Who Should Buy Long-Term Bonds? - Semantic Scholar · 2015-07-28 · Who Should Buy Long-Term Bonds? By JOHN Y. CAMPBELL AND LUIS M. VICEIRA* According to conventional wisdom, long-term

Who Should Buy Long-Term Bonds?

By JOHN Y. CAMPBELL AND LUIS M. VICEIRA*

According to conventional wisdom, long-term bonds are appropriate for conserva-tive long-term investors. This paper develops a model of optimal consumption andportfolio choice for infinite-lived investors with recursive utility who face stochasticinterest rates, solves the model using an approximate analytical method, andevaluates conventional wisdom. As risk aversion increases, the myopic componentof risky asset demand disappears but the intertemporal hedging component doesnot. Conservative investors hold assets to hedge the risk that real interest rates willdecline. Long-term inflation-indexed bonds are most suitable for this purpose, butnominal bonds may also be used if inflation risk is low.(JEL G12)

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Long-term bonds have been issued for cturies, and they remain extremely commonnancial instruments. It is natural to suppose tbonds have been popular because they meeneeds of an investor clientele. Investmentvisers and financial journalists, for exampoften say that bonds are appropriate for loterm investors who seek a stable income.

Curiously, modern financial economics hhad little to say about the demand for long-tebonds. In the early postwar period, John Hic(1946), following John M. Keynes (1930) anFrederick Lutz (1940), argued that investowould naturally prefer to hold short-term bonand would only hold long-term bonds if com

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* Campbell: Department of Economics, Littauer Center213, Harvard University, Cambridge, MA 02138, andNational Bureau of Economic Research (e-mail:[email protected]); Viceira: Harvard BusinessSchool, Morgan Hall 367, Harvard University, Boston, MA02163, National Bureau of Economic Research, and Centrefor Economic Policy Research (e-mail: [email protected]).Campbell acknowledges the financial support of the Na-tional Science Foundation and Viceira acknowledges thefinancial support of the Bank of Spain. We are grateful forhelpful comments on earlier drafts by anonymous refereesAndrew Abel, Qiang Dai, Sanjiv Das, Ravi Jagannathan,Robert Merton, Mark Rubinstein, Stephen Schaefer, CostisSkiadas, and seminar participants at the University of Chi-cago, Columbia University, Duke University, FAME, theMassachusetts Institute of Technology, Northwestern Uni-versity, The Wharton School, the NBER Summer Institute,the WFA meetings, and the CEPR European Summer Symposium in Financial Markets at Studienzentrum Gerzensee

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pensated by a term premium. Franco Modigliand Richard Sutch (1966) countered that soinvestors might have a preference for long-tebonds (a long-term “preferred habitat”), asuch investors would require a premium toshort, not a premium to go long. HoweveModigliani and Sutch were vague about tcharacteristics of investors that would lead tlong-term preferred habitat. They took it asgiven that some investors would desire stawealth at a long, rather than a short, horizon1

Since the 1960’s there has been a vast increin the sophistication of bond-pricing models, blittle further progress has been made in undstanding the demand for long-term bonds. Recauthors, building on the seminal contributionsOldrich Vasicek (1977) and John C. Cox et(1985), have related term premia to the covances of bond returns with an exogenously sp

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1 They wrote:

Suppose that a person has ann period habitat; that is,he has funds which he will not need forn periodsand which, therefore, he intends to keep invested inbonds forn periods. If he invests inn period bonds,he will know exactly the outcome of his investmentsas measured by the terminal value of his wealth ... .If, however, he stays short, his outcome is uncer-tain ... . Thus, if he has risk aversion, he will preferto stay long

unless compensated by a term premium (pp. 183–84). Seealso Joseph E. Stiglitz (1970) and Mark Rubinstein (1976).

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100 THE AMERICAN ECONOMIC REVIEW MARCH 2001

ified stochastic discount factor, but have not aswhat bond portfolios are optimal for differetypes of investors.2

One reason for this gap in the literature mbe that it is extremely hard to characterizetimal portfolio strategies for long-term invetors. Robert C. Merton (1969, 1971) and PauSamuelson (1969) obtained some explicitsults under the assumption that real asset retare independently and identically distributover time; but this assumption implies that rinterest rates are constant, so in the absencinflation uncertainty—or with full indexation obond payments to inflation—bond returnsnonrandom and all bonds are perfect substitfor cash. Stanley Fischer (1975), Zvi Bodieal. (1985), and Alan D. Viard (1993) have nontheless used this assumption to study bondmand. In Fischer’s model there is one nomibond with a fixed nominal interest rate, and oinflation-indexed bond with a fixed real intererate. The maturity of these bonds need nospecified, since bonds of all maturitiesperfect substitutes for each other.3 Bodie et al.use historical data to estimate the variancovariance matrix of real returns on nominbonds, assuming that this matrix and meanbond returns are constant over time. In thmodel, random inflation allows imperfect sustitutability among nominal bonds of differematurities, but constant real interest rates imthat long-term and short-term indexed bondsperfect substitutes. Viard uses the same frawork as Bodie et al. and derives some furtanalytical results.

Merton (1969, 1971, 1973) studied the inttemporal portfolio choice problem with timvarying investment opportunities, introducing

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2 Robert J. Shiller (1990), Campbell et al. (1997), andQiang Dai and Kenneth J. Singleton (2000) review therecent bond-pricing literature.

3 Fischer also considers multiple goods whose relativeprices may change; this allows him to introduce multipleindexed bonds, but the bonds are distinguished by the pricesto which they are indexed, and not by maturity. Fischerrecognizes that his assumptions may be problematic, concluding: “It is possible that too little uncertainty about thereturns from holding nominal bonds and equity over longperiods is reflected in the basic model of the paper and thasuch uncertainty would result in portfolio holders beingwilling to pay a substantial premium for a long-term in-dexed bond” (p. 528). This paper explores Fischer’s con-jecture.

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important concept of intertemporal hedging dmand for financial assets, but he did not obtaexplicit solutions for portfolio weights. Recentlynumber of authors such as Michael J. Brennanal. (1997), Pierluigi Balduzzi and Anthony WLynch (1999), Michael W. Brandt (1999), anNicholas C. Barberis (2000) have used numerimethods to solve particular long-run portfolchoice problems, while Tong Suk Kim and Edward Omberg (1996) and Campbell and Vice(1999) have derived some analytical results, bthese papers generally concentrate on the chbetween cash and equities rather than the demfor long-term bonds.4

A shared assumption of this research is thinvestors live exclusively off financial wealthand do not have any labor income. Bodie et(1992) show that if labor income is riskless,if shocks to labor income can be perfecthedged with risky assets, then labor income cbe treated as an implicit holding of financiaassets. In this case the results of the literatapply to the total of explicit and implicit asseholdings. When labor income has unhedgearisk, things become more complicated and nmerical or approximate analytical solutiomethods are required (John Heaton and Dorah J. Lucas, 1996; Joao Cocco et al., 19Viceira, 2001).

In this paper we study intertemporal portfolichoice in an environment with random real interest rates. For simplicity we assume thatvestors have only financial wealth and no labincome. We use an approximation techniqdeveloped in our earlier papers (Campbe1993; Campbell and Viceira, 1999) to replathe intractable portfolio choice problem with aapproximate problem that can be solved usithe method of undetermined coefficients. Wuse the approximate solution to understanddemand for long-term bonds.

We calibrate our model to historical data othe U.S. term structure of interest rates, areport optimal portfolios for investors with awide range of different attitudes towards risk.

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4 Since circulating the first version of this paper we havebecome aware of numerical work by Brennan et al. (1996)exploring long-term investors’ demand for long-term bonds.More recent analytical work on this topic includes Brennanand Yihong Xia (1998), Jun Liu (1998), and Jessica A.Wachter (1998).

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101VOL. 91 NO. 1 CAMPBELL AND VICEIRA: WHO SHOULD BUY LONG-TERM BONDS?

order to study the effects of inflation risk ooptimal bond portfolios and investor utility, wcompare the solutions to our model when onindexed bonds are available with the solutiowhen only nominal bonds, or both nominal aindexed bonds, are available. We also allowborrowing and short-sales constraints, andthe possibility of investment in equities.

We begin by specifying a simple two-factmodel of the term structure of interest rates, amented to fit equity as well as bond returns. Ttwo factors are the log real interest rate and theexpected rate of inflation. Each factor followsnormal first-order autoregressive [AR(1)] procewith constant variance. This implies that log boyields are linear in the factors and the model isthe tractable “affine yield” class (Darrell Duffiand Rui Kan, 1996; Dai and Singleton, 2000). Tmodel for the real term structure is a discrete-tiversion of Vasicek (1977), while the model for thnominal term structure is a discrete-time versof Terence Langetieg (1980). Closely relatmodels are discussed in Campbell et al. (19Ch. 11).5

Next we consider the portfolio choice problefor an infinite-lived investor who has only financial wealth and must choose consumption aoptimal portfolio weights in each period. Becauthe investor is infinite-lived, she does not valstability of wealth at any unique horizon; rathshe cares about the long-run properties ofconsumption path. We assume that the investpreferences are of the form suggested by Larence Epstein and Stanley Zin (1989, 1991) aPhilippe Weil (1989); the investor has constarelative risk aversion and constant intertempoelasticity of substitution in consumption, but theparameters need not be related to one anoEpstein-Zin preferences nest the traditional powutility specification in which relative risk aversiois the reciprocal of the intertemporal elasticitysubstitution.

We show that the investor’s demand for lonterm bonds can be decomposed into a “myopdemand and a “hedging” demand. Myopic dmand depends positively on the term premiuand inversely on the variance of long-term bo

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5 Brennan and Xia (1998), Liu (1998), and Wachter(1998) also use affine yield models but do not distinguishbetween real and nominal bonds.

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returns and the investor’s risk aversion. As riaversion increases, myopic demand shrinkszero. Hedging demand, on the other hand,proportional to one minus the reciprocal of risaversion. It is zero when risk aversion is onbut accounts for all bond demand when riaversion is infinitely large. We show that ainfinitely risk-averse investor with zero intertemporal elasticity of substitution in consumption will choose an indexed bond portfolio thais equivalent to an indexed perpetuity, that isportfolio that delivers a riskless stream of reconsumption. In this way we are able to suppthe commonsense view that long-term bonare appropriate for long-lived investors whdesire stability of income.

Our analysis delivers explicit solutions foportfolio weights, consumption rules, and investor welfare. We can compare investor behior under alternative assumptions about tavailable menu of assets. We find that whindexed bonds are not available, investors faing a given interest-rate process shorten thbond portfolios and increase their precautionasavings. This has serious utility costs for coservative investors, who are much better owhen they have the opportunity to buy indexebonds. In general equilibrium, of course, suchanges in asset demands are likely to alterinterest-rate process itself, but we do not eplore this effect.

