CHAPTER 1
The Equity Premium: ABCs
Rajnish Mehra
University of California, Santa Barbara, and NBER
and
Edward C. Prescott
Arizona State University and Federal Reserve Bank of Minneapolis
1. Introduction 2
1.1. An Important Preliminary Issue 2
1.2. Data Sources 3
1.3. Estimates of the Equity Premium 6
1.4. Variation in the Equity Premium over Time 9
2. Is the Equity Premium due to a Premium for Bearing Non-Diversiﬁable Risk? 11
2.1. Standard Preferences 14
References 25
Appendix A 29
Appendix B 29
Appendix C 35
Appendix D 35
JEL Classiﬁcation: G10, G12, D9
Keywords: asset pricing, equity risk premium, CAPM, consumption CAPM,
risk free rate puzzle
We thank George Constantinides, John Donaldson and Viral Shah for helpful comments and
Francisco Azeredo for excellent research assistance.
HANDBOOK OF THE EQUITY RISK PREMIUM
Copyright c 2008 by Rajnish Mehra and Edward C. Prescott. All rights of reproduction in any form reserved. 1
2 Chapter 1 • The Equity Premium: ABCs
1. INTRODUCTION
The year 1978 saw the publication of Robert Lucas’ seminal paper “Asset Prices in
an Exchange Economy” in Econometrica. Its publication transformed asset pricing
and substantially raised the level of discussion, providing a theoretical construct to
study issues that could not be addressed within the dominant paradigm at the time, the
Capital Asset Pricing Model.1
A crucial input parameter for using the latter is the equity
premium2
(the return earned by a broad market index in excess of that earned by a
relatively risk-free security). Lucas’ asset pricing model allowed one to pose questions
about the magnitude of the equity premium.3
In our paper, “The Equity Premium:
A Puzzle,”4
we decided to address this issue.
In this chapter we take a retrospective look at our original paper and show why we
concluded that the equity premium is not a premium for bearing non-diversiﬁable risk.5
We critically evaluate the data sources used to document the puzzle and touch on other
issues that may be of interest to the researcher who did not have a ringside seat 20 years
ago. We stress that the perspective here captures the spirit of our original paper and not
necessarily our current thinking on these issues.6
This and the subsequent two chapters are motivated by the intention to make this
volume a self-contained reference for the beginning researcher in the ﬁeld. The chapters
that follow address the research eﬀorts that have preoccupied the profession in an eﬀort
to explain the equity premium.
This chapter is organized into two parts. Part 1 documents the historical equity premium
in the United States and in selected countries with signiﬁcant capital markets (in
terms of market value) and comments on data sources. Part 2 examines the question, “Is
the equity premium a premium for bearing non-diversiﬁable risk?”
1.1. An Important Preliminary Issue
Any discussion of the equity premium raises the question of whether arithmetic or
geometric returns should be used for summarizing historical return data. In Mehra and
Prescott (1985), we used arithmetic averages. If returns are uncorrelated over time, the
appropriate statistic is the arithmetic average because the expected future value of a $1
investment is obtained by compounding the mean returns. Thus, this is the appropriate
statistic to report if one is interested in the mean terminal value of the investment.7
1See Mossin (1966) for a lucid articulation.
2This was generally assumed to be in the 6–8 percent range.
3To put this advance in perspective, an equivalent contribution in physics would be to come up with a
model that enabled one to address the question of whether the value of Newton’s gravitational constant
G (6.672 × 10−11Nm2
/kg2
) is reasonable, given other cosmological observations. Why is the value of G what
we observe?
4See Mehra and Prescott (1985).
5This chapter draws on material in Mehra and Prescott (2003). All of the acknowledgements in that chapter
continue to apply.
6For an elaboration, see McGrattan and Prescott (2003, 2005) and Mehra and Prescott (2008) in this volume.
7We present a simple proof in Appendix A.
Rajnish Mehra and Edward C. Prescott 3
The arithmetic average return exceeds the geometric average return. If returns are
log-normally distributed, the diﬀerence between the two is one-half the variance of the
returns. Since the annual standard deviation of the equity returns is about 20 percent,
there is a diﬀerence of about 2 percent between the two measures. Using geometric
averages signiﬁcantly underestimates the expected future value of an investment. In
this chapter, as in our 1985 paper, we report arithmetic averages. In instances where
we cite the results of research when arithmetic averages are not available, we clearly
indicate this.8
1.2. Data Sources
A crucial consideration in a discussion of the historical equity premium has to do with
the reliability of early data sources. The data we used in documenting the historical
equity premium in the United States can be subdivided into three distinct subperiods,
1802–1871, 1871–1926, and 1926–present, with wide variation in the quality of the data
over each subperiod. Data on stock prices for the 19th century is patchy, often necessarily
introducing an element of arbitrariness to compensate for its incompleteness.
1.2.1. Subperiod 1802–1871
Equity Return Data
The equity return data prior to 1871 is not particularly reliable. To the best of our knowledge,
the stock return data used by all researchers for the period 1802–1871 is due to
Schwert (1990), who gives an excellent account of the construction and composition of
early stock market indexes. Schwert (1990) constructs a “spliced” index for the period
1802–1987; his index for the period 1802–1862 is based on the work of Smith and Cole
(1935), who constructed a number of early stock indexes. For the period 1802–1820,
their index was constructed from an equally weighted portfolio of seven bank stocks,
and another index for 1815–1845 was composed of six bank stocks and one insurance
stock. For the period 1834–1862, the index consisted of an equally weighted portfolio
of (at most) 27 railroad stocks.9
They used one price quote, per stock, per month,
from local newspapers. The prices used were the average of the bid and ask prices,
rather than transaction prices, and the computation of returns ignores dividends. For the
period 1863–1871, Schwert uses data from Macaulay (1938), who constructed a valueweighted
index using a portfolio of 25 Northeast and Mid-Atlantic railroad stocks;10
this index also excludes dividends. Needless to say, it is diﬃcult to assess how well this
data proxies the “market,” since undoubtedly there were other industry sectors that were
not reﬂected in the index.
8In this case an approximate estimate of the arithmetic average return can be obtained by adding one-half
the variance of the returns to the geometric average.
9“They chose stocks in hindsight . . . the sample selection bias caused by including only stocks that survived
and were actively quoted for the whole period is obvious” (Schwert (1990)).
10“It is unclear what sources Macaulay used to collect individual stock prices but he included all railroads
with actively traded stocks.” Ibid.
4 Chapter 1 • The Equity Premium: ABCs
Return on a Risk-Free Security
Since there were no Treasury bills extant at the time, researchers have used the data
set constructed by Siegel (2002) for this period, using highly rated securities with an
adjustment for the default premium. Interestingly, based on this data set, the equity
premium for the period 1802–1862 was zero. We conjecture that this may be due to the
fact that since most ﬁnancing in the ﬁrst half of the 19th century was done through debt,
the distinction between debt and equity securities was not very clear-cut.11
1.2.2. Subperiod 1871–1926
Equity Return Data
Shiller (1989) is the deﬁnitive source for the equity return data for this period. His data is
based on the work of Cowles (1939), which covers the period 1871–1938. Cowles used
a value-weighted portfolio for his index, which consisted of 12 stocks12
in 1871 and
ended with 351 in 1938. He included all stocks listed on the New York Stock Exchange,
whose prices were reported in the Commercial and Financial Chronicle. From 1918
onward he used the Standard and Poor’s (S&P) industrial portfolios. Cowles reported
dividends, so that, unlike the earlier indexes for the period 1802–1871, a total return
calculation was possible.
Return on a Risk-Free Security
There is no deﬁnitive source for the short-term risk-free rate in the period before 1920,
when Treasury certiﬁcates were ﬁrst issued. In our 1985 study, we used short-term commercial
paper as a proxy for a riskless short-term security prior to 1920 and Treasury
certiﬁcates from 1920–1930.13
Our data prior to 1920 was taken from Homer (1963).
