Kocherlakota: The Equity Premium The Equity Premium: It’s ... · Kocherlakota: The Equity Premium The Equity Premium: It’s Still a Puzzle NARAYANA R. KOCHERLAKOTA Federal Reserve

Journal of Economic Literature,Vol. XXXIV (March 1996), pp. 42–71

Kocherlakota: The Equity Premium

The Equity Premium: It’s Still a Puzzle

NARAYANA R. KOCHERLAKOTAFederal Reserve Bank of Minneapolis

I thank Rao Aiyagari, John Campbell, Dean Corbae, John Heaton, Mark Huggett, Beth In-gram, John Kennan, Deborah Lucas, Barbara McCutcheon, Rajnish Mehra, Gene Savin, SteveWilliamson, Kei-Mu Yi, and the referees for their input in preparing this manuscript. I thankEdward R. Allen III, Ravi Jagannathan, and Deborah Lucas for our many interesting andenlightening conversations on the subject of asset pricing over the years. Finally, I thank LarsHansen for teaching me how to think about the issues addressed in this paper. None of thepreceding should be viewed as responsible for the opinions and errors contained within. Theviews expressed herein are those of the authors and not necessarily those of the FederalReserve Bank of Minneapolis or the Federal Reserve System.

I. Introduction

OVER THE LAST one hundred years,the average real return to stocks in

the United States has been about six per-cent per year higher than that on Trea-sury bills. At the same time, the averagereal return on Treasury bills has beenabout one percent per year. In this pa-per, I discuss and assess various theore-tical attempts to explain these two differ-ent empirical phenomena: the large“equity premium” and the low “risk freerate.” I show that while there are severalplausible explanations for the low levelof Treasury returns, the large equity pre-mium is still largely a mystery to econo-mists.

In order to understand why the samplemeans of the equity premium and therisk free rate represent “puzzles,” it isuseful to review the basics of modern as-set pricing theory. It is well known thatthe real returns paid by differentfinancial securities may differ consider-ably, even when averaged over longperiods of time. Financial economiststypically explain these differences in av-erage returns by attributing them to dif-

ferences in the degree to which a secu-rity’s return covaries with the typical in-vestor’s consumption. If this covarianceis high, selling off the security wouldgreatly reduce the variance of the typicalinvestor’s consumption stream; in equi-librium, the investor must be deterredfrom reducing his risk in this fashion bythe security’s paying a high average return.

There is a crucial problem in makingthis qualitative explanation of cross-sec-tional differences in asset returns opera-tional: what exactly is “the typical inves-tor’s consumption”? The famous CapitalAsset Pricing Model represents one an-swer to this question: it assumes that thetypical investor’s consumption stream isperfectly correlated with the return tothe stock market. This allows financialanalysts to measure the risk of a financialsecurity by its covariance with the returnto the stock market (that is, a scaled ver-sion of its “beta”). More recently,Douglas Breeden (1979) and Robert Lu-cas (1978) described so-called “repre-sentative” agent models of asset returnsin which per capita consumption is per-fectly correlated with the consumptionstream of the typical investor. (See also

42

Mark Rubinstein 1976; and Breeden andRobert Litzenberger 1978.) In this typeof model, a security’s risk can be mea-sured using the covariance of its returnwith per capita consumption.

The representative agent model of as-set pricing is not as widely used as theCAPM in “real world” applications (suchas project evaluation by firms). However,from an academic economist’s perspec-tive, the “representative agent” model ofasset pricing is more important than theCAPM. Representative agent modelsthat imbed the Lucas-Breeden paradigmfor explaining asset return differentialsare an integral part of modern macro-economics and international economics.Thus, any empirical defects in the repre-sentative agent model of asset returnsrepresent holes in our understanding ofthese important subfields.

In their seminal (1985) paper, RajnishMehra and Edward Prescott describe aparticular empirical problem for the rep-resentative agent paradigm. As men-tioned above, over the last century, theaverage annual real return to stocks hasbeen about seven percent per year, whilethe average annual real return to Trea-sury bills has been only about one per-cent per year. Mehra and Prescott (1985)show that the difference in the covari-ances of these returns with consumptiongrowth is only large enough to explainthe difference in the average returns ifthe typical investor is implausibly averseto risk. This is the equity premium puz-zle: in a quantitative sense, stocks arenot sufficiently riskier than Treasurybills to explain the spread in their re-turns.

Philippe Weil (1989) shows that thesame data presents a second anomaly.According to standard models of individ-ual preferences, when individuals wantconsumption to be smooth over states(they dislike risk), they also desiresmoothness of consumption over time

(they dislike growth). Given that thelarge equity premium implies that inves-tors are highly risk averse, the standardmodels of preferences would in turn im-ply that they do not like growth verymuch. Yet, although Treasury bills offeronly a low rate of return, individuals de-fer consumption (that is, save) at a suffi-ciently fast rate to generate average percapita consumption growth of aroundtwo percent per year. This is what Weil(1989) calls the risk free rate puzzle: al-though individuals like consumption tobe very smooth, and although the riskfree rate is very low, they still saveenough that per capita consumptiongrows rapidly.

There is now a vast literature thatseeks to resolve these two puzzles. I be-gin my review of this literature by show-ing that the puzzles are very robust: theyare implied by only three assumptionsabout individual behavior and asset mar-ket structure. First, individuals havepreferences associated with the “stan-dard” utility function used in macro-economics: they maximize the expecteddiscounted value of a stream of utilitiesgenerated by a power utility function.Second, asset markets are complete: indi-viduals can write insurance contractsagainst any possible contingency. Finally,asset trading is costless, so that taxes andbrokerage fees are assumed to be insig-nificant. Any model that is to resolve thetwo puzzles must abandon at least one ofthese three assumptions.

My survey shows that relaxing thethree assumptions has led to plausibleexplanations for the low value of the riskfree rate. Several alternative preferenceorderings are consistent with it; also, theexistence of borrowing constraints pushinterest rates down in equilibrium. How-ever, the literature provides only two ra-tionalizations for the large equity pre-mium: either investors are highly averseto consumption risk or they find trading

Kocherlakota: The Equity Premium 43

stocks to be much more costly than trad-ing bonds. Little auxiliary evidence existsto support either of these explanations,and so (I would say) the equity premiumpuzzle is still unresolved.

The Lucas-Breeden representativeagent model is apparently inconsistentwith the data in many other respects.Sanford Grossman and Robert Shiller(1981) (and many others) argue that as-set prices vary too much to be explainedby the variation in dividends or in percapita consumption. Hansen and Ken-neth Singleton (1982, 1983) point outthat in forming her portfolio, the repre-sentative investor appears to be ignoringuseful information available in laggedconsumption growth and asset returns.Robert Hall (1988) argues that consump-tion does not respond sufficiently tochanges in expected returns. John Coch-rane and Hansen (1992) uncover severalanomalies, including the so-called De-fault-Premium and Term Structure Puz-zles.

This paper will look at none of theseother puzzles. Instead, it focuses exclu-sively on the risk free rate and equitypremium puzzles. There are two ratio-nalizations for limiting the discussion inthis way. The first is simple: my conver-sations with other economists have con-vinced me that these puzzles are themost widely known and best under-stood of the variety of evidence usuallyarrayed against the representative agentmodels.

The second rationalization is perhapsmore controversial: I claim that thesetwo puzzles have more importance formacroeconomists. The risk free ratepuzzle indicates that we do not knowwhy people save even when returns arelow: thus, our models of aggregate sav-ings behavior are omitting some crucialelement. The equity premium puzzledemonstrates that we do not knowwhy individuals are so averse to the

highly procyclical risk associated withstock returns. As Andrew Atkeson andChristopher Phelan (1994) point out,without this knowledge we cannot hopeto give a meaningful answer to R. Lucas’(1987) question about how costly in-dividuals find business cycle fluctua-tions in consumption growth. Thus,the two puzzles are signs of large gapsin our understanding of the macroecon-omy.

Given the lack of a compelling expla-nation for the large equity premium, thearticle will at times read like a litany offailure. I should say from the outset thatthis is misleading in many ways. We havelearned much from the equity premiumpuzzle literature about the properties ofasset pricing models, about methods ofestimating and testing asset pricing mod-els, and about methods of solving for theimplications of asset pricing models. Ibelieve that all of these contributions aresignificant.

For better or for worse, though, thisarticle is not about these innovations. In-stead, I focus on what might be calledthe bottom line of the work, “Do weknow why the equity premium is sohigh? Do we know why the risk free rateis so low?” It is in answering the first ofthese questions that the literature fallsshort.

The rest of the paper is structured asfollows. The next section describes thetwo puzzles and lays out the fundamentalmodelling assumptions that generatethem. Section III explores the potentialfor explaining the two puzzles by chang-ing the preferences of the representativeagent. Section IV looks at the implica-tions of market frictions for asset re-turns. Section V concludes.1

1 In writing this paper, I have benefited greatlyfrom reading four other review articles by AndrewAbel (1991), Aiyagari (1993), Cochrane and Han-sen (1992), and Heaton and D. Lucas (1995b). Irecommend all of them highly.

44 Journal of Economic Literature, Vol. XXXIV (March 1996)

II. What Are the Puzzles?

1. Aspects of the Data

The equity premium and risk freerate puzzles concern the co-movementsof three variables: the real return tothe S & P 500, the real return to shortterm nominally risk free bonds,2 andthe growth rate of per capita real con-sumption (more precisely, nondurablesand services). In this paper, I use theannual United States data from 1889–1978 originally studied by Mehra andPrescott (1985) (see the Appendix fordetails on the construction of the data).3

Plots of the data are depicted in Figures1–3; all of the series appear station-ary and ergodic (their statistical proper-ties do not appear to be changing overtime).

Table 1 contains some summary statis-tics for the three variables. There arethree features of the table that give riseto the equity premium and risk free ratepuzzles. First, the average real rate of re-turn on stocks is equal to seven percent

2 Ninety day Treasury bills from 1931–1978,Treasury certificates from 1920–31, and 60–day to90–day Commercial Paper prior to 1920.

