robust with respect to different parameter settings and our choices are not always the
For OLMAR, in all cases, we empirically set the mean reversion parameter,
that is, = 10, which provides consistent results. Individually, we set w = 5 for
OLMAR-1 and α = 0.5 for OLMAR-2. As the results show, it is easy to choose
satisfying parameters for the proposed OLMAR algorithms.
12.3.1 Comparison Approaches and Their Setups
We compare the proposed algorithms with a number of benchmarks and representative
strategies. Below we summarize a list of compared algorithms, all of which provide
extensive empiricalevaluations in theirrespectivestudies. Focusing onempiricalstud-
ies, we ignore certain algorithms that focus on theoretical analysis and lack thorough
empirical evaluations.
All parameters are set following their original studies.
1. Market: Market strategy, that is, uniform BAH strategy
2. Best-Stock: Best stock in the market, which is a strategy in hindsight
3. BCRP: Best constant rebalanced portfolios strategy in hindsight
4. UP: Covers universal portfolios implemented according to Kalai and Vempala
(2002), where the parameters are set as δ
= 0.004, δ = 0.005, m = 100, and
S = 500
5. EG: Exponential gradient algorithm with the best learning rate η = 0.05 as
suggested by Helmbold et al. (1998)
6. ONS: Online Newton step with the parameters suggested by Agarwal et al. (2006),
that is, η = 0, β = 1, γ =
7. Anticor: BAH
(Anticor(Anticor)) as a variant of Anticor to smooth the perfor-
mance, which achieves the best performance among the three solutions proposed
by Borodin et al. (2004)
8. B
: Nonparametric kernel-based moving window strategy with W = 5, L = 10,
and threshold c = 1.0, which has the best empirical performance according
to Györfi et al. (2006)
9. B
: Nonparametric nearest-neighbor-based strategy with parameters W = 5,
L = 10, and p
= 0.02 +0.5
, as the authors suggested (Györfi et al. 2008)
12.4 Performance Metrics
We adopt the most common metric, cumulative wealth, to primarily compare different
trading strategies. In addition to the cumulative wealth, we also adopt the annual-
ized Sharpe ratio (SR) to compare the performance of different trading algorithms.
In general, higher values of the cumulative wealth and annualized SR indicate bet-
ter algorithms. Besides, we also adopt maximum drawdown (MDD) and the Calmar
ratio (CR) for analyzing a strategy’s downside risk. The lower the MDD values,
Our OLPS platform provides these algorithms.
We can tune their parameters for better performance, but it is beyond the scope of this book.
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Table 12.2 Summary of the performance metrics used in the evaluations
Criteria Performance Metrics
Absolute return Cumulative wealth (S
) Annualized percentage yield
Risk Annualized standard deviation Maximum drawdown
Risk-adjusted return Annualized Sharpe ratio (SR) Calmar ratio (CR)
the less the strategy’s (downside) risk. The higher the CR values, the better the strat-
egy’s (downside) risk-adjusted return. We summarize them in Table 12.2 and present
their details as follows.
12.5 Summary
A strategy has to be back-tested using historical market data, such that we have
confidence that it will continue to be effective in the unseen future markets. This
chapter introduces some implementation issues for the empirical studies, including the
platform, data, and various setups. In future, we can further extend the online portfolio
selection (OLPS) system using real-market feeds and execute the orders using a
paper trading account or real trading account. The next chapter will demonstrate the
empirical results obtained from the implementation and corresponding back-tests.
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