- Email: [email protected]
The forward bias: is it a money tree?
Economics Letters 61 (1998) 373–379
The forward bias: is it a money tree? a
Kerk L. Phillips , Karl Snow
a ,b ,
*
a
b
Department of Economics, Brigham Young University, Provo, UT 84602 -2363, USA Department of Finance, Stockholm School of Economics, Box 6501, S-113 83 Stockholm, Sweden Received 26 January 1998; accepted 10 September 1998
Abstract This paper examines the forward discount anomaly, i.e. the fact that the forward exchange rate is a biased predictor of the future spot rate. We run a series of rolling regressions which we use to predict the value of the future spot rate based upon this bias. We show that the average return from an investment strategy based on the bias in forward exchange rates is in many cases insignificantly different from zero. In other cases, however, the return is significantly positive. Hence the in-sample bias does not necessarily lead to a money-making strategy for all currencies. 1998 Elsevier Science S.A. All rights reserved. Keywords: Exchange rates; Forecasting; Forward bias; International investment; Trading strategies JEL classification: F31; G14; G15
1. Introduction There is a wealth of literature documenting the bias in the forward exchange rate. One implication of this bias is that speculators in foreign currency markets ought to be able to use the forward premium to help forecast the return on taking a long or short position in foreign currencies. This ability to predict should lead to zero-cost strategies which yield significantly positive returns on average. In effect, the predictive power of the forward rate should give speculators a sort of money tree. Let ft be the one-period ahead forward exchange rate as of date t and e t be the spot exchange rate at date t, denote the forward premium as ft , the appreciation in the spot rate from t to t 1 1 as ´t11 and the return on a short investment in foreign exchange as Dt 11 :
ft ; ln ft 2 ln e t ,
(1a)
´t11 ; ln e t11 2 ln e t ,
(1b)
*Corresponding author. Correspondence address: Department of Finance, Stockholm School of Economics, Box 6501, S-113 83 Stockholm, Sweden. Tel.: 146-8-736-91-59; fax: 146-8-31-23-27; e-mail: [email protected] 0165-1765 / 98 / $ – see front matter 1998 Elsevier Science S.A. All rights reserved. PII: S0165-1765( 98 )00192-X
K.L. Phillips, K. Snow / Economics Letters 61 (1998) 373 – 379
374
Dt 11 ; ln ft 2 ln e t11 5 ft 2 ´t 11 .
(1c)
Tests of the bias in the forward rate are based on three key assumptions: covered interest rate parity (CIRP), uncovered interest rate parity (UIRP), and rational expectations (RE). These assumptions can be written as follows: i t 2 i t* 5 ft , CIRP,
(2a)
i t 2 i t* 5 Et h´t11 j, UIRP,
(2b)
´t 5 Et 21 h´t j 1 u t ; u t | iid(0,s u2 ), RE,
(2c)
where i t and i t* are the expected returns as of date t on risk-free investments between date t and t 1 1 in the home and foreign countries, respectively. Et h j is the conditional expectations operator using information as of date t. Combining Eqs. (2a)–(2c) gives:
´t 5 ft21 1 u t .
(3)
Using (1c) we can rewrite (3) as:
Dt 5 ft 21 2 ´t 5 u t .
(4)
Eq. (4) shows that under our three hypotheses the return on shorting foreign exchange follows a white noise process; there should be no information that can be used to predict its time-series behavior. The empirical literature on the bias in forward rates focuses on estimating the following:
´t 5 a 1 bft21 1 u t .
(5)
We subtract ft21 from both sides of Eq. (5) to get the return from shorting foreign exchange:
Dt 5 a 1 ( b 2 1)ft 21 1 u t .
(6)
The null hypothesis of [a 5 0 and b 5 1] is usually rejected because estimates of b are significantly less than one. There is a wealth of literature attesting to the in-sample bias of the forward exchange rate – see Engel (1996) and Froot and Thaler (1990) for good explanations and summaries of this literature. Eq. (6) shows that rejection of the null implies there is a forecastable component to the return. Specifically, if a is close to zero and b is substantially lower than one, then the return will be expected to be positive next period when the forward premium is negative today. Hence the bias in the forward exchange rate implies that investors ought to be able to systematically make money by taking either a short or long position in the forward exchange market based on the value of the forward premium. Note that this money-making opportunity does not depend upon assumptions (2a)–(2c), it is simply a property of the bias. We confirm the bias empirically by examining the return to taking either a short or long position in foreign exchange based upon the value of the forward premium. We obtain data on both spot and 3-month forward exchange rates for 16 currencies vis-a-vis the U.S. dollar. The data are reported monthly over a sample period that runs roughly from January 1972 through April 1996.