3 In what follows, I use only the Mehra-Prescottdata set. However, it is important to realize thatthe puzzles are not peculiar to these data. For ex-ample, Kocherlakota (1994) shows that adding tenmore years of data to the Mehra-Prescott series

0.14

0.10

0.06

0.02

–0.02

–0.06

–0.10

cons

umpt

ion

grow

th

1880 1900 1920 1940 1960 1980time

Figure 1. Annual Real Per Capita Consumption Growth

does not eliminate the puzzles. Hansen and Sin-gleton (1983) and Aiyagari (1993) find that similarphenomena characterize post-World War IImonthly data in the United States. Amlan Roy(1994) documents the existence of the two puzzlesin post-World War II quarterly data in Germanyand Japan. It may not be too strong to say that theequity premium and risk free rate puzzles appearto be a general feature of organized asset markets.(Jeremy Siegel, 1992, points out that the equitypremium was much lower in the 19th century.However, consumption data is not available forthis period.)


0.6

0.4

0.2

–0.0

–0.2

–0.4

stoc

k re

turn

s

1880 1900 1920 1940 1960 1980time

Figure 2. Annual Real Return to S & P 500

0.25

0.20

0.15

0.10

–0.00

–0.10

–0.20

bill

retu

rns

1880 1900 1920 1940 1960 1980time

Figure 3. Annual Real Return to Nominally Risk Free Short Term Debt


per year while the average real rate ofreturn on bonds is equal to one percentper year. Second, by long-term historicalstandards, per capita consumptiongrowth is high: around 1.8 percent peryear. Finally, the covariance of per cap-ita consumption growth with stock re-turns is only slightly bigger than the co-variance of per capita consumptiongrowth with bond returns.

In some sense, there is a simple expla-nation for the higher average return ofstocks: stock returns covary more withconsumption growth than do Treasurybills. Investors see stocks as a poorerhedge against consumption risk, and sostocks must earn a higher average return.Similarly, while individuals must havesaved a lot to generate the high con-sumption growth described in Table 1,there is an explanation of why they didso: as long as individuals discount thefuture at a rate lower than one percentper year, we should expect their con-sumptions to grow. The data does notcontradict the qualitative predictions ofeconomic theory; rather, as we shall see,it is inconsistent with the quantitativeimplications of a particular economicmodel.

2. The Original Statement of the Puzzles

In their 1985 paper, Mehra andPrescott construct a simple model thatmakes well-defined quantitative predic-tions for the expected values of the realreturns to the S & P 500 and the threemonth Treasury bill rate. There arethree critical assumptions underlyingtheir model. The first is that in periodt, all individuals have identical prefer-ences over future random consumptionstreams represented by the utility func-tion:

Et Σs=0∞ βs(ct+s)1−α/(1−α), α ≥ 0 (1)

where {ct}t=1∞ is a random consumption

stream. In this expression, and through-out the remainder of the paper, Et repre-sents an expectation conditional on infor-mation available to the individual inperiod t. Thus, individuals seek to maxi-mize the expectation of a discountedflow of utility over time.

In the formula (1), the parameter β isa discount factor that households applyto the utility derived from future con-sumption; increasing β leads investors tosave more. In contrast, increasing α hastwo seemingly distinct implications forindividual attitudes toward consumptionprofiles. When α is large, individualswant consumption in different states tobe highly similar: they dislike risk. Butindividuals also want consumption in dif-ferent dates to be similar: they dislikegrowth in their consumption profiles.

The second key assumption of theMehra-Prescott model is that people cantrade stocks and bonds in a frictionlessmarket. What this means is that an indi-vidual can costlessly (that is, without sig-nificant taxes or brokerage fees) buy andsell any amount of the two financial as-sets. In equilibrium, an individual mustnot be able to gain utility at the marginby selling bonds and then investing theproceeds in stocks; the individual should

TABLE 1SUMMARY STATISTICS

UNITED STATES ANNUAL DATA, 1889–1978

Sample MeansRs

t 0.070Rb

t 0.010Ct /Ct-1 0.018

Sample Variance-CovarianceRs

t Rbt Ct/Ct-1

Rst 0.0274 0.00104 0.00219

Rbt 0.00104 0.00308 −0.000193

Ct/Ct-1 0.00219 −0.000193 0.00127

In this table, Ct/Ct-1 is real per capita consumptiongrowth, Rs

t is the real return to stocks and Rbt is the real

return to Treasury bills.


not be able to gain utility by buying orselling bonds. Hence, given the costless-ness of performing these transactions, anindividual’s consumption profile mustsatisfy the following two first order con-ditions:

Et{(ct+1/ct)−α(Rt+1s − Rt+1

b )} = 0. (2a)

βEt{(ct+1/ct)−αRt+1b } = 1. (2b)

In these conditions, Rts is the gross return

to stocks from period (t − 1) to period t,while Rt

b is the gross return to bondsfrom period (t − 1) to period t.

These first order conditions imposestatistical restrictions on the comove-ment between any person’s pattern ofconsumption and asset returns. The thirdcritical feature of Mehra and Prescott’sanalysis, though, is that they assume theexistence of a “representative” agent. Ac-cording to this assumption, the aboveconditions (2a) and (2b) are satisfied notjust for each individual’s consumption,but also for per capita consumption.

There is some confusion about whatkinds of assumptions underlie this substi-tution of per capita for individual con-sumption. It is clear that if all individualsare identical in preferences and theirownership of production opportunities,then in equilibrium, all individuals willin fact consume the same amount; thus,under these conditions, substituting percapita consumption into conditions (2a)and (2b) is justified. Many economistsdistrust representative agent models be-cause they believe this degree of homo-geneity is unrealistic.

However, while this degree of homo-geneity among individuals is sufficient toguarantee the existence of a repre-sentative agent, it is by no means neces-sary. In particular, George Constan-tinides (1982) shows that even ifindividuals are heterogeneous in prefer-ences and levels of wealth, it may be pos-sible to find some utility function for the

representative individual (with coeffi-cient of relative risk aversion no largerthan the most risk averse individual andno smaller than the least risk averse indi-vidual) which satisfies the first orderconditions (2a) and (2b). The key is thatasset markets must be complete: indi-viduals in the United States must have asufficiently large set of assets availablefor trade that they can diversify any idio-syncratic risk in consumption. Whenmarkets are complete, we can constructa “representative” agent because aftertrading in complete markets, individualsbecome marginally homogeneous eventhough they are initially heterogeneous.

We can summarize this discussion asfollows: Mehra and Prescott assume thatasset markets are frictionless, that assetmarkets are complete, and that the resul-tant representative individual has prefer-ences of the form given by (1). However,Mehra and Prescott also make threeother, more technical, assumptions.First, they assume that per capita con-sumption growth follows a two stateMarkov chain constructed in such a waythat the population mean, variance, andautocorrelation of consumption growthare equivalent to their correspondingsample means in United States data.They also assume that in period t, theonly variables that individuals know arethe realizations of current and past con-sumption growth. Finally, they assumethat the growth rate of the total divi-dends paid by the stocks included in theS & P 500 is perfectly correlated withthe growth rate of per capita consump-tion, and that the real return to the(nominally risk free) Treasury bill is per-fectly correlated with the return to abond that is risk free in real terms.

These statistical assumptions allowMehra and Prescott to use the first orderconditions (2a) and (2b) to obtain ananalytical formula that expresses thepopulation mean of the real return to the


S & P 500 and the population mean realreturn to the three month Treasury billin terms of the two preference parame-ters β and α. Using evidence from mi-croeconometric data and introspection,they restrict β to lie between 0 and 1,and α to lie between 0 and 10. Theirmain finding is that for any value of thepreference parameters such that the ex-pected real return to the Treasury bill isless than four percent, the difference inthe two expected real returns (the “eq-uity premium”) is less than 0.35 percent.This stands in contrast to the facts de-scribed earlier that the average real re-turn to Treasury bills in the UnitedStates data is one percent while the aver-age real return to stocks is nearly six per-cent higher.

Mehra and Prescott conclude fromtheir analysis that their model of assetreturns is inconsistent with United Statesdata on consumption and asset returns.Because the model was constructed bymaking several different assumptions,presumably the fit of the model to thedata could be improved by changing anyof them. In the next section, I restatethe equity premium puzzle to showwhy the three final assumptions aboutthe statistical behavior of consumptionand asset returns are relatively unimpor-tant.

3. A More Robust Restatement of the Puzzles

As we saw above, the crux of theMehra-Prescott model is that the firstorder conditions (2a) and (2b) must besatisfied with per capita consumptiongrowth substituted in for individual con-sumption growth. Using the Law of Iter-ated Expectations, we can replace theconditional expectation in (2a) and (2b)with an unconditional expectation sothat:

E{(Ct+1Ct)−α(Rt+1s − Rt+1

b )} = 0 (2a′)

βE{(Ct+1/Ct)−αRt+1b } = 1. (2b′)

(The variable Ct stands for per capitaconsumption.) We can estimate the ex-pectations or population means on theleft hand side of (2a′) and (2b′) using thesample means of:

et+1s = {(Ct+1/Ct)−α(Rt+1

s − Rt+1b )}

et+1b = β{(Ct+1/Ct)−αRt+1

b }given a sufficiently long time series ofdata on Ct+1 and the asset returns.4

Table 2 reports the sample mean of ets

for various values of α; for all values of αless than or equal to 8.5, the samplemean of et

s is significantly positive. Wecan conclude that the population meanof et

s is in fact positive for any value of αless than or equal to 8.5; hence, for anysuch α, the representative agent can gainat the margin by borrowing at the Treas-ury bill rate and investing in stocks.5 This

4 Throughout this paper, I will be evaluatingvarious representative agent models by using ver-sions of (2a′) and (2b′) as opposed to (2a) and(2b). While (2a,b) certainly imply (2a′,b′), thereare a host of other implications of (2a,b): the lawof iterated expectations “averages” over all of theimplications of (2a) and (2b) to arrive at (2a′) and(2b′). Focusing on (2a) and (2b) is in keeping withmy announced goal in the introduction of justlooking at the equity premium and risk free ratepuzzles.

Hansen and Jagannathan (1991) and Cochraneand Hansen (1992) reinterpret the equity pre-mium puzzle and the risk free rate puzzles usingthe variance bounds derived in the former paper.Note that, as Cochrane and Hansen (1992) empha-size, the variance bound is an even weaker impli-cation of (2a) than (2a′) is (in the sense that (2a′)implies the variance bound while the converse isnot true).

