FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2007 305

Arbitrage-Free Bond Pricing withDynamic Macroeconomic Models

Michael F. Gallmeyer, Burton Hollifield, Francisco J. Palomino, and Stanley E. Zin

The authors examine the relationship between changes in short-term interest rates induced bymonetary policy and the yields on long-maturity default-free bonds. The volatility of the long endof the term structure and its relationship with monetary policy are puzzling from the perspectiveof simple structural macroeconomic models. The authors explore whether richer models of riskpremiums, specifically stochastic volatility models combined with Epstein-Zin recursive utility,can account for such patterns. They study the properties of the yield curve when inflation is anexogenous process and compare this with the yield curve when inflation is endogenous anddetermined through an interest rate (Taylor) rule. When inflation is exogenous, it is difficult tomatch the shape of the historical average yield curve. Capturing its upward slope is especiallydifficult because the nominal pricing kernel with exogenous inflation does not exhibit any negativeautocorrelation—a necessary condition for an upward-sloping yield curve, as shown in Backusand Zin. Endogenizing inflation provides a substantially better fit of the historical yield curvebecause the Taylor rule provides additional flexibility in introducing negative autocorrelationinto the nominal pricing kernel. Additionally, endogenous inflation provides for a flatter termstructure of yield volatilities, which better fits historical bond data. (JEL G0, G1, E4)

Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 305-26.

Then in his February 16, 2005, testimony,Chairman Greenspan (2005) expressed a com-pletely different concern about long rates:

[L]ong-term interest rates have trended lowerin recent months even as the Federal Reservehas raised the level of the target federal fundsrate by 150 basis points…Historically, though,even these distant forward rates have tendedto rise in association with monetary policytightening...For the moment, the broadlyunanticipated behavior of world bond marketsremains a conundrum.

Chairman Greenspan’s comments reflect thefact that we do not yet have a satisfactory under-

The response of long-term interest ratesto changes in short-term interest ratesis a feature of the economy that oftenpuzzles policymakers. For example, in

remarks made on May 27, 1994, Alan Greenspan(1994, p. 5) expressed concern that long ratesmoved too much in response to an increase inshort rates:

In early February, we had thought long-termrates would move a little higher temporarilyas we tightened…The sharp jump in [long]rates that occurred appeared to reflect thedramatic rise in market expectations of eco-nomic growth and associated concerns aboutpossible inflation pressures.

Michael F. Gallmeyer is an assistant professor of finance at the Mays Business School, Texas A&M University; Burton Hollifield is an associateprofessor of financial economics at the David A. Tepper School of Business, CarnegieMellon University; Francisco J. Palomino is a PhD candidateat the David A. Tepper School of Business, Carnegie Mellon University; and Stanley E. Zin is a professor of economics at the David A. TepperSchool of Business, Carnegie Mellon University, and a research fellow at the National Bureau of Economic Research. The authors thank DavidBackus and Pamela Labadie for valuable comments and suggestions, Bill Gavin for suggesting the topic, Monika Piazzesi and Chris Telmerfor providing data, and Zhanhui Chen for excellent research assistance.

© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted intheir entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be madeonly with prior written permission of the Federal Reserve Bank of St. Louis.

standing of how the yield curve is related to thestructural features of the macroeconomy, such asinvestors’ preferences, the fundamental sourcesof economic risk, and monetary policy.

Figure 1 plots the nominal yield curve for avariety of maturities from 1 quarter—which werefer to as the short rate—up to 40 quarters forU.S. Treasuries, starting in the first quarter of 1970and ending in the last quarter of 2005.1 Figure 2plots the average yield curve for the entire sampleand for two subsamples. Figure 3 plots the stan-dard deviation of yields against their maturities.Two basic patterns of yields are clear from thesefigures: (i) On average, the yield curve is upward

sloping and (ii) there is substantial volatility inyields at all maturities. Chairman Greenspan’scomments, therefore, must be framed by the factthat long yields are almost as volatile as shortrates. The issue, however, is the relationship ofthe volatility at the long end to the volatility atthe short end, and the correlation between changesin short-term interest rates and changes in long-term interest rates.

We can decompose forward interest rates intoexpectations of future short-term interest rates andinterest rate risk premia. Because long-term inter-est rates are averages of forward rates, long-runinterest rates depend on expectations of futureshort-term interest rates and interest rate risk pre-miums. A significant component of long rates is

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306 JULY/AUGUST 2007 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW

1 Yields up to 1991 are from McCulloch and Kwon (1993) thenDatastream from 1991 to 2005.

10

20

30

40

19701975198019851990199520002005

2

4

6

8

10

12

14

Nominal Yields

Maturity (Quarters)

Yie

lds

(%C

ont

inuo

usly

Co

mp

oun

ded

Ann

ualR

etur

n[C

CA

R])

Figure 1

Time-Series Properties of the Yield Curve, 1970:Q1–2005:Q4

the risk premium, and there is now a great dealof empirical evidence documenting that the riskpremiums are time-varying and stochastic. Move-ments in long rates can therefore be attributed tomovements in expectations of future nominalshort rates, movements in risk premiums, or somecombination of movements in both.

Moreover, if monetary policy is implementedusing a short-term interest rate feedback rule—forexample, a Taylor rule—then inflation rates mustadjust so that the bond market clears. The result-ing endogenous equilibrium inflation rate willthen depend on the same risk factors that driverisk premiums in long rates. Monetary policyitself, therefore, could be a source of fluctuationsin the yield curve in equilibrium.

We explore such possibilities in a model oftime-varying risk premiums generated by therecursive utility model of Epstein and Zin (1989)combined with stochastic volatility of endowment

growth. We show how the model can be easilysolved using now-standard affine term-structuremethods. Affine term-structure models have theconvenient property that yields are maturity-dependent linear functions of state variables. Weexamine some general properties of multi-perioddefault-free bonds in our model, assuming firstthat inflation is an exogenous process and byallowing inflation to be endogenous and deter-mined by an interest rate feedback rule. We showthat the interest rate feedback rule—the form ofmonetary policy—can have significant impacts onproperties of the term structure of interest rates.

