TCER Working Paper Series

Government Debt Maturity in Japan: 1965 to the Present
Junko Koeda
Yosuke Kimura
August 2021
Working Paper E-163
https://www.tcer.or.jp/wp/pdf/e163.pdf

TOKYO CENTER FOR ECONOMIC RESEARCH
1-7-10-703 Iidabashi, Chiyoda-ku, Tokyo 102-0072, Japan©2021 by Junko Koeda and Yosuke Kimura.
All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without
explicit permission provided that full credit, including ©notice, is given to the source.

Abstract
This study constructs a dataset of Japanese government bonds' maturity structure forthe fiscal years 1965–2020. Using the maturity structure data at the end of eachfiscal year for the past three decades, this study structurally estimates acanonical preferred-habitat term structure model extracting the bond supply factor.
The results provide a debt maturity equation in the fiscal-year cycle anddemonstrate that two yield factors (bond supply factor and short-term interest rate)
can account for annual-frequency variations in Japanese bond yields. The supply
factor also explains the continued decline in the long-term interest rate for the
past two decades.

Junko Koeda
TCER
and
Waseda University
School of Political Science and Economics
1-6-1 Nishiwaseda Shinjuku-ku, Tokyo 169
jkoeda@waseda.jp

Yosuke Kimura
Tokyo Institute of Technology
School of Engineering, Department ofIndustrial Engineering and Economics
2-12-1-W9-54 Ookayama, Meguro-ku, Tokyo,152-8552 Japankimura.y.bq@m.titech.ac.jp

Government Debt Maturity in Japan: 1965 to thePresent∗
Junko Koeda†and Yosuke Kimura‡First draft: April 15, 2021

Abstract
This study constructs a dataset of Japanese government bonds’ maturity structure forthe fiscal years 1965–2020. Using the maturity structure data at the end of each fiscal
year for the past three decades, this study structurally estimates a canonical preferredhabitat term structure model extracting the bond supply factor. The results provide adebt maturity equation in the fiscal-year cycle and demonstrate that two yield factors(bond supply factor and short-term interest rate) can account for annual-frequency
variations in Japanese bond yields. The supply factor also explains the continued
decline in the long-term interest rate for the past two decades.

JEL Classification: E43, E52, G11, G12, H63
Keywords: maturity structure, yield curve, debt management, Japan, supply factor, bond
yield

∗

The authors thank Dimitri Vayanos, Kazuhiko Ohashi, Fumio Hayashi, and the participants of seminars
at the Bank of Japan, the Development Bank of Japan, Hitotsubashi University, Keio University, the Ministryof Finance of Japan, and the University of Tokyo for their helpful comments. The authors acknowledge
financial support from Grants-in-Aid for Scientific Research (grant number 18K01710). This research was
conducted mainly when the authors were affiliated with the Policy Research Institute at the Ministry of
Finance, Japan. The views expressed in this paper are those of the authors and do not necessarily reflect
the official views of the ministry.
†
Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050 Japan, Email: jkoeda@waseda.jp‡
Tokyo Institute of Technology, Email: kimura.y.bq@m.titech.ac.jp

Government debt has been expanding in the prevailing environment of low interest–rates
environment worldwide. Japan is an outstanding case of soaring government debt, with theratio of government debt to gross domestic product (GDP) increasing from over 60 percent
to well-over 200 percent in the past three decades. The ratio of maturity-weighted debt toGDP, which is often used as a proxy for the bond supply factor, has continued to increasein Japan while interest rates have fallen (Figure 1). Does this “conundrum” indicate that apositive relationship between the supply factor and bond yields as discussed by, for example,
Krishnamurthy and Vissing-Jorgensen (2012) and Greenwood and Vayanos (2014), does nothold for Japan? Alternatively, should a different measure for the supply factor be used? We
argue for the latter using a term structure framework.

15
MWD/GDP
LTD/GDP10-year yield (%)

10

5

0
1990

1995

2000

2005

2010

2015

2020

Figure 1: Maturity-weighted debt to gross domestic product and bond yields in Japan
Notes: This figure plots the ratio of maturity-weighted debt to GDP in Japan following Greenwood and
Vayanos(2014). M W D/GDP and LT D/GDP stand for the ratio of maturity weighted debt to GDP and
the ratio of long-term debt to GDP respectively. The Bank of Japan’s government bond holdings are excluded
from this debt. The 10-year bond yield is zero coupon yield (also shown in Figure 4).

Moving away from Ricardian equivalence (Barro, 1974), the term structure literature
has started to analyze the effects of the bond supply and maturity structure of government
bonds on bond yields. Vayanos and Vila (2021) develop a no-arbitrage term structure modelwith preferred habitat investors and arbitrageurs. Greenwood and Vayanos (2014) present1

a special case of Vayanos and Vila (2021) assuming that the demand and supply for eachmaturity in the absence of arbitrageurs are price inelastic, thereby downplaying the role ofpreferred–habitat investors in absorbing supply shocks from the government. These authors
then provide empirical support for their model for the United States (US) by regressing bondyields on the ratio of maturity-weighted debt to GDP, their proxy for the supply factor.
Hayashi (2018) provides an algorithm to solve a discrete version of Greenwood and Vayanos(2014) allowing for many supply factors. Hamilton and Wu (2012) estimate a discrete version
of Vayanos and Vila (2021),1 assuming the preferred–habitat investors’ bond demand is adecreasing affine function of the yield, and thus, at equilibrium, the supply factor is an affine
function of the level, slope, and curvature factors. As a result, a bond supply shock is acombination of three-factor shocks that may be difficult to identify.
More recently, additional studies identify bond demand or supply shocks based on thepreferred habitat theory formalized by Vayanos and Vila (2021). The theoretical model has
been extended by introducing the effective lower bound on nominal rates (King, 2019b)2
and incorporating macroeconomic variables (King, 2019a). Gorodnichenko and Ray (2017)
identify bond demand shocks by high-frequency changes in futures prices before and after
each treasury auction, exploiting the primary market structure in the US. These authors
then examine how shocks spread to other maturities. Kaminska and Zinna (2020) propose
and estimate a state-space representation of Vayanos and Vila (2021) on the term structureof real rates in which the US bond supply factor is a linear function of selected observed
supply factors, that is, reserves by foreign officials, large-scale bond purchases by the Fed, andTreasury supply. The authors note that these supply factors better capture the low-frequency
behavior of the supply factor. This study extracts the bond supply factor exploiting the bondmaturity structure information at an annual frequency in Japan and structurally estimates
1

Fukunaga, Kato and Koeda (2015) and Koeda (2017) estimate a discrete version of Hamilton and Wu (2012)
for Japan.
2This study does not introduce the zero lower bound (ZLB) or the effective lower bound (ELB), becausebond yields often took negative values in Japan, and did so even before the introduction of a negativeinterest rate policy in early 2016; and thus, bond yields were not necessarily bounded by the short rate’s
ELB in Japan.

