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Federal Reserve Bank of Chicago

A Portfolio-Balance Approach to the
Nominal Term Structure
Thomas B. King

UPDATED
January 2015
WP 2013-18

A Portfolio-Balance Approach to the Nominal
Term Structure
Thomas B. King*
January 31, 2015

Abstract
I examine how the maturity structure of outstanding government liabilities a¤ects the nominal yield curve under a variety of assumptions
about investor objectives. In the class of models I consider, equilibria are arbitrage free, expectations are rational, and assets are valued
only for their pecuniary returns. Nonetheless, a portfolio-balance mechanism arises through the dependence of the pricing kernel on the return
on wealth. This mechanism results in a positive relationship between
the duration of bondholders’portfolios and the price of interest-rate risk.
Quantitatively, the models suggest that the e¤ects of shifting Treasury
supply on yields can be substantial— for example, the increase in the average maturity of U.S. government debt that occurred between 1976 and
1988 may have raised the ten-year yield by 50 basis points. On the other
hand, partly re‡
ecting an attenuation of portfolio-balance e¤ects when interest rates are near zero, the Federal Reserve’ asset purchase programs
s
likely had a fairly small impact on the yield curve by removing duration
from the market.

Federal Reserve Bank of Chicago. Contact: thomas.king@chi.frb.org. I thank Stefania
D’
Amico, Jonathan Goldberg, Robin Greenwood, Francisco Vazquez-Grande, Andrea Vedolin,
and seminar participants at the Federal Reserve Bank of Chicago, the Federal Reserve Board,
the ECB/Bank of England workshop on Understanding the Yield Curve, and the Federal
Reserve Day-Ahead Conference for helpful comments and discussions. Roger Fan provided
excellent research assistance. The views expressed here do not re‡ect o¢ cial positions of the
Federal Reserve.

1

1

Introduction

Recent empirical work has been nearly universal in concluding that ‡
uctuations
in the structure of government debt have signi…cant e¤ects on the term structure
of interest rates and, most likely, on other asset prices.1 This issue has received
particular attention following the recent e¤orts of several central banks to reduce long-term interest rates by purchasing large quantities of government debt
and other securities. Yet, although much of the recent empirical evidence comes
from studies of these asset purchases themselves, enough of it derives from other
episodes to demonstrate that the mechanism transmitting debt ‡
uctuations to
…nancial conditions is not unique to recent experience or, for that matter, to
monetary-policy interventions. Perhaps the most commonly cited explanation
for these results is that a reduction in the quantity of longer-term bonds that
investors must hold leads them to require less compensation for bearing the remaining interest-rate risk in their portfolios. Consequently, expected returns,
term premiums, and yields on bonds fall. Since it supposedly works by shifting investors’ exposure to long-term interest rates, this outcome is sometimes
2
referred to as the “duration channel.”
Despite its intuitive appeal, the duration channel is all but absent from the
extant theoretical literature, and several authors argue that the empirical results noted above probably re‡ some other mechanism.3 A common refrain
ect
is that, in frictionless markets, asset quantities should be irrelevant for asset
prices, a critique exempli…ed by Eggertsson and Woodford’ (2003) proof that,
s
in a particular class of general-equilibrium models, the structure of government
debt available to the public makes no di¤erence for either asset prices or macroeconomic outcomes. For their part, advocates of the duration channel frequently
point to the model of Vayanos and Vila (2009) for theoretical support. In that
model, shifts in the demand for Treasury bonds on the part of preferred-habitat
investors change the equilibrium exposures of arbitrageurs to interest-rate risk,
causing ‡
uctuations in term premia. However, the Vayanos-Vila model is both
analytically intractable and quite stylized— for example, investors only hold
Treasury bonds, in‡
ation and consumption risk are not priced, and policy actions by the government are not explicitly modeled. These features make it
di¢ cult to assess the economic importance of its central mechanism and to reconcile it with other macro-…nance literature, including Eggertsson and Woodford’ neutrality proposition.
s
This paper notes that the potential for asset quantities to a¤ect asset prices
exists in any model in which the pricing kernel depends on the return on wealth.
1 See Bernanke et al. (2004), Kuttner (2006), Gagnon et al. (2010), Greenwood and
Vayanos (2010, 2014), Krishnamurthy and Vissing-Jorgensen (2011, 2013), Meaning and Zhu
(2011), Swanson (2011), D’
Amico et al., (2012), Hamilton and Wu (2012), Ihrig et al. (2012),
Joyce et al. (2011), Li and Wei (2012), Rosa (2012), Cahill et al. (2013), D’
Amico and King
(2013), Bauer and Rudebusch (2014), Chabot and Herman (2014), and Rogers et al. (2014).
2 For example, Gagnon et al. (2010) appeal strongly to this idea. Bernanke (2010) and
Yellen (2011) refer to a similar mechanism in discussing how LSAPs work.
3 See Woodford (2012), Krishnamurthy and Vissing-Jorgensen (2013), and Bauer and Rudebusch (2014).

2

In such models, changes in the relative quantities of assets that are held by investors a¤ect the distribution of the wealth return and therefore a¤ect all asset
prices. In the spirit of an older literature that studied how asset supply mattered for prices, I refer to these as “portfolio balance e¤ects.” However, unlike
classical appications of portfolio-balance concept, the equilibria considered here
are arbitrage free and obey rational expectations. They do not require that
there be anything "special" about money or short-term debt. In the case of
Treasury bonds, no-arbitrage portfolio balance is essentially equivalent to the
duration channel. Preferences that result in pricing kernels that depend on the
return on wealth have long been common in the …nance literature, and (since
Epstein-Zin-Weil utility has this property) are increasingly used in macroeconomic models as well. The Vayanos-Vila model is a member of this class, while
the Eggertsson-Woodford model is not.
I study how debt structure a¤ects interest rates in several calibrated models
of this type, spanning a variety of assumptions about investor preferences and
portfolios. In particular, the models allow variously for in‡
ation and consumption, diversi…cation of investor portfolios into non-Treasury assets, and a zero
lower bound on nominal interest rates. Under parameterizations that match
the historical data, portfolio-balance e¤ects on the yield curve can be signi…cant. For example, in some cases, an increase in the outstanding duration of
Treasury securities of the magnitude that occurred between the mid-1970s and
mid-1980s leads to an increase in the 10-year term premium of more than 50
basis points. This …nding could explain a portion of the empirical evidence
(for example, estimates based on the Kim and Wright (2005) model) that term
premiums did increase around this time. However, a critical assumption here
is the extent to which the holders of Treasury bonds are exposed only to Treasury bonds. For given preferences, portfolio-balance e¤ects of shifting Treasury
maturity structure are largest when bondholders do not hold any other assets,
implying a high degree of market segmentation.
Moreover, even assuming complete isolation of the Treasury market, these
models cannot generate duration e¤ects that are anywhere near empirical estimates of the impact of the Federal Reserve’ large-scale asset purchases on
s
longer-term interest rates. In part, this impotence has to do with the nonlinearity created by the zero lower bound (ZLB) on nominal interest rates. The
truncation of the lower tail of the rate distribution at zero reduces volatility
along the entire yield curve, and— since duration e¤ects depend on the product
of duration and volatility— render this channel less e¤ective. (Doh, 2010, also
makes this point.) The reduction in interest-rate volatility, and therefore in
portfolio-balance e¤ects, is even greater when the central bank broadcasts its
medium-term intentions for short-term interest rates through so-called "forward
guidance."
These …ndings lead to a number of implications for the interpretation of
recent policy actions. The conclusion that duration removal may not be the
primary mechanism involved in asset purchases is consistent with the recent
event-study analyses of Krishnamurthy and Vissing-Jorgensen (2013) and Cahill
et al. (2013). As those authors note, the e¤ects of Treasury purchases might
3

at least partly be explained through a scarcity channel (as in D’
Amico and
King, 2013) or through the signals that unconventional monetary policy sends
about future short-term rates (as in Bauer and Rudebusch, 2014). The result
that greater certainty about the course of short rates dampens duration e¤ects
implies that forward guidance and asset purchases, two tools of monetary policy
that are often viewed as complements at the ZLB, actually o¤set each other
somewhat in terms of their e¤ects on longer-term yields. More broadly, the
structure of government debt is likely to matter most for interest rates when
interest-rate volatility is high. This observation should inspire caution if the
Federal Reserve needs to sell assets at some point, since, to the extent that the
path of policy is likely to be less certain in that environment, the duration e¤ect
could be asymmetrically large relative to the period of asset purchases. Similar
considerations may also be worth policymakers’ attention when designing the
pattern of Treasury debt issuance.
The relationship of this paper to previous literature is discussed in some
detail in Section 2. Brie‡ it builds on a long line of models in which the
y,
focus is on the adjustment of prices to make investors willing to hold whatever
securities are outstanding in each period. The …rst generation of such models, in
the tradition of Tobin (1958), mostly had to do with substitution between money
and bonds. These models have been criticized by more-recent authors because
they modeled demand for those instruments in an arbitrary, reduced-form way
(see for example, Woodford, 2012). A second generation of papers, beginning
with Frankel (1985), derived asset-demand functions from investor optimization
conditions. While this approach is more consistent with modern …nance theory,
in the sense that it essentially imposes no-arbitrage conditions, implementations
of it have typically had to assume that investors form expectations of future asset
prices in an arbitrary, non-rational way in order to obtain analytical solutions.
What distinguishes the class of models discussed here from these previous e¤orts
is that investors’ demand for assets is determined from their optimization in
each period, assuming that they realize that security prices will be determined
in the same way in the next period. Unlike Tobin-style models, investors do
not care about quantities of particular securities, only about the overall risk
of their portfolios. Unlike Frankel-style models, expectations are fully rational.
The only paper to date that has successfully characterized the equilibrium in a
structural yield-curve model with both of these properties is Vayanos and Vila
(2009), which, as noted, is a special case of the class of portfolio-balance models
I consider.
Finally, a technical contribution of the paper is to develop a numerical algorithm, iterating on a version of Tauchen and Hussey (1991), to solve for statecontingent prices in this class of models. Solving no-arbitrage portfolio-balance
models is a nontrivial computational task because, even under simplifying assumptions about funcitonal forms, equilibrium asset prices involve a nonlinear
recursion in multidimensional function space. (This is why Vayanos-Vila can
only be solved analytically in limiting cases.) One advantage of the approach I
propose is that it requires only weak conditions on the functional form of the
pricing kernel and the dynamics of the state of the economy. In particular, the
4

approach allows for departures from linearity, normality, and homoscedasticity,
and there are essentially no restrictions in modeling the dynamics of the assetsupply distribution itself. An important example of a relevant nonlinearity is
the ZLB.
The paper proceeds as follows. Section 2 lays out the general properties
of portfolio-balance models and explains the conditions under which changing
quantities can a¤ect asset prices under no-arbitrage. It also puts these models in
the context of previous literature. Section 2.3 sketches the solution algorithm I
use to solve such models, with the details contained in an appendix. Section 3
shows how portfolio-balance applies speci…cally to the term structure of interest
rates and illustrates how it results in duration e¤ects in a calibrated one-factor
model, both with and without the ZLB imposed. Section 4 extends this model
to several more realistic cases with in‡
ation risk and stochastic bond supply.
Section 5 applies the model to study the Federal Reserve’ asset purchases, and
s
Section 6 concludes.

