Inside the Yield Book E-Book

Martin L. Leibowitz

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Bloomberg Professional
• Sprache: Englisch

Seitenzahl: 407

Veröffentlichungsjahr: 2013

Beschreibung

A completely updated edition of the guide to modern bond analysis First published in 1972, Inside the Yield Book revolutionized the fixed-income industry and forever altered the way investors looked at bonds. Over forty years later, it remains a standard primer and reference among market professionals. Generations of practitioners, investors, and students have relied on its lucid explanations, and readers needing to delve more deeply have found its explication of key mathematical relationships to be unmatched in clarity and ease of application. This edition updates the widely respected classic with new material from Martin L. Leibowitz. Along the way, it skillfully explains and makes sense of essential mathematical relationships that are basic to an understanding of bonds, annuities, and loans--in fact, any securities or investments that involve compound interest and the determination of present value for future cash flows. The book also includes a new foreword. * Contains information that is more instructive, important, and useful than ever for mastering the crucial concepts of time, value, and return * Combines the clear fixed-income insights found in the original edition with completely new knowledge to help you navigate today's dynamic market * Includes over one hundred pages of new material on the role of bonds within the total portfolio In an era of calculators and computers, some of the important underlying principles covered here are not always grasped thoroughly by market participants. Investors, traders, and analysts who want to sharpen their ability to recall and apply these fundamentals will find Inside the Yield Book the perfect resource.

Leseprobe

Contents

Preface to the 2013 Edition

Acknowledgments

Part I: Duration Targeting: A New Look at Bond Portfolios (2013 Edition)

Introduction

Chapter 1: Duration Targeting and the Trendline Model

Duration Targeting

Trendlines

Generality of the TL Model

A Jump Yield Path

Mirror-Image Paths

Random Paths

Random Paths to the Same Ending Yield

Trendline Duration

Horizon Effects

Conclusion

Chapter 2: Volatility and Tracking Error

Introduction

TL Volatility

Non-Trendline Yield Paths

Tracking Errors

Total Volatility

Random Yield Walks with Drift

Chapter 3: Historical Convergence to Yield

Introduction

Historical Yield Paths

Historical Yield Volatilities

Historical Tracking Errors

Conclusion

Chapter 4: Barclays Index and Convergence to Yield

Introduction

Historical Data

Holding Period Returns for a December 2000 Investment

Holding Period Returns for Three Different Entry Points

Holding Period Returns for All Entry Points

Total Return Volatility

Barclays Returns and Tracking Error

Individual Credit and Government Index Analysis

Conclusion

Chapter 5: Laddered Portfolio Convergence to Yield

Introduction

Laddered Portfolio with a Stable Flat Yield Curve

Laddered Portfolio Rebalancing After a Parallel Curve Shift

Laddered Portfolio Yield Pathways

Laddered Portfolio Duration

Laddered Portfolio Convergence to Yield

Laddered Portfolio Barbell versus Single Bond Bullet

Conclusion

Appendix: Path Return and Volatility

References

Part II: Some Topics That Didn’t Make it into the 1972 Edition (2004 Edition)

Contents of the 2004 Edition

Foreword

Preface to the 2004 Edition: A Historical Perspective

Technical Appendix to “Some Topics”

Part III: Inside the Yield Book (Original Edition)

Preface to the 1972 Edition

Contents of the 1972 Edition

List of Tables

About the Authors

Index

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Copyright © 2013 by Sidney Homer and Martin L. Leibowitz. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

The first edition of Inside the Yield Book was published by Prentice-Hall, Inc. in 1972. The second edition of Inside the Yield Book was published by Bloomberg Press in 2004.

Published simultaneously in Canada.

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ISBN 978-1-118-39013-9 (Hardcover); ISBN 978-1-118-42191-8 (ebk); ISBN 978-1-118-61159-3 (ebk); ISBN 978-1-118-41758-4 (ebk)

To all the the portfolio managers, analysts, traders,

and even competitors in the financial community

who have been so generous in sharing their thoughts,

their concerns, and their enthusiasm with the

authors over the years.

Preface to the 2013 Edition

The earlier editions of Inside the Yield Book which represent Parts II and III of this edition focused on a single bond that was continuously held either to maturity or to some specified horizon. The bond was analyzed in terms of the impact of various coupon reinvestment rates and the effect of capital gains or losses associated with horizon yield changes.

In contrast to the single-bond model, most actual bond investments take the form of portfolios composed of multiple bond holdings and a continually changing bond composition. The return/risk character of bond portfolios generally differs markedly from the return pattern associated with a continuously held single bond.

Both institutional and individual bond portfolios generally involve some active rebalancing process that maintains certain key characteristics. The sensitivity to yield movements, as measured by the portfolio’s duration, is one of the most critical of these characteristics. Duration stability can be achieved either explicitly by specifying a duration target, or implicitly by tracking a stable-duration bond index or by preserving some prescribed maturity structure.

Institutional duration targeting (DT) typically entails the sale of aging bonds with shortened terms to fund the purchase of longer-term bonds. In contrast, individual investors who maintain ladderlike portfolios may rebalance using new cash inflows, coupon payments, and/or the proceeds from maturing bonds.

