The Equity Premium: Why Is It a Puzzle?

(as corrected February 2003) Rajnish Mehra This article takes a critical look at the equity premium puzzle—the inability of standard intertemporal economic models to rationalize the statistics that have characterized U.S. financial markets over the past century. A summary of historical returns for the United States and other industrialized countries and an overview of the economic construct itself are provided. The intuition behind the discrepancy between model prediction and empirical data is explained. After detailing the research efforts to enhance the model’s ability to replicate the empirical data, I argue that the proposed resolutions fail along crucial dimensions.

lmost two decades ago, Edward Prescott Furthermore, this pattern of excess returns to and I (see Mehra and Prescott 1985) chal- equity holdings is not unique to U.S. capital mar- lenged the profession with a poser: The kets. Table 2 confirms that equity returns in other A historical U.S. equity premium (the return developed countries also exhibit this historical reg- earned by a risky security in excess of that earned ularity when compared with the return to debt by a relatively risk free U.S. T-bill) is an order of holdings. The annual return on the U.K. stock mar- magnitude greater than can be rationalized in the ket, for example, was 5.7 percent in the post-WWII context of the standard neoclassical paradigm of period, an impressive 4.6 pp premium over the financial economics. This regularity, dubbed “the average bond return of 1.1 percent. Similar statisti- equity premium puzzle,” has spawned a plethora cal differences have been documented for France, of research efforts to explain it away. In this article, Germany, and Japan. And together, the United I take a retrospective look at the puzzle and criti- States, the United Kingdom, Japan, Germany, and cally evaluate the various attempts to solve it.1 France account for more than 85 percent of capital- ized global equity value. Empirical Facts The dramatic investment implications of the differential rates of return can be seen in Table 3, Historical data provide a wealth of evidence docu- which maps the enormous disparity in capital menting that for more than a century, U.S. stock appreciation of $1 invested in different assets for returns have been considerably higher than returns 1802–1997 and for 1926–2000.2 for T-bills. As Table 1 shows, the average annual real return (that is, the inflation-adjusted return) on This kind of long-term perspective under- the U.S. stock market for the past 110 years has been scores the remarkable wealth-building potential of about 7.9 percent. In the same period, the real the equity premium and explains why the equity return on a relatively riskless security was a paltry premium is of central importance in portfolio allo- 1.0 percent. The difference between these two cation decisions, in making estimates of the cost of returns, 6.9 percentage points (pps), is the equity capital, and in the current debate about the advan- premium. This statistical difference has been even tages of investing Social Security funds in the stock more pronounced in the post-World War II period. market. Siegel’s (1998) data on U.S. stock and bond returns going back to 1802 reveal a similar, although some- A Premium for Bearing Risk? what smaller, premium for the past 200 years. Why has the rate of return on stocks been significantly higher than the rate of return on relatively risk free assets? One intuitive answer is that stocks Rajnish Mehra is professor of finance at the University are “riskier” than bonds and investors require a of California, Santa Barbara, a research associate of the premium for bearing this additional risk. Indeed, National Bureau of Economic Research, and senior the standard deviation of the returns to stocks investment advisor to Vega Asset Management, New (about 20 percent a year historically) is larger than York. that of the returns to T-bills (about 4 percent a year),

Table 1. U.S. Returns, 1802–2000 Mean Real Return Relatively Riskless Period Market Index Security Risk Premium 1802–1998 7.0% 2.9% 4.1 pps 1889–2000 7.9 1.0 6.9 1926–2000 8.7 0.7 8.0 1947–2000 8.4 0.6 7.8 Sources: Data for 1802–1998 are from Siegel (1998); for 1889–2000, from Mehra and Prescott (1985), updated by the author. The rest are the author’s estimates.

Table 2. Returns for Selected Countries, 1947–98 Mean Real Return Relatively Riskless Country Period Market Index Security Risk Premium United Kingdom 1947–99 5.7% 1.1% 4.6 pps Japan 1970–99 4.7 1.4 3.3 Germany 1978–97 9.8 3.2 6.6 France 1973–98 9.0 2.7 6.3 Sources: Data for the United Kingdom are from Siegel; the rest of the data are from Campbell (forthcoming 2003).

Table 3. Terminal Value of $1 Invested Stocks T-Bills Investment Period Real Nominal Real Nominal 1802–1997 $558,945 $7,470,000 $276 $3,679 1926–2000 266.47 2,586.52 1.71 16.56 Sources: Ibbotson Associates (2001); Siegel (1998).

so obviously, stocks are considerably riskier than the concept should be differentiated from incre- bills. mental consumption itself. The same amount of But are they? Figure 1 illustrates the variability consumption may result in different degrees of in the annual real rate of return on the S&P 500 well-being at different times. (For example, a five- Index (Panel A) and a relatively risk free security course dinner after a heavy lunch yields consider- (Panel B) over the 1889–2000 period. ably less satisfaction than a similar dinner when To enhance and deepen our understanding of one is hungry!) the risk–return trade-off in the pricing of financial As a consequence, assets that pay off when assets, we make a detour into modern asset-pricing times are good and consumption levels are high theory and look at why different assets yield differ- (i.e., when the incremental value of additional con- ent rates of return. The deus ex machina in asset- sumption is low) are less desirable than those that pricing theory is that assets are priced in such a way pay off an equivalent amount when times are bad that, ex ante, the loss in marginal utility incurred by and additional consumption is both desirable and sacrificing current consumption and buying an more highly valued. Thus, assets that pay off when asset at a certain price is equal to the expected gain times are good must offer a premium to induce in marginal utility contingent on the anticipated investors to hold them. increase in consumption when the asset pays off in Let me illustrate this principle in the context of the future. the standard popular paradigm, the capital asset The operative concept here is “incremental loss pricing model. The CAPM postulates a linear rela- or gain in well-being because of consumption,” and tionship between an asset’s beta (a measure of

January/February 2003 55 Financial Analysts Journal

Figure 1. Real Return on S&P 500 and Relatively Riskless Asset, 1889–2000

A. S&P 500 Return (%) 60

−20

−40 1889 1899 1909 1919 1929 1939 1949 1959 1969 1979 1989 1999 2000

B. Relatively Riskless Asset Return (%) 20

−20 1889 1899 1909 1919 1929 1939 1949 1959 1969 1979 1989 1999 2000 systematic risk) and expected return. Thus, high- valuable and thus require a lower rate of return to beta stocks yield a high expected rate of return. In induce investors to hold them. (Insurance policies the CAPM, good times and bad times are captured are a classic example of assets that smooth con- by the return on the market. The performance of the sumption. Individuals willingly purchase and hold market, as captured by a broad-based index, acts as them in spite of their very low rates of return.) a surrogate indicator for the relevant state of the To return to the original question: Are stocks economy. A high-beta security thus tends to pay off so much riskier than T-bills that a 7 pp differential more when the market return is high (when times in their rates of return is justified? What came as a are good and consumption is plentiful), but as just surprise to many economists and researchers in noted, such a security provides less incremental finance was the conclusion of a research paper that utility than a security that pays off when consump- Prescott and I wrote in 1979. Stocks and bonds pay tion is low. It is less valuable to investors and, off in approximately the same states of nature or consequently, sells for less. Thus, assets that pay off economic scenarios, and hence, as argued earlier, in states of low-marginal utility will sell for a lower they should command approximately the same rate price than similar assets that pay off in states of high marginal utility. Since rates of return are inversely of return. In fact, using standard theory to estimate proportional to asset prices, a high-beta asset will, risk-adjusted returns, we found that stocks, on on average, have a higher rate of return than a low- average, should command, at most, a 1 pp return beta security. premium over bills. Because for as long as we had Another perspective on asset pricing empha- reliable data (about a hundred years), the mean sizes that economic agents prefer to smooth pat- premium on stocks over bills was considerably and terns of consumption over time. Assets that pay off consistently higher, we realized that we had a puz- a relatively larger amount at times when consump- zle on our hands. Our paper detailing this research, tion is already high “destabilize” these patterns of “The Equity Premium: A Puzzle,” was published consumption, whereas assets that pay off when in 1985. consumption levels are low “smooth out” con- To illustrate the puzzle, consider a frictionless sumption. Naturally, these latter assets are more economy that has a single representative, “stand-