K.L. Phillips, K. Snow / Economics Letters 61 (1998) 373 – 379
375
We estimate a three-period forward version of (5) to obtain estimates of a and b :
´ 3t 5 a 1 bf 3t23 1 u t ,
(7)
where the superscript indicates the length of the forward contract. We use the estimates to form a forecast of D 3t based on the value of f 3t23 : Et23 hD t j 5 (1 2 bˆ )f t23 2 aˆ . 3
3
(8)
If the forecast is positive we take a short position, if it is negative we take a long one. This gives us a series of returns that for any particular observation is equal to either D 3t or 2 D 3t :
H
D t3 R 5 2 D 3t 3 t
if (1 2 bˆ )f t323 2 aˆ . 0,
(9)
otherwise.
The bias should and does manifest itself by making the average return significantly greater than zero. Table 1, which reports the mean values for the return series for all 16 countries and indicates significance from t-statistics for tests that the mean value is zero, confirms the bias in sample.1 Table 1 also reports the average value for 2 D 3t – the average return from always taking a long position in Table 1 Sample means Country
Sample
Means 2D
Great Britain Germany France Italy Canada Japan Austria Belgium Denmark Netherlands Norway Sweden Switzerland Finland Spain Australia a
1972:01–1994:12 1972:01–1996:04 1972:01–1994:12 1976:08–1996:04 1972:01–1994:10 1972:01–1996:04 1972:01–1996:04 1972:01–1996:04 1973:01–1996:04 1972:01–1990:12 1972:01–1996:04 1973:01–1996:02 1972:01–1996:04 1972:01–1989:10 1973:05–1996:04 1976:01–1996:04
Significant at 99% level of Significant at 95% level of c Significant at 90% level of Confidence levels are based b
1
3 t
0.0021 0.0030 0.0022 0.0077 20.0006 0.0075 0.0059 0.0064 0.0077 c 0.0035 0.0090 b 0.0023 0.0030 0.0096 b 0.0060 0.0049
t-Statistic R
3 t
0.0122 a 0.0160 a 0.0091 b 0.0116 b 0.0069 a 0.0241 a 0.0183 a 0.0116 b 0.0159 a 0.0247 a 0.0105 b 0.0036 0.0196 a 0.0141 a 0.0205 a 0.0103 b
20.6024 a 26.5930 a 14.3379 a 6.2114 a 41.2881 a 34.9111 a 27.5265 a 11.0049 a 17.2095 a 36.7539 a 3.7224 a 2.8158 a 31.7290 a 8.7279 a 30.2653 a 11.7339 a
confidence. confidence. confidence. on one-tailed tests using standard errors with a Newey–West correction for three lags.
t-Statistics are based on one-tailed tests using standard errors with a Newey–West correction for three lags.
