ńņš. 15 |

ciation, or does some of it still represent a high expected return from holding debt in that

countryā™s currency? Furthermore, while expected depreciation is clearly a large part of the

story for high interest rates in countries that have constant high inļ¬‚ation or that may suffer

spectacular depreciation of a pegged exchange rate, how does the story work for, say, the U.S.

vs. Germany, where inļ¬‚ation rates diverge little, yet exchange rates ļ¬‚uctuate a surprisingly

large amount?

Table 6 presents the facts, as summarized by Hodrick (2000) and Engel (1996). The ļ¬rst

row of Table 6 presents the average appreciation of the dollar against the indicated currency

over the sample period. The dollar fell against DM, yen and Swiss Franc, but appreciated

against the pound. The second row gives the average interest rate differential ā“ the amount

by which the foreign interest rate exceeds the US interest rate. According to the expectations

hypothesis, these two numbers should be equal ā“ interest rates should be higher in countries

whose currencies depreciate against the dollar.

The second row shows roughly the right pattern. Countries with steady long-term inļ¬‚ation

have steadily higher interest rates, and steady depreciation. The numbers in the ļ¬rst and

second rows are not exactly the same, but exchange rates are notoriously volatile so these

averages are not well measured. Hodrick shows that the difference between the ļ¬rst and

second rows is not statistically different from zero. This fact is exactly analogous to the

fact of Table 4 that the expectations hypothesis works well āon averageā for US bonds and

is the tip of an iceberg of empirical successes for the expectations hypothesis as applied to

currencies.

As in the case of bonds, however, we can also ask whether times of temporarily higher

or lower interest rate differentials correspond to times of above and below average depre-

394

SECTION 20.1 TIME-SERIES PREDICTABILITY

ciation as they should. The third and ļ¬fth rows of Table 6 address this question, updating

Hansen and Hodrickā™s (1980) and Famaā™s (1984) regression tests. The number here should

be +1.0 in each case ā“ an extra percentage point interest differential should correspond to

one extra percentage point expected depreciation. As you can see, we have exactly the op-

posite pattern: a higher than usual interest rate abroad seems to lead, if anything to further

appreciation. It seems that the old fallacy of confusing interest rate differentials across coun-

tries with expected returns, forgetting about depreciation, also contains a grain of truth. This

is the āforward discount puzzle,ā and takes its place alongside the forecastability of stock

and bond returns. Of course it has produced a similar avalanche of academic work dissect-

ing whether it is really there and if so, why. Hodrick (1987), Engel (1996), and Lewis (1995)

provide surveys.

The R2 shown in Table 6 are quite low. However, like D/P, the interest differential is a

slow-moving forecasting variable, so the return forecast R2 build with horizon. Bekaert and

Hodrick (1992) report that the R2 rise to the 30-40% range at six month horizons and then

decline again. Still, taking advantage of this predictability, like the bond strategies described

above, is quite risky.

U

DM Ā£ SF

Mean appreciation -1.8 3.6 -5.0 -3.0

Mean interest differential -3.9 2.1 -3.7 -5.9

b, 1975-1989 -3.1 -2.0 -2.1 -2.6

.026 .033 .034 .033

R2

b, 1976-1996 -0.7 -1.8 -2.4 -1.3

Table 6. The ļ¬rst row gives the average appreciation of the dollar against the indi-

cated currency, in percent per year. The second row gives the average interest dif-

ferential ā“ foreign interest rate less domestic interest rate, measured as the forward

premium ā“ the 30 day forward rate less the spot exchange rate. The third through

ļ¬fth rows give the coefļ¬cients and R2 in a regression of exchange rate changes on

the interest differential = forward premium,

f d

st+1 ā’ st = a + b(ft ā’ st ) + Īµt+1 = a + b(rt ā’ rt ) + Īµt+1

where s = log spot exchange rate, f = forward rate, rf = foreign interest rate, rd =

domestic interest rate. Source: Hodrick (1999) and Engel (1996).

The puzzle does not say that one earns more by holding bonds from countries with higher

interest rates than others. Average inļ¬‚ation, depreciation, and interest rate differentials line

up as they should. If you just buy bonds with high interest rates, you end up with debt from

Turkey and Brazil, whose currencies inļ¬‚ate and depreciate steadily. The puzzle does say that

one earns more by holding bonds from countries whose interest rates are higher than usual

relative to U.S. interest rates.

However, the fact that the āusualā rate of depreciation and āusualā interest differential

395

CHAPTER 20 EXPECTED RETURNS IN THE TIME-SERIES AND CROSS-SECTION

varies through time, if they are well-deļ¬ned concepts at all, may diminish if not eliminate the

out-of-sample performance of trading rules based on these regressions.

The foreign exchange regressions offer a particularly clear-cut case in which āPeso prob-

lemsā can skew forecasting regressions. Lewis (1995) credits Milton Friedman for coining

the term to explain why Mexican interest rates were persistently higher than U.S. interest

rates in the early 1970ā™s even though the currency had been pegged for more than a decade.

A small probability of a huge devaluation each period can correspond to a substantial interest

differential. You will see long stretches of data in which the expectations hypothesis seems

not to be satisļ¬ed, because the collapse does not occur in sample. The Peso subsequently col-

lapsed, giving substantial weight to this view. Since āPeso problemsā have become a generic

term for the effects of small probabilities of large events on empirical work. Rietz (1988) of-

fered a Peso problem explanation for the equity premium that investors are afraid of another

great depression which has not happened in sample. Selling out of the money put options and

earthquake insurance in Los Angeles are similar strategies whose average returns in a sample

will be severely affected by rare events.

20.2 The Cross-section: CAPM and Multifactor Models

Having studied how average returns change over time, now we study how average returns

change across different stocks or portfolios.

20.2.1 The CAPM

For a generation, portfolios with high average returns also had high betas. I illustrate with

the size-based portfolios.

The ļ¬rst tests of the CAPM such as Lintner (1965) were not a great success. If you plot

or regress the average returns versus betas of individual stocks, you ļ¬nd a lot of dispersion,

and the slope of the line is much too ļ¬‚at ā“ it does not go through any plausible riskfree rate.

Miller and Scholes (1972) diagnosed the problem. Betas are measured with error, and

measurement error in right hand variables biases down regression coefļ¬cients. Fama and

MacBeth (1973) and Black, Jensen and Scholes (1972) addressed the problem by grouping

stocks into portfolios. Portfolio betas are better measured because the portfolio has lower

residual variance. Also, individual stock betas vary over time as the size, leverage, and risks

of the business change. Portfolio betas may be more stable over time, and hence easier to

measure accurately.

There is a second reason for portfolios. Individual stock returns are so volatile that you

ā

cannot reject the hypothesis that all average returns are the same. Ļ/ T is big when Ļ =

396

SECTION 20.2 THE CROSS-SECTION: CAPM AND MULTIFACTOR MODELS

40 ā’ 80%. By grouping stocks into portfolios based on some characteristic (other than ļ¬rm

name) related to average returns, you reduce the portfolio variance and thus make it possible

to see average return deferences. Finally, I think much of the attachment to portfolios comes

from a desire to more closely mimic what actual investors would do rather than simply form

a test.

Fama and MacBeth and Black Jensen and Scholes formed their portfolios on betas. They

found individual stock betas, formed stocks into portfolios based on their betas, and then

estimated the portfolioā™s beta in the following period. More recently, size, book/market,

industry, and many other characteristics have been used to form portfolios.

Ever since, the business of testing asset pricing models has been conducted in a simple

loop:

1. Find a characteristic that you think is associated with average returns. Sort stocks into

portfolios based on the characteristic, and check that there is a difference in average

returns between portfolios. Worry here about measurement, survival bias, ļ¬shing bias,

and all the other things that can ruin a pretty picture out of sample.

2. Compute betas for the portfolios, and check whether the average return spread is

accounted for by the spread in betas.

3. If not, you have an anomaly. Consider multiple betas.

This is the traditional procedure, but econometrics textbooks urge you not to group data

in this way. They urge you to use the characteristic as an instrument for the poorly measured

right hand variable instead. It is an interesting and unexplored idea whether this instrumental

variables approach could fruitfully bring us back to the examination of individual securities

rather than portfolios.

The CAPM proved stunningly successful in empirical work. Time after time, every strat-

egy or characteristic that seemed to give high average returns turned out to also have high

betas. Strategies that one might have thought gave high average returns (such as holding very

volatile stocks) turned out not to have high average returns when they did not have high betas.

To give some sense of that empirical work, Figure 44 presents a typical evaluation of

the Capital Asset Pricing Model. (Chapter 15 presented some of the methodological issues

surrounding this evaluation; here I focus on the facts.) I examine 10 portfolios of NYSE

stocks sorted by size (total market capitalization), along with a portfolio of corporate bonds

and long-term government bonds. As the spread along the vertical axis shows, there is a

sizeable spread in average returns between large stocks (lower average return) and small

stocks (higher average return), and also a large spread between stocks and bonds. The ļ¬gure

plots these average returns against market betas. You can see how the CAPM prediction

ļ¬ts: portfolios with higher average returns have higher betas. In particular, notice that the

long term and corporate bonds have mean returns in line with their low betas, despite their

standard deviations nearly as high as those of stocks. Comparing this graph with the similar

Figure 5 of the consumption-based model back in Chapter 2, the CAPM ļ¬ts very well.