2
2.1

No-Arbitrage Portfolio Balance
General Asset-Pricing Relations

I use the term “portfolio balance”to encompass any model in which the equilibrium prices of …nancial assets are related to the quantities of those assets that
investors must hold and in which the quantities can be considered to have an
exogenous or policy-dependent component. In particular, suppose there exists a
…nite number of assets N , with time-t prices pt = (p1t ; :::; pN t ), and par values
Xt = (X1t ; :::; XN t ). In equilibrium, investor wealth must be equal to the value
uctuations
of all assets: Wt = X0 pt . The idea behind portfolio balance is that ‡
t
in the state of the economy that change the value of Xt will change pt because
expected returns— and therefore current prices— must adjust to make investors
willing to hold the outstanding supply of securities at each point in time. In
the most straightforward cases, the supply of each security is treated as an exogenous quantity issued by a price-insensitive entity, such as the government.
More generally, Xt may itself depend on prices.
The absence of equilibrium arbitrage opportunities is equivalent to the existence of a stochastic discount factor (SDF) Mt;t+s that prices all assets in the
economy. In particular, the price of any asset n at time t is given by
pnt = Et [Mt;t+s qnt+s ]

(1)

where qnt+s is the asset’ payo¤ s periods hence and Et indicates the expectation
s
conditioned on information at time t. This condition must hold for all horizons
s > 0. It can be rewritten as
2
2 s
33
Z
pnt = Et 4exp 4
rt+v dv 55 Et [qnt+s ] + covt [Mt;t+s qnt+s ]
(2)
0

5

where rt is the time-t instantaneous risk-free rate of interest. (In this paper,
I restrict attention to discrete-time models, so rt will be constant within each
period.)
Given dynamics for the state of the economy, no-arbitrage asset-pricing models are distinguished soley by the assumptions they make about the determinants
of the SDF. In order for asset quantities to a¤ect asset prices, therefore, Mt;t+s
must depend on Xt in some way. While this dependence could in principle take
a number of forms, I restrict attention to cases in which investors do not care
about the quantities of the particular securities that they hold per se. This
rules out, for example, models with convenience yields, monetary services, or
other special bene…ts that might attach to certain assets beyond their pecuniary
returns. Instead, I consider models in which Mt;t+s can be written as a function
of the return on wealth Rt;t+s and, possibly, of the state of the economy in time
t and t + s. That is,
Mt;t+s = M (st ; st+s ; Rt;t+s ; )

(3)

where the vector st is the time-t state vector, which is assumed to be exogenous
with respect to Xt . By de…nition,
Rt;t+s =

X0 qt+s
t
X0 pt
t

(4)

where qt+s = q1t ::: qN t .
Without further restrictions, the response of the price of asset n to an incremental change in the quantity outstanding of asset m is
@pnt
@Xmt

@qnt+1
Mt;t+s
@Xmt
"
qnt+1 @M
+Et
Wt @Rt;t+s

= Et

qmt+s

Rt;t+s pmt +

N
X

Xkt

k=1

@qkt+s
@Xmt

Rt+1

While it is not possible to sign this reaction without specifying how payo¤s are
determined and the additional properties of the pricing kernel, it is clear that
the derivative in equation (5) will not be zero in general. Indeed, a su¢ cient
condition for quantities to matter for prices is that the covariance of Mt;t+s and
qkt+s is nonzero for some k. For example, if we further assume that asset payo¤s
do not depend on the current portfolio allocation Xt , then it is straightforward
to show that @pnt =@Xmt = 0 for all n only if qmt+s =pmt = Rt;t+s — that is,
unless the return on asset m is equal to the return on the market portfolio in
all states of the world, changing the quantity of m will a¤ect all prices.
The Eggertsson and Woodford (2003) "neutrality proposition" mentioned
@p
in the previous section states, in essence, that @Xnt = 0 in a certain class
mt
of models so long as assets are valued only for their pecuniary returns. The
reason that this proposition fails here is that the class of models that Eggertsson
and Woodford consider are ones in which investor utility is a time-separable
6

@pkt
@Xmt

(5)
!#

function of consumption. Consequently, in their analysis, Mt;t+s depends only
on consumption in periods t and t + s; it does not involve the market return
Rt;t+s . Thus, condition (3) does not hold, and there are no portfolio-balance
e¤ects.4 Yet, while time-separable preferences are clearly an important case
to consider, there are equally important models in which (3) does hold. The
following subsection considers some of these models that have been applied by
previous studies.

2.2

Examples from the Literature

One simpli…cation that allows portfolio-balance models to be solved analytically
is to suppose that the distribution of asset payo¤s is …xed. In this case (further
assuming that wealth is predetermined), equation (5) reduces to
1
@M
@pnt
=
Et qnt+s qmt+s
@Xmt
Wt
@Rt;t+s

(6)

Thus, for example, if the one-period risk-free rate is exogenous and the SDF is
linear in Rt;t+s with slope coe¢ cient , then a change Xmt in the quantity of
asset m induces a contemporaneous price response in asset n of5
pnt = covt [qnt+s qmt+s ]

Xmt
Wt

(7)

Since is typically negative (states of the world with higher returns are discounted by more), raising the quantity of asset m lowers the price of asset n
whenever the two assets have positively correlated payo¤s.
In a classic application of this type of model, Frankel (1985) considered a
case in which agents solve a Markowitz portfolio choice problem over a variety
of asset types:
a
max Et R0 wt
vart R0 wt
(8)
t+1
t+1
wt
2
where Rt+1 = R1t;t+1 ::: RN t;t+1 is the vector of one-period asset returns, a is the coe¢ cient of relative risk aversion, and wt is the vector of dollar
0
Wt .
values allocated to each asset. The maximization is subject to wt 1
Mean-variance problems of this type result in an SDF that is linear in the market return. Consequently, one can write:
Et [Rt+1 ] = exp [rt ] + a
4 Eggertsson

t xt

(9)

and Woodford are explicit about the consequences of this assumption: “The
marginal utility to the representative household of additional income in a given state of the
world depends on the household’ consumption in that state, not on the aggregate payo¤
s
of its asset portfolio in that state. And changes in the composition of the securities in the
hands of the public do not change the state-contingent consumption of the representative
household. . . ” [Emphasis in original.]
5 The assumption that the short rate is exogenous implies @E [M
t
t;t+1 ] =@Rt;t+1 = 0, which
allows the product term in equation (6) to be reduced to the covariance term in equation (7).

7

where
xt =

is the covariance matrix of Rt+1 conditional on time-t information and
N
P
x1t ::: xN t is the vector of asset shares. (I.e., xnt Xnt =
Xmt
t

m=1

.) Since (9) is just an equilibrium condition, it must be true for any value of xt .
Thus, if a policymaker can set xt to an arbitrary value, he can determine the
expected return on any given asset Et [Rnt+1 ]. In the case of bonds with known
terminal payo¤s, this is su¢ cient to determine period-t prices. Friedman (1986),
Engle et al. (1995), and Reinhart and Sack (2000) are among the subsequent
studies to exploit this relationship. More recently, Gromb and Vayanos (2010)
illustrate the e¤ects of quantities on prices in version of this problem in which
there are only two assets, and Neeley (2010) and Joyce et al. (2011) apply it
speci…cally to the study of central bank asset purchases.
While this approach is useful for studying certain problems, its main di¢ culty is that treating one-period-ahead asset payo¤s as exogenous is not realistic.
In most cases of practical interest, the payo¤ of an asset s periods ahead includes
the price of that asset s periods ahead, and that price should be determined in
the same way as prices today. In terms of equation (5), there is no reason
to expect @qnt+1 to be equal to zero, particularly if there is persistence in the
@Xmt
value of Xt across periods. One way this dependence manifests itself is that the
conditional covariance matrix of returns t is endogenous. But, for example,
when taking these models to the data, Frankel (1985) and subsequent authors
calibrate t to a covariance matrix based on the historical returns data. This is
tantamount to assuming that investors do not change their beliefs about future
prices when xt changes, even though the model itself implies that these prices
will be di¤erent. While it is possible that investors do not account for the e¤ects
of quantities on return variances when determining asset prices, doing so in the
context of these models represents a violation of rational expectations. There
is nothing logically wrong with performing policy hypotheticals using equation
(9), but rational expectations requires us to solve for t as a function of xt
before doing so.
Another important class of models in which portfolio-balance e¤ects are
potentially operative are those involving recursive preferences. In particular,
consider a representative investor with Epstein-Zin-Weil utility over real consumption. As Epstein and Zin (1989) showed, the SDF in this case can be
written as
ct+s 1
1
Mt;t+s =
(Rt;t+s
(10)
t;t+s )
ct
where ct is consumption, t;t+s is the gross rate of in‡
ation between periods t
and t + s, and and are utility parameters. This SDF has the form of (3),
and so the portfolio-balance channel applies. Indeed, Piazessi and Schneider
(2008) study the relationship between bond quantities and prices in this type of
model. Again, however, they do not model expectations as rational. Instead,
to solve the model, Piazzesi and Schneider assume that investors have adaptive
expectations over the asset-return distribution, which they estimate using survey
data and vector autoregressions. As they argue, there may well be good reasons

8

to believe that investors form expectations in this way. However, it is also of
interest to consider the case in which expectations are set rationally. For a given
vector of asset shares xt , it is evidently not possible to solve analytically for the
state-contingent price vector that satis…es both equation (4) and equation (10)
in all states of the world.