Most researchers have used either our data set or Siegel’s.
1.2.3. Subperiod 1926–Present
Equity Return Data
This period is the “Golden Age” with regard to accurate ﬁnancial data. The NYSE
database at the Center for Research in Security Prices (CRSP) was initiated in 1926
and provides researchers with high-quality equity return data. The Ibbotson Associates
Yearbooks14
are also a very useful compendium of post-1926 ﬁnancial data.
11The ﬁrst actively traded stock was ﬂoated in the U.S. in 1791 and by 1801 there were over 300 corporations,
although less than 10 were actively traded (Siegel (2002)).
12It was only from February 16, 1885, that Dow Jones began reporting an index, initially composed of 12
stocks. The S&P index dates back to 1928, though for the period 1928–1957 it consisted of 90 stocks. The
S&P 500 debuted in March 1957.
13See Mehra and Prescott (2008) in this volume for a discussion on the choice of a proxy for the risk-free
asset.
14Ibbotson Associates, 2006. Stocks, bonds, bills and inﬂation. 2005 Yearbook. Ibbotson Associates, Chicago.
Rajnish Mehra and Edward C. Prescott 5
TABLE 1
U.S. Annual Real Growth Rate of Per Capita Consumption of Non-durables and Services
1889–1978 1889–2004 1889–1929 1930–1978 1930–2004
Mean 0.018 0.018 0.021 0.016 0.018
Std. Dev. 0.036 0.032 0.044 0.028 0.022
Serial Correlation –0.140 –0.135 –0.463 0.520 0.450
Return on a Risk-Free Security
Since the advent of Treasury bills in 1931, short-maturity bills have almost universally
been used as proxy for a “real” risk-free security15
since the innovation in inﬂation
is orthogonal to the path of real GNP growth.16
With the debut of Treasury Inﬂation
Protected Securities (TIPS) on January 29, 1997, the return on these securities is the
real risk-free rate.17
1.2.4. Consumption Data
In our study, we used the Kuznets–Kendrik–NIA per capita real consumption of nondurables
and services for the period 1889–1978. Our data source was Grossman and
Shiller (1981). An updated version of this series is available in Shiller (1989).18
The initial series (the ﬂow of perishable and semi-durable goods to consumers)
for the period 1889–1919 was constructed by William Shaw.19
Simon Kuznets (1938,
1946) modiﬁed Shaw’s measure by incorporating transportation and distribution costs,
and created a series (the ﬂow of perishable and semi-durable goods to consumers)
for the period 1919–1929. The ﬁnal version of these series is available in an unpublished
mimeograph underlying Tables R-27 and R-28 in Kuznets (1961). Kendrick
(1961) made further adjustments to these series in order to make them comparable to
the Department of Commerce’s personal consumption expenditure series. Kendrick’s
adjustments are available in Tables A-IIa and A-IIb in Kendrick (1961). This is the data
source that Grossman and Shiller (1981) used in constructing the1889–1929 subset of
their series on per capita real consumption of non-durables and services. The post-1929
data is from the National Income and Product Accounts of the United States.
Table 1 details the statistics on the growth rate of real per capita consumption of
non-durables and services, while Figure 1 is a time-series plot. We used the statistics in
the 1889–1978 column in our original study.
15Mehra and Prescott. Ibid.
16Litterman (1980) documents that in postwar data the innovation in inﬂation had a standard deviation of
one-half of one percent.
17Mehra and Prescott. Ibid.
18Further updates are available on Robert Shiller’s website: www.econ.yale.edu/∼shiller.
19See Shaw (1947).
6 Chapter 1 • The Equity Premium: ABCs
20.15
20.10
20.05
0.00
0.05
0.10
0.15
1889
1893
1897
1901
1905
1909
1913
1917
1921
1925
1929
1933
1937
1941
1945
1949
1953
1957
1961
1965
1969
1973
1977
1981
1985
1989
1993
1997
2001
FIGURE 1 U.S. annual real growth rate of per capita consumption of non-durables and services
1889–2004.
Source: Mehra and Prescott (1985), updated by the authors.
While the serial correlation of consumption growth for the entire sample is negative,
Azeredo (2007) points out that for more than the last 70 years it has been positive.
In addition, we note that the standard deviation has declined.20
We discuss the
implications for the equity premium in Section 2.1.2 and further in Appendix B.
1.3. Estimates of the Equity Premium
Historical data provides us with a wealth of evidence documenting that for over a century,
stock returns have been considerably higher than those for Treasury bills. This is
illustrated in Table 2, which reports the unconditional estimates21
for the U.S. equity
premium based on the various data sets used in the literature, going back to 1802. The
average annual real return (the inﬂation adjusted return) on the U.S. stock market over
the last 116 years has been about 7.67 percent. Over the same period, the return on a
relatively riskless security was a paltry 1.31 percent. The diﬀerence between these two
returns, the “equity premium,” was 6.36 percent.
20Christina Romer (1999) points out that the larger pre-1929 estimates may be an artifact of the early
methodology rather than due to a change in the underlying stochastic process.
21To obtain unconditional estimates, we use the entire data set to form our estimate. The Mehra–Prescott data
set spans the longest time period for which both consumption and stock return data is available; the former is
necessary to test the implications of consumption-based asset pricing models.
Rajnish Mehra and Edward C. Prescott 7
TABLE 2
U.S. Equity Premium Using Diﬀerent Data Sets
Real return on a relatively
Real return on riskless
a market index (%) security (%) Equity premium (%)
Data set Mean Mean Mean
1802–2004
(Siegel) 8.38 3.02 5.36
1871–2005
(Shiller) 8.32 2.68 5.64
1889–2005
(Mehra–Prescott) 7.67 1.31 6.36
1926–2004
(Ibbotson) 9.27 0.64 8.63
TABLE 3
Equity Premium for Selected Countries
Mean real return
Relatively
riskless Equity
Country Period Market index (%) security (%) premium (%)
United Kingdom 1900–2005 7.4 1.3 6.1
Japan 1900–2005 9.3 −0.5 9.8
Germany 1900–2005 8.2 −0.9 9.1
France 1900–2005 6.1 −3.2 9.3
Sweden 1900–2005 10.1 2.1 8.0
Australia 1900–2005 9.2 0.7 8.5
India 1991–2004 12.6 1.3 11.3
Source: Dimson et al. (2002) and Mehra (2007) for India.
Furthermore, this pattern of excess returns to equity holdings is not unique to the U.S.
but is observed in every country with a signiﬁcant capital market. The U.S. together
with the U.K., Japan, Germany, and France accounts for more than 85 percent of the
capitalized global equity value.
The annual return on the British stock market was 7.4 percent over the last 106
years, an impressive 6.1 percent premium over the average bond return of 1.3 percent.
Similar statistical diﬀerentials are documented for France, Germany, and Japan. Table 3
documents the equity premium for these countries.
8 Chapter 1 • The Equity Premium: ABCs
The dramatic investment implications of this diﬀerential rate of return can be seen
in Table 4, which maps the capital appreciation of $1 invested in diﬀerent assets from
1802 to 2004 and from 1926 to 2004.
One dollar invested in a diversiﬁed stock index yields an ending wealth of $655,348
versus a value of $293, in real terms, for $1 invested in a portfolio of T-bills for the
period 1802–2004. The corresponding values for the 78-year period, 1926–2004, are
$238.30 and $1.54. It is assumed that all payments to the underlying asset, such as
dividend payments to stock and interest payments to bonds, are reinvested and that no
taxes are paid.
This long-term perspective underscores the remarkable wealth-building potential of
the equity premium. It should come as no surprise, therefore, that the equity premium
is of central importance in portfolio allocation decisions and estimates of the cost of
capital and is front and center in the current debate about the advantages of investing
Social Security Trust funds in the stock market.