THE DUFFIE-KAN AFFINE TERM-STRUCTURE MODEL

The Duffie and Kan (1996) class of affine term-structure models, translated into discrete time

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FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2007 307

0 5 10 15 20 25 30 35 404

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

1970:Q1–2005:Q4

1990:Q1–2005:Q4

1970:Q1–1989:Q4

Maturity (Quarters)

Mean Nominal Yields (% CCAR)

Figure 2

Average Yield-Curve Behavior

by Backus, Foresi, and Telmer (2001), is basedon a k-dimensional vector of state variables zthat follows a “square root” model:

where {ε t} ~ NID�0,1�,Σ�z� is a diagonal matrixwith a typical element given by σi�z� = ai + bi′z,where bi has nonnegative elements, andΦ is stablewith positive diagonal elements. The process forz requires that the volatility functions, σi�z�, bepositive, which places additional restrictions onthe parameters.

The asset-pricing implications of the modelare given by the pricing kernel,mt+1, a positiverandom variable that prices all financial assets.That is, if a security has a random payoff, ht+1, atdate t+1, then its date-t price is Et[mt+1ht+1]. Thepricing kernel in the affine model takes the form

z I z zt t t t+ += −( ) + + ( )11 2

1Φ Φ Σθ ε/,

where the k × 1 vector γ is referred to as the“factor loadings” for the pricing kernel, the k × 1vector λ is referred to as the “price of risk” vectorbecause it controls the size of the conditionalcorrelation of the pricing kernel and the under-lying sources of risk, and the k × kmatrix Σ�zt� isthe stochastic variance-covariance matrix of theunforecastable shock.

Let bt�n� be the price at date t of a default-free

pure-discount bond that pays 1 at date t+n, withbt

�0� = 1. Multi-period default-free discount bondprices are built up using the arbitrage-free pricingrestriction,

(1) b E m btn

t t tn( )

+ +−( )=

1 1

1 .

− = + ′ + ′ ( )+ +log ,m z zt t t t11 2

1δ γ λ εΣ /

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308 JULY/AUGUST 2007 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

Maturity (Quarters)

Standard Deviation of Nominal Yields (% CCAR)

1970:Q1–1989:Q4

1970:Q1–2005:Q4

1990:Q1–2005:Q4

Figure 3

Volatility of Yields of Various Maturities

Bond prices of all maturities are log-linear func-tions of the state:

whereA�n� is a scalar and B�n� is a 1 × k row vector.The intercept and slope parameters, which we

often refer to as “yield-factor loadings,” of thesebond prices can be found recursively according to

(2)

where Bj�n� is the j th element of the vector B�n�.

Because b�0� = 1, we can start these recursionsusing A�0� = 0 and Bj

�0� = 0, j = 1,2,…,k.Continuously compounded yields, yt

�n�, aredefined by bt

�n� = exp�–nyt�n��, which impliesyt

�n� = –�logbt�n��/n. We refer to the short rate, it, as

the one-period yield: it � yt�1�.

This is an arbitrage-free model of bond pricingbecause it satisfies equation (1) for a given pricingkernelmt. It is not yet a structural equilibriummodel, because the mapping of the parameters ofthe pricing model to deeper structural parametersof investors’ preferences and opportunities hasnot yet been specified. The equilibrium structuralmodels we consider will all lie within this generalclass, hence, can be easily solved using thesepricing equations.

A TWO-FACTOR MODEL WITHEPSTEIN-ZIN PREFERENCES

We begin our analysis of structural modelsof the yield curve by solving for equilibriumreal yields in a representative-agent exchangeeconomy. Following Backus and Zin (2006), weconsider a representative agent who choosesconsumption to maximize the recursive utilityfunction given in Epstein and Zin (1989). Givena sequence of consumption, {ct,ct+1,ct+2,…}, wherefuture consumptions can be random outcomes,

A A B I Bn n nj j

n

j

k+( ) ( ) ( ) ( )

== + + −( ) − +( )∑1

2

1

12

δ θ λΦ aa

B B B b

j

n nj j

n

j

k

j

,

+( ) ( ) ( )

== ′ +( ) − +( ) ′∑1 2

1

12

γ λΦ ,,

− = +( ) ( ) ( )log ,b A B ztn n n

t

the intertemporal utility function, Ut, is the solu-tion to the recursive equation,

(3)

where 0 < β < 1 characterizes impatience (themarginal rate of time preference is 1–1/β ), ρ ≤ 1measures the preference for intertemporal sub-stitution (the elasticity of intertemporal substi-tution for deterministic consumption paths is1/�1 – ρ�), and the certainty equivalent of randomfuture utility is

(4)

where α ≤ 1 measures static risk aversion (thecoefficient of relative risk aversion for staticgambles is 1 – α). The marginal rate of intertem-poral substitution,mt+1, is

Time-additive expected utility corresponds tothe parameter restriction ρ = α.

In equilibrium, the representative agent con-sumes the stochastic endowment, et, so that,log �ct+1/ct� = log �et+1/et� = xt+1, where xt+1 is thelog of the ratio of endowments in t+1 relative to t .The log of the equilibrium marginal rate of sub-stitution, referred to as the real pricing kernel, istherefore given by

(5)

whereWt is the value of utility in equilibrium.The first two terms in the marginal rate of

substitution are standard expected utility terms:the pure time preference parameter, β, and a con-sumption growth term times the inverse of thenegative of the intertemporal elasticity of substitu-tion. The third term in the pricing kernel is a newterm coming from the Epstein-Zin preferences.

The endowment-growth process evolves sto-chastically according to

x x vt x x x t t tx

+ += −( ) + +11 2

11 φ θ φ ε/ ,

log� log�

log� log�

m x

W

t t

t

+ +

+

= + −( )+ −( ) −

1 1

1

1β ρ

α ρ µµt tW +( ) 1 ,

mcc

UUt

t

t

t

t t+

+−

+

+

−

=

( )

11

11

1

βµ

ρ α ρ

.

