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.).
At a glance
- Grade: 1.3 (1.0 is the best grade)
- Empirical CAPM comparison using a Tesla single-stock portfolio
- DCC GARCH(1,1) estimation supplies conditional covariances for the conditional CAPM
- 23 sources in the bibliography
- 12 figures
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
Objectives and Topics
This paper investigates whether a conditional capital asset pricing model with a GARCH-in-mean representation can offer advantages over the traditional CAPM. It compares the models empirically using a single-stock Tesla portfolio and examines how time-varying volatility and covariance affect the modelling of risk and expected returns.
- Traditional CAPM assumptions and their limitations
- Conditional expectations and time-varying systematic risk
- ARCH, GARCH and GARCH-in-mean models
- Empirical estimation using Tesla and a US market proxy
- Return distributions, volatility clustering and model diagnostics
Excerpt from the Book
1 Literature review and introduction
The Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) is one of the most popular models to evaluate the pricing of risky assets, in particular stocks. Doolan/Smith (2017) conclude, 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 many times by several authors, beginning with Merton (1973) and his inter-temporal and multi-beta CAPM version, Bodie et al. (1983, 1984) and, among others, the articles by Jagannathan/Wang (1996), Bollerslev et al. (1988) and Harvey (1989, 1991) where C-CAPM models are described, estimated and tested. 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 extension has been proposed by Engle, Lilien, Robins (1987) and Bollerslev, Engle, Wooldridge (1988). The GARCH-M can model time-varying conditional moments, but also time-varying risk premia and the implied beta factor. Linnenbrink (1998) and Hansson/Hördahl (1998) present this approach appealingly, also various tests and extensions have been made by e.g. Hodgson/Vorkink (2003) and Doolan/Smith (2017).
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.The goal is to compare its performance to a traditional CAPM with a single stock portfolio of an investor (Tesla, Inc.). Throughout this seminar paper, I will use a consistent notation of symbols, no matter what the original notation of the cited source is.
Chapter Summaries
1 Literature review and introduction: Introduces the development of the CAPM, conditional asset pricing and GARCH models, and sets out the comparison using Tesla stock.
2 Theoretical foundation: Explains the traditional and conditional CAPM, the limitations of constant-risk assumptions, and the foundations of ARCH, GARCH and GARCH-M models.
3 Empirical analysis: Describes the data, diagnostic tests and staged estimation procedure before presenting estimates for the competing models and time-varying risk measures.
4 Conclusions: Interprets the comparison and discusses the influence of modelling choices and possible extensions.
Keywords
capital asset pricing model, conditional CAPM, GARCH-in-mean, dynamic conditional correlation, time-varying covariance, volatility clustering, systematic risk, beta, equity risk premium, Tesla, Wilshire 5000, financial econometrics
Frequently Asked Questions
What is this paper about?
It examines whether a conditional CAPM incorporating time-varying risk can offer advantages over the traditional CAPM.
What are its main topics?
The paper covers asset pricing, conditional volatility and covariance, GARCH models, risk premia and empirical model comparison.
What is the main research question?
The central question is whether a conditional CAPM with a GARCH-in-mean representation can outperform the traditional CAPM.
What scientific method does the author use?
The author combines a literature-based theoretical discussion with empirical time-series analysis, ordinary least squares and staged DCC GARCH estimation.
What does the main analysis cover?
It presents the modelling approach and data, tests return characteristics, and compares estimates of the traditional and conditional models.
Which keywords characterize the work?
Key terms include conditional CAPM, GARCH-in-mean, time-varying covariance, volatility clustering, beta and equity risk premium.
Why is Tesla included in the analysis?
Tesla provides the single-stock portfolio used to compare the traditional CAPM with the conditional specification.
Which market proxy and risk-free rate are used?
The analysis uses the Wilshire 5000 as a proxy for the US market portfolio and the three-month Treasury bill rate as the risk-free rate.
How are conditional covariances incorporated?
A DCC GARCH model estimates conditional covariances, which are then used in the conditional CAPM and to calculate time-varying betas.
- Quote paper
- M.Sc. Sebastian Wilde (Author), 2021, The CAPM with time-varying covariances, Munich, GRIN Verlag, https://www.grin.com/document/1267030