The new material included in Part I of this edition of Inside the Yield Book is devoted to an analysis of the often surprising return behavior exhibited by these DT portfolios over multiyear horizons.

Acknowledgments

The authors would like to first acknowledge the seminal role of our late colleague Terry Langetieg, who was the lead author of the 1990 Financial Analysts Journal paper that first raised the issue of Duration Targeting as the most common form of bond management and provided some initial insights into the very distinctive return characteristics of such funds.

We would also like to express our gratitude to the firm of Morgan Stanley for their support and encouragement of much of the research that formed the basis for this study of Duration-Targeting funds.

And we would be remiss not to acknowledge the role of Wiley’s “master editor” Bill Falloon, who played a key role in bringing our past three Wiley books to fruition and who first broached the idea that the time was ripe for a truly new edition of Inside the Yield Book.

Part I

Duration Targeting: A New Look at Bond Portfolios (2013 Edition)

Introduction

The standard approach to the analysis of prospective returns and risks of any portfolio combines some estimate of expected returns with a measure of interim volatility. For bonds, volatility is approximated by the product of the yield volatility and the duration. The yield move (and corresponding return) in any one period usually is presumed to be statistically independent of previous yield moves.

At first, the standard return/risk approach appears to provide a reasonable basis for projecting multiperiod returns and risks. However, with duration-targeted (DT) portfolios, where the same duration is maintained over time, returns converge back toward the initial yield, so the multiyear volatility turns out to be far less than that suggested by the initial duration. Perhaps surprisingly, this convergence and volatility reduction holds regardless of whether yields have high volatility or exhibit a steady rising or falling trend over the investment horizon.

This theoretical “gravitational pull” toward the initial yield was examined in terms of the actual returns of the Barclays index as well as to the returns of a hypothetical 10-year laddered portfolio. Both portfolios have durations in the five-year range. Our theoretical model of DT suggests that annualized returns for five-year duration portfolios should approach the initial yield in six to nine years. A historical analysis covering the period from 1977 to 2011 showed that such convergence does indeed occur.

Accrual Offsets of Price Effects

The DT rebalancing process will result in capital gains or losses, depending on whether yields have fallen or risen during the time between rebalancing. After rebalancing, the bond portfolio will reflect current market yields and will be positioned to capture the new prevailing yields as going-forward accruals. Such accruals always act in the opposite direction of price changes and, at least partially, offset duration-based price effects.

The importance of accruals is largely underappreciated because portfolio risk and return are usually analyzed in the context of relatively short holding periods. Accruals become significant over longer holding periods when accruals can build and ultimately dominate price effects.

In order to see how accruals and price effects interact, we start with simple trendline paths to terminal yields. Later we consider more general non-trendline paths.

Trendline Model

At the outset, we assume a multiyear investment horizon and a corresponding hypothetical terminal yield distribution.

From the myriad of paths to any terminal yield, we initially focus on a simple trendline (TL) along which yields change by the same amount each year. The simplicity of this idealized TL model enables us to derive a compact formula for the DT returns of zero coupon bonds. This TL return depends only on the initial yield, the duration target, the horizon, and the terminal yield. Because there is only one TL path to each terminal yield, there is a one-to-one correspondence between terminal yields and TL returns.

The TL model returns are based on a linear pricing model that is reasonably accurate for moderate yield changes. Because all DT rebalancing transactions involve the same duration and the same yield change, the annual price effects are always equal. In contrast to the constant pace of TL price changes, the importance of annual accruals accelerates over time. For example, the first-year accrual is equal to the initial yield, the second-year accrual is the initial yield plus the first-year yield change, the third-year accrual is the second-year accrual plus the second-year yield change, and so on. These accruals accumulate at rate that is roughly proportional to the square of the investment horizon.

The Effective Maturity

Because accruals along TL paths grow (or decline) at a faster rate than price changes, there is an effective maturity point at which the cumulative accruals will fully offset the cumulative price losses (or gains). This effective maturity turns out to be approximately twice the targeted duration.

If the investment horizon is less than the effective maturity, the total price effect will be greater than the total accrual effect. At the effective maturity, the net price/accrual effect will be zero. Consequently, the annualized return to the effective maturity will equal the initial yield for every TL path, regardless of whether the terminal yield is higher or lower than the initial yield. This “gravitational pull” forces all such TL returns back to the initial yield level.

Terminal Yield Distributions

We now turn to the case where a terminal probability distribution is specified. One simple example is scenario analysis in which estimates/forecasts of future yields are projected based on a range of expectations. Each yield forecast may be assigned a distinct probability weight and the weighted average of future yields can be viewed as the expected yield. Each projected yield can then be paired with a corresponding TL return and an expected return can be computed using the same weights as for the yield projections.

More generally, the standard deviation of TL returns can also be found by applying the TL return formula to the standard deviation of the terminal yields. As the horizon approaches the effective maturity, the expected TL return will converge on the starting yield—no matter how much the expected terminal yield may differ from the starting yield. The standard deviation of TL returns will then also compress down to zero, no matter how wide the standard deviation of terminal yields.

