bonds, interest rate regimes & trading strategies

Transcription

1 RAMSEY QUANTITATIVE SYSTEMS INC. bonds, interest rate regimes & trading strategies By Theo Athanasiadis Introduction The past 30 years have been an ideal environment for bond investors in developed countries, as interest rates have been in a secular downtrend (generating steady capital appreciation for bondholders) and the yield curve has generally remained upward sloping (generating consistent positive carry). Today s multi-decade lows in interest rates have awakened the bond market vigilantes from their long sleep. For a few years now, talk of a bond bubble has been popular, and from all sides it seems we hear prognostications of the inevitable catastrophe that is coming Introduction 1 for all bond investors when interest rates inevitably start to rise. Many pundits argued in the past few years that bonds are doomed for a collapse and investors should Historical Perspective 2 avoid having any duration risk in their portfolios. However, as is often the case, investors who approached the bond market with a variant perception performed Interest Rate Regimes 4 well in Last year s strong bond performance (U.S. Treasury Bond futures returned nearly 20%) proved those proclaiming the death of the bond bull to Reverse Historical Path 6 themselves be dead wrong (for now). CONTENTS Trading Strategies 7 Conclusion 12 References 13 Appendices 14 At the same time, in the world of CTAs, most of which have spent their entire existence in a secular bond bull market, we see widespread confidence (some might say complacency) that trendfollowing strategies will weather a rising rate environment, and their models will not only avoid getting hurt, but will excel in that environment, capturing the up-trend in rates and down-trend in bond futures. These managers have profited from a multi-decade bull market, and their assumption seems to be that the inverse of that environment would prove equally profitable, since most trendfollowing models have no explicit directional bias. In this paper we take a more cautious approach and attempt to address these issues by letting the data speak for itself. Most of the papers written on the subject either focus on a relatively short historical sample or study only the U.S. Treasury bond market. To reduce sample biases associated with the long and exceptional period that began in the early 1980s, or with focusing on a single country, we start our analysis from early 1970s and examine the four largest world economies (U.S., U.K., Germany and Japan). Extending the historical sample and adding cross-sectional data lessens those biases. Even though our sample covers a long historical period across countries and includes many instances of rising rates, it does not include a scenario of rising rates coupled with an upward-slopping yield curve. Motivated by the paper written by Roy Niederhoffer et. al. CTAs and Rising Interest Rates: Is the Party Over?, we simulate the interest rates for each country by reversing their historical path of the last 30 years while keeping their cross-sectional relation constant. In this way, we simulate an environment of rising interest rates where rates rise following the same (reverse) historical path, while the yield curve remains upward sloping (most of the time), reflecting the term premium that investors in long-term bonds require. We believe these are reasonable assumptions and are true to the data with which we have to work. It is of course unlikely that a secular bear market in bonds would follow the exact inverse path of the 1982-present bull market. The shape and curvature of the yield curve, the volatility structure of the market, and the correlation structure of the various global bond markets to one another will almost certainly be different in the future. As Mark Twain is claimed to have said, history doesn t repeat itself, but it does rhyme. As we show, the answers to the questions of whether bonds are doomed for a collapse and whether trend following models in fixed income will excel in a rising rate environment are not as easy as they may at first seem and there are other important parameters that should be considered. The first section presents a historical perspective on interest rates, emphasizing on the historical performance of longterm (10-year) government bonds of the countries considered, along with a decomposition of their returns those coming from changes in rates and those coming from rolling down the yield curve. The second section illustrates the historical performance of bonds, broken down by interest rate regimes and (again) return composition. In section three, we reverse the historical path of rates, keeping the cross-sectional relationship stable, and examine the performance of bonds in that hypothetical environment. In section four, we backtest and compare two well-known strategies: Yield Curve Carry and Trend Following (Time-Series Momentum), along with Buy and Hold, applied to both historical and reverse historical data. In the final section we attempt to draw some conclusions from the data. In the appendices we describe our data collection process and research methodology, and also examine the predictive power of carry. 1

