Strucutre Term of Interest Rates

8/6/2019 Strucutre Term of Interest Rates

1/37

The Term Structure ofReal Interest Rates:

Theory and Evidence from

UK Index-Linked Bonds

Juha Seppala

Department of Economics, University of Illinois,

1206 South Sixth Street, Champaign, IL 61820, USAtel: (217) 244-9506 fax: (217) 244-6678

email: [email protected]

October 15, 2003

I am grateful to Fernando Alvarez, Marco Bassetto, Marco Cagetti, John Cochrane, Bernard Dumas, SylvesterEijffinger, Alex Monge, Casey Mulligan, Marcelo Navarro, Antti Ripatti, Thomas J. Sargent, Pietro Veronesi, AnneVillamil, Jouko Vilmunen, and especially Lars Peter Hansen, Urban Jermann, and an anonymous referee for commentsand discussions. In addition, Lili Xie provided excellent research assistance for the latest revision of the paper. Ialso received useful comments from seminar participants at the HansenSargent Macro lunch group, J.P. Morgan,Bank of Finland, University of Chicago Financial Engineering workshop, Lehman Brothers, Stockholm School ofEconomics, University of Michigan, McGill University, CEPR/LIFE/Wharton Conference on International Financeand Economic Activity (Vouliagmeni, 2000), Society for Economic Dynamics Meetings (Costa Rica, 2000), the WorldCongress of the Econometric Society (Seattle, 2000), and the University of Illinois. Juha Niemela and Ulla Rantalahtihelped me obtain the macro data. Martin Evans kindly shared his data on nominal and real yields. The BritishHousehold Panel Survey data were made available through The Data Archieve. The data were originally collectedby the ESRC Research Center on Micro-Social Change at the University of Essex. I would like to thank the YrjoJahnsson Foundation for financial support. Parts of this paper were written while I was visiting the Bank of FinlandResearch Department. I would like to thank the organization for its support. All errors are mine.


2/37

Abstract

This paper studies the behavior of the default-risk-free real term structure and term premiain two general equilibrium endowment economies with complete markets but without money.In the first economy there are no frictions as in Lucas (1978) and in the second risk-sharing islimited by the risk of default as in Alvarez and Jermann (2000, 2001). Both models are solvednumerically, calibrated to UK aggregate and household data, and the predictions are comparedto data on real interest rates constructed from the UK index-linked data. While both modelsproduce time-varying risk or term premia, only the model with limited risk-sharing can generateenough variation in the term premia to account for the rejections of expectations hypothesis.

JEL classification: E43, E44, G12.Keywords: Term structure of interest rates, General equilibrium, Default risk, Term pre-

mia, Index-linked bonds.


3/37

1 Introduction

One of the oldest problems in economic theory is the interpretation of the term structure of interestrates. It has been long recognized that the term structure of interest rates conveys information

about economic agents expectations about future interest rates, inflation rates, and exchange rates.In fact, it is widely agreed that the term structure is the best source of information about economicagents inflation expectations for one to four years ahead.1 Since it is generally recognized thatmonetary policy can only have effect with long and variable lags as Friedman (1968) put it,the term structure is an invaluable source of information for monetary authorities.2 Moreover,empirical studies indicate that the slope of the term structure predicts consumption growth betterthan vector autoregressions or leading commercial econometric models.3

Empirical research on the term structure of interest rates has concentrated on the (pure) expec-tations hypothesis. That is, the question has been whether forward rates are unbiased predictors offuture spot rates. The most common way to test the hypothesis has been to run a linear regression(error term omitted):

rt+1 rt = a + b(ft rt),

where rt is the one-period spot rate at time t and ft is the one-period-ahead forward rate at time t.The pure expectations hypothesis implies that a = 0 and b = 1. Rejection of the first restrictiona = 0 is consistent with the expectations hypothesis with a term premium that is nonzero butconstant.

By and large the literature rejects both restrictions.4 Rejection of the second restriction, b =1, requires a risk or term premium that varies through time and is correlated with the forwardpremium, ft rt. Many studiese.g., Fama and Bliss (1987) and Fama and French (1989)takethis to indicate the existence of time-varying risk or term premium.5 In order for policy makersto extract information about market expectations from the term structure, they need to have a

general idea about the sign and magnitude of the term premium. Obviously, as emphasized byCochrane (1999), the sign and the magnitude of the term premium are also interesting for marketprofessionals. Therefore, it is interesting to ask if there are models that are capable of generatingterm premia that are similar to the ones observed in actual time series. Unfortunately, as Soderlindand Svensson (1997) note in their review,

We have no direct measurement of this (potentially) time-varying covariance [term

1See, e.g., Fama (1975, 1990) and Mishkin (1981, 1990a, 1992) for studies on inflation expectations and the termstructure of interest rates using U.S. data. Mishkin (1991) and Jorion and Mishkin (1991) use international data.Abken (1993) and Blough (1994) provide surveys of the literature.

2Svensson (1994ab) and Soderlind and Svensson (1997) discuss monetary policy and the role of the term structureof interest rates as a source of information.

3See, e.g., Harvey (1988), Chen (1991), and Estrella and Hardouvelis (1991).4The literature is huge. Useful surveys are provided by Melino (1988), Shiller (1990), Mishkin (1990b), and

Campbell, Lo, and MacKinlay (1997). Shiller (1979), Shiller, Campbell, and Schoenholtz (1983), Fama (1984, 1990),Fama and Bliss (1987), Froot (1989), Campbell and Shiller (1991), and Campbell (1995) are important contributionsto this literature.

5The literature is somewhat inconsistent in the definitions of risk and term premium. In the discussion below,both terms are used interchangeably. In Section 3.3, I will define the risk premium as the term that accounts forthe rejections of expectations hypothesis in prices and term premium as the term that accounts for the rejections ofexpectations hypothesis in rates.

3


4/37

premium], and even ex post data is of limited use since the stochastic discount factor isnot observable. It has unfortunately proved to be very hard to explain (U.S. ex post)term premia by either utility based asset pricing models or various proxies for risk.

The question whether utility-based asset pricing models are capable of generating risk premiasimilar to the ones observed in actual time series was originally posed in Backus, Gregory, andZin (1989). They use a complete markets model first presented by Lucas (1978), and their answeris that the model can account for neither sign nor magnitude of average risk premia in forwardprices and holding-period returns.6 In addition, they show that one cannot reject the expectationshypothesis with data generated via the Lucas model. Donaldson, Johnsen, and Mehra (1990) obtainthe same result for a general equilibrium production economy, and den Haan (1995) investigatesthe issue further.

The problem is that given the variability in U.S. aggregate consumption series, the stochasticdiscount factor derived from frictionless utility-based asset pricing models is not sufficiently volatile.This is closely related to the equity premium puzzle first posed by Mehra and Prescott (1985) and

the risk-free rate puzzle posed by Weil (1989). Mehra and Prescott (1985) conjecture that the mostpromising way to resolve the equity premium puzzle is to introduce features that make certaintypes of intertemporal trades among agents infeasible. Usually this has meant that markets areexogenously incomplete. Economic agents are allowed to trade only in certain types of assets, andthe set of available assets is exogenously predetermined.

Heaton and Lucas (1992) use a three-period incomplete markets model to address the termpremium puzzle. Their answer is that uninsurable income shocks may help explain one of themore persistent term structure puzzles but the question remains whether the prediction of arelatively large forward premium will obtain in a long horizon model.

Alvarez and Jermann (2000, 2001) study the asset pricing implications of an endowment econ-omy when agents can default on contracts. They show how endogenously determined solvency

constraints that prevent the agent from defaulting on his own contracts help explain the equitypremium and the risk-free rate puzzles. For the purpose of dynamic asset pricing, this frameworkhas three advantages over the standard incomplete markets approach described above.7

First, allocations do not depend on a particular arbitrary set of available securities. Second,the markets are complete and hence any security can be priced. This is particularly important inaddressing questions related to the term structure of interest rates. Finally, finding the solutionto an incomplete markets problem involves solving a very difficult fixed point problem, whereassolving the model in Alvarez and Jermann can be very fast and easy to implement.

However, it should be noted that, while empirical research has concentrated on the nominalterm structure, both Lucas (1978) and Alvarez and Jermann (2000, 2001) are models of endowmenteconomies without money. This paper studies the implications of the Lucas and Alvarez-Jermann

models on the behavior of the default-risk-free ex-ante real term structure and term premia. Bothmodels are solved numerically and calibrated to UK aggregate and household data, and the predic-tions are compared to data on ex-ante real interest rates constructed from UK index-linked data.

6LeRoy (1973), Rubinstein (1976), and Breeden (1979) were important early contributions to the literature ondynamic general equilibrium asset pricing models.

7The work by Alvarez and Jermann builds on contributions by Kehoe and Levine (1993), Kocherlakota (1996),and Luttmer (1996). See also Aiyagari (1994) and Zhang (1997) for studies on endogenous solvency constraints.

4


5/37

While both models produce time-varying risk premia, only the model with limited risk-sharingcan generate enough variation in the term premia to account for the rejections of the expectationshypothesis.8

It should also be emphasized that in both the Lucas and Alvarez-Jermann economies the objectof the study is the term structure ofdefault-risk-free real interest rates. Even though in the Alvarez-Jermann economy risk-sharing is limited by the risk of default, the instruments that are tradedin state-contingent markets are default-risk-free.9 Therefore, it is natural to compare the Lucasand Alvarez-Jermann term structures with the term structure of real bonds issued by the UKgovernment. At least in principle, these bonds are default-risk free. Hence, the thesis of this paperis not that default risk directly explains the term premium in the real term structure, but ratherthat default risk limits risk-sharing in such a way as to produce a realistic term premium.

