Chapter 13
Empirical Results
This chapter introduces the empirical results of the algorithms using the historical
market data. These results will demonstrate the effectiveness of these strategies
and provide confidence on their practicability in real trading. We also relax some
constraints to evaluate their capability in real trading scenarios.
This chapter is organized as follows. Section 13.1 conducts the experiments
to evaluate the cumulative wealth for all the algorithms. Section 13.2 shows the
experimental results of risk-adjusted returns. Section 13.3 measures the sensitivity
of parameters for these algorithms. Section 13.4 relaxes transaction costs and mar-
gin buying constraints. Section 13.5 compares the computational times for different
algorithms. Section 13.6 further analyzes the behaviors of the proposed algorithms.
Finally, Section 13.7 summarizes this chapter and proposes some future directions.
13.1 Experiment 1: Evaluation of Cumulative Wealth
First, we compare the performance of the competing approaches based on their cumu-
lative return, which is the main metric of this study. From the experimental results
shown in Table 13.1, we can draw several observations.
First of all, we observe that most online portfolio selection (OLPS) strategies gen-
erally perform better than the market and the best stock in a market, which indicates
that it is promising to investigate learning algorithms for portfolio selection. Second,
although the follow the winner approaches (UP, EG, and ONS) achieve higher cumu-
lative wealth than the market strategy, their performance is significantly less than that
of the follow the loser approach (Anticor) or the pattern matching–based strategies
(B
K
and B
NN
). Thus, to achieve better investment return, it is more powerful and
promising to exploit the latter two approaches. Third, on all original datasets (except
the DJIA dataset), the proposed strategies significantly outperform most competitors,
including Anticor, B
K
, and B
NN
, which are the state of the art. In particular, the pro-
posed algorithms sequentially beat existing strategies. For example, on the benchmark
dataset NYSE (O), the state-of-the-art performance is 3.35E+11 achieved by B
NN
.
Our proposed algorithms achieve much better performances of 1.48E+13, 5.14E+15,
6.51E+15, and 3.68E+16 for CORN, PAMR, CWMR, and OLMAR, respectively.
103
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104 EMPIRICAL RESULTS
Table 13.1 Cumulative wealth achieved by various trading strategies on the six datasets and
their reversed datasets
Algorithms NYSE (O) NYSE (N) TSE SP500 MSCI DJIA
Market 14.50 18.06 1.61 1.34 0.91 0.76
Best-stock 54.14 83.51 6.28 3.78 1.50 1.19
BCRP 250.60 119.81 6.78 4.07 1.51 1.24
UP 26.68 31.49 1.60 1.62 0.92 0.81
EG 27.09 31.00 1.59 1.63 0.93 0.81
ONS 109.19 21.59 1.62 3.34 0.86 1.53
Anticor 2.41E+08 6.21E+06 39.36 5.89 3.22 2.29
B
K
1.08E+09 4.64E+03 1.62 2.24 2.64 0.68
B
NN
3.35E+11 6.80E+04 2.27 3.07 13.47 0.88
CORN-U 1.48E+13 5.37E+05 3.56 6.35 26.10 0.84
CORN-K1 3.19E+13 1.94E+05 1.65 4.64 16.32 0.79
CORN-K2 6.10E+13 4.86E+05 1.74 9.12 80.41 0.82
PAMR 5.14E+15 1.25E+06 264.86 5.09 15.23 0.68
PAMR-1 5.13E+15 1.26E+06 260.26 5.08 15.51 0.69
PAMR-2 4.88E+15 1.36E+06 249.95 5.00 16.87 0.71
CWMR-Var 6.51E+15 1.44E+06 328.61 5.94 17.27 0.69
CWMR-Stdev 6.49E+15 1.41E+06 332.62 5.90 17.28 0.68
OLMAR-1 3.68E+16 2.54E+08 424.80 5.83 16.39 2.12
OLMAR-2 1.02E+18 4.69E+08 732.44 9.59 22.51 1.16
Algorithms NYSE (O)
−1
NYSE (N)
−1
TSE
−1
SP500
−1
MSCI
−1
DJIA
−1
Market 0.12 1.27 1.67 0.88 1.26 1.44
Best-stock 0.33 24.59 37.65 1.65 3.45 2.77
BCRP 2.86 56.60 58.61 1.91 3.45 2.98
UP 0.23 0.3 1.18 1.10 1.26 1.54
EG 0.22 0.38 1.21 1.08 1.27 1.53
ONS 0.84 1.01 1.62 2.97 1.73 2.35
Anticor 1.38E+03 4.26E+04 7.24 9.64 6.31 4.58
B
K
2.77E+07 162.74 8.81 1.01 4.47 1.43
B
NN
4.60E+09 3.57E+04 66.09 1.89 30.06 1.85
CORN-U 1.74E+10 8.01E+03 53.06 1.81 36.05 1.83
CORN-K1 4.99E+09 6.79E+03 12.88 1.67 17.87 1.75
CORN-K2 3.19E+10 7.27E+03 40.87 2.63 66.64 1.66
PAMR 2.03E+04 3.07E+04 2.67 7.42 40.33 6.61
PAMR-1 2.02E+04 3.09E+04 2.68 7.43 39.82 6.62
PAMR-2 2.11E+04 3.21E+04 2.75 7.32 39.83 6.65
CWMR-Var 1.67E+04 6.35E+04 4.04 8.09 40.46 6.90
CWMR-Stdev 1.66E+04 6.49E+04 4.05 8.07 40.42 6.91
OLMAR-1 2.07E+04 3.99E+07 2.90 18.40 42.25 9.56
OLMAR-2 3.41E+04 2.26E+07 7.04 40.99 51.51 8.80
Note: Numbers in bold indicate the best results on the corresponding datasets.
