Asset Price Dynamics in Partially Segmented Markets ∗
Robin Greenwood Samuel G. Hanson Gordon Y. Liao
Harvard University and NBER Harvard University and NBER Cambridge Square Capital
September 2017
Abstract
We develop a model in which capital moves quickly within an asset class, but slowly between
asset classes. While most investors specialize in a single asset class, a handful of generalists can
gradually reallocate capital across markets. Upon the arrival of a large supply shock, prices of risk
in the directly impacted asset class become disconnected from those in others. Over the long run,
capital flows between markets and prices of risk become more closely aligned. While prices in the
directly impacted market initially overreact to the supply shock, we show that prices in related asset
classes underreact under plausible conditions. We use the model to assess event-study evidence on
the impact of recent large-scale asset purchases by central banks.
∗We thank Daniel Bergstresser, Darrell Duffi e, Yueran Ma, Andrei Shleifer, Jeremy Stein, Adi Sunderam, and seminar
participants at Berkeley Haas, Brandeis, MIT Sloan, NYU Stern, University of Minnesota Carlson, University of North
Carolina Kenan-Flager, and University of Texas McCombs for useful feedback. We thank David Biery for research
assistance. Greenwood and Hanson gratefully acknowledge funding from the Division of Research at Harvard Business
School.
1 Introduction
How do large supply shocks in one financial market affect asset prices in other markets? For example,
in recent years the Federal Reserve and other major central banks have purchased large quantities of
long-term government bonds in an attempt to drive down long-term borrowing rates throughout the
economy. In response, there has been an active debate amongst academics and policymakers about the
extent too which these “quantitative easing”policies have impacted the prices of financial assets– such
as corporate bonds and equities– outside of the market for government bonds.
If all investors can frictionlessly trade each different asset class, then financial markets are fully
integrated and all investors will price exposure to common risk factors in the same way. Furthermore, a
shock to the supply of one asset class will lead all investors to identically adjust the way they price risk.
However, if markets are more segmented– i.e. if some assets are held by specialists who are unable to
trade across different markets, then in the wake of a large supply shock the way that specialists in one
market price risk may become disconnected from the way it is priced by those in other markets. For
example, interest rate risk may be priced differently by specialists in the government and corporate
bond markets following a large shock to the supply of government bonds.
Over the longer run, however, the forces of arbitrage should ensure that capital will flow from mar-
kets where prices of risk are low to markets where prices of risk are high, narrowing these differences.
But the process of market integration can be slow, because generalist investors with the flexibility to
trade across asset classes can do so only gradually. For example, investment committees at pension
funds and endowments– who have this flexibility– typically only reallocate capital annually or bian-
nually. And, cross-market arbitrage is risky because risks cannot be easily unbundled from assets, so
generalist investors may not aggressively trade across markets, even when they can. As a result, it
may take time for a large supply shock in one market to fully propagate into other markets.
In this paper, we explore how the pricing of two risky asset classes, denoted assets A and B,
responds to a large shock to the supply of one asset class, say asset A. In our model, risk-averse
investors absorb shocks to the supply of these two assets, which are exposed to a common risk factor.
Since the two assets are close substitutes, in the absence of asset-pricing frictions, their prices would be
tightly linked by cross-market arbitrage. Our key contribution is to show how a shock to the supply of
asset A is reflected over time in the price of asset B in a setting where there are two key asset pricing
frictions. First, the markets for the two asset classes are partially segmented as in Gromb and Vayanos
(2002). Specifically, a first group of investors (“A-specialists”) can only trade asset A, a second group
(“B-specialists”) can only trade asset B, and only a final group (“generalists”) can trade both asset
A and asset B. Second, the generalist investors can only rebalance their portfolios gradually over
1
time as in Duffi e (2010), so capital moves slowly between the two markets. In combination, these
two frictions imply a rich and realistic set of asset price dynamics following a supply shock that only
directly impacts one market.
For concreteness, we focus our main analysis on a setting where asset A represents long-term
default-free government bonds and asset B represents long-term defaultable corporate bonds. We
have chosen this setting since we believe that our model provides a useful framework for understanding
how quantitative easing policies might impact asset classes not directly purchased by central banks.
However, we show that the insights from our model are quite general and emerge so long as the two
assets are exposed to a common risk factor. For example, we show that our model can be easily adapted
to understand how a shock to the demand for U.S. stocks would impact the prices of European stocks.
In the setting we have just described, what happens when there is an unanticipated supply shock
in one market? Suppose, for example, that the Federal Reserve announces that it will sell a large
portfolio of long-term U.S. Treasury bonds, permanently expanding the amount of interest rate risk
that investors must bear in equilibrium. Treasury market specialists react immediately to this shock,
absorbing the increased supply into their inventories. The risk premium on long-term Treasury bonds
will rise, lifting their yields. However, Treasury yields will overreact– the short-run price impact will
exceed the long-run impact– because the amount of capital that can initially accommodate the shock
is limited to specialists and a handful generalists. Over the long-run, generalist investors will allocate
additional capital to the Treasury market, reducing the price impact of the supply shock at longer
horizons. Price dynamics of this sort are similar to those described in Duffi e (2010), who explores the
pricing implications of slow-moving capital for a single asset.
Our key contribution is to show how prices evolve in the second related market that is not directly
impacted by the supply shock. Specifically, consider how corporate bond prices will react to a large
shock to the supply of long-term Treasuries. Corporate bonds are indirectly affected by this shock
because generalist investors respond by increasing their holdings of long-term Treasuries and reducing
their holdings of long-term corporate bonds. These cross-market capital flows drive down the prices
of corporate bonds and push up corporate bond yields. In this way, the trading of generalist investors
transmits the supply shock across markets, serving to increase market integration.
Under plausible parameter values, we show that corporate bond yields will underreact to this shock
to the supply of government bonds: the short-run price impact is less than the long-run impact. This
stands in contrast to the initial overreaction of Treasury yields. The simultaneous overreaction and
underreaction in different markets is driven by the fact that generalists only reallocate capital slowly.
As a result, it takes time for financial markets to fully digest large supply shocks.
The price dynamics in our model depend critically on the fractions of specialists in each mar-
2
ket, the number of time periods it takes generalists to rebalance their portfolios, and the degree of
substitutability between the two assets. The fraction of specialists and generalist investors play an
especially important role. When there are a small number of slow-moving generalists, the Treasury
market overreacts while the corporate market underreacts to the shock to Treasury supply. However,
if there are many slow-moving generalists, both markets overreact to the Treasury supply shock.
Importantly, we show that both partial segmentation and slow-moving capital between markets
are needed to generate short-run underreaction in the indirectly affected market. Specifically, if there
is only slow-moving capital– i.e., if all investors can trade both assets, but some investors can only
rebalance their portfolios gradually– then we always obtain short-run overreaction in both markets.1
This is because, in this case, prices in both markets reflect a common set of factor risk prices charged by
fast-moving generalist investors. To be sure, these prices of risk will initially overreact to a large supply
shock because some investors are slow-moving, but the overreaction is the same in both markets.
A natural application of our model– and one that motivates our analysis– is the assessment of
large-scale asset purchases by central banks. A key question about these “quantitative easing”policies
is whether they impacted prices outside the market in which the central bank is directly transacting.
The favored methodology for answering this question– and one used in dozens of recent papers–
has been a short-run event study approach that examines price changes on days when purchases are
announced. Some studies have concluded that the impact is most pronounced in the directly impacted
market, with only modest spillovers to other related markets (Krishnamurthy and Vissing-Jorgensen
[2011, 2013] and Woodford [2012]). Others have suggested that, at longer horizons, the spillovers to
other markets are more significant (Mamaysky [2014]).
The short-run event study methodology was originally developed in the 1960s to assess whether
prices rapidly impound news about fundamentals (Fama, Fisher, Jensen, and Roll [1969]). However,
in recent years, event studies have increasingly been used to measure changes in risk premia. Our
model crystallizes the limitations of this approach, particularly when one is interested in assessing
price impact across partially segmented markets. Specifically, even if prices immediately impound
fundamental news– as they do in our model– it may take time for risk premia to normalize across
markets following a large supply shock.
Our model is closely related to three strands of research in financial economics. The idea that front-
line arbitrageurs in financial markets are highly specialized traces back to Merton (1987) and Grossman
and Miller (1988), and is a central tenet of the theory of limited arbitrage (De Long et al [1990], Shleifer
and Vishny [1997], and Gromb and Vayanos [2002]). A small literature in finance describes asset prices
and returns in partially segmented markets (Stapleton and Subrahmanyam [1977], Errunza and Losq
1This corresponds to a multi-asset version of Duffi e’s (2010) model which is developed in Bogousslavsky (2016).
3
[1985], Merton [1987]). More recently, a number of researchers have demonstrated downward-sloping
demand curves for individual financial asset classes, which would be puzzling if markets were fully
integrated (Gabaix, Krishnamurthy, and Vigneron [2007], Gârleanu, Pedersen, and Poteshman [2009],
Greenwood and Vayanos [2014], and Hanson [2014]). These researchers often motivate their analysis
by positing an extreme form of market segmentation, in which a different pricing model is used to price
the securities in each distinct asset class. In our model, trading by slow-moving generalists links these
market-specific pricing models together over time, offering a middle ground between models positing
extreme segmentation and traditional models featuring perfect integration.
Second, our paper is related to research on “slow-moving capital,”which is the idea that capital
does not flow as quickly towards attractive investment opportunities as textbook theories might suggest
(Mitchell, Pedersen, Pulvino [2007], Duffi e [2010], Acharya, Shin, and Yorulmazer [2013]). Here, our
model draws most heavily from Duffi e (2010), who studies the implications of slow-moving capital for
price dynamics in a single asset market. Compared to his paper, our key contribution is to analyze
the price impact of supply shocks across multiple partially-segmented asset markets as well as the
resulting patterns of cross-market arbitrage.2
Third, our model is a overlapping-generations, rational-expectations model in which mean-variance
investors with finite investment horizons trade an infinitely-lived asset that is subject to supply shocks.3
From a technical standpoint, we show how to incorporate both partial market segmentation and slow-
moving capital into this workhorse class of asset pricing models.
2 Model