The rest of the paper is organized as follows. Section 2 presents the data on nominal andindex-linked bonds. Section 3 presents the basic features of the Lucas and Alvarez-Jermann models.Section 4 calibrates the models given the data on the risk-free rate, aggregate consumption, businesscycles, and invidual incomes in the UK Section 5 presents the numerical results for both modelsand compares the models behavior to the behavior of UK nominal and real term structures.Section 6 summarizes the sensitivity analysis of the Alvarez-Jermann model. Section 7 concludes.Appendix A explains how the models are numerically solved.

2 UK Index-Linked Bonds

The main complication in analyzing ex-ante real interest rates is that in most economies theysimply are unobservable. The most important exception is the UK market for index-linked debt.It constitutes a significant proportion of marketable government debt, and its daily turnover is byfar the highest in the world. The UK market for index-linked debt was started in 1981, and by

March 1994 it accounted approximately 15% of outstanding issues by market value.10

Unfortunately, UK index-linked bonds do not provide a correct measure of ex-ante real interestrates. The reason is that the nominal amounts paid by index-linked bonds do not fully compensatethe holders for inflation; the indexation operates with a lag. Both coupon and principal paymentsare linked to the level of the RPI (Retail Price Index) published in the month seven months priorto the payment date. In addition, the RPI number relates to a specific day in the previous month,so that the effective lag is approximately eight months. The motivation behind this procedure isthat it always allows the nominal value of the next coupon payment to be known, and nominal

8There are several other interesting models that offer at least partial resolution of the equity premium and/orrisk-free rate puzzles. A few examples are Constantinides (1990), Constantinides and Duffie (1996), Bansal andColeman (1996), Campbell and Cochrane (1999), and Abel (1999). In principle, any of these models could be used

to study the Backus-Gregory-Zin term premium puzzle in UK data. Since the previous version of this paper wascompleted, Wachter (2002) and Buraschi and Jiltsov (2003) have successfully applied habit formation to the termpremium puzzle. Duffee (2002) and Dai and Singleton (2002) study the issue in the context of affine term structuremodels.

9In the Alvarez-Jermann economy, the agents have an incentive to default on their contracts if honoring theircontracts would leave them worse off than they would be in autarky. Since everyone knows this and there is noprivate information, nobody is willing to lend more than the debtor is willing to pay back. Therefore, in equilibriumnobody defaults but risk-sharing is limited by the risk of default.

10See Brown and Schaefer (1996) for more details.

5


6/37

accrued interest can always be calculated.Different authors have made different assumptions in order to overcome the indexation lag

problem. Woodard (1990), Deacon and Derry (1994) and Brown and Schaefer (1994) impose theFisher hypothesis: nominal yields move one for one with changes in inflation, which means thatthere is no inflation risk premia. On the other hand, Kandel, Ofer, and Sarig (1996) and Barr andCampbell (1997) assume that different versions of the expectations hypothesis hold. Finally, usingthe properties of stochastic discount factors, Evans (1998) isolates the indexation lag problemto a conditional covariance term between the future (maturity less the inflation lag) inflation andnominal bond prices. He then estimates this term using a VAR model and derives the real interestrates.

Evans also tests for the versions of expectations hypothesis used by Kandel, Ofer, and Sarig (1996)and Barr and Campbell (1997) and rejects both versions at the 1 percent significance level.11 Sincethis paper is mainly concerned on the expectations hypothesis, the methodology of Evans seemedbest suited for my purposes. The data discussed below was provided by him.

Evans (1998) estimates of the term structure of real interest rates in the UK from January 1984until August 1995 reveal the following stylized facts:

1. While the average term structure is upward sloping, the average real term structure is down-ward sloping.

2. While the long-term of nominal term structure is quite volatile, the long-term of the real termstructure appears to be highly stable.12 However, the reverse is true for the short-term: theshort-term of the real term structure is highly volatile compared with that of the nominalterm structure.

3. Both the short-term and and long-term of the real term structure seem to be less autocorre-lated than the corresponding maturities in the real term structure.

4. Shapes of the real term structure are relatively simple compared with the nominal termstructure.

Similar observations have been obtained by Woodward (1990) and Brown and Schaefer (1994,1996) who made different assumptions to overcome the indexation lag problem and estimatedthe yield curve using different methods than Evans used. With the exception of the average shapeof real term structure, the results in Brown and Schaefer are consistent with results presentedhere. Brown and Schaefer (1994, 1996) estimated the term structure of real interest rates to beupward-sloping on average. Following the suggestion by an anonymous referee, I investigated thisissue using real rates estimated from the nominal data as in den Haan (1995) and Chapman (1997).

These data also produce on average downward-sloping term structures. More details are availableupon request.13

11He also rejects the Fisher hypothesis. He finds that expected inflation is negatively correlated with real yields.12This observation is very interesting since Dybvig, Roll, and Ross (1996) showed that under very general conditions

the limiting forward rate, if it exists, can never fall. In affine-yield models, such as Cox, Ingersoll, and Ross (1985),this means that the long-term of the term structure should converge, and these models have been criticized on thegrounds that the long-term of the nominal term structure does not appear to be stable.

13One interesting observation should be mentioned. If one calculates the average term structure using only the

6


7/37

3 The Lucas and Alvarez-Jermann Models

3.1 The Environment

Alvarez and Jermann (1996, 2000, 2001) consider a pure exchange economy with two agents.14

Agents have identical preferences represented by time-separable expected discounted utility. Theirendowments follow a finite-state first-order Markov process. The difference between the Alvarez-Jermann economy and the Lucas economy is that in the former agents cannot commit to theircontracts. The agents have an incentive to default on their contracts if honoring their contractswould leave them worse off than they would be in autarky. Since everyone knows this and thereis no private information, nobody is willing to lend more than the debtor is willing to pay back.Therefore, in equilibrium nobody defaults but risk-sharing is limited by the risk of default.

In the planning problem, the possibility of default is prevented by participation constraints. Theeconomy can be decentralized through complete asset markets where the positions that the agentscan take are endogenously restricted by person, state, and time-dependent solvency constraints.

From now on, the discussion concentrates on the Alvarez-Jermann economy, the Lucas economybeing a special case of the Alvarez-Jermann economy where participation or solvency constraintsdo not ever bind.

Let i = 1, 2 denote each agent and {zt} denote a finite-state Markov process z Z ={z1, . . . , zN} with transition matrix . zt determines the aggregate endowment et, and the in-dividual endowments, eit, i = 1, 2 as follows:

et+1 = (zt+1)et and eit =

i(zt)et for i = 1, 2,

where (zt+1) is the growth rate of aggregate endowment between t and t + 1 when the state inperiod t + 1 is zt+1 and

i(zt) determines agent is share of the aggregate endowment when thestate in period t is zt.

Let zt = (z1, . . . , zt) denote the history of z up to time t. The matrix determines theconditional probabilities for all histories (zt|z0). Households care only about their consumptionstreams, {c} = {ct(z

t) : t 0, zt Zt}, and rank them by discounted expected utility,

U(c)(zt) =

j=0

zt+jZt+j

ju(ct+j(zt+j))(zt+j |zt),

where (0, 1) is a constant discount factor and the one-period utility function is of the constantrelative risk aversion type

u(c) =c1

1 , 1 < < ,

where is the agents constant coefficient of relative risk-aversion.In the planning problem, the participation constraints force the allocations to be such that

under no history will the expected utility be lower than that in autarky

U(ci)(zt) U(ei)(zt) t 0, zt Zt, i = 1, 2. (1)

data after the UK left the European Exchange Rate Mechanism in October 1992, the term structure has beenupward-sloping on average regardless whether one uses Evans data or real rates estimated from nominal data.

14Alvarez and Jermann (2000) consider the more general case with I 2 agents.

7


8/37

In other words, in no time period and in no state of the world can either agents expected discountedutility be less than what the agent would obtain in autarky. If this were the case, the agent wouldnot commit to his contract and would choose to default. The punishment for default would bepermanent exclusion from the risk-sharing arrangement. The motivation for autarky constraintsof the form (1) is quite natural. In the real world, it is sometimes difficult to make a debtor pay.People do not always keep their promises and debt collection can be costly and possibly useless.

One can argue that the model is unrealistic in two respects. On the one hand, the punishment istoo severe; typically an agent who declares bankruptcy can return to the risk-sharing arrangementafter a finite number of periods. On the other hand, the agent who reverts to autarky can keephis endowment stream; there is no confiscation of assets as punishment for default. Trying torelax both of these unrealistic features of the model probably would not change the asset pricingimplications too much. Relaxing the first assumption would reduce risk-sharing, since punishmentfor default would not be so serious. Relaxing the second assumption would increase risk-sharing,since punishment for default would be more serious.

Note that the model abstracts completely from game-theoretic issues related to bargaining andrenegotiation. Punishing one agent by forever excluding him from the risk-sharing arrangementmakes the punishing agent much worse off than with finite punishment. The justification is that asin Alvarez and Jermann (2000), there are actually many more agents than just one and each agenthas a very small weight in the collective bargaining problem. The model is easiest to solve whenthere are only two agents. Section 3.3 shows that adding more agents need not change the assetpricing implications of the model.

