The Pricing of Forward Exchange Rates

International Finance Discussion Papers Number 312

September 1987

The Pricing of Forward Exchange Rates by

Ross Levine

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an

acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

This paper addresses the question: do risk premia account for the observed time-varying discrepancies between forward and corresponding future spot exchange rates? A simple theoretical framework is used to derive testable restrictions on the parameters of a multivariate regression model. Using various econometric procedures and different estimation periods, the data reject the restrictions. In contrast to past investigations, the empirical results are inconsistent with a world in which time-varying risk premia are the sole determinants of observed deviations from the unbiased expectations hypothesis. Anticipated

real exchange rate movements may explain the rejection.

The Pricing of Forward Exchange Rates

by

Ross Levine

Although an expanding empirical literature finds that forward exch.ange rates systematically differ from corresponding future spot prices, the source of this bias has not been convincingly identified. Frankel and Froct (1987) argue that agents systematically make mistakes in predicting exchange rates, and reject rational expectations. Krasker (1980), Lewis (1986) and Obstfeld (1987) suggest that even if expectations are fully rational ex ante, exchange rate forecasts may appear biased and serially correlated in the ex post sample if there is the possibility of a major policy change .* The most frequently advanced and thoroughly studied theoretical explanation for the observed discrepancies between forward and future spot exchange rates is the risk premia hypothesis. Empirical

. . : soos . 3 support for the risk premia hypothesis, however, is inconclusive.

1. The author is a staff economist in the International Finance Division. This paper represents the views of the author and should not be interpreted as reflecting those of the Board of Governors of the Federal Reserve System or other members of its staff. I have benefited from the comments of Maria Carkovic, Michael Darby, Sebastian Edwards, Edward Leamer, and workshop participants at the Board of Governors, Johns Hopkins University, the Kellogg Graduate School of Management, and the Wha:cton Finance Department.

2. This explanation is called the "peso problem" because there was a long time interval before the actual Mexican peso devaluation where an expectation of a devaluation persisted, and the forward exchange rate consistently undervalued the peso.

3. See Levich'’s (1985) review of the literature.

This paper uses an approach developed and implemented by Korajczyk (1985) to examine the validity of the risk premia hypothesis. This approach tests the predictions of a wide class of international asset pricing models concerning forward exchange rates. In particular, a diverse set of asset pricing paradigms [see Kouri (1977), Lucas (1982), and Solnik (1983), for example] predicts that forecastable deviations between forward and future spot exchange rates equal expected real return differentials on default free nominal bonds, where real returns are evaluated in the currency of issuance. This prediction imposes testable restrictions cn the parameters of a multivariate regression system. In examining these restrictions, this paper tests whether the data are consistent with a world in which only time-varying risk premia on nominally riskless bonds explain the deviations between forward and future spot exchange rates. Although the tests performed in this paper are not tests of a particular asset pricing model, they do test the implications of a broad class of asset: pricing models.

Korajcezyk (1985) cannot reject the hypothesis that risk premia cause the observed deviations between forward and future spot prices. He, however, uses incorrectly matched forward and future spot exchange rate data which lead to unwarranted econometric procedures. More importantly, Korajezyk fails to test an important restriction suggested by theory and past research into the pricing of forward exchange rates. Using a nunber of econometric procedures and different estimation periods, the data reject this restriction and reverse Korajezyk’s conclusion.

After deriving the testable restrictions and discussing the estimation methodology in Section I, Section II describes the data and

presents the empirical results. Although ex ante real interest rate

differentials are often significant explanatory variables of ex post differences between forward and future spot prices, the empirical results are inconsistent with a world in which only time-varying risk premia on default free bonds explain the forward exchange rate bias. Anticipated real exchange rate movements may explain this conclusion. Section III

summarizes the results and suggests future research endeavors.

I. The Unbiased Expectations Hypothesis and

Expected Real Interest Rates

The unbiased expectations hypothesis (UEH) states that the forward exchange rate equals the market prediction of the future spot exchange

rate. In logarithmic form this suggests that

<1> E(s.43- f,1¢,) = 0,

where

Sia] is the logarithm of the spot exchange rate at time t+l expressed, for example, in dollars per foreign currency;

fe is the logarithm of the forward exchange rate set at time t, payable at t+l, and also expressed in dollars per foreign currency;

ve is the information set available at time t; and

E(.) is the expected value operator.

This UEH implies that information available at the time the forward

contract is set should not be useful in explaining ex post discrepancies

between forward and corresponding future spot exchange rates. This hypothesis can be tested by analyzing whether or not B = O in the

regression equation:

<2> s

t+1 ~ fe = XB + €

t+1’

where E(e.4,) = 0, x, is a subset of oe» and a tilde indicates a random variable at time t. The hypothesis that B = 0 has been rejected by many authors [see, for example, Hansen and Hodrick 1980, 1983 and Hodrick and Srivastava 1984] using elements of x. such as the forward premium, i, - s

“t

and lagged values of the forward rate forecast error, S.- feo. As

discussed above, identifying the source of this rejection has proved

t’

difficult. In order to resolve this interpretational problem, I combine the Covered Interest Rate Parity (CIRP) condition, the Fisher equation, and the assumptions of rational expectations and ex ante relative purchasing power parity to derive an estimable expression for E(si417 £.1¢,) which is

fully consistent with the asset pricing models listed above. CIRP states

that

<3> Rest 7 Reta + fe > St where Read = the continuously compounded yield on a 1-period nominally riskless bond denominated in (for example) dollars (from t to t+l); and Revi the continuously compounded yield on a foreign (denominated)

l-period nominally riskless bond (from t to t+1).

The CIRP condition merely states that two nominally riskless investments must: have the same nominal rate of return when evaluated in the same currency. If CIRP did not hold, profitable arbitrage opportunities would exist. Frenkel and Levitch (1977) and McCormick (1979) present empirical support for the CIRP condition.