56 ©2003, AIMR® The Equity Premium in” household. This household unit orders its pref- resulting additional consumption next period. To erences over random consumption paths by carry over one additional unit of equity, pt units of the consumption good must be sacrificed and the ∞ ′ βt (), <<β resulting loss in utility is ptU (ct). By selling this E0 ∑ Uct 0 1, (1) + t =0 additional unit of equity next period, pt+1 yt+1 additional units of the consumption good can be β ′ where consumed and Et[(pt+1 + yt+1)U (ct+1)] is the ⋅ E0( ) = expectation operator conditional on expected value of the incremental utility next information available at time zero period. At an optimum, these quantities must be (which denotes the present time) equal. The result is the fundamental pricing rela- 4 β = subjective time discount factor tionship: ′ = β + ′ U = an increasing, continuously differen- ptU (ct) Et[(pt+1 yt+1)U (ct+1)]. (3) tiable concave utility function Equation 3 is used to price both stocks and ct = per capita consumption riskless one-period bonds. For equity, The utility function is further restricted to be of the U′()c constant relative risk aversion (CRRA) class: β ------t+1 1 = Et ′()Ret, +1 , (4) U ct 1–α c Uc(), α = ------, 0 <<α∞, (2) 1 – α where Re,t+1 is equal to (pt+1 + yt+1)/pt. For the riskless one-period bonds, the relevant pricing where the parameter α measures the curvature of expression is the utility function. When α = 1, the utility function U′()c is defined to be logarithmic, which is the limit of β ------t+1 1 = Et ′()Rft, +1 , (5) Equation 2 as α approaches 1. U ct The feature that makes Equation 2 the “prefer- The gross rate of return on the riskless asset, Rf, is, ence function of choice” in much of the literature by definition, on growth and in Real Business Cycle theory is that it is scale invariant. Although the levels of aggre- ----1 Rft, +1 = , (6) gate variables, such as capital stock, have increased qt over time, the equilibrium return process is station- with q as the price of the bond. ary. A second attractive feature is that it is one of t We can rewrite Equation 4 as only two preference functions that allow for aggre- β gation and a “stand-in” (representative) agent for- 1 = Et(Mt+1Re,t+1), (7) mulation that is independent of the initial = ′ ′ where Mt+1 U (ct+1)/U (ct). Since U(c) is assumed distribution of endowments. One disadvantage of to be increasing, Mt+1 is a strictly positive stochastic this representation is that it links risk preferences discount factor. This definition guarantees that the with time preferences. With CRRA preferences, economy will be arbitrage free and the law of one agents who like to smooth consumption across var- price will hold. ious states of nature also prefer to smooth con- A little algebra demonstrates that the expected sumption over time; that is, they dislike growth. gross return on equity is5 Specifically, the coefficient of relative risk aversion is the reciprocal of the elasticity of intertemporal –U′()c , R ()= + ------t+1------et, +1 substitution. There is no fundamental economic Et Ret, +1 Rft, +1 covt []′() . (8) Et U ct+1 reason why this must be so. I revisit the implications of this issue later, in examining preference The equity premium, Et(Re,t+1)–Rf,t+1, can structures that do not impose this restriction.3 thus be easily computed. Expected asset returns For this illustration of the puzzle, assume one equal the risk-free rate plus a premium for bearing productive unit that produces in period t output yt, risk, which depends on the covariance of the asset which is the period dividend. There is one equity returns with the marginal utility of consumption. share with price pt (denominated in consumption Assets that covary positively with consumption— units) that is competitively traded; it is a claim on that is, assets that pay off in states when consump- the stochastic process {yt}. tion is high and marginal utility is low—command Consider the intertemporal choice problem of a high premium because these assets “destabilize” a typical investor at time t. He equates the loss in consumption. utility associated with buying one additional unit The question now is: Is the magnitude of the of equity to the discounted expected utility of the covariance between the assets and the marginal

January/February 2003 57 Financial Analysts Journal utility of consumption large enough to justify the The other terms involving z and Re are defined observed 6 pp equity premium in U.S. equity mar- analogously. Furthermore, because the growth rate kets? In addressing this issue, we make some addi- of consumption is i.i.d., the conditional and uncon- tional assumptions. Although these assumptions ditional expectations are the same. are not necessary and were not part of the original Similarly, Mehra–Prescott (1985) paper, they facilitate expo- 6 1 sition and result in closed-form solutions. These Rf = ------(13a) –αµ +1⁄α 2 2 α2 assumptions are as follows: βe x x ≡ • the growth rate of consumption, xt+1 ct+1/ct, and is identically and independently distributed 1 2 2 (i.i.d.), ln R = – ln β + αµ – ---α σ . (13b) ≡ ⁄ f x 2 x • the growth rate of dividends, zt+1 yt+1 yt, is i.i.d., and Therefore, •(x , z ) are jointly lognormally distributed. t t ln E(R ) – ln R = ασ .(14) A consequence of these assumptions is that the e f x,z gross return on equity, Re,t, is i.i.d. and (xt,Re,t) are In this model, it also follows that jointly lognormally distributed. () ασ –α ln ER – ln R = , ,(15) ′ = e f xRe Substituting U (ct) ct in the fundamental pricing relationship, where σ = cov() ln x, ln R . xR, e e U′()c The (log) equity premium in this model is the p = βE ()p + y ------t+1 , (9a) product of the coefficient of risk aversion and the t t t+1 t+1 U′()c t covariance of the (continuously compounded) we get growth rate of consumption with the (continuously compounded) return on equity or the growth rate β []()–α pt = Et pt+1 + yt+1 xt+1 . (9b) of dividends. If the model equilibrium condition is imposed that x = z (a consequence of which is the It can be easily shown that the expected return on 7 restriction that the return on equity be perfectly the risky asset is correlated with the growth rate of consumption), E ()z we get ()= ------t t+1 Et Ret, +1 –α . (10) βE ()z x = ασ2 t t+1 t+1 lnE(Re) – lnRf x (16) Analogously, the gross return on the riskless asset and the equity premium is then the product of the can be written as coefficient of relative risk aversion, α, and the vari- 1 1 ance of the growth rate of consumption. As we see = ------2 Rft, +1 β –α . (11) later, this variance, σ , is 0.00125, so unless α is E ()x x t t+1 large, a high equity premium is impossible. The Because the growth rates of consumption and growth rate of consumption just does not vary dividends are assumed to be lognormally distrib- enough! uted, Table 4 contains the sample statistics for the U.S. economy for the 1889–1978 period that we 2 µ +1⁄ 2α e z z reported in Mehra and Prescott (1985). E ()R = ------(12a) t et, +1 µ αµ ⁄ ()α2 α2 α2 ασ In our calibration, we are guided by the tenet z– x +1 2 z + x–2 xz, βe that model parameters should meet the criteria of and