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CHAPTER 20 EXPECTED RETURNS IN THE TIME-SERIES AND CROSS-SECTION

Figure 44. The CAPM. Average returns vs. betas on the NYSE value-weighted portfolio

for 10 size-sorted stock portfolios, government bonds, and corporate bonds, 1947-1996. The

solid line draws the CAPM prediction by ļ¬tting the market proxy and treasury bill rates

exactly (a time-series test). The dashed line draws the CAPM prediction by ļ¬tting an OLS

cross-sectional regression to the displayed data points. The small ļ¬rm portfolios are at the top

right. The points far down and to the left are the government bond and treasury bill returns.

In fact, Figure 44 captures one of the ļ¬rst signiļ¬cant failures of the CAPM. The smallest

ļ¬rms (the far right portfolio) seem to earn an average return a few percent too high given their

betas. This is the celebrated āsmall-ļ¬rm effectā (Banz 1981). Would that all failed economic

theories worked so well! It is also atypical in that the estimated market line through the stock

portfolios is steeper than predicted, while measurement error in betas usually means that the

estimated market line is too ļ¬‚at.

20.2.2 Fama-French 3 factors

Book to market sorted portfolios show a large variation in average returns that is unrelated

to market betas. The Fama and French 3 factor model successfully explains the average

returns of the 25 size and book to market sorted portfolios with a 3 factor model, consisting

of the market, a small minus big (SMB) portfolio and a high minus low (HML) portfolio.

398

SECTION 20.2 THE CROSS-SECTION: CAPM AND MULTIFACTOR MODELS

In retrospect, it is surprising that the CAPM worked so well for so long. The assumptions

on which it is built are very stylized and simpliļ¬ed. Asset pricing theory recognized at least

since Merton (1971a,b) the theoretical possibility, indeed probability, that we should need

factors, state variables or sources of priced risk beyond movements in the market portfolio in

order to explain why some average returns are higher than others.

The Fama - French model is one of the most popular multi-factor models that now dom-

inate empirical research. Fama and French (1993) presents the model; Fama and French

(1996) gives an excellent summary, and also shows how the 3 factor model performs in eval-

uating expected return puzzles beyond the size and value effects that motivated it.

āValueā stocks have market values that are small relative to the accountantā™s book value.

(Book values essentially track past investment expenditures.) This category of stocks has

given large average returns. āGrowthā stocks are the opposite of value and have had low

average returns. Since low prices relative to dividends, earnings or book value forecast times

when the market return will be high, it is natural to suppose that these same signals forecast

categories of stocks that will do well; the āvalue effectā is the cross-sectional analogy to

price-ratio predictability in the time-series.

High average returns are consistent with the CAPM, if these categories of stocks have

high sensitivities to the market, high betas. However, small and especially value stocks seem

to have abnormally high returns even after accounting for market beta. Conversely āgrowthā

stocks seem to do systematically worse than their CAPM betas suggest. Figure 45 shows this

value-size puzzle. It is just like Figure 44, except that the stocks are sorted into portfolios

based on size and book-market ratio9 rather than size alone. As you can see, the highest port-

folios have three times the average excess return of the lowest portfolios, and this variation

has nothing at all to do with market betas.

Figures 46 and 47 dig a little deeper to diagnose the problem, by connecting portfolios

that have different size within the same book/market category, and different book/market

within size category. As you can see, variation in size produces a variation in average returns

that is positively related to variation in market betas, as we had in Figure 45. Variation in

book/market ratio produces a variation in average return is negatively related to market beta.

Because of this value effect, the CAPM is a disaster when confronted with these portfolios.

(Since the size effect disappeared in 1980, it is likely that almost the whole story can be told

with book/market effects alone.)

To explain these patterns in average returns, Fama and French advocate a multifactor

model with the market return, the return of small less big stocks (SMB) and the return of

high book/market minus low book/market stocks (HML) as three factors. They show that

variation in average returns of the 25 size and book/market portfolios can be explained by

varying loadings (betas) on the latter two factors. (All their portfolios have betas close to one

on the market portfolio. Thus, market beta explains the average return difference between

I thank Gene Fama for providing me with these data.

9

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CHAPTER 20 EXPECTED RETURNS IN THE TIME-SERIES AND CROSS-SECTION

Figure 45. Average returns vs. market beta for 25 stock portfolios sorted on the basis of

size and book/market ratio.

stocks and bonds, but not across categories of stocks.)

Figures 48 and 49 illustrate Fama and Frenchā™s results. The vertical axis is still the average

return of the 25 size and book/market portfolios. Now, the horizontal axis is the predicted

values from the Fama-French three factor model. The points should all lie on a 45ā—¦ line if

the model is correct. The points lie much closer to this prediction than they do in Figures 46

and 47. The worst ļ¬t is for the growth stocks (lowest line, left hand panel), for which there is

little variation in average return despite large variation in size beta as one moves from small

to large ļ¬rms.

20.2.3 What are the size and value factors?

What are the macroeconomic risks for which the Fama-French factors are proxies or

mimicking portfolios? There are hints of some sort of ādistressā or ārecessionā factor at

work.

A central part of the Fama French model is the fact that these three pricing factors also

explain a large part of the ex-post variation in the 25 portfolios ā“ the R2 in time-series regres-

sions are very high. In this sense, one can regard it as an APT rather than a macroeconomic

400

SECTION 20.2 THE CROSS-SECTION: CAPM AND MULTIFACTOR MODELS

Figure 46. Average excess returns vs. market beta. Lines connect portfolios with different

size category within book to market categories.

factor model.

The Fama-French model is not a tautology, despite the fact that factors and test portfolios

are based on the same set of characteristics.

We would like to understand the real, macroeconomic, aggregate, nondiversiļ¬able risk

that is proxied by the returns of the HML and SMB portfolios. Why are investors so con-

cerned about holding stocks that do badly at the times that the HML (value less growth) and

SMB (small-cap less large-cap) portfolios do badly, even though the market does not fall?

Fama and French (1995) note that the typical āvalueā ļ¬rm has a price that has been driven

down from a long string of bad news, and is now in or near ļ¬nancial distress. Stocks bought

on the verge of bankruptcy have come back more often than not, which generates the high

average returns of this strategy. This observation suggests a natural interpretation of the value

premium: If a credit crunch, liquidity crunch, ļ¬‚ight to quality or similar ļ¬nancial event comes

along, stocks in ļ¬nancial distress will do very badly, and this is just the sort of time at which

one particularly does not want to hear that oneā™s stocks have become worthless! (One cannot

count the ādistressā of the individual ļ¬rm as a ārisk factor.ā Such distress is idiosyncratic and

can be diversiļ¬ed away. Only aggregate events that average investors care about can result

in a risk premium.) Unfortunately, empirical support for this theory is weak, since the HML

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CHAPTER 20 EXPECTED RETURNS IN THE TIME-SERIES AND CROSS-SECTION

Figure 47. Average excess returns vs. market beta. Lines connect portfolios with different

book to market categories within size categories.

portfolio does not covary strongly with other measures of aggregate ļ¬nancial distress. Still,

it is a possible and not totally tested interpretation, since we have so few events of actual

systematic ļ¬nancial stress in recent history.

Heaton and Lucasā™ (1997) results add to this story for the value effect. They note that the

typical stockholder is the proprietor of a small, privately held business. Such an investorā™s

income is of course particularly sensitive to the kinds of ļ¬nancial events that cause distress

among small ļ¬rms and distressed value ļ¬rms. Such an investor would therefore demand

a substantial premium to hold value stocks, and would hold growth stocks despite a low

premium.

Lettau and Ludvigson (2000) (also discussed in the next section) document that HML has

a time-varying beta on both the market return and on consumption. Thus, though there is

(unfortunately) very little unconditional correlation between HML and recession measures,

Lettau and Ludvigson document that HML is sensitive to bad news in bad times.

Liew and Vassalou (1999) are an example of current attempts to link value and small ļ¬rm

returns to macroeconomic events. They ļ¬nd that in many countries counterparts to HML and

SMB contain information above and beyond that in the market return for forecasting GDP

growth. For example, they report a regression

GDPtā’t+1 = a + 0.065 M KTtā’1ā’t + 0.058 HM Ltā’1ā’t + Īµt+1

402

SECTION 20.2 THE CROSS-SECTION: CAPM AND MULTIFACTOR MODELS

Figure 48. Average excess return vs. prediction of the Fama-French 3 factor model. Lines

connect portfolios of different size categories within book to market category.

GDPtā’t+1 denotes the next yearā™s GDP growth and MKT, HML denote the previous

yearā™s return on the market index and HML portfolio. Thus, a 10% HML return reļ¬‚ects

a 1/2 percentage point rise in the GDP forecast.

