EXPERIMENT 6 123
Table 13.5 Some descriptive statistics on the NYSE (O) dataset
Stat. 1234567 8 9
Cum 13.10 4.35 16.10 16.90 13.36 52.02 8.76 3.07 13.71
Mean 1.0005 1.0004 1.0006 1.0006 1.0006 1.0010 1.0005 1.0003 1.0011
Std 0.0135 0.0171 0.0128 0.0170 0.0140 0.0257 0.0156 0.0136 0.0371
Ac 0.1344 0.1206 0.0817 0.0952 0.0927 0.0378 0.0615 0.0479 0.0217
10 11 12 13 14 15 16 17 18
Cum 14.16 10.70 6.85 7.86 6.75 7.64 32.65 30.61 12.21
Mean 1.0005 1.0006 1.0005 1.0005 1.0004 1.0004 1.0009 1.0008 1.0005
Std 0.0115 0.0175 0.0159 0.0138 0.0137 0.0130 0.0224 0.0202 0.0134
Ac 0.1114 0.1312 0.0455 0.0766 0.0637 0.0744 0.0449 0.0626 0.0064
19 20 21 22 23 24 25 26 27
Cum 4.81 8.92 17.22 10.36 4.13 6.21 4.31 22.92 14.43
Mean 1.0004 1.0010 1.0006 1.0005 1.0015 1.0004 1.0005 1.0010 1.0006
Std 0.0146 0.0346 0.0151 0.0148 0.0505 0.0149 0.0230 0.0313 0.0142
Ac 0.1301 0.0243 0.0956 0.1047 0.2089 0.0042 0.0226 0.0915 0.1002
28 29 30 31 32 33 34 35 36
Cum 5.98 15.21 54.14 6.98 16.20 43.13 4.25 6.54 5.39
Mean 1.0004 1.0006 1.0008 1.0004 1.0006 1.0008 1.0004 1.0005 1.0004
Std 0.0139 0.0161 0.0153 0.0117 0.0159 0.0174 0.0143 0.0178 0.0143
Ac 0.0858 0.0697 0.1004 0.0870 0.0880 0.1024 0.0873 0.0626 0.0257
Note: “Cum” denotes the cumulative return (product of all price relatives) of an asset. “Mean” refers to one asset’s
arithmetic mean, and “Std” denotes the asset’s standard deviation. “Ac” denotes the autocorrelation (with lag 1) of an
asset. Numbers in bold denote the top five in the corresponding rows.
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124 EMPIRICAL RESULTS
The three pattern matching–based approaches (B
K
,B
NN
, and CORN) have similar
patterns in their allocation weights, while their top five allocations vary. In general,
their volatilities are much higher than EG and ONS. Their concentration on asset
#23, which has the highest weight, confirms the observation (Györfi et al. 2006;
–0.2
0
0.2
0.4
0.6
0
10
(a)
20 30
Portfolio mean and std
Asset #
0.025
0.028
0.03
0.032
0.035
0
10
(b)
20 30
Portfolio mean and std
Asset #
–0.2
0
0.2
0.4
0.6
010
(c)
20 30
Portfolio mean and std
Asset #
–0.2
0
0.2
0.4
0.6
0.8
0
10 20 30
Portfolio mean and std
Asset #
(d)
–0.2
0
0.2
0.4
0.6
0.8
0 10
(e)
20 30
Portfolio mean and std
Asset #
–0.2
0
0.2
0.4
0.6
0.8
0
10
(f)
20 30
Portfolio mean and std.
Asset #
Figure 13.12 Distributions of portfolio weights. The x-axis denotes indices of assets, and the
y-axis is each asset’s average weight. For each asset, the center of an error bar denotes its
portfolio mean (over 5651 trading days), and vertical lines denote its standard deviations:
(a) BCRP; (b) EG; (c) ONS; (d) B
K
; (e) B
NN
; and (f) CORN. (Continued)
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EXPERIMENT 6 125
–0.2
0
0.2
0.4
0.6
0
10
(g)
20 30
Portfolio mean and std
Asset #
–0.2
0
0.2
0.4
0.6
0
10 20 30
Portfolio mean and std
Asset #
(h)
–0.2
0
0.2
0.4
0.6
010
(i)
20 30
Portfolio mean and std
Asset #
Figure 13.12 (Continued) Distributions of portfolio weights. The x-axis denotes indices of
assets, and the y-axis is each asset’s average weight. For each asset, the center of an error
bar denotes its portfolio mean (over 5651 trading days), and vertical lines denote its standard
deviations: (g) Anticor; (h) PAMR; and (i) OLMAR.
