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Fig. 1
The equity lines of the strategies selected by all the methods for SPXDAX - in-sampleSPX - S&P500 Index, DAX - Deutscher Aktienindex, ES, EHC, GD, DEM - equity line of the median strategy resulted from respectively Exhaustive Search, Extended Hill Climbing, Grid Method and Differential Evolution Method. Prices of both assets have been normalized in order to have initial value equal to 1000. The equity line has been calculated for the strategy working on daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 1998 to the end of 2013 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 2
The histograms of the reached optimization criterion and the execution time of EHC for SPXDAX – in-sampleOC - optimization criterion calculated as 100 * (%ARC * %ARC) / (%ASD * %MDD). The optimization criterion have been calculated from the sample of 1000 independent algorithm executions. The strategies have been working on the daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 1998 to the end of 2013 has been simulated, with assumption of fee equal to 0.25% of the position value.
Fig. 3
The histograms of the reached optimization criterion and the execution time of DEM for SPXDAX - in-sampleOC - optimization criterion calculated as 100 * (%ARC * %ARC) / (%ASD * %MDD). The equity lines have been calculated for the strategy working on daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 1998 to the end of 2013 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 4
The equity lines of the strategies selected by the all the methods for SPXDAX – out-of-sampleES, EHC, GD, DEM - equity line of the median strategy resulted from respectively Exhaustive Search, Extended Hill Climbing, Grid Method and Differential Evolution Method. Prices of both assets have been normalized in order to have initial value equal to 1000. The equity line has been calculated for the strategy working on daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 2014 to the end of 2017 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 5
The equity line of the strategy selected by all the methods for AAPLMSFT – in-sampleAAPL - Apple Inc. stock, MSFT - Microsoft Corp. stock, ES, EHC, GD, DEM - equity line of the median strategy resulted from respectively Exhaustive Search, Extended Hill Climbing, Grid Method and Differential Evolution Method. Prices of both assets have been normalized in order to have initial value equal to 1000. The equity line has been calculated for the strategy working on daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 1998 to the end of 2013 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 6
The histograms of the reached optimization criterion and the execution time of EHC for AAPLMSFT – in-sampleOC - optimization criterion calculated as 100 * (%ARC * %ARC) / (%ASD * %MDD). The optimization criterion has been calculated from the sample of 1000 independent algorithm executions. The strategies have been working on the daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 1998 to the end of 2013 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 7
The histograms of the reached optimization criterion and the execution time of DEM for AAPLMSFT – in-sample OC - optimization criterion calculated as 100 * (%ARC * %ARC) / (%ASD * %MDD). The optimization criterion have been calculated from the sample of 1000 independent algorithm executions. The strategies have been working on the daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 1998 to the end of 2013 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 8
The equity lines of the strategies selected by all the methods for AAPLMSFT – out-of-sampleAAPL - Apple Inc. stock, MSFT - Microsoft Corp. stock, ES, EHC, GD, DEM - equity line of the median strategy resulted from respectively Exhaustive Search, Extended Hill Climbing, Grid Method and Differential Evolution Method. Prices of both assets have been normalized in order to have initial value equal to 1000. The equity line has been calculated for the strategy working on daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 2014 to the end of 2017 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 9
The equity lines of the strategy selected by all the methods for HGFCBF – in-sampleHGF - High Grade Copper Futures, CBF - Crude Oil Brent Futures, ES, EHC, GD, DEM - equity line of the median strategy resulted from respectively Exhaustive Search, Extended Hill Climbing, Grid Method and Differential Evolution Method. Prices of the both assets have been normalized in order to have initial value equal to 1000. The equity line has been calculated for the strategy working on daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 1998 to the end of 2013 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 10
The histograms of the reached optimization criterion and the execution time of EHC for HGFCBF – in-sampleOC - optimization criterion calculated as 100 * (%ARC * %ARC) / (%ASD * %MDD). The optimization criterion has been calculated from the sample of 1000 independent algorithm executions. The strategies have been working on the daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 1998 to the end of 2013 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 11
The histograms of the reached optimization criterion and the execution time of DEM for HGFCBF – in-sampleOC - optimization criterion calculated as 100 * (%ARC * %ARC) / (%ASD * %MDD). The optimization criterion has been calculated from the sample of 1000 independent algorithm executions. The strategies have been working on the daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 1998 to the end of 2013 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 12
The equity lines of the strategies selected by the different methods for HGFCBF – out-of-sampleSPX - S&P500 Index, DAX - Deutscher Aktienindex, ES, EHC, GD, DEM - equity line of the median strategy resulted from respectively Exhaustive Search, Extended Hill Climbing, Grid Method and Differential Evolution Method. Prices of the both assets have been normalized in order to have initial value equal to 1000. The equity line has been calculated for the strategy working on daily frequency, investing 20% of capital in position on each asset with rebalancing every 5 trading days. Trading from the beginning of 2014 to the end of 2017 has been simulated, with the assumption of fee equal to 0.25% of the position value.