µ α αt t t tU E U+ +( ) 1 1

1

; ,

U c Ut t t t= −( ) + ( )

+1 1

1β βµρ ρ ρ/

,

Gallmeyer, Hollifield, Palomino, Zin

FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2007 309

where

is the process for the conditional volatility ofendowment growth. We will refer to vt as sto-chastic volatility. The innovations εt

x and εtv are

distributed NID�0,1�.Note that the state vector in this model con-

forms with the setup of the Duffie-Kan modelabove. Define the state vector zt � [xt vt]′, whichimplies parameters for the Duffie-Kan model:

Following the analysis in Hansen, Heaton, andLi (2005), we will work with the logarithm of thevalue function scaled by the endowment:

(6)

where we have used the linear homogeneity of µt(see equation (4)). Take logarithms of (6) to obtain

where wt � log�Wt/et� and ut � log �µt�exp�wt+1 +xt+1���. Consider a linear approximation of theright-hand side of this equation as a function ofut around the pointm

–:

where κ < 1. For the special case with ρ = 0, thatis, a log time aggregator, the linear approximationis exact, implying κ– = 1 – β and κ = β (see Hansen,Heaton, and Li, 2005). Similarly, approximatingaroundm– = 0, results in κ– = 0 and κ = β.

w m

m

t ≈ −( ) + ( )

+( )

− +

−ρ β β ρ

β ρβ β

1 1

1

log� exp

eexp

xxp

,

ρκ κ

mu m

u

t

t

( )

−( )

+;

w ut t= −( ) + ( ) −ρ β β ρ1 1log� exp ,�

W e W et t t t t/ /

–

/

= −( ) + ( )( )

= ( ) +

+1

1

1

1

β β µ

β

ρ ρ

ββ µρ

tt

t

t

t

We

ee

+

+

+×

1

1

1

1// ρ

,

a b a bv1 1 22

20 0 1 0 0= = [ ]′ = = [ ]′, , ,� .σ

Σ z a b z a b zt t t( ) = + ′ + ′{ }diag ,�1 1 2 2

Φ = { }diag ,φ φx v

θ θ θ= [ ]′x v

v vt v v v t v tv

+ += −( ) + +1 11 φ θ φ σ ε

Given the state variables and the log-linearstructure of the model, we conjecture a solutionfor the log value function of the form

where ω–, ωx, and ωv are constants to be deter-mined. By substituting,

Because xt+1 and vt+1 are jointly normally distrib-uted, the properties of normal random variablescan be used to solve for ut:

We can use the above expression to solve for thevalue-function parameters and verify its log-linear solution:

The solution allows us to simplify the term[logWt+1 – logµt�Wt+1�] in the real pricing kernelin equation (5):

ω κ ω φ

ω κκφ

φ

ω κ ω φ α ω

x x x

xx

x

v v v x

= +( )

⇒ =−

= +

1

1

2++( )

⇒ =−

−

1

1 21

1

2

2

ω κκφ

ακφv

v x

=−

+−

+( ) −( )+ −

ω κκ

κκ

ω φ θ

ω φ�

1 1

1 1

1

x x x

v v(( ) +

θ α ω σv v v2

2 2.

u w x

E w x

t t t t

t t

; log� exp

log� exp

µ + +

+

+( )( )( )= +

1 1

1 tt

t t tE w x

+

+ +

( )

= + +

1

1

1 1 2

α α

αVartt t t

x x x v v v

w x+ ++

= + +( ) −( ) + −( )1 1

1 1 1ω ω φ θ ω φ θ

++ +( ) +

+ +( ) +

ω φ ω φα ω α ω σ

x x t v v t

x t v v

x v

v

1

21

22 2 2.

w x x vt t x t v t+ + + ++ = + +( ) +1 1 1 11ω ω ω .

w x vt x t v t= + +ω ω ω ,

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310 JULY/AUGUST 2007 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW

The real pricing kernel, therefore, is a memberof the Duffie-Kan class with two factors andparameters:

(7)

We can now use the recursive formulas in equa-tion (2) to solve for real discount bond prices andthe real yield curve.

Note how the factor loadings and prices ofrisk depend on the deeper structural parametersand the greatly reduced dimensionality of theparameter space relative to the general affinemodel. Also, for the time-additive expected utilityspecial case, α = ρ, the volatility factor does notenter the conditional mean of the pricing kernelbecause γv = 0; and the price of risk for the volatil-ity factor is zero because λv = 0. Finally, we cansee from the expressions for bond prices that thetwo key preference parameters, ρ and α, providefreedom in controlling both the factor loadingsand the prices of risk in the real pricing kernel.

NOMINAL BOND PRICINGTo understand the price of nominal bonds,

we need a nominal pricing kernel. If we assume

δ β ρ φ θ α α ρ ω σ

γ γ γ

= − ( ) + −( ) −( ) + −( )

=

log 1 12

2 2x x v v

x v[[ ]′

= −( ) −( )−

′1

21

1

2

ρ φ α α ρκφx

x

λλ λ λ

α α ρ κφκφ

α

= [ ]′

=

−( ) − −( )−

−

x v

x

x

11

2

−( )−

−

κ α ρκφ κφ1

11

2

v x

.

log� log� log�W W w x wt t t t t t t+ + + + +− ( ) = + − +1 1 1 1 1µ µ xx

x E x v E v

t

x t t t v t t t

+

+ + + +

( )= +( ) − + −

1

1 1 1 11ω ω

− +( ) − + +α ω α ω2

12

21

21x t t v t tx vVar Var

= +( ) + − +( ) −+ +ω ε ω σ ε α ωx t tx

v v tv

x tv v12

11 21 1

2/ αα ω σ2

2 2v v .

that there is a frictionless conversion of moneyfor goods in this economy, the nominal kernel isgiven by

(8)

where pt+1 is the log of the money price of goodsat time t+1 relative to the money price of goodsat time t, that is, the inflation rate between t andt+1. Clearly then, the source of inflation, its ran-dom properties, and its relationship to the realpricing kernel is of central interest for nominalbond pricing. We next consider two differentspecifications for equilibrium inflation.