1 2 2 ln E ()R = –lnβαµ++– ---α α ασ , (12b) t et, +1 x 2 x xz, Table 4. U.S. Economy Sample Statistics, 1889–1978 where Statistic Value µ = x E(lnx) Risk-free rate, Rf 1.008 2 σ = Mean return on equity, E(Re) 1.0698 x var(lnx) Mean growth rate of consumption, E(x)1.018 σ = x, z cov(lnx, lnz) Standard deviation of growth rate of lnx = the continuously compounded growth consumption, σ(x)0.036 rate of consumption Mean equity premium, E(Re) – Rf 0.0618

58 ©2003, AIMR® The Equity Premium cross-model verification. Not only must they be  2 var()x σ = ln 1 + ------consistent with the observations under consider- x []()2 ation, but they should not be grossly inconsistent Ex with other observations in growth theory, business = 0.00125 cycle theory, labor market behavior, and so on. and There is a wealth of evidence from various studies that the coefficient of risk aversion, α, is a small 1 2 µ = lnEx()– ---σ number, certainly less than 10.8 We can thus pose x 2 x the question: If we set risk-aversion coefficient α to = 0.0172, be 10 and β to be 0.99, what are the expected rates which implies that of return and the risk premium using the parameterization just described? lnER()– lnR α = ------f Using the expressions derived earlier, Appen- σ2 dix A, and Table 4,9 we have x = 47.6. 1 2 2 lnR = –lnβαµ+ – ---α σ f x 2 x Because = 0.120 β αµ 1α2σ2 ln = –lnRf + x – --- x or 2 = –0.60, Rf = 1.127, the implication is that β equals 0.55. which is a risk-free rate of 12.7 percent! Besides postulating an unacceptably high α, Because another problem is that this is a “knife edge” solu- () ασ2 tion. No other set of parameters will work, and a lnERe = lnRf + x small change in α will lead to an unacceptably high = 0.132, risk-free rate.11 we have = βαµ ⁄α2σ2 The relationship lnRf –ln + x – 12 x E(R ) = 1.141, shows why an extremely high α can be consistent e 2 2 with a “low” risk-free rate. The term 12⁄ασ or a return on equity of 14.1 percent, which implies x dominates when α is very large;12 however, then a an equity risk premium of 1.4 pps, far lower than the 6.18 pps historically observed. small change in the growth rate of consumption Note that in this calculation, I was very liberal will have a large impact on interest rates. This is in choosing the values for α and β. Most studies inconsistent with a cross-country comparison of indicate a value for α that is close to 2. If I were to real risk-free rates and their observed variability. pick a lower value for β, the risk-free rate would be For example, throughout the 1980s, South Korea even higher and the premium lower. So, the 1.4 pp had a much higher growth rate than the United value represents the maximum equity risk pre- States but real rates were not appreciably higher. mium that can be obtained, given the constraints Nor does the risk-free rate vary considerably over α β on and , in this class of models. Because the time, as would be expected if α were large. observed equity premium is more than 6 pps, we An alternative perspective on the puzzle has have a puzzle on our hands that risk considerations been provided by Hansen and Jagannathan (1991). alone cannot account for. The fundamental pricing equation can be written as Weil (1989) dubbed the high risk-free rate obtained in the preceding analysis “the risk-free M , R () ------t+1 et, +1 Et Ret, +1 = Rft, +1 – covt ().(17) rate puzzle.” The short-term real rate in the United Et Mt+1 States has averaged less than 1 percent, so the high value of α required to generate the observed equity Equation 17 also holds unconditionally, so premium results in an unacceptably high risk-free R – σ()σM ()ρR ER()= ------ft, +1 ------t+1 et, ------+1 RM, (18a) rate. et, +1 EM() The late Fischer Black proposed that α=55 t+1 would solve the puzzle.10 Indeed, it can be shown or that the U.S. experience from 1889 through 1978 () σ()ρ α = ER, – R , Mt+1 RM, reported here can be reconciled with 48 and ------et+1 ------ft+1 = –------, (18b) σ() EM() β=0.55. To see this, observe that Ret, +1 t+1

January/February 2003 59 Financial Analysts Journal

ρ where R,M is the correlation of the return on the 2000; Bansal and Coleman 1996; Constantinides, security and the stochastic discount factor M. And Donaldson, and Mehra 2002; Heaton and Lucas ≤ ρ ≤ because –1 R,M 1, 1996; McGrattan and Prescott 2001; Storesletten et σ() al.). None has fully resolved the anomalies. In the ER()– R Mt+1 ------et, +1 ------ft, +1 ≤ ------. (19) next section, I examine some of these efforts to solve σ()EM() 13 Ret, +1 t+1 the puzzle. This inequality is referred to as the “Hansen– Jagannathan lower bound on the pricing kernel.” Alternative Preference Structures For the U.S. economy, the long-term Sharpe The research attempting to solve the equity pre- σ ratio, defined as [E(Re,t+1) – Rf,t+1]/ (Re,t+1), can be mium puzzle by modifying preferences can be calculated to be 0.37. Because E(Mt+1) is the grouped into two broad approaches—one that calls expected price of a one-period risk-free bond, its for modifying the conventional time-and-state- value must be close to 1. In fact, for the parameter- separable utility function and another that incorpo- = α= ization discussed earlier, E(Mt+1) 0.96 when 2. rates habit formation. Thus, if the Hansen–Jagannathan bound is to be satisfied, the lower bound on the standard devia- Modifying the Time-and-State-Separable tion for the pricing kernel must be close to 0.3. Utility Function. The analysis in the preceding When this lower bound is calculated in the Mehra– section shows that the isoelastic (CRRA) prefer- Prescott framework, however, an estimate of 0.002 ences used in Mehra and Prescott (1985) can be σ is obtained for (Mt+1), which is off by more than made consistent with the observed equity pre- an order of magnitude. mium only if the coefficient of relative risk aversion I want to emphasize that the equity premium is implausibly large. A restriction imposed by this puzzle is a quantitative puzzle; standard theory is class of preferences is that the coefficient of risk consistent with our notion of risk that, on average, aversion is rigidly linked to the elasticity of inter- stocks should return more than bonds. The puzzle temporal substitution; one is the reciprocal of the arises from the fact that the quantitative predictions other. The implication is that if an individual is of the theory are an order of magnitude different averse to variation of consumption in different from what has been historically documented. The states at a particular point in time, then she or he puzzle cannot be dismissed lightly because much will be averse to consumption variation over time. of our economic intuition is based on the very class There is no a priori reason that this must be so. Since of models that fall short so dramatically when con- on average, consumption grows over time, the fronted with financial data. It underscores the fail- agents in the Mehra–Prescott setup have little ure of paradigms central to financial and economic incentive to save. The demand for bonds is low and modeling to capture the characteristic that appears the risk-free rate, as a consequence, is counter- to make stocks comparatively so risky. Hence, the factually high. viability of using this class of models for any quan- To deal with this problem, Epstein and Zin titative assessment—say, to gauge the welfare presented a class of preferences, which they termed implications of alternative stabilization policies— “generalized expected utility” (GEU), that allows is thrown open to question. independent parameterization for the coefficient of For this reason, over the past 15 years or so, risk aversion and the elasticity of intertemporal attempts to resolve the puzzle have become a major substitution. research impetus in finance and economics. Several In these preferences, utility is recursively generalizations of key features of the Mehra– defined by ⁄ ρ Prescott (1985) model have been proposed to rec- ()ρ ⁄ ()α 11– 1–ρ β()1–α 1– 1– oncile observations with theory, including alterna- Ut = ct + EtUt+1 . (20) tive assumptions about preferences (Abel 1990; Benartzi and Thaler 1995; Campbell and Cochrane The usual isoelastic preferences follow as a special 1999; Constantinides 1990; Epstein and Zin 1991), case when α = ρ. modified probability distributions to admit rare The major advantage of this class of models is but disastrous events (Rietz 1988), survivorship that a high coefficient of risk aversion, α, does not bias (Brown, Goetzmann, and Ross 1995), incom- necessarily imply that agents will want to smooth plete markets (Constantinides and Duffie 1996; consumption over time. This modification has the Heaton and Lucas 1997; Mankiw 1986; Storesletten, potential to at least resolve the risk-free rate puzzle. Telmer, and Yaron 1999), and market imperfections The main difficulty in testing this alternative (Aiyagari and Gertler 1991; Alvarez and Jermann preference structure stems from the fact that the