2 Historical Perspective Figures 1 and 2 below plot the historical 3-month and 10-year zero-coupon rates for each country while Table 1 presents the correlation matrix of all rates and countries since It is obvious that: Both short- and long-term rates have been in secular downtrend since early 1980s. Failing to examine the environment before 1980 would create a bias in any analysis. The down-trend is more pronounced in the long-end of the curve driven primarily by lower inflation expectations and falling risk premium across the developed countries, whereas the shortend is more reflective of central bank policies. This also evident from the correlation matrix. Co-movements between short-term rates have been significantly weaker than those between long-term rates. There have been different market regimes and considerable dispersion among the countries both in short and long-term rates. The dispersion, measured as the spread between the minimum and maximum rates, among long-term rates has been lower than the dispersion among short-term rates with both being in a secular downtrend. However, since 2008 with zero/negative short-term rates becoming common policy across developed countries, the dispersion has moved from the short-end to the long-end of the curve. Historically, the U.K. had consistently the highest short- and long-term rates whereas Japan had the lowest. This was the case until 2008, with the spread reflecting different macro-economic conditions and central bank policies between the countries. Japan has been a special case with both its short and long-term rates exhibiting very low correlation with the rest of the countries due to its unique macro-economic situation (multi-year deflation). Figures 1 & 2: Short- & Long-Term Interest Rates Table 1: Correlation Matrix of Changes in Rates ( ) 3M U.S. 3M U.K. 3M Germany 3M Japan 10Y U.S. 10Y U.K. 10Y Germany 10Y Japan 3M U.S. 100% 3M U.K. 26% 100% 3M Germany 35% 45% 100% 3M Japan 11% 41% 16% 100% 10Y U.S. 31% 10% 18% 0% 100% 10Y U.K. 20% 43% 26% 13% 56% 100% 10Y Germany 27% 27% 34% 12% 66% 69% 100% 10Y Japan 16% 40% 21% 85% 11% 21% 23% 100% I. 2

3 Historical Performance Figures 3-6 plot the historical cumulative (log) returns for the 10-year synthetic bond futures of each country, along with their returns constituents. Table 2 provides the descriptive statistics while Table 3 decomposes the returns to their constituents. We can see that the past 30 years have been a great environment for bonds futures, delivering returns of more than 100%. But this has not always been the case. All bonds, except for Japanese Government Bonds (JGBs), suffered dramatic losses from mid-1970 to early-1980 due to rising rates and yield curve flattening. Specifically, an investor holding U.S. 10-Year Notes would have lost more than 70% from 1977 to 1981, whereas an investor holding German Euro-Bunds would have lost almost 50% during the same period. Except for the U.K. 10-Year Gilt where the yield curve spent a lot of time flat/inverted, the returns coming from the roll-carry are of equal importance with the returns from change in rates. We also see that the returns from change in exchange rates are of secondary importance over the long term. Table 2: Performance Statistics U.S. 10-Year Note U.K. 10-Year Gilt German 10-Year Euro-Bund Japanese 10-Year JGB Annualized Geometric Excess Return 2.8% 3.2% 3.5% 4.2% Annualized Excess Return 3.4% 3.8% 3.9% 4.5% Annualized Volatility 10.7% 10.8% 7.9% 7.5% Sharpe Ratio Max Drawdown -72.9% -45.3% -42.2% -39.7% Return/Max Drawdown Table 3: Return Decomposition (Average Annualized Excess Return) Roll-Carry Change in Rates Change in Exchange Rates U.S. 10-Year Note 2.0% 1.4% - U.K. 10-Year Gilt 0.1% 3.6% 0.1% German 10-Year Euro-Bund 1.6% 2.4% -0.2% Japanese 10-Year JGB 2.0% 2.3% 0.1% Figures 3 & 4: U.S. & U.K. Synthetic 10-Year Futures Cumulative Returns 3

