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Estimation of Some Technology Stock Betas |
The Capital Asset Pricing Model (CAPM) is a venerable but often maligned tool to characterize comovements between asset and market prices. Although many issues arise in CAPM implementation and interpretation, one problem that practitioners face is to estimate the coefficients of the CAPM with incomplete stock price data.
This example shows how to use the missing data regression functions to estimate the coefficients of the CAPM. You can run the example directly with the M-file CAPMdemo.m.
Given a host of assumptions that can be found in the references (see Sharpe [11], Lintner [6], Jarrow [5], and Sharpe, et. al. [12]), the CAPM concludes that asset returns have a linear relationship with market returns. Specifically, given the return of all stocks that constitute a market denoted as M and the return of a riskless asset denoted as C, the CAPM states that the return of each asset Ri in the market has the expectational form
for assets i = 1, ..., n,
where βi is a parameter
that specifies the degree of comovement between a given asset and
the underlying market. In other words, the expected return of each
asset is equal to the return on a riskless asset plus a risk-adjusted
expected market return net of riskless asset returns. The collection
of parameters
is
called asset betas.
Note that the beta of an asset has the form
which is the ratio of the covariance between asset and market returns divided by the variance of market returns. If an asset has a beta = 1, the asset is said to move with the market; if an asset has a beta > 1, the asset is said to be more volatile than the market. Conversely, if an asset has a beta < 1, the asset is said to be less volatile than the market.
The standard CAPM model is a linear model with additional parameters
for each asset to characterize residual errors. For each of n assets
with m samples of observed asset returns
, market returns
, and riskless asset returns
, the estimation model has the form
for samples k = 1, ..., m and
assets i = 1, ..., n, where
is a parameter that specifies the
nonsystematic return of an asset,
is the asset beta, and
is the residual error for each
asset with associated random variable
.
The collection of parameters
are called asset alphas. The strict form of the
CAPM specifies that alphas must be zero and that deviations from zero
are the result of temporary disequilibria. In practice, however, assets
may have nonzero alphas, where much of active investment management
is devoted to the search for assets with exploitable nonzero alphas.
To allow for the possibility of nonzero alphas, the estimation model generally seeks to estimate alphas and to perform tests to determine if the alphas are statistically equal to zero.
The residual errors Vi are assumed to have moments
and
for assets
,
where the parameters
are called residual or nonsystematic
variances/covariances.
The square root of the residual variance of each asset, for
example,
for
, is said to be the residual or
nonsystematic risk of the asset since it characterizes the residual
variation in asset prices that are not explained by variations in
market prices.
Although betas can be estimated for companies with sufficiently long histories of asset returns, it is difficult to estimate betas for recent IPOs. However, if a collection of sufficiently observable companies exists that can be expected to have some degree of correlation with the new company's stock price movements, that is, companies within the same industry as the new company, it is possible to obtain imputed estimates for new company betas with the missing-data regression routines.
To illustrate how to use the missing-data regression routines, estimate betas for 12 technology stocks, where a single stock (GOOG) is an IPO.
Load dates, total returns, and ticker symbols for the 12 stocks from the MAT-file CAPMuniverse.
load CAPMuniverse whos Assets Data Dates Name Size Bytes Class Assets 1x14 952 cell array Data 1471x14 164752 double array Dates 1471x1 11768 double array Grand total is 22135 elements using 177472 bytes
The assets in the model have the following symbols, where the last two series are proxies for the market and the riskless asset:
Assets(1:7) Assets(8:14) ans = 'AAPL' 'AMZN' 'CSCO' 'DELL' 'EBAY' 'GOOG' 'HPQ' ans = 'IBM' 'INTC' 'MSFT' 'ORCL' 'YHOO' 'MARKET' 'CASH'
The data covers the period from January 1, 2000 to November 7, 2005 with daily total returns. Two stocks in this universe have missing values that are represented by NaNs. One of the two stocks had an IPO during this period and, consequently, has significantly less data than the other stocks.
