Affine Term Structure Pricing with Bonds Supply As Factors
byFumio Hayashi
National Graduate Institute for Policy Studies, Japan, and National Bureau of Economic Research

March 2016

Abstract
This paper presents a theoretical model for analyzing the effect of the maturity structureof government debt on the yield curve. It is an ATSM (affine term structure model)
in which the factors for the yield curve include, in addition to the short rate, thegovernment bond supply for each maturity. The supply shock is not restricted to be
perfectly correlated across maturities. The effect on the yield curve of a bond supply
shock that is local to a maturity is largest at the maturity. This hump-shaped responseof the yield curve obtains in spite of the absence of preferred-habitat investors.
Keywords: ATSM, yield curve, portfolio balance channel, supply of government bonds,
impulse responses.

I thank Junko Koeda for posing the question studied in this paper and participants at a Kyoto University
workshop. The research reported here was supported by grants-in-aid from the Ministry of Education, Culture,
Sports, Science, and Technology of the Japanese government (grant number 25285097 and 26870124).

1

1

Introduction

How does the maturity structure of government debt affect the yield curve? The question has
recently gained prominence due to the arrival of unconventional monetary policy, whose main aim
is to lower the long-term interest rates by making the supply of long-term bonds scarce throughreserve-financed open-market purchases. The theoretical justification often cited by central bankers
is the portfolio balance channel.1 This paper provides a formulation of it.
To be sure, such a channel cannot exist according to the irrelevance theorem of Wallace (1981).
However, recent theoretical studies show that it can be reconnected if suitable transactions costs are
considered,2 and recent empirical studies on the effect of unconventional monetary policy find thatthe maturity structure matters for the yield curve.3
This paper proceeds under the assumption that relevant transactions costs exist so that thebond market can be studied in isolation. More specifically, it is closely related to the recent explicit
models of the bond market by Vayanos and Villa (2009) and, especially, Greenwood and Vayanos(2014). The model in the former has two types of bond investors: arbitrageurs with themean-variance preferences and preferred-habitat investors for each maturity. The net supply ofbonds is zero for each maturity. An attractive feature of the model is that the equilibrium yields areaffine functions of the factors that drive the short rate and the preferred-habitat investors’ demandfor bonds of various maturities, thus providing a link to the ATSM (affine term structure model), thereigning theory of the yield curve in finance.
Greenwood and Vayanos (2014, GV henceforce) simplify Vayanos and Vila (2009) by assumingthat the preferred-habitat investors’ demand is price-inelastic, or equivalently, by replacing thepreferred habitats by an entity that sets the supply of bonds of various maturities inelastically. That
entity can be interpreted as the government composed of the treasury and the central bank. Let bn
be the loading (the coefficient in the affine equation) on the supply factor for the yield of maturity n.
GV show that (i) an increase in the supply factor raises all yields (i.e., bn > 0 for all n) (ii) the profile
1

To cite Bernanke (2010): “I see the evidence as most favorable to the view that such purchases work
primarily through the so-called portfolio balance channel [italics by the author]... Specifically, the Fed’s strategy[the operation twist] relies on the presumption that different financial assets are not perfect substitutes ininvestors’ portfolios, so that changes in the net supply of an asset available to investors affect its yield andthose of broadly similar assets.”2

See Andres, Lopez-Salido, and Nelson (2004) and Chen, Curdia, and Ferrero (2012).

3

For a list of recent empirical studies, see the survey in Woodford (2012) and also Bauer and Rudebusch(2014).

2

of this supply effect can be hump-shaped (i.e., bn is increasing and then decreasing in n), (iii) for therisk premium, its profile is increasing across all maturities (i.e., the bn for the risk premium is anincreasing function of n), and (iv) it lies above the yield profile (i.e., the bn for the risk premium isgreater than the bn for the yield for all n).
Our paper differs from GV in two respects.4 First, the supply factor in GV is one-dimensional,
while in our model there can be as many supply factors as there are maturities. Put differently, thematurity structure in GV is controlled by the global factor, while in our model the supply shock can
be local, not restricted to be perfectly correlated across maturities. Despite the vastly expanded listof factors, the yields are still affine in the factors. Second, instead of characterizing the factor
loadings, we employ the IR (impulse response) analysis to describe the dynamic version of the four
results mentioned above (suitably modified to accommodate by local supply factors). We can thus
examine the dynamic effect on the yield curve of what we call the “legacy” feature of bond supply
that n-period bonds purchased by the government form part of the stock of n − 1 period bonds next
period barring an offsetting market operation.
The rest of the paper is organized as follows. Section 2 is a restatement in discrete time of GV’s
continuous-time model. Section 3 and the two appendixes show how multiple supply shocks can beincorporated into the model of GV. Section 4 conducts the IR analysis. Section 5 is a brief conclusion.

