# Liquidity, information and the size of the forward exchange rate bias.

We first develop the theoretical rationale of the forward exchange
rate unbiasedness hypothesis (FUH) for buyers of forward exchange under
an assumption of risk neutrality and interest rate parity. Then by using
Jensen's Inequality, we show that FUH cannot simultaneously hold
true for both the sellers and buyers of the same forward currency
contracts. Because of the symmetric nature of relationships among
foreign exchange currency pairs, we conjecture that forward rate biases
should be small. We introduce a new test statistic that averts unit root
problems. This test statistic helps verify our conjectures in the
empirical data. We analyze the liquidity effect of the informational
content of forward exchange rates using our test statistic. We show that
whether unbiasedness holds or not is driven by market conditions of
crisis or non-crisis.

Introduction

Forward exchange rate contracts are used, among other things, to eliminate future spot exchange rate risk. (1) Currency markets are unique in the sense that there are several symmetry features among currency pairs and even among currency triplets. For example, let [S.sub.1] be the number of units of EUR per dollar; if [S.sub.2] is a model for the number of pounds per EUR, then ([S.sub.1])([S.sub.2]) is a model for pounds per dollar. Similarly, [([S.sub.1]).sup.-1] is a model for dollars per EUR, and [([S.sub.2]).sup.-1] is a model for EUR per pound. (2)

Forward rates are expected to neutralize future exchange rate risk for both parties (sellers and buyers of the same currency) and, to be fair, unbiased estimators of corresponding future spot rates. (3) However, existing empirical research fails to support FUH, and such a phenomenon is referred to as the forward rate bias puzzle. There have been many attempts to unravel this puzzle, yet to our knowledge, none appears completely satisfactory. Alongside the eguity premium puzzle, the forward rate bias puzzle remains one of the unsolved mysteries of financial economics.

The nature of the puzzle is succinctly spelled out by many authors. Not only is the forward rate estimator inefficient, but it predicts the future spot rate in the opposite direction. Of course because the forward rate seems to be systematically biased, it permits the existence of the carry trade. Contrary to interest rate parity theory, future spot exchange rates do not usually depreciate for high interest rate currencies and low interest rate currencies do not appreciate by as much as is expected. (4) However, the empirical evidence varies across countries, by economy type (advanced, developing and emergent economies) and by business cycle conditions. See Bansal and Dahlquist (2000) for a further discussion of these issues.

Fama (1984) (5) first popularized this problem even though it had been noted by many authors like Bilson (1981), Hodrick (1987), Hansen and Hodrick(1980), Frenkel (1980), Cornell (1977) and others. (6) Fama (1984), in a study of nine industrialized countries, attributed the existence of the puzzle to the fact that the volatility of the risk premium is greater than the volatility of the realized spot rates. Bilson (1981) analyzed the speculative efficiency hypothesis wherein the null was that there were no profits to be made from pure speculation. The analyzed data led to non-acceptance of the null. Hansen and Hodrick (1980) had also rejected the simple null hypothesis of zero returns to speculation using different modeling technigues.

Bansal (1997) deepened the puzzle by postulating that the puzzle was interest rate dependent. It existed only in certain environments where U.S. interest rates exceeded foreign interest rates. He also showed that dependent of the interest rate regime, the volatility of the forward premium could be greater or less than the realized future spot variation in exchange rates. Nevertheless, carry traders have continued to profit in currency markets. Goodhart, McMahon and Ngama (1992) attributed the failure of the unbiasedness hypothesis in their study to the existence of outlier data and structural breaks. All of these studies are based on data from the 1990s at best and thus have little relevance to current currency markets.

Recently authors have attributed the existence of the puzzle to the existence of non-linearities (Sarno, Valente and Leon 2006); to fads in the markets (Sercu and Vinaimont 2006) and to perpetual learning (Chakraboty and Haynes 2008). Burnside et al. (2011a) attributed the existence of the carry trade to over-optimism, and Burnside et al. (2011b) conjectured that it might be the result of peso problems. More recently Sarno et al. (2012) proposed two different models to solve the problem: firstly, a global model that uses the U.S. pricing kernel to measure all term structures and currency premiums; and secondly, a series of local currency pricing kernels that are used to evaluate local term structures and the related currency premiums.

Sarno et al. (2012) concluded that neither set of models was completely satisfactory. The global model matched depreciation rates but fitted the interest rate data badly. The local models fit the term structures of interest rates well but did poorly on observed depreciation rates. So the most cutting edge research has concluded that the puzzle is still basically unresolved.

We address major elements that the cited studies do not tackle: namely, the size of the bias and the impact of liquidity on the bias. In this paper, we provide a new insight on this old puzzle by examining both theoretically and empirically its root and trigger causes. We first discuss the theoretical rationale of FUH underthe assumption of rational expectation and risk neutrality by examining the forward and spot rate behavior. (7) By using Jensen's Inequality, we show that forward rates cannot be expected to be unbiased estimators of corresponding future spot rates for both buyers and sellers of the same forward currency contracts. (8) Because of the symmetry of the foreign exchange markets, we expect the sizes of forward rate biases to be fairly small. (9) We further empirically examine the informational content of forward exchange rates and discuss the role of liquidity in the predictive power of the forward exchange rates.

The reminder of this paper is organized as follows. Section 2 discusses forward rate and spot rate determination and the rationale of FUH under a risk neutrality assumption for buyers of forward exchange. Section 3 examines the rationale for FUH under a risk neutrality assumption for buyers and sellers of forward exchange by using Jensen's Inequality. In a relatively preference free setting, we show that FUH cannot simultaneously hold true for both buyers and sellers of a particular currency. Section 4 presents the informational content of forward rates using alternative testing methods and discusses the liquidity effect on the predictive power of the forward rates. Section 5 concludes.

The Rationale of Forward Unbiasedness Hypothesis (FUH)

FORWARD EXCHANGE RATE DETERMINATION

It is well known that foreign exchange rates can be quoted in two ways, direct and indirect. In a direct quote, the exchange rates are quoted as domestic currency per unit of foreign currency, and in an indirect quote, the exchange rates are quoted as foreign currency per unit of domestic currency; the direct quote is the reciprocal of the indirect quote. (10) To fix the notation that we use, assume country i is the home country and country j the foreign country.

Let [S.sub.t], number of units of currency j per unit currency i, and [F.sub.t;t+T], number of units of currency j per unit currency i, denote the indirect quotes of spot exchange rate and 365T-day forward exchange rate at time t, respectively. Furthermore let [r.sub.D], the interest rate in country i, and [r.sub.F], the rate in country j, denote the annual (continuous) interest rate on deposit in country i and the annual (continuous) interest rate on foreign deposit in country j, respectively.

A rational investor in country i, starting with one unit of her home currency, would compare the following two alternatives: firstly, keeping her home currency and earning an annual (continuous) domestic interest rate, [r.sub.D], on their domestic deposits and ending up with [e.sub.rDT] units home currency after 365T days; or secondly, converting her home currency at the spot exchange rate, [S.sub.t], earning an annual (continuous) foreign interest, [r.sub.F], on deposit in country j, and then after 365T days exchange currency j for currency i at the previously negotiated forward exchange rate, [F.sub.t;t+T]. [F.sub.t;t+T] is a forward contract negotiated at time t, for a period of time T. The forward contract negotiated at time t will expire at time t+T. If the second alternative is adopted, the investor will end up with [S.sub.t][e.sub.rFt]/[F.sub.t;t+T] units of currency j, to eliminate the arbitrage opportunity, the results in the two alternatives should be the same. That is, the following condition [e.sub.rDT] = [S.sub.t][e.sub.rFT]/[F.sub.t;t+T] must hold. Thus, 365T-day forward rate at time t should be determined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

We should notice that Equation (1) does not involve the future spot exchange rate.