In Table 5 we document the premium for some interesting historical subperiods:
1889–1933, when the United States was on a gold standard; 1934–2005, when it was oﬀ
the gold standard; and 1946–2005, the postwar period. Table 6 presents 30-year moving
averages, similar to those reported by the U.S. meteorological service to document
“normal” temperature.
TABLE 4
Terminal Value of $1 Invested in Stocks and Bonds
Stocks T-bills
Investment period Real Nominal Real Nominal
1802–2004 $655,348.00 $10,350,077.00 $293.00 $4,614.00
1926–2004 $238.30 $2,533.43 $1.54 $17.87
Source: Ibbotson (2006) and Siegel (2002).
TABLE 5
Equity Premium in Diﬀerent Subperiods
Real return on Real return on a
a market index relatively riskless Equity premium
(%) security (%) (%)
Time period Mean Mean Mean
1889–1933 7.01 3.39 3.62
1934–2005 8.08 0.01 8.07
1946–2005 8.19 0.71 7.48
Source: Mehra and Prescott (1985). Updated by the authors.
Rajnish Mehra and Edward C. Prescott 9
TABLE 6
Equity Premium 30-Year Moving Averages
Real return on Real return on a
a market index relatively riskless Equity premium
(%) security (%) (%)
Time period Mean Mean Mean
1900–1950 7.45 2.95 4.50
1951–2005 8.53 1.11 7.42
Source: Mehra and Prescott (1985). Updated by the authors.
Although the premium has been increasing over time, this is primarily due to the
diminishing return on the riskless asset, rather than a dramatic increase in the return
on equity, which has been relatively constant. The low premium in the 19th century is
largely due to the fact that the equity premium for the period 1802–1861 was zero.22
If
we exclude this period, we ﬁnd that diﬀerence in the premium in the second half of the
19th century relative to average values in the 20th century is less striking.
We see a dramatic change in the equity premium in the post-1933 period—the premium
rose from 3.62 percent to 8.07 percent, an increase of more than 125 percent.
Since 1933 marked the end of the period when the U.S. was on the gold standard, this
break can be seen as the change in the equity premium after the implementation of the
new policy.
1.4. Variation in the Equity Premium Over Time
The equity premium has varied considerably over time, as illustrated in Figures 2 and 3.
Furthermore, the variation depends on the time horizon over which it is measured. There
have even been periods when it has been negative.
The low-frequency variation has been countercyclical. This is shown in Figure 4,
where we have plotted the stock market value as a share of national income23
and the
mean equity premium averaged over certain time periods. We have divided the time
period from 1929 to 2005 into subperiods where the ratio market value of equity to
national income (MV/NI) was greater than and when it was less than the mean value24
over the sample period. Historically, as the ﬁgure illustrates, subsequent to periods when
this ratio was high, the realized equity premium was low. A similar result holds when
22See the earlier discussion on data.
23In Mehra (1998) it is argued that the variation in this ratio is diﬃcult to rationalize in the standard neoclassical
framework since, over the same period, after-tax cash ﬂows to equity as a share of national income
are fairly constant. Here we do not address this issue and simply utilize the fact that this ratio has varied
considerably over time.
24Mean MV/NI for the period 1929–2005 was 0.91.
10 Chapter 1 • The Equity Premium: ABCs
260
240
220
0
20
40
60
1926
1929
1932
1935
1938
1941
1944
1947
1950
1953
1956
1959
1962
1965
1968
1971
1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
EquityRiskPremium(Percent)
FIGURE 2 Realized equity risk premium per year: 1926–2004.
0
2
4
6
8
10
12
14
16
18
1945
1947
1949
1951
1953
1955
1957
1959
1961
1963
1965
1967
1969
1971
1973
1975
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
AverageEquityRiskPremium(Percent)
FIGURE 3 Equity risk premium over the 20-year period 1926–2004. (Source: Ibbotson (2006))
Rajnish Mehra and Edward C. Prescott 11
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
1929
1932
1935
1938
1941
1944
1947
1950
1953
1956
1959
1962
1965
1968
1971
1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
AverageEquityPremium
0.00
0.50
1.00
1.50
2.00
2.50
RatioofMarketValuetoNationalIncome
FIGURE 4 Market value to national income ratio and average equity premium (average of subperiods
when the MV/NI ratio is > or < avg. MV/NI ratio).
stock valuations are low relative to national income. In this case the subsequent equity
premium is high.
Since after-tax corporate proﬁts as a share of national income are fairly constant
over time, this translates into the observation that the realized equity premium was low
subsequent to periods when the price/earnings ratio is high, and vice versa. This is the
basis for the returns predictability literature in ﬁnance.
In Figure 5 we have plotted stock market value as a share of national income and the
subsequent three-year mean equity premium. This provides further conformation that,
historically, periods of relatively high market valuation have been followed by periods
when the equity premium was relatively low.
2. IS THE EQUITY PREMIUM DUE TO A PREMIUM FOR
BEARING NON-DIVERSIFIABLE RISK?
Why have stocks been such an attractive investment relative to bonds? Why has the rate
of return on stocks been higher than that on relatively risk-free assets? One intuitive
answer is that since stocks are “riskier” than bonds, investors require a larger premium
for bearing this additional risk; and indeed, the standard deviation of the returns to
stocks (about 20 percent per annum historically) is larger than that of the returns to
T-bills (about 4 percent per annum), so, obviously they are considerably more risky
than bills! But are they?
12 Chapter 1 • The Equity Premium: ABCs
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
1929
1932
1935
1938
1941
1944
1947
1950
1953
1956
1959
1962
1965
1968
1971
1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
AverageEquityPremium
0.00
0.50
1.00
1.50
2.00
2.50
RatioofMarketValuetoNationalIncome
FIGURE 5 Market value to national income ratio and average 3-year ahead equity premium (average of
subperiods when the MV/NI ratio is > or < avg. MV/NI ratio).
Figures 6 and 7 illustrate the variability of the annual real rate of return on the S&P
500 index and a relatively risk-free security over the period 1889–2005.25
To enhance and deepen our understanding of the risk-return trade-oﬀ in the pricing
of ﬁnancial assets, we take a detour into modern asset pricing theory and look at why
diﬀerent assets yield diﬀerent rates of return. The deus ex machina of this theory is that
assets are priced such that, ex-ante, the loss in marginal utility incurred by sacriﬁcing
current consumption and buying an asset at a certain price is equal to the expected gain
in marginal utility, contingent on the anticipated increase in consumption when the asset
pays oﬀ in the future.
The operative emphasis here is the incremental loss or gain of utility of consumption
and should be diﬀerentiated from incremental consumption. This is because the same
amount of consumption may result in diﬀerent degrees of well-being at diﬀerent times.
As a consequence, assets that pay oﬀ when times are good and consumption levels are
high—when the marginal utility of consumption is low—are less desirable than those
that pay oﬀ an equivalent amount when times are bad and additional consumption is
more highly valued. Hence, consumption in period t has a diﬀerent price if times are
good than if times are bad.
Let us illustrate this principle in the context of the standard, popular paradigm,
the Capital Asset Pricing Model (CAPM). The model postulates a linear relationship
between an asset’s “beta,” a measure of systematic risk, and its expected return. Thus,
high-beta stocks yield a high expected rate of return. That is because in the CAPM,
25The index did not consist of 500 stocks for the entire period.
Rajnish Mehra and Edward C. Prescott 13
260
240
220
0
20
40
60
1889
1893
1897
1901
1905
1909
1913
1917
1921
1925
1929
1933
1937
1941
1945
1949
1953
1957
1961
1965
1969
1973
1977
1981
1985
1989
1993
1997
2001
2005
FIGURE 6 Real annual return on S&P 500 Index (%) 1889–2005.
Source: Mehra and Prescott (1985). Data updated by the authors.
225
220
215
210
25
0
5
10
15
20
25
1957
1961
1893
1897
1901
1905
1909
1913
1917
1921
1925
1929
1933
1937
1941
1945
1949
1953
1965
1969
1973
1977
1981
1985
1989
1993
1997
2001
2005
1889
FIGURE 7 Real annual return on T-bills (%) 1889–2005.