2

The Greenwood-Vayanos Model in Discrete TimeWe first present the discrete-time version of the bond pricing model of Greenwood and Vayanos(n)(2014) (hereafter, GV). Denote by Pt the price of the zero-coupon bond with maturity n in period t.
(0)

(n)

By convention, Pt = 1. The (continuously compounded) yield to maturity yt is therefore
1
(n)
(n)
yt = − log Pt .
n

(2.1)
(n)

The nominal share of the n-period bond supplied is denoted by st . The longest maturity of bonds
P(n)supplied is N. The shares add up to 1 (i.e., N
n=1 st = 1) but some shares can be negative. The profile
(2)

(3)

(N)

(st , st , ..., st ) will be referred to as the maturity structure of bonds supplied.
(n)

Those bonds are traded by arbitrageurs. If zt is the nominal share of their n-period bond4

Like Vayanos and Vila (2009) and GV, our model ignores the zero lower bound for the interest rates. See
Hamilton and Wu (2012) and Koeda (2015) for how the lower bound can be incorporated in the model ofVayanos and Vila (2009).

3

holdings, the (one plus) holding period return on their bond portfolio is

Rt+1 ≡

(n−1)
(n)
N
X
P
−P
t

t+1

(n)
Pt

n=1

(n)

zt .

(2.2)

The arbitrageurs’ decision problem is to maximize this risk-adjusted portfolio return subject to theadding-up constraint:5

max

(n)
{zt }Nn=1

Et (Rt+1 ) −γVart (Rt+1 )2NX

subject to(n)zt = 1.

(2.3)

t=1

Here, γ is a risk-aversion coefficient.
(n)

(n)

The bond market equilibrium is that zt = st for n = 1, 2, ..., N. With the arbitrageurs caring
only about the portfolio return and with the maturity structure exogenously given by thegovernment, there are no preferred-habitat investors in the model. Because of the adding-up
P(n)constraint N
t=1 zt = 1, only N − 1 of these equilibrium conditions are independent. In GV and(1)many others, the shortest yield yt is exogenous, being determined outside the model.
The equilibrium bond prices are assumed to be a time-invariant function of a vector, ft , offactors whose dynamics is given by a Gaussian VAR(1):ft+1 = c + Φ ft + εt+1 , εt ∼ N(0, Ω).

(2.4)
(1)

The state vector ft in GV is two-dimensional. The first factor determines the shortest yield yt . The
second factor is a single variable that drives the maturity structure.6 It is a global supply factor
because bonds supplied at two different maturities are perfectly correlated. GV show that, as in thestandard ATSM (affine term structure model), the time-invariant function is exponential affine, that
is,
(n)

(n)

log Pt = ān + b̄0n ft , so by (2.1), yt = an + b0n ft where an ≡ −ānb̄n, bn ≡ − , n = 1, 2, ..., N. (2.5)
nn

5

We follow the discrete-time formulation by Hamilton and Wu (2012) of Vayanos and Vila (2009) to take(n)
Rt+1 to be the portfolio return rather than the increase in wealth. In the latter case, zt is the level of nominal(n)investment in bonds of maturity n. We are thus normalizing Rt+1 and zt to the initial wealth. We do thisbecause the model’s calibration later in the paper is more straightforward.
6

More precisely,(n)st = ζ(n) + θ(n) f2t , n = 1, 2, ..., N,
where ζ(n) and θ(n) are deterministic function of n and f2t is the second factor.

4

3

Affine Bond Pricing with Bonds Supply as Factors

We modify the model of GV and allow the bond supply to have local actors that are specific to eachmaturity. This is accomplished by treating the maturity structure as additional factors. We showthat the bond prices are exponential affine as in (2.5), even with this expanded state vector ft .
As in Vayanos and Vila (2009), GV, and many others, we proceed in two steps:(a) Derive the arbitrageurs’ FOCs (first-order conditions) that involve the affine coefficients (ān , b̄n )under the VAR dynamics (2.4) and the conjecture of the affine bond pricing (2.5).
(n)

(n)

(b) Impose on the FOC the bond market equilibrium conditions, zt = st for n = 2, 3, ..., N, toderive a set of equations for the affine coefficients.
Regarding (a), the FOCs for n-period bonds are (see Appendix 1 for a derivation) (n−1)(1) Pt+1 − P(n)t  − 1 − Pt = γ 1 ∂ Vart (Rt+1 ) , n = 2, 3, ..., N,Et 


(n)
(1)
(n)
2
Pt
Pt
∂zt
|
{z
}
| {z }
holding-period return

|

(3.1)

short rate{z}risk premium on n-period bonds

which captures the trade-off between risk and return: the higher the risk-aversion coefficient γ, themore risk premium demanded by arbitrageurs. Using (2.4) and (2.5), we can rewrite the risk
premium (the left-hand side of (3.1)) and 12∂ Vart (Rt+1 )(the right-hand side of (3.1)) as(n)∂zt (n−1) Pt+1 − P(n)
 1 − P(1)
1
t 
(n−1)
(n)
(1)
t

 −
Et 
≈ Et log Pt+1 − log Pt + b̄0n−1 Ωb̄n−1 − yt

(n)
(1)
2
Pt
Pt
1
= ān−1 + b̄0n−1 (c + Φft ) − ān − b̄0n ft + b̄0n−1 Ωb̄n−1 + ā1 + b̄01 ft ,
2