3.2 Constrained Optimal Allocations

The constrained optimal allocations are defined as processes, {ci}, i = 1, 2, that maximize agent 1sexpected utility subject to feasibility and participation constraints at every date and every history,

given agent 2s initial promised expected utility. The recursive formulation of the constrainedoptimal allocations is given by the functional equation

T V(w,z ,e) = maxc1,c2,{wz}

u(c1) + zZ

V(wz , z, e)(z|z)

subject to

c1 + c2 e

u(c2) + zZ

wz(z|z) w

V(wz , z, e) U1(z, e) z Z (2)

wz U2(z, e) z Z, (3)

where primes denote next-period values, V(w,z ,e) is agent 1s value function, w is agent 2spromised utility this period, wz is agent 2s promised utility when the next-period state of theworld is z, and (2) and (3) are the participation constraints for agent 1 and agent 2, respectively.The second welfare theorem holds for the economy. Hence, one can solve for the allocations bysolving the planning problem and read the prices off the first-order conditions of the competitiveequilibrium.

8


9/37

3.3 Asset Pricing

To analyze asset prices, the economy must be decentralized. The markets are complete, so thatin every state z there are one-period Arrow securities for each next period state of nature, z.

However, the solvency constraints prevent agents from holding so much debt in any state that theywould like to default on their debt contracts. The solvency constraints affect each state differently,since the relative value of autarky compared to honoring the contract is different in every state.

Let q(z, z) be the price of an Arrow security that pays one unit of consumption good at thebeginning of the next period when the next-period state is z and the current-period state is z.Agent is holdings of this asset are denoted by aiz . Finally, let B

i(z, z) be the minimum positionagent i can take in the asset that pays off when the next-period state is z and the current-periodstate is z. Hence,

Definition 1. The households problem given the current state (a, z) is to maximize the expectedutility

Hi

(a, z) = maxc,{az}zZ u(c) + zZ

Hi

(az

, z

)(z

|z)

subject to solvency and budget constraints

az Bi(z, z) z Z

zZ

q(z, z)az + c a + ei(z).

The equilibrium can now be defined.

Definition 2. The equilibrium is a set of solvency constraints {Bit}, prices {qt}, and alloca-tions {cit, a

i

t+1

} such that

1. Taking the constraints and prices as given, the allocations solve both households optimizationproblems.

2. Markets clear:a1(zt+1) + a2(zt+1) = 0 t 0,zt+1 Zt+1.

3. When the solvency constraints are binding, the continuation utility equals autarchy utility:

Hi(Bit+1(zt+1), zt+1) = Ui(ei)(zt+1) t 0,zt+1 Zt+1. (4)

Condition (4) means that solvency constraints are endogenously generated. This ensures thatthe solvency constraints prevent default and it allows as much insurance as possible. A bindingsolvency constraint means that the agent is indifferent between defaulting and staying in the risk-sharing regime. A slack solvency constraint means that the agents expected discounted utilityis strictly higher than what he would obtain in autarky. Hence, the model provides endogenous

justification for debt, solvency, short-selling, and other exogenous constraints that are commonlyused in the literature on incomplete markets.

9


10/37

From the perspective of asset pricing, the crucial point of the model is that the prices of Arrowsecurities are given by the maximum of the marginal rates of substitution for agents 1 and 2.15

That is, if the current state is zt, the price of an Arrow security that pays one unit of consumptiongood at the beginning of the next period if the next-period state is zt+1 is given by

q(zt+1, zt) = maxi=1,2

u(cit+1)

u(cit)(zt+1|zt). (5)

The economic intuition is that the unconstrained agent in the economy does the pricing. SinceB(zt+1) gives the minimum amount of an asset one can buy, the constrained agent would like to sellthat asset and hence his marginal valuation of the asset is lower. In other words, the constrainedagent has an internal interest rate that is higher than the market rate. Therefore, he would liketo borrow more than is feasible, to keep the autarky constraints satisfied. In the full risk-sharingregime (Lucas economy), the two marginal rates of substitution are equalized.

Notice that one can introduce as many new agents into the economy as desired without changing

the asset pricing implications, provided that the new agents marginal valuations are always lessthan or equal to market valuations. A corollary of this is that each new agent whose income processis perfectly correlated with aggregate income has no effect on asset prices.

In addition, let q(zt+j , zt) be the price of an Arrow security from state zt to state zt+j , whichis given by

q(zt+j , zt) =

t+j1k=t

q(zk+1, zk), (6)

and let q(zk+1, zk) be given by (5).Let mt+1 denote the real stochastic discount factor

mt+1 maxi=1,2 u(c

i,t+1)

u(ci,t) .

The price of an n-period zero-coupon bond is given by

pbn,t =

zt+nZt+n

q(zt+n, zt) = Et

n

j=1

mt+j

. (7)

Using (5), (6), and the equation above, note that

pbn,t = zt+1Zt+1maxi=1,2

u(ci,t+1)

u(ci,t)pbn1,t+1(z

t+1|zt) = Et[mt+1pbn1,t+1]. (8)

The bond prices are invariant with respect to time, and hence equation (8) gives a recursive formulafor pricing zero-coupon bonds of any maturity.

Forward prices are defined by

pfn,t =pbn+1,t

pbn,t,

15This result was also derived by Cochrane and Hansen (1992) and Luttmer (1996).

10


11/37

and the above prices are related to interest rates (or yields) by

fn,t = log(pfn,t) and rn,t = (1/n)log(p

bn,t). (9)

To define the risk premium as in Sargent (1987), write (8) for a two-period bond using theconditional expectation operator and its properties:

pb2,t = Et[mt+1pb1,t+1]

= Et[mt+1]Et[pb1,t+1] + covt[mt+1, p

b1,t+1]

= pb1,tEt[pb1,t+1] + covt[mt+1, p

b1,t+1],

which implies that

pf1,t =pb2,t

pb1,t= Et[p

b1,t+1] + covt

mt+1,

pb1,t+1

pb1,t

. (10)

Since the conditional covariance term is zero for risk-neutral investors, I will call it the risk premiumfor the one-period forward contract, rp1,t, given by

rp1,t covt

mt+1,

pb1,t+1

pb1,t

= pf1,t Et[p

b1,t+1],

and similarly rpn,t is the risk premium for the n-period forward contract:

rpn,t covt

n

j=1

mt+j ,pb1,t+n

pb1,t

= pfn,t Et[p

b1,t+n].

In addition, I will call a difference between the one-period forward rate and the expected value ofone-period interest rate next period the term premium for the one-period forward contract, tp1,t:

tp1,t f1,t Et[r1,t+1]

and similarly tpn,t is the term premium for the n-period forward contract:

tpn,t fn,t Et[r1,t+n].

4 Calibration

4.1 Free Parameters

In the spirit of Backus, Gregory, and Zin (1989), I will solve the model for the simplest possiblecase that produces nonconstant interest rates and has income heterogeneity.16 This is obtained byintroducing uncertainty in the growth rate of aggregate endowment while treating the agents in asymmetric fashion. Similar techniques can be applied for more complicated cases.

16Without income heterogeneity, the Alvarez-Jermann and Lucas models would be identical.

11


12/37

In particular, there will be three exogenous states: zhl, ze, and zlh. The states zhl and zlh areassociated with a recession, and following Mankiw (1986) and Constantinides and Duffie (1996),the recessions are associated with a widening of inequality in earnings.17 During the expansionboth agents have the same endowment ze. The states are ordered so that

1(zhl) = 2(zlh)

1(ze) = 2(ze)

1(zlh) = 2(zhl)

and the transition matrix preserves symmetry between the agents.Thus, there are five free parameters associated with aggregate and individual endowment. These

are 1hl for individual incomes,

1 =

1hl0.5

1 1hl

and 2 =

1 1hl0.5

1hl

,

e

and r

for the growth rates of aggregate endowment,

=

re

r

,

and r and e for the transition matrix

=

r 1 r 0(1 e)/2 e (1 e)/2

0 1 r r

.

The order of calibration is as follows. First, the transition matrix for the aggregate states canbe expressed as a function of the fraction of time spent in the expansion state, , and the first-orderautocorrelation of aggregate consumption, :18

=

(1 ) + (1 )(1 )

(1 ) (1 )(1 ) +

.

Clearly,

=1 r

2 (e + r)

= e + r 1.

Next, given the transition matrix for the aggregate states, e and r determine the average growthrate of aggregate consumption and its standard deviation. Finally, 1hl determines the standarddeviation of individual income. Note that with this parameterization one cannot pin down thepersistence of individual income. An alternative would be to allow individual incomes to takedifferent values during expansions. I discuss in Section 4.2.2 why I chose not to do so.

17Storesletten, Telmer, and Yaron (1999) document this for U.S. data.18See Barton, David, and Fix (1962).

12


13/37

In addition, there are two free parameters, and , associated with preferences. Section 4.2explains the five equations that determine endowment parameters, and Section 4.3 shows how thepreference parameters are pinned down to match the first and second moments of the risk-free ratein the UK Appendix A gives details on how the model is solved numerically, and Appendix B ofSeppala (2000) shows that the main results are highly robust with respect to measurement errorin moment conditions.

4.2 Aggregate and Individual Endowment

4.2.1 Aggregate Consumption and Business Cycles

The first step is to calibrate the law of motion for the aggregate endowment so that it matchesa few facts about aggregate consumption (in the model, aggregate endowment equals aggregateconsumption) and business cycles. Campbell (1998) reports that in the annual UK data for 18911995, the average growth rate of aggregate consumption is 1.443%, the standard deviation of the

growth rate of aggregate consumption is 2.898%, and the first-order autocorrelation is 0.281.Since the UK does not publish official definitions of expansions and recessions, I followed Chap-man (1997), who used the following definitions in his study of the cyclical properties of U.S. realterm structure. Business cycle expansions are defined as at least two consecutive quarters of posi-tive growth19 in a three quarter equally-weighted, centered moving average of the real GDP/capita(output). Cycle contractions are at least two consecutive quarters of negative growth in themoving average of output. A peak is the last quarter prior to the beginning of a contraction, and atrough is the last quarter prior to the beginning of a expansion. Business cycles thusly defined arepresented in Figure 1 for the UK from the first quarter of 1957 until the last quarter of 1997. Thequarterly observations on the GDP at 1990 prices and annual observations on the population wereobtained from the CD-ROM July 1999 version of the International Monetary Funds International

Financial Statistics. The quarterly population series were constructed by assuming that populationgrows at constant rate within the year and that the original annual data are as at December 31of each year. According to the definitions, expansions are 3.8824 times more likely to occur thanrecessions in the UK from the first quarter of 1957 until the last quarter of 1997.