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EXPERIMENT 2 105
This observation supports the effectiveness of proposed algorithms, and, to the best
of our knowledge, no one has ever claimed such a fantastic performance.
Fourthly, it is promising to see that all pattern matching–based algorithms, espe-
cially CORN, have better performance than the benchmarks on most datasets. And
the proposed CORN significantly outperforms the existing pattern matching–based
algorithms, including B
K
and B
NN
, validating its motivating idea of improving the
matching process. Fifthly, the encouraging results achieved by the last three strategies
(PAMR, CWMR, and OLMAR) validate the importance of exploiting mean reversion
in financial markets via an effective learning algorithm.
In addition, we can see that most algorithms perform poorly on the DJIA
dataset, including CORN, PAMR, and CWMR. While the failure of CORN is still
unexplainable, the failure of the two mean reversion algorithms indicates that the
motivating (single-period) mean reversion may not exist in the dataset, as analyzed in
Sections 10.1 and 11.1.2. While OLMAR is proposed to explore (multiple-period)
moving average reversion, it can achieve much better performance, which thus
validates its motivating idea.
On the reversed datasets, though not as shiny as the original datasets, the pro-
posed algorithms also perform excellently. In all cases, the proposed algorithms not
only beat the benchmarks, including the market and BCRP, but also achieve the best
performance. Note that these reversed datasets are artificial datasets, which never
exist in real markets. However, algorithms’ behaviors on these datasets still provide
strong evidence that the proposed algorithms can effectively exploit the markets and
outperform the benchmarks and the state of the art.
Besides the final cumulative wealth, we are also interested in examining how the
cumulative wealth changes over the entire trading periods. Figure 13.1 shows the
trends of cumulative wealth by the proposed algorithms and four existing algorithms
(two benchmarks and two state-of-the-art algorithms). Note that PAMR and CWMR
almost overlap on the figures; thus, we only present the trends of PAMR. Clearly, the
proposed strategies consistently surpass the benchmarks and the competing strategies
over the entire trading periods on most datasets (except DJIA dataset), which again
validates the efficacy of the proposed techniques.
Finally, to measure whether such excess returns can be obtained by simple luck,
we conduct a statistical t-test as described in Section 12.4. Table 13.2 shows the
statistical results on the four proposed algorithms. The results clearly show that the
observed excess return is impossible to obtain by simple luck on most datasets. To be
specific, on datasets except DJIA, the probabilities for achieving the excess return by
luck are almost 0. On the DJIA dataset, though PAMR and CWMR have a probability
of 40% to achieve the excess return by luck, OLMAR has a probability of only 1.69%.
Nevertheless, the results show that the proposed strategies are promising and reliable
to achieve high returns with high confidence.