 

1
= ān−1 + b̄0n−1 c − ān + b̄0n−1 Ωb̄n−1 + ā1 + b0n−1 Φ − b̄0n + b̄01 ft ,
2
 (2)

1 ∂ Vart (Rt+1 )
(N)
0
Ω
b̄
z
+
·
·
·
+
b̄
z
.
≈
b̄
1
N−1
n−1
t
t
(n)
2
∂zt

(3.2)

(3.3)
(3.4)

These expressions, while exact in continuous time, are only approximations in discrete time.7
Turning to (b), upon the imposition of the market equilibrium conditions, the expression (3.4) of
7

The derivation can be found in Hamilton and Wu (2012). For the sake of completeness, Appendix 1 has aderivation.

5

1 ∂ Vart (Rt+1 )can be written as (with F denoting the dimension of ft )
(n)
2
∂z
t

 (2)

1 ∂ Vart (Rt+1 )(N)≈ b̄0n−1 Ω b̄1 st + · · · + b̄N−1 st(n)2∂z(1×F) (F×F) |
{z
}

(i)

(i)

(by replacing zt by st , i = 2, ..., N)

t

(F×1)

= b̄0n−1 Ω  s(2)  t    s(3)  t 
b̄2 · · · b̄N−1  .  . {z}  .  (N) (F×(N−1))
s 


b̄1

(1×F) (F×F) |

(3.5)

t

|{z}
((N−1)×1)= b̄0n−1 Ω





b̄1

b̄2

(1×F) (F×F) |

S
ft .
b̄N−1
((N−1)×F) (F×1)
}

···
{z

(F×(N−1))

(2)

(3)

(N)

The key here is to use the selection matrix S to extract the maturity structure (st , st , ..., st ) from ft .8
Substituting (3.3) and (3.5) into (3.1), we obtain an equation that is affine in ft . This has to holdfor any ft . Setting both the constant term and the coefficients of ft to zero, we obtain"b̄0n = b̄0n−1 Φ − γ b̄0n−1 Ω(1×F)(1×F) (F×F)(1×F) (F×F)

#
b̄1
(F×1)

b̄2
(F×1)

|

···

b̄N−1
(F×1)

{z

S

((N−1)×F)+ b̄01 , n = 2, 3, ..., N,

(3.7)

(1×F)

}

(F×(N−1))ān = ān−1 + b̄0n−1 c(1×F) (F×1)+

1 0b̄n−1 Ω b̄n−1 + ā1 , n = 2, 3, ..., N.
(F×F) (F×1)
2 (1×F)(3.8)

Given an initial condition about b̄1 , equation (3.7) determines the factor loading coefficients b̄n ,n = 2, 3, ..., N. Given an initial condition about ā1 and given the factor loadings (b̄0 s) thus
determined, the recursion (3.8) generates ān , n = 2, 3, ..., N. Unlike in the standard textbook ATSM
and in contrast to (3.8), equation (3.7) is not a recursion thanks to the quadratic term (the second
term on the right-hand side of the equation); it is a QVE (quadratic vector equation). Appendix 2
shows how we solved this QVE.
Before closing this section, we note for later reference that there is a link between yields and risk(n)premia. Define the risk premium rpt by the right-hand side of (3.2). The usual forward iteration
8

(2)

(3)

(N)

If (st , st , ..., st ) are placed last in ft , the S matrix can be written as
S((N−1)×F)=

0((N−1)×(F−(N−1)))6IN−1 .

(3.6)

argument then yields:9

(k)

yt =

k−1
 1
1 X  (1)(k−i)
Et yt+i + rpt+i + Ak , k = 1, 2, ..., N,
k
k

(3.9)

i=0

where Ak ≡ − 12

4

Pk−1

0
i=0 b̄k−i−1 Ωb̄k−i−1 .

Impulse Responses

Our model’s parameters are: γ (the arbitrageurs’ risk-aversion coefficient), N (the maximum
maturity), and (c, Φ, Ω) (the VAR parameters describing the factor dynamics). The previous section
and Appendix 2 were about how to map the model parameters to the affine yield curve coefficients(an , bn , n = 1, 2, ..., N) through the QVE (3.7) and the recursion (3.8) in {an }. In this section, we
calculate the IR (impulse response) of the yield curve to shocks originating in the maturity structure.
Specifying the Factor Dynamics
The example we consider has only one factor besides the maturity structure. Without loss of(1)

generality, we can take that non-supply factor to be the short rate yt . Thus F (the number of factors)
equals N, and the factors are given by (1) (2) (3)(N) 0ft = yt , st , st , ..., st
.

(4.1)

(N×1)

With this choice of the factor vector, the initial condition to be fed to the recursion (3.8) and the QVE(quadratic vector equation) (3.7) is given byā1 = 0, b̄01 = (1, 0, ..., 0).