On the other hand, one can ignore Fama and Frenchā™s motivation and regard the model

as an arbitrage pricing theory. If the returns of the 25 size and book/market portfolios could

be perfectly replicated by the returns of the 3 factor portfolios ā“ if the R2 in the time-series

regressions were 100% ā“ then the multifactor model would have to hold exactly, in order to

preclude arbitrage opportunities. In fact the R2 of Fama and Frenchā™s time-series regressions

are all in the 90%-95% range, so extremely high Sharpe ratios for the residuals (which are

portfolios) would have to be invoked for the model not to ļ¬t well. Equivalently, given the

average returns from HML and SMB and the failure of the CAPM to explain those returns,

there would be near-arbitrage opportunities if value and small stocks did not move together

in the way described by the Fama-French model.

One way to assess whether the three factors proxy for real macroeconomic risks is by

checking whether the multifactor model prices additional portfolios, and especially portfo-

lios that do not have high R2 values. Fama and French (1996) extend their analysis in this

direction: They ļ¬nd that the SMB and HML portfolios comfortably explain strategies based

on alternative price multiples (P/E, B/M), strategies based on 5 year sales growth (this is es-

pecially interesting since it is the only strategy that does not form portfolios based on price

variables) and the tendency of 5 year returns to reverse. All of these strategies are not ex-

403

CHAPTER 20 EXPECTED RETURNS IN THE TIME-SERIES AND CROSS-SECTION

Figure 49. Average excess returns vs. predictions of the Fama-French 3 factor model.

Lines connect portfolios of different book to market category within the same size category.

plained by CAPM betas. However they all also produce portfolios with high R2 values in a

time-series regression on the HML and SMB portfolios! This is good and bad news. It might

mean that the model is a good APT; that the size and book/market characteristics describe the

major sources of priced variation in all stocks. On the other hand it might mean that these ex-

tra sorts just havenā™t identiļ¬ed other sources of priced variation in stock returns. (Fama and

French also ļ¬nd that HML and SMB do not explain āmomentum,ā despite large R2 values.

More on momentum later.)

Oneā™s ļ¬rst reaction may be that explaining portfolios sorted on the basis of size and book

to market by factors sorted on the same basis is a tautology. This is not the case. For exam-

ple, suppose that average returns were higher for stocks whose ticker symbols start later in

the alphabet. (Maybe investors search for stocks alphabetically, so the later stocks are āover-

looked.ā) This need not trouble us if Z stocks happened to have higher betas. If not ā“ if letter

of the alphabet were a CAPM anomaly like book to market ā“ however, it would not necessar-

ily follow that letter based stock portfolios move together. Adding A-L and M-Z portfolios

to the right hand side of a regression of the 26 A,B,C, etc. portfolios on the market portfolio

need not (and probably does not) increase the R2 at all. The size and book to market pre-

mia are hard to measure, and seem to have declined substantially in recent years. But even

if they decline back to CAPM values, Fama and French will still have found a surprisingly

large source of common movement in stock returns.

More to the point, in testing a model It is exactly the right thing to do to sort stocks

404

SECTION 20.2 THE CROSS-SECTION: CAPM AND MULTIFACTOR MODELS

into portfolios based on characteristics related to expected returns. When Black Jensen and

Scholes and Fama and MacBeth ļ¬rst tested the CAPM, they sorted stocks into portfolios

based on betas, because betas are a good characteristic for sorting stocks into portfolios that

have a spread in average returns. If your portfolios have no spread in average returns ā“ if you

just choose 25 random portfolios ā“ then there will be nothing for the asset pricing model to

test.

In fact, despite the popularity of the Fama French 25, there is really no fundamental reason

to sort portfolios based on 2 way or larger sorts of individual characteristics. You should use

all the characteristics at hand that (believably!) indicate high or low average returns and

simply sort stocks according to a one-dimensional measure of expected returns.

The argument over the status of size and book/market factors continues, but the important

point is that it does so. Faced with the spectacular failure of the CAPM documented in Figures

and 4, one might have thought that any hope for a rational asset pricing theory was over. Now

we are back where we were, examining small anomalies and arguing over reļ¬nements and

interpretations of the theory. That is quite an accomplishment!

20.2.4 Macroeconomic factors

Labor income, industrial production, news variables and conditional asset pricing models

have also all had some successes as multifactor models.

I have focused on the size and value factors since they provide the most empirically suc-

cessful multifactor model to date, and have therefore attracted much attention.

Several authors have used macroeconomic variables as factors in order to examine di-

rectly the story that stock performance during bad macroeconomic times determines average

returns. Jagannathan and Wang (1996) and Reyfman (1997) use labor income; Chen Roll

and Ross (1986) use industrial production and inļ¬‚ation among other variables. Cochrane

(1996) uses investment growth. All these authors ļ¬nd that average returns line up against

betas calculated using these macroeconomic indicators. The factors are theoretically easier

to motivate, but none explains the value and size portfolios as well as the (theoretically less

solid, so far) size and value factors.

Lettau and Ludvigson (2000) specify a macroeconomic model that does just as well as

the Fama-French factors in explaining the 25 Fama-French portfolios. Their plots of actual

average returns vs. model predictions show a relation as strong as those of Figures 48 and

49. Their model is

mt+1 = a + b(cawt )āct+1

where caw is a measure of the consumption-wealth ratio. This is a āscaled factor modelā of

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CHAPTER 20 EXPECTED RETURNS IN THE TIME-SERIES AND CROSS-SECTION

the sort advocated in Chapter 8. You can think of it as capturing a time-varying risk aversion.

This is a stunning result.

Though Mertonā™s (1971a,b) theory says that variables which predict market returns should

show up as factors which explain cross-sectional variation in average returns, surprisingly few

papers have actually tried to see whether this is true, now that we do have variables that we

think forecast the market return. Campbell (1996) and Ferson and Harvey (1999) are among

the few exceptions.

20.2.5 Momentum and reversal

Sorting stocks based on past performance, you ļ¬nd that a portfolio that buys long-term

losers and sells long-term winners does better than the opposite ā“ individual stock long-term

returns mean-revert. This āreversalā effect makes sense given return predictability and mean-

reversion, and is explained by the Fama-French 3 factor model. However, a portfolio that

buys short-term winners and sells short-term losers also does well ā“ āmomentum.ā This

effect is a puzzle.

Since a string of good returns gives a high price, it is not surprising that stocks that do

well for a long time (and hence build up a high price) subsequently do poorly, and stocks that

do poorly for a long time (and hence dwindle down to a low price, market value, or market

to book ratio) subsequently do well Table 3, taken from Fama and French (1996) reveals that

this is in fact the case. (As usual, this table is the tip of an iceberg of research on these effects,

starting with DeBont and Thaler 1985 and Jagadeesh and Titman 1993.)

Portfolio Average

Formation Return, 10-1

Strategy Period Months (Monthly %)

Reversal 6307-9312 60-13 -0.74

Momentum 6307-9312 12-2 +1.31

Reversal 3101-6302 60-13 -1.61

Momentum 3101-6302 12-2 +0.38

Table 3. Average monthly returns from reversal and momentum strategies. Each

month, allocate all NYSE ļ¬rms on CRSP to 10 portfolios based on their perfor-

mance during the āportfolio formation monthsā interval. For example, 60-13 forms

portfolios based on returns from 5 years ago to 1 year, 1 month ago. Then buy

the best-performing decile portfolio and short the worst-performing decile portfo-

lio. Source: Fama and French (1996) Table VI.

Reversal

406

SECTION 20.2 THE CROSS-SECTION: CAPM AND MULTIFACTOR MODELS

Here is the āreversalā strategy. Each month, allocate all stocks to 10 portfolios based

on performance in year -5 to year -1. Then, buy the best-performing portfolio and short the

worst-performing portfolio. The ļ¬rst row of Table 3 shows that this strategy earns a hefty

-0.74% monthly return10 . Past long-term losers come back and past winners do badly. This

is a cross-sectional counterpart to the mean-reversion that we studied in section 1.4. Fama

and French (1998a) already found substantial mean-reversion ā“ negative long-horizon return

autocorrelations ā“ in disaggregated stock portfolios, so one would expect this phenomenon.

Spreads in average returns should correspond to spreads in betas. Fama and French verify

that these portfolio returns are explained by their 3 factor model. Past losers have a high

HML beta; they move together with value stocks, and so inherit the value stock premium.

Momentum

The second row of Table 3 tracks the average monthly return from a āmomentumā strat-

egy. Each month, allocate all stocks to 10 portfolios based on performance in the last year.

Now, quite surprisingly, the winners continue to win, and the losers continue to lose, so that

buying the winners and shorting the losers generates a positive 1.31% monthly return.

At every moment there is a most-studied anomaly, and momentum is that anomaly as I

write. It is not explained by the Fama French 3 factor model. The past losers have low prices

and tend to move with value stocks. Hence the model predicts they should have high average

returns, not low average returns. Momentum stocks move together, as do value and small

stocks so a āmomentum factorā works to āexplainā momentum portfolio returns. This is so

obviously ad-hoc (i.e. an APT factor that will only explain returns of portfolios organized on

the same characteristic as the factor) that nobody wants to add it as a risk factor.