2

a discrete version of the Greenwood and Vayanos (2014) model. The structural estimation
enables us to identify the supply shock and its effect on bond yields without worrying about
the endogeneity addressed by Greenwood and Vayanos (2014) using instrumental variable
estimation. To the best of our knowledge, no previous study in the literature has estimatedthe bond supply factor directly using maturity structure information besides the ratio of
maturity-weighted debt to GDP. Furthermore, the evolution of the government debt maturity
structure has not been fully analyzed for many countries.
To fill these research gaps for the case of Japan, this study constructs and analyzes amaturity structure database for Japanese government bonds (JGBs) and bills. Japan is aninteresting case to study because it developed the world’s second-largest government bondmarket (OECD, 2019). Additionally, and data on issue-level government bond characteristics (e.g., coupon and maturity) have been available since fiscal year (FY) 1965 when the
post-World War II (WWII) de facto debt management policy in Japan started. Moreover,the country actively implements a debt maturity policy. The Ministry of Finance (MOF)
announces a detailed debt maturity plan when the budget for the upcoming fiscal year is approved by the Cabinet in December, several months before the new FY begins in April. Furthermore, the Bank of Japan (BOJ), the largest government bondholder currently, influences
the maturity structure of marketable bonds through its asset purchases under quantitativeeasing. The pre-announced debt maturity plan includes which specific bond maturities toissue and by how much. Thus, the plan is more detailed than those of other countries. For
example, in the United Kingdom, the issuance plan categorizes bond maturities into three
types: short, medium, and long term. In the US, the issuance plan is revised every six
months. In Japan, based on the plan, government bonds are issued through “communications with the markets” until the maturity structure is finalized at the fiscal-year end. To
analyze the maturity structure consistently with the FY cycle, we construct the maturity
structure of marketable bonds at the fiscal-year end (end-March), which is free from noises
reflecting temporary changes and adjustments in the maturity structure within the year.

3

The constructed maturity structure variable is used for model estimation via the maximum
likelihood method.
This study makes several contributions. First, it proposes a novel way to extract the
supply factor, which resolves the conundrum that interest rates are seemingly negatively
correlated with the bond supply factor. While the existing supply proxy, maturity–weighted
debt to GDP, prefixes the loading on debt in each maturity with increasing weights, theloading of our supply factor has a higher weight on 2–10 year remaining maturities thanother maturities, particularly 6–8 year remaining maturities possibly reflecting a close linkwith the futures markets. Furthermore, our supply factor declined for the past three decades
despite the continued expansion of the ratio of government debt to GDP. We find that
this decline accounts for the continued fall in the long-term interest rate in the two-decade
long zero lower bound (ZLB) interest rate environment. Second, this study is the first to
construct maturity structure data for Japan using issue-level data since FY1965. Thus,
the data help us understand how debt management policy has evolved from the aspectsof maturity structure. Third, the structural estimation enables us to identify the supply
shock and its effect on bond yields without worrying about they endogeneity addressed by
Greenwood and Vayanos (2014) using instrumental variable estimation. Further, it enables
us to link the supply factor to the maturity structure, which clarifies a debt maturity policy
function.
The rest of the paper is organized as follows. Section I documents the maturity structure
data construction and discusses the evolution of the structure. Section II presents the model.
Section III explains the estimation strategy and results. Section IV conducts an impulse
response exercise. Section V provides an additional discussion. Section VI concludes.

I

Maturity Structure and Debt Management Policy

This section explains our maturity structure data construction and documents how thesevariables have evolved in changing debt management policies in Japan.

4

A

Maturity structure data

JGBs are debt securities issued by the central government of Japan. This study constructs
maturity structure data with annual frequency at the end of the FY (the Japanese FY starts
in April and ends in March). Our sample begins from March 1966 (end of FY1965) and ends
in March 2021 (end of FY2020).
As in Fukunaga, Kato and Koeda (2015),3 we collected data from the Japanese Bond
Handbook (Ko-Shasai Binran) on every JGB issued. The handbook is published semiannually (end-March and end-September) by the Japan Securities Dealers Association (JSDA).
The handbook provides data on the bond characteristics of each bond, including bond type,
series number, issue date, coupon rate, maturity, direct underwriter (if any), and semi-annual
observations of face value outstanding. Outstanding marketable JGBs reflect changes due tobuybacks, liquidity operations, and early redemption. Since the JSDA stopped publishing
this information after March 2019, the last two years of the sample period were constructedby combining publicly available data on JGBs and T-bill issuance and bond holdings by the
BOJ. We provide a detailed description of the data construction in the Online Appendix. We
break the stream of each bond’s cash flows into principal and coupon payments to construct
the future cash flows, as in Greenwood and Vayanos (2014).4
We group JGBs5 and T-bills into marketable and non-marketable types.6 The nonmarketable bonds include the following:• Bonds underwritten by the Trust Fund Bureau (TFB), Postal Savings, and Postal Life
Insurance or Pension Reserves;
3

However, there are several differences (i) our definitions of non-marketable bonds differ, and (ii) we use acash flow based calculation rather than a principal based one.
4
We apply a principal based calculation for inflation-indexed bonds and flexible interest rate bonds.
5
Officially, JGBs are government bonds issued with maturity of one year or longer.
6
Legally, JGBs can be categorized into three types: (i) general bonds, (ii) Fiscal Investment and Loan
Program (FILP) bonds, and (iii) other bonds (MoF, 2004). General bonds are to be repaid by tax revenues.
FILP bonds provide funding for government investment and lending operations under the FILP and are to
be repaid by returns from FILP operations. Other bonds include subsidy, subscription, and contribution
bonds. The marketable general and FILP bonds are treated as the same financial instruments in the JGB
markets.

5

• JGBs for individual investors7 ; and• Other small amounts (i.e., subsidy, subscription, and contribution bonds).
Figure 2 shows outstanding marketable and non-marketable JGBs in trillion yen. Outstanding non-marketable JGB is shown by the difference between the blue dashed and green
dash-dotted lines. In early years of the investigated period, non-marketable bonds were
mostly those underwritten by the TFB. The TFB is a branch of the MOF that manages
funds collected by the government through postal savings, pensions and other systems in
JGBs. It became active in FY1965 and was later abolished in FY2001. Bonds underwritten
by postal savings, postal life insurance, and pension reserves increased particularly amid thereform of the Fiscal Loan Fund Special Account in the 2000s.8 Since FY2013, the BOJ’spurchases of long-term bonds have reduced the size of marketable bonds, excluding BOJ
holdings (black solid line in Figure 2).

Outstanding Amount (in trillion yen)

1200

1000

marketable + non-marketable
marketable excl. BOJ holdings
marketable

800

600

400

200

0
1960

1970

1980

1990

2000

2010

2020

Figure 2: Marketable vs. non-marketable Japanese government bonds, FY1965-2020
Notes: This figure plots the face-value outstanding of marketable bonds (green dash-dotted line), marketablebonds excluding BOJ holdings (black solid line), and marketable bonds plus non-marketable bonds (bluedashed line) in trillion Japanese yen.