We also consider optimal portfolios wheequities, as well as bonds, are available. We fithat the ratio of bonds to equities in the optimportfolio increases with the coefficient of relative risk aversion. As Niko Canner et al. (1997have pointed out, this is consistent with convetional portfolio advice but inconsistent withstatic mean-variance analysis. The static mevariance model with a riskless one-period as(“cash”) predicts that all investors should holdsingle mutual fund of risky assets; more consvative investors should increase the ratio of cato the risky mutual fund, but should not changtheir relative holdings of risky assets. Oumodel helps to resolve the asset-allocation pzle identified by Canner et al.; more generallyunderscores the dangers of using static portfochoice theory to study the dynamic problemfaced by long-term investors.

The organization of the paper is as followSection I presents the two-factor term-structu

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model, and shows how it can be solved for boprices at all maturities. Section II sets upinvestor’s intertemporal consumption and portlio choice problem, explains our approximationthe problem, and discusses the approximate stion in the case where only indexed bondsavailable. Section III asks how things chanwhen only nominal bonds, or both nominal aindexed bonds, are available. This section aconsiders the consumption and portfolio choproblem in the presence of equities, and shhow to impose borrowing and short-sales cstraints. Section IV concludes.

I. A Two-Factor Model of the Term Structureof Nominal Interest Rates

A. Specification of the Model

Our focus in this paper is the microeconomproblem of portfolio choice for a long-term invetor facing exogenous bond returns. In ordergenerate empirically reasonable and theoreticwell-specified bond returns, however, we startwriting down a general-equilibrium bond-pricinmodel. We consider a discrete-time, two-fachomoskedastic model that allows for nonzero crelation between innovations in the short-term rinterest rate and innovations in expected inflat

This two-factor, homoskedastic model is tsimplest model of the term structure of interrates that allows us to distinguish betweenflation-indexed and nominal bonds. It wouldpossible to consider more factors and to allfor conditional heteroskedasticity—for example, using square-root factor processes. Hever, while we could fit the data better in thway, we would add mathematical complexwithout gaining substantial economic insighinto the long-run portfolio choice problem.

The real part of the model is determinedthe stochastic discount factor (SDF)Mt 1 1 thatprices all assets in the economy. In a represtative-agent framework the SDF can be relato the marginal utility of a representative invetor, but here we simply use it as a devicegenerate a complete set of bond prices.assume thatMt 1 1 has the following lognormastructure, a discrete-time version of Vasic(1977):

(1) 2mt 1 1 5 xt 1 vm,t 1 1,

-

-

xt 1 1 5 ~1 2 fx!mx 1 fxxt 1 «x,t 1 1 ,

vm,t 1 1 5 bmx«x,t 1 1 1 «m,t 1 1 ,

where mt 1 1 5 log(Mt 1 1) and xt, the one-period-ahead conditional expectation ofmt 1 1,follows an AR(1) process.

The nominal part of the model is also chaacterized by a lognormal, conditionally homoskedastic structure:

(2) p t 1 1 5 zt 1 vp,t 1 1 ,

zt 1 1 5 ~1 2 fz!mz 1 fzzt 1 vz,t 1 1 ,

vz,t 1 1 5 bzx«x,t 1 1 1 bzm«m,t 1 1 1 «z,t 1 1 ,

vp,t 1 1 5 bpx«x,t 1 1 1 bpm«m,t 1 1 1 bpz«z,t 1 1

1 «p,t 1 1 ,

wherept 1 1 is the log inflation rate andzt is theone-period-ahead conditional expectation of tinflation rate.

The system is subject to four normally distributed, white noise shocks«m,t 1 1, «p,t 1 1,«x,t 1 1, and «z,t 1 1 that determine the innova-tions to the log SDF, the log inflation rate, antheir conditional means. These shocks are crosectionally uncorrelated, with variancessm

2 ,sp

2 , sx2, and sz

2. It is important to note thatzt 1 1, the expected inflation rate, is affected bboth a pure expected-inflation shock«z,t 1 1 andthe shocks to the expected and unexpectedSDF«x,t 1 1 and«m,t 1 1. That is, innovations toexpected inflation can be correlated with innvations in the log SDF, and hence with innovtions in the short-term real interest rate. Thecorrelations mean that nominal interest ratneed not move one-for-one with expected infltion—that is, the Fisher hypothesis need nhold—and nominal bond prices can includeinflation risk premium as well as a real termpremium.

We have written the model with a selfcontained real sector (1) and a nominal sec(2) that is affected by shocks to the resector. But this is merely a matter of notational convenience. Our model is a reducform rather than a structural model, so

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captures correlations among shocks to rand nominal interest rates but does not hanything to say about the true underlyisources of these shocks.

Campbell et al. (1997) note that«m,t11 onlyaffects the average level of the real term strucand not its average slope or time-series behaAccordingly, we can either drop it or identify ivariance with an additional restriction. We follothe second approach and introduce equities inmodel. We assume that the unexpected log reon equities is affected by shocks to both thepected and unexpected log SDF:

(3) r e,t 1 1 2 Et r e,t 1 1 5 bex«x,t 1 1

1 bem«m,t 1 1.

Campbell (1999) shows that this decompositof the unexpected log equity return into a linecombination of the shocks to the expected aunexpected log SDF is consistent with a repsentative-agent endowment model wherepected aggregate consumption growth folloan AR(1). From the fundamental pricing eqution 1 5 Et[Mt 1 1Rt 1 1] and the lognormastructure of the model it is easy to show thatrisk premium on equities, over a one-peririskless returnr1,t 1 1, is given by

(4)

Et @r e,t 1 1 2 r 1,t 1 1# 112

Vart ~r e,t 1 1 2 r 1,t 1 1!

5 Covt ~r e,t 1 1 2 r 1,t 1 1, 2mt 1 1!

5 bmxbexsx2 1 bemsm

2 .

The variance term on the left-hand side of (4a Jensen’s Inequality correction that appebecause we are working in logs, and the teon the right-hand side relate the risk premiuon equities to the covariance of equity retuwith innovations in the SDF. This specificatioimplies that the equity premium, like all othrisk premia in the model, is constant over timThus it ignores the time variation in the equpremium that is the subject of our earlier papon long-run portfolio choice (Campbell and Vceira, 1999).

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B. Derivation of the Term Structure

We now show how our model can price dfault-free zero-coupon bonds that are eithinflation-indexed (paying one unit of consumption at maturity) or nominal (paying one dollaat maturity). There is a direct link between thstochastic discount factor and the log return,equivalently the log yield, on a one-period indexed bond:r1,t 1 1 5 2log Et[Mt 1 1]. Be-causeMt 1 1 is lognormal, we have that

(5) r 1,t 1 1 5 Et @2mt 1 1# 212

Vart @mt 1 1#

5 xt 212

~bmx2 sx

2 1 sm2 !.

Our assumptions onmt 1 1 imply that the short-term real interest rate is stochastic, though risless one period in advance. It inherits thstochastic properties ofxt 1 1, and follows anAR(1) process with meanm 2 (bmx

2 sx2 1

sm2 )/ 2 and persistencefx.Campbell et al. (1997), following Singleto

(1990), Tong-sheng Sun (1992), and DavBackus (1993), show that a lognormal, condtionally homoskedastic stochastic discount fator implies a pricing structure for log indexebond yields which is affine inxt 1 1. The logyield on ann-period indexed zero-coupon bonynt, times bond maturityn, which equals minusthe log price of the bond,pnt, is given by

(6) n z ynt 5 2pnt 5 An 1 Bnxt ,

whereAn andBn are functions of bond maturityn but not of timet, and satisfy the followingrecursive equations:

(7) Bn 5 1 1 fxBn 2 1 51 2 fx

n

1 2 fx,

An 2 An 2 1 5 ~1 2 fx!mxBn 2 1

2 12

@~bmx 1 Bn 2 1!2sx2 1 sm

2 #,

andA0 5 B0 5 0. An implication of (6) is thatyields on indexed bonds of different maturitieare perfectly correlated with each other.

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The one-period log return on ann-periodindexed zero-coupon bond is by definiti( pn2 1,t 1 1 2 pn,t). Combining this expressiowith (6) and (7), the excess return over tone-period log interest rate is

(8) r n,t 1 1 2 r 1,t 1 1 5 2 12

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so then-period bond is risky, with a sensitivitBn2 1 to real interest rate shocks and a rpremium given by

(9)

Et @r n,t 1 1 2 r 1,t 1 1# 112

Vart ~r n,t 1 1 2 r 1,t 1 1!

5 Covt ~r n,t 1 1 2 r 1,t 1 1, 2mt 1 1!

5 2bmxBn 2 1sx2.

The variance term on the left-hand side of (9a Jensen’s Inequality correction that appebecause we are working in logs. The conditiocovariance of the excess bond return withlog SDF determines the risk premium. In ohomoskedastic model the conditional covaance is constant through time but dependenthe bond maturity; thus the expectations hpothesis of the term structure holds for indexbonds. It is important to realize that constarisk premia do not imply constant investmeopportunities because real interest rates arechastic in our model.

SinceBn2 1 . 0, the risk premium has thopposite sign tobmx. With a positive bmx,long-term indexed bonds pay off when the mginal utility of consumption for a representatiinvestor is high, that is, when wealth is modesirable. In equilibrium, these bonds havenegative risk premium. With a negativebmx, onthe other hand, long-term indexed bonds paywhen the marginal utility of consumption forrepresentative investor is low, and so in eqlibrium they have a positive risk premium.

Equations (8) and (9) imply that the Sharratio for indexed bonds is2bmxsx, which is

-

independent of bond maturity. The invarianof the Sharpe ratio to bond maturity followfrom the one-factor structure of the real sectof the model. The ratio of the risk premium tthe variance of the excess return, which detmines a myopic investor’s allocation to longterm bonds, is2bmx/Bn2 1. This does dependon bond maturity but not on the volatility of threal interest rate.

The pricing of nominal bonds follows thesame steps as the pricing of indexed bonds. Trelevant stochastic discount factor for nominbonds is the nominal SDFMt 1 1

$ , the log ofwhich is given by:

(10) mt 1 1$ 5 mt 1 1 2 p t 1 1 .

Since bothMt11 andPt11 are jointly lognormaland homoskedastic,Mt11

$ is also lognormal. Thelog nominal return on a one-period nominal bois r1,t11

$ 5 2log Et[Mt11$ ], which implies that

r1,t11$ is a linear combination of the expected lo

real SDF and expected inflation given in aAppendix available from the authors uporequest.6

The log price of ann-period nominal zero-coupon bond,pn,t

$ , also has an affine structureIt is a linear combination ofxt andzt the coef-ficients of which are time-invariant, though thevary with the maturity of the bond:

(11) 2pn,t$ 5 An

$ 1 B1,n$ xt 1 B2,n

$ zt .

The Appendix gives expressions for the coefcientsAn

$, B1,n$ , andB2,n

$ .Since nominal bond prices are driven b

shocks to both real interest rates and inflatiothey have a two-factor structure rather than tsingle-factor structure of indexed bond priceInflation affects the excess return on ann-periodnominal bond over the one-period nominal iterest rate, so risk premia in the nominal ter

6 To economize space, throughout the article we omsome proofs and empirical results of secondary importanHowever, we have included all of them in an Appendwhich is readily available from the authors or from thepersonal web pages: http://www.economics.harvard.efaculty/jcampbell/campbell.html and http://www.peoplhbs.edu/lviceira.