4 Figures 5 & 6: German & Japanese Synthetic 10-Year Futures Cumulative Returns Historical Bond Performance by Interest Rate Regime Instead of breaking the historical sample into sub-periods based on discretion after eyeballing the data, a better way to study the different market environments for bonds is through the use of specific rules. Knowing that the bond market returns depend on the shape of the yield curve and the movements in rates, we use those variables to define four unique market environments/regimes: Interest Rate Regimes Regime 1: Yield Curve Flat or Inverted and Rates Falling Regime 2: Yield Curve Upward Sloping and Rates Falling Regime 3: Yield Curve Flat or Inverted and Rates Rising Regime 4: Yield Curve Upward Sloping and Rates Rising We measure the yield curve as the difference between the 10-year zero-coupon rates and 3-month T-Bill rates for each country. Looking at the historical distribution of the 3-month average yield curve for each country, we selected the average cross-country level corresponding to the 25th percentile as the yield curve threshold. If the 3-month average yield curve is above 0.3, we define the yield curve as upward sloping; otherwise it is flat or inverted. Table 4: 3-Month Average Yield Curve Percentiles Japan Germany U.S. U.K. Average Cross-Country To characterize the direction of movement in rates we used a 3 and 6 month moving average cross-over. When the 3- month average of the 10-year yield is above (below) the 6-month average, the rates are rising (falling). Table 4 presents the descriptive statistics for each bond broken down by each quadrant. 4

5 Table 4: Interest Rate Regimes and Bond Performance Regimes 10-Year U.S. Note ( ) # months % of total months 12.6% 38.8% 12.4% 36.2% Average Duration (months) Annualized Excess Return 6.6% 10.1% -3.6% -2.5% Annualized Volatility 6.7% 11.1% 14.3% 9.5% Return Decomposition (Annualized) Roll-Carry -0.4% 2.9% -0.8% 2.8% Change in Rates 7.0% 7.2% -2.9% -5.2% 10-Year U.K. Gilt ( ) # months % of total months 27.2% 31.4% 21.5% 19.9% Average Duration (months) Annualized Excess Return 4.6% 9.0% -2.5% -0.7% Annualized Volatility 9.4% 10.2% 12.2% 9.6% Return Decomposition (Annualized) Roll-Carry -2.1% 2.2% -2.2% 2.5% Change in Rates 6.7% 6.9% -0.6% -3.3% 10-Year German Euro-Bund ( ) # months % of total months 13.9% 46.6% 9.7% 29.8% Average Duration (months) Annualized Excess Return 6.3% 10.5% -13.9% -0.6% Annualized Volatility 7.2% 7.0% 11.6% 6.8% Return Decomposition (Annualized) Roll-Carry -0.8% 2.5% -1.6% 2.4% Change in Rates 7.2% 8.2% -11.8% -2.9% 10-Year Japanese JGB ( ) # months % of total months 5.4% 59.3% 5.1% 30.2% Average Duration (months) Annualized Excess Return 6.0% 6.1% -4.4% 2.1% Annualized Volatility 10.5% 6.6% 13.5% 7.3% Return Decomposition (Annualized) Roll-Carry -1.5% 2.5% -2.4% 2.7% Change in Rates 7.3% 3.5% -2.3% -0.6% The information in Table 4 is helpful for understanding how roll-carry and change in rates are affected by movements in rates and the slope of the yield curve. A sharp rise in rates similar to early 1980 s is enough to cancel out any benefits that comes from the roll-carry while a flatter/inverted yield curve can significantly reduce and cancel the roll-carry. Looking at the table, although there is enough differentiation among countries, we can identify some common characteristics: 1. The yield curve for the most part has been upward sloping in all countries reflecting the term premium that investors require for duration and liquidity risk, with regimes 2 and 4 covering more than 70% of the historical period except for the U.K. where the yield curve has been relatively flat for almost half the historical sample. 2. Regime 2 has also been the best environment for bonds, coinciding with the highest returns driven mainly by falling rates (75%) and less by the roll-carry (25%). It has been the most prevalent historically in every country ranging from 32% in U.K. to 60% in Japan, and also the stickiest with the longest duration. 3. Regime 3 has been the worst and most volatile environment with the lowest and negative bond returns coming from both the roll-carry and rising rates. 4. Although the return from the roll-carry is relatively small compared to the return from the change in rates in any given regime, when we combine the regimes over all of time, a big part of the return from change in rates cancels out, whereas the roll-carry return remains, reflecting the asymmetric nature of the yield curve. 5