2.3

Solution Method

As illustrated in the cases above, it is not generally possible to solve portfoliobalance models analytically for asset prices as functions of quantities under
rational expectations. The special cases in which it is possible (for example,
strictly exogenous one-period payo¤s or risk neutrality) are typically not of
practical interest. The central di¢ culty is that the solution for prices involves
the moments of future prices, and, under rational expectations, these moments
themselves must be endogenous. While it is common in asset-pricing models for
today’ asset prices to depend on the distribution of tomorrow’ asset prices, the
s
s
particular di¢ culty with portfolio-balance models is that the SDF itself depends
upon both of these objects.
I propose to solve these models numerically for the time-t vector of asset
prices pt using an iterative, discrete–
state approximation method.. This approach has the added advantage that it places very few constraints on either the
functional form of the pricing kernel or the dynamics of the state vector. Consequently, it is straightforward to consider models with potentially important
nonlinearities, such as the zero lower bound.
I …rst make explicit that prices and quantities depend on the state of the
economy. Namely, let xn (st ) describe how the quantity of asset n depends
on the state. Let st be Markov on the support S with transition density
(st+1 jst ). It is assumed that the form of the pricing kernel in equation (3),
the laws of motion for the states, and the dependence of quantities on the
states are known— that is, we (and investors) have knowledge of the functions
(st+1 jst ), M (st ; st+1 ; Rt;t+1 ), and xn (st ). We seek a vector-valued function
p(st ) = ( p1 (st ) ::: pN (st ) ) that describes how all asset prices depend on
st .
The price of asset n is given by
Z
pn (st ) =
(s0 jst )M (st ; s0 ; Rt;t+1 )qn (s0 )ds0
(11)
S

where the integral is taken over all dimensions of the state and q(st ) = (q1 (st )
::: qN (st )) is the vector of functions determining asset payo¤s in state st . This
function may be known (if payo¤s are strictly exogenous) or, more generally,
dependent upon the unknown pricing funciton p(st ). While the algorithm described below applies in principle to any asset, the case of zero-coupon Treasury
bonds considered in this paper is equivalent to the further assumption on the
payo¤ function:

9

qn (st )

pn

1 (st ) for n > 1
1 for n = 1

(12)

where the index n is then interpretable the bond’ maturity.
s
Given a distribution for the market return, Rt;t+1 , (11) is a system of linear
Fredholm equations of the …rst kind, which in principle can be discritized and
solved by quadrature in one step (see Tauchen and Hussey, 1991). However,
the fact that Rt;t+1 is de…ned as in (4) requires us to iterate by, …rst, solving
(11) using a given distribution of Rt;t+1 , and, second, given the resulting pricing
fuctions …nding the updated distribution of Rt;t+1 . These steps can be repeated
to convergence.
The speci…cs of this algorithm are discussed in Appendix A. Loosely speaking, it can be summarized as follows:
1. Guess a function pi (:) such that pt = pi (st ) on a discretization of S.
2. Based on this function and the known law of motion for st , compute
(the discrete approximation to) the distribution of Rt;t+1 , including its
covariance with st+1 and pt+1 .
3. Using that distribution, solve for the updated function pi+1 (st ) via equation (1) and return to step 2.
As illustrated in the appendix, the algorithm converges quickly for the type
of model considered in this paper. An additional modi…cation, also discussed
in the appendix, is necessary to handle the models that involve recursive preferences with endogenous consumption, since in those cases the pricing kernel
depends on the state in a way that is unknown ex ante.

3
3.1

Portfolio Balance in the Term Structure
Preliminaries

While the above discussion applies to any set of assets in principle, I now focus
the analysis on the study of zero-coupon, default-free bonds. This is an important case to consider, not just because of its policy relevance with respect
to asset purchases and debt management, but also because these bonds, with
known terminal payo¤s and being uncontaminated by credit and liquidity risk,
help to isolate the workings of the portfolio-balance channel. As noted, restricting attention to the pricing of default-free bonds does not necessarily imply that
we assume that investors hold only those bonds.
Henceforth, I maintain assumption (12) so that pnt indicates the time-t price
of a bond with n periods to maturity. By de…nition, p1t = exp[ rt ] is the
price of the one-period bond, and equation (2) determines the rest of the yield
cuve. In particular, perfect certainty corresponds to the “strong form” of the
expectations hypothesis— long-term interest rates are the average of expected

10

short-term interest rates (up to a Jensen’ inequality term). Otherwise, the
s
covariance term in equation (2) adds a risk premium to expected returns and
a term premium to bond yields. The SDF M only appears in this covariance;
thus, since I assume that rt is set exogenously, quantity changes operate by
a¤ecting term premia in the presence of uncertainty.6
Restricting attention to bonds in this way, portfolio-balance e¤ects take on a
special meaning— bond quantities a¤ect bond prices through a duration channel.
That is, tilting the distribution of outstanding assets toward lower maturities
alters the term structure by "removing duration from the market," precisely the
e¤ect that is often pointed to, in less formal terms, as the primary mechanism
through which asset purchases operate. (See, for example, Gagnon et al., 2010.)
To date, the only paper to capture such duration e¤ects in a theoretical model
that features both no-arbitrage restrictions and rational expectations is Vayanos
and Vila (2009). In that model, arbitrage investors solve a problem like (8),
but they interact with “preferred-habitat” agents who have downward-sloping
demand curves for assets of particular maturities. Thus, the portfolio shares xt
that they hold are endogenous. Assuming a linear-Gaussian factor structure,
Vayanos and Vila obtain an a¢ ne solution in which demand shocks that remove
long-term debt from the hands of arbitrageurs push down long-term interest
rates.
As the only model to incorporate supply e¤ects into an otherwise standard
representation of the term structure, Vayanos-Vila has been highly in‡
uential in
the way that economists have designed and interpreted recent empirical studies.7
However, the presence of the preferred-habitat agents and the endogeneity of
xt in that model can obscure the fundamentally simpler relationship between
prices and quantities. The example below strips this mechanism to its essentials
to show how duration e¤ects work. (Appendix B shows how Vayanos and Vila’
s
model …ts into the broader portfolio-balance framework.)

3.2

Example: A linear, one-factor model

Consider a model in which the short rate rt is the only source of stochastic variation and in which investors solve the mean-variance portfolio-choice problem
(8), where the available assets consist only of Treasury bonds with maturities
1; :::; N . The relative supply of bonds is …xed over time, xt = x. Bond prices
then solve the system of quadratic equations
pt = exp [ rt ] Et [qt+1 ]

a

tx

pt 0x

(13)

at each point in time, where t is the conditional covariance matrix of the asset
payo¤s qt+1 . (This is the equivalent of equation (2) for the Markowitz model.)
Since the payo¤ on a one-period bond is always 1, the …rst row and column
6 It is not necessary to assume r exogenous in order to generate portfolio-balance e¤ects
t
or to apply the solution method. However, this assumption is common in both the termstructure and monetary-policy literatures.
7 See, for example, Doh (2010), Hamilton and Wu (2011), and Li and Wei (2012).

11

of t are zeros. The conditional variance of the price of a one period bond,
denoted ! 2 , is the second diagonal element of t . Since p1t = exp[ rt ], this
1t
variance is a determinsitic function of the short-rate process and is a known and
exogenous quantity.
In this model, the expectations hypothesis holds if x = ( 1 0 ::: 0 ).
That is, eliminating all duration from investors’ portfolios reduces the term
premium to zero. On the other hand, as long as a > 0 and x1 < 1, we have
pnt < p1t Et [pnt+1 ], for n > 1, so that all risky assets require a discount and
all risky returns include a risk premium. Indeed, since rt is the only source of
variation, the term premium in this example re‡
ects only the compensation for
bearing duration risk. Speci…cally, there is a single market price of risk given
by the Sharpe ratio
Et [Rnt;t+1 ] R1t;t+1
(14)
t =
nt

where nt is the standard deviation of the return on asset n. This, in turn,
implies that all asset prices and returns are perfectly correlated, regardless of
the value of x or the dynamics of rt . In particular, there exist coe¢ cients
Ant and Bnt such that we can write Rnt;t+1 = Ant + Bnt p1t+1 for any n.8
The coe¢ cient Bnt represents the response of the return on a risky asset to
the risk-free rate— the analogue of the “spot-rate duration” that appears in
continuous-time term-structure models. The standard deviation of the return
on the market portfolio can be expressed as ! 1t x0 Bt , where Bt is the vector of
Bnt coe¢ cients. Combining this with equation (14), we have
t

= a! 1t x0 Bt

(15)

analogous to a result produced by Vayanos and Vila (2009). Reducing the
quantity x0 Bt is the model’ version of “removing duration from the market.”
s
Doing so will lower both the total risk of the Treasury portfolio and the price
of that risk.9
To implement the model numerically, take periods to be one year in length
and suppose that there are N = 30 bonds. Suppose that the short rate follows
a linear process
rt = 0 + 1 rt 1 + "t
(16)
where "t has variance 2 . For parsimony, I approximate the maturity structure
of government debt with an exponential distribution:
xn / exp [ zn]

(17)

8 Speci…cally, B
(1 p2t Bnt )=p1t . Because rt follows an AR(1)
nt
nt =! 1t and Ant
process, the function Bnt has a similar shape across n at each value of the short rate as in
the Vasicek (1977) model.
9 Since (13) is the result of a Markowitz portfolio problem it also has a conditional CAPM
w
representation. Speci…cally, since E t [Rt;t+1 ] =E t [qt+1 ]0 x=p0 x is the expected return on
t
wealth, we can write

Et [Rt+1 ]
where t = t x=x0
R1t;t+1 = ax0 t x.