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structure include compensation for inflatirisk. Like all other risk premia in the modehowever, the risk premia on nominal bondsconstant over time; thus the expectationspothesis of the term structure holds for nomias well as for real bonds.

C. The Term Structure of Interest Ratesin the United States

We estimate the two-factor term-structumodel using data on U.S. nominal intererates, equities, and inflation. We use nomizero-coupon yields at maturities three montone year, three years, and ten years fromHuston McCulloch and Heon-Chul Kwo(1993), updated by Frank F. Gong and EliRemolona (1996a, b). We take data on equifrom the Indices files on the CRSP tapes, usthe value-weighted return, including dividendon the NYSE, AMEX, and NASDAQ marketFor inflation, we use a Consumer Price Indthat retrospectively incorporates the rentequivalence methodology, thereby avoiding adirect effect of nominal interest rates on mesured inflation. The Appendix shows that esmation results are extremely similar if winstead use the personal consumption expeture (PCE) deflator to measure inflation. Athough the raw data are available monthly,construct a quarterly data set in order to redthe influence of high-frequency noise in infltion and short-term movements in interest raWe begin our sample in 1952, just after tFed-Treasury Accord that dramatically alterthe time-series behavior of nominal intererates. Our data end in 1996.

To avoid the implication of the model that boreturns are driven by only two common factors,that all bond returns can be perfectly explainedany two bond returns, we assume that bond yiare measured with error. The errors in yieldsnormally distributed, serially uncorrelated, auncorrelated across bonds. Then the term-stture model becomes a classic state-space modwhich unobserved state variablesxt andzt follow alinear process with normal innovations andobserve linear combinations of them with normerrors. The model can be estimated by maximlikelihood using a Kalman filter to construct thlikelihood function (Andrew C. Harvey, 1989George Pennacchi, 1991; Gong and Remolo

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1996a, b; Andrea Berardi, 1997; Silverio Foresial., 1997). This is an attractive alternative to tGeneralized Method of Moments used to estimterm-structure models by Michael R. Gibbons aKrishna Ramaswamy (1993) and others.

In Table 1 we report parameter estimatesthe period 1952–1996 and the period 1981996. Interest rates were unusually high avolatile in the 1979–1982 period, during whicthe Federal Reserve Board under Paul Volckwas attempting to reestablish the credibilityanti-inflationary monetary policy and was experimenting with monetarist operating procdures. Many authors have argued that rinterest rates and inflation have behaved diffently in the monetary-policy regime establishesince 1982 by Federal Reserve chairmVolcker and Alan Greenspan (see, for exampRichard Clarida et al., 1998). Accordingly wreport separate estimates for the period startin 1983 in addition to the full sample period.

The parameter values in Table 1 are restricmaximum-likelihood estimates of the modeUnrestricted maximum-likelihood estimatesthe data well in the 1952–1996 sample periobut they deliver implausibly low means foshort-term nominal and real interest rates in t1983–1996 sample period. (The model doesnecessarily fit the sample means becausesame parameters are used to fit both time-seand cross-sectional behavior; thus the mocan trade off better fit elsewhere for worse fitmean short-term interest rates.) Accordingly wrequire that the model exactly fit the sampmeans of nominal interest rates and inflatioThis restriction hardly reduces the likelihoodall in 1952–1996 and even in 1983–1996cannot be rejected at conventional significanlevels.

The first two columns of Table 1 report parameters and asymptotic standard errors forperiod 1952–1996. All parameters are in natuunits, so they are on a quarterly basis. Westimate a moderately persistent process forreal interest rate; the persistence coefficientfxis 0.87, implying a half-life for shocks to reainterest rates of about five quarters. The epected inflation process is much more perstent, with a coefficientfz of 0.9992 that impliesa half-life for expected inflation shocks of ovetwo centuries! Of course, the model also allowfor transitory noise in realized inflation.

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The bottom of Table 1 reports the implictions of the estimated parameters for the meand standard deviations of real interest ranominal interest rates, and inflation, measuin percent per year. The implied mean log yieon an indexed three-month bill is 1.39 percefor the 1952–1996 sample period. Takengether with the mean log yield on a nominthree-month bill of 5.50 percent and the melog inflation rate of 3.77 percent (both restrictto equal the sample means over this period),adjusting for Jensen’s Inequality using one-hthe conditional variance of log inflation, thimplied inflation risk premium in a three-montnominal Treasury bill is 35 basis points. Thfairly substantial risk premium is explained bthe significant positive coefficientbpx and thesignificant negative coefficientbpm in Table 1.7

Risk premia on long-term indexed bonds, rative to a three-month indexed bill, are det

TABLE 1—TERM-STRUC

Parameter

1952.I–1996

Estimate

mx 0.0573mz 0.0094fx 0.8688fz 0.9992bmx 274.9797bzx 0.0752bzm 20.0012bpx 0.5198bpm 20.0088bpz 1.4320bex 23.4957bem 0.3013sx 0.0025sm 0.2694sz 0.0013sp 0.0071Log-likelihood 26.3327Number of observations 179E[ r1,t 1 1] 1.39 perceE[ r1,t 1 1

$ ] 5.50 perces (r1,t 1 1) 1.01 perces (r1,t 1 1

$ ) 6.70 perceE[pt 1 1] 3.77 percest(pt 1 1) 1.57 perce

m-ftic

ndrl-

7 This result is somewhat sensitive to the sample period.If we estimate the model over the period 1952–1979, wefind a smaller inflation risk premium at the short end of theterm structure, consistent with the results of Foresi et al.(1997).

,

mined by the parameterbmx. This is negativeand significant, implying positive risk premion long-term indexed bonds and an upwarsloping term structure of real interest rates. Rpremia on nominal bonds, relative to indexebonds, are determined by the inflation-risk prametersbzx and bzm. The former is positivebut statistically insignificant, while the latter inegative and significant. Both point estimatimply positive inflation risk premia on nominabonds relative to indexed bonds.

The parameters in Table 1 can also be usedcalculate the volatility of the log stochastic discount factor. From (1), the variance ofmt 1 1 issx

2/(1 2 fx2) 1 bmx

2 sx2 1 sm

2 . The estimates inTable 1 imply a large quarterly standard devition of 0.33, consistent with the literature ovolatility bounds for the stochastic discount fator (Shiller, 1982; Lars P. Hansen and RaJagannathan, 1991; John H. Cochrane and Hsen, 1992). When financial markets are coplete, the discounted marginal utility growth oeach investor must be equal to the stochasdiscount factor. Therefore the consumption aportfolio solutions we report later in the papefor the complete-markets case imply highly vo

E MODEL ESTIMATION

1983.I–1996.III

ndard error Estimate Standard er

0.0298 0.0194 0.06930.0087

0.0057 0.9862 0.00420.0012 0.8599 0.02161.6949 228.6919 114.00250.0516 20.4114 0.18860.0006 0.0008 0.00240.3050 20.0267 0.97900.0034 0.0008 0.01930.2940 21.5412 1.50473.4123 29.3629 6.30140.0979 0.5089 1.35280.0001 0.0027 0.00060.0927 0.1351 0.35790.0001 0.0016 0.00020.0004 0.0072 0.0018

26.822255

2.93 percent6.40 percent3.25 percent3.09 percent3.49 percent1.52 percent

r

5

)2

)0

)7

)5

)

)

)

)

)

)


TABLE 2—SAMPLE AND IMPLIED MOMENTS OF THETERM STRUCTURE

Moment

1952.I–1996.III 1983.I–1996.III

1 year 3 year 10 year 1 year 3 year 10 yea

A: Nominal Term Structure

(1) E[ rn,t 1 1$ 2 r1,t 1 1

$ ] 1 s2(rn,t 1 1$ 2 r1,t 1 1

$ )/2 sample 0.397 0.651 0.915 0.706 2.111 5.67implied 0.559 1.290 1.967 0.155 0.658 2.278(s.e.) (0.272) (0.655) (1.071) (0.570) (2.373) (8.307

(2) s (rn,t 1 1$ 2 r1,t 1 1

$ ) sample 1.615 4.615 11.365 1.135 4.220 12.61implied 1.634 4.501 11.566 1.312 4.627 14.896(s.e.) (0.084) (0.263) (0.707) (0.296) (1.171) (3.664

(3) SR$ 5 (1)/(2) sample 0.246 0.141 0.080 0.622 0.500 0.45implied 0.342 0.287 0.170 0.118 0.142 0.153(s.e.) (0.172) (0.148) (0.091) (0.450) (0.535) (0.579

(4) E[yn,t 1 1$ 2 y1,t 1 1

$ ] sample 0.440 0.802 1.185 0.527 1.267 2.06implied 0.294 0.742 1.174 0.071 0.276 0.766(s.e.) (0.142) (0.392) (0.748) (0.283) (1.148) (4.373

(5) s (yn,t 1 1$ 2 y1,t 1 1

$ ) sample 0.222 0.409 0.613 0.177 0.341 0.54implied 0.182 0.488 0.826 0.140 0.380 0.803(s.e.) (0.010) (0.026) (0.048) (0.019) (0.045) (0.123

B: Real Term Structure

(6) E[ rn,t 1 1 2 r1,t 1 1] 1 s2(rn,t 1 1 2 r1,t 1 1)/2 implied 0.490 1.075 1.345 0.245 0.851 2.513(s.e.) (0.247) (0.563) (0.710) (0.904) (3.139) (9.289

(7) s (rn,t 1 1 2 r1,t 1 1) implied 1.309 2.994 3.788 1.590 5.521 16.295(s.e.) (0.071) (0.184) (0.275) (0.369) (1.263) (3.708

(8) SR 5 (6)/(7) implied 0.374 0.374 0.374 0.154 0.154 0.154(s.e.) (0.198) (0.198) (0.198) (0.590) (0.590) (0.590

(9) E[yn,t 1 1 2 y1,t 1 1] implied 0.253 0.665 1.100 0.118 0.381 0.858(s.e.) (0.130) (0.347) (0.581) (0.455) (1.620) (5.177

(10) s (yn,t 1 1 2 y1,t 1 1) implied 0.182 0.486 0.816 0.067 0.235 0.738(s.e.) (0.010) (0.026) (0.045) (0.020) (0.070) (0.204

C: Equities

(11) E[ re,t 1 1 2 (r1,t 1 1$ 2 pt 1 1)]

1 s2(re,t 1 1 2 (r1,t 1 1$ 2 pt 1 1))/2

sampleimplied

6.9108.988

8.7384.527

(s.e.) (3.131) (9.212)(12) s (re,t 1 1 2 (r1,t 1 1

$ 2 pt 1 1)) sample 15.917 14.646implied 15.896 14.748(s.e.) (0.708) (1.069)

(13) SR 5 (11)/(12) sample 0.434 0.597implied 0.565 0.307(s.e.) (0.184) (0.621)

Note: Standard errors (s.e.) are in parentheses.

ea

of5

us.a-blerm

perim-lta

plealbyra-ndnts

)

-

structure variables, measured in percentyear. It also reports standard errors for theplied moments, calculated using the demethod. Panel A of Table 2 reports sammoments for returns and yields on nominbonds, together with the moments impliedour estimated model; Panel B shows compable implied moments for indexed bonds, aPanel C reports sample and implied mome

atile marginal utilities, due either to volatilconsumption or high risk aversion. This ismanifestation of the equity premium puzzleRajnish Mehra and Edward C. Prescott (198in our microeconomic model with exogenoasset returns and endogenous consumption

Table 2 explores the term-structure implictions of our estimates in greater detail. The tacompares implied and sample moments of te

n’soni-

rdre-

nal)(5)eldesere-i-

tlyativin-

innalnthileumnt35-m-lye-earhisla-

dsthey,ardin-es(8)fornddelen-forrt-forearersrs-

theeo-

elseess

pee–heiedrshisho-

s

heite-inoflenddsti-

thendofct

a-hite-

odree

s-

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andrd

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eatiod


for equities. Row (1) of the table gives JenseInequality-corrected average excess returnsn-period nominal bonds over one-period nomnal bonds, while row (2) gives the standadeviations of these excess returns. Row (3)ports annualized Sharpe ratios for nomibonds, the ratio of row (1) to row (2). Row (4reports mean nominal yield spreads and rowreports the standard deviations of nominal yispreads. Rows (6) through (10) repeat thmoments for indexed bonds. Note that theported risk premia and Sharpe ratios for nomnal and indexed bonds are not direccomparable because they are measured relto different short-term assets, nominal anddexed respectively.