Li et al. 2011a) that the asset is important in all these approaches. Moreover, the
increasing top five weights, which indicate more active exploitations, may lead to
their increased performance. However, their volatilities also show that the subsets of
assets are changing from day to day, which is inconvenient from the point of view
of transaction costs. Anyway, such observations confirm that their pattern-matching
process is improving and validate CORN’s motivation.
The three mean reversion algorithms (Anticor, PAMR, and OLMAR) generally
concentrate on the top five volatile stocks, as shown in Figure 13.12g through i and
Table 13.6, while their orders may vary. SinceAnticor, PAMR/CWMR, and OLMAR,
in general, achieve the best performance on most other datasets, we also plot their
average allocations in Table C.5,
in Appendix C. From the figure and tables, we can
have several observations. First, similar to the pattern matching–based approaches,
these algorithms have much higher volatilities than EG or ONS. However, different
from the pattern matching–based algorithms, which only have higher volatilities on
We ignore their corresponding figures, which are similar to Figure 13.12.
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126 EMPIRICAL RESULTS
Table 13.6 Top five (average) allocation weights of some strategies on NYSE (O)
Asset # 6 23 9 26 20 Asset # 23 6 20 9 16
BCRP 0.28 0.25 0.20 0.18 0.09 EG 0.032 0.030 0.029 0.029 0.029
Asset # 8 35 2 3 22 Asset # 23 20 26 33 9
ONS 0.25 0.17 0.13 0.07 0.06 B
K
0.21 0.11 0.08 0.07 0.06
Asset # 23 20 9 6 26 Asset # 23 9 26 6 20
B
NN
0.21 0.15 0.08 0.08 0.08 CORN 0.38 0.09 0.09 0.09 0.08
Asset # 20 23 9 26 6 Asset # 23 20 9 26 6
Anticor 0.11 0.10 0.10 0.06 0.05 PAMR 0.19 0.11 0.11 0.08 0.06
top five weighted assets, the three algorithms also have much higher volatilities on
other assets. Concerning their performance, it is possible that to achieve better per-
formance, a portfolio has to be frequently rebalanced, not only on certain assets as
the pattern matching–based algorithms do but also on all assets.
Second, most average weights of the state-of-the-art algorithms are assigned to
the assets with the highest volatilities (highest Std values). It is common knowl-
edge that high return is often associated with high risk,
while the reverse is not
always true. That is, although a portfolio has to be rebalanced among volatile assets,
such that the portfolio can gain profits from market volatility, high volatility can-
not guarantee high profit. For example, on the NYSE (O) dataset, although Anticor
and PAMR have the same top five average allocation pool, their performances are
drastically different.
Third, PAMR, which systematically exploits the mean reversion property, rebal-
ances more actively than Anticor, and OLMAR rebalances even more actively.
Connecting the rebalance activities to their performance, we may conclude that even
though both are based on the same principle, more active rebalance leads to better
performance, as it can better exploit market volatility. PAMR’s concentration on asset
#23, which has the highest negative autocorrelation, sheds lights on the possible con-
nection between mean reversion algorithms and the autocorrelation among assets (Lo
and MacKinlay 1990; Conrad and Kaul 1998; Lo 2008). Moreover, from Table C.5,
we can observe that most of the top average allocation weights of the mean reversion
algorithms are assets with negative autocorrelations, except DJIA.
13.7 Summary
In this chapter, we empirically evaluated the four proposed algorithms. The empiri-
cal results clearly validate the effectiveness of the proposed algorithms. In terms of
cumulative wealth, which is the main performance metric, our proposed algorithms
sequentially beat the state-of-the-art algorithms. In terms of (volatility/drawdown)
risk-adjusted return, the proposed algorithms achieve high risk-adjusted returns,
Such a statement is true in traditional finance. However, in recent years, some arbitrage strategies,
which can earn return without high risk, have emerged.
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SUMMARY 127
although they also have higher risk. The evaluations of parameter sensitivity show that
the proposed algorithms are always robust to their parameters and have a wide range
of satisfying choices such that they have good performance. The proposed algorithms
are also scalable to two practical issues, that is, margin buying and transaction costs.
Finally, although correlation-driven nonparametric learning (CORN) takes similar
time as the state of the art, the three mean reversion costs significantly less time,
which thus is suitable for practical large-scale applications, such as high-frequency
trading.
In the future, we plan to study the sources of profits among online portfolio
selection (OLPS). One way is to remove the possible “bid–ask bounce” by using a
different methodology for computing the closing prices, such as averaging the prices
of several transactions, which would reduce or even eliminate the bid–ask bounce.
Moreover, incorporating other sources of information, such as volume information,
is also possible to improve the proposed algorithms.
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