The Risk Premium, Exchange Rate Expectations, and the Forward Exchange Rate: Estimates for the Yen-Dollar Rate
Summary (3 min read)
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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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.
Q2. What have the authors stated for future works in "The risk premium, exchange rate expectations, and the forward exchange rate: estimates for the yen-dollar rate" ?
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.
Q3. How many lags of the risk premium variables were initially included in the forward premium estimating?
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.
Q4. What is the rational expectations hypothesis rejected for the exchange rate forecast?
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.
Q5. What is the reason why the no risk premium hypothesis is rejected?
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.
Q6. Why is the forward rate unbiasedness rejected?
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.