*This post is available as a PDF download here.*

# Summary

- Yield curve changes over time can be decomposed into Level, Slope, and Curvature changes, and these changes can be used to construct portfolios.
- Market shocks, monetary policy, and preferences of different segments of investors (e,g. pensions) may create trends within these portfolios that can be exploited with absolute and relative momentum.
- In this commentary, we investigate these two factors in long/short and long/flat implementations and find evidence of success with some structural caveats.
- Despite this, we believe the results have potential applications as either a portable beta overlay or for investors who are simply trying to figure out how to position their duration exposure.
- Translating these quantitative signals into a forecast about yield-curve behavior may allow investors to better position their fixed income portfolios.

It has been well established in fixed income literature that changes to the U.S. Treasury yield curve can be broken down into three primary components: a level shift, a slope change, and a curvature twist.

A level change occurs when rates increase or decrease across the entire curve at once. A slope change occurs when short-term rates decrease (increase) while long-term rates increase (decrease). Curvature defines convexity and concavity changes to the yield curve, capturing the bowing that occurs towards the belly of the curve.

Obviously these three components do not capture 100% of changes in the yield curve, but they do capture a significant portion of them. From 1962-2019 they explain 99.5% of the variance in daily yield curve changes.

We can even decompose longer-term changes in the yield curve into these three components. For example, consider how the yield curve has changed in the three years from 6/30/2016 to 6/30/2019.

*Source: Federal Reserve of St. Louis.*

We can see that there was generally a positive increase across the entire curve (i.e. a positive level shift), the front end of the curve increased more rapidly (i.e. a flattening slope change) and the curve flipped from concave to convex (i.e. an inverted bowing of the curve).

Using the historical yield curve changes, we can mathematically estimate these stylized changes using principal component analysis. We plot the loadings of the first three components below for this three-year change.

*Source: Federal Reserve of St. Louis. Calculations by Newfound Research.*

We can see that **–PC1–** has generally positive loadings across the entire curve, and therefore captures our level shift component. **–PC2–** exhibits negative loadings on the front end of the curve and positive loadings on the back, capturing our slope change. Finally, **–PC3–** has positive loadings from the 1-to-5-year part of the curve, capturing the curvature change of the yield curve itself.

Using a quick bit of linear algebra, we can find the combination of these three factors that closely matches the change in the curve from 6/30/2016 to 6/30/2019. Comparing our model versus the actual change, we see a reasonably strong fit.

*Source: Federal Reserve of St. Louis. Calculations by Newfound Research.*

So why might this be useful information?

First of all, we can interpret our principal components as if they are portfolios. For example, our first principal component is saying, “buy a portfolio that is long interest rates across the entire curve.” The second component, on the other hand, is better expressed as, “go short rates on the front end of the curve and go long rates on the back end.”

Therefore, insofar as we believe changes to the yield curve may exhibit absolute or relative momentum, we may be able to exploit this momentum by constructing a portfolio that profits from it.

As a more concrete example, if we believe that the yield curve will generally steepen over the next several years, we might buy 2-year U.S. Treasury futures and short 10-year U.S. Treasury futures. The biggest wrinkle we need to deal with is the fact that 2-year U.S. Treasury futures will exhibit very different sensitivity to rate changes than 10-year U.S. Treasury futures, and therefore we must take care to duration-adjust our positions.

Why might such changes exhibit trends or relative momentum?

- During periods where arbitrage capital is low, trends may emerge. We might expect this during periods of extreme market shock (e.g. recessions) where we might also see the simultaneous influence of monetary policy.
- Effects from monetary policy may exhibit autocorrelation. If investors exhibit any anchoring to prior beliefs, they might discount future policy changes.
- Segmented market theory suggests that different investors tend to access different parts of the curve (e.g. pensions may prefer the far end of the curve for liability hedging purposes). Information flow may therefore be segmented, or even impacted by structural buyers/sellers, creating autocorrelation in curve dynamics.

In related literature, Fan et al (2019) find that the net hedging or speculative position has strong cross-sectional explanatory power for agricultural and currency futures returns, but *not *in fixed income markets. To quote,

“In sharp contrast, we find no evidence of a significant speculative pressure premium in the interest rate and fixed income futures markets. Thus, albeit from the lens of different research questions, our paper reaffirms Bessembinder (1992) and Moskowitz et al. (2012) in establishing that fixed income futures markets behave differently from other futures markets as regards the information content of the net positions of hedgers or speculators. A hedgers-to-speculators risk transfer in fixed income futures markets would be obscured if agents choose to hedge their interest rate risk with other strategies (i.e. immunization, temporary change in modified duration).”