4.2.2 Individual Endowment

The next step is to calibrate the process for the individual endowments. As mentioned above, thecurrent specification only allows one to match the standard deviation of individual income. Analternative would be to allow individual incomes to take different values also during expansions.Unfortunately, in the UK there are only two data sets that provide information on household income.The first is the Family Expenditure Survey (FES), which covers the years 19681992 and the secondis the British Household Panel Survey (BHPS), which covers the period September 1990 to January1998. Since I am interested in household income heterogeneity for particular households and theFES is a survey, I chose to use the BHPS. This approach is similar to that taken in previous studies,such as Heaton and Lucas (1996) and Storesletten, Telmer, and Yaron (1999), which calibrate asset

19The growth (rate) is the difference in the logarithm of the series.

13


14/37

Growth Rate of Real GDP/Capita

1957 1962 1967 1972 1977 1982 1987 1992 1997

-3

-2

-1

0

1

2

3

4

5

Figure 1: The growth rate of real GDP/capita in the UK 1957:11997:4 and its moving average(quarterly observations).

pricing models to the U.S. data on households.20

The BHPS provides a panel of monthly observations of individual and household income and

other variables from September 1990 until January 1998 for more than 5,000 households, giving atotal of approximately 10,000 individuals.21 I use the subset of the panel that has reported positiveincome in every year since 1991. This gives a total of 2,391 individuals in the panel. For householdswith more than one member with reported income, I constructed the individual income as the totalhousehold income divided by the number of the members of the household with reported income.Following Heaton and Lucas (1996), the individual annual income dynamics are assumed to followan AR(1)-process:

log(it) = i + i log(it1) +

it, (11)

where it = eit/n

i=1 eit. This provides a close connection for the standard deviation of

it in the

data with the standard deviation of s in the model. Note that it also imposes a cointegrationrelationship between aggregate and individual income. Unfortunately, given the length of data it

seems unlikely that one could test whether this relationship is reasonable.

20Heaton and Lucas (1996) and Storesletten, Telmer, and Yaron (1999) also provide examples on how to estimatethe standard deviation of individual income. To implement the Storesletten, Telmer, and Yaron method, one needsto know how the cross-sectional variance in household income varies over business cycles. The BHPS was startedonly in September 1990 when the UK was in depression, and the depression ended in the first quarter of 1992. Thus,the data does not cover a full cycle and it has only one data point for the recession. Therefore, I concentrate on thesimplest possible specification (the least free parameters) that would provide different results from the Lucas model.

21For more details on the BHPS, see Taylor (1998).

14


15/37

Table 1: Cross-sectional means and standard deviations of the coefficient estimates in the regres-sion (11). Data: British Household Panel Survey.

Coefficient Cross-Sectional Mean Cross-Sectional Standard Deviation

i 6.4331 5.2376i 0.2035 0.6315i 0.2830 0.2853

Table 2: Cross-sectional means and standard deviations of the coefficient estimates in the regres-sion (11) obtained by Heaton and Lucas (1996). Data: Panel Study of Income Dynamics

Coefficient Cross-Sectional Mean Cross-Sectional Standard Deviationi 3.354 2.413i 0.529 0.332i 0.251 0.131

Table 1 reports cross-sectional means and standard deviations of the coefficient estimates in theregression (11). The relevant numbers are the first-order autocorrelation coefficient, i, and thestandard deviation of the error term, i =

E[(it)

2]. Heaton and Lucas (1996) estimate the samecoefficients using a sample of 860 U.S. households in the Panel Study of Income Dynamics (PSID)

that have annual incomes for 1969 to 1984. Table 2 reports cross-sectional means and standarddeviations of the coefficient estimates obtained by Heaton and Lucas. The estimated autocorrelationcoefficient in the BHPS, 0.2035, is significantly smaller than that in the PSID, 0.529. On the otherhand, the standard deviation of the error term in the BHPS, 0.2830, is roughly the same as that inthe PSID, 0.251. Storesletten, Telmer, and Yaron (1999) obtain a higher number for the standarddeviation of individual income.22

The question of the persistence of individual income is, unfortunately, an unresolved issue.Heaton and Lucas (1996) estimate a relatively low number while Storesletten, Telmer, and Yaron(1999) obtain estimates much closer to one. Finally, Blundell and Preston (1998) impose the pres-ence of a random walk component. Baker (1997) provides analysis that questions the assumptionof a unit root. In order to reasonably pin down the persistence of individual income one would

need to have very long panels of data, which are currently unavailable.22For previous estimates of the standard deviation of individual income using the UK data, see Meghir and White-

house (1996) and Blundell and Preston (1998). Both studies use the FES data. Blundell and Preston (1998), whichis closer to exercise here, decompose the variance in income into permanent and transitory components, and assumeis that the process is composed of a pure transitory component and a random walk. They show strong growth intransitory inequality toward the end of this period, while young cohorts are shown to face significantly higher levelsof permanent inequality. At the beginning of the sample, their estimates are close to mine, but at the end theirestimates are higher, as in Storesletten, Telmer, and Yaron (1999).

15


16/37

23

45

67

0.2

0.40.6

0.8

1

0

5

10

15

20

25

30

35

Standard Deviation of (Individual Consumption/Aggregate Consumption)

Risk aversionDiscount Factor

%

Figure 2: Standard deviation of (individual consumption/aggregate consumption) as a function ofthe discount factor and coefficient of relative risk aversion.

4.2.3 Calibrated Values

Solving the system of five unknowns in five equations leads to the endowment vectors and growth

rates23

1 =

0.77060.5

0.2294

, 2 =

0.22940.5

0.7706

, and =

0.95731.0291

0.9573

,

and the transition matrix

=

0.4283 0.5717 00.0736 0.8527 0.0736

0 0.5717 0.4283

.

4.3 Preferences

The next step is to match agents preference parameters, the discount factor , and the coefficientof relative risk aversion, , to the key statistics of the asset market data. To illustrate the featuresof the model, Figure 2 plots the standard deviation of individual consumption, std(log( c

ij c

j )), as

a function of and .

23Due to highly nonlinear nature of the model, the results are quite sensitive to the parameter values. Therefore,I will report all the parameter values with four decimal precision.

16


17/37

23

45

67

0.2

0.4

0.6

0.8

1

50

0

50

100

150

200

250

300

Expected Riskfree Rate


%

Figure 3: Average risk-free rate as a func-tion of the discount rate and risk aversionin an Alvarez-Jermann economy.

23

45

67

0.2

0.4

0.6

0.8

1

50

0

50

100

150

200

250

300

Expected Riskfree Rate


%

Figure 4: Average risk-free rate as a func-tion of the discount rate and risk aversionin a Lucas economy.

The flat segment in the upper-left corner of the Figure corresponds to autarky and the flatsegment in the lower-right corner to perfect risk-sharing. The parameter values which are interestingfor the purpose of asset pricing are those that generate allocations between these two extremes.To accomplish this, one has to choose either relatively low risk aversion and a relatively highdiscount factor or relatively high risk aversion and a relatively low discount factor. In Section 5, Ireport numerical results only for one pair of coefficients of risk aversion and discount factor, andin Appendix B of Seppala (2000) I show that these results are very robust to measurement error

in risk-free rates, aggregate consumption, business cycles, and individual income. In addition, Ishow how increasing the coefficient of risk-aversion, the discount factor, or the standard deviationof individual income moves the results in the direction of more risk-sharing (the Lucas model).

Figures 36 present the average and standard deviations of the risk-free rate in the Alvarez-Jermann and Lucas economies. Note that in the Alvarez-Jermann economy in autarky no trade isallowed because one of the agents would default on his contract, and hence all the statistics havebeen set to zero. The shapes in the Lucas economy are relative easy to understand using log-linearapproximations, as was done in Section 2.

Recall that the price of an n-period zero-coupon bond in a Lucas economy is given by

pbn,t

= nEtet+n

et

.

17


18/37

23

45

67

0.2

0.4

0.6

0.8

1

0

20

40

60

80

100

Standard Deviation of Riskfree Rate


%

Figure 5: Standard deviation of the risk-free rate as a function of the discount rateand risk aversion in an Alvarez-Jermanneconomy.

23

45

67

0.2

0.4

0.6

0.8

1

0

20

40

60

80

100

Standard Deviation of Riskfree Rate


%

Figure 6: Standard deviation of the risk-free rate as a function of the discount rateand risk aversion in a Lucas economy.

Taking a log-linear approximation of the n-period interest rate,24 one obtains

rn,t = log() +

nEt

log

et+n

et

2

2nvart

log

et+n

et

, (12)

E[rn,t] = log() +

nE

log

et+n

et

2

2nvar

log

et+n

et . (13)

Therefore, the interest rate decreases exponentially in the discount factor and increases slowly inthe risk-aversion coefficient.