7
8

JGBs for individual investors were established in 2003 to diversify JGB products.
For more discussion, see for example, Cargill and Yoshino (2003).

6

The net face-value of the bond outstanding with remaining maturity greater than n − 1
years and less than or equal to n years (SFn V,t ) is shown in Figure 3a. “Net” means thenet bond supply by the government (i.e., MOF issuance subtracting the BOJ holdings)
constructed by aggregating cash flows across individual bonds. The nominal share of thePn-year bond supplied, snF V,t for n = 1, ..., N , is defined as SFn V,t / i SFi V,t where N is thePmaximum maturity in years. The denominator ( i SFi V,t ) is the total net face-value ofbonds supplied by the government in period t, which corresponds to the black solid line in
Figure 2. By construction, snF V,t adds up to one over N -year maturities. Figure 3b plots snF V,t
with N = 40. We discuss the evolution of snF V,t and the related debt management policychanges9 in the next subsection.

200

150

1
2
3
4
5
6
7
8
9
10

11
12
13
14
15
16
17
18
19
20

21
22
23
24
25
26
27
28
29
30

31
32
33
34
35
36
37
38
39
40

100

50

0
1960

1
2
3
4
5
6
7
8
9
10

0.5

Share of n-year bond, face value

Amount of n-year bond, face value (in trillion yen)

250

0.4

11
12
13
14
15
16
17
18
19
20

21
22
23
24
25
26
27
28
29
30

31
32
33
34
35
36
37
38
39
40

0.3

0.2

0.1

1970

1980

1990

2000

2010

2020

0
1960

1970

1980

1990

2000

2010

2020

(a) Bonds outstanding by maturity (SFn V,t , in trillion yen)(b) The share of bond supply by maturity (snF V,t )Figure 3: Maturity structure
Notes: Figure 3a plots bonds outstanding by maturity in trillion yen and Figure 3b plots the bond-supply
share. FV in the subscript of snF V,t stands for “face value.”

9

Koeda (2021) reviews the related literature on Japanese debt management policies (most of them are written
in Japanese) and presents the following four phases of the debt management policy: Phase I (FY1965–
1975, early market development); Phase II (FY1976–1998, stabilizing the maturity structure); Phase III(FY1999–2012, further market developments); and Phase IV (FY2013–2020, increased BOJ long-term bondpurchases).

7

B

The evolution of debt management policy

The post-WWII debt management policy in Japan de facto started in FY1965. In that year,
JGBs were issued for the first time after the war under a supplementary budget to cover arevenue shortfall . In the early years of the investigated period, the share of bond supply
snF V,t (Figure 3b) was unstable, as there was only one type of bond (7-years). Syndication
underwriting was the only issuance method for marketable JGBs, and the secondary markets were underdeveloped (Section 1.1, MoF, 2004).10 In the late 1970s, the market became
thicker (Figure 3a) and the maturity structure gradually stabilized (Figure 3b). The development of the secondary market and the diversification of bond type gained more policy
attention (Sections 1.1 and 2.2, MoF, 2004). Amid financial liberalization and internationalization, there were notable market developments, such as the introduction of an auction in
1978, the opening of the futures markets in 1986, and the introduction of a “partial” auction
system for 10-year bonds in 1989 by Syndicate, the main underwriting body at that time.
In 1999, the auction was introduced for a 1-year financial bill, and the market size notably
increased for bonds and bills with remaining maturity of 1 year or less (Figure 3a). The
resulting shorting of the maturity possibly reflects the introduction of BOJ’s zero interest
rate policy (McCauley and Ueda, 2012). “Communications with the markets” (the phrase
repeated used by the MOF) gained more importance and the TBF was finally abolishedin 2001. Furthermore, after the end of the syndication underwriting system in 2006 in response to some market participants’ views that the system became outdated and lacked
market efficiency (pp. 207–209, MoF, 2012), the auction method became the only issuancemethod of marketable JGBs. Since April 2013, the BOJ has been purchasing bonds withlong maturity under its qualitative and quantitative easing policy with an explicit target on
the average maturity of its bond holdings. In September 2016, the BOJ introduced a yield
curve control targeting a10-year JGB yield of around 0 percent by committing to necessary
10

Syndication underwriting was participated by the most of financial institutions, and the underwrittenbonds were usually bought and held until maturity (p. 7 and p. 132, MoF, 2012).

8

JGB purchases. As a result, the marketable bonds outstanding excluding BOJ holdings has
notably declined since FY2013 (Figure 2).

II

Model

The model is a discrete version of Greenwood and Vayanos (2014), which has a setting thatallows us to focus on the effect of government bond supply on bond yields and risks. In themodel, bond supply changes affect bond yields by changing the amount of interest-rate risk
held by arbitrageurs. There are bonds with different maturities in the economy. Demandand supply for each maturity in the absence of arbitrageurs are assumed to be price inelastic.
As in Hamilton and Wu (2012) and Hayashi (2018), in our model, the decision problem
of arbitrageurs is to maximize the risk-adjusted portfolio return subject to an adding-up
constraint. The maximization problem of arbitrageurs is
hiγ
max Et (Rt+1 ) − Vart (Rt+1 )2{ztn }Nn=1

subject toNX

ztn = 1,(1)n=1where ztn is the nominal share of arbitrageurs’ n-period bond holdings, γ captures the degreeof risk aversion, and N is the maximum maturity that arbitrageurs hold. The holding–period
return on arbitrageurs’ portfolio (Rt+1 ) is defined by

Rt+1 ≡

N
(n−1)
(n)
X
Pt+1 − Pt
(n)
Pt

n=1

ztn .

(2)

Accordingly, the first-order condition is derived as follows:

Et

A

"

(n−1)

(n)

Pt+1

− Pt
(n)

Pt

#

(1)

−

1 − Pt(1)Pt=γ

1 ∂Vart (Rt+1 ), n = 2, 3, . . . , N.
2
∂ztn

Factor dynamics

The vector of state variables X consists of the supply factor (β) and the short rate (r),

9

(3)





 βt 
Xt = 
,rt

It is assumed to follow a Gaussian VAR(1):(4)

Xt+1 = µ + ρXt + Σεt+1 , εt ∼ N (0, I),

where














β
 εt



 σβ 0  ρβ 0  µβ 
µ=.
 , εt = ,Σ = ,ρ = εrt0 σr
ρrβ ρrrµr

Thus, the supply factor dynamics is assumed to be unaffected by the short rate (as assumed
by Greenwood and Vayanos, 2014), and the short rate is assumed to be unaffected by contemporaneous supply shocks. This implies that the central bank can control the short ratein the short run regardless of the contemporaneous maturity structure for n = 2, 3, . . . , N .11
This implies that the short rate r depends on its lag and the lagged vector of the supply
factor, as follows:
rt = µr + [ρrβ , ρrr ]Xt−1 + σr εrt .

