14 PROBLEM FORMULATION
on every buy and sell operation, based upon a transaction cost rate of γ ∈ (0, 1).
At the beginning of period t, the portfolio manager rebalances his or her wealth to
a new portfolio b
t
, from last close price adjusted portfolio
ˆ
b
t−1
, each component of
which is calculated as
ˆ
b
t−1,i
=
b
t−1,i
×x
t−1,i
b
t−1
x
t−1
. Such rebalance incurs a transaction cost
of
γ
2
×
m
i=1
b
t,i
−
ˆ
b
t−1,i
, where the initial portfolio is set to (0,...,0). Thus, the
cumulative wealth after n periods can be expressed as
S
γ
n
= S
0
n
t=1
(b
t
·x
t
) ×
1 −
γ
2
×
m
i=1
b
t,i
−
ˆ
b
t−1,i
.
Another practical issue is margin buying, which allows the portfolio managers
to buy securities with cash borrowed from securities brokers, using their own equity
positions as collateral. Following existing studies (Cover 1991; Helmbold et al. 1998;
Agarwal et al. 2006), we relax this constraint and evaluate it empirically. We assume
the margin setting to be 50% down and a 50% loan,
∗
at an annual interest rate
of 6% (equivalently, the corresponding daily interest rate of borrowing, c, is set to
0.000238). With such a setting, a new asset named “margin component” is generated
for each asset, and its price relative for period t equals 2 ×x
t,i
−1 −c. In the case
of x
t,i
≤
1+c
2
, which means the stock drops more than half, we simple set its mar-
gin component to 0 (Li et al. 2012).
†
As a result, if margin buying is allowed, the
total number of assets becomes 2m. By adding such a “margin component,” we can
magnify both the potential profit or loss on the i-th asset.
‡
2.3 Evaluation
One standard criterion to evaluate an OLPS strategy is its portfolio cumulative wealth
at the end of trading periods. As we set the initial wealth, S
0
= 1 and thus S
n
also
denote the portfolio cumulative return, which is the ratio of final portfolio cumulative
wealth divided by its initial wealth. Another equivalent criterion, which considers
compounding effect, is annualized percentage yield (APY), that is, APY =
y
√
S
n
−1,
where y is the number of years corresponding to n periods.
§
APYmeasures the average
wealth increment that a strategy could achieve in a year. Typically, the higher the
portfolio cumulative wealth or annualized percentage yield, the better the strategy’s
performance is.
Besides the absolute return metrics, it is also important to evaluate a strategy’s
risk and risk-adjusted return (Sharpe 1963, 1994). One common criterion is the annu-
alized standard deviation of portfolio period returns to measure volatility risk and
∗
That is, if one has $100 stock (down or collateral) one can borrow at most $100 cash (loan).
†
Such a measure is not perfect since it manually changes the margin component, although less than
5 per dataset. One may refer to Györfi et al. (2012, Chapter 4) for other solutions to the possibility of ruin.
‡
For example, assume two assets with price relatives of (1.1, 0.9). After adjustment, the price relative
vector becomes (1.1, 0.9, 1.2, 0.8). Putting wealth on the latter two margin components, the portfolio’s
profit or loss magnifies. That is, 10% profit (1.1) becomes 20% (1.1 ×2 −1 −c) and 10% loss (0.9) also
becomes 20% (0.9 ×2 −1 −c). Note that the portfolio vector representing the proportions of capital is
still a simplex.
§
One year consists of 252 trading days or 50 trading weeks.
T&F Cat #K23731 — K23731_C002 — page 14 — 9/28/2015 — 21:06