9(k)

By the definition of rpt , we have


(k−1)
(k)
(1)
(k) 1
Et log Pt+1 − log Pt − yt = rpt − b̄0n−1 Ωb̄k−1 .
2(0)

The usual forward iteration trick with the terminal condition log Pt+k = 0 yields(k)− log Pt =

k−1
X
i=0

(k)

 (1)

(k−i) 1Et yt+i + rpt+i − b̄0k−i−1 Ωb̄k−i−1 .
2(k)(3.9) obtains because yt = − 1k log Pt .

7

(4.2)

The VAR dynamics are specified as(short rate)(maturity structue)

(1)

(1)

yt = c1 + ρyt−1 + ε1t , ε1t ∼ N(0, σ21 ),

(4.3)



(n+1)

2


 cn + θst−1 + εnt , εnt ∼ N(0, σn )(n)st = 
 cN + εNt , εNt ∼ N(0, σ2 )N

if n = 2, 3, ..., N − 1,

(4.4)

if n = N.

Here, the supply shocks (ε1t ,...,εNt ) are uncorrelated.10 Four points about the dynamics.
• Since the bond supply shocks (ε2t , ..., εNt ) are uncorrelated across maturities, each of them
represent a purely local supply shock in the maturity spectrum.
• The short rate is exogenous because there is no feedback to the short rate from the maturity
structure and the short rate shock ε1t is uncorrelated with the bond supply shocks. Therefore, theresponse of the yield curve to a bond supply shock would represent what is referred to as theportfolio balance effect.
• The maturity structure dynamics (4.4) captures the “legacy” property of bonds that an n-periodbond becomes an n − 1-period bond next period. The strength of legacy is represented by θ.
Calibration
To avoid heavy demand on CPU time, we choose the unit interval to be a quarter, not a month. So(1)

yt is the 3-month interest rate. We set N to be 80 quarters (20 years). For calibration purposes we
use the U.S. Treasury zero-coupon yield curve calculated by Gurkaynak, Sack, and Wright (2007).11
10

The implied restrictions on the factor dynamics parameters (c, Φ, Ω) are as follows.
c = (c1 , c2 , ..., cN−1 , cN )0 , Φ = 
(N×N)
(N×1)


ρ

00

(1×(N−1))

0((N−1)×1)

Φ̃

((N−1)×(N−1))







 , Ω =  (N×N) σ210((N−1)×1)

00(1×(N−1))Ω̃((N−1)×(N−1))




 ,


where Ω̃ = diag{σ22 , ..., σ2N } and Φ̃ is given by

0

0

 .
 .

Φ̃
=  .
0((N−1)×(N−1))


0


0

θ
0
..
.
0
0
0

11

0
θ
..
.

0
0
..
.

···
···
..
.

···
···
···

0
···
···

θ
0
···


0 

0 


0  .

0 

θ

0

The parameterized function used to fit the yields of various observed maturities is given as equation (22)in Gurkaynak et. al. (2007). The daily parameter values can be downloaded from the Federal Reserve Board

8

The values for (c1 , ρ, σ1 ) are obtained from an AR1 estimation on the short rate. Our results tobe reported below are sensitive to the calibrated values for those parameters, particularly for σ1 ,which in turn depend on the choice of the sample period. We report results based on the period
under Alan Greenspan’s chairmanship (1987:Q3 - 2005:Q4). The estimated AR1 model is (withstandard errors in parentheses)
(1)

(1)

yt = 0.0003829 + 0.9592 yt−1 , S.E.R = 0.001291, t = 1987:Q4-2005:Q4.
(0.00037) (0.029)

(4.5)

(1)

(yt is expressed as a rate per quarter.) This estimated AR1 determiens ρ and σ1 : ρ = 0.9592, and
σ1 = 0.001291. The value of c1 is set so that the AR1 process’s steady-state value, c1 /(1 − ρ), equals(1)

the sample mean over 1987:Q3-2005:Q4 of y1 .12(n)

We pick (c2 , ..., cN ) so that the steady-state value of st is 1/N for n = 2, ..., N.13 We set σn = λ/Nwith λ = 0.01 for n = 2, 3, ..., N. The affine coefficients turned out to be fairly insensitive to λ; forboth the yield curve and the risk premium, the results about the level and the IR to a supply shock(n)of a fixed size (we will consider a 1 percentage point shock to st ) are very similar for a wide rangeof λ including [0.001, 0.1].
This leaves two parameters, γ (the risk aversion parameter) and θ (which controls the maturity
structure dynamics). As it turns out, the latter has no effect on the steady-state yield curve (the yield
curve given by the affine equation (2.5) when ft equals to its steady-state value).14 We can therefore(n)website. From this we can calculate yt for any maturity n. We take the yield for the quarter to be for the lastbusiness day of the quarter. For the short rate we use the value implied by the parameterized yield curve at
n = 1 (3 months). This short rate series is very similar to the constant-maturity 3-month rate available fromFederal Reserve Board’s Table H-15.
(1)

