GPIF working paper

Bond investment strategy based on factor analysis: Revisited
Application to carry and roll-down strategy in the U.S. Treasury market

December 27,2021
Koichi Miyazaki
Yuki Abe
Kazuhiro Shimada

EXECUTIVE SUMMARY

As one of the traditional bond investment strategy, there is "carry and roll-down strategy",which assumes that the future yield curve remains unchanged.

In the U.S. Treasury market, where the volatility of yields is reasonably high, it is mentioned that"the assumption that future yield curves will remain unchanged is hard to be satisfied".

This study revisited Litterman and Scheinkman [1991] for the first time in 30 years to explore thepotential of an improved carry and roll-down strategy in the U.S. Treasury market, adding scenario
returns to carry and roll-down strategy based on the factor analysis.

1. Introduction
1.1 Literature Review
Factor analysis is one method of multivariate analysis, which is a statistical analysis method. According to
Noguchi (2018), "a method of explaining information with many variables by a small number of implicit
common factors." Factor analysis is often used in the field of social sciences, for example in the field of
education, to express the performance of various courses in schools by weighting implicit common factors suchas logical thinking, memory, and computational capability, and in the field of marketing, to extract factors
concerning consumer preferences from each indicator of purchasing behavior. In addition to factor analysis,
principal component analysis is also a statistical analysis technique similar to factor analysis among
multivariate analysis. Principal components analysis differs from factor analysis in that principal components
are determined in descending order of their contribution to the variance of observed data without placing
common factors. Putting strict definition aside, we regard factor analysis as principal component analysis in
the sense of the statistical method and review preceding researches applied to bond market analysis in thischapter.

1(NOTE)This working paper summarizes the research results of GPIF staffs, and the content and opinions expressed in the
text do not represent GPIF's official views.

To best of our knowledge, Litterman and Scheinkman [1991] is the first paper utilizing factor analysis toshed some light on bond market. They applied factor analysis to changes in U.S. zero-coupon government bondyields, decomposed the changes in yields into three factors, and showed the impact (derived from factor
loading of each factor) of each factor on the yield curve with the remaining maturity as the horizontal axis andthe change in yields as the vertical axis. From the shape of the impact of each factor on the maturity of
government bonds, the first factor is called the "level" factor, the second factor is called the "slope" factor, andthe third factor is called the "curvature" factor. They showed that approximately 98% of the yield movement
could be explained by these three factors. In addition, they proposed a method for hedging the risk of the yield
curve using the sensitivity of each factor to various government bonds.
Since Litterman and Scheinkman [1991] was published, the application of factor analysis to bond market
analysis have been extensively studied by financial institutions. However, such kinds of studies applied toJapanese bond market analysis is quite limited, Hayashida (2005), Hoshika and Miyazaki (2007), and Sakudo(2010) are scarce examples.
Hayashida (2005) generated the future return paths of fixed-rate government bonds, floating-rate
government bonds, and short-term assets, and determined expected returns and risks of those bonds, by way of
three-factor model that used principal component analysis for the changes in spot rates in constructing modelsof yield curve movements. Hoshika and Miyazaki (2007) discussed an effective bond investment strategy
based on the forecast of changes in bond yields using a two-factor model, considering the speed of yieldchanges and the trade-off between trading opportunities and transaction costs, which were overlooked inReisman and Zohar (2004 a, 2004 b, 2004 c). Sakudo (2010) confirmed a linear relationship between forward
rates and excess returns in the Japanese government bond market via tent-type regression coefficients similar
to those of Cochrane and Piazzesi (2005). Moreover, he applied factor analysis to the forward rates in theJapanese government bond market to evaluate the three factors: the level, slope, and curvature, and found thatthe curvature factor had the greatest impact on the risk premium of bond investment.
Although “carry and roll-down strategy” is a bit old-fashioned bond investment strategy, it has regained
popularity since the early stage of the quantitative easing by the Bank of Japan. Yamada (2000) is thepioneering literature to provide an essential discussion of carry and roll-down strategy for the Japanese
government bond market. Yamada (2000) calculated the excess return in carry and roll-down strategy from thegovernment bond index for the period from February 1988 to September 2000 and concluded that “the average
excess return is high at 0.88% and the sharp ratio adjusted for tracking error (=0.72%) is also quite high at
1.2.” Yamada (2000) also found that the optimal portfolio of carry and roll-down strategy was bullet-type
except for just a brief period of time, and gave up convexity relative to the government bond index, but heanalyzed that the excess return was remained due to the small negative convexity to the index coming fromchanges in yields over the same period.
In recent years, Kikugawa et al. (2017) identified carry and roll-down strategy as one factor and mentionedthat its premium is large and its correlation with credit and stocks is low. They noted consequently theinvestment in this factor improves the performance of bond investments. In addition, they presented "The2

central bank has announced a policy of intervening in the shape of the yield curve. If it makes the curve moreskewed, the effect of the carry and roll-down factor will be enhanced."
In contrast, in the U.S. Treasury market where the yield level is relatively high even in the global low-yieldenvironment (please refer to Kimura (2021) for more detailed analysis), volatility fluctuates considerably byreflecting the monetary policy of the U.S. Federal Reserve System (Fed), and it is difficult to satisfy thecondition that the future yield curve is unchanged, which carry and roll-down strategy relies on.
1.2 The purpose of this study and the organization of this paper
In this study, we first conduct an empirical analysis to what extent excess returns, tracking errors, andinformation ratios could be obtained when using the U.S. Treasury index as a benchmark for carry and rolldown strategy in the U.S. Treasury market. The analysis here is long-only, therefore it differs from the analysis
by Koijen et al. (2015) that compared carry trades which included shorting U.S. Treasuries, to an equallyweighted benchmark. To the best of our knowledge, there is no studies that attempt to apply factor analysis tocarry and roll-down strategy. Therefore, the purpose of this study is to revisit Litterman and Scheinkman[1991] for the first time in 30 years to explore effective applications of factor analysis to carry and roll-down
strategy in order to improve the performance of carry and roll-down strategy in the U.S. Treasury market.
The organization of this study is as follows. In Chapter 2, we review returns, expected returns, and returns
under the yield curve unchanged scenario on bond investments, and analyze the performance of carry and rolldown strategy in the U.S. Treasury market. In Chapter 3 we report the results of applying factor analysis to
U.S. Treasury yields. In Chapter 4 we report on the effective application of factor analysis to U.S. Treasury
yields to carry and roll-down strategy considering the results of Chapter 3 and examples of the application ofthis approach. In the final chapter we summarize the research and clarify future challenges.