5 Breeden, Michael Gibbons, and Litzenberger(1989) and Grossman, Angelo Melino, and Shiller(1987) point out that the first order conditions(2a) and (2b) are based on the assumption that allconsumption within a given year is perfectly sub-stitutable; if this assumption is false, there is thepossibility of time aggregation bias. However,Hansen, Heaton, and Amir Yaron (1994) show thatas long as the geometric average of consumptiongrowth rates is a good approximation to the arith-metic average of consumption growth rates, thepresence of time aggregation should only affect


is the equity premium puzzle. Intui-tively, while the covariance betweenstock returns and per capita consump-tion growth is positive (see Table 1), it isnot sufficiently large to deter the repre-sentative investor with a coefficient ofrelative risk aversion less than 8.5 fromwanting to borrow and invest in stocks.Note that the marginal benefit of sellingTreasury bills and buying stocks falls as

the investor’s degree of risk aversionrises.6

Table 3 reports the sample mean of etb

for various values of α and β = 0.99. Itshows that for β equal to 0.99 and αgreater than one, the representativehousehold can gain at the margin bytransferring consumption from the fu-ture to the present (that is, reducing itssavings rate). This is the risk free ratepuzzle. High values of α imply that indi-viduals view consumption in differentperiods as complementary. Such indi-viduals find an upwardly sloped con-sumption profile less desirable becauseconsumption is not the same in every pe-riod of life. As a result, an individualwith a high value of α realizes more of autility gain by reducing her savings rate.Note that the risk free rate puzzle comesfrom the equity premium puzzle: there isa risk free rate puzzle only if α is re-quired to be larger than one so as tomatch up with the high equity premium.7

TABLE 2THE EQUITY PREMIUM PUZZLE

a e– t-stat

0.0 0.0594 3.345 0.5 0.0577 3.260 1.0 0.0560 3.173 1.5 0.0544 3.082 2.0 0.0528 2.987 2.5 0.0512 2.890 3.0 0.0496 2.790 3.5 0.0480 2.688 4.0 0.0464 2.584 4.5 0.0449 2.478 5.0 0.0433 2.370 5.5 0.0418 2.262 6.0 0.0403 2.153 6.5 0.0390 2.044 7.0 0.0372 1.934 7.5 0.0357 1.824 8.0 0.0341 1.715 8.5 0.0326 1.607 9.0 0.0310 1.501 9.5 0.0295 1.39510.0 0.0279 1.291

In this table, e– is the sample mean of et = (Ct+1/Ct)−α

(Rst+1 − Rb

t+1) and α is the coefficient of relative riskaversion. Standard errors are calculated using theimplication of the theory that et is uncorrelated with et-k

for all k; however, they are little changed by allowing et

to be MA(1) instead. This latter approach to calculatingstandard errors allows for the possibility of timeaggregation (see Hansen, Heaton, and Yaron 1994).

Tables 2–3 through the calculation of the standarderrors. Using their suggested correction makeslittle difference to the t-statistics reported ineither Table. (Indeed, Corr(et, et-k) is small for allk ≤ 10.)

6 This intuition becomes even more clear if oneassumes that ln(Ct+1/Ct), ln(Rt+1

s ), and ln(Rt+1b ) are

jointly normally distributed. Then, equations (2a)and (2b) become:

E(rts − rt

b ) − αCov(gt,rts−rt

b)

+ 0.5Var(rts) + 0.5Var(rt

b) = 0 (2a)

ln(β) − αE(gt) + E(rtb) + 0.5{α2Var(gt)

− 2αCov(gt,rtb) + Var(rt

b)} = 0 (2B)

where (gt, rts, rt

b) ≡ (ln(Ct/Ct−1), ln(Rts), ln(Rt

b)). In thedata, Cov(gt, rt

b) is essentially zero. BecauseE(rt

s − rtb) is so large, either α or Cov(gt,rt

s) must belarge in order to satisfy (2a). As with (2a) and (2b), the sample analogs of (2a)and (2b) can only be satisfied by setting β equal toa value larger than one and α equal to a valuegreater than 15. See N. Gregory Mankiw andStephen Zeldes (1991) for more details.

7 See Edward Allen (1990), Cochrane and Han-sen (1992), and Constantinides (1990) for similarpresentations of the two puzzles. Stephen Cecchetti, Pok-sang Lam, and NelsonMark (1993) present the results of joint tests of(2a′,b′). I present separate t-tests because I thinkdoing so provides more intuition about the failingsof the representative agent model. Of course,there is nothing wrong statistically with doing two


It is possible to find parameter set-tings for the discount factor β and thecoefficient of relative risk aversion α thatexactly satisfy the sample versions ofequations (2a′) and (2b′). In particular,by setting α = 17.95, it is possible to sat-isfy the equity premium first order con-dition (2a′); by also setting β = 1.08, it ispossible to satisfy the risk free rate first

order condition (2b′). (See Kocherlakota,1990a, for a demonstration of how com-petitive equilibria can exist even when β> 1.) It is necessary to set α to such alarge value because stocks offer a hugepremium over bonds, and aggregate con-sumption growth does not covary greatlywith stock returns; hence, the repre-sentative investor can be marginally in-different between stocks and bonds onlyif he is highly averse to consumptionrisk. Similarly, the high consumptiongrowth enjoyed by the United Statessince 1890 can be consistent with highvalues of α and the risk free rate only ifthe representative investor is so patientthat his “discount” factor β is greaterthan one.

Thus, the equity premium and riskfree rate puzzles are solely a product ofthe parametric restrictions imposed byMehra and Prescott on the discount fac-tor β and the coefficient of relative riskaversion α. Given this, it is important tounderstand the sources of these restric-tions. The restriction that the discountfactor β is less than one emerges fromintrospection: most economists, includ-ing Mehra and Prescott, believe that anindividual faced with a constant con-sumption stream would, on the margin,like to transfer some consumption fromthe future to the present.

The restriction that α should be lessthan ten is more controversial. Mehraand Prescott (1985) quote several micro-econometric estimates that bound αfrom above by three. Unfortunately, theonly estimate that they cite from finan-cial market data has been shown to beseverely biased downwards (Kocherlak-ota 1990c). In terms of introspection, ithas been argued by Mankiw and Zeldes(1991) that an individual with a coeffi-cient of relative risk aversion above tenwould be willing to pay unrealisticallylarge amounts to avoid bets. However,they consider only extremely large bets

TABLE 3THE RISK FREE RATE PUZZLE

a e– t-stat

0.0 0.0033 0.0567 0.5 −0.0081 −1.296 1.0 −0.0162 −2.233 1.5 −0.0239 −2.790 2.0 −0.0313 −3.100 2.5 −0.0382 −3.263 3.0 −0.0448 −3.339 3.5 −0.0510 −3.360 4.0 −0.0569 −3.348 4.5 −0.0624 −3.312 5.0 −0.0675 −3.259 5.5 −0.0723 −3.195 6.0 −0.0768 −3.123 6.5 −0.0808 −3.043 7.0 −0.0846 −2.959 7.5 −0.0880 −2.871 8.0 −0.0910 −2.779 8.5 −0.0937 −2.685 9.0 −0.0960 −2.590 9.5 −0.0980 −2.492 10.0 −0.0997 −2.394

In this table, e– is the sample mean of et =β(ct+1/ct)−α

(Rbt+1 − 1) and α is the coefficient of relative risk

aversion. The discount factor β is set equal to 0.99. Thestandard errors are calculated using the implication ofthe theory that et is uncorrelated with et-k for all k;however, they are little changed by allowing et to beMA(1) instead. This latter approach to calculatingstandard errors allows for the possibility of timeaggregation (see Hansen, Heaton, and Yaron 1994).

separate t-tests, each with size α, as long as onekeeps in mind that the size of the overall testcould be as low as zero or it could be as high as2α. (There may be a loss of power associated withusing sequential t-tests but power does not seem tobe an issue in tests of representative agent models!)


(ones with potential losses of 50 percentof the gambler’s wealth). In contrast,Shmuel Kandel and Robert Stambaugh(1991) show that even values of α as highas 30 imply quite reasonable behaviorwhen the bet involves a maximal poten-tial loss of around one percent of thegambler’s wealth.

Because of the arguments offered byKandel and Stambaugh (1991) and Ko-cherlakota (1990c), some economists(see, among others, Craig Burnside1994; Campbell and Cochrane 1995;Cecchetti and Mark 1990; Cecchetti,Lam, and Mark 1993; and Hansen,Thomas Sargent, and Thomas Tallarini1994) believe that there is no equity pre-mium puzzle: individuals are more riskaverse than we thought, and this high de-gree of risk aversion is reflected in thespread between stocks and bonds. How-ever, it is clear to me from conversationsand from knowledge of their work that avast majority of economists believe thatvalues for α above ten (or, for that mat-ter, above five) imply highly implausiblebehavior on the part of individuals. (Forexample, Mehra and Prescott (1985,1988) clearly chose an upper bound aslarge as ten merely as a rhetorical flour-ish.) D. Lucas (1994, p. 335) claims thatany proposed solution that “does not ex-plain the premium for α ≤ 2.5 . . . is . . .likely to be widely viewed as a resolutionthat depends on a high degree of riskaversion.” She is probably not exaggerat-ing the current state of professionalopinion.

Tables 2 and 3 show that any modelthat leads to (2a′,b′) will generate thetwo puzzles. This helps make clear whatthe puzzles are not about. For example,while Mehra and Prescott ignore sam-pling error in much of their discussion,Tables 2 and 3 do not. This allays theconcerns expressed by Allan Gregory andGregor Smith (1991) and Cecchetti,Lam, and Mark (1993) that the puzzles

could simply be a result of sampling er-ror. Similarly, Mehra and Prescott as-sume that the growth rate of the divi-dend to the S & P 500 is perfectlycorrelated with the growth rate of percapita consumption. Simon Benningaand Aris Protopapadakis (1990) arguethat this might be responsible for theconflict between the model and the data.However, in Tables 2 and 3, the equitypremium and risk free rate puzzles arestill present when the representative in-vestor is faced with the real return to theS & P 500 itself, and not some imaginaryportfolio with a dividend that is perfectlycorrelated with consumption.

