The Equity Premium the Equity Premium: It’S Still a Puzzle

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. KOCHERLAKOTA Federal 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, Steve Williamson, Kei-Mu Yi, and the referees for their input in preparing this manuscript. I thank Edward R. Allen III, Ravi Jagannathan, and Deborah Lucas for our many interesting and enlightening conversations on the subject of asset pricing over the years. Finally, I thank Lars Hansen for teaching me how to think about the issues addressed in this paper. None of the preceding should be viewed as responsible for the opinions and errors contained within. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.

I. Introduction ferences in the degree to which a security’s return covaries with the typical in- VER THE LAST one hundred years, vestor’s consumption. If this covariance Othe average real return to stocks in is high, selling off the security would the United States has been about six per- greatly reduce the variance of the typical cent per year higher than that on Trea- investor’s consumption stream; in equi- sury bills. At the same time, the average librium, the investor must be deterred real return on Treasury bills has been from reducing his risk in this fashion by about one percent per year. In this pa- the security’s paying a high average return. per, I discuss and assess various theore- There is a crucial problem in making tical attempts to explain these two differ- this qualitative explanation of cross-sec- ent empirical phenomena: the large tional differences in asset returns opera- “equity premium” and the low “risk free tional: what exactly is “the typical inves- rate.” I show that while there are several tor’s consumption”? The famous Capital plausible explanations for the low level Asset Pricing Model represents one an- of Treasury returns, the large equity pre- swer to this question: it assumes that the mium is still largely a mystery to econo- typical investor’s consumption stream is mists. perfectly correlated with the return to In order to understand why the sample the stock market. This allows financial means of the equity premium and the analysts to measure the risk of a financial risk free rate represent “puzzles,” it is security by its covariance with the return useful to review the basics of modern as- to the stock market (that is, a scaled ver- set pricing theory. It is well known that sion of its “beta”). More recently, the real returns paid by different Douglas Breeden (1979) and Robert Lu- financial securities may differ consider- cas (1978) described so-called “repre- ably, even when averaged over long sentative” agent models of asset returns periods of time. Financial economists in which per capita consumption is per- typically explain these differences in av- fectly correlated with the consumption erage returns by attributing them to dif- stream of the typical investor. (See also 42 Kocherlakota: The Equity Premium 43

Mark Rubinstein 1976; and Breeden and (they dislike growth). Given that the Robert Litzenberger 1978.) In this type large equity premium implies that inves- of model, a security’s risk can be mea- tors are highly risk averse, the standard sured using the covariance of its return models of preferences would in turn im- with per capita consumption. ply that they do not like growth very The representative agent model of as- much. Yet, although Treasury bills offer set pricing is not as widely used as the only a low rate of return, individuals de- CAPM in “real world” applications (such fer consumption (that is, save) at a suffi- as project evaluation by firms). However, ciently fast rate to generate average per from an academic economist’s perspec- capita consumption growth of around tive, the “representative agent” model of two percent per year. This is what Weil asset pricing is more important than the (1989) calls the risk free rate puzzle: al- CAPM. Representative agent models though individuals like consumption to that imbed the Lucas-Breeden paradigm be very smooth, and although the risk for explaining asset return differentials free rate is very low, they still save are an integral part of modern macro- enough that per capita consumption economics and international economics. grows rapidly. Thus, any empirical defects in the repre- There is now a vast literature that sentative agent model of asset returns seeks to resolve these two puzzles. I be- represent holes in our understanding of gin my review of this literature by show- these important subfields. ing that the puzzles are very robust: they In their seminal (1985) paper, Rajnish are implied by only three assumptions Mehra and Edward Prescott describe a about individual behavior and asset mar- particular empirical problem for the rep- ket structure. First, individuals have resentative agent paradigm. As men- preferences associated with the “stan- tioned above, over the last century, the dard” utility function used in macro- average annual real return to stocks has economics: they maximize the expected been about seven percent per year, while discounted value of a stream of utilities the average annual real return to Trea- generated by a power utility function. sury bills has been only about one per- Second, asset markets are complete: indi- cent per year. Mehra and Prescott (1985) viduals can write insurance contracts show that the difference in the covari- against any possible contingency. Finally, ances of these returns with consumption asset trading is costless, so that taxes and growth is only large enough to explain brokerage fees are assumed to be insig- the difference in the average returns if nificant. Any model that is to resolve the the typical investor is implausibly averse two puzzles must abandon at least one of to risk. This is the equity premium puz- these three assumptions. zle: in a quantitative sense, stocks are My survey shows that relaxing the not sufficiently riskier than Treasury three assumptions has led to plausible bills to explain the spread in their re- explanations for the low value of the risk turns. free rate. Several alternative preference Philippe Weil (1989) shows that the orderings are consistent with it; also, the same data presents a second anomaly. existence of borrowing constraints push According to standard models of individ- interest rates down in equilibrium. How- ual preferences, when individuals want ever, the literature provides only two ra- consumption to be smooth over states tionalizations for the large equity pre- (they dislike risk), they also desire mium: either investors are highly averse smoothness of consumption over time to consumption risk or they find trading 44 Journal of Economic Literature, Vol. XXXIV (March 1996) stocks to be much more costly than trad- highly procyclical risk associated with ing bonds. Little auxiliary evidence exists stock returns. As Andrew Atkeson and to support either of these explanations, Christopher Phelan (1994) point out, and so (I would say) the equity premium without this knowledge we cannot hope puzzle is still unresolved. to give a meaningful answer to R. Lucas’ The Lucas-Breeden representative (1987) question about how costly in- agent model is apparently inconsistent dividuals find business cycle fluctua- with the data in many other respects. tions in consumption growth. Thus, Sanford Grossman and Robert Shiller the two puzzles are signs of large gaps (1981) (and many others) argue that as- in our understanding of the macroecon- set prices vary too much to be explained omy. by the variation in dividends or in per Given the lack of a compelling expla- capita consumption. Hansen and Ken- nation for the large equity premium, the neth Singleton (1982, 1983) point out article will at times read like a litany of that in forming her portfolio, the repre- failure. I should say from the outset that sentative investor appears to be ignoring this is misleading in many ways. We have useful information available in lagged learned much from the equity premium consumption growth and asset returns. puzzle literature about the properties of Robert Hall (1988) argues that consump- asset pricing models, about methods of tion does not respond sufficiently to estimating and testing asset pricing mod- changes in expected returns. John Coch- els, and about methods of solving for the rane and Hansen (1992) uncover several implications of asset pricing models. I anomalies, including the so-called De- believe that all of these contributions are fault-Premium and Term Structure Puz- significant. zles. For better or for worse, though, this This paper will look at none of these article is not about these innovations. In- other puzzles. Instead, it focuses exclu- stead, I focus on what might be called sively on the risk free rate and equity the bottom line of the work, “Do we premium puzzles. There are two ratio- know why the equity premium is so nalizations for limiting the discussion in high? Do we know why the risk free rate this way. The first is simple: my conver- is so low?” It is in answering the first of sations with other economists have con- these questions that the literature falls vinced me that these puzzles are the short. most widely known and best under- The rest of the paper is structured as stood of the variety of evidence usually follows. The next section describes the arrayed against the representative agent two puzzles and lays out the fundamental models. modelling assumptions that generate The second rationalization is perhaps them. Section III explores the potential more controversial: I claim that these for explaining the two puzzles by chang- two puzzles have more importance for ing the preferences of the representative macroeconomists. The risk free rate agent. Section IV looks at the implica- puzzle indicates that we do not know tions of market frictions for asset re- why people save even when returns are turns. Section V concludes.1 low: thus, our models of aggregate savings behavior are omitting some crucial 1 In writing this paper, I have benefited greatly element. The equity premium puzzle from reading four other review articles by Andrew Abel (1991), Aiyagari (1993), Cochrane and Han- demonstrates that we do not know sen (1992), and Heaton and D. Lucas (1995b). I why individuals are so averse to the recommend all of them highly. Kocherlakota: The Equity Premium 45

0.14

0.10

0.06

0.02 consumption growth –0.02

–0.06

–0.10 1880 1900 1920 1940 1960 1980 time Figure 1. Annual Real Per Capita Consumption Growth

II. What Are the Puzzles? Plots of the data are depicted in Figures 1–3; all of the series appear station- 1. Aspects of the Data ary and ergodic (their statistical proper- The equity premium and risk free ties do not appear to be changing over rate puzzles concern the co-movements time). of three variables: the real return to Table 1 contains some summary statis- the S & P 500, the real return to short tics for the three variables. There are term nominally risk free bonds,2 and three features of the table that give rise the growth rate of per capita real con- to the equity premium and risk free rate sumption (more precisely, nondurables puzzles. First, the average real rate of re- and services). In this paper, I use the turn on stocks is equal to seven percent annual United States data from 1889– 1978 originally studied by Mehra and Prescott (1985) (see the Appendix for does not eliminate the puzzles. Hansen and Sin- details on the construction of the data).3 gleton (1983) and Aiyagari (1993) find that similar phenomena characterize post-World War II monthly data in the United States. Amlan Roy (1994) documents the existence of the two puzzles 2 Ninety day Treasury bills from 1931–1978, in post-World War II quarterly data in Germany Treasury certificates from 1920–31, and 60–day to and Japan. It may not be too strong to say that the 90–day Commercial Paper prior to 1920. equity premium and risk free rate puzzles appear 3 In what follows, I use only the Mehra-Prescott to be a general feature of organized asset markets. data set. However, it is important to realize that (Jeremy Siegel, 1992, points out that the equity the puzzles are not peculiar to these data. For ex- premium was much lower in the 19th century. ample, Kocherlakota (1994) shows that adding ten However, consumption data is not available for more years of data to the Mehra-Prescott series this period.) 46 Journal of Economic Literature, Vol. XXXIV (March 1996)

0.6

0.4

0.2

stock returns –0.0

–0.2

–0.4 1880 1900 1920 1940 1960 1980 time Figure 2. Annual Real Return to S & P 500