SPOT EXCHANGE RATE BEHAVIOR

Since 1970, the central banks have adopted a floating exchange rate system and allowed market forces to determine exchange rates. In the simplest model, the spot exchange rate [S.sub.t] behaves like a geometric Brownian motion with a constant drift. That is, it follows a stochastic differential equation of the form

d[S.sub.t] = [mu][S.sub.t]dt + [S.sub.t]d[W.sub.t] (2)

where, [W.sub.t] is a Wiener process. Equation (2) is widely used in practice and front office systems and mainly serves as a tool to communicate prices in foreign exchange options. Solving the above stochastic differential equation, we have the explicit formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

At first glance, we do not see any connections between the forward rate (1) and the future spot rate (3).

Proposition: Let [Q.sub.B] be a risk-neutral probability measure. If the exchange rate obeys a stochastic differential equation of the form (2), and if the riskless rates of return for domestic investors (country i) and foreign investors (country j) are [r.sub.D] and [r.sub.F], respectively, then under [Q.sub.B], it must be the case that [mu] = [r.sub.F] - [r.sub.D].

Remark: We want to point out the risk neutrality assumption in the above proposition is only imposed on the market participants who are initially long currency i (spot market) and short currency j in the forward market, after they have hedged an overseas investment. These are future forward sellers of the currency j and future forward buyers of currency i.

Applying the above proposition, substitute [mu] = [r.sub.F] - [r.sub.D] into (3) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Taking expectation on both sides of equation (4), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

and from (1) and (5), we have

[F.sub.o,t] = E[S.sub.t] (6)

(6) can be generalized to

E[S.sub.t+T] = [F.sub.t;t+T] (7)

for any t and T. Thus, we have shown the theoretical rationale of FUH, coupled with IRP theory holding, under the assumption of risk neutrality for buyers of currency i in the forward market participants. Under this set of assumptions, [F.sub.O,T], is an unbiased estimator of the expected future spot rate at time t. Recursively we can extend this to all future time periods and contract lengths.

WHAT ABOUT FORWARD SELLERS OF CURRENCY I (FORWARD BUYERS OF CURRENCY J)?

One distinctive feature of foreign exchange market is its symmetry. Assume that an investor from country j and currency j has the same two choices as investors from country i. The investor in country j's market has two choices: make a home market investment in country j's bond markets, or convert the currency in the spot exchange market then invest in country i's bond market and currency and sell the proceeds in the forward market. Both strategies should have equivalent payoffs to avoid any arbitrage opportunities. Now, let us impose the assumption of risk neutrality on future-forward buyers of the currency j (future forward sellers of currency i) and interest parity. By using the same analysis as before, we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

We have thus shown the theoretical rationale of FUH under an assumption of risk neutrality and interest parity. However, after we apply Jensen's Inequality to strictly concave function, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, we immediately learn that (7) and (8) cannot co-exist. This is referred to Siegel's paradox (Siegel 1972).

Siegel (1972) has argued that the forward price of a currency is a biased estimator of the expected future spot price. We have gone one step further in resolving the forward rate bias puzzle by showing the theoretical rationale of FUH for forward market buyers of currency i, under an assumption of risk neutrality and interest rate parity. We show it is theoretically possible for the forward exchange rates to be unbiased estimators of future corresponding spot rates for forward market buyers of currency i but not for both sellers and buyers. This indicates that the risk neutrality assumption, for both contract buyers and sellers, cannot simultaneously hold.

Informational Content of Forward Rates and Currency Liquidity

Though we have shown that it is impossible for FUH to hold true for both buyers and sellers of forward exchange contracts, a natural question arises: do forward exchange rates contain any information about corresponding future spot rates? As mentioned earlier, the symmetries of the foreign exchange market are the key features that distinguish this market from all others because forward market buyers and sellers are always simultaneously long in one currency (or forward contract) and short in another currency (or forward contract), with the objective of minimizing their risks or securing certain expected future cash flows.

Because of symmetry, we expect both sides of the foreign exchange market participants to express themselves symmetrically. We have shown that the assumption of risk neutrality for both buyers and sellers of forward exchange is not an adequate assumption because it leads to contradictory results according to Jensen's Inequality. Because of the symmetry of the markets, we therefore expect both sides of the market participants to be either risk-averse or risk-loving. Risk aversion is an oft-cited assumption in finance, namely that an investor will always choose the most attractive risk-reward package, all things being equal.

Assume the forward rate [F.sub.t+T] is significantly downward (upward) biased as an estimate of the corresponding future spot rate [S.sub.t] for future buyers of currency I (forward sellers of currency j). The same currency pair forward rate from the perspective of country j's residents, forward sellers of currency I, which is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] would be significantly upward (downward) biased as an estimate of the corresponding future spot rates quoted. That is, if

[S.sub.t+T]>(<)[F.sub.t;t+T] (9)

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

which is in contradiction with the symmetric feature of the market. Thus, we expect the bias of forward rate as estimators of corresponding future spot rates is small even for less liquid currency pairs.

The informational content of forward exchange rates can be evaluated by using the univariate regression:

[S.sub.t+T]= [[alpha].sup.+] [beta][F.sub.t,r+T] + [[epsilon].sub.t], (11)

to see whether the coefficient [alpha] is significantly different from zero. If so, then we examine whether forward rates are unbiased, upward biased or downward biased estimators of corresponding future spot rates.

Early empirical researchers like Cornell (1977) and Chiang (1998) were concerned about the efficiency of forward markets. Tauchen (2001) investigated the small sample properties of the Fama (1984) estimators and concluded that the evidence against the unbiasedness hypothesis was much strongerthan originally believed. From Maynard and Phillips (2001) we quote: "The difference in persistence between the short-memory spot return and long-memory forward premium does not admit a valid regression relation in returns and the slope coefficient in the Fama (1984) regression is found to converge to zero." Applying limit theory to the slope coefficient in this regression, they uncover a biased and negatively skewed distribution. They attribute the rejection of unbiasedness to the differences in persistence between the series.

Maynard (2003) reported that recent empirical studies have rejected exact 1:1 cointegration between spot and forward exchange rates. He posits that the non-cointegration of the spot and forward exchange rates is a possible explanation for the puzzling forward bias puzzle. He uses limit theory to show that the coefficient in the regression in returns form has a unit root component in its limit distribution. This unit root imparts a bias and skewness to the forward estimator. The empirical evidence suggests that the implied Dickey-Fuller-type terms do exhibit a downward bias yet are of insufficient magnitude to fully account for the observed negative betas.

Chakraborty and Evans (2008) appealed to irrationality on the part of traders to explain the puzzle and stated that the typical model specification may be inappropriate. They claim that the model in levels is super consistent, and the puzzle does not exist in this format. A small deviation of the beta parameter from unity is the cause of the adverse finding that changes in the spot rate are negatively related with the forward premium. They, however, do not advance any reasons why the beta parameter should deviate from unity. In summary the existing econometric evidence also does not provide a resolution of the puzzle.

