# Forward risk premia and the maturity of contracts: a note.

The growing importance of the forward exchange markets, following the
establishment of the flexible exchange rate system in 1973, triggered
interest in the study of the relationship between the spot and forward
rates of exchange. The earlier works tended to conclude that the forward
exchange rate is an unbiased predictor of the future spot rate, albeit
not an accurate one in the sense that it does not exhibit minimum
forecast error. The subsequent employment of more sophisticated testing
methods has resulted in the accumulation of a body of evidence which
cast serious doubt on the conclusions of the earlier works and the
unbiased predictor hypothesis. Included are the works of Domowitz and
Hakkio (1985), Hansen and Hodrick (1980), Chiang (1986,1988), and
Baillie and Bollerslev (1990).

Along with the empirical evidence, a number of explanations have been posited as to why the forward rate may not be an unbiased predictor of the future spot rate. Among these, the time- varying risk premia has received more attention: Lucas (1982), Hodrick and Srivastava (1984), and Frankel (1982) provided theoretical justification for the presence of the risk premia while Fama (1984), Wolff (1987) and Hodrick and Srivastava (1986) made significant inroad toward its empirical measurement. The essence of this explanation is that the observed forward rate ([F.sub.t]) contains not only the market expectation of the future spot rate ([S.sub.t+n]) but also a time- varying risk premium component ([P.sub.t]).(1)

More formally, following Fama (1984), the forward forecast error ([F.sub.t] - [S.sub.t+1]) and the change in the spot rate ([S.sub.t+1] - [S.sub.t]) can be expressed as:

[F.sub.t] - [S.sub.t+1] = [[Alpha].sub.1] + [[Beta].sub.1]([F.sub.t] - [S.sub.t]) + [[Epsilon].sub.1], and (1)

[S.sub.t+1] - [S.sub.t] = [[Alpha].sub.2] + [[Beta].sub.2]([F.sub.t] - [S.sub.t]) + [[Epsilon].sub.2] (2)

It can then be shown that if the two components of the forward rate are uncorrelated, [[Beta].sub.1] and [[Beta].sub.2] will measure the proportion of the variance of the forward premium which is due to the variance of the risk premium and that of the expected change in spot rate, respectively. If the two components are correlated, [[Beta].sub.1] and [[Beta].sub.2] will still indicate the proportions, but their precision will depend on the size of the covariance between the risk premium and the expected appreciation/depreciation of the currency. Assuming that the forward rate is rationally determined, [[Beta].sub.1] will be a direct measure of variation of the premium in the forward rate. Fama estimated this specification using the exchange rates of nine major industrial countries vis-a-vis the U.S. dollar during the 1973-82 period, and concluded that the covariance between the expected spot rate and the risk premium components of the forward price is negative, and that the variance of the risk premium component is much larger than its counterpart.(2)

This study is designed to extend Fama's work to investigate the impact of forward contract maturity on the proportion of risk premium in the total variation of the forward premium. We use OLS to estimate equation (1) for our sample of six currencies with three contract maturities for each to observe the effect of contract maturity on risk premium. Since the error terms across currencies may be correlated, we also use Zellner's Seemingly Unrelated Regression (SUR) technique, as a test of robustness of the results.(3)

The data used consists of the spot, 30-day, 60-day, and 90-day rates for the British pound (BP), Canadian dollar (CD), French franc (FF), Swiss franc (SF), German mark (DM), and Japanese yen (JY) for the 12-year period, January 2, 1976, through December 31, 1987.(4) To avoid serial correlation, due to the overlap of forward rate observations, we delete all observations between each spot rate and its matching forward rate. This procedure reduced our usable number of observations to 132, 64, and 41 for one-, two-, and three-month forward rates respectively.

Table 1 contains the sample autocorrelations and standard deviations for the changes in exchange rate ([S.sub.t+n]-[S.sub.t]), the forward prediction error ([F.sub.t]-[S.sub.t+n]), and the contemporaneous forward-spot differential ([F.sub.t]-[S.sub.t]). Autocorrelations for changes in exchange rate and spot-forward spread are close to zero for all currencies and maturities. However, the forward prediction error, ([F.sub.t]-[S.sub.t+n]), consisting of the risk premium and the rational forecast error does exhibit autocorrelation. Given that the latter component is a white noise, the observed autocorrelation can be attributed to the risk premium component. Consistent with Fama's results, we find the standard deviations of the forward perdiction errors are larger than those of the changes in the spot rates for similar maturity comparisons. Further, variabilities of both the forward prediction error and the change in spot increase with maturity. This, of course, is in line with the notion that both the current spot and the current forward rate lose their predictive capability with the increase in maturity length.

