cochrane 1997

FEDERAL RESERVE BANK OF CHICAGO 3

John H. Cochrane is the Sigmund E. EdelstoneProfessor of Finance in the Graduate School ofBusiness at the University of Chicago and a con-sultant to the Federal Reserve Bank of Chicago.His research is supported by a grant from theNational Science Foundation, administered by theNational Bureau of Economic Research, and bythe University of Chicago. The author would liketo thank George Constantinides, Andrea Eisfeldt,and David Marshall for many helpful comments.

Where is the market going?Uncertain facts and novel theories

John H. Cochrane

Over the last century, thestock market in the UnitedStates has yielded impressivereturns to its investors. Forexample, in the postwar period,

stock returns have averaged 8 percentage pointsabove Treasury bills. Will stocks continue togive such impressive returns in the future? Arelong-term average stock returns a fundamentalfeature of advanced industrial economies? Or arethey the opposite of the old joke on Soviet agri-culture—100 years of good luck? If not puregood luck, perhaps they result from features ofthe economy that will disappear as financialmarkets evolve.

How does the recent rise in the stock mar-ket affect our view of future returns? Do highprices now mean lower returns in the future?Or have stocks finally achieved Irving Fisher’sbrilliantly mistimed 1929 prediction of a “per-manently high plateau?” If stocks have reacheda plateau, is it a rising plateau, or is the marketlikely to bounce around its current level for manyyears, not crashing but not yielding returns muchgreater than those of bonds?

These questions are on all of our minds aswe allocate our pension plan monies. They arealso important to many public policy questions.For example, many proposals to reform socialsecurity emphasize the benefits of moving toa funded system based on stock market invest-ments. But this is a good idea only if the stockmarket continues to provide the kind of returnsin the future that it has in the past.

In this article, I summarize the academic,and if I dare say so, scientific, evidence on

these issues. I start with the statistical analysisof past stock returns. The long-term averagereturn is in fact rather poorly measured. Thestandard statistical confidence interval extendsfrom 3 percent to 13 percent. Furthermore,average returns have been low following timesof high stock prices, such as the present. There-fore, the statistical evidence suggests a periodof quite low average returns, followed by slowreversion to a poorly measured long-term average,and it cautions us that statistical analysis aloneleaves lots of uncertainty.

Then, I survey economic theory to see ifstandard models that summarize a vast amountof other information shed light on stock returns.Standard models do not predict anything likethe historical equity premium. After a decadeof effort, a range of drastic modifications to thestandard models can account for the historicalequity premium. But it remains to be seenwhether the drastic modifications and a highequity premium, or the standard models and alow equity premium will triumph in the end. Insum, economic theory gives one further reasonto fear that long-term average excess returnswill not return to 8 percent, and it details the kindof beliefs one must have about the economy toreverse that pessimistic view.

ECONOMIC PERSPECTIVES4

However, I conclude with a warning thatlow average returns do not imply one shouldchange one’s portfolio. Someone has to holdevery stock on the market. An investor shouldonly hold less stocks than average if that investoris different from the average investor in someidentifiable way, such as risk exposure, attitude,or information.

Average returns and riskThe most obvious place to start thinking

about future stock returns is a statistical analysisof past stock returns.

Average real returnsTable 1 presents several measures of average

real returns on stocks and bonds in the postwarperiod. The value weighted NYSE portfolioshows an impressive annual return of 9 percentafter inflation. The S&P 500 is similar. Theequally weighted NYSE portfolio weightssmall stocks more than the value weightedportfolio. Small stock returns have been evenbetter than the market on average, so the equallyweighted portfolio has earned more than 11percent. Bonds by contrast seem a disaster.Long-term government bonds earned only 1.8percent after inflation, despite a standard devi-ation (11 percent) more than half that of stocks(about 17 percent). Corporate bonds earned aslight premium over government bonds, but at2.1 percent are still unappealing compared tostocks. Treasury bills earned only 0.8 percenton average after inflation.

A reward for riskTable 1 highlights a crucially important

fact. High average returns are only earned as acompensation for risk. High stock returns can-not be understood merely as high “productivityof the American economy” (or high marginal

productivity of capital) or impatience by con-sumers. Such high productivity or impatiencewould lead to high returns on bonds as well.To understand average stock returns, and toassess whether they will continue at these levels,it is not necessary to understand why the econ-omy gives such high returns to saving—itdoesn’t—but why it gives such high compensa-tion for bearing risk. The risk is substantial.A 17 percent standard deviation means themarket is quite likely to decline 9 – 17 = 8%or rise 9 + 17 = 26% in a year. (More precisely,there is about a 30 percent probability of adecline bigger than –8 percent or a rise biggerthan 26 percent.)

Risk at short and long horizonsIt is a common fallacy to dismiss this risk

as “short-run price fluctuation” and to argue thatstock market risk declines in the long run.

The most common way to fall into this trapis to confuse the annualized or average returnwith the actual return. For example, the two-yearlog or continuously compounded return is thesum of the one-year returns, r

0→2 = r

0→1+ r

1→2 .

Then, if returns were independent over time, likecoin flips, the mean and variance would scale thesame way with horizon: E(r

0→2 ) = 2E(r

0→1 )

and σ2(r 0→2

) = 2 σ2(r0→1

). Investors who caredabout mean and variance would invest the samefraction of their wealth in stocks for any returnhorizon. The variance of annualized returnsdoes stabilize; σ(1/2 r

0→2) = 1/2 σ2(r

0→1). But

the investor cares about the total, not annual-ized return. An example may clarify the dis-tinction. Suppose you are betting $1 on a coinflip. This is a risky bet, you will either gain orlose $1. If you flip the coin 1,000 times, the aver-age number of heads (annualized returns) willalmost certainly come out quite near 50 percent.

Annual real returns 1947–96TABLE 1

VW S&P500 EW GB CB TB

(- - - - - - - - - - - - - - - - - - percent - - - - - - - - - - - - - - - - - - -)

Average return E(R) 9.1 9.5 11.0 1.8 2.1 0.8

Standard deviation σ( )R 16.7 16.8 22.2 11.1 10.7 2.6

Standard error σ( ) /R T 2.4 2.4 3.0 1.6 1.5 0.4

Notes: VW = value weighted NYSE, EW = equally weighted NYSE, GB = ten-year government bond,CB = corporate bond, TB = three-month Treasury bills. All less CPI inflation.

Source: All data for this and subsequent figures and tables in this article are from the Center for Researchon Security Prices (CRSP) at the University of Chicago.


0 4 8 12 16 20 24 28 320

4

8

12

16average return, percent

standard deviation of return, percent

Treasury bill Government bonds

Corporate bonds

Value weightedS&P500

Equally weighted

However, the risk of the bet (total return) ismuch larger: It only takes an average numberof heads equal to 0.499 (that is, 499/1,000) tolose a dollar; if the average number of heads is0.490, still very close to 0.5, you lose $10. Justas we care about dollars, not the fraction ofheads, we care about total returns, not annual-ized rates.

To address the short-run price fluctuationfallacy directly, table 2 shows that mean returnsand standard deviations scale with horizon justabout as this independence argument suggests,out to five years.

(In fact, returns are not exactly indepen-dent over time. Estimates in Fama and French[1988a] and Poterba and Summers [1988]suggest that the variance grows a bit less slowlythan the horizon for the first five to ten years,and then grows with horizon as before, sostocks are in fact a bit safer for long horizonsthan the independence assumption suggests.However, this qualification does not rescue theannualized return fallacy. Also bear in mindthat long-horizon statistics are measured evenless well than annual statistics; there are onlyfive nonoverlapping ten-year samples in thepostwar period.)

The stock market is like a coin flip, but itis a biased coin flip. Thus, even though meanand variance may grow at the same rate withhorizon, the probability that oneloses money in the stock marketdoes decline over time. (For exam-ple, for the normal distribution, tailprobabilities are governed by E(r)/σ(r), which grows at the squareroot of horizon.) However, portfolioadvice is not based on pure proba-bilities of making or losing money;but on measures such as the meanand variance of return. Based onsuch measures, there is not muchpresumption that stocks are dramati-cally safer for long-run investments.

I cannot stress enough that thehigh average returns come only ascompensation for risk. Our taskbelow is to understand this risk andpeople’s aversion to it. Many dis-cussions, including those surroundingthe move to a funded social securitysystem, implicitly assume that onegets the high returns without taking

on substantial risk. What happens to a fundedsocial security system if the market goes down?

Means versus standarddeviations—Sharpe ratio

Figure 1 presents mean returns versus theirstandard deviations. In addition to the portfoli-os listed in table 1, I include ten portfolios ofNYSE stocks sorted by size. This pictureshows that average returns alone are not aparticularly useful measure. By taking on morerisk, one can achieve very high average re-turns. In the picture, the small stock portfolio

How risk and return vary withinvestment horizon

TABLE 2

1 8.6 17.1 0.50

2 9.1 17.9 0.51

3 9.2 16.8 0.55

5 10.5 21.9 0.48

Notes: Re = value weighted return less T-bill rate.Column one shows average excess return dividedby horizon. Column two shows standard deviationof excess return divided by square root of horizon.Column three shows Sharpe ratio divided by squareroot of horizon. All statistics in percent.

Horizon h (years)

FIGURE 1

Mean vs. standard deviation of real returns, 1947–96

Notes: Triangles are equally weighted and value weighted NYSE; S&P 500;three-month Treasury bill; ten-year government bond; and corporate bondreturns. Unmarked squares are NYSE size portfolios.

E R

h

e( ) σ(R

h

e) 1 (

( )h

E R

R

e

e

)

σ


earns over 15 percent per year average realreturn, though at the cost of a huge standarddeviation. Furthermore, one can form portfolioswith even higher average returns by leveraging—borrowing money to buy stocks—or investingin securities such as options that are very sensi-tive to stock returns. Since standard deviation(and beta or other risk measures) grow exactlyas fast as mean return, the extra mean returngained in this way exactly corresponds to theextra risk of such portfolios. When consideringeconomic models, it is easy to get them toproduce higher mean returns (along with higherstandard deviations) by considering claims toleveraged capital.

In sum, excess returns of stocks over Trea-sury bills are more interesting than the level ofreturns. This is the part of return that is a com-pensation for risk, and it accounts for nearly allof the amazingly high average stock returns.Furthermore, the Sharpe ratio of mean excessreturn to standard deviation, or the slope of aline connecting stock returns to a risk-freeinterest rate in figure 1, is a better measure ofthe fundamental characteristic of stocks thanthe mean excess return itself, since it is invari-ant to leveraging. The stock portfolios listed intable 1 all have Sharpe ratios near 0.5.

Standard errorsThe average returns and Sharpe ratios look

impressive. But are these true or just chance?One meaning of chance is this: Suppose that theaverage excess return really is low, say 3 percent.How likely is it that a 50-year sample has anaverage excess return of 8 percent? Similarly, ifthe next 50 years are “just like” the last 50, in thesense that the structure of the economy is the samebut the random shocks may be different, what isthe chance that the average return in the next 50years will be as good as it was in the last 50?

Since we only see one sample, these ques-tions are really unanswerable at a deep level.Statistics provides an educated guess in thestandard error. Assuming that each year’sreturn is statistically independent, our bestguess of the standard deviation of the averagereturn is σ / T , where σ is the standard devia-tion of annual returns and T is the data size.

This formula tells us something important:Stock returns are so volatile that it is very hardto statistically measure average returns. Table 1includes standard errors of stock returns mea-sured in the last 50 years, and table 3 shows

standard errors for a variety of horizons.The confidence interval, mean +/–2 standarddeviations, represents the 95 percent probabilityrange. As the table shows, even very long-termaverages leave a lot of uncertainty about meanreturns. For example, with 50 years of data, an8 percent average excess return is measuredwith a 2.4 percentage point standard error. Thus,the confidence interval says that the true aver-age excess return is between 8 – 2 × 2.4 = 3%and 8 + 2 × 2.4 = 13% with 95 percent proba-bility.1 This is a wide band of uncertainty aboutthe true market return, given 50 years of data.One can also see that five- or ten-year averagesare nearly useless; it takes a long time to sta-tistically discern that the average return hasincreased or decreased. As a cold winter neednot presage an ice age, so even a decade of badreturns need not change one’s view of the trueunderlying average return.

The standard errors are also the standarddeviations of average returns over the next Tyears, and table 3 shows that there is quite a lotof uncertainty about those returns. For example,if the true mean excess return is and will con-tinue to be 8 percent, the five-year standard errorof 7.6 percent is almost as large as the mean. Thismeans that there is still a good chance that thenext five-year return will average less than theTreasury bill rate.

On the other hand, though the averagereturn on stocks is not precisely known, the 2.4percent standard error means that we can confi-dently reject the view that the true mean excessreturn was zero or even 2–3 percent. The argu-ment that all the past equity premium was luckdoesn’t hold up well against this simple statis-tical argument.

Standard error of average returnat various horizons

TABLE 3

Horizon T Standard error σσ // T(years) (percentage points)

5 7.6

10 5.4

25 3.8

50 2.4

Note: Returns assumed to be statisticallyindependent with standard deviationσ = 17 percent.


Selection and crashesTwo important assumptions behind the

standard error calculation, however, suggest waysin which the postwar average stock return mightstill have been largely due to luck. Argentinaand the U.S. looked very similar at the middleof the last century. Both economies were un-derdeveloped relative to Britain and Germanyand had about the same per capita income. IfArgentina had experienced the U.S.’s growthand stock returns, and vice versa, this articlewould be written in Spanish from the BuenosAires Federal Reserve Bank, with high Argentinestock returns as the subject.

The statistical danger this story points to isselection or survival bias. If you flip one cointen times, the chance of seeing eight heads islow. But if you flip ten coins ten times, thechance that the coin with the greatest numberof heads exceeds eight heads is much larger.Does this story more closely capture the 50-yearreturn on U.S. stocks? Brown, Goetzmann, andRoss (1995) present a strong case that the uncer-tainty about true average stock returns is muchlarger than σ / T suggests. As they put it,“Looking back over the history of the Londonor the New York stock markets can be extraor-dinarily comforting to an investor—equitiesappear to have provided a substantial premiumover bonds, and markets appear to have recov-ered nicely after huge crashes. . . . Less com-forting is the past history of other major mar-kets: Russia, China, Germany, and Japan. Eachof these markets has had one or more majorinterruptions that prevent their inclusion inlong term studies” [my emphasis].

