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1

TESTS OF THE CAPM MODEL WITH APPLICATION OF THE GARCH MODELS - ANALYSIS FOR THE WSE

100%
Fiszeder P.
Przegląd Statystyczny
|
2006
|
vol. 53
|
issue 3
36-56
EN
The CAPM model tries to describe the behaviour of capital markets. It is often used by financial market practitioners for example to estimate expected returns, to estimate the cost of capital or to evaluate portfolio managers. In this article tests of the CAPM model with the application of the multivariate GARCH models are presented. It has been shown by numerous studies that conditional variances, covariances of returns and betas are time-varying. Most of the CAPM tests ignore those properties of financial time series. If conditional variance of the error term is time-varying then OLS estimators of the parameters in the mean equation are less efficient. Variability of betas may significantly influence the results of CAPM tests. The multivariate GARCH-M model is able to capture those properties of asset returns and simultaneously describes relations between expected returns and conditional covariances of returns. That is why it can be applied to test the restrictions of the CAPM model. Two new specifications of the GARCH-M model are proposed in the paper: the GARCH-M model with the price of market risk evolving over time according to the random walk process and the GARCH model with the asymmetric GARCH-M effect. In the empirical part of the paper tests of the CAPM model are performed for sectors quoted on the Warsaw Stock Exchange. Estimated market risk premia are changing over time. The highest estimates of risk premia are for financial crisis in Russia and Brazil. Estimated conditional betas are also time-varying. Some of the results support the restrictions of the CAPM model; however, the relation between expected returns and covariances of sector returns and market portfolio analysed for standard specifications of the GARCH-M model is not significant. Moreover there are strong relations between the S&P 500 index and all analysed sector portfolios. The WIG index is not a proper market portfolio or the CAPM model should include additional factors like a portfolio of the overall stock market (for instance the S&P 500 index). Results of the estimation for the proposed model with an asymmetric GARCH-M effect suggest that the relation between expected returns and covariances is significant, but the price of market risk is positive when return of the S&P 500 index is positive and negative when return of the S&P 500 is negative. During the bull market the price of market risk is positive and during the bear market negative. It seems that the GARCH-M model with price of risk dependent on the sign of a portfolio of the overall stock market gives a better description of relations between expected returns and conditional covariances of returns.
2

APT MODEL WITH FACTOR GARCH MODEL - ANALYSIS FOR THE WARSAW STOCK EXCHANGE (WSE)

100%
Fiszeder P.
Przegląd Statystyczny
|
2008
|
vol. 55
|
issue 1
45-66
EN
Systematic risk has been measured traditionally using the CAPM beta estimated by applying the market model developed by Sharpe. However, it is generally accepted by practitioners as well as researchers that stock prices are influenced by a number of different economic factors. Thus, the Arbitrage Pricing Theory (APT) assuming the stock return to be a linear function of a certain number of economic factors has received increased attention in recent years. It has been shown by numerous studies that conditional variances and covariances of returns are time-varying. Most of the APT tests ignore those properties of financial time series. The variability of variances of factors and variances and covariances of asset returns may significantly influence the results of the APT tests. The APT model with the factor GARCH covariance structure, which is able to capture those properties of asset returns, is presented in the paper. In the empirical part of the paper, a test of the APT model is performed for sectors quoted on the WSE. Factors were extracted by principal component analysis for economic variables: stock indices - WIG, S&P 500, DAX, BUX, currency rates -USD/PLN, EUR/PLN, yield on the 52-week Polish Treasury Bills, 1 month WIBOR rate, US 10-year note yield, prices of raw materials - crude oil, copper, gold, CRB index. A two-stage estimation procedure with a multivariate GARCH model in the second stage is used and the number of factors is tested. Risk connected with seven factors is priced in the market. Presented model significantly better explained variability of conditional variances and covariances of stock returns than variability of returns. Two new modifications of procedures for construction of the APT with factor GARCH model are proposed. Many financial processes have common conditional volatility, which can be described by factor GARCH model, however it is not very probable, that considered factors explained the whole variability of conditional variances and covariances. The first proposed modification is extension of the model, which captures unexplained variability of asset returns conditional covariance matrix. The second one is estimation of the market risk premia and conditional variances of factors in the first stage by applying a model with the threshold GARCH-M effect instead of applying the traditional GARCH-M model. The threshold GARCH-M model assumes that the relation between expected return and variance can be different, depending on the sign of the exogenous variable, in this analysis returns of the S&P 500 index.The results of tests indicate the advantage of the model constructed by this procedure
3

GARCH-M MODEL WITH TIME-VARYING PARAMETER - ANALYSIS OF SELECTED STOCKS AND INDICES QUOTED ON THE WSE

63%
Fiszeder P., Kwiatkowski J.
Przegląd Statystyczny
|
2005
|
vol. 52
|
issue 3
73-88
EN
The authors analyse relation between expected return and conditional variance, mainly whether it is time-varying or constant (and also linear). It is assumed that for specified period of time investors expect higher returns from assets with higher risk. However there is no agreement whether positive relation between expected returns and variance is 'dynamic'. Investment over short horizons may sometimes be influenced by portfolio balance and transaction cost consideration or by unexpected immediate consumption needs. All these factors may obscure the risk and return relation in the short horizon. The risk and return relation may also be nonlinear or time varying. The authors analyse this relation for 26 stocks and 2 indices quoted on the Warsaw Stock Exchange with the aim to provide additional insight into the nature of stocks volatility and its relation to expected returns. The GARCH-M models with constant and time-varying parameter are implemented. For most stocks there are no reasons to reject the hypothesis of no autocorrelation of returns. Observable higher serial correlations in portfolio returns are in agreement with the results of other investigations. According to the estimates of the parameters in the conditional variance, current information remains important for forecasts of the conditional variance for very long horizons. Large persistence in variance in financial time series is perplexing because currently no theory predicts that this should be the case. Estimates of the GARCH-M model with constant parameter indicate that for most assets the relation between expected, return and conditional variance is not significant. However the results are sensitive to specification of conditional mean. Only for two stocks the relation between expected return and conditional variance is time-varying. Estimates of the GARCH-M model with time-varying parameter can explain different empirical results concerning the GARCH-M model with constant parameter. Wide range of confidence intervals for time-varying parameter may explain insignificance of analyzed relation for the GARCH-M model with constant parameter.
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