A momentum factor is more palatable as a performance attribution factor ā“ to say that a

fund did well by following a momentum strategy rather than by stock picking ability, leaving

aside why a momentum strategy should work. Carhart (1997) uses it in this way to show that

similar momentum behavior in fund returns is due to momentum in the underlying stocks

rather than persistent stock-picking skill.

Momentum may be explained as just a new way of looking at an old phenomenon, the

small apparent predictability of monthly individual stock returns. A tiny regression R2 for

forecasting monthly returns of 0.0025 (1/4%) is more than adequate to generate the momen-

tum results of Table 3. The key is the large standard deviation of individual stock returns,

typically 40% or more at an annual basis. The average return of the best performing decile

of a normal distribution is 1.76 standard deviations above the mean11 , so the winning mo-

Fama and French do not provide direct measures of standard deviations for these portfolios. One can infer

10

however from the betas, R2 values and standard deviation of market and factor portfolios that the standard

deviations are roughly 1-2 times that of the market return, so that Sharpe ratios of these strategies are comparable to

that of the market return.

11 Weā™re looking for

Rā

x rf (r)dr

E(r|r ā„ x) = R ā

x f (r)dr

407

CHAPTER 20 EXPECTED RETURNS IN THE TIME-SERIES AND CROSS-SECTION

mentum portfolio typically went up about 80% in the previous year, and the typical losing

portfolio went down about 60% per year. Only a small amount of continuation will give a 1%

monthly return when multiplied by such largeā returns. To be precise, the monthly indi-

past

vidual stock standard deviation is about 40%/ 12 ā 12% . If the R2 is 0.0025, the standard

ā

deviation of the predictable part of returns is 0.0025 Ć— 12% = 0.6%. Hence, the decile pre-

dicted to perform best will earn 1.76 Ć— 0.6% ā 1% above the mean. Since the strategy buys

the winners and shorts the losers, an R2 of 0.0025 implies that one should earn a 2% monthly

return by the momentum strategy ā“ more even than the 1.3% shown in Table 3. Lewellen

(2000) offers a related explanation for momentum coming from small cross-correlations of

returns.

We have known at least since Fama (1965) that monthly and higher frequency stock re-

turns have slight, statistically signiļ¬cant predictability with R2 in the 0.01 range. However,

such small though statistically signiļ¬cant high frequency predictability, especially in small

stock returns, has also since the 1960s always failed to yield exploitable proļ¬ts after one

accounts for transactions costs, thin trading, high short sale costs and other microstructure is-

sues. Hence, one naturally worries whether momentum is really exploitable after transactions

costs.

Momentum does require frequent trading. The portfolios in Table 3 are reformed every

month. Annual winners and losers will not change that often, but the winning and losing

portfolio must still be turned over at least once per year. Carhart (1996) calculates transac-

tions costs and concludes that momentum is not exploitable after those costs are taken into

account. Moskowitz and Grinblatt (1999) note that most of the apparent gains come from

short positions in small, illiquid stocks, positions that also have high transactions costs. They

also ļ¬nd that a large part of momentum proļ¬ts come from short positions taken November,

anticipating tax-loss selling in December. This sounds a lot more like a small microstructure

glitch rather than a central parable for risk and return in asset markets.

Table 3 already shows that the momentum effect essentially disappears in the earlier data

sample, while reversal is even stronger in that sample. Ahn, Boudoukh, Richardson, and

Whitelaw (1999) show that apparent momentum in international index returns is missing

from the futures markets, also suggesting a microstructure explanation.

Of course, it is possible that a small positive autocorrelation is there and related to some

risk. However, it is hard to generate real positive autocorrelation in realized returns. As

we saw extensively in section 20.335, a slow and persistent variation in expected returns

most naturally generates negative autocorrelation in realized returns. News that expected

returns are higher means future dividends are discounted at a higher rate, so todayā™s price and

return declines. The only way to overturn this prediction is to suppose that expected return

where x is deļ¬ned as the top 10th cutoff,

Z ā 1

f (r)dr = .

10

x

With a normal distribution, x = 1.2816Ļ and E(r|r ā„ x) = 1.755Ļ.

408

SECTION 20.3 SUMMARY AND INTERPRETATION

shocks are positively correlated with shocks to current or expected future dividend growth.

A convincing story for such correlation has not yet been constructed. On the other hand, the

required positive correlation is very small and not very persistent.

20.3 Summary and interpretation

While the list of new facts appears long, similar patters show up in every case. Prices reveal

slow-moving market expectations of subsequent excess returns, because potential offsetting

events seem sluggish or absent. The patterns suggest that there are substantial expected return

premia for taking on risks of recession and ļ¬nancial stress unrelated to the market return.

Magnifying glasses

The effects are not completely new. We knew since the 1960s that high frequency returns

are slightly predictable, with R2 of 0.01 to 0.1 in daily to monthly returns. These effects were

dismissed because there didnā™t seem to be much that one could do about them. A 51/49 bet

is not very attractive, especially if there is any transactions cost. Also, the increased Sharpe

ratio one can obtain by exploiting predictability is directly related to the forecast R2 , so tiny

R2 , even if exploitable, did not seem like an important phenomenon.

Many of the new facts amount to clever magnifying glasses, ways of making small facts

economically interesting. For forecasting market returns, we now realize that R2 rise with

horizon when the forecasting variables are slow-moving. Hence small R2 at high frequency

can mean really substantial R2 , in the 30-50% range, at longer horizons. Equivalently, we re-

alize that small expected return variation can add up to striking price variation if the expected

return variation is persistent. For momentum effects, the ability to sort stocks and funds into

momentum-based portfolios means that incredibly small predictability times portfolios with

huge past returns gives important subsequent returns.

Dogs that did not bark

In each case, an apparent difference in yield should give rise to an offsetting movement,

but seems not to do so. Something should be predictable so that returns are not predictable,

and it isnā™t.

The d/p forecasts of the market return were driven by the fact that dividends should be

predictable, so that returns are not. Instead, dividend growth seems nearly unpredictable. As

we saw, this fact and the speed of the d/p mean reversion imply the observed magnitude of

return predictability.

The term structure forecasts of bond returns were driven by the fact that bond yields

should be predictable, so that returns are not. Instead, yields seem nearly unpredictable at the

one year horizon. This fact means that the forward rate moves one for one with expected

returns, and that a one percentage point increase in yield spread signals as much as a 5

percentage point increase in expected return.

Exchange rates should be forecastable so that foreign exchange returns are not. Instead,

409

CHAPTER 20 EXPECTED RETURNS IN THE TIME-SERIES AND CROSS-SECTION

a one percentage point increase in interest rate abroad seems to signal a greater than one

percentage point increase in expected return.

Prices reveal expected returns

If expected returns rise, prices are driven down, since future dividends or other cash ļ¬‚ows

are discounted at a higher rate. A ālowā price, then, can reveal a market expectation of a high

expected or required return.

Most of our results come from this effect. Low price/dividend, price/earnings, price/book

values signal times when the market as a whole will have high average returns. Low market

value (price times shares) relative to book value signals securities or portfolios that earn high

average returns. The āsmall ļ¬rmā effect derives from low prices ā“ other measures of size

such as number of employees or book value alone have no predictive power for returns (Berk

1997). The ā5 year reversalā effect derives from the fact that 5 years of poor returns lead to

a low price. A high long-term bond yield means that the price of long term bonds is ālow,ā

and this seems to signal a time of good long-term bonds returns. A high foreign interest rate

means a low price on foreign bonds, and this seems to indicate good returns on the foreign

bonds.

The most natural intepretatation of all these effects is that the expected or required return

ā“ the risk premium ā“ on individual securities as well as the market as a whole varies slowly

over time. Thus we can track market expectations of returns by watching price/dividend,

price/earnings or book/market ratios.

Macroeconomic risks

The price-based patterns in time-series and cross-sectional expected returns suggest a pre-

mium for holding risks related to recession and economy-wide ļ¬nancial distress. All of the

forecasting variables are connected to macroeconomic activity (Fama and French 1989). The

dividend price ratio is highly correlated with the default spread and rises in bad times. The

term spread forecasts bond and stock returns, and is also one of the best recession forecasters.

It rises steeply at the bottoms of recessions, and is inverted at the top of a boom. Thus, return

forecasts are high at the bottom of business cycles and low at the top of booms. āValueā and

āsmall-capā stocks are typically distressed. Formal quantitative and empirically successful

economic models of the recession and distress premia are still in their infancy (I think Camp-

bell and Cochrane 1999 is a good start), but the story is at least plausible, and the effects have

been expected by theorists for a generation.