A comparison of the model implicationsrows (1) and (6) shows that ten-year nomibonds have a risk premium over three-monominal bills of 1.97 percent per year, whten-year indexed bonds have a risk premiover three-month indexed bills of 1.35 perceper year. These numbers, together with thebasis-point risk premium on three-month noinal bills over three-month indexed bills, impa ten-year inflation risk premium (the risk prmium on ten-year nominal bonds over ten-yindexed bonds) slightly above 1.1 percent. Testimate is consistent with the rough calcutions in Campbell and Shiller (1996).

Rows (2) and (7) show that nominal bonare much more volatile than indexed bonds;difference in volatility increases with maturitso that ten-year nominal bonds have a standdeviation three times greater than ten-yeardexed bonds. This difference in volatility makthe Sharpe ratio for indexed bonds in rowconsiderably higher than the Sharpe rationominal bonds in row (3). Since indexed boreturns are generated by a single-factor mothe Sharpe ratio for indexed bonds is indepdent of maturity at 0.37. The Sharpe rationominal bonds declines with maturity; shoterm nominal bonds have a ratio close to thatindexed bonds, but the Sharpe ratio for ten-ynominal bonds is only 0.17. These numbimply that in our portfolio analysis, investowith low risk aversion will have a strong myopic demand for indexed bonds.

Table 2 can also be used to evaluateempirical fit of the model. A comparison of thmodel’s implied moments with the sample m

e

,

ments for nominal bonds shows that the modfits the volatility of excess nominal bond returnand changes in yields extremely well. Thmodel somewhat overstates the average excnominal bond return and the nominal Sharratio, but this can be attributed in part to thupward drift in interest rates over the 19521996 sample period which biases downward tsample means. The standard errors for implvolatilities are small, while the standard erroof implied mean excess returns are large. Treflects the well-known result that it is mucharder to obtain precise estimates of first mments than of second moments.

Another way to judge the fit of the model ito ask how much of the variability of bondyields, or bond returns, is accounted for by tstructural parameters as opposed to the whnoise measurement errors we have allowedeach bond yield. The estimated variancesmeasurement errors (not reported in Tab1) are zero for one-year and ten-year bonds aare extremely small for three-month bills anthree-year bonds. Measurement errors are emated to account for less than 0.5 percent ofvariance of three-month and three-year boyields and less than 5 percent of the variancethree-year bond returns. This reflects the fathat bond yields are highly persistent at all mturities, so the model fits them primarily witpersistent structural processes rather than whnoise measurement errors.

Overall the model appears to provide a godescription of the nominal U.S. term structuconsidering its parsimony and the fact that whave forced it to fit both time-series and crossectional features of the data.

Rows (11), (12), and (13) of Table 2 reposummary statistics for equities: the annualizJensen’s Inequality-corrected average excreturns on equities relative to nominal bills, thstandard deviation of these excess returns,their Sharpe ratio. The model fits the standadeviation of equities extremely well but ovepredicts the equity premium and the Sharratio for equities. The implied Sharpe ratio o0.57 implies that investors with low risk avesion will have an extremely large myopic demand for equities; this is again a manifestatiof the equity premium puzzle.

The right-hand sides of Tables 1 and 2 repthese estimates for the Volcker-Greenspan per

ared

withrcteore

rs.thahaostr-in

prelsonds isweull-buriod

rvi-he

util-nd

ble-

usetti-

sn-t-e

ed

e

n3),sr

tn

efof


1983–1996. Many of the parameter estimatesquite similar; however, we find that in this perioreal interest rates are much more persistent,fx 5 0.986 and an implied half-life for real inte-est rate shocks of about 12 years. The expeinflation process now mean-reverts much mrapidly, with fz 5 0.860 implying a half-life forexpected inflation shocks of about five quarteThese results are consistent with the notionsince the early 1980’s the Federal Reservemore aggressively controlled inflation at the cof greater long-term variation in the real inteest rate (Clarida et al., 1998). The increasereal-interest-rate persistence increases the riskmia on indexed and nominal bonds, but it agreatly increases the volatility of indexed boreturns so the Sharpe ratio for indexed bondlower at 0.15. In the remainder of the paperpresent portfolio choice results based on our fsample estimates for the period 1952–1996,we also discuss results for the 1983–1996 pewhere they are importantly different.

II. The Demand for Indexed Bonds

A. Investor Preferences andthe Budget Constraint

The investor is infinite-lived, lives off hefinancial wealth, and faces the investment enronment described above. We assume thatpreferences are described by the recursiveity proposed by Epstein and Zin (1989) aWeil (1989):

(12) U~Ct , Et Ut 1 1!

5 $~1 2 d!Ct~1 2 g!/u

1 d~Et Ut 1 11 2 g!1/u} u/~1 2 g!,

whered , 1 is the discount factor,g . 0 is thecoefficient of relative risk aversion,c . 0 is theelasticity of intertemporal substitution, andu [(1 2 g)/(1 2 1/c). The recursive utility function(12) reduces to the standard time-separapower utility function with relative risk aversion g whenc 5 1/g. In this caseu 5 1.

Recursive preferences are useful becathey allow us to separate the investor’s a

d

ts

-

t

r

,

tude towards risk from her attitude towardintertemporal substitution of consumptioover time. This separation is particularly important in our framework, where the shorterm interest rate moves over time giving thinvestor an incentive to change her plannconsumption growth rate.

The investor maximizes (12) subject to thintertemporal budget constraint

(13) Wt 1 1 5 Rp,t 1 1~Wt 2 Ct !,

whereRp,t 1 1 is the gross return at timet 1 1on her portfolio at timet.

Epstein and Zin (1989, 1991) have showthat when the budget constraint is given by (1the optimal portfolio and consumption policiemust satisfy the following Euler equation foany asseti :

(14) 15 EtFHdSCt 1 1

CtD2~1/c!J u

Rp,t 1 12~1 2 u!Ri ,t 1 1G .

When i 5 p, (14) reduces to:

(15) 15 EtFHdSCt 1 1

CtD2~1/c!

Rp,t 1 1J uG .

Dividing (12) by Wt and using the budgeconstraint we obtain the following expressiofor utility per unit of wealth:

(16) Vt ; Ut /Wt 5 H ~1 2 d!S Ct

WtD 1 2 ~1/c!

1 dS1 2Ct

WtD 1 2 ~1/c!

3 ~Et @Vt 1 11 2 gRp,t 1 1

1 2 g #!1/uJ 1/~1 2 c!

.

Epstein and Zin (1989, 1991) show that thmaximizedVt, the value function per unit owealth, can be written as a power function(1 2 d) and the consumption-wealth ratio:

sase

-

llype-e

ldndsedeedtorthis

tto

alox-is

eog-

aralthendx-Fi-edp-hetheent

ellst

ssiona-

gss-

ith

tsf

ainy-n

flth.-

inb--th

ln

urlike


(17) Vt 5 ~1 2 d!2@c/~1 2 c!#S Ct

WtD 1/~1 2 c!

.

Two special cases are worth noting. First, acapproaches one, the exponents in (17) increwithout limit. The value function has a finitlimit, however, because the ratioCt /Wt ap-proaches (12 d) as shown by Alberto Giovannini and Weil (1989). Second, asc approacheszero,Vt approachesCt /Wt. A consumer who isextremely reluctant to substitute intertemporaconsumes the annuity value of wealth eachriod, and this consumer’s utility per dollar is thannuity value of the dollar.

B. Loglinear Approximation of the Model

At this point, to simplify the analysis and buiintuition we assume that there are only two boavailable to the investor, a one-period indexbond and ann-period indexed bond. Given thone-factor structure of our model for indexbonds, this is equivalent to providing the inveswith a complete indexed term structure. Underassumption,Rp,t11 is equal to

(18) Rp,t 1 1 5 an,t ~Rn,t 1 1 2 R1,t 1 1! 1 R1,t 1 1,

wherean,t is the fraction of the investor’s sav-ings allocated to then-period indexed bond atime t. In Section III we generalize the modelinclude nominal bonds and equities.

In order to find optimal savings and the optimallocations to the two bonds, we adopt an apprimate analytical solution method. The first stepto characterizean,t, the optimal allocation to thn-period bond, by combining a second-order llinear approximation to the Euler equation withfirst-order approximation to the intertempobudget constraint. We then guess a form foroptimal consumption and portfolio policies ashow that policies of this form satisfy the approimate Euler equation and budget constraint.nally we use the method of undetermincoefficients to identify the coefficients of the otimal policies from the primitive parameters of tmodel. By using a second-order expansion oflog Euler equation we account for second-momeffects in the model.

e

Following Fernando Restoy (1992), Campb(1993), and Campbell and Viceira (1999), we firloglinearize the Euler equation (14) fori 5 n andi 5 1, where asset 1 is the short-term riskleasset. Subtracting the loglinearized Euler equatfor the riskless asset from the loglinearized eqution for assetn, we find:

(19) Et @r n,t 1 1 2 r 1,t 1 1# 11

2Vart ~r n,t 1 1!

5 S u

cDCovt ~Dct 1 1, r n,t 1 1!

1 ~1 2 u!Covt ~r p,t 1 1, r n,t 1 1!,

where lowercase letters denote variables in loandD is the first-difference operator. This expresion says that the risk premium on assetn is aweighted average of the asset’s covariance wconsumption growth divided byc, and the asset’scovariance with the portfolio return. The weighare u and (12 u), respectively; in the case opower utility, u 5 1 and only the consumptioncovariance enters the Euler equation. We obt(19) from (14) by using both a second-order Talor approximation around the conditional meaof { rp,t11, Dct11} and the approximation log(11x) ' x for smallx. Equation (19) holds exactly iconsumption growth and the return on weahave a joint conditional lognormal distributionWe show later that our approximate solution implies just such a distribution.