Fig. 13
The boxplot of the optimization criterion of strategies selected by the machine learning methods, as a percentage of the global maxima found by the Exhaustive SearchThe samples were denoted by the algorithm acronym and the number of trading case, so 1, 2 and 3 stands for respectively SPXDAX, AAPLMSFT and HGFCBF. The box plots present the empirical distribution quartiles and highlight the outliers. Half of the observations are inside the corresponding box, when the line inside marks the median. The observation was considered as an outlier and marked by a circle if the distance from both first and third quartile (from the nearest side of the box) was higher than 1.5 interquartile range. The range of observations, without outliers was marked by the whiskers. That type of box plot was often called the Turkey Box Plot. It was worth to notice that the box plots of the Grid Method results were just a line because the results of that method were deterministic.
Fig. 14
The boxplot of machine learnings methods’ computation time empirical distributionThe samples were denoted by the algorithm acronym and the number of trading case, so 1, 2 and 3 stands for respectively SPXDAX, AAPLMSFT and HGFCBF. The box plots presents the empirical distribution quartiles and highlight the outliers. A half of the observations are inside the corresponding box, when the line inside marks the median. The observation was considered as an outlier and marked by a circle if the distance from both first and third quartile (from the nearest side of the box) was higher than 1.5 interquartile range. The range of observations, without the outliers was marked by the whiskers. That type of box plot was often called the Turkey Box Plot. It was worth to notice that the box plots of Grid Method results were just a line because the results of that method were deterministic.
The summary of the reached optimization criterion and the execution time of methods for SPXDAX – in-sample
ES
EHC
GM
DEM
OC
Time [sec]
OC
Time [sec]
OC
Time [sec]
OC
Time [sec]
Minimum
77.79
35562.17
65.58
11.87
77.16
128.04
71.93
13.11
1st Quantile
77.79
35562.17
74.39
13.93
77.16
128.04
77.16
24.84
Median
77.79
35562.17
77.16
30.97
77.16
128.04
77.16
31.15
Mean
77.79
35562.17
75.94
43.1
77.16
128.04
77.34
42.73
2nd Quantile
77.79
35562.17
77.16
65.32
77.16
128.04
77.79
61.5
Max
77.79
35562.17
77.79
569.39
77.16
128.04
77.79
141.08
Standard deviation
0.00
0.00
2.61
48.77
0.00
0.00
0.36
24.06
The median strategies parameters and statistics resulted from all the methods for SPXDAX
In-sample
Out-of-sample
ES
EHC
GM
DEM
ES
EHC
GM
DEM
k1
60.00
100.00
100.00
100.00
60.00
100.00
100.00
100.00
k2
45.00
35.00
35.00
35.00
45.00
35.00
35.00
35.00
k1.2
65.00
45.00
45.00
45.00
65.00
45.00
45.00
45.00
k2.2
75.00
85.00
85.00
85.00
75.00
85.00
85.00
85.00
%ARC
4.27
3.92
3.92
3.92
-0.03
-0.62
-0.62
-0.62
%ASD
5.17
4.63
4.63
4.63
4.02
3.74
3.74
3.74
IR
0.83
0.85
0.85
0.85
-0.01
-0.17
-0.17
-0.17
%MDD
4.53
4.30
4.30
4.30
7.20
6.34
6.34
6.34
OC
77.79
77.16
77.16
77.16
0.00
-1.62
-1.62
-1.62
The descriptive statistics of the considered assets
In-sample
Out-of-sample
SPX