We develop the model in three steps. First, we develop a stylized, linear model for pricing long-term
default-free bonds and long-term defaultable bonds. Second, we introduce our two key asset pricing
frictions, namely, partial market segmentation and slow-moving capital across markets. Finally, we
solve for the rational expectations equilibrium in this setting. While we focus on two fixed-income
asset classes in our main analysis, the insights from our model are completely general and emerge so
long as the two assets are exposed to a common risk factor. Indeed, in the Internet Appendix, we
develop a model for pricing two risky equity claims that shares the same formal structure and delivers
the same insights as the model we develop below.
2Duffi e and Strulovici (2012) present a model of the movement of capital across two partially segmented markets, but
their focus is on the endogenous speed of capital mobility, which we take as exogenous.
3Thus, many of the technical issues that we must address are foreshadowed in this literature, including De Long,
Shleifer, Summers, and Waldmann (1990), Spiegel (1998), Bacchetta and van Wincoop (2010), Watanabe (2008), Banerjee
(2011), Greenwood and Vayanos (2014), and Albagli (2015).
4
2.1 Long-term bonds
We consider a model with two perpetual risky bonds, A and B. Both A and B bonds are exposed to
interest rate risk. The B bonds are also exposed to default risk, making them a partial substitute for
the A bonds.
A-bonds The A perpetuities are default-free and pay a coupon of CA each period, so the gross
return on A bonds from time t to t+ 1 is
PA,t+1 + CA
1 +RA,t+1 = , (1)
PA,t
where PA,t in the price of A bonds at time t.
To maintain tractability, we must assume that asset prices (or yields) and expected returns are
linear functions of a vector of state variables. Although stylized, this linearity assumption is common
in rational expectations models like ours where mean-variance investors absorb stochastic supply
shocks. In order to apply our model to fixed income assets, we (i) substitute log returns for simple
returns throughout and (ii) use a Campbell-Shiller (1988) linearization of log returns. This modelling
approach is also used in Hanson (2014) and Hanson and Stein (2015). We view (i) and (ii) as linearity-
generating modelling devices that do not impact the qualitative conclusions that we draw from the
model. Together these modelling devices allow us to develop a discrete-time analog to the continuous-
time term structure models in Vayanos and Vila (2009) and Greenwood and Vayanos (2014).
Consider the Campbell-Shiller (1988) log-linear approximation to the return on A perpetuities.
Specifically, defining θA ≡ 1/ (1 + CA) < 1, the one-period log return on A bonds is approximately
︷ D︸︸A ︷ ︷DA︸︸−1︷
1 θA
rA,t+1 ≡ ln (1 +RA,t+1) ≈ − yA,t − − yA,t+1, (2)1 θA 1 θA
where yA,t is the log yield-to-maturity on A bonds at time t and
1 CA + 1
DA = − = (3)1 θA CA
is the Macaulay duration when the bonds are trading at par.4 Thus, the log excess return on A bonds
4This approximation for default-free coupon-bearing bonds appears in Chapter 10 of Campbell, Lo, and MacKinlay
(1997) and is an approximate generalization of the fact that the log-return on an n-period zero-coupon bond from t to
t+ 1 is exactly rn,t+1 = nyn,t − (n− 1) yn−1,t+1 where, for instance, yn,t is the log yield on n-period zero-coupon bonds
at time t. We review the derivation in the Internet Appendix. Our approximation for defaultable bonds in equation (6)
then follows trivially since we assume that default losses are a random fraction of market value.
5
over the short-term interest rate from time t to t+ 1, denoted rt, is
rxA,t+1 ≈
1 θA
− yA,t −1 θA 1−
yA,t+1 − rt. (4)
θA
B-bonds The B bonds are also exposed to default risk. Specifically, consider a homogenous portfolio
of perpetual, defaultable bonds each of which promises to pay a coupon of CB each period. Suppose
that a random fraction ht+1 of the B bonds default at t + 1 and are worth (1 − Lt+1) (PL,t+1 + C)
where 0 ≤ Lt+1 < 1 is the loss-given-default as a fraction of market value. The remaining fraction
(1−ht+1) of the B bonds do not default and are worth (PB,t+1 + C). Thus, the return on the portfolio
of B bonds is
(1− Zt+1) (PB,t+1 + CB)
1 +RB,t+1 = , (5)
PB,t
where Zt+1 = ht+1Lt+1, satisfying 0 ≤ Zt+1 < 1, is the random portfolio default realization at time
t+1. This stylized formulation of default risk follows Duffi e and Singleton’s (1999) “recovery of market
value”assumption which has become standard in the credit risk literature.
As a above, we use a Campbell-Shiller (1988) log-linear approximation to the return on these
perpetual bonds. Specifically, the log excess return on B bonds from time t to t+ 1 is
≈ 1 − θBrxB,t+1 − yB,t yB,t+1 − zt+1 − rt, (6)1 θB 1− θB
where θB ≡ 1/ (1 + CB) and zt ≡ − ln (1− Zt) is the log default loss at time t. Relative to equation
(4), the additional zt+1 term in equation (6) reflects the default realization that is specific to B bonds.
Risk factors There are three different sources of risk in our model: interest rate risk, default risk,
and supply risk. First, both the A and B bonds are exposed to interest rate risk. There is an exogenous
short-term interest rate that evolves randomly, so both long-term bonds will suffer a capital loss if
short-term rates rise unexpectedly. Second, the B bonds are exposed to default risk: the future period-
by-period default realization is unknown and evolves randomly. Finally, both bonds are exposed to
supply risk: there are random supply shocks which impact the prices and yields on long-term bonds,
holding fixed the expected future path of short-term interest rates and expected future defaults. Thus,
using Campbell’s (1991) terminology, interest rate risk and default risk are forms of fundamental “cash
flow”risk, whereas supply risk is a form of “discount rate”risk.
Concretely, we make the following assumptions:
• Short-term interest rates: The log short-term riskless rate available to investors between
time t and t + 1, denoted rt, is known at time t. The short-term rate rt follows an exogenous
6
AR(1) process
rt+1 = r + ρr (rt − r) + εr,t+1, (7)
where V art [ε 2r,t+1] = σ . One can think of the short-term rate as being determined outsider
the model either by monetary policy or by a stochastic short-term storage technology that is
available in perfectly elastic supply.
• Default losses: The default process zt also follows an AR(1) process
zt+1 = z + ρz (zt − z) + εz,t+1, (8)
where V art [ε 2z,t+1] = σz. The variance of zt+1 influences the degree of substitutability between
the A and B bonds.
• Supply: Both bonds are available in an exogenous, time-varying supply. The two bonds are
subject to different supply shocks which also limits their substitutability for generalists investors.
Specifically, the supply that investors must hold of A bonds evolves according to
sA,t+1 = sA + ρs (sA,t − sA) + εsA,t+1, (9)A
where V art [ε 2sA,t+1] = σs . Similarly, the supply that investors must hold of B bonds evolves asA
sB,t+1 = sB + ρs (sB,t − sB) + εB sB ,t+1, (10)
where V ar 2t [εsB ,t+1] = σs .B
Note that we have made no assumptions about the correlation structure among the four underlying
shocks εr,t+1, εz,t+1, εsA,t+1, and εsB ,t+1. Indeed, our model can be solved for any arbitrary correlation
structure among these shocks. However, in our numerical illustrations in Section 3, for simplicity we
will assume that the four underlying shocks are mutually orthogonal. Because of the clarity that it
affords, this assumption is common in asset pricing models with noisy asset supply.
2.2 Market structure
There are three types of investors– A-specialists, B-specialists, and generalists– all with identical risk
tolerance τ . Investors are distinguished by their ability to transact in different assets and the frequency
with which they can rebalance their portfolios.5 Because only a subset of generalist investors can trade
5Allowing for heterogeneity in risk aversion does not change the asset-pricing implications of our model. Indeed, an
economy with arbitrary heterogeneity in risk aversion is isomorphic to the economy we describe below. With hetero-
7
both the A and B bonds, the markets are partially segmented. Furthermore, capital moves slowly
between the two markets because the generalists can only rebalance their portfolios gradually.
A-specialists Fast-moving A-specialists are free to adjust their holdings of A bonds and the riskless
short-term asset each period; however, A-specialists cannot hold B bonds. A-specialists are present
in mass qA and we denote their demand for A bonds by bA,t. Fast-moving A-specialists have mean-
variance preferences over 1-period portfolio log returns. Thus, their demand for A bonds is
Et [rxA,t+1]
bA,t = τ . (11)
V art [rxA,t+1]
B-specialists Fast-moving B-specialists can adjust their holdings of B bonds and the riskless short-
term asset each period, but cannot hold A bonds. B-specialists are present in mass qB and their
demand for B bonds is denoted bB,t. B-specialists also have mean-variance preferences over 1-period
portfolio log returns, so their demand for B bonds is
Et [rxB,t+1]
bB,t = τ . (12)
V art [rxB,t+1]
Generalists Slow-moving generalists can adjust their holdings of both A and B bonds, as well as
the riskless short-term asset, but can do so only every k periods. Generalists are present in mass
1 − qA − qB. Fraction 1/k of these generalists investors are active each period and can reallocate
their portfolios between the A and B bonds. However, they must then maintain this same portfolio
allocation for the next k periods. First introduced in Duffi e (2010), this is a simple way to model
the frictions that limit the speed of capital flows across markets and is reminiscent of New Keynesian
models of nominal price rigidity in which firms only adjust their nominal prices every k periods (e.g.,
Fischer [1977] and Taylor [1979]).
Slow-moving generalist investors have mean-∑variance preferences over their k∑-period cumulative
portfolio excess return. Defining rx k kA,t→t+k ≡ i=1 rxA,t+i and rxB,t→t+k ≡ i=1 rxB,t+i as the
cumulative k-period returns from t to t+ k on A and B bonds, the k-period portfolio excess return of
generalists who are active at t is
rxdt,t→t+k = dA,t × rxA,t→t+k + dB,t × rxB,t→t+k. (13)
Thus, generalist investors who are active at time t choose their holdings of A and B bonds, denoted
geneous risk aversion, one should interpret τ as the aggregate risk tolerance of all investors in the economy, qA as the
fraction of aggregate risk tolerance attributable to A-specialists, qB as the fraction attributable to B-specialists, and
1− qA − qB as the fraction attributable to generalists.
8
dA,t and dB,t, to solve
{ }
max Et [rxdt,t→t+k]− (2τ)
−1 (V art [rxdt,t→t+k]) . (14)
dA,t,dB,t
This implies that
   −1  dA,t V art [rxA,t→t+k] Covt [rxA,t→t+k, rxB,t→t+k] Et [rxA,t→t+k]= τ (15)
dB,t Covt [rxA,t→t+k, rxB,t→t+k] V art [rxB,t→t+k] Et [rxB,t→t+k]
Et[rxA,t→t+k] − (k) Et[rxβ B,t→t+k]τ
= V art[rxA,t→t+k] B|A V art[rxB,t→t+k] ,
1−R2(k) Et[rxB,t→t+k] − (k) Et[rxA,t→t+k]AB V art[rx βB,t→t+k] A|B V art[rxA,t→t+k]
(k)
where, for example, (k)βA|B is the coeffi cient from a linear regression of rx
2
A,t→t+k on rxB,t→t+k and RAB
is the goodness of fit from this regression.6 ,7
Equation (15) says that, all else equal, generalist investors allocate more capital to market B
when the expected return on B bonds is higher. Furthermore, assuming that A and B bonds comove
positively (i.e., (k)βA|B > 0), generalists allocate less capital to market B when the expected return on
A bonds is higher. In this way, the response of generalist investors transmits shocks to the supply of
A bonds to the B market, promoting cross-market integration over time. Cross-market capital flows
become more responsive to differences in expected returns between the two markets when the two
(k)
bonds are closer substitutes (i.e., when R2AB is higher). In the limit as the two bonds become perfect
(k)
substitutes (i.e., as R2AB approaches 1), generalist investors become extremely aggressive in exploiting
any cross-market pricing differences.
Remarks on market structure The market structure we have described is a natural way to cap-
ture the industrial organization of real world asset management. Due to agency and informational
problems, savers are only willing to give asset managers the discretion to adjust their portfolios quickly
if the manager accepts a narrow, specialized mandate. These same agency and informational frictions
6All investors in our model are myopic– i.e., they make a one-time decision to maximize expected utility over terminal
wealth at some future date– which means that we are shutting off intertemporal hedging motives that stem from time
variation in risk premia. However, introducing these intertemporal hedging motives would not qualitatively alter our
main conclusions. This is because, in settings like ours where risk premia are mean-reverting (here due to mean-reverting