12

Let ȳ, ȳ¯ , and ỹ be the sample mean of yt over 1987:Q3-2005:Q4, 1987:Q4-2005:Q4, and 1987:Q3-2005:Q3,
respectively. Let ĉ1 and ρ̂ be the AR1 estimates shown in (4.5) of c1 and ρ (so ĉ1 = 0.0003829 and r̂ = 0.9592).
Then ȳ¯ = ĉ1 + ρ̂ ŷ, or ĉ1 /(1 − ρ̂) = ŷ − ( ŷ − ȳ¯ )/(1 − ρ̂). In our sample, the three sample means are different enoughto make ĉ1 /(1 − ρ̂) deviate substantially from the sample mean ȳ. (ĉ1 /(1 − ρ̂) = 3.75% in APR while ȳ = 4.64%
in APR.) So, instead of setting c1 = ĉ1 , we choose c1 so that c1 /(1 − ρ̂) = ȳ.
13

Let

(2)
(N)
st
≡ (st , ..., st )0 be the maturity structure, c̃ ≡ (c2 , ..., cN−1 , cN )0 , and let Φ̃ be as in footnote 10.
((N−1)×1)

Then the steady-state value of st , call it s, satisfies: (I − Φ̃)s = c̃.
As noted in the text and in footnote 13, we chose (c2 , ..., cN ) so that the maturity structure in the steadystate is flat. Thus the factor vector in the steady state is
14

factor vector in the steady state =

 c1
1
1
, , ...,.
1−ρ NN| {z }
N − 1 terms

If the staedy-state maturity structure is flat, as here, then θ does not affect the steady-state yield curve given

9

calibrate γ to the steady-state yield curve without specifying the value of θ. We set γ = 52/4 (orγ = 52 if the quarterly portfolio return Rt+1 in (2.3) is expressed at an annual rate) because, as shownin Figure 1, the steady-state yield curve and the corresponding sample mean for the sample period(1987:Q3 - 2005:Q4) are reasonably close to each other under this value of γ.15
Impulse Responses
The N-dimensional factor dynamics consisting of (4.3) and (4.4) has the IR (impulse response)
function of ft to the bond supply shocks (ε2 , ..., εN ) given by
(0, 0, ..., 0, θ j , 0, ..., 0)
(n− j)
=


∂εnt

|{z} 
 00

for j = 0, 1, ..., n − 2, (with θ j = 1 for θ = 0 and j = 0),∂f0t+ j

(1×N)

for j = n − 1, n, ....

(1×N)

(4.6)

for n = 2, 3, ..., N.
That is, a unit supply shock originating at maturity n becomes a supply shock of size θ j at maturityn − j in j periods, and gets absorbed at maturity 1 in n − 1 periods (recall that the short rate isexogenous to supply shocks).
The factor IR readily translates to the yield-curve IR via the affine equation (2.5). Thanks to thespecial feature of the factor IR, there is a tight connection between the factor loadings (bn ’s in (2.5))
and the yield IR profile. The latter is a profile of N elements given by:



 
 (1) (2) (3)(0, b2,n− j , b3,n− j ..., bN,n− j ) θ j

(N)


∂ yt+ j , yt+j , yt+ j , ..., yt+ j
{z
}

|
=
(1×N)


∂εnt

|
{z
} 



00
(1×N)(1×N)

for j = 0, 1, ..., n − 2,for j = n − 1, n, ....

(4.7)

for n = 2, 3, ..., N,
where bn,k is the k-th element of bn . To understand this formula, imagine an N × N matrix whose(N×1)

columns are [0 b2 ... bN ]. The impact response profile (the profile for j = 0) is the n-th row of thismatrix. The fifth row up from this row, if it is scaled by the legacy parameter θ5 , is the profile for
j = 5, for example. The factor loadings for the short rate, b1 , do not enter the profile because theshort rate is exogenous. The legacy parameter θ, while irrelevant for the steady-state analysis,by the affine equation (2.5), despite the fact that θ affects the affine coefficients (an , bn ) (n = 2, 3, ..., N).
This value of γ = 52 is close to the value for the same parameter calibrated by GV (Greenwood and
Vayanos (2014)) of 57.
15