376
K.L. Phillips, K. Snow / Economics Letters 61 (1998) 373 – 379
forward exchange regardless of any information contained in the forward premium. As can be seen this strategy yields lower average means which are in many cases significantly lower than the averages for R 3t . This apparent ability to make money is not a strategy that a speculator in foreign currency markets could use, however, since the estimates are based on the full sample period. This means that information from the future is used to help in predicting the appreciation in the exchange rate. A speculator would need to adopt a strategy of predicting out-of-sample. We test the out-of-sample money making ability of the forward bias by running a series of rolling regressions. We have time series observations from t 5 1 to t 5 T. To do this we estimate a and b using OLS over a fixed sample period between periods t 2 S and t, where S is the size of the fixed sample period. We then form an estimate of the 3-month ahead return to shorting the exchange rate for the last period in the sample. Our estimate is given by Eq. (8) replacing t with t 1 3. As before, if the estimate is negative we take a long position of foreign exchange for period t 1 3, if it is positive we go short. We then move the sample ahead one period to t 2 S 1 1 and t 1 1 and repeat the estimation. We do this for t 5 120 to t 5 T. We run these rolling regressions using sample sizes of 2, 3, 5 and 10 years. Similar to the rolling regressions, we also try an expanding sample size where we run the regression using observations 1 through t for t 5 120 to t 5 T. This methodology gives a series of returns the mean of which we analyze as before. The major difference between this series and the one evaluated in Table 1 is that the in-sample forecasts require data from the whole sample period in order to calculate estimates of a and b, whereas the rolling regressions use only the data available up through period t to form these estimates. As before we test to see if the means of these returns series are non-zero. The results of these tests are reported in Table 2. We start each set of rolling regressions at the same starting date (120 months after the first observation) so that the sample period for the returns is the same across window sample sizes. As can be seen the means and their significance vary considerably across countries and size of the sample window. Canada, Japan and the Netherlands are the only three countries where our investment strategy yields a significantly positive return regardless of the sample window. France, Italy, Belgium and Australia do not have any significantly positive returns. Wu and Hua (1996) show that the size of the forward bias depends on whether the forward premium is positive or negative. In order to incorporate this possible asymmetry we used the following regression to forecast and determine the appropriate position in the forward market:
´ 3t 5 a 1 b1 d t 23 f 3t 23 1 b2 (1 2 d t23 )f 3t23 1 u 3t ,
(10)
where d t is a dummy variable that is 1 when ft is positive and 0 otherwise. The results are not reported here since they are virtually identical to those in Table 2. This indicates that, although allowing for asymmetry may yield better forecasts of ´t , it does not yield a substantively different average return. The series of returns analyzed in Table 2 showed a significant degree of autocorrelation in most cases. Autocorrelation is to be expected given we have 3-month forward contracts observed at monthly frequencies. An unexpected shock occurring at any given point in time will disturb three different investments. Our returns appear to display autocorrelation over longer periods. We control for investors using any autocorrelation beyond the third lag to predict by including past values of Dt as regressors in the forecasting equation. The resultant regression is:
K.L. Phillips, K. Snow / Economics Letters 61 (1998) 373 – 379
377
Table 2 Means of returns based on forecasts from rolling regressions. Model: ´ 3t 5 a 1 bf t323 1 u 3t Country
Great Britain Germany France Italy Canada Japan Austria Belgium Denmark Netherlands Norway Sweden Switzerland Finland Spain Australia a
Nobs
150 150 150 93 150 150 166 166 153 102 166 151 166 87 150 117
Significant at 99% level of Significant at 95% level of c Significant at 90% level of Confidence levels are based b
Sample window (months) 24
36
60
120
0.0090 0.0176 a 0.0016 0.0004 0.0077 a 0.0102 c 0.0145 b 0.0032 0.0024 0.0207 a 0.0176 a 0.0023 0.0081 0.0127 c 0.0128 b 20.0003
0.0079 0.0161 b 0.0054 0.0042 0.0074 a 0.0148 b 0.0088 20.0028 0.0026 0.0274 a 0.0139 b 0.0042 0.0011 0.0111 0.0084 0.0022
0.0024 0.0108 c 0.0004 0.0067 0.0072 a 0.0232 a 0.0108 c 20.0019 0.0072 0.0240 a 0.0103 c 0.0084 0.0050 20.0047 0.0132 b 20.0007
0.0164 b 0.0091 20.0030 20.0079 0.0054 b 0.0218 a 0.0091 c 0.0038 0.0102 c 0.0297 a 0.0052 0.0026 0.0038 20.0063 0.0164 a 0.0038
Expanding
In-sample
0.0038 0.0144 b 20.0087 0.0088 0.0068 a 0.0191 a 0.0110 c 0.0027 0.0172 b 0.0276 a 0.0062 0.0171 a 0.0098 c 0.0018 0.0184 a 0.0049
0.0169 a 0.0242 a 0.0097 0.0126 c 0.0085 a 0.0273 a 0.0200 a 0.0146 b 0.0164 b 0.0353 a 0.0104 c 0.0118 c 0.0197 a 0.0145 b 0.0255 a 0.0183 a
confidence. confidence. confidence. on one-tailed tests using standard errors with a Newey–West correction for three lags.
´ 3t 5 a 1 b1 d t 23 f 3t 23 1 b2 (1 2 d t23 )f 3t23 1 b3 D 3t23 1 u 3t .