supply shocks), these intertemporal hedging motives tend to push in the same direction as myopic, mean-variance motives
for holding risky assets (see e.g., Campbell and Viceira (1999) and Albagli (2015)).
7We obtain similar results if we alter equati∑on (14) to reflect the fact that the cumulative return from rolling over
an investment at the short rate for k periods, k−1i=0 rt+i, is unknown at time t.{As in Campbell and Viceira (2001),this adds a hedging motive for holding long-duration assets that have high excess returns when short rates turn out t}o
be lower than expe∑cted. Formally, this∑means that generalists∑solve maxdA,t,dB,t Et [r 1P,t→t+k]− (V ar [r ])2τ t P,t→t+k
where rP,t→t+∑k = ( k−1i=0 rt+i)+dA,t∑× ( k ki=1 rxA,t+i)+dB,t× (∑ i=1 rxB,t+∑i). The solution takes the same form as (15),
replacing E [ k rx k −1 k A k−1t i=1 A,t+i] with Et[ i=1 rxA,t+i]− τ Covt[ i=1 rxt+i, i=0 rt+i] and similarly for the B bonds.
9
also mean that savers are only willing to give managers the discretion to adjust quickly if the manager
gives them an open-ended claim (Stein [2005]). As a result, fast-moving investors often have endoge-
nously short horizons. By contrast, most institutions, such as endowments and pensions, that have
longer horizons and possess greater flexibility to reallocate capital across asset classes are subject to
governance mechanisms– themselves a response to informational and agency frictions– that limit the
speed of any such capital movements.
In this paper, we regard the parameters governing market structure– i.e., qA, qB, and k– as fixed
and exogenously given. To be sure, the relevant empirical values for these parameters depend crucially
on the two asset classes being considered. Taking these parameters as fixed and given allows us to
draw out the asset-pricing implications arising from partial segmentation and slow-moving capital.
In this regard, we are following Bacchetta and van Wincoop (2010, 2017), Duffi e (2010), Chien,
Cole, and Lustig (2012), and Bogousslavsky (2016) who all assume a fixed, exogenous rebalancing
frequency. Endogenizing these parameters is challenging and is beyond the scope of the present paper.
In principle, one could consider an equilibrium in which qA, qB, and k take on a set of fixed values such
that A-specialists, B-specialists, and generalists all have the same expected utility in the long-run (see
Oehmke [2009] for such an analysis). Relatedly, one could also allow generalists’rebalancing frequency
and, hence, the speed of capital flow across markets could vary over time as in Duffi e and Strulovici
(2012).
2.3 Equilibrium
To solve for a rational expectations equilibrium of our model, we need to clear the market for both
the A and B bonds in a way that is consistent with (i) optimization on the part of A-specialists,
B-specialists, and generalists and (ii) where all agents correctly perceive the laws of motions for all
exogenous and endogenous variables.
Market clearing In market A at time t, there is a mass qA of fast-moving specialists, each with
demand bA,t, and a mass (1− qA − qB) × (1/k) of active slow-moving generalists, each with demand
dA,t. These investors must accommodate the active supply, which is the total∑supply of sA,t less any
supply held off the market by inactive generalist investors, (1− qA − qB) k−1 k−1j=1 dA,t−j . Thus, the
market-clearing condition for A bonds is
Specialist Active generalist
Inactive generalist
Total bond
d︷em︸︸an︷d ︷ dem︸︸and ︷ holdingssu︷︸p︸p︷ly ︷ ︸︸ ∑ ︷
q −1 −1
k−1
AbA,t + (1− qA − qB)k dA,t = sA,t − (1− qA − qB)(k dA,t−i). (16)
i=1
10
The market-clearing condition for B bonds is analogous.
Equilibrium conjecture We solve for a rational expectations equilibrium in which equilibrium
yields and generalist demands are linear functions of a state vector, xt, that includes the steady-state
deviations of the short-term interest rate, the default realization, the supply of A bonds, the supply
of B bonds, inactive generalist holdings of A bonds, and inactive generalist holdings of B bonds.
Formally, we conjecture that the yields on A and B bonds are
y ′A,t = αA0 +αA1xt, (17)
yB,t = αB0 +α
′
B1xt, (18)
and that the demands of slow-moving generalists are
d ′A,t = δA0 + δA1xt, (19)
d ′B,t = δB0 + δB1xt, (20)
where the 2 (1 + k)× 1 dimensional state vector, xt, is given by
xt= [rt−r, zt−z, sA,t−sA, s
′
B,t−sB, dA,t−1−δA0, · · · , dA,t−(k−1)−δA0, dB,t−1−δB0, · · · , dB,t−(k−1)−δB0] .
(21)
These assumptions imply that the state vector follows an AR(1) process
xt+1 = Γxt + t+1, (22)
where t+1 ≡ [ε ′r,t+1, εz,t+1, εsA,t+1, εsB ,t+1, 0, · · · , 0] , V ar [t+1] ≡ Σ, and the transition matrix Γ
depends on generalists’demand functions (i.e., Γ depends on δA1 and δB1).
Equilibrium bond yields Equilibrium bond yields in our model take a natural form. To understand
the equilibrium yield on asset A, rewrite equation (4) as
(1)
yA,t = Et [(1− θ ) (r + rx ) + θ y ] = E [(1− θ ) (r + τ−1A t A,t+1 A A,t+1 t A t VA bA,t) + θAyA,t+1],
where the second equality follows from equation (11) and (1)VA ≡ V art [rxA,t+1] = V art [(θA/ (1− θA))× yA,t+1] =
(θA/ (1− θ ))2α′A A1ΣαA1 is the equilibrium variance of 1-period excess returns on asset A. Iterating
11
this expression forward, we find that
∑ ︷︸︸︷ ︷ Risk prem iumShort rate ︸︸ ︷(1)
yA,t = (1− θ ) ∞ θi −1A i=0 AEt[ rt+i + τ VA bA,t+i]. (23)
The equilibrium yield on the default-free A perpetuity is a weighted average of expected future short
rates and future risk premia, where expected risk premia depend on the expected future bond holdings
of A-specialists. (The equilibrium bond holdings of A-specialists are a linear function of the state
vector, xt.) Proceeding similarly for asset B, we obtain
∑ ︷︸ Risk prem iumShort︸r︷ate D︷efa︸ul︸t loss ︷ ︸︸ ︷
y = (1− θ ) ∞ ︷ (1)B,t B i=0 θiBE [ r −1t t+i + zt+i+1 + τ VB bB,t+i]. (24)
The equilibrium yield on the defaultable B perpetuity is a weighted average of expected future short
rates, future default losses, and future risk premia.
Finally, making use the assumed AR(1) dynamics for the exogenous state variables and the market-
clearing condition for asset A, the equilibrium yield on asset A satisfies
{︷ Exp(ected fut︸u︸re s)hort rates }︷
1− θA
yA,t = r + − (rt − r) (25)1 ρrθA
[︷ Uncondition︸al︸term premia ]︷
+ (q τ)−1
(1)
A VA (sA − (1− qA − qB) δA0)
︷  Conditional︸︸term premia ︷ 1−θA(1) 1−θ ρ (sA,t − sA)+ (q −1Aτ) V  A sAA ∑ ∑ .
− (1− θA) (1− qA − q −1 ∞B) k i=0 θiAEt[
k−1
j=0(dA,t+i−j − δA0)]
Thus, for instance, the unconditional, steady-state term premium on the A market perpetuity, depends
on the steady-state bond holdings of A-specialists, namely (sA − (1− qA − qB) δA0) /qA. The yield for
12
asset B has an extra term relating to expected future defaults, but is otherwise similar:
{︷ Exp(ected fut︸u︸re s)hort rates }︷ ︷{Expected futu︸r︸e default losse}s︷
1− θB 1− θ
yB,t = r + − (rt − r) + z +1 ρrθB 1−
ρz (zt − z) (26)ρ θ
[︷
z
Unconditional t︸er︸m/credit premia ]︷
−1 (1)+ (qBτ) VB (sB − (1− qA − qB) δB0)
︷  Conditional ter︸m︸/credit premia ︷
1−θB
−1 (1)  1−θ ρ ∑(sB,t − sB)+ (q B sBτ) V B ∑ B .
− (1− θB) (1− qA − q ) k−1 ∞B i=0 θiBEt[
k−1
j=0(dB,t+i−j − δB0)]
Although equations (25) and (26) show that yields in markets A and B take a similar algebraic form,
the risk premia in the two markets will not be the same because of the different risks that market
specialists must bear in equilibrium.
Equilibrium solution As shown in the Internet Appendix, a rational expectations equilibrium of
our model is a fixed point of a specific operator involving the “price-impact”coeffi cients, (α′ ′A1,αB1),
which show how bond supply and inactive generalist demand impact bond yields, and the “demand-
impact”coeffi cients, (δ′ ′A1, δB1), which show how bond supply and inactive generalist demand impact
active generalist demands. Specifically, let ω = (α′A1,α
′
B1, δ
′
A1, δ
′
B1) and consider the operator f (ω0)
which gives (i) the price-impact coeffi cients that will clear the two markets and (ii) the demand-impact
coeffi cients consistent with optimization on the part of generalists when agents conjecture that ω = ω0
at all future dates. Thus, a rational expectations equilibrium of our model is a fixed point ω∗ = f (ω∗).
In any rational expectations equilibrium of our model, bond yields always reflect the expected
path of future fundamentals (short rates and default losses). As a result, equilibrium bond holdings
do not depend on fundamentals. This implies that an equilibrium of our model is a solution to a
system of 8k nonlinear equations in 8k unknowns. Specifically, we need to determine how equilibrium
yields and active generalist demand in markets A and B respond to shifts in the supply of A and
B bonds: this generates 8 unknowns and 8 corresponding equations. We also need to determine
how equilibrium yields and active generalist demand in markets A and B respond to the holdings
of inactive generalists: this generates 8 (k − 1) unknowns and 8 (k − 1) corresponding equations. We
solve the relevant system of 8k nonlinear equations numerically using the Powell hybrid algorithm
which performs a quasi-Newton search to find solutions to a system of nonlinear equations starting
from an initial guess.8 To find all of the solutions, we apply this algorithm by sampling over 10, 000
8Rational expectations models with noisy asset supply can only be solved in closed-form in very simple, special cases.
For instance, we have only been able to obtain closed-form solutions to our model if we turn off the two key asset pricing
13
different initial guesses.
As we detail in the Internet Appendix, when asset supply is stochastic, an equilibrium solution only
exists if investors are suffi ciently risk tolerant (i.e., for τ suffi ciently large). When an equilibrium exists,
there are multiple equilibrium solutions. Equilibrium non-existence and multiplicity of this sort arise
in overlapping-generations, rational-expectations models such as ours where risk-averse investors with
finite investment horizons trade an infinitely-lived asset that is subject to supply shocks.9 Different
equilibria correspond to different self-fulfilling beliefs that investors can hold about the price-impact
of supply shocks and, hence, the risks associated with holding the two assets.
The intuition for equilibrium multiplicity can be understood most clearly in the relatively simple
case when there is just a single risky asset and when there are only fast-moving investors. To clarify
the relevant issues, we treat this special case in the Internet Appendix. If investors are suffi ciently
risk tolerant there are two equilibria in this special case: a low price impact (or low return volatility)
equilibrium and a high price impact (or high return volatility) equilibrium. If investors believe that
supply shocks will have a large impact on prices, they will perceive the asset as being highly risky. As
a result, investors will only absorb a positive supply shock if they are compensated by a large decline
in prices, making the initial belief self-fulfilling. However, if investors believe that prices will be less
sensitive to supply shocks, they will perceive the asset as being less risky and will absorb a supply
shock even if they are only compensated by a modest decline in prices.
Things are slightly more complicated in our more general model. Specifically, the addition of
multiple risky assets, the partial segmentation of markets, and the introduction of slow-moving capital,
give rise to additional unstable equilibria. However, we always find a unique equilibrium that is
stable in the sense that equilibrium is robust to a small perturbation in investors’beliefs regarding
the equilibrium that will prevail in the future. Formally, letting ω(1) = ω∗ + ξ for some small ξ
and defining ω(n) = f(ω(n−1)), an equilibrium ω∗ is stable if lim →∞ω(n) = ω∗n and is unstable if
lim →∞ω(n)n 6= ω∗. Let {λi} denote the eigenvalues of the Jacobian D f (ω∗ω ). If maxi |λi| < 1, then
ω∗ is stable; if maxi |λi| > 1, then ω∗ is unstable.
Consistent with Samuelson’s (1947) “correspondence principle,”which says that the comparative
statics of stable equilibria have certain properties, this unique stable equilibrium has comparative
statics that accord with standard economic intuition. By contrast, the unstable equilibria have com-
parative statics that run contrary to standard intuition.10 Since we will be interested in comparative
frictions– partial segmentation and slow-moving capital– that are the focus our of paper.
9For previous treatments of these issues, see Spiegel (1998), Bacchetta and van Wincoop (2003), Watanabe (2008),
Banerjee (2011), Greenwood and Vayanos (2014), and Albagli (2015).
10For instance, in the simple case discussed above, the low price-impact equilibrium is stable and the high price-impact
equilibrium is unstable. At the stable equilibrium, an increase in the volatility of fundamental shocks or the volatility of
supply shocks is associated with an increase in the price-impact coeffi cient and an increase in the volatility of returns.
14
statics, we focus on this unique stable equilibrium in our numerical illustrations in Section 3. However,