2. Performance Analysis of carry and roll-down strategy in the U.S.
Treasury market
2.1 Returns, expected returns, and returns under the yield curve unchanged scenario on government bonds
We illustrate the return, expected return, and return under the yield curve unchanged scenario of holding a 5-yearbond for 0.5 years and selling it with a coupon, as an example here. 𝑌𝑌0,5 , 𝑃𝑃5 �𝑌𝑌0,5 � express the yield and price of the

5-year bond at time zero (assuming that the start of the investment is time zero), respectively. These are the values

that are fixed at time zero. At the end of 0.5 years, the 5-year bond becomes a 4.5-year bond, at which time half of the
coupon (𝐶𝐶5 ) on the 5-year bond is paid. 𝑌𝑌�0.5,4.5 𝑃𝑃4.5 �𝑌𝑌�0.5,4.5 � express the yield and price of a 4.5-year bond in 0.5 years

past, respectively. We place tiled (~) above to show that these are random variables that are not fixed at time zero. The
expected value at time zero for the yield on a 4.5-year bond in 0.5 years past is equal to the forward yield 𝐹𝐹0,0.5,4.5 on

3

a 4.5-year bond starting in 0.5 years from time zero. Here, 𝑃𝑃0,5 (𝑥𝑥), 𝑃𝑃0.5,4.5 (𝑥𝑥) represents the following functions.

𝑃𝑃0,5 (𝑥𝑥) =

𝐶𝐶5⁄2
𝐶𝐶5⁄2
𝐶𝐶5⁄2
𝐶𝐶5⁄2 + 100
+
+ ⋯+
+
1
2
9
𝑥𝑥
𝑥𝑥
𝑥𝑥
𝑥𝑥 10
�1 + 2��1 + 2��1 + 2��1 + 2�

𝑃𝑃0.5,4.5 (𝑥𝑥) = 𝐶𝐶5⁄2 +

𝐶𝐶5⁄2
𝐶𝐶5⁄2
𝐶𝐶5⁄2
𝐶𝐶5⁄2 + 100
+
+ ⋯+
+
1
2
8
𝑥𝑥
𝑥𝑥
𝑥𝑥
𝑥𝑥 9
�1 + 2��1 + 2��1 + 2��1 + 2�

The return, expected return, and carry-roll-down over the next six months from time zero could be derived by Taylor
expansion of the above functions to a second-order term in the vicinity of the current 5-year bond yield 𝑌𝑌0,5, thenreformed into a simple equation. (see Appendix for return)

Return:�

�𝑌𝑌
�−𝑃𝑃0,5 �𝑌𝑌0,5 �
𝑃𝑃
𝑅𝑅�5→4.5 = 0.5,4.5 𝑃𝑃0.5,4.5
≈
�𝑌𝑌 �
0,5

0,5

1

𝐶𝐶

2

1

′
′′
� 5 + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � +𝑃𝑃0,5 �𝑌𝑌0,5 � 2

2
1 ′′
′
𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,4.5 � + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,4.5 � �

2

2

(1)

In square brackets, the first term represents carry profit, the second term (static duration) and the third term (staticconvexity) together represent roll-down profit, and the fourth term (dynamic duration) and the fifth term (dynamic
convexity) together represent capital gains and losses other than roll-down, respectively.

Expected returns:
�𝑌𝑌�

𝑃𝑃

𝐸𝐸0 �𝑅𝑅�5→4.5 � = 𝐸𝐸0 � 0.5,4.5 0.5,4.5

�−𝑃𝑃0,5 �𝑌𝑌0,5 �

𝑃𝑃0,5 �𝑌𝑌0,5 �

1

�≈

1

𝑃𝑃0,5 �𝑌𝑌0,5 �

𝐶𝐶

2

1

� 5 + 𝑃𝑃′0.5,4.5 �𝑌𝑌0,5 ��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � + 𝑃𝑃′′0.5,4.5�𝑌𝑌0,5 ��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � +2

2

𝑃𝑃′0.5,4.5�𝑌𝑌0,5 ��𝐹𝐹0,0.5,4.5 − 𝑌𝑌0,4.5 � + 2 𝑃𝑃′′0.5,4.5�𝑌𝑌0,5 �(𝜎𝜎4.5 )2 �

(2)Here 𝜎𝜎4.5 expressed the volatility of the semiannual yield change on a 4.5-year bond.

Carry and roll-down (defined as the return under the yield curve unchanged scenario, hereafter CA&RD):
𝐶𝐶𝐶𝐶＆𝑅𝑅𝑅𝑅5→4.5 ≈

1

𝐶𝐶

1

2

′
′′
� 5 + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � �

𝑃𝑃0,5 �𝑌𝑌0,5 � 2

2(3)

Under the yield curve unchanged scenario, 𝑌𝑌�0.5,4.5 = 𝑌𝑌0,4.5 , and the capital gain and loss other than the roll-down inparentheses in the return equation (1) are set at 0.
2.2 Carry and roll-down strategy

We divided the U.S. Treasury market to which the investment is made into 13 buckets according to the remaining
period as follows, and named bucket 1 to bucket 13 in order; 1 year (remaining maturity 1 year or less), 2 years(remaining maturity more than 1 year and 2 years or less), 3 years (remaining maturity more than 2 years and 3 years4

or less), 4 years (remaining maturity more than 3 years and 4 years or less), 5 years (remaining maturity more than 4
years and 5 years or less), 6 years (remaining maturity more than 5 years and 6 years or less), 7 years (remaining
maturity more than 6 years and 7 years or less), 8 years (remaining maturity more than 7 years and 8 years or less), 9
years (remaining maturity more than 8 years and 9 years or less), 10 years (remaining maturity more than 9 years and
10 years or less), 15 years (remaining maturity more than 10 years and 15 years or less), 20 years (remaining maturity
more than 15 years and 25 years or less), 30 years (remaining maturity more than 25 years and 30 years or less)𝑀𝑀, the durations of each bucket by
We denoted respectively the market weights of each bucket by 𝑤𝑤1𝑀𝑀 ⋯ 𝑤𝑤13

𝑀𝑀
𝐷𝐷1𝑀𝑀 ⋯ 𝐷𝐷13
, the total market weight by 1 and the duration of the market as a whole by 𝐷𝐷 (in this study, we used FTSE