Source: Mehra and Prescott (1985). Data updated by the authors.
14 Chapter 1 • The Equity Premium: ABCs
good times and bad times are captured by the return on the market. The performance
of the market, as captured by a broad-based index, acts as a surrogate indicator for the
relevant state of the economy. A high-beta security tends to pay oﬀ more when the market
return is high—when times are good and consumption is plentiful; it provides less
incremental utility than a security that pays oﬀ when consumption is low, is less valuable,
and consequently sells for less. Thus, higher-beta assets that pay oﬀ in states of
low marginal utility will sell for a lower price than similar assets that pay oﬀ in states
of high marginal utility. Since rates of return are inversely proportional to asset prices,
the lower beta assets will, on average, give a lower rate of return than the former.
Another perspective on asset pricing emphasizes that economic agents prefer to
smooth patterns of consumption over time. Assets that pay oﬀ a larger amount at times
when consumption is already high “destabilize” these patterns of consumption, whereas
assets that pay oﬀ when consumption levels are low “smooth” out consumption. Naturally,
the latter are more valuable and thus require a lower rate of return to induce
investors to hold these assets. (Insurance policies are a classic example of assets that
smooth consumption. Individuals willingly purchase and hold them, despite their very
low rates of return.)
To return to the original question: are stocks that much riskier than T-bills so as to
justify a 7-percentage diﬀerential in their rates of return?
What came as a surprise to many economists and researchers in ﬁnance was the conclusion
of our paper, written in 1979. Stocks and bonds pay oﬀ in approximately the
same states of nature or economic scenarios and, hence, as argued earlier, they should
command approximately the same rate of return. In fact, using standard theory to estimate
risk-adjusted returns, we found that stocks on average should command, at most,
a 1 percent return premium over bills. Since, for as long as we had reliable data (about
100 years), the mean premium on stocks over bills was considerably and consistently
higher, we realized that we had a puzzle on our hands. It took us six more years to convince
a skeptical profession and for our paper “The Equity Premium: A Puzzle” to be
published (Mehra and Prescott (1985)).
2.1. Standard Preferences
The neoclassical growth model and its stochastic variants are a central construct in
contemporary ﬁnance, public ﬁnance, and business cycle theory. It has been used extensively
by, among others, Abel et al. (1989), Auerbach and Kotlikoﬀ (1987), Becker and
Barro (1988), Brock (1979), Cox, Ingersoll, and Ross (1985), Donaldson and Mehra
(1984), Kydland and Prescott (1982), Lucas (1978), and Merton (1971). In fact, much
of our economic intuition is derived from this model class. A key idea of this framework
is that consumption today and consumption in some future period are treated as
diﬀerent goods. Relative prices of these diﬀerent goods are equal to people’s willingness
to substitute between these goods and businesses’ ability to transform these goods
into each other.
The model has had some remarkable successes when confronted with empirical
data, particularly in the stream of macroeconomic research referred to as Real Business
Cycle Theory, where researchers have found that it easily replicates the essential
Rajnish Mehra and Edward C. Prescott 15
macroeconomic features of the business cycle. See, in particular, Kydland and Prescott
(1982). Unfortunately, when confronted with ﬁnancial market data on stock returns,
tests of these models have led, without exception, to their rejection. Perhaps the most
striking of these rejections is our 1985 paper.26
To illustrate this we employ a variation of Lucas’ (1978) endowment economy rather
than the production economy studied in Prescott and Mehra (1980). This is an appropriate
abstraction to use if it is the equilibrium relation between the consumption and asset
returns that are being used to estimate the premium for bearing non-diversiﬁable risk,
which is what we were doing. Introducing production would only complicate the selection
of exogenous processes, which resulted in the observed process for consumption.27
To examine the role of other factors for mean asset returns, it would be necessary to
introduce other features of reality such as taxes and intermediation costs as has recently
been done.28
If the model had accounted for diﬀerences in average asset returns, the
next step would have been to use the neoclassical growth model, which has intertemporal
transformation opportunities through variations in the rate at which the capital stock
is accumulated, to see if this abstraction accounted for the observed large diﬀerences in
average asset returns.
Since per capita consumption has grown over time, we assume that the growth rate
of the endowment follows a Markov process. This is in contrast to the assumption in
Lucas’ model that the endowment level follows a Markov process. Our assumption,
which requires an extension of competitive equilibrium theory,29
enables us to capture
the non-stationarity in the consumption series associated with the large increase in per
capita consumption that occurred over the last century.
We consider a frictionless economy that has a single representative “stand-in”
household. This unit orders its preferences over random consumption paths by
E0
∞
t=0
βt
U(ct) , 0 < β < 1, (1)
where ct is the per capita consumption and the parameter β is the subjective time discount
factor, which describes how impatient households are to consume. If β is small,
people are highly impatient, with a strong preference for consumption now versus consumption
in the future. As modeled, these households live forever, which implicitly
means that the utility of parents depends on the utility of their children. In the real world,
this is true for some people and not for others. However, economies with both types
of people—those who care about their children’s utility and those who do not—have
essentially the same implications for asset prices and returns.30
26The reader is referred to McGrattan and Prescott (2003) and Mehra and Prescott (2007 and 2008) for
an alternative perspective.
27In a production economy, consumption would be endogenously determined, restricting the class of
consumption processes that could be considered. See Appendix C.
28See McGrattan and Prescott (2003) and Mehra and Prescott (2007 and 2008).
29This is accomplished in Mehra (1988).
30See Constantinides, Donaldson, and Mehra (2002, 2005). Constantinides et al. (2007) explicitly model
bequests.
16 Chapter 1 • The Equity Premium: ABCs
We use this simple abstraction to build quantitative economic intuition about what
the returns on equity and debt should be. E0{·} is the expectations operator conditional
upon information available at time zero (which denotes the present time), and
U : R+ → R is the increasing, continuously diﬀerentiable concave utility function. We
further restrict the utility function to be of the constant relative risk aversion (CRRA)
class
U(c, α) =
c1−α
− 1
1 − α
, 0 < α < ∞, (2)
where the parameter α measures the curvature of the utility function. When α = 1, the
utility function is deﬁned to be logarithmic, which is the limit of the above representation
as α approaches 1. The feature that makes this the “preference function of choice”
in much of the literature in Growth and Real Business Cycle Theory is that it is scaleinvariant.
This means that a household is more likely to accept a gamble if both its
wealth and the gamble amount are scaled by a positive factor. Hence, although the
level of aggregate variables such as capital stock have increased over time, the resulting
equilibrium return process is stationary. A second attractive feature is that it is one of
only two preference functions that allows for aggregation and a “stand-in” representative
agent formulation that is independent of the initial distribution of endowments. One
disadvantage of this representation is that it links risk preferences with time preferences.
With CRRA preferences, agents who like to smooth consumption across various states
of nature also prefer to smooth consumption over time, that is, they dislike growth.
Speciﬁcally, the coeﬃcient of relative risk aversion is the reciprocal of the elasticity of
intertemporal substitution. There is no fundamental economic reason why this must be
so. We will revisit this issue in the next chapter, where we examine preference structures
that do not impose this restriction.31
We assume there is one productive unit, which produces output yt in period t, which
is the period dividend. There is one equity share with price pt that is competitively
traded; it is a claim to the stochastic process {yt}.
Consider the intertemporal choice problem of a typical investor at time t. He equates
the loss in utility associated with buying one additional unit of equity to the discounted
expected utility of the resulting additional consumption in the next period. To carry
over one additional unit of equity, pt units of the consumption good must be sacriﬁced,
and the resulting loss in utility is ptU (ct). By selling this additional unit of equity in the
next period, pt+1 + yt+1 additional units of the consumption good can be consumed and
βEt{(pt+1 + yt+1)U (ct+1)} is the expected value of the incremental utility next period.