B(5)Maturity structure

As in Greenwood and Vayanos (2014), the share of the net bond supply with maturity nrelative to the total net bond supply at time t (snt ) is described as a factor model. The net
bond supply is defined by the bond outstanding in the government bond markets subtracting
bonds held by private preferred habitat investors. The net bond supply is an affine functionof the vector of supply factors βt so that the maturity structure equation is given by

11

Greenwood and Vayanos (2014) mainly focus on the case where the supply factor and the short rate are
independent ρrβ = σrβ = 0.

10snt = κn +ψn βt ,
where

C

(6)

PN

n
n=1 st = 1.

Equilibrium term structure

The bond market clears for all maturities at the equilibrium. The market-clearing condition
is defined by ztn = snt . In equilibrium, the n-period log bond price can be expressed as′

pnt = ān + b̄n Xt ,
where





e N + b̄′ ,b̄1 b̄2 · · · b̄N −1 Ψ
1


1 ′
′
′
′
′
ān = ān−1 + b̄n−1 µ + b̄n−1 ΣΣ b̄n−1 − γ b̄n−1 ΣΣ b̄1 b̄2 · · · b̄N −1 κN + ā1 ,2b̄′n = b̄′n−1 ρ − γ b̄′n−1 ΣΣ′

where





 κ2  . . κN̄ =  . ,


κN





 ψ2 
 . . ΨN = 
 . ,

ψNΨ̃ N =



ΨN, 0



.

(7)
(8)

(9)

Thus, the n-period log bond yield is given by′

rtn = an + bn Xt ,(10)an = −ān /n and bn = −b̄n /n. The appendix provides the derivation of Eqs. (7) and (8).

III

Estimation

This section describes the data used in the estimation and estimation strategy, and presents
the results. Given the data availability of bond yields, the sample period starts from the11

end of FY1989 and ends at the end of FY2020 (end-March). Despite the short sample sizeof 32 years, we estimate the model based on annual frequency for the following reasons.
First, this study analyzes the debt maturity policy consistent with the fiscal-year cycle.
Higher-frequency maturity structure data help increase the sample size while containing
noise reflecting temporary changes and adjustments in the maturity structure within theyear. Second, we exploit cross-sectional information in addition to time-series informationin our estimation. In the benchmark estimation, we use cross sectional information on 20
maturity-structure variables and 20 bond yield variables. Thus we use 1280 data points intotal, which may be sufficient. We discuss the implied small sample behavior in Section V.

A

Data

To estimate the model, we need annual data on (i) bond yields with different maturitiesand (ii) the maturity structure of the net bond supply in the market value. For (i), we
use the FY averages of the end-of-month Bloomberg’s zero yield curve data from FY1989
to FY2020,12 as presented in Figure 4, using the bond yields of 1, 2, .., 20-year maturities(denoted as Rt1 , Rt2 , ..., Rt20 ) for the estimation. Thus, N = 20. Table .1 in the appendix
provides summary statistics on the bond yields and maturities.
For (ii), we compute the maturity structure variable in the market values defined byP
snt = (Ptn SFn V,t − Htn )/ i (Pti SFi V,t − Hti ), where Ptn is the n-year bond price implied by theend of FY yield curve information and SFn V,t is the net face-value of bond supply as definedin Subsection 2.1. Htn is the private preferred-habitat investors’ demand on n-year bond in
the market value.
Since full data on Htn are unavailable, we compute Htn based on recent disclosure information on life insurance companies.13 Our analysis focuses on life insurance companies because
12

Bloomberg’s zero coupon bond yield data are available from April 1989. We apply Nelson–Siegel curves
to zero-coupon yields obtained from Bloomberg to obtain bond yield for all maturities.
13
As Fukunaga, Kato and Koeda (2015) indicate, insurance business law in Japan requires every insurancecompany to disclose the amount outstanding of JGBs and T-bills by remaining maturity at least once eachbusiness year. The amount of bond holding by insurance companies is reported mostly in the face value
under the “held to maturity” purpose applying the amortized cost method. The disclosure information
usually reports the end of fiscal year values.

12they already hold three quarters of the JGBs held by insurance companies, according to the
BOJ’s flow-of-funds data14 . We denote their bond holdings by the specific range of maturities
over i years and less than or equal to j years as Hi<τ ≤j , where τ is the remaining maturity
in years. In the annual reports of life insurance companies, the remaining maturities aregrouped by “1 year or less,” “over 1 year and less than or equal to 3 years,” “over 3 years andless than or equal to 5 years,” “over 5 years and less than or equal to 7 years,” “over 7 years
and less than or equal to 10 years,” and “over 10 years.” Thus, there are six maturity groups.
We construct an unbalanced panel data on JGB holdings by maturity for about 40 life insurance companies in Japan for 2016-2019 (37, 39, 39, 40 companies respectively). The shareof life insurance companies’ holdings in the total net bond supply in the six maturity groups
are 2, 5, 8, 18, 28, 48 percent on average respectively. The share in each maturity year (hnt )is determined via cubic Hermite interpolation of the average years of each maturity groupand the corresponding shares15 . We use these shares to compute Htn where Htn =hnt Ptn SFn V,t .

14

The flows-of-fund data provided by the BOJ show that over 20 % of the volume of government bonds andbills is held by private insurance companies and pension funds. These private preferred-habitat investors,
especially life insurance companies, increased their holdings of JGBs with maturities over 10 years to matchthe duration of assets to the long duration of their liabilities under regulations and accounting standards
that force them to reduce their risky asset holdings.
15
Specifically, we choose 0.5, 2, 4, 6, 8.5, 15, 20 and 30 years for the average years and 0.02, 0.05, 0.08, 0.18,
0.28, 0.48, 0.48, and 0.48 for the shares in the benchmark estimation.

13

8
r11

r

7

Bond Yield (%)

6
5
4
3

r

2

r12

r

3

r13

r

4

r

14

r

5

r

15

r

6

r

16

r

7

r17

r

8

r18

r

9

r19

r

10

r20

2
1
0
1990

1995

2000

2005

2010

2015

2020

Figure 4: Bond yield data
Notes: This figure plots Japanese bond yields with different maturities in annualized rate in percent.

B

State space representation and estimation strategy

The model can be rewritten in a state space representation. The transition equations of thelatent state variables Xt = [βt , rt ]’ are given by

Xt+1 = ρXt + Σεt+1 , εt ∼ N (0, I),(11)where the constant term is zero for identification.16 The measurement equations consist of
the short-rate equation, the yield equations and the maturity-structure equations given as
follows.
Short rate equation:
Rt1 = δ + [0, 1]Xt ,
16

(12)

The values of snt are unchanged with different nonzero values of µ as the values of κ and Ψ can adjust
accordingly. The same argument holds for the values of rtn .

14

Yield equations:


Rt2













′



  a2   b2  .   .   . R ..  =  ..  +  ..  Xt + σ R uRt , ut ∼ N (0, IN −1 ),

 
 

  ′ 
 

bN
aN
RtN

(13)

Maturity-structure equations:


s2t








  κ2   [ψ2 , 0]  .   .  ..
 ..  =  ..  + 
 Xt + σ s ust , ust ∼ N (0, IN −1 ),
.