60 ©2003, AIMR® The Equity Premium counterparts of Equations 3 and 4 when GEU is An alternate approach, expounded by Camp- used depend on the unobserved utility at time t + 1, bell and Cochrane, incorporates the possibility of which makes calibration tricky. One would need to recession—that is, a major economic downturn— make specific assumptions about the consumption as a state variable. In this model, the risk aversion process to obtain first-order conditions in terms of of investors rises dramatically when the chances of observables. a recession increase; thus, the model can generate Although Epstein and Zin claimed that their a high equity premium. Since risk aversion framework offers a solution to the equity premium increases precisely when consumption is low, it puzzle, I believe that the proxies they used for the generates a precautionary demand for bonds that unobservables overstate their claim. The frame- helps lower the risk-free rate. This model is consis- work does ameliorate the risk-free rate puzzle, but tent with both consumption and asset market data. it does not solve the equity premium puzzle. Whether investors actually display the huge, time- varying, countercyclical variations in risk aversion Habit Formation. A second approach to postulated in this model is, however, open to ques- modifying preferences, initiated by Constantinides tion. (1990), incorporates habit formation. This formula- In summary, models with habit formation, rel- tion assumes that utility is affected not only by ative consumption, or subsistence consumption current consumption but also by past consump- have had success in addressing the risk-free rate tion. It captures the notion that utility is a decreas- puzzle but only limited success with resolving the ing function of past consumption and marginal equity premium puzzle because in these models, utility is an increasing function of past consump- effective risk aversion and prudence become tion. improbably large. Utility is defined as

∞ ()λ 1–α Idiosyncratic and Uninsurable s c – c Uc()= E ∑ β ------t+s t------+s–1 , λ > 0, (21) t 1 – α Income Risk s =0 In infinite-horizon models, agents, when faced where λ is a parameter that captures the effect of with uninsurable income shocks, dynamically self- past consumption. This preference ordering makes insure; agents simply stock up on bonds when the agent extremely averse to consumption risk times are good and deplete them in bad times, even when the risk aversion is small. For small thereby effectively smoothing consumption. changes in consumption, changes in marginal util- Hence, the difference between the equity premium ity can be large. So, although this approach cannot in incomplete markets and in complete markets is solve the equity premium puzzle without invoking small (Heaton and Lucas 1996, 1997). The analysis extreme aversion to consumption risk, it can changes, however, when a shock is permanent. address the risk-free rate puzzle because the Constantinides and Duffie developed a model that induced aversion to consumption risk increases the incorporates heterogeneity by capturing the notion demand for bonds, thereby reducing the risk-free that consumers are subject to idiosyncratic income rate. shocks that cannot be insured away. A modification of the preceding approach is to Simply put, the model recognizes that consum- define utility of consumption relative to average ers face the risk of job loss or other major personal per capita consumption. Abel termed this approach disasters that cannot be hedged away or insured “keeping up with the Joneses.” The idea is that against. Equities and related procyclical invest- one’s utility depends not on the absolute level of ments (assets whose payoffs are contingent on the consumption but on how one is doing relative to business cycle) exhibit the undesirable feature that others. The effect is that, once again, an individual they drop in value when the probability of job loss can become extremely sensitive and averse to con- increases—as it does in recessions, for instance. In sumption variation. Equity may have a negative economic downturns, consumers thus need an rate of return, which can result in personal con- extra incentive to hold equities and similar invest- sumption falling relative to others’ consumption. ment instruments; the equity premium can then be Equity thus becomes an undesirable asset relative rationalized as the added inducement needed to to bonds. Since average per capita consumption is make equities palatable to investors. This model rising over time, the induced demand for bonds can generate a high risk premium, but whether the with this modification helps mitigate the risk-free required degree of consumption variation can be rate puzzle. generated in an economy populated with agents