In the Alvarez-Jermann model, the relationship between parameter values and interest rates ismore complicated because the interest rates depend not only on aggregate endowment but also onthe consumption share of the unconstrained agent:

exp(rt) = pbt = Et

et+1

et

maxi=1,2

cit+1

cit

, (14)

where cit is agent is consumption share in period t. When the discount factor is lowered, the

incentive to participate in the risk-sharing arrangement is reduced. This leads to an increase invariability in consumption shares and hence an increase in the max operator, thereby increasingbond prices and reducing one-period interest rates.

Campbell (1998) reports that in annual UK data for 18911995, the average real risk-free ratewas 1.198 and its standard deviation was 5.446. In the Alvarez-Jermann model, matching thesevalues leads to = 0.3378 and = 3.514. = 0.3378 is considerably less than what either

24Again, the approximation is exact only if consumption growth has a log-normal distribution. See Campbell (1986).

18


19/37

complete markets or incomplete markets literature typically use. The reason is that, in order to beable to match asset market data, one has to reduce risk-sharing. In the Alvarez-Jermann endowmenteconomy, an incentive to participate in risk-sharing is very high so that only by lowering the discountfactor can autarky become tempting. In addition, note that the persistence of individual incomeis lower than the values estimated by either Heaton and Lucas (1996) or Storesletten, Telmer, andYaron (1999). The quantitative results in Alvarez and Jermann (2001) indicate that increasing thepersistence does not change the asset pricing implications, provided that one is allowed to increasethe discount factor. That is, the more persistent the individual income, the more tempting isdefault (the agents would like to accumulate state-contingent debt during bad times and defaultduring good times) and hence the risk-sharing is reduced for more patient agents.

Obviously, in the Lucas economy risk-sharing is never limited, so it is not possible to match boththe average risk-free rate and the standard deviation simultaneously. Since the standard deviationis more important for explaining the behavior of the term premium, I chose the discount factor andrisk-aversion coefficients in the Lucas economy

(0.99, 6.1149) = arg min(0,0.99],(1,100)

{|E[r(, )] 1.198| s.t. std[r(, )] = 5.446} .

Table 3 summarizes the main statistics for the models, given the above discount factor andrisk aversion values, and in the data. Notice that, in the Alvarez-Jermann economy, the standarddeviation of individual consumption is close to the standard deviation of individual income. Inother words, the allocations are close to autarky allocations. In Appendix B of Seppala, I showthat this result is also very robust. In order to be able to match the basic asset pricing data, one hasto reduce risk-sharing considerably from the full risk-sharing benchmark. It is difficult to say howreasonable this result is: In a recent paper, Brav, Constantinides, and Geczy (1999) conclude thatthe observation error in the consumption data makes it impossible to test the complete consumption

insurance assumption against the assumption of incomplete consumption insurance.The mechanism for limited risk-sharing works as follows. Both agents are off the solvencyconstraint when the current state is the same as the previous state. The average duration ofexpansion is about six years and the average duration of depression is about two years. Only whenexpansion changes to recession or vice versa is somebody constrained. When expansion changesto recession, the agent who got the lower share during the recession has the higher consumptiongrowth rate. Hence, he would like to default. When recession changes to expansion, the agentwho got the lower share during the recession has the higher consumption growth and would liketo default.25 The next section shows how this mechanism translates to the behavior of the termstructure of interest rates.

5 Results

5.1 The Term Structure of Interest Rates

Figures 710 present the interest rates for maturities of 1 to 30 years and the forward rates of 1to 30-year forward contracts during expansions and recessions in the Alvarez-Jermann and Lucas

25Clearly, with only two agents, one cannot interpret the agents literally, but rather how the fraction of populationis affected by solvency constraints over the business cycle.

19


20/37

Table 3: Selected statistics for the Lucas model when = 0.99 and = 6.1149, for the Alvarez-Jermann model when = 0.3378 and = 3.514, and for the data.

Lucas Model Alvarez-Jermann Model Data

E[r] (%) 7.93 1.198 1.198std[r] (%) 5.446 5.446 5.446E(c) (%) 1.443 1.443 1.443std[c] (%) 2.898 2.898 2.898corr[c] 0.281 0.281 0.281Pr(exp.)/ Pr(rec.) 3.882 3.882 3.882std(log(ci/

j(c

j))) (%) 0.0 26.71

corr(log(ci/

j(cj)) 1.0 0.4336

std(log(ei/j(ej))) (%) 28.3 28.3 28.3corr(log(ei/

j(ej)) 0.4193 0.4193 0.2035

economies. A few things are worth noting from the Figures. First, both models produce bothupward and downward-sloping term structures. However, the models cyclical behavior is exactlythe opposite. In the Alvarez-Jermann model, the term structure of interest rates is downward-sloping in recessions and upward-sloping during expansions. In the Lucas model, the term structureof interest rates is upward-sloping in recessions and downward-sloping during expansions. Also, inboth models upward-sloping term structures are always uniformly below downward-sloping termstructures.26 The cyclical behavior of the term structure is of particular interest since empiricaland theoretical results from previous studies have been contradictory.

Fama (1990) reports that

A stylized fact about the term structure is that interest rates are pro-cyclical. (. . .)[I]n every business cycle of the 19521988 period the one-year spot rate is lower at thebusiness trough than at the preceding or following peak. (. . .) Another stylized fact isthat long rates rise less than short rates during business expansions and fall less duringcontractions. Thus spreads of long-term over short-term yields are counter-cyclical.(. . .) [I]n every business cycle of the 19521988 period the five-year yield spread (thefive-year yield less the one-year spot rate) is higher at the business trough than at thepreceding or following peak.

Notice that this statement applies to the term structure of nominal interest rates. On the otherhand, Donaldson, Johnsen, and Mehra (1990) report that in a stochastic growth model with fulldepreciation the term structure of (ex-ante) real interest rates is rising at the top of the cycle andfalling at the bottom of the cycle. In addition, at the top of the cycle the term structure liesuniformly below the term structure at the bottom of the cycle.

26See Proposition 1 below for the explanation.

20


21/37

0 5 10 15 20 25 302

0

2

4

6

8

10

12

Years

%

Interest Rates and Forward Rates (state 1)

Figure 7: Interest rates (solid line) andforward rates in the Alvarez-Jermanneconomy during recessions.

0 5 10 15 20 25 30

3

2

1

0

1

2

3

4

5

6

7

Years

%


Figure 8: Interest rates (solid line) andforward rates in the Lucas economy duringrecessions.

0 5 10 15 20 25 302

0

2

4

6

8

10

Years

%


Figure 9: Interest rates (solid line) andforward rates in the Alvarez-Jermanneconomy during expansions.

0 5 10 15 20 25 306

6.5

7

7.5

8

8.5

9

9.5

10

10.5

11

Years

%


Figure 10: Interest rates (solid line) andforward rates in the Lucas economy duringexpansions.

21


22/37

In the Lucas economy, the cyclical behavior of the term structure will depend on the autocor-relation of consumption growth. Recall equation (12):

rn,t = log() +

n Et

loget+n

et

2

2n vart

loget+n

et

.

It implies that if consumption growth is positively autocorrelated, then a good shock today willforecast good shocks in the future and consequently high interest rates for the near future. As thematurity increases, however, the autocorrelation decreases and the maturity term in the denom-inator starts to kick in reducing the interest rates. The interest rates move one-for-one with thebusiness cycle, exactly as Donaldson, Johnsen, and Mehra (1990) report.27

In the Alvarez-Jermann economy, the prices of multiple-period bonds are determined by

n exp(rn,t) = pbn,t =

nEt

et+n

et

max

i=1,2

cit+1

cit

max

i=1,2

cit+n

cit+n1

.

However, to study the slope of the term structure it is sufficient to know whether interest ratesare procyclical or countercyclical. To see this, note that the following version of Dybvig, Roll, andRoss (1996) holds in the Alvarez-Jermann economy.

Proposition 1. If the transition matrix for the economy is ergodic, then as n approaches infinity,the forward price, pfn,t, converges to a constant.

Proof. Let mij denote the pricing kernel between states i and j. In state i, the price of a one-periodbond is

j ijmij . This can be expressed as

j bij, where bij defines a matrix B. Similarly, the

price of a two-period bond is

jk

ijmijjkmjk

or

j b(2)ij , where b

(2)ij denotes the (i, j) element of B

2. In general, the price of an n-period bond is

given by

j b(n)ij . Since the transition matrix is ergodic, the Perron-Frobenius theorem guarantees

that the dominant eigenvalue of B is positive and that any positive vector operated on by powersof B will eventually approach the associated eigenvector and grow at the rate of this eigenvalue.Recall that the n-period forward price is the ratio of the price of an (n + 1)-period bond to theprice of an n-period bond. As n gets large, the ratio converges to the dominant eigenvalue of Bregardless of the current state.

An immediate corollary of Proposition 1 is that the risk premia will converge. Since the tran-sition matrix is ergodic, the expected spot price, Et[p

b1,t+n], will converge and the limiting risk

premium is the difference between the limiting forward price and the limiting expected spot price.Therefore, it is enough to study one-period bonds that are determined by equation (14)

exp(rt) = pbt = Et

et+1

et

maxi=1,2

cit+1

cit

.

27However, the question of the cyclical behavior of the term structure is more complicated for production economies.See den Haan (1995) and Vigneron (1999).

22


23/37

0 5 10 15 20 25 305

4

3

2

1

0

1

2

3

4

5

Years

%

Risk and Term Premium (state 1)

Figure 11: Risk premium (solid line) andterm premium in the Alvarez-Jermanneconomy during recessions.