We want to point out that most of the heavily cited existing literature, such as Hansen and Hodrick (1980), Bilson (1981), Fama (1984), Froot and Frankel (1989) and Sarno et al. (2012), uses the logarithmic transformed data to test FUH. As Osborne (2002) points out, using transformed data can fundamentally alter the nature of the variable, making the interpretation of the results somewhat more complex. By using logarithmic transformed data, the existing literature actually tests whether the logarithmic forward rates are unbiased estimators of logarithmic corresponding future spot rates, that is, whether Elog[S.sub.t+T] = log[F.sub.t;t+T] holds or not.

Again, by applying Jensen's Ineguality to the strictly concave function f(x) =log(x), we have Elog([S.sub.t]) < logE([S.sub.t]). Thus, whether Elog([S.sub.t+T]) = log[F.sub.t;t+T] holds or not gives no indication of whether E[S.sub.t+T]= [F.sub.t;t+T], the true FUH, holds or not.

Table 1 provides descriptive statistics for the spot exchange rate for five currencies against U.S. dollar: euro (EUR), Japanese yen (JPY), Canadian dollar (CAD), Australian dollar (AUD) and New Zealand dollar (NZD). Obviously the normality assumption for the spot rate is violated; thus, tests based on a regression of the future spot rate on the forward rate can result in spurious regression problems, or the so-called unit root problem.

To overcome the spurious regression problems, we develop and use the following alternative

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

to test the following three hypotheses:

1) Hypothesis HO: [alpha] = 0, to see whether forward rates are unbiased estimators,

2) Hypothesis H-: [alpha] < 0, to see whether forward rates are upward biased estimators, and

3) Hypothesis H+: [alpha] > 0, to see whether forward rates are downward biased estimators.

Figure 1 shows the histograms of the spot-forward ratio [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the five currencies, from which we can see the spot-forward ratio [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is conformable with a normal distribution even though a normality assumption is not required to use our testing method.

[FIGURE 1 OMITTED]

Though long-term forward rates are available, due to the fact that the money market interest rate does change over time, we only test the information content of one-month forward rates based on the fact that interest rates are relatively stable over the short term. We employ daily data of one-month forward and spot exchange rates using both direct and indirect quotes. Our data spanned from February 1995 to August 2010 with nearly 4,000 observations. Our data were obtained from Bloomberg. We examine the following five foreign exchange markets: euro (EUR), Japanese yen (JPY), Canadian dollar (CAD), New Zealand dollar (NZD) and Australian dollar (AUD), which are among the most active traded currencies against the U.S. dollar.

We report results of the tests in Tables 2 and 3. As can be seen in Tables 2 and 3, in the EUR-USD market, FUH cannot be rejected at any conventional confidence levels whether direct or indirect quotes are used. In the other four markets, FUH is rejected. As argued earlier, it is impossible for forward rates to be unbiased estimators of future corresponding spot rates for both sides of the market participants. The fact that we cannot reject FUH in the EUR-USD market allows us to draw the conclusion that the EUR-USD forward rates may provide good information on corresponding future spot rates.

One would naturally attribute the results concerning EUR-USD forward rates' informational content about their corresponding future spot rates to the high liquidity of the currency. Though there is no central source of information from which we can figure out how the liquidity of the various currency pairs rank, we can look to the periodic surveys done by the central banks and monetary authorities of the major global regions to get an idea. According to these sources and Mancini et al (2013), EUR-USD is the most liquid exchange rate pair, which is in line with the perception of market participants and the fact that it has by far the largest market share in terms of turnover.

Due to the fact that the euro and U.S. dollar are the world's two largest currencies, representing the world's two largest economic and trading blocs, many multinational corporations conduct business in both the United States and Europe. These corporations have an almost constant need to hedge their exchange rate risk by using forward rate contracts. Some firms, such as international financial institutions, have offices in both the United States and Europe. Firms that fit this description are also constantly involved in trading the euro and the U.S. dollar. Because the EUR-USD is such a popular currency pair, arbitrage opportunities are next to impossible to find.

As mentioned earlier, in the other four markets FUH was rejected. This is not surprising according to our theoretical argument made in Section 2, which states that forward rates cannot be unbiased estimators as corresponding future spot rates for both contract buyers and sellers. From Tables 2 and 3, we can see that in JYP-USD, NZD-USD and AUD-USD markets, forward rates are downward biased estimators of future corresponding spot rates when indirect quotations are used and upward biased estimators of future corresponding spot rates when direct quotations are used. We find the opposite results in CAD-USD market.

Though FUH was rejected in these four markets, we find the biases are small in all four markets, as indicated in the small a values (the absolute value of all [alpha] are far less than 0.5 %) and the standard errors (all the standard errors are less than 0.0006, or 0.06%). Thus, with a 95% confidence level, we can say that the biases of forward rates as corresponding future spot rates in these four markets are within two standard errors from estimated [alpha] values, which are Less than 1%. Thus, even though we reject FUH in these four exchange markets, forward exchange rates still provide fairly good estimators of the corresponding future spot rates.

Our empirical studies were conducted using data spanning from 1995 to 2010. As indicated in Mancini et al. (2013), the liquidity of foreign exchange markets declined significantly during the 2007-2009 financial crisis, especially after the bankruptcy of Leman Brothers. To examine the liquidity effect on the informational content of forward rates about their corresponding future spot rate, we separate data periods into two phases. Phases I (pre-crisis) spans Feb. 1995 to Dec. 2007. Phase II (during crisis) runs from Dan. 2008 to Oct.2010. We then repeat the following three hypothesizes:

1) Hypothesis HO: [alpha] = 0, to see whether forward rates are unbiased estimators;

2) Hypothesis H-: [alpha] < 0, to see whether forward rates are upward biased estimators and

3) Hypothesis H+: [alpha] > 0, to see whether forward rates are downward biased estimators.

If liquidity plays a role in the informational content of forward exchange rates, we would expect the forward rates to contain less information about their corresponding future spot rates during the 2007-2009 financial crisis, as reflected in larger [alpha] values. However, the absolute values of [alpha] from two phases testing results reported in Table 4 are not noticeably different. For some currency pairs, such as CAD/USD, EUR/USA and USD/NZD, the absolute a values are even smaller during Phase II. Thus, the testing results fail to support the liquidity effect on the informational content of forward rates.

To further examine the role of liquidity, we perform the above three hypothesis tests using the data of a less liquid (popular) currency pair ISK (Icelandic krona)-USD spanning from Sep. 2004 to Sep. 2010. (11) We first use the entire data set to perform the three hypothesis tests and then use the data of Phase I and the data of Phase II. The test results are reported in Table 5, from which we can see when the aggregate data of the entire period is used. FUH cannot be rejected for both direct and indirect quotes,

However, when disaggregated data sets are used, FUH was rejected, and we do see the larger values of [alpha] are positively correlated with the [alpha] values in more liquid currency pairs. Though FUH was rejected in most cases in ISK-USD market, from the results in Table 5, we can see the biases of forward rates, as estimators of corresponding future spot rates are less than 2.2 % (ISK/USD Phase II) at 95% confidence level.

Our empirical results show that in most cases forward rates are likely biased estimators of corresponding future spot rates. However, the biases are fairly small, which is consistent with our conjecture based on the symmetry feature of the foreign exchange markets. Our empirical testing results do not strongly support the liquidity effect on the informational content of forward rates.