Non-Trendline Paths

The DT model can be extended beyond TL paths to the full range of pathways generated by random walks. As an example, consider a jump path where yields immediately move to a high yield level and then remain there throughout the investment period. The total price change along the jump path (and along any other non-TL path) will be the same as for the TL because the price effect depends only on the beginning and ending yields, not on the path between those yields. In contrast, accruals beyond the first year are highly path dependent and may differ significantly from the TL accrual. In the case of the jump path, all accruals beyond the first year will be at the higher yield and will therefore exceed the TL accrual.

Among the infinitely many other paths to the terminal yield, one path will be a mirror image of the jump path with each yield gap relative to the TL having the same magnitude but with the opposite sign. Thus, the yield accruals for the jump path and its mirror will offset each other, so that the average accrual for the mirror pair will be the same as the TL accrual. Because the price effects are the same for all paths to a given terminal yield, the average of the annualized returns for the mirror pair will just equal the TL return.

This concept of mirror image pairs turns out to have broad generality because we can almost always find a mirror image for any non-TL path. Because the annualized return for each pair equals the annualized TL return, the average of the annualized returns across all non-TL paths will equal the TL return, provided each mirror has a symmetric probability of occurrence.

Tracking Error and Total Volatility

The average return from the full array of paths to a given terminal yield will just match the TL return. However, each path will have a unique return based on the accruals along its specific yield pathway. This resulting dispersion of returns leads to tracking errors around the TL return. In the Appendix, a formula for this tracking error is developed. By combining the tracking error with the standard deviation of TL returns, a total volatility can be found.

This total volatility incorporates the spread of all pathway returns relative to the expected TL return. For short horizons, this total volatility can be quite large, but it declines to a minimal level for horizons approaching the effective maturity. For example, with a five-year duration and a 100 bps yield change volatility, the total DT volatility declines to about 90 bps over a window of six to nine years. Within this minimal volatility window, returns are projected to be with ± 90 bps of the starting yields.

These theoretical projections are consistent with historical results using 1977 to 2011 Treasury par bonds and, as indicated earlier, actual Barclays index returns.

Key Findings

1. For any given yield move, the TL path return will converge back toward the starting yield.

2. Once the horizon reaches an effective maturity that is approximately twice the DT duration, the TL path return will coincide with starting yield (e.g., a five-year duration DT has a nine-year effective maturity).

3. For any horizon yield distribution, the TL return to any yield point is a good ex ante estimate of the expected return across all TL paths to that point. The standard deviation of the TL returns can be determined in a similar fashion.

4. As the horizon approaches the effective maturity, the expected TL return will converge to the starting yield and the standard deviation of TL returns will converge to zero.

5. Non-TL paths to a given terminal yield will have a return that differs from the TL path return. However, each such non-TL can be paired with a mirror image non-TL path, so that the average of the paired returns is the same as the TL path return. By extension, the average return from a full array of non-TL paths to a given terminal yield will have the same return as the average return across all TL paths.

6. The array of non-TL paths surrounding each TL path leads to a dispersion of returns surrounding each TL path return. The total volatility measure incorporates both the volatility of the TL returns and the dispersion derived from the non-TL paths. The total volatility may be large at first, but then decreases and remains relatively flat for several years prior to the effective maturity. For example, with a 100 bp random walk volatility and a five-year DT, the tracking stabilizes at around 90 bp for horizons between six and nine years. A similar stabilization is also observed for the Barclays index and for constant maturity Treasuries.

7. Within this theoretical stability window, the starting yield, plus or minus the tracking error, can serve as a good ex ante estimate of returns across the full array of random paths.

8. Thus, within the stability window, the DT process can be viewed as providing a form of “statistical immunization.”

9. More generally, the DT process implies that the impact of a subsequent rise or fall in yields will be relatively quickly eroded as returns converge back toward the starting yield.

10. On one hand, the DT process can be viewed as creating a statistical yield trap because an investor who is not satisfied with the current level of yields cannot count on subsequent higher (or lower) yields to provide any significant improvement in multiyear returns.

11. On the other hand, an investor who is comfortable with the current level of yields need not fear that adverse yield moves are likely to generate multiyear returns that permanently fall much below the starting yield.

12. The only way to escape from this yield trap is to depart from the existing DT target and to materially shorten or lengthen the duration.

13. Within an asset allocation framework, volatility estimates are typically based on an instantaneous or a short-term horizon. Such estimates can lead to a serious overstatement of the multiyear risk of DT bond funds relative to other asset classes.

14. When the preceding theoretical results were tested against the 1977 to 2011 historical Barclays index returns, annualized six-year returns were found to match the starting yields.

Chapter 1

Duration Targeting and the Trendline Model

Duration Targeting

Most bond portfolios can be broadly classified as (1) buy and hold, (2) immunized, or (3) duration targeted (DT).

Buy and hold strategies are typically aimed at securing returns that closely match the initial yield value. In this case, they can be viewed as primarily having absolute return objectives.

Immunization strategies are intended to generate returns that match liabilities as rates shift, coupons reinvest, time passes, and the liability duration evolves. In essence, immunization also acts as an ultimate absolute return strategy in that it tries to immunize the initially promised return against changes in the structure of interest rates.