(11)
Table 3 reports the means of returns based on this forecasting equation. As can be seen the results do not differ appreciably from those in Table 2. The reasons for the predictability of returns in some countries and unpredictability in others are not clear. It is true that the estimates, aˆ and bˆ , are very unstable over time. However, this instability does not seem to differ between those countries with predictable returns and those without. Fig. 1 plots the estimates of bˆ for Canada, Japan and the Netherlands, the countries with robustly positive average returns. Fig. 2 plots them for France, Italy, Belgium and Australia, the countries which have no significantly positive average returns. In both cases the estimates of bˆ vary widely over time. We also note that this paper has made no attempt to control for riskiness. We can say that it is possible to make money on a risk-unadjusted basis for some countries. It may be, however, that this is a time-varying risk premium. If we could price the riskiness associated with the investment strategies we consider, then it is possible that significantly positive risk-adjusted average returns would not be observed. Alternatively, it could be that speculators do not form rational expectations of the future spot rate. This is the position that Frankel and Froot (1987) take. We leave the resolution of this interesting question for further research (see Phillips and Snow, 1998). To summarize, we find that the bias in the forward exchange rate is very strong as long as it is analyzed in-sample. Out-of-sample analysis indicates, however, that this bias does not lead to significant money-making opportunities in nearly as many countries. Despite the fact that all 16 of our
K.L. Phillips, K. Snow / Economics Letters 61 (1998) 373 – 379
378
Table 3 Means of returns based on forecasts from rolling regressions. Model: ´ 3t 5 a 1 b1 d t 23 f 3t 23 1 b2 (1 2 dt 23 )f t323 1 b3 D t323 1 u 3t Country
Great Britain Germany France Italy Canada Japan Austria Belgium Denmark Netherlands Norway Sweden Switzerland Finland Spain Australia a
Nobs
150 150 150 93 150 150 166 166 153 102 166 151 166 87 150 117
Significant at 99% level of Significant at 95% level of c Significant at 90% level of Confidence levels are based b
Sample window (months)
Expanding
24
36
60
120
0.0053 0.0117 c 0.0075 20.0106 0.0091 a 0.0070 0.0122 b 0.0030 20.0006 0.0128 c 0.0146 a 0.0040 0.0048 0.0121 c 0.0097 20.0016
0.0067 0.0117 c 0.0045 20.0118 0.0080 a 0.0118 b 0.0069 20.0042 0.0015 0.0186 b 0.0121 b 0.0083 0.0015 0.0135 b 0.0139 b 20.0007
20.0008 0.0137 b 0.0057 0.0019 0.0071 a 0.0134 b 0.0090 c 0.0006 0.0081 0.0153 b 0.0107 b 0.0100 c 0.0056 20.0018 0.0152 b 20.0061
0.0197 a 0.0121 c 0.0061 20.0032 0.0046 b 0.0232 a 0.0075 0.0058 0.0119 b 0.0313 a 0.0067 0.0050 0.0054 0.0056 0.0197 a 0.0079
20.0004 0.0080 0.0016 0.0051 0.0063 a 0.0167 a 0.0110 c 0.0062 0.0142 b 0.0227 a 0.0044 0.0112 c 0.0051 20.0035 0.0156 b 20.0036
confidence. confidence. confidence. on one-tailed tests using standard errors with a Newey–West correction for three lags.
Fig. 1.
K.L. Phillips, K. Snow / Economics Letters 61 (1998) 373 – 379
379
Fig. 2.
countries have significantly positive returns based on in-sample forecasts, only three, Canada, Japan and the Netherlands, appear to consistently yield significantly positive returns out-of-sample. Another four, France, Italy, Belgium and Australia, have no significantly positive returns. Our results indicate that if there is a money tree inherent in the forward exchange rate, it grows only in certain countries.
References Engel, C., 1996. The forward discount anomaly and the risk premium: a survey of recent evidence. Journal of Empirical Finance 3, 123–192. Frankel, J., Froot, K., 1987. Using survey data to test standard propositions regarding exchange rate expectations. American Economic Review 77, 133–153. Froot, K., Thaler, R., 1990. Anomalies: foreign exchange. Journal of Economic Perspectives 4, 179–192. Phillips, K., Snow, K., 1998. The forward bias: is it risk? Brigham Young University, mimeo. Wu, Y., Hua, Z., 1996. Asymmetry in forward exchange bias: a puzzling result. Economics Letters 50, 407–411.