Conclusion

This study examines the root and trigger causes of the forward rate bias puzzle, both theoretically and empirically. We first show analytically the forward rate and spot rate determination and the rationale for FUH under risk neutral assumption and interest rate parity for buyers of forward currency. By using Jensen's Inequality, we demonstrate that FUH cannot hold true simultaneously for both buyers and sellers of the same currency in the forward market. However, due to symmetry feature of foreign exchange markets, we expect both sides of the foreign exchange market participants to express themselves symmetrically with the objective to minimize their risks.

We argue if forward rates are significantly upward (downward) deviated from corresponding future spot rates for contract buyers than for contract sellers, forward rates would be significantly downward (upward) deviated from corresponding future spot rates. This is in contradiction of the symmetry feature. Thus, even though we do not expect FUH to hold true for both sides of the market participants, we expect forward rate bias to be small even for less liquid currency pairs.

We use a long sample period that covers a wide range of major currencies for our empirical testing to avoid sample specific problems. Our testing results show that in most cases FUH is rejected. However, the forward rate biases are found to be within 1% range for the more liquid currency pairs, and in all cases, we find the biases are within 2.2% (largest [alpha] from ISK/USD Phase II +2 standard errors=0.015734+ 2x 0.002871=0.021476< 2.2%) range at 95% confidence level. Our testing results do not show clear evidence that the liquidity of the currency pair affect the informational content of the forward rates about their corresponding future rates.

Our study differs from existing research in several respects. We combine theoretical arguments with empirical evidence to explain and to examine the forward rate bias as opposed to the conventional purely empirical approach. We use an alternative empirical testing method as opposed to the conventional regression methods using logarithmic transformed data, which severely alters the nature of the true FUH. We examine the informational content of forward rates from the perspectives of both contract buyers and sellers by using exchange rate data quoted in both direct and indirect form. This differs from the conventional approach, which only uses one form of the exchange rate quote and thus is only from the perspective of one side of market participants.

Finally, we not only discuss the validation of FUH theoretically and empirically but also give an assessment of the estimation bias of forward rates as estimators of corresponding future spot rates. We offer a new insight to both academicians and practitioners that, in spite of rejection of FUH due to the symmetry of the foreign exchange market, forward rates are still reasonably reliable estimators of corresponding future spot rates even for less liquid currency pairs. This fact is confirmed by our empirical evidence.

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Sarno, L., G. Valente, and H. Leon. "Nonlinearity in Deviations from Uncovered Interest Parity: An Explanation of the Forward Bias Puzzle." Review of Finance, vol. 10, no. 3, 2006, pp. 443- 482.

Sercu, P., and T. Vinaimont. "The Forward Bias in the ECU: Peso Risks vs. Fads and Fashions." Journal of Banking and Finance, vol. 30, no. 8, 2006, pp. 2409-2432.

Siegel, J. "Risk, Interest Rates and The Forward Exchange." Quarterly Journal of Economics, vol. 86, no. 2,1972, pp. 303-309.

Tauchen, G. "The Bias of Tests for a Risk Premium in Forward Exchange Rates." Journal of Empirical Finance, vol. 8, no. 5, 2001, pp. 695-704.

Wystup, U., "Foreign Exchange Symmetries" http://mathfinance2.com/MF_website/UserAnonymous/Company/papers/wystup_symmetries_eqf.pdf (2008).

LLOYD BLENMAN

University of North Carolina Charlotte

GUAN JUN WANG

Savannah State University

(1) In full generality, the demand for a forward contract can be shown to contain a pure hedging component plus a speculative component, all of course depending on the type of risk tolerance assumed. Risk neutral investors place zero weight on volatility risk. This assumption then drives the general result into a setting where the demand for forward contracts is based solely on the pure hedging component. For fundamental papers dealing with pure speculation in currency markets, see Siegel (1972), Feldstein (1968) and McCulloch (1975).

(2) In addition, if my base currency is dollars and I switch to euros, I am short dollars and long euros. If the euro appreciates, the dollar depreciates and so I profit from this position. There are also other symmetries on derivatives with currency as the underlying asset. For further details on these issues, see Wystup (2008), for example.

(3) We understand that the markets are composed of many groups of players who are, respectively, risk averse, risk seeking and risk loving. However, in aggregate the market is best described as a risk-neutral one. See Sarno (2005) for a discussion on this point.

(4) For very weak currencies of emerging market countries, high interest rates presage a depreciation of their currencies. Most of those countries have no formal forward markets and for those countries IRPT mostly holds.

(5) Fama (1984) was the first to point out the significance of the estimates of beta for their implications for future spot exchange rate variability and forward premium volatility.

(6) Cornell (1977) was more interested in issues of market efficiency rather than in the specific nature of the bias.

(7) Risk neutrality and rational expectations can be consistent with preferences where volatility is given zero weight. We do not explicitly use the zero volatility weight assumption in any of our calculations

(8) Some previous studies have shown this theoretically, but we are the first to measure the size of the bias in current currency markets during normal periods and also in periods of crisis.

(9) These biases can be even smaller if we were to assume risk aversion on the part of one set of market participants and risk preferring behavior on the other set. However, it is not clear if there is any benefit to be gleaned from a higher level of modeling realism.

(10) However, what is indirect from one country's perspective is direct from the other country's perspective. So we will minimize the use of the terms indirect and direct in the discussion that follows.

(11) Franket and Poonawala (2010) show in their paper that less liquid (emerging market) currencies are less biased than those of very Liquid (advanced economies) currencies.

Introduction

Forward exchange rate contracts are used, among other things, to eliminate future spot exchange rate risk. (1) Currency markets are unique in the sense that there are several symmetry features among currency pairs and even among currency triplets. For example, let [S.sub.1] be the number of units of EUR per dollar; if [S.sub.2] is a model for the number of pounds per EUR, then ([S.sub.1])([S.sub.2]) is a model for pounds per dollar. Similarly, [([S.sub.1]).sup.-1] is a model for dollars per EUR, and [([S.sub.2]).sup.-1] is a model for EUR per pound. (2)

Forward rates are expected to neutralize future exchange rate risk for both parties (sellers and buyers of the same currency) and, to be fair, unbiased estimators of corresponding future spot rates. (3) However, existing empirical research fails to support FUH, and such a phenomenon is referred to as the forward rate bias puzzle. There have been many attempts to unravel this puzzle, yet to our knowledge, none appears completely satisfactory. Alongside the eguity premium puzzle, the forward rate bias puzzle remains one of the unsolved mysteries of financial economics.

The nature of the puzzle is succinctly spelled out by many authors. Not only is the forward rate estimator inefficient, but it predicts the future spot rate in the opposite direction. Of course because the forward rate seems to be systematically biased, it permits the existence of the carry trade. Contrary to interest rate parity theory, future spot exchange rates do not usually depreciate for high interest rate currencies and low interest rate currencies do not appreciate by as much as is expected. (4) However, the empirical evidence varies across countries, by economy type (advanced, developing and emergent economies) and by business cycle conditions. See Bansal and Dahlquist (2000) for a further discussion of these issues.

Fama (1984) (5) first popularized this problem even though it had been noted by many authors like Bilson (1981), Hodrick (1987), Hansen and Hodrick(1980), Frenkel (1980), Cornell (1977) and others. (6) Fama (1984), in a study of nine industrialized countries, attributed the existence of the puzzle to the fact that the volatility of the risk premium is greater than the volatility of the realized spot rates. Bilson (1981) analyzed the speculative efficiency hypothesis wherein the null was that there were no profits to be made from pure speculation. The analyzed data led to non-acceptance of the null. Hansen and Hodrick (1980) had also rejected the simple null hypothesis of zero returns to speculation using different modeling technigues.