R1t;t+1 1 =

t

(Et [Rt;t+1 ]

R1t;t+1 )

t x. The expected excess return on the market portfolio is Et [Rt+1 ]

12

where the parameter z is the average maturity outstanding. The exponential
approximation is likely to be a good one because the distribution of government
liabilities typically consists overwhelmingly of shorter maturities. For example,
as shown in the top panel of Figure 1, liabilities with less than …ve years of
duration (including currency and reserves) account, on average, for about 80
percent of total face value. The exponential functional form implies that,
as in the data, the distribution is highly skewed toward securities with very
low duration.10 However, as discussed further below, the exact shape of the
distribution beyond its …rst moment is actually of little importance.
Let 0 = 0:003, 1 = 0:95, = 0:015, and a = 8. These values are calibrated
roughly to match the dynamics of the short-rate and the average value of the
15-year yield in the data. The black line in Figure 2 demonstrates how the
duration of the Treasury portfolio a¤ects the behavior of asset prices in the
model by plotting the equilibrium values of the (negative) price of risk t across
possible values of z in this model. The relationship between duration and risk
prices is monotonic but nonlinear. Near the historical mean duration of 2.7
years (see the bottom panel of Figure 1), increasing duration by one year raises
the price of risk by about 50%.
Figure 3 illustrates how these risk prices translate into yields. I compare two
cases: z = 2:7 years and z = 2:0 years. As discussed in Section 5, the di¤erence
of 0:7 years is approximately equivalent to the e¤ects of the cumulative asset
purcahses by the Fed through 2012. The di¤erence in the maturity distribution
of securities is illustrated by the red bars in the top panel of the …gure. The
e¤ect on the yield curve, evaluated at the sample average value of the short rate,
is shown in the middle panel. For the calibration used here, the ten-year yield
is 56 basis points lower under the lower-duration distribution. Again, this is
due to reductions both in the price of risk illustrated in the previous …gure and
in risk itself. The latter is shown in the bottom panel of Figure 3, which plots
the standard deviation of one-year returns (the square-root of the diagonal of
t ) under the two supply distributions.
Finally, we can demonstrate the earlier claim that the shape of the distribution x is of little importance for the shape of the yield curve. Analytically,
equations (13) and (15) show that the individual asset share xn does not matter
for the individual asset price pnt . Only the weighted sum of asset shares x0 Bt
is relevant, and it a¤ects all prices in the same way. Consequently, this model
cannot produce local-supply e¤ects from large quantity gluts or shortages in
particular sectors of the market. To illustrate this, consider a stark alternative
to the exponential shape used for the maturity distribution above. In particular, suppose that the distribution was nearly degenerate around the same mean
of z = 2:7, as shown in the top panel of Figure 4. This adjustment does slightly
increase the duration risk of the portfolio, primarily because it completely eliminates the risk-free one-year bonds from investors’ portfolios. However, after
recalibrating a to match the average slope of the previous yield curve, there
1 0 The data in this …gure are compiled from CRSP, taking the duration of all Treasury debt
in the hands of the public, weighted by its face value. In addition, I include the monetary
base, as reported by the Federal Reserve, as a "Treasury" security with zero duration.

13

is virtually no di¤erence between the term structures generated under the two
distributions. This can be seen in the bottom panel of the …gure, where the
solid blue line is the same yield curve shown in Figure 3, and the dashed red
line is the (recalibrated) yield curve at the same value of the short rate under
the degenerate supply distribution. Similar results obtain for other possible
choices of the shape of x.

3.3

E¤ect of the Zero Lower Bound

Since risk prices in portfolio-balance models are themselves a function of the
quantity of risk held by investors, heteroskedasticity can cause risk prices—
and therefore term premiums— to di¤er signi…cantly across states of the world.
A particularly important case of this is the zero lower bound on the nominal
short rate. The presence of this bound, all else equal, implies that there is
less uncertainty about short-term interest rates in the near future when the
current value of those rates is near zero. Consequently, investors in all bond
maturities face less near-term interest-rate risk and therefore demand lower risk
premiums. Furthermore, equation (15) shows that the e¤ect on risk prices
of removing duration from the market (in the sense of reducing the weighted
average x0 Bt ) is directly proportional to the variance of the one-period bond
! 1t . Thus, the reduction in the volatility of short-term interest rates induced
by the ZLB will also dampen duration e¤ects.
To illustrate these outcomes in the simple one-factor model above, suppose
that, rather than allowing the short rate to follow an unrestricted AR process,
we append the condition
Pr ["t < 0] = 0 8t
(18)
to equation (16). That is, let the distribution of the short-rate shock "t be
a truncated normal, where the truncation point varies with rt in such as way
that it always enforces the zero lower bound. The conditional distribution of
the short rate under this process is illustrated in Figure 5. The reduction in
volatility near zero is evident.
Refering back to Figure 2, the blue line illustrates risk prices calculated at
the ZLB for di¤erent values of duration; on average, they are roughly half of
the linear case considered previously. Furthermore, the slope of the blue line is
considerably ‡
atter than that of the black line, indicating that a given change in
duration has a smaller e¤ect on t when the ZLB binds. To see these results in
terms of the yield curve, the solid lines in the top panel Figure 6 correspond to
the same maturity distributions that were used in Figure 3. (2.7 years for the
blue line and 2.0 years for the red.) The same duration–
reduction experiment
that caused a 56 basis point reduction when the short rate was at 5.8% causes
only a 29 basis point reduction when the short rate is at 0% with a binding
ZLB.
The reduction in duration e¤ects becomes even more pronounced if the short
rate is anticipated to stay near zero with a high degree of certainty for multiple
periods, because this results in a larger reduction in the variance of rates across

14

the term structure. This type of thought experiment is relevant because the
forward-guidance language that has been included in most FOMC statements
since 2008 is widely viewed as having essentially committed the Fed to keeping
rates near zero for a signi…cant time into the future. To illustrate, the dashed
lines in the …gure show the model’ outcomes when rt is constrained to equal
s
0 for both periods t and t+1 (and investors know this with certainty at time
t) and only then follows its truncated AR(1) process. The forward guidance
itself has a large e¤ect on long-term rates; in this example, it reduces the 10year yield by 52 basis points (the di¤erence between the solid and dashed blue
lines). However, with the forward guidance in place, the e¤ect of the reduction
in duration on the ten-year yield is only 23 basis points, rather than 29. The
reason is that, as illustrated in the bottom panel, forward guidance in the onefactor model implies that the returns on all bonds in period t+1 are known with
certainty. Consequently, the risk premium for the period over which the forward
guidance is in place is zero, regardless of the amount of duration outstanding—
i.e., ! 1t is zero, and so, by equation (15), the price of risk t is zero. Thus,
although they both serve to reduce longer-term yields, forward guidance and
asset purchases are somewhat o¤setting, at least if duration e¤ects are the only
channel through which the supply distribution matters for prices.

4

Multi-Factor Models

While the simple one-factor model is useful for illustrating portfolio-balance effects, it is not particularly realistic. This section considers several extensions of
that model. In particular, I consider models that include in‡
ation risk, recursive preferences over consumption, and the possibility that investors maintain
wealth in assets other than Treasury securities. In addition, I allow the duration
of Treasury securities itself to be a stochastic variable over which investors must
form expectations. The purpose of analyzing these models is to get a sense of
the range of possible magnitudes for the e¤ects of Treasury supply changes, as
well as to illustrate the ‡
exibility of the general approach.
To more cleanly compare the e¤ects of di¤erent assumptions about the pricing kernel, I take the dynamics of the state of the economy to be identical
across all …ve models. Speci…cally, I assume that in‡
ation and the short-term
interest rate follow a VAR(1) process truncated at zero for the short rate. I
also assume that the distribution of outstanding Treasury securities changes
over time. To allow for such changes in a parsimonious way, I maintain the
exponential maturity distribution (17), but I allow the parameter z to follow a
…rst-order autoregressive process. This induces (nonlinear) dynamics into all
elements of the asset-share vector xt .

15

4.1
4.1.1

Model Speci…cation
Model 1 - Quadratic nominal preferences with Treasury-only
portfolio

The …rst model retains the quadratic nominal preferences from the example of
the previous section (maintaining the pricing equation (13)); it di¤ers only in
the dimension and dynamics of the state vector. In this model, investors do not
care about in‡
ation per se, since their objective function is over nominal returns.
However, in‡
ation may still be important to the extent that the nominal short
rate responds to it. The VAR dynamics I assume for the short rate and in‡
ation
allow for this possibility. In the other models below, in‡
ation itself will play
a direct role in investors’payo¤s, and so this process maintains comparability
across models in the exogenous dynamics of the economy.
Model 1 is similar in most respects to that of Greenwood and Vayanos (2014).
The principal di¤erences are the addition of the in‡
ation term and the imposition of the ZLB.11
4.1.2

Model 2 - Quadratic real preferences with Treasury-only portfolio

Model 2 allows investors to care about real returns, rather than nominal. In
this case, maintaining quadratic preferences, the investors’ decision problem
becomes
maxEt R0
t;t+1 wt
wt

a
vart R0
t;t+1 wt
2

t;t+1

t;t+1

(19)

t+1 ]

(20)

Prices then solve
pt = exp [ rt ] Et [qt+1 ]

a

t xt

pt 0xt

covt [qt+1 ;

and the nominal pricing kernel is conditionally linear in the real return (Rt;t+1
Note that changes in xt have no direct e¤ect on in‡
ation-risk premia through
portfolio rebalancing. While they may change the covariance of prices with in‡
ation through their e¤ect on the overall volatility of asset prices, these e¤ects
will generally be second-order.
4.1.3

Model 3 - Recursive preferences with Treasury-only portfolio

I next consider a model with Epstein-Zin-Weil preferences over consumption
(Model 3). Such models have the pricing kernel given in equation (10). Unlike
1 1 There are three other, minor di¤erences between this model and Greenwood-Vayanos.
First, the Greenwood-Vayanos model is in continuous time. Second, I use the exponential
form in (17) for the supply distribution, whereas they use a linear-factor structure. Finally,
they assume that market-value weights wnt are exogenous, while I assume that the par values
xnt are exogenous. None of these di¤erences in approach is likely to have substantial e¤ects
on the model outcomes.