6 What if Rates Rise Following the Reverse Historical Path? As we saw, the worst environment for bonds was during late 1970 s and 1980 s when rates rose and the yield curve flattened/inverted. Except for that short time period there is not enough data to test the impact of rising rates on bonds. To address that issue, we simulate a rising rate environment by reversing historical path of rates from while keeping their cross-sectional relationship constant. 1 Using the same formulas for the estimation of the synthetic futures price and return, we estimated the simulated performance of a 10-year synthetic bond for each country assuming that rates follow the reverse historical path. Figures 7-10 plot the cumulative log return of each synthetic 10-year bond along with its return constituents, while Tables 5 and 6 present the descriptive statistics and return decomposition. In most cases the return from roll-carry cancels out with the return from change in rates creating a range- bound price series for each bond, except for the U.K. 10-Year Gilt whose price falls due to zero roll-carry and rising rates. In other words, if rates increase following the reverse historical path, assuming that the shape of the yield curve remains the same with the historical, bond futures prices will not necessarily go down (except for the U.K.) but they will oscillate. Table 5: Performance Statistics (Reverse Historical Order) U.S. 10-Year Note U.K. 10-Year Gilt German 10-Year Euro-Bund Japanese 10-Year JGB Annualized Geometric Return -0.1% -3.1% -1.0% -0.3% Annualized Return -0.4% -2.6% -0.7% -0.1% Annualized Volatility 10.5% 10.1% 7.3% 6.2% Sharpe Ratio Max Drawdown % % -42.8% -42.2% Return/Max Drawdown -0.4% -2.6% -2.3% -0.8% Table 6: Return Decomposition (Average Annualized Excess Return) Roll-Carry Change in Rates Change in Exchange Rates U.S. 10-Year Note 2.4% -2.7% - U.K. 10-Year Gilt 0.4% -3.0% 0.0% German 10-Year Euro-Bund 1.8% -2.3% -0.1% Japanese 10-Year JGB 2.5% -2.7% 0.0% Figures 7 & 8: U.S. & U.K. Synthetic 10-Year Futures Cumulative Returns (Reverse Path) 1 We chose this period from the whole sample because if we included the period and reversed the history it would result in an inverted yield curve with rates falling. 6

7 Figures 9 & 10: German & Japanese Synthetic 10-Year Futures Cumulative Returns (Reverse Path) Trading Strategies on Bond Futures In the following pages we compare two well-known trading strategies: Trend Following (Time-Series Momentum) and Yield Curve Carry, along with Buy & Hold on both historical and reverse historical data. For the purposes of this paper we avoid any system parameterization and focus on plain vanilla systems that should capture the general effects we wish to isolate. Transaction costs are not considered. To test a trend following system, we create a synthetic bond futures price index and choose a simple moving average crossover system of 1 and 10 months which is widely cited in literature. When the 1 month price is above (below) the 10 month average price, the model establishes a long (short) position for that month. The Yield Curve Carry model is also kept simple, generating a long (short) signal if the 10-year zero-coupon rate is higher (lower) than the short-term risk-free rate. In other words, as long as the yield curve is upward slopping the model is long; if the yield curve inverts the model goes short. Historical System Performance Tables 7-8 and Figures present the historical system performance of an equal weight portfolio composed of all bonds, and of each bond separately broken down by long and short signals. We can observe that: 1. Over the whole period, Yield Curve Carry and Buy & Hold strategies have outperformed Trend Following. This is true not only for the equal weighted portfolio but for most individual bonds. More specifically, Yield Curve Carry was the best strategy for Japanese and U.S. bonds, Buy & Hold was the best strategy in U.K. bonds and Trend Following was the best in German bonds. 2. Buy & Hold performed better than Trend Following (both long only and long/short), even though it suffered a deep drawdown in late 1970 s and 1980 s where rates spiked in most countries. The only case that Buy & Hold performed better than Yield Curve Carry was in U.K. bonds due to the flat/inverted yield curve that was dominant in that historical sample. 3. Trend Following was the best performer during the late 1970 s and 1980 s where rates rose sharply. During those periods YC Carry was mediocre because the yield curve flattened and Buy & Hold struggled because of higher rates and flatter/inverted yield curve. However, since 1990 s increases in rates were smoother and the yield curve remained relatively steep in most countries, causing Trend Following to underperform the other two strategies. 4. The correlation between Buy & Hold and the other two strategies is relatively low, with Yield Curve Carry having the highest and the Trend Following strategy having the lowest. Yield Curve Carry and Trend Following are highly correlated (50%) driven primarily by the correlation of their long signals (80%). 7

