In the first three articles of our Dynamic Asset Allocation for Practitioners series (article 1, article 2, article 3), we focused on the first half of the total process. We specified a universe of global asset classes and sorted it on relative strength with 21 different raw and risk-adjusted momentum indicators, each subjected to a battery of robustness testing – 250,000 tests in total. We now move on to the second half of our decomposition of the Dynamic Asset Allocation framework: Optimization.

To be crystal clear, our discussion will NOT focus on trying to identify which signals or parameters, or combinations of signals and parameters, are better or worse than others. While most quants – being tinkerers at heart – waste most of their time fine-tuning the parameters of their strategies to maximize performance in backtests, we are acutely aware of the futility of these efforts. The few decades of data that most quants have at their disposal for simulation are nowhere near enough to tease out differentiating features given the vast dimensionality of this exercise.

Rather, our discussion of optimization will focus on how to use the signals at our disposal to assemble a portfolio of assets that is most likely to deliver the maximum return with minimal risk. We will begin our discussion with an exploration of whether the magnitude of the momentum signal itself provides useful information for building portfolios. To begin, we will examine the strength of this signal by constructing portfolios such that asset weights are in proportion to the magnitude of their momentum relative to other assets.

First, let’s look at what happens when we don’t incorporate any momentum tilts in the portfolio: a base portfolio where all 12 assets are held in equal weight (see article 1 for our investment universe). This base portfolio will serve to gauge the benefit of our optimization efforts. Consistent with the process we applied in previous posts, we will iterate over holding periods of 5, 10, 15, and 20 days, including all different trading days. For example, for simulations with 5 day holding periods, we will run 5 independent tests, which rebalance on day 1, day 2, day 3, day 4 and day 5.

*Table 1: Equal weight portfolio.*

*Source: ReSolve Asset Management. Data from CSI data and underlying index providers where data for investable funds have been extended prior to their date of inception. The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained.*

**METHODOLOGY**

Now let’s investigate the performance of portfolios that are tilted toward the assets with the highest momentum. First, we rank our assets from weakest to strongest at each lookback according to our various momentum metrics (discussed in detail here and here). The lowest scoring asset at each lookback will receive a 1 and the highest will receive a 12 (the total number of assets in our universe). The ranks are then averaged across lookbacks, and then percentile-ranked for a final score. We detail the steps below.

**Formula 1: Rank**

Where is the vector of ranks of, , the proportional momentum scores at lookback . For Excel, assume the scores at lookback horizon , for assets 1 through 12, are in cells A1 through A12 (the first lookback). We’ll use the RANK.EQ function as we’re not concerned with tied scores here.

=RANK.EQ(A1,A12

Now that we have vectors of rank averages we will convert them into a percentile rank, from 0 to 1, by applying the percentile rank formula from our second article, . We will use this final transformation as a means of weighting (and eventually filtering) – a heuristic optimization processes.

**Formula 2: Percentile transform**

Where is the percentile ranks of the vector of averaged rank scores . The following Excel code assumes that the vector of averaged ranks is stored in cells F13:F24.

=PERCENTRANK.INC(13:24,F13)

**MOMENTUM WEIGHTING**

Our first test will be to use our percentile ranks as a means of weighting the portfolio. This is accomplished by dividing each asset’s percentile score at time by the sum of the scores at time . The result is a vector which distributes portfolio capital in proportion to an asset’s momentum rank score at each rebalance period. Note that for this test the portfolio will always hold some positive weight in every asset, with highest momentum assets earning the most weight.

**Formula 3: Non-parametric momentum weight**

**RESULTS**

*Tables 2 and 3: Momentum-weight performance of aggregate models.*

*Source: ReSolve Asset Management. Data from CSI data and underlying index providers where data for investable funds have been extended prior to their date of inception. The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained.*

Momentum weighting markedly improves upon our equal-weight baseline, enhancing every metric across all 21 indicators. We observe an average boost of about 1.5% per year in annualized returns, an almost 40% improvement in Sharpe ratio, and a 25% reduction in maximum drawdown. Clearly there is some signal in the rank order of assets by momentum.

**CONCENTRATING ON THE BEST ASSETS**

You may recall from our previous posts that we held the top 2-5 assets at each lookback as a way to filter the strongest from the weakest. This is a reasonable approach, but it does not account for changes in the distribution of momentum across assets through time. Sometimes many assets will have strong returns, while just one or two will have extreme negative returns that drag the cross-sectional average down. At such times, it may be useful to hold all of the assets that are doing relatively well, and simply ignore the ones in extreme negative trends. Many other situations are possible.

For these next tests we are going to filter by 50^{th} percentile (0.5), so that we will hold all assets that exhibit momentum scores above the median momentum score. This will allow the number of assets that we hold in our portfolios to change dynamically, in consideration of the changing cross-sectional distribution of momentum.

