MARTIN L. LEIBOWITZ is a managing director with MorganStanley Research Department's global strategy team. Prior to MorganStanley, he was vice chairman and chief investment officer ofTIAA-CREF. He is the author of numerous books, and over 200articles on various investment analysis topics, ten of which havewon the prestigious Graham and Dodd Award. Leibowitz has receivedthree of the CFA Institute's highest awards: the NicholasMolodovsky Award, the James R. Vertin Award, and the ProfessionalExcellence Award. In 1995, he became the first inductee into theFixed Income Analysts Society's Hall of Fame.

The late SIDNEY HOMER was the founder and general partnerin charge of Salomon Brothers' bond market research department.Best known for his pioneering and analytical studies of bond markethistory and relative values and the economic forces that createbond market trends, he is the author of several books includingA History of Interest Rates, The Bond Buyer's Primer, andThe Price of Money. He was inducted posthumously into theFixed Income Analysts Society's Hall of Fame in 1997.

ANTHONY BOVA is an Executive Director with Morgan StanleyEquity Research's Global Strategy team. He won Best Article in theninth annual Bernstein Fabozzi/Jacobs Levy Awards presented by theJournal of Portfolio Management for his coauthoring of"Gathering Implicit Alphas in a Beta World." In 2008, Mr. Bovacoauthored Modern Portfolio Management (Wiley), whichfocused on active equity strategies. In 2010, he coauthored TheEndowment Model of Investing (Wiley) with Martin Leibowitz andBrett Hammond of TIAA-CREF.

STANLEY KOGELMAN is President of Delft StrategicAdvisors, LLC, and a Principal at Advanced Portfolio Management.Prior to founding Delft Strategic Advisors, he served as a managingdirector at JPMorgan Private Bank and Goldman, Sachs & Co. Heis the author and coauthor of more than sixty articles on financeand received two Graham and Dodd Awards and published two booksjointly with Martin Leibowitz. In an earlier academic career, heheld senior mathematics faculty positions and wrote severalpioneering books on mathematics education, including Mind OverMath and The Only Math Book You'll Ever Need.

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...