Bansal (1997) deepened the puzzle by postulating that the puzzle was interest rate dependent. It existed only in certain environments where U.S. interest rates exceeded foreign interest rates. He also showed that dependent of the interest rate regime, the volatility of the forward premium could be greater or less than the realized future spot variation in exchange rates. Nevertheless, carry traders have continued to profit in currency markets. Goodhart, McMahon and Ngama (1992) attributed the failure of the unbiasedness hypothesis in their study to the existence of outlier data and structural breaks. All of these studies are based on data from the 1990s at best and thus have little relevance to current currency markets.

Recently authors have attributed the existence of the puzzle to the existence of non-linearities (Sarno, Valente and Leon 2006); to fads in the markets (Sercu and Vinaimont 2006) and to perpetual learning (Chakraboty and Haynes 2008). Burnside et al. (2011a) attributed the existence of the carry trade to over-optimism, and Burnside et al. (2011b) conjectured that it might be the result of peso problems. More recently Sarno et al. (2012) proposed two different models to solve the problem: firstly, a global model that uses the U.S. pricing kernel to measure all term structures and currency premiums; and secondly, a series of local currency pricing kernels that are used to evaluate local term structures and the related currency premiums.

Sarno et al. (2012) concluded that neither set of models was completely satisfactory. The global model matched depreciation rates but fitted the interest rate data badly. The local models fit the term structures of interest rates well but did poorly on observed depreciation rates. So the most cutting edge research has concluded that the puzzle is still basically unresolved.

We address major elements that the cited studies do not tackle: namely, the size of the bias and the impact of liquidity on the bias. In this paper, we provide a new insight on this old puzzle by examining both theoretically and empirically its root and trigger causes. We first discuss the theoretical rationale of FUH underthe assumption of rational expectation and risk neutrality by examining the forward and spot rate behavior. (7) By using Jensen's Inequality, we show that forward rates cannot be expected to be unbiased estimators of corresponding future spot rates for both buyers and sellers of the same forward currency contracts. (8) Because of the symmetry of the foreign exchange markets, we expect the sizes of forward rate biases to be fairly small. (9) We further empirically examine the informational content of forward exchange rates and discuss the role of liquidity in the predictive power of the forward exchange rates.

The reminder of this paper is organized as follows. Section 2 discusses forward rate and spot rate determination and the rationale of FUH under a risk neutrality assumption for buyers of forward exchange. Section 3 examines the rationale for FUH under a risk neutrality assumption for buyers and sellers of forward exchange by using Jensen's Inequality. In a relatively preference free setting, we show that FUH cannot simultaneously hold true for both buyers and sellers of a particular currency. Section 4 presents the informational content of forward rates using alternative testing methods and discusses the liquidity effect on the predictive power of the forward rates. Section 5 concludes.

The Rationale of Forward Unbiasedness Hypothesis (FUH)

FORWARD EXCHANGE RATE DETERMINATION

It is well known that foreign exchange rates can be quoted in two ways, direct and indirect. In a direct quote, the exchange rates are quoted as domestic currency per unit of foreign currency, and in an indirect quote, the exchange rates are quoted as foreign currency per unit of domestic currency; the direct quote is the reciprocal of the indirect quote. (10) To fix the notation that we use, assume country i is the home country and country j the foreign country.

Let [S.sub.t], number of units of currency j per unit currency i, and [F.sub.t;t+T], number of units of currency j per unit currency i, denote the indirect quotes of spot exchange rate and 365T-day forward exchange rate at time t, respectively. Furthermore let [r.sub.D], the interest rate in country i, and [r.sub.F], the rate in country j, denote the annual (continuous) interest rate on deposit in country i and the annual (continuous) interest rate on foreign deposit in country j, respectively.

A rational investor in country i, starting with one unit of her home currency, would compare the following two alternatives: firstly, keeping her home currency and earning an annual (continuous) domestic interest rate, [r.sub.D], on their domestic deposits and ending up with [e.sub.rDT] units home currency after 365T days; or secondly, converting her home currency at the spot exchange rate, [S.sub.t], earning an annual (continuous) foreign interest, [r.sub.F], on deposit in country j, and then after 365T days exchange currency j for currency i at the previously negotiated forward exchange rate, [F.sub.t;t+T]. [F.sub.t;t+T] is a forward contract negotiated at time t, for a period of time T. The forward contract negotiated at time t will expire at time t+T. If the second alternative is adopted, the investor will end up with [S.sub.t][e.sub.rFt]/[F.sub.t;t+T] units of currency j, to eliminate the arbitrage opportunity, the results in the two alternatives should be the same. That is, the following condition [e.sub.rDT] = [S.sub.t][e.sub.rFT]/[F.sub.t;t+T] must hold. Thus, 365T-day forward rate at time t should be determined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

We should notice that Equation (1) does not involve the future spot exchange rate.

SPOT EXCHANGE RATE BEHAVIOR

Since 1970, the central banks have adopted a floating exchange rate system and allowed market forces to determine exchange rates. In the simplest model, the spot exchange rate [S.sub.t] behaves like a geometric Brownian motion with a constant drift. That is, it follows a stochastic differential equation of the form

d[S.sub.t] = [mu][S.sub.t]dt + [S.sub.t]d[W.sub.t] (2)

where, [W.sub.t] is a Wiener process. Equation (2) is widely used in practice and front office systems and mainly serves as a tool to communicate prices in foreign exchange options. Solving the above stochastic differential equation, we have the explicit formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

At first glance, we do not see any connections between the forward rate (1) and the future spot rate (3).

Proposition: Let [Q.sub.B] be a risk-neutral probability measure. If the exchange rate obeys a stochastic differential equation of the form (2), and if the riskless rates of return for domestic investors (country i) and foreign investors (country j) are [r.sub.D] and [r.sub.F], respectively, then under [Q.sub.B], it must be the case that [mu] = [r.sub.F] - [r.sub.D].

Remark: We want to point out the risk neutrality assumption in the above proposition is only imposed on the market participants who are initially long currency i (spot market) and short currency j in the forward market, after they have hedged an overseas investment. These are future forward sellers of the currency j and future forward buyers of currency i.

Applying the above proposition, substitute [mu] = [r.sub.F] - [r.sub.D] into (3) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Taking expectation on both sides of equation (4), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

and from (1) and (5), we have

[F.sub.o,t] = E[S.sub.t] (6)

(6) can be generalized to

E[S.sub.t+T] = [F.sub.t;t+T] (7)

for any t and T. Thus, we have shown the theoretical rationale of FUH, coupled with IRP theory holding, under the assumption of risk neutrality for buyers of currency i in the forward market participants. Under this set of assumptions, [F.sub.O,T], is an unbiased estimator of the expected future spot rate at time t. Recursively we can extend this to all future time periods and contract lengths.

WHAT ABOUT FORWARD SELLERS OF CURRENCY I (FORWARD BUYERS OF CURRENCY J)?

One distinctive feature of foreign exchange market is its symmetry. Assume that an investor from country j and currency j has the same two choices as investors from country i. The investor in country j's market has two choices: make a home market investment in country j's bond markets, or convert the currency in the spot exchange market then invest in country i's bond market and currency and sell the proceeds in the forward market. Both strategies should have equivalent payoffs to avoid any arbitrage opportunities. Now, let us impose the assumption of risk neutrality on future-forward buyers of the currency j (future forward sellers of currency i) and interest parity. By using the same analysis as before, we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

We have thus shown the theoretical rationale of FUH under an assumption of risk neutrality and interest parity. However, after we apply Jensen's Inequality to strictly concave function, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, we immediately learn that (7) and (8) cannot co-exist. This is referred to Siegel's paradox (Siegel 1972).