To make this point come to life, think concretely about what you have to do to take

advantage of the value or predictability strategies. You have to buy stocks or long-term bonds

at the bottom, when stock prices are low after a long and depressing bear market; in the

bottom of a recession or ļ¬nancial panic; a time when long-term bond prices and corporate

bond prices are unusually low. This is a time when few people have the guts (the risk-

tolerance) or the wallet to buy risky stocks or risky long-term bonds. Looking across stocks

rather than over time, you have to invest in āvalueā or small market capitalization companies,

dogs by any standards. These are companies with years of poor past returns, years of poor

410

SECTION 20.3 SUMMARY AND INTERPRETATION

sales, companies on the edge of bankruptcy, far off of any list of popular stocks to buy.

Then, you have to sell stocks and long term bonds in good times, when stock prices are high

relative to dividends, earnings and other multiples, when the yield curve is ļ¬‚at or inverted so

that long term bond prices are high. You have to sell the popular āgrowthā stocks with good

past returns, good sales and earnings growth.

Iā™m going on a bit here to counter the widespread impression, best crystallized by Shiller

(2000) that high price earnings ratios must signal āirrational exuberance.ā Perhaps, but is

it just a coincidence that this exuberance comes at the top of an unprecedented economic

expansion, a time when the average investor is surely feeling less risk averse than ever, and

willing to hold stocks despite historically low risk premia? I donā™t know the answer, but

the rational explanation is surely not totally impossible! Is it just a coincidence that we are

ļ¬nding premia just where a generation of theorists said we ought to ā“ in recessions, credit

crunches, bad labor markets, investment opportunity set variables, and so forth?

This line of explanation for the foreign exchange puzzle is still a bit farther off, though

there are recent attempts to make economic sense of the puzzle (See Engelā™s 1996 survey;

Atkeson, Alvarez and Kehoe 1999 is a recent example.) At a verbal level, the strategy leads

you to invest in countries with high interest rates. High interest rates are often a sign of

monetary instability or other economic trouble, and thus may mean that the investments are be

more exposed to the risks of global ļ¬nancial stress or a global recession than are investments

in the bonds of countries with low interest rates, who are typically enjoying better times.

Overall, the new view of ļ¬nance amounts to a profound change. We have to get used to

the fact that most returns and price variation come from variation in risk premia, not variation

in expected cash ļ¬‚ows, interest rates, etc. Most interesting variation in priced risk comes from

non-market factors. These are easy to say, but profoundly change our view of the world.

Doubts

Momentum is, so far, unlike all the other results. The underlying phenomenon is a small

predictability of high frequency returns. However, the price-based phenomena make this pre-

dictability important by noting that, with a slow-moving forecasting variable, the R2 build

over horizon. Momentum is based on a fast-moving forecast variable ā“ the last yearā™s return.

Therefore the R2 decline with horizon. Instead, momentum makes the tiny autocorrelation

of high frequency returns signiļ¬cant by forming portfolios of extreme winners and losers,

so a small continuation of huge past returns gives a large current return. All the other re-

sults are easily digestible as a slow, business-cycle related time-varying expected return. This

speciļ¬cation gives negative autocorrelation (unless we add a distasteful positive correlation

of expected return and dividend shocks) and so does not explain momentum. Momentum

returns have also not yet been linked to business cycles or ļ¬nancial distress in even the infor-

mal way that I suggested for the price-based strategies. Thus, it still lacks much of a plausible

economic interpretation. To me, this adds weight to the view that it isnā™t there, it isnā™t ex-

ploitable, or it represents a small illiquidity (tax-loss selling of small illiquid stocks) that will

be quickly remedied once a few traders understand it. In the entire history of ļ¬nance there

has always been an anomaly-du-jour, and momentum is it right now. We will have to wait to

411

CHAPTER 20 EXPECTED RETURNS IN THE TIME-SERIES AND CROSS-SECTION

see how it is resolved.

Many of the anomalous risk premia seem to be declining over time. The small ļ¬rm effect

completely disappeared in 1980; you can date this as the publication of the ļ¬rst small ļ¬rm

effect papers or the founding of small ļ¬rm mutual funds that made diversiļ¬ed portfolios of

small stocks available to average investors. To emphasize this point, Figure 50 plots size

portfolio average returns vs. beta in the period since 1979. You can see that not only has the

small ļ¬rm premium disappeared, the size-related variation in beta and expected return has

disappeared.

Figure 50. Average returns vs. market betas. CRSP size portfolios less treasury bill rate,

monthly data 1979-1998.

The value premium has been cut roughly in half in the 1990s, and 1990 is roughly the date

ā

of widespread popularization of the value effect, though Ļ/ T leaves a lot of room for error

here. As you saw in Table RR, the last 5 years of high market returns have cut the estimated

return predictability from the dividend-price ratio in half.

These facts suggest an uncomfortable implication: that at least some of the premium the

new strategies yielded in the past was due to the fact that they were simply overlooked or are

artifacts of data-dredging.

Since they are hard to measure, one is tempted to put less emphasis on these average

412

SECTION 20.4 PROBLEMS

returns. However, they are crucial to our interpretation of the facts. The CAPM is perfectly

consistent with the fact that there are additional sources of common variation. For example,

it was long understood that stocks in the same industry move together; the fact that value or

small stocks also move together need not cause a ripple. The surprise is that investors seem to

earn an average return premium for holding these additional sources of common movement,

whereas the CAPM predicts that (given beta) they should have no effect on a portfolioā™s

average returns.

20.4 Problems

1. Does equation (20.308) condition down to information sets coarser than that observed by

agents? Or must we assume that whatever VAR is used by the econometrician contains

all information seen by agents?

2. Show that the two regressions in Table 5 are complementary ā“ that the coefļ¬cients add

up to one, mechanically, in sample.

3. Derive the return innovation decomposition (20.319), directly. Write the return

rt = ādt + Ļ (pt ā’ dt ) ā’ (ptā’1 ā’ dtā’1 )

Apply Et ā’ Etā’1 to both sides,

(340)

rt ā’ Etā’1 rt = (Et ā’ Etā’1 ) ādt + Ļ (Et ā’ Etā’1 ) (pt ā’ dt ) .

Use the price-dividend identity and iterate forward to obtain (20.308).

4. Find the univariate representation and mean-reversion statistics for prices implied by the

simple VAR and the three dividend examples.

5. Find the univariate return representation from a general return forecasting VAR.

rt+1 = axt + Īµrt+1

xt+1 = bxt + Īµxt+1

Find the correlation between return and x shocks necessary to generate uncorrelated

returns.

6. Show that stationary xt ā’ yt , āxt , āyt imply that xt and yt must have the same variance

ratio and long-run differences must become perfectly correlated. Start by showing that

the long run variance limkā’ā var(xt+k ā’ xt )/k for any stationary variable must be

zero. Apply that fact to xt ā’ yt .

7. Compute the long-horizon regression coefļ¬cients and R2 in the VAR (20.311)-(20.317).

Show that the R2 do indeed rise with horizon. Do coefļ¬cients and R2 rise forever, or do

they turn around at some point?

413

Chapter 21. Equity premium puzzle and

consumption-based models

The original speciļ¬cation of the consumption-based model was not a great success, as we

saw in Chapter 1. Still, it is in some sense the only model we have. The central task of

ļ¬nancial economics is to ļ¬gure out what are the real risks that drive asset prices and expected

returns. Something like the consumption-based model ā“ investorsā™ ļ¬rst order conditions for

savings and portfolio choice ā“ has to be the starting point.

Rather than dream up models, test them and reject them, ļ¬nancial economists since the

work of Mehra and Prescott (1986) and Hansen and Jagannathan (1991) have been able to

work backwards to some extent, characterizing the properties that discount factors must have

in order to explain asset return data. Among other things, we learned that the discount factor

had to be extremely volatile, while not too conditionally volatile; the riskfree rate or condi-

tional mean had to be pretty steady. This knowledge is now leading to a much more successful

set of variations on the consumption-based model.

21.1 Equity premium puzzles

21.1.1 The basic equity premium/riskfree rate puzzle

The postwar US market Sharpe ratio is about 0.5 ā“ an 8% return and 16% standard devi-

ation. The basic Hansen-Jagannathan bound

E(Re ) Ļ(m)

ā¤ ā Ī³Ļ(āc)

e)

Ļ(R E(m)

implies Ļ(m) ā„ 50% on an annual basis, requiring huge risk aversion or consumption growth

volatility.

The average risk free rate is about 1%, so E(m) ā 0.99. High risk aversion with power

utility implies a very high riskfree rate, or requires a negative subjective discount factor.

Interest rates are quite stable over time and across countries, so Et (m) varies little. High

risk aversion with power utility implies that interest rates are very volatile.

In Chapter 1, we derived the basic Hansen-Jagannathan (1991) bounds. These are char-

acterizations of the discount factors that price a given set of asset returns. Manipulating

414

SECTION 21.1 EQUITY PREMIUM PUZZLES

0 = E(mRe ) we found

|E(Re )|

Ļ(m)

(341)

ā„ .

Ļ(Re )

E(m)

In continuous time, or as an approximation in discrete time, we found that time-separable

utility implies

|E(Re )|

(342)

Ī³Ļ(āc) ā„

Ļ(Re )

where Ī³ = ā’cu00 /u0 is the local curvature of the utility function, and risk aversion coefļ¬cient

for the power case.