8 Table 7: Performance Statistics Trading Strategies Buy & Hold Carry-Long Carry-Short Carry-Total TF-Long TF-Short TF-Total 10-Year U.S. Note ( ) Annualized Excess Return 2.9% 3.1% 0.2% 3.2% 2.7% -0.2% 2.5% Annualized Volatility 10.7% 9.3% 5.2% 10.7% 8.0% 7.1% 10.7% Sharpe Ratio Max Drawdown -72.8% -40.7% -34.9% -41.2% -24.8% -72.6% -44.8% Return/max DD Year U.K. Gilt ( ) Annualized Excess Return 3.5% 2.2% -1.2% 1.0% 2.0% -1.4% 0.5% Annualized Volatility 10.2% 7.6% 6.8% 10.3% 7.8% 6.6% 10.3% Sharpe Ratio Max Drawdown -41% -27% -53% -63% -34% -61% -70% Return/max DD Year German Euro-Bund ( ) Annualized Excess Return 3.8% 4.0% 0.2% 4.2% 4.1% 0.3% 4.4% Annualized Volatility 7.8% 6.4% 4.5% 7.8% 5.9% 5.1% 7.8% Sharpe Ratio Max Drawdown -42.2% -18.4% -24.2% -38.3% -20.9% -23.0% -36.7% Return/max DD % Year Japanese JGB ( ) Annualized Excess Return 4.2% 4.4% 0.2% 4.6% 3.7% -0.5% 3.1% Annualized Volatility 7.6% 6.7% 3.5% 7.6% 6.1% 4.5% 7.6% Sharpe Ratio Max Drawdown -39.7% -18.8% -24.3% -27.9% -19.1% -40.2% -29.2% Return/max DD Equal Weight Portfolios ( ) Annualized Excess Return 2.5% 2.4% 0.0% 2.4% 2.3% -0.2% 2.1% Annualized Volatility 5.6% 4.4% 2.8% 4.8% 4.1% 3.5% 5.1% Sharpe Ratio Max Drawdown -33.3% -15.6% -19.9% -12.3% -13.6% -34.8% -24.3% Return/max DD Correlation Matrix of Equal Weight Portfolios ( ) Buy & Hold 100% Carry-Long 87% 100% Carry-Short -64% -18% 100% Carry-Total 42% 81% 42% 100% TF-Long 79% 83% -29% 59% 100% TF-Short -69% -43% 70% 2% -9% 100% TF-Total 17% 37% 25% 49% 74% 60% 100% 8