January/February 2003 61 Financial Analysts Journal displaying a relatively low level of risk aversion During the 1920s administration of Henri Poincaré remains to be seen. in France, bondholders lost nearly 90 percent of the value invested in nominal debt. In the 1980s, Mex- Disaster States and Survivorship ican holders of dollar-denominated debt lost a siz- able fraction of the debt’s value when the Mexican Bias government, in a period of rapid inflation, con- Reitz proposed a solution to the equity premium verted the debt to pesos and limited the rate at puzzle that incorporates a small probability of a which the funds could be withdrawn. Bondholders large drop in consumption. He found that the risk- in czarist Russia and holders of Chinese debt hold- free rate in such a scenario is much lower than the ings after the fall of the Nationalist government return on an equity security. suffered a similar fate under the new communist This model requires a 1 in 100 chance of a 25 regimes. percent decline in consumption to reconcile the These examples demonstrate that in times of equity premium with a risk-aversion parameter of financial crises, bonds are as likely to lose value as 10. Such a scenario has not been observed in the stocks. In every instance when, because of political United States for the past 100 years (the time for upheavals or other disasters, trading in equity has which we have data). Nevertheless, one can evalu- been suspended, governments have either reneged ate the implications of the model. on their debt obligations or expropriated much of One implication is that the real interest rate and the real value of nominal debt through unantici- the probability of the occurrence of the extreme pated inflation. Thus, although survivorship bias event move inversely. For example, the perceived may have an impact on the levels of the return on probability of a recurrence of a depression was both equity and debt, researchers have no evidence probably high just after WWII but, subsequently, that crises affect the returns to stocks and bonds declined over time. If real interest rates had risen differentially. Hence, the equity premium is not significantly as the war years receded, that evi- affected. dence would support the Reitz hypothesis. Simi- larly, if the low-probability event precipitating the Borrowing Constraints large decline in consumption is interpreted to be a nuclear war, the perceived probability of such an In models that impose borrowing constraints and event has surely varied over the past 100 years. It transaction costs, these features force investors to must have been low before 1945, the first and only hold an inventory of bonds (precautionary year the atom bomb was used, and it must have demand) to smooth consumption. Hence, in been higher before the Cuban missile crisis than infinite-horizon models with borrowing con- after it. If real interest rates moved with these sen- straints, agents come close to equalizing their mar- timents as predicted, that evidence would support ginal rates of substitution with little effect on the 14 Rietz’s disaster scenario. But interest rates did not equity premium. Some recent attempts to resolve move as predicted. the equity premium puzzle incorporating both bor- Another attempt at resolving the puzzle, pro- rowing constraints and consumer heterogeneity posed by Brown et al., focuses on survivorship bias. appear promising. One approach, which departs The central thesis is that the ex post measured from the representative agent model, was proposed returns reflect the premium in the United States on in Constantinides et al. a stock market that has successfully weathered the The novelty of this paper is the incorporation vicissitudes of fluctuating financial fortunes. Many of a life-cycle feature to study asset pricing. The other exchanges have been unsuccessful; hence, the idea is appealingly simple: The attractiveness of ex ante equity premium was low because no one equity as an asset depends on the correlation knew a priori which exchanges would survive. between consumption and equity income. If equity For this explanation to work, however, stock pays off in states of high marginal utility of con- and bond markets must be differently affected by sumption, it will command a higher price (and, financial crises. One reason the effects might not be consequently, a lower rate of return) than if its different is that governments have expropriated payoff occurs in states of low marginal utility. much of the real value of nominal debt by the Because the marginal utility of consumption varies mechanism of unanticipated inflation. Five histor- inversely with consumption, equity will command ical instances come readily to mind: During the a high rate of return if it pays off in states when post-WWI period of German hyperinflation, hold- consumption is high, and vice versa.15 ers of bonds denominated in Reich marks lost vir- The key insight of the paper is as follows: As tually all of the value invested in those assets. the correlation of equity income with consumption

62 ©2003, AIMR® The Equity Premium changes over the life cycle of an individual, so does in the demand for equity by the young and the the attractiveness of equity as an asset. Consump- decrease in the demand for equity by the middle- tion can be decomposed into the sum of wages and aged would work in opposite directions. On bal- equity income. A young person, looking forward, ance, the effect in the Constantinides et al. model is has uncertain future wage and equity income. Fur- to increase both the equity and the bond return thermore, the correlation of equity income with while simultaneously shrinking the equity pre- consumption at this stage is not particularly high, mium. Furthermore, the relaxation of the borrow- as long as stock and wage income are not highly ing constraint reduces the net demand for bonds— correlated. This relationship is empirically the case, and the risk-free rate puzzle re-emerges. as documented by Davis and Willen (2000). Equity will thus be a hedge against fluctuations in wages and a “desirable” asset to hold as far as the young Liquidity Premium are concerned. Bansal and Coleman developed a monetary model The same asset (equity) has a very different that offers an explanation of the equity premium. characteristic for middle-aged investors. Their In their model, assets other than money play a key wage uncertainty has largely been resolved. Their role by facilitating transactions, which affects the future retirement wage income is either zero or rate of return they offer in equilibrium. deterministic, and the innovations (fluctuations) in To motivate the importance of considering the their consumption occur from fluctuations in role of a variety of assets in facilitating transactions, equity income. At this stage of the life cycle, equity Bansal and Coleman argued that, on the margin, income is highly correlated with consumption. the transaction-service return of money relative to Consumption is high when equity income is high interest-bearing checking accounts should be the and equity is no longer a hedge against fluctuations interest rate paid on these accounts. They estimated in consumption; hence, for this group, equity this rate, based on the rate offered on NOW requires a higher rate of return. accounts for the period they analyzed, to be 6 per- The characteristics of equity as an asset, there- cent. Because this number is substantial, they sug- fore, change depending on who the predominant gested that other money-like assets may also holder of the equity is. Life-cycle considerations implicitly include a transaction-service component thus become crucial for asset pricing. If equity is a in their return. Insofar as T-bills and equity have “desirable” asset for the marginal investor in the different service components built into their economy, then the observed equity premium will returns, the Bansal–Coleman argument may offer be low relative to an economy in which the mar- an explanation for the observed equity premium. ginal investor finds holding equity unattractive. In fact, if this service-component differential were The deus ex machina is the stage in the life cycle of about 5 percent, there would be no equity premium the marginal investor. puzzle. Constantinides et al. argued that the young, This approach can be challenged, however, on who should (in an economy without frictions and three accounts. First, the bulk of T-bill holdings are with complete contracting) be holding equity, are effectively shut out of this market because of bor- concentrated in institutions, which do not use them rowing constraints. They have low wages; so, ide- as compensatory balances for checking accounts; ally, they would like to smooth lifetime thus, it is difficult to accept that they have a signif- consumption by borrowing against future wage icant transaction-service component. Second, the income (consuming a part of the loan and investing returns on NOW and other interest-bearing the rest in higher-returning equity). They are pre- accounts have varied over time. For most of the vented from doing so, however, because human 20th century, checking accounts were not interest capital alone does not collateralize major loans in bearing, and returns were higher after 1980 than in modern economies (for reasons of moral hazard earlier periods. Yet, contrary to the implications of and adverse selection). this model, the equity premium did not diminish In the presence of borrowing constraints, in the post-1980 period, when (presumably) the equity is thus exclusively priced by middle-aged implied transaction-service component was the investors and the equity premium is high. If the greatest. Finally, this model implies a significant borrowing constraint were to be relaxed, the young yield differential between T-bills and long-term would borrow to purchase equity, thereby raising government bonds, which (presumably) do not the bond yield. The increase in the bond yield have a significant transaction-service component. would induce the middle-aged to shift their port- However, such a yield differential has not been folio holdings from equity to bonds. The increase observed.

