68 PASSIVE–AGGRESSIVE MEAN REVERSION
the weights to be transferred by scaling the directional vector. One common term
in τ
t
is
t
x
t
−¯x
t
1
2
. Its numerator denotes the -insensitive loss for period t, which
equals the t-th portfolio return minus a mean reversion threshold, or zero. Assuming
other variables are constant, if the return is high (low), it leads to a large (small)
value of τ
t
, which would aggressively transfer more (less) wealth from outperforming
assets to underperforming assets. The denominator is essentially the market quadratic
variability, that is, the number of assets times market variance of period t. In modern
portfolio theory (Markowitz 1952), the variance of assets returns typically measures
volatility risk for a portfolio. As indicated by the denominator, if the risk is high (low),
the step size τ
t
would be small (large). Consequently, the weight transfer made by the
update scheme will be weakened (strengthened). This is consistent with our intuition
that prediction would not be accurate in drastically dropping markets, and we opt to
make less transfer to reduce risk. Moreover, PAMR-1 caps the step size by a constant
C, while PAMR-2 decreases the step size by adding a constant
1
2C
to its denominator.
Both mechanisms can prevent drastic weight transfers in case of noisy price relatives,
which is consistent with their motivations.
From the above analysis on the updates of portfolio and step size, we can conclude
that PAMR nicely balances between return and risk and clearly reflects the mean rever-
sion trading idea. To the best of our knowledge, such an important trade-off has only
been considered by nonparametric kernel-based Markowitz-type strategy (Ottucsák
and Vajda 2007). While the strategy trades off return and risk with respect to a set of
similar historical price relatives, the proposed PAMR explicitly trades off return and
risk with respect to last price relatives. This nice property distinguishes the proposed
approach from most existing approaches that often cater to return, but ignore risk,
and are therefore undesirable.
One objective for PAMR-1 and PAMR-2 is to prevent a portfolio from being
affected too much from noisy price relatives, which might drastically change the
portfolio. In this part, let us exemplify the benefits of PAMR’s variants. Let x
t
=
(1.00, 0.01), whose second value is a noise, and b
t
= (1, 0). Setting = 0.30 and
C = 1.00, we can calculate the next portfolio b
t+1
. This market sequence describes
that certain stocks drop significantly, which is common during the financial crisis.
Without tuning, PAMR would transfer a large proportion to the second asset. This can
be verified by calculating PAMR’s portfolio; in other words, PAMR calculates the
update step size τ
t
= 1.43 and obtains the subsequent portfolio b
t+1
= (0.29, 0.71).
However, to avoiding such noises, a natural choice is to transfer less proportion
to the second asset. On the other hand, PAMR-1 and PAMR-2 obtain the step
sizes of τ
t
= 1.00 and τ
t
= 0.71, respectively, which are smaller than the origi-
nal PAMR’s. Accordingly, we obtain the next portfolios b
t+1
= (0.50, 0.50) and
b
t+1
= (0.65, 0.35) for PAMR-1 and PAMR-2, respectively. Clearly, the variants
transfer less wealth to the second asset than the original PAMR does. Thus, PAMR-1
and PAMR-2 suffer less from noisy price relatives, though they cannot completely
avoid such suffering situations.
Finally, let us analyze PAMR’s time complexity. Besides a normalization/
projection step (Step 7 in Algorithm 9.1), PAMR takes O(m) per period. In our
T&F Cat #K23731 — K23731_C009 — page 68 — 9/29/2015 — 18:26