The very nature of both buy and hold and immunization leads to durations that decrease over time. In contrast, duration-targeted portfolios deliberately maintain a relatively stable duration.

Apart from liability-driven immunizations, some form of the stable DT approach is characteristic of virtually all actively and passively managed institutional portfolios. Institutions typically develop a policy portfolio based on a set of assumed return/risk parameters for relevant asset classes. In this process, the risk level assumed for the bond component is basically equivalent to specifying a given duration. Once selected, the policy portfolio serves as a baseline in the face of market movements, with the portfolio being periodically rebalanced back toward the policy structure. For the high-grade bond component of the fund, this common rebalancing process tends to maintain a stable duration and hence is essentially tantamount to duration targeting.

Duration targeting can also be viewed as providing relative returns versus a specific benchmark. In some cases, a fund’s mandate may be to match an index’s returns as closely as possible. In more active strategies, the portfolio’s incremental performance will be gauged relative to some bond index such as the Barclays U.S. Government/Credit index. Such a bond index will have a specific duration value that is fairly stable and only changes gradually over time.

Thus, both active bond management and bond indexing can be viewed as implicitly employing a strategy that approximates duration targeting.

To maintain the required duration, DT portfolios utilize periodic rebalancing. If yields rise, the rebalancing transaction will result in a price loss. But, going forward, the new higher yield implies a higher accrual that partially offsets the price loss. Conversely, if yields fall, lower accruals tend to offset price gains. Over time, the accumulated accruals will tend to offset the duration-based price effects.

Our key finding is that DT bond returns tend to converge toward the initial yield over time. This convergence to yield is independent of the future path of interest rates. Over a sufficiently long holding period, the initial yield turns out to be a surprisingly accurate predictor of annualized returns, regardless of whether rates rise or fall.

For clarity of exposition, this chapter makes a number of simplifying assumptions: zero-coupon bonds, no compounding, zero transaction costs, and duration values that both age with time and can be used as linear measures of price sensitivity. For modest horizons and reasonable yield changes, both compounding and convexity effects are relatively minimal. For large yield changes and longer time periods, compounding and convexity effects should ideally be taken into account.

A section in the Appendix presents a more comprehensive analysis of how duration-based measures actually relate to a bond’s convex price sensitivity.

To illustrate the DT process, we begin with a plot of five-year constant maturity yield paths for three five-year periods. Exhibit 1.1 includes two paths, from 1999 to 2004 and 2005 to 2010, that exhibit falling rates and one path from 1978 to 1983 that exhibits rising rates.

EXHIBIT 1.1 Three historical rate paths

Source: Morgan Stanley Research.

To show how DT works, in Exhibit 1.2 we focus on the 1978–1983 path and assume an initial five-year zero coupon bond investment at the 9.32 percent yield that prevailed in 1978. By 1979, yields had risen by 106 basis points (we assume a flat yield curve) to 10.38 percent and the bond’s duration has shortened to four years.

EXHIBIT 1.2 Historical example of DT returns

Source: Morgan Stanley Research.

To maintain a five-year duration target, we rebalance by first selling the four-year bond and then using the sale proceeds to buy a new five-year bond at the new 10.38 percent yield. This bond sale results in a price loss that is approximately the negative of the duration times the yield change. That is, the price loss of −4.24 percent is −4 × 106 bp.

Over the course of the first year, interest accrues at the 9.32 percent purchase yield. The total return of 5.08 percent is the sum of the 9.32 percent accrual and the −4.24 percent price return.

At the end of 1980 (and all subsequent years), the same rebalancing process is repeated. Over five years, yields rise by a total of 2.25 percent to an ending yield of 11.57 percent. This total yield rise results in a cumulative price loss of −9.00 percent (= −4 × 2.25 percent), or −1.80 percent per year.

The total accrual depends on the timing of the individual yield increases, with earlier moves tending to have a greater cumulative impact over time. In this example, the total accrual is 56.35 percent, an annualized 11.27 percent per year. In comparison to the initial 9.32 percent yield, accruals provide an incremental 1.95 percent per year. This 1.95 percent excess accrual largely offsets the annualized −1.80 percent price loss, so that the excess return is only 0.2 percent over the initial 9.3 percent yield.

The preceding results are summarized in Exhibit 1.3. The excess returns in Exhibit 1.3 also show that similar offsets are obtained for the 1999–2004 path with its declining yields. There may, of course, be some paths that have excess returns that differ significantly from zero. For example, in the 2005–2010 period, the incremental accruals dominated the price effects, so that the annualized 5.2 percent return exceeded the initial 4.4 percent yield by a more significant 0.8 percent.

EXHIBIT 1.3 Historical examples of DT return convergence

Source: Morgan Stanley Research.

Trendlines

This section examines the dynamics of DT in the context of a trendline (TL) model of interest rate changes. Along a TL path, yields move in equal increments until the terminal yield is reached. Although the TL represents only one of an unlimited number of paths to a given end point, the TL model turns out to provide readily generalizable results.