Finally, we can see that the exact na-ture of the process generating consump-tion growth is irrelevant. For example,Cecchetti, Lam, and Mark (1993), andKandel and Stambaugh (1990) have bothproposed that allowing consumptiongrowth to follow a Markov switching pro-cess (as described by James Hamilton1989) might explain the two puzzles. Ta-bles 2 and 3, though, show that this con-jecture is erroneous:8 the standard t-testis asymptotically valid when consumptiongrowth follows any stationary and er-godic process, including one that isMarkov switching.9

8 See Abel (1994) for another argument alongthese lines.

9 Some readers might worry that the justifica-tion for Tables 2–3 is entirely asymptotic. I amsympathetic to this concern. I have run a MonteCarlo experiment in which the pricing errors est

and etb are assumed to be i.i.d. over time and havea joint distribution equal to their empirical jointdistribution. In this setting, the Monte Carlo re-sults show that the asymptotic distribution of thet- statistic and its distribution with 100 observa-tions are about the same.

However, there is a limit to the generality ofTables 2 and 3. One could posit, as in Kocher-lakota (1994) and Thomas Rietz (1988), that thereis a low probability of consumption falling by anamount that has never been seen in United Statesdata. With power utility, investors are very averseto these large falls in consumption, even if theyare highly unlikely. Hence, investors believe thatstocks are riskier and bonds are more valuable


Thus, Tables 2 and 3 show that thereare really only three crucial assumptionsgenerating the two puzzles. First, assetmarkets are complete. Second, assetmarkets are frictionless. (As discussedabove, these two assumptions imply thatthere is a representative consumer.) Fi-nally, the representative consumer’s util-ity function has the power form assumedby Mehra and Prescott (1985) and satis-fies their parametric restrictions on thediscount factor and coefficient of relativerisk aversion. Any model that seeks to re-solve the two puzzles must weaken atleast one of these three assumptions.

III. Preference Modifications

The paradigm of complete and fric-tionless asset markets underlies some ofthe fundamental insights in both financeand economics. (For example, theModigliani-Miller Theorems and the ap-peal of profit maximization as an objec-tive for firms both depend on the com-pleteness of asset markets.) Hence, it isimportant to see whether it is possible toexplain the two puzzles without aban-doning this useful framework. To do so,we must consider possible alterations inthe preferences of the representative in-dividual; in this section, I consider threedifferent modifications to the prefer-ences (1): generalized expected utility,habit formation, and relative consump-tion effects.

1. Generalized Expected Utility

In their (1989) and (1991) articles,10

Larry Epstein and Stanley Zin describe a

generalization of the “standard” prefer-ence class (1). In these Generalized Ex-pected Utility preferences, the period tutility Ut received by a consumer from astream of consumption is described re-cursively using the formula:

Ut = {ct1−ρ + β{EtUt+1

1−α}(1−ρ)/(1−α)} 1/(1−ρ). (3)

Thus, utility today is a constant elasticityfunction of current consumption and fu-ture utility. Note that (1) can be ob-tained as a special case of these prefer-ences by setting α = ρ.

Epstein and Zin point out a crucialattribute of these preferences. In thepreferences (1), the coefficient of rela-tive risk aversion α is constrained to beequal to the reciprocal of the elasticityof intertemporal substitution. Thus,highly risk averse consumers must viewconsumption in different time periodsas being highly complementary. This isnot true of the GEU preferences (3). Inthis utility function, the degree of riskaversion of the consumer is governedby α while the elasticity of inter-temporal substitution is governed by1/ρ. Epstein and Zin (1991) argue thatdisentangling risk aversion and inter-temporal substitution in this fashionmay help explain various aspects of as-set pricing behavior that appear anoma-lous in the context of the preferences(1).

To see the usefulness of GEU prefer-ences in understanding the equity pre-mium puzzle and the risk free rate puz-zle, suppose there is a representativeinvestor with preferences given by (3)who can invest in stocks and bonds.Then, the investor’s optimal consump-than appears to be true when one looks at infor-

mation in the 90–year United States sample; thet-statistics are biased upwards. While this explana-tion is consistent with the facts, it has the defectof being largely nontestable using only returns andconsumption data. (See Mehra and Prescott, 1988,for another critique of this argument.)

10 Weil (1989), Kandel and Stambaugh (1991),and Hansen, Sargent, and Tallarini (1994) provide

complementary analyses of the implications of thistype of generalized expected utility for asset pric-ing.

There are other ways to generalize expectedutility so as to generate interesting implications forasset prices—see Epstein and Tan Wang (1994)and Epstein and Zin (1990) for examples.


tion profile must satisfy the two first or-der conditions:

Et{Ut+1ρ−α(Ct+1/Ct)−ρ(Rt+1

s − Rt+1b )} = 0 (4a)

βEt{(EtUt+11−α)(α−ρ)/(1−α)

Ut+1ρ−α(Ct+1/Ct)−ρRt+1

b } = 1. (4b)

Note that the marginal rates of substitu-tion depend on a variable that is funda-mentally unobservable: the period (t + 1)level of utility of the representativeagent (and the period t expectations ofthat utility). This unobservability makesit difficult to verify whether the data areconsistent with these conditions.

To get around this problem, we canexploit the fact that it is difficult to pre-dict future consumption growth usingcurrently available information (see Hall1978). Hence, I assume in the discussionof GEU preferences that future con-sumption growth is statistically indepen-dent of all information available to theinvestor today.11 Under this restriction,it is possible to show that utility in pe-riod (t + 1) is a time and state invariantmultiple of consumption in period (t +1). We can then apply the Law of Iter-ated Expectations to (4a) and (4b) to ar-rive at:

βE{(Ct+1/Ct)−α(Rt+1s − Rt+1

b )} = 0 (4a′)

β[E(Ct+1/Ct)1−α](α−ρ)/(1−α)

E{(Ct+1/Ct)−αRt+1b } = 1. (4b′)

See Kocherlakota (1990b) for a precisederivation of these first order condi-tions.12

The first order condition (4a′) tells usthat using GEU preferences will notchange Table 2: for each value of α, themarginal utility gain of reducing bondholdings and investing in stocks is thesame whether the preferences lie in theexpected utility class (1) or in the gener-alized expected utility class (3). In con-trast, Table 4 shows that it is possible toresolve the risk free rate puzzle usingthe GEU preferences. In particular, forβ = 0.99, and various values of α in theset (0, 18], Table 4 presents the value ofρ that exactly satisfies the first ordercondition (4b).

The intuition behind these results issimple. The main benefit of modellingpreferences using (3) instead of (1) isthat individual attitudes toward risk andgrowth are no longer governed by thesame parameter. However, the equitypremium puzzle arises only because ofeconomists’ prior beliefs about risk aver-sion; hence, this puzzle cannot be re-solved by disentangling attitudes towardrisk and growth. On the other hand, theconnection between risk aversion and in-tertemporal substitution in the standardpreferences (1) is the essential element

11 This assumption of serial independence isalso employed by Abel (1990), Epstein and Zin(1990), and Campbell and Cochrane (1995). Kan-del and Stambaugh (1991), Kocherlakota (1990b),and Weil (1989) have shown that the thrust of thediscussion that follows is little affected by allowingfor more realistic amounts of dependence in con-sumption growth.

12 Epstein and Zin (1991) derive a different rep-resentation for the first order conditions (4a) and(4b) of the representative investor:

Et{(Ct+1/Ct)−γρ(Rt+1m )γ−1(Rt+1

s − Rt+1b )} = 0 (∗)

Et{βγ(Ct+1/Ct)−γρ(Rt+1m )γ−1(Rt+1

b )} + 1 (∗∗)

where γ = (1 − α)/(1 − ρ). In (∗) and (∗∗), Rtm is

the gross real return to the representative inves-tor’s entire portfolio of assets (including humancapital, housing, etc.). This representation has thedesirable attribute of being valid regardless of theinformation set of the representative investor (sowe don’t have to assume that consumption growthfrom period t to period (t + 1) is independent ofall information available to the investor in periodt). Unfortunately, the variable Rt

m is not observ-able. Epstein and Zin (1991) use the value-weighted return to the NYSE as a proxy, but ofcourse this understates the true level of diversifi-cation of the representative investor and (poten-tially) greatly overstates the covariability of hermarginal rate of substitution with the return to thestock market. As a result, using this proxy variableleads one to the spurious conclusion that GEUpreferences can resolve the equity premium puz-zle with low levels of risk aversion.


of the risk free rate puzzle. To matchthe equity premium, risk aversion has tobe high. But in the standard prefer-

ences (1), this forces intertemporalsubstitution to be low. The GEU prefer-ences can explain the risk free ratepuzzle by allowing intertemporal substi-tution and risk aversion to be high simul-taneously.13

2. Habit Formation

We have seen that the risk free ratepuzzle is a consequence of the demandfor savings being too low when individu-als are highly risk averse and have pref-erences of the form (1). The GEU pref-erence class represents one way togenerate high savings demand when indi-viduals are highly risk averse. There isanother, perhaps more intuitive, ap-proach. The standard preferences (1) as-sume that the level of consumption inperiod (t − 1) does not affect the mar-ginal utility of consumption in period t.It may be more natural to think that anindividual who consumes a lot in period(t − 1) will get used to that high level ofconsumption, and will more stronglydesire consumption in period t; mathe-matically, the individual’s marginal util-ity of consumption in period t is anincreasing function of period (t −1) con-sumption.

This property of intertemporal prefer-ences is termed habit formation. The ba-sic implications of habit formation forasset returns (as explained by Constan-

TABLE 4 RESOLVING THE RISK FREE RATE PUZZLE

WITH GEU PREFERENCES

a 1/α 1/ρ

0.500 2.000 23.93 1.000 1.000 14.91 1.500 0.6667 10.62 2.000 0.5000 8.126 2.500 0.4000 6.493 3.000 0.3333 5.345 3.500 0.2857 4.494 4.000 0.2500 3.840 4.500 0.2222 3.322 5.000 0.2000 2.903 5.500 0.1818 2.557 6.000 0.1667 2.267 6.500 0.1538 2.022 7.000 0.1429 1.811 7.500 0.1333 1.629 8.000 0.1250 1.469 8.500 0.1176 1.330 9.000 0.1111 1.206 9.500 0.1053 1.096 10.00 0.1000 0.997410.50 0.09524 0.909111.00 0.09091 0.829511.50 0.08696 0.757412.00 0.08333 0.692012.50 0.08000 0.632513.00 0.07692 0.578113.50 0.07407 0.528314.00 0.07143 0.482614.50 0.06897 0.4406

15.00 0.06667 0.401815.50 0.06452 0.366016.00 0.06250 0.332916.50 0.06061 0.302217.00 0.05882 0.273717.50 0.05714 0.247218.00 0.05556 0.2226

In this table, the first column is the coefficient ofrelative risk aversion α, the second column is thecorresponding elasticity of intertemporal substitution ifpreferences are of the form (1), and the third column isthe value of 1/ρ that satisfies the risk free rate firstorder condition (4b′) given that preferences are of theform (3), β = 0.99 and α has the value specified in thefirst column.