13.2 Experiment 2: Evaluation of Risk and Risk-Adjusted Return
We now evaluate the volatility risk and drawdown risk, and the risk-adjusted return in
terms of an annualized Sharpe ratio (SR) and Calmar ratio (CR). Figure 13.2 shows
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106 EMPIRICAL RESULTS
10
0
10
4
10
8
10
12
10
16
1 1500
(a)
3000 4500
Total wealth achieved
Trading days
Β
ΝΝ
Market
CORN
BCRP
PAMR
Anticor
OLMAR
10
0
10
3
10
6
10
9
1 1500 3000
(b)
4500 6000
Total wealth achieved
Trading days
Β
ΝΝ
Market
CORN
BCRP
PAMR
Anticor
OLMAR
10
0
10
1
10
2
10
3
1 300 600
(c)
900 1200
Total wealth achieved
Trading days
Β
ΝΝ
Market
CORN
BCRP
PAMR
Anticor
OLMAR
1
5
1 300 600 900 1200
Total wealth achieved
Trading days
(d)
Β
ΝΝ
Market
CORN
BCRP
PAMR
Anticor
OLMAR
1
3
9
27
1 300
(e)
600 900
Total wealth achieved
Trading days
Β
ΝΝ
Market
CORN
BCRP
PAMR
Anticor
OLMAR
1
2
1 250 500
Total wealth achieved
Trading days
(f)
Β
ΝΝ
Market
CORN
BCRP
PAMR
Anticor
OLMAR
Figure 13.1 Trends of cumulative wealth achieved by various strategies on the six datasets:
(a) NYSE (O); (b) NYSE (N); (c) TSE; (d) SP500; (e) MSCI; and (f) DJIA.
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EXPERIMENT 2 107
Table 13.2 Statistical t-test of the proposed algorithms on the six datasets
Algorithms Statistics NYSE (O) NYSE (N) TSE SP500 MSCI DJIA
Market Size 5651 6431 1259 1276 1043 507
MER 0.0005 0.0005 0.0004 0.0003 0.0000 −0.0004
CORN WR 53.78% 52.64% 51.55% 52.82% 62.22% 51.08%
MER 0.0058 0.0023 0.0014 0.0018 0.0033 −0.0002
α 0.0052 0.0017 0.0008 0.0014 0.0032 0.0002
β 1.2351 1.0552 1.5050 1.3096 0.8622 0.9047
t-statistics 13.8069 8.2069 1.2119 2.9073 9.8834 0.3466
p-value 0.0000 0.0000 0.1129 0.0019 0.0000 0.3645
PAMR WR 55.87% 51.75% 56.87% 53.37% 59.25% 51.87%
MER 0.0069 0.0026 0.0054 0.0017 0.0029 −0.0003
α 0.0063 0.0021 0.0049 0.0013 0.0029 0.0002
β 1.2095 1.1241 1.4982 1.2375 1.1177 1.2393
t-statistics 15.7829 5.9979 3.9241 2.0020 6.1358 0.2195
p-value 0.0000 0.0000 0.0000 0.0227 0.0000 0.4132
CWMR WR 56.17% 52.08% 56.00% 53.92% 59.44% 51.08%
MER 0.0070 0.0027 0.0057 0.0019 0.0030 −0.0003
α 0.0064 0.0021 0.0051 0.0015 0.0030 0.0002
β 1.2139 1.1325 1.5139 1.2512 1.1161 1.2476
t-statistics 15.9510 5.9496 3.9190 2.1806 6.4078 0.2482
p-value 0.0000 0.0000 0.0000 0.0147 0.0000 0.4020
OLMAR WR 56.91% 53.13% 55.12% 51.49% 58.39% 52.47%
MER 0.0074 0.0036 0.0061 0.0019 0.0030 0.0020
α 0.0068 0.0030 0.0056 0.0015 0.0030 0.0025
β 1.2965 1.1768 1.5320 1.2854 1.1763 1.2627
t-statistics 15.2405 7.3704 3.4583 1.9423 1.1763 1.2627
p-value 0.0000 0.0000 0.0003 0.0262 0.0000 0.0169
Note: MER denotes mean excess return, which equals the mean of daily returns over a risk-
free return. WR denotes winning ratio, which is the ratio of trading periods with a higher
return than the market.
the evaluation results on the six datasets. In addition to the proposed four algorithms,
we also plot two benchmarks (Market and BCRP) and two state-of-the-art algorithms
(Anticor and B
NN
). In particular, Figure 13.2a and 13.2b depicts the volatility risk
(standard deviation of daily returns) and the drawdown risk (maximum drawdown)
on the six stock datasets. Figure 13.2c and 13.2d compares their corresponding SRs
and CRs.
In the preceding results on cumulative wealth, we find that the proposed methods
achieve the highest cumulative return on most original datasets. However, high return
is associated with high risk, as no real financial instruments can guarantee high return
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