9 Figures 11-15: Historical Strategy Performance on 10-Year Bond Futures 9

10 Reverse Historical Path Performance We repeat the same analysis using the reverse historical path of rates. Tables 9-10 and Figures present the system performance of an equal weighted portfolio and of each bond broken by long and short signals. We see that if rates increase following the reverse historical path: 1. Yield Curve Carry will outperform Trend Following and Buy & Hold strategies. This holds both for the equal weighted portfolio and for each bond individually. As we noticed before, if the curve remains relatively steep and rates don t spike violently (similar to 1980 s or 1990 s), the return from roll-carry will be more than enough to offset any drag in performance from higher rates. 2. Buy & Hold and Trend Following will struggle because of range bound futures prices driven by the offset of rising rates by positive carry. 3. Although Yield Curve Carry will remain highly correlated with the Buy & Hold, the relative outperformance will be massive. On the contrary, the correlation of Trend-Following with the other two will become negative driven by its shorts. 4. As we show in the appendix, part of the strong performance of Yield Curve Carry is driven in part by the forecasting power of the yield curve. Historically, a steep yield curve not only contained the term premia required by investors, but also (accurately) forecasted lower rates for the next month. Table 7: Performance Statistics Trading Strategies Buy & Hold Carry-Long Carry-Short Carry-Total TF-Long TF-Short TF-Total 10-Year U.S. Note ( ) Annualized Excess Return 2.9% 3.1% 0.2% 3.2% 2.7% -0.2% 2.5% Annualized Volatility 10.7% 9.3% 5.2% 10.7% 8.0% 7.1% 10.7% Sharpe Ratio Max Drawdown -72.8% -40.7% -34.9% -41.2% -24.8% -72.6% -44.8% Return/max DD Year U.K. Gilt ( ) Annualized Excess Return 3.5% 2.2% -1.2% 1.0% 2.0% -1.4% 0.5% Annualized Volatility 10.2% 7.6% 6.8% 10.3% 7.8% 6.6% 10.3% Sharpe Ratio Max Drawdown -41% -27% -53% -63% -34% -61% -70% Return/max DD Year German Euro-Bund ( ) Annualized Excess Return 3.8% 4.0% 0.2% 4.2% 4.1% 0.3% 4.4% Annualized Volatility 7.8% 6.4% 4.5% 7.8% 5.9% 5.1% 7.8% Sharpe Ratio Max Drawdown -42.2% -18.4% -24.2% -38.3% -20.9% -23.0% -36.7% Return/max DD % Year Japanese JGB ( ) Annualized Excess Return 4.2% 4.4% 0.2% 4.6% 3.7% -0.5% 3.1% Annualized Volatility 7.6% 6.7% 3.5% 7.6% 6.1% 4.5% 7.6% Sharpe Ratio Max Drawdown -39.7% -18.8% -24.3% -27.9% -19.1% -40.2% -29.2% Return/max DD

11 Trading Strategies Buy & Hold Carry-Long Carry-Short Carry-Total TF-Long TF-Short TF-Total Equal Weight Portfolios ( ) Annualized Excess Return 2.5% 2.4% 0.0% 2.4% 2.3% -0.2% 2.1% Annualized Volatility 5.6% 4.4% 2.8% 4.8% 4.1% 3.5% 5.1% Sharpe Ratio Max Drawdown -33.3% -15.6% -19.9% -12.3% -13.6% -34.8% -24.3% Return/max DD Correlation Matrix of Equal Weight Portfolios ( ) Buy & Hold 100% Carry-Long 87% 100% Carry-Short -64% -18% 100% Carry-Total 42% 81% 42% 100% TF-Long 79% 83% -29% 59% 100% TF-Short -69% -43% 70% 2% -9% 100% TF-Total 17% 37% 25% 49% 74% 60% 100% Figures 15-17: Strategy Performance on 10-Year Bond Futures (Reverse Path) 11

