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20.00%

Regression line

10.00%

Disney

0.00%

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

-10.00%

-20.00%

-30.00%

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-Î²).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.

27

the regression are monthly returns) between 1999 and 2003 averaged 0.313%, resulting in

the following estimate for the performance:

Rf (1-Î²) = 0.313% (1-1.01) = -.0032%

Intercept - Rf (1-Î²) = 0.0467% - (-0.0032%) = 0.05%

This analysis suggests that Disneyâ€™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 marketâ€™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 â€˜industry-wideâ€™ 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 Disneyâ€™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 itâ€™s peer group by

0.73% between 1999 and 2003, after adjusting for risk. (Firm-specific Jensenâ€™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. Disneyâ€™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

28

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.

â˜ž 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 itâ€™s higher R-squared suggests that it is less risky

b. Amgen, because itâ€™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 Poorâ€™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):

29

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.

30

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

period:

Figure 4.4: Estimating Aracruzâ€™s Beta: Choice of Indices

Aracruz ADR vs S&P 500 Aracruz vs Bovespa

80 1 40

1 20

60

1 00

40 80

Aracruz ADR

Aracruz

60

20

40

0 20

0

-20

-20

-40 -40

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

BOVESPA

S&P

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.

31

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 â€“ 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

DAX FTSE Euro 300 MSCI

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

32

holds only domestic stocks â€“ just Brazilian stocks in the case of Aracruz or German

stocks in the case of Deutsche â€“ 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

assets.

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

data.

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

stock.

j=k

Cost of Equity = Rf + ! ! j (E(Rj ) - Rf)

j=1

where,

Rf = Risk-free rate

Î²j = Beta specific to factor j

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