At an optimum, these quantities must be equal. Hence, the fundamental relation that
prices assets is ptU (ct) = βEt{(pt+1 + yt+1)U (ct+1)}. Versions of this expression can be
found in Rubinstein (1976), Lucas (1978), Breeden (1979), and Prescott and Mehra
(1980), among others. Excellent textbook treatments can be found in Cochrane (2005),
Danthine and Donaldson (2005), Duﬃe (2001), and LeRoy and Werner (2001).
31See Epstein and Zin (1991) and Weil (1989).
Rajnish Mehra and Edward C. Prescott 17
We use it to price both stocks and risk-less one-period bonds.
For equity we have
1 = βEt
U (ct+1)
U (ct)
Re,t+1 , (3)
where
Re,t+1 =
pt+1 + yt+1
pt
. (4)
For the risk-less one-period bonds, the relevant expression is
1 = βEt
U (ct+1)
U (ct)
Rf,t+1, (5)
where the gross rate of return on the riskless asset is by deﬁnition
Rf,t+1 =
1
qt
, (6)
with qt being the price of the bond. Since U(c) is assumed to be increasing, we can
rewrite (3) as
1 = βEt Mt+1Re,t+1 , (7)
where Mt+1 is a strictly positive stochastic discount factor. This guarantees that the
economy will be arbitrage-free and the law of one price holds. A little algebra shows
that
Et(Re,t+1) = Rf,t+1 + Covt
−U (ct+1), Re,t+1
Et(U (ct+1))
. (8)
The equity premium Et(Re,t+1) − Rf,t+1 thus can be easily computed. Expected asset
returns equal the risk-free rate plus a premium for bearing risk, which depends on the
covariance of the asset returns with the marginal utility of consumption. Assets that
co-vary positively with consumption—that is, they pay oﬀ in states when consumption
is high and marginal utility is low—command a high premium since these assets
“destabilize” consumption.
The question we need to address is the following: is the magnitude of the covariance
between the marginal utility of consumption large enough to justify the observed
6 percent equity premium in U.S. equity markets?
To address this issue, we make some additional assumptions. While they are not
necessary and were not, in fact, part of our original paper on the equity premium, we
include them to facilitate exposition and because they result in closed-form solutions.32
32The exposition below is based on Abel (1988), his unpublished notes, and Mehra (2003). See Appendix B
for the analysis in our 1985 paper.
18 Chapter 1 • The Equity Premium: ABCs
These assumptions are
1. the growth rate of consumption xt+1 ≡
ct+1
ct
is i.i.d.
2. the growth rate of dividends zt+1 ≡
yt+1
yt
is i.i.d.
3. (xt, zt) are jointly log-normally distributed.
The consequences of these assumptions are that the gross return on equity Re,t (deﬁned
above) is i.i.d. and that (xt, Re,t) are jointly log-normal.
Substituting U (ct) = c−α
t in the fundamental pricing relation33
pt = βEt (pt+1 + yt+1)
U (ct+1)
U (ct)
, (9)
we get
pt = βEt (pt+1 + yt+1)x−α
t+1 . (10)
As pt is homogeneous of degree 1 in y, we can represent it as
pt = wyt,
and hence Re,t+1 can be expressed as
Re,t+1 =
w + 1
w
·
yt+1
yt
=
w + 1
w
· zt+1. (11)
It is easily shown that
w =
βEt{zt+1x−α
t+1}
1 − βEt{zt+1x−α
t+1}
; (12)
hence,
Et{Re,t+1} =
Et{zt+1}
βEt{zt+1x−α
t+1}
. (13)
Analogously, the gross return on the riskless asset can be written as
Rf,t+1 =
1
β
1
Et{x−α
t+1}
. (14)
Since we have assumed the growth rate of consumption and dividends to be lognormally
distributed,
Et{Re,t+1} =
eμz+ 1
2 σ2
z
βeμz−αμx+1/2(σ2
z +α2σ2
x−2ασx,z)
(15)
33In contrast to our approach, which is in the applied general equilibrium tradition, there is another tradition
of testing Euler equations (such as Eq. (9)) and rejecting them. Hansen and Singleton (1982) and Grossman
and Shiller (1981) exemplify this approach. See Appendix D for an elaboration.
Rajnish Mehra and Edward C. Prescott 19
and
ln Et{Re,t+1} = − ln β + αμx −
1
2
α2
σ2
x + ασx,z, (16)
where μx = E(ln x), σ2
x = Var(ln x), σx,z = Cov(ln x, ln z), and ln x is the continuously
compounded growth rate of consumption. The other terms involving z and Re are
deﬁned analogously. Similarly,
Rf =
1
βe−αμx+ 1
2 α2σ2
x
(17)
and
ln Rf = − ln β + αμx −
1
2
α2
σ2
x. (18)
∴ ln E{Re} − ln Rf = ασx,z. (19)
From (11) it also follows that
ln E{Re} − ln Rf = ασx,Re
,
where σx,Re
= Cov(ln x, ln Re). (20)
The (log) equity premium in this model is the product of the coeﬃcient of risk aversion
and the covariance of the (continuously compounded) growth rate of consumption
with the (continuously compounded) return on equity or the growth rate of dividends. If
we impose the equilibrium condition that x = z, a consequence of which is the restriction
that the return on equity is perfectly correlated to the growth rate of consumption,
we get
ln E{Re} − ln Rf = ασ2
x, (21)
and the equity premium then is the product of the coeﬃcient of relative risk aversion
and the variance of the growth rate of consumption. As we see ahead, this variance is
0.001369, so unless the coeﬃcient of risk aversion α is large, a high-equity premium is
impossible. The growth rate of consumption just does not vary enough!
In Mehra and Prescott (1985) we report the following sample statistics for the U.S.
economy over the period 1889–1978:
Risk-free rate Rf = 1.0080
Mean return on equity E{Re} = 1.0698
20 Chapter 1 • The Equity Premium: ABCs
Mean growth rate of consumption E{x} = 1.0180
Standard deviation of the growth rate of consumption σ{x} = 0.0360
Mean equity premium E{Re} − Rf = 0.0618
In our calibration, we are guided by the tenet that model parameters should meet the
criteria of cross-model veriﬁcation: not only must they be consistent with the observations
under consideration, but they should not be grossly inconsistent with other
observations in growth theory, business cycle theory, labor market behavior, and so on.
There is a wealth of evidence from various studies that the coeﬃcient of risk aversion
α is a small number, certainly less than 10.34
We can then pose a question: if we set
the risk aversion coeﬃcients α to be 10 and β to be 0.99, what are the expected rates of
return and the risk premia using the parameterization above?
Using the expressions derived earlier, we have
ln Rf = − ln β + αμx −
1
2
α2
σ2
x = 0.124
or
Rf = 1.132,
that is, a risk-free rate of 13.2 percent!
Since
ln E{Re} = ln Rf + ασ2
x
= 0.136,
we have E{Re} = 1.146, or a return on equity of 14.6 percent. This implies an equity
risk premium of 1.4 percent, far lower than the 6.18 percent historically observed equity
premium. In this calculation we have been liberal in choosing the values for α and β.
Most studies indicate a value for α that is close to 3. If we pick a lower value for
β, the risk-free rate will be even higher and the premium lower. So the 1.4 percent
value represents the maximum equity risk premium that can be obtained in this class
of models given the constraints on α and β. Since the observed equity premium is
over 6 percent, we have a puzzle on our hands that risk considerations alone cannot
account for.
2.1.1. The Risk-Free Rate Puzzle
Philippe Weil (1989) has dubbed the high risk-free rate obtained above “the risk-free rate
puzzle.” The short-term real rate in the U.S. averages less than 1 percent, while the high
value of α required to generate the observed equity premium results in an unacceptably
34A number of these studies are documented in Mehra and Prescott (1985).