The generality of the TL model stems from the observation that any non-trendline path must have a mirror-image path relative to the TL. The average of the two returns for a pair of mirror-image paths to a given terminal yield can be shown to always equal the TL return. The trendline model will therefore represent the average return across virtually all paths to the given terminal yield. (The only exceptions to mirror imaging occur when negative returns are excluded from consideration.)

For a TL path and a zero-coupon bond with duration D, we define a trendline duration DTL that reflects the sensitivity of the annualized bond return to the total yield change over an N-year holding period. DTL depends only on D and N and represents the combined sensitivity of both accruals and price to yield changes. When DTL is zero, accrual gains precisely offset price losses and the average return converges to the initial yield.

This full convergence to yield will be shown to occur for a holding period N* that is one year less than twice the bond duration. We could view N* as an effective maturity corresponding to the targeted duration, D.

Despite its simplicity, the TL model turns out to have fairly general applicability because it provides quite reasonable estimates of actual market convergence. We demonstrate these convergence properties through both simulation and analysis of historical data.

In our zero-coupon bond model of DT, we assume a D-year bond is purchased at time zero. At the end of the year, the bond is sold and the proceeds are invested in a new D-year bond. Price losses are estimated from the Macaulay duration, D, which is equal to the zero-coupon bond maturity.

To illustrate how DT bond sales and reinvestment impact returns, we begin with a D = five-year bond over the simple five-year trendline (TL) shown in Exhibit 1.4, with yields rising from 3 percent to a 5.5 percent at the rate of 50 basis points per year. At the end of the first year, yields have risen by 0.5 percent to 3.5 percent. At that time, the duration of the aged bond will be four years.

EXHIBIT 1.4 A five-year yield trendline

Source: Morgan Stanley Research.

To maintain a five-year duration, we engage in a rebalancing transaction, as in the previous section. When selling the aged bond, the price loss is approximately equal to the negative of the four-year duration times the yield change, −4 × 0.5 percent = −2 percent (see Appendix for derivation).

Exhibit 1.5 shows that the 1 percent total return for the first year is the sum of a 3 percent yield accrual, based on the initial yield, and the price loss of −2 percent.

EXHIBIT 1.5 Trendline accruals and price effects (initial yield = 3%; duration target = 5)

Source: Morgan Stanley Research.

The second part of the rebalancing transaction requires reinvesting in a new five-year bond with a 3.5 percent yield. The 3.5 percent yield then becomes the new accrual. In each subsequent year, yields increase by 0.5 percent so the price loss is the same −2 percent. The accruals over the subsequent year also increase by 0.5 percent. However, although the annual price loss remains the same, the annual accruals escalate each year with the rising yields. Thus, the annual rate of accruals increasingly comes to dominate the constant annual price effect.

The average accrual over the five years is 4 percent, 1 percent higher than the 3 percent initial yield. The average return of 2 percent is the sum of the average accrual and the average price loss.

For the five-year horizon, price losses dominate accrual gains. Over longer investment periods, net accrual gains continue to increase and ultimately dominate price losses. Net accrual gains will just balance the price loss if the holding period is extended to nine years; that is, one year less than twice the duration (see Appendix).

Exhibit 1.5 also includes the calculation of the TL accrual factor: the excess average accrual (relative to the initial yield) divided by the total yield change. In the Appendix, we show that for a trendline the accrual factor = (1 − 1/N)/2. The value of this factor depends only on the investment horizon and increases toward ½ as the holding period (or rebalancing frequency) increases. Using this formula with N = 5, the accrual factor is (1 − 1/5)/2 = 0.4, as shown in Exhibit 1.5.

The accrual factor makes it easy to calculate the average TL accrual. For example, with a five-year horizon and total yield change of 2.5 percent, the excess accrual is 40 percent of 2.5 percent = 1 percent.

Generality of the TL Model

On the surface, it might appear that the TL model is overly simplistic because yields generally do not move uniformly along a simple trendline path. In fact, there are an unlimited number of potential upward and downward yield moves that can lead to the same final destination.

To illustrate the generality of the TL model, we will show that each TL represents an average result across an array of all yield pathways leading to the same end point. The average accrual, capital gain (or loss), and total return of all such random paths thus each turn out to be very close to the corresponding TL values.

This averaging property of trendlines facilitates the use of scenario analysis because investors can more easily make terminal yield forecasts than anticipate precise yield paths.

A Jump Yield Path

In this section, we focus on a non-TL jump path with the same 5.5 percent terminal yield as the TL but with higher accruals. Exhibits 1.6 and 1.7 illustrate a path where the yield starts at 3 percent and, at the end of the first year, jumps by 2.5 percent and stays at 5.5 percent for the next four years.

EXHIBIT 1.6 Jump path to final yield

Source: Morgan Stanley Research.

EXHIBIT 1.7 Jump path deviation from trendline

Source: Morgan Stanley Research.

The jump path yields generate a 5.5 percent average accrual over the four post-jump years, 2.5 percent higher than the initial yield. Exhibit 1.8 shows that the annualized excess accrual over the full five years is 2 percent. The excess accrual for the jump path is twice the 1 percent trendline accrual, and so the accrual factor of 0.8 is twice the 0.4 TL accrual factor.