In addition, think of the things that didn’thappen in the last 50 years. There were no

banking panics, no depressions, no civil wars,no constitutional crises, the cold war was notlost, and no missiles were fired over Berlin,Cuba, Korea, or Vietnam. If any of these thingshad happened, there undoubtedly would havebeen a calamitous decline in stock values. Thestatistical problem is nonnormality. Taking thestandard deviation from a sample that did notinclude rare calamities, and calculating averagereturn probabilities from a normal distributionmay understate the true uncertainty. But inves-tors, aware of that uncertainty, discount pricesand hence leave high returns on the table.

We can cast the issue in terms of funda-mental beliefs about the economy. Was it clearto people in 1945 (or 1871, or whenever thesample starts) and throughout the period thatthe average return on stocks would be 8 percentgreater than that of bonds? If so, one wouldexpect them to have bought more stocks, evenconsidering the risk described by the 17 per-cent year-to-year variation. But perhaps it wasnot in fact obvious in 1945, that rather thanslipping back into depression, the U.S. wouldexperience a half century of growth never be-fore seen in human history. If so, much of theequity premium was unexpected; good luck.

Time varying expected returns

Regressions of returns on price/dividend ratiosWe are not only concerned with the aver-

age return on stocks but whether returns areexpected to be unusually low at a time of highprices, such as the present. The first and mostnatural thing one might do to answer this ques-tion is to look at a regression forecast. To thisend, table 4 presents regressions of returns onthe price/dividend (P/D) ratio.

OLS regressions of excess returns and dividend growth on VW P/D ratioTABLE 4

Horizon k(years) b σσσσσ(b) R 2 b σσσσσ(b) R 2

1 –1.04 (0.33) 0.17 –0.39 (0.18) 0.07

2 –2.04 (0.66) 0.26 –0.52 (0.40) 0.07

3 –2.84 (0.88) 0.38 –0.53 (0.43) 0.07

5 –6.22 (1.24) 0.59 –0.99 (0.47) 0.15

Notes: Rt→t +k indicates the k year return on the value weighted NYSE portfolio less the k yearreturn from continuously reinvesting in Treasury bills; b = regression slope coefficient(defined by the regression equation above); σ(b) = standard error of regression coefficient.Standard errors in parentheses use GMM to correct for heteroscedasticity and serial correlation.

Rt→→→→→t+k

= a + b(Pt/D

t) D

t+k/D

t = a + b(P

t/D

t)


The regression at a one-year horizonshows that excess returns are in fact predictablefrom P/D ratios, though the 0.17 R2 is not par-ticularly remarkable. However, at longer andlonger horizons, the slope coefficients increaseand larger and larger fractions of return varia-tion can be forecasted. At a five-year horizon,60 percent of the variation in stock returns canbe forecasted ahead of time from the P/D ratio.(Fama and French, 1988b, is a famous earlysource for this kind of regression.)

One can object to dividends as the divisorfor prices. However, price divided by justabout anything sensible works about as well,including earnings, book value, and movingaverages of past prices. There seems to be anadditional business-cycle component of expectedreturn variation that is tracked by the term spreador other business cycle forecasting variables,including the default spread and investment–capital ratio, the T-bill rate, the ratio of the T-bill rate to its moving average, and the dividend/earnings ratio. (See Fama and French, 1989,for term and default spreads, Campbell, 1987,for term spread, Cochrane, 1991c, for investment–capital ratios, Lamont, 1997, for dividend/earn-ings, and Ferson and Constantinides, 1991, foran even more exhaustive list with references).However, price ratios such as P/D are the mostimportant forecasting variables, especially at longhorizons, so I focus on the P/D ratio to keep the

analysis simple. In a similar fashion, cross-sectional variation in expected returns can bevery well described by the P/D ratio or (better)the ratio of market value to book value, whichcontains the price in its numerator. Portfoliosof “undervalued” or “value” stocks with lowprice ratios outperform portfolios of “overvalued”or “growth” stocks with high price ratios. (SeeFama and French, 1993).

Slow moving P/D and P/EFigure 2 presents P/D and price/earnings

(P/E) ratios over time. This graph emphasizesthat price ratios are very slow moving variables.This is why they forecast long-horizon move-ments in stock returns.

The rise in forecast power with horizon isnot a separate phenomenon. It results from theability to forecast one period returns and theslow movement in the P/D ratio.2 As an analogy,if it is ten degrees below zero in Chicago (lowP/D ratio), one’s best guess is that it will warmup a degree or so per day. Spring does come,albeit slowly. However, the weather varies a lot;it can easily go up or down 20 degrees in a day,so this forecast is not very accurate (low R2).But the fact that it is ten degrees below zerosignals that the temperature will rise a bit onaverage per day for many days. By the time welook at a six-month horizon, we forecast a 90degree rise in temperature. The daily variationof 20 degrees is still there, but the change in

temperature (90 degrees) that canbe forecasted is much largerrelative to the daily variation,implying a high R2.

The slow movement in theP/D ratio also means that theability to forecast returns is notthe fabled alchemists’ stone thatturns lead into gold. A high P/Dratio means that prices willgrow more slowly than divi-dends for a long time until theP/D ratio is reestablished, andvice versa. Trading on thesesignals—buying more stocks intimes of low prices, and less intimes of high prices—can raise(unconditional) average returnsa bit, but not much more than1 percent for the same standard

0

20

40

60

1948 ’56 ’64 ’72 ’80 ’88 ’96

VW price/dividendratio

2 x S&P500price/earnings ratio

FIGURE 2

P/D and P/E ratios

Notes: VW is value weighted NYSE portfolio. Two times the S&P 500 P/Eratio is plotted so that the lines can be more easily compared.


deviation. If there were a 50 percent R2 at adaily horizon, one could make a lot of money;but not so at a five-year horizon.

The slow movement of the P/D ratio alsomeans that on a purely statistical basis, one cancast doubt on whether the P/D ratio reallyforecasts returns. What we really know, look-ing at figure 2 (figure 4 also makes this point),is that low prices relative to dividends andearnings in the 1950s preceded the boom mar-ket of the early 1960s; that the high P/D ratiosof the mid-1960s preceded the poor returns ofthe 1970s; and that the low price ratios of themid-1970s preceded the current boom. We alsoknow that price ratios are very high now. In anyreal sense, there really are three data points. Ido not want to survey the extensive statisticalliterature that formalizes this point, but it isthere. Most importantly, it shows that the t-statis-tics one might infer from regressions such astable 4 are inflated; with more sophisticatedtests, return predictability actually has about a10 percent probability value before one startsto worry about fishing and selection biases.

What about repurchases? P/Eand other forecasts

Is the P/D ratio still a valid signal? Per-haps increasing dividend repurchases meanthat the P/D ratio will not return to its histori-cal low values; perhaps it has shifted to a newmean so today’s high ratio is notbad for returns. To address thisissue, figure 2 plots the S&P 500 P/Eratio along with the P/D ratio. Thetwo measures line up well. The P/Eratio forecasts returns almost aswell as the P/D ratio. The P/E ratio,price/book value, and other ratiosare also at historic highs, forecastinglow returns for years to come. Yetthey are of course immune to thecriticism that the dividend–earningsrelationship might be fundamentallydifferent from the past.

Return forecastsWhat do the regressions of table

4 say, quantitatively, about futurereturns? Figure 3 presents one-yearreturns and the P/D ratio forecast.Figure 4 presents five-year returnsand the P/D ratio forecast. I includein-sample and out-of-sample fore-casts in figures 3 and 4. To form

the out-of-sample forecasts, I paired the regres-sions from table 4 with an autoregression of P/D

t,

P D P Dt t t+ += + +1 1µ ρ δ .

Then, for example, since my data runthrough the end of 1996, the forecast returnsfor 1997 and 1998 are

E R a b

E R a b

( ) ( )

( ) ( ),1997 1996

1998 1996

= += + +

P D

P Dµ ρ

and so on.3

The one-year return forecast is extraordi-narily pessimistic. It starts at a –8 percent excessreturn for 1997, and only very slowly returns tothe estimated unconditional mean excess returnof 8 percent. In ten years, the forecast is still –5percent, in 25 years it is –1.75 percent, and it isstill only 2.35 percent in 50 years. The five-yearreturn forecasts are similarly pessimistic.

Of course, this forecast is subject to lotsof uncertainty. There is uncertainty about whatactual returns will be, given the forecasts. Thiswill always be true: If one could preciselyforecast the direction of stock prices, stockswould cease to be risky and would cease topay a risk premium. There is also a great dealof uncertainty about the forecasts themselves.

-50

-25

0

25

50

1950 ’60 ’70 ’80 ’90 2000 ’10 ’20

percent, one-year excess return

In-sample forecast

Actual returns

Out-of-sample forecast

FIGURE 3

Actual and forecast one-year excess returns

Notes: One-year excess returns on the value weighted NYSE andforecast from a regression on the P/D ratio. Returns are plotted on theday of the forecast. For example, 1995 plots a + b × P/D

1995 and the

1996 return. The out-of-sample forecast is made by joining Rt+1 = a + bP/Dt

with P/Dt+1 = µ + ρP/Dt.


The forecasts attempt to measure expectedreturns, the quantity that investors must tradeoff against unavoidable risk in deciding howattractive an investment is, and they undoubt-edly measure expected returns with error.

The plots of actual returns on top of thein-sample, one-year-ahead forecasts in figures 3and 4 give one measure of the forecast uncertainty.One can see that year-to-year returns are quite

likely to vary a lot given theforecast. Five-year returns trackthe forecast more closely, buthere the chance of over-fittingis greater.

To get a handle on howreliable or robust the pessimisticforecast is, figure 5 gives ascatter plot of one-year returnsand their forecasts based onP/D, together with the fittedregression line. The scatterplotindicates that the regressionresults are not spurious, or theresult of a few outlying years.

The point marked “97?” isthe P/D ratio at the end of 1996together with the forecast returnfor 1997. We see immediatelyone source of trouble with thepoint forecast: the P/D ratio hasnever in the postwar period beenas high as it is now. Extendinghistorical experience to never-

before seen values is always dangerous. One isparticularly uncomfortable with a predictionthat the market should earn less than the T-billrate, given the strong theoretical presumptionfor a positive expected excess return.4 Onecould easily draw a downward sloping linethrough the points, flattening out on the right,predicting a zero excess return for P/D ratiosabove 30 to 35, and never predicting a negative

excess return. A nonlinear re-gression that incorporates thisidea will fit about as well as thelinear regression I have run.However, the scatterplot doesnot demand such a nonlinearrelation either, so this is largelya matter of choice. In sum,while the scatterplot does sug-gest that the current forecastshould be low, it does not giverobust evidence that the forecastexcess return should be negative.

What about the last few years ofhigh returns?

The P/D ratios also pointedto low returns in 1995 and1996. Anyone who took thatadvice missed out on a dramaticsurge in the market, and somefund managers who took that

-60

-30

0

30

60

90

120

150

180

1950 ’60 ’70 ’80 ’90 2000 ’10 ’20

percent, five-year excess return

In-sample forecastActual returns

Out-of-sample forecast

FIGURE 4

Actual and forecast five-year excess returns

Notes: Five-year excess returns on value weighted NYSE (five-year returnsminus five-year T-bill returns) and forecast from a regression of five-yearreturns on the P/D ratio. Returns are plotted on the last day; for example,1996 plots the forecast a + b × P/D

1991 and the return from 1991 to 1996.

The out-of-sample forecast is formed for 1997–2001 from the observedP/D ratios, 1992–96, and for 2002 onward from P/Dt = µ + 0.98P/Dt–1

.

10 15 20 25 30 35 40 45-40

-20

0

20

40

60excess return, percent

price/dividend ratio

97?94

96

95

FIGURE 5

One-year excess returns vs. one-year aheadforecast from P/D


advice are now unemployed. Doesn’t thismean that the P/D signal should no longerbe trusted?

To answer this criticism, look at the figuresagain. They make clear that the returns for 1995and 1996 and even another 20 percent or soreturn for 1997 are not so far out of line, despitea pessimistic P/D forecast, that we should throwaway the regression based on the previous 47years of experience. To return to the analogy,if it is ten degrees below zero in Chicago, thatmeans spring is coming. But we can easily havea few weeks of 20 degree below weather beforespring finally arrives. The graphs make vividhow large a 17 percent standard deviationreally is, and to what extent the forecasts basedon the P/D ratio mark long-term tendenciesthat are still subject to lots of short-term swingsrather than accurate forecasts of year-to-yearbooms or crashes.

Another source of uncertainty about theforecast is how persistent the P/D ratio reallyis. If, for example, the P/D ratio had no persis-tence, then the low return forecast would onlylast a year. After that, it would return to theunconditional mean of 9 percent (8 percentover Treasury bills). Now, given a true valueρ = 0.98 in P/D

t = µ + ρP/D

t-1 + δ

t, the median

ordinary least squares (OLS) estimate is 0.90,as I found in sample. That is why figure 3 usesthe value ρ = 0.98. However, given this truevalue, the OLS estimate lies between 0.83 and0.94 only 50 percent of the time and between0.66 and 1.00 for 95 percent of the time. Thus,there is a huge range of uncertainty over the truevalue of ρ. The best thing that could happen tothe forecast is if the P/D ratio were really lesspersistent than it seems. In this case, the near-term return forecast would be unchanged, butthe long-term return forecast would return to 9percent much more quickly.

Variance decompositionWhen prices are high relative to dividends

(or earnings, cash flow, book value, or someother divisor), one of three things must betrue: 1) Investors expect dividends to rise in thefuture. 2) Investors expect returns to be low in thefuture. Future cash flows are discounted at alower than usual rate, leading to higher prices.3) Investors expect prices to keep rising forever,in a “bubble.” This is not a theory, it is anaccounting identity like 1=1: If the P/D ratio ishigh, either dividends must rise, prices must

decline, or the P/D ratio must never return to itshistorical average. Which of the three optionsholds for our stock market?