 
 

 

 
[ψ
,
0]
sN
κ
N
N
t

(14)

s
where uRt and ut are measurement errors in the yield- and maturity-structure equations

respectively. We estimate the model using the maximum likelihood method with Kalmanfiltering. The model parameters are Θ = [δ, ρ, Σ, κ, Ψ , γ, σ R , σ s ]. Standard errors are derived
by numerically computing the Hessian matrix.

C

Estimated results

Figure 5 shows the estimated state variables in the model. The short rate (r, black-dashedline) drops in the early 1990s and stays very low thereafter. The supply factor (β, red-solidline), the factor that governs the maturity structure, turns negative in the mid-1990s andhas been declining for the past three decades pushing down yield curves persistently despite
the continued expansion of maturity-weighted debt to GDP.

15

8supply factor
short rate

6

4

2

0

-2

-4
1990

1995

2000

2005

2010

2015

2020

Figure 5: The estimated state variables (β and r)
Notes: This figure plots the estimate β (red solid line) and r (black dashed line).

Table 1 reports the estimated model parameters. Standard errors are shown in parentheses. Despite the low-frequency data, the supply factor dynamics is very persistent with
ρβ at about 0.97. An increase in supply raises expected future short rates with a positive
ρrβ . The maturity structure equation coefficients κ and Ψ are plotted in Figure 6. The coefficients κ capture the average maturity structure and the coefficients Ψ captures systemic
debt management.17 The bond supply shares of 2-10 year remaining maturities (ψ2 , ..., ψ10 )are estimated to be positive, implying that a decrease in the supply factor involves a fall inthe shares of 2-10 year remaining maturities.

17

Vayanos and Vila (2021) assume a smoothed function for Ψ which converges to zero as the maturity
increases. Our estimated Ψ , if smoothed, can be approximated with their assumed function.

16

κ2
κ3
κ4
κ5
κ6
κ7
κ8
κ9
κ10
κ11
κ12
κ13
κ14
κ15
κ16
κ17
κ18
κ19
κ20

0.19200.11380.11090.09580.08420.07520.0712
0.06600.05610.00270.00280.00360.00430.00470.00470.00550.00600.00630.0060Maturity Structure(0.0044) ψ2 0.0013(0.0041) ψ3 0.0119(0.0057) ψ4 0.0105(0.0033) ψ5 0.0073(0.0042) ψ6 0.0167(0.0029) ψ7 0.0144(0.0023) ψ8 0.0129(0.0036) ψ9 0.0113(0.0023) ψ10 0.0084(0.0007) ψ11 -0.0041(0.0007) ψ12 -0.0038(0.0008) ψ13 -0.0033(0.0009) ψ14 -0.0030(0.0010) ψ15 -0.0027(0.0012) ψ16 -0.0019(0.0015) ψ17 -0.0016(0.0017) ψ18 -0.0014(0.0018) ψ19 -0.0014(0.0016) ψ20 -0.0015(0.0020)(0.0043)(0.0052)(0.0038)(0.0042)(0.0049)(0.0043)(0.0041)(0.0034)(0.0016)(0.0015)(0.0014)(0.0014)(0.0013)(0.0012)(0.0015)(0.0013)(0.0014)(0.0013)

Factor Dynamics
ρβ 0.9742 (0.0111)
ρrβ 0.4219 (0.1314)
ρrr 0.7820 (0.0440)
Σ11 0.1061 (0.0329)
Σ22 0.4092 (0.0710)
Short Rate
0.2455 (0.0472)

δ

Risk Aversionγ
3.7696 (0.4538)
Measurement Errors
σε 0.1074 (0.0089)
σs 0.0072 (0.0006)

Table 1: Estimated parameters
Notes: The table shows the estimated coefficients and standard errors in parentheses.

The supply factor coeff

0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
-0.005
-0.01
2

4

6

8

10

12

14

16

18

20

Figure 6: Estimated maturity structure
Notes: The estimated κN and Ψ N in the maturity structure equation (Eq. (6)) are plotted against maturity.

Figure 7 shows the estimated yield-curve coefficients, that is, an and bn in Eq. (10).
Consistent with Greenwood and Vayanos (2014), the estimated coefficient of supply in theyield equation is increasing with maturity (Figure 7, left). It is worth noting that the shape17

of this factor loading looks like that of the slope factor. The supply coefficient in the model) is computed as theimplied expected one-period holding period return (i.e., Et (pnt+1 ) − pn+1tfirst element of b̄n ρ − b̄n+1 or equivalently the first element of (n + 1)bn+1 − nbn ρ. The
value of this coefficient, which captures how much the expected return changes in response
to one-unit change in the supply factor change (ceteris paribus), is estimated to increaseagainst maturity, as shown in Figure 8.

supply-factor coeff

1.6

constant coeff

5
4

1.4

3

1.2

2
1

1

0

2

4

6

8

10

12

14

16

18

20

14

16

18

20

0.8
short-rate coeff

1

0.6

0.8

0.4
0.6

0.2

0.4

0

0.2

2

4

6

8

10

12

14

16

18

20

2

4

6

8

10

12

Figure 7: Estimated yield-equation coefficients
Notes: These figures plot the yield-equation coefficients. The left figure corresponds to the supply-factor coefficient; the top right figure corresponds to the constant coefficient; and the bottom right figure corresponds
to the short-rate coefficient.

supply-factor coeff

0.6

0.5

0.4

0.3

0.2

0.1

0
2

4

6

8

10

12

14

16

18

20

Figure 8: Estimated supply coefficient in the return equation
Notes: This figure plots the supply coefficient in the return equation against maturity.

18

The model-implied yield decomposes into the contributions from each factor, as shownin Figure 9. The figure shows how the short rate and the supply factor explain the 10year bond fluctuations. The black solid line shows the actual 10-year bond yield minus theconstant term in the yield equation (a10 ), and the red bars show the supply factor term in′

the yield equation (the first element of b̄10 Xt ). The blue bars show the short-rate term in the′yield equation (the second element of b̄10 Xt ). The supply factor term in the yield equation
captures the duration effect highlighted by Greenwood and Vayanos (2014). A more negative
supply factor accounts for the continued decline in the long-term bond yields under a ZLB
environment, while the short rate is stacked near the ZLB.

4
supply factor
short rateactual-constant

3
2
1
0
-1
-2
-3
-4
-5
1990

1995

2000

2005

2010

2015

2020

Figure 9: Decomposing the model-implied 10-year yield
Notes: This figure plots the estimated short-rate and supply-factor terms (blue and red bars, respectively)
and the actual 10-year bond yield minus the estimated constant term (black solid line) in Eq. (10).

