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19 The treasury bond rate is the sum of expected inflation and the expected real rate. If we assume that real
growth is equal to the real rate, the long term stable growth rate should be equal to the treasury bond rate.
20 The input that is most difficult to estimate for emerging markets is a long term expected growth rate. For
Brazilian stocks, I used the average consensus estimate of growth in earnings for the largest Brazilian
companies which have listed ADRs . This estimate may be biased, as a consequence.
21


Figure 4.2: Implied Premium for US Equity Market

7.00%



6.00%



5.00%
Implied Premium




4.00%



3.00%



2.00%



1.00%



0.00%
1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002
Year



In terms of mechanics, we used smoothed historical growth rates in earnings and
dividends as our projected growth rates and a two-stage dividend discount model.
Looking at these numbers, we would draw the following conclusions.
The implied equity premium has seldom been as high as the historical risk premium.

Even in 1978, when the implied equity premium peaked, the estimate of 6.50% is
well below what many practitioners use as the risk premium in their risk and return
models. In fact, the average implied equity risk premium has been between about 4%
over the last 40 years.
The implied equity premium did increase during the seventies, as inflation increased.

This does have interesting implications for risk premium estimation. Instead of
assuming that the risk premium is a constant and unaffected by the level of inflation
and interest rates, which is what we do with historical risk premiums, it may be more
realistic to increase the risk premium as expected inflation and interest rates increase.


histimpl.xls: This data set on the web shows the inputs used to calculate the
premium in each year for the U.S. market.
22


implprem.xls: This spreadsheet allows you to estimate the implied equity premium
in a market.


˜ 4.4: Implied and Historical Premiums
Assume that the implied premium in the market is 3%, and that you are using a historical
premium of 7.5%. If you valued stocks using this historical premium, you are likely
a. to find more under valued stocks than over valued ones
b. to find more over valued stocks than under valued ones
c. to find about as many undervalued as overvalued stocks

III. Risk Parameters
The final set of inputs we need to put risk and return models into practice are the
risk parameters for individual assets and projects. In the CAPM, the beta of the asset has
to be estimated relative to the market portfolio. In the APM and Multi-factor model, the
betas of the asset relative to each factor have to be measured. There are three approaches
available for estimating these parameters; one is to use historical data on market prices
for individual assets; the second is to estimate the betas from fundamentals and the third
is to use accounting data. We will use all three approaches in this section.


A. Historical Market Betas
This is the conventional approach for estimating betas used by most services and
analysts. For firms that have been publicly traded for a length of time, it is relatively
straightforward to estimate returns that a investor would have made on the assets in
intervals (such as a week or a month) over that period. These returns can then be related
to a proxy for the market portfolio to get a beta in the capital asset pricing model, or to
multiple macro economic factors to get betas in the multi factor models, or put through a
factor analysis to yield betas for the arbitrage pricing model.
23


Standard Procedures for Estimating CAPM Parameters - Betas and Alphas
The standard procedure for estimating betas is to regress21 stock returns (Rj)
against market returns (Rm) -
Rj = a + b Rm
where
a = Intercept from the regression
b = Slope of the regression = Covariance (Rj, Rm) / σ2m
The slope of the regression corresponds to the
Jensen™s Alpha: This is the difference between
beta of the stock and measures the riskiness of the actual returns on an asset and the return you
the stock. would have expected it to make during a past
period, given what the market did, and the
The intercept of the regression provides
asset™s beta.
a simple measure of performance during the
period of the regression, relative to the capital asset pricing model.
Rj = Rf + β (Rm - Rf)
= Rf (1-β) + β Rm ........... Capital Asset Pricing Model
Rj = a + b Rm ........... Regression Equation
Thus, a comparison of the intercept (a) to Rf (1-β) should provide a measure of the
stock's performance, at least relative to the capital asset pricing model.22
If a > Rf (1-β) .... Stock did better than expected during regression period
a = Rf (1-β) .... Stock did as well as expected during regression period
a < Rf (1-β) .... Stock did worse than expected during regression period
The difference between a and Rf (1-β) is called Jensen™s alpha, and provides a measure of
whether the asset in question under or out performed the market, after adjusting for risk,
during the period of the regression. R Squared: The R squared measures the
The third statistic that emerges from the proportion of the variability of a dependent
regression is the R squared (R2) of the regression. variable that is explained by an independent
variable or variables.