The nominal exchange rate may be expressed as <i> S, =p, - > +d,

t t t t

where Py is the logarithm of the U.S. price level, P, is the logarithm of the foreign price level, and d. is the logarithm of the deviation from PPP at time t. qd, may be referred to as the logarithm of the real exchange rate because it represents the logarithm of the relative price of a bundle of U.S. goods for a bundle of foreign goods. Since this relative price may charge intertemporally, the real return on the same nominally riskless asset may differ when evaluated in different currencies.

Combining equations <3> and <4> we obtain an expression for the ex

post difference between Sead and fi:

*

<> ttl 7 Reai)>

nn I Mh | Ht ' + Al 1 [oy

(R

t+1 t t+l Tea ttl t

where Tea is the domestic inflation rate from period t to t+l and Ta is the corresponding foreign inflation rate,

This expression can be simplified by using the Fisher equation which expresses nominal yields (on nominally riskless bonds) as the

summation of expected real returns and expected inflation:

<6a> Rey 7 E(r4 41164) + E(T. 1 19,)

<6b> Reel 7 BCE 18) + BCT |4,).

where Teed is the real return on a nominally riskless bond maturing at time ttl. Equation <6a[b]> defines the expected real return on U.S. [foreign] bonds, where the real value is evaluated in dollars [foreign currency]. It is important to notice that the expected real return on a foreign bond held by a foreign resident may differ from the expected real return on a foreign

bond held by a U.S. resident because of anticipated real exchange rate

*

changes. For example, if E(r. 41)

= E(t.) then a U.S. resident would expect a higher yield from holding foreign bonds than from holding U.S. bonds if the dollar's real value is expected to depreciate. *

Darby (1975) shows how the Fisher equations should be modified to incorporate the effect of taxes. The "Darby effect" would add an additional term to each equation. For simplicity, this effect is ignored in the theoretical presentation; however, it is considered in the empirical estimation.

Combining equations <5>, <6a>, <6b> and assuming rational expectations, the expected deviation of Sea] from fee conditional on oe»

is

</> El(Se44 ~ £9184) ~ ELE, ~ Fey 16] + BL(4.4) - 4.) 18,1.

4. Notice that implicit in this discussion is a definition of residency. A U.S. resident is someone who purchases his goods in the U.S. and

deflates nominal returns by the U.S. price level.

The CIRP condition merely states that two nominally riskless investments must have the same nominal rate of return when evaluated in the same currency. If CIRP did not hold, profitable arbitrage opportunities would exist. Frenkel and Levitch (1977) and McCormick (1979) present empirical support for the CIRP condition.

The nominal exchange rate may be expressed as <A> 4d Se 7 Pe Pe F Me:

where Py is the logarithm of the U.S. price level, Pe is the logarithm of the foreign price level, and d.. is the logarithm of the deviation from PPP at time t. d. may be referred to as the logarithm of the real exchange rate because it represents the logarithm of the relative price of a bundle of U.S. goods for a bundle of foreign goods. Since this relative price may change intertemporally, the real return on the same nominally riskless asset may differ when evaluated in different currencies.

Combining equations <3> and <4> we obtain an expression for the ex post difference between Sea] and f,: ~* ~ *

<3> s - i, -1 ~ Teen * Sead > de Qed > Bey,

~ a* where Lead is the domestic inflation rate from period t to t+l and Lead is the corresponding foreign inflation rate.

This expression can be simplified by using the Fisher equation

which expresses nominal yields (on nominally riskless bonds) as the

summation of expected real returns and expected inflation:

<ba> Rea. 7 E(ri4119,) + E(T 19.)

<6b> Real = E(Z, 41 1¢,) + E(Ty 4 1¢,).

where Tha is the real return on a nominally riskless bond maturing at time t+l. Equation <6a[b]> defines the expected real return on U.S. [foreign] bonds, where the real value is evaluated in dollars [foreign currency]. It is important to notice that the expected real return on a foreign bond held by a foreign resident may differ from the expected real return on a foreign bond held by a U.S. resident because of anticipated real exchange rate

~* changes. For example, if E(r + = E(

t Tea)? then a U.S. resident would

expect a higher yield from holding foreign bonds than from holding U.S. bonds if the dollar's real value is expected to depreciate.”

Darby (1975) shows how the Fisher equations should be modified to incorporate the effect of taxes. The "Darby effect" would add an additional term to each equation. For simplicity, this effect is ignored in the theoretical presentation; however, it is considered in the empirical estimation.

Combining equations <5>, <6a>, <6b> and assuming rational expectations, the expected deviation of Sea] from tL. conditiona:. on ¢

t’ is

</> EL(Se41 - FL) 144) ~ EL (Fey . Tear) 1O_] + EL(4. 4) - d.)14,,).

4. Notice that implicit in this discussion is a definition of residency. A U.S. resident is someone who purchases his goods in the U.S. and deflates nominal returns by the U.S. price level.

Equation <7> demonstrates that forward exchange rates are not unbiased predictors of future spot exchange rate unless (i) expected real returns on nominally riskless bonds are equal across currencies and (ii) the expected rate of change in the real exchange rate is zero.”

Since a forward foreign exchange contract is an agreement to exchange the future payoffs on nominal bonds denominated in the currencies of two countries, any factors affecting the expected real value of these bonds will be incorporated into forward prices. When real returns are evaluated in the currency of issuance, expected real returns on nominally riskless assets may differ if the two currencies have different price level risks. Thus, if country price levels have different stochastic properties and agents are risk averse, bonds will be priced to reflect those differences, and the forward exchange rate will not equal the expected future spot price. Empirical evidence suggests that ex ante real interest rates are not equal internationally [See: Mishkin 1984; Huizinga and Mishkin 1984; and Merrick and Saunders 1986]. Therefore, we should not expect the UEH to hold.