16

t;t+1 ).

the quadratic preferences of Models 1 and 2, this spe…ciation involves consumption growth. While one could adopt an exogenous process for consumption,
it turns out that the exogenous dynamics for the state variables rt , t , and xt
that we have already speci…ed are su¢ cient to pin down the dynamics of the
1
in equation (10), if we assume that consumption can be apquantity ct+s
ct
proximated as a trend-stationary process. Again, proceeding in this way has
the advantage that it maintains the same state vector and dynamics across all
models, isolating the di¤erences that are due to assumptions about the pricing
kernel alone. The solution requires an additional step the solution algoritm,
which is described in Appendix B. It does not require us to take a stand on the
values of and or the precise form of the consumption process.
4.1.4

Model 4 - Quadratic real preferences with diversi…ed portfolio

A major criticism of all of the above models is that they assume that government
debt is the only asset available to the public. Furthermore, to the extent that
participants in the bond market also bear the tax burden of repaying the bonds,
they may not function as assets at all, as discussed in Barro (1974) and Wallace
(1981). A more ‡
exible and realistic model should allow for Treasuries to receive
a weight of less than 1 in the representative investor’ portfolio. Models 4 and 5
s
consider an extension of the real-quadratic-preference and recursive-preference
models, respectively, that allows for this possiblity.
In particular, suppose that Treasuries constitute a fraction v of the market
value of investor assets. Then, rather than the de…nition (4), we have
Rt;t+1 = v(

xt 0qt+1
) + (1
xt 0pt

w
v)Rt+1

(21)

w
where Rt+1 is the return on the non-Treasury portfolio. For simplicity, I assume
w
w
that Rt+1 is iid. Thus, Rt acts as a fourth state variable in the models in which
it appears, but its time-t level contains no information about its time t + 1 level.
w
Consequently, the yield curve is not a¤ected by shocks to Rt+1 , even though
the term premium does provide compensation for the risk associated with these
shocks to the extent that they are correlated with bond returns. The previous
models, which implicitly take v = 1, essentially represent an upper bound on the
potential size of portfolio-balance e¤ects in the term structure. For any given
calibration, as v approaches zero, those portfolio-balance e¤ects must approach
zero as well.

4.1.5

Model 5 - Recursive preferences with diversi…ed portfolio

The …nal model also takes the Treasury share in the portfolio to be v < 1 so
that the return on wealth is given by (21), but it assumes recursive preferences
as in Model 3.

17

4.2

Calibration

The parameters of the truncated VAR process for in‡
ation and the short rate
are calibrated to the dynamics of the annual U.S. data over the period 1971 2008 and are shown in Table 1.12 Estimating the AR(1) process for zt on the
data shown in Figure 1 gives an intercept of 0.16, an autoregressive parameter
of 0.95, and a standard error of 0.14 (also shown in the table), and these are the
values used in the calibration. The assumption that zt is contemporaneously
and dynamically uncorrelated with rt and t is both analytically convenient —
since we will want to isolate the e¤ects of exogenous shocks to zt — and realistic.
In the data, those correlations are very close to zero.
In both the quadratic and EZW models, the pricing kernel depends on a
single parameter (either a or ). In each case, I calibrate the value of this
parameter such that the 1- to 15-year slope of the yield curve is 130 basis points
on average, given the observed values of in‡
ation and the short rate over the
1971 – 2008 period, consistent with the average yield-curve slope observed in
the data during that period.
Models 4 and 5 require a calibration for the Treasury-wealth share v. Given
the various considerations that might a¤ect segmention and exposure to Treasuries, it is unclear what the appropriate value of v is. Since models 1 through
3 give us results at one extreme (with v assumed equal to 1), for illustration
in models 4 and 5, I calibrate v to a relatively low value of 0.1. I calibrate
the distribution of Rw to the results of Lustig et al. (2013), who estimate that
the excess annual return on wealth averages 2.4% with a standard deviation of
9.2%.

4.3
4.3.1

Results
Model moments

Table 3 reports some summary statistics for the …ve models. Given the calibration to the average level of 15-year yields, the models with quadratic preferences
…t the short end of the yield curve better than the models with recursive preferences, which imply too much curvature. Both types of preferences generate
too little volatility in long-term yields, a manifestation of the "excess volatility"
puzzle for bonds, although the recursive preference models do slightly better in
this dimension than the quadratic-preference models. Partly because of low
volatility, both types of preferences understate the average excess returns on
bonds.
In the data, yields themselves are only weakly correlated with the average
duration of outstanding government debt. All …ve models also display fairly
weak correlation with duration in yields. Annual bond returns, on the other
hand, are approximately 30% correlated with duration in the data. The models
also broadly replicate this fact, with the quadratic-preference models having
1 2 Yield data are the Gurkaynak et al. (2007) zero coupon yields maintained at the Federal
Reserve Board. The sample period begins in 1971 because that is the date at which 15-year
yields become available. In‡ation is measured by the core PCE index.

18

slightly higher correlations than the recursive-preference models. The weakest
correlations of duration with both yields and returns are produced by Model 5.
Intuitively, assuming a low share of Treasury bonds in the portfolio implies that
the total Treasury duration that investors must bear is unimportant. Model
3 also assumes a small share of Treasuries in the overall portfolio, but, in that
model, risk aversion is calibrated to a very high value of a = 75 (see Table
1) which boosts the correlation between duration and returns. Indeed, the
reason for the large calibrated value of a in Model 3 is that the low amount of
duration risk that investors bear in that model requires them to be very risk
averse in order to generate a su¢ cient term premium to match the average
empirical yield-curve slope. (The recursive-preference model does not require
this because it allows the consumption process to adjust to create additional
risk for the investor.)
4.3.2

Supply e¤ects on term premia

Figure 7 displays impulse-response functions for the yield curve in all …ve models, following a one-standard-deviation positive shock to duration (increasing
the outstanding duration by 0.14 years). In each of these graphs, maturity is
plotted along the lower-left axis, and calendar time is ploted along the lowerright axis. Because the models involve nonlinearities, initial conditions matter
somewhat for these outcomes, and in each case they are evaluated at the unconditional means of the state variables implied by the parameters in Table 1.
Note that since the short rate and in‡
ation do not respond to Treasury supply
by assumption, there is no e¤ect of a shock to z on the expected path of rates.
Thus, the responses shown in the …gure are due exclusively to changes in term
premia.
In all of the models, the response functions have the same basic shape,
increasing relatively sharply at shorter maturities and peaking in the middle of
the curve. Because zt is very persistent, these responses decay only slowly over
the twenty years following the hypothetical shock, and at about the same rate
across all models. Magnitudes vary somewhat across models. Responses for
the quadratic models with preferences over real returns (models 2 and 4) are
nearly identical and are a bit smaller than the model with nominal preferences.
This re‡
ects the fact that, in the nominal model, duration risk is the only source
of risk. With real preferences, in‡
ation risk matters, and duration shocks have
less volatility to account for. (This is why the risk aversion coe¢ cient for Model
1 is calibrated to 9.5, while that for Model 2 is calibrated to 7.2.) Nonetheless,
the responses in Models 1 - 4 are quantiatively similar (around 3 basis points
for the initial response of the ten-year yeild), while the response in Model 5 is
an order of magnitude smaller.
In general, the response of the yield curve to stochastic duration shocks is
not monotonic across maturities. The reason is that it con‡
ates two e¤ects. On
the one hand, investors’duration exposure— and therefore the price of duration
risk— goes up in the period of the shock, raising the risk premium required on
long-term bonds by more than that on short-term bonds. On the other hand,
19

since zt is mean-reverting, duration is expected to be lower in future periods.
Thus, long-term bonds are likely to face lower risk prices on average over their
lifetimes than short-term bonds. (In the one-factor model, the duration shock
illustrated in Figures 3 and 6 was assumed to be permanent, and the response
therefore was monotonic across maturities.) Figure 8 illustrates the consequences of the duration persistence assumption in Model 1 by considering the
e¤ect of the same shock to duration when the autoregressive coe¢ cient for zt
is calibrated to be zero instead of 0.95 (holding the unconditional mean of the
zt process constant). The response of the two-period yield is the same in both
cases, since two-period bondholders are una¤ected by the amount of duration
in the market beyond the current period. However, overall the response to the
shock when zt is iid is much lower at all other maturities than when zt is persistent. In addition, the magnitude of the response peaks at about the four-year
maturity, as the anticipated mean-reversion dominates the temporary increase
in duration risk beyond relatively short horizons.
Consequently, it is not just the size of a shock to Treasury duration that
matters. The anticipated persistence of that shock is also very important for
both the magnitude and the shape of the term-structure response. This point
could be relevant for assessing the e¤ects of central bank asset purchases, which
may have a lifetime that is shorter than the typical shock to Treasury maturity
structure.
Table 5 puts these results into further perspective by applying the models to
the most-recent historical episode in which there was a large change in Treasury
maturity structure. In particular, as was illustrated in Figure 1, between the
years 1976 and 1988, Treasury duration (including base money) in the hands of
the public increased from about 1.5 years to about 3.5 years. The …rst column
of the table shows what each of the …ve models predict for the ten-year term
premium, given a shock to zt of this amount, assuming that in‡
ation and the
short rate are at their average values over the 1976-1988 period. In all cases
except Model 5, the predicted response is about a 50 basis point increase in the
ten-year term premium. Using the Kim and Wright (2005) model, Rudebusch
et al. (2007) do indeed document an increase in term premiums estimated from
unobseved-factor models around this time.13 Although their estimates point
to an even larger increase (about 100 basis points on net over this period), the
results in Table 5 suggest that the increase in Treasury duration could have
been a signi…cant contributing factor.
1 3 Other

models reduced-form term-structure models, such as that of Adrian et al. (2013)
also generate an increasing term premium during this period, as do estimates based on survey
data. See, for example,
http://libertystreeteconomics.newyorkfed.org/2014/05/treasury-term-premia-1961present.html#.VFEFwBawXv4.