The forward bias: is it a money tree? a
Kerk L. Phillips , Karl Snow
a ,b ,
*
a
b
Department of Economics, Brigham Young University, Provo, UT 84602 -2363, USA Department of Finance, Stockholm School of Economics, Box 6501, S-113 83 Stockholm, Sweden Received 26 January 1998; accepted 10 September 1998
Abstract This paper examines the forward discount anomaly, i.e. the fact that the forward exchange rate is a biased predictor of the future spot rate. We run a series of rolling regressions which we use to predict the value of the future spot rate based upon this bias. We show that the average return from an investment strategy based on the bias in forward exchange rates is in many cases insignificantly different from zero. In other cases, however, the return is significantly positive. Hence the in-sample bias does not necessarily lead to a money-making strategy for all currencies. 1998 Elsevier Science S.A. All rights reserved. Keywords: Exchange rates; Forecasting; Forward bias; International investment; Trading strategies JEL classification: F31; G14; G15
1. Introduction There is a wealth of literature documenting the bias in the forward exchange rate. One implication of this bias is that speculators in foreign currency markets ought to be able to use the forward premium to help forecast the return on taking a long or short position in foreign currencies. This ability to predict should lead to zero-cost strategies which yield significantly positive returns on average. In effect, the predictive power of the forward rate should give speculators a sort of money tree. Let ft be the one-period ahead forward exchange rate as of date t and e t be the spot exchange rate at date t, denote the forward premium as ft , the appreciation in the spot rate from t to t 1 1 as ´t11 and the return on a short investment in foreign exchange as Dt 11 :
ft ; ln ft 2 ln e t ,
(1a)
´t11 ; ln e t11 2 ln e t ,
(1b)
*Corresponding author. Correspondence address: Department of Finance, Stockholm School of Economics, Box 6501, S-113 83 Stockholm, Sweden. Tel.: 146-8-736-91-59; fax: 146-8-31-23-27; e-mail: [email protected] 0165-1765 / 98 / $ – see front matter 1998 Elsevier Science S.A. All rights reserved. PII: S0165-1765( 98 )00192-X
K.L. Phillips, K. Snow / Economics Letters 61 (1998) 373 – 379
374
Dt 11 ; ln ft 2 ln e t11 5 ft 2 ´t 11 .
(1c)
Tests of the bias in the forward rate are based on three key assumptions: covered interest rate parity (CIRP), uncovered interest rate parity (UIRP), and rational expectations (RE). These assumptions can be written as follows: i t 2 i t* 5 ft , CIRP,
(2a)
i t 2 i t* 5 Et h´t11 j, UIRP,
(2b)
´t 5 Et 21 h´t j 1 u t ; u t | iid(0,s u2 ), RE,
(2c)
where i t and i t* are the expected returns as of date t on risk-free investments between date t and t 1 1 in the home and foreign countries, respectively. Et h j is the conditional expectations operator using information as of date t. Combining Eqs. (2a)–(2c) gives:
´t 5 ft21 1 u t .
(3)
Using (1c) we can rewrite (3) as:
Dt 5 ft 21 2 ´t 5 u t .
(4)
Eq. (4) shows that under our three hypotheses the return on shorting foreign exchange follows a white noise process; there should be no information that can be used to predict its time-series behavior. The empirical literature on the bias in forward rates focuses on estimating the following:
´t 5 a 1 bft21 1 u t .
(5)
We subtract ft21 from both sides of Eq. (5) to get the return from shorting foreign exchange:
Dt 5 a 1 ( b 2 1)ft 21 1 u t .
(6)
The null hypothesis of [a 5 0 and b 5 1] is usually rejected because estimates of b are significantly less than one. There is a wealth of literature attesting to the in-sample bias of the forward exchange rate – see Engel (1996) and Froot and Thaler (1990) for good explanations and summaries of this literature. Eq. (6) shows that rejection of the null implies there is a forecastable component to the return. Specifically, if a is close to zero and b is substantially lower than one, then the return will be expected to be positive next period when the forward premium is negative today. Hence the bias in the forward exchange rate implies that investors ought to be able to systematically make money by taking either a short or long position in the forward exchange market based on the value of the forward premium. Note that this money-making opportunity does not depend upon assumptions (2a)–(2c), it is simply a property of the bias. We confirm the bias empirically by examining the return to taking either a short or long position in foreign exchange based upon the value of the forward premium. We obtain data on both spot and 3-month forward exchange rates for 16 currencies vis-a-vis the U.S. dollar. The data are reported monthly over a sample period that runs roughly from January 1972 through April 1996.
