker and we cannot necessarily generalize these results to major swap markets until swap transaction data becomes available in these larger markets to study. Appendix A. To better understand the third prediction from the theory outlined in Section 2.4 we can examine the various potential outcomes which depend on the value of D (which is equal to 0 when the contract duration is irregular and herders do not participate in the market) and the value of  ω,t (which is larger when the underlying exchange rate generation process is more volatile). Eqs. (16) and (17) represent the two known outcomes of ϕ which then impact the forward ask rate.







fta (ϕ) ∼ =

et 1 + ϕ + [ε˛1 (2q − 1)]/[2˛2 v(ω,t ) + ˛1 + 2˛3 Dh(ω,t )]

+ DH(ω,t )et

1 + DH(ω,t )

(16)

If ϕt = −ϕ < 0 then the forward ask rate is as follows:





fta (−ϕ) ∼ =



et 1 − ϕ + [ε˛1 (2q − 1)]/[2˛2 (1 − v(ω,t )) + ˛1 + 2˛3 Dh(ω,t )]

+ DH(ω,t )et

1 + DH(ω,t )

(17)

We can summarize the four different potential outcomes. One, D = 0 and  ω,t = 0. Two, D = 0 and  ω,t is large. Three, D is large and  ω,t = 0. Four, D is large and  ω,t is large. The volatility of the underlying exchange rate generation process,  ω,t , is important because of the other functions that  ω,t impacts. H( ω,t ) represents the price elasticity of demand for the hedgers so that H < 0 and the lim H(ω,t ) = ω,t →∞

0. The probability of a hedger buying the NTD forward is denoted by h( ω,t ). We define h( ω,t ) so that h < 0 while the lim h(ω,t ) = 1 and the lim h(ω,t ) = 0. The probability of a trader buying a ω,t →∞

ω,t →0

swap contract is v(ω,t ). If  ω,t is very large then the value of v(ω,t ) approaches 1 because the high variance of the exchange rate makes traders more likely to follow simple behavioral rules and use less discretion. Under the first scenario both D = 0 and  ω,t = 0. These are irregular swap contracts in a market with very low volatility. In this scenario the forward ask rate, from Eq. (16), is as follows: fta (ϕ) ∼ = et



1+ϕ+

ε˛1 (2q − 1) 2˛2 v(ω,t ) + ˛1

 (A1)

Because the volatility of the exchange rate process is small, v(ω,t ) is greater than zero but less than 1. However, when the volatility of the exchange rate process is large, as in the second scenario, v(ω,t ) = 1. This second scenario is shown in the equation below. fta (ϕ) ∼ = et



1+ϕ+

ε˛1 (2q − 1) 2˛2 + ˛1

 (A2)

The value of the forward ask rate is greater in Eq. (A1). When swap contracts have irregular durations and hedgers do not participate (D = 0) then increases in exchange rate volatility ( ω,t ) create a greater difference between the forward rate and the spot exchange rate. This same process can be repeated for Eq. (17) when ϕt = ϕ < 0. Again, the value of |ft − et | will be greater when D = 0 and  ω,t increases.

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What about the third and fourth scenario? In both scenarios D is large because contract durations are regular, standardized intervals. As a result, hedgers represent a large portion of the swap market. In the third scenario,  ω,t = 0 and Eq. (16) is as follows:



fta

(ϕ) ∼ =



et 1 + ϕ + [ε˛1 (2q − 1)] / [2˛2 v(ω,t ) + ˛1 ] 1 + DH(ω,t )



+ DH(ω,t )et

(A3)

Suppose, as in the fourth scenario, that the exchange rate volatility becomes really large. Then for the regular, standardized swap contracts the value of Eq. (16) is as shown below. fta (ϕ) ∼ = et



1+ϕ+

ε˛1 (2q − 1) 2˛2 + ˛1 + 2˛3 D



(A4)

The value of the forward ask rate is greater in Eq. (A3) than in Eq. (A4). Recall that D is rather large so that the fraction in Eq. (A4) becomes very small and the forward rate approaches the spot rate. However, in Eq. (A3) both D and H( ω,t ) are very large so that the latter portion of the numerator and denominator in Eq. (A3) (DH( ω,t )) dominates the parenthetical portion (1+ ϕ + · · ·). As a result the value of the forward ask rate in Eq. (A3) is closer to the spot exchange rate than it would be in Eq. (A4). Based on these four scenarios we can show the third prediction from the theory; heightened spot rate volatility is much more likely to decrease the difference between the forward rate and the current spot rate for irregular contract durations. References Akram, Q.F., Rime, D., Sarno, L., 2008. Arbitrage in the foreign exchange market: turning on the microscope. Journal of International Economics 76, 237–253. Baba, N., Packer, F., 2009. From turmoil to crisis: dislocations in the FX swap market before and after the failure of Lehman Brothers. Bank for International Settlements, Working Paper 285. Available at: http://www.bis.org/publ/ work285.pdf?noframes=1 (accessed 31.03.10). Bank for International Settlements, 2010. Triennial Central Bank Survey: Foreign Exchange and Derivatives Market Activity in 2010. Bank for International Settlements, Basel, Switzerland. Burnside, C., Eichenbaum, M., Rebelo, S., 2009. Understanding the forward premium puzzle: a microstructure approach. American Economic Journal: Macroeconomics 1, 127–154. Eun, C.S., Sabherwal, S., 2002. Forecasting exchange rates: do banks know better? Global Finance Journal 13, 195–215. Furfine, C.H., 2004. Public disclosures and calendar-related movements in risk premiums: evidence from interbank lending. Journal of Financial Markets 7, 97–116. Hui, C.-H., Genberg, H., Chung, T.-K., 2009. Funding liquidity risk and deviations from interest-rate parity during the financial crisis of 2007–2009. Hong Kong Monetary Authority, Working Paper 0913. Available at: http://www.info.gov.hk/hkma/eng/research/working/pdf/HKMAWP09 13 full.pdf (accessed 05.02.10). Moulton, P.C., 2005. You can’t always get what you want: trade-size clustering and quantity choice in liquidity. Journal of Finance & Economics 78, 89–119. Peiers, B., 1997. Informed traders, intervention, and price leadership: a deeper view of the microstructure of the foreign exchange market. Journal of Finance 52, 1589–1614. Szilagyi, P.G., Batten, J.A., 2006. Arbitrage, covered interest parity and long-term dependence between the US dollar and the yen. The Institute for International Integration Studies Discussion Paper Series, IIIS Discussion Paper No. 128. Available at: http://www.tcd.ie/iiis/documents/discussion/pdfs/iiisdp128.pdf (accessed 05.02.10). Taylor, M.P., 1989. Covered Interest Arbitrage and market turbulence. The Economic Journal 99, 376–391. Taylor, M.P., 1987. Covered interest parity: a high frequency, high quality data study. Economica 54, 429–438.