Historically, virtually all variation in P/Dratios has reflected varying expected returns.At a simple level, table 4 makes this point withregressions of long-horizon dividend growthon P/D ratios to match the regressions of returnson P/D ratios. The dividend-growth coefficientsare much smaller, typically one standard errorfrom zero, and the R2 values are tiny. Worse,the signs are wrong. To the extent that a highP/D ratio forecasts any change in dividends, itseems to forecast a small decline in dividends.

To be a little more precise, the identity

1 = +1–1

+1 +1–1R R Rt t t= ++ +t t

t

P DP

1 1

yields, with a little algebra, the approximateidentity

1

1

) ( )

lim ( ),

t tj

t j t j

jt j t j

p dj

d r

jp d

− = +=

∞− +

→ ∞−

∑ + +

+ +

const. ρ

ρ

∆

where ρ = P/D/(1 + P/D) is a constant of approxi-mation, slightly less than one and lowercaseletters denote logarithms (Campbell and Shiller,1988). Equation 1 gives a precise meaning tomy earlier statement that a high P/D ratio mustbe followed by high dividend growth ∆ d, lowreturns r, or a bubble.

Bubbles do not appear to be the reason forhistorical P/D ratio variation. Unless the P/Dratio grows faster than 1/ρ j, there is no bubble.It is hard to believe that P/D ratios can growforever. Empirically, P/D ratios do not seemto have a trend or unit root over time.5

This still leaves two possibilities: are highprices signals of high dividend growth or lowreturns? To address this issue, equation 1 implies6

21

1

) ( ) ( , )

( , ).

var cov

c

t t t tj

j

t j

t tj

j

t j

p d p d d

ov p d r

− = − −

−

=

∞

+

=

∞

+

∑

∑

ρ

ρ

∆

P/D ratios can only vary if they forecastchanging dividend growth or if they forecast


changing returns. Equation 2 has powerfulimplications. At first glance, it would seem areasonable approximation that returns cannot beforecasted (the “random walk” hypothesis) andneither can dividend growth. But if this werethe case, the P/D ratio would have to be a con-stant. Thus, the fact that the P/D ratio varies atall means that either dividend growth or returnscan be forecasted.

This observation solidifies one’s belief inP/D ratio forecasts of returns. Yes, the statisticalevidence that P/D ratios forecast returns is weak.But P/D ratios have varied. The choice is, P/Dratios forecast returns or they forecast dividendgrowth. They have to forecast something. Giventhis choice and table 3, it seems a much firmerconclusion that they forecast returns.

Table 5 presents some estimates of theprice-dividend variance decomposition (equa-tion 2), taken from Cochrane (1991b). As onemight suspect from table 4, table 5 shows thatin the past almost all variation in P/D ratioshas been due to changing return forecasts.(The rows of table 5 do not add up to exactly100 percent because equation 2 is an approxi-mation. The elements do not have to be be-tween 0 and 100 percent. For example, –34,138 occurs because high prices seem to fore-cast lower dividend growth. Therefore theymust and do forecast really low returns. Thereal and nominal rows differ because P/D fore-casts inflation in the sample.)

So much for history. What does it mean?Again, we live at a moment of historicallyunprecedented P/D, P/E, and other multiples.Perhaps this time high prices reflect high long-rundividend growth. If so, the prices have to reflectan unprecedented expectation of future dividend

growth: the P/D ratio is about double its long-term average, so the level of dividends has todouble, above and beyond its usual growth.However, if this time is at all like the past, highprices reflect low future returns.

The bottom lineStatistical analysis suggests that the long-

term average return on broad stock marketindexes is 8 percent greater than the T-bill rate,with a standard error of about 3 percent. Highprices are related to low subsequent excessreturns. Based on these patterns, the expectedexcess return (stock return less T-bill rate) isnear zero for the next five years or so, and thenslowly rising to the historical average. The largestandard deviation of excess returns, about 17percent, means that actual returns will certainlydeviate substantially from the expected return.Finally, one always gets more expected returnby taking on more risk.

Economics: Understanding theequity premium

Statistical analysis of past returns leavesa lot of uncertainty about future returns. Further-more, it is hard to believe that average excessreturns are 8 percent without knowing whythis is so. Perhaps most important, no statisticalanalysis can predict if the future will be likethe past. Even if the true expected excess returnwas 8 percent, did that result from fundamen-tal or temporary features of the economy?Thus, we need an economic understandingof stock returns.

Economic theory and modeling is oftenportrayed as an ivory tower exercise, out oftouch with the real world. Nothing could befurther from the truth, especially in this case.Many superficially plausible stories have beenput forth to explain the historically high returnon stocks and the time-variation of returns.Economic models or theories make these storiesexplicit, check whether they are internallyconsistent, see if they can quantitatively explainstock returns, and check that they do not makewildly counterfactual predictions in other dimen-sions, for example, requiring wild variation inrisk-free rates or strong persistent movementsin consumption growth. Few stories survivethis scrutiny.

We have a vast experience with economictheory; a range of model economies haveformed the backbone of our understanding of

Variance decompositionof VW P/D ratio

TABLE 5

Dividends Returns

Real –34 138

Standard error (10) (32)

Nominal 30 85

Standard error (41) (19)

Notes: VW = value weighted NYSE. Table entriesare the percent of the variance of the price/dividendratio attributable to dividend and return forecasts,100 × cov(pt–dt, Σ

15j=1 ρ

j ∆dt+j)/var(pt–dt) and similarlyfor returns.


economic growth and dynamic micro, macro,and international economics for close to 25years. Does a large equity premium make sensein terms of such standard economics? Did peoplein 1947, and throughout the period, know thatstocks were going to yield 8 percent over bondson average, yet were rationally unwilling tohold more stocks because they were afraidof the 17 percent standard deviation or someother measure of stocks’ risk? If so, we have“explained” the equity premium. If so, statis-tics from the past may well describe the future,since neither people’s preferences nor the riski-ness of technological opportunities seems tohave changed dramatically. But what if it makesno sense that people should be so scared ofstocks? In this case, it is much more likely thatthe true premium is small, and the historicalreturns were in fact just good luck.

The answer is simple: Standard economicmodels utterly fail to produce anything like thehistorical average stock return or the variationin expected returns over time. After ten yearsof intense effort, there is a range of drasticmodifications to standard models that canexplain the equity premium and return predict-ability and (harder still) are not inconsistentwith a few obvious related facts about con-sumption and interest rates. However, thesemodels are truly drastic modifications; theyfundamentally change the description of thesource of risk that commands a premium inasset markets. Furthermore, they have not yetbeen tested against the broad range of experi-ence of the standard models. These facts mustmean one of two things. Either the standardmodels are wrong and will change drastically,or the phenomenon is wrong and will disappear.

I first show how the standard model utterlyfails to account for the historical equity premium(Sharpe ratio). The natural response is to seeif perhaps we can modify the standard model.I consider what happens if we simply allow avery high level of risk aversion. The answerhere, as in many early attempts to modify thestandard model, is unsatisfactory. While onecan explain the equity premium, easy explana-tions make strongly counterfactual predictionsregarding other facts. The goal is to explain theequity premium in a manner consistent with thelevel and volatility of consumption growth (bothabout 1 percent per year), the predictability ofstock returns described above, the relative lackof predictability in consumption growth, the

relative constancy of real risk-free rates overtime and across countries, and the relativelylow correlation of stock returns with consump-tion growth. This is a tough assignment, whichis only now starting to be accomplished.

Then I survey alternative views that dopromise to account for the equity premium,without (so far) wildly counterfactual predic-tions on other dimensions. Each modificationis the culmination of a decade-long effort by alarge number of researchers. (For literaturereviews, see Kocherlakota, 1996, and Cochraneand Hansen, 1992). The first model maintainsthe complete and frictionless market simplifi-cation, but changes the specification of howpeople feel about consumption over time, byadding habit persistence in a very special waythat produces a strong precautionary savingmotive. The second model abandons the perfectmarkets simplification. Here, uninsurable indi-vidual-level risks are the key to the equity pre-mium. I will also discuss a part of an emergingview that the equity premium and time-variationof expected returns result from the fact that fewpeople hold stocks. This view is not flushedout yet to a satisfactory model, but does givesome insight.

Both modifications answer the basic ques-tion, “why are consumers so afraid of stocks?”in a similar way, and give a fundamentallydifferent answer from the standard model’sview that expected returns are driven by risksto wealth or consumption. The modificationsboth say that consumers are really afraid ofstocks because stocks pay off poorly in reces-sions. In one case a recession means a timewhen consumption has recently fallen, no matterwhat its level. In the second case a recession isa time of unusually high cross-sectional (thoughnot aggregate) uncertainty. In both cases theraw risk to wealth is not a particularly impor-tant part of the story.

The standard modelTo say anything about dynamic economics,

we have to say something about how peopleare willing to trade consumption in one momentand set of circumstances (state of nature) forconsumption in another moment and set ofcircumstances. For example, if people werealways willing to give up a dollar of consump-tion today for $1.10 in a year, then the economywould feature a steady 10 percent interest rate.It also might have quite volatile consumption,


as people accept many such opportunities. Ifpeople did not care what the circumstanceswere in which they would get $1.10, then allexpected returns would be equal to the interestrate (risk neutrality). Of course, this is an ex-treme example. There certainly comes a pointat which such willingness to substitute con-sumption becomes strained. If someone weregoing to consume $1,000 this year but $10,000next year, it might take a bit more than a 10percent interest rate to get him or her to con-sume even less this year.

To capture these ideas about people’swillingness to substitute consumption, we usea utility function that gives a numerical “happi-ness” value for every possible stream of futureconsumption,

30

) ( ) .U E e u dttt t= −=∞z ρ C

E denotes expectation; Ct denotes consumption

at date t; ρ is the subjective discount factor;and e–ρt captures the fact that consumers preferearlier consumption to later. The function u(⋅)is increasing and concave, to reflect the ideathat people always like more consumption, butat a diminishing rate. The function u(C) = C1–γ

is a common specification, with γ between 0and 5. γ = 0 or u(C) = C corresponds to riskneutrality, a constant interest rate ρ, and aperfect willingness to substitute across time.γ = 1 corresponds to u(C) = ln(C), which is avery attractive choice since it implies that eachdoubling of consumption adds the same amountof happiness. For most asset pricing problems,writing the utility function over an infinitelifespan is a convenient simplification thatmakes little difference to the results. Economicmodels are often written in discrete time, inwhich case the utility function is

U E e uCt

t

t= −

=

∞

∑ ρ

0

( ).

Dynamic economics takes this representa-tion of people’s preferences and mixes it witha representation of technological opportunitiesfor production and investment. For example,the simplest model might specify that output ismade from capital, Y = f(K), output is investedor consumed, Y = C+I, and capital depreciatesbut is increased by investment, K

t+1 = (1-δ)K

t + I

t

in discrete time. To study business cycles, one

adds detail, including at least labor, leisure,and shocks. To study monetary issues, oneadds some friction that induces people to usemoney, and so on.

Despite the outward appearance of tension,this is a great unifying moment for macroeco-nomics. Practically all issues relating to busi-ness cycles, growth, aggregate policy analysis,monetary economics, and international economicsare studied in the context of variants of thissimple model. The remaining differences con-cern details of implementation.

Since this basic economic frameworkexplains such a wide range of phenomena, whatdoes it predict for the equity risk premium? Givethe opportunity to buy assets such as stocksand bonds to a consumer whose preferencesare described by equation 3, and figure outwhat the optimal consumption and portfoliodecision is. (The appendix includes derivationsof all equations.) The following conditionsdescribe the optimal choice:

4) ( )r E cf = +ρ γ ∆

5) ( )

) ( ) ( , ),

E r r

c r corr c r

f− = γγ σ σ

cov c,r = ( )

(

∆∆ ∆

where ∆c denotes the proportional change inconsumption, r denotes a risky asset return, r f

denotes the risk-free rate, cov denotes covari-ance, corr denotes correlation, and

γ ≡ − ′′′

Cu C

u C

( )

( )

is a measure of curvature or risk aversion.Higher γ means that more consumption givesless pleasure very quickly; it implies that peopleare less willing to substitute less consumptionnow for more consumption later and to take risks.

Equation 5 expresses the most fundamentalidea in finance. It says that the average excessreturn on any security must be proportional tothe covariance of that return with marginalutility and, hence, consumption growth. Thisis because people value financial assets thatcan be used to smooth consumption over timeand in response to risks. For example, a “risky”stock, one that has a high standard deviationσ(r), may nonetheless command no greateraverage return E(r) than the risk-free rate if its


return is uncorrelated with consumption growthcorr(∆c,r) = 0. If it yields any more, the consumercan buy just a little bit of the security, and comeout ahead because the risk is perfectly diversifi-able. Readers familiar with the capital assetpricing model (CAPM) will recognize the intu-ition; replacing wealth or the market portfoliowith consumption gives the most modern andgeneral version of that theory.

The equity premium puzzleTo evaluate the equity premium, I transform

equation 5 to

6)( )

( )( ) ( , ).

E r r

rc corr c r

f− =σ

γ σ ∆ ∆

The left-hand side is the Sharpe ratio. As Ishowed above, the (unconditional) Sharpe ratiois about 0.5 for the stock market, and it is robustto leveraging or choice of assets. The right-handside of equation 6 says something very impor-tant. A high Sharpe ratio or risk premium mustbe the result of 1) high aversion to risk, γ, or 2)lots of risk, σ(∆c). Furthermore, it can only occurfor assets whose returns are correlated with therisks. This basic message will pervade the follow-ing discussion of much-generalized economicmodels. If the right-hand side of equation 6 islow, then the consumer should invest more inthe asset with return r. Doing so will make theconsumption stream more risky and more corre-lated with the asset return. Thus, as the consumerinvests more, the right-hand side of equation 6will approach the left-hand side.

The right-hand side of equation 6 is a pre-diction of what the Sharpe ratio should be. Itdoes not come close to predicting the historicalequity premium. The standard deviation ofaggregate consumption growth is about 1 percentor 0.01. The correlation of consumption growthwith stock returns is a bit harder to measuresince it depends on horizon and timing issues.Still, for horizons of a year or so, 0.2 is a prettygenerous number. γ �1 or 2 is standard; γ = 10is a very generous value. Putting this all together,10 × 0.01 × 0.2 = 0.02 rather than 0.5. At a20 percent standard deviation, a 0.02 Sharperatio implies an average excess return forstocks of 0.02 × 20 = 0.4% (40 basis points)rather than 8 percent.