Condition (ii) implies that ex ante real returns are independent of

residence, although ex post real returns on the same asset may differ

5. As Korajczyk points out, the forward forecast error would also equal zero if the expected real return differential and expected rate of change in the real exchange rate are perfectly negatively correlated. This would be the case, for example, under a fixed exchange rate regime where the the probability of adjustment is zero. Since the empirical investigation wil] cover the floating exchange rate period, it is

unlikely that EIT aa Tea! will equal “-E[d. a7 d.] for all t.

internationally.° When there are non-zero expected real exchange rate movements, expected real returns on nominally riskless bonds depend upon the currency in which real returns are evaluated. Consequently, anticipated real exchange rate movements affect the expected real value of default free bonds and will be reflected in forward exchange rates. This implies that forward exchange rates will not equal the market’s prediction of future spot rates when there are anticipated real exchange rate movements.

Empirical evidence concerning condition (ii) is inconclusive. Although it is clear that exact PPP does not hold (i.e., d, = 0, for all t), the evidence regarding Roll'’s (1979) "efficient market version of PPP" (i.e., Eld.ii- d.] = 0) is mixed. / The tests performed in this paper are designed to determine the relative importance of anticipated real exchange rate movements and expected real return differentials as explanations of

the time-varying forward exchange rate bias.

Assuming that EL (a. y4- ddd.) = 0, one obtains: <8> E[(S £_)|x E[(Z.,- [(Siyy- £1] = ((leya> Fey) |X]

where x, is a subset of o.- Equation <8> implies that the forecastable

difference between Sead and f. equals the expected real return differential

6. Condition (ii) does not rule out stochastic real exchange rates from entering the pricing of assets, even if the expected rate of change in the real exchange rate is zero. See Levine (1986).

7. Kravis and Lipsey (1978) and Isard (1977) show that exact PPP does not hold. Darby (1980), Cumby and Obstfeld (1984), Adler and Lehnan (1983), Officer (1984), and Hsieh (1982) present conflicting evidence on the question of whether Roll'’s efficient market version of PPP holds.

on default free nominal bonds deflated by their home currency price levels. This equation is similar to the expression for foreign exchange market risk premia derived by Kouri (1979), Lucas (1982), Stulz (1981), and Solnik (1983). These models, however, explicitly characterize the term Eres ~ (ena Sega) [Xe] - ~* ~ Assuming that El(riya- Cea) [Xe] is observable, the following

regression equation is testable:

~ ~xX ~ <9> Seay” fy = % + OQEL (TL - Leap) |X] + 452, + Mea

2. is a subset of X,. The theory suggests that 99 = 9 = 0 and 94= 1. The variable 2. is added in order to incorporate an additional testable implication from equation <8>. Information available at time t should not systematically explain the ex post difference between forward and future spot sates beyond the information's ability to predict real return diffe:entials. Consequently, theory predicts that variables which have been shown to be statistically significant explanatory variables of the forwa:cd exchange rate bias should enter insignificantly once

EL(E.4- Toy) |X] is included. Rejection of any of these restrictions would constitute evidence against the null hypothesis that risk premia on nominally riskless bonds are the only source of bias in the forward exchange market. Assuming rational expectations, rejection of the null

hypothesis supports the notion that anticipated real exchange rate changes

are an important cause of the observed discrepancies between forward and

future spot prices.

10

* ~ : 2 t+" Teg) |X] is unobservable, proxies must be used.

Since E[(r Wickens (1982) demonstrates that consistent and asymptotically efficient parameter estimates may be obtained using instrumental variables techniques in which expected variables are replaced by their realized values, exogenous variables are used as instruments, and the model is estimated jointly treating the auxiliary equations describing expectations formation

as part of the system. Consequently, three stage least squares (35LS) is

used to obtain consistent and asymptotically efficient estimates of 6:

~ ~x* ~ <10> s - £, =6 ea TeeD + 652, + Sea?

where Y, is the set of instrumental variables used to predict real interest rate differences, and Cee is a white noise error term. Although the implications drawn from using the instrumental variables methodology are contingent upon the instruments containing the information sets employed by economic agents in forming expectations, Wickens (1982) demonstrates that even if a subset of agents’ information sets is used to construct expectations in the first stage regressions, the 3SLS estimator will remain

consistent though not asymptotically efficient. The instruments are

described below.

11

II. Estimation II.A. The Data

Daily observations on spot exchange rates, l-month forward exchange rates, and 1-month Eurocurrency interest rates were obtained from Data Resources Incorporated. The ten countries in the sample are the United States, Belgium, Canada, France, Italy, Japan, the Netherlands, Switzerland, the United Kingdom, and West Germany. The data cover the period July 1973 through April 1986 for most time series. Eurocurrency interest rate data for Belgium, Canada, Italy, and Japan, however, do not begin until January 1981. Monthly observations of the consumer price index (CPI) are used to construct inflation series, and are obtained from the International Monetary Fund's IFS tape (line 64).

Past investigations into the nature of the forward exchange rate bias have typically matched forward exchange rates sampled on the last Friday of the month with spot exchange rate sampled on the last Friday of the following month to obtain the forward forecast error, or forward rate bias, Sia.” fh. This is not, however, the way the foreign exchange market matches forward and corresponding future spot exchange rates.

In order to correctly match forward and future spot exchange rates, it is important to understand the mechanics of the spot exchange market.

It takes two working days to settle a spot contract. Thus, a spot contract written on contract day Thursday, March 25 is for settlement on the spot value day Monday, March 29. A forward contract written on contract day

Thursday, March 25 is dated in the following manner. First, find the spot

value date (Monday, March 29) corresponding to the contract date (Thursday,

12

March 25). Second, go one month forward from the spot value date. If April 29 is a working day then April 29 is the future value date. If, however, April 29 is a holiday in either country then go forward unt:il the first working day is found in order to identify the future value day. If, however, by going forward from April 29 we go into the next month, <hen go backward from April 29 until the first working day is identified in order to find the future value date. The future value day is the day on which the one-month forward contract written on March 25 is settled. Note, however, that the future value day is not the date of the expected future spot rate corresponding to the forward contract. The reason is that it takes two working days to clear spot transactions. Therefore, to find the corresponding future spot exchange rate, the day two working days before

the future value day must be chosen. This paper uses correctly matched

data.