12 Conclusion Figures 18 & 19: Strategy Performance on 10-Year German & Japanese Bond Futures (Reverse Path) The last 30 years have been an ideal environment for bond investors in developed countries, as interest rates have been in a secular downtrend and the yield curve has generally remained upward sloping. Using the short- and longterm interest rates of the world s four largest economies (U.S., U.K., Germany and Japan) since the 1970 s, we show that there have been time periods (such as the late 1970 s and 1980 s) where interest rates rose and where the yield curve flattened or inverted in one or more countries. Failing to examine those historical periods and look across countries would create a bias in any fixed-income analysis. The performance of the 10-year government bond futures of those countries over this 45-year period has been solid, driven almost equally by the carry from rolling down the yield curve and falling rates. Breaking down the historical sample for each country into regimes, as defined by the shape of the yield curve and the movement in rates, we see that: The yield curve has been upward sloping for most (70%) of the historical period in all four countries, reflecting the term premium that investors require for duration and liquidity risk. The regime of falling rates and upward sloping yield curve has been the most prevalent and stickiest in all countries historically. It has also, logically, been the best environment for bond investors. On the contrary, the regime of a flat or inverted yield curve and rates rising has been the worst and most volatile regime for bonds, although this has been fairly rare. Although the return from the roll-carry is relatively small compared to the return from the change in rates in any given regime, when we combine the regimes over all of time, a big part of the return from change in rates cancels out, whereas the roll-carry return remains, reflecting the asymmetric nature of the yield curve. The historical sample doesn t contain a long time period where rates rose smoothly and the yield curve remained upward-sloping. We simulate this environment reversing the historical path of rates of the last 30 years for each country. We see that if rates rise following the reverse path, the prices of the bond futures will move in a rangebound fashion with a small negative drift. Examining the historical performance of different strategies we see that Buy & Hold and Yield Curve Carry strategies have outperformed Trend Following over the past 45 years. However, both strategies underperformed Trend Following during the late 1970 s and early 1980 s when rates spiked violently and the yield curve inverted in most countries. Performing the same analysis using the reverse historical path of rates, we see that Yield Curve Carry outperforms both Buy & Hold and Trend Following strategies. If the yield curve remains upward-sloping and the roll-carry is enough to cancel out the losses from the increase in interest rates, Trend Following strategies will struggle. Trend Following as a strategy has been massively aided by the bull market in bonds that began in As we show, much of the return we associated with Trend Following strategies in the past few decades is really related to Yield Curve Carry, not price momentum. While this may seem like a semantic or academic distinction, it has a powerful implication for the ability of Trend Following strategies in bonds to perform well in the future. Simply put, in a bear market for bonds with a steep yield curve, we would expect trendfollowers to perform poorly. The strong performance of the Yield Curve Carry strategies is strengthened by the positive correlation between the slope of the yield curve and the future rate change. If this relationship breaks down (e.g. upward-sloping yield curves followed by higher rates) the returns of the Yield Curve Carry strategy would be reduced. 12

13 References Ilmanen, Antti Understanding the Yield Curve: Part 6 A Framework for Analyzing Yield Curve Trades.. New York: Salomon Brothers Ilmanen, Antti Expected Returns (John Wiley: United Kingdom) Koijen, Ralph S. J. and Moskowitz, Tobias J. and Pedersen, Lasse Heje and Vrugt, Evert B., Carry (November 7, 2013). Fama-Miller Working Paper. Available at SSRN: or ssrn Niederhoffer, Roy and Weddepohl, Coen CTAs and Rising Interest Rates: Is the Party Over?. White paper 13

