Exchange Rate Models
Signal achievement

Research on the behaviour of exchange rates has highlighted the discrepancy between the forward rate and the corresponding realization of the spot rate. This difference can be interpreted as the sum of a premium reflecting risk and inflation and the market's error in expectations concerning the future spot rate. If the market forecasts spot rates efficiently, errors in expectations will be purely random and uncorrelated with their own past values. The remainder of the discrepancy, the premium, will vary over time in a partly systematic and partly stochastic fashion. The systematic component of the premium is of considerable interest for both theory and policy. Recent attempts to explain these premia have regressed the discrepancy between the forward and realized spot rates on a set of explanatory variables which represent the information available to the market, but the choice of explanatory variables in these studies has been somewhat arbitrary.

In Discussion Paper No. 189, Research Fellow Christian Wolff uses a different technique to analyse the discrepancies between 30-day forward rates and the corresponding spot rates for the dollar/Deutschmark, dollar/yen and dollar/sterling over the period 1973-84. Wolff separates the two components of the discrepancy using the Kalman filter, a signal-extraction technique widely used in engineering and elsewhere. This approach treats the observed series as a mixture of a systematic 'signal' and random 'noise' - the engineering analogy is a radio signal overlaid with hiss - and decomposes the observed series into an estimate of the signal (in this case the premia) and an estimate of the noise (the errors in expectations). The errors in expectations can be treated as noise provided that expectations are formed rationally. In order to apply the Kalman filter it is also necessary to specify the process governing the behaviour of the premium term, and Wolff assumes that these are generated by an autoregressive moving average (ARMA) process, whose structure he determines by examining the data.

Wolff then applies the Kalman filter to the series in order to extract the 'signal', i.e. the estimates of the premia. These estimates reveal premia of between 4% and 6% for the three series studied, and the dollar/sterling premium in particular shows substantial volatility. The premia for different currencies have apparently moved together: correlations between the estimated premia lie in the range 0.40-0.55. In addition, more than half the variation in the difference between the forward rate and the corresponding realization of the spot rate can be attributed to the premia.

In Discussion Paper No. 187, Wolff applies this technique to the analysis of a related question: the superiority of the current spot rate over the current forward rate as a predictor of the future spot rate. Having distinguished between expectational errors and risk premia, as in Discussion Paper No. 189, Wolff can then model the premia; predictions of the premia may in turn allow improved forecasts of the future spot rate. Wolff applies this approach to the same monthly data set used in his earlier paper. Each choice of parameter values in the Kalman filter generates a sequence of estimated premia, which can then be combined with the forward rate to forecast the future spot rate. Optimal values of the parameters in the Kalman filter can be chosen to minimize the root mean square of the errors in these forecasts.

Wolff finds that this technique does reduce the one-step-ahead errors in forecasting the future spot rate, even though the current spot rate is still a better predictor of the future spot rate. In the most successful premium models, changes in the premia account for between 9% and 16% of the variance of the one- step-ahead errors of forecasts using the unadjusted forward rate. Wolff concludes that although the premium models are inferior in predictive power to the simple random walk model, the contest is close: useful information concerning the spot exchange rates is captured by modelling the time-series properties of the premia. He notes that this methodology can be applied in a straightforward fashion to other financial markets, such as futures markets and markets for government debt instruments.

Forward Exchange Rates and Expected Future Spot Rates
Forward Foreign Exchange Rates, Expected Spot Rates, and Premia: A Signal-Extraction Approach
Christian Wolff

Discussion Paper Nos. 187 and 189, July 1987 (ATE)