Exogenous Inflation

If we expand the state space to include anexogenous inflation process, pt, the state vectorbecomes zt = [xt vt pt]′. The stochastic processfor exogenous inflation is

where ε Pt+1 is also normally distributed independ-

ently of the other two shocks. In this case, theparameters for the affine nominal pricing kernelare

In the exogenous inflation model, the price ofinflation risk is always exactly 1 and does notchange with the values of any of the other struc-tural parameters in the model. In addition, thefactor loadings and prices of risk for output growthand stochastic volatility are the same as in thereal pricing kernel. We will refer to this nominalpricing kernel specification as the exogenousinflation economy.

Monetary Policy and EndogenousInflation

We begin by assuming that monetary policyfollows a simple nominal interest rate rule. Wewill abuse conventional terminology and often

δ δ φ θ

γ γ γ φ

λ λ λ

$

$

$

= + −( )=

′

= [ ]′

1

1

p p

x v p

x v .

p pt p p p t p tp

+ += −( ) + +1 11 φ θ φ σ ε ,

log log ,m m pt t t+ + +( ) = ( ) −1 1 1$

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FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2007 311

refer to the interest rate rule as a Taylor rule.Although there are a variety of ways to specify aTaylor rule—see Ang, Dong, and Piazzesi (2004)—we will consider a rule in which the short-terminterest rate depends on contemporaneous output,inflation, and a policy shock:

(9)

where the monetary policy shock satisfies

and where εts ~ NID�0,1� is independent of the

other two real shocks.Because this nominal interest rate rule must

also be consistent with equilibrium in the bondmarket, that is, it must be consistent with thenominal pricing kernel in equation (8) as well asequation (9), we can use these two equations tofind the equilibrium process for inflation. Con-jecture a log-linear solution for pt,

(10)

with π–, πx, and πs constants to be solved.To solve for a rational expectations solution

to the model, we substitute the guess for theinflation rate into both the Taylor rule and thenominal pricing kernel and solve for the parame-ters π–, πx, πv, and πs that equate the short ratedetermined by the pricing kernel with the shortrate determined by the Taylor rule.

From the dynamics of xt+1, vt+1, and st+1,inflation, pt+1, is given by

Substituting into the nominal pricing kernel,

− ( ) = − ( ) +

= + + +

+ + +log logm m p

x v v

t t t

x t v t x

1 1 1$

δ γ γ λ tt tx

v v tv

t

x x x v

p1 21 1 1

1

/ ε λ σ εδ π π φ θ π

+ + ++ +

= + + −( ) + 11 −( )+ +( ) + +( ) +

+

φ θ

γ π φ γ π φ π φ

λ

v v

x x x t v v v t s s t

x

x v s

++( ) + +( ) ++ + +π ε λ π σ ε π σ εx t tx

v v v tv

s s tsv1 2

1 1 1/ .

p x v st x t v t s t

x x x v

+ + + += + + +

= + −( ) +1 1 1 1

1

π π π ππ π φ θ π 11

1 2

−( )+ + +

+ +

φ θπ φ π φ π φ

π ε

v v

x x t v v t s s t

x t t

x v s

v /11 1 1

xv v t

vs s t

s+ ++ +π σ ε π σ ε .

p x v st x t v t s t= + + +π π π π ,

s st s t s ts= +−φ σ ε1

i x p st x t p t t= + + +τ τ τ ,

From these dynamics, the nominal one-periodinterest rate, it = –log �Et[m

$t+1]�, is

Comparing this with the interest rate rule,

gives the parameter restrictions consistent withequilibrium:

(11)

These expressions form a recursive system weuse to solve for the equilibrium parameters ofthe inflation process. See Cochrane (2006) for amore detailed account of this rational expecta-tions solution method.

It is clear from these expressions that theequilibrium inflation process will depend on thepreference parameters of the household generallyand attitudes toward risk specifically.

In a similar fashion, we can extend the analy-sis to any Taylor-type rule that is linear in thestate variables, including (i) lagged short rates,(ii) other contemporaneous yields at any maturity,and (iii) forward-looking rules, such as those inClarida, Galí, and Gertler (2000). Such extensionsare possible because, in the affine framework,interest rates are all simply linear functions ofthe current state variables. See Ang, Dong, andPiazzesi (2004) and Gallmeyer, Hollifield, andZin (2005) for some concrete examples.

π γ ττ φ

πγ λ π

τ φ

πτ φ

xx x

p x

v

v x x

p v

sp

= −−

=− +( )

−

= −−

12

1

2

ss

p

x x x v v v

v v

πτ

δ τ π φ θ π φ θ

λ π=

−

− + −( ) + −( )− +(

11

1 1

12

)) −

2 2 2 21

2σ π σv s s

.

i x x v s st x p x t v t s t t= + + + + +( ) +τ τ τ π π π π ,

i

xt x x x v v v

x x x t v

= + + −( ) + −( )+ +( ) +

δ π π φ θ π φ θ

γ π φ γ

1 1

++( ) +

− +( ) − +( )π φ π φ

λ π λ π σ

v v t s s t

x x t v v v

v s

v12

12

2 2 22 2 212

− π σs s .

Gallmeyer, Hollifield, Palomino, Zin

312 JULY/AUGUST 2007 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW

A Monetary Policy–Consistent PricingKernel

Substituting the equilibrium inflation processfrom equations (10) and (11) into the nominalpricing kernel, we obtain an equilibrium three-factor affine term-structure model that is consis-tent with the nominal interest rate rule. Thestate space is

and the parameters of the pricing kernel are

We will often refer to this nominal pricing kernelspecification as the endogenous inflationeconomy.

The Taylor rule parameters, through theirdetermination of the equilibrium inflation process,affect both the factor loadings on the real factorsas well as their prices of risk. Monetary policythrough its effects on endogenous inflation, there-fore, can result in risk premiums in the term struc-ture that are significantly different from those inthe exogenous inflation model. We explore sucha possibility through numerical examples.