We can loglinearize the Euler equation (15)a similar fashion. After reordering terms, we otain the well-known equilibrium linear relationship between expected log consumption growand the expected log return on wealth:

(20) EtDct 1 1 5 c log d 1 vp,t 1 cEt rp,t 1 1,

where the termvp,t is an intercept proportionato the conditional variance of log consumptiogrowth in relation to log portfolio returns:

(21) vp,t 51

2 S u

cDVart ~Dct 1 1 2 cr p,t 1 1!.

In general this intercept is time-varying, but in omodel it becomes a constant. These equations,

rns

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(19), hold exactly if consumption and asset retuare jointly conditionally lognormal.

Taking the return on wealth as given, we can aloglinearize the intertemporal budget constraint (around the mean log consumption-wealth rati

(22) Dwt11 < rp,t111 S1 21

rD ~ct 2 wt ! 1 k,

wherek 5 log(r) 1 (1 2 r)log(1 2 r)/r, andr 5 1 2 exp{E(ct 2 wt)} is a loglinearizationparameter. Note thatr is endogenous in thatdepends on the average log consumption-weratio, which is unknown until the model hbeen solved. Campbell (1993) proposes theproximation (22) and shows that it holds exacif the consumption-wealthratio is constanover time. Giovannini and Weil (1989) shothat if the elasticity of intertemporal substittion c 5 1, then the consumption-wealth rais constant at (12 d); in this special case ousolution is exact and we know thatr 5 d.

Finally, equation (18) allows us to approximarp,t11, the log return on wealth, as follows:

(23) r p,t 1 1 5 an,t ~r n,t 1 1 2 r 1,t 1 1! 1 r 1,t 1 1

1 12

an,t ~1 2 an,t !Vart ~r n,t 1 1!.

This is a discrete-time version of the log returnwealth in continuous time, where Ito’s Lemma cbe applied to equation (18). The accuracy ofapproximation increases as the time interval shri

Combining the trivial equality

(24) Dct11 5 ~ct11 2 wt11! 2 ~ct 2 wt!1 Dwt11

with equations (19), (22), and (23) and the dinition of u we find that

(25)

an,t 51

g

Et @r n,t 1 1 2 r 1,t 1 1# 112

Vart ~r n,t 1 1!

Vart ~r n,t 1 1!

2 S 1

1 2 cDSg 2 1

g D3

Covt ~r n,t 1 1, ct 1 1 2 wt 1 1!

Vart ~r n,t 1 1!.

-

.

This equation was first derived by Resto(1992). The first term is the myopic componeof asset demand; it is proportional to the ripremium on then-period bond and the reciprocal of the coefficient of relative risk aversionThe second term is Merton’s (1969, 1971, 197intertemporal hedging demand. It reflects tstrategic behavior of the investor who wishto hedge against future adverse changesinvestment opportunities, as summarized byconsumption-wealth ratio. In our setup the ivestment opportunity set is time-varying bcause interest rates are time-varying (althouexpected excess returns are constant); accingly the investor may want to hedge her cosumption against adverse changes in interrates. Intertemporal hedging demand is zewhen risk aversiong 5 1, but asg increasesmyopic demand shrinks to zero and hedgidemand does not. In the limit asg becomesarbitrarily large, hedging demand accountsall the demand for the risky asset.

An important special case arises when telasticity of intertemporal substitution is unityAs c 3 1, the log consumption-wealth ratibecomes constant so the covariance of asreturns with this ratio approaches zero. Hoever the covariance is divided by 12 c, whichalso approaches zero. Giovannini and W(1989), by taking appropriate limits, havshown that portfolio choice is not myopic in thcase even though the consumption-wealth rais constant. The solution presented in this pais exact for the casec 5 1.

C. An Explicit Solution

Equation (25) is recursive in the sense tharelates current portfolio decisions to future cosumption and portfolio decisions. In order to ga complete solution to the model we needderive consumption and portfolio rules that dpend only on current state variables. We do tby guessing that the consumption function takthe form

(26) ct 2 wt 5 b0 1 b1xt .

Calculations summarized in the Appendix veify this guess and show that the coefficients agiven by

ebe

g-

eral

se

nde-

athe

ogrt-

eitye-eic

--

ith

--t,ts.

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s.ds


(27) b0 5r

1 2 r F ~1 2 c!~1 2 g!

2g

3 Sbmx 1r

1 2 rfxD 2

sx2

21

2~1 2 c!sm

2 2 c log d 1 k

1 m~1 2 f!r~1 2 c!

1 2 rfxG

and

(28) b1 5 ~1 2 c!r

1 2 rfx.

In addition, the optimal portfolio share in thrisky asset is constant over time and canwritten as

(29) ant 5 an 521

gBn 2 1

3 Fbmx 1 ~1 2 g!r

1 2 rfxG .

These solutions are analytical, given the lolinearization parameterr. But r itself is a non-linear function of the coefficientsb0 and b1,sincer 5 1 2 exp{E[ct 2 wt]} 5 1 2 exp{ b0 1 b1mx}. Equations (27), (28), and thexpression forr define implicitly a nonlineamapping ofr onto itself which has an analyticsolution only in the casec 5 1, whenr 5 d. Inall other cases we solve forr numerically usinga simple recursive algorithm. We setr to someinitial value (typicallyr 5 d) and compute thecoefficients of the optimal policies; given thecoefficients we compute a new value forr, fromwhich we obtain a new set of coefficients, aso forth. We continue until the difference btween two consecutive values ofr is less than1024. This recursion converges very rapidly tonumber between zero and one whenevervalue function of the model is finite.

Equations (26) and (28) show that the lconsumption-wealth ratio is linear in the sho

term real interest rate (sincext is linearly relatedto r1,t 1 1). The response of consumption to thinterest rate depends on the investor’s elasticof intertemporal substitution, but does not dpend directly on her relative risk aversion. Thrisk aversion coefficient affects the dynambehavior of consumption only indirectlythrough its effect on the loglinearization parameterr. In the Appendix we show that this indirect effect is empirically very small.

The consumption-wealth ratio increases wthe interest rate ifc , 1 and falls with theinterest rate ifc . 1. An increase in the shortterm real interest rate is equivalent to an improvement in the investment opportunity seand it has both income and substitution effecAn investor with lowc is reluctant to substituteintertemporally, and for her the income effedominates, leading her to increase her consumtion relative to her wealth. Conversely, the sustitution effect dominates for an investor withigh c . 1. This investor will reduce presenconsumption when the interest rate increasBoth income and substitution effects on cosumption are stronger, the more persistent isimprovement in investment opportunities—thcloser is fx to one. The consumption-wealtratio is constant only whenc 5 1, in which casect 2 wt equals log(12 d).

Equation (29) shows that the optimal portfolio allocation to the long-term bond is constaover time and independent of the level of thshort-term interest rate. The portfolio allocatiodepends on the bond maturity, on the perstence of the short-term interest rate, and oninvestor’s relative risk aversion, but does ndepend directly on her elasticity of intertempral substitution. The elasticity of intertemporsubstitution affects portfolio choice only indirectly through its effect on the loglinearizatioparameterr, and we show in the Appendix thathis indirect effect is empirically very small.

The first term inside the brackets in (29represents the myopic demand for long-tebonds, while the second term representsintertemporal hedging demand. The myopic dmand depends on the ratio of mean excess breturn to variance,2bmx/Bn2 1, divided by thecoefficient of risk aversiong. Thus myopic de-mand goes to zero as risk aversion increase

The intertemporal hedging demand for bonis zero wheng 5 1. Investors with unit relative

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risk aversion are “myopic investors” who dmand long-term bonds exclusively for their ripremium. Investors withg . 1, however, havea positive intertemporal hedging demand. Loterm bonds have high returns when interest rafall, that is, when future investment opportunties deteriorate, so conservative investorsthem to hedge investment-opportunity risk.8

It is interesting to consider what happensthe limit as the investor becomes infinitely riaverse. The optimal portfolio for an infiniteconservative investor is one natural way tofine a “riskless” asset. Financial economiconventionally take a short-term perspectand treat a one-period indexed bond as risklA one-period nominal bond is a good substitfor a one-period indexed bond (Viard, 199and thus by extension the riskless asset is oidentified with a short-term nominal asset suas a Treasury bill. In a world with time-varyininterest rates, however, only the current shterm real interest rate is riskless; future shoterm real interest rates are uncertain. Tmeans that a strategy of investing in one-perbonds is risky for an infinite-lived investor.

In the Appendix, we show that an infinitlived investor who is infinitely risk averse aninfinitely reluctant to substitute consumptiointertemporally chooses a portfolio of indexed bonds that is equivalent to an indexperpetuity. Such a portfolio finances a risless, constant stream of consumption, so isuitable for an infinitely conservative lonterm investor despite its unstable capivalue. In this sense an indexed perpetuity,a one-period indexed bond, is the riskleasset in our model.9

An investor who is infinitely risk averse buwhose elasticity of intertemporal substitutionconsumption is nonzero holds a slightly diffe

utheit.hee

thealhealt in

8 Investors withg , 1, on the other hand, prefer to holdassets that pay off when investment opportunities are goodthey “reverse hedge” the risk of adverse shifts in investmentopportunities.

9 This result does not depend on our use of a loglinearapproximate solution method. Brennan and Xia (1998) andWachter (1998) derive the result without approximation in acontinuous-time, finite-horizon model assuming a Vasicek(1977) term structure and power utility defined over termi-nal wealth and consumption, respectively. Wachter alsoproves the result in a more general class of models. Stiglitz(1970) makes a similar point in a two-period model.

.

ent bond portfolio, because this investor wishto tilt the consumption growth path in responto interest-rate incentives and therefore allosome variability in consumption growth. Thispossible in a recursive-utility framework because marginal utility is determined not only bcurrent consumption but also by discounted fture utility. Hence even an infinitely risk-aversinvestor may be willing to accept some variabity in current consumption if it is compensateby movements in future utility that leave curremarginal utility unchanged. The empirical results in the Appendix show that the deviatiofrom the real-perpetuity portfolio is very smalconsistent with the general result that the elaticity of intertemporal substitution has onlsmall effects on the optimal portfolio policy.

A noteworthy feature of the portfolio solutionin (29) is that it does not restrict the portfolishare of the risky asset to lie between zero aone. In discrete time with lognormally distributed risky asset returns, portfolios with shopositions (a , 0) or leverage (a . 1) permitbankruptcy which would produce infinitely negative utility. In continuous time with diffusionsfor asset prices, this problem disappears asinvestor is able to continually rebalance hportfolio and can adjust her position at any timto avoid bankruptcy. Our approximate solutioimplicitly relies on this property of continuoustime models as we have replaced the exdiscrete-time portfolio return by the approximation (29) which holds exactly in continuoutime.

While the possibility of bankruptcy mayseem troubling, in Section III we show how trule it out by imposing short-sales and leveraconstraints. Also, Campbell and Viceira (200study our consumption and portfolio choicproblem in a continuous-time setting. They finthat continuous time rules out bankruptcy bchanges neither the form of the solution nor tnature of the approximation needed to findOur discrete-time solution (29) converges to tcontinuous-time solution as the model’s timinterval shrinks.