0.25

0.20

0.15

0.10

bill returns –0.00

–0.10

–0.20 1880 1900 1920 1940 1960 1980 time Figure 3. Annual Real Return to Nominally Risk Free Short Term Debt Kocherlakota: The Equity Premium 47

The Original Statement of the Puzzles TABLE 1 2. SUMMARY STATISTICS In their 1985 paper, Mehra and UNITED STATES ANNUAL DATA, 1889–1978 Prescott construct a simple model that Sample Means makes well-defined quantitative predic- s Rt 0.070 tions for the expected values of the real b R t 0.010 returns to the S & P 500 and the three Ct /Ct-1 0.018 month Treasury bill rate. There are Sample Variance-Covariance three critical assumptions underlying s b Rt Rt Ct/Ct-1 s their model. The first is that in period Rt 0.0274 0.00104 0.00219 b − t, all individuals have identical prefer- R t 0.00104 0.00308 0.000193 Ct/Ct-1 0.00219 −0.000193 0.00127 ences over future random consumption streams represented by the utility func-

In this table, Ct/Ct-1 is real per capita consumption tion: growth, Rs is the real return to stocks and Rb is the real t t Σ∞ βs( )1−α ( −α) α ≥ return to Treasury bills. Et s=0 ct+s / 1 , 0 (1) ∞ where {ct}t=1 is a random consumption stream. In this expression, and through- per year while the average real rate of out the remainder of the paper, Et repre- return on bonds is equal to one percent sents an expectation conditional on infor- per year. Second, by long-term historical mation available to the individual in standards, per capita consumption period t. Thus, individuals seek to maxi- growth is high: around 1.8 percent per mize the expectation of a discounted year. Finally, the covariance of per cap- flow of utility over time. ita consumption growth with stock re- In the formula (1), the parameter β is turns is only slightly bigger than the co- a discount factor that households apply variance of per capita consumption to the utility derived from future con- growth with bond returns. sumption; increasing β leads investors to In some sense, there is a simple expla- save more. In contrast, increasing α has nation for the higher average return of two seemingly distinct implications for stocks: stock returns covary more with individual attitudes toward consumption consumption growth than do Treasury profiles. When α is large, individuals bills. Investors see stocks as a poorer want consumption in different states to hedge against consumption risk, and so be highly similar: they dislike risk. But stocks must earn a higher average return. individuals also want consumption in dif- Similarly, while individuals must have ferent dates to be similar: they dislike saved a lot to generate the high con- growth in their consumption profiles. sumption growth described in Table 1, The second key assumption of the there is an explanation of why they did Mehra-Prescott model is that people can so: as long as individuals discount the trade stocks and bonds in a frictionless future at a rate lower than one percent market. What this means is that an indi- per year, we should expect their con- vidual can costlessly (that is, without sig- sumptions to grow. The data does not nificant taxes or brokerage fees) buy and contradict the qualitative predictions of sell any amount of the two financial as- economic theory; rather, as we shall see, sets. In equilibrium, an individual must it is inconsistent with the quantitative not be able to gain utility at the margin implications of a particular economic by selling bonds and then investing the model. proceeds in stocks; the individual should 48 Journal of Economic Literature, Vol. XXXIV (March 1996) not be able to gain utility by buying or representative individual (with coeffi- selling bonds. Hence, given the costless- cient of relative risk aversion no larger ness of performing these transactions, an than the most risk averse individual and individual’s consumption profile must no smaller than the least risk averse indi- satisfy the following two first order con- vidual) which satisfies the first order ditions: conditions (2a) and (2b). The key is that −α s b asset markets must be complete: indi- E {(c + /c ) (R + − R + )} = 0. t t 1 t t 1 t 1 (2a) viduals in the United States must have a β ( )−α b = sufficiently large set of assets available Et{ ct+1/ct Rt+1} 1. (2b) for trade that they can diversify any idio- s In these conditions, Rt is the gross return syncratic risk in consumption. When to stocks from period (t − 1) to period t, markets are complete, we can construct b while Rt is the gross return to bonds a “representative” agent because after from period (t − 1) to period t. trading in complete markets, individuals These first order conditions impose become marginally homogeneous even statistical restrictions on the comove- though they are initially heterogeneous. ment between any person’s pattern of We can summarize this discussion as consumption and asset returns. The third follows: Mehra and Prescott assume that critical feature of Mehra and Prescott’s asset markets are frictionless, that asset analysis, though, is that they assume the markets are complete, and that the resul- existence of a “representative” agent. Ac- tant representative individual has prefer- cording to this assumption, the above ences of the form given by (1). However, conditions (2a) and (2b) are satisfied not Mehra and Prescott also make three just for each individual’s consumption, other, more technical, assumptions. but also for per capita consumption. First, they assume that per capita con- There is some confusion about what sumption growth follows a two state kinds of assumptions underlie this substi- Markov chain constructed in such a way tution of per capita for individual con- that the population mean, variance, and sumption. It is clear that if all individuals autocorrelation of consumption growth are identical in preferences and their are equivalent to their corresponding ownership of production opportunities, sample means in United States data. then in equilibrium, all individuals will They also assume that in period t, the in fact consume the same amount; thus, only variables that individuals know are under these conditions, substituting per the realizations of current and past con- capita consumption into conditions (2a) sumption growth. Finally, they assume and (2b) is justified. Many economists that the growth rate of the total divi- distrust representative agent models be- dends paid by the stocks included in the cause they believe this degree of homo- S & P 500 is perfectly correlated with geneity is unrealistic. the growth rate of per capita consump- However, while this degree of homo- tion, and that the real return to the geneity among individuals is sufficient to (nominally risk free) Treasury bill is per- guarantee the existence of a repre- fectly correlated with the return to a sentative agent, it is by no means neces- bond that is risk free in real terms. sary. In particular, George Constan- These statistical assumptions allow tinides (1982) shows that even if Mehra and Prescott to use the first order individuals are heterogeneous in prefer- conditions (2a) and (2b) to obtain an ences and levels of wealth, it may be pos- analytical formula that expresses the sible to find some utility function for the population mean of the real return to the Kocherlakota: The Equity Premium 49

S & P 500 and the population mean real β ( )−α b = E{ Ct+1/Ct Rt+1} 1. (2b′) return to the three month Treasury bill in terms of the two preference parame- (The variable Ct stands for per capita ters β and α. Using evidence from mi- consumption.) We can estimate the ex- croeconometric data and introspection, pectations or population means on the they restrict β to lie between 0 and 1, ′ ′ α left hand side of (2a ) and (2b ) using the and to lie between 0 and 10. Their sample means of: main finding is that for any value of the s = ( )−α( s − b ) preference parameters such that the ex- et+1 { Ct+1/Ct Rt+1 Rt+1 } pected real return to the Treasury bill is b = β ( )−α b less than four percent, the difference in et+1 { Ct+1/Ct Rt+1} the two expected real returns (the “eq- given a sufficiently long time series of uity premium”) is less than 0.35 percent. data on Ct+1 and the asset returns.4 s This stands in contrast to the facts de- Table 2 reports the sample mean of et scribed earlier that the average real re- for various values of α; for all values of α turn to Treasury bills in the United less than or equal to 8.5, the sample s States data is one percent while the aver- mean of et is significantly positive. We age real return to stocks is nearly six per- can conclude that the population mean s α cent higher. of et is in fact positive for any value of Mehra and Prescott conclude from less than or equal to 8.5; hence, for any their analysis that their model of asset such α, the representative agent can gain returns is inconsistent with United States at the margin by borrowing at the Treas- data on consumption and asset returns. ury bill rate and investing in stocks.5 This Because the model was constructed by 4 Throughout this paper, I will be evaluating making several different assumptions, various representative agent models by using ver- presumably the fit of the model to the sions of (2a′) and (2b′) as opposed to (2a) and data could be improved by changing any (2b). While (2a,b) certainly imply (2a′,b′), there are a host of other implications of (2a,b): the law of them. In the next section, I restate of iterated expectations “averages” over all of the the equity premium puzzle to show implications of (2a) and (2b) to arrive at (2a′) and why the three final assumptions about (2b′). Focusing on (2a) and (2b) is in keeping with my announced goal in the introduction of just the statistical behavior of consumption looking at the equity premium and risk free rate and asset returns are relatively unimpor- puzzles. tant. Hansen and Jagannathan (1991) and Cochrane and Hansen (1992) reinterpret the equity pre- 3. A More Robust Restatement of the mium puzzle and the risk free rate puzzles using the variance bounds derived in the former paper. Puzzles Note that, as Cochrane and Hansen (1992) empha- size, the variance bound is an even weaker impli- As we saw above, the crux of the cation of (2a) than (2a′) is (in the sense that (2a′) Mehra-Prescott model is that the first implies the variance bound while the converse is not true). order conditions (2a) and (2b) must be 5 Breeden, Michael Gibbons, and Litzenberger satisfied with per capita consumption (1989) and Grossman, Angelo Melino, and Shiller growth substituted in for individual con- (1987) point out that the first order conditions (2a) and (2b) are based on the assumption that all sumption growth. Using the Law of Iter- consumption within a given year is perfectly sub- ated Expectations, we can replace the stitutable; if this assumption is false, there is the conditional expectation in (2a) and (2b) possibility of time aggregation bias. However, Hansen, Heaton, and Amir Yaron (1994) show that with an unconditional expectation so as long as the geometric average of consumption that: growth rates is a good approximation to the arith- metic average of consumption growth rates, the ( )−α( s − b ) = E{ Ct+1Ct Rt+1 Rt+1 } 0 (2a′) presence of time aggregation should only affect 50 Journal of Economic Literature, Vol. XXXIV (March 1996)

