Introduction

Academic literature recognizes a large set of indicators or factors that are connected with the various assets. These indicators can be utilized in a variety of trading strategies, which means that such indicators are popular among practitioners who seek to invest their funds. Usually, the indicators are connected with some evaluation period.

For example, we have a trend following strategy using a moving average, either simple or exponential, and we are creating a trading rule that we buy if the price of the asset is above the average. What is the ideal period for the evaluation of the moving average?

Academic papers that examine strategies usually suggest some fixed period or present a set of possible periods. It is easy to over-fit a trading strategy with some back-testing and optimizing the performance. Moreover, it is common that some indicator has a fixed period for the evaluation and this is widely accepted as something unchangeable. For example, 12 months momentum in stocks, 1-month short-term reversal, Stochastic Oscillator with K equal to 13 and D to 3, etc. It is even possible to guess a period and do the back-test. Also, a practitioner’s judgment and intuition could be used. But the key question is whether the period would be reliable also in the future.

This paper aims to show some possible approaches to find the optimal evaluation periods of indicators. This is a key question among practitioners and therefore we see it as crucial to shed a light on this topic. Although we are focused on momentum strategies, the information in this paper is widely applicable also in the construction of any other trading strategy where the investor has to decide indicator’s period – whether he would use 1 month, 2 months, 3 months… Furthermore, it does not need to be months, it could be days, years, etc.

This paper is focused on a cross-sectional momentum asset allocation strategy on 5 ETFs based on the Mebane Faber’s “Relative Strength Strategies for Investing“. The aim is to find the best performers according to the various momentum periods from 3 months to 15 months. Momentum is chosen because it is a notoriously known strategy that is proved by academics and favorite among practitioners.

The idea is simple, the past winners are expected to perform favorably and past losers are expected to continue to underperform. Each month, ETFs are sorted according to their t-months momentum and we long the 3 top performers. The key question is that, if we have 13 momentum indicators, what is the best way to build a trading strategy? Or in other words, which indicator is the best?

Video summary:

Two approaches how to pick an indicator’s period

It is important that the picked indicator or indicators would work also in the future. We can easily optimize the strategy and find the best performing indicator if we are doing a back-test, but we need to think about the performance in the future. We show two approaches, but we do not state that there are only two solutions to this problem.

• Firstly, there is an option to pick a set of momentum indicators instead of using only one. Naturally, not all the momentum periods would perform as good as during the back-test, but the aim is to find a reliable set of indicators that would outperform the average also in the out of sample. For example, if we pick 3 momentum indicators based on the back-test and in the out-of-sample one indicator would not be as good, there are still two other. As a result, the strategy would remain reasonably profitable. This can be simply understood as a trade-off, we reduce both performance and the risk that there would be under-performance.
• Secondly, one can optimize the period by evaluating the performance of every momentum strategy. This approach is simple, the strategies are evaluated for some pre-fixed period and the best performing (or the set of best performing) strategy is picked to be traded the next month. This process is repeated each month to find the optimal strategy.

We would call the first approach as the average approach and the second as the walk forward approach.

Last but not least, the results of this paper are informative about the momentum strategy in this article. Therefore, if we pick 3 momentum indicators it does not mean that three indicators are the best to pick in a different reversal strategy, for example. If we evaluate the strategy for three years to make a decision which strategy to trade in the next month, it does not mean that this period should be universally used. It does mean that these numbers may be good „first guesses“, but everything should be understood as some basic principles that need to be slightly refined in every strategy. The key idea is to pick a larger number of indicators to find the „well-performing average” based on the longer back-test or to optimize the choice of indicators each month based on past performance. To sum it up, this article should be rather taken as a useful manual to find the solution and not the universal solution.

Data and methodology

The investment universe consists of 5 various and diversified ETFs, namely the Vanguard Real Estate ETF (VNQ), Invesco DB Commodity Index (DBC), iShares 7-10 Year Treasury Bond ETF (IEF), SPDR S&P 500 (SPY) and iShares MSCI EAFE ETF (EFA). If the ETF was not available, the corresponding index was used instead.

Firstly, we construct momentum factors for 3 to 15 months. Nextly, we rank each ETF according to their momentum and we fix a cut point that indicates how many assets we invest in. In this study, we would use 3 ETFs. Therefore, each month, 3 winners are picked for a long position and the portfolio is rebalanced every month. As a result, we have 13 momentum strategies that are candidates for the final strategy. 

