TABLE 8.6 Forecasting performance for USD-DEM.

Forecasting accuracy of various models in predicting short-term market volatility. The performance is measured every hour over 5 years, from January 1, 1992, to December 31,1996, with 43,230 observations. In parentheses, the accuracy of rescaled forecasts is shown.

The summations (including AO in Equations 8.40, 8.41, and 8.43 are over all hours in the out-of-sample period. The number of independent observations is large so that the degrees of freedom of the calculated tests are sufficiently large. Performance measures based on squares such as the signal correlation or squared errors are not used because our interest is in squared returns and the fourth moment of the distribution of returns may not be finite, as discussed in Section 5.4.2.

8.4.2 Performance of ARCH-Type Models

In Table 8.6, the results for the different performance measures are presented for the most traded FX rate, USD-DEM, for the static and dynamic optimizations. In parentheses, the results for the scaled forecasts are presented. For all measures, three parameter models perform better than the benchmark and the EMA-HARCH performs the best. The forecast accuracy is remarkable for all ARCH-type models. In more than two-thirds of the cases, the forecast direction is correctly predicted and the mean absolute errors are smaller than the benchmark errors for all models. The realized potential measure shows that the forecast of volatility change is accurate not only for small .sy but also for large ones. The condition expressed in Equation 8.42 is always satisfied for all models. Neither the scaled forecast nor the dynamic optimization seems to significantly improve the forecasting accuracy. The realized potential Qr is the only measure that consistently improves with

dynamic optimization. Examining the model coefficients computed in moving samples shows that they oscillate around mean values. No structural changes in the coefficients were detected. The accuracy improvement in Qr together with the loss in Qf in the case of dynamic optimization indicates that the prediction of large movements is improved at the cost of the prediction of direction of small real movements. From the point of view of forecasting short-term volatility, the EMA-HARCH is the best of the models considered here and compares favorably to HARCH. Similar conclusions can be drawn from the results for four other FX rates.11 The cross rate JPY-DEM presents results slightly less accurate than the other currencies, but it should be noted that the early half of the sample has been synthetically computed from USD-DEM and USD-JPY. This may lead to noise in the computation of hourly volatility and affect the forecast quality.

1 The interested reader will find them in Dacorogna et al. (1998b), where similar tables are listed for USD-JPY, GBP-USD, USD-CHF, and DEM-JPY.

FORECASTING RISK AND RETURN

9.1 introduction to forecasting

This section examines forecasting models for different variables. The predicted variable should be observable, so the forecasts and the true variable values can be compared in the future to allow for statistical forecast quality tests. The following variables can be predicted:

The absolute size of future returns. This can be done in different mathematical forms, one of them using realized volatility as defined in Section 3.2.4. We ignore the direction of the future price returns here and assume their probability distribution function to be symmetric by default. Under this assumption, the chosen forecast variable also measures the risk of holding a position in the financial asset.

The future return over the forecast period, including its sign. This also implies a price forecast because the future price is the price now plus the future return.

A full probability distribution function of future returns. This is the most comprehensive goal. Given the natural uncertainty of forecasting, we are rarely able or willing to forecast details of the distribution function, and we are more than happy to have good forecasts of its center and its width.