13 Weil (1989) emphasizes that GEU prefer-ences cannot explain the risk free rate puzzle ifone forces the coefficient of relative risk aversionto be “reasonable” (say, approximately 1). Heshows that under this restriction, then the elastic-ity of intertemporal substitution must be verylarge to be consistent with the risk free rate. (SeeTable 4 for a confirmation.) This large elasticity ofintertemporal substitution appears to contradictthe results of Hall (1988), who shows that the esti-mated slope coefficient is very small in a regres-sion of consumption growth on expected real in-terest rates (although Weil, 1990, points out that itis difficult to see a direct linkage between Hall’sregression coefficients and the parameters that de-scribe the GEU preferences).


tinides 1990; and Heaton 1995) can becaptured in the following simple model.Suppose that there is a representativeagent with preferences in period t repre-sented by the utility function:

Et Σs=0∞ βs(ct+s − λct+s−1)1 − α/(1 − α),λ > 0. (5)

Thus, the agent’s momentary utility is adecreasing function of last period’s con-sumption: it takes more consumption to-day to make him happy if he consumedmore yesterday.

The individual’s marginal utility withrespect to period t consumption is givenby the formula:

MUt = (ct − λct−1)−α − βλEt(ct+1 − λct)−α. (6)

Thus, the marginal utility of period tconsumption is an increasing function ofperiod (t − 1) consumption. Note thatthere are two pieces to this formula formarginal utility. The first term capturesthe fact that if I buy a BMW rather thana Yugo today, then I am certainly betteroff today; however, the second piecemodels the notion that having the BMW“spoils” me and reduces my utility fromall future car purchases.

The individual’s optimal consumptionportfolio must satisfy the two first orderconditions that make him indifferent tobuying or selling more stocks or bonds:

βEt{(MUt+1/MUt)(Rt+1s − Rt+1

b )} = 0 (7a)

βEt{(MUt+1/MUt)Rt+1b )} = 1 (7b)

where MUt is defined as in (6). We canapply the Law of Iterated Expectationsto obtain:

βE{(MUt+1/MUt)(Rt+1s − Rt+1

b )} = 0 (7a′)

βE{(MUt+1/MUt)Rt+1b )} = 1. (7b′)

The problem with trying to satisfy(7a′) and (7b′) is that MUt depends onthe investor’s ability to predict futureconsumption growth. As with GEU pref-

erences, this means that the formula forthe marginal utility depends on the indi-vidual’s (possibly unobservable) informa-tion. As in that context, it is convenientto assume that consumption growth fromperiod t to period (t + 1) is unpre-dictable. Then the ratio of marginal utili-ties is given by:

MUt+1/MUt = {(ϕt+1ϕt − λϕt)−α

- λβ(ϕt+1ϕt)−αE(ϕ − λ)−α}/

{(ϕt − λ)−α − λβ(ϕt)−αE(ϕ − λ)−α} (8)

where ϕt ≡ Ct/Ct−1. This allows us to eas-ily estimate MUt+1/MUt by using the sam-ple mean to form an estimate of the un-conditional expectations in the formula(8).

With the preference class (1), it is notpossible to simultaneously satisfy equa-tions (2a′) and (2b′) without driving βabove one. This is not true of prefer-ences that exhibit habit formation. Forexample, suppose we set β = 0.99. Un-like the preference class (1), it is thenpossible to find preference parameters tosatisfy both (7a′) and (7b′). In particular,if we set α = 15.384 and λ = 0.174, thenthe sample analogs of both first orderconditions are exactly satisfied in thedata.14

14 Instead of (7a′,b′), Wayne Ferson and Con-stantinides (1991) work with an alternative impli-cation of (7a,b):

E{MUtZt/(Ct−λCt−1)−α}

= βE{MUt+1Rt+1Zt/(Ct−λCt−1)−α} (∗)

where Rt+1 is the gross return to an arbitrary asset,and Zt is an arbitrary variable observable to theeconometrician and to the agent at time t. Thiscondition has the desirable property that it doesnot involve terms that are unobservable to theeconometrician. However, (*) is a different type ofimplication of optimal behavior from (4a,b) or(2a,b) because it is not just an averaged version of(7a,b) (even when Zt is set equal to 1). In order tobe consistent across preference orderings, I workwith (7a,b).

In the context of a calibrated general equilib-rium model, Constantinides (1990) generates a


There are two implications of thisresult. First, habit formation does notresolve the equity premium puzzle: itis still true that the representative in-vestor is indifferent between stocks andbonds only if she is highly averse toconsumption risk. On the other hand,habit formation does help resolve therisk free rate puzzle—it is possible tosatisfy (7a′) and (7b′) without setting βlarger than one. The intuition behindthis finding is simple. For any given levelof current consumption, the individualknows that his desire for consumptionwill be higher in the future because con-sumption is habit forming; hence, the in-dividual’s demand for savings increasesrelative to the preference specification(1).

3. Relative Consumption: “Keeping up with the Joneses”

The standard preferences (1) assumethat individuals derive utility only fromtheir own consumption. Suppose,though, that as James Duesenberry(1949) posits, an individual’s utility is afunction not just of his own consumptionbut of societal levels of consumption.Then, his investment decisions will beaffected not just by his attitudes towardhis own consumption risk, but also by hisattitudes toward variability in societalconsumption. Following up on this intui-tion, Abel (1990) and Jordi Gali (1994)examine the asset pricing implications ofvarious classes of preferences in whichthe utility an individual derives from a

given amount of consumption dependson per capita consumption.15

I will work with a model that combinesfeatures of Abel’s and Gali’s formula-tions. Suppose the representative indi-vidual has preferences in period t:

Et Σs=0∞ βsct+s

1−αCt+s−1λ Ct+s−1

λ /(1−α), α ≥ 0 (9)

where ct+s is the individual’s level of con-sumption in period (t + s) while Ct+s isthe level of per capita consumption inperiod (t + s) (in equilibrium, of course,the two are the same). Thus, the individ-ual derives utility from how well she isdoing today relative to how well the av-erage person is doing today and how wellthe average person did last period. If theindividual is a jealous sort, then it isnatural to think of γ and λ as being nega-tive: the individual is unhappy when oth-ers are doing well. If the individual ispatriotic, then it is natural to think of γand λ as being positive: the individual ishappy when per capita consumption forthe nation is high.

The representative individual treatsper capita consumption as exogenouswhen choosing how much to invest instocks and bonds. Hence, in equilibrium,the individual’s first order conditionstake the form:

Et{(Ct+1/Ct)γ−α(Ct/Ct−1)λ(Rt+1s

− Rt+1b )} = 0 (10a)

βEt{(Ct+1/Ct)γ−α(Ct/Ct−1)λRt+1

b } = 1. (10b)

We can apply the law of iterated expec-tations to derive the two conditions:

E{(Ct+1/Ct)γ−α(Ct/Ct−1)λ(Rt+1s

− Rt+1b )} = 0 (10a′)

βE{(Ct+1/Ct)γ−α(Ct/Ct−1)λRt+1b } = 1. (10b′)

large equity premium and a low risk free rate byassuming that the representative agent’s utilityfunction has a large value of λ and a low value ofα. However, it is important to keep in mind thatwhen λ is large, the individual requires a largelevel of consumption in order to just survive; hewill pay a lot to avoid small consumption gambleseven if α is low. Thus, Constantinides’ proposedresolution of the puzzles requires individuals to bevery averse to consumption risk (although, as heshows, not to wealth risk).

15 See Campbell and Cochrane (1995) andJames Nason (1988) for models which feature bothrelative consumption effects and time-varying riskaversion.


For any specification of the discountfactor β and the coefficient of relativerisk aversion α, it is possible to find set-tings for γ and λ such that the puzzlesare resolved in that the sample analogsof (10a′) and (10b′) are satisfied. For ex-ample, suppose β = 0.99. Then, if (α − γ)= 19.280, and λ = 3.813, the sample ana-logs of (10a′) and (10b′) are satisfied ex-actly.

This model offers the following expla-nation for the seemingly large equitypremium. When γ is large in absolutevalue, an individual’s marginal utility ofown consumption is highly sensitive tofluctuations in per capita consumptionand therefore strongly negatively relatedto stock returns. Thus, even if α is smallso that the “representative” investor isnot all that averse to individual consump-tion risk, she does not find stocks attrac-tive because she is highly averse to percapita consumption risk.

However, the presence of contempora-neous per capita consumption in the rep-resentative investor’s utility functiondoes not help explain the risk free ratepuzzle: if λ is set equal to zero, (10a′)and (10b′) become equivalent to (2a′)and (2b′). We know from our analysis ofthose equations that if λ = 0, it is neces-sary to drive β larger than one in orderto satisfy (10b′) given that (10a′) is satis-fied. The positive effect of lagged percapita consumption on current marginalutility increases the individual’s demandfor savings and allows (10a′) to be satis-fied with a relatively low value of β.

4. Summary

Mehra and Prescott (1985) require thepreferences of the representative indi-vidual to lie in the “standard” class (1).We have seen that these preferences canbe made consistent with the large equitypremium only if the coefficient of rela-tive risk aversion is pushed near 20.Given the low level of the average Trea-

sury bill return, individuals who see con-sumption in different periods as so com-plementary will be happy with the steepconsumption profile seen in UnitedStates data only if their “discount” factoris above one.

Several researchers have explored theconsequences of using broader classes ofpreferences. Broadly speaking, there aretwo lessons from this type of sensitivityanalysis. First, the risk free rate puzzlecan be resolved as long as the link be-tween individual attitudes toward riskand growth contained in the standardpreferences (1) is broken. Second, theequity premium puzzle is much more ro-bust: individuals must either be highlyaverse to their own consumption risk orto per capita consumption risk if they areto be marginally indifferent between in-vesting in stocks or bonds.

It seems that any resolution to the eq-uity premium puzzle in the context of arepresentative agent model will have toassume that the agent is highly averse toconsumption risk. Per capita consump-tion is very smooth, and therefore doesnot covary greatly with stock returns. Yetpeople continue to demand a high ex-pected return for stocks relative tobonds. The only possible conclusion isthat individuals are extremely averse toany marginal variation in consumption(either their own or societal).