This devastating calculation is the celebrated“equity premium puzzle” of Mehra and Prescott

(1985), as reinterpreted by Hansen and Jagan-nathan (1991). The failure is quantitative notqualitative, as Kocherlakota (1996) points out.Qualitatively, the right-hand side of equation 6does predict a positive equity premium. Theproblem is in the numbers. This is a strongadvertisement for quantitative rather than justqualitative economics.

Can we change the numbers?The correlation of consumption growth

with returns is the most suspicious ingredientin this calculation. While the correlation isundeniably low in the short run, a decade-longrise in the stock market should certainly lead tomore consumption. In fact, the low correlationis somewhat of a puzzle in itself: Standard (one-shock) models typically predict correlations of0.99 or more. Marshall and Daniel (1997) findcorrelations in the data up to 0.4 at a two-yearhorizon, and by allowing lags. But even pluggingin a correlation of corr(∆c,r) = 1, σ(∆c) = 0.01and γ < 10 implies a Sharpe ratio less than 0.1, orone-fifth the sample value.

A large literature has tried to explain theequity premium puzzle by introducing frictionsthat make T-bills “money-like,” which artifi-cially drive down the interest rate (for example,Aiyagari and Gertler, 1991). The highest Sharperatio occurs in fact when one considers short-term risk-free debt and money, since the latterpays no interest. Perhaps the same mechanismcan be invoked for the spread between stocksand bonds. However, a glance at figure 1 showsthat this will not work. High Sharpe ratios arepervasive in financial markets. One can recovera high Sharpe ratio from stocks alone, or fromstocks less long-term bonds.

Time-varying expected returnsThe consumption-based view with

u′(C) = C-γ also has trouble explaining the factthat P/D ratios forecast stock returns. Considerthe conditional version of equation 6

7)( )

( )( ) ( , ),

E r r

rc corr c rt t

f

tt t

− =σ

γ σ ∆ ∆

where Et, σ

t , corr

t represent conditional moments.

I showed above that the P/D ratio gives a strongsignal about mean returns, E

t(r). It does not

however give much information about thestandard deviation of returns. Figure 5 doessuggest a slight increase in return standard


deviation along with the higher mean returnwhen P/D ratios decline—the leverage effectof Black (1976). However, the increase instandard deviation is much less than the increasein mean return. Hence, the Sharpe ratio of meanto standard deviation varies over time andincreases when prices are low.

How can we explain variation in the Sharperatio? Looking at equation 7, it could happen ifthere were times of high and low consumptionvolatility, variation in σ

t(∆c). But that does not

seem to be the case; there is little evidence thataggregate consumption growth is much morevolatile at times of low prices than high prices.The conditional correlation of consumptiongrowth and returns could vary a great dealover time, but this seems unlikely, or moreprecisely like an unfathomable assumptionon which to build the central understandingof time-varying returns.

What about the CAPM?Finance researchers and practitioners often

express disbelief (and boredom) with consump-tion-based models such as the above. Even theCAPM performs better: Expected returns ofdifferent portfolios such as those in figure 1 lineup much better against their covariances withthe market return than against their covarianceswith consumption growth. Why not use theCAPM or other, better-performing finance modelsto understand the equity premium?

The answer is that the CAPM and relatedfinance models are useless for understanding themarket premium. The CAPM states that the ex-pected return of a given asset is proportional to its“beta” times the expected return of the market,

E R R E R Ri fi m

m f( ) ( ) .,= + −β

This is fine if you want to think about anindividual stock’s return given the marketreturn. But the average market return—thething we are trying to explain, understand, andpredict—is a given to the CAPM. Similarly,multifactor models explain average returns onindividual assets, given average returns on“factor mimicking portfolios,” including themarket. Option pricing models explain optionprices, given the stock price. To understand themarket premium, there is no substitute for eco-nomic models such as the consumption-basedmodel outlined above and its variants.

Highly curved power utilitySince we have examined all the other

numbers on the right-hand side of equation 6,perhaps we should raise curvature γ. This is acentral modification. All of macroeconomicsand growth theory considers values of γ nolarger than 2–3. To generate a Sharpe ratio of0.5, γ = 250 is needed in equation 6. Even ifcorr = 1, γ must still equal 50. What’s wrongwith γ = 50 to 250? Although a high curvatureγ explains the equity premium, it runs quicklyinto trouble with other facts.

Consumption and interest ratesThe most basic piece of evidence for low γ

is the relationship between consumption growthand interest rates. Real interest rates are quitestable over time (see the standard deviation intable 1) and roughly the same the world over,despite wide variation in consumption growthover time and across countries. A value of γ = 50to 250 implies that consumers are essentiallyunwilling to substitute consumption over time;equivalently that variation in consumptiongrowth must be accompanied by huge variationsin interest rates that we do not observe.

Look again at the first basic relationshipbetween consumption growth and interestrates, equation 4, reproduced here:

r E ctf

t= +ρ γ ( ).∆

High values of γ are troublesome even inunderstanding the level of interest rates. Aver-age consumption growth and real interest ratesare both about 1 percent. Thus, γ = 50 to 250requires ρ = –0.5 to –2.5, or a –50 percent to–250 percent subjective discount rate. That’sthe wrong sign: People should prefer currentto future consumption, not the other wayaround (Weil, 1989).7

The absence of much interest rate variationacross time and countries is an even biggerproblem. People save more and defer consump-tion in times of high interest rates, so consump-tion growth rises when interest rates are higher.However, γ = 50 means that a country withconsumption growth 1 percentage point higherthan normal must have real interest rates 50percentage points higher than normal, andconsumption 1 percentage point lower thannormal must have real interest rates 50 percent-age points lower than normal, implying hugenegative interest rates—consumers pay financial


intermediaries 48 percent to keep their money.This just does not happen.

This observation can also be phrased asa conceptual experiment, suitable for thinkingabout one’s own preferences or for surveyevidence on others’ preferences. For example,what does it take to convince someone to skipa vacation? Take a family with $50,000 per yearincome, consumption equal to income, whichspends $2,500 (5 percent) on an annual vaca-tion. If interest rates are good enough, though,the family can be persuaded to skip this year’svacation and go on two vacations next year.What interest rate does it take to persuade thefamily to do this? The answer is ($52,500/$47,500)γ – 1. For γ = 250 that is an interestrate of 3 × 1011. For γ = 50, we still need aninterest rate of 14,800 percent. I think most ofus would defer the vacation for somewhatlower interest rates.

The standard use of low values for γ inmacroeconomics is also important for deliveringrealistic quantity dynamics in macroeconomicmodels, including relative variances of invest-ment and output, and for delivering reasonablespeeds of adjustment to shocks.

Risk aversionEconomists have also shied away from high

curvature γ on the basis that people do notseem that risk averse. After examining theargument, I conclude that there are fewer solidreasons to object to high risk aversion than toobject to high aversion to intertemporal substi-tution via the consumption-interest rate relation-ships I examined above.

Surveys and thought experimentsSince Sharpe ratios are high for many

assets, much analysis of risk aversion comesfrom simple thought experimentsrather than data. For example, howmuch would a family pay per yearto avoid a bet that led with equalprobability to a $y increase or de-crease in annual consumption forthe rest of their lives? Table 6presents some calculations of howmuch our family with $50,000 peryear of income and consumptionwould pay to avoid various bets ofthis form.8 For bets that are reason-ably large relative to wealth, high γmeans that families are willing topay almost the entire amount of the

bet to avoid taking it. For example, in the low-er right-hand corner, the family with γ = 250would rather pay $9,889 for sure than take a50 percent chance of a $10,000 loss. This predic-tion is surely unreasonable, and has led mostauthors to rule out risk aversion coefficientsover ten. Survey evidence for this kind of betalso finds low risk aversion, certainly below γ = 5(Barsky, Kimball, Juster, and Shapiro, 1997), andeven negative risk aversion if the survey is takenin Las Vegas.

Yet the results for small bets are not so unrea-sonable. The family might reasonably pay 5 centsto 25 cents to avoid a $10 bet. We are all riskneutral for small enough bets. For small bets,

amount willing to pay to avoid bet

size of bet

size of bet

consumption≈ γ .

Thus, for any γ, the amount one is willingto pay is an arbitrarily small fraction of the betfor small enough bets. For this reason, it iseasy to cook numbers of conceptual experi-ments like table 6 by varying the size of the betand the presumed wealth of the family. Signifi-cantly, I only used local curvature above; γrepresented the derivative γ = –Cu″(C)/u′(C).In asking how much the family would pay toavoid a $10,000 bet, we are asking for theresponse to a very, very non-local event.

The main lesson of conceptual experimentsand laboratory and survey evidence of simplebets is that people’s answers to such questionsroutinely violate expected utility. This observa-tion lowers the value of this source of evidenceas a measurement of risk aversion. As a similarcautionary note, Barsky et al. report that whetheran individual partakes in a wide variety ofrisky activities correlates poorly with the level

Amount family would pay to avoid an even betTABLE 6

Risk aversion γγγγγBet 2 10 50 100 250

$10 $0.00 $0.01 $0.05 $0.10 $0.25

100 0.20 1.00 4.99 9.94 24

1,000 20 99 435 665 863

10,000 2,000 6,921 9,430 9,718 9,889

Notes: I assume the family has a constant $50,000 per yearconsumption and an even chance of winning or losing theindicated net, per year.


of risk aversion inferred from a survey. In theend, surveys about hypothetical bets that arefar from the range of everyday experience arehard to interpret.

Microeconomic evidenceMicroeconomic observations might be a

more useful measure of risk aversion, that is,evidence from people’s actual behavior in theirdaily activities. For example, the numbers intable 6 could be matched with insurance data.People are willing to pay substantially morethan actuarially fair values to insure against cartheft or house fires. What is the implied riskaversion? But even if there are other marketswhose prices reflect less risk aversion thanstocks, this leaves open the question: If peopleare risk-neutral in other markets, why do theybecome risk averse in the stockbroker’s office?Perhaps the risk aversion people display in thestock broker’s office should be the fact and the(possibly) low risk aversion displayed in otheroffices should be the puzzle.

Portfolio calculationsA common calibration of risk aversion

comes from simple portfolio calculations(see Friend and Blume, 1975). Following theprinciple that the last dollar spent should givethe same increase in happiness in any alterna-tive use, the marginal value of wealth shouldequal the marginal utility of consumption,9

VW(W,.) = u

c(C). Therefore, if we assume returns

are independent over time and no other vari-ables are important for the marginal value ofwealth, V

W(W), equation 6 can also be written as

8)( )

( )( ) ( , ).

E r r

r

WV

Vw corr w r

fWW

W

− = −σ

σ ∆ ∆

The quantity –WWWW

/VW is in fact a better

measure of risk aversion than –Cucc/u

c, since it

represents aversion to bets over wealth ratherthan bets over consumption; most bets observedare paid off in dollars. For a consumer whoinvests entirely in stocks, σ(∆w) is the standarddeviation of the stock return and corr(∆w,r) =1. To generate a Sharpe ratio of 0.5, it seemsthat we only need risk aversion equal to 3,

−= ≅

WV

VWW

W

0 5

0 173

.

..

The Achilles heel of this calculation is thehidden simplifying assumption that returns areindependent over time, so no variables otherthan wealth show up in V

W. If this were the case,

consumption would move one-for-one withwealth, and σ(∆c) = σ(∆w). If wealth doublesand nothing else has changed, the consumerwould double consumption. The calculation,therefore, hides a model with the drasticallycounterfactual implication that consumptiongrowth has a 17 percent standard deviation!

The fact that consumption has a standarddeviation so much lower than that of stockreturns suggests that returns are not indepen-dent over time (as is already known from thereturn on P/D regressions) and/or that otherstate variables must be important in drivingstock returns. If some other state variable, z,is allowed—representing subsequent expectedreturns, labor income, or some other measureof a consumer’s overall opportunities—thesubstitution V

W(W,z) = u

c(C) in equation 6 adds

another term to equation 8,

E r r

r

WV

Vw corr w r

zW

Vz corr z r

fWW

W

Wz

W

( )

( )( ) ( , )

( ) ( , ).

− = −

+

σσ

σ

∆ ∆

The Sharpe ratio may be driven not by consum-ers’ risk aversion and the wealth-riskiness ofstocks, but by stocks’ exposure to other risks.

In the current context, this observation justtells us that portfolio-based calibrations of riskaversion do not work, because they implicitlyassume independent returns and, hence, con-sumption growth as volatile as returns. Below,I introduce plausible candidates for the variable zthat can help us to understand high Sharperatios. The fact that consumption is so muchless volatile than shock returns indicates thatthe other state variables must account for thebulk of the equity premium.

Overall, the evidence against high riskaversion is not that strong, and it is at least apossibility to consider. This argument does notrescue the power utility model with γ = 50 to250—that ship sank on the consumption-interestrate shoals. However, other models with highrisk aversion can be contemplated.


New utility functions and state variablesIf changing the parameter γ in u′(C) = C-γ

does not work, perhaps we need to change thefunctional form. Changing the form of u(C)is not a promising avenue. As I have stressedby using a continuous time derivation, only thederivatives of u(C) really matter; hence quitesimilar results are achieved with other functionalforms. A more promising avenue is to considerother arguments of the utility function, ornonseparabilities.

Perhaps how people feel about eating moretoday is affected not just by how much they arealready eating, but by other things, such as howmuch they ate yesterday or how much theyworked today. Then, the covariance of stockreturns with these other variables will alsodetermine the equity premium. Fundamentally,consumers use assets to smooth marginal utility.Perhaps today’s marginal utility is related tomore than just today’s consumption.

