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asset prices over very long time periods. In the CAPM, the premium is defined as the
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difference between average returns on stocks and average returns on risk-free securities
over an extended period of history.

Basics
In most cases, this approach is composed of the following steps. It begins by
defining a time period for the estimation, which can range to as far back as 1871 for U.S.
data. It then requires the calculation of the average returns on a stock index and average
returns on a riskless security over the period. Finally, it calculates the difference between
the returns on stocks and the riskless return and uses it as a risk premium looking
forward. In doing so, we implicitly assume that
1. The risk aversion of investors has not changed in a systematic way across
time. (The risk aversion may change from year to year, but it reverts back to
historical averages.)
2. It assumes that the average riskiness of the “risky” portfolio (stock index) has
not changed in a systematic way across time.

Estimation Issues
While users of risk and return models may have developed a consensus that historical
premium is, in fact, the best estimate of the risk premium looking forward, there are
surprisingly large differences in the actual premiums we observe being used in practice.
For instance, the risk premium estimated in the US markets by different investment
banks, consultants and corporations range from 4% at the lower end to 12% at the upper
end. Given that we almost all use the same database of historical returns, provided by
Ibbotson Associates6, summarizing data from 1926, these differences may seem
surprising. There are, however, three reasons for the divergence in risk premiums.
Time Period Used: While there are many who use all the data going back to 1926,

there are almost as many using data over shorter time periods, such as fifty, twenty or
even ten years to come up with historical risk premiums. The rationale presented by
those who use shorter periods is that the risk aversion of the average investor is likely
to change over time and that using a shorter and more recent time period provides a


6 See "Stocks, Bonds, Bills and Inflation", an annual edition that reports on the annual returns on stocks,
treasury bonds and bills, as well as inflation rates from 1926 to the present. (http://www.ibbotson.com)
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more updated estimate. This has to be offset against a cost associated with using
shorter time periods, which is the greater noise in the risk premium estimate. In fact,
given the annual standard deviation in stock prices7 between 1928 and 2002 of 20%,
the standard error8 associated with the risk premium estimate can be estimated as
follows for different estimation periods in Table 4.1.
Table 4.1: Standard Errors in Risk Premium Estimates
Estimation Period Standard Error of Risk Premium Estimate
20
5 years
= 8.94%
5
20
10 years
= 6.32%
10
20
25 years
= 4.00%
25
20
50 years
= 2.83%
50

Note that to get reasonable standard errors, we need very long time periods of
historical returns. Conversely, the standard errors from ten-year and twenty-year
estimates are likely to be almost as large or larger than the actual risk premium
estimated. This cost of using shorter time periods seems, in our view, to overwhelm
any advantages associated with getting a more updated premium.
Choice of Riskfree Security: The Ibbotson database reports returns on both treasury

bills and treasury bonds and the risk premium for stocks can be estimated relative to
each. Given that the yield curve in the United States has been upward sloping for
most of the last seven decades, the risk premium is larger when estimated relative to
shorter term government securities (such as treasury bills). The riskfree rate chosen in
computing the premium has to be consistent with the riskfree rate used to compute
expected returns. For the most part, in corporate finance and valuation, the riskfree
rate will be a long term default-free (government) bond rate and not a treasury bill




7 For the historical data on stock returns, bond returns and bill returns, check under "updated data" in
www.stern.nyu.edu/˜adamodar.
8 These estimates of the standard error are probably understated because they are based upon the
assumption that annual returns are uncorrelated over time. There is substantial empirical evidence that
returns are correlated over time, which would make this standard error estimate much larger.
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rate. Thus, the risk premium used should be the premium earned by stocks over
treasury bonds.
• Arithmetic and Geometric Averages: The final sticking point when it comes to
estimating historical premiums relates to how the average returns on stocks, treasury
bonds and bills are computed. The arithmetic average return measures the simple
mean of the series of annual returns, whereas the geometric average looks at the
compounded return9. Conventional wisdom argues for the use of the arithmetic
average. In fact, if annual returns are uncorrelated over time and our objectives were
to estimate the risk premium for the next year, the arithmetic average is the best
unbiased estimate of the premium. In reality, however, there are strong arguments
that can be made for the use of geometric averages. First, empirical studies seem to
indicate that returns on stocks are negatively correlated10 over time. Consequently,
the arithmetic average return is likely to over state the premium. Second, while asset
pricing models may be single period models, the use of these models to get expected
returns over long periods (such as five or ten years) suggests that the single period
may be much longer than a year. In this context, the argument for geometric average
premiums becomes even stronger.
In summary, the risk premium estimates vary across users because of differences in time
periods used, the choice of treasury bills or bonds as the riskfree rate and the use of
arithmetic as opposed to geometric averages. The effect of these choices is summarized
in table 4.2, which uses returns from 1928 to 2003. 11
Table 4.2: Historical Risk Premia for the United States “ 1928- 2003
Stocks “ Treasury Bills Stocks “ Treasury Bonds