TABLE 2 the investor’s degree of risk aversion 6 THE EQUITY PREMIUM PUZZLE rises. b Table 3 reports the sample mean of et a e– t-stat for various values of α and β = 0.99. It β α 0.0 0.0594 3.345 shows that for equal to 0.99 and 0.5 0.0577 3.260 greater than one, the representative 1.0 0.0560 3.173 household can gain at the margin by 1.5 0.0544 3.082 transferring consumption from the fu- 2.0 0.0528 2.987 ture to the present (that is, reducing its 2.5 0.0512 2.890 3.0 0.0496 2.790 savings rate). This is the risk free rate 3.5 0.0480 2.688 puzzle. High values of α imply that indi- 4.0 0.0464 2.584 viduals view consumption in different 4.5 0.0449 2.478 periods as complementary. Such indi- 5.0 0.0433 2.370 viduals find an upwardly sloped con- 5.5 0.0418 2.262 6.0 0.0403 2.153 sumption profile less desirable because 6.5 0.0390 2.044 consumption is not the same in every pe- 7.0 0.0372 1.934 riod of life. As a result, an individual 7.5 0.0357 1.824 with a high value of α realizes more of a 8.0 0.0341 1.715 utility gain by reducing her savings rate. 8.5 0.0326 1.607 9.0 0.0310 1.501 Note that the risk free rate puzzle comes 9.5 0.0295 1.395 from the equity premium puzzle: there is 10.0 0.0279 1.291 a risk free rate puzzle only if α is re- quired to be larger than one so as to – −α In this table, e is the sample mean of et = (Ct+1/Ct) match up with the high equity premium.7 s − b α (Rt+1 Rt+1) and is the coefficient of relative risk aversion. Standard errors are calculated using the implication of the theory that et et-k is uncorrelated with 6 This intuition becomes even more clear if one for all k; however, they are little changed by allowing et ( ) ( s ) ( b ) assumes that ln Ct+1/Ct , ln Rt+1 , and ln Rt+1 are to be MA(1) instead. This latter approach to calculating jointly normally distributed. Then, equations (2a) standard errors allows for the possibility of time and (2b) become: aggregation (see Hansen, Heaton, and Yaron 1994). s b s b E(rt − rt )− αCov(gt,rt−rt ) s b + 0.5Var(rt) + 0.5Var (r t ) = 0 (2a) b 2 ln(β) − αE(gt) + E(rt ) + 0.5{α Var(gt) is the equity premium puzzle. Intui- − α ( b) + ( b) = ( ) tively, while the covariance between 2 Cov gt,rt Var rt } 0 2B s b s b where (gt, rt, rt ) ≡ (ln(Ct/Ct−1), ln(Rt), ln(Rt )). In the stock returns and per capita consump- b data, Cov(gt, rt ) is essentially zero. Because s b s tion growth is positive (see Table 1), it is E(rt − rt ) is so large, either α or Cov(gt,rt) must be not sufficiently large to deter the repre- large in order to satisfy (2a). sentative investor with a coefficient of As with (2a) and (2b), the sample analogs of (2a) and (2b) can only be satisfied by setting β equal to relative risk aversion less than 8.5 from a value larger than one and α equal to a value wanting to borrow and invest in stocks. greater than 15. See N. Gregory Mankiw and Note that the marginal benefit of selling Stephen Zeldes (1991) for more details. 7 See Edward Allen (1990), Cochrane and Han- Treasury bills and buying stocks falls as sen (1992), and Constantinides (1990) for similar presentations of the two puzzles. Stephen Cecchetti, Pok-sang Lam, and Nelson Mark (1993) present the results of joint tests of Tables 2–3 through the calculation of the standard ′ ′ errors. Using their suggested correction makes (2a ,b ). I present separate t-tests because I think little difference to the t-statistics reported in doing so provides more intuition about the failings of the representative agent model. Of course, either Table. (Indeed, Corr(et, et-k) is small for all k ≤ 10.) there is nothing wrong statistically with doing two Kocherlakota: The Equity Premium 51 ′ TABLE 3 order condition (2b ). (See Kocherlakota, THE RISK FREE RATE PUZZLE 1990a, for a demonstration of how com- petitive equilibria can exist even when β a e– t-stat > 1.) It is necessary to set α to such a 0.0 0.0033 0.0567 large value because stocks offer a huge 0.5 −0.0081 −1.296 premium over bonds, and aggregate con- 1.0 −0.0162 −2.233 sumption growth does not covary greatly 1.5 −0.0239 −2.790 − − with stock returns; hence, the repre- 2.0 0.0313 3.100 sentative investor can be marginally in- 2.5 −0.0382 −3.263 3.0 −0.0448 −3.339 different between stocks and bonds only 3.5 −0.0510 −3.360 if he is highly averse to consumption 4.0 −0.0569 −3.348 risk. Similarly, the high consumption 4.5 −0.0624 −3.312 growth enjoyed by the United States 5.0 −0.0675 −3.259 − − since 1890 can be consistent with high 5.5 0.0723 3.195 α 6.0 −0.0768 −3.123 values of and the risk free rate only if 6.5 −0.0808 −3.043 the representative investor is so patient 7.0 −0.0846 −2.959 that his “discount” factor β is greater 7.5 −0.0880 −2.871 − − than one. 8.0 0.0910 2.779 Thus, the equity premium and risk 8.5 −0.0937 −2.685 9.0 −0.0960 −2.590 free rate puzzles are solely a product of 9.5 −0.0980 −2.492 the parametric restrictions imposed by 10.0 −0.0997 −2.394 Mehra and Prescott on the discount factor β and the coefficient of relative risk – β −α In this table, e is the sample mean of et = (ct+1 /ct) aversion α. Given this, it is important to (Rb − 1) and α is the coefficient of relative risk t+1 understand the sources of these restric- aversion. The discount factor β is set equal to 0.99. The tions. The restriction that the discount standard errors are calculated using the implication of β the theory that et is uncorrelated with et-k for all k; factor is less than one emerges from however, they are little changed by allowing et to be introspection: most economists, includ- MA(1) instead. This latter approach to calculating ing Mehra and Prescott, believe that an standard errors allows for the possibility of time individual faced with a constant con- aggregation (see Hansen, Heaton, and Yaron 1994). sumption stream would, on the margin, like to transfer some consumption from the future to the present. It is possible to find parameter set- The restriction that α should be less tings for the discount factor β and the than ten is more controversial. Mehra coefficient of relative risk aversion α that and Prescott (1985) quote several micro- exactly satisfy the sample versions of econometric estimates that bound α equations (2a′) and (2b′). In particular, from above by three. Unfortunately, the by setting α = 17.95, it is possible to sat- only estimate that they cite from finan- isfy the equity premium first order con- cial market data has been shown to be dition (2a′); by also setting β = 1.08, it is severely biased downwards (Kocherlak- possible to satisfy the risk free rate first ota 1990c). In terms of introspection, it has been argued by Mankiw and Zeldes separate t-tests, each with size α, as long as one (1991) that an individual with a coeffi- keeps in mind that the size of the overall test cient of relative risk aversion above ten could be as low as zero or it could be as high as would be willing to pay unrealistically 2α. (There may be a loss of power associated with using sequential t-tests but power does not seem to large amounts to avoid bets. However, be an issue in tests of representative agent models!) they consider only extremely large bets 52 Journal of Economic Literature, Vol. XXXIV (March 1996)

(ones with potential losses of 50 percent could simply be a result of sampling er- of the gambler’s wealth). In contrast, ror. Similarly, Mehra and Prescott as- Shmuel Kandel and Robert Stambaugh sume that the growth rate of the divi- (1991) show that even values of α as high dend to the S & P 500 is perfectly as 30 imply quite reasonable behavior correlated with the growth rate of per when the bet involves a maximal poten- capita consumption. Simon Benninga tial loss of around one percent of the and Aris Protopapadakis (1990) argue gambler’s wealth. that this might be responsible for the Because of the arguments offered by conflict between the model and the data. Kandel and Stambaugh (1991) and Ko- However, in Tables 2 and 3, the equity cherlakota (1990c), some economists premium and risk free rate puzzles are (see, among others, Craig Burnside still present when the representative in- 1994; Campbell and Cochrane 1995; vestor is faced with the real return to the Cecchetti and Mark 1990; Cecchetti, S & P 500 itself, and not some imaginary Lam, and Mark 1993; and Hansen, portfolio with a dividend that is perfectly Thomas Sargent, and Thomas Tallarini correlated with consumption. 1994) believe that there is no equity pre- Finally, we can see that the exact na- mium puzzle: individuals are more risk ture of the process generating consump- averse than we thought, and this high de- tion growth is irrelevant. For example, gree of risk aversion is reflected in the Cecchetti, Lam, and Mark (1993), and spread between stocks and bonds. How- Kandel and Stambaugh (1990) have both ever, it is clear to me from conversations proposed that allowing consumption and from knowledge of their work that a growth to follow a Markov switching pro- vast majority of economists believe that cess (as described by James Hamilton values for α above ten (or, for that mat- 1989) might explain the two puzzles. Ta- ter, above five) imply highly implausible bles 2 and 3, though, show that this con- behavior on the part of individuals. (For jecture is erroneous:8 the standard t-test example, Mehra and Prescott (1985, is asymptotically valid when consumption 1988) clearly chose an upper bound as growth follows any stationary and er- large as ten merely as a rhetorical flour- godic process, including one that is ish.) D. Lucas (1994, p. 335) claims that Markov switching.9 any proposed solution that “does not explain the premium for α ≤ 2.5 . . . is . . . 8 See Abel (1994) for another argument along likely to be widely viewed as a resolution these lines. 9 Some readers might worry that the justifica- that depends on a high degree of risk tion for Tables 2–3 is entirely asymptotic. I am aversion.” She is probably not exaggerat- sympathetic to this concern. I have run a Monte t Carlo experiment in which the pricing errors es ing the current state of professional b and et are assumed to be i.i.d. over time and have opinion. a joint distribution equal to their empirical joint Tables 2 and 3 show that any model distribution. In this setting, the Monte Carlo re- that leads to (2a′,b′) will generate the sults show that the asymptotic distribution of the t- statistic and its distribution with 100 observa- two puzzles. This helps make clear what tions are about the same. the puzzles are not about. For example, However, there is a limit to the generality of while Mehra and Prescott ignore sam- Tables 2 and 3. One could posit, as in Kocher- lakota (1994) and Thomas Rietz (1988), that there pling error in much of their discussion, is a low probability of consumption falling by an Tables 2 and 3 do not. This allays the amount that has never been seen in United States concerns expressed by Allan Gregory and data. With power utility, investors are very averse to these large falls in consumption, even if they Gregor Smith (1991) and Cecchetti, are highly unlikely. Hence, investors believe that Lam, and Mark (1993) that the puzzles stocks are riskier and bonds are more valuable Kocherlakota: The Equity Premium 53