U.S. Treasury Bond Index as a benchmark representing the market as a whole). Also, we denoted carry and roll-down

for each bucket by 𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅1→0.5 , ⋯ , 𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅13→(13−0.5) . The subscript "1-10" indicates the period, 11 represents 15
years, 12 represents 20 years and 13 represents 30 years, and -0.5 in parentheses represents that a half year has passedin the bucket). Here, in order to ensure that the excess returns from carry and roll-down strategy do not depend on the
direction of the market, we made constraints that the duration of the portfolio and the duration of the benchmark arematched when attempting to maximize carry and roll-down (the weights of each bucket 𝑤𝑤1 , ⋯ , 𝑤𝑤13 are determinedso that carry and roll-down of the portfolio becomes maximum). The optimization model and excess return (α), which
express carry and roll-down strategy, are as follows:[Optimization model]
Maximize the objective function:

𝑀𝑀𝑀𝑀𝑀𝑀
𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅1→0.5 ∙ 𝑤𝑤1 + ⋯ 𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅13→(13−0.5) ∙ 𝑤𝑤13
𝑤𝑤1 , ⋯ 𝑤𝑤13

𝑀𝑀
Constraints:𝐷𝐷1𝑀𝑀 ∙ 𝑤𝑤1 + ⋯ + 𝐷𝐷13
∙ 𝑤𝑤13 = 𝐷𝐷

:𝑤𝑤1 + ⋯ + 𝑤𝑤13 = 1:𝑤𝑤1 ≥ 0, ⋯ , 𝑤𝑤13 ≥ 0∗
𝑤𝑤1∗ , ⋯ , 𝑤𝑤13

denotes optimal weights obtained by implementing the optimization model.

[Excess gain and loss on carry and roll-down strategy (α)]∗
𝑀𝑀 )− 𝑤𝑤13𝛼𝛼 = 𝑅𝑅�1→0.5 (𝑤𝑤1∗ − 𝑤𝑤1𝑀𝑀 ) + ⋯ + 𝑅𝑅�13→(13−0.5) (𝑤𝑤13(4)Here we confirm the excess gain and loss (α) of carry and roll-down strategy. α is a random variable and takes both
positive and negative values. When it is positive it becomes excess gain, and when it is negative it becomes excess
loss. If the yield curve unchanged scenario is realized six months later, the return on each government bond
𝑅𝑅�𝑖𝑖→(𝑖𝑖−0.5) becomes 𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅𝑖𝑖→(𝑖𝑖−0.5) whose maturity is changed from each maturity of 𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅5→4.5 given in equation(3) in the case of the 5-year bond, therefore α is to be 𝛼𝛼𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅 as follows.

∗
𝑀𝑀 )𝛼𝛼𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅 = 𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅1→0.5 (𝑤𝑤1∗ − 𝑤𝑤1𝑀𝑀 ) + ⋯ + 𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅13→(13−0.5) (𝑤𝑤13− 𝑤𝑤13(5)∗
Since 𝑤𝑤1∗ , ⋯ , 𝑤𝑤13
in equation (5) is the optimal weight that maximizes the objective function under the yield curve5

𝑀𝑀
unchanged scenario, it always exceeds the value of the objective function using the market weight 𝑤𝑤1𝑀𝑀 ⋯ 𝑤𝑤13.

Consequently, 𝛼𝛼𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅 , is always the positive value and becomes the excess return. Thus, whether carry and roll-down

strategy produces excess gain and loss (α) depends on the feasibility of the yield curve unchanged scenario.

In terms of the Japanese government bond market, as the Bank of Japan's quantitative and qualitative monetaryeasing with yield curve control has been pervasive, the volatility of the yield curve is small, and the realization of theyield curve unchanged scenario is highly likely. Therefore, as Kikugawa et al. (2017) point out, carry and roll-down
strategy could improve the performance of bond investments. On the other hand, the U.S. Treasury market is in anenvironment in which the Fed is expected to raise interest rates in the future at the time of writing. In the U.S. Treasury
market as well, as pointed out by Yamada (2000), which analyzed the Japanese government bond market, it is assumed
that the optimal portfolio of carry and roll-down strategy will often become bullet-type. However, in an environment
where the Fed is expected to raise interest rates going forward, it is expected that an upward curvature will appear in
the medium-term maturity buckets, and bullet-type performance will be relatively weakened. In addition, the U.S.
Treasury market is much efficient than the Japanese government bond market, and it is possible that the expected
profit from carry and roll-down could be easily eliminated due to the dynamics of the yield curve. We therefore
examine the performance of carry and roll-down strategy in the U.S. Treasury market.
2.3 Performance Analysis of carry and roll-down strategy in the U.S. Treasury market
This Chapter attempts the performance analysis of carry and roll-down strategy using U.S. Treasury yields (constantmaturity par yields) for approximately four and a half years from the end of April 2017 to the end of October 2021.
The estimation period is set from the end of April 2017 to the end of March 2019 since the estimation period for themodel (factors and factor loading) is required in Chapter 3 and 4. The performance analysis is conducted coveringover the period of the out-of-sample from the end of March 2019 to the end of October 2021. Firstly, at the end of
March 2019, the U.S. Treasury market is divided into 13 buckets, as described in Section 2.2 and we calculate theoptimal weights that maximize carry and roll-down. CA&RD return in April 2019 is given by the return on investment
with these weights for a month. Then we compare the benchmark returns with CA&RD returns calculated in the sameway for October 2021.
To start with the analysis, we review the dynamics of U.S. Treasury yields over the period covered by the analysis.
Figure 1 shows the yield movements for 3-year bond, 5-year bond, 7-year bond, 10-year bond, 20-year bond, and 30year bond. Because carry and roll-down strategy generally adopts bullet or barbell-type portfolios, performance could
be inferred by the 5-year and 7-year yield relative to the combination of 3-year and 30-year yield. Firstly, we review
the dynamics of the yield curve from the end of April 2017 to the end of March 2019, which is the estimation period
for the model in Chapter 3 and 4. This period, even though it is only two years, consists of variety of yield regimes
such as ① the flattening rally from the end of April 2017 to the end of July 2017, ② the flattening sell-off from theend of July 2017 to the end of August 2018, ③ the steepening sell-off from the end of August 2018 to the end of
October 2018, and ④ the steepening rally from the end of October 2018 to the end of March 2019.

Next, we review the period from the end of March 2019 to the end of October 2021 covered by this performance6

analysis in this Chapter. This period consists of the following periods; ⑤ parallel rally from the end of March 2019
to the end of August 2019, ⑥ modest steepening sell-off from the end of August 2019 to the end of December 2019,⑦ steepening rally from the end of December 2019 to the end of July 2020, ⑧ modest steepening sell-off from theend of July 2020 to the end of December 2020, ⑨ intense steepening sell- off from the end of December 2020 to theend of March 2021, ⑩ flattening rally from the end of March 2021 to the end of July 2021, ⑪ flattening sell-off
from the end of July 2021 to the end of October 2021.