Such a modification is a fundamental changein how we view stock market risk. For example,perhaps more leisure raises the marginal utilityof consumption. Stocks are then feared becausethey pay off badly in recessions when employ-ment is lower and leisure is higher, not becauseconsumers are particularly averse to the riskthat stocks decline per se. Formally, our funda-mental equation 6 is derived from

E r r u rt tf

t c( ) ( , ),− = cov ∆

and substituting uc = ∂u(c)/∂c. If I substitute u

c

= ∂u(c,x)/∂c instead, then uc will depend on

other variables x as well as c, and

E r r c ru

urf cx

c

x( ) ( , ) ( , ).− = +γ cov cov∆

Since the first covariance does not account formuch premium, we will have to rely heavily onthe latter term to explain the premium.

There is a practical aim to generalizing theutility function as well. As illustrated in thelast section, one parameter γ did two thingswith power utility: It controlled how muchpeople are willing to substitute consumptionover time (consumption and interest rates) andit controlled their attitudes toward risk. Thechoice γ = 50 to 250 was clearly a crazy repre-sentation of how people feel about consump-tion variation over time, but perhaps not so

bad a representation of risk aversion. Maybea modification of preferences can disentanglethe two attitudes.

State separability and leisureWith the latter end in mind, Epstein and Zin

(1989) started an avalanche of academic researchon utility functions that relax state-separability.The expectation E in the utility function of equa-tion 3 sums over states of nature, for example

U prob u C prob

u C

= × + ×(rain) if it rains

if it shines

( ) (shine)

( ).

“Separability” means that one adds utilityacross states, so the marginal utility of con-sumption in one state is unaffected by whathappens in another state. But perhaps the mar-ginal utility of a little more consumption in thesunny state of the world is affected by the levelof consumption in the rainy state of the world.

Epstein and Zin and Hansen, Sargent, andTallarini (1997) propose recursive utility func-tions of the form

U C e f E f Ut t t t= +− − −+

1 11

γ ρ ( ) .

If f(x) = x, this expression reduces to powerutility. These utility functions are not state-separable, and do conveniently distinguish riskaversion from intertemporal substitution amongother modifications. However, this research isonly starting to pay off in terms of plausiblemodels that explain the facts (Campbell, 1996,is an example) so I will not review it here.

Perhaps leisure is the most natural extravariable to add to a utility function. It is notclear a priori whether more leisure enhancesthe marginal utility of consumption (why botherbuying a boat if you are at the office all dayand cannot use it?) or vice versa (if you have towork all day, it is more important to comehome to a really nice big TV), but we can letthe data speak on this matter. However, explicitversions of this approach have not been verysuccessful to date. (Eichenbaum, Hansen, andSingleton, 1988, for example). On the otherhand, recent research has found that addinglabor income growth as an extra ad-hoc factorcan be useful in explaining the cross section ofaverage stock returns (Jagannathan and Wang,1996; Reyfman, 1997). Though not motivated


by an explicit utility function, the facts inthis research may in the future be interpretableas an effect of leisure on the marginal utilityof consumption.

Force of habitNonseparabilities over time have been more

useful in empirical work. Anyone who has had alarge pizza dinner knows that yesterday’s con-sumption can have an impact on today’s appetite.Might a similar mechanism apply over a longertime horizon? Perhaps people get accustomedto a standard of living, so a fall in consumptionhurts after a few years of good times, even thoughthe same level of consumption might have seemedvery pleasant if it arrived after years of bad times.This view at least explains the perception thatrecessions are awful events, even though arecession year may be just the second or thirdbest year in human history rather than the abso-lute best. Law, custom, and social insuranceinsure against falls in consumption as much aslow levels of consumption.

Following this idea, Campbell and Cochrane(1997) specify that people slowly develop habitsfor higher or lower consumption. Technically,they replace the utility function u(C) with

9 1) ( ) ( ) ,u C X C X− = − −η

where X represents the level of habits. In turn,habit X adjusts slowly to the level of consump-tion.10 (I use the symbol η for the power, becausecurvature and risk aversion no longer equal η.)This specification builds on a long tradition inthe microeconomic literature (Duesenberry, 1949,and Deaton, 1992) and recent asset-pricing litera-ture (Constantinides, 1990; Ferson and Constan-tinides, 1991; Heaton, 1995; and Abel, 1990).

When a consumer has such a habit, localcurvature depends on how far consumption isabove the habit, as well as the power η,

γ ηtt cc

c

t

t t

C u

u

C

C X≡ − =

−.

As consumption falls toward habit, peoplebecome much less willing to tolerate furtherfalls in consumption; they become very riskaverse. Thus, a low power coefficient η(Campbell and Cochrane use η = 2) can stillmean a high and time-varying curvature.

Recall our fundamental equation 6 for theSharpe ratio,

E r r

rc corr c rt t

f

tt t t

( )

( )( ) ( , ).

− =σ

γ σ ∆ ∆

High curvature γt means that the model canexplain the equity premium, and curvature thatvaries over time, as consumption risesin booms and falls toward habit in recessions,means that the model can explain a time-varyingand countercyclical (high in recessions, low inbooms) Sharpe ratio, despite constant con-sumption volatility, σt(∆c), and correlation,corrt(∆c,r).

So far so good, but doesn’t raising curva-ture imply high and time-varying interest rates?In the Campbell-Cochrane model, the answeris no. The reason is precautionary saving. Sup-pose times are bad and consumption is lowrelative to habit. People want to borrow againstfuture, higher consumption and this shoulddrive up interest rates. However, people arealso much more risk averse in bad times; theywant to save more in case tomorrow might beeven worse. These two effects balance.

The precautionary saving motive alsomakes the model more plausibly consistentwith variation in consumption growth acrosstime and countries. The interest rate in themodel adds a precautionary savings motiveterm to equation 4,

r E cS

cf = + − FH

IKρ η η σ( ) ( ),∆ ∆1

2

22

where S– denotes the steady state value of

(C–X)/C, about 0.05. The power coefficient,η = 2, controls the relationship between con-sumption growth and interest rates, while thecurvature coefficient, γ = η C/(C–X), controlsthe risk premium. Thus, this habit model allowshigh risk aversion with low aversion to inter-temporal substitution and is consistent withthe consumption and interest rate data and asensible value of ρ.

Campbell and Cochrane create a simpleartificial economy with these preferences.Consumption growth is independent over timeand real interest rates are constant. They calcu-late time series of stock prices and interest rates


in the artificial economy and subject them tothe standard statistical analysis reviewed above.The artificial data replicate the equity premium(0.5 Sharpe ratio). The ability to forecast returnsfrom the P/D ratio and the P/D variance decom-position are both quite like the actual data. Thestandard deviation of returns rises a bit whenprices decline, but less than the rise in meanreturns, so a low P/D ratio forecasts a higherSharpe ratio. Artificial data from the modelalso replicate much of the low observed correla-tion between consumption growth and returns,and the CAPM and ad-hoc multifactor modelsperform better than the power utility consump-tion-based model in the artificial data.

The model also provides a good accountof P/D fluctuations over the last century, basedentirely on the history of consumption. Howev-er, it does not account for the currently highP/D ratio. This is because the model generatesa high P/D ratio when consumption is veryhigh relative to habit and, therefore, risk aver-sion is low. Measured consumption has beenincreasing unusually slowly in the 1990s.

Like other models that explain the equitypremium and return predictability, this onedoes so by fundamentally changing the storyof why consumers are afraid of holding stocks.From equation 9, the marginal utility of con-sumption is proportional to

u CC X

Cc tt t

t

= −FHG

IKJ

−−

ηη

.

Thus, consumers dislike low consumptionas before, but they are also afraid of recessions,times when consumption, whatever its level, islow relative to the recent past as described byhabits. Consumers are afraid of holding stocksnot because they fear the wealth or consumptionvolatility per se, but because bad stock returnstend to happen in recessions, times of a recentbelt-tightening.

This model fulfills a decade-long searchkicked off by Mehra and Prescott (1985). It isa complete-markets, frictionless economy thatreplicates not only the equity premium but alsothe predictability of returns, the nearly constantinterest rate, and the near-random walk behaviorof consumption.

Habit models with low risk aversionThe individuals in the Campbell-Cochrane

model are highly risk averse. They would respondto surveys about bets on wealth much as theγ = 50 column of table 6. The model does notgive rise to a high equity premium with lowrisk aversion; it merely disentangles risk aver-sion and intertemporal substitution so that ahigh risk aversion economy can be consistentwith low and constant interest rates, and itgenerates the predictability of stock returns.

Constantinides (1990) and Boldrin,Christiano, and Fisher (1995) explore habitpersistence models that can generate a largeequity premium without large risk aversion.That is, they create artificial economies inwhich consumers simultaneously shy awayfrom stocks with a very attractive Sharpe ratioof 0.5, yet would happily take bets with muchlower rewards.

Suppose a consumer wins a bet or enjoysa high stock return. Normally, the consumerwould instantly raise consumption to matchthe new higher wealth level. But consumptionis addictive in these models: Too much currentconsumption will raise the future habit leveland blunt the enjoyment of future consumption.Therefore, the consumer increases consumptionslowly and predictably after the increase inwealth. Similarly, the consumer would borrowto slowly decrease consumption after a declinein wealth, avoiding the pain of a sudden loss atthe cost of lower long-term consumption.

The fact that the consumer will choose tospread out the consumption response to wealthshocks means that the consumer is not averseto wealth bets. If consumption responds littleto a wealth shock, then marginal utility ofconsumption, u

c(C), also responds little, as does

the marginal value of wealth, VW(W,⋅) = u

c. Risk

aversion to wealth bets is measured by thechange of marginal utility when wealth changes(∂ln V

W/∂lnW = –WV

WW/V

W).

The argument is correct, but shows theproblem with these models. The change inconsumption in response to wealth is not elimi-nated, it is simply deferred. Thus, these modelshave trouble with long-run behavior of con-sumption and asset returns.

If consumption growth is considered inde-pendent over time (formally, an endowmenteconomy), which is a good approximation tothe data, the model must feature strong interest


rate variation to keep consumers from trying toadapt consumption smoothly to wealth shocks.For example, consumers all want to save ifwealth goes up, thereby slowly increasingconsumption. For consumption growth to remainunpredictable, we must have a strong declinein the interest rate at the same time as the wealthincrease. Of course, interest rates are in factquite stable and, if anything, slightly positivelycorrelated with stock returns.

Alternatively, interest rates may be fixedto be constant over time as in Constantinides(1990) (formally, linear technologies). Butthen there is no force to stop consumers fromslowly and predictably raising consumptionafter a wealth shock. Thus, such models predictcounterfactually that consumption growth ispositively auto-correlated over time, and thatlong-run consumption growth shares the highvolatility of long- and short-run wealth.

The Campbell-Cochrane habit modelavoids these long-run problems with precau-tionary savings. In response to a wealth shock,consumers with the Campbell-Cochrane versionof habit persistence would also like to savemore for intertemporal substitution reasons,but they also feel less risk averse and so wantto save less for precautionary savings reasons.The balance means that consumption can be arandom walk with constant interest rates, con-sistent with the data. However, it means thatconsumption does move right away, so u

c and

VW are affected by the wealth shock, and the

consumers are highly risk averse. In this model,wealth (stock prices) comes back toward con-sumption after a shock, so that long-run wealthshares the low volatility of long- and short-runconsumption, and high stock prices forecastlow subsequent returns.

A finance perspectiveTo get a high equity premium with low

risk aversion, we need to find some crucialcharacteristic that separates stock returns fromwealth bets. This is a difficult task. After all,what are stocks if not a bet? The answer mustbe some additional state variable. Having astock pay off badly must tell you additionalbad news that losing a bet does not; therefore,people shy away from stocks even though theywould happily take a bet with the same meanand variance.

In the context of perfect-markets modelswithout leisure or other goods, the only real

candidates for extra state variables are vari-ables that describe changes in expected returns.If stock prices rise, the consumer learns some-thing important that is not learned from winninga bet: The consumer learns that future stockreturns are going to be lower. The trouble isthe sign of this relationship. Lower returns inthe future are bad news.11 Stocks act as a hedgefor this bad news; they go up just at the timeone gets bad news about future returns. Thisconsideration makes stocks more desirable thanpure bets. Thus, considering time variation inexpected returns requires even more risk aver-sion to explain the equity premium.

Consistency with individualconsumption behavior

The low risk aversion models face onemore important hurdle: microeconomic data.Suppose an individual receives a wealth shock(wins the lottery), not shared by everyone else.For aggregate wealth shocks, we could appealto interest rate variation to avoid the predictionthat consumption would grow slowly and pre-dictably. However, interest rates can’t changein response to an individual wealth shock.Thus, we are stuck with the prediction that theindividual’s consumption will increase slowlyand predictably. The huge literature on individ-ual consumption (see Deaton, 1992, for surveyand references) almost unanimously finds theopposite. People who receive windfalls consumetoo much, too soon, and have typically spent itall in a few years. The literature abounds with“liquidity constraints” to explain the “excesssensitivity” of consumption. The Campbell-Cochrane model avoids the prediction of slowlyincreasing consumption by specifying an externalhabit; each person’s habit responds to everyoneelse’s consumption, related to the need to“keep up with the Joneses,” as advocated byAbel (1990). Though the external specificationhas little impact on aggregate consumption andprices, it means that individual consumptionresponds fully and immediately to individualwealth shocks, because there is no need forindividuals to worry about habit formation.The downside is, again, high risk aversion.

In the end, there is currently no representa-tive agent model with low risk aversion that isconsistent with the equity premium, the stabilityof real interest rates, nearly unpredictable con-sumption growth, and return predictability ofthe correct sign.


Heterogeneous agents andidiosyncratic risks

In the above discussion, I did not recog-nize any difference between people. Everyoneis different, so why bother looking at represen-tative agent or complete market models? Whilemaking an assumption such as “all people areidentical” seems obviously foolish, it is notfoolish to hope that we can use aggregatebehavior to make sense of aggregate data,without explicitly taking account of the differ-ences between people. While differences arethere, one hopes they are not relevant to thebasic story. However, seeing the difficultiesthat representative agent models face, perhapsit is time to see if the (aggregate) equity premiumdoes in fact surface from differences betweenpeople rather than common behavior.

The empirical hurdleIdiosyncratic risk explanations face a big

empirical challenge. Look again at the basicSharpe ratio equation 6,

E r r

rc corr c r

f( )

( )( ) ( , ).