Interestingly, Markowitz et al. (2012) suggest that speculators may be profiting from time-series momentum at the expense of hedgers, suggesting that they earn a premium for providing liquidity. Such does not appear to be the case for fixed income futures, however.

As far as we are aware, it has not yet been tested in the literature whether the net speculator versus hedger position has been tested for yield curve trades, and it may be possible that a risk transfer does not exist at the individual maturity basis, but rather exists for speculators willing to bear level, slope, or curvature risk.

**Stylized Component Trades**

While we know the exact loadings of our principal components (i.e. which maturities make up the principal portfolios), to avoid the risk of overfitting our study we will capture level, slope, and curvature changes with three different stylized portfolios.

To implement our portfolios, we will buy a basket of 2-, 5-, and 10-year U.S. Treasury futures contracts (“UST futures”). We will assume that the 5-year contract has 2.5x the duration of the 2-year contract and the 10-year contract has 5x the duration of the 2-year contract.

To capture a level shift in the curve, we will go long across all the contracts. Specifically, for every dollar of 2-year UST futures exposure we purchase, we will buy $0.4 of 5-year UST futures and $0.20 of 10-year UST futures. This creates equal duration exposure across the entire curve.

To capture slope change, we will go short 2-year UST futures and long the 10-year UST futures, holding zero position in the 5-year UST futures. As before, we will duration-adjust our positions such that for each $1 short of the 2-year UST futures position, we are $0.20 long the 10-year UST futures.

Finally, to capture curvature change we will construct a butterfly trade where we short the 2- and 10-year UST futures and go long the 5-year UST futures. For each $1 long in the 5-year UST futures, we will short $1.25 of 2-year UST futures and $0.25 of 10-year UST futures.

Note that the slope and curvature portfolios are implemented such that they are duration neutral (based upon our duration assumptions) so a level shift in the curve will generate no profit or loss.

An immediate problem with our approach arises when we actually construct these portfolios. Unless adjusted, the volatility exhibited across these trades will be meaningfully different. Therefore, we target a constant 10% volatility for all three portfolios by adjusting the notional exposure of each portfolio based upon an exponentially-weighted estimate of prior 3-month realized volatility.

*Source: Stevens Futures. Calculations by Newfound Research. **Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.*

It appears, at least to the naked eye, that changes in the yield curve – and therefore the returns of these portfolios – may indeed exhibit positive autocorrelation. For example, **–Slope–** appears to exhibit significant trends from 2000-2004, 2004-to 2007, and 2007-2012.

Whether those trends can be identified and exploited is another matter entirely. Thus, with our stylized portfolios in hand, we can begin testing.

**Trend Signals**

We begin our analysis by exploring the application of time-series momentum signals across all three of the portfolios. We evaluate lookback horizons ranging from 21-to-294 trading days (or, approximately 1-to-14 months). Portfolios assume a 21-trading-day holding period and are implemented using 21 overlapping portfolios to control for timing luck.

*Source: Stevens Futures. Calculations by Newfound Research. **Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.*

Some observations:

- Time-series momentum appears to generate positive returns for the Level portfolio. Over the period tested, longer-term measures (e.g. 8-to-14-month horizons) offer more favorable results.
- Time-series momentum on the Level portfolio does, however, underperform naïve buy-and-hold. The returns of the strategy also do not offer a materially improved Sharpe ratio or drawdown profile.
- Time-series momentum also appears to capture trends in the Slope portfolio. Interestingly, both short- and long-term lookbacks are less favorable over the testing period than intermediate-term (e.g. 4-to-8 month) ones.
- Finally, time-series momentum appeared to offer no edge in timing curvature trades.

Here we should pause to acknowledge that we are blindly throwing strategies at data without much forethought. If we consider, however, that we might reasonably expect duration to be a positively compensated risk premium, as well as the fact that we would expect the futures to capture a generally positive roll premium (due to a generally upward sloping yield curve), then explicitly shorting duration risk may not be a keen idea.

In other words, it may make more sense to implement our level trade as a long/flat rather than a long/short. When implemented in this fashion, we see that the annualized return versus buy-and-hold is much more closely maintained while volatility and maximum drawdown are significantly reduced.

*Source: Stevens Futures. Calculations by Newfound Research. **Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.*

Taken together, it would appear that time-series momentum may be effective for trading the persistence in Level and Slope changes, though not in Curvature.

**Momentum Signals**

If we treat each stylized portfolio as a separate asset, we can also consider the returns of a cross-sectional momentum portfolio. For example, each month we can rank the portfolios based upon their prior returns. The top-ranking portfolio is held long; the 2^{nd} ranked portfolio is held flat; and the 3^{rd} ranked portfolio is held short.