K.L. Phillips, K. Snow / Economics Letters 61 (1998) 373 – 379
375
We estimate a three-period forward version of (5) to obtain estimates of a and b :
´ 3t 5 a 1 bf 3t23 1 u t ,
(7)
where the superscript indicates the length of the forward contract. We use the estimates to form a forecast of D 3t based on the value of f 3t23 : Et23 hD t j 5 (1 2 bˆ )f t23 2 aˆ . 3
3
(8)
If the forecast is positive we take a short position, if it is negative we take a long one. This gives us a series of returns that for any particular observation is equal to either D 3t or 2 D 3t :
H
D t3 R 5 2 D 3t 3 t
if (1 2 bˆ )f t323 2 aˆ . 0,
(9)
otherwise.
The bias should and does manifest itself by making the average return significantly greater than zero. Table 1, which reports the mean values for the return series for all 16 countries and indicates significance from t-statistics for tests that the mean value is zero, confirms the bias in sample.1 Table 1 also reports the average value for 2 D 3t – the average return from always taking a long position in Table 1 Sample means Country
Sample
Means 2D
Great Britain Germany France Italy Canada Japan Austria Belgium Denmark Netherlands Norway Sweden Switzerland Finland Spain Australia a
1972:01–1994:12 1972:01–1996:04 1972:01–1994:12 1976:08–1996:04 1972:01–1994:10 1972:01–1996:04 1972:01–1996:04 1972:01–1996:04 1973:01–1996:04 1972:01–1990:12 1972:01–1996:04 1973:01–1996:02 1972:01–1996:04 1972:01–1989:10 1973:05–1996:04 1976:01–1996:04
Significant at 99% level of Significant at 95% level of c Significant at 90% level of Confidence levels are based b
1
3 t
0.0021 0.0030 0.0022 0.0077 20.0006 0.0075 0.0059 0.0064 0.0077 c 0.0035 0.0090 b 0.0023 0.0030 0.0096 b 0.0060 0.0049
t-Statistic R
3 t
0.0122 a 0.0160 a 0.0091 b 0.0116 b 0.0069 a 0.0241 a 0.0183 a 0.0116 b 0.0159 a 0.0247 a 0.0105 b 0.0036 0.0196 a 0.0141 a 0.0205 a 0.0103 b
20.6024 a 26.5930 a 14.3379 a 6.2114 a 41.2881 a 34.9111 a 27.5265 a 11.0049 a 17.2095 a 36.7539 a 3.7224 a 2.8158 a 31.7290 a 8.7279 a 30.2653 a 11.7339 a
confidence. confidence. confidence. on one-tailed tests using standard errors with a Newey–West correction for three lags.
t-Statistics are based on one-tailed tests using standard errors with a Newey–West correction for three lags.