Compute separate regressions for each stock, where the stocks with missing data will have estimates that reflect their reduced observability.
[NumSamples, NumSeries] = size(Data);
NumAssets = NumSeries - 2;
StartDate = Dates(1);
EndDate = Dates(end);
fprintf(1,'Separate regressions with ');
fprintf(1,'daily total return data from %s to %s ...\n', ...
datestr(StartDate,1),datestr(EndDate,1));
fprintf(1,' %4s %-20s %-20s %-20s\n','','Alpha','Beta','Sigma');
fprintf(1,' ---- -------------------- ');
fprintf(1,'-------------------- --------------------\n');
for i = 1:NumAssets
% Set up separate asset data and design matrices
TestData = zeros(NumSamples,1);
TestDesign = zeros(NumSamples,2);
TestData(:) = Data(:,i) - Data(:,14);
TestDesign(:,1) = 1.0;
TestDesign(:,2) = Data(:,13) - Data(:,14);
% Estimate CAPM for each asset separately
[Param, Covar] = ecmmvnrmle(TestData, TestDesign);
% Estimate ideal standard errors for covariance parameters
[StdParam, StdCovar] = ecmmvnrstd(TestData, TestDesign, ...
Covar, 'fisher');
% Estimate sample standard errors for model parameters
StdParam = ecmmvnrstd(TestData, TestDesign, Covar, 'hessian');
% Set up results for output
Alpha = Param(1);
Beta = Param(2);
Sigma = sqrt(Covar);
StdAlpha = StdParam(1);
StdBeta = StdParam(2);
StdSigma = sqrt(StdCovar);
% Display estimates
fprintf(' %4s %9.4f (%8.4f) %9.4f (%8.4f) %9.4f (%8.4f)\n', ...
Assets{i},Alpha(1),abs(Alpha(1)/StdAlpha(1)), ...
Beta(1),abs(Beta(1)/StdBeta(1)),Sigma(1),StdSigma(1));
endThis code fragment generates the following table.
Separate regressions with daily total return data from 03-Jan-2000
to 07-Nov-2005 ...
Alpha Beta Sigma
-------------------- -------------------- --------------------
AAPL 0.0012 ( 1.3882) 1.2294 ( 17.1839) 0.0322 ( 0.0062)
AMZN 0.0006 ( 0.5326) 1.3661 ( 13.6579) 0.0449 ( 0.0086)
CSCO -0.0002 ( 0.2878) 1.5653 ( 23.6085) 0.0298 ( 0.0057)
DELL -0.0000 ( 0.0368) 1.2594 ( 22.2164) 0.0255 ( 0.0049)
EBAY 0.0014 ( 1.4326) 1.3441 ( 16.0732) 0.0376 ( 0.0072)
GOOG 0.0046 ( 3.2107) 0.3742 ( 1.7328) 0.0252 ( 0.0071)
HPQ 0.0001 ( 0.1747) 1.3745 ( 24.2390) 0.0255 ( 0.0049)
IBM -0.0000 ( 0.0312) 1.0807 ( 28.7576) 0.0169 ( 0.0032)
INTC 0.0001 ( 0.1608) 1.6002 ( 27.3684) 0.0263 ( 0.0050)
MSFT -0.0002 ( 0.4871) 1.1765 ( 27.4554) 0.0193 ( 0.0037)
ORCL 0.0000 ( 0.0389) 1.5010 ( 21.1855) 0.0319 ( 0.0061)
YHOO 0.0001 ( 0.1282) 1.6543 ( 19.3838) 0.0384 ( 0.0074)
The Alpha column contains alpha estimates for each stock that are near zero as expected. In addition, the t-statistics (which are enclosed in parentheses) generally reject the hypothesis that the alphas are nonzero at the 99.5% level of significance.
