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117 The parameters and the normalized residuals of the GARCH(1,1) process are estimated using the maximum likelihood procedure presented in Chapter 8. The simulated returns are generated from the simulated normalized residuals and the estimated parameters. The estimated parameters of the AR(p)GARCH(l,l) processes together with the simulated residuals are used to generate the simulated returns for this process. As before, half of the average spread is subtracted (added) from the simulated price process to obtain the simulated bid (ask) prices. For each replication we start by generating the simulated data a year before the model is tested. This year is 1989 and it is used to create the history dependency in returns and to initialize the different trading model indicators. Empirical Results The simulated data are the 5min #time series,11 from January 1,1990, to December 31,1996, for three major foreign exchange rates, USDDEM, USDCHF (Swiss Franc), USDFRF (French Franc), and the most liquid crossrate DEMJPY (Deutsche Mark  Japanese Yen). From Chapter 6, we know that highfrequency data inherits intraday seasonalities and require deseasonalization. We use for this study the deseasonalization methodology presented in Chapter 6. Our data set contains 671,040 observations per currency. The simulations for each currency and process are done for 1000 replications. Before discussing the details of different studies, we present in Table 11.6 results that substantiate the claims made at the beginning of this section. We give a summary of the pvalues of the main performance measures for the USDDEM, USDCHF, USDFRF, and DEMJPY. The pvalue12 represents the fraction of simulations generating a performance measure larger than the original. The methodology of this study places a historical realization in the simulated distribution of the performance measure under the assumed process and calculates its onesided pvalue.13 This indicates whether the historical realization is likely to be generated from this particular distribution or not. More important, it indicates The realtime system uses tickbytick data for its trading recommendations. The simulations in this study arc carried out with 5min data as it is computationally expensive to use the tickbytick data for the simulations. The historical performance of the currency pairs from the 5min series are within a few tenths of a percent for all performance measures with the performance of the realtime trading models utilizing the tickbytick data. Therefore, there is no loss of generality from the use of the 5min frequency for the simulations instead of the tickbytick feed. 12 The / value represents a decreasing index of the reliability of a result. The higher the pvalue, the less we can believe that the observed relation between variables in the samples is a reliable indicator of the relation between the respective variables in the population. Specifically, the /rvalue represents the probability of error that is involved in accepting our observed result as valid, that is, as representative of the population. For example, a pvalue of 0.05 indicates that there is a 5% probability that the relation between the variables found in our sample is purely coincidental. In other words, assuming that in the population there was no relation between those variables whatsoever, and by repeating the experiment, we could expect that in approximately every 20 replications of the experiment there would be one in which the relation between the variables in question would be equal or stronger than ours. In many areas of research, the pvalue of 5% is treated as a borderline acceptable level. 13 pvalue calculations reported in this study are the simulated pvalucs obtained from the distribution of 1000 replications of a given performance measure. For brevity, we simply refer to it as pvalue in the text.
TABLE 11.6 pvalue Comparisons. pvalue comparisons with random walk (RW), GARCH( 1,1), and AR(4)GARCH( 1,1). The pvalues are expressed in percentage. The definitions of the three performance measures are presented in Section 11.3. Currency   GARCH(l.l)  AR(4)GARCH(1,1)  Annual return     USDDEM     USDCHF     USDFRF     DEMJPY     Xeffective     USDDEM     USDCHF     USDFRF     DEMJPY     Reffective     USDDEM     USDCHF     USDFRF     DEMJPY    
whether the historical performance is likely to occur in the future. A small revalue (less than 5%) indicates that the historical performance lies in the tail of the distribution and the studied performance distribution is not representative of the datagenerating process, given that the trading model is a good one. If the process that generates the performance distribution is close to the datagenerating process of the foreign exchange returns, the historical performance would lie within two standard deviations of the performance distribution, indicating that the studied process may be retained as representative of the datagenerating process. Random Walk Process The results for the random walk process for USDDEM time series are reported in Table 11.7. The first and the second columns are the historical realization and the pvalue of the corresponding performance measures. The remaining columns report the 5th and the 95th percentiles, mean, standard deviation, skewness, and the kurtosis of the simulations. After the transaction costs, actual data with the USDDEM, USDCHF, USDFRF and DEMJPY yield an annualized total return of 9.63,3.66, 8.20, and 6.43%, respectively. The USDCHF has the weakest performance relative to the other three currencies. The Xeg and Reg performance of the USDDEM, USDFRF, and DEMJPY are all positive and range between 3 and 4%. For the USDCHF, the Xeff and Reg are 1.68 and 4.23%, reflecting the weakness of its performance.
TABLE 11.7 Random Walk Simulations for USDDEM. The second column presents the performance of the trading model with the actual data. The results under columns pvalue, percentile, mean, standard deviation, skewness, and kurtosis present the values of these statistics from 1000 replications with the random walk process computed every 5min for a period from 1990 to 1996. The pvalues are reported in percentage terms (e.g., 0.3 refers to 0.3%). The definitions of the performance measures are presented in Sections 11.2.2 and I 1.3. Description  Historical  pvalue  Percentile  Mean  St.Dev  Skew.  Kurt.   realization  (in %)  (5%, 95%)      Annual return  9.63   11.38, 4.03  3.44  4.74  0.09  0.13  Xcffcctivc  3.78   20.25, 4.14  12.11  5.09  0.13  0.23  Reffective  4.43   26.42, 7.70  16.80  5.90  0.03  0.20  Max drawdown  11.02  100.0  25.26, 94.86  53.79  21.36  0.71  0.21  Deal frequency  1.68  100.0  2.20, 2.71  2.46  0.16  0.10  0.19  Horizon:  7 days        Xeffective  3.47   19.65,4.24  11.83  4.76  0.08  0.15  Reffective  1.80   24.14, 7.21  15.51  5.20  0.05  0.15  Horizon:  29 days        Xeffective  3.27   20.21,4.36  12.10  4.95  0.07  0.23  Reffective  2.16   27.05, 8.07  17.45  5.91  0.02  0.28  Horizon:  117 days        Xeffective  4.07   20.85, 3.42  12.21  5.44  0.10  0.32  Reffective  5.10   31.01,6.53  18.10  7.49  0.26  0.25  Horizon:  301 days        Xeffective  4.62   23.37, 2.42  11.89  6.32  0.39  0.02  Reffective  6.83   27.85, 3.25  14.56  7.49  0.35  0.16 
The pvalues of the annualized return for the USDDEM, USDCHF, USDFRF, and DEMJPY are 0.3, 8.9, 1.2, and 2.1%, respectively. For the USDDEM and USDFRF, as reported in Table 11.6, the pvalues are less than the 2% level and it is about 2% for the USDCHF. In the case of the USDCHF, the /jvalue for the annualized return is 8.9, which is well above the 5% level. As indicated in Section 11.3, the annualized return only utilizes two points of the equity curve leaving a large degrees of freedom to infinitely many paths that would be compatible with a given total return. Xeg and Reg are more stringent performance measures, which utilize the entire equity curve in their calculations. The pvalues of Xeff and Reff are 0.0,0.0% for USDDEM, 0.7 and 0.6% for USDCHF, 0.2 and 0.1% for USDFRF, and 0.2 and 0.1% for DEMJPY. The pvalues for the Xeff and Reg are all less than 1 %, rejecting the null hypothesis that the random walk process is consistent with the datagenerating process of exchange rate returns. The maximum drawdowns for the USDDEM, USDCHF, USDFRF and DEMJPY are 11.02, 16.08, 11.36, and 12.03%. The mean maximum drawdowns
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