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`

.

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

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. 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 *R _{k,i}*,
market returns

$${R}_{k,i}={\alpha}_{i}+{C}_{k}+{\beta}_{i}({M}_{k}-{C}_{k})+{V}_{k,i}$$

for samples *k* = 1, ..., *m* and
assets *i* = 1, ..., *n*, where *α _{i}* is
a parameter that specifies the nonsystematic return of an asset,

The collection of parameters *α*_{1}, ..., *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 *S*_{11}, ..., *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

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

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. So, 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.

`convert2sur`

| `ecmlsrmle`

| `ecmlsrobj`

| `ecmmvnrfish`

| `ecmmvnrfish`

| `ecmmvnrmle`

| `ecmmvnrobj`

| `ecmmvnrstd`

| `ecmmvnrstd`

| `ecmnfish`

| `ecmnhess`

| `ecmninit`

| `ecmnmle`

| `ecmnobj`

| `ecmnstd`

| `mvnrfish`

| `mvnrmle`

| `mvnrobj`

| `mvnrstd`

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