Our model assumes that the state variablextfollows a homoskedastic process. This hasadvantage that the Kalman filter is an optimestimator of the model’s parameters even if tunderlying model has a shorter time intervthan the data measurement interval or is se

;

p-eel

mame

onise

nsseessio,e

nalare

ant

l-ot torm-wethet,m-theofhewn

p-heisis

heetean

ter-S), it

meedhis

ofs-

nr-

ein-ceintefy-a

ixa

p-sx-

of-hei-e

iesc,lee-blein

-(4e

00e-

ef-

10 In the case of power utility (c [ 1/g), the discrete-time equivalent of the Cox-Huang PDE is given by theexpectational difference equation

W*tCt

5 EtFS1 1W*t 1 1

Ct 1 1Dd1/gMt 1 1

1 2 ~1/g!G ,

where W*t [ Wt 2 Ct is invested wealth. This equationfollows from the Euler equation (15), after substituting inRp,t 1 1 5 (W*

t 1 1 1 Ct 1 1)/W*t and noting that under

complete markets we haved(Ct 1 1/Ct)2g 5 Mt 1 1. When

loglinearized around the mean log consumption-wealth ra-tio, this expectational difference equation has the samesolution that we have already derived.


continuous time (A. R. Bergstrom, 1984; Cambell and Albert S. Kyle, 1993). It would bstraightforward, however, to solve the modunder the alternative assumption thatxt followsa square-root process. In this case the opticonsumption function would still have the for(26), and the optimal portfolio share in thrisky asset would still be constant. Equati(25) shows whyan remains constant under thform of conditional heteroskedasticity. Thnumerator ofan in (25) is a linear combinatioof the expected excess return on the risky aand the conditional covariance of the excreturn with the log consumption-wealth ratwhile the denominator is proportional to thconditional variance of the excess return. Ifxtfollows a square-root process, the conditiomean and variance of the excess returnproportional toxt, while (26) implies that theconditional covariance is also proportional toxt.Therefore, the ratio (25) must be a constindependent ofxt.

Some further properties of the solution folow from the fact that, even with only twassets, markets are complete with respecreal-interest-rate risk because our real-testructure model has only one factor. Herestate these properties briefly, and we referreader to the Appendix for full details. Firswith complete markets the investor can cobine short- and long-term bonds so thatreturn on her bond portfolio is independentthe maturity of the long-term bond traded in tmarket. That is, she can synthesize her ooptimal long-term bond, with the maturity otimal for her given her risk preferences. Tinterest-rate sensitivity of the portfolioanBn2 1, and equation (29) shows that thisindependent of bond maturityn.

Second, if real-interest-rate variation is tonly source of risk, then markets are complwith respect to all sources of risk. We cexplore this case by settingsm

2 5 0 so that«m,t 1 1 drops from the definition ofmt 1 1 in (1).In this case the SDF is unique. Since the intemporal marginal rate of substitution (IMRof any investor can be used as a valid SDFfollows that all investors must have the saIMRS which must equal the SDF we specifiexogenously for our term-structure model. Tprovides a check on the internal consistencyour solution. Using (14) to express the inve

l

t

tor’s IMRS as a function of consumptiogrowth and the portfolio return, it is straightfoward to show that log(IMRSt 1 1) 5 mt 1 1.

Third, Cox and Chi-fu Huang (1989) havproposed an alternative solution method fortertemporal consumption and portfolio choiproblems with complete markets. They workcontinuous time and show that with complemarkets, optimally invested wealth must satisa partial differential equation (PDE). Unfortunately this PDE does not generally haveclosed-form solution. We show in the Appendthat our solution methodology is equivalent todiscrete-time version of the Cox-Huang aproach; our loglinear approximation allows uto solve the discrete-time equivalent of the CoHuang PDE in closed form.10

D. Empirical Properties of the Solution

Table 3 explores the empirical propertiesthe portfolio solution (29) using the bondpricing parameters estimated in Table 1 for tperiod 1952–1996. Qualitative results are simlar for the 1983–1996 period reported in thAppendix. An understanding of these propertwill help us interpret solutions in more realistibut also more complex, settings with multiplong-term assets and portfolio constraints. Rsults for these settings are also reported in Ta3 and in Table 4, and they are discussedSection III.

We compute optimal portfolio rules for investors with the same time discount ratepercent) but different coefficients of relativrisk aversion of 0.75, 1, 2, 5, 10, and 5,0(effectively almost infinite). To save space wpresent results only for an elasticity of intertemporal substitution equal to one, since this co

iore-of

er-ponracwfor

estndAsxederoosirmelowheto-

ses

e-alithes-er-b-lthalthx-

ort-sode-erb-es.theheio.v-byorsrt-ngeses-ge

ng

tothelth

tly)

-

g-elywenddsion-dern-

-

alrstoelat

rastyof--

he,her

sois

becan


ficient has only a negligible effect on portfolallocation. The Appendix presents completesults for elasticities equal to the reciprocalsthe risk-aversion coefficients we consider.

The second column of Table 3 reports the pcentage portfolio share of a ten-year zero-couindexed bond. Since indexed bonds have atttive Sharpe ratios, we find that investors with lorisk aversion have a very large myopic demandlong-term indexed bonds; they want to invmany times their total wealth in these bonds aborrow at the short-term riskless interest rate.risk aversion increases, the demand for indebonds gradually declines, but it does not go to zbecause highly risk-averse investors have a ptive intertemporal hedging demand for long-teindexed bonds. The numbers in parentheses beach allocation clarify this point by reporting tshare of intertemporal hedging demand in thetal demand for long-term bonds. This share rifrom zero wheng 5 1 to 100 percent wheng 55,000.

Results for the optimal consumption rule rported in the Appendix show that the optimconsumption behavior that is associated wthese portfolio rules depends on both the invtor’s coefficient of relative risk aversion and helasticity of intertemporal substitution. An investor with zero elasticity of intertemporal sustitution consumes the annuity value of weaeach period, so the average consumption-weratio for this investor is just the average epected return on the portfolio. The average pfolio return declines with risk aversion, andthe average consumption-wealth ratio alsoclines with risk aversion. Investors with highelasticities of consumption are willing to sustitute intertemporally in response to incentivThe direction of the substitution depends onaverage return on the portfolio in relation to ttime discount rate and the risk of the portfolInvestors with low risk aversion have high aerage portfolio returns so they substitutereducing present consumption, while investwith high risk aversion have low average pofolio returns so they substitute by increasipresent consumption. The magnitude of theffects is such that all investors with unit elaticity of substitution have the same averaconsumption-wealth ratio equal to (12 d), re-gardless of their risk aversion and resultiportfolio composition.

-

-

The accuracy of our loglinear approximationthe intertemporal budget constraint depends onstandard deviation of the log consumption-wearatio. This standard deviation is zero forc 5 1(the case where our approximation holds exacand is roughly proportional to (c 2 1). (It wouldbe exactly proportional if the loglinearization parameterr were fixed.) Forc close to zero, thestandard deviation is about 3 percent. This sugests that our approximation should be extremaccurate for a term-structure model of the sorthave estimated in 1952–1996. Campbell aHyeng Keun Koo (1997) use numerical methoto solve a model with an exogenous portfolreturn that follows an AR(1) process like the edogenous portfolio return in our model; they finthat approximation error is very small whenevthe standard deviation of the log consumptiowealth ratio is 5 percent or below.

The volatility of consumption growth depends on the volatility of the optimal portfolioreturn, together with the volatility of the logconsumption-wealth ratio. We find that optimconsumption is highly volatile unless investoare extremely risk averse. A simple wayunderstand this result is as follows. The modparameters estimated for 1952–1996 imply ththe standard deviation ofmt 1 1 is 0.19 if wecounterfactually setsm

2 5 0. This is the stan-dard deviation of marginal utility for an investowho can trade freely in indexed bonds but hno access to equities or nominal bonds. If utilitakes the power form, the standard deviationmarginal utility is the standard deviation of consumption growth times the coefficient of relative risk aversion. Thus with power utility wemust have either volatile consumption or higrisk aversion, just as in the literature on thequity premium puzzle. With recursive utilityconsumption can be somewhat smoother if telasticity of intertemporal substitution is lowethan the reciprocal of risk aversion (a point alnoted by Campbell, 1999), but this effectempirically quite small.

III. Multiple Assets and Portfolio Constraints

A. Extended Solution Procedure

The results in the previous section can easilygeneralized to the case where the investor

dstho

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-

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d

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hold multiple long-term assets: indexed bonnominal bonds, and equities. We can assumethe short-term asset is indexed or nominal,allow both types of short-term assets. For realihowever, and since inflation risk is modest atshort end of the term structure, we now assuthat the one-period asset is nominal.

The Appendix shows that even in the preence of nominal assets, the log consumptiwealth ratio still depends only on the stavariable xt, and not on expected inflationzt.Furthermore this ratio is still a linear functionxt, and the slope coefficient is still given bb1 5 r(1 2 c)/(1 2 rf) as in equation (28)The menu of available assets affects this coficient only indirectly by affecting the intercepb0 of the consumption function, which in turdetermines the loglinearization parameterr.

We write the vector of allocations to longterm assets asa. The residual allocation (12i9a), where i is a vector of ones, is to thshort-term asset. Then we have

(30) a 51

gS21a,

where S is the variance-covariance matrixexcess returns over the short-term asset an

(31) a 5 m 1 gp 11 2 g

1 2 ch.

Here m is a vector of Jensen’s Inequalitcorrected mean excess returns whosei th ele-ment is

(32)

mi 5 Et @r i ,t 1 1 2 ~r 1,t 1 1$ 2 p t 1 1!#

1 12

Vart @r i ,t 1 12 ~r 1,t 1 1$ 2 pt 1 1!#,

p is a vector of conditional covariances with treal return on the short-term bond whosei thelement is

(33) pi 5 Covt @r i ,t 1 1 2 ~r 1,t 1 1$ 2 pt 1 1!,

r 1,t 1 1$ 2 pt 1 1#,

,atr,

e

-

-

andh is a vector of conditional covariances witthe consumption-wealth ratio whosei th elementis

(34) hi 5 Covt @r i ,t 1 1 2 ~r 1,t 1 1$ 2 pt 1 1!,

ct 1 1 2 wt 1 1#,

5r

1 2 rfx~1 2 c!Covt @r i ,t 1 1

2 ~r 1,t 1 1$ 2 pt 1 1!, xt 1 1#.

The vectorm gives the standard one-periomean-variance analysis, while the vectorpappears because we have assumed thatshort-term asset is nominal so that it is riskin real terms. In practice the elements ofp areall extremely small and have little impact othe portfolio allocation. The vectorh repre-sents the intertemporal hedging componentdemand.

The solution of the model is analytical givethe loglinearization parameterr. We findr us-ing the same recursive procedure as before;assume a value forr, solve the model, get a newvalue forr, and so on until convergence.

