EXPERIMENT 3 113
10
0
10
4
10
8
10
12
10
16
50 500 5000
Total wealth achieved
C
(a)
BCRP
PAMR
Market
10
0
10
3
10
6
50 500 5000
Total wealth achieved
C
(b)
BCRP
PAMR
Market
10
0
10
1
10
2
10
3
50 500 5000
Total wealth achieved
C
(c)
BCRP
PAMR
Market
1
4
50 500 5000
Total wealth achieved
C
(d)
BCRP
PAMR
Market
1
4
16
50 500
(e)
5000
Total wealth achieved
C
BCRP
PAMR
Market
1
2
50 500
(f)
5000
Total wealth achieved
C
BCRP
PAMR
Market
Figure 13.6 Parameter sensitivity of PAMR-1 (or PAMR-2) with respect to C with fixed
( = 0.5): (a) NYSE (O); (b) NYSE (N); (c) TSE; (d) SP500; (e) MSCI; and (f) DJIA.
cumulative wealth by varying C from 50 to 5000 on PAMR-1, with fixed = 0.5. We
only show the figures on PAMR-1, as the effects on PAMR-2 are similar to those on
PAMR-1. Clearly, the proposed PAMR is insensitive to C, and a wide range of values
correspond to the highest cumulative wealth. This again exhibits that the proposed
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114 EMPIRICAL RESULTS
PAMR is robust with respect to its parameters. Similarly, the figure on DJIA again
indicates that the mean reversion effect may not exist on the dataset, in the sense of
our motivation.
13.3.3 CWMR’s Parameter Sensitivity
The proposed CWMR algorithms contain two parameters, that is, the confidence
parameter φ and the mean reversion sensitivity parameter . Throughout the algo-
rithms, the mean reversion sensitivity parameter decisively influences the final
performance, while the confidence parameter has little effect on the final perfor-
mance, whose effects are similar to C on PAMR-1/2. Figure 13.7 depicts CWMR’s
robustness with respect to , plus the final cumulative wealth achieved by Market and
BCRP. The results first show that the final cumulative wealth increases as the sen-
sitivity parameter decreases and stabilizes after falls below certain data-dependent
thresholds, which means that the mean reversion idea has been completely exploited.
Then, the results again verify that the mean reversion trading idea works in the finan-
cial markets and the proposed CWMR algorithm can successfully exploit it, which
generates significant final cumulative wealth on most datasets. Moreover, as ana-
lyzed in Section 10.4, CWMR degrades to uniform CRP strategy when is larger
than 1. Needless to say, our empirical setting = 0.5 is not the best one; however, the
proposed CWMR still significantly surpasses existing approaches. Finally, similar to
PAMR, CWMR fails on DJIA, which still indicates that the motivating (single-period)
mean reversion may not exist on the dataset.
13.3.4 OLMAR’s Parameter Sensitivity
Now we evaluate OLMAR’s sensitivity to its parameters, that is, for both OLMAR-1
and OLMAR-2, and w and α for OLMAR-1 and OLMAR-2, respectively. Figure 13.8
shows OLMAR-1’s sensitivityto with fixed w = 5. Since OLMAR-2andOLMAR-1
have the similar figures for , we only list its effects on OLMAR-1. Figure 13.9 shows
its sensitivity to w with fixed = 10, and Figure 13.10 shows OLMAR-2’s sensitivity
to α with fixed = 10.