Siegel (1972) has argued that the forward price of a currency is a biased estimator of the expected future spot price. We have gone one step further in resolving the forward rate bias puzzle by showing the theoretical rationale of FUH for forward market buyers of currency i, under an assumption of risk neutrality and interest rate parity. We show it is theoretically possible for the forward exchange rates to be unbiased estimators of future corresponding spot rates for forward market buyers of currency i but not for both sellers and buyers. This indicates that the risk neutrality assumption, for both contract buyers and sellers, cannot simultaneously hold.

Informational Content of Forward Rates and Currency Liquidity

Though we have shown that it is impossible for FUH to hold true for both buyers and sellers of forward exchange contracts, a natural question arises: do forward exchange rates contain any information about corresponding future spot rates? As mentioned earlier, the symmetries of the foreign exchange market are the key features that distinguish this market from all others because forward market buyers and sellers are always simultaneously long in one currency (or forward contract) and short in another currency (or forward contract), with the objective of minimizing their risks or securing certain expected future cash flows.

Because of symmetry, we expect both sides of the foreign exchange market participants to express themselves symmetrically. We have shown that the assumption of risk neutrality for both buyers and sellers of forward exchange is not an adequate assumption because it leads to contradictory results according to Jensen's Inequality. Because of the symmetry of the markets, we therefore expect both sides of the market participants to be either risk-averse or risk-loving. Risk aversion is an oft-cited assumption in finance, namely that an investor will always choose the most attractive risk-reward package, all things being equal.

Assume the forward rate [F.sub.t+T] is significantly downward (upward) biased as an estimate of the corresponding future spot rate [S.sub.t] for future buyers of currency I (forward sellers of currency j). The same currency pair forward rate from the perspective of country j's residents, forward sellers of currency I, which is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] would be significantly upward (downward) biased as an estimate of the corresponding future spot rates quoted. That is, if

[S.sub.t+T]>(<)[F.sub.t;t+T] (9)

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

which is in contradiction with the symmetric feature of the market. Thus, we expect the bias of forward rate as estimators of corresponding future spot rates is small even for less liquid currency pairs.

The informational content of forward exchange rates can be evaluated by using the univariate regression:

[S.sub.t+T]= [[alpha].sup.+] [beta][F.sub.t,r+T] + [[epsilon].sub.t], (11)

to see whether the coefficient [alpha] is significantly different from zero. If so, then we examine whether forward rates are unbiased, upward biased or downward biased estimators of corresponding future spot rates.

Early empirical researchers like Cornell (1977) and Chiang (1998) were concerned about the efficiency of forward markets. Tauchen (2001) investigated the small sample properties of the Fama (1984) estimators and concluded that the evidence against the unbiasedness hypothesis was much strongerthan originally believed. From Maynard and Phillips (2001) we quote: "The difference in persistence between the short-memory spot return and long-memory forward premium does not admit a valid regression relation in returns and the slope coefficient in the Fama (1984) regression is found to converge to zero." Applying limit theory to the slope coefficient in this regression, they uncover a biased and negatively skewed distribution. They attribute the rejection of unbiasedness to the differences in persistence between the series.

Maynard (2003) reported that recent empirical studies have rejected exact 1:1 cointegration between spot and forward exchange rates. He posits that the non-cointegration of the spot and forward exchange rates is a possible explanation for the puzzling forward bias puzzle. He uses limit theory to show that the coefficient in the regression in returns form has a unit root component in its limit distribution. This unit root imparts a bias and skewness to the forward estimator. The empirical evidence suggests that the implied Dickey-Fuller-type terms do exhibit a downward bias yet are of insufficient magnitude to fully account for the observed negative betas.

Chakraborty and Evans (2008) appealed to irrationality on the part of traders to explain the puzzle and stated that the typical model specification may be inappropriate. They claim that the model in levels is super consistent, and the puzzle does not exist in this format. A small deviation of the beta parameter from unity is the cause of the adverse finding that changes in the spot rate are negatively related with the forward premium. They, however, do not advance any reasons why the beta parameter should deviate from unity. In summary the existing econometric evidence also does not provide a resolution of the puzzle.

We want to point out that most of the heavily cited existing literature, such as Hansen and Hodrick (1980), Bilson (1981), Fama (1984), Froot and Frankel (1989) and Sarno et al. (2012), uses the logarithmic transformed data to test FUH. As Osborne (2002) points out, using transformed data can fundamentally alter the nature of the variable, making the interpretation of the results somewhat more complex. By using logarithmic transformed data, the existing literature actually tests whether the logarithmic forward rates are unbiased estimators of logarithmic corresponding future spot rates, that is, whether Elog[S.sub.t+T] = log[F.sub.t;t+T] holds or not.

Again, by applying Jensen's Ineguality to the strictly concave function f(x) =log(x), we have Elog([S.sub.t]) < logE([S.sub.t]). Thus, whether Elog([S.sub.t+T]) = log[F.sub.t;t+T] holds or not gives no indication of whether E[S.sub.t+T]= [F.sub.t;t+T], the true FUH, holds or not.

Table 1 provides descriptive statistics for the spot exchange rate for five currencies against U.S. dollar: euro (EUR), Japanese yen (JPY), Canadian dollar (CAD), Australian dollar (AUD) and New Zealand dollar (NZD). Obviously the normality assumption for the spot rate is violated; thus, tests based on a regression of the future spot rate on the forward rate can result in spurious regression problems, or the so-called unit root problem.

To overcome the spurious regression problems, we develop and use the following alternative

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

to test the following three hypotheses:

1) Hypothesis HO: [alpha] = 0, to see whether forward rates are unbiased estimators,

2) Hypothesis H-: [alpha] < 0, to see whether forward rates are upward biased estimators, and

3) Hypothesis H+: [alpha] > 0, to see whether forward rates are downward biased estimators.

Figure 1 shows the histograms of the spot-forward ratio [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the five currencies, from which we can see the spot-forward ratio [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is conformable with a normal distribution even though a normality assumption is not required to use our testing method.

[FIGURE 1 OMITTED]

Though long-term forward rates are available, due to the fact that the money market interest rate does change over time, we only test the information content of one-month forward rates based on the fact that interest rates are relatively stable over the short term. We employ daily data of one-month forward and spot exchange rates using both direct and indirect quotes. Our data spanned from February 1995 to August 2010 with nearly 4,000 observations. Our data were obtained from Bloomberg. We examine the following five foreign exchange markets: euro (EUR), Japanese yen (JPY), Canadian dollar (CAD), New Zealand dollar (NZD) and Australian dollar (AUD), which are among the most active traded currencies against the U.S. dollar.

We report results of the tests in Tables 2 and 3. As can be seen in Tables 2 and 3, in the EUR-USD market, FUH cannot be rejected at any conventional confidence levels whether direct or indirect quotes are used. In the other four markets, FUH is rejected. As argued earlier, it is impossible for forward rates to be unbiased estimators of future corresponding spot rates for both sides of the market participants. The fact that we cannot reject FUH in the EUR-USD market allows us to draw the conclusion that the EUR-USD forward rates may provide good information on corresponding future spot rates.