376
K.L. Phillips, K. Snow / Economics Letters 61 (1998) 373 – 379
forward exchange regardless of any information contained in the forward premium. As can be seen this strategy yields lower average means which are in many cases significantly lower than the averages for R 3t . This apparent ability to make money is not a strategy that a speculator in foreign currency markets could use, however, since the estimates are based on the full sample period. This means that information from the future is used to help in predicting the appreciation in the exchange rate. A speculator would need to adopt a strategy of predicting out-of-sample. We test the out-of-sample money making ability of the forward bias by running a series of rolling regressions. We have time series observations from t 5 1 to t 5 T. To do this we estimate a and b using OLS over a fixed sample period between periods t 2 S and t, where S is the size of the fixed sample period. We then form an estimate of the 3-month ahead return to shorting the exchange rate for the last period in the sample. Our estimate is given by Eq. (8) replacing t with t 1 3. As before, if the estimate is negative we take a long position of foreign exchange for period t 1 3, if it is positive we go short. We then move the sample ahead one period to t 2 S 1 1 and t 1 1 and repeat the estimation. We do this for t 5 120 to t 5 T. We run these rolling regressions using sample sizes of 2, 3, 5 and 10 years. Similar to the rolling regressions, we also try an expanding sample size where we run the regression using observations 1 through t for t 5 120 to t 5 T. This methodology gives a series of returns the mean of which we analyze as before. The major difference between this series and the one evaluated in Table 1 is that the in-sample forecasts require data from the whole sample period in order to calculate estimates of a and b, whereas the rolling regressions use only the data available up through period t to form these estimates. As before we test to see if the means of these returns series are non-zero. The results of these tests are reported in Table 2. We start each set of rolling regressions at the same starting date (120 months after the first observation) so that the sample period for the returns is the same across window sample sizes. As can be seen the means and their significance vary considerably across countries and size of the sample window. Canada, Japan and the Netherlands are the only three countries where our investment strategy yields a significantly positive return regardless of the sample window. France, Italy, Belgium and Australia do not have any significantly positive returns. Wu and Hua (1996) show that the size of the forward bias depends on whether the forward premium is positive or negative. In order to incorporate this possible asymmetry we used the following regression to forecast and determine the appropriate position in the forward market:
´ 3t 5 a 1 b1 d t 23 f 3t 23 1 b2 (1 2 d t23 )f 3t23 1 u 3t ,
(10)
where d t is a dummy variable that is 1 when ft is positive and 0 otherwise. The results are not reported here since they are virtually identical to those in Table 2. This indicates that, although allowing for asymmetry may yield better forecasts of ´t , it does not yield a substantively different average return. The series of returns analyzed in Table 2 showed a significant degree of autocorrelation in most cases. Autocorrelation is to be expected given we have 3-month forward contracts observed at monthly frequencies. An unexpected shock occurring at any given point in time will disturb three different investments. Our returns appear to display autocorrelation over longer periods. We control for investors using any autocorrelation beyond the third lag to predict by including past values of Dt as regressors in the forecasting equation. The resultant regression is:
K.L. Phillips, K. Snow / Economics Letters 61 (1998) 373 – 379
377
Table 2 Means of returns based on forecasts from rolling regressions. Model: ´ 3t 5 a 1 bf t323 1 u 3t Country
Great Britain Germany France Italy Canada Japan Austria Belgium Denmark Netherlands Norway Sweden Switzerland Finland Spain Australia a
Nobs
150 150 150 93 150 150 166 166 153 102 166 151 166 87 150 117
Significant at 99% level of Significant at 95% level of c Significant at 90% level of Confidence levels are based b
Sample window (months) 24
36
60
120
0.0090 0.0176 a 0.0016 0.0004 0.0077 a 0.0102 c 0.0145 b 0.0032 0.0024 0.0207 a 0.0176 a 0.0023 0.0081 0.0127 c 0.0128 b 20.0003
0.0079 0.0161 b 0.0054 0.0042 0.0074 a 0.0148 b 0.0088 20.0028 0.0026 0.0274 a 0.0139 b 0.0042 0.0011 0.0111 0.0084 0.0022
0.0024 0.0108 c 0.0004 0.0067 0.0072 a 0.0232 a 0.0108 c 20.0019 0.0072 0.0240 a 0.0103 c 0.0084 0.0050 20.0047 0.0132 b 20.0007
0.0164 b 0.0091 20.0030 20.0079 0.0054 b 0.0218 a 0.0091 c 0.0038 0.0102 c 0.0297 a 0.0052 0.0026 0.0038 20.0063 0.0164 a 0.0038
Expanding
In-sample
0.0038 0.0144 b 20.0087 0.0088 0.0068 a 0.0191 a 0.0110 c 0.0027 0.0172 b 0.0276 a 0.0062 0.0171 a 0.0098 c 0.0018 0.0184 a 0.0049
0.0169 a 0.0242 a 0.0097 0.0126 c 0.0085 a 0.0273 a 0.0200 a 0.0146 b 0.0164 b 0.0353 a 0.0104 c 0.0118 c 0.0197 a 0.0145 b 0.0255 a 0.0183 a
confidence. confidence. confidence. on one-tailed tests using standard errors with a Newey–West correction for three lags.
´ 3t 5 a 1 b1 d t 23 f 3t 23 1 b2 (1 2 d t23 )f 3t23 1 b3 D 3t23 1 u 3t .
