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Journal Article•DOI•

The Risk Premium, Exchange Rate Expectations, and the Forward Exchange Rate: Estimates for the Yen-Dollar Rate

Stuart Landon1, Constance E. Smith1•
University of Alberta1
01 Feb 2003-Review of International Economics (Blackwell Publishing Ltd.)-Vol. 11, Iss: 1, pp 144-158
TL;DR: In this paper, the forward rate is an unbiased predictor of the future spot rate and the rejection of this hypothesis could occur because market behavior is inconsistent with rational-expectations or because there exists a risk premium.
Abstract: The hypothesis that the forward rate is an unbiased predictor of the future spot rate has been rejected in many empirical studies. The rejection of this hypothesis could occur because market behavior is inconsistent with rational-expectations or because there exists a risk premium. Equations describing the forward premium and the change in the exchange rate are estimated jointly, and tests of both the rationalexpectations and no-risk-premium hypotheses are conducted. Empirical estimates, obtained using quarterly data for the yen‐dollar exchange rate, reject the rational-expectations hypothesis and suggest that there exists a time-varying risk premium.

Summary (3 min read)

Jump to: [1. Introduction] – [2. Theoretical Background] – [3. Empirical Implementation] – [Estimates of the Exchange Rate Forecasting Equations] – [4.4.2 Tests of the No Risk Premium Hypothesis] – [5. Concluding Comments] – [B. Rational Expectations Not Imposed] – [A: The Forecasting Equation] and [B: The Forward Premium Equation]

1. Introduction

  • The assumption of uncovered interest rate parity is widely used in both theoretical and empirical studies.
  • While these studies often find evidence of a time-varying risk premium, they do not attempt to model the risk premium as a function of observed economic variables.
  • The empirical results obtained, using quarterly data for the Japanese yen-U.S. dollar exchange rate, reject the restrictions associated with the rational expectations hypothesis and confirm that some information useful in predicting the exchange rate is not incorporated in the forward premium.
  • The data and empirical results are described in Section 4.

2. Theoretical Background

  • To determine whether forward rate unbiasedness has been rejected due to the presence of a time-varying risk premium or because market behaviour is inconsistent with rational expectations, it is necessary to employ a model that incorporates a time-varying risk premium.
  • The expected return on domestic and foreign currency bonds differs by a risk premium as a result of investor risk aversion.
  • The covered interest rate parity condition implies that the forward premium equals the difference between the returns on domestic and foreign currency bonds: ft - st = it d - it f, (2) where f is the log of the one-period forward exchange rate.

3. Empirical Implementation

  • As explained above, the hypothesis of forward exchange rate unbiasedness involves two assumptions: that expectations are rational, in the sense that the forecast of the change in the exchange rate utilizes all information that is useful in forecasting the exchange rate, and that there is no risk premium.
  • Hence, rational expectations can be tested by estimating equations (7) and (8) jointly and testing the cross-equation restrictions: β = b. (9) If this restriction is rejected, the expected exchange rate that enters the forward premium equation is not consistent with the process determining the exchange rate.
  • If the cross-equation restriction is not rejected, but this orthogonality condition is violated, market 8If the rational expectations cross-equation restrictions are imposed (b=β), the no risk premium restriction (γ=ι0) can be tested even if the elements of Vt and Zt are identical.
  • 11 considering for possible inclusion in the vector.
  • As with the simple forecast model that incorporates only lagged values of the exchange rate, a random walk forecast is unlikely to satisfy the orthogonality criteria of rational expectations.

Estimates of the Exchange Rate Forecasting Equations

  • Estimates of the first two exchange rate forecasting models described above, the "general" and "simple" models, respectively, are presented in Table 1.
  • As discussed in Section 3, it is possible to test the hypotheses of rational expectations and no risk premium by testing restrictions on the parameters of equations (7) and (8), the exchange rate forecasting and forward premium equations, respectively.
  • 14 As indicated in Table 2, the estimates explain a large proportion of the variation in the forward premium and the estimates provide no evidence of serial correlation (at the one percent significance level), heteroscedasticity, or structural change.
  • These restrictions imply that the elements of the parameter vector b in the exchange rate forecasting equation, equation (7), are equal to the elements of the parameter vector β in the forward premium equation, equation (8).
  • 17A test of the joint hypothesis of rational expectations and no time-varying risk premium is also easily rejected for both forecasting models.