IV

Impulse Responses to the Supply ShockHow do bond yields and the maturity structure respond to a supply shock? The impulse
response of a model-implied variable from the term structure model, yt to a yield-factor
19

shock can be defined by the difference between the following conditional expectations:

Et [yt+k | Xt + νt ; Θ] − Et [yt+k | Xt ; Θ] ,(15)

where νt represents the vector of shocks. As in Diebold, Rudebusch and Aruoba (2006),
the yield-curve factors are treated as the endogenous variables in the VAR estimation. We
numerically compute Eq. (15) given the model parameter estimates. The error bands are
obtained by drawing parameter vectors from the asymptotic distribution and picking the84th and 16th percentiles.18
Figure 10 shows the impulse responses of the model-implied variables to a positive supply
shock that increases the supply factor by 1 upon impact (note that 1 standard deviation of theestimated β is about 0.9). It is expected to take nearly a decade for the effect on the supply
factor to be halved, as the supply factor is persistent. The model-implied 10-year bond yield
jumps up by about 135 basis points upon impact, with large risk of further interest hikes.
The short rate is likely to increase. One caveat regarding this impulse response exercise isthat it assumes the shock is fully absorbed by arbitrageurs. If the shock were fully absorbed
by preferred-habitat investors, there would be no change in the maturity structure of the net
bond supply that arbitrageurs face in the model.

V

Discussion

A

Are private preferred-habitat investors price elastic?

One important model assumption is that preferred-habitat investors are assumed to be priceinelastic, as in Greenwood and Vayanos (2014). We investigate this assumption’s validity via
a simple regression analysis using the panel data for life-insurance companies (see Subsection
A for data construction). Table 2 reports the firm-fixed effect regression of the log of their18

See Hayashi and Koeda (2019) for an application.

20

supply factor

1

short rate

2

0.9

1.5

0.8

1

0.7

0.5

0.6

0
0

5

10

0

10y yield

10

5y yield

2

2

5

1.8
1.8
1.6
1.6

1.4
1.2

1.4

1
1.2
0

5

10

0

5

10

Figure 10: Impulse responses to a supply shock
Notes: This figure plots the impulse responses of the supply factor (top left), short rate (top right, annualized
rate in percent), 10-year yield (bottom left, annualized rate in percent), and 5-year yield (bottom right,
annualized rate in percent).

bond holdings by specific range of maturities (logHi<τ ≤j ) on the average of bond yields overthe corresponding maturity range (ri<τ ≤j ), where τ is the remaining maturity in years. The
estimated results show that all the coefficients were statistically insignificant, not rejecting
the model assumption that the private preferred-habitat investors are price inelastic in Japan.

21

Dependent Variable:log Hτ ≤1

log H1<τ ≤3

log H3<τ ≤5log H5<τ ≤7log H7<τ ≤10

(1)

(2)

(3)

(4)

(5)

r

-0.026log H10<τ(6)(7)(1.454)-0.007r1<τ ≤3(1.669)-1.001

r3<τ ≤5(1.478)1.181

r5<τ ≤7(0.914)-0.518r7<τ ≤10(0.635)-0.135r10(0.355)-0.024r20(0.176)
Observations

111

121

122

124

130

139

139

Adj. R2

0.854

0.844

0.833

0.886

0.945

0.989

0.989

Table 2: Demand Elasticity of Life Insurance Companies
Notes: The table regresses life insurance companies’ bond holdings for specific maturities on the corresponding bond yields. All specifications use firm fixed effects. Standard errors are shown in parentheses.

B

Is there an appropriate proxy for the supply factor?

Our supply factor presented in the previous section is a latent variable. However, for the
supply factor, some may prefer to explicitly use a proxy that does not require model estimation. One natural candidate for such a proxy is the first principal component obtained byapplying the principal component analysis (PCA) to the shares of the net bond supply bymaturity. The PCA-based supply proxy chooses the loading to capture variations in the maturity structure, whereas the existing supply proxy, maturity-weighted debt to GDP, prefixes
the loading on debt in each maturity with increasing weights. Figure 11 plots the PCA-based
supply proxy in the face value and in the market value denoted as β̂F V and β̂M V respectively.
The corresponding PCA score is standardized for easier interpretation of the results. In thecase of N = 20, β̂F V and β̂M V explain 72% and 68% of the variations in the corresponding
maturity structure variables.19 Figure 11 shows that the PCA-based supply factor declined
19

Specifically, β̂F V and β̂M V are the first principal components obtained by applying the PCA to s1F V , ..., s20
FV
and s1 , ..., s20 ,respectively.

22

for the past three decades despite the continued expansion of the ratio of government debt
to GDP. Thus, the declining PCA-based supply factor is theoretically consistent with thefall in the long-term interest rate in Japan. We also find that the PCA-based loading has ahigher weight on 6-10 year remaining maturities than other maturities, possibly reflecting aclose link with futures markets.

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5
1990

1995

2000

2005

2010

2015

2020

Figure 11: PCA-based supply factor
Notes: This figure plots the first principal component analysis (PCA) component obtained by applying the
PCA on snt for n = 1, ..., N .

23

Panel A : Maturity-weighted debt to GDP
r 10

Dependent variable:
Estimator:

OLS
(1)

M W D/GDP

(2)

2SLS
(3)

(4)

(5)

-0.158-0.058-0.047-0.146-0.050-0.032(0.120)(0.033)(0.0531)(0.081)(0.031)(0.026)

0.563

β̂M V

0.586

(0.192)(0.157)
0.642

β̂F V

0.691

(0.213)r

Constant(6)(0.160)

0.747

0.691

0.656

0.759

0.705

0.671

(0.081)(0.046)(0.059)(0.080)(0.049)(0.057)

2.210

1.616

1.578

2.120

1.559

1.483

(0.731)(0.278)(0.263)(0.654)(0.290)(0.249)Observations

32

32

32

32

31

31

Adj. R2

0.944

0.969

0.968

0.943

0.966

0.965

Panel B: Long-term debt to GDP
r 10

Dependent variable:OLS

LT D/GDP

2SLS

(7)

(8)

(9)

(10)

(11)

(12)

-0.201-0.082-0.072-0.195-0.084-0.057(0.074)(0.055)(0.056)(0.087)(0.068)(0.046)

0.516

β̂M V

0.508

(0.255)β̂F VrConstant(0.257)
0.586

0.629

(0.272)(0.194)

0.757

0.702

0.668

0.761

0.715

0.680

(0.069)(0.047)(0.062)(0.079)(0.049)(0.057)

1.880

1.516

1.514

1.855

1.520

1.463

(0.469)(0.239)(0.242)(0.538)(0.345)(0.248)Observations

32

32

32

32

31

31

Adj. R2

0.953

0.969

0.969

0.953

0.966

0.966

Table 3: Bond Yields and Supply-Factor Proxies
Notes: The table regresses 10-year bond yields on the variables of interest, the supply factor and the short
rate using our annual data from FY1989 to FY2020. The variable of interest is an alternative supply factor
measure, where M W D/GDP stands for the fraction of maturity weighted debt to GDP and LT D/GDP
stands for the fraction of long-term debt to GDP. We use two supply factors: β̂M V stands for the supply
factor computed using maturity structure based on market value, and β̂M V stands for the supply factor basedon face value. OLS stands for ordinary least squares. The two stage least squares (2SLS) estimation columns
show the estimation results from second-stage regressions. We instrument for the alternative supply factors
by the ratio of debt-to-GDP, D/GDP , and for β̂ by lagged β̂. Both the first- and second-stage regressions
include the short-rate as a control. Newey–West standard errors with 1-year lag are shown in parentheses.