IV. Incomplete Markets and Trading Costs

In this section, I examine the ability ofmodels with incomplete markets andvarious sorts of trading frictions to ex-plain the asset returns data.16 Unlike the

16 Throughout my discussion, I ignore the exist-ence of fiat money. Pamela Labadie (1989) andAlberto Giovannini and Labadie (1991) show thatmotivating a demand for money via a cash-in-advance constraint does not significantly alter theasset pricing implications of the Mehra-Prescott(1985) model. However, a richer model of moneydemand might have more dramatic effects on assetprices.


previous section, which focused solely onthe decision problem of a “representa-tive” agent, the analysis will (necessarily)be fully general equilibrium in nature.Because of the complexity of these dy-namic general equilibrium models, theirimplications are generated numerically; Ifind that this makes any intuition morespeculative in nature.

Throughout this section, all investorsare assumed to have identical prefer-ences of the form (1). The “game” in thisliterature is to explain the existence ofthe puzzles solely through the presenceof incompleteness and trading costs thatMehra and Prescott assume away (just asin Section III, the “game” was to explainthe puzzle without allowing such fric-tions).

1. Incomplete Markets

Underlying the Mehra-Prescott modelis the presumption that the behavior ofper capita consumption growth is a goodguide to the behavior of individual con-sumption growth. We have seen that thisbelief is warranted if asset markets arecomplete so that individuals can writecontracts against any contingency: indi-viduals will use the financial markets todiversify away any idiosyncratic differ-ences in their consumption streams. As aresult, their consumption streams willlook similar to each other and to per cap-ita consumption.

However, in reality, it does not appeareasy for individuals to directly insurethemselves against all possible fluctua-tions in their consumption streams. (Forexample, it is hard to directly insure one-self against fluctuations in labor income.)For this reason, most economists believethat insurance markets are incomplete.Intuitively, in the absence of these kindsof markets, individual consumptiongrowth will feature risk not present inper capita consumption growth and soindividual consumption growth will be

more variable than per capita consump-tion growth. As a result, models with in-complete markets offer the hope thatwhile the covariance of per capita con-sumption growth with stock returns issmall, individual consumption growthmay covary enough with stock returns toexplain the equity premium.

Dynamic Self-insurance and the RiskFree Rate. The intuitive appeal of in-complete markets for explaining asset re-turns data is supported by the work ofWeil (1992). He studies a two periodmodel in which financial markets are in-complete. He shows that if individualpreferences exhibit prudence (that is,convex marginal utility—see Miles Kim-ball 1990), the extra variability in indi-vidual consumption growth induced bythe absence of markets helps resolve therisk free rate puzzle. Without completemarkets, individuals must save more inorder to self-insure against the random-ness in their consumption streams; theextra demand for savings drives down therisk free rate. Weil also shows that if in-dividuals exhibit not just prudence butdecreasing absolute prudence (see Kim-ball 1990), the additional variability inconsumption growth induced by marketincompleteness also helps to explain theequity premium puzzle. The extra riski-ness in individual consumption makesstocks seem less attractive to an individ-ual investor than they would to a “repre-sentative” consumer.17

Unfortunately, two period modelscannot tell the full story because theyabstract from the use of dynamic trad-ing as a form of insurance against risk.For example, suppose my salary nextyear is equally likely to be $40,000 or

17 Mankiw (1986) shows that if the conditionalvariability of individual income shocks is higherwhen the aggregate state of the economy is lower,then prudence alone is sufficient to generate ahigher equity premium in the incomplete marketsenvironment.


$50,000. If I die at the end of the yearand do not care about my descendants,then the variability in income has to befully reflected in my expenditure pat-terns. This is the behavior that is cap-tured by two period models. On theother hand, if I know I will live for manymore years, then I need not absorb myincome risk fully into current consump-tion—I can partially offset it by reducingmy savings when my income is high andincreasing my savings when my income islow.

To understand the quantitative impli-cations of dynamic self-insurance for therisk free rate, it is best to consider a sim-ple model of asset trade in which thereare a large number (in fact, a continuum)of ex ante identical infinitely lived con-sumers. Each of the consumers has arandom labor income and their labor in-comes are independent of one another(so there is no variability in per capitaincome). The consumers cannot directlywrite insurance contracts against thevariability in their individual incomestreams. In this sense, financial marketsare incomplete. However, the individualsare able to make risk free loans to oneanother, although they cannot borrowmore than B in any period.

For now, I want to separate the ef-fects of binding borrowing constraintsfrom the effects of incomplete markets.To eliminate the possibility that thedebt ceiling B is ever a binding con-straint, I assume throughout the follow-ing discussion that B is larger thanymin/r, where ymin is the lowest possiblerealization of income. No individual willborrow more than ymin/r; doing so wouldmean that with a sufficiently long run ofbad income realizations, the individualwould be forced into default, which is as-sumed to be infinitely costly (see Aiya-gari 1994). Market clearing dictates thatfor every lender, there must be a bor-rower; hence, ymin must be larger than

zero if there is to be trade in equilib-rium.18

As in the two period context, an indi-vidual in this economy faces the possibil-ity of an uninsurable consumption fall.The desire to guard against this possibil-ity (at least on the margin) generates agreater demand to transfer resourcesfrom the present to the future than in acomplete markets environment. This ex-tra demand for saving forces the equilib-rium interest rate below the completemarkets interest rate.19

However, the ability to dynamicallyself-insure in the infinite horizon settingmeans that the probability of an uninsur-able consumption fall is much smallerthan in the two period context. In equi-librium, the “typical” individual has nosavings (because net asset holdings arezero), but does have a “line of credit”ymin /r. The individual can buffer anyshort-lived falls in consumption by bor-rowing; his line of credit can be ex-hausted only by a relatively unlikely long“run” of bad income realizations.(Equivalently, in a stationary equilib-rium, few individuals are near theircredit limit at any given point in time.)

18 Christopher Carroll (1992) argues using datafrom the Panel Study in Income Dynamics thatymin equals zero. In this case, no individual canever borrow (because their debt ceiling equalszero). There can be trade of risk free bonds only ifthere is an outside supply of them (say, by thegovernment). Then, the risk free rate puzzle be-comes an issue of how low the amount of outsidedebt must be in order to generate a plausibly lowvalue for the risk free rate. There is no clean an-swer to this question in the existing literature.

19 Marilda Sotomayor (1984) proves that the in-dividual demand for asset holdings and for con-sumption grows without bound over time if theinterest rate equals the rate of time preference—the complete markets interest rate. (Her result isvalid without the assumption of convex marginalutility as long as the consumption set is boundedfrom below.) This implies that the equilibrium in-terest rate must fall below the rate of time prefer-ence in order to deter individuals from saving somuch. See Zeldes (1989b) and Angus Deaton(1992) for numerical work along these lines in afinite horizon setting.


Thus, the extra demand for savings gen-erated by the absence of insurance mar-kets will generally be smaller in the infi-nite horizon economy than in a twoperiod model.

This reasoning suggests that in the in-finite horizon setting, the absence of in-come insurance markets may have littleimpact on the interest rate. The numeri-cal work of Huggett (1993) and Heatonand D. Lucas (1995b) confirms this in-tuition. They examine infinite horizoneconomies which are calibrated to ac-cord with aspects of United States dataon individual labor income and find thatthe difference between the incompletemarkets interest rate and the completemarkets interest rate is small.20

The above discussion assumes that in-dividual income shocks die out eventu-ally (that is, the shocks are stationary).Constantinides and Darrell Duffie(1995) point out that the story is very

different when shocks to individual laborincome are permanent instead of transi-tory. Under the assumption of perma-nence, dynamic self-insurance can playno role: income shocks must be fully ab-sorbed into consumption. For example,if my salary falls by $10,000 perma-nently, then the only way to keep myconsumption smooth over time is reduceit by the same $10,000 in every period: Icannot just temporarily run down mysavings account to smooth consumption.Because income shocks must be fully ab-sorbed into consumption, the resultsfrom the two period model becomemuch more relevant. Indeed, Constan-tinides and Duffie (1995) show that theabsence of labor income insurance mar-kets, combined with the permanence oflabor income shocks, has the potential togenerate a risk free rate that may bemuch lower than the complete marketsrisk free rate.

The issue, then, becomes an empiricalone: how persistent (and variable) areotherwise undiversifiable shocks to indi-vidual income? This is a difficult matterto sort out because of the paucity of timeseries evidence available. However, Hea-ton and D. Lucas (1995a) use data fromthe Panel Study of Income Dynamicsand estimate the autocorrelation of idio-syncratic21 labor income shocks to bearound 0.5. When they examine the pre-dictions of the above model using thisautocorrelation for labor income (oreven an autocorrelation as high as 0.8),they find that the equilibrium interest

20 In fact, they obtain these results in econo-mies in which some fraction of the agents facebinding borrowing constraints in equilibrium. Re-laxing these borrowing constraints or adding out-side debt will serve to further increase the equilib-rium interest rate toward the complete marketsrate (Aiyagari 1994).

There is a caveat associated with the results ofHuggett (1993) and Heaton and D. Lucas(1995b)—and with most of the other numericalwork described later in this paper. For computa-tional reasons, these papers assume that the pro-cess generating individual income is discrete. Thisdiscretization may impose an artificially high lowerbound on income—that is, an artificially highvalue of ymin. Thus, Heaton and D. Lucas (forth-coming) assume that income cannot fall below 75percent of its mean value. Huggett (1993) is moreconservative in assuming that income cannot fallbelow about ten percent of its mean value. In bothcases, finer and more accurate discretizations maydrive ymin lower.

This technical detail may have important conse-quences for the numerical results because no indi-vidual can ever borrow more than ymin/r. As yminfalls, there is an increased probability of a fall inconsumption that cannot be “buffered” by usingaccumulated savings or by borrowing, and thedownward pressure on the equilibrium interestrate rises. In fact, as ymin goes to zero, the equilib-rium interest rate must go to zero (if there is nooutside debt).

21 What matters is the persistence of the undi-versifiable component of labor income. Heatonand Lucas (1995a) implicitly assume that individu-als can hedge any labor income risk that is corre-lated with aggregate consumption. Hence, theyobtain their autocorrelation estimate of 0.5 onlyafter first controlling for aggregate effects uponindividual labor income (this may explain whytheir estimate of persistence is so much lower thanthat of Glenn Hubbard, Jonathan Skinner, andZeldes, 1994, who control for aggregate effectsonly through the use of year dummies).


rate is close to the complete marketsrate.