One would naturally attribute the results concerning EUR-USD forward rates' informational content about their corresponding future spot rates to the high liquidity of the currency. Though there is no central source of information from which we can figure out how the liquidity of the various currency pairs rank, we can look to the periodic surveys done by the central banks and monetary authorities of the major global regions to get an idea. According to these sources and Mancini et al (2013), EUR-USD is the most liquid exchange rate pair, which is in line with the perception of market participants and the fact that it has by far the largest market share in terms of turnover.

Due to the fact that the euro and U.S. dollar are the world's two largest currencies, representing the world's two largest economic and trading blocs, many multinational corporations conduct business in both the United States and Europe. These corporations have an almost constant need to hedge their exchange rate risk by using forward rate contracts. Some firms, such as international financial institutions, have offices in both the United States and Europe. Firms that fit this description are also constantly involved in trading the euro and the U.S. dollar. Because the EUR-USD is such a popular currency pair, arbitrage opportunities are next to impossible to find.

As mentioned earlier, in the other four markets FUH was rejected. This is not surprising according to our theoretical argument made in Section 2, which states that forward rates cannot be unbiased estimators as corresponding future spot rates for both contract buyers and sellers. From Tables 2 and 3, we can see that in JYP-USD, NZD-USD and AUD-USD markets, forward rates are downward biased estimators of future corresponding spot rates when indirect quotations are used and upward biased estimators of future corresponding spot rates when direct quotations are used. We find the opposite results in CAD-USD market.

Though FUH was rejected in these four markets, we find the biases are small in all four markets, as indicated in the small a values (the absolute value of all [alpha] are far less than 0.5 %) and the standard errors (all the standard errors are less than 0.0006, or 0.06%). Thus, with a 95% confidence level, we can say that the biases of forward rates as corresponding future spot rates in these four markets are within two standard errors from estimated [alpha] values, which are Less than 1%. Thus, even though we reject FUH in these four exchange markets, forward exchange rates still provide fairly good estimators of the corresponding future spot rates.

Our empirical studies were conducted using data spanning from 1995 to 2010. As indicated in Mancini et al. (2013), the liquidity of foreign exchange markets declined significantly during the 2007-2009 financial crisis, especially after the bankruptcy of Leman Brothers. To examine the liquidity effect on the informational content of forward rates about their corresponding future spot rate, we separate data periods into two phases. Phases I (pre-crisis) spans Feb. 1995 to Dec. 2007. Phase II (during crisis) runs from Dan. 2008 to Oct.2010. We then repeat the following three hypothesizes:

1) Hypothesis HO: [alpha] = 0, to see whether forward rates are unbiased estimators;

2) Hypothesis H-: [alpha] < 0, to see whether forward rates are upward biased estimators and

3) Hypothesis H+: [alpha] > 0, to see whether forward rates are downward biased estimators.

If liquidity plays a role in the informational content of forward exchange rates, we would expect the forward rates to contain less information about their corresponding future spot rates during the 2007-2009 financial crisis, as reflected in larger [alpha] values. However, the absolute values of [alpha] from two phases testing results reported in Table 4 are not noticeably different. For some currency pairs, such as CAD/USD, EUR/USA and USD/NZD, the absolute a values are even smaller during Phase II. Thus, the testing results fail to support the liquidity effect on the informational content of forward rates.

To further examine the role of liquidity, we perform the above three hypothesis tests using the data of a less liquid (popular) currency pair ISK (Icelandic krona)-USD spanning from Sep. 2004 to Sep. 2010. (11) We first use the entire data set to perform the three hypothesis tests and then use the data of Phase I and the data of Phase II. The test results are reported in Table 5, from which we can see when the aggregate data of the entire period is used. FUH cannot be rejected for both direct and indirect quotes,

However, when disaggregated data sets are used, FUH was rejected, and we do see the larger values of [alpha] are positively correlated with the [alpha] values in more liquid currency pairs. Though FUH was rejected in most cases in ISK-USD market, from the results in Table 5, we can see the biases of forward rates, as estimators of corresponding future spot rates are less than 2.2 % (ISK/USD Phase II) at 95% confidence level.

Our empirical results show that in most cases forward rates are likely biased estimators of corresponding future spot rates. However, the biases are fairly small, which is consistent with our conjecture based on the symmetry feature of the foreign exchange markets. Our empirical testing results do not strongly support the liquidity effect on the informational content of forward rates.

Conclusion

This study examines the root and trigger causes of the forward rate bias puzzle, both theoretically and empirically. We first show analytically the forward rate and spot rate determination and the rationale for FUH under risk neutral assumption and interest rate parity for buyers of forward currency. By using Jensen's Inequality, we demonstrate that FUH cannot hold true simultaneously for both buyers and sellers of the same currency in the forward market. However, due to symmetry feature of foreign exchange markets, we expect both sides of the foreign exchange market participants to express themselves symmetrically with the objective to minimize their risks.

We argue if forward rates are significantly upward (downward) deviated from corresponding future spot rates for contract buyers than for contract sellers, forward rates would be significantly downward (upward) deviated from corresponding future spot rates. This is in contradiction of the symmetry feature. Thus, even though we do not expect FUH to hold true for both sides of the market participants, we expect forward rate bias to be small even for less liquid currency pairs.

We use a long sample period that covers a wide range of major currencies for our empirical testing to avoid sample specific problems. Our testing results show that in most cases FUH is rejected. However, the forward rate biases are found to be within 1% range for the more liquid currency pairs, and in all cases, we find the biases are within 2.2% (largest [alpha] from ISK/USD Phase II +2 standard errors=0.015734+ 2x 0.002871=0.021476< 2.2%) range at 95% confidence level. Our testing results do not show clear evidence that the liquidity of the currency pair affect the informational content of the forward rates about their corresponding future rates.

Our study differs from existing research in several respects. We combine theoretical arguments with empirical evidence to explain and to examine the forward rate bias as opposed to the conventional purely empirical approach. We use an alternative empirical testing method as opposed to the conventional regression methods using logarithmic transformed data, which severely alters the nature of the true FUH. We examine the informational content of forward rates from the perspectives of both contract buyers and sellers by using exchange rate data quoted in both direct and indirect form. This differs from the conventional approach, which only uses one form of the exchange rate quote and thus is only from the perspective of one side of market participants.

Finally, we not only discuss the validation of FUH theoretically and empirically but also give an assessment of the estimation bias of forward rates as estimators of corresponding future spot rates. We offer a new insight to both academicians and practitioners that, in spite of rejection of FUH due to the symmetry of the foreign exchange market, forward rates are still reasonably reliable estimators of corresponding future spot rates even for less liquid currency pairs. This fact is confirmed by our empirical evidence.

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LLOYD BLENMAN

University of North Carolina Charlotte

GUAN JUN WANG

Savannah State University

(1) In full generality, the demand for a forward contract can be shown to contain a pure hedging component plus a speculative component, all of course depending on the type of risk tolerance assumed. Risk neutral investors place zero weight on volatility risk. This assumption then drives the general result into a setting where the demand for forward contracts is based solely on the pure hedging component. For fundamental papers dealing with pure speculation in currency markets, see Siegel (1972), Feldstein (1968) and McCulloch (1975).

(2) In addition, if my base currency is dollars and I switch to euros, I am short dollars and long euros. If the euro appreciates, the dollar depreciates and so I profit from this position. There are also other symmetries on derivatives with currency as the underlying asset. For further details on these issues, see Wystup (2008), for example.