20

5

Duration E¤ects of the Federal Reserve’ Ass
set Purchases

I now consider the e¤ects of the Federal Reserve’ large-scale asset purchases
s
(LSAPs) that took place between 2008 and 2012 within the context of the …ve
models developed in the previous section. While calculating the overall effect of these operations on the oustanding duration of government debt is not
straightforward, I approximate the magnitude based on the aspects of the programs are summarized in Table 6. One di¢ culty is that a large component
of the Federal Reserve’ purchases consisted of agency mortgage-backed secus
rities. Throughout the paper I have ignored agency MBS when calculating
the duration of government liabilities, but this exclusion may be inappropriate
given that these securities have typically been perceived to carry an implicit
government guarantee. Two technical measurement problems arise when trying to extend the analysis to agency MBS. The …rst is that comprehensive data
on the maturity structure of outstanding MBS are not available. The second
is that, unlike the case of zero-coupon Treasuries, duration for MBS depends
on economic conditions— in particular, it depends on the level of interest rates
through the negative convexity induced by the prepayment option. However,
according to the Barclays MBS index, the average duration of MBS since 1989
is about three years, only slightly lower than that of Treasury debt. (See Hanson, 2012.) In performing the calculations below, I assume that asset purchases
did not materially change the average duration of MBS outstanding, but I do
incorporate the amount of MBS outstanding, so that Fed purchases swap their
duration for zero-duration reserves.14
The net e¤ect of the LSAP programs, as of December 2012, was to reduce the
outstanding supply of Treasury and MBS securities by $2 trillion and increase
reserves by a similar amount. Moreover, the Treasury securities removed from
the market were, after the duration twist induced by the Maturity Extension
Program (MEP), almost all of maturity greater than …ve years. Consequently,
the average duration of Treasury securities in the hands of the public after
all was said and done was about 0.5 years lower than it otherwise would have
been. Taking all of these facts together, we can estimate how investors’duration
changed as a result of the programs. Speci…cally, as of Q4 2008, Treasury debt
held by the U.S. public was $5.3 trillion, agency MBS was $5.0 trillion, and the
monetary base was $1.4 trillion. The average duration of Treasuries (excluding
the monetary base) in the hands of the public was 3.1 years as of the beginning
of the program. Therefore, assuming that MBS have an average duration of
three years (and again counting the base as zero duration), the weighted average
of Treasuries, MBS, and the monetary base in the hands of the public was 2.7
years when the …rst LSAP was announced. Holding all else constant, this value
would have fallen to 2.0 years as a result of the programs.15
1 4 Data

on MBS quantities come from SIFMA (www.sifma.org/research/statistics.aspx).
ignore the e¤ect of the Fed’ $165 billion of agency debt purchases on the outstanding
s
duration of government securities, although the resulting increase in reserves is included.
15 I

21

The …nal column of Table 6 presents the range of values for the empirical
e¤ects of the LSAPs on the ten-year term premium, culled from the literature
that has examined this question. In particular, this range draws on estimates
from Gagnon et al. (2011), Krishnamurthy and Vissing-Jorgensen (2011), Ihrig
et al (2012), D’
Amico et al. (2012), and Rosa (2013). (In some cases, additional
minor calculations were required to make the results comparable.) Of course, all
of these estimates are subject to a high degree of uncertainty surrounding both
parameter values and speci…cation. Furthermore, since most of these papers
employ an event-study methodology using program-announcement dates, they
are more likely to understate than overstate the e¤ects, to the extent that they
do not adequately capture expectations of purchases prior to announcement
dates or any additional e¤ects that may be priced in when the purchases actually
occurred. Nonetheless, they provide a rough guide to what we should expect
for the combined e¤ects of the programs— likely on the order of 100 to 250 basis
points altogether. To be clear, this is (according to the authors of the studies)
only the e¤ect on the term premium, controlling for changes in expectations of
the short rate, and it only re‡
ects the initial impact of the programs, ignoring
any subsequent dynamic e¤ects.
How do these estimates compare to what would be predicted by portfoliobalance models? The second column of Table 5 shows the e¤ect of removing 0.7
years of duration from the hands of investors, starting from a level of duration
that is equal to what was observed prior to the LSAP’ introduction (2.7 years)
s
and a con…guration of short rates and in‡
ation that approximates the situation
on average since that time. Speci…cally, in‡
ation is set at a level of 1.4 percent
(its average over the period 2009 – 2012), and the short-term interest rate is
taken to be near the zero lower bound with two years of forward guidance in
place. Although the market’ perception of the length of time that the ZLB
s
would bind ‡
uctuated over this period, the assumption of two years is likely
close to the average that prevailed (see Femia et al., 2013).
The e¤ect of the LSAPs on the ten-year yield of the LSAP shock in this
environment is at most 13 basis points, in the model with quadratic, nominal
preferences (Model 1). In the recursive-preference model with diversi…cation
(Model 5), it is a mere 1 basis point. Note that this is the initial impact of the
shock— given the estimated dynamics of zt , the e¤ect would decay to zero over
time, similarly to the response shown in Figure 7.16 Of course, as noted above,
if investors anticipated LSAP shocks to be less persistent that conventional
duration shocks, the responses would be even smaller than this.
Because this exercise involved some approximation and educated guessing,
one might worry that the apparent weak impact of LSAPs simply re‡
ects inaccuracies in the model inputs. To address this possibility, I consider adjusting
the model in a number of ways that might make these e¤ects larger. First,
I double the size of the LSAP shock (assuming a reduction from 3.05 years to
1.65 years). Second, I weaken the e¤ects of forward guidance by assuming that
1 6 The decay in the e¤ect of the LSAP shock over time is qualitatively consistent with the
evidence presented in Ihrig et al. (2012) and Wright (2012).

22

investors anticipate the short rate to stay at zero for only one year, rather than
two. Third, I allow for the possibility that non-Treasury asset-return uncertainty may have permanently increased in the wake of the crisis by doubling
the standard deviation of Rv (in the models where that quantity is relevant).
Finally, I allow for the possiblity that risk aversion was higher than usual during
the …nancial crisis by doubling the value of the preference parameter (a or ) in
each model. Any one of these modi…cations by itself would generally increase
the impact of the LSAP shock. To ensure that I am indeed capturing an upper
bound on the plausible magnitude of the duraiton e¤ects of LSAPs, I consider
all four of them simultaneously. As shown in the …nal column of Table 5, these
modi…cations collectively do increase the impact of the LSAP shock by a factor
of between 3 and 5. However, even the largest is still only about half of the
bottom end of the range of empirical estimates. These results suggest that the
Federal Reserve’ programs could not have generated e¤ects on long-term yields
s
of the magnitude that has been observed in empirical studies through duration
e¤ects alone.

6

Conclusion

This paper has presented a new method for studying the ways in which the term
structure of interest rates and other asset prices are a¤ected by ‡
uctuations in
the quantities of assets outstanding. The type of model considered is a rationalexpectations version of portfolio-balance models that have been in use, with
varying degrees of formality, for decades. It may be viewed as a generalization
of the preferred-habitat model of Vayanos and Vila (2009), with investors that
potentially have a broader class of objective functions and in which nonlinearities
may be important. The key aspect of the model class that allows portfoliobalance e¤ects to exist under no-arbtrage is that the pricing kernel depends on
the return on wealth.
I examined portfolio-balance e¤ects on the yield curve under a variety of
assumptions about investor objectives. Some sets of assumptions can generate
sizeable e¤ects of Treasury maturity structure on term premia in some states of
the world. However, these e¤ects can be attenuated by various factors, including
nonlinearities associated with the zero lower bound, diversi…caiton of investor
portfolios into assets that are uncorrelated with Treasury returns, and weak
persistence of duration shocks. Many of these factors seem to be in play when
considering the Federal Reserve’ asset purchases. Even under rather extreme
s
assumptions, I was unable to come close to even the lower bound of the empirical
e¤ects of these programs through the duration channel alone. I interpret these
results, not as evidence that the asset purchases were ine¤ective, but as evidence
that they probably had their e¤ects primarily through mechanisms other than
the removal of duration risk. It seems possible that phenomena not easily
captured by a no-arbitrage model, such as market dislocations, liquidity shocks,
and capital constraints, were important during and after the …nancial crisis
and that LSAPs had additional e¤ects through those channels. This reading is

23

broadly consistent with the empirical conclusions of Krishnamurthy and VissingJorgensen (2011), Cahill et al. (2013), and D’
Amico and King (2013). More
structural modeling of the behavior driving such scarcity e¤ects is needed to
determine whether they would signi…cantly alter the results presented above.