DAX
AAPL
MSFT
HGF
CBF
SPX
DAX
AAPL
MSFT
HGF
CBF
%ARC
3.92
4.79
35.22
6.32
9.41
12.14
9.67
8.07
22.68
25.70
-0.62
-11.01
%ASD
20.39
24.97
46.69
33.06
28.48
34.54
11.94
18.37
22.27
21.43
19.25
33.07
IR
0.19
0.19
0.75
0.19
0.33
0.35
0.81
0.44
1.02
1.2
-0.03
-0.33
%MDD
56.78
72.68
43.80
71.65
68.37
73.48
14.16
29.27
30.45
18.05
42.47
75.83
Mean and median optimization criterion reached by the different methods, referred to the ES method in percent – in-sample
ES
Grid
EHC median
DEM median
EHC mean
DEM mean
SPXDAX
100
99.19
99.19
99.19
97.62
99.42
AAPLMSFT
100
100.00
100.00
100.00
99.55
99.92
HGFCBF
100
88.94
88.94
88.94
89.65
90.74
The median strategy parameters and statistics resulted from all the methods for AAPLMSFT
In-sample
Out-of-sample
ES
EHC
GM
DEM
ES
EHC
GM
DEM
k1
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
k2
60.00
60.00
60.00
60.00
60.00
60.00
60.00
60.00
k1.2
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
k2.2
40.00
40.00
40.00
40.00
40.00
40.00
40.00
40.00
%ARC
17.79
17.79
17.79
17.79
1.37
1.37
1.37
1.37
%ASD
11.13
11.13
11.13
11.13
5.68
5.68
5.68
5.68
IR
1.60
1.60
1.60
1.60
0.24
0.24
0.24
0.24
%MDD
7.71
7.71
7.71
7.71
9.54
9.54
9.54
9.54
OC
368.83
368.83
368.83
368.83
3.49
3.49
3.49
3.49
The median strategies parameters and statistics resulted from all the methods for HGFCBF
In-sample
Out-of-sample
k1
ES 60.00
EHC 60.00
GM 60.00
DEM 60.00
ES 60.00
EHC 60.00
GM 60.00
DEM 60.00
k2
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
k1.2
50.00
30.00
30.00
30.00
50.00
30.00
30.00
30.00
k2.2
25.00
95.00
95.00
95.00
25.00
95.00
95.00
95.00
%ARC
8.18
9.53
9.53
9.53
-1.59
6.60
6.60
7.38
%ASD
8.16
9.83
9.83
9.83
7.09
8.17
8.17
8.07
IR
1.00
0.97
0.97
0.97
-0.22
0.81
0.81
0.91
%MDD
7.51
9.52
9.52
9.52
15.86
12.16
12.16
12.16
OC
109.11
97.04
97.04
97.04
-2.24
43.81
43.81
55.49
The summary of the reached optimization criterion and the execution time of the methods for HGFCBF – in-sample
ES
EHC
GM
DEM
OC
Time [sec]
OC
Time [sec]
OC
Time [sec]
OC
Time [sec]
Minimum
109.11
42193.57
77.29
12.29
97.04
113.75
97.04
9,76
1st Quantile
109.11
42193.57
93.62
14.73
97.04
113.75
97.04
19.92
Median
109.11
42193.57
97.04
33.09
97.04
113.75
97.04
23.08
Mean
109.11
42193.57
97.82
42.69
97.04
113.75
99.01
27.34
2nd Quantile
109.11
42193.57
109.11
36.85
97.04
113.75
97.04
29.15
Max
109.11
42193.57
109.11
622.17
97.04
113.75
109.11
110.93
Standard deviation
0.00
0.00
8.59
51.28
0.00
0.00
4.46
12.5
The summary of the reached optimization criterion and the execution time of methods for AAPLMSFT – in-sample
ES
EHC
GM
DEM
OC
Time [sec]
OC
Time [sec]
OC
Time [sec]
OC
Time [sec]
Minimum
368.83
32821.18
301.34
11.96
368.83
150.66
274.97
11.82
1st Quantile
368.83
32821.18
368.83
14
368.83
150.66
368.83
19.67
Median
368.83
32821.18
368.83
18.18
368.83
150.66
368.83
22.19
Mean
368.83
32821.18
367.16
27.4
368.83
150.66
368.55
22.71
2nd Quantile
368.83
32821.18
368.83
32.97
368.83
150.66
368.83
25.27
Max
368.83
32821.18
368.83
174.06
368.83
150.66
368.83
45.8
Standard deviation
0.00
0.00
5.51
18.71
0.00
0.00
5.14
4.52
Mean and median computation time of the methods, referred to the ES method in percent