Table 2 contains both the OLS and the SUR regression results. [[Beta].sub.1] estimates, indicating the proportion of variation due to risk premium, are universally positive and reliably different from zero. Estimates exceeding unity imply that the variance of the premium is larger than the variance of expected change in the exchange rate. With OLS estimates, this is the case for all but two maturities of the CD and the 30-day maturities of the FF and the DM. Further, all OLS estimates indicate that the variability of the premium is significantly greater than the variability of the expected rate change. Additionally, except for the CD, the magnitude of [[Beta].sub.1] always increases with maturity, which indicates that the relative significance of the premium increases with the length of the contract's maturity. Finally, note that the utilization of the SUR methodology results in improved test results: [[Beta].sub.1] estimates exceed 1.0 for all maturities of the DM and the FF. However, the CD remains the exception with all estimates less than 1.0. Note also that with both the OLS and the SUR the standard errors of the [[Beta].sub.1] estimates increase with maturity.

[TABULAR DATA OMITTED]

Endnotes

1. The latter is a market-determined compensation for the risk-taking behavior of speculators who stand ready to absorb the excess supply of (demand for) the forward contract, above and beyond the matching of the hedgers on both sides of the market. The sign of this risk premium will depend on whether there exists an excess supply or demand. Accordingly, both the sign and the magnitude of the premium may change over time with the changes in the currency's characteristics and those of the markets in which it trades.

2. Subsequent studies used more sophisticated measures, only to reach similar conclusions. Wolff (1987) used a signal extraction approach to conclude that more than half of the variation in forward spot differential is accounted for by variation in risk premium component. Hodrick and Srivastava (1986) used three statistical methods including the Generalized Method of Moment to confirm Fama's basic findings that risk premium is negatively correlated with expected rate of depreciation.

3. The correlatedness may arise due to the fact that the U.S. dollar is common to all of these exchange rates, and also due to the fact that four of the currencies included are European and under the ERM arrangement.

4. The data were collected from various issues of the IMM Yearbook and its predecessor, The Chicago Mercantile Exchange Statistical Yearbook.

References

Baillie, R.T. and T. Bollerslev. 1990. A Multivariate Generalized ARCH Approach to Modeling Risk Premia in Forward Foreign Exchange Rate Markets. Journal of International Money and Finance 9 (September): 309-324.

Chiang, T.C. 1986. Empirical analysis on the Predictors of Future Spot Rates. The Journal of Financial Research 9 (Summer): 153-162.

Chiang, T.C. 1988. The Forward Rate as a Predictor of the Future Spot Rate - A Stochastic Coefficient Approach. Journal of Money, Credit and Banking 20 (May): 212-231.

Domowitz, I. and C.S. Hakkio. 1985. Conditional Variance and the Risk Premium in the Foreign Exchange Market. Journal of International Economics 14 (August): 47-66.

Fama, E.F. 1984. Forward and Spot Exchange Rates. Journal of Monetary Economics 14 (November): 319-338.

Frankel, J.A. 1982. In Search of the Exchange Risk Premium: A Six Currency Test Assuming Mean Variance Optimization. Journal of International Money and Finance 1 (December): 255-274.

Hansen, L.P. and R.J. Hodrick. 1980. Forward Exchange Rates as Optimal Predictors of Future Spot Rates: An Econometric Analysis. Journal of Political Economy 88 (October): 829-853.

Hodrick R.J. and S. Srivastava. 1984. An Investigation of Risk and Return in Forward Foreign Exchange. Journal of International Money and Finance 3 (April): 5-29.

Hodrick, R.J. and S. Srivastava. 1986. The Covariation of Risk Premiums and Expected Future Spot Exchange Rates. Journal of International Money and Finance 5 (March): 5-21.

Lucas, R.E., Jr. 1982. Interest Rates and Currency Prices in a Two- Country World. Journal of Monetary. Economics 10 (November): 335-359.

Wolff, C.C.P. 1987. Forward Foreign Exchange Rates, Expected Spot Rates and Premia: A Signal Extraction Approach. Journal of Finance 42 (June): 42-88.

