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Valuation with Missing Data

Introduction

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 using CAPMdemo.m located at matlabroot/toolbox/finance/findemos.

Capital Asset Pricing Model

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 β1, ..., βn 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.

Estimation of the CAPM

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 Rk,i, market returns Mk, and riskless asset returns Ck, the estimation model has the form

for samples k = 1, ..., m and assets i = 1, ..., n, where αi is a parameter that specifies the nonsystematic return of an asset, βi is the asset beta, and Vk,i is the residual error for each asset with associated random variable Vi.

The collection of parameters α1, ..., αn 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 i,j = 1, ..., n, where the parameters S11, ..., Snn are called residual or nonsystematic variances/covariances.

The square root of the residual variance of each asset, for example, sqrt(Sii) for i = 1, ..., n, 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.

Estimation with Missing Data

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.

Estimation of Some Technology Stock Betas

To illustrate how to use the missing-data regression routines, estimate betas for 12 technology stocks, where a single stock (GOOG) is an IPO.

  1. 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.

  2. 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));
    end

    This 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.

Grouped Estimation of Some Technology Stock Betas

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));
end

This 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).

References

[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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