January/February 2003 63 Financial Analysts Journal

Taxes constraint represents the condition that the present discounted value of expenditures must be less than McGrattan and Prescott proposed an explanation or equal to the discounted value of after-tax for the equity premium based on changes in tax income. Expenditures of the household are con- rates. (An important aspect to keep in mind is that sumption and purchases of stocks, V (s – s ). their thesis is not a risk-based explanation. They t t+1 t Income for the households is received from three can account for an equity premium but not as an sources—dividends, wages, and government equity risk premium.) McGrattan and Prescott transfers. Households pay personal taxes, τ , on found that, at least in the post-WWII period, the pers dividend and wage income. equity premium is not puzzling. They argued that Companies own capital and hire labor to pro- the large reduction in individual income tax rates duce output with a constant-return-to-scale pro- and the increased opportunity to shelter income duction technology, from taxation led to a doubling of equity prices = ξ between 1960 and 2000. And this increase in equity yt f(km,t,ku,t, tnt). (24) prices led, in turn, to much higher ex post returns where on equity than on debt. = km tangible assets This argument can be illustrated by use of a = ku intangible assets simple one-sector (a corporate sector) model that = includes only taxes on corporate distributions and n labor services ξ = taxes on corporate profits. The authors extended t level of technology in period t the model to include sufficient details from the U.S. Equation 24 thus assumes that companies use both economy—especially in relation to the tax code— tangible assets and intangible assets to produce to allow them to calibrate the model. They matched output y. Tangible assets include structures, equip- up the model with data from the National Income ment, inventory, and land. Intangible assets are the and Product Accounts (NIPA) and the Statistics of result of on-the-job training, research and develop- Income (SOI) from the Internal Revenue Service. ment, and organization and company-specific The model is detailed as follows. Consider a know-how. In addition to capital, labor services, n, ξ representative-agent economy of infinite life with are required. The level of technology in period t, t, γ household preferences defined over consumption is assumed to grow at rate . and leisure. Each household chooses sequences of Companies choose capital and labor to maxi- consumption and leisure to maximize utility, mize the present value of dividends. In this framework, therefore, the value of corporate equity, Vt, is ∞ = τ + τ β′ ()<<β Vt (1 – pers)[km,t+1 (1 – corp) ku,t+1]. (25) max ∑ Uct, lt , 0 1, (22) t =0 Equation 25 makes clear that a large drop in the personal tax rate with little change in the corporate subject to the household’s budget constraint: τ tax rate, corp, will indeed raise the value of equity. ∞ However, to show that the market value of U.S. []() ∑ pt ct + Vt st+1 – st corporate equity relative to GNP, V/y, rises with a t =0 decline in the personal tax rate with little change in ∞ (23) the corporate tax rate, the capital-to-output ratio = ∑ p []()1 – τ ()κd s + w n + , τ t pers t t t t t must not change with changes in pers, only with t =0 τ changes in corp. McGrattan and Prescott demonstrate this in their paper. where In the United States, corporate income tax rates = ct per capita consumption at time t have changed little since 1960 whereas personal = lt fraction of productive time allocated income tax rates have declined significantly. In to nonmarket activities, such as leisure particular, personal tax rates in the period prior to = the tax cuts initiated by the John F. Kennedy admin- Vt price per share = istration were considerably higher than in the st number of shares held in period t = period after the Tax Reform Act of 1986. McGrattan dt dividends = and Prescott argued that this reduction in personal wt wages tax rates was largely unforeseen and that the large κ = t government transfers unanticipated increase in equity prices had a signif- τ = pers personal taxes icant effect on equity returns. The fraction of time allocated by households to As an illustration, the authors proposed the = market activities is denoted by nt 1 – lt. The budget following hypothetical example. Suppose the tax

64 ©2003, AIMR® The Equity Premium rate on dividends were to fall from 50 percent to 0 level out, everything else being equal, equity in 40 years. This fall in the tax rate implies that returns will decline for the following reason. The equity prices would grow by approximately 1.8 pps real before-tax return on equity is the sum of three a year higher than the growth in the GNP. Growth returns—the dividend yield, the anticipated capital in per capita GNP over the post-WWII period has gain, and the unanticipated capital gain. The divi- been 2 percent a year; the growth in population was dend yield has been high (more than 3 percent) for roughly 1.5 percent in the early part of the postwar much of the post-WWII period because high tax period and fell to about 1 percent a year toward the rates have implied a low price of equity. Recently, end of the period. Thus, their model would predict it has declined, and it is presently just over 1 per- an average growth in equity prices of 4.8–5.3 percent. The anticipated capital gain is the growth rate cent a year over the 40-year period. To compute a of productive assets, which is roughly 3 percent. return, a dividend yield must be added to this price This growth rate has not changed. The unantici- rise. The rise in equity prices implies a fall in the pated capital gain is the growth in the price of dividend yield because the dividend output ratio equity as a result of unanticipated changes in tax stays roughly constant. If the dividend yield were, rates. This growth rate has changed—falling from a on average, in the range of 3–4 percent, then equity range of 1.5–2.0 percent to 0. Adding these rates returns would be in the range of 7.8–9.3 percent, a leads to about an expected (3 + 3 + 2 percent) 8 figure that is of the same magnitude as documented percent real, before-tax stock return in the early in Table 2 for the post-WWII period. post-WWII period and, barring any further unex- Additionally, if households are assumed to pected changes in tax rates, a (1 + 3 + 0 percent) 4 have a liquidity motive for holding debt, bond percent return in the future. returns would be low and the resulting equity premium would be large. The low yield on debt would No Premium? be reinforced by the additional demand for debt induced by constraints on individuals to hold their An alternative point of view held by a group of retirement assets in debt securities. Indeed, in the academicians and professionals is that currently first half of the post-WWII period, pension fund there is no equity premium and, by implication, no assets were almost entirely invested in debt securi- equity premium puzzle. To address these claims, ties because of institutional restrictions on pension two different interpretations of the term “equity fund managers. Before 1974, when ERISA made premium” must be distinguished. One is the ex post pension plan fiduciaries personally liable for or realized equity premium. This figure is the imprudence or misconduct, fiduciary breaches typ- actual, historically observed difference between the ically resulted in a loss of tax qualification for the return on the market, as captured by a stock index, pension fund. Such penalties were likely to be and the risk-free rate, as proxied by the return on avoidable if pension fund managers held debt government bills. This premium is what Prescott assets of various maturities so as to avoid large and I addressed in our 1985 paper. The other fund losses and to facilitate the timing of the distri- (related) concept is the ex ante equity premium. This bution of benefits. figure is a forward-looking measure of the pre- A potential calibration problem with the mium—that is, the equity premium that is expected McGrattan–Prescott model arises because of the to prevail in the future or the conditional equity difficulty in identifying the marginal investor. premium given the current state of the economy. Although the marginal tax rate has dramatically For example, after a bull market, stock valuations declined in the post-WWII period, the same cannot are high relative to fundamentals, the market has be said about the rates that apply to the marginal risen sharply, and the ex post (realized) equity pre- investor unless the marginal investor can be identi- mium is high, but such a time is precisely when the fied. The marginal tax rate and the rate that applies ex ante (expected) equity premium is likely to be to the marginal investor may be quite different, and low. Conversely, after a major downward correc- of course, it is the rate that applies to the marginal tion, the realized premium will be low whereas the investor that is relevant for pricing. To estimate the expected premium is likely to be high. This rela- tax rate that applies to the marginal investor, tionship should not come as a surprise, because McGrattan and Prescott computed a weighted- returns to stock have been documented to be mean average rate, averaged across income groups, for reverting. each year in the 1947–96 period. The accuracy of this Which of these interpretations of the equity estimate, however, is open to question. premium is relevant for an investment advisor? The McGrattan–Prescott model predicts that, Clearly, the choice depends on the planning hori- eventually, as the dividend yield falls and tax rates zon. The equity premium documented in our 1985

January/February 2003 65 Financial Analysts Journal paper reflects very long investment horizons. It has Market watchers and other professionals who little to do with what the premium is going to be in are interested in short-term investment planning the next couple of years. The ex post equity pre- will wish to project the conditional, expected equity mium is the realization of a stochastic process over premium over their planning horizon. This task is a certain period, and it has varied considerably over by no means simple. Even if the equity premium in time. current market conditions is small (and the general Furthermore, the variation in the realized pre- consensus is that it is), it does not imply that either mium depends on the time horizon over which it is the historical premium was too high or that the measured. In some periods, it has even been nega- equity premium has diminished. tive, as illustrated in Figure 2.