QUANTITATIVE EXERCISESWe calibrate the exogenous processes in our

model to quarterly postwar U.S. data as follows:

1. Endowment growth: φx = 0.36, θx = 0.006,σx = 0.0048�1 – φx

2�1/2

δ δ π π φ θ π φ θ

γ γ φ π γ φ

$

$

= + + −( ) + −( )= + +

x x x v v v

x x x v v

1 1

ππ φ π

λ λ π λ π π

v s s

x x v v s

′

= + + ′$ .

z x v s

z

t t t t

x v s

x v

t

; ′

= { }= [ ]′

Φ

Σ

diag , ,φ φ φ

θ θ θ 0

(( ) = + ′ + ′ + ′{ }=

diag ,� ,�

,

a b z a b z a b z

a

t t t1 1 2 2 3 3

1 0 bb

a b

a b

v

s

1

22

2

32

3

0 1 0

0 0 0

0 0 0

= [ ]′

= = [ ]′

= = [ ]′σ

σ

,

, ,

2. Inflation: φp = 0.8471, θp = 0.0093,σp = 0.0063�1 – φp

2�1/2

3. Stochastic volatility: φv = 0.973,θv = 0.0001825, σv = 0.9884 × 10–5

4. Policy shock: φs = 0.922,σs = �0.023 × 10–4�1/2

The endowment growth process is calibratedto quarterly per capita consumption of durablegoods and services, and inflation is calibrated tothe nondurables and services deflator, similarlyto Piazzesi and Schneider (2007). The volatilityprocess is taken from Bansal and Yaron (2004),who calibrate their model to monthly data. Weadjust their parameters to deal with quarterlytime-aggregation. We take the parameters for thepolicy shock fromAng, Dong, and Piazzesi (2004),who estimate a Taylor rule using an affine term-structure model with macroeconomic factorsand an unobserved policy shock.

Figures 4 though 7 depict the average yieldcurves and yield volatilities for different prefer-ence parameters for the exogenous and endoge-nous inflation models. In the top panel of eachfigure, asterisks denote the empirical averagenominal yield curve, a blue dashed-dotted linedenotes the average real yield curve commonacross both inflation models, a dashed linedenotes the average nominal yield curve in theexogenous inflation economy, and a solid linedenotes the average nominal yield curve in theendogenous inflation economy. The bottom paneldepicts yield volatilities for the same cases as theaverage yield curve in the top panel. (Asterisksin Figures 4 through 9 are the moments—meansand standard deviations—of the data in Figure 1.)

Each figure is computed using a different setof preference parameters. We fix a level of theintertemporal elasticity parameter, ρ, for eachpanel and pick the remaining preference param-eters—the risk aversion coefficient, α, and the rateof time preference, β—to minimize the distancebetween the average nominal yields and yieldvolatilities in the data and those implied by theexogenous inflation economy. We pick theTaylor rule parameters to minimize the distancebetween the average nominal yields and yieldvolatilities in the data and those implied by theendogenous inflation economy. Table 1 reports

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Figure 4

Average Yield Curve and Volatilities for the Epstein-Zin Model with Stochastic Volatility

NOTE: The parameters are ρ = –0.5, α = –4.835, β = 0.999, τ– = 0.003, τx = 1.2475, and τp = 1.000. The empirical moments for the fullsample (1970:Q1–2005:Q4) are plotted with asterisks, properties of the real yield curve are plotted with a dashed-dotted blue line,properties of the yield curve in the exogenous inflation economy are plotted with a dashed black line, and properties of the yieldcurve in the economy with endogenous inflation are plotted with a solid black line.

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Figure 5

Average Yield Curve and Volatilities for the Epstein-Zin Model with Stochastic Volatility

NOTE: The parameters are ρ = 0.0, α = –4.061, β = 0.998, τ– = 0.003, τx = 0.973, and τp = 0.973. The empirical moments for the fullsample (1970:Q1–2005:Q4) are plotted with asterisks, properties of the real yield curve are plotted with a dashed-dotted blue line,properties of the yield curve in the exogenous inflation economy are plotted with a dashed black line, and properties of the yieldcurve in the economy with endogenous inflation are plotted with a solid black line.

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Figure 6

Average Yield Curve and Volatilities for the Epstein-Zin Model with Stochastic Volatility

NOTE: The parameters are ρ = 0.5, α = –4.911, β = 0.994, τ– = –0.015, τx = 3.064, and τp = 2.006. The empirical moments for the fullsample (1970:Q1–2005:Q4) are plotted with asterisks, properties of the real yield curve are plotted with a dashed-dotted blue line,properties of the yield curve in the exogenous inflation economy are plotted with a dashed black line, and properties of the yieldcurve in the economy with endogenous inflation are plotted with a solid black line.

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Figure 7

Average Yield Curve and Volatilities for the Epstein-Zin Model with Stochastic Volatility

NOTE: The parameters are ρ = 1.0, α = –6.079, β = 0.990, τ– = –0.004, τx = 1.534, and τp = 1.607. The empirical moments for the fullsample (1970:Q1–2005:Q4) are plotted with asterisks, properties of the real yield curve are plotted with a dashed-dotted blue line,properties of the yield curve in the exogenous inflation economy are plotted with a dashed black line, and properties of the yieldcurve in the economy with endogenous inflation are plotted with a solid black line.

the factor loadings and the prices of risk for eacheconomy corresponding to the figures. Table 2reports the coefficients on the equilibrium infla-tion rate and properties of the equilibrium infla-tion rate in the endogenous inflation economy.

Figure 4 reports the results with ρ = –0.5; herethe representative agent has a low intertemporalelasticity of substitution. The remaining preferenceparameters are α = –4.835 and β = 0.999. Withthis choice of parameters, the average real termstructure is slightly downward sloping.