the Internet Appendix discusses the main properties of the model’s unstable equilibria.
2.4 Defining market integration
What do we mean by “market integration”? In well-integrated financial markets where all investors
can frictionlessly adjust their holdings of each asset, each investor prices exposure to common risk
factors in the exact same way.11 For instance, if a set of investors differ in their risk preferences and
their exposures to background risks, then, so long as they all can frictionlessly buy or sell each risky
asset, they will all price exposure to common risk factors in the same way.12 However, all investors will
not price risk in the same way if, as in our model, different investors are bound by different frictional
constraints– e.g., different constraints on the assets they can trade. Below we clarify what it means
for markets to be integrated at different horizons.
Thus, we define markets to be fully integrated in the short run if, at each date, all investors price
exposures to common risk factors in the same way. For example, the pricing of interest rate risk is
integrated in the short run if, at each date, all investors demand the same the expected return per
unit of marginal exposure to short-term interest rate shocks. Similarly, we define markets to be fully
integrated in the long-run if all investors price exposures to common risk factors in the same way on
average– i.e., in the model’s long-run, steady-state. Long-run integration is therefore a weaker form
of market integration than short-run integration. Since investors have the same risk preferences and
face no background risks in our model, investors will price exposures to common risk factors in the
same way if and only if they all hold the same portfolios, which is only possible if all investors hold
the market portfolio.13
The degree of market integration depends on which investors can bear risk at different horizons.
By contrast, these comparitive statics take the opposite sign at the unstable equilibrium.
11 In frictionless markets the only binding constraint that investors face are their budget constraints. Investors cannot
face binding short-sales constraints, borrowing constraints, or constraints on the set of assets they can trade.
12More generally, if markets are complete (the set of traded assets spans the set of risk factors) and all investors can
frictionlessly adjust their asset holdings, then all investors must share the same stochastic discount factor (SDF)– risk
sharing will be perfect– and this unique SDF will price all assets. If markets are incomplete and all investors can
frictionlessly adjust their asset holdings, then each investor’s SDF will price all assets. While investors’SDFs may not
be equated state-by-state in incomplete markets– risk sharing may be imperfect, every investor’s SDF must share the
same linear projection onto the space of excess returns– i.e., there is a unique “SDF-mimicking portfolio.” Thus, any
investor’s SDF equals this unique SDF-mimicking portfolio plus a residual term that is orthogonal to all of the excess
returns. This implies that all agents price risk in the same way– i.e., all agents charge the same prices of risk for bearing
additional factor risk at the margin, irrespective of their risk preferences or exposures to other background risks.
13To see the factor risk prices charged by A-specialists, write rxA,t+1 −Et [rx ′A,t+1] = φAt+1 where t+1 are the four
factor innovations and φA =(− [θA/ (1− θ)A)]αA1 are the equilibrium factor loadings for A bonds. Since Et [rxA,t+1] =
τ−1V art [rxA,t+1] bA,t = φ
′ τ−1Σφ b = φ′A A A,t AλA,t, the risk prices charged by A-specialists( are λ ≡ τ)−1A,t ΣφAbA,t.
Proceeding similarly for B-specialists, we have Et [rxB,t+1] = τ−1V art [rx ′ −1B,t+1] bB,t = φB τ ΣφBbB,t = φ
′
BλB,t
where φB = − [θB/ (1− θB)]αB1 − ez. Thus, the risk prices charged by B-specialists are λB,t ≡ τ−1ΣφBbB,t.
In partially segmented markets the risk prices charged by A-specialists are not consistent with the pricing of asset B
and vice versa (i.e., when (1− q ′ ′A − qB) < 1, Et [rxB,t+1] 6= φBλA,t and Et [rxA,t+1] 6= φAλB,t).
15
It is driven by two parameters: (1− qA − qB) and k.14 The first parameter, (1− qA − qB), is the
population share of generalists. This parameter determines the degree of long-run integration between
markets. For instance, if (1− qA − qB) = 1, markets will be well-integrated in the long-run even if
k is large. When (1− qA − qB) < 1, specialists in the two markets price risk differently on average;
this is the sense in which the two markets are partially segmented in the long-run. By contrast, active
generalists price risk consistently across the two markets on average; this is the sense in which the two
markets are partially integrated in the long-run.
The second parameter, k, indexes the speed with which generalist capital can flow between markets.
Thus, k determines the degree of short-run integration between the two markets, holding fixed the
degree of long-run integration. Markets are perfectly segmented if (1− qA − qB) = 0 or k → ∞. If
either of these conditions holds, the two markets operate independently of each other.
Formally, collect all of the 1-period returns in a vector rxt+1 ≡ [rxA,t+1, rxB,t+1]′ and the asset
supplies in a vector st ≡ [sA,t, s ]′B,t . Letting rx ′Mt,t+1 = strxt+1 denote the excess returns on the
current market portfolio, we define markets to be fully integrated in the short-run if
E [rx ] = τ−1t t+1 V art [rxt+1] st = βt [rxt+1, rxMt,t+1]Et [rxMt,t+1] , (27)
where βt [rxt+1, rxMt,t+1] = V art [rxt+1] st/ (s
′
tV art [rxt→t+j ] st). In other words, markets are fully
integrated in the short-run if, at each date, a conditional-CAPM based on the current market portfolio
prices both the A and B assets. In our model, markets are fully integrated in the short-run if and only
if (1− qA − qB) = 1 and k = 1. By contrast, when (1− qA − qB) =6 1 or k =6 1, this conditional-CAPM
will not price the 1-period returns on the A and B assets.15
Similarly, we define markets to be fully integrated in the long-run if
E [rx −1t→t+k] = τ V art [rxt→t+k]E [st] = β[rxt→t+k, rxM,t→t+k]E[rxM,t→t+k], (28)
where β[rxt→t+k, rxM,t→t+k] = V art [rxt→t+k]E [st] / (E [s
′
t]V art [rxt→t+k]E [st]) and E[rxM,t→t+k] =
E [s′t]E [rxt→t+k]. In other words, markets are fully integrated in the long-run if the same unconditional-
14 In our model, news about fundamentals– short-term rates and default losses– is reflected immediately in both
markets. Thus, market inte[gration has noth]ing [to do wit]h the speed at which fundamental news is reflected in prices.15Of course, there is always a unique SDF-mimicking portfolio c∗t = τ (V art [rxt+1])−1Et [rxt+1] su[ch that Et [rxt+]1] =
τ−1V ar [rx ] c∗ = β ∗ ∗ ∗ ∗′t t+1 t t rxt+1, rxC ,t+1 Et rxt Ct ,t+1 where rxCt ,t+1 = ct rxt+1 and βt rxt+1, rxC∗t ,t+1 =
V art [rxt+1] c
∗
t / (c
∗′
t V art [rxt+1] c
∗
t ). However, unless k = 1 and (1− qA − qB) = 1, we will have c∗t 6= st (i.e., the
SDF-mimicking portfolio will differ from the market portfolio) and different agents will price risk in different ways.
When k > 1 and (1− qA − qB) < 1, no individual investor will hold the portfolio c∗t . When k = 1 and (1− qA − qB) < 1,
generalists will hold c∗t (i.e., d
∗ ∗
t = ct ) but A- and B-specialists will clearly not hold ct .
To see the prices of risk implied by c∗t , let(Φ = [(φ ′A φB ] denote the matrix of equilibrium factor loadings and
note that E [rx ] = τ−1V ar [rx ] c∗ = Φ τ−1Σ c∗ ∗
))
t t+1 t t+1 t A,tφA + cB,tφB = ΦλC∗,t. Thus, the implied risk prices are
λC∗,t ≡ τ−1Σ (cA,tφA + cB,tφB).
16
CAPM based on the average market portfolio (rxM,t→t+k = E [s
′
t] rxt→t+k) prices k-period returns
on both the A and B assets on average– i.e., in the model’s steady state where asset supply equals
the long-run average E [st]. In our model, markets are fully integrated in the long-run if and only if
(1− qA − qB) = 1, irrespective of k. By contrast, when (1− qA − qB) < 1, this unconditional CAPM
will not price the k-period returns on the A and B assets in the model’s steady state.
The reason markets are not integrated in our model is because cross-market “arbitrage” is risky
for generalists. Unless (1− qA − qB) = 1 and k = 1, full short-run integration fails because generalists
demand compensation for the risk associated with the short-run trades they place to exploit the
differences in the way that A-specialists and B-specialists price risk. Similarly, unless (1− qA − qB) =
1, full long-run integration fails because generalists are engaged in a risky cross-market arbitrage trade
even in the model’s long-run steady state. Specifically, when (1− qA − qB) < 1, generalists will not
hold the market portfolio in the steady state (i.e., E [dA,t] 6= E [sA,t] and E [dB,t] 6= E [sB,t]). Relative
to the market portfolio, generalists’portfolio will incorporate a tilt that reflects average cross-market
pricing differences. And, generalists will demand compensation for bearing the risks stemming from
this portfolio tilt.16 Thus, in the general case where (1− qA − qB) < 1 and k > 1, we obtain neither
full short-run nor long-run market integration.
3 Market integration following large supply shocks
In this section, we use our model to illustrate the asset price dynamics following a large supply shock
in one asset market. We relate these dynamics to the model’s two key asset pricing frictions, namely
partial market segmentation and slow-moving capital.
3.1 Parameters for baseline illustration
Table 1 lists the set of parameter values that we use in our numerical illustrations. For the purposes
of these illustrations, it will be helpful to think of market A as the U.S. Treasury market and market
B as the corporate bond market and we have picked parameter values to approximate the behavior
of these markets. Thus, for example, we choose σr and ρr to approximately match the volatility and
persistence of short-term nominal interest rates. Each period in our numerical illustrations corresponds
to one quarter of a year. However, we report annualized values for both risk premia and bond yields
by multiplying the corresponding quarterly risk premia and yields by four.
16For instance, when qA = qB and sA = sB , generalists will be overweight B bonds and underweight A bonds relative
to the steady-state market portfolio. The reason is that B bonds are riskier than A bonds since the former are exposed
to default risk. As a result, B-specialists will hold less of the B bonds than A -specialists hold of the A bonds.
17
Our goal is to use the model to draw structural impulse response functions that trace out the
causal impact on asset prices following a shock to asset supply. Thus, for simplicity and clarity of
interpretation, we assume that the four exogenous shocks in the model– εr,t+1, εz,t+1, εsA,t+1, and
εsB ,t+1– are mutually orthogonal.
We begin our analysis by choosing k = 8 quarters and qA = qB = 45%, but later show comparative
statics for these parameters. Based on these values, our illustrations assume that most risk-bearing
capital is controlled by specialists, with 10% being controlled by flexible generalist investors, one-eighth
of whom reallocate their portfolios each quarter. Our choice of k = 8 quarters is somewhat arbitrary,
but we think of this as capturing the empirically relevant case of pension funds or endowments who
typically review their asset allocations on an annual or biannual basis. As shown below in Table 2, all
the main results carry through if we instead assume that k = 4 quarters, but the dynamics are easier
to see in a graphical format when we assume k = 8 quarters.
The parameter τ should be interpreted as the aggregate risk tolerance of all investors in the economy
and should not be equated with the risk tolerance (i.e., inverse CARA parameter) of any individual
investor. We normalize the volatility of shocks to the supply of A and B to one (σsA = σsB = 1)
and then choose an aggregate risk tolerance of τ = 1.75 to obtain realistic values for equilibrium risk
premia.17 We assume a steady-state supply of both A and B of five units (sA = sB = 5).
3.2 An unanticipated supply shock
3.2.1 Baseline illustration
We first consider the impact of an unanticipated supply shock that doubles the supply of asset A in
quarter 5– i.e., sA,t jumps from 5 to 10 units between quarters 4 and 5. To make the illustration as
stark as possible, we study a near-permanent supply shock and set ρs = 0.999.A
Figure 1 illustrates the impact of this shock, plotting the evolution of expected annual returns
on both bonds in Panel A, the holdings of specialists and generalist investors in Panel B, the active
supply of both bonds in Panel C, and bond yields in Panel D. Prior to the supply shock, Panel A
shows that the steady-state risk premium in market B is 1.07% per annum versus a risk premium
of 0.79% in market A. The additional risk premium of 0.28% arises because market B is subject to