24

r 10

Dependent variable:
Estimator:

D/GDP

OLS
(2)

(3)

(4)

(5)

(6)

-1.322-0.381-0.241-1.467-0.565-0.431(1.101)(0.216)(0.188)(1.399)(0.298)(0.256)

β̂M V

0.633

0.575

(0.139)(0.134)β̂F VrConstant

2SLS

(1)

0.725

0.658

(0.162)(0.159)

0.746

0.683

0.647

0.746

0.690

0.661

(0.078)(0.048)(0.058)(0.077)(0.052)(0.060)

2.386

1.593

1.502

2.534

1.765

1.675

(0.770)(0.259)(0.238)(0.883)(0.314)(0.292)

Observations

32

32

32

31

31

31

Adj. R2

0.926

0.967

0.967

0.917

0.963

0.963

Table 4: Size versus Composition
Notes: The table regresses 10-year bond yields on the variables of interest, the supply factor and the short
rate using our annual data from FY1989 to FY2020. The variable of interest is a debt to GDP, D/GDP . We
use two supply factors: β̂M V stands for supply factor computed using maturity structure based on market
value and β̂F V stands for supply factor based on face value. The columns of 2SLS estimation show theestimation results from second-stage regressions. We instrument for the debt-to-GDP by the lagged debtto-GDP, D/GDP , and for β̂ by lagged β̂. Both the first- and second-stage regressions include the short-rate
as a control. Newey-West standard errors with 1-year lag are shown in parentheses.

To examine how well our supply proxies (β̂F V and β̂M V ) accounts for bond yield fluctuations compared with the existing proxies, we conduct a simple model-free regression of a
10-year bond yield on β̂F V and β̂M V and an existing supply proxy controlling for the shortrate. Table 3 includes M W D/GDP or LT D/GDP as the existing proxies, and shows thatβ̂F V or β̂M V outstandingly explains the 10-year bond yield fluctuations. Since β̂F V or β̂M V
is a linear function of the shares of bond supply by maturity, it captures the compositionof government debt. Table 4 adds the size of government debt, specifically, total face-value
bonds outstanding to GDP (D/GDP ) as the additional supply measure. The estimated results show that the coefficient for β̂F V or β̂M V was statistically significantly positive, whereas
the coefficient for D/GDP is statistically insignificant. Thus only the composition effect isconfirmed in the regression. Going forward, however, increasing the size of the outstanding JGBs while keeping the same maturity structure may significantly impact bond yields,
particularly in light of the shrinking absorbing capacity of JGBs (Hoshi and Ito (2014)).
25

C

Short sample simulations

To examine the implied small sample behavior of the model coefficients, we used the estimated model to generate 10,000 samples of the same length as our sample period (31observations) using a similar approach to that applied by Ang, Piazzesi and Wei (2006). In
each simulated sample, we re-estimated the model via maximum likelihood. We find thatnone of the simulation results suggested that the yield curve coefficient regarding the supply factor is negative, thereby supporting Proposition 1 in Greenwood and Vayanos (2014).
We also examine the small sample distributions of ρβ from the re-estimated model. The
population coefficient from the estimated model (“truth”) is 0.97. The average and mediancoefficients across all the simulations from re-estimating the model are both around 0.97,
thus they were reasonably close to the truth, with standard deviations of the coefficients
across the simulations of 0.02.

VI

Conclusions

Using the constructed maturity structure data, we analyzed the evolution of the maturity
structure for JGB markets over the past five decades. We also structurally estimated a bondsupply factor that focuses on the composition rather than the size of the outstanding JGB for
the past three decades. We concluded that the composition of government debt explains thecontinued fall in nominal long-term interest rates in Japan, and thus, Japanese government
debt management plays an important role in managing a low-interest rate environment.
The structural estimation results indicated that the supply-factor effect on bond yields
is significant. In particular, despite the continued expansion of the ratio of government debt
to GDP, the estimated supply factor has been declining, pushing down the long-term bondyields. This decline has accompanied a notable increase in the share of bond remaining
maturities of less than 1 year and also a decrease in the share of bond remaining maturities
between 2 and 10 years. The impulse response exercise implies that a positive supply shock,if it occurs, could persistently raise bond yields, heightening risks in JGB markets that have26

a large rollover size of about 20 percent of GDP each year. The supply factor shock may
have been reflecting various events, such as the development of money markets, the BOJ’s
long-term government bond purchases, and the maturity diversification of issuance policy.
In future research, it would be worthwhile to further analyze the circumstances and policies
that drive supply factors to change.

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28

Appendix
Appendix Tables and Figures
Variable

Mean

SD

Min

MedianMax

Bond Yield (%)
r

1.011

2.039

-0.282

0.181

7.507

r2

0.998

1.984

-0.282

0.115

7.431

r3

1.087

1.975

-0.271

0.252

7.417

r4

1.214

1.982

-0.251

0.397

7.419

r5

1.351

1.990

-0.242

0.558

7.420

r6

1.486

1.996

-0.228

0.738

7.417

r7

1.614

1.998

-0.208

0.910

7.407

r8

1.731

1.996

-0.183

1.073

7.393

r9

1.839

1.991

-0.153

1.223

7.374

r 10

1.937

1.982

-0.122

1.360

7.352

β̂M V

0.000

1.000

-2.164-0.262

1.422

Supply Factor

β̂F V

0.000

1.000

-1.817-0.317

1.422

M W D/GDP

6.518

3.994

2.044

5.290

12.833

LT D/GDP

3.513

3.209

0.354

2.280

8.357

Maturity Structure
s1

0.212

0.050

0.137

0.212

0.302

s2

0.100

0.008

0.088

0.098

0.117

s3

0.077

0.013

0.044

0.075

0.104

s4

0.080

0.015

0.031

0.080

0.108

s5

0.077

0.013

0.042

0.078

0.099

s6

0.057

0.022

0.027

0.047

0.096

s7

0.056

0.020

0.027

0.050

0.096

s8

0.058

0.020

0.027

0.052

0.096

s9

0.059

0.020

0.027

0.057

0.098

s10

0.0567

0.020

0.027

0.051

0.100

D/GDP

0.910

0.436

0.349

0.879

1.664

Inflation

0.516

1.192

-1.700

0.200

3.300

Other Variables

Table .1: Summary Statistics
Notes: The table shows the summary statistics of our data. Our sample is 32 years with cross-sectional information on the
different maturities of bond yields and maturity structure. Bond yields are zero coupon rates obtained from Bloomberg. The
maturity structure variable sn is the shares of bonds with remaining maturity less than or equal to n years but greater thann − 1 years divided by the total net value of bonds for all government bonds. Supply factor β̂ is the first principal component
derived from the maturity structure variables. The maturity-weighted debt to GDP ratio, M W D/GDP , and the long-term debt
to GDP ratio, LT D/GDP , are computed following Greenwood and Vayanos (2014) and Krishnamurthy and Vissing-Jorgensen(2012), respectively.

