In the first two articles of our Dynamic Asset Allocation for Practitioners series (Article 1 and Article 2), we explored a wide variety of ways to measure the raw, and risk adjusted, momentum of a universe of global asset classes for the purpose of ranking and allocation. We described the mathematics behind each metric, and subjected each approach to a variety of robustness checks.

Most studies of momentum based portfolios focus on a certain level of portfolio concentration, such that the portfolio always holds the top 2, 3, 4, or n assets. This is a source of potential curve fitting, as the choice of the optimal number of assets is usually made by evaluating ex post test results, and what worked best in the past may not work best in the future. In our first two posts we examined the distribution of results across portfolio concentrations ranging from 2 through 5 assets out of a 10 asset universe to provide a more comprehensive and realistic illustration of what investors might expect out of sample.

Another potential source of curve fitting relates to the asset class universe. To address this, we ran tests on 11 variations of our 10 asset class universe, whereby one asset was removed from the universe in each run to avoid cases where one asset happens to contribute an outsized amount to total performance over the test period. We showed the distribution of results across asset universes, again to give a better sense of what out of sample results might look like.

Another approach to reduce the risk of curve fitting by specifying a fixed number of portfolio holdings is to examine the distribution of momentum across asset classes at each rebalance period, and simply eliminate assets where their momentum score falls below a certain threshold relative to all of the other assets. For example, an asset might be eliminated if its momentum score falls below the average momentum score across all assets in the universe. Or we might screen for assets in the top quartile, etc.

By filtering assets based on their score relative to the other assets at each rebalance, the number of portfolio holdings is not fixed through time. Some periods may have a narrow concentration of strong assets with a majority of very weak assets, which would result in a portfolio that was more heavily concentrated in the few strong assets. In other periods, many assets might be performing well at the same time while a couple of assets lag dramatically. In this case the portfolio would hold many assets and only eliminate extreme laggards. This approach allows the portfolio the ‘breathe’ based on the concentration of momentum in the asset universe.

Chart 1. illustrates how the number of assets held at each rebalance changes through time in response to the distribution of momentum across asset classes at each rebalance date when we filter assets with a Guassian score <0.5. The number of holdings ranges from 7 during a rebalance in 2003 to as few as 2 assets in several periods. Not surprisingly, the portfolio averages 5 assets over the full test period.

Data source: Bloomberg

In addition to its role as a filtering mechanism, momentum score can also be used as a weighting tool. This constitutes a more direct approach to tests of portfolio concentration because each asset contributes to the portfolio in proportion to its observed momentum relative to other assets in the universe. We will call this approach ‘momentum weighting’. We will explore filtering with equal weighting, as well as momentum weighting below, and provide performance tests for each.

## Momentum Weights

There are many ways to apportion momentum weight, but in our testing we have found that applying a Gaussian transform to the cross sectional momentum across lookback horizons is quite effective. Essentially, this process uses a Gaussian transform to standardize relative momentum scores at each lookback horizon, such that the final momentum score for each asset is the average of the Gaussian score at each lookback. The following formulas describe the process we use to determine the weights of each asset in the portfolio.

In the above equations, G is the Gaussian value of asset *i* using lookback horizon *l*. Θ signifies that we are imposing a Gaussian transform on the value in brackets, which is simply the momentum of asset *i* at lookback horizon *l* minus the average momentum of all assets at that horizon, all divided by the standard deviation of momentum values at that horizon. The function in brackets is a z-score calculation, and the Gaussian transform translates the z score into a percentile value between 0 and 1.

Once we have transformed the raw momentum vectors for each lookback horizon into vectors of Gaussian values, we average the vectors across lookbacks to derive the final Gaussian score vector. Finally, we need to releverage the Gaussian weights so that the final weights add up to 100%. The second function accomplishes this by dividing each average Gaussian asset score by the sum of all average Gaussian scores across all assets.