default risk, which exposes investors to an additional source of fundamental risk and amplifies the
supply risk facing B bond holders. The initial yield in market B is 5.47% per annum versus a yield of
4.79% in market A. The steady-state yield in market A equals the average short-term riskless rate of
17As is common in models where mean-variance investors face stochastic supply shocks, equilibrium prices and risk
premia only depend on the ratio of asset supply to investors’aggregate risk tolerance. Thus, papers studying models of
this sort either (i) normalize τ = 1 and then choose the volatility of supply shocks to obtain realistic results or (ii) they
normalize the volatility of supply shocks to 1 and then choose τ . We opted for the latter approach.
18
r = 4.00% plus a steady-state risk premia of 0.79%. The 0.68% steady-state yield spread between the
B and A markets equals the difference in steady-state risk premia of 0.28% plus market B’s expected
default losses of z = 0.40% per annum.
When the supply shock hits the market A at t = 5, expected returns and yields in both markets
react immediately. Panel A shows that expected returns in market A overreact: they jump from 0.79%
to 1.59% before ultimately falling back to a new long-run level of 1.35%. The overreaction of expected
returns on A bonds reflects the relative steepness of short-run demand curves and relative flatness of
long-run demand curves. Unlike textbook theories in which asset prices are determined solely by the
stock of risky assets supplied, our approach suggests that supply flows– the rate at which the stock is
changing– also matter in the short run. As explained in Duffi e (2010), these are general properties of
models featuring slow-moving capital and are not unique to our model.
In contrast, Panel A shows the key novel implication of our model: expected returns in market
B actually underreact to the shock to the supply of A, rising slowly from 1.07% prior to the shock
to a new long-run level of 1.30%. Why does market A overreact to the supply shock while market
B underreacts? Panel B shows how the positions of different market participants evolve over time.
Following the initial supply shock in market A, both specialist holdings of A (bA,t) and active generalist
holdings of A (dA,t) spike upwards. As a partial hedge against their increased holdings of A, active
generalists reduce their holdings in market B. This reduction in generalists’B holdings is motivated
by a need to reduce the common short-term interest rate risk across their holdings in both markets.
As time passes and more generalists reallocate their portfolios in response to the shock, B-specialists
must hold more of the B bonds to fill the void left by the generalists. Thus, the expected returns on
B bonds rise gradually over time.
The dynamics of risk premia are closely tied to the dynamics of the “active supply”of A and B
that must be absorbed by active market participants each period. For instance, th∑e active supply of
A bonds is given by the right-hand-side of equation (16): sA,t − (1− qA − qB)(k−1 k−1i=1 dA,t−i)– i.e.,
the total asset supply less the assets that are being held off the market by inactive generalists. The
evolution of the active supplies is shown in Panel C and mirrors the path of bond risk premia shown
in Panel A. The initial supply shock to A at t = 5 immediately increases the active supply of A bonds
but has no immediate effect on the active supply of B bonds. This is because slow-moving generalists
have yet to reduce their holdings in market B. Over the ensuing periods, generalists gradually increase
their holdings of A and reduce their holdings of B. Therefore, the active supply in A gradually declines
while the active supply in B gradually rises, paralleling the movements in bond risk premia.
By t = 12 all generalist investors have reallocated their portfolios in response to the supply shock
at t = 5 (recall that k = 8 in this illustration). However, the gradual adjustment of generalists gives
19
rise to modest echo effects after t = 12, generating a series of damping oscillations that converge to
the new long-run equilibrium. As in Duffi e (2010), these oscillations arise because generalists who
reallocate soon after the supply shock lands take large opportunistic positions. These large positions
temporarily reduce the active supply of A and then need to be absorbed in later periods.18
In market A, risk premia charged by A-specialists are the sum of risk premia related to shocks
to short-term interest rates, shocks to the supply of A, and shocks to the supply of B. Changes
in total risk premia are primarily driven by the pricing of interest rate risk. In market B, the risk
premia charged by B-specialists can be similarly decomposed into its components, which also includes
a premium for default risk. Following the supply shock, the risk premia associated with interest
rate risk differs significantly between the two sets of specialists. As generalists react to this pricing
discrepancy, the difference in interest rate risk premia charged by the two sets of specialists gradually
narrows. However, the difference does not vanish in the long run because of the permanent risks
associated with cross-market arbitrage.
In Panel D, we shift from risk premia to bond yields. The overreaction of the A market and the
underreaction of the B market are more muted in yield space than in risk premium space: market A
yields overreact by 7% and market B yields underreact by 12%. This is natural since, as shown in
equations (23) and (24), bond yields reflect the weighted average expected risk premia over the life of
the bond with risk premia in the near future having a larger effect on the weighted average than those
in the distant future.
Panel D also shows that shock to the supply of A bonds leads the yield spread between default-
able and default-free bonds, yB − yA, to decline. However, because yield A overreacts and yield B
underreacts to the shock, the yield spread overreacts even more (19%) than the yield on A bonds.
3.2.2 Unpacking the role of the two key frictions
Our model combines two key asset pricing frictions, namely partial market segmentation and slow-
moving capital. In this subsection, we show that both of these frictions are necessary to generate
underreaction in market B following a shock to the supply of the A bonds.
To understand how these two frictions interact, Figure 2 shows asset price dynamics following the
same near-permanent shock to the supply of the A bonds considered above: the supply of the A bonds
unexpectedly doubles in quarter t = 5. The four panels of Figure 2 show the risk premium dynamics
as we switch these two asset pricing frictions on and off.
• Panel A: Benchmark with no frictions (Upper left): Here all investors are fast-moving
18Exploiting this feature of Duffi e’s (2010) model, Bogousslavsky (2016) argues that infrequent rebalancing can explain
the echo effects found intra-day and daily stock returns.
20
generalists who can invest in both assets and can rebalance their portfolios each period. This is
a special case of our model where qA = qB = 0 and k = 1– i.e., where both markets are fully
integrated in the short run. In this case, there is no overreaction and the risk premia on the A
and B bonds move in near-perfect lockstep following the shock to the supply of the A bonds.
This is because the price of interest rate risk jumps and both bonds have the same exposure to
this risk factor.19
• Panel B: Only partial segmentation (Upper right): Here all investors can rebalance each
period, so capital is not slow-moving. However, markets are partially segmented: there are
specialists who can trade each bond and as well as a subset of generalists who can trade both
bonds. This is a special case of our general model when qA > 0, qB > 0, (1− qA − qB) > 0, and
k = 1. Relative to the frictionless case, partial segmentation raises the steady-state risk premium
in both markets because a large fraction of total risk is now born by undiversified specialists.
And, the shock to the supply of A bonds has a larger impact on risk premia in market A and
a smaller impact on those in market B– i.e., the cross-market spillover is smaller– than in the
benchmark case with full short-run integration. However, since all agents can rebalance each
period, there is no difference between short-run and long-run demand curves. As a result, the
short-run price impact is the same as the long-run price impact.
• Panel C: Only slow-moving capital (Lower left): This case corresponds to a multi-asset
version of Duffi e’s (2010) model and is considered Bogousslavsky (2016). Here all investors can
trade both the A and B bonds; there is no market segmentation. However, fraction q of investors
are fast-moving and can trade each period, while fraction (1− q) are slow-moving and can only
trade every k periods. This is not a special case of our general model. When the two assets
are exposed to a common risk factor, the two-asset Duffi e model will always deliver short-run
overreaction in both assets A and B. Factor risk prices overreact in the short-run because short-
run demand curves are steeper than long-run demand curves. However, since fast-moving agents
price risk similarly in both markets, we will always see similar short-run overreaction in both
the directly impacted market (A) and the market that is not directly impacted (B).20
19The risk premia do not move in perfect lock step because (i) the prices of A-supply risk and B-supply risk also jump
and (ii), when σ2z > 0, the B bonds have a larger exposure to these two risk factors than the A bonds. Thus, the risk
premium on the B bonds rises slightly more than that on the A bonds following the supply shock. However, this effect
is tiny when σ2z is small. (i.e., when A and B bonds are close substitutes) and is not visible in the plot.
20To see this formally, note that we can always write rxt+1 − Et [rxt+1] = Φt+1 for some set of equilibrium factor
loadings Φ = [φ ′A φB ] , so that V art [rxt+1] = ΦΣΦ
′. I∑n the multi-asset version of the Duffi e (2010) model, the market
clearing conditions can be written as qb + (1− q) k−1 k−1t j=0 dt−j = st where bt = τ (V art [rxt+1])∑−1Et [rxt+1] is the
demand of fast-moving investors. This im∑plies that E [rx ] = (qτ)−1 V ar [rx ] (s − (1− q) k−1 k−1t t+1 t t+1 t j=0 dt−j) = Φλt
where λ = (qτ)−1 ΣΦ′(s − (1− q) k−1 k−1t t j=0 dt−j) are factor risk prices charged by fast-moving investors.
21
• Panel D: Both partial segmentation and slow-moving capital (Lower right): Our
model features both partial market segmentation (qA > 0, qB > 0, (1− qA − qB) > 0) and slow-
moving capital between markets (k > 1). Our model will always generate short-run overreaction
in the A market that is directly impacted by the supply shock. However, our model can either
generate short-run overreaction or underreaction in the B market that is not directly hit by
the supply shock. Specifically, as we show below, when there is minimal segmentation between
the two markets– i.e., when (1− qA − qB) is suffi ciently large– our model delivers short-run
overreaction in both markets as in the multi-asset version of the Duffi e (2010) model. However,
when the two markets are suffi ciently segmented– i.e., when (1− qA − qB) is suffi ciently small,
our model delivers short-run underreaction in the indirectly impacted market.
One additional contrast between the models in Panels C and D is worth noting. In Panel C where
there is only slow-moving capital, there is almost no abnormal trading in the B bonds following the
shock to the supply of the A bonds.21 In other words, the response of B’s price to the shock to the
supply of A is not mediated by trade. By contrast, in Panel D where there is also partial segmentation,
the only reason that B’s price responds to increase in the supply of A is because generalist investors
are engaged in a long-A, short-B trade– i.e., the response of B’s price to the shock is entirely mediated
by trade.
In summary, both partial segmentation and slow-moving capital are necessary to generate short-
run underreaction in market B. However, these two frictions are not jointly suffi cient. Instead, to
guarantee short-run underreaction in market B one needs both slow-moving capital and a meaningful
degree of segmentation between the two markets.
3.2.3 Comparative statics
In Table 2, we perform a variety of comparative statics exercises to illustrate how the price dynamics
following a large shock to the supply of A depend on the parameters of our model. We primarily focus
on the parameters governing our two key asset pricing frictions: the population shares of A-specialists,
B-specialists, and generalists as well as the frequency at which generalists can rebalance (k).
We again study the same supply shock considered in Figure 1: the supply of asset A unexpectedly
doubles at time t. For each different set of model parameters, Table 2 summarizes the impact of this
21There is a tiny amount of abnormal trading in B bonds in Panel C following the shock to the supply of A bonds.
