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Regression line


-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%



S&P 500

The regression statistics for Disney are as follows:28
(a) Slope of the regression = 1.01. This is Disney's beta, based on returns from 1999 to
2003. Using a different time period for the regression or different return intervals (weekly
or daily) for the same period can result in a different beta.
(b) Intercept of the regression = 0.0467%. This is a measure of Disney's performance,
when it is compared with Rf (1-β).29 The monthly risk-free rate (since the returns used in

28 The regression statistics are computed in the conventional way. The appendix explains the process in
more detail.
29 In practice, the intercept of the regression is often called the alpha and compared to zero. Thus, a positive
intercept is viewed as a sign that the stock did better than expected and a negative intercept as a sign that
the stock did worse than expected. In truth, this can be done only if the regression is run in terms of excess
returns, i.e., returns over and above the riskfree rate in each month for both the stock and the market index.

the regression are monthly returns) between 1999 and 2003 averaged 0.313%, resulting in
the following estimate for the performance:
Rf (1-β) = 0.313% (1-1.01) = -.0032%
Intercept - Rf (1-β) = 0.0467% - (-0.0032%) = 0.05%
This analysis suggests that Disney™s stock performed 0.05% better than expected, when
expectations are based on the CAPM, on a monthly basis between January 1999 and
December 2003. This results in an annualized excess return of approximately 0.60%.
= (1 + Monthly Excess Return)12 - 1
Annualized Excess Return
= (1+0.0005)12 -1 =.0060 or 0.60%
By this measure of performance, Disney did slightly better than expected during the
period of the regression, given its beta and the market™s performance over the period.
Note, however, that this does not imply that Disney would be a good investment looking
forward. It also does not provide a breakdown of how much of this excess return can be
attributed to ˜industry-wide™ effects, and how much is specific to the firm. To make that
breakdown, the excess returns would have to be computed over the same period for other
firms in the entertainment industry and compared with Disney™s excess return. The
difference would be then attributable to firm-specific actions. In this case, for instance,
the average annualized excess return on other entertainment firms between 1999 and
2003 was 1.33%. This would imply that Disney stock underperformed it™s peer group by
0.73% between 1999 and 2003, after adjusting for risk. (Firm-specific Jensen™s alpha =
0.60% - 1.33% = - 0.73%)
(c) R squared of the regression = 29%. This statistic suggests that 29% of the risk
(variance) in Disney comes from market sources (interest rate risk, inflation risk etc.),
and that the balance of 71% of the risk comes from firm-specific components. The latter
risk should be diversifiable, and therefore unrewarded. Disney™s R squared is slightly
higher than the median R squared of companies listed on the New York Stock Exchange,
which was approximately 21% in 2003.
(d) Standard Error of Beta Estimate = 0.20. This statistic implies that the true beta for
Disney could range from 0.81 to 1.21 (subtracting adding one standard error to beta
estimate of 1.01) with 67% confidence and from 0.61 to 1.41 (subtracting adding two
standard error to beta estimate of 1.01) with 95% confidence. While these ranges may

seem large, they are not unusual for most U.S. companies. This suggests that we should
consider regression estimates of betas from regressions with caution.

˜ 4.5: The Relevance of R-squared to an Investor
Assume that, having done the regression analysis, both Disney and Amgen, a
biotechnology company, have betas of 1.01. Disney, however, has an R-squared of
31%, while Amgen has an R-squared of only 15%. If you had to pick between these
investments, which one would you choose?
a. Disney, because it™s higher R-squared suggests that it is less risky
b. Amgen, because it™s lower R-squared suggests a greater potential for high returns
c. I would be indifferent, because they both have the same beta
Would your answer be any different if you were running a well-diversified fund?

In Practice: Using a Service beta
Most analysts who use betas obtain them from an estimation service; Merrill
Lynch, Barra, Value Line, Standard and Poor™s, Morningstar and Bloomberg are some of
the well known services. All these services begin with regression betas and make what
they feel are necessary changes to make them better estimates for the future. While most
of these services do not reveal the internal details of this estimation, Bloomberg is an
honorable exception. The following is the beta calculation page from Bloomberg for
Disney, using the same period as our regression (January 1999 to December 2003):

While the time period used is identical to the one used in our earlier regression, there are
subtle differences between this regression and the earlier one in Figure 4.1. First,
Bloomberg uses price appreciation in the stock and the market index in estimating betas
and ignores dividends.30 This does not make much of a difference for a Disney, but it
could make a difference for a company that either pays no dividends or pays significantly
higher dividends than the market. Second, Bloomberg also computes what they call an
adjusted beta, which is estimated as follows:
Adjusted Beta = Raw Beta (0.67) + 1 (0.33)
These weights do not vary across stocks, and this process pushes all estimated betas
towards one. Most services employ similar procedures to adjust betas towards one. In
doing so, they are drawing on empirical evidence that suggests that the betas for most
companies, over time, tend to move towards the average beta, which is one. This may be
explained by the fact that firms get more diversified in their product mix and client base
as they get larger.

30 This is why the intercept in the Bloomberg print out (0.03%) is slightly different from the intercept
estimated earlier in the chapter (0.05%). The beta and R-squared are identical.