24

Appendices
A. Solution Algorithm
Let pi (st ) be a proposal for the pricing function on a discretization of
d
the state space D = (d1 ; : : : ; dG ) 2 SG , where G is the number of nodes and
i = 0; : : : ; I indexes iterations, and let qi be the corresponding discritiztion of
d
q(st ). Suppose that the nodes are uniformly distributed over the state space,
so that the conditional transition probability from node j to node h can be
approximated by
b(dh jdj )

(dh jdj )

"

G
X
g=1

#

1

(dg jdj )

pi and b are used to generate a proposal for the joint distribution of period
d
t + 1 states and prices. That distribution, in turn, generates an updated pricing
function pi+1 (st ) through the analogue to equation (11)— that is, by solving
d
G
X
pi+1 (dj ) =
b(dg jdj )M
dn
g=1

0

x (dj ) qi (dg )
d
dj ; dg ;
0
x (dj ) pi (dj )
d

!

pi+1 1 (dg )
dn

(A1)

for the vector pi+1 (dj ) at each node j = 1; :::; G.
d
This procedure constitutes a contraction mapping on D so long as the moments of the pricing kernel are well behaved. The Banach Theorem then guarantees for any given discretization D, pi (dj ) ! pd (dj ) 8 dj 2 D, where pd
d
is the (unique) pricing function that obtains if b is the data-generating process.
But continuity of ensures that, for any node j,
!#
"
0
x (dj ) qi (st+1 )
d
lim pdn (dj ) = Et pn 1 (st+1 ) M dj ; st+1 ;
0
G !1
x (dj ) pi (dj )
d
i.e., in the limit, the pricing function solves the no-arbitrage condition (1).
Finally, by construction, if the algorithm converges, any point of convergence
is a rational-expectations equilibrium. This follows immediately, since convergence is de…ned as the …xed point at which the joint distribution of pt+1 and
Mt+1 is consistent with the vector pt , for each point in the state space.
It is important to note that, although the algorithm only solves for the
vector of prices at G points in the state space, once these solutions are in hand
it is straightforward to calculate equilibrium prices at any point through the
Nystrom extension. In particular, take an arbitrary state value st . For G large
enough, we have
pn (st )

" G
X
g=1

(dg jst )M

0

x (st ) qi (dg )
d
st ; dg ;
0
x (st ) pi (st )
d

25

!

pn

#" G
#
X
(dg jst )
1 (dg )
g=1

1

Once the algorithm has converged, the quantities on the right-hand side are all
known. Thus, securities can be priced in at any point in S.
Figure A1 displays some results on the convergence of the solution algorithm
for the one-factor model discussed in Section 3. The top panel shows the
computed 5-, 10-, 15-, and 30-year yields, shown for a short rate at its average
value of 5.8%, across the …rst 50 iterations (i = 1; : : : 50). The algorithm is
initialized at a price vector p0 (dj ) = (1; : : : ; 1) for all values of dj and uses
d
G = 65 nodes across the state space. It is evident from this …gure that, for each
maturity n, the solution converges very quickly once i > n.
The middle panel shows convergence in the number of gridpoints by displaying the computed yield curve (after I = 50 iterations), again using rt = 0:058
for illustration. Yield curves are shown for G = 5, 9, 17, 33, and 65, in each
case spaced equally across possible values of p1t . The space is assumed bounded
between 0 and 0.80, so this partitioning corresponds roughly to increments of
between 38 basis points and 5 percentage points. While 5 nodes is clearly too
few to achieve convergence, the solutions using 17 or more nodes are indistinguishable from each other.
For brevity, these results were shown for the average value of the short
rate. Similar convergence results obtain for other points in the state space,
although solutions will not be accurate near the bounds if the underlying state
process itself is not actually bounded. For example, in the above case, we
would not expect the procedure to generate correct solutions near p1t = 0:80
(corresponding to a risk-free return of 25%). However, so long as the bound on
the state space is imposed far enough away from the values of the states that
are actually realized in practice, this limitation has a negligible e¤ect on the
results. The bottom panel of the …gure illustrates this claim by comparing the
yield curve computed above with the yield curve computed when the grid for p1t
is extended over the entire range (0; 1), again at the average value of the short
rate. (The latter computation used G = 100.) The two curves are virtually
identical, di¤ering by less than 1 basis point across maturities.
As noted in the text, the model with Epstein-Zin preferences involves the
additional variable of consumption growth, and, under a further assumption of
trend-stationarity, this can be solved for within the context of the model itself,
via an additional step in the algorithm. In particular, suppose consumption
has a deterministic long-run trend with growth rate g, and denote the time-t
level of consumption relative to this trend as e(st ). Then, we can rewrite the
c
…rst term in equation (10) as
(ct+1 =ct )

1=

=

1

i
=

i

(st+1 )
i (s )
t

where i
g 1=
and i (st ) e(st )
c
.
At iteration i of the algorithm, de…ne Ai as the G G matrix whose h; j th
element is
!1
0
x (dj ) qi (dh )
i
d
ah;j = b(dh jdj )
(dh )
0
x (dj ) pi (dj )
d
26

From equation (10), i is an eigenvalue of Ai , with corresponding eigenveci
i
(d1 ) :::
(dG ) . But, since Ai is necessarily a non-negative
tor i =
square matrix, the Perron-Frobenius Theorem implies that the eigenvector corresponding to the (unique) maximum eigenvalue is the only eigenvector that
has all real and positive elements. Consequently, it is the only viable solution. Thus, for given pricing and payo¤ functions pi (dj ) and qi (dh ), it is
d
d
straightforward to …nd i and i . With these in hand, equation (A1) for the
Epstein-Zin model is
pi+1
dn

G
X
1
(dj ) =
b(dg jdj ) i
g=1

i

0

x (dj ) qi (dh )
d
0
x (dj ) pi (dj )
d

(dt+1 )
i (d )
t

!1

(dh )

pi+1 1 (dg )
dn

Note that we need not solve explicitly for consumption, nor the parameters
and , in order to compute asset prices or …nd the distribution of M within this
model.

B. Preferred Habitat
In the models considered in the text, the supply of debt that must be held
by investors is exogenous. More generally, this distribution itself could depend
on bond prices, as in the model of Vayanos and Vila (2009). This appendix
brie‡ shows how the class of models discussed in the text can incorporate this
y
type of endogeneity.
Apart from its a¢ ne structure, the essential feature of Vayanos-Vila that
di¤ers from the models considered in the text is that investors (“arbitraguers”
in their terminology) face an elastic supply curve at each maturity. This supply
curve is assumed to arise from the presence of preferred-habitat agents, each of
whom deals in debt of only one maturity. Speci…cally, for each bond, preferredhabitat agents demand a (market-value) quantity that is a function h of maturity
and price:
pnt nt = h (n; pnt )
(B1)
where nt is the par value demanded. The par value left in the hands of the
arbitrageurs is simply xnt
nt . The models in the text e¤ectively assumed
h(n; pnt ) = 0 for all n. Equation (B1) can be substituted directly into (4) to
give the return on the arbitrageurs’portfolio:

Rt =

N
P

xnt (pnt )pn 1t+1
n=1
N
P
xnt (pnt )pnt

n=1

where

xnt (pnt )

xnt

27

h (n; pnt )
pnt

(B2)

For any given assumption of the form of h(:), and an assumption on the pricing
kernel, the algorithm can solve for asset prices under this modi…ed de…nition of
the return on wealth.
In the Vayanos-Vila model, bonds are in zero net supply— a positive position
by preferred-habitat agents implies a negative position for arbitrageurs, and vice
versa. In the models discussed here, Treasury bonds are assumed to be issued by
the government and, from the point of view of investors, constitute net wealth.
That is, they are in positive net supply at all maturities. (One rationale for
this assumption is that only the government, not private investors, can create
perfectly default-free debt.) We can further impose that both arbitrageurs and
preferred-habitat agents can only hold positive quantities of bonds by modifying
(B2) as follows:
xnt (pnt )

min xnt ; max 0; xnt

28

h (n; pnt )
pnt

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31

Table 1. Investor Assumptions for Portfolio-Balance
Models
Model
1
2
3
4
5

Preferences
Quadratic - Nominal returns
Quadratic - Real returns
EZW
Quadratic - Real returns
EZW

Treas. weight (v)
1
1
1
0.1
0.1

Preference calibration
a = 9:5
a = 7:3
= 7:3
a = 74:9
= 7:6

Notes: The table shows the salient features of the …ve three-factor portfolio-balance
models described in the text. Models are characterized by (1) the form of investors
preferences (quadratic over real or nominal returns or Epstein-Zin-Weil), (2) the weight
v that Treasury bonds receive in their overall portfolio, and (3) a single preference
parameter, which is calibrated to match the average 1- to 15-year yield curve slope in
the data.

Table 2. Calibration of Economic Dynamics
t

rt

zt

0.26
0.72
0
0.006
0.017

0
0
0.96
0.15
0.14

coef
t 1

rt 1
zt 1
intercept
stdev err

0.83
0.04
0
0.004
0.013

Notes: The table shows the parameters of the VAR dynamics for the three state
variables– ation, the short-term interest rate, and the average Treasury duration
in‡
()— used in the …ve portfolio-balance models described in the text. Parameter values
are estimated based on data over the 1971-2008 period. The dynamics further assume
that the short rate is truncated at zero.

32

Table 3. Yield Curve Statistics
Maturity
5y
10y
15y

Data
7.1
7.5
7.7

Model 1
7.0
7.4
7.7

Maturity
5y
10y
15y

Data
2.6
2.4
1.5

Model 1
2.2
1.7
1.3

Maturity
5y
10y
15y

Data
0.09
0.07
0.07

Means

Model 2
6.9
7.4
7.7

Model 3
8.5
8.0
7.7

Model 4
7.0
7.4
7.7

Model 5
8.5
8.1
7.7

Model 4
2.3
1.7
1.3

Model 5
3.4
2.5
1.8

Correlation with Treasury duration
Model 1 Model 2 Model 3 Model 4
0.09
0.08
0.08
0.08
0.13
0.12
0.10
0.12
0.17
0.15
0.12
0.15

Model 5
0.05
0.05
0.05

Standard deviations
Model 2
2.3
1.7
1.3

Model 3
3.3
2.3
1.6

Notes: The table shows certain summary statistics for nominal yields at various
maturities generated by the …ve three-factor portfolio-balance models described in the
text and compares them to the values observed in the data between 1971 and 2008.
The model results are calculated by feeding in the actual time series for the three
state variables (core PCE in‡
ation, the short-term interest rate, and average Treasury
duration outstanding) over this period.

33

Table 4. Excess Return Statistics
Maturity
5y
10y
15y

Data
2.2
3.8
4.9

Model 1
1.9
3.0
3.6

Maturity
5y
10y
15y

Data
6.2
12.1
17.7

Model 1
5.6
9.3
11.4

Maturity
5y
10y
15y

Data
0.29
0.31
0.30

Means

Model 2
1.8
3.0
3.7

Model 3
3.1
2.6
2.4

Model 4
1.9
3.1
3.8

Model 5
3.2
2.8
2.7

Model 4
5.5
9.3
11.4

Model 5
8.5
13.6
15.4

Correlation with Treasury duration
Model 1 Model 2 Model 3 Model 4
0.25
0.24
0.09
0.24
0.28
0.27
0.18
0.27
0.29
0.28
0.19
0.28

Model 5
0.05
0.15
0.16

Standard deviations
Model 2
5.5
9.3
11.3

Model 3
8.4
13.2
14.9

Notes: The table shows certain summary statistics for returns on bonds of various
maturities, in excess of the one-year return, generated by the …ve three-factor portfoliobalance models described in the text and compares them to the values observed in the
data between 1971 and 2008. The model results are calculated by feeding in the actual
time series for the three state variables (core PCE in‡
ation, the short-term interest
rate, and average Treasury duration outstanding) over this period.