As before, we will evaluate lookback horizons ranging from 21-to-294 trading days (approximately 1-to-14 months) and assuming a 21-trading-day holding period, implemented with 21 overlapping portfolios.

Results – as well as example allocations from the 7-month lookback portfolio – are plotted below.

*Source: Stevens Futures. Calculations by Newfound Research. *

Here we see very strong performance results except in the 1- and 2-month lookback periods. The allocation graph appears to suggest that results are not merely the byproduct of consistently being long or short a particular portfolio and the total return level appears to suggest that the portfolio is able to simultaneously profit from both legs.

If we return back to the graph of the stylized portfolios, we can see a significant negative correlation between the Level and Slope portfolios from 1999 to 2011. The negative correlation appears to disappear after this point, almost precisely coinciding with a 6+ year drawdown in the cross-sectional momentum strategy.

This is due to a mixture of construction and the economic environment.

From a construction perspective, consider that the Level portfolio is long the 2-, the 5-, and the 10-year UST futures while the Slope portfolio is short 2-year and long the 10-year UST futures. Since the positions are held in a manner that targets equivalent duration exposure, when the 2-year rate moves more than the 10-year rate, we end up in a scenario where the two trades have negative correlation, since one strategy is short and the other is long the 2-year position. Conversely, if the 10-year rate moves more than the 2-year rate, we end up in a scenario of positive correlation, since both strategies are long the 10-year.

Now consider the 1999-2011 environment. We had an easing cycle during the dot-com bust, a tightening cycle during the subsequent economic expansion, and another easing cycle during the 2008 crisis. This caused significantly more directional movement in the 2-year rate than the 10-year rate. Hence, negative correlation.

After 2008, however, the front end of the curve became pinned to zero. This meant that there was significantly more movement in the 10-year than the 2-year, leading to positive correlation in the two strategies. With positive correlation there is less differentiation among the two strategies and so we see a considerable increase in strategy turnover – and effectiveness – as momentum signals become less differentiated.

With that in mind, had we designed our Slope portfolio to be long 2-year UST futures and short 10-year UST futures (i.e. simply inverted the sign of our allocations), we would have seen *positive *correlation between Level and Slope from 1999 to 2011, resulting in a very different set of allocations and returns. In actually testing this step, we find that the 1999-2011 period is no longer dominated by Level versus Slope trades, but rather Slope versus Curvature. Performance of the strategy is still largely positive, but the spread among specifications widens dramatically.

Taken all together, it is difficult to conclude that the success of this strategy was not, in essence, driven almost entirely by autocorrelation in easing and tightening cycles with a relatively stable back end of the curve.^{1} Given that there have only been a handful of full rate cycles in the last 20 years, we’d be reluctant to rely too heavily on the equity curve of this strategy as evidence of a robust strategy.

**Conclusion**

In this research note, we explored the idea of generating stylized portfolios designed to isolate and profit from changes to the form of the yield curve. Specifically, using 2-, 5-, and 10-year UST futures we design portfolios that aim to profit from level, slope, and curvature changes to the US Treasury yield curve.

With these portfolios in hand, we test whether we can time exposure to these changes using time-series momentum.

We find that while time-series momentum generates positive performance for the Level portfolio, it fails to keep up with buy & hold. Acknowledging that level exposure may offer a positive long-term risk premium, we adjust the strategy from long/short to long/flat and are able to generate a substantially improved risk-adjusted return profile.

Time-series momentum also appears effective for the Slope portfolio, generating meaningful excess returns above the buy-and-hold portfolio.

Applying time-series momentum to the Curvature portfolio does not appear to offer any value.

We also tested whether the portfolios can be traded employing cross-sectional momentum. We find significant success in the approach but believe that the results are an artifact of (1) the construction of the portfolios and (2) a market regime heavily influenced by monetary policy. Without further testing, it is difficult to determine if this approach has merit.

Finally, even though our study focused on portfolios constructed using U.S. Treasury futures, we believe the results have potential application for investors who are simply trying to figure out how to position their duration exposure. For example, a signal to be short (or flat) the Level portfolio and long the Slope portfolio may imply a view of rising rates with a flattening curve. Translating these quantitative signals into a forecast about yield-curve behavior may allow investors to better position their fixed income portfolios.

Since this study utilized U.S. Treasury futures, these results translate well to implementing a portable beta strategy. For example, if you were an investor with a desired risk profile on par with 100% equities, you could add bond exposure *on top* of the higher risk portfolio. This would add a (generally) diversifying return source with only a minor cash drag to the extent that margin requirements dictate.

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