Figure 2. Risk Premium, 1926–2000

A. Realized Equity Risk Premium per Year Equity Risk Premium (%) 60 50 40 30 20 10 0 −10 −20 −30 −40 −50 26 36 46 56 66 76 86 96 00 Year-End

B. Equity Risk Premium for 20-Year Periods Average Equity Risk Premium (%) 18

0 45 55 65 75 85 95 00 20-Year Period Ending

Source: Ibbotson (2001).

The data used to document the equity pre-  2 var()x σ = ln 1 + ------. mium over the past 100 years are as good an eco- x []()2 nomic data set as analysts have, and a span of 100 Ex years is a long series when it comes to economic 1 2 8.lnEx()= µ+ ---σ ; data. Before the equity premium is dismissed, not x 2 x only do researchers need to understand the hence, observed phenomena, but they also need a plausi- ble explanation as to why the future is likely to be µ () 1σ2 x = ln Ex – --- x. any different from the past. In the absence of this 2 explanation, and on the basis of what is currently Derivation of Equation 8. known, I make the following claim: Over the long Expanding Equa- term, the equity premium is likely to be similar to tion 7 results in what it has been in the past and returns to invest- U′()c U′()c , R , β ------t+1 ()β------t+1 et+1- Et ′() Et Ret, +1 + covt ′() = 1. ment in equity will continue to substantially dom- U ct U ct inate returns to investment in T-bills for investors with a long planning horizon. Substituting for Rf,t+1 from the following equation, U′()c β ------t+1 1 = Et ′() Rft, +1, I thank George Constantinides, Sanjiv Das, John U ct Donaldson, Mark Rubinstein, and especially, Edward Prescott for helpful discussions and Chaitanya Mehra results in for editorial assistance. This research was supported by –U′()c , R a grant from the Academic Senate of the University of () ------t+1 ------et, +1 Et Ret, +1 = Rft, +1 + covt []′() , California. Et U ct+1 which is Equation 8 in the article. Appendix A. Lognormal Properties and Derivations Derivation of Equations 10–13. Because pt is homogeneous of degree 1 in y, we can represent Some facts about the lognormal distribution follow: it as ∼ µ σ2 ∼ µ 2 σ2 1. If lnz N( z,),z then alnz N(a z, a z ). pt = wyt. ()a []() ′ = –α 2. Ez = E exp a lnz Substituting for U (ct) ct and pt in the following  fundamental pricing relationship, µ --1- 2σ2 = expa z + a z . 2 U′()c + ∼ µ + µ 2 σ2 + 2 σ2 p = βE ()p + y ------t+1 , 3. alnz blnx N(a z b x, a z b z t t t+1 t+1 ′() U ct + ρσ σ ρ = 2ab x z), where cor(lnx, lnz). results in a b 1 2 2 2 2 4. Ez()x = exp aµ ++bµ ---()a σ + b σ . z x 2 z x β []()–α wyt = Et wyt+1 + yt+1 xt+1 ; 5. If x = z, then hence, a b a+b Ez()x = Ex() β []()–α w = Et w + 1 zt+1x ()µ--1-()2σ2 t+1 = exp ab+ x + ab+ x . 2 or 2 2 α 6. var()x = Ex()– []Ex() β ()– Et zt +1x = ------t+1 2 2 w –α . = exp() 2µ + 2σ – exp() 2µ + σ – β () x x x x 1 Et zt+1xt+1 2 2 = exp() 2µ + σ []exp()σ – 1 x x x By definition, Re,t+1, the gross rate of return on []()2[]()σ2 equity, is = Ex exp x – 1 . pt+1 + yt+1 2 var()x = ------7. exp()σ = 1 + ------; Ret, +1 . x 2 p []Ex() t hence, Substituting for pt, we get

January/February 2003 67 Financial Analysts Journal

------w + 1 ------yt+1 1 1 R , = = ------et+1  Rft, +1 β –α . w yt () Et xt+1 w + 1 = ------z w t+1 Using the lognormal properties of Fact 2 and Fact 4, we get or µ ⁄ σ2 z+1 2 z ()------w + 1 () ()------e ------E R , = E z . Et Ret, +1 = t et+1 w t t+1 µ –αµ +1⁄ 2()σ2+α2σ2–2ασ βe z x z x xz, Because and w + 1 1 ------= ------, 1 –α = ------w β () Rf 2 2 . Et zt+1xt+1 –αµ +1⁄ 2α σ βe x x we have Taking logs on both sides results in E ()z ()------t t+1 1 2 2 Et Ret, +1 = α , ()= – βαµ+ – ---α σ + ασ β ()– ln Et Ret, +1 ln x x xz, Et zt+1xt+1 2 which is Equation 10 in the text. and Analogously, the gross return on the riskless 1 2 2 ln R = –ln βαµ+ – ---α σ . asset can be written as f x 2 x

Notes

1. For an elaboration of the issues presented here, see Mehra 7. The derivation of Equations 10–13 is given in Appendix A. (2002) and Mehra and Prescott (forthcoming 2003); some 8. A number of these studies are documented in Mehra and sections of this article closely follow the exposition in that Prescott (1985). paper. For current approaches to solving the equity risk σ2 9. For instance, to get x , we use Fact 7 and plug in var(x) and premium puzzle, see the presentations and discussions at E(x) from Table 4. The properties of the lognormal distribu- www.aimrpubs.org/ap/home.html from AIMR’s Equity tion are documented in Appendix A. Risk Premium Forum. 10. Private communication, 1981. 2. The calculations in Table 3 assume that all payments to the 11. An alternate approach is to experiment with negative time underlying asset, such as dividend payments to stock and preferences; however, there seems to be no empirical evi- interest payments to bonds, were reinvested and that no dence that agents do have such preferences. taxes were paid. 12. Kandel and Stambaugh (1991) suggested this approach. 3. See Epstein and Zin (1991) and Weil (1989). 13. See also the excellent surveys by Kocherlakota (1996), 4. Versions of this expression can be found in Rubinstein Cochrane (1997), and Campbell (forthcoming 2003). (1976), Lucas (1978), Breeden (1979), Prescott and Mehra 14. This is true unless the supply of bonds is unrealistically low. (1980), and Donaldson and Mehra (1984), among others. See Aiyagari and Gertler. Excellent textbook treatments of asset pricing are available 15. This argument is precisely the reason high-beta stocks in in Cochrane (2001), Danthine and Donaldson (2001), Duffie the simple CAPM framework have a high rate of return. In (2001), and LeRoy and Werner (2001). that model, the return on the market is a proxy for consump- 5. The derivation is given in Appendix A. tion. High-beta stocks pay off when the market return is 6. The exposition in the text is based on Abel (1988) and his high (i.e., when marginal utility is low); hence, their price unpublished notes. I thank him for sharing them with me. is (relatively) low and their rate of return, relatively high.