4.4.2 Tests of the No Risk Premium Hypothesis

  • Part 2 of Table 3 reports the results of likelihood ratio tests of the hypotheses of no time- varying risk premium (equation (10)) and no constant or time-varying risk premium (equation (11)).
  • For similar reasons, an increase in the U.S. current account would be expected to lead to an increase in the risk premium on the yen.
  • Since the U.S. is Japan's largest trading partner, but Japan is not the largest trading partner of the U.S., in the U.S. case, there may be third-country effects of current account changes that are not being captured in this model.
  • 17 significant parameter estimates associated with the risk premium variables reported in Table 2.
  • To illustrate the importance of the risk premium in the determination of the forward premium, Figures 3A, 3B, and 3C graph the actual forward premium and the estimated risk premium, where the latter is calculated using the parameter estimates (α̂+γ̂NZt) from Table 2 associated with the general, simple and random walk forecast models.

5. Concluding Comments

  • The findings of this paper provide evidence that may help to explain the consistent empirical rejection of the hypothesis of forward exchange rate unbiasedness.
  • This allows for a direct test of rational expectations (market efficiency).
  • If market participants have rational expectations, then information useful in predicting the exchange rate should be incorporated in the forward premium.
  • The empirical results reported above reject the hypothesis of no time-varying forward exchange rate risk premium for three different specifications of the exchange rate forecasting equation, with and without rational expectations imposed.
  • These results are not sensitive to the form of the exchange rate forecasting equation.

B. Rational Expectations Not Imposed

  • That is, their parameters are elements of the vector γ. 22The Solnik (1974) model assumes investors have a one-period investment horizon.
  • A complete markets model which endogenizes equilibrium asset pricing, of the type presented in Lucas (1982), necessarily makes many simplifying assumptions (such as identical consumption preferences of investors across countries) which, as Lewis (1995) notes, makes this type of model less useful in explaining the empirical behaviour of the foreign exchange market.
  • AUS - U.S. equity values, proxied by an index of U.S. industrial share prices produced as a Laspeyres-type index by the Standard and Poors Corporation for 400 industrials on the New York Exchange, based on daily closing quotations.
  • The log of the three-month forward yen price of one U.S. dollar.

A: The Forecasting Equation

  • For each variable, the number of lags was reduced by sequentially eliminating the variable with the smallest t-statistic until all the included variables were significant using a 20 percent confidence interval.
  • To reduce the number of variables in this forecasting equation (so as to improve its efficiency and forecast accuracy), variables were eliminated sequentially starting with the variable associated with the smallest t-statistic.
  • The final form of the forecasting equation was determined when none of the excluded variables were significant when added individually back into the forecasting equation.

B: The Forward Premium Equation

  • The following methodology was used to determine the most parsimonious specification for all three versions of the forward premium equation (the versions associated with the general and simple forecast models as well as the random walk forecast).
  • Flood, Robert P., Peter M. Garber, and Charles Kramer (1996) "Collapsing exchange rate regimes: Another linear example," Journal of International Economics 41, 223-234.
  • "Testing the efficiency of the Canadian-U.S. exchange market under the assumption of no-risk premium," Journal of Finance 36, 43-49.
  • "Exchange rate expectations, the forward exchange rate bias and risk premia in target zones," Open Economies Review, 8:2 99-136.

