The CAPM provides a single state, single factor, general equilibrium theory of the risk-return relation. However, in the 1960s, Mandelbrot (1963) already observed stock returns to have a very peaked distribution with heavy tails and also periods of persistent volatility, which contradicts the CAPM. In response to these observations, the Conditional CAPM (C-CAPM) has been discussed by several authors. In a C-CAPM investors can price an asset or portfolio conditional on the available information at a point in time. This is done by replacing the unconditional by conditional moments of returns. Statistically, processes of ”Generalized Autoregressive Conditional Heteroscedasticity” (GARCH) can capture the so called ”stylized facts”, some observed by Mandelbrot (1963). GARCH models were developed by Engle (1982) and Bollerslev (1986) and try to model time-varying second moments of asset returns. If a GARCH process is assumed for the disturbance term in a C-CAPM, a GARCH-in mean model (GARCH-M) can be estimated, where the conditional variance or covariance impacts the conditional expectation of (excess) returns. The GARCH-M can model time-varying conditional moments, but also time-varying risk premia and the implied beta factor. As for this seminar paper, I mostly follow the comprehensive dissertation ”Das CAPM mit zeitabhängigen Beta-Faktoren” of Linnenbrink (1998) and the paper of Bollerslev et al. (1988). First, the theoretical foundations of the CAPM, the C-CAPM, GARCH processes and the GARCH-M extension are presented. Then, in the empirical part, I estimate a (univariate) GARCH-M representation of the C-CAPM. I compare its performance to a traditional CAPM with a single stock portfolio of an investor (selected stock: Tesla, Inc.).
Table of Contents
1 Literature review and introduction
2 Theoretical foundation
2.1 The traditional CAPM
2.2 The flaws of the traditional CAPM
2.3 The Conditional CAPM
2.4 The (G)ARCH model foundations
2.5 The GARCH-M extension
3 Empirical analysis
3.1 Econometric approach and methods
3.2 Data description
3.3 Descriptive analysis and tests
3.4 Model estimates
4 Conclusions
Research Objectives and Themes
The academic work investigates whether the Conditional Capital Asset Pricing Model (C-CAPM), represented by a GARCH-in-mean (GARCH-M) model, provides a superior empirical fit for stock returns compared to the traditional, static CAPM by accounting for time-varying risk and volatility.
- Theoretical evaluation of the traditional CAPM and its limitations regarding "stylized facts".
- Introduction of GARCH processes to model time-varying conditional moments.
- Application of the GARCH-M framework to estimate conditional equity risk premia.
- Empirical performance comparison using Tesla, Inc. stock data and the Wilshire 5000 index.
- Calculation of implied time-varying beta factors as a measure of systematic risk.
Excerpt from the Book
The Conditional CAPM
In contrast to the CAPM, the C-CAPM allows for time-varying moments of a conditional probability distribution of returns (expected values, variances and covariances). For the C-CAPM, I will follow Linnenbrink (1998), Hansson/Hördahl (1998) and Hodgson/Vorkink (2003). As the crucial representations of the CAPM have already been displayed in the section 2.1, we simply need to enrich the formulas by replacing unconditional by conditional moments. For the purpose of this seminar paper and with respect to the empirical section I highlight the C-CAPM formulation in terms of risk premia, corresponding to equation (2):
E (Yit | It-1)= E (Ymt | It-1) Cov (Rit,Rmt | It-1) / Var (Rmt | It-1) (5)
= E (Ymt | It-1) βit . (6)
Notice, that we now account for the variability in t of the beta coefficient, which is now βi = βit. Hansson and Hördahl (1998) and also Linnenbrink (1998) comment on the C-CAPM model equation as the (conditional) expected (excess) return (i.e. the ERP) to be dependent on time-varying risk, which is the main point of the C-CAPM. The C-CAPM considers the time-varying systematic risk (Beta) and with that risk also the conditional ERP is time-varying. From the setting, we see that we need an additional assumption for the C-CAPM, namely that the returns now must be conditionally normally distributed.
Summary of Chapters
Literature review and introduction: Provides an overview of the CAPM, its development, and its limitations, identifying the need for conditional models (C-CAPM) and GARCH representations.
Theoretical foundation: Explains the mathematical derivation of the traditional CAPM, outlines the GARCH model framework for volatility, and details how the GARCH-M extension integrates these for risk premium estimation.
Empirical analysis: Details the specific econometric steps, including DCC GARCH estimation and the application of the model to TSLA stock data, followed by discussion on descriptive statistics and estimation results.
Conclusions: Summarizes the key findings, confirming that the C-CAPM with GARCH effects offers improved explanatory power over the traditional CAPM.
Keywords
CAPM, Conditional CAPM, GARCH-M, Time-varying Covariance, Equity Risk Premium, Systematic Risk, Volatility Clustering, Financial Econometrics, Beta Factor, Asset Pricing, Stylized Facts, DCC GARCH, Stock Returns, Market Proxy, Risk Aversion
Frequently Asked Questions
What is the core subject of this research paper?
The paper explores the application of the Conditional Capital Asset Pricing Model (C-CAPM) to evaluate if it performs better than the traditional CAPM in explaining risky asset pricing, specifically by using GARCH-in-mean models.
What are the primary themes discussed in the work?
The study centers on the limitations of the traditional CAPM, the role of conditional heteroscedasticity (volatility), the estimation of time-varying risk premia, and the practical implementation of GARCH-M models in financial markets.
What is the main research question?
The research seeks to determine whether the Conditional CAPM with a GARCH-M representation can outperform the traditional static CAPM in pricing stocks with time-varying risk.
Which methodology is employed in the study?
The author uses a two-stage econometric approach: first estimating conditional second moments using a DCC GARCH(1,1) model, and subsequently, a univariate C-CAPM to estimate risk premia.
What content is covered in the main body section?
The main part encompasses theoretical foundations, a detailed econometric methodology, data description of selected time series (TSLA and Wilshire 5000), descriptive statistics, and an assessment of model parameters and estimates.
Which keywords best characterize this research?
Key terms include CAPM, Conditional CAPM, GARCH, volatility clustering, systematic risk, and equity risk premium.
Why are GARCH models used in this study?
GARCH models are utilized because they effectively capture the "stylized facts" of financial return data—specifically conditional heteroscedasticity and time-varying volatility—which the static CAPM ignores.
Does the C-CAPM improve upon the traditional CAPM for the analysed stock?
The study indicates that whilst the traditional CAPM remains significant, the C-CAPM provides additional explanatory power regarding conditional variances, thus proving to be a superior approach for incorporating inter-temporal risk factors.
- Citation du texte
- M.Sc. Sebastian Wilde (Auteur), 2021, The CAPM with time-varying covariances, Munich, GRIN Verlag, https://www.grin.com/document/1267030