Figure 1: Trends of the U.S. Treasury yields (constant maturity per yield, in percentage)(Source: Prepared by the authors from FactSet data)

Figure 2 shows the cumulative CA&RD returns and cumulative benchmark returns over the period from the end of
March 2019 to the end of October 2021. In periods ⑤ (parallel rally) and ⑥ (modest steepening sell-off), both
cumulative CA&RD returns, and cumulative benchmark returns decrease after increase, but there is little divergence
between the two. In period ⑦ (steepening rally), both returns increase, but the cumulative CA&RD returns greatly
exceed the cumulative benchmark returns. The following period ⑧ (modest steepening sell-off), both returns
decrease moderately, but cumulative CA&RD returns further exceed cumulative benchmark returns. However, as
entering the period ⑨ (intense steepening sell-off), cumulative CA&RD returns are behind the cumulative
benchmark returns as both returns decrease significantly. In the period ⑩ (flattening rally), cumulative CA&RD
returns exceed cumulative benchmark returns as both returns increase. In the final period ⑪ (flattening sell-off), thecumulative CA&RD returns are far behind the cumulative benchmark returns while both returns decrease. Over the
analysis period, the excess return (α) is only 0.19% per annum and it seems that very few excess returns remain
considering the transaction cost.

7

Figure 2: Comparison of cumulative CA&RD returns and cumulative benchmark returns(Source: Prepared by the authors from FactSet data)
As we examined the optimal portfolio using carry and roll-down strategy during the performance analysis period,
with a few exceptions, the portfolio mainly consisted of 6-year or 7-year bond, which accounts for about 80% to 90%of the portfolio, and 20-year or 30-year bond, which is the rest of the portfolio as duration adjustments. This isconsistent with Yamada (2000), which assumes that the optimal portfolio of carry and roll-down strategy for theJapanese government bond market is "bullet-type except for a just brief period of time." However, a major difference
from Yamada (2000) is that carry and roll-down strategy which is optimally weighted generates only a little profit in
the efficient U.S. Treasury market. In order to effectively manage carry and roll-down strategy in the U.S. Treasury
market, it is indispensable to have a mechanism to properly switch between "bullet-type" and "barbell-type"

3. Application of factor analysis to U.S. Treasury yields
3.1 Three-Factor model
Litterman and Scheinkman [1991] decomposes yield dynamics into three factors by applying factor analyses todynamics in U.S. zero-coupon government bond yields. To analyze richness and cheapness of bonds among buckets,
we adopted a model where the yield dynamics of the 13 buckets are represented by the dynamics of the three major
factors attained from factor analysis to the constant maturity par yields (rather than changes in yields) of the 13 buckets
in the U.S. Treasury market shown in Section 2.2, and the residual is defined as an error term. We determine the fairyield is the one expressed by the linear combination of the three major factors, and recognize that the bucket ischeapness if error term is positive and richness if error term is negative.

8

𝑌𝑌𝑖𝑖 (𝑡𝑡) = 𝑎𝑎𝑖𝑖,1 𝑓𝑓1 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,2 𝑓𝑓2 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,3 𝑓𝑓3 (𝑡𝑡) + 𝜀𝜀𝑖𝑖 (𝑡𝑡)(𝑖𝑖 = 1, ⋯ ,13)(6)Here, each 𝑌𝑌𝑖𝑖 (𝑡𝑡), 𝜀𝜀𝑖𝑖 (𝑡𝑡), 𝑓𝑓1 (𝑡𝑡), 𝑓𝑓2 (𝑡𝑡), 𝑓𝑓3 (𝑡𝑡) represent the value of the constant maturity par yield of the 𝑖𝑖 bucket, theerror term of the 𝑖𝑖 bucket, and the value of the first factor, the second factor, and the third factor at time 𝑡𝑡 (as of theend of each month from the end of April 2017 to the end of March 2019), respectively. 𝑎𝑎𝑖𝑖,1 , 𝑎𝑎𝑖𝑖,2, 𝑎𝑎𝑖𝑖,3 are defined as

the value of the first factor loading, the second factor loading, and the third factor loading of the 𝑖𝑖 bucket.
(Remark)

In practice, factor analysis is applied to the standardized yields in equation (6), and an inverse change instandardization is applied to the aggregated three main factors expressed in equation (6) to obtain an appropriate levelof yield. The concept of factor analysis is described in detail in Noguchi (2018), and Python was used to implement
the model.
3.2 Analysis of U.S. Treasury yields using three-factor model
Figure 3 and Figure 4 show the factor loading and the factors, respectively, which are obtained by applying factor
analysis to constant maturity par yields in 13 buckets of the U.S. Treasury market for the period from the end of April
2017 to the end of March 2019. Figure 3 shows the shape of factor loading for the maturity of a Treasury bond. Since
the factor loading of the first factor is the same level in various maturity buckets, it could be interpreted as a "level"factor that causes a parallel shift. The second factor as a “slope” factor causes a flattening sell-off or steepening rally(it makes the twist after 15-year), since the factor loading of the second factor becomes smaller as the maturity
becomes longer and the sign changes from positive to negative after 15-year. Since the factor loading of the third
factor is downward convex mainly in the 6-8-year buckets, when the third factor decreases to negative (or increases
to positive) , an upward (or downward) “curvature” appears and decreases (or increases) the performances of eachbucket.
The results are mostly consistent with the those of Litterman and Scheinkman [1991] which apply the factor analysis
to yield dynamics. The term-structure model of interest rates in Nelson and Siegel [1987] also utilizes a functional
form that assumes these three factors.

9

Figure 3: Factor Loading(Source: Prepared by the authors from FactSet data)

Figure 4: Factors(Source: Prepared by the authors from FactSet data)
Next, we review the dynamics of each factor in Figure 4. It shows that the first factor dynamics roughly captures10

the yield dynamics for each U.S. Treasury bond maturity in Figure 1 previously shown. The rise of the second factor
flattens the yield curve, as confirmed in Figure 1, for example, in the period from the end of April 2017 to the end of
August 2018. The third factor, for example, has decreased since March 2021 and has increased yields, mainly in the
6-8-year buckets. In this course, as seen in Section 2.3, the 6-8-year buckets are underperformed, and the excess returnof carry and roll-down strategy accumulated thus far is mostly disrupted. Since the factor model can provide a detailed
and accurate representation of yield dynamics for each U.S. Treasury maturity, it seems to have a granularity that caneven express the excess return of carry roll-down strategy. Actually, Figure 5 shows both the cumulative return fromcarry and roll-down strategy based on the fair U.S. Treasury bond yields suggested by the three factor model(excluding the error term on the right-hand side of equation (6) and the cumulative return from carry and roll-down
strategy using the actual yields on U.S. Treasury bonds shown in Section 2.3, and it shows that the dynamics of them
are almost the same, although the latter exceeds by a little less than 50BP the former. This means that the optimal
portfolios applied in carry roll-down strategy for both yields are almost identical. We carefully consider the reasons
as follows.