The benchmark strategy is a simple strategy that equally weights each ETF and stays in a long position.

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Average approach

The performance of the strategy is evaluated from 30.4.1987 to 30.9.2019. As a first step, we divide the sample into two halves (1987 to 2003 and 2003 to 2019) and we examine the performance during the first half.

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The idea is to pick some number of top-performing strategies (3 in this case study) that outperform the average benchmark strategy. We can see that every momentum strategy outperforms the simple benchmark, but we choose 8, 10 and 13 months momentum strategies to build the final composite strategy. The reason is simple, we want to cover the best performing area. We equally weight these 3 strategies and examine their performance also in the second half. As we have previously mentioned, the idea is to minimize the risk that we would pick one bad performing strategy. Therefore, we do not pick just one strategy, but a whole area (or corresponding part of this area).

Naturally, in the second half (2003 to 2019), the strategies perform differently. Almost every strategy outperforms the benchmark, except for the 7 months momentum but the returns are slightly lower. Moreover, the 10-month momentum is not performing as good as in the first half, but still, there are two other strategies.

This example is one possible solution. Do not pick one strategy, there is no need to be fixed to some period. Rather than finding one optimal period, diversify the periods.

We also examine strategies during the whole back-testing period and examine the performance of the composite momentum strategy.

Naturally, we also compare the strategy with the previously mentioned simple benchmark. The benchmark strategy is a simple strategy that equally weights each ETF and stays in a long position.

Strategy	Return	Maximal drawdown	Return to drawdowns ratio
Benchmark	8,20%	43,51%	0,19
Composite momentum	9,89%	39,41%	0,25

Walk forward approach

The performance of the strategy is evaluated from 31.5.1997 to 30.9.2019. Since we have already constructed the momentum strategies, we rank their performance during a fixed period and pick the best performers. To be more precise, each month, we evaluate their performance during the past 3 years, rank them and top 3 strategies are used in the next month equally weighted strategy. Therefore, we walk forward each month when we pick the best performing strategies based on the past 3 years. Again, the period of the evaluation is another parameter, but the choice is not that crucial. For example, we compared 3 and 10 years and picked the 3 years, the performance was better, however, the differences were really only minor. We compare this approach to a strategy where we use all 13 strategies. Such strategy simply uses all momentum factors. Nextly, we compare it also to the simple benchmark strategy where we hold all ETFs. 

Strategy	Return	Maximal drawdown	Return to drawdowns ratio
Benchmark	7,07%	43,51%	0,16
Top strategies	8,47%	40,41%	0,21
All strategies	7,81%	35,04%	0,22

Additionally, since the back-testing periods are different for both strategies, we also make a comparison of both approaches during the same period. Although the average approach is, in this case, more profitable, it is not a universal truth among all strategies.

Strategy	Return	Maximal drawdown	Return to drawdowns ratio
Average approach	9,04%	39,41%	0,23
Walk forward approach	8,47%	40,41%	0,21

Conclusion

We believe that we have successfully shown two simple approaches to find optimal evaluation periods for not only momentum strategies but also for any other strategies based on trading indicators. We also believe that we have shown that although the 12 months momentum is a standard, it does not need to be the best. This can be widely applicable also in other strategies and serve as a motivation to dig deeper and to modify any academic principle for the specific situation of the practitioner. Moreover, the results show a well-known fact or principle that over-fitting is usually bad. For example, in the average approach during the first half, 10-month momentum is the best performer, but in the second half, it is far from the best. Last but not least, there are other possibilities to evaluate periods to make an informed decision on which one would be the best. We have looked only on the returns, but the choice was made purely for simplicity. We have examined also a variant where we used the return to the draw-down ratio, but the results were similar, so we have chosen the simpler variant. There could be used also the Sharpe ratio, Information ratio, and other metrics.

References

Faber, Meb, Relative Strength Strategies for Investing (April 1, 2010). Available at SSRN: https://ssrn.com/abstract=1585517 or http://dx.doi.org/10.2139/ssrn.1585517

Authors:
Radovan Vojtko, CEO, Quantpedia.com
Matus Padyšák, Analyst, Quantpedia.com

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