Dynamic Self-insurance and the Eq-uity Premium Puzzle. The above analysisshows that as long as most individualsare sufficiently far from the debt ceilingymin /r, dynamic self-insurance impliesthat the equilibrium interest rate in in-complete markets models is well-ap-proximated by the complete markets in-terest rate. In this section, I claim that asimilar conclusion can be reached aboutthe equity premium as long as the inter-est rate and the persistence of shocks areboth sufficiently low.

The following argument is a crude in-tuitive justification for this claim, basedlargely on the Permanent Income Hy-pothesis. Consider an infinitely lived in-dividual who faces a constant interestrate r and who faces income shocks thathave a positive autocorrelation equal toρ; if ρ = 1, the shocks are permanentwhile if ρ < 1, their effect eventually diesout. Financial markets are incomplete inthe sense that the individual cannot ex-plicitly insure against the income shocks.Because of concave utility, the individualwants to smooth consumption as much aspossible.

Suppose the individual’s income is sur-prisingly higher in some period by δ dol-lars. Because of the autocorrelation inthe income shocks, the present value ofhis income rises by:

δ + ρδ/(1 + r) + ρ2δ /(1 + r)2 + . . . .

= δ/(1 − ρ/(1 + r)) = δ(1 + r)/(1 + r − ρ).

The individual wants to smooth con-sumption. Hence, he increases consump-tion in every period of life by the annu-itized value of the increase in his wealth.A simple present value calculation im-plies that his consumption rises by δr/(1+ r − ρ) dollars.

If the individual were able to fully in-

sure against all income shocks, then hisconsumption would not change at all inresponse to the surprise movement in hisincome process. Thus, his ability to self-insure against income shocks using onlysavings can be measured by how closer/(1 + r − ρ) is to zero. For example, if ρ= 1, and all shocks are permanent, thenr/(1 + r − ρ) is equal to one; the incomeshock is fully absorbed into consump-tion, which is very different from the im-plications of the full insurance model. Incontrast, if ρ < 1, then as r nears zero,the implications of the full insurancemodel and the incomplete markets mod-els for movements in individual con-sumption become about the same. In theUnited States annual data, we have seenthat r is around one percent; hence, evenif ρ is as high as 0.75, consumption isfairly insensitive to surprise movementsin labor income. (Recall that Heaton andD. Lucas, 1995a, estimate the autocorre-lation of undiversifiable income shocksto be 0.5.)

This intuitive argument is borne outby the numerical work of ChristopherTelmer (1993) and D. Lucas (1994; seealso Heaton and D. Lucas 1995a, 1995b;Wouter den Haan 1994; and Albert Mar-cet and Singleton 1991). These papersexamine the quantitative predictions ofdynamic incomplete markets models,calibrated using individual income data,for the equity premium. They find thatdynamic self-insurance allows individualsto closely approximate the allocations ina complete markets environment. Theequilibrium asset prices are thereforesimilar to the predictions of the Mehra-Prescott (1985) model.

A critical assumption underlying thisnumerical work is that income shockshave an autocorrelation less than one.Constantinides and Duffie (1995) showthat if labor income shocks are insteadpermanent, then it is possible for incom-plete markets models to explain the large


size of the equity premium. As in thecase of the risk free rate puzzle, thefailure of dynamic self-insurance when ρ= 1 means that the model’s implicationsresemble those of a two period frame-work.

To sum up: the assumption that finan-cial markets are complete strikes manyeconomists as prima facie ridiculous, andit is tempting to conclude that it is re-sponsible for the failure of the Mehra-Prescott model to explain the UnitedStates data on average stock and bondreturns. However, as long as investorscan costlessly trade any financial asset(for example, risk free loans) over time,they can use the accumulated stock ofthe asset to self-insure against idiosyn-cratic risk (for example, shocks to indi-vidual income). The above discussionshows that given this ability to self-in-sure, the behavior of the risk free rateand the equity premium are largely unaf-fected by the absence of markets as longas idiosyncratic shocks are not highlypersistent.

2. Trading Costs

In the incomplete and complete mar-kets models described above, it is as-sumed that individuals can costlesslytrade any amount of the available securi-ties. This assumption is unrealistic inways that might matter for the two puz-zles. For example, in order to take ad-vantage of the possible utility gains asso-ciated with the large premium offered byequity, individuals have to reduce theirholdings of Treasury bills and buy morestocks. The costs of making these tradesmay wipe out their apparent utility bene-fits. In this subsection, I consider the im-plications for asset returns of varioustypes of trading costs.

Borrowing and Short Sales Con-straints. Consider the familiar two pe-riod model of borrowing and lending inmost undergraduate microeconomic text-

books. In this model, the individual’sbudget set of consumption choices (c1,c2) is written as:

c1 ≤ y1 + b1

c2 + b1(1 + r) ≤ y2

c1 ≥ 0, c2 ≥ 0

where b1 is the level of borrowing in pe-riod 1. The budget set allows the individ-ual to borrow up to the present value ofhis period two income.

Many economists believe that thismodel of borrowing and lending ignoresan important feature of reality: becauseof enforcement and adverse selectionproblems, individuals are generally notable to fully capitalize their future laborincome. One way to capture this view ofthe world is to add a constraint to theindividual’s decision problem, b1 ≤ B,where B is the limit on how much theindividual can borrow. If B is less thany2/(1 + r), the constraint on b1 may bebinding because the individual may notbe able to transfer as much income fromperiod two to period one as he desires.

The restriction b1 ≤ B is called a bor-rowing constraint. (Short sales con-straints are similar restrictions on thetrade of stocks.) The law of supply anddemand immediately implies that theequilibrium interest rate is generallylower if many individuals face a bindingborrowing constraint than if few indi-viduals do. Intuitively, “tighter” borrow-ing constraints exogenously reduce thesize of the borrowing side of the market;in order to clear markets, interest ratesmust fall in order to shrink the size ofthe lending side of the market.

In keeping with this intuition, thework of Huggett (1993) and Heaton andD. Lucas (1995a, 1995b) demonstratesnumerically that if a sizeable fraction ofindividuals face borrowing constraints,then the risk free rate may be substan-tially lower than the predictions of the


representative agent model. Using datafrom the Panel Study on Income Dynam-ics, Zeldes (1989a) documents that thelarge number of individuals with low lev-els of asset holdings appear to face suchconstraints.

However, Heaton and D. Lucas(1995a, 1995b) show that borrowing andshort sales constraints do not appear tohave much impact on the size of theequity premium. There is a sound intui-tion behind these results: an individualwho is constrained in the stock marketgenerally must also be constrained in thebond market (or vice versa). Otherwise,the individual could loosen the con-straint in the bond market by shifting re-sources to it from the stock market.Hence, just as the average risk free ratemust fall to clear the bond market whenmany individuals are constrained, somust the average stock return fall in or-der to clear the constrained stock mar-ket.22

Transaction Costs. Individuals who tryto engage in asset trade quickly find thatthey face all sorts of transaction costs.(These include informational costs, bro-kerage fees, load fees, the bid-askspread, and taxes.) To understand howtransaction costs affect the pricing of se-curities, consider an investor in a riskfree world who can invest in two differ-ent securities, stocks and bonds. Stockspay a constant dividend equal to d, andhave a constant price equal to p. Becausestocks are a perpetuity, their rate of re-turn rs, ignoring transaction costs, equalsd/p. Bonds are short-lived and pay a con-stant interest rate rb. There is no costassociated with trading bonds. However,stocks are costly to trade in that a buyer

of stocks must incur a cost pτ , in addi-tion to the price p, to buy a share.

In equilibrium, the cost of buying ashare of stock cannot be smaller than thepresent value of the benefits of owningthat share for N periods (or there wouldbe excess demand for stocks becauseeveryone would want to buy shares).This statement can be expressed mathe-matically as:

p(1 + τ) ≥ d/(1 + rb) + d/(1 + rb)2 + . . . + d/(1 + rb)N + p/(1 + rb)N.

This present value expression can be re-written as:

rs ≡ d/p ≤ (1 + τ)rb/{1 − (1 + rb)−N}.

Note that the right hand side getssmaller as N grows large. Intuitively, asthe investor holds the stock for a longperiod of time, he is able to more fullyamortize the initial trading cost; the indi-vidual is therefore only prevented frommaking arbitrage profits if stock returnsare extremely close to those of bonds.Indeed, when N equals infinity, the up-per bound on the premium paid bystocks is very tight:

rs − rb ≤ τrb.

Thus, if rb = 1 percent, and investors canbuy stocks and hold them forever, thenrs can be as large as seven percent inequilibrium only if τ is greater than 600percent!

In the above risk free model, an inves-tor could buy and hold a stock for an in-finite number of periods. However, ifthe investor instead faces income riskwhich cannot be directly insured, she isnot able to plan on being able to holdstocks forever. The reason is simple: ifshe is ever pushed up against a debt ceil-ing, so she cannot borrow to fully offsetbad income shocks, she will have to sellthe stocks in order to smooth her con-sumption. This cap on the investor’s“holding period” generated by potential

22 Following Hua He and David Modest (1995),Cochrane and Hansen (1992) show that this argu-ment is exactly correct when individuals face amarket wealth constraint that restricts the totalvalue of their asset portfolio not to fall below someexogenously specified (negative) number.


runs of bad luck raises the possibilitythat in models with missing income in-surance markets, transaction costs couldhave big effects on asset return spreads.

However, from our discussion of cali-brated incomplete markets models, weknow that an infinitely lived individualcan do a pretty good job of smoothingconsumption by simply buying and sell-ing the asset that is cheaper to trade.Hence, the agent faces only a low prob-ability of needing to sell stocks in orderto smooth consumption. She can conse-quently contemplate making arbitrageprofits by buying and holding stocks fora long period of time; as we have seen,this means that stocks cannot pay amuch higher return than bonds in equi-librium.

