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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 вЂњriskyвЂќ 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,

вЂў

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

вЂў

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.

вЂў 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 вЂ“ 1928- 2003

Stocks вЂ“ Treasury Bills Stocks вЂ“ 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.

11

Geometric Arithmetic Geometric

Arithmetic

1928 вЂ“ 2003 7.92% 5.99% 6.54% 4.82%

1962 вЂ“ 2003 6.09% 4.85% 4.70% 3.82%

1992 вЂ“ 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 вЂ“ 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?

вЂў

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

вЂў

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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