34

Table 5. Term Premium Experiments

Experiment
Assumptions:
State values
"Forward guidance"
Preference param.
Other asset vol.
E¤ect on term premium (pp)
Model 1
Model 2
Model 3
Model 4
Model 5

1970s-80s
Duration rise
z0 = 1:5
z1 = 3:5

Asset purchases
Baseline
Extreme values
z0 = 2:7
z0 = 3:05
z1 = 2:0
z1 = 1:65

= 0:058
r = 0:091
None
baseline
baseline

= 0:014
r = 0:0025
2 years
baseline
baseline

= 0:014
r = 0:0025
1 year
double baseline
double baseilne

0.56
0.43
0.52
0.44
0.06

-0.13
-0.10
-0.11
-0.10
-0.01

-0.37
-0.48
-0.33
-0.49
-0.05

Notes: The table shows the outcomes of experiments in which the duration of
outstanding Treasury debt is shifted within each of the …ve three-factor portfoliobalance described in the text. The top rows show the amount by which duration is
shifted relative to its starting value (z0 ), the middle rows show the assumed initial
values of the other state variables and certain other modeling assumptions, and the
bottom rows show the response of the ten-year term premium in the period when
the shift occurs. In all cases the change in duration is assumed to be unanticipated
and to subsequently follow the dynamic process described in the text. The …rst
experiment is meant to replicate the increase in outstanding duration that occurred
roughly between 1976 and 1988. The second experiment is meant to replicate the
duration extracted from the market by the Federal Reserve’ asset purchases during
s
the period 2008 - 2012. For the latter, I consider both a baseline case and a case that
uses more-extreme assumptions to maximize term-premium e¤ects.

35

Table 6. Summary of Federal Reserve Asset Purchase
Programs
N e t q u a n tity o f

LSAP I

N e t q u a n tity

A ssu m ed ch a n g e

E m p iric a l

of M BS

of reserves

in ave ra g e

e¤ect on ten-

p u rch a sed

p u rch a sed

created

Tre a su ry d u ra tio n

year term

($ b il)

D ates

N e t q u a n tity

Tre a su rie s

($ b il)

($ b il)

o u tsta n d in g

p re m iu m

5 /0 8-5 /1 0

$300

$941

$ 1 ,3 1 8

0

4 0-10 0 b p

M B S re inve stm e nt

8/ 10 -10 /1 1

$285

$0

$285

0

1 0 -2 5 b p

LSA P II

1 1 / 1 0 -7 / 1 1

$600

$0

$600

0

15 -55 b p

1 0 / 1 1 -1 2 / 1 2

$0

$0

$0

-0 .5

30 -65 b p

$ 1 ,1 8 5

$941

$ 2 ,2 0 3

-0 .5

9 5-2 45 b p

MEP
To ta l

N o te s: A ll q u a ntitie s a re n e t o f re d e m p tio n s a n d p rin c ip a l p ay m e nts th ro u g h D e c e m b e r 2 0 1 2 . T h e a ssu m e d
d u ra tio n ch a n g e is o n ly th a t in Tre a su ry se c u ritie s (i.e ., e x c lu d in g a g e n c y d e b t, M B S , a n d th e m o n e ta ry b a se ). T h e
e m p iric a l e ¤ e c t o f e a ch p ro g ra m o n th e te n -ye a r n o m in a l Tre a su ry y ie ld is ta ke n fro m th e lite ra tu re m e ntio n e d in
S e c tio n 5 o f th e te x t.

36

Figure 1. Characteristics of U.S. government liabilities in public hands
A. Percentage with duration < 5 years

2011

2009

2007

2005

2003

2001

1999

1997

1989
1989

1995

1987
1987

1993

1985
1985

1993

1983
1983

1991

1981
1981

1991

1979
1979

1977

1975

1973

1971

100
90
80
70
60
50
40
30
20
10
0

B. Average duration

2011

2009

2007

2005

2003

2001

1999

1997

1995

1977

1975

1973

1971

4
3.5
3
2.5
2
1.5
1
0.5
0

Notes: Includes coupon notes, bonds, and bills issued by the U.S. Treasury, less the amount held in the Federal
Reserve’s SOMA portfolio, plus reserves and currency in circulation, which are assumed to have a duration of zero.
Sources: CRSP, Federal Reserve.

37

Figure 2. Risk prices in the one-factor model
λ
0.40
0.35
0.30

Unrestricted

0.25
0.20
0.15

With ZLB

0.10
0.05
0.00
0

1

2 Duration 3
(years)

4

5

Notes: The figure shows the price of short-rate risk (expected excess returns divided by their standard deviation) in
the one-factor model for various values of aggregate duration (z). The black line corresponds to a model in which the
short rate follows a simple AR(1) process. The blue line corresponds to a model in which that process is truncated at
zero.

38

Figure 3. A duration shift in the one-factor model
Duration distribution of outstanding debt
45%
Ave. duration = 2.7
Ave. duration = 2.0

40%
35%
30%
25%
20%
15%
10%
5%
0%

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

Yield curve
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
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

Return Volatility (One-Year)
0.25
0.20
0.15
0.10
0.05
0.00
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
Notes: The figure shows the effect of moving from an average duration of 2.7 to 2.0 years in the one-factor model,
evaluated at a short rate of 5.8% (the sample average).

39

Figure 4. Effect of the maturity-distribution shape in the one-factor model
Duration distribution of outstanding debt
100%
90%

Exponential

80%

Degenerate

70%
60%
50%
40%
30%
20%
10%
0%
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

Yield curve
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
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

Notes: The top panel shows two possible distributions for the maturity of outstanding Treasury debt, one
exponential and one nearly degenerate, both with means of 2.7 years. The bottom panel shows the corresponding yield
curves generated by the one-factor model when the short rate is at its average value of 5.8%. In both cases, the riskaversion coefficient is calibrated to match the average value of the 15-year yield.

40

Figure 5. Short-rate processes

Time t+1
short rate (%)
10

ZLB 90%
quant.

8
6
4

AR(1) w/ ZLB

2
0

ZLB 10%
quant.

AR(1)
0

2

4
6
Time t short rate (%)

8

10

Notes: The figure shows the conditional mean of rt+1, as a function of rt near the value of zero under both a
standard AR(1) process (light-blue solid line) and the modified process enforcing the zero lower bound, given by
equation (18) (dark-blue solid line). 10% and 90% quantiles of the modified process are shown by the dashed lines.
Frequency is annual.

41

Figure 6. A duration shift in the one-factor model at the zero lower bound
Yield curve
0.06

0.05

0.04

0.03

0.02

w/forward guidance

0.01

0
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

Return Volatility (One-Year)
0.07
0.06
0.05
0.04
0.03
0.02

forward guidance implies
σnt = 0 ∀ n

0.01
0
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

Notes: The figure shows the effect of moving from an average duration of 2.7 years (blue) to 2.0 years (red) in the
one-factor model, evaluated at a short rate of 0%. The solid lines represent the case in which the short rate is expected
to evolve according to its usual truncated-AR(1) process. The dashed lines (“forward guidance”) represent the case in
which the short rate is anticipated to remain at 0% with certainty for one year and only then to follow its usual
truncated-AR(1) process.

42

Figure 7. Impulse-response functions for the three-factor models
Model 2

Model 1
0.04%

0.04%

0.03%

0.03%

0.02%

0.02%

0.01%

0.01%
0.00%

0.00%
1 3 5
7 9 11
13 15 17
Maturity

1 3 5
7 9 11
13 15 17

16

12
8
19 21 23
25 27 29 4Years after shock
0

Maturity

Model 3

16
12
8
19 21 23
25 27 29 4Years after shock
0

Model 4

0.04%

0.04%

0.03%

0.03%

0.02%

0.02%

0.01%

0.01%
0.00%

0.00%

1 3 5
7 9 11
13 15

1 3 5
7 9 11
13 15

16
12
8
17 19 21
23 25 27
4Years after shock
290
Maturity

16
12
8
17 19 21
23 25 27
4Years after shock
290
Maturity

Model 5
0.04%
0.03%
0.02%
0.01%
0.00%
1 3 5
7 9 11
13 15

16
12
8
17 19 21
23 25 27
4
290 Years after shock
Maturity

Notes: The figures show the response of the yield curve, evaluated initially at the unconditional means of the three state variables,
to a one-standard-deviation positive shock to average Treasury duration (zt) in each of the five portfolio-balance models described in
the text.
43

Figure 8. Effect of duration persistence on the response of yields
0.04%

0.03%

Persistence = 0.95

0.02%

0.01%

Persistence = 0

0.00%
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
Maturity
Notes: The figure shows the response of the yield curve in Model 1 to one-standard-deviation positive shock to
average outstanding Treasury duration (zt), in the period when the shock occurs, under two assumptions for the
persistence of the zt process. The blue line shows the response when the autoregressive coefficient for zt is equal to 0.95
(same as in Figure 7), and the green line shows the response when this autoregressive coefficeint is zero. The functions
are evaluated at the unconditional means of the state variables.

44

Figure A1. Solution convergence in the one-factor model
Yield
0.09
0.08

30 y
15 y
10 y
5y

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Iteration

Yield
0.09

G = 65
G = 33
G = 17

0.08
0.07

G=9
G=5

0.06
0.05
0.04
0.03
0.02
0.01
0
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
Maturity
Yield
0.09

inf[p1] = 0

0.08
0.07

inf[p1] = 0.80

0.06
0.05
0.04
0.03
0.02
0.01
0
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
Maturity

Notes: For the one-factor portfolio-balance model, the graphs show how the solution algorithm converges in the
number of iterations (top), number of nodes in the grid (middle), and truncation point of the state space (bottom). All
calculations are illustrated at the average value of the short rate.
45

Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
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5