Rajnish Mehra and Edward C. Prescott 21
high risk-free rate. The risk-free rate as shown in Eq. (18) can be decomposed into
three components:
ln Rf = − ln β + αμx −
1
2
α2
σ2
x.
The ﬁrst term, −ln β, is a time preference or impatience term. When β < 1, it reﬂects
the fact that agents prefer early consumption to later consumption. Thus, in a world of
perfect certainty and no growth in consumption, the unique interest rate in the economy
will be Rf = 1/β.
The second term, αμx, arises because of growth in consumption. If consumption
is likely to be higher in the future, agents with concave utility would like to borrow
against future consumption in order to smooth their lifetime consumption. The greater
the curvature of the utility function and the larger the growth rate of consumption, the
greater the desire to smooth consumption. In equilibrium, this will lead to a higher
interest rate since agents in the aggregate cannot simultaneously increase their current
consumption.
The third term, 1
2 α2
σ2
x, arises due to a demand for precautionary saving. In a world
of uncertainty, agents would like to hedge against future unfavorable consumption realizations
by building “buﬀer stocks” of the consumption good. Hence, in equilibrium,
the interest rate must fall to counter this enhanced demand for savings.
Figure 8 plots ln Rf = − ln β + αμx − 1
2 α2
σ2
x calibrated to the U.S. historical values
with μx = 0.0175 and σ2
x = 0.00123 for various values of β. It shows that the
precautionary savings eﬀect is negligible for reasonable values of α (1 < α < 5).
For α = 3 and β = 0.99, Rf = 1.65, which implies a risk-free rate of 6.5 percent—
much higher than the historical mean rate of 0.8 percent. The economic intuition is
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
a
0
280
270
260
250
240
230
220
210
10
20
30
40
50
60
70
80
b5 0.99
b 5 0.96
b 5 0.55
FIGURE 8 Mean risk-free rate vs. α.
22 Chapter 1 • The Equity Premium: ABCs
straightforward—with consumption growing at 1.8 percent a year with a standard deviation
of 3.6 percent—agents with isoelastic preferences have a suﬃciently strong desire
to borrow in order to smooth consumption that it takes a high interest rate to induce
them not to do so.
The late Fischer Black35
proposed that α = 55 would solve the puzzle. Indeed, it can
be shown that the 1889–1978 U.S. experience reported above can be reconciled with
α = 48 and β = 0.55.
To see this, observe that since
σ2
x = ln 1 +
var(x)
[E(x)]2
= 0.00123 and μx = ln E(x) −
1
2
σ2
x = 0.0175,
this implies
α =
ln E(R) − ln Rf
σ2
x
= 48.4.
Since
ln β = − ln Rf + αμx −
1
2
α2
σ2
x
= −0.60,
this implies β = 0.55.
Besides postulating an unacceptably high α, another problem is that this is a “knifeedge”
solution. No other set of parameters will work, and a small change in α will
lead to an unacceptable risk-free rate, as shown in Figure 8. An alternate approach is
to experiment with negative time preferences; however, there seems to be no empirical
evidence that agents do have such preferences.36
Figure 8 shows that for extremely high α, the precautionary savings term dominates
and results in a “low” risk-free rate.37
However, then a small change in the growth rate of
consumption will have a large impact on interest rates. This is inconsistent with a crosscountry
comparison of real risk-free rates and their observed variability. For example,
throughout the 1980s, South Korea had a much higher growth rate than the U.S., but
real rates were not appreciably higher. Nor does the risk-free rate vary considerably
over time, as would be expected if α was large. In Section 3 we show how alternative
preference structures can help resolve the risk-free rate puzzle.
35Private communication, 1981.
36In a model with growth, equilibrium can exist with β > 1. See Mehra (1988) for the restrictions on the
parameters α and β for equilibrium to exist.
37Kandel and Stambaugh (1991) have suggested this approach.
Rajnish Mehra and Edward C. Prescott 23
0.4
0.3
0.2
0.5 1.5
a
2.5210
0.1
EquityPremium(Percent)
0
20.1
correl . 5 20.14
correl . 5 0.00
correl . 5 0.45
FIGURE 9 Equity Premium vs. α.
2.1.2. The Eﬀect of Serial Correlation in the Growth Rate of Consumption
The preceding analysis has assumed that the growth rate of consumption is i.i.d over
time. However, for the sample period 1889–2004 it is slightly negative (−0.135), while
for the sample period 1930–2004 the value is 0.45. The eﬀect of this non-zero serial
correlation on the equity premium can be analyzed using the framework in Appendix
B. Figure 9 shows the eﬀect of changes in the risk aversion parameter on the equity
premium for diﬀerent serial correlations.38
When the serial correlation of consumption
is positive, the equity premium actually declines with increasing risk aversion, thus,
further exacerbating the equity premium puzzle.39
An alternative perspective on the puzzle is provided by Hansen and Jagannathan
(1991). The fundamental pricing equation can be written as
Et(Re,t+1) = Rf,t+1 − Covt
Mt+1, Re,t+1
Et(Mt+1)
. (22)
38In addition, see Figure 2B in Appendix B.
39See Azeredo (2007) for a detailed discussion.
24 Chapter 1 • The Equity Premium: ABCs
This expression also holds unconditionally, so that
E(Re,t+1) = Rf,t+1 − σ(Mt+1)σ(Re,t+1)ρR,M /E(Mt+1) (23)
or
E(Re,t+1) − Rf,t+1/σ(Re,t+1) = −σ(Mt+1)ρR,M /E(Mt+1), (24)
and since −1 ≤ ρR,M ≤ 1,
E(Re,t+1) − Rf,t+1/σ(Re,t+1) ≤ σ(Mt+1)/E(Mt+1). (25)
This inequality is referred to as the Hansen–Jagannathan lower bound on the pricing
kernel.
For the U.S. economy, the Sharpe ratio, E(Re,t+1) − Rf,t+1/σ(Re,t+1), can be calculated
to be 0.37. Since E(Mt+1) is the expected price of a one-period risk-free
bond, its value must be close to 1. In fact, for the parameterization discussed earlier,
E(Mt+1) = 0.96 when α = 2. This implies that the lower bound on the standard deviation
for the pricing kernel must be close to 0.3 if the Hansen–Jagannathan bound is to be
satisﬁed. However, when this is calculated in the Mehra–Prescott framework, we obtain
an estimate for σ(Mt+1) = 0.002, which is oﬀ by more than an order of magnitude.
We would like to emphasize that the equity premium puzzle is a quantitative puzzle;
standard theory is consistent with our notion of risk that, on average, stocks should
return more than bonds. The puzzle arises from the fact that the quantitative predictions
of theory are an order of magnitude diﬀerent from what has been historically documented.
The puzzle cannot be dismissed lightly, since much of our economic intuition
is based on the very class of models that fall short so dramatically when confronted with
ﬁnancial data. It underscores the failure of paradigms central to ﬁnancial and economic
modeling to capture the characteristic that appears to make stocks comparatively so
risky. Hence, the viability of using this class of models for any quantitative assessment,
say, for instance, to gauge the welfare implications of alternative stabilization policies,
is thrown open to question.
For this reason, over the last 20 years or so, attempts to resolve the puzzle have
become a major research impetus in ﬁnance and economics. Several generalizations
of key features of the Mehra and Prescott (1985) model have been proposed to better
reconcile observations with theory. These include alternative assumptions on prefer-
ences,40
modiﬁed probability distributions to admit rare but disastrous events,41
survival
40For example, Abel (1990), Bansal and Yaron (2004), Benartzi and Thaler (1995), Boldrin, Christiano, and
Fisher (2001), Campbell and Cochrane (1999), Constantinides (1990), Epstein and Zin (1991), and Ferson
and Constantinides (1991).
41See Rietz (1988) and Mehra and Prescott (1988).
Rajnish Mehra and Edward C. Prescott 25
bias,42
incomplete markets,43
and market imperfections.44
They also include attempts at
modeling limited participation of consumers in the stock market45
and problems of temporal
aggregation.46
We examine some of the research eﬀorts to resolve the puzzle47
in
the next two chapters.