II.B. Small Sample Properties and Regression Residuals

Korajczyk estimates equation <10> using data from 1973 through 1980 for eight currencies paired with the U.S. dollar. He points out that there may be reasons for concern regarding the distribution of various test statistics because of small sample size and possible deviations from normality in the regression residuals. Using incorrectly matched forward and future spot prices, Korajczyk presents evidence that the regression residuals have distributions significantly different from the normal distribution. This induces Korajczyk to construct alternative test

statistic distributions using a bootstrap procedure. These constructed

13

distributions yield different results from those implied by the test statistics’ asymptotic distributions. When correctly matched data are used, the Kolmogorov-Smirnov test does not reject the hypothesis of

normally distributed regression residuals. Moreover, even with the degrees

of freedom corrections suggested by Box and Watson (1962) for hypothesis testing when regression residuals are not normally distributed, or using the v- distribution when conducting cross equation tests as recommended by Judge et al. (1980), this paper's conclusions are unchanged.

Even if the regression errors are normally distributed, we know only the asymptotic distribution of the test statistics in the 3SLS models wher. the variance-covariance matrix is estimated. For this reason Korajczyk uses Monte Carlo simulations to construct alternative test statistic distributions. Again, these alternative distributions yield different conclusions from those implied by the statistics’ asymptotic distributions.

There are, however, important problems with using Monte Carlo simalations to construct alternative distributions. "Monte Carlo studies in econometrics often have been criticized for imprecision present in estimating the underlying finite sample properties investigated and for the specificity of the results from the particular parameter values and sample sizes chosen, so making any conclusions very tentative at best." (Ericsson 1984, p. 691) In simultaneous systems there is the additional problem that Monte Carlo techniques typically assume that the regressors are exogenous. In the Monte Carlo procedure, a new set of regressors is constructed by sampling from a normal distribution and adding these errors to the regressors whose coefficients are given by the null hypothesis. "If the

regressors are merely predetermined endogenous variables, such a technique

14

does not generate new future right-hand-side variables whose values would be simultaneously determined with realizations of the current errois." (Hodrick and Srivastava 1986, p.31) These problems make suspect

: 8 Korajczyk's conclusions based on Monte Carlo simulations.

II.C. Ex Ante Real Interest Rates and the Forward Bias

This subsection tests whether the data are consistent with a world in which the only source of bias in forward exchange rates arises from risk premia on Eurocurrency bonds. Assuming (i) rational expectations and (ii) E[ (4.447 di) 1¢,] = 0, Section I defined the null hypothesis, and described the 3SLS estimation procedure. The instrumental variables for predicting real interest rate differentials are the same as those used by Korajczyk and are chosen on the basis of their documented ability to predict. future real interest rates. The instruments are : (1) a constant; (2) the average real return differential over the preceding twelve months; (3) the lagged

difference in the inflation rates between the U.S. and the foreign country;

8. In addition to increasing the sample size from 88 to 149 observations, which may reduce small sample problems, this study constructed alternative test statistic distributions based on Monte Carlo simulations. Although these results are not presented because of the problems discussed in the text, it is important to note that while conclusions concerning specific parameter values based on the Monte Carlo distributions are different from those based on the asymptotic distributions, this paper’s null hypothesis is rejected using either distribution.

15

(4) a proxy for nominal interest rate variability;? (5) the lagged value of the real return differential; and (6) the forward premium at time t. Instrument two was chosen because Fama and Gibbons (1982) demonstrate that it explains U.S. real interest rates (as an approximation to an ARIMA [0,1,1]). Instruments three and six were included since Mishkin (1984) shows that they are significant explanatory variables of real interest rate differentials. Instrument four was added because nominal interest rate volatility explains risk premia in the U.S. Treasury bill market [Fama 1976]. Since I could obtain appropriate data for the time period April 1974 through December 1985 for only the United States, the United Kingdom, Wes: Germany, France, Switzerland, and the Netherlands, my sample uses these five currencies paired with the U.S. dollar. Beginning in 1981, the data set also has relevant figures for Belgium, Canada, Italy, and Japan. As will be demonstrated below, using these additional countries over the shorter time period does not alter the results.

The unadjusted R? statistics for the first stage regressions of the real return differential on the instrument set range from .24 to .49, and the mean is .38 (over the period April 1974 - December 1985). Only for the United Kingdom do the data not reject the hypothesis that all non-intercept parameters are zero.

Table 1 gives unrestricted 3SLS estimates of equation <10> without Ze For four out of the five currencies the slope coefficient is

significantly different from zero at a .05 significance level. More

9. Korajczyk uses the difference between the sample standard deviations of nominal interest rates over the preceding 26 weeks. Since I have daily observations, I use the difference between the sample standard deviations on one-month interest rates (with daily observations) over the preceding month as a measure of nominal interest rate variability.

16

TABLE 1

UNRESTRICTED 3SLS

*

s - EL = by + 6) ead > Tea? * fea

A. Estimates

Country 55 64 F(6)=0,5,=1) D.W.

NE - 001 1.45" 2.39 2.11 (003) (.20)

UK 002 2.23" 2.09 1.92 (003) (.59)

FR -.001 1.49” 0.55 2.19 (003) (46)

WG -.001 1.28” 0.56. 2.07 (.002) (.29)

SW 000 1.06 01 1.99 (004) (1.29)

B. System Tests

Test F P-Value

5 9= 0; all equations 43 71

$4: equal across equations .52 .72

64> 1; all equations 1.78 .11

5o= 0, 64> 1; all equations 1.02 42

Weighted Ro = .24 April 1974 - December 1985

Note: Standard errors in parentheses. F(69= 0,6)= 1) represents the

F-statistic for the null hypothesis for b9= 0 and 64> 1.

* Significantly different from the null at the .05 level.

17

importantly in no individual country is the null hypothesis that b9= 0 and 64= l rejected. The test statistics for joint hypotheses across equations are provided in part B of the table. The four tests in Table 1 are: (1) the constant term is zero in all equations; (2) the slope coefficients are equal across equations; (3) all the slope coefficients equal one; and (4) the intercepts are zero and the slope parameters equal one for all cross sections. None of these joint system tests is rejected by the data at the .05 significance level.