Thus, as Aiyagari and Mark Gertler(1991) and Heaton and D. Lucas (1995a)find, the only way to explain the equitypremium using transaction costs is to as-sert that there are significant differencesin trading costs across the stock andbond markets. In my view, there is littleevidence to support this proposition atpresent, although both Aiyagari andGertler (1991) and Heaton and D. Lucas(1995b) make some attempts in this di-rection. To be considered as the leadingexplanation of the puzzle, more needs tobe done to document the sizes andsources of trading costs (especially forpension funds and other institutional in-vestors that operate on behalf of stock-holders).23

Market Segmentation. A key aspect ofthe Mehra-Prescott (1985) model is that

it treats per capita consumption growthas a good proxy for individual consump-tion growth. Yet, Mankiw and Zeldes(1991) and Michael Haliassos and CarolBertaut (forthcoming) document thatonly about 30 percent of individuals inthe United States own stocks (either di-rectly or through defined contributionpension funds). This suggests the possi-bility of market segmentation: that forwhatever reason, only a subset of inves-tors are actively involved in asset trade.Campbell (1993) points out that it is theper capita consumption growth of the ac-tive traders that should be substitutedinto first order conditions like (2a) and(2b), not overall per capita consumptiongrowth as Mehra and Prescott (1985)use.

However, Mankiw and Zeldes (1991)provide direct evidence that market seg-mentation alone cannot explain the eq-uity premium. They find that the con-sumption growth of stockholders doescovary considerably more with stock re-turns than the consumption growth ofnonstockholders. However, the covari-ance is still not large enough: it is only ifthey are highly risk averse that stock-holders are marginally indifferent be-tween stocks and bonds. There is still anequity premium puzzle when we look atthe consumption of those involved in as-set trade.24

23 In their partial equilibrium settings, ErzoLuttmer (1995) and He and Modest (1995) assumethat all investors fully turn over their port- foliosrelatively frequently (once a month or once aquarter) and all at the same time. This assumptionallows them to explain the equity premium withrelatively small transaction costs; however, itseems difficult to build a general equilibriummodel in which investors would choose to incurtrading costs so frequently and so simultaneously.

24 Ingram (1990) examines the general equilib-rium implications of a model in which half theagents use all possible information to form theirportfolios, while the other half hold an optimal“nonchurning” portfolio: they are constrained notto change the split of assets between stocks andbonds in response to new information. She findsthat the equity premium is no higher than if allagents were engaged in asset trade, but the riskfree rate is significantly lower. Presumably, the“rule of thumb” traders hold too many bonds intheir portfolio because they want to be sure thatthey are insured against “runs” of bad income re-alizations. This drives the equilibrium interest ratedownwards, but has little effect on the equity pre-mium.


3. Summary

The Mehra-Prescott (1985) model as-sumes that asset trade is costless andthat asset markets are complete. Casualobservation suggests that both assump-tions are unduly strong. Unfortunately,the predictions of the model for the eq-uity premium and the risk free rate arenot greatly affected by allowing marketsto be incomplete: dynamic self-insuranceallows individuals to smooth idiosyn-cratic shocks without using explicit insur-ance contracts. It is true (by the law ofsupply and demand) that a frequentlybinding borrowing constraint forces theaverage risk free rate down toward itssample value. The only way to simultane-ously generate a sizeable equity pre-mium, though, is to make trading in thestock market substantially more costlythan in the bond market. Under this as-sumption, the premium on stocks repre-sents compensation not for risk (as in thestandard financial theory), but rather forbearing additional transactions costs.The sources of these additional costs re-main unclear.

Despite its general lack of success inexplaining the equity premium, this lit-erature has made an important contribu-tion to our understanding of trade inmarkets with frictions. Just as people inreal life are able to find ways aroundgovernment regulation of the economy,rational individuals in models are able tofind ways around barriers to trade. Ittakes an enormous amount of sand in thegears to disrupt the ability of individualsto approximate an efficient allocation ofresources.

There is, however, a continuing weak-ness in the “market frictions” literature.Usually, the incompleteness of marketsand the costliness of trade in these mod-els is motivated by adverse selection ormoral hazard considerations. Yet, thismotivation remains informal. From a

theoretical point of view (and possibly anempirical one as well), the models willbe stronger if they explicitly take into ac-count the informational problems thatlead to the trading frictions.

V. Discussion and Conclusions

This paper began by posing two puz-zles: how can we explain why averagestock returns are so much higher thanbond returns in the United States data,and why has per capita consumptiongrown so quickly given that bond returnsare so low? As it turns out, there are avariety of ways to generate more savingsdemand than in the Mehra-Prescottmodel and thereby explain the latterphenomenon.

However, the equity premium puzzleis much more challenging. Throughoutthis article, we have seen only two waysto explain the wedge in average returnsbetween stocks and bonds. The first isthat there is a large differential in thecost of trading between the stock andbond markets. To make this explanationcompelling, it is important to ascertainthe size of actual trading costs in thesemarkets, and to provide an explanationof why those costs exist. Heaton and D.Lucas (1995b) and Aiyagari and Gertler(1991) discuss some early steps in thisdirection.

The second explanation is that, con-trary to Mehra and Prescott’s (1985)original parametric restrictions, individ-ual investors have coefficients of relativerisk aversion larger than ten (either withrespect to their own consumption orwith respect to per capita consump-tion).25 As I explained earlier, the prob-lem with this explanation is that only a

25 Of course, as mentioned above, to resolve therisk free rate puzzle, we cannot only adopt a moreflexible notion of what constitutes reasonable riskaversion; we must abandon the standard prefer-ences (1) or impose borrowing constraints thatbind frequently.


handful of economists believe that indi-viduals are that risk averse. One way tosupport the “high risk aversion” view isto demonstrate that this apparently“strange” assumption about human be-havior is consistent with data other thanthe average realization of the equity pre-mium. Until now, little has been donealong these lines, but Tallarini’s (1994)analysis of a “prototypical” real businesscycle model using generalized expectedutility preferences represents a promis-ing first step.

Ten years ago, Mehra and Prescott(1985) wrote in their conclusion that therelatively low rate of return of Treasurybills, “is not the only example of someasset receiving a lower return than thatimplied by Arrow-Debreu general equi-librium theory. Currency, for example, isdominated by Treasury bills with positivenominal yields yet sizable amounts ofcurrency are held.” Thus, they regardthe equity premium puzzle as beinganalogous to the so-called “rate of returndominance” puzzle that motivates muchof modern monetary theory.

I believe that the past decade of workhas made this analogy even more persua-sive: I find that the suggested “solutions”to the equity premium puzzle closely re-semble the approaches used by econo-mists to generate a demand for money inthe face of rate of return dominance.Thus, the “high risk aversion” story ex-plains the puzzle by some peculiar aspectof individual preferences—which is ex-actly the way money-in-the-utility-func-tion models explain money demand. The“transactions costs” story explains thatbonds are held despite their low rate ofreturn because they are less costly totrade—which is exactly the way transac-tions costs models explain money de-mand.

Generally, both of these types of mod-els of money demand are regarded as adhoc “reduced forms.” They fail to cap-

ture the essential technological forces(for example, lack of communication)that disrupt the process of fully central-ized exchange and thereby generate ademand for money. This same complaintof superficiality can also be made of thetwo types of stories explaining the equitypremium.

In one sense, the accuracy of themoney demand analogy is depressing:monetary theorists are a long way fromdelivering a definitive model of moneydemand. On the other hand, the work ofRobert Townsend (1987) and others inmonetary theory is exciting because itshows so clearly what it will take to maketrue progress in explaining the size ofthe equity premium. Like fiat money,the equity premium appears to be awidespread and persistent phenomenonof market economies. The universality ofthe equity premium tells us that, likemoney, the equity premium mustemerge from some primitive and ele-mentary features of asset exchange thatare probably best captured through ex-tremely stark models. With this in mind,we cannot hope to find a resolution tothe equity premium puzzle by continuingin our current mode of patching thestandard models of asset exchange withtransactions costs here and risk aversionthere. Instead, we must seek to identifywhat fundamental features of goods andasset markets lead to large risk adjustedprice differences between stocks andbonds. While I have no idea what these“fundamental features” are, it is my be-lief that any true resolution to the equitypremium puzzle lies in finding them.

Appendix

Apart from two exceptions, the follow-ing italicized description is taken directlyfrom Mehra and Prescott (1985, pp.147–48).

The data used in this study consists of


five basic series for the period 1889–1978. The first four are identical to thoseused by Grossman and Shiller (1981) intheir study. The series are individuallydescribed below.

(i) Series P: Annual average Standard& Poor’s Composite Stock Price Index di-vided by the Consumption Deflator, aplot of which appears in Grossman andShiller (1981, p. 225, fig. 1).

(ii) Series D: Real annual dividends forthe Standard & Poor’s series.

(iii) Series C: Kuznets-Kendrik-USNIAper capita real consumption on non-dur-ables and services.

(iv) Series PC: Consumption deflatorseries, obtained by dividing real con-sumption in 1972 dollars on non-dur-ables and services by the nominal con-sumption on non-durables and services.

(v) Series RF: Nominal yield on rela-tively riskless short-term securities overthe 1889–1978 period; the securities usedwere ninety-day government Treasurybills in the 1931–1978 period, TreasuryCertificates for the 1920–1930 periodand sixty-day to ninety-day Prime Com-mercial Paper prior to 1920 [the datawas obtained from Homer (1963) andIbbotson and Sinquefeld (1978)].

These series were used to generate theseries actually utilized in this paper. Se-ries P and D above were used to deter-mine the average annual real return onthe Standard & Poor’s 500 Composite In-dex over the ninety-year period of study.The annual return for year t was com-puted as (Pt+1 + Dt+1 − Pt)/Pt. The returnsare plotted in fig. 2. Series C was usedto determine the process on the growthrate of consumption over the same period. . . A plot of the percentage growth rateof real consumption appear in fig. 1. Todetermine the real return on a relativelyriskless security we used the series RFand PC. For year t, this is calculatedto be (1 + RFt)PCt/PCt+1. This series isplotted in fig. 3.

This description differs from that ofMehra and Prescott (1985) in two ways.First, the figure numbers have beenchanged to match the numbering in thispaper. Second, Mehra and Prescott(1985) used the formula RFt − (PCt+1 −PCt)/PCt for the real return to the rela-tively riskless security. This formula ex-presses the real rate as being the differ-ence between the nominal rate and theinflation rate. Of course, their formula isonly an approximation; I use the correctratio formula instead.

This difference in the formulae createsonly small differences in results. For ex-ample, they find that the average rela-tively riskless rate is 0.8 percent. I find itinstead to be one percent. They find thatthe variance of the relatively riskless rateis 0.0032; I find it to be 0.00308.

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