Thus, Tables 2 and 3 show that there generalization of the “standard” prefer- are really only three crucial assumptions ence class (1). In these Generalized Ex- generating the two puzzles. First, asset pected Utility preferences, the period t markets are complete. Second, asset utility Ut received by a consumer from a markets are frictionless. (As discussed stream of consumption is described re- above, these two assumptions imply that cursively using the formula: there is a representative consumer.) Fi- = 1−ρ + β 1−α (1−ρ)/(1−α) 1/(1−ρ). nally, the representative consumer’s util- Ut {ct {EtUt+1 } } (3) ity function has the power form assumed by Mehra and Prescott (1985) and satis- Thus, utility today is a constant elasticity fies their parametric restrictions on the function of current consumption and fu- discount factor and coefficient of relative ture utility. Note that (1) can be obtained as a special case of these prefer- risk aversion. Any model that seeks to re- α ρ solve the two puzzles must weaken at ences by setting = . least one of these three assumptions. Epstein and Zin point out a crucial attribute of these preferences. In the III. Preference Modifications preferences (1), the coefficient of relative risk aversion α is constrained to be The paradigm of complete and fric- equal to the reciprocal of the elasticity tionless asset markets underlies some of of intertemporal substitution. Thus, the fundamental insights in both finance highly risk averse consumers must view and economics. (For example, the consumption in different time periods Modigliani-Miller Theorems and the ap- as being highly complementary. This is peal of profit maximization as an objec- not true of the GEU preferences (3). In tive for firms both depend on the com- this utility function, the degree of risk pleteness of asset markets.) Hence, it is aversion of the consumer is governed important to see whether it is possible to by α while the elasticity of inter- explain the two puzzles without aban- temporal substitution is governed by doning this useful framework. To do so, 1/ρ. Epstein and Zin (1991) argue that we must consider possible alterations in disentangling risk aversion and inter- the preferences of the representative in- temporal substitution in this fashion dividual; in this section, I consider three may help explain various aspects of as- different modifications to the prefer- set pricing behavior that appear anoma- ences (1): generalized expected utility, lous in the context of the preferences habit formation, and relative consump- (1). tion effects. To see the usefulness of GEU prefer- 1. Generalized Expected Utility ences in understanding the equity premium puzzle and the risk free rate puz- In their (1989) and (1991) articles,10 zle, suppose there is a representative Larry Epstein and Stanley Zin describe a investor with preferences given by (3) who can invest in stocks and bonds. than appears to be true when one looks at infor- Then, the investor’s optimal consump- mation in the 90–year United States sample; the t-statistics are biased upwards. While this explana- complementary analyses of the implications of this tion is consistent with the facts, it has the defect type of generalized expected utility for asset pric- of being largely nontestable using only returns and ing. consumption data. (See Mehra and Prescott, 1988, There are other ways to generalize expected for another critique of this argument.) utility so as to generate interesting implications for 10 Weil (1989), Kandel and Stambaugh (1991), asset prices—see Epstein and Tan Wang (1994) and Hansen, Sargent, and Tallarini (1994) provide and Epstein and Zin (1990) for examples. 54 Journal of Economic Literature, Vol. XXXIV (March 1996) tion profile must satisfy the two first or- The first order condition (4a′) tells us der conditions: that using GEU preferences will not α ρ−α( )−ρ( s − b ) = change Table 2: for each value of , the Et{Ut+1 Ct+1/Ct Rt+1 Rt+1 } 0 (4a) marginal utility gain of reducing bond −α (α−ρ) ( −α) holdings and investing in stocks is the βE {(E U1+ ) / 1 t t t 1 same whether the preferences lie in the ρ−α( )−ρ b = ( ) Ut+1 Ct+1/Ct Rt+1} 1. 4b expected utility class (1) or in the generalized expected utility class (3). In con- Note that the marginal rates of substitu- trast, Table 4 shows that it is possible to tion depend on a variable that is funda- resolve the risk free rate puzzle using mentally unobservable: the period (t + 1) the GEU preferences. In particular, for level of utility of the representative β = 0.99, and various values of α in the agent (and the period t expectations of set (0, 18], Table 4 presents the value of that utility). This unobservability makes ρ that exactly satisfies the first order it difficult to verify whether the data are condition (4b). consistent with these conditions. The intuition behind these results is To get around this problem, we can simple. The main benefit of modelling exploit the fact that it is difficult to pre- preferences using (3) instead of (1) is dict future consumption growth using that individual attitudes toward risk and currently available information (see Hall growth are no longer governed by the 1978). Hence, I assume in the discussion same parameter. However, the equity of GEU preferences that future con- premium puzzle arises only because of sumption growth is statistically indepen- economists’ prior beliefs about risk aver- dent of all information available to the sion; hence, this puzzle cannot be re- investor today.11 Under this restriction, solved by disentangling attitudes toward it is possible to show that utility in pe- risk and growth. On the other hand, the riod (t + 1) is a time and state invariant connection between risk aversion and in- multiple of consumption in period (t + tertemporal substitution in the standard 1). We can then apply the Law of Iter- preferences (1) is the essential element ated Expectations to (4a) and (4b) to arrive at: ( )−γρ( m )γ−1( s − b ) = Et{ Ct+1/Ct Rt+1 Rt+1 Rt+1 } 0 (∗) β ( )−α( s − b ) = ′ E{ Ct+1/Ct Rt+1 Rt+1 } 0 (4a ) βγ( )−γρ( m )γ−1( b ) + Et{ Ct+1/Ct Rt+1 Rt+1 } 1 (∗∗) −α (α−ρ) ( −α) γ − α − ρ ∗ ∗∗ m β[E(C + /C )1 ] / 1 where = (1 )/(1 ). In ( ) and ( ), Rt is t 1 t the gross real return to the representative inves- ( )−α b = ( ′) tor’s entire portfolio of assets (including human E{ Ct+ /Ct Rt+ } 1. 4b 1 1 capital, housing, etc.). This representation has the See Kocherlakota (1990b) for a precise desirable attribute of being valid regardless of the information set of the representative investor (so derivation of these first order condi- we don’t have to assume that consumption growth tions.12 from period t to period (t + 1) is independent of all information available to the investor in period 11 m This assumption of serial independence is t). Unfortunately, the variable Rt is not observ- also employed by Abel (1990), Epstein and Zin able. Epstein and Zin (1991) use the value- (1990), and Campbell and Cochrane (1995). Kan- weighted return to the NYSE as a proxy, but of del and Stambaugh (1991), Kocherlakota (1990b), course this understates the true level of diversifi- and Weil (1989) have shown that the thrust of the cation of the representative investor and (poten- discussion that follows is little affected by allowing tially) greatly overstates the covariability of her for more realistic amounts of dependence in con- marginal rate of substitution with the return to the sumption growth. stock market. As a result, using this proxy variable 12 Epstein and Zin (1991) derive a different rep- leads one to the spurious conclusion that GEU resentation for the first order conditions (4a) and preferences can resolve the equity premium puz- (4b) of the representative investor: zle with low levels of risk aversion. Kocherlakota: The Equity Premium 55

ences (1), this forces intertemporal TABLE 4 RESOLVING THE RISK FREE RATE PUZZLE substitution to be low. The GEU prefer- WITH GEU PREFERENCES ences can explain the risk free rate puzzle by allowing intertemporal substi- a 1/α 1/ρ tution and risk aversion to be high simul- 13 0.500 2.000 23.93 taneously. 1.000 1.000 14.91 1.500 0.6667 10.62 2. Habit Formation 2.000 0.5000 8.126 2.500 0.4000 6.493 3.000 0.3333 5.345 We have seen that the risk free rate 3.500 0.2857 4.494 puzzle is a consequence of the demand 4.000 0.2500 3.840 for savings being too low when individu-

4.500 0.2222 3.322 als are highly risk averse and have pref- 5.000 0.2000 2.903 5.500 0.1818 2.557 erences of the form (1). The GEU pref- 6.000 0.1667 2.267 erence class represents one way to 6.500 0.1538 2.022 generate high savings demand when indi- 7.000 0.1429 1.811 viduals are highly risk averse. There is