This is because mean-reverting supply shocks generate negative serial correlation in returns, so the variance ratio
V art[rxA,t→t+k]/k is decreasing in k. As a result, longer-horizon investors worry less about transitory discount rate
risk, leading them to take larger positions than short-horizon investors in both risky bonds following the supply shock.
However, this is a second order effect in our illustration.
22
supply shock on both the A and B markets by listing the yields and expected annual returns in (i)
at time t − 1 before the shock arrives (labeled as “pre-shock level”), (ii) the short-run change from
time t− 1 to time t when the shock arrives (labeled as “short-run ∆”), and (iii) the long-run change
from time t− 1 to time t+ 2k (labeled as “long-run ∆”). We define the degree to which bond yields
overreact in the short run as the difference between the short-run change and the long-run change,
expressed as a percentage of the long-run change22
(yt − yt−1)− (yt+2k − yt−1)
%Short-Run-Over-React(y) ≡ . (29)
(yt+2k − yt−1)
Our measure of short-run overreaction for risk premia is defined analogously.
The first row in Table 2 shows the results for the baseline set of parameters used in Figure 1. Risk
premia in market A overreact by 42% in the short-run using our baseline set of parameters, while risk
premia in market B underreact by 83% in the short-run. Similarly, yields in market A overreact by
7%, while yields in market B underreact by 12% in the short-run.
The second row in Table 2 shows that, if market participants are more risk tolerant, this reduces
the price impact of the supply shock on both market A and market B. However, changing investor
risk tolerance has a similar impact on the short- and long-run response to the supply shock. Thus,
the degree of overreaction or underreaction in each market is unchanged in percentage terms.
Recall that qA and qB indicate the relative fraction of specialists in market A and B, respectively.
In row 3, we set qA = qB = 0.5, so there are no generalists and the two markets are completely
segmented. In this case, the supply shock in market A is no longer transmitted to the market B.
We next change the mix between A-specialists and B-specialists, holding fixed the overall mix
between generalists and specialists. Row 4 shows that if we hold the total number of specialists fixed
at qA+qB = 0.9, then as we increase the share of B-specialists and decrease the share of A-specialists,
we get more overreaction in A. Market B is only modestly affected by this change because the supply
shock is primarily being absorbed by generalists. Similarly, row 5 shows that if we increase the share
of A-specialists, holding fixed the total number of specialists, we see less overreaction in A.
Recall that k is the number of quarters that it takes for generalists to fully reallocate their portfolios
following a supply shock and that k = 8 quarters in in our baseline example. In row 6 we instead set
k = 12 quarters. Naturally, this larger value of k increases the short-run overreaction in market A
and increases the short-run underreaction in market B. Similarly, when we set k = 4 in row 7, there
22Since our supply shock is not quite permanent, we subtract off the constant (1− ρ2kS )/ρ2kS from %Short-Run-Over-A A
R[ eact to ensure that o]ur measure is zero if markets are perfectly integrated in the short run (1 − qA − qB = k = 1).In this case, there is no “over-reaction”but only “reaction”and we have [(yt − yt−1)− (yt+2k − yt−1)] / [yt+2k − yt−1] =
α s − α 2k 2k 2k 2ksA A,t sAsA,tρs /αsAsA,tρs = (1− ρs )/ρs . For k = 8 and ρs = 0.999, this constant is 1.6%.A A A A A
23
is less short-run overreaction in market A and less short-run underreaction in market B.
In row 8, we continue to assume that k = 4 quarters, but we now assume that qA = qB = 0.2
as opposed to qA = qB = 0.45 as we did in row 7. In this case there are many generalists and few
specialists, so the markets are well-integrated in the long-run. Both the A and B markets overreact
to the shock to the supply of asset A in this case. Intuitively, the two markets behave like two
well-integrated markets where the only asset pricing friction is slow-moving capital. As a result, the
dynamics are similar to those in the multi-asset version of Duffi e’s (2010) model discussed above.
3.3 A pre-announced increase in supply
We next study asset price dynamics following the announcement of a large future change in asset
supply. As an example of a large pre-announced supply change, consider the large-scale asset purchase
programs initiated by the Federal Reserve between 2008 and 2013. The Fed’s initial announcement
of long-term bond purchases occurred on November 25, 2008, but asset purchases did not begin until
January 2009 and continued in the months and years thereafter.
To mimic the announcement at time t of a future increase in the supply of asset A, we assume that
sA,t jumps up and that the amount of asset A held by inactive generalists also jumps up such that
the active supply of asset A does not change at time t.23 (Duffi e (2010) uses a similar approach to
model price dynamics following a pre-announced supply shock.) This means that, unlike the case of
an unanticipated supply shock, it would be possible to clear the market at time t without any increase
in the holdings of A-specialists or active generalists. Thus, the only reason that prices change when a
future supply change is announced is because the announcement leads active long-horizon generalists
to opportunistically adjust their asset holdings. Formally, letting t [Xt] = Xt − Et−1 [Xt] denote the
time t innovation or surprise to some random process Xt, an anticipated supply shock is defined so
that the innovation to the right-hand of equation (16) is zero when it is announced at time t:
∑
 [s ]− (1− q − q ) k−1
k−1
t A,t A B t[dA,t−j ] = 0. (30)
j=1
We examine the pre-announcement of a one-time, near-permanent jump in the supply of asset A:
market participants learn in quarter t = 5 that the supply of asset A will double in quarter t = 8.
We continue to assume that k = 8 quarters which means that, by the time the supply of A jumps at
t = 8, half of all generalists investors have had the opportunity to adjust their portfolios following the
23Normally, the holdings of inactive generalist investors evolve deterministically and do not jump. Thus, a pre-
announced supply increase is an event that agents assume is impossible, which is a drawback of this analysis. Greenwood,
Hanson, and Vayanos (2016) develop a term structure model where agents receive news about the likely future path of
bond supply. However, their model does not feature slow-moving capital.
24
announcement at t = 5.24
Figure 3 shows the dynamics of risk premia, investor positions, active asset supplies, and bond
yields following the pre-announced supply increase. Panel A shows that annual risk premia in market
A actually decline slightly after the announcement at t = 5 and before the supply rises at t = 8. And
risk premia in market A still jump up when bond supply jumps to its new level at t = 8. The risk
premium in market B , which is only indirectly affected due to cross-market arbitrage by generalists,
rises gradually over time.
What drives these dynamics? Panel B shows how the positions of market participants evolve
over time in response to this pre-announced supply increase. Upon the announcement, generalists
begin to opportunistically adjust their portfolios in direction of the anticipated shock, increasing their
holdings of A (dA) and decreasing their holdings of B (dB). A-specialists provide temporary liquidity
to generalists, planning to replenish their inventories once the supply shock actually lands.25 The
gradual build up of generalist demand for asset A is responsible for the slight decline in the risk
premium on asset A from t = 5 to 7 before its upward jump at t = 8. Similarly, the gradual reduction
of generalists’demand for B results in a slow increase in the risk premium on asset B.
In contrast to the generalists, the specialists’demand in market A (bA) decreases initially then
increases. This is because specialists can adjust quickly, and thus they have the ability to front-run the
anticipated change in supply– specialists reduce their portfolio holdings of A just before the positive
supply shock and increase holding of A immediately after the shock.
Panel C summarizes the evolution of the active supplies of assets A and B. By construction, the
active supplies of both assets do not change at quarter t = 5 when the increase in the supply of asset
A is pre-announced. Instead, the active supply of asset A jumps at t = 8. However, the active supply
of A does not jump by the full 5 units at t = 8 because pre-announcing the supply shock mobilizes
slow-moving generalists beginning at t = 5.
Panel D shows the evolution of bond yields in response to the pre-announced supply shock. Follow-
ing the announcement, there is now short-run underreaction in both markets: yields in both markets
gradually rise to their new steady-state levels. The yields in market A underreact by 13% and the
24To implement this announcement, we assume that sA,t and (1− qA − qB) dA,t−5/8 both jump by 5 units at t = 5
relative to their steady-state values. Thus, if the generalists who are active at t = 5, 6, and 7 did not react to this
pre-announced supply increase, the active supply of asset B would jump by 5 units at t = 8.
However, active generalists do react to the pre-announced supply increase: we have d̃A,t ≡ dA,t−δA0 > 0 at t = 5, 6, and
7. As a result, when the supply shock lands at t = 8, the active supply that must be absorbed is 5−(1− qA − qB) (d̃A,5+
d̃A,6 + d̃A,7)/8 < 5. As discussed below, this early mobilization of generalist investors reduces the active supply of asset
A that must be absorbed when the shock lands at t = 8, damping the overreaction of market A.
25Why do generalists buy A bonds in advance of the supply increase? Generalists have long-horizons (k = 8 in this
example), but can only adjust their portfolios slowly. Although generalists expect A bond yields to rise over the next
several quarters and, therefore, expect to suffer a capital loss on A bonds, they expect this capital loss to be more than
offset by an increase in income from holding A bonds. As a result, the expected cumulative 8-period excess return from
holding A bonds rises when a future supply increase is announced.
25
yields in market B underreact by 16%. Why do A yields rise gradually when the supply shock is pre-
announced but overreacted in Figure 1 when the same shock was unanticipated? First, risk premia in
the near future have a larger effect on bond yields than those in the distant future. Second, risk-premia
in the A market are only expected to rise significantly once the supply actually increases at quarter
t = 8. In combination, these two facts imply that yields in market A must rise gradually over time
following a pre-announced supply increase.
We can compare the dynamics of prices in the case of pre-announced increase in supply in Figure 3
to the unanticipated increase in supply in Figure 1. The long-run impact on yields and risk premia is
the same in either case. However, the short-run effects can be quite different whether or not a shock is
anticipated. When the shock is a unanticipated, yields and risk premia in the A market overreact more
strongly at announcement. Pre-announcing the supply shock mobilizes slow-moving generalists before
the supply of A actually rises. As shown in Figure 3 Panel C, this early mobilization reduces the active
supply of asset A that must be absorbed when the shock lands, damping the overreaction of market
A. While introducing a delay between announcement and the increase in supply limits overreaction,
this also lengthens the amount of time it takes for the full impact to be reflected in prices.
To put this in more concrete terms, consider a central bank that is seeking to reduce long-term
interest rates by purchasing long-term government bonds. Allowing for a long period of time between
announcement and implementation of the asset purchases will result in lower profits to market spe-
cialists who must accommodate the shock in the short-run. Of course, doing so also delays the desired
impact on asset prices. In this way, our model highlights an important trade-off that central bankers
face when designing asset purchase programs.
4 Multiple assets per market
In describing the model so far, we have made no distinction between “asset” A and “market” A.
This distinction takes on meaning when market A contains many different securities. For example, in
the Treasury market, there are many different securities with different maturities and thus different
exposures to interest rate risk.
In the Internet Appendix, we show how to extend our model to a more complex setting in which
there are multiple risky securities trading in each market. Specifically, suppose there are multiple
