Improving Moving Average Trading Rules with Boosting and Statistical Learning Methods, страница 5

The idea of using a fi lter to reduce trading frequency and to obtain higher returns is very old in fi nance. Alexander (1961) and Fama and Blume (1966) show that several fi lters are able to produce a higher return than normal, if we do not take into account the commissions and transaction costs. The success of the fi lters would justify the existence of some systematic trends in the prices which are not explained by the random walk model.

Besides, in order to avoid overactive trading strategies, which are very expensive in transaction costs, we have considered an additional fi lter in the signals provided by all the learning methods shown above. In order to explain the fi ltering process, let us consider the case of the Boosting algorithm. The implementation of a fi lter for the rest of the learning methods is identical. Thus the formula (4) in the boosting algorithm provides a binary fi nal output which points out a (buy or sell) signal depending on the sign. As the coeffi cient a_t in (2) represents the strength of the signal h_t(x), T

it is also possible to consider the expression α_th_t ( )x as the strength of the signal H(x) in (4).

t=1T

  Consequently, we fi lter the signal (4) as follows: given a number e > 0, if ε α^≤ ∑ th_t ( )x H x( ) =1

t=1 and we are out of the market, a buy signal is generated. If we are in the market, the trading rule suggests we should continue holding the market.

If ∑t^T1α_th_t ( )x ≤ −ε H x( ) = −1 and we are in the market, a sell signal is generated. If we are out T

=

of the market, we continue holding the risk-free security. − <^{ε α}∑ th_t ( )x <ε H x( ) = 0, no signal

t=1

is generated and we maintain the previous position.

The threshold e is selected as the percentile 5% of the empirical distribution of the quantities

∑^T α_th_t ( )x for a sample of the previous 100 observations to x In what follows we will call this

t=1

procedure the ‘fi ltered Boosting model’. The same idea provides a ‘fi ltered Committee moving average model’, and a ‘fi ltered Bayesian moving average model’.

As well as the net returns, we also consider other two profi tability indicators: the ideal profi t ratio and, given that individuals are generally risk averse, a version of the Sharpe ratio (Sharpe, 1966). The ideal profi t ratio measures the net returns of the trading strategies against a perfect predictor and is calculated by

                                          I = ∑ ∑tT=1 r It ⋅ btT + tT=1 rft ⋅Ist + n⋅log11+−cc)                            (8)

                                          R

∑t=1 rt + n′⋅log11+−cc)

In accordance with the last equation, R_I = 1 if the indicator variable takes the correct trading position for all observations in the sample. If all trade positions are wrong, then the value of this measure is R_I = −1. An R_I = 0 value is considered as a benchmark to evaluate the performance of an investment strategy. Regarding the Sharpe ratio version, it is simply the mean net return of the trading strategy divided by its standard deviation:

                                                                       RS = ^r    (9) σ

The higher the Sharpe risk-adjusted ratio, the higher the mean net return, and the lower the volatility.

Finally, it is necessary to point out that all of our computations in this paper have been carried out using MATLAB codex.

EMPIRICAL RESULTS

We have studied the improving capability of boosting and other statistical learning methods over technical trading rules, out of sample, in the daily closing values of the NYSE Composite Index, which refl ects the price of all common stocks listed on the New York Stock Exchange.