Using the Kolmogorov-Smirnov normality test, one cannot reject the > null hypothesis of normally distributed regression residuals. The largest Kolmogorov-Smirnov statistic of the five equations is .09 with most falling below .07. These statistics do not reject normality at a .05 significance level [See Powell 1982].

Although real interest rate differentials explain only 18 percent of the intertemporal variation in the forward forecast error, the results presented in Table 1 are consistent with a world in which the risk premia on Eurocurrency bonds are the sole determinant of expected deviations between forward and future spot exchange rates.

If, however, risk premia on default free nominal bonds are the cause of the forward bias, information available at the signing of the forward contract should not systematically explain the forward forecast error beyond the information’s usefulness in predicting real interest rate differentials. Consequently, even variables which have been shown to significantly explain the forward forecast error should enter insignificantly once the real return differential is incorporated. Lagged values of the forward forecast error, s, - f£ and the forward premium,

t t-1’

f. - S, are two elements of the information set available at the setting of

18

the forward price which have traditionally been used in studying the UEH. Korajczyk considers lagged values of the dependent variable but not: the forward premium. This turns out to have important implications.

The results from estimating equation <10> where Z, = s, - Teed are presented in Table 2. For no individual country is 2. significantly different from zero, and only for the United Kingdom, do the data ‘reject the null hypothesis that o= 0, $4= 1, and b5= 0. In addition, the coefficient on the real return differential is significantly different from zero in four out of the five currencies considered. The system tests tell a similar story. Although inclusion of lagged forecast errors results in rejection of the joint hypothesis that the real return differential coefficients are one for all cross sections, the hypothesis that b= 0, $= 1, and §o= 0 for all currencies is not rejected. Table 2 shows that (s,- f..) has little explanatory power beyond its influence through real return differentials.

Table 3, however, tells a different story. Table 3 gives the results from estimating equation <10> with 2. set equal to the forward premium (f£,- S,)- The results strongly reject the hypothesis that 2, enters with a zero coefficient. For all five currencies 55 is significantly different from zero and the data reject the hypothesis that 6,= 0, 6,= 1, and 6

7 0 for each individual currency. In addition, once

0 1

the forward premium is included, the slope coefficient on the real]. return differential is no longer significant for the Netherlands. The system tests also strongly reject the hypothesis that information available at the signing of the forward contract is only useful in explaining the ‘orward exchange rate bias to the extent that that information is useful in

forecasting real return differentials. Neither of the models, whose

19

TABLE 2

UNRESTRICTED 3SLS

~* ~ Steet Fe = 8 + Oy (heyy > Keg) + 890SE- Fey) + ead A. Estimates Cour,try 6 9 5 6, (5950, 5.-1, D.W. 65-0 * NE -.001 1.36 .09 2.30 2.27 (.003) (.21) (.05) * * UK 003 2.60 .07 3.12 2.02 (.003) (.56) (.07) FR -.000 1.68" 02 0.85 2.25 (.003) (.46) (.05) WG -.001 1.72% ~—.05 2.23 2.19 (.003) (.32) (.05) sw 000 1.24 .03 17 2.07 (.004) (.77) (.06) B. System Tests Test F P-Value $9= 0; all equations .39 71 b= 0; all equations 1.00 -42 64= 1; all equations 2.90 .O1 §9=9, §,-1, §5=0 all equations 44 .82 Weighted R* = .26 April 1974 - December 1985

See note in Table l.

20

TABLE 3

UNRESTRICTED 3SLS

* ~ t+1 7 Tea? * OoCFe- Se) + fea

A. Estimates

Sia. 7 f. = 56 + 6, (r

Country 55 $4 $5 F(69=0,6 =1, D.W. 6-0}

NE 002 40 -1.76" 11.9" 2.12 (.002) (.23) (.30)

UK -.002 2.15% -2.30" 5.13" 2.02 (.003) (.54) (.70)

FR .003 2.95" 1,33" 2.75" 2.14 (.003) (.69) (.55)

* *

WG 002 75" -1.25 2.58 2.08 (.003) (.33) (.47)

SW 013" -.78 -3.36" 5.53" 1.94 (.005) (.78) (.82)

B. System Tests

Test F P-Value

b9= 0; all equations 2.20 .05

b5= 0; all equations 12.00 .000

$= 1; all equations 5.59 .000

5970, $,=1, §5=0 all equations 7.17 .000

Weighted R* = .24 April 1974 - December 1985

See Note in Table 1.

21

resultts are presented in Tables 2 and 3, exhibits Significant departures from the assumption of normally distributed regression residuals.

Taken together, Tables 1 - 3 demonstrate that risk premia on Eurocurrency bonds are incorporated into forward exchange rates. The data also suggest that risk premia on Eurocurrency bonds are not the only systematic component of the forward rate forecast errors. Past information is useful in explaining the time-varying discrepancy between forward and future spot exchange rates beyond that information's usefulness in

predicting real return differentials.

If.D. Interest Rates, The Forward Bias, and Different Time Periods

As discussed above, Korajczyk conducts a similar study over the period April 1974 through December 1980. For completeness, Tables 4 - 6 present the 3SLS estimates and system tests for this period. None of these equations has regression residuals significantly different from the normal distribution. Note that when equation <10> is estimated without 2. (Table 5), tne system tests do not reject the null hypothesis that b9= 0, §\= 1, and 55= O for all currencies. Once lagged values of the dependent variables, or the forward premia, are included for Ze. however, the joint null hypothesis that b= 0, 4= 1, and b5= 0 is rejected. In particular, Table 6 shows that the forward premium enters significantly in three out of five currencies. Omission of this variable biased Korajczyk'’s (1985) conclusions.