We can also allow for borrowing and shorsales constraints. Unconstrained portfolio alcations are often highly leveraged; but thpermits the possibility of bankruptcy indiscrete-time model, and many investors aconstrained in their use of leverage. Becauseunconstrained optimal portfolio policy is constant over time, we can impose constraints usresults in Lucie Tepla´ (2000). Following JaksaCvitanic and Ioannis Karatzas (1992), Tep´(2000) shows that standard results in static pofolio choice with borrowing and short-saleconstraints extend to intertemporal models tunconstrained optimal portfolio policies owhich are constant over time. The optimal pofolio allocations under borrowing constrainare the unconstrained allocations with a highshort-term interest rate, and the optimal portflio allocations under short-sales constraints afound by reducing the dimensionality of thasset space until the optimal unconstrainedlocations imply no short sales. These andother results given in this section are explainin detail in the Appendix.

)

wn in


TABLE 3—OPTIMAL PERCENTAGE ALLOCATION TO n-PERIOD BOND AND PERCENTAGE HEDGING

DEMAND OVER TOTAL DEMAND

Relative riskaversion

Indexedonly

Nominalonly

10-yearindexed

10-yearnominal

3-yearnominal

10-yearnominal

0.75 1,286 193 841 108 2,602 2744(22) (22) (24) (0) (22) (23)

1 988 147 654 81 1,998 2572(0) (0) (0) (0) (0) (0)

2 541 78 374 40 1,091 2315(9) (6) (13) (0) (8) (9)

5 272 37 206 16 547 2160(27) (20) (37) (22) (27) (29)

10 183 23 150 8 365 2109(46) (36) (57) (25) (45) (47)

5,000 94 9 95 0 184 257(100) (100) (100) (104) (100) (100

Notes:The allocations shown on the table assumec 5 1. Percentage share of hedging demand in total demand is shoparentheses.

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nominal bond. Risk-tolerant investors do thbecause they are attracted by the high Sharatio of the three-year nominal bond relativethe Sharpe ratio of the ten-year nominal boand they short ten-year bonds to reduce thportfolio risk. Conservative investors explothe fact that three-year bonds have a grereal-interest-rate sensitivity than ten-yebonds. By shorting ten-year bonds they chedge the expected-inflation exposure of thryear bonds and create bond portfolios with siilar properties to long-term indexed bonds, evthough indexed bonds are not directly availain the marketplace. This strategy relies onability to sell bonds short and is ruled out if wimpose short-sales constraints.

We can use our solutions to consider tpartial-equilibrium effects on investors’ utilitof switching from a world where only nominabonds are available to a world where all deinstruments are indexed, or to a world wheboth nominal and indexed bonds are availabThe analysis is partial equilibrium in that whold constant the empirical model for intererates even as we vary the menu of availaassets. We study this question using equa(17), which gives the value function per unwealth as a monotonic transformation of toptimal consumption-wealth ratio. We cacompute an approximate value function withoany need for further approximations justsubstituting into this expression our appro

B. The Demand for Nominal Bonds

The third column in Table 3 reports thoptimal unconstrained portfolio rules implieby the nominal term-structure model esmated over the period 1952–1996, assumthat the only assets available to investorsone-quarter and ten-year nominal zecoupon bonds. Nominal bonds have slighhigher average returns than indexed bonbut are subject to inflation risk. Table 3 shothat when investors are forced to bear this rthey shorten the maturities of their bond pofolios.

The fourth and fifth columns in Table 3 reposolutions when both indexed and nominal bonare available. We allow investors to hold thremonth nominal, ten-year nominal, and ten-yindexed bonds. Investors with low risk aversihold a mix of both indexed and nominal bondseeking to earn both the real term premium athe inflation risk premium, and exploiting thimperfect correlations between the real anominal sources of risk. More conservativevestors concentrate their portfolios on indexbonds.

The last two columns in Table 3 report sotions when the assets available to investorsthree-month, three-year, and ten-year nombonds. Investors hold highly leveraged portlios, with long positions in the three-year nominal bond and short positions in the ten-ye

mte

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mate consumption-wealth ratio.11 The log valuefunction is a linear function of the short-terinterest rate, so it is time-varying; we calculaits unconditional mean.

Results reported in the Appendix for th1952–1996 sample show that bond indexatcan have substantial benefits for conservainvestors. At the short end of the term struture, the replacement of a nominal short-teasset with an indexed short-term asset elinates the risk of unexpected inflation;though this risk is small, it does affect thwelfare of extremely conservative investoMore importantly, indexation eliminates thrisk in long-term bonds caused by changesexpected future inflation. Investors who ctrade freely in both intermediate-term along-term nominal bonds can hedge this ribut short-sales-constrained investors canThese investors benefit greatly from the intduction of indexed bonds. A constrainedvestor with g 5 10 (5) and c 5 1, forexample, gains utility equivalent to a 3(11-) percent increase in wealth from tradithree-month nominal, ten-year nominal, aten-year indexed bonds rather than thrmonth, three-year, and ten-year nominbonds, and gains almost as much by tradonly three-month and ten-year indexed bonResults for the 1983–1996 sample are qutatively similar but imply considerablsmaller benefits of bond indexation. TVolcker-Greenspan monetary regime hgreatly reduced long-run uncertainty aboinflation, and has correspondingly reducedbenefits of eliminating inflation risk entirely

These findings contradict the claim of Vard (1993) that indexation has only mineffects on investors’ utility. Viard models indexation as elimination of the inflation riska one-period asset, and studies the benefione-period investors. Since there is little riin inflation over one period, Viard’s resultnot surprising. We get much larger benefitsindexation because we model indexationelimination of the inflation risk in long-term

e

11 For very low c, the value function per unit wealthactually equals the consumption-wealth ratio. We handlethe casec 5 1 by taking appropriate limits in (17). Camp-bell and Viceira (1999) provide a more detailed discussionin a related model.

-

t.

-

.-

o

assets, and study the benefits to long-teinvestors.12

Advocates of bond indexation have somtimes argued that the availability of indexeassets will stimulate saving. However this effedepends on the elasticity of intertemporal sustitution, c. If c 5 1, then the consumptionwealth ratio is constant regardless of thavailable asset menu. Ifc is close to zero, asmany empirical estimates suggest, thenconsumption-wealth ratio approximately equathe value function. Thus the utility gain fromindexation is accompanied by an increaseconsumption and a decline in saving.

All these calculations depend on the assumtion that the process driving interest ratesinvariant to the presence or absence of inflatioindexed bonds. This assumption might be dfensible in a small open economy, butunlikely to hold in the general equilibrium of aclosed economy. Thus our results are notreliable guide to the effects of bond indexatioon social welfare. They do, however, makehard to argue the irrelevance of indexation.bond indexation does not change the stochaprocess driving interest rates, it must have uity benefits for constrained conservative invetors; if it does not help such investors, it muhave effects on interest rates.

C. Bond Demand in the Presence of Equitie

The realism of the preceding analysis is limited by the fact that we have not allowed invetors to hold equities. We now considerscenario in which both bonds and equities aavailable to the investor.

Table 4 reports optimal demands for equitiand for three-month and ten-year indexednominal bonds by investors who are uncostrained or subject to borrowing and short-saconstraints. For simplicity we assume either thshort- and long-term bonds are all indexed,that they are all nominal; we do not allow investors to hold equities, indexed bonds, anominal bonds simultaneously. Panel A reporesults for the 1952–1996 sample, which wconsider first.

12 Campbell and Shiller (1996) also emphasize the ben-efits of indexation to long-term investors, but they do notpresent a formal analysis.

al


TABLE 4—OPTIMAL PERCENTAGE ALLOCATION TO EQUITIES AND TO n-PERIOD BOND

Relative riskaversion

Unconstrained Constrained Unconstrained Constrained

Equity Indexed Equity Indexed Equity Nominal Equity Nomin

(A) Sample Period: 1952–1996

0.75 443 1,082 100 0 470 25 100 01 332 835 100 0 352 21 100 02 166 464 100 0 175 15 100 05 66 242 60 40 69 12 69 1210 33 168 30 70 33 11 33 115,000 0 94 0 94 22 10 0 10

(B) Sample Period: 1983–1996

0.75 262 21 100 0 259 1 100 01 196 21 94 6 195 24 96 42 98 54 53 47 99 58 52 485 39 74 28 72 41 78 25 7510 20 81 19 81 22 85 16 845,000 0 88 0 88 3 92 3 92

Note: The allocations shown on the table assumec 5 1.

-ed100r-xeex-arpuc

foall,theioaveaniskx-

ingoldonlstin-

et-

In-ore

r ra-fo-tureoute-totiotive

al-y-69,in-79),dingesarest-orsnesntsrmt for

13 A myopic investor facing independent risks allocates ashare to each risk that is proportional to its mean divided byits variance, or equivalently its Sharpe ratio divided by itsstandard deviation. Although equities have a higher Sharperatio than indexed bonds, their standard deviation is muchhigher so the optimal equity share is lower.

d

eh

r

d

y

allocation puzzle” of Canner et al. (1997).vestment advisers often suggest that mconservative investors should have a highetio of long-term bonds to stocks in their portlios. Canner et al. (1997) document this feaof conventional investment advice and pointthat it is inconsistent with the mutual-fund thorem of static portfolio analysis, accordingwhich risk aversion should affect only the raof cash to risky assets and not the relaweights on different risky assets.

Our analysis shows that static portfolio anysis is not just theoretically incorrect in a dnamic setting, as pointed out by Merton (191971, 1973), Bruno H. Solnik (1974), Rubstein (1976), and Douglas T. Breeden (19among others, but can be seriously misleaempirically. The portfolio allocations to equitiand indexed bonds in Panel A of Table 4strikingly consistent with conventional invement advice. Aggressive long-term investshould hold stocks, while conservative oshould hold long-term bonds and small amouof cash. The explanation is that long-tebonds, and not cash, are the riskless asselong-term investors.14

In a world with full indexation, the unconstrained demand for both long-term indexbonds and equities is positive and often abovepercent, implying that the investor optimally borows to finance purchases of equities and indebonds. The portfolio share of indexed bondsceeds that of equities, despite the higher Shratio of equities, because indexed bonds are mless risky than equities.13 As the coefficient ofrelative risk aversion increases, the demandsboth long-term indexed bonds and equities fbut the share of equities falls faster. In the limitinfinitely risk-averse investor holds a portfolequivalent to an indexed perpetuity as we halready discussed. When there are borrowingshort-sale constraints, investors with low raversion invest fully in equities as a way to maimize their risk and expected return without usleverage, while more risk-averse investors hboth indexed bonds and equities. Cash playsa minor role and only in the portfolios of the morisk-averse investors, who are almost fullyvested in indexed bonds.

These findings are related to the “ass

ortione

14 Canner et al. (1997) are aware of the potential imp-tance of intertemporal hedging demand for the asset-allocapuzzle. They write: “In principle, intertemporal hedging of th

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15 Canner et al. argue in the NBER working paper ver-sion of their paper (1994) that money illusion might help toresolve the asset-allocation puzzle. However they considermoney illusion in the context of short-term mean-varianceanalysis and do not relate it to intertemporal hedging as wedo here.