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Rajnish Mehra and Edward C. Prescott 29
APPENDIX A
Suppose the distribution of returns period by period is independently and identically
distributed. Then, as the number of periods tends to inﬁnity, the future value of the
investment, computed at the arithmetic average of returns, tends to the expected value
of the investment with probability 1. To see this, let
VT =
T
Π
t=1
(1 + rt) ,
where rt is the asset return in period t and VT is the terminal value of $1 at time T. Then
E(VT ) = E
T
Π
i=1
(1 + rt) .
Since the rts are assumed to be uncorrelated, we have
E(VT ) =
T
Π
i=1
E (1 + rt)
or
E(VT ) =
T
Π
i=1
(1 + E (rt)) .
Let the arithmetic average
AA =
1
T
T
t=t
rt.
Then, by the strong law of large numbers (Theorem 22.1, Billingsley (1995)),
E(VT ) →
T
Π
i=1
(1 + AA) as T → ∞
or
E(VT ) → (1 + AA)T
as the number of periods T becomes large.
APPENDIX B
The Original Analysis of the Equity Premium Puzzle
In this appendix we present our original analysis of the equity premium puzzle. Needless
to say, it draws heavily from Mehra and Prescott (1985).
30 Chapter 1 • The Equity Premium: ABCs
The Economy, Asset Prices and Returns
We employ a variation of Lucas’ (1978) pure exchange model. Since per capita consumption
has grown over time, we assume that the growth rate of the endowment
follows a Markov process. This is in contrast to the assumption in Lucas’ model that
the endowment level follows a Markov process. Our assumption, which requires an
extension of competitive equilibrium theory, enables us to capture the non-stationarity
in the consumption series associated with the large increase in per capita consumption
that occurred in the 1889–1978 period.
The economy we consider was judiciously selected so that the joint process governing
the growth rates in aggregate per capita consumption and asset prices would be
stationary and easily determined. The economy has a single representative “stand-in”
household. This unit orders its preferences over random consumption paths by
E0
∞
t=0
βt
U(ct) , 0 < β < 1, (1B)
where ct is per capita consumption, β is the subjective time discount factor, E{·} is the
expectation operator conditional upon information available at time zero (which denotes
the present time), and U: R+ → R is the increasing concave utility function. To ensure
that the equilibrium return process is stationary, we further restrict the utility function
to be of the constant relative risk aversion (CRRA) class
U(c, α) =
c1−α
− 1
1 − α
, 0 < α < ∞. (2B)
The parameter α measures the curvature of the utility function. When α is equal to
one, the utility function is deﬁned to be the logarithmic function, which is the limit of
the above function as α approaches one.
We assume there is one productive unit that produces output yt in period t, which is
the period dividend. There is one equity share with price pt, that is competitively traded;
it is a claim to the stochastic process {yt}.
The growth rate in yt is subject to a Markov chain; that is,
yt+1 = xt+1yt, (3B)
where xt+1 ∈ {λ1, . . . , λn} is the growth rate, and
Pr{xt+1 = λj; xt = λi} = φij. (4B)
It is also assumed that the Markov chain is ergodic. The λis are all positive and
y0 > 0. The random variable yt is observed at the beginning of the period, at which time
dividend payments are made. All securities are traded ex-dividend. We also assume that
the matrix A with elements aij ≡ βφijλ1−α
j for i, j = 1, . . . , n is stable; that is, lim Am
as
m → ∞ is zero. In Mehra (1988) it is shown that this is necessary and suﬃcient for the
expected utility to exist if the stand-in household consumes yt every period. The paper
Rajnish Mehra and Edward C. Prescott 31
also deﬁnes and establishes the existence of a Debreu (1954) competitive equilibrium
with a price system having a dot product representation under this condition.
Next we formulate expressions for the equilibrium time t price of the equity share
and the risk-free bill. We follow the convention of pricing securities ex-dividend or
ex-interest payments at time t, in terms of the time t consumption good. For any security
with process {ds} on payments, its price in period t is
Pt = Et
∞
s=t+1
βs−t U (ys)ds
U (yt)
, (5B)
as the equilibrium consumption is the process {ys} and the equilibrium price system has
a dot product representation.
The dividend payment process for the equity share in this economy is {ys}.
Consequently, using the fact that U (c) = c−α
,
Pe
t = Pe
(xt, yt)
= E
∞
s=t+1
βs−t yα
t
yα
s
ys|xt, yt . (6B)
The variables xt and yt are suﬃcient relative to the entire history of shocks up to,
and including, time t for predicting the subsequent evolution of the economy. They thus
constitute legitimate state variables for the model. Since ys = ytxt+1 . . . xs, the price of
the equity security is homogeneous of degree one in yt, which is the current endowment
of the consumption good. As the equilibrium values of the economies being studied
are time-invariant functions of the state (xt, yt), the subscript t can be dropped. This is
accomplished by redeﬁning the state to be the pair (c, i), if yt = c and xt = λi. With this
convention, the price of the equity share from (6B) satisﬁes
pe
(c, i) = β
n
j=1
φij(λjc)−α
[pe
(λjc, j) + λjc]cα
. (7B)
Using the result that pe
(c, i) is homogeneous of degree one in c, we represent this
function as
pe
(c, i) = wic, (8B)
where wi is a constant. Making this substitution in (7B) and dividing by c yields
wi = β
n
j=1
φijλ
(1−α)
j (wj + 1) for i = 1, . . . , n. (9B)
This is a system of n linear equations in n unknowns. The assumption that guaranteed
the existence of equilibrium guarantees the existence of a unique positive solution to this
system.
32 Chapter 1 • The Equity Premium: ABCs
The period return if the current state is (c, i) and next period state (λjc, j) is
re
ij =
pe
(λjc, j) + λjc − pe
(c, i)
pe(c, i)
=
λj(wj + 1)
wi
− 1. (10B)
The equity’s expected period return if the current state is i is
Re
i =
n
j=1
φijre
ij. (11B)
Capital letters are used to denote the expected return. With the subscript i, it is the
expected return conditional upon the current state being (c, i). Without this subscript,
it is the expected return with respect to the stationary distribution. The superscript
indicates the type of security.
The other security considered is the one-period real bill or riskless asset, which pays
one unit of the consumption good next period with certainty. From (6B),
p
f
i = pf
(c, i)
= β
n
j=1
φij
U (λjc)
U (c)
(12B)
= β φijλ−α
j .
The certain return on this riskless security is
R
f
i =
1
p
f
i − 1
(13B)
when the current state is (c, i).
As mentioned earlier, the statistics that are probably most robust to the modeling
speciﬁcation are the means over time. Let π ∈ Rn
be the vector of stationary probabilities
on i. This exists because the chain on i has been assumed to be ergodic. The vector
π is the solution to the system of equations π = φT
π, with
n
i=1
πi = 1 and φT
= {φji}.
The expected returns on the equity and the risk-free security are, respectively,
Re
=
n
i=1
πiRe
i and Rf
=
n
i=1
πiR
f
i . (14B)
Rajnish Mehra and Edward C. Prescott 33
Time sample averages will converge in probability to these values given the ergodicity
of the Markov chain. The risk premium for equity is, Re
− Rf
, a parameter that is used
in the test.
The parameters deﬁning preferences are α and β, while the parameters deﬁning technology
are the elements of [φij] and [λi]. Our approach is to assume two states for the
Markov chain and to restrict the process as follows:
λ1 = 1 + μ + δ, λ2 = 1 + μ − δ,
φ11 = φ22 = φ, φ12 = φ21 = (1 − φ).
The parameters μ, φ, and δ now deﬁne the technology. We require δ > 0 and 0 < φ < 1.
This particular parameterization was selected because it permitted us to independently
vary the average growth rate of output by changing μ, the variability of consumption by
altering δ, and the serial correlation of growth rates by adjusting φ.