Tables 7 - 8 provide the system tests for the period January 1981

through December 1985 for all nine countries paired with the U.S. The

22

TABLE 4

UNRESTRICTED 3SLS

*

Stay > fe = 80 + Oy ead > Tea) + fea A. Estimates Country §5 by F(6)=0,6,=1) D.W. *

NE .001 .96 .1l 2.23 (.004) (.20)

UK 005 96" 1.38 1.76 (.003) (.47)

FR 002 1.97" 1.76 2.28 (.003) (.56)

WG 002 .09 3.32" 2.12 (.002) (.35)

SW .002 -.17 49 1.98 (.005) (1.29)

B. System Tests

Test F P-Value

bo= 0; all equations 71 .62

64: equal across equations 2.74 .03

64= 1; all equations 2.20 .05

o> 0, 64= 1; all equations 1.52 .13

eee

Weighted R? = .18

See Note in Table 1.

April 1974 - December 1980

23

TABLE 5

UNRESTRICTED 3SLS

~* ~ Seay 7 Fe = 80 + Oy ear > Tea) + 8p 6Se> Fe + fea A. Estimates Count:ry 89 $4 $5 F(55=0,6 =1, D.W. 6=0 * NE .002 . 36 .10 3.77 2.39 (.002) (.20) (.07) * * UK .004 . 83 24 3.27 2.16 (.003) (.44) (.09) FR 0022.27" ~—.03 2.14 2.22 (.003) (.55) (.07) wc 001 1.13% ~—.07 50 2.29 (.004) (.34) (.06) SW .002 .58 -.02 .12 1.99 (.005) (1.34) (.08) B. System Tests Test F P-Value 9= 0; all equations .55 .69 b5= (); all equations 2.02 .07 64= L; all equations 3.87 .002 5-0, 64-1, 55=0 all equations 2.20 .005 Weighted Rt = .16— April 1974 - December 1980

See Note in Table 1.

24

TABLE 6

UNRESTRICTED 3SLS

~* ~ Seay > fe 7 5g t+ 81% ear > Tear? + OoCFem Se) * feta A. Estimates Country $5 64 $5 POO oy D.W. 5 ee NE 004 47" -1.61" 8.05" 2.24 (.004) (.20) (.33) UK 001 1.02" -1.34 1.61 1.78 ¢(.005) (.45) (.92) FR .001 1.16 -.50 .60 2.28 (.003) (.68) (.65) * * WG .007 -.50 -1.73 5.23 2.11 ¢(.004) (.40) (.60) sw 023" -3,19% -4.21" 6.01" 1.89 (.007) (1.37) (1.11) a B. System Tests Test E P-Value 8 9= 0; all equations 2.20 .05 b5= 0; all equations 6.83 .000 $4= 1; all equations 4.06 .001 970, §,=1, §5=0 all equations 3.06 .000 a Weighted R? = .23 April 1974 - December 1980

See Note in Table 1.

25

results in these tables confirm the rejection of the hypothesis that only risk premia on Eurocurrency interest rates explain the deviations of forward rates from future spot prices.

The system of equations was tested using alternative estimation methodologies in order to examine the robustness of the conclusions. Since 3SLS yields inconsistent estimates of parameters in all equations if any equation is mis-specified, the system is estimated using two-stage least squaces (2SLS). "2SLS is not as efficient as 3SLS, but only the incorrectly specified equation is inconsistently estimated if misspecification is present in the system." (Hausman 1978, p. 1265) The conclusions from the 2SLS estimations do not differ from the 3SLS results in any estimation period. I also consider tax effects. Using Darby's (1975) modified Fisher equation instead of equation <6> adds two terms to

1

equation <10>, 0 but does not change the conclusions.

10. Darby (1975) modifies the Fisher equations so that: Ream EG yyy) + EGiyy) + [77-1 ECL)

(2.6") * ~* =* * * =k

* where r and r are the marginal tax rates in the U.S. and foreign country respectively. Given these equations, <l0> becomes

~ ~* ~ * * ~* <10"> Seuae Fe= Yo = WELT ead > Tega] + Wglt 7-7 VIEL)

- ygit/C-r) IEG) + 26 + Star:

The estimation is performed using an additional set of auxiliary

equations to generate inflation predictions (an AR(6) process is used). The null hypothesis on y,, 14: and Vy is unchanged, and the data reject the null on these parameters when the inflation variables are included.

26

TABLE 7 UNRESTRICTED 3SLS

All Nine Currencies Against the Dollar

Sea - fe 7 80 + 81 tar > Feat) + fee System Tests Test E P-Value b9= 0; all equations 1.50 L144 647 1; all equations 17.32 .0001 5=0, 6,=1, all equations 8.94 .0001 Weighted R’ = .35... January 1981 - December 1985

TABLE 8 UNRESTRICTED 3SLS

All Nine Currencies Against the Dollar *

Seat 7 Fe = 8 + 8, ear ~ Fear) + 8p (fer Se) + ead System Tests Test F P-Value 9= 0; all equations 3.67 .0002 65> 0; all equations 6.36 0001 64> 1; all equations 30.89 .0001 §5=0, 64-1, §5=0 all equations 12.94 .0001

>? > 2 or OTF TO oR Weighted RT = .41 January 1981 - December 1985

27

TABLE 9 UNRESTRICTED 3SLS

All Nine Currencies Against the Dollar

~* ~ St41 ~ P= 8 + 6 CELL, - Tea1) + 89 (8,- £4.) + S41 eee System Tests

Test F P-Value

$o= 0; all equations 1.49 .145

b5= 0; all equations 2.37 .048

64> :.; all equations 16.74 .0001

5570, 64-1, §5=0 all equations 5.95 .0001 eee

Weighted R? = .36 January 1981 - December 1985

28

IIIT. Summary and Conclusions

This paper investigates the observed time-varying discrepancy between forward and future spot exchange rates. In order to determine whether rejection of the UEH is due to risk premia, a theoretical framework is developed which yields restrictions on the parameters of a multivariate regression model. This framework is consistent with a host of international asset pricing models which assume that expected real exchange rate changes are zero. These models predict that the forecastable deviations of the forward exchange rate from the future spot exchange rate equals the expected real return differential on default free nominal bonds, and that past information, beyond the information's ability to forecast real interest rates, should not significantly explain the intertemporal variation of the forward forecast error. Using various econometric procedures and different estimation periods, the data reject these predictions. More specifically, the data demonstrate that while anticipated real interest rate differentials are reflected in forward exchange rates, expected real interest rate differences are not the only systematic component of forward forecast errors. In particular, forward premia are useful in explaining differences between forward and future spot prices beyond that information's ability to predict real return differentials.