10

affects the IR profile in two ways: it scales the factor loadings (as just mentioned) and it is adeterminant of the factor loadings themselves. Below we consider two values for θ: 0 and 1.
Assume first θ = 1. The set of across-maturity profiles shown in the left panel of Figure 2 are
the yield IR profiles for j = 0 (the impact response), j = 20 (5 years out), j = 40 (10 years), and j = 60
(15 years). The shock considered here originates at the longest maturity with n = N (= 80), with ashock size of 1 percentage point. With θ = 1, (4.7) shows that the yield IR profile depends on n and j
only thorough n − j. Therefore, for example, the IR profile after 15 years to a shock originating at
n = 80 quarters (20 years) (the dashed profile in the panel) is also the impact response profile (the IR
profile for j = 0) to a shock originating at n = 20 quarters (5 years).
As the panel shows, the IR profiles are hump-shaped, peaking near the maturity where thelocal supply shock originates. To explain why, it is useful to temporarily turn our attention to therisk premium.16 Since, as seen from (3.3), the risk premium, too, is an affine function of the factor
vector, the IR formula (4.7) can be used to calculate the IR of the risk premium if the factor loading
coefficients bn,k are interpreted as the coefficients of ft in (3.3). The right panel of Figure 2 shows therisk-premium IR profiles for different horizons thus calculated. Two points to note.
• Unlike the yield IR profiles, which are hump-shaped, the risk-premium IR profiles are
upward-sloping. The intuition is the same as given in GV (Greenwood and Vayanos (2014)) forfactor loadings: since long-term bonds are more sensitive to duration risk than short-term bonds,
the risk premium demanded by arbitrageurs must increase with the maturity. This explains why
a local shock, no matter where in the maturity spectrum it originates, raises the risk premium forlong-term bonds. This is illustrated for the impact response profile with the originating maturity
n of 20 (5 years) by the dashed profile in the right panel.
• For all j, the risk-premium IR profile lies above the yield IR profile.
This last point follows from the relationship shown in (3.9) that the current yield is the averageof the risk premium over the bond’s life. Replacing t by t + j in the formula and differentiating bothsides with respect to the bond supply shock at the originating maturity n, εnt , and noting that theshock does not affect the future short rates, we obtain a link between the yield and the risk-premium
IRs:



(2)
(1)
(k)
(k−1)
∂ rpt+ j+k−2 ∂ rpt+ j+k−1 
1  ∂ rpt+j ∂ rpt+ j+1

= 
+
+ ··· ++
 , k = 1, 2, ..., N.
k  ∂εnt
∂εnt
∂εnt
∂εnt
∂εnt 
(k)∂yt+ j16(4.8)

A similar argument can be found in GV (Greenwood and Vayanos (2014), see the introduction) for factor
loadings.

11

∂y

(k)

(k)

The yield IR, ∂εt+j, is less than the risk-premium IR,nt∂ rpt+j∂εnt, because it is the average of a declining

sequence whose first term is the risk-premium IR.17
A similar reasoning explains the hump-shaped yield IR profiles. If the risk-premium IR profile(k)were flat so that

∂ rpt+j∂εnt

did not depend on the maturity k, then the average in (4.8) would be declining

in k and the yield IR profile would be downward-sloping. But the risk-premium IR profile isupward sloping, which changes the sign of the slope for short maturities.
Finally, turn to the case θ = 0. Since supply shocks are now temporary, only the impact
response is non-zero for both the yield and the risk premium. The impact response profile is givenby (4.7) for j = 0, which shows that it is invariant to θ (θ j = 1 for j = 0 regardless of θ). For example,
in the right panel of Figure 2, the impact response profile for the risk premium to a supply shock
originating at 20 quarters (5 years) is the dotted profile in the panel. Since the effect on the risk
premium is temporary, as one can surmise from the formula (4.8), a local supply shock originating
at any maturity has very little effect on the yield curve.

5

Conclusion

We have developed a model that provides a mapping from an arbitrary change in the maturity
structure to the yield curve. The model can be viewed as a modern incarnation of Tobin’s (1969)
formulation of portfolio balance channel. Unlike in Tobin’s, the mapping is explicitly tied to the
dynamics of the supply factors, so that the strength of the portfolio channel can be made a functionof the persistence of the supply shock.

(k)

17

The sequence is declining because, as shown in the right panel of Figure 2,
and as j increases given k.

12∂ rpt+j∂εnt

declines as k falls given j

Figure 1: Average Yield Curve
Yield Curve
10
data
model
9

8

annual percentage points

7

6

5

4

3

2

1

0

0

10

20

30

40
50
maturity in quarters

60

70

80

Note: The solid line is the steady-state yield curve implied by the model. The filled-in squares are the sample
means from data.

Figure 2: Impulse Responses, θ = 1
Yield Curve

Risk Premium Profile

88
impact5 years out
10 years out
15 years out

7

6

6

5

5

annual basis points

annual basis points

7

4

3

4

3

2

2

1

1

0

0

20
40
60
maturity in quarters

impact5 years out
10 years out
15 years out

0

80

0

20
40
60
maturity in quarters

80

Note: The profiles across maturities of the impulse responses of the yield and the risk premium. The shock is a
1 percentage point increase in the share of 20-year bonds in the maturity structure.

13

Appendix 1

Derivation of (3.1), (3.2), and (3.4)

The arbitrageurs’s decision problem is described by (2.3) with (2.2) of the text. If φt is the LagrangeP(n)multiplier for the constraint N
n=1 zt = 1, the FOCs (first-order conditions) are
 (n−1) Pt+1 − P(n)t − γ 1 ∂ Vart (Rt+1 ) = φ , n = 1, 2, ..., N,
Et 
t

(n)
(n)
2
Pt
∂zt
N
X

(A1.1)(n)zt = 1.

