PRELIMINARIES 61
constitutes the CRP strategy. Here, the reason to choose a passive CRP strategy is that
these “mine” stocks are usually known only in hindsight, thus identifying them a priori
is almost impossible. Thus, to avoid suffering too much from such situations, the pro-
posed approach alternates between “aggressive” and “passive” reversion depending
on market conditions. The passive mean reversion avoids the high risk of aggressive
mean reversion, which would put most wealth on these “mine” stocks.
In the following, we propose a novel trading strategy named “passive–aggressive
mean reversion,” or PAMR for short. On the one hand, the underlying assumption
is that better-performing assets would perform worse than others in the next period.
On the other hand, if the market drops too much, we would stop actively rebalanc-
ing portfolios to avoid certain “mine” stocks and their associated risk. To exploit
these intuitions, we suggest adopting passive–aggressive (PA) online learning
(Crammer et al. 2006), which was originally proposed for classification. The basic
idea of PA is that it passively keeps the previous solution if the loss is zero, while it
aggressively updates the solution whenever the suffering loss is nonzero.
We now describe the proposed PAMR strategy in detail. Firstly, if the portfolio
period return is below a threshold, we will try to keep the previous portfolio such
that it passively reverts to the mean to avoid potential “mine” stocks. Secondly, if the
portfolio period return is above the threshold, we will actively rebalance the portfolio
to ensure that the expected portfolio daily return is below the threshold, in the belief
that the next price relatives will revert. This sounds a bit counterintuitive, but it is
indeed reasonable, because if the price relative reverts, keeping the expected port-
folio return below the threshold enables one to maintain a high portfolio return in the
next period. Here, the expected portfolio return is calculated with respect to historical
price relatives, for example, in our study, the last price relative (Helmbold et al. 1998).
To further illustrate that aggressive reversion to the mean can be more effective
than a passive one, let us continue the example that has a market going nowhere but
actively fluctuating. In such a market, the proposed strategy is much more powerful
than best constant rebalanced portfolio (BCRP), a passive mean reversion trading
strategy in hindsight, as shown in Table 9.1. As the motivating example shows,
BCRP grows to
5
4
n
for a n-trading period, while at the same time, PAMR grows
to
5
4
×
3
2
n−1
(the details of the calculation/algorithm will be presented in the next
section). We intuitively explain the success of PAMR below.
Assume the threshold for a PAMR update is set to 1, that is, if the portfolio period
return is below 1, we do nothing but keep the existing portfolio. Our strategy begins
with a portfolio
1
2
,
1
2
. For period 1, the return is
5
4
> 1. Then, at the beginning
of period 2, we rebalance the portfolio such that an approximate portfolio return
based on last price relatives is below the threshold of 1, and the resulting portfolio is
2
3
,
1
3
. As the mean reversion principle suggests, although we are building a portfolio
performing below the threshold in the current period, we are actually maximizing the
next portfolio return. As we can observe, the return for period 2 is
3
2
> 1. Then,
following the same rule, we will rebalance the portfolio to
1
3
,
2
3
. As a result, in such
a market, PAMR’s growth rate is
5
4
×
3
2
n−1
for a n-period, which is superior to
BCRP’s
5
4
n
.
T&F Cat #K23731 — K23731_C009 — page 61 — 9/29/2015 — 18:26