Figure 5: Comparison of cumulative CA&RD returns with cumulative CA&RD (model) returns(Source: Prepared by the authors from FactSet data)
If the optimal portfolios are the same, does this indicate that the actual yields on U.S. Treasuries do not get rich or
cheap and it would be always mostly the same as the fair yields on U.S. Treasuries based on the three-factor model?
Figure 6 shows the error term (𝜀𝜀𝑖𝑖 (𝑡𝑡); on the right-hand side of equation (6)) representing richness and cheapness of

the U.S. Treasuries with various kinds of maturities for the periods shown in Figure 1. For the in-sample period fromthe end of April 2017 to the end of March 2019, which is the estimation period for the model, Figure 6 shows theerror term just in the factor analysis, and for the out-of-sample period thereafter, the error term is updated by thecross-sectional (across the yield curve) least square regression using the yield curve data at the end of the month and11the same factor loading in the in-sample period. Because the factor loading in the in-sample period is also used in
the out-of-sample period, the error terms in the out-of-sample period are larger than those in the in-sample period.
Figure 6 shows that during the out-of-sample period in which carry and roll-down strategy is activated, richness andcheapness among maturity buckets appear appropriately as the sign of the error term takes both plus and minus, there
were periods when each bucket becomes rich and cheap.
Figure 6: Changes in the error terms (in percentage) between the actual yields of each U.S. Treasury and the yieldsof the model(Source: Prepared by the authors from FactSet data)
However, there are many cases where the signs of the error terms are the same between adjacent buckets, forexample, in the case of a roll-down from the actual yield of an cheap (rich) bucket to the yield of an cheap (rich)bucket, there is not much difference from the roll-down from the fair yield to the fair yield based on the three-factor
model, and the effects of the error terms on carry could be literally within an error interval. In other words, when the
return of carry and roll-down derived from the fair yield curve under the yield curve unchanged scenario is simply
adopted in determining the optimal portfolio for carry and roll-down strategy, then the optimal portfolio would bedetermined in a manner that does not reflect the richness and cheapness of each maturity bucket, and in most periods,
it would be the bullet-type centered on 6–7-year buckets. By adopting expected returns reflecting richness andcheapness in each bucket, we explore the possibility of the improved carry and roll-down strategy in which optimal
portfolios switch appropriately to barbell-type at some point of time.

124. Effective Application of Factor Analysis to carry and roll-down
strategies
4.1 Effective application of the three-factor model to carry and roll-down strategies
To start with this Chapter, we revisit the scenario returns for the two kinds of carry and roll-down strategies in
Section 3.2, taking 5-year bond as an example.
(i.) The scenario return using the actual yields for each maturity of U.S. Treasury Bonds(Equation (3): Please refer to above)

𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅5→4.5 ≈

1

𝐶𝐶

1

2

′
′′
� 5 + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � �

𝑃𝑃0,5 �𝑌𝑌0,5 � 22(3)(ii.) The scenario return using fair yields for each maturity of U.S. Treasury Bonds based on the three-factor model(Excluding the error term on the right side of equation (6))

𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅5→4.5 ≈

1

𝐶𝐶

′
� 5 + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��∑3𝑗𝑗=1 𝑎𝑎4.5,𝑗𝑗 𝑓𝑓𝑗𝑗 (0) − ∑3𝑗𝑗=1 𝑎𝑎5,𝑗𝑗 𝑓𝑓𝑗𝑗 (0)� +𝑃𝑃0,5 �𝑌𝑌0,5 � 22𝑃𝑃′′ �𝑌𝑌 ��∑3𝑗𝑗=1 𝑎𝑎4.5,𝑗𝑗 𝑓𝑓𝑗𝑗 (0) − ∑3𝑗𝑗=1 𝑎𝑎5,𝑗𝑗 𝑓𝑓𝑗𝑗 (0)� �
2 0.5,4.5 0,51(7)

Next, as an effective application of the three-factor model to carry and roll-down strategy, we would like to propose
a scenario return that the error term contributes to the return of the roll-down in a way that a certain portion γ(0 <𝛾𝛾 ≤ 1) of the error term 𝜀𝜀5 (0) which is representing richness /cheapness, that is γ ∙ 𝜀𝜀5 (0) is reverting back to 0 overa half-year period.

(iii.) The scenario return based on the proposed method

𝐶𝐶𝐶𝐶&𝑅𝑅𝑅𝑅5→4.5 ≈
1
2

1

𝐶𝐶

′
� 5 + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5� �∑3𝑗𝑗=1 𝑎𝑎4.5,𝑗𝑗 𝑓𝑓𝑗𝑗 (0) − �∑3𝑗𝑗=1 𝑎𝑎5,𝑗𝑗 𝑓𝑓𝑗𝑗 (0) + γ ∙ 𝜀𝜀𝑖𝑖 (0)�� +𝑃𝑃0,5 �𝑌𝑌0,5 � 2

2′′
𝑃𝑃0.5,4.5
�𝑌𝑌0,5 � �∑3𝑗𝑗=1 𝑎𝑎4.5,𝑗𝑗 𝑓𝑓𝑗𝑗 (0) − �∑3𝑗𝑗=1 𝑎𝑎5,𝑗𝑗 𝑓𝑓𝑗𝑗 (0) + γ ∙ 𝜀𝜀𝑖𝑖 (0)�� �(8)(Remark)In the expansion of equation (8), it is assumed that the effect of the third term (static convexity) on the return is smallenough to be neglected, so no error term is added to the third term here to simplify the way of calculation.

13

4.2 Performance analysis on the proposed carry and roll-down strategy
For the period from the end of March 2019 to the end of October 2021, Figure 7 shows the cumulative excess
returns from the investment to U.S. Treasury bond index, with the optimal portfolio based on the three scenarios
shown in Section 4.1 (i.) through (iii.) being updated monthly. In Figure7, the cumulative excess returns of carry androll-down strategy based on scenario returns of (i.), (ii.), and (iii.) correspond to (1) CA&RD, (2) CA&RD (model),and (3) CA&RD+α (model), respectively. The difference between (1) CA&RD and (2) CA&RD (model) in Figure 7
is the same as that between (1) CA&RD and (2) CA&RD (model) in Figure 5. We mentioned in Section 3.2, that
Figure.5 suggested, "(1) CA&RD exceeds (2) CA&RD (model) by a bit less than 50BP, but the dynamics of both arealmost the same." However, according to Figure 7, which is more detailed, (1) CA & RD outperforms (2) CA & RD
(model) by nearly 50BP in period ⑥ (modest steepening sell-off), and then repeats some outperforming andunderperforming from the subsequent period⑦ to the final period⑪. After all, the cumulative excess return of (1)CA & RD exceeds the cumulative excess return of (2) CA & RD (model) by only less than 50BP.