7.500 0.1333 1.629 another, perhaps more intuitive, ap- 8.000 0.1250 1.469 8.500 0.1176 1.330 proach. The standard preferences (1) as- 9.000 0.1111 1.206 sume that the level of consumption in 9.500 0.1053 1.096 period (t − 1) does not affect the mar- 10.00 0.1000 0.9974 ginal utility of consumption in period t. 10.50 0.09524 0.9091 It may be more natural to think that an 11.00 0.09091 0.8295 individual who consumes a lot in period 11.50 0.08696 0.7574 − 12.00 0.08333 0.6920 (t 1) will get used to that high level of 12.50 0.08000 0.6325 consumption, and will more strongly 13.00 0.07692 0.5781 desire consumption in period t; mathe- 13.50 0.07407 0.5283 matically, the individual’s marginal util- 14.00 0.07143 0.4826 ity of consumption in period t is an 14.50 0.06897 0.4406 − 15.00 0.06667 0.4018 increasing function of period (t 1) con- 15.50 0.06452 0.3660 sumption. 16.00 0.06250 0.3329 This property of intertemporal prefer- 16.50 0.06061 0.3022 ences is termed habit formation. The ba- 17.00 0.05882 0.2737 17.50 0.05714 0.2472 sic implications of habit formation for 18.00 0.05556 0.2226 asset returns (as explained by Constan- 13 Weil (1989) emphasizes that GEU prefer- In this table, the first column is the coefficient of ences cannot explain the risk free rate puzzle if relative risk aversion α, the second column is the one forces the coefficient of relative risk aversion corresponding elasticity of intertemporal substitution if to be “reasonable” (say, approximately 1). He preferences are of the form (1), and the third column is shows that under this restriction, then the elastic- the value of 1/ρ that satisfies the risk free rate first ity of intertemporal substitution must be very order condition (4b′) given that preferences are of the large to be consistent with the risk free rate. (See form (3), β = 0.99 and α has the value specified in the Table 4 for a confirmation.) This large elasticity of intertemporal substitution appears to contradict first column. the results of Hall (1988), who shows that the esti- mated slope coefficient is very small in a regression of consumption growth on expected real in- of the risk free rate puzzle. To match terest rates (although Weil, 1990, points out that it is difficult to see a direct linkage between Hall’s the equity premium, risk aversion has to regression coefficients and the parameters that de- be high. But in the standard prefer- scribe the GEU preferences). 56 Journal of Economic Literature, Vol. XXXIV (March 1996) tinides 1990; and Heaton 1995) can be erences, this means that the formula for captured in the following simple model. the marginal utility depends on the indi- Suppose that there is a representative vidual’s (possibly unobservable) informa- agent with preferences in period t repre- tion. As in that context, it is convenient sented by the utility function: to assume that consumption growth from period t to period (t + 1) is unpre- Σ∞ βs( − λ )1 − α ( − α) λ > Et s=0 ct+s ct+s−1 / 1 , 0. (5) dictable. Then the ratio of marginal utili- Thus, the agent’s momentary utility is a ties is given by: decreasing function of last period’s con- = (ϕ ϕ − λϕ )−α MUt+1/MUt { t+1 t t sumption: it takes more consumption to- λβ(ϕ ϕ )−α (ϕ − λ)−α day to make him happy if he consumed - t+1 t E }/ more yesterday. (ϕ − λ)−α − λβ(ϕ )−α (ϕ − λ)−α The individual’s marginal utility with { t t E } (8) respect to period t consumption is given ϕ ≡ by the formula: where t Ct/Ct−1. This allows us to eas- ily estimate MUt+1/MUt by using the sam- = ( − λ )−α − βλ ( − λ )−α MUt ct ct−1 Et ct+1 ct . (6) ple mean to form an estimate of the unconditional expectations in the formula Thus, the marginal utility of period t (8). consumption is an increasing function of With the preference class (1), it is not − period (t 1) consumption. Note that possible to simultaneously satisfy equa- there are two pieces to this formula for tions (2a′) and (2b′) without driving β marginal utility. The first term captures above one. This is not true of prefer- the fact that if I buy a BMW rather than ences that exhibit habit formation. For a Yugo today, then I am certainly better example, suppose we set β = 0.99. Un- off today; however, the second piece like the preference class (1), it is then models the notion that having the BMW possible to find preference parameters to “spoils” me and reduces my utility from satisfy both (7a′) and (7b′). In particular, all future car purchases. if we set α = 15.384 and λ = 0.174, then The individual’s optimal consumption the sample analogs of both first order portfolio must satisfy the two first order conditions are exactly satisfied in the conditions that make him indifferent to data.14 buying or selling more stocks or bonds: 14 Instead of (7a′,b′), Wayne Ferson and Con- β ( )( s − b ) = Et{ MUt+1/MUt Rt+1 Rt+1 } 0 (7a) stantinides (1991) work with an alternative implication of (7a,b): β ( ) b ) = −α Et{ MUt+1/MUt Rt+1 } 1 (7b) E{MUtZt/(Ct−λCt−1) } = β ( −λ )−α (∗) where MUt is defined as in (6). We can E{MUt+1Rt+1Zt/ Ct Ct−1 } apply the Law of Iterated Expectations where Rt+1 is the gross return to an arbitrary asset, to obtain: and Zt is an arbitrary variable observable to the econometrician and to the agent at time t. This β ( )( s − b ) = condition has the desirable property that it does E{ MUt+1/MUt Rt+1 Rt+1 } 0 (7a′) not involve terms that are unobservable to the econometrician. However, (*) is a different type of β ( ) b ) = E{ MUt+1/MUt Rt+1 } 1. (7b′) implication of optimal behavior from (4a,b) or (2a,b) because it is not just an averaged version of The problem with trying to satisfy (7a,b) (even when Zt is set equal to 1). In order to ′) ′ be consistent across preference orderings, I work (7a and (7b ) is that MUt depends on with (7a,b). the investor’s ability to predict future In the context of a calibrated general equilib- consumption growth. As with GEU pref- rium model, Constantinides (1990) generates a Kocherlakota: The Equity Premium 57

There are two implications of this given amount of consumption depends result. First, habit formation does not on per capita consumption.15 resolve the equity premium puzzle: it I will work with a model that combines is still true that the representative in- features of Abel’s and Gali’s formula- vestor is indifferent between stocks and tions. Suppose the representative indi- bonds only if she is highly averse to vidual has preferences in period t: consumption risk. On the other hand, Σ∞ βs 1−α λ λ ( −α) α ≥ habit formation does help resolve the Et s=0 ct+s Ct+s−1Ct+s−1/ 1 , 0 (9) risk free rate puzzle—it is possible to where ct+s is the individual’s level of con- ′ ′ β + satisfy (7a ) and (7b ) without setting sumption in period (t s) while Ct+s is larger than one. The intuition behind the level of per capita consumption in this finding is simple. For any given level period (t + s) (in equilibrium, of course, of current consumption, the individual the two are the same). Thus, the individ- knows that his desire for consumption ual derives utility from how well she is will be higher in the future because con- doing today relative to how well the av- sumption is habit forming; hence, the in- erage person is doing today and how well dividual’s demand for savings increases the average person did last period. If the relative to the preference specification individual is a jealous sort, then it is (1). natural to think of γ and λ as being nega- 3. Relative Consumption: “Keeping tive: the individual is unhappy when oth- up with the Joneses” ers are doing well. If the individual is patriotic, then it is natural to think of γ The standard preferences (1) assume and λ as being positive: the individual is that individuals derive utility only from happy when per capita consumption for their own consumption. Suppose, the nation is high. though, that as James Duesenberry The representative individual treats (1949) posits, an individual’s utility is a per capita consumption as exogenous function not just of his own consumption when choosing how much to invest in but of societal levels of consumption. stocks and bonds. Hence, in equilibrium, Then, his investment decisions will be the individual’s first order conditions affected not just by his attitudes toward take the form: his own consumption risk, but also by his ( )γ−α( )λ( s attitudes toward variability in societal Et{ Ct+1/Ct Ct/Ct−1 Rt+1 consumption. Following up on this intui- b − R + )} = 0 (10a) tion, Abel (1990) and Jordi Gali (1994) t 1 examine the asset pricing implications of γ−α λ b βE {(C + /C ) (C /C − ) R + } = 1. various classes of preferences in which t t 1 t t t 1 t 1 (10b) the utility an individual derives from a We can apply the law of iterated expectations to derive the two conditions: large equity premium and a low risk free rate by γ−α λ E{(C + /C ) (C /C − ) (Rs+ assuming that the representative agent’s utility t 1 t t t 1 t 1 λ function has a large value of and a low value of − Rb+ )} = 0 (10a′) α. However, it is important to keep in mind that t 1 when λ is large, the individual requires a large β ( )γ−α( )λ b = level of consumption in order to just survive; he E{ Ct+1/Ct Ct/Ct−1 Rt+1} 1. (10b′) will pay a lot to avoid small consumption gambles even if α is low. Thus, Constantinides’ proposed 15 See Campbell and Cochrane (1995) and resolution of the puzzles requires individuals to be James Nason (1988) for models which feature both very averse to consumption risk (although, as he relative consumption effects and time-varying risk shows, not to wealth risk). aversion. 58 Journal of Economic Literature, Vol. XXXIV (March 1996)