Backus and Zin (1994) show that a necessarycondition for the average yield curve to be upwardsloping is negative autocorrelation in the pricing

kernel.2 Consider an affine model with independ-ent factors zt

1,zt2,…,zt

k with an innovation εtj on

the j th factor, a factor loading γ j on the j th factor,and a price of risk λj on the j th factor. In such amodel, the j th factor contributes

(12)

to the autocovariance of the pricing kernel.In our calibration, the exogenous factors in

γ γ λ εj tj

j j tj

tjz z2Autocov Cov ,( ) + ( )

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2 Piazzesi and Schneider (2007) argue that an upward-slopingnominal yield curve can be generated if inflation is bad news forconsumption growth. Such a structure leads to negative autocor-relation in the nominal pricing kernel.

Table 1Factor Loadings and Prices of Risk

Factor loadings (γ s) Prices of risk (λ s)

Constant xt vt pt st ε xt+1 ε v

t+1 ε pt+1 ε s

t+1

A. ρ = –0.5, α = –4.835, β = 0.999, τ– = 0.003, τx = 1.2475, τp = 1.000

Real kernel 0.01 0.54 25.45 — — 8.25 –902.98 — —

Exogenous inflation 0.01 0.54 25.45 0.85 — 8.25 –902.98 1.00 —

Endogenous inflation 0.02 0.14 21.63 — –1.44 7.15 –906.90 — –1.56

B. ρ = 0.0, α = –4.061, β = 0.998, τ– = 0.003, τx = 0.973, τp = 0.973

Real kernel 0.01 0.36 20.07 — — 7.34 –677.11 — —

Exogenous inflation 0.01 0.36 20.07 0.85 — 7.34 –677.11 1.00 —

Endogenous inflation 0.02 0.00 33.56 — –1.51 6.34 –663.24 — –1.63

C. ρ = 0.5, α = –4.911, β = 0.994, τ– = –0.015, τx = 3.064, τp = 2.006

Real kernel 0.01 0.18 32.23 — — 8.93 –972.61 — —

Exogenous inflation 0.01 0.18 32.23 0.85 — 8.93 –972.61 1.00 —

Endogenous inflation 0.02 –0.45 38.34 — –0.56 7.18 –966.33 — –0.61

D. ρ = 1.0, α = –6.079, β = 0.990, τ– = 0.004, τx = 1.534, τp = 1.607

Real kernel 0.02 0.00 51.82 — — 10.99 –1,398.00 — —

Exogenous inflation 0.02 0.00 51.82 0.85 — 10.99 –1,398.00 1.00 —

Endogenous inflation 0.03 –0.44 58.30 — –0.74 9.76 –1,391.30 — –0.80

NOTE: The table reports the affine term-structure parameters for the real term structure, the nominal term structure in the exogenousinflation economy, and the nominal term structure in the endogenous inflation economy. The parameters in each panel are computedusing a different set of preference parameters. We fix a level of the intertemporal elasticity parameter, ρ, and choose the remainingpreference parameters—the risk aversion coefficient, α, and the rate-of-time preference, β, to minimize the distance between theaverage nominal yields and yield volatilities in the data and those implied by the exogenous inflation economy. We pick the Taylorrule parameters to minimize the distance between the average nominal yields and yield volatilities in the data and the those impliedby the endogenous inflation economy.

the real economy—output growth and stochasticvolatility—all have positive autocovariances andthe factor innovations have positive covariances tothe factor levels. This implies that γ j

2Autocov�ztj �

and Cov�ztj,εt

j � are both positive. For a factor tocontribute negatively to the autocorrelation of thepricing kernel, the factor loading γ j and the priceof risk λj must have opposite signs. Additionally,the price of risk λj must be large enough relativeto the factor loading γ j to counteract the positiveautocovariance term γ j

2Autocov�ztj �.

Output growth has a lower autocorrelationcoefficient than stochastic volatility in our cali-bration, but because output growth has a muchhigher unconditional volatility, it has a muchhigher autocovariance than stochastic volatility.In the real economy, the factor loading γx on thelevel of output growth is equal to �1 – ρ�φx, whichis nonnegative for all ρ ≤ 1. Also, the price of riskfor output growth, λx, is positive at the parametervalues used in Figure 4 because a sufficient con-dition for it to be positive is α ≤ 0 and |ρ|≤|α|.From (12), output growth contributes positivelyto the autocovariance of the pricing kernel.

From the real pricing kernel parameters givenin (7), the price of risk for volatility is related tothe factor loading on the level of volatility by

Because 1 – βφv > 0, the volatility price of risk,λv, and the volatility factor loading, γv, have oppo-site signs, implying that the volatility factor cancontribute a negative autocovariance to the pric-ing kernel. But output growth has the strongesteffect on the autocovariance of the pricing kernel,leading to positive autocovariance in the pricingkernel. As a consequence, the average real yieldcurve is downward sloping. The numerical valuesfor the real pricing kernel’s factor loadings andprices of risk from Figure 4 are reported in panel Aof Table 1.

In the exogenous inflation economy, shocksto inflation are uncorrelated to output growth andstochastic volatility—the factor loadings andprices of risk on output growth and stochasticvolatility in the nominal pricing kernel are thesame as in the real pricing kernel. Average nomi-nal yields in the exogenous inflation economy areequal to the real yields plus expected inflationand inflation volatility with an adjustment forproperties of the inflation process. The inflationshocks are positively autocorrelated, with a factor

λ ββφ

γvv

v= −−1

.

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Table 2Properties of pt in the Endogenous Inflation Economy

π– πx πv πs E(pt) σ (pt) AR(1)

A. ρ = –0.5, α = –4.835, β = 0.999, τ– = 0.003, τx = 1.2475, τp = 1.000

0.01 –1.11 –3.92 –1.56 0.01 0.02 0.37

B. ρ = 0.0, α = –4.061, β = 0.998, τ– = 0.003, τx = 0.973, τp = 0.973

0.01 –1.00 13.87 –1.63 0.01 0.02 0.44

C. ρ = 0.5, α = –4.911, β = 0.994, τ– = –0.015, τx = 3.064, τp = 2.006

0.02 –1.75 6.28 –0.61 0.01 0.03 0.37

D. ρ = 1.0, α = –6.079, β = 0.990, τ– = 0.004, τx = 1.534, τp = 1.607

0.01 –1.23 6.66 –0.80 0.01 0.02 0.37

NOTE: The first four columns are coefficients of the inflation rate: the equilibrium inflation rate coefficients on a constant, output,stochastic volatility, and the monetary policy shock, respectively. The last two columns are properties of inflation: the unconditionalstandard deviation and the first-order autocorrelation of inflation, respectively.

loading and a price of risk that are both positive.The average nominal yield curve has approxi-mately the same shape as the real yield curve—it is downward sloping.