Figure 7: Cumulative excess returns against the U.S. Treasury Index(Source: Prepared by the authors from FactSet data)
Next, we compare (3) CA&RD+α(model), the proposed model of this study with (1) CA&RD. In period⑤(parallel
rally), period⑥(modest steepening sell-off), and period⑦(steepening rally), the performances of the two are notsignificantly different, however from period⑧(modest steepening sell-off) to period⑨(intense steepening sell-off),(3) CA&RD+α(model) outperforms (1) CA&RD by nearly 100BP, and by nearly 35BP in period⑪(flattening selloff), as a result (3) CA&RD+α(model) outperforms (1) CA&RD by nearly 135BP in total.
In comparison with the portfolio returns of the U.S. Treasury Index (BM), Table 1 shows annualized portfolio

14

returns, excess returns, tracking errors (TE), and information ratios (IR) for (1) CA&RD, (2) CA&RD (model), and(3) CA&RD+α (model) over the performance analysis period. Interestingly, (3) CA&RD+α(model), the proposedmodel of this study, has an excess return of 0.67%, which is more than three times that of (1) CA & RD, and not
only has an excellent such excess return, but also has the smallest TE. For this reason, the IR is 1.10, showing afairly good performance. The underlying reason to this can be considered as the following. By incorporating acertain portion of the error term γ ∙ 𝜀𝜀𝑖𝑖 (0) representing richness and cheapness into the returns of the roll-down of 6-

7-year bonds, the scenario return of the carry and roll-down of 6-7-year bonds is reduced when the error term isnegative and the barbell-type portfolio is selected instead of the bullet-type, even when the bullet-type portfolio of
6-7-year bonds is chosen without the effect. Due to the appropriate switch from the bullet-type portfolio to thebarbell-type portfolio, the excess returns could be increased even in the period ⑧ (modest steepening sell-off) andperiod ⑪ (flattening sell-off) .

Table 1: Annualized returns for each portfolio
Port nameBM①CA&RD②CA&RD(model)③CA&RD＋α(model)Port
Return

ExcessReturnTEIR

win-rate(vsBM)

3.71%

-

-

-

-

3.90%

0.19%

0.78%

0.24

61.29%

3.73%

0.01%

0.75%

0.02

54.84%

4.39%

0.67%

0.61%

1.10

70.97%(Source: Prepared by the authors from FactSet data)
4.3 Primary reason why the proposed method generates excess returns
In this section we consider the reasons why the proposed method produces reasonable excess returns. Since thefeature of the proposed method is to use a certain portion of the error term γ ∙ 𝜀𝜀𝑖𝑖 (0) representing richness andcheapness in the model, which is corrected over a half-year period, the error term 𝜀𝜀𝑖𝑖 (𝑡𝑡) seems to play a key role togenerate excess returns. Therefore, we re-examine equation (6).

𝑌𝑌𝑖𝑖 (𝑡𝑡) = 𝑎𝑎𝑖𝑖,1 𝑓𝑓1 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,2 𝑓𝑓2 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,3 𝑓𝑓3 (𝑡𝑡) + 𝜀𝜀𝑖𝑖 (𝑡𝑡)(𝑖𝑖 = 1, ⋯ ,13)

(6)

Since in factor analysis it is theoretically assumed that the fair yield portion composed by the first to third terms of
the right-hand side of equation (6) and the error term are independent, we obtain equation (9) by taking the varianceof both sides of equation (6).

𝑉𝑉𝑉𝑉𝑉𝑉�𝑌𝑌𝑖𝑖 (𝑡𝑡)� = 𝑉𝑉𝑉𝑉𝑉𝑉 �𝑎𝑎𝑖𝑖,1 𝑓𝑓1 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,2 𝑓𝑓2 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,3 𝑓𝑓3 (𝑡𝑡)� + 𝑉𝑉𝑉𝑉𝑉𝑉�𝜀𝜀𝑖𝑖 (𝑡𝑡)�

Here, 𝑉𝑉𝑉𝑉𝑉𝑉(∙) express the variance of the random variable in parentheses.

(𝑖𝑖 = 1, ⋯ ,13).(9)Computing the variances of the time-series data of actual yields, fair yields from the first to third terms, and theerror term, we empirically examine whether equation (9) is established or not.

15

Theoretically, we assume that equation (9) holds, but in practice, there seems to be a correlation between the fairyield and the error term.

𝑉𝑉𝑉𝑉𝑉𝑉�𝑌𝑌𝑖𝑖 (𝑡𝑡)� = 𝑉𝑉𝑉𝑉𝑉𝑉 �𝑎𝑎𝑖𝑖,1 𝑓𝑓1 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,2 𝑓𝑓2 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,3 𝑓𝑓3 (𝑡𝑡)� + 𝑉𝑉𝑉𝑉𝑉𝑉�𝜀𝜀𝑖𝑖 (𝑡𝑡)� +2Cov ��𝑎𝑎𝑖𝑖,1 𝑓𝑓1 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,2 𝑓𝑓2 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,3 𝑓𝑓3 (𝑡𝑡)� , �𝜀𝜀𝑖𝑖 (𝑡𝑡)��

. (10)(𝑖𝑖 = 1, ⋯ ,13)

In equation(10), in case that Cov ��𝑎𝑎𝑖𝑖,1 𝑓𝑓1 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,2 𝑓𝑓2 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,3 𝑓𝑓3 (𝑡𝑡)� , �𝜀𝜀𝑖𝑖 (𝑡𝑡)�� > 0(𝑖𝑖 = 1, ⋯ ,13) holds, the correlation

between the fair yield and the error term is positive. While Cov ��𝑎𝑎𝑖𝑖,1 𝑓𝑓1 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,2 𝑓𝑓2 (𝑡𝑡) + 𝑎𝑎𝑖𝑖,3 𝑓𝑓3 (𝑡𝑡)� , �𝜀𝜀𝑖𝑖 (𝑡𝑡)�� < 0 holds,

the correlation is negative.