The Beta column contains beta estimates for each stock that also have t-statistics enclosed in parentheses. For all stocks but GOOG, the hypothesis that the betas are nonzero is accepted at the 99.5% level of significance. It seems, however, that GOOG does not have enough data to obtain a meaningful estimate for beta since its t-statistic would imply rejection of the hypothesis of a nonzero beta.
The Sigma column contains residual standard deviations, that is, estimates for nonsystematic risks. Instead of t-statistics, the associated standard errors for the residual standard deviations are enclosed in parentheses.
To estimate stock betas for all 12 stocks, set up a joint regression model that groups all 12 stocks within a single design. (Since each stock has the same design matrix, this model is actually an example of seemingly unrelated regression.) The routine to estimate model parameters is ecmmvnrmle, and the routine to estimate standard errors is ecmmvnrstd.
Because GOOG has a significant number of missing values, a direct use of the missing data routine ecmmvnrmle takes 482 iterations to converge. This can take a long time to compute. For the sake of brevity, the parameter and covariance estimates after the first 480 iterations are contained in a MAT-file and are used as initial estimates to compute stock betas.
load CAPMgroupparam whos Param0 Covar0 Name Size Bytes Class Covar0 12x12 1152 double array Param0 24x1 192 double array Grand total is 168 elements using 1344 bytes
Now estimate the parameters for the collection of 12 stocks.
fprintf(1,'\n');
fprintf(1,'Grouped regression with ');
fprintf(1,'daily total return data from %s to %s ...\n', ...
datestr(StartDate,1),datestr(EndDate,1));
fprintf(1,' %4s %-20s %-20s %-20s\n','','Alpha','Beta','Sigma');
fprintf(1,' ---- -------------------- ');
fprintf(1,'-------------------- --------------------\n');
NumParams = 2 * NumAssets;
% Set up grouped asset data and design matrices
TestData = zeros(NumSamples, NumAssets);
TestDesign = cell(NumSamples, 1);
Design = zeros(NumAssets, NumParams);
for k = 1:NumSamples
for i = 1:NumAssets
TestData(k,i) = Data(k,i) - Data(k,14);
Design(i,2*i - 1) = 1.0;
Design(i,2*i) = Data(k,13) - Data(k,14);
end
TestDesign{k} = Design;
end
% Estimate CAPM for all assets together with initial parameter
% estimates
[Param, Covar] = ecmmvnrmle(TestData, TestDesign, [], [], [],...
Param0, Covar0);
% Estimate ideal standard errors for covariance parameters
[StdParam, StdCovar] = ecmmvnrstd(TestData, TestDesign, Covar,...
'fisher');
% Estimate sample standard errors for model parameters
StdParam = ecmmvnrstd(TestData, TestDesign, Covar, 'hessian');
% Set up results for output
Alpha = Param(1:2:end-1);
Beta = Param(2:2:end);
Sigma = sqrt(diag(Covar));
StdAlpha = StdParam(1:2:end-1);
StdBeta = StdParam(2:2:end);
StdSigma = sqrt(diag(StdCovar));
% Display estimates
for i = 1:NumAssets
fprintf(' %4s %9.4f (%8.4f) %9.4f (%8.4f) %9.4f (%8.4f)\n', ...
Assets{i},Alpha(i),abs(Alpha(i)/StdAlpha(i)), ...
Beta(i),abs(Beta(i)/StdBeta(i)),Sigma(i),StdSigma(i));
endThis code fragment generates the following table.
Grouped regression with daily total return data from 03-Jan-2000
to 07-Nov-2005 ...