29

Panel A: OLS
r 10

Dependent Variable:

Inflation Rate (π)

(1)

(2)

(3)

(4)

1.020

-0.229-0.1767-0.149(0.496)(0.126)(0.034)(0.036)

1.006

0.775

0.721

(0.111)(0.050)(0.060)

r

0.726

β̂M V

(0.146)
0.786

β̂F V

(0.133)
Constant

1.412

1.039

1.240

1.286

(0.497)(0.496)(0.128)(0.130)Observations

32

32

32

32

Adj. R2

0.355

0.875

0.969

0.970

(7)

(8)

Panel B: 2SLSr 10

Dependent Variable:(5)
Inflation Rate (π)

(6)

1.739

-0.682-0.543-0.594(0.412)(0.545)(0.441)(0.429)

1.203

0.948

0.938

(0.188)(0.140)(0.149)

r

0.680

β̂M V

(0.133)
0.709

β̂F V

(0.134)
Constant

1.055

1.078

1.260

1.299

(0.562)(0.321)(0.147)(0.162)

Observations

31

31

31

31

Adj. R2

0.052

0.816

0.935

0.923

Table .2: Bond Yield, Inflation Rate and Supply Factors
Notes: The table regresses 10-year bond yield on the variables of interest, the inflation rate, supply factor,
the short rate using our annual data from FY1989 to FY2020. We use two supply factors: β̂M V stands
for supply factor computed using maturity structure based on market value and β̂M V stands for supply
factor based on face value. The columns of 2SLS estimation show the estimation results from second-stage
regressions. We instrument for the inflation rate by the lagged inflation rate and for β̂ by lagged β̂. Both thefirst- and second-stage regressions for columns 2-4 and 6-8 include the short-rate as a control. Newey-Weststandard errors with 1-year lag are shown in parentheses.

30Derivation of factor-loadings equations
The FOCs for n-period bonds are

Et




n−1
Pt+1
− Ptn
1 − Pt1
1 ∂Var (Rt+1 )
−
=γ
.
n
1
Pt
Pt
2
∂ztn

Although the left-hand side is approximated by the same form as Eq. (A1.3) of Hayashi(2018), the right-hand side is slightly different because of the constant term in maturity
structure equations:e N XtSN,t = κN + Ψ N βt = κN + Ψ

Assume(n)log Pt

Et

"

(n−1)

(n)

Pt+1

− Pt
(n)

Pt

#

′

= ān + b̄n Xt

i
h

(n−1)
(n)
−1
= Et exp log Pt+1 − log Pt

 1



(n−1)
(n)
(n−1)
(n)
+ V art log Pt+1 − log Pt≈ Et log Pt+1 − log Pt
2
 1



′
′
′
′
= Et ān−1 + b̄n−1 Xt+1 − ān + b̄n Xt + V ar ān−1 + b̄n−1 Xt+1 − ān + b̄n Xt
2
 ′

1
′
′
= ān−1 + b̄n−1 (µ + ρXt ) − ān + b̄n Xt + V ar b̄n−1 Σεt+1
2

 ′
1 ′
′
′
= ān−1 + b̄n−1 µ − ān + b̄n−1 ρ + b̄n Xt + b̄n−1 ΣΣ′ b̄n−12where Ω = ΣΣ′ . The second term of the left-hand side is approximated by(1)

1 − Pt(1)
Pt(1)≈ − log Pt31′= ā1 + b̄1 Xt .

Thus, the left-hand side becomes

Et

"

(n−1)

(n)

Pt+1

− Pt
(n)

Pt

#

(1)

−

1 − Pt
(1)

Pt


 ′
1 ′
′
′
′
= ān−1 + b̄n−1 µ − ān + b̄n−1 ρ + b̄n Xt + b̄n−1 ΣΣ′ b̄n−1 − ā1 + b̄1 Xt
2



1 ′
′
′
= ān−1 + b̄n−1 µ − ān + b̄n−1 ΣΣ b̄n−1 + ā1 + b̄′n−1 ρ − b̄′n + b̄′1 Xt .
2

Next, we approximate the portfolio return to derive the variance in the right-hand side
as follows:

Rt+1 ≡

N̄
(n−1)
(n)
X
Pt+1 − Pt
(n)

Pt

n=1

(1)

= − log Pt zt1 +(1)= − log Pt zt1 +

ztn ≈

N
X

= Bt +

′

n=2
N 
X

b̄n−1 ztn

n=2

(n−1)

log Pt+1

n=1

N 
X

n=2

N 
X

(n−1)

log Pt+1

− log Ptn ztn

− log Ptn ztn′ān−1 + b̄n−1 (µ + ρXt ) − ān + b̄n Xt ztn +

!

N
X
n=2

′

b̄n−1 ztn

!

εt+1

εt+1

′

= Bt + dt εt+1
where dt ≡PN′nn=2 b̄n−1 zt . Thus, the variance of arbitrager’s portfolio return is approximated′by V ar(Rt+1 ) ≈ dt Ωdt . The right-hand side is32
1 ∂Var(Rt+1 )≈ b̄′n−1 ΣΣ′ b̄1 s2t + · · · + b̄N −1 sN
t
n
2
∂zt

= b̄′n−1 ΣΣ′ b̄1 zt2 + · · · + b̄N −1 ztN= b̄′n−1 ΣΣ′b̄1 b̄2 · · ·= b̄′n−1 ΣΣ′b̄1 b̄2 · · ·(By market clearing conditions)2 st  s3  t b̄N −1  . 
 .. 




N
st


κ
+
Ψ̃
X
N
N t
b̄N −1

Using the above expressions, the FOCs for n-period bonds can be approximated as:+ b̄′n−1 ρ − b̄′n + b̄′1 Xt



′
′
κN + Ψ̃ N Xt
= γ b̄n−1 ΣΣ b̄1 b̄2 · · · b̄N −11
ān−1 + b̄′n−1 µ − ān + b̄′n−1 ΣΣ′ b̄n−1 + ā1
2

Accordingly, we can derive the factor loading equations by comparing the coefficients.
b̄′n = b̄′n−1 ρ − γ b̄′n−1 ΣΣ′b̄1 b̄2 · · ·b̄N −11ān = ān−1 + b̄′n−1 µ + b̄′n−1 Ωb̄n−1 − γ b̄′n−1 ΣΣ′
233Ψ̃ N + b̄′1b̄1 b̄2 · · ·b̄N −1κ + ā1