Equity premium puzzle

The postwar mean value weighted NYSE is about 8% per year over the T-bill rate, with a

standard deviation of about 16%. Thus, the market Sharpe ratio E(Re )/Ļ(Re ) is about 0.5

for an annual investment horizon. If there were a constant risk free rate,

E(m) = 1/Rf

would nail down E(m). The T-bill rate is not very risky, so E(m) is not far from the inverse

of the mean T-bill rate, or about E(m) ā 0.99. Thus, these basic facts about the mean and

variance of stocks and bonds imply Ļ(m) > 0.5. The volatility of the discount factor must

be about 50% of its level in annual data!

Per capita consumption growth has standard deviation about 1% per year. With log utility,

that implies Ļ(m) = 0.01 = 1% which off by a factor of 50. To match the equity premium

we need Ī³ > 50,which seems a huge level of risk aversion. Equivalently, a log utility investor

with consumption growth of 1% and facing a 0.5 Sharpe ratio should be investing dramati-

cally more in the stock market, borrowing to do so. He should invest so much that his wealth

and hence consumption growth does vary by 50% each year.

Correlation puzzle

The bound takes the extreme possibility that consumption and stock returns are perfectly

correlated. They are not, in the data. Correlations are hard to measure, since they are sensitive

to data deļ¬nition, timing, time-aggregation, and so forth. Still, the correlation of annual stock

returns and nondurable plus services consumption growth in postwar U.S. data is no more

than about 0.2. If we use this information as well ā“ if we characterize the mean and standard

deviation of all discount factors that have correlation less than 0.2 with the market return ā“

the calculation becomes

|E(Re )|

Ļ(m) 1 1

ĀÆ ĀÆ

ā„ĀÆ = 0.5 = 2.5

Ļm,Re ĀÆ Ļ(Re )

E(m) 0.2

with Ļ(m) ā Ī³Ļ(āc), we now need a risk aversion coefļ¬cient of 250!

415

CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

Here is a classier way to state the correlation puzzle. Remember that proj(m|X) should

price assets just as well as m itself. Now, m = proj(m|X)+Īµ and Ļ 2 (m) = Ļ2 (proj (m|X))+

Ļ2 (Īµ). Some of the early resolutions of the equity premium puzzle ended up adding noise

uncorrelated with asset payoffs to the discount factor. This modiļ¬cation increased discount

factor volatility and satisļ¬ed the bound. But as you can see, adding Īµ increases Ļ 2 (m) with

no effect whatsoever on the modelā™s ability to price assets. As you add Īµ, the correlation be-

tween m and asset returns declines. A bound with correlation, or equivalently comparing

Ļ2 (proj(m|X)) rather than Ļ 2 (m) to the bound avoids this trap.

Average interest rates and subjective discount factors

It has been traditional to use risk aversion numbers of 1 to 5 or so, but perhaps this is

tradition, not fact. Whatā™s wrong with Ī³ = 50 to 250?

The most basic piece of evidence for low Ī³ comes from the relation between consumption

growth and interest rates.

"Āµ Ā¶ā’Ī³ #

Ct+1

Rf = Et (mt+1 ) = Et Ī²

t

Ct

or, in continuous time,

f

(343)

rt = Ī“ + Ī³Et (āc) .

We can take unconditional expectations to compare these equations with average interest

rates and consumption growth.

Average real interest rates are also about 1% Thus, Ī³ = 50 to 250 with a typical Ī“ such as

Ī“ = 0.01 implies a very high riskfree rate, of 50 ā’ 250%. To get a reasonable interest rate, we

have to use a subjective discount factor Ī“ = ā’0.5 to ā’2.5, or ā’50% to ā’250%. Thatā™s not

impossible ā“ present values can converge with negative discount rates (Kocherlakota 1990) ā“

but it does not seem reasonable. People prefer earlier consumption, not later consumption.

Interest rate variation and the conditional mean of the discount factor

Again, however, maybe weā™re being too doctrinaire. What evidence is there against Ī³ =

50 ā’ 250 with corresponding Ī“ = ā’0.5 to ā’2.5?

Real interest rates are not only low on average, they are also relatively stable over time

and across countries. Ī³ = 50 in equation (21.343) means that a country or a boom time with

consumption growth 1 percentage point higher than normal must have real interest rates 50

percentage points higher than normal, and consumption 1 percentage point lower than normal

should be accompanied by real interest rates of 50 percentage points lower than normalā“ you

pay them 48% to keep your money. We donā™t see anything like this.

Ī³ = 50 to 250 in a time-separable utility function implies that consumers are essentially

unwilling to substitute (expected) consumption over time, so huge interest rate variation must

force them to make the small variations in consumption growth that we do see. This level

of aversion to intertemporal substitution is too large. For example, think about what interest

416

SECTION 21.1 EQUITY PREMIUM PUZZLES

rate you need to convince someone to skip a vacation. Take a family with $50,000 per year

consumption, and which spends $2,500 (5%) on an annual vacation. If interest rates are

good enough, though, the family can be persuaded to skip this yearā™s vacation and go on a

much more lavish vacation next year. The required interest rate is ($52, 500/$47, 500)Ī³ ā’ 1.

For Ī³ = 250 that is an interest rate of 3 Ć— 1011 ! For Ī³ = 50, we still need an interest

rate of 14, 800%. I think most of us would give in and defer the vacation for somewhat

lower interest rates! A reasonable willingness to substitute intertemporally is central to most

macroeconomic models that try to capture output, investment, consumption, etc. dynamics.

As always, we can express the observation as a desired characteristic of the discount

f

factor. Though mt+1 must vary a lot, its conditional mean Et (mt+1 ) = 1/Rt must not vary

much. You can get variance in two ways ā“ variance in the conditional mean and variance in

the unexpected component; var(x) = var [Et (x)] + var [x ā’ Et (x)]. The fact that interest

rates are stable means that almost all of the 50% or more unconditional variance must come

from the second term.

The power functional form is really not an issue. To get past the equity premium and these

related puzzles, we will have to introduce other arguments to the marginal utility function ā“

some non-separability. One important key will be to introduce some non-separability that

distinguishes intertemporal substitution from risk aversion.

21.1.2 Variations

Just raising the interest rate will not help, as all-stock portfolios have high Sharpe ratios

too.

Uninsured individual risk is not an obvious solution. Individual consumption is not

volatile enough to satisfy the bounds, and is less correlated with stock returns than aggre-

gate consumption.

The average return in postwar data may overstate the true expected return; a target of

3-4% is not unreasonable.

Is the interest rate ātoo lowā?

A large literature has tried to explain the equity premium puzzle by introducing frictions

that make treasury bills āmoney-likeā and so argue that the short-term interest rate is arti-

ļ¬cially low. (Aiyagari and Gertler 1991 is an example). However, high Sharpe ratios are

pervasive in ļ¬nancial markets. Portfolios long small stocks and short big stocks, or long

value (high book/market) and short growth stocks, give Sharpe ratios of 0.5 or more as well.

Individual shocks

Maybe we should abandon the representative agent assumption. Individual income shocks

417

CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

are not perfectly insured, so individual income and consumption is much more volatile than

aggregate consumption. Furthermore, through most of the sample, only a small portion of

the population held any stocks at all.

This line of argument faces a steep uphill battle. The basic pricing equation applies to

each consumer. Individual income growth may be more volatile than the aggregate, but itā™s

not credible that any individualā™s consumption growth varies by 50% -250% per year! Keep

in mind, this is nondurable and services consumption and the ļ¬‚ow of services from durables,

not durables purchases.

Furthermore, individual consumption growth is likely to be less correlated with stock

returns than is aggregate consumption growth, and the more volatile it is, the less correlated.

As a simple example, write individual consumption equal to aggregate consumption plus an

idiosyncratic shock, uncorrelated with economywide variables,

āci = āca + Īµi .

t t t

Hence,

Ā” Ā¢

cov(āci , rt ) = cov āca + Īµi , rt = cov (āca , rt ) .

t t t t

As we add more idiosyncratic variation, the correlation of consumption with the any aggre-

gate such as stock returns declines in exact proportion so that the asset pricing implications

are completely unaffected.

Luck and a lower target

One nagging doubt is that a large part of the U.S. postwar average stock return may

represent good luck rather than ex-ante expected return.

First of all, the standard deviation of stock returns is so high that standard errors are

ā

surprisingly large. Using the standard formula Ļ/ T , the standard error of average stock

ā

returns in 50 years of data is about 16/ 50 ā 2.3. This fact means that a two-standard error

conļ¬dence interval for the expected return extends from about 3% to about 13%!

This is a pervasive, simple, butāsurprisingly under-appreciated problem in empirical asset

pricing. In 20 years of data, 16/ 20 = 3.6 so we can barely say that an 8% average re-

turn is above zero. 5 year performance averages of something like a stock return are close to

ā

meaningless on a statistical basis, since 16/ 5 = 7. 2. (This is one reason that many funds

are held to tracking error limits relative to a benchmark. You may be able to measure perfor-

mance relative to a benchmark, even if your return and the benchmark are both very volatile.