Anticipated real exchange rate movements may explain the rejection of the hypothesis that risk premia on default free nominal bonds are the only source of bias in forward exchange rates. If agents anticipated real

exchange rate changes, these expectations will be incorporated into forward

29

1 . exchange rates. Future research into the nature of time-varying differences between forward and future spot prices should concentrate on constructing and estimating an intertemporal, international asset pricing

model which includes expected real exchange rate movements.

ll. At first glance, one may suggest incorporating ex ante real exchange rate movements into equation <10>, and testing whether or not it enters with a coefficient of one. This is done in Levine (1986). The data do not reject the hypothesis that the coefficients on expected real exchange rate changes and expected real interest rate differentials are equal to one and that past information, beyond its influence through these two variables, is useless in predicting the forward forecast error. There is, however, an econometric problem with this study because the coefiiicients on expected real return differences and anticipated real exchange changes are biased toward the null. The finding that past information does not enter significantly once these variables are incorporated, however, does support the belief that forward markets are efficient and that it is anticipated real exchange rate changes which leads) to rejecting the hypothesis that 65 = 0 in this paper.

30

REFERENCES

Adler, M. and B. Lehman, "Deviations from Purchasing Power Parity in the Long Run," Journal of Finance, 38, (December 1983), 1471-87.

Box, G.E.P. and G.S. Watson, "Robustness to Non-Normality of Regression Tests," Biometrika, 49, (1962), 93-106.

Darby, M.R., "Movements in Purchasing Power Parity: The Short and Long Runs," in M.R. Darby et al., The International Transmission of Inflation, NBER Monograph Series, Chicago: University of Chicago Press, 1983.

, "The Financial and Tax Effects of Inflation." Economic. Inquiry, 8, (June 1975), 266-76.

Cumby, R.E. and M. Obstfeld, "International Interest-Rate Linkages Under Flexible Exchange Rates: A Review of Recent Evidence," in J.F.0. Bilson and R.C. Marston, eds., Exchange Rates Theory and Practice, Chicago: University of Chicago Press for the NBER 1983.

Ericsson, N.R., "Post-simulation Analysis of Monte Carlo Experiments Interpreting Pesaran’s (1974) Study of Non-nested Hypothesis Test Statistics," Review of Economic Studies, 53, (1986), 691-707.

Fama, E.F., "Forward and Spot Exchange Rates," Journal of Monetary Economics, 14, (November 1984), 319-38.

"Inflation Uncertainty and Expected Returns on Treasury Bills," Journal of Political Economy, 84 (June 1976), 427-48.

, and M.R. Gibbons, "A Comparison of Inflation Forecasts," Journal of Monetary Economics, 13, (May 1984), 327-48.

Frankel, J., and K.A. Froot, "Using Survey Data to Test Standard Propositions Regarding Exchange Rate Expectations," American Zconomic Review, 77, (March 1987), 133-53.

Frenkel, J.A., and R.M. Levich, "Transactions Costs and Interest Arbitrage Tranquil Versus Turbulent Periods," Journal of Political Econony, 85, (December 1977), 1209-26.

Hansen, L.P., and R.J. Hodrick, "Risk Averse Speculation in the Forward Foreign Exchange Market: An Econometric Analysis of Linear Models," in Exchange Rates and International Macroeconomics, ed. J.A. Frenkel. Chicago: University of Chicago Press, 1983.

, and R.J. Hodrick, "Forward Exchange Rates and Optimal Predictors of Future Spot Rates: An Econometric Analysis," Journal of Political Economy, 88, (October 1980), 829-53.

31

Hausman, J.A., "Specification Tests in Econometrics," Econometrica, 46, (November 1978), 1251-71.

Hodrick, R.J., "International Asset Pricing with Time-Varying Risk

Premia," Journal of International Economics, 11, (November 1981), 573-87.

» and S. Srivastava, "An Investigation of Risk and Return

ir. Forward Foreign Exchange, "Journal of International Monetary ard Finance, 3, (April 1984), 5-29.

, "Foreign Currency Futures," urpublished ms., (September 1985).

Hsieh, D.A., "The Determination of the Real Exchange Rate: The

Productivity Approach," Journal of International Economics, (May 1982), 355-62.

Huizinga, J., and F. Mishkin, "Inflation and Real Interest Rates on Assets with Different Risk Characteristics," Journal of Finance, 39, (July 1984), 699-712.

Isard, P., "How Far Can We Push the Law of One Price?" American Economic Review, 67, (December 1977), 942-948.

Judge, G.G., and W.E. Griffiths, R.C. Hill, and T-C. Lee, The Theory and Practice of Econometrics, New York: John Wiley & Sons, 1980.

Koraczyk, R.A., "The Pricing of Forward Contracts for Foreign Exchange," Journal of Political Economy, 93, (April 1985), 346-68.

Kouri, P., "International Investment and Interest Rate Linkages under

Flexible Exchange Rates," in The Political Economy of Monetary Reform, ed. R. Aliber. London: Macmillan & Co., 1977.

Krasker, W.S., "The ‘Peso’ Problem in Testing the Efficiency of

Forward Exchange Markets," Journal of Monetary Economics, 6, (April 1980), 269-276.

Kravis, I. and R. Lipsey, "Price Behavior in the Light of Balance of Payment Theories," Journal of International Economics, 8,

(May 1978), 193-246.

Levich, R.M., "Empirical Studies of Exchange Rates: Price Behavior, Rate Determination and Market Efficiency," in Handbook of international Economics, Vol: 2, eds. R.W. Jones and P.B Kenen, Elsevier Science Publishers B.V., 1985.