For any specification of the discount sury bill return, individuals who see con- factor β and the coefficient of relative sumption in different periods as so com- risk aversion α, it is possible to find set- plementary will be happy with the steep tings for γ and λ such that the puzzles consumption profile seen in United are resolved in that the sample analogs States data only if their “discount” factor of (10a′) and (10b′) are satisfied. For ex- is above one. ample, suppose β = 0.99. Then, if (α − γ) Several researchers have explored the = 19.280, and λ = 3.813, the sample ana- consequences of using broader classes of logs of (10a′) and (10b′) are satisfied ex- preferences. Broadly speaking, there are actly. two lessons from this type of sensitivity This model offers the following expla- analysis. First, the risk free rate puzzle nation for the seemingly large equity can be resolved as long as the link be- premium. When γ is large in absolute tween individual attitudes toward risk value, an individual’s marginal utility of and growth contained in the standard own consumption is highly sensitive to preferences (1) is broken. Second, the fluctuations in per capita consumption equity premium puzzle is much more ro- and therefore strongly negatively related bust: individuals must either be highly to stock returns. Thus, even if α is small averse to their own consumption risk or so that the “representative” investor is to per capita consumption risk if they are not all that averse to individual consump- to be marginally indifferent between in- tion risk, she does not find stocks attrac- vesting in stocks or bonds. tive because she is highly averse to per It seems that any resolution to the eq- capita consumption risk. uity premium puzzle in the context of a However, the presence of contempora- representative agent model will have to neous per capita consumption in the rep- assume that the agent is highly averse to resentative investor’s utility function consumption risk. Per capita consump- does not help explain the risk free rate tion is very smooth, and therefore does puzzle: if λ is set equal to zero, (10a′) not covary greatly with stock returns. Yet and (10b′) become equivalent to (2a′) people continue to demand a high ex- and (2b′). We know from our analysis of pected return for stocks relative to those equations that if λ = 0, it is neces- bonds. The only possible conclusion is sary to drive β larger than one in order that individuals are extremely averse to to satisfy (10b′) given that (10a′) is satis- any marginal variation in consumption fied. The positive effect of lagged per (either their own or societal). capita consumption on current marginal utility increases the individual’s demand IV. Incomplete Markets and Trading Costs for savings and allows (10a′) to be satis- In this section, I examine the ability of fied with a relatively low value of β. models with incomplete markets and 4. Summary various sorts of trading frictions to explain the asset returns data.16 Unlike the Mehra and Prescott (1985) require the preferences of the representative indi- 16 Throughout my discussion, I ignore the exist- vidual to lie in the “standard” class (1). ence of fiat money. Pamela Labadie (1989) and Alberto Giovannini and Labadie (1991) show that We have seen that these preferences can motivating a demand for money via a cash-in- be made consistent with the large equity advance constraint does not significantly alter the premium only if the coefficient of rela- asset pricing implications of the Mehra-Prescott (1985) model. However, a richer model of money tive risk aversion is pushed near 20. demand might have more dramatic effects on asset Given the low level of the average Trea- prices. Kocherlakota: The Equity Premium 59 previous section, which focused solely on more variable than per capita consump- the decision problem of a “representation growth. As a result, models with in- tive” agent, the analysis will (necessarily) complete markets offer the hope that be fully general equilibrium in nature. while the covariance of per capita con- Because of the complexity of these dy- sumption growth with stock returns is namic general equilibrium models, their small, individual consumption growth implications are generated numerically; I may covary enough with stock returns to find that this makes any intuition more explain the equity premium. speculative in nature. Dynamic Self-insurance and the Risk Throughout this section, all investors Free Rate. The intuitive appeal of in- are assumed to have identical prefer- complete markets for explaining asset re- ences of the form (1). The “game” in this turns data is supported by the work of literature is to explain the existence of Weil (1992). He studies a two period the puzzles solely through the presence model in which financial markets are in- of incompleteness and trading costs that complete. He shows that if individual Mehra and Prescott assume away (just as preferences exhibit prudence (that is, in Section III, the “game” was to explain convex marginal utility—see Miles Kim- the puzzle without allowing such fric- ball 1990), the extra variability in indi- tions). vidual consumption growth induced by the absence of markets helps resolve the 1. Incomplete Markets risk free rate puzzle. Without complete Underlying the Mehra-Prescott model markets, individuals must save more in is the presumption that the behavior of order to self-insure against the random- per capita consumption growth is a good ness in their consumption streams; the guide to the behavior of individual con- extra demand for savings drives down the sumption growth. We have seen that this risk free rate. Weil also shows that if in- belief is warranted if asset markets are dividuals exhibit not just prudence but complete so that individuals can write decreasing absolute prudence (see Kim- contracts against any contingency: indi- ball 1990), the additional variability in viduals will use the financial markets to consumption growth induced by market diversify away any idiosyncratic differ- incompleteness also helps to explain the ences in their consumption streams. As a equity premium puzzle. The extra riski- result, their consumption streams will ness in individual consumption makes look similar to each other and to per cap- stocks seem less attractive to an individ- ita consumption. ual investor than they would to a “repre- However, in reality, it does not appear sentative” consumer.17 easy for individuals to directly insure Unfortunately, two period models themselves against all possible fluctua- cannot tell the full story because they tions in their consumption streams. (For abstract from the use of dynamic trad- example, it is hard to directly insure one- ing as a form of insurance against risk. self against fluctuations in labor income.) For example, suppose my salary next For this reason, most economists believe year is equally likely to be $40,000 or that insurance markets are incomplete. Intuitively, in the absence of these kinds 17 Mankiw (1986) shows that if the conditional of markets, individual consumption variability of individual income shocks is higher growth will feature risk not present in when the aggregate state of the economy is lower, then prudence alone is sufficient to generate a per capita consumption growth and so higher equity premium in the incomplete markets individual consumption growth will be environment. 60 Journal of Economic Literature, Vol. XXXIV (March 1996)

$50,000. If I die at the end of the year zero if there is to be trade in equilib- and do not care about my descendants, rium.18 then the variability in income has to be As in the two period context, an indi- fully reflected in my expenditure pat- vidual in this economy faces the possibil- terns. This is the behavior that is cap- ity of an uninsurable consumption fall. tured by two period models. On the The desire to guard against this possibil- other hand, if I know I will live for many ity (at least on the margin) generates a more years, then I need not absorb my greater demand to transfer resources income risk fully into current consump- from the present to the future than in a tion—I can partially offset it by reducing complete markets environment. This ex- my savings when my income is high and tra demand for saving forces the equilib- increasing my savings when my income is rium interest rate below the complete low. markets interest rate.19 To understand the quantitative impli- However, the ability to dynamically cations of dynamic self-insurance for the self-insure in the infinite horizon setting risk free rate, it is best to consider a sim- means that the probability of an uninsur- ple model of asset trade in which there able consumption fall is much smaller are a large number (in fact, a continuum) than in the two period context. In equi- of ex ante identical infinitely lived con- librium, the “typical” individual has no sumers. Each of the consumers has a savings (because net asset holdings are random labor income and their labor in- zero), but does have a “line of credit” comes are independent of one another ymin /r. The individual can buffer any (so there is no variability in per capita short-lived falls in consumption by bor- income). The consumers cannot directly rowing; his line of credit can be ex- write insurance contracts against the hausted only by a relatively unlikely long variability in their individual income “run” of bad income realizations. streams. In this sense, financial markets (Equivalently, in a stationary equilib- are incomplete. However, the individuals rium, few individuals are near their are able to make risk free loans to one credit limit at any given point in time.) another, although they cannot borrow more than B in any period. 18 Christopher Carroll (1992) argues using data from the Panel Study in Income Dynamics that For now, I want to separate the ef- ymin equals zero. In this case, no individual can fects of binding borrowing constraints ever borrow (because their debt ceiling equals from the effects of incomplete markets. zero). There can be trade of risk free bonds only if there is an outside supply of them (say, by the To eliminate the possibility that the government). Then, the risk free rate puzzle be- debt ceiling B is ever a binding con- comes an issue of how low the amount of outside straint, I assume throughout the follow- debt must be in order to generate a plausibly low value for the risk free rate. There is no clean an- ing discussion that B is larger than swer to this question in the existing literature. ymin/r, where ymin is the lowest possible 19 Marilda Sotomayor (1984) proves that the in- realization of income. No individual will dividual demand for asset holdings and for consumption grows without bound over time if the borrow more than ymin/r; doing so would interest rate equals the rate of time preference— mean that with a sufficiently long run of the complete markets interest rate. (Her result is bad income realizations, the individual valid without the assumption of convex marginal utility as long as the consumption set is bounded would be forced into default, which is as- from below.) This implies that the equilibrium in- sumed to be infinitely costly (see Aiya- terest rate must fall below the rate of time prefer- gari 1994). Market clearing dictates that ence in order to deter individuals from saving so much. See Zeldes (1989b) and Angus Deaton for every lender, there must be a bor- (1992) for numerical work along these lines in a rower; hence, ymin must be larger than finite horizon setting. Kocherlakota: The Equity Premium 61