default-free bonds in market A which have different durations. Similarly, there are multiple bonds in
market B which differ in terms of their duration or their exposure to default risk. Fast-moving A-
specialists are free to adjust their holdings of all securities in the A market (and the riskless short-term
asset) each period, but cannot hold securities in market B. The B-specialists face an analogous set of
26
constraints. Generalists can invest capital in both markets, but, as in our baseline model, generalists
can only gradually reallocate their portfolios over time.
Subject to some mild conditions which guarantee that cross-market arbitrage remains risky for
generalist investors, the intuitions from the two asset model carry over to this richer setting with more
securities.26 Specifically, a CAPM-like pricing model based on the portfolio of A-specialists prices
all assets in the A market and a different CAPM-like pricing model– with different prices of factor
risk– based on the portfolio of B-specialists prices all assets in the B market. These two market-
specific pricing models are linked over time by the cross-market arbitrage activities of the slow-moving
generalists, who take steps to align the way that risk is priced by the two groups of specialists. And,
much as before, the degree of market integration over time depends on the risks faced by cross-market
arbitrageurs.
We illustrate the impact of a supply shock on two different markets that each contain multiple
assets using an example. Here we use an annual calibration– i.e., one period corresponds to a year–
and solve the model in the case where generalists reallocate their portfolios every k = 2 year and
with N = 2 assets in both the A and B markets. The two assets in each market differ solely in their
durations. The short-term bonds, denoted A1 and B1, have a duration of 5 years (i.e., DA1 = DB1 = 5).
The long-term bonds, denoted A2 and B2, have a duration of 10 years (i.e., DA2 = DB2 = 10). As
before, the two bonds in market A are default free whereas the two bonds in market B are exposed
to default risk. We assume that B1 and B2 have the same exposure to the common default process,
zt, and that there is no idiosyncratic default risk.
Figure 4 shows the evolution of risk premia following an unexpected shock in year 10 which per-
manently increases the supply of long-term default-free bonds (A2) by 3 units and reduces the supply
of short-term default-free bonds (A1) by the same amount. This scenario corresponds to a “reverse
Operation Twist” in which the Federal Reserve sells long-term Treasuries and reinvests the proceeds
in short-term Treasuries. Figure 4 shows that the risk premia for all four assets rise after impact.
Naturally, this supply shock has a larger impact on the risk premium for long-maturity bonds in
each market, leading both the A and B yield curves to steepen. These patterns are consistent with
existing term structure models in which bond supply shocks impact bond yields because the demand
for bonds is downward-sloping (e.g., Greenwood and Vayanos [2014] and Greenwood, Hanson, and
Vayanos [2016]).
26The key condition is that generalists are unable or unwilling to use only the securities in market A and only securities
in market B to construct factor-mimicking portfolios that isolate exposure to each of the underlying risk factors present
in each market. If the generalists can do this in both markets– i.e., if both markets are complete, then there is a riskless
arbitrage unless all agents charge the same prices of factor risk. The Internet Appendix spells out these conditions more
formally.
27
While risk premia on all securities in each market move in lockstep, risk premia in the A and B
markets do not move in lockstep. As in our simpler model, it takes time for the slow-moving generalists
to integrate the A and B markets following the shock, leading the risk premia of the two bonds in
market A to initially overreact and the risk premia of the two bonds in market B to initially underreact
in Figure 4. Furthermore, market integration remains imperfect even in the long run. Specifically, even
many period after the supply shock, Figure 4 shows that the yield curve in market A has steepened
more than the yield curve in market B.
5 Application: Event studies and changes in the price of risk
In response to the rapidly evolving financial crisis and global recession in late 2008 and early 2009,
central banks around the world announced their intentions to purchase substantial quantities of gov-
ernment bonds and other long-term debt securities. A central question in assessing the effectiveness
of these large-scale asset purchase programs is whether they impacted assets prices beyond the bonds
specifically targeted by central banks. Suppose, for example, that the impact of asset purchase pro-
grams was limited to asset classes in which the purchases were being made (Treasury bonds and
mortgage-backed securities in the case of the Federal Reserve), perhaps because these markets are
almost completely segmented from other financial markets. Such a finding should dampen central
bankers’enthusiasm for these programs, and cast doubt on the idea that asset purchases could affect
broader economic activity.
Our model provides a natural framework for understanding how these large-scale asset purchase
programs should spill across financial markets over time. According to our model, the largest short-run
effects of these quantitative easing (QE) programs should be felt in the asset classes that are directly
targeted. In the long run, however, changes in risk premia should spill over to non-targeted markets.
Differences between the short-run and long-run price impact should reflect the degree to which the
programs were anticipated, the length of time between the announcement and implementation, the ef-
fective degree of segmentation between different markets, and the speed at which capital flows between
markets.
Most empirical studies of these asset purchase programs have used an event study methodology,
focusing on the one day or intraday impact on bond yields following announcements of future asset
purchases. In one of the first of these event studies, Gagnon, Raskin, Remache, and Sack (2011) report
the cumulative interest rate changes on a set of dates between November 2008 and January 2010 when
the Federal Reserve made major announcement dates associated with its first round of quantitative
easing (QE1). Looking at the two asset classes that the Fed was directly targeting, they report a
28
91 basis points decline in 10-year Treasury yields and a 113 basis points decline in mortgage-backed
security yields. However, the find that Baa-rated corporate bond yields, which the Fed was not directly
targeting, declined by only 67 basis points on these same days. Swanson (2017) also finds that the
Fed’s asset purchase announcements had a larger impact on Treasury yields than on Baa corporate
bond yields.
Krishnamurthy and Vissing-Jorgensen (2011) extend this analysis to the Fed’s second round of
quantitative easing (QE2) and also discuss the impact on other assets, including high yield corporate
bonds. After controlling for other factors, Krishnamurthy and Vissing-Jorgensen conclude that the
effects of asset purchases were most pronounced among the assets being purchased (MBS and Trea-
suries in QE1 and Treasuries in QE2), suggesting a high degree of segmentation between different fixed
income markets.
At the same time, some researchers have recognized that short-horizon announcement returns may
not capture the full impact of these asset purchase programs. In their empirical assessment of the
Bank of England’s quantitative easing program, Joyce, Lasoasa, Stevens and Tong (2011) suggest that
these programs may have gradually impacted corporate bonds and equities. Fratzcher, Lo Duca, and
Straub (2013) suggest that the Fed’s asset purchase programs triggered portfolio flows that ultimately
impacted emerging market asset prices and foreign exchange rates.
Researchers have used different approaches to measure the long-run effects of large-scale asset
purchases. Joyce, Lasoasa, Stevens and Tong (2011) report the cumulative change in asset prices
for the longer period between March 4, 2009 and May 31, 2010 in addition to 1-day announcement
returns. They show that corporate bond yields fall by a cumulative 70 basis points around the Bank of
England’s asset purchase announcements, but by 400 basis points over the longer period. Mamaysky
(2014) takes a more tailored approach to each asset market, choosing an announcement window that
maximizes the statistical power of measured returns. Using this approach, he shows that the impact
of quantitative easing on both equity and high yield bond markets is much larger after 15 days than
what one would measure using a 1-day window. But even this approach may significantly understate
the long-run effect, because, as we have noted, it may take several quarters or even years for the full
impact of a large supply shocks to register.
Our model clarifies the broader issue at stake: event studies are a useful methodology for detecting
short-run price changes, but often lack the statistical power to detect changes in risk premia occurring
at longer horizons. Indeed, the event study methodology was originally developed in the 1970s to tackle
questions of informational effi ciency of stock prices, not changes in risk premia, but has increasingly
been used in other settings, such as in event studies assessing quantitative easing.
The limitations of event studies are particularly severe when there is noise from “cash flow”news.
29
For instance, consider the effects of σz– the volatility of fundamental cash flow shocks in market
B– on our ability to detect the impact on prices in market B stemming from a supply shock that
hits market A . When σz is large relative to σr, there is insuffi cient power to detect changes in risk
premia in market B. The statistical power would increase with the number of events, but power is
nonetheless decreasing in σz. Thus, our model suggests that short-run event studies may have a hard
time detecting spillover effects on markets, such as equities and high yield bonds, where there can
be significant confounding news. More generally, our framework suggests that event studies are an
inappropriate methodology for measuring cross-market price impact, especially at long horizons.
In summary, our model suggests that researchers should be cautious in using event studies to assess
the long-run impact of large supply shocks on market prices and risk premia. Measuring the long-run
impact of supply shocks across markets is inherently diffi cult because the full economic impact may
occur over such a long time that it is swamped by other factors.
6 Conclusion
We have studied how security prices react when there is a large shock to the supply of one asset
class, such as when central banks embark on a large-scale purchase of government bonds. Our model
incorporates two key asset pricing frictions: partial segmentation between markets and slow-moving
capital. The presence of these two frictions means that when one market is hit with a supply shock,
prices of risk in that market become disconnected from prices in other related markets. In the longer
run, however, there is convergence in the pricing of risk as capital flows across markets. In summary,
it takes time to digest large supply shocks and for the impact to fully spill over to related markets.
Our findings sound a note of caution for researchers who seek to measure the impact of quantitative
easing policies on asset prices, a topic that has attracted considerable attention in recent years. We have
shown that the standard technique for measuring price impact– through an event study of short-run
price responses to central bank announcements– is poorly suited to measuring long-run changes in risk
premia across markets. While prices in neighboring markets will often change at short horizons, these
near-term changes are likely to reveal more about the short-run capacity of specialists to accommodate
asset flows than about the ultimate long-term changes in risk premia.
30
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Figure 1: Impact of an unanticipated shock to the supply of A bonds. This figure shows the impact 
of an unanticipated shock that doubles the supply of A bonds in quarter 5. The model parameters 
for this illustration are shown in Table 1. Panel A shows the evolution of annual bond risk premia 
in market A, Et[rxA,t+1], and market B, Et[rxB,t+1], over time. Panel B shows the evolution of specialists 
holdings in markets A and B (bA,t and bB,t) as well as the positions of active generalists (dA,t and dB,t). 
Panel C shows the evolution of the “active supplies” of A and B bonds. The active supply of A is 
𝑠𝑠 − (1 − 𝑞𝑞 − 𝑞𝑞 )𝑘𝑘−1 ∑𝑘𝑘−1𝐴𝐴,𝑡𝑡 𝐴𝐴 𝐵𝐵 𝑖𝑖=1 𝑑𝑑𝐴𝐴,𝑡𝑡−𝑖𝑖 and the active supply of B is defined analogously. Panel D shows 
the evolution of bond yields in market A, yA,t, and market B, yB,t, over time. 
 