(3) We understand that the markets are composed of many groups of players who are, respectively, risk averse, risk seeking and risk loving. However, in aggregate the market is best described as a risk-neutral one. See Sarno (2005) for a discussion on this point.

(4) For very weak currencies of emerging market countries, high interest rates presage a depreciation of their currencies. Most of those countries have no formal forward markets and for those countries IRPT mostly holds.

(5) Fama (1984) was the first to point out the significance of the estimates of beta for their implications for future spot exchange rate variability and forward premium volatility.

(6) Cornell (1977) was more interested in issues of market efficiency rather than in the specific nature of the bias.

(7) Risk neutrality and rational expectations can be consistent with preferences where volatility is given zero weight. We do not explicitly use the zero volatility weight assumption in any of our calculations

(8) Some previous studies have shown this theoretically, but we are the first to measure the size of the bias in current currency markets during normal periods and also in periods of crisis.

(9) These biases can be even smaller if we were to assume risk aversion on the part of one set of market participants and risk preferring behavior on the other set. However, it is not clear if there is any benefit to be gleaned from a higher level of modeling realism.

(10) However, what is indirect from one country's perspective is direct from the other country's perspective. So we will minimize the use of the terms indirect and direct in the discussion that follows.

(11) Franket and Poonawala (2010) show in their paper that less liquid (emerging market) currencies are less biased than those of very Liquid (advanced economies) currencies.

TABLE 1 Descriptive Statistics of Spot Rate (Direct Quotation) EUR JPY AUD CAD Mean 1.192906 111.6076 0.713409 1.320392 Standard Error 0.002782 0.188229 0.001795 0.002859 Median 1.2166 111.91 0.735 1.36 Mode 1.333 107.82 0.7653 1.3685 Standard 0.177329 12.00396 0.11488 0.18006 Deviation Sample 0.031445 144.095 0.013198 0.032422 Variance Kurtosis -0.67208 -0.02124 -0.68651 -1.12112 Skewness -0.16372 -0.17962 -0.00985 -0.31061 Range 0.7724 66.725 0.5004 0.6985 Minimum 0.8267 80.63 0.4789 0.9202 Maximum 1.5991 147.355 0.9793 1.6187 Sum 4846.776 453908 2921.409 5237.995 Count 4063 4067 4095 3967 Confidence 0.005454 0.369033 0.00352 0.005605 Level (95.0%) NZD Mean 0.609591 Standard Error 0.001722 Median 0.637 Mode 0.691 Standard 0.106789 Deviation Sample 0.011404 Variance Kurtosis -0.99842 Skewness -0.35644 Range 0.4262 Minimum 0.3922 Maximum 0.8184 Sum 2345.096 Count 3847 Confidence 0.003376 Level (95.0%) TABLE 2 Estimated Equation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Indirect Quotation) Note: exchange rates are expressed as per unit of US dollar. (*) and (**) indicate standard errors and P-values, respectively. Currency [alpha] [alpha]=0 [alpha]>0 [alpha]<0 EUR/USD 0.0009 accept reject reject (0.00057) (*) (0.132) (**) JPY/USD 0.002952 reject accept reject (0.00053) (*) (0.000) (**) CAD/USD -0.0012 reject reject accept (0.00038) (*) (0.001) (**) NZD/USD 0.0036 reject accept reject (0.00060) (*) (0.000) (**) AUD/USD 0.0031 reject accept reject (0.00056) (*) (0.000) (**) TABLE 3 Estimated Equation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Direct Quotation) Note: exchange rates are expressed as per unit of foreign currency (*) and (**) indicate standard errors and P-values, respectively. Currency [alpha] [alpha]=0 [alpha]>0 [alpha]<0 USD/EUR 0.0001 accept reject reject (0.00057) (*) (0.833) (**) USD/DPY -0.0018 reject reject accept (0.00054) (0.001) USD/CAD 0.0018 reject accept reject (0.00037) (*) (0.000) (**) USD/NZD -0.0021 reject reject accept (0.00061) (*) (0.000) (**) USD/AUD -0.0018 reject reject accept (0.00058) (*) (0.002) (**) TABLE 4 Estimated Equation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Currency [alpha] Standard P-value Pair Error EUR/USD 0.002027 0.000582 0.000506 (Phase I) EUR/USD 0.00000 0.001165 0.954552 (Phase II) USD/EUR -0.00125 0.000576 0.029954 (Phase I) USD/EUR 0.00126 0.001179 0.285563 (Phase II) JPY/USD 0.004946 0.000578 0.00000 (Phase I) JPY/USD -0.00672 0.001306 0.00000 (Phase II) USD/JPY -0.00379 0.000586 0.00000 (Phase I) USD/JPY 0.007965 0.001318 0.00000 (Phase II) CAD/USD -0.00192 0.000329 0.00000 (Phase I) CAD/USD 0.001972 0.001504 0.190318 (Phase II) USD/CAD 0.002279 0.000332 0.00000 (Phase I) USD/CAD -0.00048 0.00143 0.737343 (Phase II) NZD/USD 0.003918 0.000587 0.000000 (Phase I) NZD/USD 0.001905 0.002043 0.351478 (Phase II) USD/NZD -0.00282 0.000589 0.000000 (Phase I) USD/NZD 0.000996 0.002065 0.629758 (Phase II) AUD/USD 0.002602 0.000519 0.00000 (Phase I) AUD/USD 0.005515 0.002089 0.008487 (Phase II) USD/AUD -0.00171 0.000519 0.001016 (Phase I) USD/AUD -0.00224 0.002241 0.317797 (Phase II) Currency [alpha]=0 [alpha]>0 [alpha]<0 Pair EUR/USD reject accept reject (Phase I) EUR/USD accept reject reject (Phase II) USD/EUR accept reject reject (Phase I) USD/EUR accept reject reject (Phase II) JPY/USD reject accept reject (Phase I) JPY/USD reject reject accept (Phase II) USD/JPY reject reject accept (Phase I) USD/JPY reject accept reject (Phase II) CAD/USD reject reject accept (Phase I) CAD/USD accept reject reject (Phase II) USD/CAD reject accept reject (Phase I) USD/CAD accept reject accept (Phase II) NZD/USD reject accept reject (Phase I) NZD/USD accept reject reject (Phase II) USD/NZD reject reject accept (Phase I) USD/NZD accept reject reject (Phase II) AUD/USD reject accept reject (Phase I) AUD/USD reject accept accept (Phase II) USD/AUD reject reject accept (Phase I) USD/AUD accept reject accept (Phase II) TABLE 5 Estimated Equation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Currency [alpha] Standard Pair Error ISK/USD (whole 0.002546 0.001533 period) USD/ISK (whole 0.00078 0.00143 period) ISK/USD -0.00847 0.001361 (Phase I) USD/ISK 0.010087 0.001349 (Phase I) ISK/USD 0.015734 0.002871 (Phase II) USD/ISK -0.01036 0.002635 (Phase II) Currency P-value [alpha]=0 [alpha]>0 [alpha]<0 Pair ISK/USD (whole 0.09699 accept reject reject period) USD/ISK (whole 0.585706 accept reject reject period) ISK/USD 0.000000 reject reject accept (Phase I) USD/ISK 0.000000 reject accept reject (Phase I) ISK/USD 0.000000 reject accept reject (Phase II) USD/ISK 0.000000 reject reject accept (Phase II)

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Author: | Blenman, Lloyd; Wang, Guan Jun |
---|---|

Publication: | Quarterly Journal of Finance and Accounting |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Mar 22, 2017 |

Words: | 7157 |

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