ā

If Ļ(Ri ā’ Rm ) is small, then Ļ(Ri ā’ Rm )/ T can be small, even if Ļ(Ri ) and Ļ(Rm ) are

large.)

However, large standard errors can argue that the equity premium is really higher than

the postwar return. Several other arguments suggest a bias ā“ that a substantial part of the 8%

average excess return of the last 50 years was good luck, and that the true equity premium is

more like 3-4%.

418

SECTION 21.1 EQUITY PREMIUM PUZZLES

Brown, Goetzmann and Ross (1995) suggest that the U.S. data suffer from selection bias.

One of the reasons that I write this book in the U.S., and that the data has been collected from

the U.S., is precisely because U.S. stock returns and growth have been so good for the last 50

- 100 years.

One way to address this question is to look at other samples. Average returns were a lot

lower in the U.S. before WWII. In Shillerā™s (1989) annual data from 1871-1940, the S&P500

average excess return was only 4.1% However, Campbell (1999, table 1) looks across coun-

tries for which we have stock market data from 1970-1995, and ļ¬nds the average equity

premium practically the same as that for the U.S. in that period. The other countries averaged

a 4.6% excess return while the U.S. had a 4.4% average excess return in that period.

On the other hand, Campbellā™s countries are Canada, Japan, Australia and Western Eu-

rope. These probably shared a lot of the U.S. āgood luckā in the postwar period. There are

lots of countries for which we donā™t have data, and usually because returns were very low in

those countries. As Brown, Goetzmann and Ross (1995) put it, āLooking back over the his-

tory of the London or the New York stock markets can be extraordinarily comforting to an

investor ā“ equities appear to have provided a substantial premium over bonds, and markets

appear to have recovered nicely after huge crashes. ... Less comforting is the past history of

other major markets: Russia, China, Germany and Japan. Each of these markets has had one

or more major interruptions that prevent their inclusion in long term studiesā [my emphasis].

Think of the things that didnā™t happen in the last 50 years. We had no banking panics,

and no depressions; no civil wars, no constitutional crises; we did not lose the cold war, no

missiles were ļ¬red over Berlin, Cuba, Korea or Vietnam. If any of these things had happened,

we might well have seen a calamitous decline in stock values, and I would not be writing

about the equity premium puzzle.

A view that stocks are subject to occasional and highly non-normal crashes ā“ world wars,

great depressions, etc. ā“ makes sampling uncertainty even larger, and means that the average

return from any sample that does not include a crash will be larger than the actual average

return ā“ the Peso Problem again (Reitz 1988).

Fama and French (2000) notice that the price/dividend ratio is low at the beginning of the

sample and high at the end. Much of that is luckā“the dividend yield is stationary in the very

long run, with slow-moving variation through good and bad times. We can understand their

alternative calculation most easily using the return linearization,

rt+1 = ādt+1 + (dt ā’ pt ) ā’ Ļ(dt+1 ā’ pt+1 ).

Then, imposing the view that the dividend price ratio is stationary, we can estimate the aver-

age return as

E (rt+1 ) = E (ādt+1 ) + (1 ā’ Ļ)E(dt ā’ pt ).

The right hand expression gives an estimate of the unconditional average return on stocks

equal to 3.4%. This differes from the sample average return of 9% because, the d/p ratio

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CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

declined dramatically in the postwar sample.

Here is the fundamental issue: Was it clear to people in 1947 (or 1871, or whenever

one starts the sample) and throughout the period that the average return on stocks would be

8% greater than that of bonds, subject only to the 16% year to year variation? Given that

knowledge, would investors have changed their portfolios, or would they have stayed pat,

patiently explaining that these average returns are earned in exchange for risk that they are

not prepared to take? If people expected these mean returns, then we face a tremendous

challenge of explaining why people did not buy more stocks. This is the basic assumption

and challenge of the equity premium puzzle. But phrased this way, the answer is not so

clear. I donā™t think it was obvious in 1947 that the United States would not slip back into

depression, or another world war, but would instead experience a half century of economic

growth and stock returns never before seen in human history. 8% seems like an extremely ā“

maybe even irrationally ā“ exuberant expectation for stock returns as of 1947, or 1871. (You

can ask the same question, by the way, about value effects, market timing, or other puzzles

we try to explain. Only if you can reasonably believe that people understood the average

returns and shied away because of the risks does it make sense to explain the puzzles by risk

rather than luck. Only in that case with the return premia continue anyway!)

This consideration mitigates, but cannot totally solve the equity premium puzzle. Even a

3% equity premium is tough to understand with 1% consumption volatility. If the premium

is 3%, the Sharpe ratio is 3/16 ā 0.2, so we still need risk aversion of 20, and 100 if we

include correlation. 20-100 is a lot better than 50-250, but is still quite a challenge.

21.1.3 Predictability and the equity premium

The Sharpe ratio varies over time. This means that discount factor volatility must vary

over time. Since consumption volatility does not seem to vary over time, this suggests that

risk aversion must vary over time ā“ a conditional equity premium puzzle.

Conventional portfolio calculations suggest that people are not terribly risk averse. These

calculations implicitly assume that consumption moves proportionally to wealth, and inherits

the large wealth volatility.

If stock returns mean-revert, E(Re )/Ļ(Re ) and hence Ļ(m)/E(m) rises faster than the

square root of the horizon. Consumption growth is roughly i.i.d., so Ļ(āc) rises about with

the square root of horizon. Thus, mean-reversion means that the equity premium puzzle is

even worse for long-horizon investors and long-horizon returns.

We have traced the implications of the unconditional Sharpe ratio, and of low and rela-

tively constant interest rates. The predictability of stock returns also has important implica-

tions for discount factors.

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SECTION 21.1 EQUITY PREMIUM PUZZLES

Heteroskedasticity in the discount factorā“conditional equity premium puzzle

The Hansen-Jagannathan bound applies conditionally of course,

Ā”eĀ¢

Et Rt+1 1 Ļ t (mt+1 )

=ā’ .

e e

Ļt (Rt+1 ) Ļt (Rt+1 , mt+1 ) Et (mt+1 )

Mean returns are predictable, and the standard deviation of returns varies over time. So

far, however, the two moments are forecasted by different sets of variables and at different

horizons ā“ d/p, term premium, etc. forecast the mean at long horizons; past squared returns

and implied volatility forecast the variance at shorter horizons ā“ and these variables move at

different times. Hence, it seems that the conditional Sharpe ratio on the left hand side moves

over time. (Glosten, Jagannathan and Runkle 1993, French Schwert and Stambaugh 1987,

Yan 2000 ļ¬nd some co-movements in conditional mean and variance, but do not ļ¬nd that all

movement in one moment is matched by movement in the other.)

On the right hand side, the conditional mean discount factor equals the risk free rate and

so must be relatively stable over time. Time-varying conditional correlations are a possibility,

but hard to interpret. Thus, the predictability of returns strongly suggests that the discount

factor must be conditionally heteroskedastic ā“ Ļt (mt+1 ) must vary through time. Certainly

the discount factors on the volatility bound, or the mimicking portfolios for discount factors,

both of which have Ļ = 1, must have time-varying volatility.

In the standard time-separable model, Ļt (mt+1 ) = Ī³ t Ļ t (āct+1 ). Thus, we need either

time-varying consumption risk or time-varying curvature; loosely speaking a time-varying

risk aversion. The data donā™t show much evidence of conditional heteroskedasticity in con-

sumption growth, leading one to favor a time-varying risk aversion. However, this is a case in

which high risk aversion helps: if Ī³ is sufļ¬ciently high, a small and perhaps statistically hard

to measure amount of consumption heteroskedasticity can generate a lot of discount factor

heteroskedasticity. (Kandel and Stambaugh 1990 follow this approach to explain predictabil-

ity.)

Capm, portfolios and consumption

The equity premium puzzle is centrally about the smoothness of consumption. This is

why it was not noticed as a major puzzle in the early development of ļ¬nancial theory. In turn,

the smoothness of consumption is centrally related to the predictability of returns.

In standard portfolio analyses, there is no puzzle that people with normal levels of risk

aversion do not want to hold far more stocks. From the usual ļ¬rst order condition and with

Ī = VW (W ) we can also write the Hansen-Jagannathan bound in terms of wealth, analo-

gously to (21.342),

ĀÆ ĀÆ

ĀÆE(r) ā’ rf ĀÆ ā’W VW W

(344)

ā¤ Ļ (āw)

Ļ(r) VW

The quantity ā’W WW W /VW is in fact the measure of risk aversion corresponding to most

survey and introspection evidence, since it represents aversion to bets on wealth rather than

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CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

to bets on consumption. (They can be the same for power utility, but not in general.)

For an investor who holds the market, Ļ (āw) is the standard deviation of the stock return,

about 16%. With a market Sharpe ratio of 0.5, we ļ¬nd the lower bound on risk aversion,

ā’W VW W 0.5

= ā 3.