The parameters were selected so that the average growth rate of per capita consumption,
the standard deviation of the growth rate of per capita consumption, and the
ﬁrst-order serial correlation of this growth rate, all with respect to the model’s stationary
distribution, matched the sample values for the U.S. economy between 1889–1978. The
sample values for the U.S. economy were 0.018, 0.036, and −0.14, respectively. The
resulting parameters’ values were μ = 0.018, δ = 0.036, and φ = 0.43. Given these values,
the nature of the test is to search for parameters α and β for which the model’s
averaged risk-free rate and equity risk premium match those observed for the U.S.
economy over this 90-year period.
The parameter α, which measures people’s willingness to substitute consumption
between successive yearly time periods, is an important one in many ﬁelds of economics.
As mentioned in the text, there is a wealth of evidence from various studies that
the coeﬃcient of risk aversion α is a small number, certainly less than 10. A number
of these studies are documented in Mehra and Prescott (1985). This is an important
restriction, for with a large α virtually any pair of average equity and risk-free returns
can be obtained by making small changes in the process on consumption.
Given the estimated process on consumption, Figure 1B depicts the set of values of
the average risk-free rate and equity risk premium, which are both consistent with the
model and result in average real risk-free rates between zero and four percent. These
are values that can be obtained by varying preference parameters α between 0 and 10
and β between 0 and 1. The observed real return of 0.80 percent and equity premium of
6 percent are clearly inconsistent with the predictions of the model. The largest premium
obtainable with the model is 0.35 percent, which is not close to the observed value.
An advantage of our approach is that we can easily test the sensitivity of our results
to such distributional assumptions. With α less than 10, we found that our results were
essentially unchanged for very diﬀerent consumption processes, provided that the mean
and variances of growth rates equaled the historically observed values. We use this fact
in motivating the discussion in the text.
34 Chapter 1 • The Equity Premium: ABCs
AverageRiskPremia(Percent)
2
1
0 1 2 3 4
Admissible Region
Rf
Average Risk-Free Rate (Percent)
Re
2Rf
FIGURE 1B Set of admissible average equity risk premia and real returns.
0.3
0.25
0.2
1.5 2.5 3.5 4.5432
Risk-Free Rate (Percent)
0
0.15
EquityPremium(Percent)
0.1
0.05
correl. 5 20.14
correl. 5 0.45
FIGURE 2B Set of admissible average equity risk premia and real returns.
As mentioned earlier in the text, the serial correlation of the growth rate of consumption
for the period 1930–2004 is 0.45. Figure 2B shows the resulting feasible region in
this case (we have also included the region from Figure 1B for comparison). The conclusion
that the premium for bearing non-diversiﬁable aggregate risk is small remains
unchanged.
Rajnish Mehra and Edward C. Prescott 35
APPENDIX C
Expanding the set of technologies in a pure exchange, Arrow–Debreu economy to admit
capital accumulation and production as in Brock (1979), Prescott and Mehra (1980), or
Donaldson and Mehra (1984) does not increase the set of joint equilibrium processes
on consumption and asset prices. Since the set of equilibria in a production company
is a subset of those in an exchange economy, it follows immediately that if the equity
premium cannot be accounted for in an exchange economy, modifying the technology
to incorporate production will not alter this conclusion.48
To see this, let θ denote preferences, τ technologies, E the set of the exogenous
processes on the aggregate consumption good, P the set of technologies with production
opportunities, and m(θ, τ) the set of equilibria for economy (θ, τ).
Theorem
∪
τεE
m(θ, τ) ⊃ ∪
τεP
m(θ, τ)
Proof. For θ0εθ and τ0εP, let (a0, c0) be a joint equilibrium process on asset prices and
consumption. A necessary condition for equilibrium is that the asset prices a0 be consistent
with c0, the optimal consumption for the household with preferences θ0. Thus, if
(a0, c0) is an equilibrium, then
a0 = g(c0, θ),
where g is deﬁned by the ﬁrst-order necessary conditions for household maximization.
This functional relation must hold for all equilibria, regardless of whether they are for a
pure exchange or a production economy.
Let (a0, c0) be an equilibrium for some economy (θ0, τ0) with τ0εP. Consider the
pure exchange economy with θ1 = θ0 and τ1 = c0. Our contention is that (a0, c0) is a
joint equilibrium process for asset prices and consumption for the pure exchange economy
(θ1, τ1). For all pure exchange economies, the equilibrium consumption process is
τ, so c1 = τ1 = c0, given that more is preferred to less. If c0 is the equilibrium process,
the corresponding asset price must be g(c0, θ1). But θ1 = θ0 so g(c0, θ1) = g(c0, θ0) =
a0. Hence, a0 is the equilibrium for the pure exchange economy (θ1, τ1), proving the
theorem.
APPENDIX D
Estimating the Equity Risk Premium Versus Estimating the Risk
Aversion Parameter
Estimating or measuring the relative risk aversion parameter using statistical tools is
very diﬀerent than estimating the equity risk premium. Mehra and Prescott (1985), as
48The discussion below is based on Mehra (1998).
36 Chapter 1 • The Equity Premium: ABCs
discussed earlier, use an extension of Lucas’ (1978) asset pricing model to estimate
how much of the historical diﬀerence in yields on Treasury bills and corporate equity
is a premium for bearing aggregate risk. Crucial to their analysis is their use of micro
observations to restrict the value of the risk aversion parameter. They did not estimate
either the risk aversion parameter or the discount rate parameters. Mehra and Prescott
(1985) reject extreme risk aversion based upon observations on individual behavior.
These observations include the small size of premia for jobs with uncertain income and
the limited amount of insurance against idiosyncratic income risk. Another observation
is that people with limited access to capital markets make investments in human capital
that result in very uneven consumption over time.
A sharp estimate for the magnitude of the risk aversion parameter comes from
macroeconomics. The evidence is that the basic growth model, when restricted to be
consistent with the growth facts, generates business cycle ﬂuctuations if and only if
this risk aversion parameter is near zero. (This corresponds to the log case in standard
usage.) The point is that the risk aversion parameter comes up in a wide variety of
observations at both the household and the aggregate level and is not found to be large.
For all values of the risk aversion coeﬃcient less than 10, which is an upper bound
number for this parameter, Mehra and Prescott ﬁnd that a premium for bearing aggregate
risk accounts for little of the historic equity premium. This ﬁnding has stood the test
of time.
Another tradition is to use consumption and stock market data to estimate the degree
of relative risk aversion parameter and the discount factor parameter. This is what
Grossman and Shiller report they did in their American Economic Review Papers and
Proceedings article (1981, p. 226). In a paper in which they develop “a method for estimating
nonlinear rational expectations models directly from stochastic Euler equations,”
Hansen and Singleton illustrate their methods by estimating the risk aversion parameter
and the discount factor using stock dividend consumption prices (1981, p. 1269).
What the work of Grossman and Shiller (Ibid.) and Hansen and Singleton (1982,
1983) establish is that using consumption and stock market data and assuming frictionless
capital markets is a bad way to estimate the risk aversion and discount factor
parameters. It is analogous to estimating the force of gravity near the earth’s surface by
dropping a feather from the top of the Leaning Tower of Pisa, under the assumption that
friction is zero.
A tradition related to statistical estimation is to statistically test whether the stochastic
Euler equation arising from the stand-in household’s intertemporal optimization
holds. Both Grossman and Shiller (1983) and Hansen and Singleton (1982) reject this
relation. The fact that this relation is inconsistent with the U.S. time-series data is no
reason to conclude that the model economy used by Mehra and Prescott to estimate how
much of the historical equity premium is a premium for bearing aggregate risk is not
a good one for that purpose. Returning to the analogy from physics, it would be silly
to reject Newtonian mechanics as a useful tool for drawing scientiﬁc inference because
the distance traveled by the feather did not satisfy 1/2gt2
.