9 The compounded return is computed by taking the value of the investment at the start of the period
(Value0) and the value at the end (ValueN) and then computing the following:
1/ N
! Value N $
Geometric Average = # '1
&
" Value0 %
10 In other words, good years are more likely to be followed by poor years and vice versa. The evidence on
negative serial correlation in stock returns over time is extensive and can be found in Fama and French
(1988). While they find that the one-year correlations are low, the five-year serial correlations are strongly
negative for all size classes.
11 The raw data on treasury bill rates, treasury bond rates and stock returns was obtained from the Federal
Reserve data archives maintained by the Fed in St. Louis.
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Geometric Arithmetic Geometric
Arithmetic
1928 “ 2003 7.92% 5.99% 6.54% 4.82%
1962 “ 2003 6.09% 4.85% 4.70% 3.82%
1992 “ 2003 8.43% 6.68% 4.87% 3.57%
Note that the premiums can range from 3.57% to 8.43%, depending upon the choices
made. In fact, these differences are exacerbated by the fact that many risk premiums that
are in use today were estimated using historical data three, four or even ten years ago. If
we follow the propositions about picking a long-term geometric average premium over
the long term treasury bond rate, the historical risk premium that makes the most sense is
4.82%.

Historical Premiums in other markets
While historical data on stock returns is easily available and accessible in the
United States, it is much more difficult to get this data for foreign markets. The most
detailed look at these returns estimated the returns you would have earned on 14 equity
markets between 1900 and 2001 and compared these returns with those you would have
earned investing in bonds.12 Figure 4.1 presents the risk premiums “ i.e., the additional
returns - earned by investing in equity over treasury bills and bonds over that period in
each of the 14 markets:




12 Dimson, E., P. March and M. Staunton, 2002, Triumph of the Optimists, Princeton University Prsss.
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Figure 4.1: Equity Risk Premiums - By Country

8%


7%


6%
Compounded Annual Risk Premium




5%


4%


3%


2%


1%


0%
Australia

Belgium

Canada

Denmark

France

Germany

Ireland

Italy

Japan

Netherlands

South Africa

Spain

Sweden

Switzerland

UK

USA

World
Country

Stocks - Short term Government Return Stocks - Long Term Government Return


Data from Dimson et al. The differences in compounded annual returns between stocks and short
term governments/ long term governments is reported for each country.
While equity returns were higher than what you would have earned investing in
government bonds or bills in each of the countries examined, there are wide differences
across countries. If you had invested in Spain, for instance, you would have earned only
3% over government bills and 2% over government bonds on an annual basis by
investing in equities. In France, in contrast, the corresponding numbers would have been
7.1% and 4.6%. Looking at 40-year or 50-year periods, therefore, it is entirely possible
that equity returns can lag bond or bill returns, at least in some equity markets. In other
words, the notion that stocks always win in the long term is not only dangerous but does
not make sense. If stocks always beat riskless investments in the long term, stocks should
be riskless to an investor with a long time horizon.



histretSP.xls: This data set has yearly data on treasury bill rates, treasury bond
rates and returns and stock returns going back to 1928.
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A Modified Historical Risk Premium
In many emerging markets, there is very little historical data and the data that
exists is too volatile to yield a meaningful estimate of the risk premium. To estimate the
risk premium in these countries, let us start with the basic proposition that the risk
premium in any equity market can be written as:
Equity Risk Premium = Base Premium for Mature Equity Market + Country Premium
The country premium could reflect the extra risk in a specific market. This boils down
our estimation to answering two questions:
What should the base premium for a mature equity market be?

How do we estimate the additional risk premium for individual countries?

To answer the first question, we will make the argument that the US equity market is a
mature market and that there is sufficient historical data in the United States to make a
reasonable estimate of the risk premium. In fact, reverting back to our discussion of
historical premiums in the US market, we will use the geometric average premium earned
by stocks over treasury bonds of 4.82% between 1928 and 2003. We chose the long time
period to reduce standard error, the treasury bond to be consistent with our choice of a
riskfree rate and geometric averages to reflect our desire for a risk premium that we can

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