Therefore, if the equality does not hold in equation (9) and [>] ([<]) holds, then the fair yield and error term arethought to be positively (negatively) correlated. The inequality [>] means that "the error term becomes larger when
the fair yield increases, and the error term becomes smaller when the fair yield decrease." Thus, when [>] holds insome bucket the bucket will likely to underperform in bearish market and to outperform in bullish market due to theerror term. Conversely, The inequality [<] means that "the error term becomes smaller when the fair yield increases,
and the error term becomes larger when the fair yield decreases." This suggests that the sector will likely to outperformin bearish market and to underperform in bullish market due to the error term.
Here, Table 2 shows the ratio obtained by dividing the sum of the variances on the right-hand side of equation (9)by the variance on the left-hand side of equation (9). The value in Table 2 shows "negative correlation" if it is larger
than 100%, or "positive correlation" if it is less than 100%. In Table 2 for the 1-3- and 15-20-year buckets the indicators
are larger than 100% and "negative correlations", and for the 4–10-year buckets the indicators are 100% or less and"positive correlations". In other words, by considering the error term in the proposed method, a barbell-type portfolio
consisting of buckets of 1-3 years and 15-20 years was selected for the period ⑧ (modest steepening sell-off) andthe period ⑪ (flattening sell-off), and this selection of the portfolio has led to higher excess returns.
Table 2: Variance of actual yields, fair yields, and error terms
A
B
C

1Y

2Y

3Y

4Y

5Y

6Y

7Y

8Y

9Y

10Y

15Y

20Y

30Y

Var of actual yields
Var of fair yields(model)
Var of error terms
Var of fair yields + Var of error terms

0.94000.95970.00020.95990.93820.95000.0044
0.95440.89880.89510.00520.90030.85070.83180.00280.8346

0.79150.76290.00100.76390.74190.71740.00080.71810.68780.66540.00040.66580.67220.65660.00040.65700.64270.63560.00050.63610.60800.60640.00090.60740.5321
0.54240.00130.54370.41250.42750.00300.43040.3876
0.37950.00220.3817

C/A

102.1%

101.7%

100.2%

98.1%

96.5%

96.8%

96.8%

97.7%

99.0%

99.9%

102.2%

104.4%

98.5%(Source: Prepared by the authors from FactSet data)

5. Summary and future issues
In this study, we first conduct an empirical analysis to what extent excess returns, tracking errors, and information
ratios can be obtained when carry and roll-down strategy is adopted in the U.S. Treasury market relative to the U.S.
Treasury index, and found that, unlike the Japanese government bond market where the yield curve seems to becontrolled by the Bank of Japan’s monetary policy, a simple carry and roll strategy is not always an effective strategy
in the U.S. Treasury market, which is fairly priced with scarce arbitrage opportunity. Based on the results of this16

empirical analysis, we considered that in order for carry and roll-down strategy in the US Treasury market to work
effectively, a mechanism is necessary for construction of the optimal portfolio to properly switch between "bullettype" and "barbell-type." To construct this mechanism, we revisited Litterman and Scheinkman [1991] for the firsttime in 30 years to incorporate the error term obtained by applying factor analysis to the yield itself into the scenario
returns of carry and roll-down strategy. Carry and roll-down strategy based on the proposed model was confirmed tohave some potential to obtain excess returns possibly even when U.S. Treasuries are bearish.
There are three main issues to be addressed in the future as follows. (I) In this study, we could only show the resultsof empirical analysis in a relatively short-term period after the end of April 2017, when index weights for each maturityof U.S. Treasuries are easily available for GPIF. Long-term empirical analysis with an analysis period of the past 20
years will be necessary. (II) When conducting the empirical analyses based on the yield data in the past long-term
periods, the optimality of the setting itself should be examined. Specifically, answers to the following questions need
to be explored. Here the factor loading obtained by applying factor analysis to the monthly yield data for the past two
years was fixed and they were used for the performance analysis period, but is the length of the past two years
appropriate when estimating the factor loading? When calculating scenario returns, we adopted γ = 1⁄2 in theempirical analysis based on the assumption that a certain portion of the error term γ ∙ 𝜀𝜀𝑖𝑖 (0) representing richness andcheapness would be corrected in six months, but to what extent should it be appropriate to be set? (III) It is necessary
to consider the transaction costs required to rebalance the portfolio.
Although there are many challenges to be overcome as described above, we believe that it is possible to utilize theapproach of this study to a certain extent in the effective management of carry and roll-down strategy in the U.S.

Treasury market, which is fairly priced in a manner that generally meets no arbitrage condition.

Appendix: Derivation of equation (1)𝑃𝑃0,5 (𝑥𝑥) =𝐶𝐶5 ⁄2

𝑥𝑥 1
�1+ �
2

+

𝐶𝐶5 ⁄2

𝑥𝑥 2
�1+ �
2

𝑃𝑃0.5,4.5 (𝑥𝑥) = 𝐶𝐶5⁄2 ++ ⋯+𝐶𝐶5 ⁄2

𝑥𝑥 1
�1+ �
2

+

𝐶𝐶5 ⁄2

𝑥𝑥 9
�1+ �
2

𝐶𝐶5 ⁄2

𝑥𝑥 2
�1+ �
2

𝐶𝐶 ⁄2+100

+ 5

𝑥𝑥 10
2

�1+ �

+ ⋯+

𝐶𝐶5 ⁄2

𝑥𝑥 8
�1+ �
2

(A.1)
𝐶𝐶 ⁄2+100

+ 5

l

𝑥𝑥 9
2

�1+ �

(A.2)

The Taylor expansion of equation (A.2) to the second order around the 5-year bond yield 𝑌𝑌0,5 is as follows.
2
1 ′′′𝑃𝑃0.5,4.5 �𝑌𝑌�0.5,4.5 � ≈ 𝑃𝑃0.5,4.5 (𝑌𝑌0.5 ) + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,5 � + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,5 �
2(A.2’)

The change in bond prices over half a year from the present (the numerator on the left side of equation (1)) is
attained by substituting (A.1) and (A.2'),′
𝑃𝑃0.5,4.5 �𝑌𝑌�0.5,4.5 � − 𝑃𝑃0,5 �𝑌𝑌0,5 � ≈ �𝑃𝑃0.5,4.5 (𝑌𝑌0.5) − 𝑃𝑃0,5 �𝑌𝑌0,5 �� + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,5 � +2𝑃𝑃′′ �𝑌𝑌 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,5 � .
2 0.5,4.5 0,5
1

17

(A.3)

And expand each of the first to third terms on the right side of equation (A.3),〔term 1〕= �𝑃𝑃0.5,4.5 (𝑌𝑌0.5 ) − 𝑃𝑃0,5 �𝑌𝑌0,5 ��

Substituting 𝑌𝑌0,5 into (𝑥𝑥) in equations (A.1) and (A.2) to organize them,= �𝐶𝐶5 ⁄2 += 𝐶𝐶5⁄2 +𝐶𝐶5 ⁄2

�1+

𝑌𝑌0,5 1
2

�

100

+

𝐶𝐶5 ⁄2

�1+

9−𝑌𝑌�1 + 0,5
2 �≈ 𝐶𝐶5⁄2 + 100 �1 − 9 ∙𝑌𝑌0,5 22

�

+ ⋯+𝐶𝐶5⁄2 + 100

𝐶𝐶5 ⁄2

�1+

𝑌𝑌0,5 82

�𝐶𝐶 ⁄2+100+ 5 𝑌𝑌
�1+

9
0,5
�
2

�−�

𝐶𝐶5 ⁄2

�1+

𝑌𝑌0,5 12

�+𝐶𝐶5 ⁄2

�1+

𝑌𝑌0,5 𝑥𝑥 2
2

�

+ ⋯+

𝐶𝐶5 ⁄2

�1+

𝑌𝑌0,5 92

�+𝐶𝐶5 ⁄2+100
𝑌𝑌�1+ 0,5 �
2

10

�.

𝑌𝑌 10
�1 + 0,5
2 �

𝑌𝑌0,5
𝑌𝑌0,5
� − (𝐶𝐶5⁄2 + 100) �1 − 10 ∙�22

= 𝐶𝐶5⁄2 + 100 − 450𝑌𝑌0,5 − 𝐶𝐶5⁄2 +
5= 50 ∙ 𝑌𝑌0,5 + 𝐶𝐶5 ∙ 𝑌𝑌0,5.
2

10
𝑌𝑌0,5
𝐶𝐶5 ∙ 𝑌𝑌0,5 − 100 + 1000 ∙24And, it is considered that 𝐶𝐶5 ∙ 𝑌𝑌0,5 is sufficiently small and negligible, and considering the assumption (𝐶𝐶5 =𝑌𝑌0,5 ∙ 100）of Per bond, the following could be obtained.

≈ 100 ∙

𝑌𝑌0,5
= 𝐶𝐶5⁄2
2′〔term 2〕= 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,5 �′= 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,4.5 + 𝑌𝑌0,4.5 − 𝑌𝑌0,5 �
2
1 ′′〔term 3〕= 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,5 �
2

1 ′′
2
= 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,4.5 + 𝑌𝑌0,4.5 − 𝑌𝑌0,5 �
2

1 ′′
2
�𝑌𝑌0,5 � ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,4.5 )2 + 2(𝑌𝑌�0.5,4.5 − 𝑌𝑌0,4.5 )(𝑌𝑌0,4.5 − 𝑌𝑌0,5 ) + (𝑌𝑌0,4.5 − 𝑌𝑌0,5 � �= 𝑃𝑃0.5,4.5
2

1 ′′
2
�𝑌𝑌0,5 � ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,4.5 )2 + (𝑌𝑌0,4.5 − 𝑌𝑌0,5 � �≈ 𝑃𝑃0.5,4.5
2
In the final approximation, 2�𝑌𝑌�0.5,4.5 − 𝑌𝑌0,4.5 ��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � is assumed to be small enough.
Substituting [term1 to term3] into equation (A.3), the price change (numerator on the left side of equation (1))
for half a year from the present is as follows.
2
𝐶𝐶
1 ′′
′
′
� 5 + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌0,4.5 − 𝑌𝑌0,5 � + 𝑃𝑃0.5,4.5
�𝑌𝑌0,5 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,4.5 � +22𝑃𝑃′′ �𝑌𝑌 ��𝑌𝑌�0.5,4.5 − 𝑌𝑌0,4.5 � �
2 0.5,4.5 0,51

2

Therefore, the right side of equation (1) is obtained.

18

References
Cochrane, J. H. and M. Piazzesi [2005] “Bond Risk Premia,” The American Economic Review 95 (1), pp. 138160.
Koijen, R. S. J., T. J. Moskowitz, L. H. Pedersen, and E. B. Vrugt [2015]” Carry,” Working Paper.
Litterman, R. and J. Scheinkman [1991] Common Factors Affecting Bond Returns, The Journal of Fixed Income,Vol. 1, No. 1, pp. 54-61.
Nelson, C. R. and A. F. Siegel [1987] Parsimonious Modeling of Yield Curves, Journal of Business, Vol. 60, No.
4, pp. 473-489.
H. Reisman and G. Zohar [2004a] “Instantaneous Mean-Variance Analysis of Bond Returns,” The Journal of
Fixed Income, Vol. 14, No. 1, pp.32-39.
H. Reisman and G. Zohar [2004b] “Short-Term Predictability of the Term Structure,” The Journal of Fixed
Income, Vol. 14, No. 3, pp.7-14.
H. Reisman and G. Zohar [2004c] “Excess yields in bond hedging,” Risk, Vol. 17, No. 12, pp.93-97.
Tadashi Kikugawa, Tomonori Uchiyama, Mamoru Motohiro, Shou Nishiuchi [2017] "Bond Fund Performanceand a Smart Beta Strategy in Japan," "Securities Analysts Journal" 55 (2), pp. 69-80. (published in Japanese)
Genzo Kimura [2021] "What Makes Interest Rates So Low in Japan, Euro Area, and the US? " Securities Analysts
Journal, 59 (11), pp. 77-85. (published in Japanese)
Toshio Sakudo [2010] "Risk premium of Bonds," "Securities Analysts Journal" 48 (8), pp. 5-13. (published inJapanese)
Edition by the Securities Analysts Association of Japan/Supervised by Yukihiro Asano and Shigeki
Sakakibara/Keisuke Ito, Seiji Ogishima and Takashi Suwabe (2020), "New - Securities Investment Theory II:
Practical Edition," Nikkei Business Publications, Inc. (published in Japanese)
Hiroshi Noguchi [2018] "Introduction to Multivariate Analysis Learned from Illustrations and Numerical
Examples", The Japanese Standards Association Group (published in Japanese)
Masashi Hayashida [2005] "Risks and Returns of Bonds," Securities Analysts Journal, 43 (9), pp. 18-28.
(published in Japanese)
Hirofumi Hoshika, Koichi Miyazaki [2007] "Investment Strategy for Japanese Government Bonds Based on a
Two-Factor Model" "Operations Research" 52, (2) pp. 108-114. (published in Japanese)
Satoshi Yamada [2000] "Empirical Study of Risk Premiums in the JGB Market and Its Application to Investment
Strategies," Securities Analysts Journal, 38 (12), pp. 32-62. (published in Japanese)19