(A1.2)

n=1
(0)

(1)

(1)

Since Pt+1 = 1, the holding period return on the 1-period bond is (1 − Pt )/Pt , which is known indate t. Hence∂ Vart (Rt+1 )(1)(1)= 0 for n = 1 and the FOC for n = 1 is: (1 − Pt )/Pt = φt . Substituting this(n)∂zt

into the rest of the FOCs in (A1.1), we obtain (3.1) of the text, which is reproduced here: (n−1)(1) Pt+1 − P(n)t  − 1 − Pt = γ 1 ∂ Vart (Rt+1 ) , n = 2, 3, ..., N,Et 


(n)
(1)
(n)
2
Pt
Pt
∂zt
| {z }|{z}short rateholding-period return

|

(3.1)

{z

}

risk premium on n-period bonds

In the rest of this appendix, we derive (3.2) and (3.4) under the VAR dynamics (2.4) and affine
pricing (2.5). This can be done easily if we show the following.
(1)

(1)

(1)

• Show: (1 − Pt )/Pt ≈ yt . This immediately follows from the approximation 1/x ≈ 1 − log(x) for
(1)

(1)

x ≈ 1 and the relation yt ≡ − log Pt .
(n)

(n)

(n−1)

(n)

• Let pt ≡ log Pt . Show: the distribution conditional on date t of pt+1 − pt is normal with meanān−1 + b̄0n−1 (c + Φft ) − ān − b̄0n ft and variance b̄0n−1 Ωb̄n−1 . This follows immediately from
(n−1)

(n)

pt+1 − pt = ān−1 + b̄0n−1 (c + Φft ) − ān − b̄0n ft + b̄0n−1 εt+1 ,(A1.3)

which can be obtained by combining (2.4) and (2.5).
• Show: (n−1)
h
i





 Pt+1 − P(n)t = E exp(p(n−1) − p(n) ) − 1 ≈ E p(n−1) − p(n) + 1 Var p(n−1) − p(n) .
Et 
t
t
t

t
t
t
t+1
t+1
t+1

(n)
2
Pt

(A1.4)

This can be obtained easily from two well-known formulas. One is that, for a normally
distributed random variable X, E[exp(X)] = exp[E(X) + 12 Var(X)], and the other is exp(x) ≈ 1 + x
(n−1)

(n)

(n−1)

(n)

for x ≈ 0. In the former, we can set X = pt+1 − pt because, as shown above, pt+1 − pt isnormally distributed conditional on date t information. In the latter formula, set (n−1) (n−1)(n)(n)
x = Et pt+1 − pt + 21 Vart pt+1 − pt .

14

• Show:
Vart (Rt+1 ) ≈ d0t Ωdt where dt ≡

N
X

(i)

b̄i−1 zt .

i=2

So

1 ∂ Vart (Rt+1 )= b̄0n−1 Ωdt .
(n)
2
∂z

(A1.5)t

To derive this, we again use the approximation exp(x) ≈ 1 + x to derive
(n−1)

(n)

Pt+1 − Pt(n)Pt (n−1)
(n−1)
(n)
(n)
= exp pt+1 − pt − 1 ≈ pt+1 − pt .

(A1.6)(n)Multiply both sides by zt and sum over n = 1, 2, ..., N to obtain

Rt+1 ≡

(n−1)
(n)
N
X
P
−P
t

t+1

n=1

(n)
Pt

(n)

zt ≈

N 
X

(n−1)

(n)

pt+1 − pt

 (n)
zt

n=1

=

N h
X

ān−1 + b̄0n−1 (c + Φft ) − ān − b̄0n ft

n=1

 N

i (n) X

(n) 
0

zt + 
b̄n−1 zt  εt+1
n=1

|

{z

}

≡At

 N

X

(n) 
0

= At + b̄n−1 zt  εt+1
n=1
 N

X


(n) 
0

= At + b̄n−1 zt  εt+1

(by (A1.3))(by the definition of At right above)(since b̄0 = 0)

n=2

= At + d0t εt+1

(by the definition of dt in (A1.5)).

(A1.7)

Since At defined here is known in date t, the source of the date t conditional variance of Rt+1 is thelast term, d0t εt+1 .

15

Appendix 2

Solving the Quadratic Vector Equation (3.7)

To emphasize that we are dealing with a QVE (quadratic vector equation) in (3.7), define the
NF-dimensional stacked vector b̄ as (recall: F is the number of factors) b̄1 (F×1)
 b̄2 (F×1)
 .
b̄ ≡ 
(NF×1)
 ..  .  b̄N (A2.1)(F×1)

Then it is straightforward to show that (3.7) and the initial condition b̄1 = δ together can be written
as
M

b̄

(NF×NF) (NF×1)=

d(NF×1)− γ g(b̄) ,(A2.2)(NF×1)

where

 IF

− Φ0 (F×F)
M ≡  0(NF×NF)
 .
 ..


 0
d

(NF×1)

0

0

...

IF

0

...

− Φ0

IF

...

..

.

..

0

− Φ0

(F×F)

...

≡ 1 ⊗ δ
(N×1)

(F×1)

.