EXHIBIT 1.8 Jump path accrual and price effects (initial yield = 3 percent; duration target = 5)

Source: Morgan Stanley Research.

In contrast to the accruals, the overall duration effect is approximately the same for the jump path as it is for the TL. The entire jump yield change occurs during the first year and there is an early price loss of −10 percent (4 × 2.5 percent). The stability of yields in years 2 to 5 means there are no subsequent price losses. The five-year annualized price loss is −2 percent (–10 percent/5), just as in the TL case. The difference between the jump path and TL annualized returns is due solely to the difference in accruals.

Exhibits 1.9 and 1.10 illustrate the TL and jump path annual returns and annualized cumulative returns. As in Exhibit 1.4, the TL annual returns increase at a steady 0.5 percent per year from 1 percent to 3 percent. In the first year, the jump path’s initial yield increase and corresponding price loss leads to a return that is far below the TL return. In subsequent years, the high yield keeps the jump path accruals at 5.5 percent.

EXHIBIT 1.9 Jump path and TL annual returns (initial yield = 3 percent, final yield = 5.5%)

Source: Morgan Stanley Research.

EXHIBIT 1.10 Jump path and TL annualized returns (initial yield = 3 percent, final yield = 5.5 percent)

Source: Morgan Stanley Research.

On an annualized return basis, the early gap between the jump path and TL quickly closes and the jump path rises above the TL. At the end of five years, the annualized jump path return is 1 percent higher than the TL return.

Mirror-Image Paths

For any non-TL yield path from 3 percent to 5.5 percent, one can almost always find a mirror-image path relative to the TL. Along this mirror path, year-end deviations from the TL will have the same magnitude but opposite sign from the original non-TL path deviations.

Exhibits 1.11 and 1.12 illustrate the mirror image of the jump path in Exhibit 1.7. The actual yield changes along the mirror path are quite different from the original jump path. The mirror path’s first-year yield of 1.5 percent is −1.5 percent below the initial yield and −2 percent below the TL’s 3.5 percent yield. To reach 5.5 percent by the end of five years, the mirror yield must increase by 4 percent; that is, by +1 percent per year for each of the last four years.

EXHIBIT 1.11 Mirror image of jump path relative to trendline

Source: Morgan Stanley Research.

EXHIBIT 1.12 Trendline reflection of jump path

Source: Morgan Stanley Research.

Exhibit 1.13 shows that the mirror path’s initial yield decline and subsequent slow climb toward 5.5 percent results in no average excess accrual and the mirror accrual factor is zero. The first-year −1.5 percent yield decline results in a 6 percent price gain. But, that gain is steadily eroded by the price losses that accompany the rising yields of years 2 through 5. The average price loss of −2 percent is the same as it was for the jump path. Along the mirror path, the lack of accrual offsets to the capital losses results in an average return of 1 percent, 2 percent below the initial yield.

EXHIBIT 1.13 Accrual and price effects for the mirror image of the jump path

Source: Morgan Stanley Research.

Exhibit 1.14 shows that the 0.8 accrual factor for the original path and the 0 percent accrual factor for its mirror path average to 0.4, the same 0.4 as the TL accrual factor! The average of the annualized accruals for the two mirror paths is also equal to the TL value. This averaging property of the TL is quite general. For any mirror pair, the average accrual factors and the average accrual will always be approximately equal to the corresponding TL values.

EXHIBIT 1.14 Comparison of trendline, jump path, and mirror image

Source: Morgan Stanley Research.

In contrast to the difference in their accruals, the jump path and the mirror image all exhibit equal price losses. The total price effect depends only on the beginning and ending yield and all paths to the same terminal yield will have the same average price loss (or gain). A non-TL path may have different total accruals than the TL, but the combined average accrual of a non-TL and its mirror will always equal the TL accrual. Thus, the average return of all mirror-image pairs will always just be equal to the TL return.

The overwhelming majority of paths leading to moderate yield changes will tend to have accessible mirror images. In this case, the trendline accrual will approximate the average accrual for all the mirror-image paths leading to the given terminal yield. The yield change over the horizon and hence the price effect will also be approximately the same. Thus, the TL return can represent the average return for all potential paths to a given terminal yield.

More generally, as we will later demonstrate, this pairing does not necessarily require precise mirror imaging. For large but necessarily finite simulations, the “cloud” of paths to a given terminal yield will tend to be symmetric and the averages across this cloud of paths will fairly quickly converge to the corresponding TL values. (One exception to the availability of mirror-image paths is when the requirement for positive yield paths eliminates certain theoretical mirror images).

Random Paths

We now move beyond simple TLs and jump paths to a random walk of annual yield moves. In simulating such annual yield moves, we begin by assuming a mean yield change of 0 with a standard deviation of 1 percent. As the horizon increases, the mean yield change remains zero while the volatility increases—leading to a wider distribution of yield changes over the longer horizons.

At the outset, our simulation is based on a zero mean; that is, no drift. However, in later sections, we shall show that our findings remain intact even with positive or negative yield drifts over time.

We begin our analysis of random paths by focusing on just one five-year terminal yield and a mirror-image pair of paths to that end point. In Exhibit 1.15, the random path begins at 3 percent and ends at 5.5 percent. Because the simulation includes all possible paths, we can also find a mirror path relative to the TL.