VW 0.16

Furthermore, the correlation between wealth and the stock market is one in this calculation,

so no correlation puzzle crops up to raise the required risk aversion. This is the heart of the

oft-cited Friend and Blume (1975) calculation of risk aversion, one source of the idea that

3-5 is about the right level of risk aversion rather than 50 or 250.

The Achilles heel is the hidden simplifying assumption that returns are independent over

time, and the investor has no other source of income, so no variables other than wealth show

up in its marginal value VW . In such an i.i.d. world, consumption moves one-for-one with

wealth, and Ļ (āc) = Ļ (āw). If your wealth doubles and nothing else has changed, you

double consumption. This calculation thus hides a consumption-based āmodel,ā and the

model has the drastically counterfactual implication that consumption growth has a 16%

standard deviation!

All this calculation has done is say that āin a model in which consumption has a 16%

volatility like stock returns, we donā™t need high risk aversion to explain the equity premium.ā

Hence the central point ā“ the equity premium is about consumption smoothness. Just looking

at wealth and portfolios, you do not notice anything unusual.

In the same way, retreating to the CAPM or factor models doesnā™t solve the puzzle either.

The CAPM is a specialization of the consumption-based model, not alternatives, and thus

hide an equity premium puzzle. For example, I derived the CAPM above as a consequence of

log utility. With log utility, you have to believe that properly measured consumption growth

has a 50% per year standard deviation! That testable implication is right there in the model,

though often ignored. Most implementations of the CAPM take the market premium as

given (ignoring the link to consumption in the modelā™s derivation) and estimate the market

premium as a free parameter. The equity premium puzzle asks whether the market premium

itself makes any sense.

The long-run equity premium puzzle

The fact that annual consumption is much smoother than wealth is an important piece of

information. In the long-run, consumption must move one-for-one with wealth, so consump-

tion and wealth volatility must be the same. Therefore, we know that the world is very far

from i.i.d., so predictability will be an important issue in understanding risk premia.

Predictability can imply mean reversion and Sharpe ratios that rise faster than the square

root of horizon. Thus,

E(Re

tā’t+k ) Ļ(mtā’t+k )

Ā”e Ā¢ā¤ ā Ī³Ļ(āctā’t+k ).

E(mtā’t+k )

Ļ Rtā’t+k

422

SECTION 21.2 NEW MODELS

If stocks do mean-revert, then discount factor volatility must increase faster than the square

root of the horizon. Consumption growth is close to i.i.d.,so the volatility of consumption

growth only increases with the square root of horizon. Thus mean-reversion implies that the

equity premium puzzle is even worse at long investment horizons.

21.2 New models

We want to end up with a model that explains a high market Sharpe ratio, and the high

level and volatility of stock returns, with low and relatively constant interest rates, roughly

i.i.d. consumption growth with small volatility, and that explains the predictability of excess

returns ā“ the fact that high prices today correspond to low excess returns in the future. Even-

tually, we would like the model to explain the predictability of bond and foreign exchange

returns as well, the time-varying volatility of stock returns and the cross-sectional variation

of expected returns, and it would be nice if in addition to ļ¬tting all of the facts, people in the

models did not display unusually high aversion to wealth bets.

I start with a general outline of the features that most models that address these puzzles

share. Then, I focus on two models, the Campbell-Cochrane (1999) habit persistence model

and the Constantinides and Dufļ¬e (1996) model with uninsured idiosyncratic risks. The

mechanisms we uncover in these models apply to a large class. The Campbell-Cochrane

model is a representative from the literature that attacks the equity premium by modifying the

representative agentā™s preferences. The Constantinides and Dufļ¬e model is a representative

of the literature that attacks the equity premium by modeling uninsured idiosyncratic risks,

market frictions, and limited participation.

21.2.1 Outlines of new models

Additional state variables are the natural route to solving the empirical puzzles. Investors

must not be particularly scared of the wealth or consumption effects of holding stocks, but

of the fact that stocks do badly at particular times, or in particular states of nature. Broadly

speaking, most solutions introduce something like a ārecessionā state variable. This fact

makes stocks different, and more feared, than pure wealth bets, whose risk is unrelated to the

state of the economy.

In the ICAPM way of looking at things, we get models of this sort by specifying things

so there is an additional recession state variables z in the value function V (W, z). Then,

expected returns are

Āµ Ā¶

ā’W VW W dW zVW z

(345)

E(r) ā’ rf = cov ,r + cov(z, r).

VW W VW

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CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

In a utility framework, we add other arguments to the utility function u(C, z), so

Āµ Ā¶

ā’CuCC dC zuCz

(346)

E(r) ā’ rf = cov ,r + cov(z, r).

uC C uC

The extra utility function arguments must enter non-separably. If u(C, z) = f (C) + g(z),

then uCz = 0. All utility function modiļ¬cations are of this sort ā“ they add extra goods like

leisure, nonseparability over time in the form of habit persistence, or nonseparability across

states of nature so that consumption if it rains affects marginal utility if it shines.

The lesson of the equity premium literature is that the second term must account for

essentially all of the market premium. Since the cross-sectional work surveyed in Chap-

ter 20 seemed to point to something like a recession factor as the primary determinant of

cross-sectional variation in expected returns, a gratifying unity seems close at hand ā“ and a

fundamental revision of the CAPM-i.i.d. view of the source of risk prices.

The predictability of returns ā“ emphasized by the dramatic contrast between consumption

and wealth volatility at short horizons ā“ suggests a natural source of state variables. Unfortu-

nately, the sign is wrong. The fact that stocks go up when their expected subsequent returns

are low means that stocks, like bonds, are good hedges for shocks to their own opportunity

sets. Therefore, adding the effects of predictability typically lowers expected returns. (The

ātypicallyā in this sentence is important. The sign of this effect ā“ the sign of zVW z ā“ does de-

pend on the utility function and environment. For example, there is no risk premium for log

utility.)

Thus, we need an additional state variable, and one strong enough to not only explain

the equity premium, given that the ļ¬rst terms in (21.345) and (21.346) are not up to the job,

but one stronger still to overcome the effects of predictability. Recessions are times of low

prices and high expected returns. We want a model in which recessions are bad times, so that

investors fear bad stock returns in recessions. But high expected returns are good times for

a pure Merton investor. Thus, the other state variable(s) that describe a recession ā“ high risk

aversion, low labor income, high labor income uncertainty, liquidity, etc. ā“ must overcome

the āgood timesā of high expected returns and indicate that times really are bad after all.

21.2.2 Habits

A natural explanation for the predictability of returns from price/dividend ratios is that people

get less risk averse as consumption and wealth increase in a boom, and more risk averse

as consumption and wealth decrease in a recession. We canā™t tie risk aversion to the level

of consumption and wealth, since that increases over time while equity premia have not

declined. Thus, to pursue this idea, we must specify a model in which risk aversion depends

on the level of consumption or wealth relative to some ātrendā or the recent past.

Following this idea, Campbell and Cochrane (1999) specify that people slowly develop

habits for higher or lower consumption. Thus, the āhabitsā form the ātrendā in consumption.

424

SECTION 21.2 NEW MODELS

The idea is not implausible. Anyone who has had a large pizza dinner or smoked a cigarette

knows that what you consumed yesterday can have an impact on how you feel about more

consumption today. Might a similar mechanism apply for consumption in general and at a

longer time horizon? Perhaps we get used to an accustomed standard of living, so a fall in

consumption hurts after a few years of good times, even though the same level of consumption

might have seemed very pleasant if it arrived after years of bad times. This thought can at

least explain the perception that recessions are awful events, even though a recession year

may be just the second or third best year in human history rather than the absolute best. Law,

custom and social insurance also insure against falls in consumption as much as low levels of

consumption.

The Model

We model an endowment economy with i.i.d. consumption growth.

āct+1 = g + vt+1 ; vt+1 ā¼ i.i.d. N (0, Ļ 2 ).

We replace the utility function u(C) with u(C ā’ X) where X denotes the level of habits.

ā

X ā’ Xt )1ā’Ī³ ā’ 1

t (Ct

E Ī“ .

1ā’Ī³

t=0

Habits should move slowly in response to consumption, something like

ā

X

Ļj ctā’j (347)

xt ā Ī»

j=0

or, equivalently

(348)

xt = Ļxtā’1 + Ī»ct .

(Small letters denote the logs of large letters throughout this section, ct = ln Ct , etc.)

Rather than letting habit itself follow an AR(1) we let the āsurplus consumption ratioā of

consumption to habit follow an AR(1):

Ct ā’ Xt

St =

Ct

(349)

st+1 = (1 ā’ Ļ)ĀÆ + Ļst + Ī» (st ) (ct+1 ā’ ct ā’ g)

s

Since s contains c and x, this equation also speciļ¬es how x responds to c, and it is locally the

same as (21.347). We also allow consumption to affect habit differently in different states by

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CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

specifying an square root type process rather than a simple AR(1),

1p

ńņš. 15 |