0 5 10 15 20 25 302

1.5

1

0.5

0

0.5

1

1.5

2

Years

%


Figure 12: Risk premium (solid line) andterm premium in the Lucas economy dur-ing recessions.

Recall that the variability of consumption shares increases in recessions and that aggregate incomegrowth is positively autocorrelated. Hence, the two terms inside the conditional expectation workopposite to each other. In the current parameterization, the dominant term is consumption het-erogeneity. The greatest variability inside the max operator occurs when one moves from theexpansion state to the recession state. This means that bond prices increase during expansions orthat interest rates decline. Interest rates are countercyclical even though consumption growth ispositively autocorrelated.28

Next, Figures 1114 present the risk and term premia for maturities of 1 to 30 years duringexpansions and recessions in the Alvarez-Jermann and Lucas economies. Note how the signs ofrisk and term premia are opposite in the Lucas vs. Alvarez-Jermann economies. A positive signon the term premium means that forward rates tend to overpredict future interest rates and anegative sign means that the term premium underpredicts. In addition, in the Lucas model theterm premium is very stable, and in the Alvarez-Jermann model, although the term premium isalso stable over the cycle for long maturities, it varies considerably and negatively with the level ofinterest rates for short maturities. This result indicates that the Alvarez-Jermann model may beuseful in accounting for rejections of the expectations hypothesis.

Figures 1520 present the mean and the standard deviation of the interest rates, forward rates,

28In the British data presented in Section 2, the correlation between one-year real interest rate and the cyclicalcomponent of real GDP/capita (obtained using a Hodrick-Prescott filter with a smoothing parameter of 1,600) is0.17. The correlation between yield spread (five-year yield minus one-year yield) and the cyclical component is 0.43.The correlations between nominal data and the cyclical component are 0.19 and 0.48, respectively. In other words,the Alvarez-Jermann model seems to be consistent with the real data and the Lucas model with the nominal data.These estimates, unfortunately, are not very reliable, as the Britain had time to go through only one business cycle inthe sample. It is interesting to note that King and Watson (1996) obtained the same result for the cyclical behaviorof nominal and real interest rates in U.S. data. They obtained real interest rates by estimating expected inflationusing VAR.

23


24/37

0 5 10 15 20 25 308

6

4

2

0

2

4

6

8

Years

%


Figure 13: Risk premium (solid line) andterm premium in the Alvarez-Jermanneconomy during expansions.

0 5 10 15 20 25 302

1.5

1

0.5

0

0.5

1

1.5

2

Years

%


Figure 14: Risk premium (solid line) andterm premium in the Lucas economy dur-ing expansions.

risk, and term premia. The average term structure is upward-sloping in the Alvarez-Jermanneconomy and downward-sloping in the Lucas economy. In both economies, the standard deviationsdecrease with the maturity as in the UK nominal and real data. In the Lucas economy, the averageshape of the term structure is easy to explain. Recall equation (13):

E[rn,t] = log() +

nE

log

et+n

et

2

2nvar

log

et+n

et

.

If the E

log

et+net

ne, then the average shape of the term structure is determined by the

ratio of the variance term to maturity. If consumption growth is positively autocorrelated, thevariance term grows faster than the maturity since shocks in the growth rate are persistent. 29 Inthe Alvarez-Jermann economy, unconditional expectations are more difficult to obtain, but it issufficient to note that the term structure is upward-sloping during the expansions and the economyis growing most of the time.

The relationship between the term premium and the shape of the term structure is as follows.Yields can be expressed as averages of forward rates:

rn,t

=1

n

n1

j=0

fj,t

.

Therefore, the average term structure can be expressed as

E[rn,t] E[r1,t] =1

n

n1j=0

E[fj,t Et[r1,t+j ]] =1

n

n1j=0

tpj,t.

29See den Haan (1995).

24


25/37

0 5 10 15 20 25 301

2

3

4

5

6

7

Years

%

Average Interest Rates and Forward Rates

Figure 15: Average interest rates (solidline) and forward rates in the Alvarez-Jermann economy.

0 5 10 15 20 25 306

6.2

6.4

6.6

6.8

7

7.2

7.4

7.6

7.8

8

Years

%

Average Interest Rates and Forward Rates

Figure 16: Average interest rates (solidline) and forward rates in the Lucas econ-omy.

0 5 10 15 20 25 306

4

2

0

2

4

6

Years

%

Average Risk and Term Premium

Figure 17: Average risk premium (solidline) and term premium in the Alvarez-Jermann economy.

0 5 10 15 20 25 302

1.5

1

0.5

0

0.5

1

1.5

2

Years

%

Average Risk and Term Premium

Figure 18: Average risk premium (solidline) and term premium in the Lucas econ-omy.

25


26/37

0 5 10 15 20 25 300

1

2

3

4

5

6

Years

%

Standard Deviation of Interest Rates, Forward Rates, and Premia

Figure 19: Standard deviation of in-terest rates (solid line), forward rates(dashed line), risk premium (dash-dotline), and term premium (dotted line) inthe Alvarez-Jermann economy.

0 5 10 15 20 25 300

1

2

3

4

5

6

Years

%

Standard Deviation of Interest Rates, Forward Rates, and Premia

Figure 20: Standard deviation of interestrates (solid line), forward rates (dashedline), risk premium (dash-dot line), andterm premium (dotted line) in the Lucaseconomy.

Using the log-linear approximation and assuming homoscedastic errors, the term premium can beexpressed as

tpn,t = 1

2vart[log(p

b1,t+n)] covt[log(p

b1,t+n), log(mt+n)].

In the Lucas economy, this reduces to

tpn,t = 2

2vart

Et+n log

et+n+1

et+n

2 covt

Et+n log

et+n+1

et+n

, log(et+n)

.

Suppose that log(et) = log(et1) + t, where t N(0, 2 ). Then

tp1,t = 2

222

22 = 22

2

1

2+

1

,

which is less that zero, if > 0. For example, in the British data

tp1,t = (6.1149)2(0.02898)2(0.281)2 1

2

+1

0.281 = 1%.

Alvarez and Jermann (2001) show that in the Alvarez-Jermann economy the sign of the termpremium depends on one-period ahead individual income variance, conditional on current aggregatestate or heteroscedasticity ex-ante:

re

std[log(i(zt+1))|t = r]

std[log(i(zt+1))|t = e].

26


27/37

0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Years

Autocorrelation of Interest Rates

Figure 21: Autocorrelation of interestrates in the Alvarez-Jermann Economy.

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

Years

Autocorrelation of Interest Rates

Figure 22: Autocorrelation of interestrates in the Lucas Economy.

When re

> 1, the term premium is negative and the average term structure is downward-sloping;when r

e< 1, the term premium is positive and the average term structure is upward-sloping.

re

< 1 means that in expansions one expects more idiosyncratic risk in the future. From (14)it follows that the max-term in the stochastic discount factor becomes more volatile and hencebond prices are higher (interest rates lower) in expansions. Therefore, a positive term premiumis required to compensate b ond holders. The next section studies whether this term premium isvolatile enough to account for rejections of the expectations hypothesis.

Finally, Figures 21 and 22 present autocorrelations for interest rates in both economies. In the

Alvarez-Jermann economy, autocorrelations are U-shaped between 0.08 and 0.2, as in the empiricalterm structure of real interest rates. However, in the Lucas economy the autocorrelation is constantand determined by the autocorrelation of consumption growth (0.281).

5.2 The Expectations Hypothesis

There are two main versions of the expectations hypothesis. While most of the empirical litera-ture has concentrated on the expectations hypothesis in rates, it is pedagogical to start with theexpectations hypothesis in prices. Recall equation (10)

pf1,t = Et[pb1,t+1] + covtmt+1,

pb1,t+1

p

b

1,t.

Backus, Gregory, and Zin (1989) tested the expectations model in the Lucas economy by startingwith (10), assuming that the risk premium was constant, i.e.,

Et[pb1,t+1] p

f1,t = a,

and regressingpb1,t+1 p

f1,t = a + b(p

f1,t p

b1,t) (15)

27


28/37

Table 4: The number of rejects with regressions in the Lucas economy.

yt+1 pb1,t+1 p

f1,t p

b1,t+1 p

f1,t rp1,t p

b1,t+1 p

f1,t p

b1,t+1 p

f1,t rp1,t

xt pf1,t pb1,t p

f1,t pb1,t pb1,t p

f1,t pb1,t p

f1,t

Wald(a = b = 0) 648 71 651 68Wald(b = 0) 109 67 105 62Wald(b = 1) 1000 1000 1000 1000

Table 5: The number of rejects with regressions in the Alvarez-Jermann economy.

yt+1 pb1,t+1 p

f1,t p

b1,t+1 p

f1,t rp1,t p

b1,t+1 p

f1,t p

b1,t+1 p

f1,t rp1,t

xt pf1,t p

b1,t p

f1,t p

b1,t p

b1,t p

f1,t p

b1,t p

f1,t

Wald(a = b = 0) 1000 54 1000 62Wald(b = 0) 1000 65 1000 62Wald(b = 1) 1000 1000 1000 1000

to see if b = 0. They generated 200 observations 1000 times and used the Wald test withWhite (1980) standard errors to check if b = 0 at the 5% significance level. They could rejectthe hypothesis only roughly 50 times out of 1000 regressions, which is what one would expect fromchance alone. On the other hand, for all values of b except 1, the forward premium is still usefulin forecasting changes in spot prices. The hypothesis b = 1 was rejected every time.