From Figure 13.8, we can observe that, in general, the cumulative wealth sharply
increases if approaches 1 and flattens if crosses a threshold. From Figure 13.9,
we can see that as w increases, the performance initially increases, spikes at a data-
dependent value, and then decreases. Regardless, its performance with most choices
of and w is much better than that of the market and BCRP. To smooth the volatility of
its performance, Figure 13.9 also shows OLMAR’s BAH versions (Li and Hoi 2012)
by combining a set of OLMAR experts with varying w’s. We can see that the BAH
version provides a much smoother cumulative return than its underlying experts. Note
that on DJIA, OLMAR performs much better than PAMR/CWMR as varies, which
validates the motivating multiperiod mean reversion. From Figure 13.10, we can
find that on most datasets, α provides significant high cumulative wealth in a wide
range of values, except the two extreme endpoints, that is, 0 and 1. If α = 1, then
all expected price relatives are always 1 and OLMAR-2 outputs b
t+1
= b
t
, which is
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EXPERIMENT 3 115
10
0
10
4
10
8
10
12
10
16
0
ε
(a)
0.5 1 1.5
Total wealth achieved
CWMR Market BCRP
10
6
0 0.5 1 1.5
Total wealth achieved
ε
CWMR
Market
BCRP
10
3
10
0
(b)
10
0
10
1
10
2
10
3
0
(c)
0.5 1 1.5
Total wealth achieved
ε
CWMR
Market
BCRP
1
4
16
0
ε
0.5 1 1.5
Total wealth achieved
(d)
CWMR Market BCRP
1
4
16
00.5
(e)
1 1.5
Total wealth achieved
ε
CWMR
BCRPMarket
1
3
00.5
(f)
11.5
Total wealth achieved
ε
CWMR
BCRPMarket
Figure 13.7 Parameter sensitivity of CWMR with respect to : (a) NYSE (O); (b) NYSE (N);
(c) TSE; (d) SP500; (e) MSCI; and (f) DJIA.
initialized to uniform portfolio. If α = 0, then its expected price relative vector equals
˜
x
t+1
=
1
t
i=1
x
i
. Such price relatives inversely relate to all of an asset’s historical price
relatives and produce bad results.
Nevertheless, all the above observations show that OLMARs’ performance is
robust to their parameters, and it is convenient to choose satisfying parameters.
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116 EMPIRICAL RESULTS
10
16
1
(a)
10 100 1000
Total wealth achieved
ε
OLMAR Market
BCRP
10
12
10
8
10
4
10
0
10
9
10
6
1 10
(b)
100 1000
Total wealth achieved
ε
OLMAR Market
BCRP
10
3
10
0
10
0
10
1
10
2
10
3
110
(c)
100 1000
Total wealth achieved
ε
OLMAR
Market
BCRP
1
10
1
(d)
10
ε
100 1000
Total wealth achieved
OLMAR Market BCRP
1
10
20
30
110
(e)
100 1000
Total wealth achieved
ε
OLMAR
BCRPMarket
1
2
3
110
(f)
100 1000
Total wealth achieved
ε
OLMAR
BCRPMarket
Figure 13.8 Parameter sensitivity of OLMAR-1 with respect to with fixed w (w = 5):
(a) NYSE (O); (b) NYSE (N); (c) TSE; (d) SP500; (e) MSCI; and (f) DJIA.
13.4 Experiment 4: Evaluation of Practical Issues
For real-world portfolio management, there are some important practical issues,
including transaction costs and margin buying. In this section, we examine how the
two issues affect the proposed strategies.
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EXPERIMENT 4 117
10
16
25
(a)
50 75 100
Total wealth achieved
W
OLMARBAH (OLMAR)
Market
BCRP
10
12
10
8
10
4
10
0
OLMARBAH (OLMAR)
Market
BCRP
(b)
10
0
10
3
10
6
10
9
25 50 75 100
Total wealth achieved
W
10
0
10
1
10
2
10
3
25 50
(c)
75 100
Total wealth achieved
W
OLMARBAH (OLMAR)
Market
BCRP
OLMARBAH (OLMAR)
Market
BCRP
1
10
20
30
25 50 75 100
Total wealth achieved
W
(d)
1
10
20
30
25 50 75 100
Total wealth achieved
W
(e)
OLMARBAH (OLMAR)
Market
BCRP
1
2
3
4
25
(f)
50 75 100
Total wealth achieved
W
OLMARBAH (OLMAR)
Market
BCRP
Figure 13.9 Parameter sensitivity of OLMAR-1 with respect to w with fixed ( = 10):
(a) NYSE (O); (b) NYSE (N); (c) TSE; (d) SP500; (e) MSCI; and (f) DJIA.
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