Ali M. Fatemi Professor, Department of Finance, Kansas State University, Manhattan, KS 66506-0503

Amir Tavakkol Assistant Professor, Department of Finance, Kansas State University, Manhattan, KS 66506-0503

Along with the empirical evidence, a number of explanations have been posited as to why the forward rate may not be an unbiased predictor of the future spot rate. Among these, the time- varying risk premia has received more attention: Lucas (1982), Hodrick and Srivastava (1984), and Frankel (1982) provided theoretical justification for the presence of the risk premia while Fama (1984), Wolff (1987) and Hodrick and Srivastava (1986) made significant inroad toward its empirical measurement. The essence of this explanation is that the observed forward rate ([F.sub.t]) contains not only the market expectation of the future spot rate ([S.sub.t+n]) but also a time- varying risk premium component ([P.sub.t]).(1)

More formally, following Fama (1984), the forward forecast error ([F.sub.t] - [S.sub.t+1]) and the change in the spot rate ([S.sub.t+1] - [S.sub.t]) can be expressed as:

[F.sub.t] - [S.sub.t+1] = [[Alpha].sub.1] + [[Beta].sub.1]([F.sub.t] - [S.sub.t]) + [[Epsilon].sub.1], and (1)

[S.sub.t+1] - [S.sub.t] = [[Alpha].sub.2] + [[Beta].sub.2]([F.sub.t] - [S.sub.t]) + [[Epsilon].sub.2] (2)

It can then be shown that if the two components of the forward rate are uncorrelated, [[Beta].sub.1] and [[Beta].sub.2] will measure the proportion of the variance of the forward premium which is due to the variance of the risk premium and that of the expected change in spot rate, respectively. If the two components are correlated, [[Beta].sub.1] and [[Beta].sub.2] will still indicate the proportions, but their precision will depend on the size of the covariance between the risk premium and the expected appreciation/depreciation of the currency. Assuming that the forward rate is rationally determined, [[Beta].sub.1] will be a direct measure of variation of the premium in the forward rate. Fama estimated this specification using the exchange rates of nine major industrial countries vis-a-vis the U.S. dollar during the 1973-82 period, and concluded that the covariance between the expected spot rate and the risk premium components of the forward price is negative, and that the variance of the risk premium component is much larger than its counterpart.(2)

This study is designed to extend Fama's work to investigate the impact of forward contract maturity on the proportion of risk premium in the total variation of the forward premium. We use OLS to estimate equation (1) for our sample of six currencies with three contract maturities for each to observe the effect of contract maturity on risk premium. Since the error terms across currencies may be correlated, we also use Zellner's Seemingly Unrelated Regression (SUR) technique, as a test of robustness of the results.(3)

The data used consists of the spot, 30-day, 60-day, and 90-day rates for the British pound (BP), Canadian dollar (CD), French franc (FF), Swiss franc (SF), German mark (DM), and Japanese yen (JY) for the 12-year period, January 2, 1976, through December 31, 1987.(4) To avoid serial correlation, due to the overlap of forward rate observations, we delete all observations between each spot rate and its matching forward rate. This procedure reduced our usable number of observations to 132, 64, and 41 for one-, two-, and three-month forward rates respectively.

Table 1 contains the sample autocorrelations and standard deviations for the changes in exchange rate ([S.sub.t+n]-[S.sub.t]), the forward prediction error ([F.sub.t]-[S.sub.t+n]), and the contemporaneous forward-spot differential ([F.sub.t]-[S.sub.t]). Autocorrelations for changes in exchange rate and spot-forward spread are close to zero for all currencies and maturities. However, the forward prediction error, ([F.sub.t]-[S.sub.t+n]), consisting of the risk premium and the rational forecast error does exhibit autocorrelation. Given that the latter component is a white noise, the observed autocorrelation can be attributed to the risk premium component. Consistent with Fama's results, we find the standard deviations of the forward perdiction errors are larger than those of the changes in the spot rates for similar maturity comparisons. Further, variabilities of both the forward prediction error and the change in spot increase with maturity. This, of course, is in line with the notion that both the current spot and the current forward rate lose their predictive capability with the increase in maturity length.