EXHIBIT 1.15 Random path and its mirror image

Source: Morgan Stanley Research.

Exhibit 1.16 shows the average of the mirror pair’s accruals is the same as the TL accrual. Similarly, the paired accrual factors average to the TL accrual factor.

EXHIBIT 1.16 Random path + Mirror image = Trendline

Source: Morgan Stanley Research.

The example of the jump path in Exhibit 1.12 and the random path in Exhibit 1.15 illustrate an important general TL characteristic. For any terminal yield, a simulated random walk will include numerous paths to that yield. Within the set of all such paths, we almost always can find an appropriate mirror image. (The only exception is when the mirror-image path would have to include a negative return).

We can view all paths to any terminal yield as roughly equivalent to the set of all mirror-image pairs. Averages across all paths are essentially equivalent to averages of paired averages. Because paired averages always equal TL values, the average accrual factor across all paths must also coincide with the TL value. Thus, over a five-year horizon, the average accrual factor across all paths is 0.4. Because the price effect is the same across all such paths, the total return will also converge to the TL value.

Random Paths to the Same Ending Yield

Exhibit 1.17 shows 11 simulated five-year paths beginning at 3 percent and ending at 5.5 percent. The paths are not precise mirror images but they do create a kind of cloud that hovers above and below the trendline. In Exhibit 1.18, we see that, for individual paths, average accruals range from a low of 2.68 percent (path 8) to a high of 4.36 percent (path 5). The cumulative average 3.95 percent accrual across all 11 paths is close to the TL average of 4.00 percent. Thus, it appears that even without precise mirror pairing, the TL average accrual is a reasonable proxy for the average across paths.

EXHIBIT 1.17 Eleven random paths leading to 5.5 percent terminal yield

Source: Morgan Stanley Research.

EXHIBIT 1.18 Random paths vs. trendline (initial yield = 3 percent, terminal yield = 5.5 percent)

Source: Morgan Stanley Research.

The individual path accrual factors in Exhibit 1.18 also show wide variation, ranging from −0.13 to 0.56. However, the last column of the Exhibit 1.18 shows that the cumulative average accrual factor quickly converges to 0.4, the TL accrual factor.

Exhibits 1.19 and 1.20 focus on 13 random paths to a five-year terminal yield of 4.5 percent, 1 percent lower than in the previous example. Because the horizon is five years, the TL accrual factor remains 0.4. Multiplying that factor by the total yield change of 1.5 percent (4.5 percent − 3.0 percent) gives the average TL accrual of 0.6. For a random set of 13 paths to 4.5 percent, the average accruals and accrual factors are seen to be quite close to the TL values.

EXHIBIT 1.19 13 Random paths vs. trendline to 4.5 percent terminal yield

Source: Morgan Stanley Research.

EXHIBIT 1.20 Random paths vs. trendline (initial yield = 3 percent, terminal yield = 4.5 percent)

Source: Morgan Stanley Research.

Trendline Duration

To this point, the illustrative examples only show rising rates, their associated capital losses, and the offsetting accrual gains. Conversely, with falling rates, capital gains will be reduced by negative excess accruals relative to the initial yield.

Exhibit 1.21 shows that for five-year yield changes ranging from −3 percent to +3 percent, the annualized excess returns remain reasonably close to the initial 3 percent yield. Regardless of the direction of rate moves, the net result of the accrual offsets is a reduction in return volatility.

EXHIBIT 1.21 TL returns vs. yield changes for D = 5, N = 5

Source: Morgan Stanley Research.

Note that the ratio of excess return to total yield change is a constant −0.4. The implication of this constant ratio is that there is an underlying trendline duration factor DTL that relates returns to yield changes.

Exhibit 1.22 plots the results of Exhibit 1.21, illustrating the steady rise in average accruals offset by a declining price return as the total yield change increases. The total return line has slope = −0.4.

EXHIBIT 1.22 Annualized excess accrual return + Price return = Excess total return (N = 5, D = 5)

Source: Morgan Stanley Research.

The price return adds (capital gains) to the total return for negative yield moves and depletes returns (capital losses) when the yield change is positive. When the total yield change is 0 percent, the accrual, price, and excess total return are all zero; that is, the average excess total return is the initial yield.

The trendline duration DTL incorporates both accrual and duration effects by combining the accrual factor introduced earlier in this chapter with a duration factor (see Appendix).

The formula for the duration factor is (D − 1)/N. The numerator, D −1, reflects the one-year duration reduction associated with year-end zero-coupon bond rebalancing. The denominator, N, reflects an annualization of the total price effect. The negative of the TL duration factor multiplied by the total yield change is the annualized return loss (or gain).

The duration factor decreases toward zero as the horizon increases. The intuition behind this decrease is that a given total yield change generates a fixed total price effect, so that the annualized price effect decreases over longer investment horizons.