Table 4 presents the number of rejections of different Wald tests in the regressions

yt+1 = a + bxt

in the Lucas economy calibrated to UK data. Table 5 presents the same tests for the Alvarez-Jermann model. Unlike in the Lucas model, the results with the Alvarez-Jermann model areconsistent with empirical evidence on the expectations hypothesis. The model can generate enoughvariation in the risk premia to account for rejections of the expectations hypothesis. When the riskpremium is subtracted from pb1,t+1 p

f1,t, b is equal to zero with 5% significance level.

In Table 6 the results of regression (15) are presented for one realization of 200 observationsfor the Lucas model and for the Alvarez-Jermann model, and for UK real interest rate data. InTable 6, Wald rows refer to the marginal significance level of the corresponding Wald test. It isworth noting how close the values of the regression coefficients are for the Alvarez-Jermann model

and the real UK data.Recent empirical literature has concentrated on the Log Pure Expectations Hypothesis. Ac-

cording to the hypothesis, the n-period forward rate should equal the expected one-period interestrate n periods ahead:

fn,t = Et[r1,t+n].

To test the hypothesis, one can run the regression

(n 1) (rn1,t+1 rn,t) = a + b(rn,t r1,t) for n = 2, 3, 4, 5, 6, 11. (16)

28


29/37

Table 6: Tests of the expectations hypothesis with a single regression.

Variable/Test Lucas Alvarez-Jermann Data

a 0.0118 0.0279 0.0047se(a) 0.0041 0.0032 0.002b 0.0070 0.4970 0.4982se(b) 0.2711 0.0481 0.1046R2 0.0067 0.4036 0.3457Wald(a = b = 0) 0.0013 2e104 3e7Wald(b = 0) 0.9569 4e19 2e6Wald(b = 1) 5e15 1e19 2e6

According to the Log Pure Expectations Hypothesis, one should find that b = 1.30 Table 7 sum-

marizes the results from this regression for the models and data. The expectations hypothesis isclearly rejected in all cases except for the Lucas model.

6 Sensitivity Analysis

Seppala (2000, Appendix B) provides extensive sensitivity analysis on the Alvarez-Jermann modelwith respect to estimated and calibrated parameters of the model. The calibrate parameters arethe discount factor, , and the coefficient of risk aversion, . The results are not surprising. Whenthe discount factor is increased, the allocations move closer to full sharing, and hence rejecting theexpectations hypothesis becomes more and more difficult. Similarly, the allocations move closer to

full sharing as the coefficient of risk aversion is increased. In addition, the term premium changessign and becomes less and less volatile, so that rejecting the expectations hypothesis becomes moreand more difficult.

Sensitivity analysis are provided with respect to estimated parameters can be done in twodifferent ways. Once one parameter has been changed, the calibrated parameters can be eitherrecalibrated or not. Both results are reported in Seppala (2000). Without recalibration, the allo-cations move closer to full sharing when the standard deviation of individual income is increased.On the other hand, when the preference parameters are recalibrated, it is possible to explainthe rejections of the expectations hypothesis when the standard deviation of individual income(std(log(ei/

j(e

j)))) is as high as 65%, expansions are four times more likely to occur than re-cessions, expansions and recessions are equally likely to take place, autocorrelation of consumption

varies between 0.45 and 0.15, the standard deviation of consumption growth varies between 1 and7%, and the average consumption growth varies between 0.5 and 3.5%.31

30See, e.g., Campbell, Lo, and McKinley (1997).31There is some evidence in the U.S. data that average consumption growth has become less volatile after the

WWII, see Chapman (2001) and Otrok, Ravikumar, and Whiteman (2002). Following a suggestion by an anonymousreferee, I divided the UK data into several subperiods, 19001914, 19151938, 19391945 (WWII), 19461969 (post-WWII), 19701979 (the British Disease), 19801990, (Thatcherism), and 1991-2001. With the expectation of WWIIperiod, the unconditional mean and standard deviation of consumption growth during those periods varied between

29


30/37

Table 7: Expectations hypothesis regressions in rates.

Regression a se(a) b se(b) R2

Lucas (n = 2) 1.3841 0.0007 1.1830 0.0003 0.1443Lucas (n = 3) 1.9277 0.0009 1.1609 0.0002 0.1808Lucas (n = 4) 2.1219 0.0010 1.1449 0.0002 0.2120Lucas (n = 5) 2.1896 0.0010 1.1339 0.0001 0.2370Lucas (n = 6) 2.2131 0.0010 1.1264 0.0001 0.2564Lucas (n = 11) 2.2265 0.0010 1.1103 0.0001 0.3056Alvarez-Jermann (n = 2) 2.5150 0.0009 0.0586 0.0001 0.0024Alvarez-Jermann (n = 3) 0.8688 0.0004 0.5663 0.0001 0.5629Alvarez-Jermann (n = 4) 1.8223 0.0007 0.2910 0.0001 0.1152Alvarez-Jermann (n = 5) 1.3232 0.0005 0.4714 0.0001 0.3786Alvarez-Jermann (n = 6) 1.6003 0.0006 0.3983 0.0001 0.2522

Alvarez-Jermann (n = 11) 1.5077 0.0006 0.4567 0.0001 0.3517Data (n = 2) 0.2752 0.2336 0.2931 0.1448 0.0365Data (n = 3) 0.2316 0.2543 0.4033 0.1113 0.1Data (n = 4) 0.2055 0.2820 0.4199 0.1029 0.1121Data (n = 5) 0.2056 0.3135 0.3135 0.4168 0.1060Data (n = 6) 0.2225 0.3470 0.4059 0.1034 0.0942Data (n = 11) 0.3891 0.5198 0.3891 0.1336 0.0292

30


31/37

7 Conclusions and Further Research

With risk-averse agents, the term structure contains expectations plus term premia. In order forpolicy makers to extract information about market expectations from the term structure, they need

to have a general idea about the sign and magnitude of the term premium. But as S oderlind andSvensson (1997) note in their review

We have no direct measurement of this (potentially) time-varying covariance [termpremium], and even ex post data is of limited use since the stochastic discount factor isnot observable. It has unfortunately proved to be very hard to explain (U.S. ex post)term premia by either utility based asset pricing models or various proxies for risk.

This paper studied the behavior of the default-risk free real term structure and term premia intwo general equilibrium endowment economies with complete markets but without money. In thefirst economy there were no frictions, as in Lucas (1978) and in the second the risk-sharing was

limited by the risk of default, as in Alvarez and Jermann (2000, 2001). Both models were solvednumerically, calibrated to UK aggregate and household data, and the predictions were compared tothe data on real interest rate constructed from UK index-linked data. While both models producetime-varying term premia, only the model with limited risk-sharing can generate enough variationin the term premia to account for the rejections of expectations hypothesis.

I conclude that the Alvarez-Jermann model provides one plausible explanation for the Backus-Gregory-Zin term premium puzzle in real term structure data. What is needed now is a theory toexplain the behavior of the term structure of nominal interest rates. Since it is usually recognizedthat monetary policy can only have effect with long and variable lags as Friedman (1968) putit, it is crucial to understand what are the inflation expectations that drive the market. Once weunderstand the behavior of both nominal and real interest rates, we can get correct estimates of

these inflation expectations. An interesting topic for further research is whether a nominal versionof the Alvarez-Jermann model is consistent with the nominal data.Another interesting topic would be to analyze the cyclical behavior of nominal and real term

structures, both in data and in theory. King and Watson (1996) provide an example of how todo this, but they had to use real interest rates that they constructed using a VAR framework.In my opinion, the British data would provide a better approximation for ex-ante real interestrates. However, since the British data are still relatively short and neither the Lucas model nor theAlvarez-Jermann model were built to confront this question, this topic is left for further research.

A Algorithm

This section explains how the Alvarez-Jermann model can be solved numerically. The aggregateendowment is growing over time, but the CRRA utility function implies that the value function,V(), autarky values of utility, Ui(), and the policies, {C1(), C2(), W()}, satisfy the followinghomogeneity property.

0.63 and 3.45 (the mean) and 1.12 and 4.38 (the standard deviation). During the WWII, the average consumptiongrowth was negative (0.89). Unfortunately, the model produces strange results with negative growth, but that ishardly typical for technologically advanced countries during the last 150 years.

31


32/37

Proposition 2. For any y > 0 and any (w,z ,e),

V(y1w,z,ye) = y1V(w,z ,e)

Ui(z,ye) = y1Ui(z, e) for i = 1, 2

Ci(y1w,z,ye) = yCi(w,z ,e) for i = 1, 2

W(y1w,z,ye) = y1W(w,z ,e).

Proof. See the proof of Proposition 3.9 in Alvarez and Jermann (1996).

Defining a new set of hat variables as

u(c) = e1u(c

e) = e1u(c)

Ui(z, e) = (e)1Ui(z, 1) = Ui(z, 1) for i = 1, 2

w =

e1

e1w = e1

w

w =(e)1

(e)1w = (e)1w,

and using the above proposition in the following way:

V(w,z ,e) = V(e1

e1w,z,

e

ee) = e1V(w,z, 1)

V(w, z, e) = (e)1V(w

(e)1, z, 1) = ((z)e)1V(w,z, 1),

the functional equation can be rewritten with stationary variables as

T V(w,z, 1) = maxc1,c2,{w(z)}

u(c1) + zZ

V(w(z), z, 1)(z)1(z|z)

subject to

c1 + c2 1 (17)

u(c2) + zZ

w(z)(z)1(z|z) w (18)

V(w(z), z, 1) U1(z, 1) z Z,

w(z) U2(z, 1) z Z. (19)

In order to guarantee the above maximization problem is well-defined, it is needed to assume that

maxzZ

zZ

(z)1(z|z)

< 1.