References

Abel, A.B. 1988. “Stock Prices under Time Varying Dividend Aiyagari, S.R., and M. Gertler. 1991. “Asset Returns with Risk: An Exact Solution in an Infinite-Horizon General Transactions Costs and Uninsured Individual Risk.” Journal of Equilibrium Model.” Journal of Monetary Economics, vol. 22, no. Monetary Economics, vol. 27, no. 3 (June):311–331. 3 (November):375–394. Alvarez, F., and U. Jermann. 2000. Efficiency, Equilibrium, and ———. 1990. “Asset Prices under Habit Formation and Catching Asset Pricing with Risk of Default. Econometrica, vol. 68, no. 4 (July):775–797. Up with the Joneses.” American Economic Review, vol. 80, no. 2 (May):38–42.

Bansal, R., and J.W. Coleman. 1996. “A Monetary Explanation Hansen, L.P., and R. Jagannathan. 1991. “Implications of of the Equity Premium, Term Premium, and Risk-Free Rate Security Market Data for Models of Dynamic Economies.” Puzzles.” Journal of Political Economy, vol. 104, no. 6 Journal of Political Economy, vol. 99, no. 2 (April):225–262. (December):1135–71. Heaton, J., and D.J. Lucas. 1996. “Evaluating the Effects of Barberis, N., M. Huang, and T. Santos. 2001. “Prospect Theory Incomplete Markets on Risk Sharing and Asset Pricing.” Journal and Asset Prices.” Quarterly Journal of Economics, vol. 116, no. 1 of Political Economy, vol. 104, no. 3 (June):443–487. (February):1–53. ———. 1997. “Market Frictions, Savings Behavior and Portfolio Benartzi, S., and R.H. Thaler. 1995. “Myopic Loss Aversion and Choice.” Journal of Macroeconomic Dynamics, vol. 1, no. 1:76–101. the Equity Premium Puzzle.” Quarterly Journal of Economics, vol. Ibbotson Associates. 2001. “Stocks, Bonds, Bills, and Inflation.” 110, no. 1 (February):73–92. 2000 Yearbook. Chicago, IL: Ibbotson Associates. Breeden, D. 1979. “An Intertemporal Asset Pricing Model with Kandel, S., and R.F. Stambaugh. 1991. “Asset Returns and Stochastic Consumption and Investment Opportunities.” Intertemporal Preferences.” Journal of Monetary Economics, vol. Journal of Financial Economics, vol. 7, no. 3 (September):265–296. 27, no. 1 (February): 39–71. Brown, S., W. Goetzmann, and S. Ross. 1995. “Survival.” Journal Kocherlakota, N.R. 1996. “The Equity Premium: It’s Still a of Finance, vol. 50, no. 2 (June):853–873. Puzzle.” Journal of Economic Literature, vol. 34, no. 1 (March):42– Campbell, J.Y. Forthcoming 2003. “Consumption Based Asset 71. Pricing.” In Handbook of the Economics of Finance. Edited by G.M. LeRoy, S.-F., and J. Werner. 2001. Principles of Financial Constantinides, M. Harris, and R. Stulz. Amsterdam, Economics. Cambridge, U.K.: Cambridge University Press. Netherlands: North-Holland. Lucas, R.E., Jr. 1978. “Asset Prices in an Exchange Economy.” Campbell, J.Y., and J.H. Cochrane. 1999. “By Force of Habit: A Econometrica, vol. 46, no. 6 (November):1429–45. Consumption-Based Explanation of Aggregate Stock Market Behavior.” Journal of Political Economy, vol. 107, no. 2 (April):205– Mankiw, N.G. 1986. “The Equity Premium and the 251. Concentration of Aggregate Shocks.” Journal of Financial Economics, vol. 17, no. 1 (September):211–219. Cochrane, J.H. 1997. “Where Is the Market Going? Uncertain Facts and Novel Theories.” Economic Perspectives, vol. 21, no. 6 McGrattan, E.R., and E.C. Prescott. 2001. “Taxes, Regulations, (November):3–37. and Asset Prices.” Working Paper No. 610, Federal Reserve Bank of Minneapolis. ———. 2001. Asset Pricing. Princeton, NJ: Princeton University Press. Mehra, R. 2002. “Current Estimates and Prospects for Change II.” In Equity Risk Premium Forum. Charlottesville, VA: AIMR Constantinides, G.M. 1990. “Habit Formation: A Resolution of (www.aimrpubs.org/ap/issues/v2002n1/toc.html). the Equity Premium Puzzle.” Journal of Political Economy, vol. 98, no. 3 (June):519–543. Mehra, R., and E.C. Prescott. 1985. “The Equity Premium: A Puzzle.” Journal of Monetary Economics, vol. 15, no. 2 ———. 2002. “Rational Asset Prices.” Journal of Finance, vol. 57, (March):145–161. no. 4 (August):1567–91. ———. 1988. “The Equity Premium: A Solution?” Journal of Constantinides, G.M., and D. Duffie. 1996. “Asset Pricing with Monetary Economics, vol. 22, no. 1 (July):133–136. Heterogeneous Consumers.” Journal of Political Economy, vol. 104, no. 2 (April):219–240. ———. Forthcoming 2003. “The Equity Premium in Retrospect.” In Handbook of the Economics of Finance. Edited by Constantinides, G.M., J.B. Donaldson, and R. Mehra. 2002 G.M. Constantinides, M. Harris, and R. Stulz. Amsterdam, “Junior Can’t Borrow: A New Perspective on the Equity Netherlands: North-Holland. Premium Puzzle.” Quarterly Journal of Economics, vol. 117, no. 1 (February):269–296. Prescott, E.C., and R. Mehra. 1980. “Recursive Competitive Equilibrium: The Case of Homogeneous Households.” Danthine, J.-P., and J.B. Donaldson. 2001. Intermediate Financial Econometrica, vol. 48, no. 6 (September):1365–79. Theory. Upper Saddle River, NJ: Prentice Hall. Davis, S.J., and P. Willen. 2000. “Using Financial Assets to Hedge Rietz, T.A. 1988. “The Equity Risk Premium: A Solution.” Journal Labor Income Risk: Estimating the Benefits.” Working paper. of Monetary Economics, vol. 22, no. 1 (July):117–131. Chicago, IL: University of Chicago. Rubinstein, M. 1976. “The Valuation of Uncertain Income Donaldson, J.B., and R. Mehra. 1984. “Comparative Dynamics Streams and the Pricing of Options.” Bell Journal of Economics, of an Equilibrium Intertemporal Asset Pricing Model.” Review vol. 7, no. 2 (Autumn):407–425. of Economic Studies, vol. 51, no. 1 (July):491–508. Siegel, J. 1998. Stocks for the Long Run. 2nd ed. New York: Irwin. Duffie, D. 2001. Dynamic Asset Pricing Theory. Princeton, NJ: Storesletten, K., C.I. Telmer, and A. Yaron. 1999. “Asset Pricing Princeton University Press. with Idiosyncratic Risk and Overlapping Generations.” Epstein, L.G., and S.E. Zin. 1991. “Substitution, Risk Aversion, Working paper. Pittsburgh: Carnegie Mellon University. and the Temporal Behavior of Consumption and Asset Returns: Weil, P. 1989. “The Equity Premium Puzzle and the Risk-Free An Empirical Analysis.” Journal of Political Economy, vol. 99, no. Rate Puzzle.” Journal of Monetary Economics, vol. 24, no. 3 2 (April):263–286. (November):401–421.

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