(F×F)


0 

0 
0  ,
0 IF (A2.3)(recall: δ is the initial condition for b̄1 ),





0


(F×1)




0
0
 S

B̄
Ω
b̄
1 
 (F×(N−1)) ((N−1)×F) (F×F) (F×1)









0
 |



{z
}
 (F×(N−1)2 ) 













 =
(F×1)
g(b̄) ≡ P
vec  B̄0 Ω B̄ , P ≡  IN−1 ⊗ S0 ,
2
 | {z }  | {z } ..

(NF×(N−1) )(NF×1)






.
((N−1)×(N−1))
((N−1)F×(N−1)2 )

0
0
 S
|
{z
}
B̄
Ω b̄N−1 |{z}(F×(N−1)) ((N−1)×F)(F×F) (F×1) ((N−1)2 ×1)
(NF×(N−1)2 )
|


{z
}


(A2.4)(A2.5)(F×1)

B̄

(F×(N−1))


≡ b̄1

(F×1)

b̄2

(F×1)

...


b̄N−1 .

(A2.6)(F×1)

16

If γ = 0, the equation (3.7) or (A2.2) has a unique solutionb̄∗ ≡ M−1 d.

(A2.7)

Although the square matrix M, which is of size NF, can be large, computing b̄∗ is fast because there
is a well-known analytical expression for the inverse of the large matrix M.18 If γ > 0, because of thequadratic term g(b̄), there can be multiple solutions if a solution exists. Following GV (Greenwood
and Vayanos (2014)), we seek the solution that converges to the unique solution b̄∗ as γ → 0. To
study such a solution for given γ > 0, write the equation asf(b̄, γ) =0 , f(b̄, γ) ≡ Mb̄ − d + γ g(b̄).

(NF×1)(A2.9)

By the implicit function theorem, there exists an interval U including 0 as an interior point and avector-valued function of a single variable, b̄(.): U → RNF , such that f(b̄(γ̃), γ̃) = 0 for all γ̃ ∈ U and
its derivative b̄0 (.) is given by"b̄0 (γ̃) = −∂f(b̄(γ̃), γ̃)#−1∂f(b̄(γ̃), γ̃)
∂γ

∂b̄0

"
#−1
∂g(b̄(γ̃))= − M + γ̃g(b̄(γ̃))∂b̄0| {z }|{z}(NF×1)(by the definition in (A2.9) of f(b̄, γ)).

(A2.10)(NF×NF)The solution b̄(γ) can be obtained by numerically solving the differential equation (A2.10) on
the interval [0, γ] with the initial condition b̄(0) = b̄∗ . This method, however, is computationallydemanding if the dimension of the matrix to be inverted, M + γ̃∂g(b̄(γ̃)), is large.19 20 An alternative is∂b̄0

to utilize the fixed-point iteration
18

Here is the expression for M−1 :

 IF

 Φ0


0 2
M−1 =  (Φ )

.

..


(Φ0 )N−1

19

0
IF
Φ0

0
0
IF
..
.

...
...
...
..
.

...

(Φ0 )2

Φ0


0 

0 

0  .

0 

IF(A2.8)The next section’s example has N = 80 and F = 1 + N, so the size of the matrix is 6480 × 6480.

∂g(b̄(γ̃))
is not CPU-intensive because we have an analytical expression of it.
∂b̄0∂g(b̄)
0
Since g(b̄) = P vec(B̄ ΩB̄) is quadratic, the Jacobian ∂b̄0 can be written as (recall:B̄= [b̄1 , ..., b̄N−1 ] and(F×(N−1))0000b̄ = [b̄1 , ..., b̄N−1 , b̄N ])20

Computation of the Jacobian∂g(b̄)

∂b̄0
|{z}

=


P

(NF×(N−1)2 )A((N−1)2 ×(N−1)F)

|

17



((N−1)2 ×F){z((N−1)2 ×NF)

(NF×NF)0}(A2.11)b̄(k+1) = M−1 [d − γg(b̄(k) )], k = 0, 1, 2, ...

(A2.13)starting from b̄(0) = b̄∗ . This method, with no need to do numerical matrix inversion, is faster by
several orders of magnitude. I have not been able to characterize a condition under which thisiteration converges, though.21 However, for numerical examples that I experimented and certainly
for the example of the next section, this iteration quickly converges to the solution to the differential
equation (A2.10).

where A here is0 IN−1 ⊗ b̄1 Ω 
(F×F)


(1×F)


 |

{z
}


 ((N−1)×(N−1)F) 


..
 + IB̄0Ω .
A≡ 
N−1 ⊗.

((N−1)×F)(F×F)

((N−1)2 ×(N−1)F)

I

0
| {z }Ω 
 N−1 ⊗ b̄N−1 (F×F)


(1×F)
((N−1)×F)

|
{z
}


(A2.12)((N−1)×(N−1)F)21

The sufficient condition identified in Poloni (2013, Theorem 4) requires that both Φ and Ω be nonnegative.

18

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19