(11)
Table 3 reports the means of returns based on this forecasting equation. As can be seen the results do not differ appreciably from those in Table 2. The reasons for the predictability of returns in some countries and unpredictability in others are not clear. It is true that the estimates, aˆ and bˆ , are very unstable over time. However, this instability does not seem to differ between those countries with predictable returns and those without. Fig. 1 plots the estimates of bˆ for Canada, Japan and the Netherlands, the countries with robustly positive average returns. Fig. 2 plots them for France, Italy, Belgium and Australia, the countries which have no significantly positive average returns. In both cases the estimates of bˆ vary widely over time. We also note that this paper has made no attempt to control for riskiness. We can say that it is possible to make money on a risk-unadjusted basis for some countries. It may be, however, that this is a time-varying risk premium. If we could price the riskiness associated with the investment strategies we consider, then it is possible that significantly positive risk-adjusted average returns would not be observed. Alternatively, it could be that speculators do not form rational expectations of the future spot rate. This is the position that Frankel and Froot (1987) take. We leave the resolution of this interesting question for further research (see Phillips and Snow, 1998). To summarize, we find that the bias in the forward exchange rate is very strong as long as it is analyzed in-sample. Out-of-sample analysis indicates, however, that this bias does not lead to significant money-making opportunities in nearly as many countries. Despite the fact that all 16 of our
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Table 3 Means of returns based on forecasts from rolling regressions. Model: ´ 3t 5 a 1 b1 d t 23 f 3t 23 1 b2 (1 2 dt 23 )f t323 1 b3 D t323 1 u 3t Country
Great Britain Germany France Italy Canada Japan Austria Belgium Denmark Netherlands Norway Sweden Switzerland Finland Spain Australia a
Nobs
150 150 150 93 150 150 166 166 153 102 166 151 166 87 150 117
Significant at 99% level of Significant at 95% level of c Significant at 90% level of Confidence levels are based b
Sample window (months)
Expanding
24
36
60
120
0.0053 0.0117 c 0.0075 20.0106 0.0091 a 0.0070 0.0122 b 0.0030 20.0006 0.0128 c 0.0146 a 0.0040 0.0048 0.0121 c 0.0097 20.0016
0.0067 0.0117 c 0.0045 20.0118 0.0080 a 0.0118 b 0.0069 20.0042 0.0015 0.0186 b 0.0121 b 0.0083 0.0015 0.0135 b 0.0139 b 20.0007
20.0008 0.0137 b 0.0057 0.0019 0.0071 a 0.0134 b 0.0090 c 0.0006 0.0081 0.0153 b 0.0107 b 0.0100 c 0.0056 20.0018 0.0152 b 20.0061
0.0197 a 0.0121 c 0.0061 20.0032 0.0046 b 0.0232 a 0.0075 0.0058 0.0119 b 0.0313 a 0.0067 0.0050 0.0054 0.0056 0.0197 a 0.0079
20.0004 0.0080 0.0016 0.0051 0.0063 a 0.0167 a 0.0110 c 0.0062 0.0142 b 0.0227 a 0.0044 0.0112 c 0.0051 20.0035 0.0156 b 20.0036
confidence. confidence. confidence. on one-tailed tests using standard errors with a Newey–West correction for three lags.
Fig. 1.
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Fig. 2.
countries have significantly positive returns based on in-sample forecasts, only three, Canada, Japan and the Netherlands, appear to consistently yield significantly positive returns out-of-sample. Another four, France, Italy, Belgium and Australia, have no significantly positive returns. Our results indicate that if there is a money tree inherent in the forward exchange rate, it grows only in certain countries.
References Engel, C., 1996. The forward discount anomaly and the risk premium: a survey of recent evidence. Journal of Empirical Finance 3, 123–192. Frankel, J., Froot, K., 1987. Using survey data to test standard propositions regarding exchange rate expectations. American Economic Review 77, 133–153. Froot, K., Thaler, R., 1990. Anomalies: foreign exchange. Journal of Economic Perspectives 4, 179–192. Phillips, K., Snow, K., 1998. The forward bias: is it risk? Brigham Young University, mimeo. Wu, Y., Hua, Z., 1996. Asymmetry in forward exchange bias: a puzzling result. Economics Letters 50, 407–411.