14 Appendices APPENDIX A1: Data Our sample covers short- and long-term interest rates of the world s largest economies (U.S., U.K., Germany and Japan) starting as far back as January 1971 through December Specifically, our data spans through the following periods: for the U.S., for the U.K., for Germany and for Japan. We estimate the monthly risk-free rate for the U.S. using the 1-month T-Bill from Ibbotson Associates (available from the Professor Kenneth French s online data library 1 ) and we append it with the 1-month LIBOR rate for the period For the risk-free rates of Germany, Japan and the U.K. we use their 3-month T-Bill rates. We derive synthetic futures prices based on data on zero-coupon rates, using 10-year and 9-year and 11-month zerocoupon rates. We estimate a 9-year and 11-month zero-coupon rate by interpolating (linearly) a 9-year and a 10-year zero-coupon rates. The data for the 3-month T-Bills, 9 and 10-year zero coupon rates for the period up until 12/1994 is available on the website of Professor Jonathan Wright 2 and we append it with data from Bloomberg for the period APPENDIX A2: Synthetic Bond Futures Price and Return Estimation The first futures contract on the 10-year government bonds started trading in 1980 s, so we synthetically create the futures prices using a 9- and 11-month and a 10-year zero-coupon rates in order to go back further in time using the methodology outlined in Ilmanen (1995) and Koijen et al. (2013). The synthetic futures price of a 10-year bond is equal to: The price of a 9 year and 11 month cash bond is equal to: S = 100 *( 1+ RiskFreeRate / 12) F = ( 1+ 10YearZeroCouponRate) ( 1+ 9Year11MonthZeroCouponRate) The monthly return, assuming the last day of each month the futures contract converges to the spot, is: 100 Return(t)= St () 1 Ft ( 1) For foreign bonds that are denominated in the foreign currency, we estimate the US-dollar return. Letting e(t) be the exchange rate measured in number of local currency per unit of foreign currency, the hedged dollar return in excess of the local risk-free rate is: DollarReturn t ()= () 10 () ( ) St et - * F( t- ) 1 1 et-1 We use the dollar return for foreign bonds but it is very close to local currency return since the last term in the equation above is of second-order importance. We can further decompose the futures return into the expected carry that comes from rolling down the yield curve (assuming that the curve remains stable) and the change from rates. Taking the logarithms of prices we have: LogReturn t ()= () ( ) St log F t 1 St = 1 log F( t 1 ) + St log St ( 1) = Roll-Carry + Change in rates We validated the returns of the synthetic bond futures by comparing them to actual futures returns. In all cases the correlation between the two series is above 96%. Any small differences can be attributed to the use of linear interpolation, the rolling policy and the delivery options of the futures (cheapest to deliver bond). The results are available upon request ( ) ()

15 APPENDIX A3: The Predictive Power of Carry Historically Yield Curve Carry has been the best performing strategy helped by a persistently steep yield curve. To examine whether this strong performance is a result of the slope of the yield curve and/or if there is any forecasting power embedded in the carry, we run the following predictive regression for each bond futures contract: Bond Excess Return (t+1) = a+ b*carry(t)+e Where Carry is the carry of each bond at time t estimated as the log ratio of the 10-year bond to the 9- & 11-month bond (where the bond is going to settle if nothing happens) and b is the coefficient that measures carry s ability to predict future bond returns. Depending on the value of the beta coefficient we have the following scenarios: b=1: Prices follow a Random Walk and the whole return comes from carry (Risk Premium Hypothesis) b>1: Carry is not only a part of the return but it also helps forecast future price movements 0<b<1: Carry is a part of the return but the prices move in the opposite direction canceling out some of the carry benefits (mix of Expectations and Risk Premium Hypotheses) b=0: Returns follow a Random Walk and carry does not offer any benefit (Expectations Hypothesis) b<0: Price movements are in the opposite direction from carry and they more than cancel out the carry Monthly Predictive Regressions U.S. 10-Year Note U.K. 10-Year Gilt German 10-Year Euro-Bund Japanese 10-Year JGB Sample Beta Coefficient Standard Error t-statistic Looking at the Table 11 we see that the beta coefficients, except for the U.K. bond, have been above 1 and statistically significant. The results support that historically the roll-carry not only offered a risk-premium but it also had forecasting power for the movement in rates over the next month. In other words, a steep yield curve contained a term premium and forecasted a lower move in rates over the next month. In the case of U.K. bonds, the beta coefficient is very close to 1, meaning that U.K. bond prices behavior was very close to a random walk and almost all the return came from the carry piece. Part of the reason why the beta coefficients have been considerably above 1 was the secular down trend in rates driven by lower inflation expectations, strong demand for bonds and investors confidence in the central banks. This exceptional environment might change going forward with the rates and inflation close to their lowest historical levels. In that environment, a fair expectation for the beta coefficient would be closer to 1 as has been for the U.K. bonds historically. This means that the future returns from Yield Curve Carry strategies would still be positive but lower. 15

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