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Munich Personal RePEc Archive
The risk premium, exchange rate
expectations, and the forward exchange
rate: Estimates for the Yen-Dollar rate
Landon, Stuart and Smith, Constance
University of Alberta, Department of Economics
6 April 1999
Online at https://mpra.ub.uni-muenchen.de/9775/
MPRA Paper No. 9775, posted 31 Jul 2008 05:47 UTC
The Risk Premium, Exchange Rate Expectations, and the
Forward Exchange Rate: Estimates for the Yen-Dollar Rate*
Stuart Landon and C.E. Smith
Department of Economics
8-14 Tory Building
University of Alberta
Edmonton, Alberta
Canada T6G 2H4
6 April 1999
Corresponding author:
C. E. Smith
Department of Economics
8-14 Tory Building
University of Alberta
Edmonton, Alberta
Canada T6G 2H4
Phone: (780) 492-5940
FAX: (780) 492-3300
e-mail: csmith@econ.ualberta.ca
*We thank an anonymous referee for helpful comments on an earlier draft of this paper.
The Risk Premium, Exchange Rate Expectations, and the
Forward Exchange Rate: Estimates for the Yen-Dollar Rate
Abstract
The forward rate is often used as the market's prediction of the future spot exchange rate
even though the hypothesis that the forward rate is an unbiased predictor of the future spot rate has
been rejected in a large number of empirical studies using data for different countries and time
periods. The rejection of this hypothesis could occur because market behaviour is inconsistent
with rational expectations or because of the existence of a risk premium. Existing studies test for
one or the other, but not both, of these factors. In this paper, equations describing the forward
premium and the change in the exchange rate are estimated jointly, and tests of both the rational
expectations and no risk premium hypotheses are conducted. The empirical estimates, obtained
using quarterly data for the yen-dollar exchange rate, reject the rational expectations hypothesis
and suggest that there exists a time-varying risk premium.
JEL Classification: F31, G14, G11
Keywords: risk premium, forward exchange rate, forward premium
1
See, for example, Dornbusch (1976), Bilson (1979), Frankel (1979), Driskill (1981),
Obstfeld and Rogoff (1984), Meese (1986), Papell (1989), Krugman (1991), Svensson (1992),
Goldberg (1994), Mark (1995), and Flood, Garber and Kramer (1996).
2
See the surveys of Boothe and Longworth (1986), Froot and Thaler (1990), Engel (1995) and
Lewis (1995).
1
1. Introduction
The assumption of uncovered interest rate parity is widely used in both theoretical and
empirical studies.
1
In conjunction with the covered interest rate parity condition, this assumption
implies that the forward exchange rate is equal to the market's prediction of the future spot
exchange rate. However, the hypothesis that the forward rate is an unbiased predictor of the future
spot rate has been rejected in a large number of studies using data for many different countries and
time periods.
2
Underlying the forward rate unbiasedness hypothesis are the assumptions of no risk
premium and rational expectations, where the latter implies that all information useful in
predicting the exchange rate is incorporated in the forward premium. The rejection of the
unbiasedness hypothesis, as reported in previous studies, indicates that one or both of these
assumptions is not consistent with market behaviour. Which of these assumptions does not hold
has different implications for the foreign exchange market. The rejection of rational expectations
suggests that markets are inefficient while the presence of a time-varying risk premium implies
that changes in macroeconomic variables, such as asset supplies, can alter relative asset yields
even if expectations are rational.
The aim of the analysis presented in this paper is to determine whether the forward rate
unbiasedness hypothesis has been rejected because market behaviour is inconsistent with rational
expectations or because there exists a time-varying risk premium. The existing literature on
3
Canova and Ito (1991) use a VAR model and Hai, Mark and Wu (1997) use the Kalman filter
to generate an exchange rate forecast. Using data for four European countries, Nessén (1997)
combines an estimate of movements of the exchange rate within a target zone with expectations
of changes in the target zone itself.
2
unbiasedness has generally examined the rational expectations hypothesis or the hypothesis of a
time-varying risk premium, but not both. For example, a large portion of the unbiasedness
literature employs empirical tests that are conditional on the assumption that there is no time-
varying risk premium. (See Bilson (1981), Longworth (1981), Fama (1984), and the surveys of
Boothe and Longworth (1986), Froot and Thaler (1990), Lewis (1995) and Engel (1995).)
Another strand of the unbiasedness literature imposes an exchange rate expectations
formation mechanism and then determines whether there exists a risk premium, where the risk
premium is calculated as the difference between the forward premium and the forecast change in
the exchange rate. Studies that employ survey data to obtain an exchange rate forecast include
Froot and Frankel (1989), MacDonald and Torrance (1990), Liu and Maddala (1992), Frankel and
Chinn (1993), Cavaglia, Verschoor and Wolff (1994) and Nieuwland, Verschoor and Wolff
(1998). Studies that use an estimating equation to derive expectations of the future spot exchange
rate include Canova and Ito (1991), Hai, Mark and Wu (1997) and Nessén (1997).
3
While these
studies often find evidence of a time-varying risk premium, they do not attempt to model the risk
premium as a function of observed economic variables. As a consequence, they do not provide
any information on whether movements in the risk premium are systematic, the reasons for these
movements, or the consistency of the calculated risk premium with theoretical models of the risk
premium. In addition, these studies cannot test the cross-equation restrictions that implicitly link
the exchange rate expectations process and the forward premium equation.
A related group of studies addresses some of these issues by examining whether
movements in the risk premium vary systematically with observed economic variables. The
Citations
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Abstract: A Rational Expectations Approach to Macroeconometrics pursues a rational expectations approach to the estimation of a class of models widely discussed in the macroeconomics and finance literature: those which emphasize the effects from unanticipated, rather than anticipated, movements in variables In this volume, Fredrick S Mishkin first theoretically develops and discusses a unified econometric treatment of these models and then shows how to estimate them with an annotated computer program