Alpha Beta Sigma
---------------------- ----------------------------------------
AAPL 0.0012 ( 1.3882) 1.2294 ( 17.1839) 0.0322 ( 0.0062)
AMZN 0.0007 ( 0.6086) 1.3673 ( 13.6427) 0.0450 ( 0.0086)
CSCO -0.0002 ( 0.2878) 1.5653 ( 23.6085) 0.0298 ( 0.0057)
DELL -0.0000 ( 0.0368) 1.2594 ( 22.2164) 0.0255 ( 0.0049)
EBAY 0.0014 ( 1.4326) 1.3441 ( 16.0732) 0.0376 ( 0.0072)
GOOG 0.0041 ( 2.8907) 0.6173 ( 3.1100) 0.0337 ( 0.0065)
HPQ 0.0001 ( 0.1747) 1.3745 ( 24.2390) 0.0255 ( 0.0049)
IBM -0.0000 ( 0.0312) 1.0807 ( 28.7576) 0.0169 ( 0.0032)
INTC 0.0001 ( 0.1608) 1.6002 ( 27.3684) 0.0263 ( 0.0050)
MSFT -0.0002 ( 0.4871) 1.1765 ( 27.4554) 0.0193 ( 0.0037)
ORCL 0.0000 ( 0.0389) 1.5010 ( 21.1855) 0.0319 ( 0.0061)
YHOO 0.0001 ( 0.1282) 1.6543 ( 19.3838) 0.0384 ( 0.0074)
Although the results for complete-data stocks are the same, note that the beta estimates for AMZN and GOOG (the two stocks with missing values) are different from the estimates derived for each stock separately. Since AMZN has few missing values, the differences in the estimates are small. With GOOG, however, the differences are more pronounced.
The t-statistic for the beta estimate of GOOG is now significant at the 99.5% level of significance. Note, however, that the t-statistics for beta estimates are based on standard errors from the sample Hessian which, in contrast to the Fisher information matrix, accounts for the increased uncertainty in an estimate due to missing values. If the t-statistic is obtained from the more optimistic Fisher information matrix, the t-statistic for GOOG is 8.25. Thus, despite the increase in uncertainty due to missing data, GOOG nonetheless has a statistically significant estimate for beta.
Finally, note that the beta estimate for GOOG is 0.62 —a value that may require some explanation. Although the market has been volatile over this period with sideways price movements, GOOG has steadily appreciated in value. Consequently, it is less tightly correlated with the market, implying that it is less volatile than the market (beta < 1).
[1] Caines, Peter E. Linear Stochastic Systems. John Wiley & Sons, Inc., 1988.
[2] Cramér, Harald. Mathematical Methods of Statistics. Princeton University Press, 1946.
[3] Dempster, A.P, N.M. Laird, and D.B Rubin. "Maximum Likelihood from Incomplete Data via the EM Algorithm,"Journal of the Royal Statistical Society, Series B, Vol. 39, No. 1, 1977, pp. 1-37.
[4] Greene, William H. Econometric Analysis, 5th ed., Pearson Education, Inc., 2003.
[5] Jarrow, R.A. Finance Theory, Prentice-Hall, Inc., 1988.
[6] Lintner, J. "The Valuation of Risk Assets and the Selection of Risky Investments in Stocks," Review of Economics and Statistics, Vol. 14, 1965, pp. 13-37.
[7] Little, Roderick J. A and Donald B. Rubin. Statistical Analysis with Missing Data, 2nd ed., John Wiley & Sons, Inc., 2002.
[8] Meng, Xiao-Li and Donald B. Rubin. "Maximum Likelihood Estimation via the ECM Algorithm," Biometrika, Vol. 80, No. 2, 1993, pp. 267-278.
[9] Sexton, Joe and Anders Rygh Swensen. "ECM Algorithms that Converge at the Rate of EM," Biometrika, Vol. 87, No. 3, 2000, pp. 651-662.
[10] Shafer, J. L. Analysis of Incomplete Multivariate Data, Chapman & Hall/CRC, 1997.
[11] Sharpe, W. F. "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance, Vol. 19, 1964, pp. 425-442.
[12] Sharpe, W. F., G. J. Alexander, and J. V. Bailey. Investments, 6th ed., Prentice-Hall, Inc., 1999.
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