16 During the 1983–1996 period the interest-rate sensi-


A weakness in this resolution of the assallocation puzzle is that it assumes that lonterm bonds are indexed, or equivalently, ththere is no inflation uncertainty. Panel ATable 4 shows that nominal bonds play a musmaller role in optimal portfolios. In a worlwith no indexation, unconstrained investowith low risk aversion hold modest nominbond positions, but constrained investors honly equities. As risk aversion increases, invtors move into cash rather than long-term noinal bonds. Figure 1 illustrates this point. Thtop panel of the figure plots constrained alloctions to equities, a ten-year nominal bond, anthree-month nominal bill, while the bottompanel plots constrained myopic allocationThe horizontal axis measures relative risk terance (1/g) rather than relative risk aversiobecause both total and myopic allocationslinear in risk tolerance when portfolio constraints are not binding. Infinitely conservatiinvestors with 1/g 5 0 are plotted at the righedge of the figure. As in the tables we setc 51, but the choice ofc has very little effect on theresults.

Risk-tolerant investors at the left of Figu1 are fully invested in equities. Highly riskaverse investors hold most of their portfolioscash, although they also hold some ten-ynominal bonds. The bottom panel shows tlong-term bonds are held purely for hedgipurposes. The myopic demand for long-tebonds is extremely close to zero at all levelsrisk aversion.

The portfolio allocations to nominal bondin Panel A of Table 4 and Figure 1 do ncorrespond well with conventional invesment advice. In order to rationalize the coventional wisdom about long-term nominbonds, one must assume that future interrates will be generated by a different procethan the one estimated in 1952–1996, a pcess with less uncertainty about future infl

... .t aingperand

-

rt

tion.15 Interestingly, we have estimated jusuch a process over the Volcker-Greenspsample period 1983–1996. Panel B of Tab4 repeats Panel A using our 1983–1996 emates and finds that even when only nominbonds are available, aggressive long-termvestors should hold stocks, while conservtive ones should hold primarily long-termnominal bonds along with small quantitiesstocks.16

Figure 2, whose structure is identical to Fiure 1, emphasizes this result. Panel A shothat almost all investors should be fully investin equities and long-term nominal bonds whthey face a term structure like the one estimafor the Volcker-Greenspan era. Only extremerisk-averse investors should hold some cashtheir portfolios. Panel B shows that intertempral hedging motives account for most of thdemand for long-term nominal bonds. If invetors behaved myopically and ignored the heding properties of long-term bonds, theportfolios would contain mostly equities ancash. The top panel of Figure 2 also shows tthe ratio of nominal bonds to equities in thoptimal portfolio increases with risk aversiojust as recommended by conventional invement advice. If investors behaved myopicalthis ratio would be constant when portfolio costraints are not binding, as shown in the bottopanel.

Although our 1983–1996 model replicateimportant features of conventional investmeadvice, it still falls short in one respect. Thoptimal portfolios in Figure 2 contain verlittle cash relative to the recommended po

tivity of a ten-year indexed zero-coupon bond is consider-ably less than that of an indexed perpetuity. Therefore aninfinitely risk-averse investor would like to hold a leveragedposition in ten-year indexed zeros, which was not the casein our 1952–1996 model. To maintain comparability withthat model, in Panel B of Table 4 and Figure 2 we replacethe ten-year zero-coupon bond with a twenty-year zero-coupon bond. This ensures that the optimal indexed portfo-lio for an infinitely risk-averse investor is available evenwhen borrowing and short-sales constraints are imposed.

sort discussed by Merton could point in the right directionUnfortunately, the magnitude of this effect is not evidenpriori, and the empirical literature on intertemporal hedglags far behind the theoretical literature” (p. 187). This paattempts to bridge the gap they identify between empiricaltheoretical work on intertemporal hedging.


B. Constrained Myopic Allocation to 10-Year and 3-Month Nominal Bond and Equity

FIGURE 1. PORTFOLIO CHOICE BASED ON THE 1952–1996 SAMPLE

A. Constrained Allocation to 10-Year and 3-Month Nominal Bond and Equity

doreneer-

ves

e

mediate between the two processes we haestimated, or by modeling liquidity motivefor holding cash.

Finally, we note that our resolution of th

folios reported by Canner et al. (1997). Wenot attempt to match those portfolios moaccurately, but suspect that it can be doeither by using a term-structure process int


B. Constrained Myopic Allocation to 20-Year and 3-Month Nominal Bond and Equity

FIGURE 2. PORTFOLIO CHOICE BASED ON THE 1983–1996 SAMPLE

A. Constrained Allocation to 20-Year and 3-Month Nominal Bond and Equity

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risk aversion or volatile consumption growthEven in the presence of portfolio constrainta risk-tolerant investor who holds a 100percent equity portfolio will have consump

asset-allocation puzzle does not escapeequity premium puzzle. That is, uncostrained investors in our model have highvolatile marginal utility, implying either high

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tion volatility similar to the volatility of stockreturns and much higher than the volatilityaggregate consumption growth. To reconcour results with aggregate consumption daone would need to assume either that minvestors are highly risk averse, or that maconsumers face tighter portfolio constrainthan those modeled here, with low risk expsure that reduces the volatility of aggregconsumption.

IV. Conclusion

In this paper we have considered infinilived investors who use their wealth to financstream of consumption. We have shown tsuch investors may hold long-term bondstwo reasons. First, if long-term bonds offerterm premium then investors may hold themspeculative purposes, to increase their expeportfolio return even at the cost of some exshort-term risk. This “myopic demand” folong-term bonds can be large when risk avsion is small, because long-term bonds hattractive Sharpe ratios. Second, long-termvestors may hold long-term bonds for hedgpurposes. Long-term bonds can finance a stlong-run consumption stream even in the facetime-varying short-term interest rates, and tis attractive to risk-averse long-term investoIn the extreme cases where there is no tepremium, or where investors are infinitely riaverse, the myopic demand for long-term bois zero and all bond demand is accounted forthe hedging demand.

Inflation-indexed bonds are particularly suitafor hedging purposes because they do not impextraneous inflation risk on long-term investoseeking a stable real consumption path. Wlong-term indexed bonds are available, an initely risk-averse long-term investor with zerotertemporal elasticity of substitution holds a boportfolio that is equivalent to an indexed perpeity. There is a sense in which the indexed pertuity is the riskless asset for a long-term invessince it finances a constant consumption streforever. When only nominal bonds are availabhighly risk-averse investors shorten their boportfolios in order to reduce their exposureinflation risk. Less risk-averse investors hold lonterm nominal bonds for speculative purposethere is a positive inflation risk premium.

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We have extended our approach to solveintertemporal portfolio choice problem imposing short-sales and borrowing constraints. This possible because our solution takes the saform as the solution to a static portfolio choicproblem for which standard mean-variananalysis is appropriate. Therefore we can soour constrained problem using methods thhave been developed to solve static problewith portfolio constraints.

Our constrained solution enables usstudy the utility effects of bond indexation ia partial-equilibrium framework with a givenprocess for interest rates. When portfolio costraints are in place, and both nominal aindexed bonds are available to investors, coservative investors benefit substantially frothe consumption insurance provided by lonterm indexed bonds. This should not be intepreted as a social welfare analysis, howevas we hold interest-rate behavior fixed andnot consider possible general-equilibrium efects of bond indexation on real and nomininterest rates.

We have also studied the demand for bonwhen equities are available as an alternatinvestment. We find that the ratio of bondsstocks in the optimal portfolio increases witrisk aversion, very much in line with populainvestment advice but contrary to the mutuafund theorem of static portfolio analysis. However the demand for long-term bonds is onlarge when these bonds are indexed, or whinflation uncertainty is low as it has beenthe Volcker-Greenspan monetary-policy regimsince 1983.

In this paper we have taken asset-return pcesses as given and have not asked how areturns are determined in general equilibriuThe return parameters we have estimated frfinancial data display the well-known equitpremium puzzle. That is, the high estimaterewards for bearing risk in bond and stock makets imply that investors’ marginal utilitiesmust be very volatile. This is particularly trueinvestors are able to hold leveraged portfoliobut is even true in the presence of borrowinand short-sales constraints. Volatile marginutilities are hard to reconcile with the smoothness of aggregate consumption.

One response to the equity premium puzzleto argue that historical estimates of risk prem

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on bonds and stocks are biased upwardssurvivorship bias or peso problems. We coof course calculate portfolio solutions wilower risk premia; the myopic componentsbond and equity demand would be smaller,the intertemporal hedging components wobe similar to those reported here since thdepend on the dynamics of interest rates rathan the levels of risk premia.

A limitation of our work is that we do noconsider labor income. It would be straightfoward to allow for riskless labor income,income perfectly correlated with asset returAs Bodie et al. (1992) have pointed out, sulabor income is equivalent to an implicit holing of financial assets; the results of our pathen apply to total asset holdings rather thexplicit holdings. In our model with timevarying real interest rates, it is worth notinthat riskless real labor income is an impliholding of a long-term inflation-indexed bonnot an implicit holding of short-term bills. Alsoshort-sales and borrowing constraints apto explicit holdings of financial assets raththan total holdings, so the presence of laincome would alter the impact of portfolio costraints on our solutions. Portfolio-constraininvestors with riskless labor income, for exaple, would have large implicit inflation-indexebond holdings and would not be able to reduthese holdings through explicit short-sellinginflation-indexed bonds. Such investors wohave smooth consumption and constraineduity holdings; this might help to explain thequity premium puzzle as suggested by GeoM. Constantinides et al. (1998).

Our approach can be extended in seveways. We can explore alternative term-struture models, adding factors or allowing fchanging interest-rate volatility. A particularly tractable possibility is a discrete-timversion of the Cox et al. (1985) model,which interest-rate volatility rises with thlevel of the interest rate. Since term premand bond return variances move in proportto one another, this model delivers constportfolio allocations.

We can consider the effect of investors’ hrizons more explicitly by solving a finitehorizon version of our model—an approataken by Wachter (1998)—or by varying thtime discount factord. In a model with a con

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stant probability of death each period, an invetor with a high death probability has a lowd.Our model predicts that this investor has a hioptimal consumption-wealth ratio, a low valufor the loglinearization parameterr, and an op-timal portfolio that is close to the optimal portfolio for a myopic single-period investor.

We can allow for multiple consumptiongoods, and consider assets that are indexethe price of one of these goods. A house, fexample, can be regarded as an asset that ders a constant flow of housing services, in tsame way that an indexed bond delivers a costant flow of consumption. This perspectivmight explain why conservative investors awilling to own houses despite the short-run vaability of house prices (Marjorie Flavin anTakashi Yamashita, 1998).

Our analysis also has interesting implicatiofor the design of pension plans and annuitieOur results suggest that conservative investwill favor indexed defined-benefit plans, whilmore risk-tolerant investors should be willing taccept some inflation or equity risk in theretirement income in exchange for higher aveage payments.

Our ultimate goal is to build a more fullyrealistic model of portfolio choice by combining several of the effects emphasized in trecent literature. This paper explores the impof interest-rate risk on long-term portfoliochoice, while Campbell and Viceira (1999studies time variation in the equity premiumand Viceira (1999) considers uninsurable risklabor income. A complete model accounting fall these effects offers the exciting prospect thfinancial economists will at last be able to offerealistic but scientifically grounded investmeadvice.

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Who Should Buy Long-Term Bonds? - Semantic Scholar · 2015-07-28 · Who Should Buy Long-Term Bonds? By JOHN Y. CAMPBELL AND LUIS M. VICEIRA* According to conventional wisdom, long-term