Thus, the extra demand for savings gen- different when shocks to individual labor erated by the absence of insurance mar- income are permanent instead of transi- kets will generally be smaller in the infi- tory. Under the assumption of perma- nite horizon economy than in a two nence, dynamic self-insurance can play period model. no role: income shocks must be fully ab- This reasoning suggests that in the in- sorbed into consumption. For example, finite horizon setting, the absence of in- if my salary falls by $10,000 perma- come insurance markets may have little nently, then the only way to keep my impact on the interest rate. The numeri- consumption smooth over time is reduce cal work of Huggett (1993) and Heaton it by the same $10,000 in every period: I and D. Lucas (1995b) confirms this in- cannot just temporarily run down my tuition. They examine infinite horizon savings account to smooth consumption. economies which are calibrated to ac- Because income shocks must be fully ab- cord with aspects of United States data sorbed into consumption, the results on individual labor income and find that from the two period model become the difference between the incomplete much more relevant. Indeed, Constan- markets interest rate and the complete tinides and Duffie (1995) show that the markets interest rate is small.20 absence of labor income insurance mar- The above discussion assumes that in- kets, combined with the permanence of dividual income shocks die out eventu- labor income shocks, has the potential to ally (that is, the shocks are stationary). generate a risk free rate that may be Constantinides and Darrell Duffie much lower than the complete markets (1995) point out that the story is very risk free rate. The issue, then, becomes an empirical 20 In fact, they obtain these results in econo- one: how persistent (and variable) are mies in which some fraction of the agents face binding borrowing constraints in equilibrium. Re- otherwise undiversifiable shocks to indi- laxing these borrowing constraints or adding out- vidual income? This is a difficult matter side debt will serve to further increase the equilib- to sort out because of the paucity of time rium interest rate toward the complete markets rate (Aiyagari 1994). series evidence available. However, Hea- There is a caveat associated with the results of ton and D. Lucas (1995a) use data from Huggett (1993) and Heaton and D. Lucas the Panel Study of Income Dynamics (1995b)—and with most of the other numerical work described later in this paper. For computa- and estimate the autocorrelation of idio- tional reasons, these papers assume that the pro- syncratic21 labor income shocks to be cess generating individual income is discrete. This around 0.5. When they examine the pre- discretization may impose an artificially high lower bound on income—that is, an artificially high dictions of the above model using this value of ymin. Thus, Heaton and D. Lucas (forth- autocorrelation for labor income (or coming) assume that income cannot fall below 75 even an autocorrelation as high as 0.8), percent of its mean value. Huggett (1993) is more conservative in assuming that income cannot fall they find that the equilibrium interest below about ten percent of its mean value. In both cases, finer and more accurate discretizations may 21 What matters is the persistence of the undi- drive ymin lower. versifiable component of labor income. Heaton This technical detail may have important conse- and Lucas (1995a) implicitly assume that individu- quences for the numerical results because no indi- als can hedge any labor income risk that is corre- vidual can ever borrow more than ymin/r. As ymin lated with aggregate consumption. Hence, they falls, there is an increased probability of a fall in obtain their autocorrelation estimate of 0.5 only consumption that cannot be “buffered” by using after first controlling for aggregate effects upon accumulated savings or by borrowing, and the individual labor income (this may explain why downward pressure on the equilibrium interest their estimate of persistence is so much lower than rate rises. In fact, as ymin goes to zero, the equilib- that of Glenn Hubbard, Jonathan Skinner, and rium interest rate must go to zero (if there is no Zeldes, 1994, who control for aggregate effects outside debt). only through the use of year dummies). 62 Journal of Economic Literature, Vol. XXXIV (March 1996) rate is close to the complete markets sure against all income shocks, then his rate. consumption would not change at all in Dynamic Self-insurance and the Eq- response to the surprise movement in his uity Premium Puzzle. The above analysis income process. Thus, his ability to self- shows that as long as most individuals insure against income shocks using only are sufficiently far from the debt ceiling savings can be measured by how close + − ρ ρ ymin/r, dynamic self-insurance implies r/(1 r ) is to zero. For example, if that the equilibrium interest rate in in- = 1, and all shocks are permanent, then complete markets models is well-ap- r/(1 + r − ρ) is equal to one; the income proximated by the complete markets in- shock is fully absorbed into consump- terest rate. In this section, I claim that a tion, which is very different from the im- similar conclusion can be reached about plications of the full insurance model. In the equity premium as long as the inter- contrast, if ρ < 1, then as r nears zero, est rate and the persistence of shocks are the implications of the full insurance both sufficiently low. model and the incomplete markets mod- The following argument is a crude in- els for movements in individual con- tuitive justification for this claim, based sumption become about the same. In the largely on the Permanent Income Hy- United States annual data, we have seen pothesis. Consider an infinitely lived in- that r is around one percent; hence, even dividual who faces a constant interest if ρ is as high as 0.75, consumption is rate r and who faces income shocks that fairly insensitive to surprise movements have a positive autocorrelation equal to in labor income. (Recall that Heaton and ρ; if ρ = 1, the shocks are permanent D. Lucas, 1995a, estimate the autocorre- while if ρ < 1, their effect eventually dies lation of undiversifiable income shocks out. Financial markets are incomplete in to be 0.5.) the sense that the individual cannot ex- This intuitive argument is borne out plicitly insure against the income shocks. by the numerical work of Christopher Because of concave utility, the individual Telmer (1993) and D. Lucas (1994; see wants to smooth consumption as much as also Heaton and D. Lucas 1995a, 1995b; possible. Wouter den Haan 1994; and Albert Mar- Suppose the individual’s income is sur- cet and Singleton 1991). These papers prisingly higher in some period by δ dol- examine the quantitative predictions of lars. Because of the autocorrelation in dynamic incomplete markets models, the income shocks, the present value of calibrated using individual income data, his income rises by: for the equity premium. They find that dynamic self-insurance allows individuals δ + ρδ/(1 + r) + ρ2δ /(1 + r)2 + . . . . to closely approximate the allocations in = δ/(1 − ρ/(1 + r)) a complete markets environment. The = δ(1 + r)/(1 + r − ρ). equilibrium asset prices are therefore similar to the predictions of the Mehra- The individual wants to smooth con- Prescott (1985) model. sumption. Hence, he increases consump- A critical assumption underlying this tion in every period of life by the annu- numerical work is that income shocks itized value of the increase in his wealth. have an autocorrelation less than one. A simple present value calculation im- Constantinides and Duffie (1995) show δ plies that his consumption rises by r/(1 that if labor income shocks are instead + r − ρ) dollars. permanent, then it is possible for incom- If the individual were able to fully in- plete markets models to explain the large Kocherlakota: The Equity Premium 63 size of the equity premium. As in the books. In this model, the individual’s case of the risk free rate puzzle, the budget set of consumption choices (c1, ρ failure of dynamic self-insurance when c2) is written as: = 1 means that the model’s implications ≤ + resemble those of a two period frame- c1 y1 b1 work. c2 + b1(1 + r) ≤ y2 To sum up: the assumption that finan- c1 ≥ 0, c2 ≥ 0 cial markets are complete strikes many economists as prima facie ridiculous, and where b1 is the level of borrowing in pe- it is tempting to conclude that it is re- riod 1. The budget set allows the individ- sponsible for the failure of the Mehra- ual to borrow up to the present value of Prescott model to explain the United his period two income. States data on average stock and bond Many economists believe that this returns. However, as long as investors model of borrowing and lending ignores can costlessly trade any financial asset an important feature of reality: because (for example, risk free loans) over time, of enforcement and adverse selection they can use the accumulated stock of problems, individuals are generally not the asset to self-insure against idiosyn- able to fully capitalize their future labor cratic risk (for example, shocks to indi- income. One way to capture this view of vidual income). The above discussion the world is to add a constraint to the ≤ shows that given this ability to self-in- individual’s decision problem, b1 B, sure, the behavior of the risk free rate where B is the limit on how much the and the equity premium are largely unaf- individual can borrow. If B is less than + fected by the absence of markets as long y2/(1 r), the constraint on b1 may be as idiosyncratic shocks are not highly binding because the individual may not persistent. be able to transfer as much income from period two to period one as he desires. 2. Trading Costs ≤ The restriction b1 B is called a bor- In the incomplete and complete mar- rowing constraint. (Short sales con- kets models described above, it is as- straints are similar restrictions on the sumed that individuals can costlessly trade of stocks.) The law of supply and trade any amount of the available securi- demand immediately implies that the ties. This assumption is unrealistic in equilibrium interest rate is generally ways that might matter for the two puz- lower if many individuals face a binding zles. For example, in order to take ad- borrowing constraint than if few indi- vantage of the possible utility gains asso- viduals do. Intuitively, “tighter” borrow- ciated with the large premium offered by ing constraints exogenously reduce the equity, individuals have to reduce their size of the borrowing side of the market; holdings of Treasury bills and buy more in order to clear markets, interest rates stocks. The costs of making these trades must fall in order to shrink the size of may wipe out their apparent utility bene- the lending side of the market. fits. In this subsection, I consider the im- In keeping with this intuition, the plications for asset returns of various work of Huggett (1993) and Heaton and types of trading costs. D. Lucas (1995a, 1995b) demonstrates Borrowing and Short Sales Con- numerically that if a sizeable fraction of straints. Consider the familiar two pe- individuals face borrowing constraints, riod model of borrowing and lending in then the risk free rate may be substan- most undergraduate microeconomic text- tially lower than the predictions of the 64 Journal of Economic Literature, Vol. XXXIV (March 1996) τ representative agent model. Using data of stocks must incur a cost p , in addi- from the Panel Study on Income Dynam- tion to the price p, to buy a share. ics, Zeldes (1989a) documents that the In equilibrium, the cost of buying a large number of individuals with low lev- share of stock cannot be smaller than the els of asset holdings appear to face such present value of the benefits of owning constraints. that share for N periods (or there would However, Heaton and D. Lucas be excess demand for stocks because (1995a, 1995b) show that borrowing and everyone would want to buy shares). short sales constraints do not appear to This statement can be expressed mathe- have much impact on the size of the matically as: 2 equity premium. There is a sound intui- p(1 + τ) ≥ d/(1 + rb) + d/(1 + rb) tion behind these results: an individual + + ( + )N + ( + )N who is constrained in the stock market . . . d/ 1 rb p/ 1 rb . generally must also be constrained in the This present value expression can be re- bond market (or vice versa). Otherwise, written as: the individual could loosen the con- r ≡ d/p ≤ (1 + τ)r /{1 − (1 + r )−N}. straint in the bond market by shifting re- s b b sources to it from the stock market. Note that the right hand side gets Hence, just as the average risk free rate smaller as N grows large. Intuitively, as must fall to clear the bond market when the investor holds the stock for a long many individuals are constrained, so period of time, he is able to more fully must the average stock return fall in or- amortize the initial trading cost; the indi- der to clear the constrained stock mar- vidual is therefore only prevented from ket.22 making arbitrage profits if stock returns Transaction Costs. Individuals who try are extremely close to those of bonds. to engage in asset trade quickly find that Indeed, when N equals infinity, the up- they face all sorts of transaction costs. per bound on the premium paid by (These include informational costs, bro- stocks is very tight: kerage fees, load fees, the bid-ask r − r ≤ τr . spread, and taxes.) To understand how s b b transaction costs affect the pricing of se- Thus, if rb = 1 percent, and investors can curities, consider an investor in a risk buy stocks and hold them forever, then free world who can invest in two differ- rs can be as large as seven percent in ent securities, stocks and bonds. Stocks equilibrium only if τ is greater than 600 pay a constant dividend equal to d, and percent! have a constant price equal to p. Because In the above risk free model, an inves- stocks are a perpetuity, their rate of re- tor could buy and hold a stock for an in- turn rs, ignoring transaction costs, equals finite number of periods. However, if d/p. Bonds are short-lived and pay a con- the investor instead faces income risk stant interest rate rb. There is no cost which cannot be directly insured, she is associated with trading bonds. However, not able to plan on being able to hold stocks are costly to trade in that a buyer stocks forever. The reason is simple: if she is ever pushed up against a debt ceil- 22 Following Hua He and David Modest (1995), ing, so she cannot borrow to fully offset Cochrane and Hansen (1992) show that this argu- bad income shocks, she will have to sell ment is exactly correct when individuals face a the stocks in order to smooth her con- market wealth constraint that restricts the total value of their asset portfolio not to fall below some sumption. This cap on the investor’s exogenously specified (negative) number. “holding period” generated by potential Kocherlakota: The Equity Premium 65 runs of bad luck raises the possibility it treats per capita consumption growth that in models with missing income in- as a good proxy for individual consump- surance markets, transaction costs could tion growth. Yet, Mankiw and Zeldes have big effects on asset return spreads. (1991) and Michael Haliassos and Carol However, from our discussion of cali- Bertaut (forthcoming) document that brated incomplete markets models, we only about 30 percent of individuals in know that an infinitely lived individual the United States own stocks (either di- can do a pretty good job of smoothing rectly or through defined contribution consumption by simply buying and sell- pension funds). This suggests the possi- ing the asset that is cheaper to trade. bility of market segmentation: that for Hence, the agent faces only a low prob- whatever reason, only a subset of inves- ability of needing to sell stocks in order tors are actively involved in asset trade. to smooth consumption. She can conse- Campbell (1993) points out that it is the quently contemplate making arbitrage per capita consumption growth of the ac- profits by buying and holding stocks for tive traders that should be substituted a long period of time; as we have seen, into first order conditions like (2a) and this means that stocks cannot pay a (2b), not overall per capita consumption much higher return than bonds in equi- growth as Mehra and Prescott (1985) librium. use. Thus, as Aiyagari and Mark Gertler However, Mankiw and Zeldes (1991) (1991) and Heaton and D. Lucas (1995a) provide direct evidence that market seg- find, the only way to explain the equity mentation alone cannot explain the eq- premium using transaction costs is to as- uity premium. They find that the con- sert that there are significant differences sumption growth of stockholders does in trading costs across the stock and covary considerably more with stock re- bond markets. In my view, there is little turns than the consumption growth of evidence to support this proposition at nonstockholders. However, the covari- present, although both Aiyagari and ance is still not large enough: it is only if Gertler (1991) and Heaton and D. Lucas they are highly risk averse that stock- (1995b) make some attempts in this di- holders are marginally indifferent be- rection. To be considered as the leading tween stocks and bonds. There is still an explanation of the puzzle, more needs to equity premium puzzle when we look at be done to document the sizes and the consumption of those involved in as- sources of trading costs (especially for set trade.24 pension funds and other institutional investors that operate on behalf of stockholders).23 24 Ingram (1990) examines the general equilibrium implications of a model in which half the Market Segmentation. A key aspect of agents use all possible information to form their the Mehra-Prescott (1985) model is that portfolios, while the other half hold an optimal “nonchurning” portfolio: they are constrained not to change the split of assets between stocks and 23 In their partial equilibrium settings, Erzo bonds in response to new information. She finds Luttmer (1995) and He and Modest (1995) assume that the equity premium is no higher than if all that all investors fully turn over their portfolios agents were engaged in asset trade, but the risk relatively frequently (once a month or once a free rate is significantly lower. Presumably, the quarter) and all at the same time. This assumption “rule of thumb” traders hold too many bonds in allows them to explain the equity premium with their portfolio because they want to be sure that relatively small transaction costs; however, it they are insured against “runs” of bad income re- seems difficult to build a general equilibrium alizations. This drives the equilibrium interest rate model in which investors would choose to incur downwards, but has little effect on the equity pre- trading costs so frequently and so simultaneously. mium. 66 Journal of Economic Literature, Vol. XXXIV (March 1996)