  
34 
 
 
 
 
 
 
Figure 2: Unpacking the two key asset-pricing frictions. This figure shows the response of annual 
bond risk premia in market A, Et[rxA,t+1], and market B, Et[rxB,t+1], to an unanticipated shock that 
doubles the supply of A bonds in quarter 5. Each panel corresponds to a different configuration of 
the two key asset pricing frictions in our model. Panel A shows the response in benchmark case with 
no frictions. This is a special case of our model where 𝑞𝑞𝐴𝐴 = 𝑞𝑞𝐵𝐵 = 0 and 𝑘𝑘 = 1. Panel B shows the 
results when there is only partial segmentation. This is a special case of our model where 𝑞𝑞𝐴𝐴 > 0,
𝑞𝑞𝐵𝐵 > 0, 1 − 𝑞𝑞𝐴𝐴 − 𝑞𝑞𝐵𝐵 > 0, and 𝑘𝑘 = 1. Panel C shows the response with only slow-moving capital. This 
case corresponds to a multi-asset version of Duffie’s (2010) model and is not a special case of our 
model. We set the fraction of fast-moving investors to 𝑞𝑞𝐴𝐴 + 𝑞𝑞𝐵𝐵. Finally, Panel D shows the response 
in our general model featuring both partial segmentation (𝑞𝑞𝐴𝐴 > 0, 𝑞𝑞𝐵𝐵 > 0, 
 1 − 𝑞𝑞𝐴𝐴 − 𝑞𝑞𝐵𝐵 > 0) and slow-moving capital (𝑘𝑘 > 1). The other model parameters for this illustration 
are shown in Table 1. 
 
35 
 
 
 
 
Figure 3: Impact of a pre-announced increase in the supply of A bonds. This figure shows the impact 
of pre-announced supply increase: there is an announcemement in quarter 5 that the supply of A 
bonds will double in quarter 8. The model parameters for this illustration are shown in Table 1. 
Panel A shows the evolution of annual bond risk premia in market A, Et[rxA,t+1], and market B, 
Et[rxB,t+1], over time. Panel B shows the evolution of specialists holdings of A and B bonds (bA,t and 
bB,t) as well as the holdings of active generalists (dA,t and dB,t). Panel C shows the evolution of the 
“active supplies” of A and B bonds. The active supply of A is 𝑠𝑠𝐴𝐴,𝑡𝑡 − (1 − 𝑞𝑞𝐴𝐴 − 𝑞𝑞𝐵𝐵)𝑘𝑘−1 ∑𝑘𝑘−1𝑖𝑖=1 𝑑𝑑𝐴𝐴,𝑡𝑡−𝑖𝑖 and 
the active supply of B is defined analogously. Panel D shows the evolution of bond yields in market 
A, yA,t, and market B, yB,t, over time.  
 
 
36 
 
Figure 4: Price impact with multiple securities in each market. This figure shows the impact on bond 
risk premia of an unexpected supply shock that permanently increases the supply of long-term 
default-free bond (𝐴𝐴2) by 3 units and decreases the supply of short-term default-free bond (𝐴𝐴1) by 
the same amount in year 10. In Figure 4, one period corresponds to a year. We choose the annual 
persistence parameters for short rates, default losses, and bond supply to be consistent with the 
quarterly persistence assumed in Table 1. The volatility of annual shocks are then chosen so that the 
unconditional volatility of each process is the same in the annual and quarterly calibrations. We 
assume 𝜏𝜏 = 2,𝑘𝑘 = 2, and 𝑞𝑞𝐴𝐴 = 𝑞𝑞𝐵𝐵 = 40% in this figure. Panel A shows the evolution of risk premia 
for short-term securities in each market (𝐴𝐴1 and 𝐵𝐵1). Panel B shows the evolution of risk premia for 
long-term securities in each market (𝐴𝐴2 and 𝐵𝐵2). 
  
37 
 
Table 1: Illustrative model parameters for numerical exercises. This table presents the illustrative 
model parameters that we use throughout our numerical exercises. One period corresponds to one 
quarter of a year. We report annualized values for the mean and standard deviation of shocks to 
both the short rate and default losses on asset B. 
 
 
Parameter Description Value 
𝑞𝑞𝐴𝐴,𝑞𝑞𝐵𝐵 Percentage of investors that are specialists in A and B 45% 
𝑘𝑘 Number of quarters between generalist portfolio rebalancing 8 
?̅?𝑟 Average annualized short-term riskless rate 4% 
𝜎𝜎𝑟𝑟 Volatility of quarterly shocks to annualized short-term riskless rate 0.64% 
𝜌𝜌𝑟𝑟 Quarterly persistence of short-term riskless rate 0.96 
𝑧𝑧̅ Expected annualized default losses on asset B 0.4% 
𝜎𝜎𝑧𝑧 Volatility of quarterly shocks to annualized default losses on asset B 0.4% 
𝜌𝜌𝑧𝑧 Quarterly persistence of default losses on asset B 0.96 
?̅?𝑠𝐴𝐴, ?̅?𝑠𝐵𝐵 Average asset supplies 5 
𝜎𝜎𝑠𝑠 ,𝜎𝜎𝑠𝑠  Volatility of quarterly supply shocks 1 𝐴𝐴 𝐵𝐵
𝜌𝜌𝑠𝑠 ,𝜌𝜌𝑠𝑠  Quarterly persistence of supply shocks 0.999 𝐴𝐴 𝐵𝐵
𝐷𝐷𝐴𝐴,𝐷𝐷𝐵𝐵 Macaulay duration in years (implies 𝜃𝜃𝐴𝐴 = 𝜃𝜃𝐵𝐵=0.95) 5 years 
𝜏𝜏 Aggregate investor risk tolerance 1.75 
 
38 
 
Table 2: Model comparative statics. This table shows how the price impact of the same supply shock—an unanticipated shock that doubles the supply 
of asset A—varies as we change key model parameters one at a time. All other parameters are held constant at the values listed in Table 1. For a 
given set of model parameters, we summarize the impact of the supply shock on both the A and B markets by listing (i) the yields and expected 
annual returns in the period before the shock arrives (labeled as “pre-shock level”), (ii) the changes in yields and expected annual returns in the period 
when the shock arrives (labeled as “short-run ∆”), and (iii) in 2k periods after the shock arrives (labeled as “long-run ∆”). Finally, we report the 
degree to which bond yields over- or underreact as the difference between the short-run change and the long-run change, expressed as a percentage of 
the long-run change 
%Short-Run-Over-React(y)= (yt − yt−1) − (yt+2k − yt−1) . 
(yt+2k − yt−1)
Our measure of over-reaction for risk premia is defined analogously. 
 
  Market A Market B 
  Risk premia, Et[rxA,t+1] Yields, yA,t Risk premia, Et[rxB,t+1] Yields, yB,t 
  Pre Short Long Over- Pre Short Long Over- Pre Short Long Over- Pre Short Long Over- 
shock 
level run ∆ run ∆ react 
shock shock shock 
level run ∆ run ∆ react level run ∆ run ∆ react level run ∆ run ∆ react 
Supply shock hits market A                 
(1) Baseline example in Figure 1  0.79 0.80 0.56 42% 4.79 0.59 0.55 7% 1.07 0.04 0.23 -83% 5.47 0.20 0.22 -12% 
(2) More risk tolerant 𝜏𝜏 = 200 0.67 0.68 0.47 43% 4.67 0.50 0.46 7% 0.90 0.04 0.19 -83% 5.30 0.17 0.19 -12% 
(3) No generalists 𝑞𝑞𝐴𝐴 = 𝑞𝑞𝐵𝐵 = 0.5 0.81 0.81 0.80 0% 4.81 0.80 0.79 0% 1.33 0.00 0.00 N/A 5.73 0.00 0.00 N/A 
(4) More B specialists 𝑞𝑞𝐴𝐴 = 0.3, 𝑞𝑞𝐵𝐵 = 0.6 1.01 1.28 0.75 68% 5.01 0.83 0.74 10% 0.92 0.05 0.24 -82% 5.32 0.21 0.23 -11% 
(5) More A specialists 𝑞𝑞𝐴𝐴 = 0.6, 𝑞𝑞𝐵𝐵 = 0.3 0.71 0.59 0.46 27% 4.71 0.48 0.45 4% 1.51 0.05 0.24 -82% 5.91 0.21 0.23 -12% 
(6) Slower-adjusting generalists 𝑘𝑘 = 12 0.80 0.82 0.55 46% 4.80 0.61 0.54 10% 1.08 0.03 0.23 -89% 5.48 0.19 0.23 -18% 
(7) Faster-adjusting generalists 𝑘𝑘 = 4 0.79 0.76 0.56 33% 4.79 0.57 0.55 3% 1.07 0.08 0.22 -66% 5.47 0.21 0.22 -5% 
 
(8) Many generalists 𝑞𝑞 = 0.2,   𝑞𝑞𝐵𝐵 = 0.2, 𝐴𝐴
𝑘𝑘 = 4 0.77 1.12 0.41 174% 4.77 0.44 0.39 12% 0.94 0.42 0.35 18% 5.34 0.38 0.36 4% 
(9) Greater default risk for B 𝜎𝜎𝑍𝑍 = 0.64% 
0.83 0.83 0.63 30% 4.83 0.66 0.62 5% 1.85 0.03 0.19 -85% 6.25 0.16 0.18 -13% 
 
 
39