First let’s have a look at the results when each asset is held in __equal weight__ given that it is above the 0.5 cutoff.

*Tables 4 and 5: Percentile-filtered, equal weight performance of aggregate models.*

*Source: ReSolve Asset Management. Data from CSI data and underlying index providers where data for investable funds have been extended prior to their date of inception. The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained.*

As with our first two articles, removing the laggards provides a powerful way to enhance portfolio performance. There is incremental outperformance of the percentile filtering versus the unfiltered momentum-weighting across all tests. Both strategies seek to respond dynamically to the distribution of momentum across assets, however it appears that allowing the portfolio to “breathe” given the percentile rank cutoff offers an additional adaptive ability, which improves results at the margin.

Finally, let’s see if there’s any benefit in momentum weighting our percentile-filtered portfolios. We’ll use Formula 3 to convert our *filtered* assets to momentum weights at each rebalance.

*Tables 12 and 13: Percentile-filtered, momentum weight performance of aggregate models.*

Further concentration of our portfolio towards favorable assets leads to marginal improvement in CAGR and maxDD, with only commensurate improvements in Sharpe and volatility.

**MEASURING THE ENSEMBLE**

Remember, the purpose of running tests using a wide variety of momentum metrics is to illustrate that momentum is robust to many different specifications. We do not want to lead readers to the conclusion that they should favor the single metric that seems to deliver the best results in backtests.

Rather, we believe all of these metrics have (for the most part) statistically indistinguishable merit, as they all capture the momentum effect from slightly different angles. As such, and consistent with previous articles in this series, we want to examine the performance of ensemble (aggregate) systems to observe the portfolio effect of using many different types of signals. First let’s examine the correlation relationships between the unfiltered momentum weight, percentile filter, and momentum-weighted percentile filter methods. We have distinguished between results for aggregate systems of raw signals and risk-adjusted signals because of their slightly different characters.

*Tables 14 and 15: Cross correlation of individual, aggregate strategies:*

*Raw Momentum*

*Risk-Adjusted Momentum*

As expected, the three methods are highly correlated for both raw and risk-adjusted methods, ranging from a low of 0.84 to a high of 0.98. However, we still expect to observe a diversification benefit when we allocate equally to the three different methods. We expect the ensemble of all model iterations to be greater than the simple average of their performances. Even iterations which are decidedly worse can contribute positively to the aggregate if they are sufficiently uncorrelated with other systems, giving us confidence that consistency and intelligent, dynamic diversification will ultimately deliver more stable results out of sample.

*Table 15: Performance comparison of ensemble models versus the simple average performance of all model iterations. The difference arises from diversification between the sub-systems.*

*Raw Momentum*

*Source: ReSolve Asset Management. Data from CSI data and underlying index providers where data for investable funds have been extended prior to their date of inception. Results reflect a strategy equally invested in all raw momentum based strategies, rebalanced monthly back to equal weight. The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained.*

*Risk-Adjusted Momentum*

*Source: ReSolve Asset Management. Data from CSI data and underlying index providers where data for investable funds have been extended prior to their date of inception. Results reflect a strategy equally invested in all risk-adjusted momentum strategies, rebalanced monthly back to equal weight. The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained.*

*Figure 1. Growth of $1 from 1991 – 2017*

*Source: ReSolve Asset Management. Data from CSI data and underlying index providers where data for investable funds have been extended prior to their date of inception. Results reflect a strategy equally invested in all raw momentum based strategies, and all risk-adjusted momentum based strategies, as indicated. Sub-strategies are equally weighted and rebalanced back to equal weight on a monthly basis. The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained.*

**CONCLUSION AND NEXT STEPS**

The purpose of this article was to introduce, and perform some simple tests to determine whether the magnitude of the momentum signature is a useful signal for the relative expected returns over the ensuing period. The results of the first test lend credence to the idea that the strength of momentum signal is correlated with better performance. However, when we compare the filtered equally-weighted results with the filtered momentum-weighted results, we see that momentum weighting is unhelpful. A more nuanced conclusion might be that it is advantageous to overweight the assets in the top half of the distribution, and underweight the assets in the bottom half. However, once we have identified the top assets, the magnitude of momentum does not provide meaningful information about relative returns between them.

We also introduced the idea of identifying the “top” assets based on a percentile filter that accounts for the cross-sectional distribution of returns, rather than using a fixed cut-off threshold for the number of assets to hold in the portfolio. This allows the size of the portfolio to “breathe” based on the distribution of momentum. The nature of the current tests does not allow for an “apples-to-apples” comparison with previous strategies, but the approach seems to have merit.

Follow-on posts will focus more specifically on the problem of portfolio optimization to target higher returns given leverage constraints, and eventually on heuristic and formal optimization methods that incorporate covariance relationships between the assets.