− =σ

γ σ ∆ ∆

This relationship should hold for every (any)consumer or household. At first sight, thinkingabout individuals seems promising. After all,individual consumption is certainly more vari-able than aggregate consumption at 1 percentper year, so we can raise σ(∆c). However, thisargument fails quantitatively. First, it is incon-ceivable that we can raise σ(∆c) enough toaccount for the equity premium. For example,even if individual consumption has a standarddeviation of 10 percent per year, and maintain-ing a generous limit γ < 10, we still predict aSharpe ratio no more than 10 × 0.1 × 0.2 = 0.2.To explain the 0.5 Sharpe ratio with risk aver-sion γ = 10, we have to believe that individualconsumption growth has a 25 percent per yearstandard deviation; for a more traditional γ = 2.5,we need 100 percent per year standard devia-tion. Even 10 percent per year is a huge standarddeviation of consumption growth. Remember,we are considering the risky or uncertain part ofconsumption growth. Predictable increases ordecreases in consumption due to age and life-cycle effects, expected raises, and so on do notcount. We are also thinking of the flow ofconsumption (nondurable goods, services) not

the much more variable purchases of durablegoods, such as cars and houses.

More fundamentally, the addition of idio-syncratic risk lowers the correlation betweenconsumption growth and returns, which lowersthe predicted Sharpe ratio. Idiosyncratic risk is,by its nature, idiosyncratic. If it happened toeveryone, it would be aggregate risk. Idiosyn-cratic risk cannot therefore be correlated withthe stock market, since the stock market returnis the same for everyone.

For a quantitative example, suppose thatindividual consumption of family i, ∆ci, is deter-mined by aggregate consumption, ∆ca, and idio-syncratic shocks (such as losing your job), ε i,

∆ ∆c ci a i= + ε .

For the risk ε i to average to zero across people,we must have E(ε i) = 0 and E(ε i∆ca) = E(εir)= 0. Then, the standard deviation of individualconsumption growth does increase with thesize of idiosyncratic risk,

σ σ σ ε2 2 2( ) ( ) ( ).∆ ∆c ci a i= +

But the correlation between individual con-sumption growth and aggregate returns de-clines in exact proportion as the standard devi-ation σ(∆ci) rises,

E r r

r

c r

r

c r

r

f a i a( )

( )

( , )

( )

( , )

( ).

− = + =σ

γ εσ

γσ

cov cov∆ ∆

Therefore, the equity premium is completelyunaffected by idiosyncratic risk.

The theoretical hurdlesThe theoretical challenge to idiosyncratic

risk explanations is even more severe. We caneasily construct models in which consumersare given lots of idiosyncratic income risk.But it is very hard to keep consumers frominsuring themselves against those risks, pro-ducing a very steady consumption stream anda low equity premium.

Start by handing out income to consum-ers; call it “labor income” and make it riskyby adding a chance of being fired. Left totheir own devices, consumers would comeup with unemployment insurance to sharethis risk, so we have to close down or limitmarkets for labor income insurance. Then,


consumers who lose their jobs will borrowagainst future income to smooth consumptionover the bad times, achieving almost the sameconsumption smoothness. We must shut downthese markets too.

However, nothing stops our borrowing-constrained consumers from saving. They buildup a stock of durable goods, governmentbonds, or other liquid assets that they can drawdown in bad times and again achieve a verysmooth consumption stream (Telmer, 1993,and Lucas, 1994). To shut down this avenuefor consumption-smoothing, we can introducelarge transactions costs and ban from the modelthe simple accumulation of durable goods.Alternatively, we can make idiosyncraticshocks permanent, because borrowing and savingcan only insure against transitory income. Iflosing your job means losing it forever, thereis no point in borrowing and planning to pay itback when you get a new job.

Heaton and Lucas (1996a) put all theseingredients together, calibrating the persistenceof labor income shocks from microeconomicdata. They also allow the cross-sectional varianceof shocks to increase in a downturn, a veryhelpful ingredient suggested by Mankiw (1986)that I discuss in detail in the next section. Despiteall of these ingredients, their model explains atbest one-half of the observed excess averagestock return (and this much only with no netdebt). It also predicts counterfactually thatinterest rates are as volatile as stock returns,and that individuals have huge (10 percent to 30percent, depending on specification) consump-tion growth uncertainty.

A model that worksConstantinides and Duffie (1996) con-

struct a model in which idiosyncratic risk canbe tailored to generate any pattern of aggregateconsumption and asset prices; it can generatethe equity premium, predictability, relativelyconstant interest rates, smooth and unpredict-able aggregate consumption growth, and soforth. Furthermore, it requires no transactionscosts, borrowing constraints, or other frictions,and the individual consumers can have anynonzero value of risk aversion.

As mentioned earlier, if consumers aregiven idiosyncratic income that is correlatedwith the market return, they will trade awaythat risk. Constantinides and Duffie thereforespecify that the variance of idiosyncratic riskrises when the market declines. Variance cannot

be traded away. In addition, if marginal utilitywere linear, an increase in variance would haveno effect on the average level of marginal utili-ty. The interaction of cross-sector variancecorrelated with the market and nonlinear mar-ginal utility produces an equity premium.

The Constantinides-Duffie model and theCampbell-Cochrane model are quite similar inspirit, though the Constantinides-Duffie modelis built on incomplete markets and idiosyncraticrisks, while the Campbell-Cochrane model isin the representative-agent frictionless andcomplete market tradition.

First, both models make a similar, funda-mental change in the description of stock marketrisk. Consumers do not fear the loss of wealthof a bad market return so much as they fear abad return in a recession, in one model a timeof heightened labor market risk and in the othera fall of consumption relative to the recent past.This recession state variable or risk factor drivesmost expected returns.

Second, both models require high risk aver-sion. While Constantinides and Duffie’s proofshows that one can dream up a labor incomeprocess to rationalize the equity premium for anyrisk aversion coefficient, I argue below that evenvaguely plausible characterizations of actuallabor income uncertainty will require high riskaversion to explain the historical equity premium.

Third, both models provide long-soughtdemonstrations that it is possible to rationalizethe equity premium in their respective classof models. This is particularly impressivein Constantinides and Duffie’s case. Manyresearchers (myself included) had come to theconclusion that the effort to generate an equitypremium from idiosyncratic risk was hopeless.

The open question in both cases is empirical.The stories are consistent; are they right? ForConstantinides and Duffie, does actual individ-ual labor income behave as their model requiresin order to generate the equity premium? Theempirical work remains to be done, but hereare some of the issues.

A simple version of the modelEach consumer i has power utility,

U E e Ct

tit= − −∑ ρ γ1 .

Individual consumption growth, Cit, is deter-

mined by an independent, idiosyncratic normal(0,1) shock, η

it,


101

21

2) ln ,,

C

Cy yit

i tit t t

−

FHG

IKJ = −η

where yt is the cross-sectional standard devia-

tion of consumption growth at any moment intime. y

t is specified so that a low market return,

Rt, gives a high cross-sectional variance of

consumption growth,

11

2

1

1

1

) ln

( )ln .

yC

CR

R

tit

itt

t

=FHG

IKJ

LNMM

OQPP

=+

+

−σ

γ γρ

Since ηit determines consumption growth,

the idiosyncratic shocks are permanent, whichI argued above was important to keep consumersfrom smoothing them away.

Given this structure, the individual isexactly happy to consume C

it and hold the

stock (we can call Cit income I

it and prove the

optimal decision rule is to set Cit = I

it.). The

first-order condition for an optimal consump-tion-portfolio decision

1 11

1=FHG

IKJ

L

NMM

O

QPP−

−

−

−

+E eC

CRt

it

itt

ργ

holds, exactly.12

The general modelThe actual Constantinides-Duffie model

is much more general than the above example.They show that the idiosyncratic risk can beconstructed to price exactly a large collectionof assets, not just one return as in the example,and they allow uncertainty in aggregate con-sumption. Therefore, they can tailor the idio-syncratic risk to exactly match the Sharperatio, return forecastability, and consumption-interest rate facts as outlined above.

In the general model, Constantinides andDuffie define

122

1 1

)( )

ln ln ,y mC

Ct tt

t

=+

+ +−γ γ

ρ γ

where mt is a strictly positive discount factor13

that prices all assets under consideration. Thatis, m

t satisfies

14 1 1) .= −E m R Rt t t tfor all

Then, they let

C C

y y

it it it

it it it t t

=

= −LNM

OQP−

δ

δ δ η121

2exp .

Following exactly the same argument in thetext, we can now show that

1 11

1=FHG

IKJ

L

NMM

O

QPP−

−

−

−

+E eC

CRt

it

itt

ργ

for all the assets priced by m.

Microeconomic evaluation and risk aversionLike the Campbell-Cochrane model, this

model could be either a new view of stockmarket (and macroeconomic) risk or just atheoretically interesting existence proof. Thefirst question is whether the microeconomicpicture painted by this model is correct, or evenplausible. Is idiosyncratic risk large enough?Does idiosyncratic risk really rise when themarket falls, and enough to account for theequity premium? Do people really shy awayfrom stocks because stock returns are low attimes of high labor market risk?

This model does not change the empiricalpuzzle discussed earlier. To get power utilityconsumers to shun stocks, they still must havetremendously volatile consumption growth orhigh risk aversion. The point of this model is toshow how consumers can get stuck with highconsumption volatility in equilibrium, alreadya difficult task.

More seriously than volatility itself, con-sumption growth variance also represents theamount by which the distribution of individualconsumption and income spreads out over time,since the shocks must be permanent and inde-pendent across people. The 10 percent to 50percent volatility (σ(∆c)) that is required toreconcile the Sharpe ratio with low risk aver-sion means that the distribution of consump-tion (and income) must also spread out by 10percent to 50 percent per year.


Constantinides and Duffie show how toavoid the implication that the overall distribu-tion of income spreads out, by limiting inherit-ance and repopulating the economy each yearwith new generations that are born equal. Butthe distribution of consumption must still spreadout within each generation to achieve the equitypremium with low risk aversion. Is this plausi-ble? Deaton and Paxson (1994) report that thecross-sectional variance of log consumptionwithin an age cohort rises from about 0.2 atage 20 to 0.6 at age 60. This means that thecross-sectional standard deviation of consump-tion rises from 0 2. = 0.45 or 45 percent at age20 to 0 6. = 0.77 or 77 percent at age 60 (77percent means that an individual one standarddeviation better off than the mean consumes77 percent more than the mean consumer).This works out to about 1 percent per year,not 10 percent or so.

Finally, the cross-sectional uncertaintyabout individual income must not only belarge, it must be higher when the market islower. This risk factor is after all the centralelement of Constantinides and Duffie’s expla-nation for the market premium. Figure 6 showshow the cross-sectional standard deviation ofconsumption growth varies with the market

return and risk aversion in my simple versionof Constantinides and Duffie’s model. If weinsist on low (γ = 1 to 2) risk aversion, thecross-sectional standard deviation of consump-tion growth must be extremely sensitive to thelevel of the market return. Looking at the γ = 2line for example, is it plausible that a year with5 percent market return would show a 10 percentcross-sectional variation in consumptiongrowth, while a mild 5 percent decline in themarket is associated with a 25 percent cross-sectional variation?

The Heaton and Lucas (1996a) model canbe regarded as an empirical assessment of theseissues. Rather than constructing a labor incomeprocess that generates an equity premium, theycalibrated the labor income process frommicroeconomic data. They found less persis-tence and less increase in cross-sectional varia-tion with a low market return than specified byConstantinides and Duffie, which is why theirmodel predicts a low equity premium with lowrisk aversion. Of course, this view is at bestpreliminary evidence. They did not test theexact Constantinides-Duffie specification as aspecial case, nor did they test whether one canreject the Constantinides-Duffie specification.

All of these empirical problems are avoidedif we allow high risk aversionrather than a large risk to drivethe equity premium. The γ = 25line in figure 6 looks possible; aγ = 50 line would look evenbetter. With high risk aversion,we do not need to specify highlyvolatile individual consumptiongrowth, spreading out of theincome distribution, or dramaticsensitivity of the cross-sectionalvariance to the market return. Asin any model, a high equitypremium must come from alarge risk, or from large riskaversion. Labor market riskcorrelated with the stock marketdoes not seem large enough toaccount for the equity premiumwithout high risk aversion.

Segmented marketsAll these models try to

answer the basic question, ifstocks are so attractive, whyhave people not bought more of

0

15

30

45

60

-25 -20 -15 -10 -5 0 5 10 15 20 25

standard deviation, percent

market return, percent

γ = 1

γ = 2

γ = 5

γ = 10

γ = 25

FIGURE 6

Cross-sectional standard deviation ofconsumption growth

Notes: Cross-sectional standard deviation of individual consumption growthas a function of the market return in the Constantinides-Duffie model.

The plot is the variable yt Rt

Ct

Ct=

++ +

−

2

1

1

1γ γρ γ

( )ln ln . Parameter

values are ρ = 0.05, lnCt /Ct–1 = 0.01, and γ and lnRt+1

as graphed.


them? So far, I have tried to find representationsof people’s preferences or circumstances, or adescription of macroeconomic risk, in whichstocks aren’t that attractive after all. Then thehigh Sharpe ratio is a compensation for risk.

Instead, we could argue that stocks reallyare attractive, but a variety of market frictionskeep people from buying them. This approachyields some important insights. First of all,stock ownership has been quite concentrated.The vast majority of American households havenot directly owned any stock or mutual funds.One might ask whether the consumption ofpeople who do own stock lines up with stockreturns. Mankiw and Zeldes (1991) find thatstockholders do have consumption that is morevolatile and more correlated with stock returnsthan non-stockholders. But it is still not volatileand correlated enough to satisfy the right-handside of equation 6 with low risk aversion.

Heaton and Lucas (1996b) look at individualasset and income data. They find that the richesthouseholds, who own most of the stocks, alsoget most of their income from proprietarybusiness income. This income is likely to bemore correlated with the stock market than isindividual labor income. Furthermore, theyfind that among rich households, those withmore proprietary income hold fewer stocksin their portfolios. This paints an interestingpicture of the equity premium: In the pastmost stocks were held by rich people, andmost rich people were proprietors whose otherincome (and consumption) was quite volatileand covaried strongly with the market. This isa hard crowd to sell stocks to, so they haverequired a high risk premium. The Campbell-Cochrane and Constantinides-Duffie modelsspecify that stock market risk is spread asevenly as possible through the population,whereas if the risk is shared among a smallgroup of people, higher rewards will have tobe offered to offset that risk.

