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In this note we explain how different factors that predict bond returns are combined in the duration model. The goal is that each factor has a similar impact on the model. We have recently enhanced the way we combine factors.
Which applications does the duration model have?
The duration model is an important performance driver in our investment process and fully determines the active positions in the funds Robeco Lux-o-rente, Robeco Flex-o-rente and Robeco Emerging Lux-o-rente. The duration model uses financial market data to capture expectations for fundamental drivers of the bond market such as economic growth and inflation. Technical factors such as trend complement these fundamental drivers.
We believe in the approach to combine multiple factors into the duration model to predict government bond returns. We want each factor to have a similar impact on the model because we cannot predict which factor will do best.
Enhancement of the model
We continuously monitor the performance of the model and strive for enhancements. As a result of this ongoing research effort we have enhanced the way to combine factors in the model. The enhancement improves the balance and robustness of the model with a more comparable impact of the model’s factors over time. The enhancement is technical. The fundamentals of the model have remained unchanged.
Comparing apples and oranges
The duration model combines the predictive power of different factors. Individual factors may underperform for a while. This is not a problem for a model combining multiple factors. It would be a problem if the model were to consist of just one factor. Ideally we would like to give more weight to the factors that we expect to perform the best in the coming period. Our research, however, has shown that we cannot predict which factor that will be. Sometimes the factor with the poorest recent performance turns out to be the best factor in the next period. We therefore decided in 1998 to apply equal weights for all factors in the model.
To combine different factors we must overcome an important hurdle, because the dimension, size and volatility of each factor are different. How to make these apples and oranges comparable?
Z-score methodology enables combination of different factors
To avoid comparing apples and oranges we need to make the different factors in the duration model comparable before we combine them. For example, the historical returns of 1-month bond and equity returns are different and can’t just be combined, because equity returns are much more volatile than bond returns.
To compare apples with apples we need to adjust the raw data. This adjustment is done by deducting the historical average and dividing the outcome by the historical volatility of the factors. When we do this normalization for the 1-month equity returns and bond returns, and produce a histogram, we obtain Figures 1 and 2. We see that the resulting equity and bond z-scores have a much more similar distribution than the original 1-month returns. The z-scores of the two factors can be compared directly and combined in the model.
Figure 1: Histogram of equity z-scores
Figure 2: Histogram of bond z-scores
Note: z-scores for 1-month equity and bond returns are computed by deducting the historical average return and dividing the outcome by the historical standard deviation of returns. From 1992 to 2013 we count how often the z-scores fall between the thresholds shown on the horizontal axis. E.g. of the 259 monthly z-scores returns for equities we see 48 months with a z-score between 0.5 and 1.0 and for bonds we see 47 months with a z-score between 0.5 and 1.0. Source: Robeco Quantitative Research, Bloomberg.
Volatility changes over time
On average the distribution of the z-scores for equity and bonds looks fine given the purpose of having equal factors weights in the model. This, however, is not always the case over shorter time periods. The volatility of equity and bond returns varies over time. A volatility measure that is adapting faster to changes of the volatility improves the balance of the model over time.
To illustrate how volatility changes over time we show the bond volatility over rolling periods in Figure 3. Figure 3 shows that the long-term bond volatility measure is slowly changing over time. A faster volatility measure illustrates that the long-term volatility is not very adaptive to changes.
For example in the most recent period we see a pick-up in the volatility of bond returns but the long-term volatility is slow to adjust. In such a period we would temporarily normalize bond returns with too low a volatility, giving bond trend more impact than equity trend.
Figure 3: Bond volatility
Note: Each month a look-back period is used to compute the volatility of the bond returns.
Volatility is either based on a long historical period (‘long-term volatility’) or a shorter
Look-back period (‘More adaptive volatility’). Source: Robeco Quantitative Research.
Enhancing the z-score methodology
We decided to adjust the normalization methodology. As a result the z-scores have become more adaptive to changes in volatility. Furthermore, the impact of extreme values is reduced. As a result of this enhancement the balance between the factors in the model has been improved.
The individual factors now have a more similar impact on the model. However, it is not our purpose for each factor to have the same impact at any point in time. E.g. if we have seen a strong negative trend in bond returns and no clear trend in equity returns, the model score will be based more on trend than on equities. But over longer periods we want the model to be balanced.
Conclusion: balance and the robustness of the model is improved
We believe in the approach to combine multiple factors into the duration model to predict government bond returns. We also want each factor to have a similar impact on the model because we cannot predict which factor will do best going forward.
To improve the equal weighting of all factors over time we have enhanced the z-score methodology. In addition this has improved the robustness of the model making z-scores more adaptive to different volatility regimes in financial markets. These changes have a small positive impact on performance and lead to a small reduction of turnover. Most importantly we have improved the balance and the robustness of the model.