For TL-based DT strategies with a target duration D, the annualized return over an N-year TL path can be expressed as:

For D = 5 and N = 5,

The accrual/pricing offset in DTL is one of the forces that tends to compress DT returns back toward the original yield. DTL is equal to zero when the accrual and duration factors are equal. At this point, the accrual and rate change effects are neutralized and the excess total return along any TL path will be zero—regardless of the terminal yield! (And because of the mirror-path averaging, the expected excess return will be zero across all equally probabilistic non-TL paths!)

Exhibit 1.23 shows the duration factor, the accrual factor, and the TL duration for a five-year horizon and a range of actual bond durations. For example, with a three-year duration, the TL duration is zero at the five-year horizon. This means that if a three-year duration DT strategy is maintained for five years, the average return will equal the initial yield. The horizontal line in Exhibit 1.24 illustrates this insensitivity to yield changes. Because the average return equals the initial yield, we can view the five-year horizon as a kind of effective maturity for a DT strategy with a three-year duration target.

EXHIBIT 1.23 Trendline duration vs. bond duration when N = 5

Source: Morgan Stanley Research.

EXHIBIT 1.24 Excess return vs. five-year yield change

Source: Morgan Stanley Research.

For a five-year horizon and D < 3, the accrual factor exceeds the duration factor, the DTL is negative, and the excess return increases in tandem with the terminal yield. Conversely, when D > 3, the duration factor exceeds the accrual factor, DTL is positive, and the excess returns decrease with increasing yield change.

Horizon Effects

To this point, we have primarily focused on DT over a five-year TL path. Exhibit 1.25 illustrates how the duration factor declines and the accrual factor increases as the horizon increases. Exhibit 1.26 illustrates the corresponding decrease in DTL.

EXHIBIT 1.25 Accrual and duration factors vs. horizon (with D = 5)

Source: Morgan Stanley Research.

EXHIBIT 1.26 Trendline duration vs. horizon (with D = 5)

Source: Morgan Stanley Research.

Over very short holding periods, there is little time for accruals to accumulate and the duration factor dominates the accrual factor. As the horizon increases, the average price impact of a given yield change decreases and accruals become increasingly important.

When the duration and accrual curves cross, DTL = 0, and price losses (or gains) are neutralized by accrual gains (or losses). In the Appendix, we show that this crossing point occurs when the horizon is one less than twice the duration. For D = 5, this implies an effective maturity of nine years.

For horizons longer than the effective maturity, accrual effects dominate price effects, DTL turns negative, and positive yield changes lead to return increases. Ultimately, the duration factor approaches zero, the accrual factor approaches a horizontal asymptote of 0.5, and the TL duration DTL approaches −0.5 (see Appendix).

Exhibit 1.27 illustrates the relationship between the total yield change and excess returns for a five-year duration target. As N increases from 3 to 5 to 9, DTL decreases from 1.0 to 0.4 to zero, and the slopes of the excess return lines flatten from −1.0 to −0.4 to zero. At the nine-year effective maturity, the excess return is always zero and the return is equal to the initial yield—again, regardless of the total yield change over the horizon.

EXHIBIT 1.27 Excess returns vs. yield change (D = 5; N = 3, 5, 9)

Source: Morgan Stanley Research.

The previous exhibits have shown that with TL paths, the effective maturity of a DT strategy depends only on N and D. Exhibit 1.28 shows that, when rates rise at a steady 50 bp, the annualized excess return steadily builds from a −2 percent initial loss. It takes a total of nine years for accruals to totally offset price losses. Similarly, when rates trend down at −25 bp per year, initial return gains are steadily depleted by ever-lower accruals, but the total offset occurs at the same nine-year horizon.

EXHIBIT 1.28 Annualized excess returns with trending yields (D = 5)

Source: Morgan Stanley Research.

Whereas Exhibit 1.28 focused on annualized returns, Exhibit 1.29 presents the annual returns for the same three cases. With upward trending rates of +50 bp per year, the price loss is always −2 percent per year. But the annual accrual increases, rising by 50 bp per year. It therefore takes only four years (4 × 50 bp = 2 percent) for the total annual accrual to equal the continuing −2 percent price loss. The fifth-year excess return of 0 percent reflects this offset.

EXHIBIT 1.29 Excess annual return with trending yields (D = 5)

Source: Morgan Stanley Research.

In later years, the excess return is always positive because accruals are always greater than price losses. It can also be seen that the +2 percent annual return is in the ninth year corresponds to the −2 percent first-year return.

On an annualized basis, it takes another four years (i.e., a total of nine years) for the accumulated positive excess returns to completely neutralize all accumulated negative excess returns.

Conversely, when yields trend downward at −25 bp per year, it is again in the fifth year that the accrual totally offsets the 1 percent price gain and the excess return is zero. And once again, the ninth-year return of −1 percent corresponds to the +1 percent first-year return.

Conclusion

High-grade bonds fundamentally differ from equities in that their initial yield accrues return over time. With duration targeting, repeated duration extension keeps end-of-year duration constant while the rebalancing process incorporates the new levels of accruals that result from rising or falling rates. These accruals offset the price effects. The relative role of accruals grows with the investment horizon, leading to bond returns having lower volatilities as the horizon lengthens. Thus, in comparison with the standard assumptions for other asset classes, the effective volatility of the fixed income component may be significantly overstated for multiyear horizons.