Chapter 5
Follow the Loser
The Best Constant Rebalanced Portfolios (BCRP) strategy is optimal if the market
is independent and identically distributed (i.i.d.; Cover 1991); however, this assump-
tion may not fit the real market and thus may lead to the inferior performance of the
“follow the winner” category. Rather than tracking the winners, the follow the loser
approach is often characterized by transferring the wealth from winners to losers. The
underlying assumption is the mean reversion (contrarian) idea (Bondt and Thaler
1985), which means that good (poor)-performing assets will perform poor (good)
in the subsequent periods. Thus, follow the loser’s approaches often are character-
ized by transferring capital from poor-performing assets (losers) to good-performing
assets (winners). Although this principle is heavily investigated in finance journals,
it has not been widely disseminated in the topic of online portfolio selection. How-
ever, some algorithms do follow this principle. One famous example is the CRP
benchmark. Moreover, Cover’s UP, which buys and holds CRP strategies, can also be
viewed as follows the loser approach from the underlying stocks’ perspective, while
we categorize it as follow the winner from the experts’ perspective.
This chapter is organized as follows. Section 5.1 illustrates the mean rever-
sion idea, which is the key underlying the “follow the loser” principle. Section 5.2
introduces a representative strategy in this category, or the Anticor strategy. Finally,
Section 5.3 summarizes the follow the loser principle.
5.1 Mean Reversion
Besides the momentum-related idea that assumes that the stock price will continue
its previous trend, there exists another different idea, or the mean reversion (con-
trarian) idea, which assumes that the assets’ prices will revert to their means. Thus,
the follow the loser algorithms will transfer the wealth from outperforming assets to
underperforming assets.
This section illustrates a simple but convincing example to show the mean rever-
sion idea. Consider a fluctuating market with two assets (A,B), and the price relative
sequence is
1
2
, 2
,
2,
1
2
,..., where each asset is not going anywhere but actively
moving within a range (Table 5.1). Obviously, in the long run, a market strategy can-
not achieve any abnormal return since the cumulative wealth of each stock remains
31
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32 FOLLOW THE LOSER
Table 5.1 Motivating example to show the mean reversion trading idea
BCRP Adjusted
Period # Market (A,B) BCRP Return Weights Notes
1 (1/2, 2)(1/2, 1/2) 5/4 (1/5, 4/5)B−→ A
2 (2, 1/2)(1/2, 1/2) 5/4 (4/5, 1/5)A−→ B
3 (1/2, 2)(1/2, 1/2) 5/4 (1/5, 4/5)B−→ A
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
the same after 2n periods. However, BCRP in hindsight can achieve a growth rate of
5
4
n
for a n-trading period.
Now let us analyze the BCRP’s behaviors to show the underlying mean rever-
sion trading idea (Table 5.1). Suppose the initial portfolio is
1
2
,
1
2
and at the end of
period 1, the close price adjusted portfolio distribution becomes
1
5
,
4
5
and cumulative
wealth increases by a factor of
5
4
.At the beginning of period 2, portfolio manager rebal-
ances to initial portfolio
1
2
,
1
2
by transferring the wealth from a better-performing
asset (B) to a worse-performing asset (A). At the beginning of period 3, the wealth
transfer with the mean reversion trading idea continues. Although the market strategy
gains nothing, BCRP can achieve a growth rate of
5
4
per period with the underly-
ing mean reversion trading idea, which assumes that if one asset performs worse,
it tends to perform better in the subsequent trading period. It actually gains profit
via the volatility of the market, or so-called volatility pumping (Luenberger 1998,
Chapter 15).
Though extensive studies in finance show that mean reversion is a plausible idea
to be used in trading (Chan 1988; Poterba and Summers 1988; Lo and MacKinlay
1990; Conrad and Kaul 1998), its counterintuitive nature hides it from the OLPS
community. While the “follow the winner” strategies are sound in theory, they often
perform poorly when using real data, which will be shown in the empirical studies
in Part IV. Perhaps the reason is that their momentum principle does not fit the real
market, especially on the tested trading frequency (such as daily). It is thus natural
to utilize the mean reversion idea in developing new strategies so as to boost the
empirical performance.
5.2 Anticorrelation
Borodin et al. (2004) proposed a follow the loser strategy named an Anticorrelation
(Anticor). Instead of making no distributional assumption like Cover’s UP, Anticor
assumes that the market follows the mean reversion principle. To exploit the property,
it statistically makes bets on the consistency of positive lagged cross-correlation and
negative autocorrelation.
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SUMMARY 33
Anticor adopts logarithmic price relatives (Hull 1997) in two specific market
windows, that is, y
1
= log(x
t−w
t−2w+1
) and y
2
= log(x
t
t−w+1
). It then calculates a cross-
correlation matrix between y
1
and y
2
,
M
cov
(i, j ) =
1
w −1
y
1,i
−¯y
1
y
2,j
−¯y
2
M
cor
(i, j ) =
M
cov
(i,j)
σ
1
(i)×σ
2
(j)
σ
1
(i), σ
2
(j) = 0
0 otherwise
.
Following the mean reversion principle, Anticor transfers weights from the assets
increased more to the assets increased less, and the corresponding amounts are
adjusted by the cross-correlation matrix. In particular, if asset i increases more than
asset j and they are positively correlated, Anticor claims a transfer from asset i to
j with the amount equaling the cross-correlation (M
cor
(i, j )) minus their negative
auto-correlation (min{0,M
cor
(i, i)} and min{0,M
cor
(j, j )}). Finally, these claims
are normalized to keep the portfolio in the simplex domain.
With the mean reversion nature, it is difficult to obtain a useful regret bound for
Anticor. Although heuristic and without theoretical guarantee, Anticor empirically
outperforms all other strategies at the time. On the other hand, though Anticor obtains
good performance, its heuristic nature cannot fully exploit mean reversion. Thus,
exploiting the property via systematic learning algorithms is highly desired, which
motivates one part of our research.
5.3 Summary
Although counterintuitive, the follow the loser principle is quite useful in obtaining
a high cumulative return in the empirical studies. This may be attributed to the fact
that many financial research studies have validated that the market behaviors follow
the mean reversion principle. Thus, to better exploit the market, a trading strategy has
to incorporate the market behaviors. We further propose three novel mean reversion-
based algorithms in Chapters 9, 10, and 11, respectively.
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