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Abstract: This paper explores the relationship between currency futures and realized spot rates for the Indian rupee US dollar exchange rate. Using futures contracts with maturities of one, two and three months, we examine the unbiasedness of futures quotes as a predictor of the spot exchange rate as well as the nature of time-varying risk premiums in this emerging market. Empirical estimates, obtained using monthly data, suggest that there is a significant time-varying risk premium in the considered futures market. We also find that the premium is of greater magnitude and more significant with increasing maturity of the futures contracts. We also examine the relationship between realized risk premiums and explanatory variables such as the spot rate, basis and realized volatility, skewness and kurtosis of the spot exchange rate. Our results show that the level of the spot rate as well as its realized volatility can be considered as significant determinants of realized risk premiums in the considered futures market.

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Frequently Asked Questions (6)
Q1. What are the contributions mentioned in the paper "The risk premium, exchange rate expectations, and the forward exchange rate: estimates for the yen-dollar rate" ?

In this paper, equations describing the forward premium and the change in the exchange rate are estimated jointly, and tests of both the rational expectations and no risk premium hypotheses are conducted. The empirical estimates, obtained using quarterly data for the yen-dollar exchange rate, reject the rational expectations hypothesis and suggest that there exists a time-varying risk premium. 

MacDonald and Taylor ( 1992 ), in their review of the forward market efficiency literature, note that there is a potential for future research that integrates exchange rate forecasts based on past trends with forecasts based on more fundamental factors. This, plus the finding that the forecast of the change in the exchange rate bears little relationship to the forward premium, suggests that most of the variation in the dollar-yen forward premium results from changes in the risk premium rather than from changes in the expected exchange rate. 

To capture the possibility of slow adjustment in the risk premium, four lags of the risk premium variables were initially included in the forward premium estimating equation. 

The rejection of the rational expectations cross-equation restrictions for the model thatemploys the general exchange rate forecast suggests that information useful in predicting the exchange rate is not being incorporated in the forward premium. 

By reporting both sets of test results it is possible to determine whether, if the no risk premium hypothesis is rejected when the rational expectations cross-equation restrictions are imposed, this rejection occurs because the rational expectations restrictions are inconsistent with the data. 

To determine whether forward rate unbiasedness has been rejected due to the presence of a time-varying risk premium or because market behaviour is inconsistent with rational expectations, it is necessary to employ a model that incorporates a time-varying risk premium.