This may be a little esoteric for many readers, so we have translated the formulas above into an Excel implementation below. If we assume the momentum metrics at lookback horizon *l* for assets 1 through 10 are in cells A20 through J20, the Gaussian transform for asset 1 is calculated as:

=NORMSDIST((A20-AVERAGE($A20:$J20))/STDEV($A20:$J20))

Once we have translated the momentum scores at each lookback horizon into Gaussian vectors, we average the vectors across all lookback horizons to find the average Gaussian momentum score for each asset. Finally, we releverage the Gaussian weights so that they add up to 100%. If the Gaussian vector is located in A21:J21, then the final weight for asset 1 is:

=A21/SUM($A21:$J21)

We now have a vector of positive momentum weights which we can use as weights in the portfolio at each rebalance.

If we do not apply any filter to the assets in the portfolio, we could theoretically hold all assets in proportion to their momentum weight. Alternatively, as discussed above we could choose to hold all assets with a Gaussian score above a certain percentile. For example, we might choose to hold all assets whose return is above the average return across all assets, in which case we would filter out assets with a score <0.5 and then releverage the final holding weights so that they total 100%. Or we could hold the top quartile (Gaussian score >0.75) for a portfolio that is more concentrated in the top assets.

For each of the raw momentum indicators from Article 1, and again for all of the risk adjusted metrics examined in Article 2, we will show results for portfolios where assets with average Gaussian momentum scores below 0.5 (the average score for all assets) are eliminated at each rebalance. While the use of a Guassian score threshold obviates the need to examine portfolio performance at different levels of portfolio concentration, we must still be conscious of the potential for one asset in our chosen universe to dominate our results. We follow the same process as in articles 1 and 2 to test each indicator against 11 different universe combinations to reduce this curve fitting risk, so performance statistics in the tables below are median values across the 11 universe tests.

Tables 1 and 2 show results from tests where portfolio assets with average Gaussian scores <0.5 are eliminated, and all remaining assets are held in equal weight.

Table 1. Raw momentum weight portfolios holding assets with Gaussian score >0.5, equal weight

Data Source: Bloomberg

Table 2. Risk adjusted momentum weight portfolios holding assets with Gaussian score >0.5, equal weight

Data Source: Bloomberg

Tables 3 and 4 show results from tests where portfolio assets with average Gaussian scores <0.5 are eliminated, and all remaining positions are weighted according to their relative momentum score.

Table 3. Risk adjusted momentum weight portfolios holding all assets, rebalanced monthly

Data Source: Bloomberg

Table 4. Risk adjusted momentum weight portfolios holdings assets with score >0.5

Data Source: Bloomberg

There are some noteworthy observations from these results. First, in comparison to the results in articles 1 and 2 where we published the median performance across portfolio concentrations ranging from 2 assets to 5 assets, we see a material improvement in Sharpe ratios and drawdowns from the use of a 0.5 Guassian score threshold. Clearly, by allowing the portfolio concentration to expand and contract in response to changes in the distribution of momentum across assets, the portfolio is more adaptive and this is reflected in improved performance metrics.

Further, in comparing tables 1 and 2 with tables 3 and 4, we observe that Sharpe ratios and drawdown character further improves when we weight assets in proportion to their momentum weight rather than holding the assets in equal weight.

## Conclusion

The purpose of this post was to introduce a methodology to avoid specifying a fixed number of asset holdings in the portfolio, and to introduce an alternative to simple equal weighting in the form of proportional asset weights. Test results indicate that, while we have reduced the potential for curve fitting and therefore have greater confidence in these results, this methodology also produces stronger risk adjusted performance.

In our next post, we will apply similar Guassian threshold techniques to filter out weak assets, but instead of weighting assets by their proportional momentum, we will introduce a variety of ways to weight portfolio asset according to their individual risk contribution in portfolios. You will see a substantial improvement in portfolio performance once we begin to manage the distribution of portfolio risk. Table 5. offers a sneak peak at the results from one of the risk sizing methods we will introduce in article 4; note Sharpe ratios near 1.6 and nearly 100% positive rolling 12 month periods.

Data Source: Bloomberg