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.

The following example shows how to use the missing data regression functions to estimate the coefficients of the CAPM.

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 R_i in the market has the expectational form

$$E[R_i]={\alpha }_i+C+{\beta }_i(E[M]-C)$$

for assets i = 1, ..., n, where β_i 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, ..., β_n is called asset betas.

The beta of an asset has the form

$${\beta }_i=\frac{\mathrm{cov}\left(R_i,M\right)}{\mathrm{var}\left(M\right)},$$

which is the ratio of the covariance between asset and market returns divided by the variance of market returns.Beta is the price volatility of a financial instrument relative to the price volatility of a market or index as a whole. Beta is commonly used with respect to equities. A high-beta instrument is riskier than a low-beta instrument. 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 R_k,i, market returns M_k, and riskless asset returns C_k, the estimation model has the form

$$R_{k,i}={\alpha }_i+C_k+{\beta }_i(M_k-C_k)+V_{k,i}$$

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

The collection of parameters α₁, ..., α_n 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 V_i are assumed to have moments

$$E\left[V_i\right]=0$$

and

$$E\left[V_i{V}_j\right]=S_{ij}$$

for assets i,j = 1, ..., n, where the parameters S₁₁, ..., S_nn are called residual or nonsystematic variances/covariances.

The square root of the residual variance of each asset, for example, sqrt(S_ii) 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     Attributes
   
     Assets         1x14              1568  cell                
     Data        1471x14            164752  double              
     Dates       1471x1              11768  double   

   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, so, has significantly less data than the other stocks.

2. Compute separate regressions for each stock, where the stocks with missing data 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     Attributes

  Covar0      12x12             1152  double              
  Param0      24x1               192  double   

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, 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. However, 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. So, 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.