21 The appendix to this chapter provides a brief overview of ordinary least squares regressions.
22 The regression can be run using returns in excess of the risk-free rate, for both the stock and the market.
In that case, the intercept of the regression should be zero if the actual returns equal the expected returns
from the CAPM, greater than zero if the stock does better than expected and less than zero if it does worse
than expected.
24


While the statistical explanation of the R squared is that it provides a measure of the
goodness of fit of the regression, the financial rationale for the R squared is that it
provides an estimate of the proportion of the risk (variance) of a firm that can be
attributed to market risk; the balance (1 - R2) can then be attributed to firm-specific risk.
The final statistic worth noting is the standard error of the beta estimate. The
slope of the regression, like any statistical estimate, is made with noise, and the standard
error reveals just how noisy the estimate is. The standard error can also be used to arrive
at confidence intervals for the “true” beta value from the slope estimate.

Estimation Issues
There are three decisions the analyst must make in setting up the regression
described above. The first concerns the length of the estimation period. The trade-off is
simple: A longer estimation period provides more data, but the firm itself might have
changed in its risk characteristics over the time period. Disney and Deutsche Bank have
changed substantially in terms of both business mix and financial leverage over the last
few years and any regression that we run using historical data will be affected by these
changes.
The second estimation issue relates to the return interval. Returns on stocks are
available on an annual, monthly, weekly, daily and even on an intra-day basis. Using
daily or intra-day returns will increase the number of observations in the regression, but it
exposes the estimation process to a significant bias in beta estimates related to non-
trading.23 For instance, the betas estimated for small firms, which are more likely to
suffer from non-trading, are biased downwards when daily returns are used. Using
weekly or monthly returns can reduce the non-trading bias significantly.24
The third estimation issue relates to the choice of a market index to be used in the
regression. The standard practice used by most beta estimation services is to estimate the
betas of a company relative to the index of the market in which its stock trades. Thus, the
betas of German stocks are estimated relative to the Frankfurt DAX, British stocks


23 The non-trading bias arises because the returns in non-trading periods is zero (even though the market
may have moved up or down significantly in those periods). Using these non-trading period returns in the
regression will reduce the correlation between stock returns and market returns and the beta of the stock.
24 The bias can also be reduced using statistical techniques suggested by Dimson and Scholes-Williams.
25


relative to the FTSE, Japanese stocks relative to the Nikkei, and U.S. stocks relative to
the S&P 500. While this practice may yield an estimate that is a reasonable measure of
risk for the parochial investor, it may not be the best approach for an international or
cross-border investor, who would be better served with a beta estimated relative to an
international index.

Illustration 4.1: Estimating CAPM risk parameters for Disney
In assessing risk parameters for Disney, the returns on the stock and the market
index are computed as follows “
(1) The returns to a stockholder in Dsiney are computed month by month from January
1999 to December 2003. These returns include both dividends and price appreciation and
are defined as follows “
Stock Returnintel, j =( PriceIntel, j - PriceIntel, j-1+Dividendsj) / PriceIntel,j-1
where Stock ReturnIntel,j = Returns to a stockholder in Disney in month j
PriceIntel, j = Price of Disney stock at the end of month j
Dividendsj = Dividends on Disney stock in month j
Dividends are added to the returns of the month in which the stock went ex-dividend. 25 If
there was a stock split26 during the month, the returns have to take into account the split
factor, since stock prices will be affected.27
(2) The returns on the S&P 500 market index are computed for each month of the period,
using the level of the index at the end of each month, and the monthly dividend yield on
stocks in the index. “
Market Returnintel, j =( Indexj - Index j-1 + Dividendst) / Indexj-1



25 The ex-dividend day is the day by which the stock has to be bought for an investor to be entitled to the
dividends on the stock.
26 A split changes the number of shares outstanding in a company without affecting any of its
fundamentals. Thus, in a three-for-two split, there will be 50% more shares outstanding after the split.
Since the overall value of equity has not changed, the stock price will drop by an equivalent amount (1 -
100/150 = 33.33%)
27 While there were no stock splits in the time period of the regression, Disney did have a 3 for 1 stock split
in July 1998. The stock price dropped significantly, and if not factored in will result in very negative
returns in that month. Splits can be accounted for as follows “
Returnintel, j =( Factorj * PriceIntel, j - PriceIntel, j-1+ Factor * Dividendsj) / PriceIntel,j-1
The factor would be set to 3 for July 1998 and the ending price would be multiplied by 3, as would the
dividends per share, if they were paid after the split.
26


where Indexj is the level of the index at the end of month j and Dividendj is the dividends
paid on the index in month j. While the S&P 500 and the NYSE Composite are the most
widely used indices for U.S. stocks, they are, at best, imperfect proxies for the market
portfolio in the CAPM, which is supposed to include all assets.
Figure 4.3 graphs monthly returns on Disney against returns on the S&P 500
index from January 1999 to December 2003.

Figure 4.3: Disney versus S&P 500: 1999 - 2003

30.00%

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