Levine, R., "The Pricing of Forward Foreign Exchange Rates," unpublished ms., UCLA (October 1986).

, "An International Arbitrage Pricing Model with PPP

Deviations," International Finance Discussion Papers, No. 294, The Board of Governors of the Federal Reserve System, (Nov. 1986).

32

Lewis, K.K., "The Persistence of the "Peso Problem" when Policy is Noisy," Unpublished Manuscript, New York University Graduate School of Business, 1986.

Lucas, R.E., "Interest Rates and Currency Prices in a Two-Country World," Journal of Monetary Economics 10 (July 1982), 335-60.

McCormick, F., "Covered Interest Arbitrage: Unexploited Profits? Comment," Journal of Political Economy, 87, (April 1979), 411-17.

Merrick, J.J., and A. Saunders, "International Expected Real Interest Rates New Tests of the Parity Hypothesis and U.S. Fiscal Policy Effects," Journal of Monetary Economics, 18, (November 1986), 313-322.

Mishkin, F.S., "Are Real Interest Rates Equal Across Countries: An Empirical Investigation of International Parity Conditions," Journal of Finance, 39, (December 1984), 1345-57.

Obstfeld, M., "Peso Problems, Bubbles, and Risk in the Empirical Assessment of Exchange-Rate Behavior," Unpublished Manuscript, University of Pennsylvania, 1986.

Officer, L., Purchasing Power Parity and Exchange Rates: Theory,

Evidence and Relevance, Greenwich Conn.: JAI Press,1982.

Powell, F.C., Statistical Tables for the Social, Biological, and Physical Sciences, Cambridge: Cambridge University Press, 1982.

Riehl, H., and R.M. Rodriguez, Foreign Exchange and Money Markets, New York: McGraw-Hill Book Co., 1983.

Roll, R., "Violations of Purchasing Power Parity and Their Implications for Efficient International Commodity Markets," in M. Sarnat and G. Szego, eds., International Finance and Trade, vol. 1, Cambridge, MA: Ballinger, 1979.

Solnik, B., "International Arbitrage Pricing Theory," Journal of Finance, 38, (May 1983), 449-57.

Stulz, R.M., "A Model of International Asset Pricing," Journal of Financial Economics, 9, (December 1981), 383-406.

Wickens, M.R., "The Efficient Estimation of Econometric Models with Rational Expectations," Review of Economic Studies, 49, (August 1986), 55-67.

IFDP NUMBER

312

311

310

309

308

307

306

305

304

303

302

301

300

- 33 -

International Finance Discussion Papers

The Pricing of Forward Exchange Rates

Realignment of the Yen-Dollar Exchange Rate: Aspects of the Adjustment Process in Japan

The Effect of Multilateral Trade Clearinghouses on the Demand for International Reserves

Protection and Retaliation: Changing the Rules of the Game International Duopoly with Tariffs

A Simple Simulation Model of International Bank Lending

A Reassessment of Measures of the Dollar's Effective Exchange Value

Macroeconomic Instability of the Less Developed Country Economy when Bank Credit is Rationed

The U.S. External Deficit in the 1980s: An Empirical Analysis

An Analogue Model of Phase-Averaging Procedures

A Model of Exchange Rate Pass-Through

The Out-of-Sample Forecasting Performance of Exchange Rate Models When Coefficients are Allowed to Change

Financial Concentration and Development: An Empirical Analysis of the Venezuelan Case

AUTHOR (s )

Ross Levine Bonnie E. Loopesko

Robert E. Johnson

Ellen E. Meade

Catherine L. Mann Eric O'N. Fisher Charles A. Wilson

Henry S$. Terrell Robert S. Dohner

B. Dianne Pauls William L. Helkie

David F. Spigelman

William L. Helkie Peter Hooper

Julia Campos

Neil R. Ericsson David F. Hendry Eric O'N. Fisher Garry J. Schinasi

P.A.V.B. Swamy

Jaime Marquez Janice Shack-Marquez

Please address requests for copies to International Finance Discussion Papers, Division of International Finance, Stop 24, Board of Governors of the

Federal Reserve System, Washington, D.C.

20551.

- 34 -

International Finance Discussion Papers

IFDP NUMBER TITLES 1986

299 Deposit Insurance Assessments on Deposits at Foreign Branches of U.S. Banks

298 The International Debt Situation

297 The Cost Competitiveness of the Europaper Market

296 Germany and the European Disease

295 The United States International Asset and Liability Position: A Comparison of Flow of Funds and Commerce Department

294 An International Arbitrage Pricing Model with PPP Deviations

293 The Structure and Properties of the FRB Multicountry Model

292 Short-term and Long-term Expectations of the Yen/Dollar Exchange Rate: Evidence from Survey Data

291 Anticipated Fiscal Contraction: The Economic Consequences of the Announcement of Gramm-Rudman-Hollings

290 Tests of the Foreign Exchange Risk Premium Using the Expected Second Moments Implied by Option Pricing

289 Deposit Risk Pooling, Irreversible Investment, and Financial Intermediation

288 The Yen-Dollar Relationship: A Recent Historical Perspective

287 Should Fixed Coefficients be Reestimated Every Period for Extrapolation?

286 An Empirical Analysis of Policy Coordination in the U.S., Japan and Europe

285 Comovements in Aggregate and Relative

Prices: Some Evidence on Neutrality

AUTHOR(s)

Jeffrey C. Marquardt

Edwin M. Truman Rodney H. Mills

John Davis

Patrick Minford Guido E. vain der Ven John E. Wilson

Ross Levine

Hali J. Edison Jaime R. Marquez Ralph W. Tryon Jeffrey A. Frankel Kenneth A. Froot

Robert A. Johnson

Richard K. Lyons

Robert A. Johnson

Manuel H. Johnson Bonnie E. loopesko

P.A.V.B. Swamy Garry J. Schinasi

Hali J. Edison Ralph Tryon

B. Dianne Pauls