Table 2 contains both the OLS and the SUR regression results. [[Beta].sub.1] estimates, indicating the proportion of variation due to risk premium, are universally positive and reliably different from zero. Estimates exceeding unity imply that the variance of the premium is larger than the variance of expected change in the exchange rate. With OLS estimates, this is the case for all but two maturities of the CD and the 30-day maturities of the FF and the DM. Further, all OLS estimates indicate that the variability of the premium is significantly greater than the variability of the expected rate change. Additionally, except for the CD, the magnitude of [[Beta].sub.1] always increases with maturity, which indicates that the relative significance of the premium increases with the length of the contract's maturity. Finally, note that the utilization of the SUR methodology results in improved test results: [[Beta].sub.1] estimates exceed 1.0 for all maturities of the DM and the FF. However, the CD remains the exception with all estimates less than 1.0. Note also that with both the OLS and the SUR the standard errors of the [[Beta].sub.1] estimates increase with maturity.

[TABULAR DATA OMITTED]

Endnotes

1. The latter is a market-determined compensation for the risk-taking behavior of speculators who stand ready to absorb the excess supply of (demand for) the forward contract, above and beyond the matching of the hedgers on both sides of the market. The sign of this risk premium will depend on whether there exists an excess supply or demand. Accordingly, both the sign and the magnitude of the premium may change over time with the changes in the currency's characteristics and those of the markets in which it trades.

2. Subsequent studies used more sophisticated measures, only to reach similar conclusions. Wolff (1987) used a signal extraction approach to conclude that more than half of the variation in forward spot differential is accounted for by variation in risk premium component. Hodrick and Srivastava (1986) used three statistical methods including the Generalized Method of Moment to confirm Fama's basic findings that risk premium is negatively correlated with expected rate of depreciation.

3. The correlatedness may arise due to the fact that the U.S. dollar is common to all of these exchange rates, and also due to the fact that four of the currencies included are European and under the ERM arrangement.

4. The data were collected from various issues of the IMM Yearbook and its predecessor, The Chicago Mercantile Exchange Statistical Yearbook.

References

Baillie, R.T. and T. Bollerslev. 1990. A Multivariate Generalized ARCH Approach to Modeling Risk Premia in Forward Foreign Exchange Rate Markets. Journal of International Money and Finance 9 (September): 309-324.

Chiang, T.C. 1986. Empirical analysis on the Predictors of Future Spot Rates. The Journal of Financial Research 9 (Summer): 153-162.

Chiang, T.C. 1988. The Forward Rate as a Predictor of the Future Spot Rate - A Stochastic Coefficient Approach. Journal of Money, Credit and Banking 20 (May): 212-231.

Domowitz, I. and C.S. Hakkio. 1985. Conditional Variance and the Risk Premium in the Foreign Exchange Market. Journal of International Economics 14 (August): 47-66.

Fama, E.F. 1984. Forward and Spot Exchange Rates. Journal of Monetary Economics 14 (November): 319-338.

Frankel, J.A. 1982. In Search of the Exchange Risk Premium: A Six Currency Test Assuming Mean Variance Optimization. Journal of International Money and Finance 1 (December): 255-274.

Hansen, L.P. and R.J. Hodrick. 1980. Forward Exchange Rates as Optimal Predictors of Future Spot Rates: An Econometric Analysis. Journal of Political Economy 88 (October): 829-853.

Hodrick R.J. and S. Srivastava. 1984. An Investigation of Risk and Return in Forward Foreign Exchange. Journal of International Money and Finance 3 (April): 5-29.

Hodrick, R.J. and S. Srivastava. 1986. The Covariation of Risk Premiums and Expected Future Spot Exchange Rates. Journal of International Money and Finance 5 (March): 5-21.

Lucas, R.E., Jr. 1982. Interest Rates and Currency Prices in a Two- Country World. Journal of Monetary. Economics 10 (November): 335-359.

Wolff, C.C.P. 1987. Forward Foreign Exchange Rates, Expected Spot Rates and Premia: A Signal Extraction Approach. Journal of Finance 42 (June): 42-88.

Ali M. Fatemi Professor, Department of Finance, Kansas State University, Manhattan, KS 66506-0503

Amir Tavakkol Assistant Professor, Department of Finance, Kansas State University, Manhattan, KS 66506-0503

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Author: | Fatemi, Ali M.; Tavakkol, Amir |
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Publication: | Review of Financial Economics |

Date: | Sep 22, 1992 |

Words: | 1587 |

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