In general, betas reported by different services for the same firm can be very
different because they use different time periods (some use 2 years and others 5 years),
different return intervals (daily, weekly or monthly), different market indices and
different post-regression estimates. While these beta differences may be troubling, the
beta estimates delivered by each of these services comes with a standard error, and it is
very likely that all of the betas reported for a firm fall within the range of the standard
errors from the regressions.

Illustration 4.2: Estimating Historical Betas for Aracruz and Deutsche Bank
Aracruz is a Brazilian company and we can regress returns on the stock against a
Brazilian index to obtain risk parameters. The stock also had an ADR listed on the U.S.
exchanges and we can regress returns on the ADR against a U.S. index to obtain
parameters. Figure 4.4 presents both graphs for the January 1999- December 2003 time
Figure 4.4: Estimating Aracruz™s Beta: Choice of Indices
Aracruz ADR vs S&P 500 Aracruz vs Bovespa
80 1 40

1 20
1 00

40 80
Aracruz ADR



0 20


-40 -40
-20 -10 0 10 20 -50 -40 -30 -20 -10 0 10 20 30


Aracruz ADR = 2.80% + 1.00 S&P Aracruz = 2.62% + 0.22 Bovespa
How different are the risk parameters that emerge from the two regressions? Aracruz has
a beta of 1.00, when the ADR is regressed against the S&P 500, and a beta of only 0.22,
when the local listing is regressed against the Bovespa.31 Each regression has its own
problems. The Bovespa is a narrow index dominated by a few liquid stocks and does not

31 The biggest source of the difference is one month (January 1999). In that month, Aracruz had a return of
133% in the Sao Paulo exchange while the ADR dropped by 9.67% in the same month. The disparity in
returns can be attributed to a steep devaluation in the Brazilian Real in that month.

represent the broad spectrum of Brazilian equities. While the S&P 500 is a broader index,
the returns on the ADR have little relevance to a large number of non-US investors who
bought the local listing.
Deutsche Bank does not have an ADR listed in the United States but we can
regress return on Deutsche against a multitude of indices. Table 4.4 presents comparisons
of the results of the regressions of returns on Deutsche against three indices “ a German
equity index (DAX), an index of large European companies (FTSE Euro 300) and a
global equity index (Morgan Stanley Capital Index (MSCI)).
Table 4.4: Deutsche Bank Risk Parameters: Index Effect
Intercept 1.24% 1.54% 1.37%
Beta 1.05 1.52 1.23
Std Error of Beta 0.11 0.19 0.25
R Squared 62% 52% 30%

Here again, the risk parameters estimated for Deutsche Bank are a function of the index
used in the regression. The standard error is lowest (and the R squared is highest) for the
regression against the DAX; this is not surprising since Deutsche is a large component of
the DAX. The standard error gets larger and the R squared gets lower as the index is
broadened to initially include other European stocks and then to global stocks.

In Practice: Which index should we use to estimate betas?
In most cases, analysts are faced with a mind-boggling array of choices among
indices when it comes to estimating betas; there are more than 20 broad equity indices
ranging from the Dow 30 to the Wilshire 5000 in the United States alone. One common
practice is to use the index that is most appropriate for the investor who is looking at the
stock. Thus, if the analysis is being done for a U.S. investor, the S&P 500 index is used.
This is generally not appropriate. By this rationale, an investor who owns only two stocks
should use an index composed of only those stocks to estimate betas.
The right index to use in analysis should be determined by the holdings of the
marginal investor in the company being analyzed. Consider Aracruz and Deutsche Bank
in the earlier illustration. If the marginal investors in.these companies are investors who

holds only domestic stocks “ just Brazilian stocks in the case of Aracruz or German
stocks in the case of Deutsche “ we can use the regressions against the local indices. If
the marginal investor is a global investor, a more relevant measure of risk may emerge by
using the global index. Over time, you would expect global investors to displace local
investors as the marginal investors, because they will perceive far less of the risk as
market risk and thus pay a higher price for the same security. Thus, one of the ironies of
our notion of risk is that Aracruz will be less risky to an overseas investor who has a
global portfolio than to a Brazilian investor with all of his or her wealth in Brazilian

Standard Procedures for Estimating Risk Parameters in the Arbitrage Pricing and Multi-
factor Models
Like the CAPM, the arbitrage pricing model defines risk to be non-diversifiable
risk, but, unlike the CAPM, the APM allows for multiple economic factors in measuring
this risk. While the process of estimation of risk parameters is different for the arbitrage
pricing model, many of the issues raised relating to the determinants of risk in the CAPM
continue to have relevance for the arbitrage pricing model.
The parameters of the arbitrage pricing model Factor Analysis: This is a statistical
are estimated from a factor analysis on historical technique, where past data is analyzed
with the intent of extracting common
stock returns, which yields the number of common
factors that might have affected the
economic factors determining these returns, the risk
premium for each factor and the factor-specific betas
for each firm.
Once the factor-specific betas are estimated for each firm, and the factor premia
are measured, the arbitrage pricing model can be used to estimated expected returns on a
Cost of Equity = Rf + ! ! j (E(Rj ) - Rf)

Rf = Risk-free rate
βj = Beta specific to factor j


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