In the endogenous inflation economy, infla-tion is a linear combination of output growth,stochastic volatility, and the monetary policyshock. As shown in panel A of Table 2, endoge-nous inflation’s loading on output, πx, is negative.This implies that the nominal pricing kernel’soutput-growth factor loading and price of risk arelower than in the exogenous inflation economy.As a consequence, output growth contributesmuch less to the autocovariance of the pricingkernel with endogenous inflation. The factor load-ing and price of risk for stochastic volatility arealso lower in the endogenous inflation economy.The policy shocks are positively autocorrelated,but the factor loading and the price of risk for thepolicy shock are of opposite sign. The averagenominal yield curve in the endogenous inflationeconomy is therefore flatter than both the realyield curve and the nominal yield curve withexogenous inflation.

Turning to the volatilities in the bottom panelof Figure 4, the exogenous inflation economyexhibits more volatility in short rates and lessvolatility in long rates than found in the data. Thisis a fairly standard finding for term-structuremodels with stationary dynamics (see Backus andZin, 1994). The volatility of long rates is mainlydriven by the loading on the factor with the largestinnovation variance and that factor’s autocorre-lation. The closer that autocorrelation is to zero,the faster that yield volatility decreases as bondmaturity increases. In our calibration, outputgrowth has the largest innovation variance and afast rate of mean reversion, equal to 0.36. Yieldvolatility drops quite quickly as bond maturityincreases. In general, the lower the loading on out-put growth, the slower that yield volatility dropsas bond maturity increases. Because endogenousinflation is negatively related to output growth,the factor loading on output growth is lower. Yieldvolatility drops at a slower rate (relative to matu-rity) in the endogenous inflation economy thanin the exogenous inflation economy.

Figure 5, panel B of Table 1, and panel B ofTable 2 report yield-curve properties with a higherintertemporal elasticity of substitution �ρ = 0� ora log time aggregator. Piazzesi and Schneider(2007) study a model with the same preferences,but without stochastic volatility. The factor load-ing on output growth in the real economy is higherthan in the economy with ρ = –0.5 reported inFigure 4 (compare panel A with panel B ofTable 1). The average real yield curve and theaverage nominal yield curve with exogenousinflation are less downward sloping when ρ = 0than when ρ = –0.5. Similarly, increasing ρ furtherto 0.5 (see Figure 6) or 1.0 (see Figure 7) leads toa real yield curve that is less downward-sloping.Because increasing ρ decreases the factor loadingon output growth, it also decreases the volatilityof real yields: See the bottom panels in Figures 4through 7.

As ρ increases, the representative agent’sintertemporal elasticity of substitution increases,implying less demand for smoothing consumptionover time. Increasing ρ decreases the representa-tive agent’s demand for long-term bonds for thepurpose of intertemporal consumption smoothingand leads to lower equilibrium prices and higheryields for real long-term bonds. The average realyield curve therefore is less downward-slopingas ρ increases. Increasing ρ also reduces the sen-sitivity of long-term real yields to output growth,leading to less volatile long-term yields: See thebottom panels in Figures 4 through 7.

Nominal yields in the economies with exoge-nous inflation are approximately equal to the realyields plus a maturity-independent constant.But in the economies with endogenous inflation,inflation and output growth have a negativecovariance, leading to a decrease in the factorloading on output growth: See panels C and D ofTables 1 and 2. For ρ ≥ 0.5 (see Figures 6 and 7),the average nominal yield curve is upward slop-ing and the shape of the volatility term structuredecays similarly to that observed in the data.

The final three columns of Table 2 reportunconditional moments of inflation in the econ-omy with endogenous inflation. There are a fewnotable features. First, the unconditional moments

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are not particularly sensitive to the intertemporalelasticity of substitution. Second, the uncondi-tional variance of inflation in the calibrated econ-omy is an order of magnitude higher than that inthe data: 0.0033 in empirical data and about 0.02in these economies. Finally, inflation is much

more autocorrelated in the data—the AR(1) coef-ficient is 0.85 in the data and about 0.4 in themodel economies.

Figure 8, Figure 9, and Table 3 show resultsfrom changing the Taylor rule parameters. Wekeep the remaining parameters fixed at the values

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τx

τx

τx

Figure 8

The Effects of Increasing τx

NOTE: The baseline parameters are ρ = 1.0, α = –6.079, β = 0.990, τ– = –0.004, τx = 1.534, and τp = 1.607. Empirical moments for the fullsample (1970:Q1–2005:Q4) are plotted with asterisks, results from the baseline parameters are plotted with a solid black line, and resultswhen the feedback from output growth to short-term interest rates is increased by 10 percent are plotted with a dashed black line.

used to generate Figure 7. Figure 8 shows thatincreasing τx, the interest rate’s responsivenessto output growth shocks, leads to a reduction inaverage nominal yields and a steepening in theaverage yield curve (top panel), as well as anincrease in yield volatility (bottom panel).

Panel A of Table 3 shows that increasing τxdecreases the constant in the nominal pricingkernel, decreases the factor loading on outputgrowth, decreases the price of risk for outputgrowth, and also increases the factor loading onstochastic volatility. The loading on output growth

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τp

τp

τp

Figure 9

The Effects of Increasing τp

NOTE: The baseline parameters are ρ = 1.0, α = –6.079, β = 0.990, τ– = –0.004, τx = 1.534, and τp = 1.607. Empirical moments for thefull sample (1970:Q1–2005:Q4) are plotted with asterisks, results from the baseline parameters are plotted with a solid black line, andresults when the feedback from inflation to short-term interest rates is increased by 10 percent are plotted with a dashed black line.

in the pricing kernel drops because the sensitivityof the inflation rate to output growth drops; inturn, the sensitivity of inflation to stochasticvolatility increases by a large amount—from 6.66to 8.55.