3. Summary theoretical point of view (and possibly an empirical one as well), the models will The Mehra-Prescott (1985) model as- be stronger if they explicitly take into ac- sumes that asset trade is costless and count the informational problems that that asset markets are complete. Casual lead to the trading frictions. observation suggests that both assumptions are unduly strong. Unfortunately, V. Discussion and Conclusions the predictions of the model for the equity premium and the risk free rate are This paper began by posing two puz- not greatly affected by allowing markets zles: how can we explain why average to be incomplete: dynamic self-insurance stock returns are so much higher than allows individuals to smooth idiosyn- bond returns in the United States data, cratic shocks without using explicit insur- and why has per capita consumption ance contracts. It is true (by the law of grown so quickly given that bond returns supply and demand) that a frequently are so low? As it turns out, there are a binding borrowing constraint forces the variety of ways to generate more savings average risk free rate down toward its demand than in the Mehra-Prescott sample value. The only way to simultane- model and thereby explain the latter ously generate a sizeable equity pre- phenomenon. mium, though, is to make trading in the However, the equity premium puzzle stock market substantially more costly is much more challenging. Throughout than in the bond market. Under this as- this article, we have seen only two ways sumption, the premium on stocks repre- to explain the wedge in average returns sents compensation not for risk (as in the between stocks and bonds. The first is standard financial theory), but rather for that there is a large differential in the bearing additional transactions costs. cost of trading between the stock and The sources of these additional costs re- bond markets. To make this explanation main unclear. compelling, it is important to ascertain Despite its general lack of success in the size of actual trading costs in these explaining the equity premium, this lit- markets, and to provide an explanation erature has made an important contribu- of why those costs exist. Heaton and D. tion to our understanding of trade in Lucas (1995b) and Aiyagari and Gertler markets with frictions. Just as people in (1991) discuss some early steps in this real life are able to find ways around direction. government regulation of the economy, The second explanation is that, con- rational individuals in models are able to trary to Mehra and Prescott’s (1985) find ways around barriers to trade. It original parametric restrictions, individ- takes an enormous amount of sand in the ual investors have coefficients of relative gears to disrupt the ability of individuals risk aversion larger than ten (either with to approximate an efficient allocation of respect to their own consumption or resources. with respect to per capita consump- There is, however, a continuing weak- tion).25 As I explained earlier, the prob- ness in the “market frictions” literature. lem with this explanation is that only a Usually, the incompleteness of markets 25 Of course, as mentioned above, to resolve the and the costliness of trade in these mod- risk free rate puzzle, we cannot only adopt a more els is motivated by adverse selection or flexible notion of what constitutes reasonable risk aversion; we must abandon the standard prefer- moral hazard considerations. Yet, this ences (1) or impose borrowing constraints that motivation remains informal. From a bind frequently. Kocherlakota: The Equity Premium 67 handful of economists believe that indi- ture the essential technological forces viduals are that risk averse. One way to (for example, lack of communication) support the “high risk aversion” view is that disrupt the process of fully central- to demonstrate that this apparently ized exchange and thereby generate a “strange” assumption about human be- demand for money. This same complaint havior is consistent with data other than of superficiality can also be made of the the average realization of the equity pre- two types of stories explaining the equity mium. Until now, little has been done premium. along these lines, but Tallarini’s (1994) In one sense, the accuracy of the analysis of a “prototypical” real business money demand analogy is depressing: cycle model using generalized expected monetary theorists are a long way from utility preferences represents a promis- delivering a definitive model of money ing first step. demand. On the other hand, the work of Ten years ago, Mehra and Prescott Robert Townsend (1987) and others in (1985) wrote in their conclusion that the monetary theory is exciting because it relatively low rate of return of Treasury shows so clearly what it will take to make bills, “is not the only example of some true progress in explaining the size of asset receiving a lower return than that the equity premium. Like fiat money, implied by Arrow-Debreu general equi- the equity premium appears to be a librium theory. Currency, for example, is widespread and persistent phenomenon dominated by Treasury bills with positive of market economies. The universality of nominal yields yet sizable amounts of the equity premium tells us that, like currency are held.” Thus, they regard money, the equity premium must the equity premium puzzle as being emerge from some primitive and ele- analogous to the so-called “rate of return mentary features of asset exchange that dominance” puzzle that motivates much are probably best captured through ex- of modern monetary theory. tremely stark models. With this in mind, I believe that the past decade of work we cannot hope to find a resolution to has made this analogy even more persua- the equity premium puzzle by continuing sive: I find that the suggested “solutions” in our current mode of patching the to the equity premium puzzle closely re- standard models of asset exchange with semble the approaches used by econo- transactions costs here and risk aversion mists to generate a demand for money in there. Instead, we must seek to identify the face of rate of return dominance. what fundamental features of goods and Thus, the “high risk aversion” story ex- asset markets lead to large risk adjusted plains the puzzle by some peculiar aspect price differences between stocks and of individual preferences—which is ex- bonds. While I have no idea what these actly the way money-in-the-utility-func- “fundamental features” are, it is my be- tion models explain money demand. The lief that any true resolution to the equity “transactions costs” story explains that premium puzzle lies in finding them. bonds are held despite their low rate of return because they are less costly to Appendix trade—which is exactly the way transactions costs models explain money de- Apart from two exceptions, the follow- mand. ing italicized description is taken directly Generally, both of these types of mod- from Mehra and Prescott (1985, pp. els of money demand are regarded as ad 147–48). hoc “reduced forms.” They fail to cap- The data used in this study consists of 68 Journal of Economic Literature, Vol. XXXIV (March 1996) five basic series for the period 1889– This description differs from that of 1978. The first four are identical to those Mehra and Prescott (1985) in two ways. used by Grossman and Shiller (1981) in First, the figure numbers have been their study. The series are individually changed to match the numbering in this described below. paper. Second, Mehra and Prescott − − (i) Series P: Annual average Standard (1985) used the formula RFt (PCt+1 & Poor’s Composite Stock Price Index di- PCt)/PCt for the real return to the rela- vided by the Consumption Deflator, a tively riskless security. This formula ex- plot of which appears in Grossman and presses the real rate as being the differ- Shiller (1981, p. 225, fig. 1). ence between the nominal rate and the (ii) Series D: Real annual dividends for inflation rate. Of course, their formula is the Standard & Poor’s series. only an approximation; I use the correct (iii) Series C: Kuznets-Kendrik-USNIA ratio formula instead. per capita real consumption on non-dur- This difference in the formulae creates ables and services. only small differences in results. For ex- (iv) Series PC: Consumption deflator ample, they find that the average rela- series, obtained by dividing real con- tively riskless rate is 0.8 percent. I find it sumption in 1972 dollars on non-dur- instead to be one percent. They find that ables and services by the nominal con- the variance of the relatively riskless rate sumption on non-durables and services. is 0.0032; I find it to be 0.00308.

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