These views are still not sorted out quanti-tatively. We don’t know why rich stockholdersdon’t buy even more stocks, given low riskaversion and the tyrannical logic of equation 6.We don’t know why only rich people heldstocks in the first place: The literature showsthat even quite high transactions costs andborrowing constraints should not be enoughto deter people with low risk aversion fromholding stocks.

If these segmented market views of thepast equity premium are correct, they suggestthat the future equity premium will be muchlower. Transaction costs are declining throughfinancial deregulation and innovation. Theexplosion in tax-deferred pension plans andno-load mutual funds means more and morepeople own stocks, spreading risks more evenly,driving up prices, and driving down prospectivereturns. Equation 6 will hold much better for theaverage consumer in the future. One would expectto see a lower equity premium. One would alsoexpect consumption that is more volatile andmore closely correlated with the stock market,which will result in a fundamental change in thenature and politics of business cycles.

Technology and investmentSo far, I have tried to rationalize stock

returns from the consumer’s point of view:Does it make sense that consumers should nothave tried to buy more stocks, driving stockreturns down toward bond returns? I can askthe same questions for the firm: Do firms’investment decisions line up with stock pricesas they should?

The relative prices of apples and orangesare basically set by technology, the relativenumber of apples versus oranges that can begrown on the same acre of land. We do notneed a deep understanding of consumers’ de-sires to figure out what the price should be. Iftechnology is (close to) linear, it will determinerelative prices, while preferences will deter-mine quantities. Does this argument work forstocks?

Again, there is a standard model that hasserved well to describe quantities in growth,macroeconomics, and international economics.The standard model consists of a productionfunction by which output, Y, is made fromcapital, K, and labor, L, perhaps with someuncertainty, θ, together with an accumulationequation by which investment I turns into newcapital in the future. In equations, togetherwith the most common functional forms,

15

1

1

1

) ( , , )

( )

.

Y f K L K L

K K I

Y C I

t t t t t t t

t t t

t t t

= == − += +

−

+

θ θδ

α α


It was well known already in the 1970sthat this standard, “neoclassical” model wouldbe a disaster at describing asset pricing facts.It predicts that stock prices and returns shouldbe extremely stable. To see this, invest an extradollar, reap the extra output that the additionalcapital will produce, and then invest a bit lessnext year. This action gives a physical or invest-ment return. For the technology described inequation 15, the investment return is

16 1

1

1 1 1 1

1

1

) ( , , )

.

R f K L

Y

K

tI

k t t t

t

t

+ + + +

+

+

= + −

= + −

θ δ

α δ

With the share of capital α ≈ 1/3, an output-capital ratio Y/K ≈ 1/3, and depreciation δ ≈ 10%,we have RI ≈ 6%, so average equity returns areeasily within the range of plausible parameters.The trouble lies with the variance. Capital isquite smooth, so even if output varies 3 percentin a year, the investment return only varies by1 percent, far below the 17 percent standarddeviation of stock returns. The basic problemis the absence of price variation. The capitalaccumulation equation shows that installedcapital, K

t, and uninstalled capital, I

t, are per-

fect substitutes in making new capital, Kt+1

Therefore, they must have the same price—theprice of stocks relative to consumption goodsmust be exactly 1.0.

The obvious modification is that theremust be some difference between installedand uninstalled capital. The most natural extraingredient is an adjustment cost or irrevers-ibility: It is hard to get any work done on theday the furniture is delivered, and it is hard totake paint back off the walls and sell it. Torecognize these sensible features of investment,we can reduce output during periods of highinvestment or make negative investment costlyby modifying equation 15 to

17) ( , ) ( , ).Y f K L c It t t t t= − ⋅θ

The dot reminds us that other variables mayinfluence the adjustment or irreversibility costterm. A common specification is

Y K La I

Kt t t tt

t

= −−θ α α12

2.

Now, there is a difference between installedand uninstalled capital, and the price of installedcapital can vary. Adding an extra unit of capitaltomorrow via extra investment costs 1–∂c(⋅)∂ Iunits of output today, while an extra unit of capi-tal would give (1–δ) units of capital tomorrow.Hence the price of capital in terms of output is

181

11

1

)/

,

Pc i

c

I

aI

K

t

t

t

= −−

≈ + −

= + −

δ∂ ∂

∂∂

δ

δ

where the last equality uses the quadratic func-tional form. (This is the q theory of investment.With an asymmetric c function, this is the basisof the theory of irreversible investment. Abeland Eberly [1996] give a recent synthesis withreferences.)

Equation 18 shows that stock prices areexpected to be high when investment is high,or firms are expected to issue stock and investwhen stock prices are high. The investmentreturn is now

19 1

1

1

12

12

1

1

1

1

1

1

11 1 1 1

1

1

12

1

1

)( )

( )( ) ( ) ( )

( )

.

Rf t c t c t

c

Y

K

a I

Ka

I

K

ai

K

Y

Ka

I

K

I

K

tI k k i

i

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t+

+

+

+

+

+

+

+

+

+

+

= −+ + − + + +

+

= −+ − +

+

≈ + − + −FHG

IKJ

δ

δα

α δ

Comparing equation 19 with equation 16,the investment return contains a new term pro-portional to the change in investment. Sinceinvestment is quite volatile, this model can beconsistent with the volatility of the market return.In equation 18, the last term adds price changesto the model of the investment return.

How does all this work? Figure 7 presentsthe investment-output ratio along with the valueweighted P/D ratio. (The results are almostidentical using an investment-capital ratio withcapital formed from depreciated past investment.)Equation 18 suggests that these two series shouldmove together. The cyclical movements ininvestment and stock prices do line up pretty


well. The longer-term variation in P/D is notmirrored in investment: This simple modeldoes not explain why investment stayed robustin the late 1970s despite dismal stock prices.However, the recent surge in the market ismatched by a surge in investment.

This kind of model has been subject to anenormous formal empirical effort, which prettymuch confirms the figure. First, the model isconsistent with a good deal of the cyclical varia-tion in investment and stock returns, both fore-casts and ex-post. (See, for example, Cochrane,1991c.) It does not do well with longer-termtrends in the P/D ratio. Second, early testsrelating investment to interest rates that imposeda constant risk premium did not work (Abel,1983). The model only works at all if one rec-ognizes that most variation in the cost of capi-tal comes from time-varying expected stockreturns with relatively constant interest rates.Third, the model in equation 18 taken literallyallows no residual. If prices deviate one iotafrom the right-hand side of equation 18, thenthe model is statistically rejected—we can saywith perfect certainty that it is not a literal descrip-tion of the data-generating mechanism. There isa residual in actual data of course, and the resid-ual can be correlated with other variables suchas cash flow that suggest the presence of finan-cial frictions (Fazzari, Hubbard, and Peterson,1988). Finally, the size of the adjustment cost, a,is the subject of the same kind of controversy

that surrounds the size of the risk aversioncoefficient, γ. From equation 19 and the factthat investment growth has standard deviationof about 10 percent, a ≈ 2 is needed to ratio-nalize the roughly 20 percent standard devia-tion of stock returns. With I/Y ≈ 15% andY/K ≈ 33%, and hence I/K ≈ 1/20, a value a ≈ 2means that adjustment costs relative to output are

a I

K

I

Y2

2

2

1

20= F

HIK x 15% = 0.75%, which does

not seem unreasonable. However, estimates ofa based on regressions, Euler equations, orother techniques often result in much highervalues, implying that implausibly large frac-tions of output are lost to adjustment costs.

This model does not yet satisfy the goal ofdetermining the equity premium by technologicalconsiderations alone. Current specifications oftechnology allow firms to transform resourcesover time but not across states of nature. If thefirm’s own stock is undervalued, it can issuemore and invest. However, if the interest rate islow, there is not much one can say about whatthe firm should do without thinking about theprice of residual risk, and hence a preferenceapproach to the equity premium. Technically,the marginal rate of transformation acrossstates of nature is undefined.

Implications of the recent surge in investmentand stock prices

The association between stock returns andinvestment in figure 7 verifies that at least one

connection between stock re-turns and the real economyworks in some respects as itshould. This argues against theview that stock market swingsare due entirely to waves ofirrational optimism and pessi-mism. It also verifies that theflow of money into the stockmarket does at least partiallycorrespond to new real assetsand not just price increases onexisting assets.

In particular, the surge instock prices since 1990 has beenaccompanied by a surge in invest-ment. If expected stock returnsand the cost of capital are low,then investment should be high.Statistically, high investment-output or investment-capital

’62 ’69 ’76 ’83 ’90 ’97

Investment-outputratio

Price/dividendratio

FIGURE 7

Value weighted portfolio P/D and investment

Notes: Investment-output ratio and P/D of value weighted NYSE.Investment = gross fixed investment, output = gross domestic output.Series are stretched to fit on the same graph.


ratios also forecast low stock returns (Cochrane,1991c). Thus, high investment provides additionalstatistical and economic evidence for the view thatexpected stock returns are quite low.

General equilibriumTo really understand the equity premium,

we need to combine the utility function andproduction function modifications to constructcomplete, explicit economic models that repli-cate the asset pricing facts. This effort shouldalso preserve, if not enhance, our understandingof the broad range of dynamic microeconomic,macroeconomic, international, and growthfacts underpinning the standard models. Thetask is challenging. Anything that affects therelationship between consumption and assetprices will affect the relationship between con-sumption and investment. Asset prices mediatethe consumption-investment decision and thatdecision lies at the heart of any dynamic macro-economic model. We have learned a bit abouthow to go about this task, but have developedno completely satisfactory model as yet.

Jermann (1997) tried putting habit persis-tence consumers in a model with a neoclassicaltechnology like equation 15, which is almostcompletely standard in business-cycle models.The easy opportunities for intertemporal trans-formation provided by that technology meantthat the consumers used it to dramatically smoothconsumption, destroying the prediction of ahigh equity premium. To generate the equitypremium, Jermann added an adjustment costtechnology like equation 17, as the production-side literature had proposed. This modificationresulted in a high equity premium, but alsolarge variation in risk-free rates.

Boldrin, Christiano, and Fisher (1995) alsoadd habit-persistence preferences to real businesscycle models with frictions in the allocation ofresources to two sectors. They generate aboutone-half of the historical Sharpe ratio. Theyfind some quantity dynamics are improvedover the standard model. However, their modelstill predicts highly volatile interest rates andpersistent consumption growth.

To avoid the implications of highly volatileinterest rates, I suspect we will need represen-tations of technology that allow easy transfor-mation across time but not across states of nature,analogous to the need for easy intertemporalsubstitution but high risk aversion in preferences.

Alternatively, the Campbell-Cochrane modelabove already produces the equity premiumwith constant interest rates, which can be inter-preted as a linear production function. Modelswith this kind of precautionary savings motivemay not be as severely affected by the additionof an explicit production technology.

Hansen, Sargent, and Tallarini (1997) usenon-state-separable preferences similar to thoseof Epstein and Zin in a general equilibriummodel. They show that a model with standardpreferences and a model with non-state-separablepreferences can predict the same path of quan-tity variables (such as output, investment, andconsumption) but differ dramatically on assetprices. This example offers some explanationof how the real business cycle and growthliterature could spend 25 years examiningquantity data in detail and miss all the modifi-cations to preferences that we seem to need toexplain asset pricing data. It also means thatasset price information is crucial to identifyingpreferences and calculating welfare costs ofpolicy experiments. Finally, it offers hope thatadding the deep modifications necessary toexplain asset pricing phenomena will not demolishthe success of standard models at describingthe movements of quantities.

ImplicationsThe standard economic models, which

have been used with great success to describegrowth, macroeconomics, international economics,and even dynamic microeconomics, do notpredict the historical equity premium, let alonethe predictability of returns. After ten years ofeffort, a range of deep modifications to thestandard models show promise in explainingthe equity premium as a combination of highrisk aversion and new risk factors. Those modi-fications are now also consistent with the broadfacts about consumption, interest rates, andpredictable returns. However, the modificationshave so far only been aimed at explaining assetpricing data. We have not yet establishedwhether the deep modifications necessary toexplain asset market data retain the models’previous successes at describing quantity data.

The modified models are drastic revisionsto the macroeconomic tradition. In the Campbell-Cochrane model, for example, strong time-varying precautionary savings motives balancestrong time-varying intertemporal substitutionmotives. Uncertainty is of first-order importance


in this model; linearizations near the steadystate and dynamics with the shocks turned offgive dramatically wrong predictions about themodel’s behavior. The costs of business cyclesare orders of magnitude larger than in standardmodels. In the Constantinides-Duffie model,one has to explicitly keep track of micro-economic heterogeneity in order to say anythingabout aggregates.

The new models are also a drastic revisionto finance. We are used to thinking of aversionto wealth risk, as in the CAPM, as a good start-ing place or first-order approximation. But thisview cannot hold. To justify the equity premium,people must be primarily averse to holdingstocks because of their exposure to some otherstate variable or risk factor, such as recessionsor changes in the investment opportunity set.To believe in the equity premium, one has tobelieve that these stories are sensible.

Finally, every quantitatively successfulcurrent story for the equity premium still requiresastonishingly high risk aversion. The alternative,of course, is that the long-run equity premiumis much smaller than the average postwar 8 per-cent excess return. The standard model wasright after all, and historically high U.S. stockreturns were largely due to luck or some othertransient phenomenon.

Faced with the great difficulty economictheory still has in digesting the equity premium,I think the wise observer shades down theestimate of the future equity premium evenmore than suggested by the statistical uncer-tainty documented above.

Portfolio implicationsIn sum, the long-term average stock return

may well be lower than the postwar 8 percentaverage over bonds, and currently high pricesare a likely signal of unusually low expectedreturns. It is tempting to take a sell recommen-dation from this conclusion. There is one veryimportant caution to such a recommendation.On average, everyone has to hold the marketportfolio. The average person does not changehis or her portfolio at all. For every individualwho keeps money out of stocks, someone elsemust have a very long position in stocks. Pricesadjust until this is the case. Thus, one shouldonly hold less stocks than the average person ifone is different from everyone else in somecrucial way. It is not enough to be bearish, onemust be more bearish than everyone else.