Figure 9 shows that increasing τp, the interestrate responsiveness to inflation, leads to a reduc-tion in average nominal yields and a flattening inthe average yield curve (top panel) and a decreasein yield volatility (bottom panel).

Panel B of Table 3 shows that increasing τpdecreases the constant in the nominal pricingkernel, increases the factor loading on outputgrowth, increases the price of risk for outputgrowth, decreases the factor loading on stochas-tic volatility, and also drops the factor loading onthe monetary policy shock. The constant in thepricing kernel drops because the constant in theinflation rate drops, the factor loading on outputgrowth increases because the sensitivity of theinflation rate to output growth increases; in turn,the sensitivity of inflation to stochastic volatilitydecreases by a large amount—from 6.66 to 3.58.

Overall, the experiments reported in Figure 8and Figure 9 show that properties of the termstructure depend on the form of the monetaryauthorities’ interest rate feedback rule. In partic-ular, the factor loading on stochastic volatility is

quite sensitive to the interest rate rule. In thiseconomy, because stochastic volatility is drivingtime-variation in interest rate risk premiums,monetary policy can have large impacts on inter-est rate risk premiums.

RELATED RESEARCHThe model we develop is similar to a version

of Bansal and Yaron’s (2004), which includesstochastic volatility; however, our simple auto-regressive state-variable process does not capturetheir richer ARMA specification. Our work is alsorelated to Piazzesi and Schneider (2007), whoemphasize that, for a structural model to gener-ate an upward-sloping nominal yield curve, itrequires joint assumptions on preferences andthe distribution of fundamentals. Our work high-lights how an upward-sloping yield curve can alsobe generated through the monetary authority’sinterest rate feedback rule.

Our paper adds to a large and growing litera-ture combining structural macroeconomic modelsthat include Taylor rules with arbitrage-free term-structure models. Ang and Piazzesi (2003), follow-ing work by Piazzesi (2005), have shown that afactor model of the term structure that imposesarbitrage-free conditions can provide a better

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Table 3Comparative Statics for the Taylor Rule Parameters

Nominal pricing kernel

Factor loadings (γ s) Prices of risk (λ s) Equilibrium inflation loadings

Constant xt vt st ε xt+1 ε v

t+1 ε st+1 π– πx πv πs

A. τx increased by 10%, from 1.53 to 1.69

Baseline 0.03 –0.44 58.30 –0.74 9.76 –1,391.30 –0.80 0.01 –1.23 6.66 –0.80

Increased τx 0.02 –0.49 60.13 –0.74 9.63 –1,389.40 –0.80 0.01 –1.35 8.55 –0.80

B. τp increased by 10%, from 1.61 to 1.77

Baseline 0.03 –0.44 58.30 –0.74 9.76 –1,391.30 –0.80 0.01 –1.23 6.66 –0.80

Increased τp 0.02 –0.39 55.30 –0.66 9.90 –1,394.40 –0.71 0.01 –1.09 3.58 –0.71

NOTE: The table reports the effect of changing the Taylor rule parameter τx or τp on the affine term-structure parameters as well asproperties of pt in the endogenous inflation economy. The equilibrium inflation rate coefficients on output, stochastic volatility, andthe monetary policy shock are reported. The baseline parameters are ρ = 1.0, α = –6.08, β = 0.990, τ– = –0.004, τx = 1.53, and τp = 1.61.

empirical model of the term structure than amodel based on unobserved factors or latentvariables alone. Estrella and Mishkin (1997),Evans and Marshall (1998 and 2001), Hördahl,Tristani, and Vestin (2004), Bekaert, Cho, andMoreno (2005), and Ravenna and Seppala (2006)also provide evidence of the benefits of buildingarbitrage-free term-structure models with macro-economic fundamentals. Rudebusch and Wu(2004) and Ang, Dong, and Piazzesi (2004) inves-tigate the empirical consequences of imposing aTaylor rule on the performance of arbitrage-freeterm-structure models.

For an alternative linkage between short- andlong-maturity bond yields, see Vayanos and Vila(2006), who show how the shape of the term struc-ture is determined in the presence of risk-aversearbitrageurs, investor clienteles for specific bondmaturities, and an exogenous short rate that couldbe driven by the central bank’s monetary policy.

CONCLUSIONSWe demonstrate that an endogenous monetary

policy that involves an interest rate feedback rulecan contribute to the riskiness of multi-periodbonds by creating an endogenous inflation processthat exhibits significant covariance risk with thepricing kernel. We explore this through a recur-sive utility model with stochastic volatility thatgenerates sizable average risk premiums. Ourresults point to a number of additional questions.First, the Taylor rule that we work with is arbi-trary, so how would the predictions of the modelchange with alternative specification of the rule?In particular, how would adding monetary non-neutralities along the lines of a New KeynesianPhillips curve as in Clarida, Galí, and Gertler(2000) and Gallmeyer, Hollifield, and Zin (2005)alter the monetary policy–consistent pricing ker-nel? Second, what Taylor rule would implementan optimal monetary policy in this context?Because preferences have changed relative to themodels in the literature, this is a nontrivial theo-retical question.

In addition, the simple calibration exercise inthis paper is not a very good substitute for a more

serious econometric exercise. Further researchwill explore the trade-offs between shock speci-fications, preference parameters, and monetarypolicy rules for empirical yield-curve modelsthat more closely match historical evidence.

Finally, it would be instructive to compareand contrast the recursive utility model withstochastic volatility with other preference speci-fications that are capable of generating realisticrisk premiums. The leading candidate on thisdimension is the external habits models ofCampbell and Cochrane (1999). We are currentlypursuing an extension of the external habitsmodel in Gallmeyer, Hollifield, and Zin (2005)to include an endogenous, Taylor rule–driveninflation process.

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