The results in Alvarez and Jermann (1996) indicate that during recession the allocations donot depend on past history. However, during the expansion there are two endogenous states: one

32


33/37

where zt = ze and zt1 = zhl, and another where zt = ze and zt1 = zlh. From now on, the statesare ordered as follows:

z1z2

z3z4z5

= (zt = zhl)

(zt = ze, zt1 = zhl)

(zt = ze, zt1 = ze)(zt = ze, zt1 = zlh)

(zt = zlh)

.

Moreover, it is possible to solve for the allocations and prices simply by solving for at most twosystems of nonlinear equations. The first system corresponds to the case in which the participationconstraints are not binding, and the second system corresponds to the case in which the agents areconstrained when entering the boom period. In addition to the nonlinear equation, there is a setof inequalities that determines which case is valid.

From feasibility (17) and non-satiation, it follows that agent 1s consumption is always theaggregate endowment less agent 2s consumption. Hence, both systems have 10 equations in 10 un-

knowns: agent 2s consumption and continuation utility in each of the five states. From now on,cn will denote c2(zn) and hats will be dropped from other variables as well.

In both the unconstrained case and the constrained case, five equations are given by (18);during the expansion neither agent has a reason to trade: c3 = 0.5; (17) and symmetry imply thatc1 + c5 = 1; and the participation constraint (19) holds with equality when agent 2 receives themost favorable shock: w(z5) = U

2(z5). The two missing equations depend on whether the agentsare constrained when entering the expansion state. In the unconstrained case, c2 = c1 and c4 = c5,and, in the constrained case, w(z2) = U

2(z2) and c2 + c4 = 1.To solve for the allocations, one needs only to solve for the unconstrained system and check

whether w(z2) U2(z2). If this is not the case, the solution is given by the constrained case.

References

Abel, A.B. (1999), Risk Premia and Term Premia in General Equilibrium, Journal of MonetaryEconomics, 43, 333.

Abken, P.A. (1993), Inflation and the Yield Curve, Federal Reserve Bank of Atlanta EconomicReview, May/June, 1331.

Aiyagari, S.R. (1994), Uninsured Idiosyncratic Risk and Aggregate Saving, Quarterly Journal ofEconomics, 109, 659684.

Alvarez, F. and U.J. Jermann (1996), Asset Pricing when Risk Sharing is Limited by Default,

Mimeo, The Wharton School.Alvarez, F. and U.J. Jermann (2000), Efficiency, Equilibrium, and Asset Pricing with Risk of

Default, Econometrica, 68, 775797.Alvarez, F. and U.J. Jermann (2001), Quantitative Asset Pricing Implications of Endogenous

Solvency Constraints, Review of Financial Studies, 14, 11171151.Backus, D.K., A.W. Gregory, and S.E. Zin (1989), Risk Premiums in the Term Structure: Evidence

from Artificial Economies, Journal of Monetary Economics, 24, 371399.

33


34/37

Baker, M. (1997), Growth-Rate Heterogeneity and the Covariance Structure of Life-Cycle Earn-ings, Journal of Labor Economics, 15, 338375.

Bansal, R. and W.J. Coleman II (1996), A Monetary Explanation of the Equity Premium, TermPremium, and Risk-Free Rate Puzzles, Journal of Political Economy, 104, 11351171.

Barr, D.G. and J.Y. Campbell (1997), Inflation, Real Rates, and the Bond Market: A Study ofUK Nominal and Index-Linked Government Bond Prices, Journal of Monetary Economics, 39,361383.

Barton, E., F. David, and E. Fix. (1962), Persistence in a Chain of Multiple Events when ThereIs Simple Persistence, Biometrika, 49, 351357.

Brav, A., G.M. Constantinides, and C.G. Geczy (1999), Asset Pricing with Heterogeneous Con-sumers and Limited Participation: Empirical Evidence, Mimeo, University of Chicago.

Blough, S.R. (1994), Yield Curve Forecasts of Inflation: A Cautionary Tale, New England Eco-nomic Review, May/June, 315.

Blundell R. and I. Preston (1998), Consumption Inequality and Income Uncertainty QuarterlyJournal of Economics, 113, 603640.

Breeden, D.T. (1979), An Intertemporal Asset Pricing Model with Stochastic Consumption andInvestment Opportunities, Journal of Financial Economics, 7, 265296.

Brown, R.H. and S.M. Schaefer (1994), The Term Structure of Real Interest Rates and the Cox,Ingersoll, and Ross Model, Journal of Financial Economics, 35, 342.

Brown, R.H. and S.M. Schaefer (1996), Ten Years of the Real Term Structure: 19841994, Journalof Fixed Income Research, March, 622.

Buraschi, A. and A. Jiltsov (2003), Habit Persistence and the Nominal Term Structure of InterestRates , Mimeo, London Business School.

Campbell, J. Y. (1986), Bond and Stock Returns in a Simple Exchange Model, Quarterly Journalof Economics, 101, 785803.

Campbell, J. Y. (1995), Some Lessons from the Yield Curve, Working Paper No. 5031, NationalBureau of Economic Research.

Campbell, J.Y. (1998), Asset Prices, Consumption, and the Business Cycle, Mimeo, HarvardUniversity.

Campbell, J.Y. and J.H. Cochrane (1999), By Force of Habit: A Consumption-Based Explanationof Aggregate Stock Market Behavior, Journal of Political Economy, 107, 205251.

Campbell, J.Y. and R.J. Shiller (1991), Yield Spreads and Interest Rate Movements: A BirdsEye View, Review of Economic Studies, 58, 495514.

Campbell, J.Y., A.W. Lo, and A.C. MacKinlay (1997), The Econometrics of Financial Markets,Princeton University Press, Princeton, New Jersey.

Chapman, D.A. (1997), The Cyclical Properties of Consumption Growth and the Real TermStructure, Journal of Monetary Economics, 39, 145172.

Chapman, D.A. (2001), Does Intrinsic Habit Formation Actually Resolve the Equity PremiumPuzzle?, Mimeo, University of Texas.

Chen, N.-F. (1991), Financial Investment Opportunities and the Macroeconomy, Journal ofFinance, 46, 529554.

Cochrane, J.H. (1999), New Facts in Finance, Mimeo, University of Chicago.Cochrane, J.H. and L.P. Hansen (1992), Asset Pricing Explorations for Macroeconomics, Working

Paper No. 4088, National Bureau of Economic Research.

34


35/37

Constantinides, G. (1990), Habit Formation: A Resolution of the Equity Premium Puzzle, Jour-nal of Political Economy, 98, 519543.

Constantinides, G. and D. Duffie (1996), Asset Pricing with Heterogeneous Consumers, Journalof Political Economy, 104, 219240.

Cox, J., J. Ingersoll, and S. Ross (1985), A Theory of the Term Structure of Interest Rates,Econometrica, 53, 385408.

Dai, Q. (2003), Term Structure Dynamics in a Model with Stochastic Internal Habit, Mimeo,New York University.

Dai, Q. and K.J. Singleton (2002), Expectations Puzzle, Time-varying Risk Premia, and AffineModels of the Term Structure, Journal of Financial Economics, 63, 415441.

Deacon, M.P. and A.J. Derry (1994), Deriving Estimates of Inflation Expectations from the Pricesof UK Government Bonds, Working Paper No. 23, Bank of England.

Donaldson, J.B., T. Johnsen, and R. Mehra (1990), On the Term Structure of Interest Rates,Journal of Economic Dynamics and Control, 14, 571596.

Duffee, G.R. (2002), Term premia and interest rate forecasts in affine models, Journal of Finance,57, 405443.

Dybvig, P., J. Ingersoll, Jr., and S. Ross (1996), Long Forward and Zero-Coupon Rates Can NeverFall, Journal of Business, 69, 125.

Estrella A. and G.A. Hardouvelis (1991), The Term Structure as a Predictor of Real EconomicActivity, Journal of Finance, 46, 555580.

Evans, M.D.D. (1998), Real Rates, Expected Inflation, and Inflation Risk Premia, Journal ofFinance, 53, 187218.

Fama, E.F. (1975), Short Term Interest Rates as Predictors of Inflation, American EconomicReview, 65, 269282.

Fama, E.F. (1984), The Information in the Term Structure, Journal of Financial Economics, 13,509528.

Fama, E.F. (1990), Term-Structure Forecasts of Interest Rates, Inflation, and Real Returns,Journal of Monetary Economics, 25, 5976.

Fama, E.F. and R.R. Bliss (1987), The Information in Long-Maturity Forward Rates, AmericanEconomic Review, 77, 246273.

Fama, E.F. and K.R. French (1989), Business Conditions and Expected Returns on Stocks andBonds, Journal of Financial Economics, 25, 2350.

Friedman, M. (1968), The Role of Monetary Policy, American Economic Review, 63, 117.Froot, K.A. (1989), New Hope for the Expectations Hypothesis of the Term Structure of Interest

Rates, Journal of Finance, 44, 283304.den Haan, W.J. (1995), The Term Structure of Interest Rates in Real and Monetary Economies,

Journal of Economic Dynamics and Control, 19, 909940.Harvey, C.R. (1988), The Real Term Structure and Consumption Growth, Journal of Financial

Economics, 22, 305333.Heaton, J. and D. Lucas (1992), The Effects of Incomplete Insurance Markets and Trading Costs

in a Consumption-Based Asset Pricing Model, Journal of Economic Dynamics and Control, 16,601620.

Heaton, J. and D. Lucas (1996), Evaluating the Effects of Incomplete Markets on Risk Sharingand Asset Pricing, Journal of Political Economy, 104, 443487.

35

Documents

Strucutre Term of Interest Rates