In the economic models that generate theequity premium, every investor is exactly happyto hold his or her share of the market portfolio,no more and no less. The point of the models isthat the superficial attractiveness of stocks isbalanced by a well-described source of risk, sothat people are just willing to hold them. Simi-larly, the time variation in the equity premiumdoes not necessarily mean one should attemptto time the market, buying more stocks at timesof high expected returns and vice versa. Everyinvestor in the Campbell-Cochrane model, forexample, holds exactly the same portfolio allthe time, while buy and sell signals come andgo. In the peak of a boom they are not feelingvery risk averse, and put their money in themarket despite its low expected returns. In thebottom of a bust, they feel very risk averse, butthe high expected returns are just enough tokeep their money in the market.

To rationalize active portfolio strategies,such as pulling out of the market at times ofhigh price ratios, you have to ask, who is therewho is going to be more in the market thanaverage now? And, what else are you going todo with the money?

More formally, it is easy to crank outportfolio advice, solutions to optimal portfolioproblems given objectives like the utility func-tion in equation 3. Assuming low risk aversion,and no labor income or other reason for time-varying risk exposure or risk aversion, solutionstypically suggest large portfolio shares in equi-ties and a strong market timing approach, some-times highly leveraged and sometimes (now)even short. (See Barberis, 1997; Brandt, 1997.)If everyone followed this advice, however, theequity premium and the predictable variation inexpected returns would disappear. Everyonetrying to buy stocks would simply drive up theprices; everyone trying to time the marketwould stabilize prices. Thus, the majority ofinvestors must be solving a different problem,deciding on their portfolios with differentconsiderations in mind, so that they are alwaysjust willing to hold the outstanding stocks andbonds at current prices. Before going againstthis crowd, it is wise to understand why thecrowd seems headed in a different direction.

Here, a good macroeconomic model ofstock market risk could be extremely useful.The models describe why average consumersare so afraid of stocks and why that fear changesover time. Then, individuals in circumstances


that make them different from everyone elsecan understand why they should behave differ-ently from the crowd. If you have no habits orare immune to labor income shocks, in otherwords if you are unaffected by the state vari-ables or risk factors that drive the stock marketpremium, by all means go your own way.However, the current state of the art is notadvanced enough to provide concrete advicealong these lines.

The last possibility is that one thinks oneis smarter than everyone else, and that theequity premium and predictability are justpatterns that are ignored by other people. This

is a dangerous stance to take. Someone must bewrong in the view that he or she is smarter thaneveryone else. Furthermore, this view alsosuggests that the opportunities are not likely tolast. People do learn. The opinions in this articleare hardly a secret. We could interpret the recentrun-up in the market as the result of peoplefinally figuring out how good an investmentstocks have been for the last century, and build-ing institutions that allow wide participation inthe stock market. If so, future returns are likelyto be much lower, but there is not much one cando about it but sigh and join the parade.

APPENDIX: DERIVATIONS

Variance decompositionMassaging an identity,

1

1

1

20

11

1

11 1 1

11 1 1

11 1

1

1

1

11 1

1

1

=

= +

= +

= +FHG

IKJ

= +

FHG

IKJ

+−

+

+− + +

+− + +

+− +

+

+

+−

==

∞+

+ −

→∞ +−

=

+

+

∏∑

∏

R R

RP D

P

P

DR

P D

D

P

DR

P

D

D

D

P

DR

D

D

RP

D

t t

tt t

t

t

tt

t t

t

t

tt

t

t

t

t

t

tt k

k

j

j

t k

t k

jt k

k

jt j

t j

)

lim .

This equation shows how price-dividendratios are exactly linked to subsequent returns,dividend growth, or a potential bubble. It isconvenient to approximate this relation. Wecan follow Cochrane (1991b) and take a Taylorexpansion now, or follow Campbell and Shiller(1986) and Taylor and expand the first equa-tion in 20 to

p d d r p dt t t t t t− = − + −+ + + +∆ 1 1 1 1ρ( )

and then iterate to

p d d r

p d

t tj

jt j t j

j

jt j t j

− = − +

−

=

∞

+ +

→∞+ +

∑ρ

ρ

1

( )

lim ( ) .

∆

Consumption-portfolio equationsI develop the consumption-portfolio prob-

lem in continuous time. This leads to a numberof simplifications that can also be derived asapproximations or specializations to the normaldistribution in discrete time. A security hasprice P, dividend Ddt and thus instantaneousrate of return dP/P + D/Pdt. The utility func-tion is

E e u C dsts

t s−

+z ρ ( ) .

The first-order condition for an optimalconsumption-portfolio choice is

′ = ′ +

′

z −+ +

−+ +

u C P E e u C D ds

E e u C P

t t ts

t s t s

tk

t k t k

( ) ( )

( ) .

ρ

ρ

Letting the time interval shrink to zero,we have

0 = +E d P D dtt t( )Λ

where

Λ tt

te u C≡ ′−ρ ( ) .

Expanding the second moment, and divid-ing by ΛP

0 = FH

IK + + F

HIK +

LNM

OQP

EdP

P

D

Pdt E

dE

d dP

Pt t tt

t

ΛΛ

ΛΛ

.


δ δ

δ

j

j

j

j

j

j

u C x u C y

u C y

∑ ∑

∑

− = + +

−

( ) ( )

( ) .

1

2

1

2

Using the power functional form,

( ) ( ) ( ) ,C x C y C y− = + + −− − −1 1 11

2

1

2γ γ γ

and solving for x,

x C C y C y= − + + −LNM

OQP

− − −1

2

1

21 1

1

1( ) ( ) .γ γ γ

This equation is easier to solve in ratioform; the fraction of consumption that thefamily would pay is related to the fractionalwealth risk by

x

C

y

C

y

C= − +F

HIK + −F

HIK

LNMM

OQPP

− − −1

1

21

1

21

1 11

1γ γ γ.

This equation is the basis for the calcula-tions in table 5.

For small risks, we can approximate

u C x u C y u C y

u C x u C y

x

C

Cu C

u C

y

C

y

C

x

C

y

C

x

y

y

C

( ) ( ) ( )

( ) ( )

( )

( )

.

− = + + −

− ′ ≈ ′′

≈ − ′′′

FH

IK = F

HIK

≈ FH

IK

≈ FH

IK

1

21

22

2 2

2

γ

γ

γ

Applying this basic condition to a risk-free asset,

r dt Ed

dt Edu C

u C

dtCu C

u CE

dC

C

r dt dt EdC

C

tf

t t

t

tf

t

= − LNM

OQP = − ′

′LNM

OQP

=

− ′′′

LNM

OQP

= + LNM

OQP

ΛΛ

ρ

ρ

ρ γ

( )

( )

( )

( )

.

This establishes equation 4. For any other asset,

0 = FH

IK + − = −

LNM

OQP

EdP

P

D

Pdt r dt E

d dP

Pt tf

tt

t

ΛΛ

.

Using Ito’s lemma on Λ, we have

Ed dP

P

Cu C

u CE

dC

C

dP

Ptt

tt

ΛΛ

LNM

OQP

= ′′′

FH

IK

( )

( ).

Finally, using the symbols

rdP

P

D

Pdt r r dt

Cu C

u Cc

dC

C

f f= + =

= − ′′′

=

, ,

( )( )

,γ ∆

we have equation 5,

E r r c r

c r c r

tf

t

t t t

( ) ,

( ) ( , ) .

− = − =

−

γ

γ σ σ ρ

cov ∆

∆ ∆a f

I drop the t subscript in the text where itis not important to keep track of the differencebetween conditional and unconditional moments.

Risk aversion calculationsWhat is the amount x that a consumer is

willing to pay every period to avoid a bet thateither increases consumption by y every periodor decreases it by the same amount? The an-swer is found from the condition


NOTES

1More formally, we can only reject hypotheses that thetrue return is less than 3 percent or greater than 13 percentwith 95 percent probability.

2A bit more formally, if you start with a regression of logreturns on the P/D ratio, r

t+1 = a + b p/d

t + ε

t+1, and a

similar autoregression of the P/D ratio, p/dt = µ + ρ p/d

t-1

+ δ t, then you can calculate the implied long-horizon

regression statistics. The fact that ρ is a very high numberimplies that long-horizon return regression coefficientsand R2 rise with the horizon, as in the table. See Hodrick(1992) and Cochrane (1991a) for calculations.

3The OLS regression estimate of ρ, is 0.90. However, thisestimate is severely downward biased. In a Monte Carloreplication of the regression, a true coefficient ρ = 0.90resulted in an estimate $ρ , with a mean of 0.82, a medianof 0.83, and a standard deviation of 0.09. Assuming a truecoefficient of 0.98 produces an OLS estimator $ρOLS, withmedian 0.90. I therefore adjust for the downward bias ofthe OLS estimate by using $ρ = 0.98.

4To generate a negative expected excess return, we haveto believe that the market return is negatively conditional-ly correlated with the state variables that drive excessreturns, for example consumption growth. This is theoreti-cally possible, but seems awfully unlikely.

5Craine (1993) does a formal test of price/dividend sta-tionarity and connects the test to bubbles. My statementsare a superficial dismissal of a large literature. A lot ofcareful attention has been paid to the bubble possibility,but the current consensus seems to be that bubbles, as Ihave defined them here, do not explain price variation.

6Eliminate the last term, multiply both sides by (pt – d

t) –

E(pt – d

t) and take expectations.

7Several ways around this argument do exist. Kocherlakota(1990) defends a preference for later consumption. Kan-del and Stambaugh (1991) note that the argument hingescritically on the definition of ∆c. If we define ∆c as theproportional change in consumption ∆c = (C

t + ∆ t – C

t)/C

t

as I have (or, more properly, ∆c = dC/C in continuoustime; see the appendix), then we obtain equation 4. How-ever, if we define ∆c as the change in log consumption,∆c = ln (C

t + ∆ t/C

t) or more properly ∆c = d(ln C), we

obtain an additional term, r f = ρ +γE(∆c) – 1/2γ2σ2(∆c).For γ < 100 or so, the choice does not matter. The lastterm is small, since E(∆c) ≈ σ(∆c) ≈ 0.01. However, sinceγ is squared, the second term can be large with γ = 250,and can take the place of a negative ρ in generating a1 percent interest rate with 1 percent consumption growth.What’s going on? The model u′(C) = C–250 is extraordinar-ily sensitive to the probability of consumption declines.The second model gives slightly higher weight to thoseprobabilities. Rather than rescue the model with γ = 250,in my evaluation, this example shows how special it is:It says that interest rates as well as all asset prices dependonly on the probabilities assigned to extremely rare events.

8I specify bets on annual consumption to sidestep theobjection that most bets are bets on wealth rather than betson consumption. As a first-order approximation, consum-ers will respond to lost wealth by lowering consumptionat every date by the same amount. More sophisticatedcalculations yield the same qualitative results.

9The value function is formally defined as the achievedlevel of expected utility. It is a function of wealth becausethe richer you are, the happier you can get, if you spendyour wealth wisely. The value can also be a function ofother variables such as labor income or expected returnsthat describe the environment. Thus,

V W E e u c ds s ttct s

ts

s t s( , ) max ( ) . .⋅ =+

−=

∞+zm r

ρ0

(constraints).

The dot reminds us that there can be other arguments tothe value function. V

W = u

c is the “envelope” condition,

and follows from this definition.

10Precisely, define the “surplus consumption ratio,”S = (C – X)/C, and denote s = ln S. Then s adapts to con-sumption following a discrete-time “square root process”

s s s sS

s s c c gt t t t t+1 111

1 2 1− = − − − + − − −LNM

OQP − −+( ) ( ) ( ).φ)(

Taking a Taylor approximation, this specification islocally the same as allowing log habit x to adjust toconsumption,

x const x ct t t+ +≈ + + −1 11. ( ) .φ φ

Campbell and Cochrane specify that habits are “external”:Your neighbor’s consumption raises your habit. This is asimplification, since it means that each consumptiondecision does not take into account its habit-formingeffect. They argue that this assumption does not greatlyaffect aggregate consumption and asset price implications,though it is necessary to reconcile the unpredictability ofindividual consumption growth.

11Technically, this assertion depends on the form of theutility function. For example, with log utility, consumersdon’t care about future returns. In this statement I am assum-ing risk aversion greater than 1. See Campbell (1996).

12To prove this assertion, just substitute for Cit and take

the expectation:

11

212

1= − − + +LNM

OQP− +E y y Rt it t t texp ln .ρ γ η γ

Since η is independent of everything else, we can useE[f(ηy)] = E[f(ηy)y)]. Now, with η normal, E (exp[–γη

ity

t]y

t) = exp [1–

2γ 2y

t2]. Therefore, we have


11

2

1

2

11

21

2

1

1

1 1

12 2 2

1

1 1

1

= − + + +LNM

OQP

= − + ++

FHG

IKJ +FHG

IKJ +

LNMM

OQPP

=

− +

− +

−

E y y R

ER

R

E

t t t t

tt

t

t

exp ln

exp ( )( )

ln ln

.

ρ γ γ

ρ γ γγ γ

ρ

13There is a possibility that the square root term in equa-tions 11 and 12 might be negative. Constantinides andDuffie rule out this possibility by assuming that thediscount factor m satisfies

131

) ln lnmC

Ctt

t

≥ − −−

ρ γ

in every state of nature, so that the square root term ispositive.

We can sometimes construct such discount factors bypicking parameters a, b in m

t = max [a + bR

t, e−ρ (C

t/C

t–1)−γ]

to satisfy equation 14. However, neither this constructionnor a discount factor satisfying equation 13 is guaranteedto exist for a given set of assets. The restriction in equation13 is a tighter form of the familiar restriction that m

t≥ 0 is

equivalent to the absence of arbitrage in the assets underconsideration. Presumably, this restriction is what rulesout markets for individual labor income risks in themodel. The example m = 1/R that I use is a positive dis-count factor that prices a single asset return 1 = E(R–1R),but does not necessarily satisfy the restriction in equation13. For high R, we can have very negative ln1/R. Thisis why the lines in figure 6 run into the horizontal axisat high R.

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